16 math questions need answered

BMIndex.indd 29 7/31/2013 7:29:35 AM

EQUITY.

CURRICULUM. -

TEACHING. - -

LEARNING. -

ASSESSMENT.

TECHNOLOGY. -

National Council of Teachers of Mathematics Principles and Standards for School Mathematics

Principles for School Mathematics

Standards for School Mathematics

NUMBER AND OPERATIONS

-

ALGEBRA

-

GEOMETRY

-

MEASUREMENT

-

DATA ANALYSIS AND PROBABILITY

-

PROBLEM SOLVING

REASONING AND PROOF

COMMUNICATION

FMEndpaper.indd 15 7/31/2013 10:58:25 AM

Curriculum Focal Points for Prekindergarten through Grade 8 Mathematics

CONNECTIONS

-

REPRESENTATION

-

-

PREKINDERGARTEN Number and Operations:

-

Geometry: -

Measurement:

KINDERGARTEN Number and Operations:

Geometry: Measurement:

GRADE 1 Number and Operations Algebra: -

-

Number and Operations:

Geometry:

GRADE 2 Number and Operations:

Number and Operations Algebra:

Measurement: -

GRADE 3 Number and Operations Algebra: -

-

Number and Operations: -

Geometry: -

GRADE 4 Number and Operations Algebra:

Number and Operations:

Measurement: -

GRADE 5 Number and Operations Algebra: -

Number and Operations:

Geometry Measurement Algebra:

GRADE 6 Number and Operations: -

Number and Operations: -

Algebra: -

GRADE 7 Number and Operations Algebra Geometry: -

Measurement Geometry Algebra: -

Number and Operations Algebra: -

GRADE 8 Algebra: -

Geometry Measurement:

Data Analysis Number and Operations Algebra: -

FMEndpaper.indd 16 7/31/2013 10:58:25 AM

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MathematicsFor Elementary Teachers TENTH EDITION A C O N T E M P O R A R Y A P P R O A C H

Gary L. Musser Blake E. Peterson William F. Burger Oregon State University Brigham Young University

FMWileyPlus.indd 3 7/31/2013 2:14:09 PM

To:

Irene, my wonderful wife of 52 years who is the best mother our son could have; Greg, our son, for his inquiring mind; Maranda, our granddaughter, for her willingness to listen; my parents who have passed away, but always with me; and Mary Burger, my initial coauthor's daughter. G.L.M.

Shauna, my beautiful eternal companion and best friend, for her continual support of all my endeavors; my four children: Quinn for his creative enthusiasm for life, Joelle for her quiet yet strong confidence, Taren for her unintimidated ap- proach to life, and Riley for his good choices and his dry wit. B.E.P.

VICE PRESIDENT & EXECUTIVE PUBLISHER Laurie Rosatone PROJECT EDITOR Jennifer Brady SENIOR CONTENT MANAGER Karoline Luciano SENIOR PRODUCTION EDITOR Kerry Weinstein MARKETING MANAGER Kimberly Kanakes SENIOR PRODUCT DESIGNER Tom Kulesa OPERATIONS MANAGER Melissa Edwards ASSISTANT CONTENT EDITOR Jacqueline Sinacori SENIOR PHOTO EDITOR Lisa Gee MEDIA SPECIALIST Laura Abrams COVER & TEXT DESIGN Madelyn Lesure

This book was set by Laserwords and printed and bound by Courier Kendallville. The cover was printed by Courier Kendallville.

Copyright © 2014, 2011, 2008, 2005, John Wiley & Sons, Inc. All rights reserved. No part of this publication may be reproduced, stored in a retrieval system or transmitted in any form or by any means, electronic, me- chanical, photocopying, recording, scanning or otherwise, except as permitted under Sections 107 or 108 of the 1976 United States Copyright Act, without either the prior written permission of the Publisher, or authoriza- tion through payment of the appropriate per-copy fee to the Copyright Clearance Center, Inc. 222 Rosewood Drive, Danvers, MA 01923, website www.copyright.com. Requests to the Publisher for permission should be addressed to the Permissions Department, John Wiley & Sons, Inc., 111 River Street, Hoboken, NJ 07030-5774, (201)748-6011, fax (201)748-6008, website http://www.wiley.com/go/permissions.

Evaluation copies are provided to qualified academics and professionals for review purposes only, for use in their courses during the next academic year. These copies are licensed and may not be sold or transferred to a third party. Upon completion of the review period, please return the evaluation copy to Wiley. Return instruc- tions and a free of charge return shipping label are available at www.wiley.com/go/returnlabel. Outside of the United States, please contact your local representative.

Library of Congress Cataloging-in-Publication Data

Musser, Gary L. Mathematics for elementary teachers : a contemporary approach / Gary L. Musser, Oregon State University,

William F. Burger, Blake E. Peterson, Brigham Young University. -- 10th edition. pages cm

Includes index. ISBN 978-1-118-45744-3 (hardback)

1. Mathematics. 2. Mathematics–Study and teaching (Elementary) I. Title. QA39.3.M87 2014 510.2’4372–dc23 2013019907

Printed in the United States of America

10 9 8 7 6 5 4 3 2 1

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Gary L. Musser is Professor Emeritus from Oregon State University. He earned both his B.S. in Mathematics Education in 1961 and his M.S. in Mathematics in 1963 at the University of Michigan and his Ph.D. in Mathematics (Radical Theory) in 1970 at the University of Miami in Florida. He taught at the junior and senior high, junior college college, and university levels for more than 30 years. He spent his final 24 years teaching prospective teachers in the Department of Mathematics at Oregon State University. While at OSU, Dr. Musser developed the mathematics component of the elementary teacher program. Soon after Profesor William F. Burger joined the OSU Department of Mathematics in a similar capacity, the two of them began to write the first edtion of this book. Professor Burger passed away during the preparation of the second edition, and Professor Blake E. Peterson was hired at OSU as his replacement. Professor Peter- son joined Professor Musser as a coauthor beginning with the fifth edition.

Professor Musser has published 40 papers in many journals, including the Pacific Journal of Mathematics, Canadian Journal of Mathematics, The Mathematics Association of America Monthly, the NCTM’s The Mathematics Teacher, the NCTM’s The Arithmetic Teacher, School Science and Mathematics, The Oregon Mathematics Teacher, and The Computing Teacher. In addition, he is a coauthor of two other college mathematics books: College Geometry—A Problem-Solving Approach with Applications (2008) and A Mathematical View of Our World (2007). He also coauthored the K-8 series Mathematics in Action. He has given more than 65 invited lectures/ workshops at a variety of conferences, including NCTM and MAA conferences, and was awarded 15 federal, state, and local grants to improve the teaching of mathematics.

While Professor Musser was at OSU, he was awarded the university’s prestigious College of Science Carter Award for Teaching. He is currently living in sunny Las Vegas, were he continues to write, ponder the mysteries of the stock market, enjoy living with his wife and his faithful yellow lab, Zoey.

Blake E. Peterson is currently a Professor in the Department of Mathematics Educa- tion at Brigham Young University. He was born and raised in Logan, Utah, where he graduated from Logan High School. Before completing his BA in secondary mathe- matics education at Utah State University, he spent two years in Japan as a missionary for The Church of Jesus Christ of Latter Day Saints. After graduation, he took his new wife, Shauna, to southern California, where he taught and coached at Chino High School for two years. In 1988, he began graduate school at Washington State Univer- sity, where he later completed a M.S. and Ph.D. in pure mathematics.

After completing his Ph.D., Dr. Peterson was hired as a mathematics educator in the Department of Mathematics at Oregon State University in Corvallis, Oregon, where he taught for three years. It was at OSU where he met Gary Musser. He has since moved his wife and four children to Provo, Utah, to assume his position at Brigham Young University where he is currently a full professor.

Dr. Peterson has published papers in Rocky Mountain Mathematics Journal, The American Mathematical Monthly, The Mathematical Gazette, Mathematics Magazine, The New England Mathematics Journal, School Science and Mathematics, The Journal of Mathematics Teacher Education, and The Journal for Research in Mathematics as well as chapters in several books. He has also published in NCTM’s Mathematics Teacher, and Mathematics Teaching in the Middle School. His research interests are teacher education in Japan and productive use of student mathematical thinking during instruction, which is the basis of an NSF grant that he and 3 of his colleagues were recently awarded. In addition to teaching, research, and writing, Dr. Peterson has done consulting for the College Board, founded the Utah Association of Mathematics Teacher Educators, and has been the chair of the editorial panel for the Mathematics Teacher.

Aside from his academic interests, Dr. Peterson enjoys spending time with his family, fulfilling his church responsi- bilities, playing basketball, mountain biking, water skiing, and working in the yard.

v

ABOUT THE AUTHORS

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vi

Are you puzzled by the numbers on the cover? They are 25 different randomly selected counting numbers from 1 to 100. In that set of numbers, two different arithmetic pro- gressions are highlighted. (An arithmetic progression is a sequence of numbers with a common difference between consecutive pairs.) For example, the sequence highlighted in green, namely 7, 15, 23, 31, is an arithmetic progression because the difference between 7 and 15 is 8, between 15 and 23 is 8, and between 23 and 31 is 8. Thus, the sequence 7, 15, 23, 31 forms an arithmetic progression of length 4 (there are 4 numbers in the sequence) with a common difference of 8. Similarly, the numbers highlighted in red, namely 45, 69, 93, form another arithmetic progression. This progression is of length 3 which has a common difference of 24.

You may be wondering why these arithmetic progressions are on the cover. It is to acknowledge the work of the mathematician Endre Szemerédi. On May 22, 2012, he was awarded the $1,000,000 Abel prize from the Norwegian Academy of Science and Letters for his analysis of such progressions. This award recognizes mathematicians for their contributions to mathematics that have a far reaching impact. One of Pro- fessor Szemerédi’s significant proofs is found in a paper he wrote in 1975. This paper proved a famous conjecture that had been posed by Paul Erdös and Paul Turán in 1936. Szemerédi’s 1975 paper and the Erdös/Turán conjecture are about finding arith- metic progressions in random sets of counting numbers (or integers). Namely, if one randomly selects half of the counting numbers from 1 and 100, what lengths of arith- metic progressions can one expect to find? What if one picks one-tenth of the numbers from 1 to 100 or if one picks half of the numbers between 1 and 1000, what lengths of arithmetic progressions is one assured to find in each of those situations? While the result of Szemerédi’s paper was interesting, his greater contribution was that the tech- nique used in the proof has been subsequently used by many other mathematicians.

Now let’s go back to the cover. Two progressions that were discussed above, one of length 4 and one of length 3, are shown in color. Are there others of length 3? Of length 4? Are there longer ones? It turns out that there are a total of 28 different arithmetic progressions of length three, 3 arithmetic progressions of length four and 1 progression of length five. See how many different progressions you can find on the cover. Perhaps you and your classmates can find all of them.

ABOUT THE COVER

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viivii

1 Introduction to Problem Solving 2

2 Sets, Whole Numbers, and Numeration 42

3 Whole Numbers: Operations and Properties 84

4 Whole Number Computation—Mental, Electronic, and Written 128

5 Number Theory 174

6 Fractions 206

7 Decimals, Ratio, Proportion, and Percent 250

8 Integers 302

9 Rational Numbers, Real Numbers, and Algebra 338

10 Statistics 412

11 Probability 484

12 Geometric Shapes 546

13 Measurement 644

14 Geometry Using Triangle Congruence and Similarity 716

15 Geometry Using Coordinates 780

16 Geometry Using Transformations 820

Epilogue: An Eclectic Approach to Geometry 877

Topic 1 Elementary Logic 881

Topic 2 Clock Arithmetic: A Mathematical System 891

Answers to Exercise/Problem Sets A and B, Chapter Reviews, Chapter Tests, and Topics Section A1

Index I1

Contents of Book Companion Web Site

Resources for Technology Problems

Technology Tutorials

Webmodules

Additional Resources

Videos

BRIEF CONTENTS

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viii

Preface xi

1 Introduction to Problem Solving 2 1.1 The Problem-Solving Process and Strategies 5 1.2 Three Additional Strategies 21

2 Sets, Whole Numbers, and Numeration 42 2.1 Sets as a Basis for Whole Numbers 45 2.2 Whole Numbers and Numeration 57 2.3 The Hindu–Arabic System 67

3 Whole Numbers: Operations and Properties 84 3.1 Addition and Subtraction 87 3.2 Multiplication and Division 101 3.3 Ordering and Exponents 116

4 Whole Number Computation—Mental, Electronic, and Written 128 4.1 Mental Math, Estimation, and Calculators 131 4.2 Written Algorithms for Whole-Number Operations 145 4.3 Algorithms in Other Bases 162

5 Number Theory 174 5.1 Primes, Composites, and Tests for Divisibility 177 5.2 Counting Factors, Greatest Common Factor, and Least

Common Multiple 190

6 Fractions 206 6.1 The Set of Fractions 209 6.2 Fractions: Addition and Subtraction 223 6.3 Fractions: Multiplication and Division 233

7 Decimals, Ratio, Proportion, and Percent 250 7.1 Decimals 253 7.2 Operations with Decimals 262 7.3 Ratio and Proportion 274 7.4 Percent 283

8 Integers 302 8.1 Addition and Subtraction 305 8.2 Multiplication, Division, and Order 318

CONTENTS

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ix

9 Rational Numbers, Real Numbers, and Algebra 338 9.1 The Rational Numbers 341 9.2 The Real Numbers 358 9.3 Relations and Functions 375 9.4 Functions and Their Graphs 391

10 Statistics 412 10.1 Statistical Problem Solving 415 10.2 Analyze and Interpret Data 440 10.3 Misleading Graphs and Statistics 460

11 Probability 484 11.1 Probability and Simple Experiments 487 11.2 Probability and Complex Experiments 502 11.3 Additional Counting Techniques 518 11.4 Simulation, Expected Value, Odds, and Conditional

Probability 528

12 Geometric Shapes 546 12.1 Recognizing Geometric Shapes—Level 0 549 12.2 Analyzing Geometric Shapes—Level 1 564 12.3 Relationships Between Geometric Shapes—Level 2 579 12.4 An Introduction to a Formal Approach to Geometry 589 12.5 Regular Polygons, Tessellations, and Circles 605 12.6 Describing Three-Dimensional Shapes 620

13 Measurement 644 13.1 Measurement with Nonstandard and Standard Units 647 13.2 Length and Area 665 13.3 Surface Area 686 13.4 Volume 696

14 Geometry Using Triangle Congruence and Similarity 716 14.1 Congruence of Triangles 719 14.2 Similarity of Triangles 729 14.3 Basic Euclidean Constructions 742 14.4 Additional Euclidean Constructions 755 14.5 Geometric Problem Solving Using Triangle Congruence

and Similarity 765

15 Geometry Using Coordinates 780 15.1 Distance and Slope in the Coordinate Plane 783 15.2 Equations and Coordinates 795 15.3 Geometric Problem Solving Using Coordinates 807

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x

16 Geometry Using Transformations 820 16.1 Transformations 823 16.2 Congruence and Similarity Using Transformations 846 16.3 Geometric Problem Solving Using Transformations 863

Epilogue: An Eclectic Approach to Geometry 877

Topic 1. Elementary Logic 881

Topic 2. Clock Arithmetic: A Mathematical System 891

Answers to Exercise/Problem Sets A and B, Chapter Reviews, Chapter Tests, and Topics Section A1

Index I1

Contents of Book Companion Web Site Resources for Technology Problems

eManipulatives Spreadsheet Activities Geometer’s Sketchpad Activities

Technology Tutorials Spreadsheets Geometer’s Sketchpad Programming in Logo Graphing Calculators

Webmodules Algebraic Reasoning Children’s Literature Introduction to Graph Theory

Additional Resources Guide to Problem Solving Problems for Writing/Discussion Research Articles Web Links

Videos Book Overview Author Walk-Through Videos Children’s Videos

FMBriefContents.indd 10 7/31/2013 12:29:55 PM

PREFACE

W elcome to the study of the foundations of ele-mentary school mathematics. We hope you will find your studies enlightening, useful, and fun. We salute you for choosing teaching as a profession and hope that your experiences with this book will help prepare you to be the best possible teacher of mathematics that you can be. We have presented this elementary mathematics material from a variety of perspectives so that you will be better equipped to address that broad range of learning styles that you will encounter in your future students. This book also encourages prospective teachers to gain the ability to do the mathematics of elementary school and to understand the underlying concepts so they will be able to assist their students, in turn, to gain a deep understand- ing of mathematics.

We have also sought to present this material in a man- ner consistent with the recommendations in (1) The Mathematical Education of Teachers prepared by the Conference Board of the Mathematical Sciences, (2) the National Council of Teachers of Mathematics’ Standards Documents, and (3) The Common Core State Standards for Mathematics. In addition, we have received valuable advice from many of our colleagues around the United States through questionnaires, reviews, focus groups, and personal communications. We have taken great care to respect this advice and to ensure that the content of the book has mathematical integrity and is accessible and helpful to the variety of students who will use it. As al- ways, we look forward to hearing from you about your experiences with our text.

GARY L. MUSSER, [email protected] BLAKE E. PETERSON, [email protected]

Unique Content Features Number Systems The order in which we present the number systems in this book is unique and most relevant to elementary school teachers. The topics are covered to parallel their evolution historically and their development in the elementary/middle school curriculum. Fractions and integers are treated separately as an extension of the whole numbers. Then rational numbers can be treated at a brisk pace as extensions of both fractions (by adjoining their opposites) and integers (by adjoining their appro- priate quotients) since students have a mastery of the concepts of reciprocals from fractions (and quotients) and opposites from integers from preceding chapters. Longtime users of this book have commented to us that this whole numbers-fractions-integers-rationals-reals

approach is clearly superior to the seemingly more effi- cient sequence of whole numbers-integers-rationals-reals that is more appropriate to use when teaching high school mathematics.

Approach to Geometry Geometry is organized from the point of view of the five-level van Hiele model of a child’s development in geometry. After studying shapes and measurement, geometry is approached more formally through Euclidean congruence and similarity, coordinates, and transformations. The Epilogue provides an eclectic approach by solving geometry problems using a variety of techniques.

Additional Topics Topic 1, “Elementary Logic,” may be used anywhere in a course.

Topic 2, “Clock Arithmetic: A Mathematical System,” uses the concepts of opposite and reciprocal and hence may be most instructive after Chapter 6, “Fractions,” and Chapter 8, “Integers,” have been completed. This section also contains an introduction to modular arithmetic.

Underlying Themes Problem Solving An extensive collection of problem- solving strategies is developed throughout the book; these strategies can be applied to a generous supply of problems in the exercise/problem sets. The depth of problem-solving coverage can be varied by the number of strategies selected throughout the book and by the problems assigned.

Deductive Reasoning The use of deduction is pro- moted throughout the book The approach is gradual, with later chapters having more multistep problems. In particular, the last sections of Chapters 14, 15, and 16 and the Epilogue offer a rich source of interesting theo- rems and problems in geometry.

Technology Various forms of technology are an inte- gral part of society and can enrich the mathematical understanding of students when used appropriately. Thus, calculators and their capabilities (long division with remainders, fraction calculations, and more) are introduced throughout the book within the body of the text.

In addition, the book companion Web site has eMa- nipulatives, spreadsheets, and sketches from Geometer’s

xi

FMPreface.indd 11 8/1/2013 12:05:27 PM

xii Preface

Sketchpad®. The eManipulatives are electronic versions of the manipulatives commonly used in the elementary classroom, such as the geoboard, base ten blocks, black and red chips, and pattern blocks. The spreadsheets contain dynamic representations of functions, statistics, and probability simulations. The sketches in Geometer’s Sketchpad® are dynamic representations of geomet- ric relationships that allow exploration. Exercises and problems that involve eManipulatives, spreadsheets, and Geometer’s Sketchpad® sketches have been integrated into the problem sets throughout the text.

Course Options We recognize that the structure of the mathematics for elementary teachers course will vary depending upon the college or university. Thus, we have organized this text so that it may be adapted to accommodate these differences.

Basic course: Chapters 1-7 Basic course with logic: Topic 1, Chapters 1–7 Basic course with informal geometry: Chapters 1–7,

12 Basic course with introduction to geometry and mea-

surement: Chapters 1–7, 12, 13

Summary of Changes to the Tenth Edition

Mathematical Tasks have been added to sections throughout the book to allow instructors more flex- ibility in how they choose to organize their classroom instruction. These tasks are designed to be investigated by the students in class. As the solutions to these tasks are discussed by students and the instructor, the big ideas of the section emerge and can be solidified through a classroom discussion.

Chapter 6 contains a new discussion of fractions on a number line to be consistent with the Common Core standards.

Chapter 10 has been revised to include a discus- sion of recommendations by the GAISE document and the NCTM Principles and Standards for School Mathematics. These revisions include a discussion of steps to statistical problem solving. Namely, (1) formulate questions, (2) collect data, (3) organize and display data, (4) analyze and interpret data. These steps are then applied in several of the examples through the chapter.

Chapter 12 has been substantially revised. Sections 12.1, 12.2, and 12.3 have been organized to parallel the first three van Hiele levels. In this way, students will be able to pass through the levels in a more meaningful fashion so that they will get a strong feeling about how

their students will view geometry at various van Hiele levels.

Chapter 13 contains several new examples to give stu- dents the opportunity to see how the various equations for area and volume are applied in different contexts.

Children’s Videos are videos of children solving math- ematical problems linked to QR codes placed in the margin of the book in locations where the content being discussed is related to the content of the prob- lems being solved by the children. These videos will bring the mathematical content being studied to life.

Author Walk-Throughs are videos linked to the QR code on the third page of each chapter. These brief videos are of an author, Blake Peterson, describing and showing points of major emphasis in each chapter so students’ study can be more focused.

Children’s Literature and Reflections from Research margin notes have been revised/refreshed.

Common Core margin notes have been added through- out the text to highlight the correlation between the content of this text and the Common Core standards.

Professional recommendation statements from the Common Core State Standards for Mathematics, the National Council of Teachers of Mathematics’ Principles and Standards for School Mathematics, and the Curriculum Focal Points, have been compiled on the third page of each chapter.

Pedagogy The general organization of the book was motivated by the following mathematics learning cube:

The three dimensions of the cube—cognitive levels, representational levels, and mathematical content—are integrated throughout the textual material as well as in the problem sets and chapter tests. Problem sets are organized into exercises (to support knowledge, skill, and understanding) and problems (to support problem solv- ing and applications).

FMPreface.indd 12 8/1/2013 12:05:27 PM

Preface xiii

We have developed new pedagogical features to imple- ment and reinforce the goals discussed above and to address the many challenges in the course.

Summary of Pedagogical Changes to the Tenth Edition

Student Page Snapshots have been updated.

Reflection from Research margin notes have been edited and updated.

Mathematical Structure reveals the mathematical ideas of the book. Main Definitions, Theorems, and Properties in each section are highlighted in boxes for quick review.

Children’s Literature references have been edited and updated. Also, there is additional material offered on the Web site on this topic.

Check for Understanding have been updated to reflect the revision of the problem sets.

Mathematical Tasks have been integrated throughout.

Author Walk-Throughs videos have been made avail- able via QR codes on the third page of every chapter.

Children’s videos, produced by Blake Peterson and available via QR codes, have been integrated through- out.

Key Features Problem-Solving Strategies are integrated throughout the book. Six strategies are introduced in Chapter 1. The last strategy in the strategy box at the top of the second page of each chapter after Chapter l contains a new strategy.

Mathematical Tasks are located in various places throughout each section. These tasks can be presented to the whole class or small groups to investigate. As the stu-

dents discuss their solutions with each other and the instructor, the big mathematical ideas of the sec- tion emerge.

FMPreface.indd 13 8/1/2013 12:05:28 PM

xiv Preface

Technology Problems appear in the Exercise/Problem sets throughout the book. These problems rely on and are enriched by the use of technology. The tech- nology used includes activities from the eManipulaties (virtual manipulatives),

spreadsheets, Geometer’s Sketchpad®, and the TI-34 II MultiView. Most of these technological resources can be accessed through the accompany- ing book companion Web site.

Student Page Snapshots have been updated. Each chapter has a page from an elementary school textbook relevant to the material being studied. Exercise/Problem Sets are separated into Part A

(all answers are provided in the back of the book and all solutions are provided in our supplement Hints and Solutions for Part A Problems) and Part B (answers are only provided in the Instructors Resource Manual). In addition, exercises and problems are distinguished so that students can learn how they differ.

Analyzing Student Thinking Problems are found at the end of the Exercise/Problem Sets. These problems are questions that elementary students might ask their teachers, and they focus on common misconceptions that are held by students. These problems give future teachers an opportunity to think about the concepts they have learned in the sec- tion in the context of teaching.

Curriculum Standards The NCTM Standards and Curriculum Focal Points and the Common Core State Standards are introduced on the third page of each chapter. In addition, margin notes involving these standards are contained throughout the book.

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Preface xv

Historical Vignettes open each chapter and introduce ideas and concepts central to each chapter.

Mathematical Morsels end every setion with an interesting historical tidbit. One of our students referred to these as a reward for completing the section.

Children’s Videos are author-led videos of children solving mathematical problems linked to QR codes in the margin of the book. The codes are placed in locations where the content being discussed is related to the content of the problems being solved by the children. These videos provide a window into how children think mathematically.

B la

ke E

. P et

er so

n

See one Live!

Reflection from Research Extensive research has been done in the mathematics education community that

focuses on the teaching and learning of elemen- tary mathematics. Many important quotations from research are given in the margins to sup- port the content nearby.

Children’s Literature These margin inserts provide many examples of books that can be used to connect reading and mathematics. They should be invaluable to you when you begin teachig.

FMPreface.indd 15 8/1/2013 12:05:34 PM

xvi Preface

People in Mathematics, a feature near the end of each chapter, high- lights many of the giants in mathemat- ics throughout history.

A Chapter Review is located at the end of each chapter.

A Chapter Test is found at the end of each chapter.

An Epilogue, following Chapter 16, provides a rich eclectic approach to geometry.

Logic and Clock Arithmetic are developed in topic sections near the end of the book.

Supplements for Students Student Activities Manual with Discussion Questions for the Classroom This activity manual is designed to enhance student learning as well as to model effective classroom practices. Since many instructors are working with students to create a personalized journal, this edition of the manual is shrink-wrapped and three-hole punched for easy customization. This supplement is an extensive revi- sion of the Student Resoure Handbook that was authored by Karen Swenson and Marcia Swanson for the first six editions of this book.

ISBN 978-1-118-67904-3

Features Include:

Hands-On Activities: Activities that help develop initial understandings at the concrete level. Discussion Questions for the Classroom: Tasks designed to engage students with mathematical ideas by stimulating communication. Mental Math: Short activities to help develop mental math skills. Exercises: Additional practice for building skills in concepts. Directions in Education: Specially written articles that provide insights into major issues of the day, including the Standards of the National Council of Teachers of Mathematics. Solutions: Solutions to all items in the handbook to enhance self-study. Two-Dimensional Manipulatives: Cutouts are provided on cardstock.

—Prepared by Lyn Riverstone of Oregon State University

The ETA Cuisenalre® Physical Manipulative Kit A generous assortment of manipulatives (including blocks, tiles, geoboards, and so forth) has been created to accompany the text as well as the Student Activity Manual. lt is available to be packaged with the text. Please contact your local Wiley representative for ordering information.

ISBN 978-1-118-67923-4

Student Hints and Solutions Manual for Part A Problems This manual contains hints and solutions to all of the Part A problems. It can be used to help students develop problem-solving profi- ciency in a self-study mode. The features include:

FMPreface.indd 16 8/1/2013 12:05:35 PM

Preface xvii

Hints: Give students a start on all Part A problems in the text.

Additional Hints: A second hint is provided for more challenging problems.

Complete Solutions to Part A Problems: Carefully written-out solutions are provided to model one correct solution.

—Developed by Lynn Trimpe, Vikki Maurer, and Roger Maurer of Linn-Benton Community College.

ISBN 978-1-118-67925-8

Companion Web site http://www.wiley.com/college/musser The companion Web site provides a wealth of resources for students.

Resources for Technology Problems These problems are integrated into the problem sets throughout the book and are denoted by a mouse icon.

eManipulatives mirror physical manipulatives as well as provide dynamic representations of other mathematical situations. The goal of using the eManipulatives is to engage learners in a way that will lead to a more in-depth understanding of the concepts and to give them experience thinking about the mathematics that underlies the manipulatives.

—Prepared by Lawrence O. Cannon, E. Robert Heal, and Joel Duffin of Utah State University, Richard Wellman of Westminster College, and Ethalinda K. S. Cannon of A415software.com.

This project is supported by the National Science Foundation.

The Geometer’s Sketchpad® activities allow students to use the dynamic capabilities of this software to investigate geometric properties and relationships. They are accessible through a Web browser so having the software is not necessary.

The Spreadsheet activities utilize the iterative properties of spreadsheets and the user friendly interface to investigate problems ranging from graphs of functions to standard deviation to simulations of rolling dice.

Technology Tutorials The Geometer’s Sketchpad® tutorial is written for those students who have access to the software and who are interested in investigating problems of their own choosing. The tutorial gives basic instruction on how to use the software and includes some sample problems that will help the students gain a better understanding of the software and the geometry that could be learned by using it.

—Prepared by Armando Martinez-Cruz, California State University, Fullerton.

The Spreadsheet Tutorial is written for students who are interested in learning how to use spreadsheets to investi- gate mathematical problems. The tutorial describes some of the functions of the software and provides exercises for students to investigate mathematics using the software.

—Prepared by Keith Leatham, Brigham Young University.

Webmodules The Algebraic Reasoning Webmodule helps students understand the critical transition from arithmetic to algebra. It also highlights situations when algebra is, or can be, used. Marginal notes are placed in the text at the appropriate locations to direct students to the webmodule.

—Prepared by Keith Leatham, Brigham Young University.

The Children’s Literature Webmodule provides references to many mathematically related examples of children’s books for each chapter. These references are noted in the margins near the mathematics that corresponds to the content of the book. The webmodule also contains ideas about using children’s literature in the classroom.

—Prepared by Joan Cohen Jones, Eastern Michigan University.

FMPreface.indd 17 8/1/2013 12:05:35 PM

xviii Preface

The Introduction to Graph Theory Webmodule has been moved from the Topics to the companion Web site to save space in the book and yet allow professors the flexibility to download it from the Web if they choose to use it.

The companion Web site also includes:

Links to NCTM Standards

Links to Common Core Standards

A Logo and TI-83 graphing calculator tutorial

Four cumulative tests covering material up to the end of Chapters 4, 9, 12, and 16

Research Article References: A complete list of references for the research articles that are mentioned in the Reflection from Research margin notes throughout the book

Guide to Problem Solving This valuable resource, available as a webmodule on the companion Web site, contains more than 200 creative problems keyed to the problem solving strategies in the textbook and includes:

Opening Problem: an introductory problem to motivate the need for a strategy.

Solution/Discussion/Clues: A worked-out solution of the opening problem together with a discussion of the strategy and some clues on when to select this strategy.

Practice Problems: A second problem that uses the same strategy together with a worked out solution and two practice problems.

Mixed Strategy Practice: Four practice problems that can be solved using one or more of the strategies introduced to that point.

Additional Practice Problems and Additional Mixed Strategy Problems: Sections that provide more practice for par- ticular strategies as well as many problems for which students need to identify appropriate strategies.

—Prepared by Don Miller, who retired as a professor of mathematics at St. Cloud State University.

Problems for Writing and Discussion are problems that require an analysis of ideas and are good opportunities to write about the concepts in the book. Most of the Problems for Writing/Discussion that preceded the Chapter Tests in the Eighth Edition now appear on our Web site.

The Geometer’s Sketchpad© Developed by Key Curriculum Press, this dynamic geometry construction and exploration tool allows users to create and manipulate precise figures while preserving geometric relationships. This software is only available when packaged with the text. Please contact your local Wiley representative for further details.

WileyPLUS WileyPLUS is a powerful online tool that will help you study more effectively, get immediate feedback when you practice on your own, complete assignments and get help with problem solving, and keep track of how you’re doing—all at one easy-to-use Web site.

Resources for the Instructor Companion Web Site The companion Web site is available to text adopters and provides a wealth of resources including:

PowerPoint Slides of more than 190 images that include figures from the text and several generic masters for dot paper, grids, and other formats.

Instructors also have access to all student Web site features. See above for more details.

Instructor Resource Manual This manual contains chapter-by-chapter discussions of the text material, student “expectations” (objectives) for each chapter, answers for all Part B exercises and problems, and answers for all of the even-numbered problems in the Guide to Problem-Solving.

—Prepared by Lyn Riverstone, Oregon State University ISBN 978-1-118-67924-1

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Preface xix

Computerized/Print Test Bank The Computerized/Printed Test Bank includes a collection of over 1,100 open response, multiple-choice, true/false, and free-response questions, nearly 80% of which are algorithmic.

—Prepared by Mark McKibben, Goucher College

WileyPLUS WileyPLUS is a powerful online tool that provides instructors with an integrated suite of resources, including an online version of the text, in one easy-to-use Web site. Organized around the essential activities you perform in class, WileyPLUS allows you to create class presentations, assign homework and quizzes for automatic grading, and track student progress. Please visit http://edugen.wiley.com or contact your local Wiley representative for a demonstration and further details.

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ACKNOWLEDGMENTS

During the development of Mathematics for Elementary Teach- ers, Eighth, Ninth, and Tenth Editions, we benefited from comments, suggestions, and evaluations from many of our col- leagues. We would like to acknowledge the contributions made by the following people:

Reviewers for the Tenth Edition

Meg Kiessling, University of Tennessee at Chattanooga Juli Ratheal, University of Texas Permian Basin Marie Franzosa, Oregon State University Mary Beth Rollick, Kent State University Linda Lefevre, SUNY Oswego

Reviewers for the Ninth Edition

Larry Feldman, Indiana University of Pennsylvania Sarah Greenwald, Appalachian State University Leah Gustin, Miami University of Ohio, Middleton Linda LeFevre, State University of New York, Oswego Bethany Noblitt, Northern Kentucky University Todd Cadwallader Olsker, California State University, Fullerton Cynthia Piez, University of Idaho Tammy Powell-Kopilak, Dutchess Community College Edel Reilly, Indiana University of Pennsylvania Sarah Reznikoff, Kansas State University Mary Beth Rollick, Kent State University

Ninth Edition Interviewees

John Baker, Indiana University of Pennsylvania Paulette Ebert, Northern Kentucky University Gina Foletta, Northern Kentucky University Leah Griffith, Rio Hondo College Jane Gringauz, Minneapolis Community College Alexander Kolesnick, Ventura College Gail Laurent, College of DuPage Linda LeFevre, State University of New York, Oswego Carol Lucas, University of Central Oklahoma Melanie Parker, Clarion University of Pennsylvania Shelle Patterson, Murray State University Cynthia Piez, University of Idaho Denise Reboli, King’s College Edel Reilly, Indiana University of Pennsylvania Sarah Reznikoff, Kansas State University Nazanin Tootoonchi, Frostburg State University

Ninth Edition Focus Group Participants

Kaddour Boukkabar, California University of Pennsylvania Melanie Branca, Southwestern College Tommy Bryan, Baylor University Jose Cruz, Palo Alto College Arlene Dowshen, Widener University Rita Eisele, Eastern Washington University Mario Flores, University of Texas at San Antonio Heather Foes, Rock Valley College

Mary Forintos, Ferris State University Marie Franzosa, Oregon State University Sonia Goerdt, St. Cloud State University Ralph Harris, Fresno Pacific University George Jennings, California State University, Dominguez Hills Andy Jones, Prince George’s Community College Karla Karstens, University of Vermont Margaret Kidd, California State University, Fullerton Rebecca Metcalf, Bridgewater State College Pamela Miller, Arizona State University, West Jessica Parsell, Delaware Technical Community College Tuyet Pham, Kent State University Mary Beth Rollick, Kent State University Keith Salyer, Central Washington University Sherry Schulz, College of the Canyons Carol Steiner, Kent State University Abolhassan Tagavy, City College of Chicago Rick Vaughan, Paradise Valley Community College Demetria White, Tougaloo College John Woods, Southwestern Oklahoma State University

In addition, we would like to acknowledge the contributions made by colleagues from earlier editions.

Reviewers for the Eighth Edition

Seth Armstrong, Southern Utah University Elayne Bowman, University of Oklahoma Anne Brown, Indiana University, South Bend David C. Buck, Elizabethtown Alison Carter, Montgomery College Janet Cater, California State University, Bakersfield Darwyn Cook, Alfred University Christopher Danielson, Minnesota State University, Mankato Linda DeGuire, California State University, Long Beach Cristina Domokos, California State University, Sacramento Scott Fallstrom, University of Oregon Teresa Floyd, Mississippi College Rohitha Goonatilake, Texas A&M International University Margaret Gruenwald, University of Southern Indiana Joan Cohen Jones, Eastern Michigan University Joe Kemble, Lamar University Margaret Kinzel, Boise State University J. Lyn Miller, Slippery Rock University Girija Nair-Hart, Ohio State University, Newark Sandra Nite, Texas A&M University Sally Robinson, University of Arkansas, Little Rock Nancy Schoolcraft, Indiana University, Bloomington Karen E. Spike, University of North Carolina, Wilmington Brian Travers, Salem State Mary Wiest, Minnesota State University, Mankato Mark A. Zuiker, Minnesota State University, Mankato

Student Activity Manual Reviewers

Kathleen Almy, Rock Valley College Margaret Gruenwald, University of Southern Indiana

xx

FMAcknowledgments.indd 20 7/31/2013 12:26:16 PM

Acknowledgments xxi

Kate Riley, California Polytechnic State University Robyn Sibley, Montgomery County Public Schools

State Standards Reviewers

Joanne C. Basta, Niagara University Joyce Bishop, Eastern Illinois University Tom Fox, University of Houston, Clear Lake Joan C. Jones, Eastern Michigan University Kate Riley, California Polytechnic State University Janine Scott, Sam Houston State University Murray Siegel, Sam Houston State University Rebecca Wong, West Valley College

Reviewers

Paul Ache, Kutztown University Scott Barnett, Henry Ford Community College Chuck Beals, Hartnell College Peter Braunfeld, University of Illinois Tom Briske, Georgia State University Anne Brown, Indiana University, South Bend Christine Browning, Western Michigan University Tommy Bryan, Baylor University Lucille Bullock, University of Texas Thomas Butts, University of Texas, Dallas Dana S. Craig, University of Central Oklahoma Ann Dinkheller, Xavier University John Dossey, Illinois State University Carol Dyas, University of Texas, San Antonio Donna Erwin, Salt Lake Community College Sheryl Ettlich, Southern Oregon State College Ruhama Even, Michigan State University Iris B. Fetta, Clemson University Marjorie Fitting, San Jose State University Susan Friel, Math/Science Education Network, University of

North Carolina Gerald Gannon, California State University, Fullerton Joyce Rodgers Griffin, Auburn University Jerrold W. Grossman, Oakland University Virginia Ellen Hanks, Western Kentucky University John G. Harvey, University of Wisconsin, Madison Patricia L. Hayes, Utah State University, Uintah Basin Branch

Campus Alan Hoffer, University of California, Irvine Barnabas Hughes, California State University, Northridge Joan Cohen Jones, Eastern Michigan University Marilyn L. Keir, University of Utah Joe Kennedy, Miami University Dottie King, Indiana State University Richard Kinson, University of South Alabama Margaret Kinzel, Boise State University John Koker, University of Wisconsin David E. Koslakiewicz, University of Wisconsin, Milwaukee Raimundo M. Kovac, Rhode Island College Josephine Lane, Eastern Kentucky University Louise Lataille, Springfield College Roberts S. Matulis, Millersville University Mercedes McGowen, Harper College Flora Alice Metz, Jackson State Community College J. Lyn Miller, Slippery Rock University Barbara Moses, Bowling Green State University

Maura Murray, University of Massachusetts Kathy Nickell, College of DuPage Dennis Parker, The University of the Pacific William Regonini, California State University, Fresno James Riley, Western Michigan University Kate Riley, California Polytechnic State University Eric Rowley, Utah State University Peggy Sacher, University of Delaware Janine Scott, Sam Houston State University Lawrence Small, L.A. Pierce College Joe K. Smith, Northern Kentucky University J. Phillip Smith, Southern Connecticut State University Judy Sowder, San Diego State University Larry Sowder, San Diego State University Karen Spike, University of Northern Carolina, Wilmington Debra S. Stokes, East Carolina University Jo Temple, Texas Tech University Lynn Trimpe, Linn–Benton Community College Jeannine G. Vigerust, New Mexico State University Bruce Vogeli, Columbia University Kenneth C. Washinger, Shippensburg University Brad Whitaker, Point Loma Nazarene University John Wilkins, California State University, Dominguez Hills

Questionnaire Respondents

Mary Alter, University of Maryland Dr. J. Altinger, Youngstown State University Jamie Whitehead Ashby, Texarkana College Dr. Donald Balka, Saint Mary’s College Jim Ballard, Montana State University Jane Baldwin, Capital University Susan Baniak, Otterbein College James Barnard, Western Oregon State College Chuck Beals, Hartnell College Judy Bergman, University of Houston, Clearlake James Bierden, Rhode Island College Neil K. Bishop, The University of Southern Mississippi,

Gulf Coast Jonathan Bodrero, Snow College Dianne Bolen, Northeast Mississippi Community College Peter Braunfeld, University of Illinois Harold Brockman, Capital University Judith Brower, North Idaho College Anne E. Brown, Indiana University, South Bend Harmon Brown, Harding University Christine Browning, Western Michigan University Joyce W. Bryant, St. Martin’s College R. Elaine Carbone, Clarion University Randall Charles, San Jose State University Deann Christianson, University of the Pacific Lynn Cleary, University of Maryland Judith Colburn, Lindenwood College Sister Marie Condon, Xavier University Lynda Cones, Rend Lake College Sister Judith Costello, Regis College H. Coulson, California State University Dana S. Craig, University of Central Oklahoma Greg Crow, John Carroll University Henry A. Culbreth, Southern Arkansas University, El Dorado Carl Cuneo, Essex Community College Cynthia Davis, Truckee Meadows Community College

FMAcknowledgments.indd 21 7/31/2013 12:26:16 PM

xxii Acknowledgments

Gregory Davis, University of Wisconsin, Green Bay Jennifer Davis, Ulster County Community College Dennis De Jong, Dordt College Mary De Young, Hop College Louise Deaton, Johnson Community College Shobha Deshmukh, College of Saint Benedict/St.

John’s University Sheila Doran, Xavier University Randall L. Drum, Texas A&M University P. R. Dwarka, Howard University Doris Edwards, Northern State College Roger Engle, Clarion University Kathy Ernie, University of Wisconsin Ron Falkenstein, Mott Community College Ann Farrell, Wright State University Francis Fennell, Western Maryland College Joseph Ferrar, Ohio State University Chris Ferris, University of Akron Fay Fester, The Pennsylvania State University Marie Franzosa, Oregon State University Margaret Friar, Grand Valley State College Cathey Funk, Valencia Community College Dr. Amy Gaskins, Northwest Missouri State University Judy Gibbs, West Virginia University Daniel Green, Olivet Nazarene University Anna Mae Greiner, Eisenhower Middle School Julie Guelich, Normandale Community College Ginny Hamilton, Shawnee State University Virginia Hanks, Western Kentucky University Dave Hansmire, College of the Mainland Brother Joseph Harris, C.S.C., St. Edward’s University John Harvey, University of Wisconsin Kathy E. Hays, Anne Arundel Community College Patricia Henry, Weber State College Dr. Noal Herbertson, California State University Ina Lee Herer, Tri-State University Linda Hill, Idaho State University Scott H. Hochwald, University of North Florida Susan S. Hollar, Kalamazoo Valley Community College Holly M. Hoover, Montana State University, Billings Wei-Shen Hsia, University of Alabama Sandra Hsieh, Pasadena City College Jo Johnson, Southwestern College Patricia Johnson, Ohio State University Pat Jones, Methodist College Judy Kasabian, El Camino College Vincent Kayes, Mt. St. Mary College Julie Keener, Central Oregon Community College Joe Kennedy, Miami University Susan Key, Meridien Community College Mary Kilbridge, Augustana College Mike Kilgallen, Lincoln Christian College Judith Koenig, California State University, Dominguez Hills Josephine Lane, Eastern Kentucky University Don Larsen, Buena Vista College Louise Lataille, Westfield State College Vernon Leitch, St. Cloud State University Steven C. Leth, University of Northern Colorado Lawrence Levy, University of Wisconsin Robert Lewis, Linn-Benton Community College Lois Linnan, Clarion University

Jack Lombard, Harold Washington College Betty Long, Appalachian State University Ann Louis, College of the Canyons C. A. Lubinski, Illinois State University Pamela Lundin, Lakeland College Charles R. Luttrell, Frederick Community College Carl Maneri, Wright State University Nancy Maushak, William Penn College Edith Maxwell, West Georgia College Jeffery T. McLean, University of St. Thomas George F. Mead, McNeese State University Wilbur Mellema, San Jose City College Clarence E. Miller, Jr. Johns Hopkins University Diane Miller, Middle Tennessee State University Ken Monks, University of Scranton Bill Moody, University of Delaware Kent Morris, Cameron University Lisa Morrison, Western Michigan University Barbara Moses, Bowling Green State University Fran Moss, Nicholls State University Mike Mourer, Johnston Community College Katherine Muhs, St. Norbert College Gale Nash, Western State College of Colorado T. Neelor, California State University Jerry Neft, University of Dayton Gary Nelson, Central Community College, Columbus Campus James A. Nickel, University of Texas, Permian Basin Kathy Nickell, College of DuPage Susan Novelli, Kellogg Community College Jon O’Dell, Richland Community College Jane Odell, Richland College Bill W. Oldham, Harding University Jim Paige, Wayne State College Wing Park, College of Lake County Susan Patterson, Erskine College (retired) Shahla Peterman, University of Missouri Gary D. Peterson, Pacific Lutheran University Debra Pharo, Northwestern Michigan College Tammy Powell-Kopilak, Dutchess Community College Christy Preis, Arkansas State University, Mountain Home Robert Preller, Illinois Central College Dr. William Price, Niagara University Kim Prichard, University of North Carolina Stephen Prothero, Williamette University Janice Rech, University of Nebraska Tom Richard, Bemidji State University Jan Rizzuti, Central Washington University Anne D. Roberts, University of Utah David Roland, University of Mary Hardin–Baylor Frances Rosamond, National University Richard Ross, Southeast Community College Albert Roy, Bristol Community College Bill Rudolph, Iowa State University Bernadette Russell, Plymouth State College Lee K. Sanders, Miami University, Hamilton Ann Savonen, Monroe County Community College Rebecca Seaberg, Bethel College Karen Sharp, Mott Community College Marie Sheckels, Mary Washington College Melissa Shepard Loe, University of St. Thomas Joseph Shields, St. Mary’s College, MN

FMAcknowledgments.indd 22 7/31/2013 12:26:16 PM

Acknowledgments xxiii

Lawrence Shirley, Towson State University Keith Shuert, Oakland Community College B. Signer, St. John’s University Rick Simon, Idaho State University James Smart, San Jose State University Ron Smit, University of Portland Gayle Smith, Lane Community College Larry Sowder, San Diego State University Raymond E. Spaulding, Radford University William Speer, University of Nevada, Las Vegas Sister Carol Speigel, BVM, Clarke College Karen E. Spike, University of North Carolina, Wilmington Ruth Ann Stefanussen, University of Utah Carol Steiner, Kent State University Debbie Stokes, East Carolina University Ruthi Sturdevant, Lincoln University, MO Viji Sundar, California State University, Stanislaus Ann Sweeney, College of St. Catherine, MN Karen Swenson, George Fox College Carla Tayeh, Eastern Michigan University Janet Thomas, Garrett Community College S. Thomas, University of Oregon Mary Beth Ulrich, Pikeville College Martha Van Cleave, Linfield College Dr. Howard Wachtel, Bowie State University Dr. Mary Wagner-Krankel, St. Mary’s University Barbara Walters, Ashland Community College Bill Weber, Eastern Arizona College Joyce Wellington, Southeastern Community College Paula White, Marshall University Heide G. Wiegel, University of Georgia Jane Wilburne, West Chester University Jerry Wilkerson, Missouri Western State College Jack D. Wilkinson, University of Northern Iowa Carole Williams, Seminole Community College Delbert Williams, University of Mary Hardin–Baylor Chris Wise, University of Southwestern Louisiana John L. Wisthoff, Anne Arundel Community College (retired) Lohra Wolden, Southern Utah University Mary Wolfe, University of Rio Grande Vernon E. Wolff, Moorhead State University

Maria Zack, Point Loma Nazarene College Stanley L. Zehm, Heritage College Makia Zimmer, Bethany College

Focus Group Participants

Mara Alagic, Wichita State University Robin L. Ayers, Western Kentucky University Elaine Carbone, Clarion University of Pennsylvania Janis Cimperman, St. Cloud State University Richard DeCesare, Southern Connecticut State University Maria Diamantis, Southern Connecticut State University Jerrold W. Grossman, Oakland University Richard H. Hudson, University of South Carolina, Columbia Carol Kahle, Shippensburg University Jane Keiser, Miami University Catherine Carroll Kiaie, Cardinal Stritch University Armando M. Martinez-Cruz, California State University, Fuller- ton Cynthia Y. Naples, St. Edward’s University David L. Pagni, Fullerton University Melanie Parker, Clarion University of Pennsylvania Carol Phillips-Bey, Cleveland State University

Content Connections Survey Respondents

Marc Campbell, Daytona Beach Community College Porter Coggins, University of Wisconsin–Stevens Point Don Collins, Western Kentucky University Allan Danuff, Central Florida Community College Birdeena Dapples, Rocky Mountain College Nancy Drickey, Linfield College Thea Dunn, University of Wisconsin–River Falls Mark Freitag, East Stroudsberg University Paula Gregg, University of South Carolina, Aiken Brian Karasek, Arizona Western College Chris Kolaczewski, Ferris University of Akron R. Michael Krach, Towson University Randa Lee Kress, Idaho State University Marshall Lassak, Eastern Illinois University Katherine Muhs, St. Norbert College Bethany Noblitt, Northern Kentucky University

We would like to acknowledge the following people for their assistance in the preparation of our earlier editions of this book: Ron Bagwell, Jerry Becker, Julie Borden, Sue Borden, Tommy Bryan, Juli Dixon, Christie Gilliland, Dale Green, Kathleen Seagraves Hig- don, Hester Lewellen, Roger Maurer, David Metz, Naomi Munton, Tilda Runner, Karen Swenson, Donna Templeton, Lynn Trimpe, Rosemary Troxel, Virginia Usnick, and Kris Warloe. We thank Robyn Silbey for her expert review of several of the features in our seventh edition, Dawn Tuescher for her work on the correlation between the content of the book and the common core standards statements, and Becky Gwilliam for her research contributions to Chapter 10 and the Reflections from Research. Our Mathematical Morsels artist, Ron Bagwell, who was one of Gary Musser’s exceptional prospective elementary teacher students at Oregon State University, deserves special recognition for his creativity over all ten editions. We especially appreciate the extensive proofreading and revision suggestion for the problem sets provided by Jennifer A. Blue for this edition. We also thank Lyn Riverstone, Vikki Maurer, and Jen Blue for their careful checking of the accuracy of the answers.

We also want to acknowledge Marcia Swanson and Karen Swenson for their creation of and contribution to our Student Resource Handbook during the first seven editions with a special thanks to Lyn Riverstone for her expert revision of the Student Activity Manual since. Thanks are also due to Don Miller for his Guide to Problem Solving, to Lyn Trimpe, Roger Maurer, and Vikki Maurer, for their long- time authorship of our Student Hints and Solutions Manual, to Keith Leathem for the Spreadsheet Tutorial and Algebraic Reasoning Web Module, Armando Martinez-Cruz for The Geometer’s Sketchpad Tutorial, to Joan Cohen Jones for the Children’s Literature mar- gin inserts and the associated Webmodule, and to Lawrence O. Cannon, E. Robert Heal, Joel Duffin, Richard Wellman, and Ethalinda K. S. Cannon for the eManipulatives activities.

FMAcknowledgments.indd 23 7/31/2013 12:26:16 PM

xxiv Acknowledgments

We are very grateful to our publisher, Laurie Rosatone, and our editor, Jennifer Brady, for their commitment and super teamwork; to our exceptional senior production editor, Kerry Weinstein, for attending to the details we missed; to Elizabeth Chenette, copyedi- tor, Carol Sawyer, proofreader, and Christine Poolos, freelance editor, for their wonderful help in putting this book together; and to Melody Englund, our outstanding indexer. Other Wiley staff who helped bring this book and its print and media supplements to fruition are: Kimberly Kanakes, Marketing Manager; Sesha Bolisetty, Vice President, Production and Manufacturing; Karoline Luciano, Senior Content Manager; Madelyn Lesure, Senior Designer; Lisa Gee, Senior Photo Editor, and Thomas Kulesa, Senior Product Designer. They have been uniformly wonderful to work with—John Wiley would have been proud of them.

Finally, we welcome comments from colleagues and students. Please feel free to send suggestions to Gary at [email protected] and Blake at [email protected]. Please include both of us in any communications.

G.L.M. B.E.P.

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1

There are many pedagogical elements in our book which are designed to help you as you learn mathematics. We suggest the following:

1. Begin each chapter by reading the Focus On on the first page of the chapter. This will give you a mathematical sense of some of the history that underlies the chapter.

2. Try to work the Initial Problem on the second page of the chapter. Since problem solving is so important in mathematics, you will want to increase your profi- ciency in solving problems so that you can help your students to learn to solve problems. Also notice the Problem Solving Strategies box on this second page. This box grows throughout the book as you learn new strategies to help you enhance your problem solving ability.

3. The third page of each chapter contains three items. First, the QR code has an Author Walk-Through narrated by Blake where he will give you a brief preview of key ideas in the chapter. Next, there is a brief Introduction to the chapter that will also give you a sense of what is to come. Finally, there are three Lists of Recommendations that will be covered in the chapter. You will be reminded of the NCTM Principles and Standards for School Mathematics and the Common Core Standards in margin notes as you work through the chapter.

4. In addition to the QR code mentioned above, there are many other such codes throughout the book. These codes lead to brief Children’s Videos where children are solving problems involving the content near the code. These will give you a feeling of what it will be like when you are teaching.

5. Each section contains several Mathematical Tasks which are designed to be solved in groups so you can come to understand the concepts in the section through your investigation of these mathematical tasks. If these tasks are not used as part of your classroom instruction, you would benefit from trying them on your own and discussing your investigation with your peers or instructor.

6. When you finish studying a subsection, work the Set A exercises at the end of the section that are suggested by the Check for Understanding. This will help you learn the material in the section in smaller increments which can be a more effec- tive way to learn. The answers for these exercises are in the back of the book.

7. As you work through each section, take breaks and read through the margin notes Reflections from Research, NCTM Standards, Common Core, and Algebraic Reasoning. These should enrich your learning experience. Of course, the Children’s Literature margin notes should help you begin a list of materials that you can use when you begin to teach.

8. Be certain to read the Mathematical Morsel at the end of each section. These are stories that will enrich your learning experience.

9. By the time you arrive at the Exercise/Problem Set, you should have worked all of the exercises in Set A and checked your answers. This practice should have helped you learn the knowledge, skill, and understanding of the material in the section (see our illustrative cube in the Pedagogy section). Next you should attempt to work all of the Set A problems. These may require slightly deeper thinking than did the exercises. Once again, the answers to these problems are in the back of the book. Your teacher may assign some of the Set B exercises and problems. These do not have answers in this book, so you will have to draw on what you have learned from the Set A exercises and problems.

10. Finally, when you reach the end of the chapter, carefully work through the Chapter Review and the Chapter Test.

A NOTE TO OUR STUDENTS

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2

G eorge Pólya was born in Hungary in 1887. He received his Ph.D. at the University of Budapest. In 1940 he came to Brown University and then joined the faculty at Stanford University in 1942.

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In his studies, he became interested in the process of discovery, which led to his famous four-step process for solving problems:

1. Understand the problem.

2. Devise a plan.

3. Carry out the plan.

4. Look back.

Pólya wrote over 250 mathematical papers and three books that promote problem solving. His most famous book, How to Solve It, which has been translated

into 15 languages, introduced his four-step approach together with heuristics, or strategies, which are helpful in solving problems. Other important works by Pólya are Mathematical Discovery, Volumes 1 and 2, and Mathematics and Plausible Reasoning, Volumes 1 and 2.

He died in 1985, leaving mathematics with the impor- tant legacy of teaching problem solving. His “Ten Commandments for Teachers” are as follows:

1. Be interested in your subject.

2. Know your subject.

3. Try to read the faces of your students; try to see their expectations and difficulties; put yourself in their place.

4. Realize that the best way to learn anything is to dis- cover it by yourself.

5. Give your students not only information, but also know-how, mental attitudes, the habit of methodical work.

6. Let them learn guessing.

7. Let them learn proving.

8. Look out for such features of the problem at hand as may be useful in solving the problems to come—try to disclose the general pattern that lies behind the present concrete situation.

9. Do not give away your whole secret at once—let the students guess before you tell it—let them find out by themselves as much as is feasible.

10. Suggest; do not force information down their throats.

C H A P T E R

1 INTRODUCTION TO PROBLEM SOLVING George Pólya—The Father of Modern Problem Solving

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3

Problem-Solving Strategies 1. Guess and Test

2. Draw a Picture

3. Use a Variable

4. Look for a Pattern

5. Make a List

6. Solve a Simpler Problem

Because problem solving is the main goal of mathematics, this chapter introduces the six strategies listed in the Problem-Solving Strategies box that are helpful in solving problems. Then, at the beginning of each chapter, an initial problem is posed that can be solved by using the strategy introduced in that chapter. As you move through this book, the Problem-Solving Strategies boxes at the beginning of each chapter expand, as should your ability to solve problems.

Initial Problem Place the whole numbers 1 through 9 in the circles in the accompanying triangle so that the sum of the numbers on each side is 17.

A solution to this Initial Problem is on page 37.

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AUTHOR

WALK-THROUGH

4

I N T R O D U C T I O N Once, at an informal meeting, a social scientist asked a mathematics professor, “What’s the main goal of teaching mathematics?” The reply was “problem solving.” In return, the mathematician asked, “What is the main goal of teaching the social sciences?” Once more the answer was “problem solving.” All successful engineers, scientists, social scientists, lawyers, accountants, doctors, business managers, and so on have to be good problem solvers. Although the problems that people encounter may be very diverse, there are common elements and an underlying structure that can help to facilitate problem

solving. Because of the universal importance of problem solving, the main professional group in mathematics educa- tion, the National Council of Teachers of Mathematics (NCTM) recommended in its 1980 Agenda for Actions that “problem solving be the focus of school mathematics in the 1980s.” The NCTM’s 1989 Curriculum and Evaluation Standards for School Mathematics called for increased attention to the teaching of problem solving in K-8 mathemat- ics. Areas of emphasis include word problems, applications, patterns and relationships, open-ended problems, and problem situations represented verbally, numerically, graphically, geometrically, and symbolically. The NCTM’s 2000 Principles and Standards for School Mathematics identified problem solving as one of the processes by which all mathematics should be taught.

This chapter introduces a problem-solving process together with six strategies that will aid you in solving problems.

Key Concepts from the NCTM Principles and Standards for School Mathematics

PRE-K-12–PROBLEM SOLVING

Build new mathematical knowledge through problem solving. Solve problems that arise in mathematics and in other contexts. Apply and adapt a variety of appropriate strategies to solve problems. Monitor and reflect on the process of mathematical problem solving.

Key Concepts from the NCTM Curriculum Focal Points

KINDERGARTEN: Choose, combine, and apply effective strategies for answering quantitative questions. GRADE 1: Develop an understanding of the meanings of addition and subtraction and strategies to solve such

arithmetic problems. Solve problems involving the relative sizes of whole numbers. GRADE 3: Apply increasingly sophisticated strategies … to solve multiplication and division problems. GRADE 4 AND 5: Select appropriate units, strategies, and tools for solving problems. GRADE 6: Solve a wide variety of problems involving ratios and rates. GRADE 7: Use ratio and proportionality to solve a wide variety of percent problems.

Key Concepts from the Common Core State Standards for Mathematics

ALL GRADES

Mathematical Practice 1: Make sense of problems and persevere in solving them. Mathematical Practice 2: Reason abstractly and quantitatively. Mathematical Practice 3: Construct viable arguments and critique the reasoning of others. Mathematical Practice 4: Model with mathematics. Mathematical Practice 7: Look for and make use of structures.

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Section 1.1 The Problem-Solving Process and Strategies 5

Pólya’s Four Steps In this book we often distinguish between “exercises” and “problems.” Unfortunately, the distinction cannot be made precise. To solve an exercise, one applies a routine procedure to arrive at an answer. To solve a problem, one has to pause, reflect, and perhaps take some original step never taken before to arrive at a solution. This need for some sort of creative step on the solver’s part, however minor, is what distinguishes a problem from an exercise. To a young child, finding 3 2+ might be a problem, whereas it is a fact for you. For a child in the early grades, the question “How do you divide 96 pencils equally among 16 children?” might pose a problem, but for you it suggests the exercise “find 96 16÷ .” These two examples illustrate how the distinction between an exercise and a problem can vary, since it depends on the state of mind of the person who is to solve it.

Doing exercises is a very valuable aid in learning mathematics. Exercises help you to learn concepts, properties, procedures, and so on, which you can then apply when solving problems. This chapter provides an introduction to the process of problem solving. The techniques that you learn in this chapter should help you to become a better problem solver and should show you how to help others develop their problem- solving skills.

A famous mathematician, George Pólya, devoted much of his teaching to helping students become better problem solvers. His major contribution is what has become known as Pólya’s four-step process for solving problems.

Step 1 Understand the Problem

Do you understand all the words? Can you restate the problem in your own words? Do you know what is given? Do you know what the goal is? Is there enough information? Is there extraneous information? Is this problem similar to another problem you have solved?

Step 2 Devise a Plan

Can one of the following strategies (heuristics) be used? (A strategy is defi ned as an artful means to an end.)

Reflection from Research Many children believe that the answer to a word problem can always be found by adding, sub- tracting, multiplying, or dividing two numbers. Little thought is given to understanding the con- text of the problem (Verschaffel, De Corte, & Vierstraete, 1999).

Common Core – Grades K-12 (Mathematical Practice 1) Mathematically proficient stu- dents start by explaining to them- selves the meaning of a problem and looking for entry points to its solution.

Common Core – Grades K-12 (Mathematical Practice 1) Mathematically proficient stu- dents analyze givens, constraints, relationships, and goals. They make conjectures about the form and meaning of the solution and plan a solution pathway rather than simply jumping into a solu- tion attempt.

Use any strategy you know to solve the next problem. As you solve this problem, pay close attention to the thought processes and steps that you use. Write down these strate-

gies and compare them to a classmate’s. Are there any similarities in your approaches to solving this problem?

Lin’s garden has an area of 78 square yards. The length of the garden is 5 less than 3 times its width. What are the dimensions of Lin’s garden?

THE PROBLEM-SOLVING PROCESS AND STRATEGIES

1. Guess and test.

2. Draw a picture.

3. Use a variable.

4. Look for a pattern.

5. Make a list.

6. Solve a simpler problem.

7. Draw a diagram.

8. Use direct reasoning.

9. Use indirect reasoning.

10. Use properties of numbers.

11. Solve an equivalent problem.

12. Work backward.

13. Use cases.

14. Solve an equation.

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6 Chapter 1 Introduction to Problem Solving

The first six strategies are discussed in this chapter; the others are introduced in subsequent chapters.

Step 3 Carry Out the Plan

Implement the strategy or strategies that you have chosen until the problem is solved or until a new course of action is suggested.

Give yourself a reasonable amount of time in which to solve the problem. If you are not successful, seek hints from others or put the problem aside for a while. (You may have a flash of insight when you least expect it!)

Do not be afraid of starting over. Often, a fresh start and a new strategy will lead to success.

Step 4 Look Back

Is your solution correct? Does your answer satisfy the statement of the problem?

Can you see an easier solution?

Can you see how you can extend your solution to a more general case?

Usually, a problem is stated in words, either orally or written. Then, to solve the problem, one translates the words into an equivalent problem using mathematical symbols, solves this equivalent problem, and then interprets the answer. This process is summarized in Figure 1.1.

Figure 1.1

Learning to utilize Pólya’s four steps and the diagram in Figure 1.1 are first steps in becoming a good problem solver. In particular, the “Devise a Plan” step is very important. In this chapter and throughout the book, you will learn the strategies listed under the “Devise a Plan” step, which in turn help you decide how to proceed to solve problems. However, selecting an appropriate strategy is critical! As we worked with students who were successful problem solvers, we asked them to share “clues” that they observed in statements of problems that helped them select appropriate strategies. Their clues are listed after each corresponding strategy. Thus, in addition to learning how to use the various strategies herein, these clues can help you decide when to select an appropriate strategy or combination of strategies. Problem solving is as much an art as it is a science. Therefore, you will find that with experience you will develop a feeling for when to use one strategy over another by recognizing certain clues, perhaps subconsciously. Also, you will find that some problems may be solved in several ways using different strategies.

In summary, this initial material on problem solving is a foundation for your success in problem solving. Review this material on Pólya’s four steps as well as the strategies and clues as you continue to develop your expertise in solving problems.

Common Core – Grades K-12 (Mathematical Practice 1) Mathematically proficient stu- dents consider analogous prob- lems and try special cases and simpler forms of the original problem in order to gain insight into its solution.

Common Core – Grades K-12 (Mathematical Practice 1) Mathematically proficient stu- dents monitor and evaluate their progress and change course if necessary.

Reflection from Research Researchers suggest that teach- ers think aloud when solving problems for the first time in front of the class. In so doing, teachers will be modeling suc- cessful problem-solving behaviors for their students (Schoenfeld, 1985).

NCTM Standard Instructional programs should enable all students to apply and adapt a variety of appropriate strategies to solve problems.

15. Look for a formula.

16. Do a simulation.

17. Use a model.

18. Use dimensional analysis.

19. Identify subgoals.

20. Use coordinates.

21. Use symmetry.

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7

From Chapter 6, Lesson “Problem Solving” from My Math, Volume 1 Common Core State Standards, Grade 2, copyright © 2013 by McGraw-Hill Education.

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8 Chapter 1 Introduction to Problem Solving

Problem-Solving Strategies The remainder of this chapter is devoted to introducing several problem-solving strategies.

Guess and Test

Problem Place the digits 1, 2, 3, 4, 5, 6 in the circles in Figure 1.2 so that the sum of the three numbers on each side of the triangle is 12.

We will solve the problem in three ways to illustrate three different approaches to the Guess and Test strategy. As its name suggests, to use the Guess and Test strategy, you guess at a solution and test whether you are correct. If you are incorrect, you refine your guess and test again. This process is repeated until you obtain a solution.

Step 1 Understand the Problem

Each number must be used exactly one time when arranging the numbers in the triangle. The sum of the three numbers on each side must be 12.

First Approach: Random Guess and Test

Step 2 Devise a Plan

Tear off six pieces of paper and mark the numbers 1 through 6 on them and then try combinations until one works.

Step 3 Carry Out the Plan

Arrange the pieces of paper in the shape of an equilateral triangle and check sums. Keep rearranging until three sums of 12 are found.

Second Approach: Systematic Guess and Test

Step 2 Devise a Plan

Rather than randomly moving the numbers around, begin by placing the smallest numbers—namely, 1, 2, 3—in the corners. If that does not work, try increasing the numbers to 1, 2, 4, and so on.

Step 3 Carry Out the Plan

With 1, 2, 3 in the corners, the side sums are too small; similarly with 1, 2, 4. Try 1, 2, 5 and 1, 2, 6. The side sums are still too small. Next try 2, 3, 4, then 2, 3, 5, and so on, until a solution is found. One also could begin with 4, 5, 6 in the cor- ners, then try 3, 4, 5, and so on.

Third Approach: Inferential Guess and Test

Step 2 Devise a Plan

Start by assuming that 1 must be in a corner and explore the consequences.

Step 3 Carry Out the Plan

If 1 is placed in a corner, we must fi nd two pairs out of the remaining fi ve numbers whose sum is 11 (Figure 1.3). However, out of 2, 3, 4, 5, and 6, only 6 5 11+ = . Thus, we conclude that 1 cannot be in a corner. If 2 is in a corner, there must be two pairs left that add to 10 (Figure 1.4). But only 6 4 10+ = . Therefore, 2 cannot

Figure 1.2

Figure 1.3

Figure 1.4

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Section 1.1 The Problem-Solving Process and Strategies 9

be in a corner. Finally, suppose that 3 is in a corner. Then we must satisfy Figure 1.5. However, only 5 4 9+ = of the remaining numbers. Thus, if there is a solu- tion, 4, 5, and 6 will have to be in the corners (Figure 1.6). By placing 1 between 5 and 6, 2 between 4 and 6, and 3 between 4 and 5, we have a solution.

Step 4 Look Back

Notice how we have solved this problem in three different ways using Guess and Test. Random Guess and Test is often used to get started, but it is easy to lose track of the various trials. Systematic Guess and Test is better because you develop a scheme to ensure that you have tested all possibilities. Gener- ally, Inferential Guess and Test is superior to both of the previous methods because it usually saves time and provides more information regarding possible solutions.

Additional Problems Where the Strategy “Guess and Test” Is Useful

1. In the following cryptarithm—that is, a collection of words where the letters represent numbers—sun and fun represent two three-digit numbers, and swim is their four-digit sum. Using all of the digits 0, 1, 2, 3, 6, 7, and 9 in place of the letters where no letter represents two different digits, determine the value of each letter.

sun

fun

swim

+

Step 1 Understand the Problem

Each of the letters in sun, fun, and swim must be replaced with the numbers 0, 1, 2, 3, 6, 7, and 9, so that a correct sum results after each letter is replaced with its associated digit. When the letter n is replaced by one of the digits, then n n+ must be m or 10 + m, where the 1 in the 10 is carried to the tens column. Since 1 1 2 3 3 6+ = + =, , and 6 6 12+ = , there are three possibilities for n, namely, 1, 3, or 6. Now we can try various combinations in an attempt to obtain the correct sum.

Step 2 Devise a Plan

Use Inferential Guess and Test. There are three choices for n. Observe that sun and fun are three-digit numbers and that swim is a four-digit number. Thus we have to carry when we add s and f . Therefore, the value for s in swim is 1. This limits the choices of n to 3 or 6.

Step 3 Carry Out the Plan

Since s = 1 and s f+ leads to a two-digit number, f must be 9. Thus there are two possibilities:

(a) b

1 3

9 3

1 6

1 6

9 6

1 2

u

u

wi

u

u

wi

+ + ( )

In (a), if u = 0 2, , or 7, there is no value possible for i among the remaining digits. In (b), if u = 3, then u u+ plus the carry from 6 6+ yields i = 7. This leaves w = 0 for a solution.

Figure 1.6

Figure 1.5

NCTM Standard Instructional programs should enable all students to monitor and reflect on the process of mathematical problem solving.

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10 Chapter 1 Introduction to Problem Solving

Step 4 Look Back

The reasoning used here shows that there is one and only one solution to this problem. When solving problems of this type, one could randomly substitute digits until a solution is found. However, Inferential Guess and Test simplifi es the solution process by looking for unique aspects of the problem. Here the natural places to start are n + +n u u, , and the fact that s f+ yields a two-digit number.

2. Use four 4s and some of the symbols + × − ÷, , , , ( ) to give expressions for the whole numbers from 0 through 9: for example, 5 4 4 4 4= × + ÷( ) .

3. For each shape in Figure 1.7, make one straight cut so that each of the two pieces of the shape can be rearranged to form a square.

(NOTE: Answers for these problems are given after the Solution of the Initial Prob- lem near the end of this chapter.)

Clues The Guess and Test strategy may be appropriate when

There is a limited number of possible answers to test.

You want to gain a better understanding of the problem.

You have a good idea of what the answer is.

You can systematically try possible answers.

Your choices have been narrowed down by the use of other strategies.

There is no other obvious strategy to try.

Review the preceding three problems to see how these clues may have helped you select the Guess and Test strategy to solve these problems.

Draw a Picture

Often problems involve physical situations. In these situations, drawing a picture can help you better understand the problem so that you can formulate a plan to solve the problem. As you proceed to solve the following “pizza” problem, see whether you can visualize the solution without looking at any pictures first. Then work through the given solution using pictures to see how helpful they can be.

Problem Can you cut a pizza into 11 pieces with four straight cuts?

Step 1 Understand the Problem

Do the pieces have to be the same size and shape?

Step 2 Devise a Plan

An obvious beginning would be to draw a picture showing how a pizza is usually cut and to count the pieces. If we do not get 11, we have to try something else (Figure 1.8). Unfortunately, we get only eight pieces this way.

Figure 1.8

Figure 1.7

Children’s Literature www.wiley.com/college/musser See “Counting on Frank” by Rod Clement.

Reflection from Research Training children in the process of using pictures to solve problems results in more improved prob- lem-solving performance than training students in any other strategy (Yancey, Thompson, & Yancey, 1989).

NCTM Standard All students should describe, extend, and make generalizations about geometric and numeric patterns.

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Section 1.1 The Problem-Solving Process and Strategies 11

Step 3 Carry Out the Plan

See Figure 1.9

Figure 1.9

Step 4 Look Back

Were you concerned about cutting equal pieces when you started? That is normal. In the context of cutting a pizza, the focus is usually on trying to cut equal pieces rather than the number of pieces. Suppose that circular cuts were allowed. Does it matter whether the pizza is circular or is square? How many pieces can you get with five straight cuts? n straight cuts?

Additional Problems Where the Strategy “Draw a Picture” Is Useful

1. A tetromino is a shape made up of four squares where the squares must be joined along an entire side (Figure 1.10). How many different tetromino shapes are possible?

Step 1 Understand the Problem

The solution of this problem is easier if we make a set of pictures of all possible arrangements of four squares of the same size.

Step 2 Devise a Plan

Let’s start with the longest and narrowest configuration and work toward the most compact.

Step 3 Carry Out the Plan

Figure 1.10

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12 Chapter 1 Introduction to Problem Solving

Step 4 Look Back

Many similar problems can be posed using fewer or more squares. The problems become much more complex as the number of squares increases. Also, new prob- lems can be posed using patterns of equilateral triangles.

2. If you have a chain saw with a bar 18 inches long, determine whether a 16-foot log, 8 inches in diameter, can be cut into 4-foot pieces by making only two cuts.

3. It takes 64 cubes to fi ll a cubical box that has no top. How many cubes are not touching a side or the bottom?

Clues The Draw a Picture strategy may be appropriate when

A physical situation is involved.

Geometric fi gures or measurements are involved.

You want to gain a better understanding of the problem.

A visual representation of the problem is possible.

Review the preceding three problems to see how these clues may have helped you select the Draw a Picture strategy to solve these problems.

Use a Variable

Observe how letters were used in place of numbers in the previous “sun fun swim+ = ” cryptarithm. Letters used in place of numbers are called variables or unknowns. The Use a Variable strategy, which is one of the most useful problem-solving strategies, is used extensively in algebra and in mathematics that involves algebra.

Problem What is the greatest number that evenly divides the sum of any three consecutive whole numbers?

By trying several examples, you might guess that 3 is the greatest such number. However, it is necessary to use a variable to account for all possible instances of three consecutive numbers.

Step 1 Understand the Problem

The whole numbers are 0 1 2 3, , , , . . . , so that consecutive whole numbers differ by 1. Thus an example of three consecutive whole numbers is the triple 3, 4, and 5. The sum of three consecutive whole numbers has a factor of 3 if 3 multiplied by another whole number produces the given sum. In the example of 3, 4, and 5, the sum is 12 and 3 4× equals 12. Thus 3 4 5+ + has a factor of 3.

Step 2 Devise a Plan

Since we can use a variable, say x, to represent any whole number, we can repre- sent every triple of consecutive whole numbers as follows: x x x, , . + +1 2 Now we can discover whether the sum has a factor of 3.

Step 3 Carry Out the Plan

The sum of x x, ,+ 1 and x + 2 is

x x x x x+ +( ) + +( ) = + = +( )1 2 3 3 3 1 .

NCTM Standard All students should represent the idea of a variable as an unknown quantity using a letter or a symbol.

Reflection from Research Given the proper experiences, children as young as eight and nine years of age can learn to comfortably use letters to represent unknown values and can operate on representations involving letters and numbers while fully realizing that they did not know the values of the unknowns (Carraher, Schliemann, Brizuela, & Earnest, 2006).

Algebraic Reasoning In algebra, the letter “x” is most commonly used for a variable. However, any letter (even Greek letters, for example) can be used as a variable.

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Section 1.1 The Problem-Solving Process and Strategies 13

Thus x x x+ + + +( ) ( )1 2 is three times x + 1. Therefore, we have shown that the sum of any three consecutive whole numbers has a factor of 3. The case of x = 0 shows that 3 is the greatest such number.

Step 4 Look Back

Is it also true that the sum of any five consecutive whole numbers has a factor of 5? Or, more generally, will the sum of any n consecutive whole numbers have a factor of n? Can you think of any other generalizations?

Additional Problems Where the Strategy “Use a Variable” Is Useful

1. Find the sum of the first 10, 100, and 500 counting numbers.

Step 1 Understand the Problem

Since counting numbers are the numbers 1 2 3 4, , , , . . . , the sum of the first 10 count- ing numbers would be 1 2 3 8 9 10+ + + + + +. . . . Similarly, the sum of the first 100 counting numbers would be 1 2 3 9 99 100+ + + + + +. . . 8 and the sum of the first 500 counting numbers would be 1 2 3 498 499 500+ + + + + +. . . .

Step 2 Devise a Plan

Rather than solve three different problems, the “Use a Variable” strategy can be used to find a general method for computing the sum in all three situa- tions. Thus, the sum of the first n counting numbers would be expressed as 1 2 3 2 1+ + + + − + − +. . . ( ) ( ) .n n n The sum of these numbers can be found by noticing that the first number 1 added to the last number n is n + 1, which is the same as ( )n − +1 2 and ( ) .n − +2 3 Adding all such pairs can be done by adding all of the numbers twice.

Step 3 Carry Out the Plan

+ + + −

+ + −

+ +

⋅ ⋅ ⋅ ⋅ ⋅ ⋅

+ +

− + +

− + +

+ + +

1 2

1

3

2

2

3

1

2 1

1

n n n

n n n

n n

( ) ( )

( ) ( )

( ) ( 11 1 1 1 1) ( ) ( ) ( ) ( )+ + + ⋅ ⋅ ⋅ + + + + + +n n n n

= +n n• ( )1

Since each number was added twice, the desired sum is obtained by dividing n n¥ ( )+ 1 by 2 which yields

1 2 3 2 1 1

2 + + + ⋅ ⋅ ⋅ + −( ) + −( ) + = +( )n n n n n¥

The numbers 10, 100, and 500 can now replace the variable n to find our desired solutions:

1 2 3 8 9 10 10 10 1

2 55

1 2 3 98 99 100 100 101

+ + + ⋅ ⋅ ⋅ + + + = +( ) =

+ + + ⋅ ⋅ ⋅ + + + = (

¥

¥ )) =

+ + + ⋅ ⋅ ⋅ + + + = =

2 5050

1 2 3 498 499 500 500 501

2 125 250

¥ ,

Reflection from Research When asked to create their own problems, good problem solvers generated problems that were more mathematically complex than those of less successful problem solvers (Silver & Cai, 1996).

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14 Chapter 1 Introduction to Problem Solving

Step 4 Look Back

Since the method for solving this problem is quite unique could it be used to solve other similar looking problems like:

i. 3 6 9 3 6 3 3 3+ + + ⋅ ⋅ ⋅ + − + − +( ) ( )n n n ii. 21 25 29 113 117 121+ + + ⋅ ⋅ ⋅ + + +

2. Show that the sum of any fi ve consecutive odd whole numbers has a factor of 5.

3. The measure of the largest angle of a triangle is nine times the measure of the smallest angle. The measure of the third angle is equal to the difference of the largest and the smallest. What are the measures of the angles? (Recall that the sum of the measures of the angles in a triangle is 180°.)

Clues The Use a Variable strategy may be appropriate when

A phrase similar to “for any number” is present or implied.

A problem suggests an equation.

A proof or a general solution is required.

A problem contains phrases such as “consecutive,” “even,” or “odd” whole numbers.

There is a large number of cases.

There is an unknown quantity related to known quantities.

There is an infi nite number of numbers involved.

You are trying to develop a general formula.

Review the preceding three problems to see how these clues may have helped you select the Use a Variable strategy to solve these problems.

Using Algebra to Solve Problems To effectively employ the Use a Variable strategy, students need to have a clear understanding of what a variable is and how to write and simplify equations contain- ing variables. This subsection addresses these issues in an elementary introduction to algebra. There will be an expanded treatment of solving equations and inequalities in Chapter 9 after the real number system has been developed.

A common way to introduce the use of variables is to find a general formula for a pattern of numbers such as 3 6 9 3, , , . . . .n One of the challenges for students is to see the role that each number plays in the expression. For example, the pattern 5 8 11, , , . . . is similar to the previous pattern, but it is more difficult to see that each term is two greater than a multiple of 3 and, thus, can be expressed in general as 3 2n + . Sometimes it is easier for students to use a variable to generalize a geometric pattern such as the one shown in the following example. This type of example may be used to introduce seventh-grade students to the concept of a variable. Following are four typical student solutions.

NCTM Standard All students should develop an initial conceptual understanding of different uses of variables.

Describe at least four different ways to count the dots in Figure 1.11.

S O L U T I O N The obvious method of solution is to count the dots—there are 16. Another student’s method is illustrated in Figure 1.12.Figure 1.11

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Section 1.1 The Problem-Solving Process and Strategies 15

4 3 4

4 5 2 4

× + × − +

⎫ ⎬ ⎭

⎯ →⎯ ( )

Figure 1.12

The student counts the number of interior dots on each side, 3, and multiplies by the number of sides, 4, and then adds the dots in the corners, 4. This method generates the expression 4 3 4 16× + = . A second way to write this expression is 4 5 2 4 16× − + =( ) since the 3 interior dots can be determined by subtracting the two corners from the 5 dots on a side. Both of these methods are shown in Figure 1.12.

A third method is to count all of the dots on a side, 5, and multiply by the number of sides. Four must then be subtracted because each corner has been counted twice, once for each side it belongs to. This method is illustrated in Figure 1.13 and generates the expression shown.

4 5 4 16× − = ⎯ →⎯}

Figure 1.13

In the two previous methods, either corner dots are not counted (so they must be added on) or they are counted twice (so they must be subtracted to avoid double counting). The following fourth method assigns each corner to only one side (Figure 1.14).

4 4 16

4 5 1 16

× = × − =

⎫ ⎬ ⎭

⎯ →⎯ ( )

Figure 1.14

Thus, we encircle 4 dots on each side and multiply by the number of sides. This yields the expression 4 4 16× = . Because the 4 dots on each side come from the 5 total dots on a side minus 1 corner, this expression could also be written as 4 5 1 16× − =( ) (see Figure 1.14). ■

Reflection from Research Sixth-grade students with no formal instruction in algebra are “generally able to solve prob- lems involving specific cases and showed remarkable ability to generalize the problem situations and to write equations using vari- ables. However they rarely used their equations to solve related problems” (Swafford & Langrall, 2000).

There are many different methods for counting the dots in the previous example and each method has a geometric interpretation as well as a corresponding arithmetic expression. Could these methods be generalized to 50, 100, 1000 or even n dots on a side? The next example discusses how these generalizations can be viewed as well as displays the generalized solutions of seventh-grade students.

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16 Chapter 1 Introduction to Problem Solving

Since each expression on the right represents the total number of dots in Figure 1.15, they are all equal to each other. Using properties of numbers and equations, each equation can be rewritten as the same expression. Learning to simplify expressions and equations with variables is one of the most important processes in mathematics. Traditionally, this topic has represented a substantial portion of an entire course in introductory algebra. An equation is a sentence involving numbers, or symbols repre- senting numbers, where the verb is equals ( ).= There are various types of equations:

3 4 7+ = True equation 3 4 9+ = False equation

2 5 7x x x+ = Identity equation x + =4 9 Conditional equation

A true or false equation needs no explanation, but an identity equation is always true no matter what numerical value is used for x. A conditional equation is an equa- tion that is only true for certain values of x. For example, the equation x + =4 9 is true when x = 5, but false when x is any other value. In this chapter, we will restrict the variables to only whole numbers. For a conditional equation, a value of the vari- able that makes the equation true is called a solution. To solve an equation means to find all of the solutions. The following example shows three different ways to solve equations of the form ax b c+ = .

Reflection from Research The more an equation varies from the standard a b c+ = format, the more difficult it is for students to work with. The most difficult problems are those with opera- tions on both sides of the equal sign (Matthews, Rittle-Johnson, McEldoon, & Taylor, 2012).

Algebraic Reasoning Variable is a central concept in algebra. Students may struggle with the idea that the letter x represents many numbers in y x= +3 4 but only one or two numbers in other situations. For example, the solution of the equation x2 9= consists of the numbers 3 and −3 since each of these numbers squared is 9. This means, that the equation is true whenever x = 3 or x = −3.

Suppose the square arrangement of dots in Example 1.1 had n dots on each side. Write an algebraic expression that would

describe the total number of dots in such a figure (Figure 1.15).

S O L U T I O N It is easier to write a general expression for those in Example 1.1 when you understand the origins of the numbers in each expression. In all such cases, there will be 4 corners and 4 sides, so the values that represent corners and sides will stay fi xed at 4. On the other hand, in Figure 1.15, any value that was determined based on the number of dots on the side will have to refl ect the value of n. Thus, the expressions that represent the total number of dots on a square fi gure with n dots on a side are generalized as shown next.

4 3 4

4 5 2 4 4 2 4

4 5 4 4 4

4 4

4

× + × −( ) +

⎫ ⎬ ⎭

⎯ →⎯ −( ) +

× − ⎯ →⎯ − ×

n

n

×× −( ) ⎫ ⎬ ⎭

⎯ →⎯ −( ) 5 1

4 1 n ■

Figure 1.15

Suppose the square arrangement of dots in Example 1.2 had 84 total dots (Figure 1.16). How many dots are there on each side?

← ⎯⎯ 84 total dots

Figure 1.16

S O L U T I O N From Example 1.2, a square with n dots on a side has 4 4n − total dots. Thus, we have the equation 4 4 84n − = . Three elementary methods that can be used to solve equations such as 4 4 84n − = are Guess and Test, Cover Up, and Work Backward.

Reflection from Research Even 6-year-olds can solve alge- braic equations when they are rewritten as a story problem, logic puzzle, or some other problem with meaning (Femiano, 2003).

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Section 1.1 The Problem-Solving Process and Strategies 17

Guess and Test As the name of this method suggests, one guesses values for the variable in the equation 4 4 84n − = and substitutes to see if a true equation results.

Try n = − = ≠10 4 10 4 36 84: ( ) Try n = − = ≠25 4 25 4 96 84: ( ) Try n = − =22 4 22 4 84: ( ) . Therefore, 22 is the solution of the equation.

Cover Up In this method, we cover up the term with the variable:

h h− =4 84 88 4. , . To make a true equation the must be Thus n == 88 Since we have 4 22 88 22• = =, .n

Work Backward The left side of the equation shows that n is multiplied by 4 and then 4 is subtracted to obtain 84. Thus, working backward, if we add 4 to 84 and divide by 4, we reach the value of n. Here 84 4 88+ = and 88 ÷ 4 = 22 so n = 22 (Figure 1.17). ■Figure 1.17

Using variables in equations and manipulating them are what most people see as “algebra.” However, algebra is much more than manipulating variables—it includes the reasoning that underlies those manipulations. In fact, many students solve algebra-like problems without equations and don’t realize that their reasoning is algebraic. For example, the Work Backward solution in Example 1.3 can all be done without really thinking about the variable n. If one just thinks “4 times something minus 4 is 84,” he can work backward to find the solution. However, the thinking that is used in the Work Backward method can be mirrored in equations as follows:

4 4 84

4 84 4

4 88

88 4

22

n

n

n

n

n

− = = + = = ÷ =

Because so much algebra can be done with intuitive reasoning, it is important to help students realize when they are reasoning algebraically. One way to better understand the underlying principles of algebraic reasoning is to look at the Algebraic Reasoning Web Module on our Web site: www.wiley.com/college/musser/

Some researchers say that arithmetic is the foundation of learning algebra. Arithmetic typically means computation with different kinds of numbers and the underlying properties that make the computation work. Much of what is discussed in Chapters 2–9 is about arithmetic and thus contains key parts of the foundation of algebra. To help you see algebraic ideas in the arithmetic that we study, there will be places throughout those chapters where these foundational ideas of algebra are called out in an “Algebraic Reasoning” margin note.

We address algebra in more depth when we talk about solving equations, relations, and functions in Chapter 9 and then again in Chapter 15 when algebraic ideas are applied to geometry.

Reflection from Research We are proposing that the teaching and learning of arithmetic be conceived as part of the foundation of learning algebra, not that algebra be conceived only as an extension of arithmetic procedures (Carpenter, Levi, Berman, & Pligge, 2005).

There is a story about Sir Isaac Newton, coinventor of the calculus, who, as a youngster, was sent out to cut a hole in the barn door for the cats to go in and out. With great pride he admitted to cutting two holes, a larger one for the cat and a smaller one for the kittens.

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18 Chapter 1 Introduction to Problem Solving

1. a. If the diagonals of a square are drawn in, how many triangles of all sizes are formed?

b. Describe how Pólya’s four steps were used to solve part a.

2. Scott and Greg were asked to add two whole numbers. Instead, Scott subtracted the two numbers and got 10, and Greg multiplied them and got 651. What was the correct sum?

3. The distance around a standard tennis court is 228 feet. If the length of the court is 6 feet more than twice the width, find the dimensions of the tennis court.

4. A multiple of 11 I be, not odd, but even, you see. My digits, a pair, when multiplied there, make a cube and a square out of me. Who am I?

5. Show how 9 can be expressed as the sum of two consecutive numbers. Then decide whether every odd number can be expressed as the sum of two consecutive counting numbers. Explain your reasoning.

6. Using the symbols +, , ,− × and ÷, fill in the following three blanks to make a true equation. (A symbol may be used more than once.)

6 6 6 6 13 =

7. In the accompanying figure (called an arithmogon), the number that appears in a square is the sum of the numbers in the circles on each side of it. Determine what numbers belong in the circles.

8. Place 10 stools along four walls of a room so that each of the four walls has the same number of stools.

9. Susan has 10 pockets and 44 dollar bills. She wants to arrange the money so that there are a different number of dollars in each pocket. Can she do it? Explain.

10. Arrange the numbers 2 3 10, , ,. . . in the accompanying tri- angle so that each side sums to 21.

11. Find a set of consecutive counting numbers whose sum is each of the following. Each set may consist of 2, 3, 4, 5, or 6 consecutive integers. Use the spreadsheet activity Consecutive Integer Sum on our Web site to assist you.

a. 84     b. 213     c. 154

12. Place the digits 1 through 9 so that you can count from 1 to 9 by following the arrows in the diagram.

13. Using a 5-minute and an 8-minute hourglass timer, how can you measure 1 minute?

14. Using the numbers 9, 8, 7, 6, 5, and 4 once each, find the following:

a. The largest possible sum:

b. The smallest possible (positive) difference:

15. Using the numbers 1 through 8, place them in the follow- ing eight squares so that no two consecutive numbers are in touching squares (touching includes entire sides or simply one point).

16. Solve this cryptarithm, where each letter represents a digit and no digit represents two different letters:

USSR

USA

PEACE

+

17. On a balance scale, two spools and one thimble balance eight buttons. Also, one spool balances one thimble and one button. How many buttons will balance one spool?

18. Place the numbers 1 through 8 in the circles on the vertices of the accompanying cube so that the difference of any two connecting circles is greater than 1.

PROBLEM SET A

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Section 1.1 The Problem-Solving Process and Strategies 19

19. Think of a number. Add 10. Multiply by 4. Add 200. Divide by 4. Subtract your original number. Your result should be 60. Why? Show why this would work for any number.

20. The digits 1 through 9 can be used in decreasing order, with + and − signs, to produce 100 as shown: 98 76 54 3 21 100− + + + = . Find two other such combina- tions that will produce 100.

21. The Indian mathematician Ramanujan observed that the taxi number 1729 was very interesting because it was the smallest counting number that could be expressed as the sum of cubes in two different ways. Find a b c, , , and d such that a b3 3 1729+ = and c d3 3 1729+ = .

22. Using the Chapter 1 eManipulative activity Number Puzzles, Exercise 2 on our Web site, arrange the numbers 1, 2, 3, 4, 5, 6, 7, 8, 9 in the following circles so the sum of the numbers along each line of four is 23.

23. Using the Chapter 1 eManipulative activity Circle 21 on our Web site, find an arrangement of the numbers 1 through 14 in the 7 circles below so that the sum of the three numbers in each circle is 21.

24. The hexagon below has a total of 126 dots and an equal number of dots on each side. How many dots are on each side?

1. Find the largest eight-digit number made up of the digits 1, 1, 2, 2, 3, 3, 4, and 4 such that the 1s are separated by one digit, the 2s by two digits, the 3s by three digits, and the 4s by four digits.

2. Think of a number. Multiply by 5. Add 8. Multiply by 4. Add 9. Multiply by 5. Subtract 105. Divide by 100. Subtract 1. How does your result compare with your original number? Explain.

3. Carol bought some items at a variety store. All the items were the same price, and she bought as many items as the price of each item in cents. (For example, if the items cost 10 cents, she would have bought 10 of them.) Her bill was $2.25. How many items did Carol buy?

4. You can make one square with four toothpicks. Show how you can make two squares with seven toothpicks (breaking toothpicks is not allowed), three squares with 10 tooth- picks, and five squares with 12 toothpicks.

5. A textbook is opened and the product of the page numbers of the two facing pages is 6162. What are the numbers of the pages?

6. Place numbers 1 through 19 into the 19 circles below so that any three numbers in a line through the center will give the same sum.

7. Using three of the symbols +, , ,− × and ÷ once each, fill in the following three blanks to make a true equation. (Parentheses are allowed.)

6 6 6 6 66 =

PROBLEM SET B

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20 Chapter 1 Introduction to Problem Solving

8. A water main for a street is being laid using a particu- lar kind of pipe that comes in either 18-foot sections or 20-foot sections. The designer has determined that the water main would require 14 fewer sections of 20-foot pipe than if 18-foot sections were used. Find the total length of the water main.

9. Mike said that when he opened his book, the product of the page numbers of the two facing pages was 7007. Without performing any calculations, prove that he was wrong.

10. The Smiths were about to start on an 18,000-mile automo- bile trip. They had their tires checked and found that each was good for only 12,000 miles. What is the smallest num- ber of spares that they will need to take along with them to make the trip without having to buy a new tire?

11. What is the maximum number of pieces of pizza that can result from 4 straight cuts?

12. Given: Six arrows arranged as follows:

↑ ↑ ↑ ↓ ↓ ↓

Goal: By inverting two adjacent arrows at a time, rear- range to the following:

↑ ↓ ↑ ↓ ↑ ↓

Can you find a minimum number of moves?

13. Two friends are shopping together when they encounter a special “3 for 2” shoe sale. If they purchase two pairs of shoes at the regular price, a third pair (of lower or equal value) will be free. Neither friend wants three pairs of shoes, but Pat would like to buy a $56 and a $39 pair while Chris is interested in a $45 pair. If they buy the shoes together to take advantage of the sale, what is the fairest share for each to pay?

14. Find digits A, B, C, and D that solve the following cryptarithm.

ABCD

DCBA

× 4

15. If possible, find an odd number that can be expressed as the sum of four consecutive counting numbers. If impos- sible, explain why.

16. Five friends were sitting on one side of a table. Gary sat next to Bill. Mike sat next to Tom. Howard sat in the third seat from Bill. Gary sat in the third seat from Mike. Who sat on the other side of Tom?

17. In the following square array on the left, the corner numbers were given and the boldface numbers were found by adding the adjacent corner numbers. Following the same rules, find the corner numbers for the other square array.

6 13

2 1

19 8 14

3

10 15 11

16

18. Together, a baseball and a football weigh 1.25 pounds, the baseball and a soccer ball weigh 1.35 pounds, and the football and the soccer ball weigh 1.9 pounds. How much does each of the balls weigh?

19. Pick any two consecutive numbers. Add them. Then add 9 to the sum. Divide by 2. Subtract the smaller of the original numbers from the answer. What did you get? Repeat this process with two other consecutive numbers. Make a con- jecture (educated guess) about the answer, and prove it.

20. An additive magic square has the same sum in each row, column, and diagonal. Find the error in this magic square and correct it.

47 56 34 22 83 7 24 67 44 26 13 75 29 52 3 99 18 48 17 49 89 4 53 37 97 6 3 11 74 28 35 19 46 87 8 54

21. Two points are placed on the same side of a square. A segment is drawn from each of these points to each of the 2 vertices (corners) on the opposite side of the square. How many triangles of all sizes are formed?

22. Using the triangle in Problem 10 in Set A, determine whether you can make similar triangles using the digits 1 2 9, , ,. . . , where the side sums are 18, 19, 20, 21, and 22.

23. Using the Chapter 1 eManipulative activity, Number Puzzles, Exercise 4 on our Web site, arrange the numbers 1, 2, 3, 4, 5, 6, 7, 8, 9 in the circles below so the sum of the numbers along each line of four is 20.

24. Using the Chapter 1 eManipulative activity Circle 99 on our Web site, find an arrangement of the numbers provided in the 7 circles below so that the sum of the three numbers in each circle is 99.

25. An arrangement of dots forms the perimeter of an equi- lateral triangle. There are 87 evenly spaced dots on each side including the dots at the vertices. How many dots are there altogether?

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Section 1.2 Three Additional Strategies 21

26. The equation y 5

12 23+ = can be solved by subtracting

12 from both sides of the equation to yield y 5

12 12+ − =

23 12− . Similarly, the resulting equation y 5

11= can be

solved by multiplying both sides of the equation by 5 to obtain y = 55. Explain how this process is related to the Work Backward method described in Example 1.3.

Analyzing Student Thinking

27. When the class was asked to solve the equation 3 8 27x − = , Wesley asked if he could use guess and test. How would you respond?

28. Rosemary said that she felt the ‘guess and test’ method was a waste of time; she just wanted to get an answer. What could you tell her about the value of using guess and test?

29. Even though you have taught your students how to ‘draw a picture’ to solve a problem, Cecelia asks if she has to draw a picture because she can solve the problems in class without it. How would you respond?

30. When Damian, a second grader, was asked to solve problems like + =3 5, he said that he had seen his older sister working on problems like x + =3 5 and wondered if these equations were different. How would you respond?

31. After the class had found three consecutive odd numbers whose sum is 99, Byron tried to find three consecutive odd numbers that would add to 96. He said he was struggling to find a solution. How could you help him understand the solution to this problem?

32. Consider the following problem:

The amount of fencing needed to enclose a rectangu- lar field was 92 yards and the length of the field was 3 times as long as the width. What were the dimensions of the field?

Vance solved this problem by drawing a picture and using guess and test. Jolie set up an equation with x being the width of the field and solved it. They got the same answer but asked you which method was better. How would you respond?

Solve the problem below using Pólya’s four steps and any strategy. Describe how you used the four steps, focusing on

any new insights that you gained as a result of looking back. How many rectangles of all shapes and sizes are in the figure at the right?

THREE ADDITIONAL STRATEGIES

Look for a Pattern

When using the Look for a Pattern strategy, one usually lists several specific instances of a problem and then looks to see whether a pattern emerges that suggests a solu- tion to the entire problem. For example, consider the sums produced by adding consecutive odd numbers starting with 1 1 1 3 4 2 2: , ( ),+ = = × 1 3 5 9 3 3+ + = = ×( ), 1 3 5 7 16 4 4+ + + = = ×( ), 1 3 5 7 9 25 5 5+ + + + = = ×( ), and so on. Based on the pat- tern generated by these five examples, one might expect that such a sum will always be a perfect square.

The justification of this pattern is suggested by the following figure.

Each consecutive odd number of dots can be added to the previous square arrange- ment to form another square. Thus, the sum of the first n odd numbers is n2.

NCTM Standard All students should represent, analyze, and generalize a variety of patterns with tables, graphs, words, and, when possible, sym- bolic rules.

Common Core – Grades K-12 (Mathematical Practice 7) Mathematically proficient stu- dents look closely to discern a pattern or structure. Young stu- dents, for example, might notice that three and seven more is the same amount as seven and three more, or they may sort a collec- tion of shapes according to how many sides the shapes have.

Algebraic Reasoning Recognizing and extending pat- terns is a common practice in algebra. Extending patterns to the most general case is a natural place to discuss variables.

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22 Chapter 1 Introduction to Problem Solving

Generalizing patterns, however, must be done with caution because with a sequence of only 3 or 4 numbers, a case could be made for more than one pattern. For example, consider the sequence 1 2 4, , , . . . . What are the next 4 numbers in the sequence? It can be seen that 1 is doubled to get 2 and 2 is doubled to get 4. Following that pattern, the next four numbers would be 8, 16, 32, 64. If, however, it is noted that the difference between the first and second term is 1 and the difference between the second and third term is 2, then a case could be made that the difference is increasing by one. Thus, the next four terms would be 7, 11, 16, 22. Another case could be made for the differences alternating between 1 and 2. In that case, the next four terms would be 5, 7, 8, 10. Thus, from the initial three numbers of 1, 2, 4, at least three different patterns are possible:

1 2 4 8 16 32 64, , , , , , , . . . Doubling 1 2 4 7 11 16 22, , , , , , , . . . Difference increasing by 1 1 2 4 5 7 8 10, , , , , , , . . . Difference alternating between 1 and 2

Problem How many different downward paths are there from A to B in the grid in Figure 1.18? A path must travel on the lines.

Step 1 Understand the Problem

What do we mean by different and downward? Figure 1.19 illustrates two paths. Notice that each such path will be 6 units long. Different means that they are not exactly the same; that is, some part or parts are different.

Step 2 Devise a Plan

Let’s look at each point of intersection in the grid and see how many different ways we can get to each point. Then perhaps we will notice a pattern (Figure 1.20). For example, there is only one way to reach each of the points on the two outside edges; there are two ways to reach the middle point in the row of points labeled 1, 2, 1; and so on. Observe that the point labeled 2 in Figure 1.20 can be found by adding the two 1s above it.

Step 3 Carry Out the Plan

To see how many paths there are to any point, observe that you need only add the number of paths required to arrive at the point or points immediately above. To reach a point beneath the pair 1 and 2, the paths to 1 and 2 are extended down- ward, resulting in 1 2 3+ = paths to that point. The resulting number pattern is shown in Figure 1.21. Notice, for example, that 4 6 10+ = and 20 15 35+ = . (This pattern is part of what is called Pascal’s triangle. It is used again in Chapter 11.) The surrounded portion of this pattern applies to the given problem; thus the answer to the problem is 20.

Step 4 Look Back

Can you see how to solve a similar problem involving a larger square array, say a 4 4× grid? How about a 10 10× grid? How about a rectangular grid?

A pattern of numbers arranged in a particular order is called a number sequence, and the individual numbers in the sequence are called terms of the sequence. The counting numbers, 1 2 3 4, , , , ,. . . give rise to many sequences. (An ellipsis, the three periods after the 4, means “and so on.”) Several sequences of counting numbers follow.

Figure 1.18

Figure 1.19

Figure 1.20

Figure 1.21

Children’s Literature www.wiley.com/college/musser See “There Was an Old Lady Who Swallowed a Fly” by Simms Taback.

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Section 1.2 Three Additional Strategies 23

SEQUENCE NAME

2, 4, 6, 8, . . . The even (counting) numbers 1, 3, 5, 7, . . . The odd (counting) numbers 1, 4, 9, 16, . . . The square (counting) numbers 1, 3, 3 3 ,2 3, . . . The powers of three 1, 1, 2, 3, 5, 8, . . . The Fibonacci sequence (after the two 1s, each

term is the sum of the two preceding terms)

Inductive reasoning is used to draw conclusions or make predictions about a large collection of objects or numbers, based on a small representative subcollection. For example, inductive reasoning can be used to find the ones digit of the 400th term of the sequence 8 12 16 20 24, , , , , . . . By continuing this sequence for a few more terms, 8 12 16 20 24 28 32 36 40 44 48 52 56 60, , , , , , , , , , , , , , ,. . . one can observe that the ones digit of every fifth term starting with the term 24 is a four. Thus, the ones digit of the 400th term must be a four.

Additional Problems Where the Strategy “Look for a Pattern” Is Useful

1. Find the ones digit in 399.

Step 1 Understand the Problem

The number 399 is the product of 99 threes. Using the exponent key on one type of

scientific calculator yields the result 1 71792506547. . This shows the first digit, but not the ones (last) digit, since the 47 indicates that there are 47 places to the right of the decimal. (See the discussion on scientific notation in Chapter 4 for further explanation.) Therefore, we will need to use another method.

Step 2 Devise a Plan

Consider 3 3 3 3 3 3 3 31 2 3 4 5 6 7 8, , , , , , , . Perhaps the ones digits of these numbers form a pattern that can be used to predict the ones digit of 399.

Step 3 Carry Out the Plan

3 3 3 3 3 24 3 72 3 218 3 6561 2 3 4 5 6 7 8= = = = = = = =3 9 7 1 3 9 7 1, , , , , , ,2 8 .. The ones digits form the sequence 3, 9, 7, 1, 3, 9, 7, 1. Whenever the exponent of the 3 has a factor of 4, the ones digit is a 1. Since 100 has a factor of 4, 3100 must have a ones digit of 1. Therefore, the ones digit of 399 must be 7, since 399 precedes 3100 and 7 precedes 1 in the sequence 3, 9, 7, 1.

Step 4 Look Back

Ones digits of other numbers involving exponents might be found in a similar fashion. Check this for several of the numbers from 4 to 9.

2. Which whole numbers, from 1 to 50, have an odd number of factors? For example, 15 has 1, 3, 5, and 15 as factors, and hence has an even number of factors: four.

3. In the next diagram, the left “H”-shaped array is called the 32-H and the right array is the 58-H.

a. Find the sums of the numbers in the 32-H. Do the same for the 58-H and the 74-H. What do you observe?

b. Find an H whose sum is 497.

c. Can you predict the sum in any H if you know the middle number? Explain.

NCTM Standard All students should analyze how both repeating and growing pat- terns are generated.

Reflection from Research In classrooms where problem solving is valued and teachers have knowledge of children’s mathematical thinking, children see mathematics as a problem- solving endeavor in which com- municating mathematical thinking is important (Franke & Carey, 1997).

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24 Chapter 1 Introduction to Problem Solving

0 1 2 3 4 5 6 7 8 9

10 11 12 13 14 15 16 17 18 19

20 21 22 23 24 25 26 27 28 29

30 31 32 33 34 35 36 37 38 39

40 41 42 43 44 45 46 47 48 49

50 51 52 53 54 55 56 57 58 59

60 61 62 63 64 65 66 67 68 69

70 71 72 73 74 75 76 77 78 79

80 81 82 83 84 85 86 87 88 89

90 91 92 93 94 95 96 97 98 99

Clues The Look for a Pattern strategy may be appropriate when

A list of data is given.

A sequence of numbers is involved.

Listing special cases helps you deal with complex problems.

You are asked to make a prediction or generalization.

Information can be expressed and viewed in an organized manner, such as in a table.

Review the preceding three problems to see how these clues may have helped you select the Look for a Pattern strategy to solve these problems.

Make a List

The Make a List strategy is often combined with the Look for a Pattern strategy to suggest a solution to a problem. For example, here is a list of all the squares of the numbers 1 to 20 with their ones digits in boldface.

1 4 9 6 5 6 9 4 1 0

1 4 9 6 5 6 9

, , , , , , , , , ,

, , , , , , ,

1 2 3 4 6 8 10

12 14 16 19 22 25 28 3244 1 0, ,36 40

The pattern in this list can be used to see that the ones digits of squares must be one of 0, 1, 4, 5, 6, or 9. This list suggests that a perfect square can never end in a 2, 3, 7, or 8.

Problem The number 10 can be expressed as the sum of four odd numbers in three ways: (i) 10 7 1 1 1= + + + , (ii) 10 5 3 1 1= + + + , and (iii) 10 3 3 3 1= + + + . In how many ways can 20 be expressed as the sum of eight odd numbers?

Step 1 Understand the Problem

Recall that the odd numbers are the numbers 1 3 5 7 9 11 13 15 17 19, , , , , , , , , , . . . Using the fact that 10 can be expressed as the sum of four odd numbers, we can form various combinations of those sums to obtain eight odd numbers whose sum is 20. But does this account for all possibilities?

Step 2 Devise a Plan

Instead, let’s make a list starting with the largest possible odd number in the sum and work our way down to the smallest.

Children’s Literature www.wiley.com/college/musser See “Math Appeal” by Greg Tang.

Reflection from Research Problem-solving abililty develops with age, but the relative dif- ficulty inherent in each problem is grade independent (Christou & Philippou, 1998).

NCTM Standard Instructional programs should enable all students to build new mathematical knowledge through problem solving.

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Section 1.2 Three Additional Strategies 25

Step 3 Carry Out the Plan

20 13 1 1 1 1 1 1 1

20 11 3 1 1 1 1 1 1

20 9 5 1 1 1 1 1 1

20

= + + + + + + + = + + + + + + + = + + + + + + + == + + + + + + + = + + + + + + + = + + + + + + + = +

9 3 3 1 1 1 1 1

20 7 7 1 1 1 1 1 1

20 7 5 3 1 1 1 1 1

20 7 33 3 3 1 1 1 1

20 5 5 5 1 1 1 1 1

20 5 5 3 3 1 1 1 1

20 5 3 3

+ + + + + + = + + + + + + + = + + + + + + + = + + ++ + + + + = + + + + + + +

3 3 1 1 1

20 3 3 3 3 3 3 1 1

Step 4 Look Back

Could you have used the three sums to 10 to help find these 11 sums to 20? Can you think of similar problems to solve? For example, an easier one would be to express 8 as the sum of four odd numbers, and a more difficult one would be to express 40 as the sum of 16 odd numbers. We could also consider sums of even numbers, expressing 20 as the sum of six even numbers.

Additional Problems Where the Strategy “Make a List” Is Useful 1. In a dart game, three darts are thrown. All hit the target (Figure 1.22). What scores

are possible?

Step 1 Understand the Problem

Assume that all three darts hit the board. Since there are four different numbers on the board, namely, 0, 1, 4, and 16, three of these numbers, with repetitions allowed, must be hit.

Step 2 Devise a Plan

We should make a systematic list by beginning with the smallest (or largest) pos- sible sum. In this way we will be more likely to find all sums.

Step 3 Carry Out the Plan

0 0 0 0 0 0 1 1 0 1 1 2

1 1 1 3

+ + = + + = + + = + + =

, ,

, 00 0 4 4 0 1 4 5

1 1 4 6 0 4 4 8 1 4

+ + = + + = + + = + + = +

, ,

, ,

++ = + + = + + =

4 9

4 4 4 12 16 16 16 48

,

, . . . ,

Step 4 Look Back

Several similar problems could be posed by changing the numbers on the dart- board, the number of rings, or the number of darts. Also, using geometric prob- ability, one could ask how to design and label such a game to make it a fair skill game. That is, what points should be assigned to the various regions to reward one fairly for hitting that region?

Reflection from Research Correct answers are not a safe indicator of good thinking. Teachers must examine more than answers and must demand from students more than answers (Sowder, Threadgill-Sowder, Moyer, & Moyer, 1983).

Figure 1.22

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26 Chapter 1 Introduction to Problem Solving

2. How many squares, of all sizes, are there on an 8 8× checkerboard? (See Figure 1.23; the sides of the squares are on the lines.)

3. It takes 1230 numerical characters to number the pages of a book. How many pages does the book contain?

Clues The Make a List strategy may be appropriate when

Information can easily be organized and presented.

Data can easily be generated.

Listing the results obtained by using Guess and Test.

Asked “in how many ways” something can be done.

Trying to learn about a collection of numbers generated by a rule or formula.

Review the preceding three problems to see how these clues may have helped you select the Make a List strategy to solve these problems.

The problem-solving strategy illustrated next could have been employed in con- junction with the Make a List strategy in the preceding problem.

Solve a Simpler Problem

Like the Make a List strategy, the Solve a Simpler Problem strategy is frequently used in conjunction with the Look for a Pattern strategy. The Solve a Simpler Problem strategy involves reducing the size of the problem at hand and making it more man- ageable to solve. The simpler problem is then generalized to the original problem.

Problem In a group of nine coins, eight weigh the same and the ninth is heavier. Assume that the coins are identical in appearance. Using a pan balance, what is the smallest number of balancings needed to identify the heavy coin?

Step 1 Understand the Problem

Coins may be placed on both pans. If one side of the balance is lower than the other, that side contains the heavier coin. If a coin is placed in each pan and the pans balance, the heavier coin is in the remaining seven. We could continue in this way, but if we missed the heavier coin each time we tried two more coins, the last coin would be the heavy one. This would require four balancings. Can we fi nd the heavier coin in fewer balancings?

Step 2 Devise a Plan

To fi nd a more effi cient method, let’s examine the cases of three coins and fi ve coins before moving to the case of nine coins.

Step 3 Carry Out the Plan

Three coins: Put one coin on each pan (Figure 1.24). If the pans balance, the third coin is the heavier one. If they don’t, the one in the lower pan is the heavier one. Thus, it only takes one balancing to fi nd the heavier coin.

Five coins: Put two coins on each pan (Figure 1.25). If the pans balance, the fi fth coin is the heavier one. If they don’t, the heavier one is in the lower pan. Remove the two coins in the higher pan and put one of the two coins in the lower pan on the other pan. In this case, the lower pan will have the heavier coin. Thus, it takes at most two balancings to fi nd the heavier coin.

Figure 1.23

Figure 1.24

Figure 1.25

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Section 1.2 Three Additional Strategies 27

Nine coins: At this point, patterns should have been identified that will make this solution easier. In the three-coin problem, it was seen that a heavy coin can be found in a group of three as easily as it can in a group of two. From the five-coin problem, we know that by balancing groups of coins together, we could quickly reduce the number of coins that needed to be examined. These ideas are combined in the nine-coin problem by breaking the nine coins into three groups of three and balancing two groups against each other (Figure 1.26). In this first balancing, the group with the heavy coin is identified. Once the heavy coin has been narrowed to three choices, then the three-coin balancing described above can be used.

The minimum number of balancings needed to locate the heavy coin out of a set of nine coins is two.

Step 4 Look Back

In solving this problem by using simpler problems, no numerical patterns emerged. However, patterns in the balancing process that could be repeated with a larger number of coins did emerge.

Additional Problems Where the Strategy “Solve a Simpler Problem” Is Useful

1. Find the sum 1 2

1 2

1 2

1 22 3 10

+ + + ⋅ ⋅ ⋅ + .

Step 1 Understand the Problem

This problem can be solved directly by getting a common denominator, here 210, and finding the sum of the numerators.

Step 2 Devise a Plan

Instead of doing a direct calculation, let’s combine some previous strategies. Namely, make a list of the first few sums and look for a pattern.

Step 3 Carry Out the Plan

1 2

3 4

7 8

15 16

, , , 1 2

1 4

1 2

1 4

1 8

1 2

1 4

1 8

1 16

+ = + + = + + + =

The pattern of sums, 1 2

3 4

7 8

15 16

, , , , suggests that the sum of the ten fractions is 2 1

2

10

10

− ,

or 1023 1024

.

Step 4 Look Back

This method of combining the strategy of Solve a Simpler Problem with Make a List

and Look for a Pattern is very useful. For example, what is the sum 1 2

1 2

1 22 100

+ + ⋅ ⋅ ⋅ + ? Because of the large denominators, you wouldn’t want to add these fractions directly.

2. Following the arrows in Figure 1.27, how many paths are there from A to B?

Figure 1.27

3. There are 20 people at a party. If each person shakes hands with each other per- son, how many handshakes will there be?

Figure 1.26

Algebraic Reasoning Using a variable, the sum to the right can be expressed more generally as follows:

1 2

1 2

1 2

2 1 22

+ + ⋅ ⋅ ⋅ + = −n n

n .

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28 Chapter 1 Introduction to Problem Solving

Clues The Solve a Simpler Problem strategy may be appropriate when

The problem involves complicated computations.

The problem involves very large or very small numbers.

A direct solution is too complex.

You want to gain a better understanding of the problem.

The problem involves a large array or diagram.

Review the preceding three problems to see how these clues may have helped you select the Solve a Simpler Problem strategy to solve these problems.

Solve the next problem and pay particular attention to the Devise a Plan step. Which strategy or strategies did you use? If you used more than one strategy, how were they

used in conjunction with each other?

In the figure below, there are chairs placed around the hexagonal tables. If 27 hexagonal tables were placed in a similar arrangement, how many chairs would it accommodate?

Combining Strategies to Solve Problems As shown in the previous four-step solution, it is often useful to employ several strate- gies to solve a problem. For example, in Section 1.1, a pizza problem similar to the following was posed: What is the maximum number of pieces you can cut a pizza into using four straight cuts? This question can be extended to the more general question: What is the maximum number of pieces you can cut a pizza into using n straight cuts? To answer this, consider the sequence in Figure 1.9: 1, 2, 4, 7, 11. To identify patterns in a sequence, observing how successive terms are related can be helpful. In this case, the second term of 2 can be obtained from the first term of 1 by either adding 1 or multiplying by 2. The third term, 4, can be obtained from the second term, 2, by add- ing 2 or multiplying by 2. Although multiplying by 2 appears to be a pattern, it fails as we move from the third term to the fourth term. The fourth term can be found by adding 3 to the third term. Thus, the sequence appears to be the following:

Extending the difference sequence, we obtain the following

Starting with 1 in the sequence, dropping down to the difference line, then back up to the number in the sequence line, we find the following:

1st Term: 1 = 1 2nd Term: 2 = +1 1

Reflection from Research The development of a disposition toward realistic mathematical modeling and interpreting of word problems should permeate the entire curriculum from the outset (Verschaffel & DeCorte, 1997).

Reflection from Research Young children, who have had little experience with standard mathematical problems, can be taught to search for more than one solution to a problem and to employ more than one method to solve that problem (Tsamir, Tirosh, Tabach, & Levenson, 2010).

Algebraic Reasoning In order to generalize patterns, one must first identify what is staying the same (the initial 1) and what is changing (the difference is increasing by one) with each step. Secondly, one must notice the relationship between what is changing and the step number.

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Section 1.2 Three Additional Strategies 29

3rd Term: 4 = + +1 1 2( ) 4th Term: 7 = + + +1 1 2 3( ) 5th Term: 11 = + + + +1 1 2 3 4( ), and so forth

Recall that earlier we saw that 1 2 3 1

2 + + + ⋅ ⋅ ⋅ + = +n n n( ). Thus, the nth term in the

sequence is 1 1 2 1 1 1

2 + + + ⋅ ⋅ ⋅ + − = + −[ ( )] ( ) .n n n Notice that as a check, the eighth

term in the sequence is 1 7 8

2 1 28 29+ = + =

•

. Hence, to solve the original problem, we

used Draw a Picture, Look for a Pattern, and Use a Variable. It may be that a pattern does not become obvious after one set of differences. Consider

the following problem where several differences are required to expose the pattern.

Problem If 10 points are placed on a circle and each pair of points is connected with a segment, what is the maximum number of regions created by these segments?

Step 1 Understand the Problem

This problem can be better understood by drawing a picture. Since drawing 10 points and all of the joining segments may be overwhelming, looking at a simpler problem of circles with 1, 2, or 3 points on them may help in further understand- ing the problem. The first three cases are in Figure 1.28.

Figure 1.28

Step 2 Devise a Plan

So far, the number of regions are 1, 2, and 4 respectively. Here, again, this could be the start of the pattern, 1 2 4 8 16, , , , , . . . . Let’s draw three more pictures to see if this is the case. Then the pattern can be generalized to the case of 10 points.

Step 3 Carry Out the Plan

The next three cases are shown in Figure 1.29.

Figure 1.29

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30 Chapter 1 Introduction to Problem Solving

Making a list of the number of points on the circle and the corresponding number of regions will help us see the pattern.

Points 1 2 3 4 5 6

Regions 1 2 4 8 16 31

While the pattern for the first five cases makes it appear as if the number of regions are just doubling with each additional point, the 31 regions with 6 points ruins this pattern. Consider the differences between the numbers in the pattern and look for a pattern in the differences.

Because the first, second, and third difference did not indicate a clear pattern, the fourth difference was computed and revealed a pattern of all ones. This observa- tion is used to extend the pattern by adding four 1s to the two 1s in the fourth dif- ference to make a sequence of six 1s. Then we work up until the Regions sequence has ten numbers as shown next.

By finding successive differences, it can be seen that the solution to our problem is 256 regions.

Step 4 Look Back

By using a combination of Draw a Picture, Solve a Simpler Problem, Make a List, and Look for a Pattern, the solution was found. It can also be seen that it is important when looking for such patterns to realize that we may have to look at many terms and many differences to be able to find the pattern.

Recapitulation When presenting the problems in this chapter, we took great care in organizing the solu- tions using Pólya’s four-step approach. However, it is not necessary to label and display each of the four steps every time you work a problem. On the other hand, it is good to get into the habit of recalling the four steps as you plan and as you work through a problem. In this chapter we have introduced several useful problem-solving strategies. In each of the following chapters, a new problem-solving strategy is introduced. These strategies will be especially helpful when you are making a plan. As you are planning to

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Section 1.2 Three Additional Strategies 31

solve a problem, think of the strategies as a collection of tools. Then an important part of solving a problem can be viewed as selecting an appropriate tool or strategy.

We end this chapter with a list of suggestions that students who have successfully completed a course on problem solving felt were helpful tips. Reread this list periodi- cally as you progress through the book.

Suggestions from Successful Problem Solvers Accept the challenge of solving a problem.

Rewrite the problem in your own words.

Take time to explore, refl ect, think. . . . Talk to yourself. Ask yourself lots of questions.

If appropriate, try the problem using simple numbers.

Many problems require an incubation period. If you get frustrated, do not hesitate to take a break—your subconscious may take over. But do return to try again.

Look at the problem in a variety of ways.

Run through your list of strategies to see whether one (or more) can help you get a start.

Many problems can be solved in a variety of ways—you only need to fi nd one solution to be successful.

Do not be afraid to change your approach, strategy, and so on.

Organization can be helpful in problem solving. Use the Pólya four-step approach with a variety of strategies.

Experience in problem solving is very valuable. Work lots of problems; your confi - dence will grow.

If you are not making much progress, do not hesitate to go back to make sure that you really understand the problem. This review process may happen two or three times in a problem since understanding usually grows as you work toward a solution.

There is nothing like a breakthrough, a small aha!, as you solve a problem.

Always, always look back. Try to see precisely what the key step was in your solution.

Make up and solve problems of your own.

Write up your solutions neatly and clearly enough so that you will be able to un- derstand your solution if you reread it in 10 years.

Develop good problem-solving helper skills when assisting others in solving prob- lems. Do not give out solutions; instead, provide meaningful hints.

By helping and giving hints to others, you will fi nd that you will develop many new insights.

Enjoy yourself! Solving a problem is a positive experience.

Reflection from Research Having children write their own story problems helps students to discern between relevant and irrelevant attributes in a problem and to focus on the various parts of a problem, such as the known and unknown quantities (Whitin & Whitin, 2008).

Reflection from Research The unrealistic expectations of teachers, namely lack of time and support, can cause young stu- dents to struggle with problem solving (Buschman, 2002).

Sophie Germain was born in Paris in 1776, the daughter of a silk merchant. At the age of 13, she found a book on the history of mathematics in her father’s library. She became enthralled with the study of mathematics. Even though her parents disapproved of this pursuit, nothing daunted her—she studied at night wrapped in a blanket, because her parents had taken her clothing away from her to keep her from getting up. They also took away her heat and light. This only hardened her resolve until her father finally gave in and she, at last, was allowed to study to become a mathematician.

© R

on B

ag w

el l

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32 Chapter 1 Introduction to Problem Solving

Use any of the six problem-solving strategies introduced thus far to solve the following.

1. a. Complete this table and describe the pattern in the ‘Answer’ column.

SUM ANSWER

1 1

1 3+ 4

1 3 5+ +

1 3 5 7+ + +

1 3 5 7 9+ + + +

b. How many odd whole numbers would have to be added to get a sum of 81? Check your guess by adding them.

c. How many odd whole numbers would have to be added to get a sum of 169? Check your guess by adding them.

d. How many odd whole numbers would have to be added to get a sum of 529? (You do not need to check.)

2. Find the missing term in each pattern. a. 256, 128, 64, _____, 16, 8

b. 1, 1 3

, 1 9

, _____, 1 81

c. 7, 9, 12, 16, _____ d. 127,863; 12,789; _____; 135; 18

3. Sketch a figure that is next in each sequence. a.

b.

4. Consider the following differences. Use your calculator to verify that the statements are true.

6 5 11

56 45 1111

556 445 111 111

2 2

2 2

2 2

− = − =

− = ,

a. Predict the next line in the sequence of differences. Use your calculator to check your answer.

b. What do you think the eighth line will be?

5. Look for a pattern in the first two number grids. Then use the pattern you observed to fill in the missing numbers of the third grid.

6. The triangular numbers are the whole numbers that are represented by certain triangular arrays of dots. The first five triangular numbers are shown.

a. Complete the following table and describe the pattern in the Number of Dots column.

NUMBER NUMBER OF DOTS

(TRIANGULAR NUMBERS)

1 1 2 3 3 4 5 6

b. Make a sketch to represent the seventh triangular number. c. How many dots will be in the tenth triangular number? d. Is there a triangular number that has 91 dots in its

shape? If so, which one? e. Is there a triangular number that has 150 dots in its

shape? If so, which one? f. Write a formula for the number of dots in the nth

triangular number. g. When the famous mathematician Carl Friedrich Gauss

was in fourth grade, his teacher challenged him to add the first one hundred counting numbers. Find this sum.

1 2 3 100+ + + ⋅ ⋅ ⋅ +

7. In a group of 12 coins identical in appearance, all weigh the same except one that is heavier. What is the minimum number of weighings required to determine the counterfeit coin? Use the Chapter 1 eManipulative activity Counterfeit Coin on our Web site for eight or nine coins to better understand the problem.

8. If 20 points are placed on a circle and every pair of points is joined with a segment, what is the total number of seg- ments drawn?

9. Find reasonable sixth, seventh, and eighth terms of the following sequences:

a. 1, 4, 9, 17, 29, _____, _____, _____ b. 3, 7, 13, 21, 31, _____, _____, _____

10. As mentioned in this section, the square numbers are the counting numbers 1 4 9 16 25 36, , , , , , . . . . Each square number

PROBLEM SET A

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Section 1.2 Three Additional Strategies 33

can be represented by a square array of dots as shown in the following figure, where the second square number has four dots, and so on. The first four square numbers are shown.

a. Find two triangular numbers (refer to Problem 6) whose sum equals the third square number.

b. Find two triangular numbers whose sum equals the fifth square number.

c. What two triangular numbers have a sum that equals the 10th square number? the 20th square number? the nth square number?

d. Find a triangular number that is also a square number.

e. Find five pairs of square numbers whose difference is a triangular number.

11. Would you rather work for a month (30 days) and get paid 1 million dollars or be paid 1 cent the first day, 2 cents the second day, 4 cents the third day, 8 cents the fourth day, and so on? Explain.

12. Find the perimeters and then complete the table.

number of triangles 1 2 3 4 5 6 10 n

perimeter 40

13. The integers greater than 1 are arranged as shown.

2 3 4 5

9 8 7 6

10 11 12 13

17 16 15 14 • • •

a. In which column will 100 fall? b. In which column will 1000 fall? c. How about 1999? d. How about 99,997?

14. How many cubes are in the 100th collection of cubes in this sequence?

15. The Fibonacci sequence is 1 1 2 3 5 8 13 21, , , , , , , , ,. . . where each successive number beginning with 2 is the sum of the preceding two; for example, 13 5 8 21 8 13= + = +, , and so on. Observe the following pattern.

1 1 1 2

1 1 2 2 3

1 1 2 3 3 5

2 2

2 2 2

2 2 2 2

+ = × + + = × + + + = ×

Write out six more terms of the Fibonacci sequence and use the sequence to predict what 1 1 2 3 1442 2 2 2 2+ + + + +. . . is without actually computing the sum. Then use your calculator to check your result.

16. Write out 16 terms of the Fibonacci sequence and observe the following pattern:

1 2 3

1 2 5 8

1 2 5 13 21

+ = + + = + + + =

Use the pattern you observed to predict the sum

1 2 5 13 610+ + + + +. . .

without actually computing the sum. Then use your calcu- lator to check your result.

17. Pascal’s triangle is where each entry other than a 1 is obtained by adding the two entries in the row above it.

a. Find the sums of the numbers on the diagonals in Pascal’s triangle as are indicated in the following figure.

b. Predict the sums along the next three diagonals in Pascal’s triangle without actually adding the entries. Check your answers by adding entries on your calculator.

18. Answer the following questions about Pascal’s triangle (see Problem 17).

a. In the triangle shown here, one number, namely 3, and the six numbers immediately surrounding it are encircled. Find the sum of the encircled seven numbers.

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34 Chapter 1 Introduction to Problem Solving

b. Extend Pascal’s triangle by adding a few rows. Then draw several more circles anywhere in the triangle like the one shown in part a. Explain how the sums obtained by adding the seven numbers inside the circle are related to one of the numbers outside the circle.

19. Consider the following sequence of shapes. The sequence starts with one square. Then at each step squares are attached around the outside of the figure, one square per exposed edge in the figure.

a. Draw the next two figures in the sequence. b. Make a table listing the number of unit squares in the

figure at each step. Look for a pattern in the number of unit squares. (Hint: Consider the number of squares attached at each step.)

c. Based on the pattern you observed, predict the number of squares in the figure at step 7. Draw the figure to check your answer.

d. How many squares would there be in the 10th figure? in the 20th figure? in the 50th figure?

20. In a dart game, only 4 points or 9 points can be scored on each dart. What is the largest score that it is not possible to obtain? (Assume that you have an unlimited number of darts.)

21. If the following four figures are referred to as stars, the first one is a three-pointed star and the second one is a six-pointed star. (NOTE: If this pattern of con- structing a new equilateral triangle on each side of the existing equilateral triangle is continued indefinitely, the resulting figure is called the Koch curve or Koch snowflake.)

a. How many points are there in the third star? b. How many points are there in the fourth star?

22. Using the Chapter 1 eManipulative activity Color Patterns on our Web site, describe the color patterns for the first three computer exercises.

23. Looking for a pattern can be frustrating if the pattern is not immediately obvious. Create your own sequence of numbers that follows a pattern but that has the capacity to stump some of your fellow students. Then write an explanation of how they might have been able to discover your pattern.

PROBLEM SET B 1. Find the missing term in each pattern. a. 10, 17, _____, 37, 50, 65

b. 1, 3 2

, _____, 7 8

, 9

16 c. 243, 324, 405, _____, 567 d. 234; _____; 23,481; 234,819; 2,348,200

2. Sketch a figure that is next in each sequence.

a.

b.

3. The rectangular numbers are whole numbers that are repre- sented by certain rectangular arrays of dots. The first five rectangular numbers are shown.

a. Complete the following table and describe the pattern in the Number of Dots column.

NUMBER NUMBER OF DOTS

(RECTANGULAR NUMBERS)

1 2

2 6

3

4

5

6

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Section 1.2 Three Additional Strategies 35

b. Make a sketch to represent the seventh rectangular number. c. How many dots will be in the tenth rectangular number? d. Is there a rectangular number that has 380 dots in its

shape? If so, which one? e. Write a formula for the number of dots in the nth rectan-

gular number. f. What is the connection between triangular numbers (see

Problem 6 in Set A) and rectangular numbers?

4. The pentagonal numbers are whole numbers that are rep- resented by pentagonal shapes. The first four pentagonal numbers are shown.

a. Complete the following table and describe the pattern in the Number of Dots column.

NUMBER

NUMBER OF DOTS (PENTAGONAL NUMBERS)

1 1 2 5 3 4 5

b. Make a sketch to represent the fifth pentagonal number. c. How many dots will be in the ninth pentagonal number? d. Is there a pentagonal number that has 200 dots in its

shape? If so, which one? e. Write a formula for the number of dots in the nth pen-

tagonal number.

5. Consider the following process: i. Choose a whole number. ii. Add the squares of the digits of the number to get a new

number.

Repeat step 2 several times.

a. Apply the procedure described to the numbers 12, 13, 19, 21, and 127.

b. What pattern do you observe as you repeat the steps over and over?

c. Check your answer for part b with a number of your choice.

6. How many triangles are in the picture?

7. What is the smallest number that can be expressed as the sum of two squares in two different ways? (You may use one square twice.)

8. How many cubes are in the 10th collection of cubes in this sequence?

9. The 2 2× array of numbers 4 5

5 6

⎡

⎣ ⎢

⎤

⎦ ⎥ has a sum of 4 5× , and

the 3 3× array 6 7 8

7 8 9

8 9 10

⎡

⎣

⎢ ⎢ ⎢

⎤

⎦

⎥ ⎥ ⎥ has a sum of 9 8× .

a. What will be the sum of the similar 4 4× array starting with 7?

b. What will be the sum of a similar 100 100× array starting with 100?

10. The Fibonacci sequence was defined to be the sequence 1 1 2 3 5 8 13 21, , , , , , , , ,. . . where each successive number is the sum of the preceding two. Observe the following pattern:

1 1 3 1

1 1 2 5 1

1 1 2 3 8 1

1 1 2 3 5 13 1

+ = − + + = − + + + = − + + + + = −

Write out six more terms of the Fibonacci sequence, and use the sequence to predict the answer to

1 1 2 3 5 144+ + + + + +. . .

without actually computing the sum. Then use your calcu- lator to check your result.

11. Write out 16 terms of the Fibonacci sequence. a. Notice that the fourth term in the sequence (called F4) is

odd: F4 3= . The sixth term in the sequence (called F6) is even: F6 8= . Look for a pattern in the terms of the sequence, and describe which terms are even and which are odd.

b. Which of the following terms of the Fibonacci sequence are even and which are odd: F F F F F38 51 150 200 300, , , , ?

c. Look for a pattern in the terms of the sequence and describe which terms are divisible by 3.

d. Which of the following terms of the Fibonacci sequence are multiples of 3: F F F F F48 75 196 379 1000, , , , ?

12. Write out 16 terms of the Fibonacci sequence and observe the following pattern:

1 3 5 1

1 3 8 13 1

1 3 8 21 34 1

+ = − + + = − + + + = −

Use the pattern you observed to predict the answer to

1 3 8 21 377+ + + + +. . .

without actually computing the sum. Then use your calculator to check your result.

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36 Chapter 1 Introduction to Problem Solving

13. Investigate the “Tower of Hanoi” problem on the Chapter 1 eManipulative activity Tower of Hanoi on our Web site to answer the following questions:

a. Determine the fewest number of moves required when you start with two, three, and four disks.

b. Describe the general process to move the disks in the fewest number of moves.

c. What is the minimum number of moves that it should take to move six disks?

14. While only 19 years old, Carl Friedrich Gauss proved in 1796 that every positive integer is the sum of at the most three triangular numbers (see Problem 6 in Set A).

a. Express each of the numbers 25 to 35 as a sum of no more than three triangular numbers.

b. Express the numbers 74, 81, and 90 as sums of no more than three triangular numbers.

15. Answer the following for Pascal’s triangle. a. In the following triangle, six numbers surrounding a

central number, 4, are circled. Compare the products of alternate numbers moving around the circle; that is, compare 3 1 10• • and 6 1 5• • .

b. Extend Pascal’s triangle by adding a few rows. Then draw several more circles like the one shown in part a anywhere in the triangle. Find the products as described in part a. What patterns do you see in the products?

16. A certain type of gutter comes in 6-foot, 8-foot, and 10-foot sections. How many different lengths can be formed using three sections of gutter?

17. Consider the sequence of shapes shown in the following figure. The sequence starts with one triangle. Then at each step, triangles are attached to the outside of the preceding figure, one triangle per exposed edge.

a. Draw the next two figures in the sequence. b. Make a table listing the number of triangles in the

figure at each step. Look for a pattern in the number of triangles. (Hint: Consider the number of triangles added at each step.)

c. Based on the pattern you observed, predict the number of triangles in the figure at step 7. Draw the figure to check your answer.

d. How many triangles would there be in the 10th figure? in the 20th figure? in the 50th figure?

18. How many equilateral triangles of all sizes are there in the 3 3 3× × equilateral triangle shown next?

19. Refer to the following figures to answer the questions. (NOTE: If this pattern is continued indefinitely, the resulting figure is called the Sierpinski triangle or the Sierpinski gasket.)

a. How many black triangles are there in the fourth figure? b. How many white triangles are there in the fourth figure? c. If the pattern is continued, how many black triangles are

there in the nth figure? d. If the pattern is continued, how many white triangles are

there in the nth figure?

20. If the pattern illustrated next is continued, a. find the total number of 1 by 1 squares in the thirtieth

figure. b. find the perimeter of the twenty-fifth figure. c. find the total number of toothpicks used to construct

the twentieth figure.

21. Find reasonable sixth, seventh, and eighth terms of the following sequences:

a. 1, 3, 4, 7, 11, _____, _____, _____ b. 0, 1, 4, 12, 29, _____, _____, _____

22. Many board games involve throwing two dice and sum- ming of the numbers that come up to determine how many squares to move. Make a list of all the different sums that can appear. Then write down how many ways each different sum can be formed. For example, 11 can be formed in two ways: from a 5 on the first die and a 6 on the second OR a 6 on the first die and a 5 on the second. Which sum has the greatest number of combinations? What conclusion could you draw from that?

23. There is an old riddle about a frog at the bottom of a 20-foot well. If he climbs up 3 feet each day and slips back 2 feet each night, how many days will it take him to reach the 20-foot mark and climb out of the well? The answer isn’t 20. Try doing the problem with a well that is only 5 feet deep, and keep track of all the frog’s moves. What strategy are you using?

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End of Chapter Material 37

Analyzing Student Thinking

24. Marietta extended the pattern 2, 4, 8 to be 2 4 8 16 32, , , , , . . . . Pascuel extended the same pattern to be 2 4 8 14 22, , , , , . . . . They asked you who was correct. How should you respond?

25. Eula is asked to find the following sum: 1 3 5 7+ + + + ⋅ ⋅ ⋅ + 97 99+ . She decided to “solve a simpler problem” but doesn’t know where to start. What would you suggest?

26. When Mickey counted the number of outcomes of roll- ing two 4-sided, tetrahedral dice (see Figure 12.89 for an example of a tetrahedron), he decided to “make a list” and got (1,1), (2,2), (3,3), (4,4), (1,2), (3,4), (2,4), (4,1). He started to get confused about which ones he had listed and which ones were left. How would you help him create a more systematic list?

27. Bridgette says that she knows how to see patterns like 3 7 11 15 19, , , , , . . . and 5 12 19 26 33, , , , , . . . because “you are just adding the same amount each time.” However, she

can’t see the pattern in 3, 5, 9, 15, 23 because the differ- ence between the numbers is changing. How could you help her see this new pattern?

28. Jeremy is asked to find the number of rectangles of all possible dimensions in the figure below and decides to “solve a simpler problem.”

What simpler problem would you suggest he use?

29. After solving several simpler problems, Janell commented that she often makes a list of solutions of the simpler prob- lems and then looks for a pattern in that list. She asks if it is okay to use more than one type of strategy when solving the same problem. How should you respond?

Place the whole numbers 1 through 9 in the circles in the accompanying triangle so that the sum of the numbers on each side is 17.

Strategy: Guess and Test

Having solved a simpler problem in this chapter, you might easily be able to conclude that 1, 2, and 3 must be in the cor- ners. Then the remaining six numbers, 4, 5, 6, 7, 8, and 9, must produce three pairs of numbers whose sums are 12, 13, and 14. The only two possible solutions are as shown.

Guess and Test

1. s u n f w i m= = = = = = =1 3 6 9 0 7 2, , , , , , 2. 0 4 4 4 4= − + −( ) ( )

1 4 4 4 4= + ÷ +( ) ( ) 2 4 4 4 4= ÷ + ÷( ) ( ) 3 4 4 4 4= + + ÷( ) 4 4 4 4 4= + × −( ) 5 4 4 4 4= × + ÷( ) 6 4 4 4 4= + ÷ +(( ) ) 7 4 4 4 4= + − ÷( ) 8 4 4 4 4= × ÷ +(( ) ) 9 4 4 4 4= + + ÷( )

There are many other possible answers.

3.

Draw a Picture

1. 5 2. Yes; make one cut, then lay the logs side by side for the sec-

ond cut. 3. 12

Use a Variable

1. 55, 5050, 125,250 2. ( ) ( ) ( ) ( ) ( )2 1 2 3 2 5 2 7 2 9m m m m m+ + + + + + + + +

= + = +10 25 5 2 5m m( ) 3. 10 80 90° ° °, ,

Look for a Pattern

1. 7 2. Square numbers 3. a. 224; 406; 518 b. 71

c. The sum is seven times the middle number.

END OF CHAPTER MATERIAL

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38 Chapter 1 Introduction to Problem Solving

Make a List

1. 48, 36, 33, 32, 24, 21, 20, 18, 17, 16, 12, 9, 8, 6, 5, 4, 3, 2, 1, 0

2. 204 3. 446

Solve a Simpler Problem

1. 1023 1024

2. 377 3. 190

R og

er V

io lle

t/ G

et ty

Im ag

es

O m

ik ro

n/ S

ci en

ce S

ou rc

e Sophie Germain (1776–1831) Sophie Germain, as a teen- ager in Paris, discovered math- ematics by reading books from her father’s library. At age 18, Germain wished to attend the prestigious Ecole Polytech-

nique in Paris, but women were not admitted. So she studied from classroom notes supplied by sympa- thetic male colleagues, and she began submitting writ- ten work using the pen name Antoine LeBlanc. This work won her high praise, and eventually she was able to reveal her true identity. Germain is noted for her theory of the vibration patterns of elastic plates and for her proof of Fermat’s last theorem in some special cases. Of Sophie Germain, Carl Gauss wrote, “When a woman, because of her sex, encounters infi nitely more obstacles than men . . . yet overcomes these fetters and penetrates that which is most hidden, she doubtless has the most noble courage, extraordinary talent, and superior genius.”

Carl Friedrich Gauss (1777–1885) Carl Friedrich Gauss, accord- ing to the historian E. T. Bell, “lives everywhere in math- ematics.” His contributions to geometry, number theory, and

analysis were deep and wide-ranging. Yet he also made crucial contributions in applied mathematics. When the tiny planet Ceres was discovered in 1800, Gauss devel- oped a technique for calculating its orbit, based on mea- ger observations of its direction from Earth at several known times. Gauss contributed to the modern theory of electricity and magnetism, and with the physicist W. E. Weber constructed one of the fi rst practical electric telegraphs. In 1807 he became director of the astronomi- cal observatory at Gottingen, where he served until his death. At age 18, Gauss devised a method for construct- ing a 17-sided regular polygon, using only a compass and straightedge. Remarkably, he then derived a general rule that pre dicted which regular polygons are likewise constructible.

Problems For each of the following, (i) determine a reasonable strategy to use to solve the problem, (ii) state a clue that suggested the strategy, and (iii) write out a solution using Pólya’s four- step process.

1. Fill in the circles using the numbers 1 through 9 once each where the sum along each of the five rows totals 17.

CHAPTER REVIEW

Review the following terms and problems to determine which require learning or relearning—page numbers are provided for easy reference.

Vocabulary/Notation Exercise  5 Problem  5 Pólya’s four-step process  5 Strategy  5 Random Guess and Test  8

Systematic Guess and Test  8 Inferential Guess and Test  8 Cryptarithm  9 Tetromino  11 Variable or unknown  12

Equation  16 Solution of an equation  16 Solve an equation  16

The Problem-Solving Process and Strategies

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Chapter Test 39

2. In the following arithmagon, the number that appears in a square is the product of the numbers in the circles on each side of it. Determine what numbers belong in the circles.

3. The floor of a square room is covered with square tiles. Walking diagonally across the room from corner to corner, Susan counted a total of 33 tiles on the two diagonals. What is the total number of tiles covering the floor of the room?

Vocabulary/Notation Pascal’s triangle  22 Sequence  22 Terms  22 Counting numbers  22

Ellipsis  22 Even numbers  23 Odd numbers  23 Square numbers  23

Powers  23 Fibonacci sequence  23 Inductive reasoning  23

Three Additional Strategies

Problems For each of the following, (i) determine a reasonable strategy to use to solve the problem, (ii) state a clue that suggested the strategy, and (iii) write out a solution using Pólya’s four-step process.

1. Consider the following products. Use your calculator to verify that the statements are true.

1 1 1

121 1 2 1 22

12321 1 2 3 2 1 333

2

2

2

× = × + + =

× + + + + =

( )

( )

( )

Predict the next line in the sequence of products. Use your calculator to check your answer.

2. a. How many cubes of all sizes are in a 2 2 2× × cube composed of eight 1 1 1× × cubes?

b. How many cubes of all sizes are in an 8 8 8× × cube com- posed of 512 1 1 1× × cubes?

3. a. What is the smallest number of whole-number gram weights needed to weigh any whole-number amount from 1 to 12 grams on a scale allowing the weights to be placed on either or both sides?

b. How about from 1 to 37 grams? c. What is the most you can weigh using six weights in this

way?

Knowledge 1. List the four steps of Pólya’s problem-solving process.

2. List the six problem-solving strategies you have learned in this chapter.

Skill 3. Identify the unneeded information in the following problem.

Birgit took her $5 allowance to the bookstore to buy some back-to-school supplies. The pencils cost $.10 each, the erasers cost $.05 each, and the clips cost 2 for $.01. If she bought 100 items altogether at a total cost of $1, how many of each item did she buy?

4. Rewrite the following problem in your own words.

If you add the square of Ruben’s age to the age of Angelita, the sum is 62; but if you add the square of Angelita’s age to the age of Ruben, the sum is 176. Can you say what the ages of Ruben and Angelita are?

5. Given the following problem and its numerical answer, write the solution in a complete sentence.

Amanda leaves with a basket of hard-boiled eggs to sell. At her first stop she sold half her eggs plus half an egg. At her second stop she sold half her eggs plus half an egg. The same thing occurs at her third, fourth, and fifth stops. When she finishes, she has no eggs in her basket. How many eggs did she start with?

Answer: 31

Understanding 6. Explain the difference between an exercise and a problem.

7. List at least two characteristics of a problem that would suggest using the Guess and Test strategy.

8. List at least two characteristics of a problem that would suggest using the Use a Variable strategy.

CHAPTER TEST

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40 Chapter 1 Introduction to Problem Solving

Problem Solving/Application For each of the following problems, read the problem care- fully and solve it. Identify the strategy you used.

9. Can you rearrange the 16 numbers in this 4 4× array so that each row, each column, and each of the two diagonals total 10? How about a 2 2× array containing two 1s and two 2s? How about the corresponding 3 3× array?

10. In three years, Chad will be three times my present age. I will then be half as old as he. How old am I now?

11. There are six baseball teams in a tournament. The teams are lettered A through F. Each team plays each of the other teams twice. How many games are played altogether?

12. A fish is 30 inches long. The head is as long as the tail. If the head was twice as long and the tail was its present length, the body would be 18 inches long. How long is each portion of the fish?

13. The Orchard brothers always plant their apple trees in square arrays, like those illustrated. This year they planted 31 more apple trees in their square orchard than last year. If the orchard is still square, how many apple trees are there in the orchard this year?

14. Arrange 10 people so that there are five rows each con- taining 4 persons.

15. A milk crate holds 24 bottles and is shaped like the one shown here. The crate has four rows and six columns. Is it possible to put 18 bottles of milk in the crate so that each row and each column of the crate has an even number of

bottles in it? If so, how? (Hint: One row has 6 bottles in it and the other three rows have 4 bottles in them.)

16. Otis has 12 coins in his pocket worth $1.10. If he only has nickels, dimes, and quarters, what are all of the possible coin combinations?

17. Show why 3 always divides evenly into the sum of any three consecutive whole numbers.

18. If 14 toothpicks are arranged to form a triangle so none of the toothpicks are broken or bent and all 14 toothpicks are used, how many different-shaped triangles can be formed?

19. Together a baseball and a football weigh 1.25 pounds, the baseball and a soccer ball weigh 1.35 pounds, and the football and the soccer ball weigh 1.6 pounds. How much does each of the balls weigh? Explain your reasoning.

20. In the figure below, there are 7 chairs arranged around 5 tables. How many chairs could be placed around a similar arrangement of 31 triangular tables?

21. Carlos’ father pays Carlos his allowance each week using patterns. He pays a different amount each day according to some pattern. Carlos must identify the pattern in order to receive his allowance. Help Carlos complete the pattern for the missing days in each week below.

a. 5¢, 9¢, 16¢, 26¢, _____, _____, _____ b. 1¢, 6¢, 15¢, 30¢, 53¢, _____, _____ c. 4¢, 8¢, 16¢, 28¢, _____, _____, _____

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c01.indd 41 7/30/2013 2:37:17 PM

42

T he Maya people lived mainly in southeastern Mexico, including the Yucatan Peninsula, and in much of northwestern Central America, includ- ing Guatemala and parts of Honduras and El Salvador. Earliest archaeological evidence of the Maya civiliza- tion dates to 9000 B.C.E., with the principal epochs of the Maya cultural development occurring between 2000 B.C.E. and 1700 C.E.

Knowledge of arithmetic, calendrical, and astronomical matters was more highly developed by the ancient Maya than by any other New World peoples. Their numeration system was simple, yet sophisticated. Their system utilized three basic numerals: a dot, •, to represent 1; a horizontal bar, —, to represent 5; and a conch shell, , to rep- resent 0. They used these three symbols, in combination, to represent the numbers 0 through 19.

For numbers greater than 19, they initially used a base twenty system. That is, they grouped in twenties and dis- played their numerals vertically. Three Mayan numerals are shown together with their values in our system and the place values initially used by the Maya.

The sun, and hence the solar calendar, was very important to the Maya. They calculated that a year con- sisted of 365.2420 days. (Present calculations measure our year as 365.2422 days long.) Since 360 had convenient factors and was close to 365 days in their year and 400 in their numeration system, they made a place-value system where the column values from right to left were 1, 20, 20 18 360 20 18 7200 20 18 144 0002 3¥ ¥ ¥( ), ( ), ( , ),= = = and so on. Interestingly, the Maya could record all the days of their history simply by using the place values through 144,000. The Maya were also able to use larger numbers. One Mayan hieroglyphic text recorded a number equiva- lent to 1,841,641,600.

Finally, the Maya, famous for their hieroglyphic writing, also used the 20 ideograms pictured here, called head variants, to represent the numbers 0 19− .

The Mayan numeration system is studied in this chap- ter along with other ancient numeration systems.

C H A P T E R

2 SETS, WHOLE NUMBERS, AND NUMERATION The Mayan Numeration System

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43

Often there are problems where, although it is not necessary to draw an actual picture to represent the problem situation, a diagram that represents the essence of the problem is useful. For example, if we wish to determine the number of times two heads can turn up when we toss two coins, we could literally draw pictures of all possible arrangements of two coins turning up heads or tails. However, in practice, a simple tree diagram is used like the one shown next.

This diagram shows that there is one way to obtain two heads out of four possible outcomes. Another type of diagram is helpful in solving the next problem.

Initial Problem A survey was taken of 150 college freshmen. Forty of them were majoring in math- ematics, 30 of them were majoring in English, 20 were majoring in science, 7 had a double major of mathematics and English, and none had a double (or triple) major with science. How many students had majors other than mathematics, English, or science?

Clues The Draw a Diagram strategy may be appropriate when

The problem involves sets, ratios, or probabilities.

An actual picture can be drawn, but a diagram is more effi cient.

Relationships among quantities are represented.

A solution of this Initial Problem is on page 79.

Draw a DiagramProblem-Solving Strategies 1. Guess and Test

2. Draw a Picture

3. Use a Variable

4. Look for a Pattern

5. Make a List

6. Solve a Simpler Problem

7. Draw a Diagram

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44

AUTHOR

WALK-THROUGH

I N T R O D U C T I O N Much of elementary school mathematics is devoted to the study of numbers. Children first learn to count using the natural numbers or counting numbers 1 2 3, , , . . . (the ellipsis, or three periods, means “and so on”). This chapter develops the ideas that lead to the concepts central to the system of whole numbers 0 1 2 3, , , , . . . (the counting numbers together with zero) and the symbols that are used to represent them. First, the notion of a one-to-one correspondence between two sets is shown to be the idea central to the formation of the concept of number. Then operations on sets are discussed. These

operations form the foundation of addition, subtraction, multiplication, and division of whole numbers. Finally, the Hindu-Arabic numeration system, our system of symbols that represent numbers, is presented after its various attributes are introduced by considering other ancient numeration systems.

Key Concepts from the NCTM Principles and Standards for School Mathematics

PRE-K-2–NUMBER AND OPERATIONS

Count with understanding and recognize “how many” in sets of objects. Use multiple models to develop initial understandings of place value and the base ten number system. Develop understanding of the relative position and magnitude of whole numbers and of ordinal and cardinal numbers

and their connections. Connect number words and numerals to the quantities they represent, using various physical models and representations.

GRADES 3-5–NUMBER AND OPERATIONS

Understand the place-value structure of the base ten number system and be able to represent and compare whole num- bers.

Key Concepts from the NCTM Curriculum Focal Points

PRE-K: Developing an understanding of whole numbers, including concepts of correspondence, counting, cardinality, and comparison.

KINDERGARTEN: Representing, comparing, and ordering whole numbers and joining and separating sets. Ordering objects by measurable attributes.

GRADE 1: Developing an understanding of whole number relationships, including grouping in tens and ones. GRADE 2: Developing an understanding of the base ten numeration system and place-value concepts. GRADE 6: Writing, interpreting, and using mathematical expressions and equations.

Key Concepts from the Common Core State Standards for Mathematics

KINDERGARTEN: Know number names and the count sequence (to 100). Count to tell the number of objects (con- nect counting to cardinality). Compare numbers (greater than, less than, equal to). Work with numbers 11–19 to gain foundations for place value.

GRADE 1: Extend the counting sequence (to 120). Understand place value (the two digits of a two-digit number represent the number of tens and ones).

GRADE 2: Extend the counting sequence (within 1000) by skip-counting. Understand place value (the three digits of a three-digit number represent the number of hundreds, tens, and ones).

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Section 2.1 Sets as a Basis for Whole Numbers 45

Sets A collection of objects is called a set and the objects are called elements or members of the set. Sets can be defined in three common ways: (1) a verbal description, (2) a listing of the elements separated by commas, with braces (“{” and “}”) used to enclose the list of elements, and (3) set-builder notation. For example, the verbal description “the set of all states in the United States that border the Pacific Ocean” can be represented in the other two ways as follows:

1. Listing: {Alaska, California, Hawaii, Oregon, Washington}.

2. Set-builder:{ |x x is a U.S. state that borders the Pacific Ocean}. (This set-builder nota- tion is read: “The set of all x such that x is a U.S. state that borders the Pacific Ocean.”)

Sets are usually denoted by capital letters such as A, B, C and so on. The symbols “∈” and “∉” are used to indicate that an object is or is not an element of a set, respec- tively. For example, if S represents the set of all U.S. states bordering the Pacific, then Alaska ∈S and Michigan ∉S. The set without elements is called the empty set (or null set) and is denoted by { } or the symbol ∅. The set of all U.S. states bordering Antarctica is the empty set.

Two sets A and B are equal, written A B= , if and only if they have precisely the same elements. Thus { is a state that borders Lake Michigan} {Illinois, Indx x| = iiana, Michigan, Wisconsin}. Notice that two sets, A and B, are equal if every element of A is in B, and vice versa. If A does not equal B, we write A Bñ .

There are two inherent rules regarding sets: (1) The same element is not listed more than once within a set, and (2) the order of the elements in a set is immate- rial. Thus, by rule 1, the set { , , }a a b would be written as { , }a b and by rule 2, { , } { , }, { , , } { , , },a b b a x y z y z x= = and so on.

The concept of a 1–1 correspondence, read “one-to-one correspondence,” is need- ed to formalize the meaning of a whole number.

Algebraic Reasoning When solving algebraic equations, we are looking for all values/ numbers that make an equation true. This set of numbers is called the solution set.

When a young child is asked to count the blocks in the set shown below, he will use his mem- orized cadence of “one, two, three, four . . . ” while pointing to the blocks. The  following

three scenarios are common:

Child 1 says “one” while pointing to the red block, “two” while pointing to the green block, and then goes back to the red block to say “three” before pointing to the yellow block and say, “four.”

Child 2 says “one” while pointing to the yellow block and says “two” while pointing to the red block and is done. Child 3 points at the green block and says “one,” the red block and says “two,” and finally points to the yellow block while

saying “three.”

What underlying concept do children need to come to understand in order to count correctly?

SETS AS A BASIS FOR WHOLE NUMBERS

One-to-One Correspondence

A 1-1 correspondence between two sets A and B is a pairing of the elements of A with the elements of B so that each element of A corresponds to exactly one ele- ment of B, and vice versa. If there is a 1-1 correspondence between sets A and B, we write A B~ and say that A and B are equivalent or matching sets.

D E F I N I T I O N 2 . 1

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46 Chapter 2 Sets, Whole Numbers, and Numeration

Figure 2.1 shows two possible 1–1 correspondences between two sets, A and B. There are four other possible 1–1 correspondences between A and B. Notice

that equal sets are always equivalent, since each element can be matched with itself, but that equivalent sets are not necessarily equal. For example, { , } ~ { , },1 2 a b but { , } { , }.1 2 ≠ a b The two sets A a b= { , } and B a b c= { , , } are not equivalent. However, they do satisfy the relationship defined next.

Figure 2.1 Subset of a Set: A B⊆

Set A is said to be a subset of B, written A B⊆ , if and only if every element of A is also an element of B.

D E F I N I T I O N 2 . 2

The set consisting of New Hampshire is a subset of the set of all New England states and { , , } { , , , , , }.a b c a b c d e f ⊆ Since every element in a set A is in A, A A⊆ for all sets A. Also, { , , } { , , }a b c a b d ⊆ because c is in the set { , , }a b c but not in the set { , , }.a b d Using similar reasoning, you can argue that ∅ ⊆ A for any set A since it is impossible to find an element in ∅ that is not in A.

If A B⊆ and B has an element that is not in A, we write A B⊂ and say that A is a proper subset of B. Thus { , } { , , },a b a b c ⊂ since { , } { , , }a b a b c ⊆ and c is in the second set but not in the fi rst.

Circles or other closed curves are used in Venn diagrams (named after the English logician John Venn) to illustrate relationships between sets. These circles are usually pictured within a rectangle, U , where the rectangle represents the universal set or universe, the set comprised of all elements being considered in a particular discussion. Figure 2.2 displays sets A and B inside a universal set U .

Set A is comprised of everything inside circle A, and set B is comprised of everything inside circle B, including set A. Hence A is a proper subset of B since x B∈ , but x A∉ . The idea of proper subset will be used later to help establish the meaning of the concept “less than” for whole numbers.Figure 2.2

Check for Understanding: Exercise/Problem Set A #1–9✔

Finite and Infinite Sets There are two broad categories of sets: finite and infinite. Informally, a set is finite if it is empty or can have its elements listed (where the list eventually ends), and a set is infinite if it goes on without end. A little more formally, a set is finite if (1) it is empty or (2) it can be put into a 1–1 correspondence with a set of the form { , , , , },1 2 3 . . . n where n is a counting number. On the other hand, a set is infinite if it is not finite.

Determine whether the following sets are finite or infinite.

a. { , , }a b c b. { , , , }1 2 3 . . . c. { , , , , }2 4 6 20. . .

S O L U T I O N

a. { , , }a b c is fi nite since it can be matched with the set { , , }1 2 3 . b. { , , , }1 2 3 . . . is an infi nite set. c. { , , , , }2 4 6 20. . . is a fi nite set since it can be matched with the set { , , , , }1 2 3 10. . . .

(Here, the ellipsis means to continue the pattern until the last element is reached.) ■

Reflection from Research While many students will agree that two infinite sets such as the set of counting numbers and the set of even numbers are equiva- lent, the same students will argue that the set of counting numbers is larger due to its inclusion of the odd numbers (Wheeler, 1987).

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Section 2.1 Sets as a Basis for Whole Numbers 47

NOTE: The small solid square (■) is used to mark the end of an example or math- ematical argument.

An interesting property of every infinite set is that it can be matched with a proper subset of itself. For example, consider the following 1-1 correspondence:

Note that B A⊂ and that each element in A is paired with exactly one element in B, and vice versa. Notice that matching n with 2n indicates that we never “run out” of elements from B to match with the elements from set A. Thus an alternative definition is that a set is infinite if it is equivalent to a proper subset of itself. In this case, a set is finite if it is not infinite.

Algebraic Reasoning Due to the nature of infinite sets, a variable is commonly used to illustrate matching elements in two sets when showing 1-1 cor- respondences.

Check for Understanding: Exercise/Problem Set A #10–11✔

After forming a group of students, use a diagram like the one shown to place the names of each member of your group in the appropriate region. All members of the group of stu-

dents in a class fit in the following rectangle.

Write a sentence that describes the attributes of a person whose name is in the red region. Write a sentence that describes the attributes of a person whose name is not in any of the circles.

Write a sentence that describes the attributes of a person whose name is in the blue region.

Operations on Sets Two sets A and B that have no elements in common are called disjoint sets. The sets { , , }a b c and { , , }d e f are disjoint (Figure 2.3), whereas { , }x y and { , }y z are not dis- joint, since y is an element in both sets.

Figure 2.3

There are many ways to construct a new set from two or more sets. The following operations on sets will be very useful in clarifying our understanding of whole num- bers and their operations.

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48 Chapter 2 Sets, Whole Numbers, and Numeration

Figure 2.5 displays A ∩ B. Observe that two sets are disjoint if and only if their intersection is the empty set. Figure 2.3 shows a Venn diagram of two sets whose intersection is the empty set.

In many situations, instead of considering elements of a set A, it is more productive to consider all elements in the universal set other than those in A. This set is defined next.

Figure 2.5 Shaded region is A B∩ .

Informally, A B∪ is formed by putting all the elements of A and B together. The next example illustrates this definition.

Union of Sets: A B∪

The union of two sets A and B, written A B∪ , is the set that consists of all elements belonging either to A or to B (or to both).

D E F I N I T I O N 2 . 3

Intersection of Sets: A B§

The intersection of sets A and B, written A B§ , is the set of all elements common to sets A and B.

D E F I N I T I O N 2 . 4

Find the union of the given pairs of sets.

a. { , } { , , }a b c d e ∪ b. { , , , ,1 2 3 4 5 } ∪ ∅ c. { , , } { , , }m n q m n p ∪

S O L U T I O N a. { , } { , , } { , , , , }a b c d e a b c d e ∪ = b. { , , , , } { , , , , }1 2 3 4 5 1 2 3 4 5 ∪ ∅ = c. { , , } { , , } { , , , }m n q m n p m n p q ∪ = ■

Notice that although m is a member of both sets in Example 2.2(c), it is listed only once in the union of the two sets. The union of sets A and B is displayed in a Venn diagram by shading the portion of the diagram that represents A B∪ (Figure 2.4).

The notion of set union is the basis for the addition of whole numbers, but only when disjoint sets are used. Notice how the sets in Example 2.2(a) can be used to show that 2 3 5+ = .

Another useful set operation is the intersection of sets.

Figure 2.4 Shaded region is A B∪ .

Thus A B∩ is the set of elements shared by A and B. Example 2.3 illustrates this definition.

Find the intersection of the given pairs of sets.

a. { , , } { , , }a b c b d f ∩ b. { , , } { , , }a b c a b c ∩ c. { , } { , }a b c d ∩

S O L U T I O N 

a. { , , } { , , } { }a b c b d f b ∩ = since b is the only element in both sets. b. { , , } { , , } { , , }a b c a b c a b c ∩ = since a b c, , are in both sets. c. { , } { , }a b c d ∩ = ∅ since there are no elements common to the given two sets. ■

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Section 2.1 Sets as a Basis for Whole Numbers 49

Complement of a Set: A

The complement of a set A, written A, is the set of all elements in the universe, U , that are not in A.

D E F I N I T I O N 2 . 5

The set A is shaded in Figure 2.6.

Figure 2.6 Shaded region is A.

Find the following sets.

a. A where U a b c d= { , , , } and A a= { } b. B where U = { , , , }1 2 3 . . . and B = { , , , }2 4 6 . . . c. A B∪ and A B∩ where U A= ={ , , , , }, { , , },1 2 3 4 5 1 2 3 and B = { , }3 4

S O L U T I O N

a. A b c d= { , , } b. B = { , , , }1 3 5 . . . c. A B∪ = ∪ ={ , } { , , } { , , , }4 5 1 2 5 1 2 4 5 A B∩ = ={ } { , , , }3 1 2 4 5 ■

The next set operation forms the basis for subtraction.

Difference of Sets: A B-

The set difference (or relative complement) of set B from set A, written A B− , is the set of all elements in A that are not in B.

D E F I N I T I O N 2 . 6

In set-builder notation, A B x x A x B− = ∈ ∉{ | }. and Also, as can be seen in Figure 2.7, A B− can be viewed as A ∩ B. Example 2.5 provides some examples of the difference of one set from another.

Figure 2.7 Shaded region is A B− .

Find the difference of the given pairs of sets.

a. { , , } { , }a b c b d − b. { , , } { }a b c e − c. { , , , } { , , }a b c d b c d −

S O L U T I O N

a. { , , } { , } { , }a b c b d a c − = b. { , , } { } { , , }a b c e a b c − = c. { , , , } { , , } { }a b c d b c d a − = ■

Use the set names of A , B, and C as well as the symbols for operations on sets to describe:

The blue region in more than one way.

The red region in more than one way.

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50 Chapter 2 Sets, Whole Numbers, and Numeration

In Example 2.5(c), the second set is a subset of the first. These sets can be used to show that 4 3 1− = .

Another way of combining two sets to form a third set is called the Cartesian prod- uct. The Cartesian product, named after the French mathematician René Descartes, forms the basis of whole-number multiplication and is also useful in probability and geometry. To define the Cartesian product, we need to have the concept of ordered pair. An ordered pair, written ( , ),a b is a pair of elements where one of the elements is designated as first (a in this case) and the other is second (b here). The notion of an ordered pair differs from that of simply a set of two elements because of the pref- erence in order. For example, { , } { , }1 2 2 1 = as sets, because they have the same ele- ments. But (1, 2) ≠ (2, 1) as ordered pairs, since the order of the elements is different. Two ordered pairs ( , )a b and ( , )c d are equal if and only if a c= and b d= .

Cartesian Product of Sets: A Bë

The Cartesian product of set A with set B, written A Bë and read “A cross B,” is the set of all ordered pairs ( , ),a b where a A∈ and b B∈ .

D E F I N I T I O N 2 . 7

Find the Cartesian product of the given pairs of sets.

a. { , , } { , }x y z m n × b. { } { , , }7 × a b c

S O L U T I O N a. { , , } { , } {( , ), ( , ), ( , ), ( , ), ( , ),x y z m n x m x n y m y n z m × = (( , )}z n b. { } { , , } {( , ), ( , ), ( , )}7 7 7 7× =a b c a b c ■

In set-builder notation, A B a b a A b B× = ∈ ∈{( , ) | }. and

Check for Understanding: Exercise/Problem Set A #12–32✔

Notice that when finding a Cartesian product, all possible pairs are formed where the first element comes from the first and the second element comes from the second set. Also observe that in Example 2.6 (a), there are three elements in the first set, two in the second, and six in their Cartesian product, and that 3 2 6.× = Similarly, in (b), these sets can be used to find the whole-number product 1 3 3.× =

All of the operations on sets that have been introduced in this subsection result in a new set. For example, A B∩ is an operation on two sets that results in the set of all elements that are common to set A and B. Similarly, C D− is an operation on set C that results in the set of elements C that are not also in set D. On the other hand, expressions such as A B⊆ or x C∉ are not operations; they are statements that are either true or false. Table 2.1 lists all such set statements and operations.

STATEMENTS OPE RATIONS

A B⊆ A B∪ A B⊂ A B∩ A B~ A

x B∈ A B− A B− A B× A B⊄ x A∉ A B≠

TABLE 2.1

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Section 2.1 Sets as a Basis for Whole Numbers 51

Venn diagrams are often used to solve problems, as shown in Example 2.7.

Thirty elementary teachers were asked which high school cours- es they appreciated: algebra or geometry. Seventeen appreciated

algebra and 15 appreciated geometry; of these, 5 said that they appreciated both. How many appreciated neither?

S O L U T I O N   Since there are two courses, we draw a Venn diagram with two over- lapping circles labeled A for algebra and G for geometry [Figure 2.8(a)]. Since 5 teach- ers appreciated both algebra and geometry, we place a 5 in the intersection of A and G [Figure 2.8(b)]. Seventeen must be in the A circle; thus 12 must be in the remaining part of A [Figure 2.8(c)]. Similarly, 10 must be in the G circle outside the intersection [Figure 2.8(d)]. Thus we have accounted for 12 5 10 27+ + = teachers. This leaves 3 of the 30 teachers who appreciated neither algebra nor geometry. ■

Figure 2.8

There are several theories concerning the rationale behind the shapes of the 10 digits in our Hindu–Arabic numeration system. One is that the number represented by each digit is given by the number of angles in the original digit. Count the “angles” in each digit below. (Here an “angle” is less than 180°.) Of course, zero is round, so it has no angles.

© R

on B

ag w

el l

EXERCISES 1. Indicate the following sets by the listing method. a. Whole numbers between 5 and 9 b. Whole numbers less than 12 and greater than 2 c. Even counting numbers less than 15 d. Even counting numbers less than 151

2. Which of the following sets are equal to { , , }?4 5 6 a. { , }5 6 b. { , , }5 4 6 c. Whole numbers greater than 3 d. Whole numbers less than 7 e. Whole numbers greater than 3 or less than 7 f. Whole numbers greater than 3 and less than 8 g. { , , }e f g

3. True or false? a. 7 6 7 8 9∈{ , , , } b. 5 2 3 4 6∉{ , , , }

c. { , , } { , , }1 2 3 1 2 3 ⊆ d. { , , } { , , }1 2 5 1 2 5 ⊂ e. { } { , }2 1 2⊆

4. Find four 1-1 correspondences between A and B other than the two given in Figure 2.1.

5. Determine which of the following sets are equivalent to the set { , , }x y z .

a. { , , , }w x y z b. { , , }1 5 17 c. { , , }w x z d. { , , }a b a e. { , , }h n s

6. Which of the following pairs of sets are equal? a. { , }a b and “First two letters of the alphabet” b. { , , , }7 8 9 10 and “Whole numbers greater

than 7” c. { , , , }7 8 a b and { , , , , }a b c 7 8 d. { , , , }x y x z and { , , , }z x y z

EXERCISE/PROBLEM SET A

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52 Chapter 2 Sets, Whole Numbers, and Numeration

7. List all the subsets of { , , }.a b c

8. List the proper subsets of { ,s n}.

9. Let A v x z B w x y= ={ , , }, { , , }, and C v w x y z= { , , , , }. In the following, choose all of the possible symbols ( , , ~ )∈ ∉ ⊂ ⊆ ⊆ =, , , , or that would make a true statement.

a. z _____ B b. B _____ C c. ∅ _____ A d. A _____ B e. v _____ C f. B _____ B

10. Determine which of the following sets are finite. For those sets that are finite, how many elements are in the set?

a. The set of ears on a typical elephant. b. { , , , , }1 2 3 99. . .         c. { , , , , , }0 1 2 3 200. . . d. Set of points belonging to a line segment e. Set of points belonging to a circle

11. Since the set A = { , , , , , }3 6 9 12 15 . . . is infinite, there is a proper subset of A that can be matched to A. Find two such proper subsets and describe how each is matched to A.

12. Given A = { , , , , , },2 4 6 8 10 . . . and B = { , , , , , },4 8 12 16 20 . . . answer the following questions.

a. Find the set A B∪ . b. Find the set A B∩ . c. Is B a subset of A ? Explain. d. Is A equivalent to B ? Explain. e. Is A equal to B ? Explain. f. Is B a proper subset of A ? Explain.

13. True or false? a. For all sets X and Y , either X Y⊆ or Y X⊆ . b. If A is an infinite set and B A⊆ , then B also is an infinite

set. c. For all finite sets A and B, if A B∩ = ∅, then the

number of elements in A B∪ equals the number of elements in A plus the number of elements in B.

14. The regions A B∩ and A B∪ can each be formed by shad- ing the set A using lines in one direction and the set B using lines in another direction. The region with shading in both directions is A B∩ and the entire shaded region is A B∪ . Use the Chapter 2 eManipulative activity Venn Diagrams on our Web site to do the following:

a. Find the region ( ) .A B C∪ ∩ b. Find the region A B C∪ ∩( ). c. Based on the results of parts a and b, what conclusions

can you draw about the placement of parentheses?

15. Draw a Venn diagram like the following one for each part. Then shade each Venn diagram to represent each of the sets indicated.

a. S         b. S T∩        c. ( ) ( )S T T S− ∪ −        d. U − S

16. A Venn diagram can be used to illustrate more than two sets. Shade the regions that represent each of the following sets. The Chapter 2 eManipulative activity

Venn Diagrams on our Web site may help in solving this problem.

a. A B C∩ ∩( )     b. A B C− ∩( )     c. A B C∪ −( )

17. Represent the following shaded regions using the symbols A B C, , , , ∪ ∩, and −. The Chapter 2 eManipulative activity Venn Diagrams on our Web site can be used in solv- ing this problem.

c.

.b.a

18. Draw Venn diagrams that represent sets A and B as shown here.

a. A B⊂        b. A B∩ = ∅      c. A B A∪ =

19. In the drawing, C is the interior of the circle, T is the interior of the triangle, and R is the interior of the rect- angle. Copy the drawing on a sheet of paper, then shade in each of the following regions.

a. C T∪ b. C R∩ c. ( )C T R∩ ∪ d. ( )C R T∪ ∪ e. ( )C R T∩ ∩ f. C R T∩ ∩( )

20. Let W = {women who have won Nobel Prizes}, A = {Americans who have won Nobel Prizes}, and C = {winners of the Nobel Prize in chemistry}. Describe the elements of the following sets.

a. W A∪      b. W A∩      c. A C∩

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Section 2.1 Sets as a Basis for Whole Numbers 53

21. Let A a b c B b c= ={ , , }, { , }, and C e= { }. Find each of the following.

a. A B∩       b. B C∪       c. A B−

22. Find each of the following differences. a. { , , } { , }s n h n h, / − b. { , , , } { , , , }0 1 2 12 13 14. . . . . .− c. {people} {married people}−

d. { , , , } { }a b c d −

23. Let A = { , , , , , }2 4 6 8 10 . . . and B = {4, 8,12,16, }.. . . a. Find A B− . b. Find B A− .

c. Based on your observations in parts a and b, describe a general case for which A B− = ∅.

24. Given A B= ={ , , , , , }, { , , , , , },0 1 2 3 4 5 0 2 4 6 8 10 and C = { , , },0 4 8 find each of the following.

a. A B∪ b. B C∪ c. A B∩ d. B C∩ e. B C− f. ( )A B C∪ − g. A ∩ ∅

25. Let M = the set of the months of the year, J = {January, June, July}, S = {June, July, August}, W = {December, January, February}. List the members of each of the following:

a. J S∪ b. J W∩ c. S W∩ d. J S W∩ ∪( ) e. M S W− ∪( ) f. J S−

26. Let A = { , , , , , , , , }3 6 9 12 15 18 21 24 . . . and B = { , , , , }.6 12 18 24 . . .

a. Is B A⊆ ? b. Find A B∪ . c. Find A B∩ . d. In general, when B A⊆ , what is true about A B∪ ?

About A B∩ ?

27. Verify that A B A B∪ = ∩ in two different ways as follows: a. Let U A= ={ , , , , , }, { , , },1 2 3 4 5 6 2 3 5 and B = { , }.1 4

List the elements of the sets A B∪ and A B∩ . Do the two sets have the same members?

b. Draw and shade a Venn diagram for each of the sets A B∪ and A B∩ . Do the two Venn diagrams look the same? NOTE: The equation A B A B∪ = ∩ is one of two laws called DeMorgan’s laws.

28. Find the following Cartesian products.

a. { } { , }a b c× b. { } { , , }5 × a b c c. { , } { , , }a b × 1 2 3 d. { , } { , }2 3 1 4 × e. { , , } { }a b c × 5 f. { , , } { , }1 2 3 × a b

29. Determine how many ordered pairs will be in the following sets.

a. { , , , } { , }1 2 3 4 × a b b. { , , } { , , , }m n o × 1 2 3 4

30. If A has two members, how many members does B have when A B× has the following number of members? If an answer is not possible, explain why.

a. 4 b. 8 c. 9 d. 50 e. 0 f. 23

31. The Cartesian product, A B× , is given in each of the fol- lowing parts. Find A and B.

a. {( , ), ( , ), ( , )}a a a 2 4 6 b. {( , ), ( , ), ( , ), ( , )}a b b b b a a a

32. True or false? a. {( , ), ( , )} {( , ), ( , )}4 5 6 7 6 7 4 5 = b. {( , ), ( , )} {( , ), ( , )}a b c d b a c d = c. {( , ), ( , )} {( , ), ( , )}4 5 7 6 7 6 5 4 = d. {( , ), ( , )} { , , } { , , }5 6 7 4 4 5 6 5 7 9 ⊆ ×

PROBLEMS 33. a. If X has five elements and Y has three elements, what is

the greatest number of elements possible in X Y∩ ? In X Y∪ ?

b. If X has x elements and Y has y elements with x greater than or equal to y, what is the greatest number of ele- ments possible in X Y∩ ? In X Y∪ ?

34. How many different 1-1 correspondences are possible between A = { , , , }1 2 3 4 and B a b c d= { , , , }?

35. How many subsets does a set with the following number of members have?

a. 0 b. 1 c. 2

d. 3 e. 5 f. n

36. If it is possible, give examples of the following. If it is not possible, explain why.

a. Two sets that are not equal but are equivalent b. Two sets that are not equivalent but are equal

37. a. When does D E D∩ = ? b. When does D E D∪ = ? c. When does D E D E∩ = ∪ ?

38. Carmen has 8 skirts and 7 blouses. Show how the con- cept of Cartesian product can be used to determine how many different outfits she has.

39. How many matches are there if 32 participants enter a single-elimination tennis tournament (one loss elimi- nates a participant)?

40. Can you show a 1-1 correspondence between the points on base AB of the given triangle and the points on the two sides AC and CB? Explain how you can or why you cannot.

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54 Chapter 2 Sets, Whole Numbers, and Numeration

41. Can you show a 1-1 correspondence between the points on chord AB and the points on arc ACB? Explain how you can or why you cannot.

42. A poll of 100 registered voters designed to find out how voters kept up with current events revealed the following facts:

65 watched the news on television. 39 read the newspaper. 39 listened to radio news. 20 watched TV news and read the newspaper. 27 watched TV news and listened to radio news. 9 read the newspaper and listened to radio news. 6 watched TV news, read the newspaper, and listened to radio news.

a. How many of the 100 people surveyed did not keep up with current events by the three sources listed?

b. How many of the 100 people surveyed read the paper but did not watch TV news?

c. How many of the 100 people surveyed used only one of the three sources listed to keep up with current events?

43. At a convention of 375 butchers ( )B , bakers ( )A , and candlestick makers ( )C , there were

50 who were both B and A but not C 70 who were B but neither A nor C 60 who were A but neither B nor C 40 who were both A and C but not B 50 who were both B and C but not A 80 who were C but neither A nor B

How many at the convention were A B, , and C?

44. A student says that A B− means you start with all the elements of A and you take away all the elements of B. So A B× must mean you take all the elements of A and multiply them times all the elements of B. Do you agree with the student? How would you explain your reasoning?

EXERCISES 1. Indicate the following sets by the listing method. a. Whole numbers greater than 8 b. Odd counting numbers between 2 and 11 c. Odd whole numbers less than 100 d. Whole numbers less than 0

2. Which of the following sets are equal to { , , , }?0 2 4 6 a. { , , , , , , }0 1 2 3 4 5 6 b. { , , , }6 2 0 4 c. Even counting numbers less than 7 d. Even whole numbers less than 7 e. Even whole numbers less than 6 f. { , , , }a c e g g. { , , , , }0 2 4 6 0

3. True or false? a. 2

3 1 2 3∈{ , , } b. 1 0 1 2∉{ , , } c. { , } { , , }4 3 2 3 4 ⊂ d. ∅ ⊆ { } e. { , }1 2 ⊆ { }2

4. Show three different 1-1 correspondences between { , , , }1 2 3 4 and { , , , }x y z w .

5. Determine which of the following sets are equivalent to the set {apple, apple, pear}.

a. {apple, pear, apple, pear} b. {apple, pear} c. {squash, eggplant} d. {pear, apple, apple} e. { , }x y

6. Which of the following pairs of sets are equal? a. { , , }0 1 2 and “first three counting numbers’’

b. { , , , }0 1 2 3 and “whole numbers less than 4’’

c. { , , , }1 2 1 2 and { , , , }2 1 2 1

d. { , }1 2 and { , , }.1 1 2

7. List all subsets of { , , }s n u . Which are proper subsets?

8. List the proper subsets of { , , }x y z

9. Let A B= ={ , , , , }, { , , },1 2 3 4 5 3 4 5 and C = { , , }.4 5 6 In the following, choose all of the possible symbols ( , , , , , ~, )∈ ∉ ⊂ ⊆ ⊆ =or that would make a true statement.

a. 2 _____ A b. B _____ A c. C _____ B d. 6 _____ C e. A _____ A f. B C∩ _____ A

10. Determine which of the following sets are finite. For those sets that are finite, how many elements are in the set?

a. The set of fingers on your hand. b. { , , , , }2 4 6 8 . . . c. { , , , , }4 8 12 100. . . d. The set of sides of a polygon.

11. Show that the following sets are infinite by matching each with a proper subset of itself.

a. { , , , , , }2 4 6 . . . . . . n b. { , , , , , , }50 51 52 53 . . . . . . n

12. Let A = { , , , , }0 10 20 30 . . . and B = { , , , , }.5 15 25 35 . . . Decide which of the following are true and which are false. Explain your answers.

a. A equals B    b. B is equivalent to A    c. A B∩ ≠ ∅ d. B is equivalent to a proper subset of A.

EXERCISE/PROBLEM SET B

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Section 2.1 Sets as a Basis for Whole Numbers 55

e. There is a proper subset of B that is equivalent to a proper subset of A.

f. A B∪ is all multiples of 5.

13. True or false? a. The empty set is a subset of every set. b. The set { , , , , }105 110 115 120 . . . is an infinite set. c. If A is an infinite set and B A⊆ , then B is a finite set.

14. Use Venn diagrams to determine which, if any, of the following statements are true for all sets A B, , and C: The Chapter 2 eManipulative Venn Diagrams on our Web site may help in determining which statements are true.

a. A B C A B C∪ ∪ = ∪ ∪( ) ( ) b. A B C A B C∪ ∩ = ∪ ∩( ) ( ) c. A B C A B C∩ ∩ = ∩ ∩( ) ( ) d. A B C A B C∩ ∪ = ∩ ∪( ) ( )

15. Draw a Venn diagram like the following for each part. Then shade each Venn diagram to represent each of the sets indicated.

a. T S−     b. S T∪     c. ( ) ( )S T T S− ∩ −

16. A Venn diagram can be used to illustrate more than two sets. Shade the regions that represent each of the fol- lowing sets. The Chapter 2 eManipulative activity Venn Diagrams on our Web site may help in solving this problem.

a. ( )A B C∪ ∩    b. A B C∩ ∩( )   c. ( ) ( )A B B C∪ − ∩

17. Represent the following shaded regions using the symbols A B C, , , , , ∪ ∩ and −. The Chapter 2 eManipulative activity Venn Diagrams on our Web site can be used in solving this problem.

b.a.

c.

18. Draw Venn diagrams that represent sets A and B as described here.

a. A B∩ ≠ ∅ b. A B A∪ = c ( )A B B∩ − = ∅

19. Make six copies of the diagram and shade regions in parts a–f.

a. ( )A B C∩ ∩ b. ( ) ( )C D A B∩ − ∪ c. ( ) ( )A B C D∩ ∩ ∩ d. ( )B C D∩ ∩ e. ( ) ( )A D C B∩ ∪ ∩ f. ( ) ( )A B C A∩ ∪ ∩

20. Let A a b c d e B c d e f g= ={ , , , , }, { , , , , }, and C a e f h= { , , , }. List the members of each set.

a. A B∪ b. A B∩ c. ( )A B C∪ ∩ d. A B C∪ ∩( )

21. If A is the set of all sophomores in a school and B is the set of students who belong to the orchestra, describe the following sets in words.

a. A B∪         b. A B∩         c. A B−        d. B A−

22. Find each of the following differences. a. { , , , } { }h i j k k − b. { , , } { }3 10 13 − c. {two-wheeled vehicles} − {two-wheeled vehicles that are

not bicycles} d. { , , , , , } { , , , , }0 2 4 6 20 12 14 16 18 20. . . −

23. In each of the following cases, find B A− . a. A B∩ = ∅     b. A B=      c. B A⊆

24. Let R a b c S c d e f T x y z= = ={ , , }, { , , , }, { , , }. List the elements of the following sets.

a. R S∪         b. R S∩         c. R T∪         d. R T∩ e. S T∪         f. S T∩         g. T ∪ ∅

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56 Chapter 2 Sets, Whole Numbers, and Numeration

25.

Let

A

B

C

= = =

{ , , , , , , }

{ , , , }

{ ,

50 55 60 65 70 75 80

50 60 70 80

60 770 80

55 65

, }

{ , }

D =

List the members of each set. a. A B C∪ ∩( )                 b. ( )A B C∪ ∩ c. ( ) ( )A C C D∩ ∪ ∩ d. ( ) ( )A C C D∩ ∩ ∪ e. ( )B C A− ∩                   f. ( ) ( )A D B C− ∩ −

26. a. If x X Y∈ ∩ , is x X Y∈ ∪ ? Justify your answer. b. If x X Y∈ ∪ , is x X Y∈ ∩ ? Justify your answer.

27. Verify that A B A B∩ = ∪ in two different ways as follows: a. Let U A= ={ , , , , , , , }, { , , , },2 4 6 8 10 12 14 16 2 4 8 16 and

B = { , , , }.4 8 12 16 List the elements of the sets A B∩ and A B∪ . Do the two sets have the same members?

b. Draw and shade a Venn diagram for each of the sets A B A B∩ = ∪ . Do the two Venn diagrams look the same?

NOTE: The equation A B A B∩ = ∪ is one of the two laws called DeMorgan’s laws.

28. Find the following Cartesian products. a. { , , } { }a b c × 1 b. { , } { , , }1 2 × p q r c. { , , } { , }p q r × 1 2 d. { } { }a × 1

29. Determine how many ordered pairs will be in A B× under the following conditions.

a. A has one member and B has four members. b. A has two members and B has four members. c. A has three members and B has seven members.

30. Find sets A and B so that A B× has the following number of members.

a. 1 b. 2 c. 3 d. 4 e. 5 f. 6 g. 7 h. 0

31. The Cartesian product, X Y× , is given in each of the fol- lowing parts. Find X and Y .

a. {( , ), ( , )}b c c c b. {( , ), ( , ), ( , ), ( , ), ( , ), ( , )}2 1 2 2 2 3 5 1 5 2 5 3

32. True or false? a. {( , ), ( , )} {( , ), ( , )}3 4 5 6 3 4 6 5 = b. {( , ), ( , )} { , , , }a c d b a c d b = c. {( , ), ( , )} {( , ), ( , )}c d a b c a d b = d. {( , ), ( , )} { , , } { , , }4 5 6 7 4 5 6 5 7 9 ⊆ ×

PROBLEMS 33. How many 1-1 correspondences are there between the fol-

lowing pairs of sets? a. Two 2-member sets b. Two 4-member sets c. Two 6-member sets d. Two sets each having m members

34. Find sets (when possible) satisfying each of the following conditions.

a. Number of elements in A plus number of elements in B is greater than number of elements in A B∪ .

b. Number of elements in I plus number of elements in J is less than number of elements in I J∪ .

c. Number of elements in E plus number of elements in F equals number of elements in E F∪ .

d. Number of elements in G plus number of elements in K equals number of elements in G K∩ .

35. Your house can be painted in a choice of 7 exterior colors and 15 interior colors. Assuming that you choose only 1 color for the exterior and 1 color for the interior, how many different ways of painting your house are there?

36. Show a 1-1 correspondence between the points on the given circle and the triangle in which it is inscribed. Explain your procedure.

37. Show a 1-1 correspondence between the points on the given triangle and the circle that circumscribes it. Explain your procedure.

38. A schoolroom has 13 desks and 13 chairs. You want to arrange the desks and chairs so that each desk has a chair with it. How many such arrangements are there?

39. A university professor asked his class of 42 students when they had studied for his class the previous weekend. Their responses were as follows:

9 had studied on Friday. 18 had studied on Saturday. 30 had studied on Sunday. 3 had studied on both Friday and Saturday. 10 had studied on both Saturday and Sunday. 6 had studied on both Friday and Sunday. 2 had studied on Friday, Saturday, and Sunday.

Assuming that all 42 students responded and answered honestly, answer the following questions.

a. How many students studied on Sunday but not on either Friday or Saturday?

b. How many students did all of their studying on one day?

c. How many students did not study at all for this class last weekend?

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Section 2.2 Whole Numbers and Numeration 57

40. At an automotive repair shop, 50 cars were inspected. Suppose that 23 cars needed new brakes and 34 cars need- ed new exhaust systems.

a. What is the least number of cars that could have needed both?

b. What is the greatest number of cars that could have needed both?

c. What is the greatest number of cars that could have needed neither?

41. If A is a proper subset of B, and A has 23 elements, how many elements does B have? Explain.

Analyzing Student Thinking

42. Kylee says that if two sets are equivalent, they must be equal. Jake says that if two sets are equal, then they must be equivalent. Who is correct? Explain.

43. Tonia determines that a set with three elements has as many proper subsets as it has nonempty subsets. Is she correct? Explain.

44. Amanda says that the sets { , , , }1 2 3 . . . and { , , , }2 4 6 . . . can- not be matched because the first set has many numbers that the second one does not. Is she correct? Explain.

45. LeNae says that A B∪ must always have more elements than either A or B. Is she correct? Explain.

46. Marshall says that A B∩ always has fewer elements than either A or B. Is he correct? Explain.

47. Michael asserts that if A and B are infinite sets, A B∩ must be an infinite set. Is he correct? Explain.

48. DeDee says that B must be a subset of A to be able to find the set A B− . How should you respond?

WHOLE NUMBERS AND NUMERATION

When a young child is asked to count the blocks in the set shown below, he will use his memorized cadence of “one, two, three, four . . . ” while pointing to the blocks. In coming

to understand the concept of number, the following two scenarios are possible:

Child 4 counts out the blocks correctly (points to each block once and only once saying “one, two, three”) and when asked to show 3 blocks, points to the third block.

Child 5 counts the blocks correctly and then when asked to show 3 blocks, indicates all three blocks.

Identify which student is incorrect and explain the possible rationale behind his reasoning?

Numbers and Numerals As mentioned earlier, the study of the set of whole numbers, W = { , , , , , },0 1 2 3 4 . . . is the foundation of elementary school mathematics. But what precisely do we mean by the whole number 3? A number is an idea, or an abstraction, that represents a quantity. The symbols that we see, write, or touch when representing numbers are called numerals. There are three common uses of numbers. The most common use of whole numbers is to describe how many elements are in a finite set. When used in this manner, the number is referred to as a cardinal number. A second use is concerned with order. For example, you may be second in line, or your team may be fourth in the standings. Numbers used in this way are called ordinal numbers. Finally, identification numbers are used to name such things as telephone numbers, bank account numbers, and social security numbers. In this case, the numbers are used in a numeral sense in that only the symbols, rather than their values, are important. Before discussing our system of numeration or symbolization, the con- cept of cardinal number will be considered.

Children’s Literature www.wiley.com/college/musser See “George’s Store at the Shore’’ by Francine Bessede.

NCTM Standard All students should develop understanding of the relative position and magnitude of whole numbers and of ordinal and cardinal numbers and their con- nections.

Common Core – Kindergarten Understand the relationship between numbers and quantities; connect counting to cardinality.

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58 Chapter 2 Sets, Whole Numbers, and Numeration

What is the number 3? What do you think of when you see the sets in Figure 2.9?

Figure 2.9

First, there are no common elements. One set is made up of letters, one of shapes, one of Greek letters, and so on. Second, each set can be matched with every other set. Now imagine all the infinitely many sets that can be matched with these sets. Even though the sets will be made up of various elements, they will all share the common attribute that they are equivalent to the set { , , }.a b c The common idea that is asso- ciated with all of these equivalent sets is the number 3. That is, the number 3 is the attribute common to all sets that match the set { , , }.a b c Similarly, the whole number 2 is the common idea associated with all sets equivalent to the set { , }.a b All other nonzero whole numbers can be conceptualized in a similar manner. Zero is the idea, or number, one imagines when asked: “How many elements are in the empty set?”

Although the preceding discussion regarding the concept of a whole number may seem routine for you, there are many pitfalls for children who are learning the concept of numerousness for the first time. Chronologically, children first learn how to say the counting chant “one, two, three, . . . .” However, saying the chant and understanding the concept of number are not the same thing. Next, children must learn how to match the counting chant words they are saying with the objects they are counting. For example, to count the objects in the set { , , }n s u , a child must correctly assign the words “one, two, three” to the objects in a 1-1 fashion. Actually, children first learning to count objects fail this task in two ways: (1) They fail to assign a word to each object, and hence their count is too small; or (2) they count one or more objects at least twice and end up with a number that is too large. To reach the final stage in understanding the concept of number, children must be able to observe several equivalent sets, as in Figure 2.9, and realize that, when they count each set, they arrive at the same word. Thus this word is used to name the attribute common to all such sets.

The symbol n A( ) is used to represent the number of elements in a finite set A. More precisely, (1) n A m( ) = if A m~ { , , , },1 2 . . . where m is a counting number, and (2) n( ) .∅ = 0 Thus

n a b c({ , , }) = 3 since { , , } ~ { , , },a b c 1 2 3 and n a b c z({ , , , , }). . . = 26 since

{ , , , , } ~ { , , , , },a b c z. . . . . .1 2 3 26

and so on, for other finite sets.

Reflection from Research There is a clear separation of development in young children concerning the cardinal and ordi- nal aspects of numbers. Despite the fact that they could utilize the same counting skills, the understanding of ordinality by young children lags well behind the understanding of cardinality (Bruce & Threlfall, 2004).

Check for Understanding: Exercise/Problem Set A #1–4✔

Ordering Whole Numbers Children may get their first introduction to ordering whole numbers through the counting chant “one, two, three, . . . .” For example, “two” is less than “five,” since “two” comes before “five” in the counting chant.

A more meaningful way of comparing two whole numbers is to use 1-1 corre- spondences. We can say that 2 is less than 5, since any set with two elements matches a proper subset of any set with five elements (Figure 2.10).

NCTM Standard All students should count with understanding and recognize “how many” in sets of objects.

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Section 2.2 Whole Numbers and Numeration 59

Figure 2.10

The general set formulation of “less than” follows.

Common Core – Kindergarten Count to answer ``how many?’’ questions about as many as 20 things arranged in a line, a rect- angular array, or a circle, or as many as 10 things in a scattered configuration; given a number from 1-20, count out that many objects.

Ordering Whole Numbers

Let a n A= ( ) and b n B= ( ). Then a b< (read “a is less than b”) or b a> (read “b is greater than a”) if A is equivalent to a proper subset of B.

D E F I N I T I O N 2 . 8

The “greater than” and “less than” signs can be combined with the equal sign to produce the following symbols: a b≤ (a is less than or equal to b) and b a≥ (b is greater than or equal to a).

A third common way of ordering whole numbers is through the use of the whole- number “line” (Figure 2.11). Actually, the whole-number line is a sequence of equally spaced marks where the numbers represented by the marks begin on the left with 0 and increase by one each time we move one mark to the right.

Figure 2.11

Common Core – Kindergarten Identify whether the number of objects in one group is greater than, less than, or equal to the number of objects in another group (e.g., by using matching and counting strategies).

Reflection from Research Most counting mistakes made by young children can be attributed to not keeping track of objects that have already been counted and objects that still need to be counted (Fuson, 1988).

Determine the greater of the two numbers 3 and 8 in three dif- ferent ways.

S O L U T I O N

a. Counting Chant: One, two, three, four, fi ve, six, seven, eight. Since “three” precedes “eight,” eight is greater than three.

b. Set Method: Since a set with three elements can be matched with a proper subset of a set with eight elements, 3 8< and 8 3> [Figure 2.12(a)].

c. Whole-Number Line: Since 3 is to the left of 8 on the number line, 3 is less than 8 and 8 is greater than 3 [Figure 2.12(b)]. ■

Figure 2.12

Check for Understanding: Exercise/Problem Set A #5–6✔

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60 Chapter 2 Sets, Whole Numbers, and Numeration

Today our numeration system has the symbol “2” to represent the number of eyes a person has. The symbols “1” and “0” combine to represent the number of toes a person has, “10.”

The Roman numeration system used the symbol “X” to represent the number of toes and “C” to represent the number of years in a century.

Using only the three symbols above, devise your own numeration system and show how you can use your system to  represent all of the quantities 0,1, 2, 3, 4, , 100.. . . What properties does your numeration system have?

Numeration Systems To make numbers more useful, systems of symbols, or numerals, have been developed to represent numbers. In fact, throughout history, many different numeration systems have evolved. The following discussion reviews various ancient numeration systems with an eye toward identifying features of those systems that are incorporated in our present system, the Hindu–Arabic numeration system.

The Tally Numeration System The tally numeration system is composed of single strokes, one for each object being counted (Figure 2.13).

Figure 2.13

The next six such tally numerals are

An advantage of this system is its simplicity; however, two disadvantages are that (1) large numbers require many individual symbols, and (2) it is difficult to read the numer- als for such numbers. For example, what number is represented by these tally marks?

The tally system was improved by the introduction of grouping. In this case, the fifth tally mark was placed across every four to make a group of five. Thus the last tally numeral can be written as follows:

Grouping makes it easier to recognize the number being represented; in this case, there are 37 tally marks.

The Egyptian Numeration System The Egyptian numeration system, which developed around 3400 B.C.E., involves grouping by ten. In addition, this system introduced new symbols for powers of 10 (Figure 2.14).

Figure 2.14

Examples of some Egyptian numerals are shown in Figure 2.15. Notice how this system required far fewer symbols than the tally system once numbers greater than

Children’s Literature www.wiley.com/college/musser See “Count on Your Fingers African Style” by Claudia Zaslavsky.

Reflection from Research Teaching other numeration systems, such as the Chinese system, can reinforce a student’s conceptual knowledge of place value (Uy, 2002).

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Section 2.2 Whole Numbers and Numeration 61

10 were represented. This system is also an additive system, since the values for the various individual numerals are added together.

Figure 2.15

Notice that the order in which the symbols are written is immaterial. A major dis- advantage of this system is that computation is cumbersome. Figure 2.16 shows, in Egyptian numerals, the addition problem that we write as 764 598 1362+ = . Here 51 individual Egyptian numerals are needed to express this addition problem, whereas our system requires only 10 numerals!

Figure 2.16

The Roman Numeration System The Roman numeration system, which developed between 500 B.C.E. and 100 C.E. also uses grouping, additivity, and many symbols. The basic Roman numerals are listed in Table 2.2.

TABLE 2.2

ROMAN NUMERAL VALUE

I  1 V  5 X 10 L 50 C 100 D 500 M 1000

Roman numerals are made up of combinations of these basic numerals, as illustrated next.

CCLXXXI (equals 281)    MCVIII (equals 1108)

Notice that the values of these Roman numerals are found by adding the values of the various basic numerals. For example, MCVIII means 1000 100 5 1 1 1+ + + + + , or 1108. Thus the Roman system is an additive system.

Two new attributes that were introduced by the Roman system were a subtractive principle and a multiplicative principle. Both of these principles allow the system to use fewer symbols to represent numbers. The Roman numeration system is a sub- tractive system since it permits simplifications using combinations of basic Roman numerals: IV (I to the left of V means five minus one) for 4 rather than using IIII, IX (ten minus one) for 9 instead of VIIII, XL for 40, XC for 90, CD for 400, and CM for 900 (Table 2.3).

Thus, when reading from left to right, if the values of the symbols in any pair of symbols increase, group the pair together. The value of this pair, then, is the value of the larger numeral less the value of the smaller. One may wonder if IC is an acceptable way to write 99. The answer is no because of the additional restriction that only I’s, X’s, and C’s may be subtracted, but only from the next two larger numerals in each case. Thus, the symbols shown in Table 2.3 are the only numerals where the subtraction principle is applied. To evaluate a complex Roman numeral, one looks to see whether any of these subtractive pairs are present, groups them together mentally, and then adds values from left to right. For example,

in    MCMXLIV think    M  CM  XL  IV, which is 1000 900 40 4+ + + .

Children’s Literature www.wiley.com/college/musser See “Fun with Roman Numerals’’ by David Adler.

TABLE 2.3

ROMAN NUMERAL VALUE

IV 4 IX 9 XL 40 XC 90 CD 400 CM 900

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62 Chapter 2 Sets, Whole Numbers, and Numeration

Notice that without the subtractive principle, 14 individual Roman numerals would be required to represent 1944 instead of the 7 numerals used in MCMXLIV. Also, because of the subtractive principle, the Roman system is a positional system, since the position of a numeral can affect the value of the number being represented. For example, VI is six, whereas IV is four.

NCTM Standard All students should develop a sense of whole numbers and rep- resent and use them in flexible ways including relating, compos- ing, and decomposing numbers.

Express the following Roman numerals in our numeration system:

a. MCCCXLIV b. MMCMXCIII c. CCXLIX

S O L U T I O N

a. Think: MCCC  XL  IV, or 1300 40 4 1344+ + = b. Think: MM  CM  XC  III, or 2000 900 90 3 2993+ + + = c. Think: CC  XL  IX, or 200 40 9 249+ + = ■

The Roman numeration system also utilized a horizontal bar above a numeral to represent 1000 times the number. For example, V meant 5 times 1000, or 5000; XI meant 11,000; and so on. Thus the Roman system was also a multiplicative system. Although expressing numbers using the Roman system requires fewer symbols than the Egyptian system, it still requires many more symbols than our current system and is cumbersome for doing arithmetic. In fact, the Romans used an abacus to perform calculations instead of paper/pencil methods as we do (see the Focus On at the beginning of Chapter 3).

The Babylonian Numeration System The Babylonian numeration system, which evolved between 3000 and 2000 B.C.E., used only two numerals, a one and a ten (Figure 2.17). For numbers up to 59, the system was simply an additive system. For example, 37 was written using 3 tens and 7 ones (Figure 2.18).

However, even though the Babylonian numeration system was developed about the same time as the simpler Egyptian system, the Babylonians used the sophisticated notion of place value, where symbols represent different values depending on the place in which they were written. The symbol could represent 1 or 1 60¥ or 1 60 60¥ ¥ depending on where it is placed. Since the position of a symbol in a place-value system affects its value, place-value systems are also positional. Thus, the Babylonian numeration system is another example of a positional system. Figure 2.19 displays three Babylonian numerals that illustrate this place-value attribute, which is based on 60. Notice the subtle spacing of the numbers in Figure 2.19(a) to assist in understanding that the represents a 1 60¥ and not a 1. Similarly, the symbol, in Figure 2.19(b) is spaced slightly to the left to indicate that it represents a 12(60) instead of just 12. Finally, the symbols in Figure 2.19(c) have 2 spaces to indicate is multiplied by 60 60¥ and is multiplied by 60.

Figure 2.19

Unfortunately, in its earliest development, this system led to some confusion. For example, as illustrated in Figure 2.20, the numerals representing 74 1 60 14( )= +¥ and 3614 1 60 60 0 60 14( )= + +¥ ¥ ¥ differed only in the spacing of symbols. Thus, there was a chance for misinterpretation. From 300 B.C.E. on, a separate symbol made up of two small triangles arranged one above the other was used to serve as a placeholder to indicate a vacant place (Figure 2.21). This removed some of the ambiguity. However, two Babylonian tens written next to each other could still be interpreted as 20, or 610, or even 3660. Although their placeholder acts much like our zero, the Babylonians did not recognize zero as a number.

Figure 2.18

Figure 2.17

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Section 2.2 Whole Numbers and Numeration 63

Figure 2.20 Figure 2.21

The Mayan Numeration System The Mayan numeration system, which developed between 300 and 900 C.E., was a vertical place-value system, and it introduced a symbol for zero. The system used only three elementary numerals (Figure 2.22).

Figure 2.22

Several Mayan numerals are shown in Figure 2.22 together with their respective val- ues. The symbol for twenty in Figure 2.23 illustrates the use of place value in that the “dot” represents one “twenty” and the represents zero “ones.”

Figure 2.23

Various place values for this system are illustrated in Figure 2.24.

Figure 2.24

The bottom section represents the number of ones (3 here), the second section from the bottom represents the number of 20s (0 here), the third section from the bottom represents the number of 18 20¥ s (6 here), and the top section represents the number of 18 20 20¥ ¥ s (1 here). Reading from top to bottom, the value of the number represented is 1 18 20 20 6 18 20 0 20 3 1( ) ( ) ( ) ( )¥ ¥ ¥+ + + or 9363. (See the Focus On at the beginning of this chapter for additional insight into this system.)

Reflection from Research Zero is an important number that should be carefully integrated into early number experiences. If children do not develop a clear understanding of zero early on, misconceptions can arise that might be detrimental to the fur- ther development of their under- standing of number properties and subsequent development of algebraic thinking (Anthony & Walshaw, 2004).

Express the following Babylonian numerals in our numeration system.

S O L U T I O N

.c.b.a

■

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64 Chapter 2 Sets, Whole Numbers, and Numeration

Notice that in the Mayan numeration system, you must take great care in the way the numbers are spaced. For example, two horizontal bars could represent 5 5+ as = or 5 20 5¥ + as , depending on how the two bars are spaced. Also notice that the place-value feature of this system is somewhat irregular. After the ones place comes the 20s place. Then comes the 18 20¥ s place. Thereafter, though, the values of the places are increased by multiplying by 20 to obtain 18 20 18 20 18 202 3 4¥ ¥ ¥, , , and so on.

Express the following Mayan numerals in our numeration system.

S O L U T I O N

■

Table 2.4 summarizes the attributes of the number systems we have studied.

TABLE 2.4

SYSTEM ADDITIVE SUBTRACTIVE MULTIPLICATIVE POSITIONAL PLACE VALUE

HAS A ZERO

Tally Yes No No No No No Egyptian Yes No No No No No Roman Yes Yes Yes Yes No No Babylonian Yes No Yes Yes Yes No Mayan Yes No Yes Yes Yes Yes

Check for Understanding: Exercise/Problem Set A #7–14✔

When learning the names of two-digit numerals, some children suffer through a “reversals” stage, where they may write 21 for twelve or 13 for thirty-one. The following story from the December 5, 1990, Grand Rapids [Michigan] Press stresses the importance of eliminating reversals. “It was a case of mistaken identity. A trans- posed address that resulted in a bulldozer blunder. City orders had called for demolition on Tuesday of a boarded-up house at 451 Fuller Ave. S.E. But when the dust settled, 451 Fuller stood untouched. Down the street at 415 Fuller Ave. S.E., only a base- ment remained.”

© R

on B

ag w

el l

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Section 2.2 Whole Numbers and Numeration 65

EXERCISES 1. In the following paragraph, numbers are used in various

ways. Specify whether each number is a cardinal number, an ordinal number, or an identification number. Linda dialed 314-781-9804 to place an order with a popular mail- order company. When the sales representative answered, Linda told her that her customer number was 13905. Linda then placed an order for 6 cotton T-shirts, stock number 7814, from page 28 of the catalog. For an extra $10 charge, Linda could have next-day delivery, but she chose the regular delivery system and was told her pack- age would arrive November 20.

2. Identify how the number is used in the sentence: cardinal, ordinal, or identification.

a. Judy came in second in the race. b. A dog is an animal with four legs. c. I live in apartment five.

3. Explain how to count the elements of the set { , , , , , }.a b c d e f

4. Which number is larger, 5 or 8? Which numeral is larger?

5. Use the set method as illustrated in Figure 2.10 to explain why 7 3> .

6. Place < or > in the blanks to make each statement true. Indicate how you would verify your choice.

a. 3 _____ 7      b. 11 _____ 9     c. 21 _____ 12

7. Change to Egyptian numerals. a. 9     b. 23     c. 453     d. 1231

8. Change to Roman numerals. a. 76     b. 49     c. 192     d. 1741

9. Change to Babylonian numerals. a. 47     b. 76     c. 347     d. 4192

10. Change to Mayan numerals. a. 17     b. 51     c. 275     d. 401

11. Change to Hindu-Arabic numerals.

d. MCMXCI e. CMLXXVI f. MMMCCXLV

12. Perform each of the following numeral conversions. a. Roman numeral DCCCXXIV to a Babylonian

numeral

b. Mayan numeral to an Egyptian numeral

c. Babylonian numeral to a Mayan numeral

d. Egyptian numeral to a Roman numeral

13. Imagine representing 246 in the Mayan, Babylonian, and Egyptian numeration systems.

a. In which system is the greatest number of symbols required?

b. In which system is the smallest number of symbols required?

c. Do your answers for parts a and b hold true for other numbers as well?

14. In 2010, the National Football League’s championship game was Super Bowl XLIV. What was the first year of the Super Bowl?

EXERCISE/PROBLEM SET A

PROBLEMS 15. Some children go through a reversal stage; that is,

they confuse 13 and 31, 27 and 72, 59 and 95. What numerals would give Roman children similar difficulties? How about Egyptian children?

16. a. How many Egyptian numerals are needed to represent the following problems? i. 59 88+ ii. 150 99− iii. 7897 934+ iv. 9698 5389−

b. State a general rule for determining the number of Egyptian numerals needed to represent an addition (or subtraction) problem written in our numeral system.

17. A newspaper advertisement introduced a new car as follows: IV Cams, XXXII Valves, CCLXXX Horsepower, coming December XXVI—the new 1999 Lincoln Mark VII. Write the Roman numeral that represents the model year of the car.

18. Linda pulled out one full page from the Sunday newspa- per. If the left half was numbered A4 and the right half was numbered A15, how many pages were in the A section of the newspaper?

19. Determine which of the following numbers is larger. 1993 1 2 3 4 1994× + + + + ⋅ ⋅ ⋅ +( ) 1994 1 2 3 4 1993× + + + + ⋅ ⋅ ⋅ +( )

20. You have five coins that appear to be identical and a balance scale. One of these coins is counterfeit and either heavier or lighter than the other four. Explain how the counterfeit coin can be identified and whether it is lighter or heavier than the others with only three weighings on the balance scale.

(Hint: Solve a simpler problem—given just three coins, can you find the counterfeit in two weighings?)

21. One system of numeration in Greece in about 300 B.C.E., called the Ionian system, was based on letters of the alphabet. The different symbols used for numbers less than 1000 are as follows:

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66 Chapter 2 Sets, Whole Numbers, and Numeration

To represent multiples of 1000 an accent mark was used. For example, ′� was used to represent 5000. The accent mark might be omitted if the size of the number being rep- resented was clear without it.

a. Express the following Ionian numerals in our numeration system.

i. mb ii. cke iii. gflg iv. ′p qwa

b. Express each of the following numerals in the Ionian numeration system. i. 85     ii. 744     iii. 2153     iv. 21,534

c. Was the Ionian system a place-value system?

PROBLEMS 15. What is the largest number that you can enter on your

calculator a. if you may use the same digit more than once? b. if you must use a different digit in each place?

16. The following Chinese numerals are part of one of the oldest numeration systems known.

The numerals are written vertically. Some examples follow

a. Express each of the following numerals in this Chinese numeration system: 80, 19, 52, 400, 603, 6031.

b. Is this system a positional system? an additive system? a multiplicative system?

EXERCISES 1. Write sentences that show the number 45 used in each of

the following ways. a. As a cardinal number b. As an ordinal number c. As an identification number

2. Identify how the number is used in the sentence: cardinal, ordinal, or identification.

a. What is behind door three? b. First turn the key, then open the lock. c. Mark has five sisters.

3. Explain why each of the following sets can or cannot be used to count the number of elements in { , , , }.a b c d

a. { }4 b. { , , , }0 1 2 3      c. { , , , }1 2 3 4

4. Which is more abstract: number or numeral? Explain.

5. Determine the greater of the two numbers 4 and 9 in three different ways.

6. Place < or > in the blanks to make each statement true. a. 6_____5 b. 100_____1000 c. 19_____91

7. Change to Egyptian numerals. a. 2431 b. 10,352

8. Change to Roman numerals. a. 79 b. 3054

9. Change to Babylonian numerals. a. 117 b. 3521

10. Change to Mayan numerals. a. 926 b. 37,865

11. Express each of the following numerals in our numeration system.

c. MCCXLVII

12. Complete the following chart expressing the given num- bers in the other numeration system.

a.

b.

c.

d.

13. Is it possible for a numeration system to be positional and not place value? Explain why or why not. If possible, give an example.

14. After the credits for a film, the Roman numeral MCMLXXXIX appears, representing the year in which the film was made. Express the year in our numeration system.

EXERCISE/PROBLEM SET B

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Section 2.3 The Hindu–Arabic System 67

17. Two hundred persons are positioned in 10 rows, each containing 20 persons. From each of the 20 columns thus formed, the shortest is selected, and the tallest of these 20 (short) persons is tagged A. These persons now return to their initial places. Next, the tallest person in each row is selected and from these 10 (tall) persons the shortest is tagged B. Which of the two tagged persons is the taller (if they are different people)?

18. Braille numerals are formed using dots in a two-dot by three-dot Braille cell. Numerals are preceded by a backwards “L” dot symbol. The following shows the basic elements for Braille numerals and two examples.

Express these Braille numerals in our numeration system.

a.

b.

19. The heights of five famous human-made structures are related as follows:

half the height of the Great Pyramid at Giza.

times the height of Big Ben.

Ben.

fact, the sum of the five heights is 2264 feet.

Find the height of each of the five structures.

20. Can an 8 8× checkerboard with two opposite corner squares removed be exactly covered (without cutting) by thirty-one 2 1× dominoes? Give details.

Analyzing Student Thinking

21. Tammie asserts that if n A n B( ) ( )≤ where A and B are finite sets, then A B⊆ . Is she correct? Explain.

22. Which way is a child most likely to first learn the order- ing of whole numbers less than 10: (i) using the counting chant; (ii) using the set method; (iii) using the number line method? Explain.

23. Anthony thinks that we should all use the tally numera- tion system instead of the Hindu–Arabic system because the Tally system is so easy. How should you respond?

24. When asked to express 999 in the Egyptian numeration system, Misti says that it will require twenty-seven numer- als and Victor says that it will require a lot more. Which student is correct? Explain.

25. When teaching the Roman numeration system, you show your students that it is additive, subtractive, and multipli- cative. Shalonda asks, “Why can’t you do division in the Roman system?” How should you respond?

26. Natalie asks if the symbol comprised of two small tri- angles, one atop the other, in the Babylonian numeration system is zero. How should you respond?

27. Blanca says that since zero means nothing, she doesn’t have to use a numeral for zero in the Mayan numeration system. Instead, she can simply leave a blank space. Is she correct? Explain.

In Coinland, their money is similar to American money, but in all coins. They have pennies and dimes (no nickels or quarters), but also have coins for dollars, ten dol-

lars, hundred dollars, and so forth. Martina, who lives in Coinland, likes to carry as few coins as possible. What is the minimum number of coins Martina could carry for each of the following amounts of money? How many of each coin would she have in each case?

68¢ 123¢ $4.52 $307.31

If Martina always exchanges her money to have a minimum number of coins, what is the maximum number of any type of coin that she would have after an exchange? Why?

THE HINDU–ARABIC SYSTEM

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68 Chapter 2 Sets, Whole Numbers, and Numeration

The Hindu–Arabic Numeration System The Hindu–Arabic numeration system that we use today was developed about 800 C.E. The following list features the basic numerals and various attributes of this system.

1. Digits, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9: These 10 symbols, or digits, can be used in combina- tion to represent all possible numbers.

2. Grouping by tens (decimal system): Grouping into sets of 10 is a basic principle of this system, probably because we have 10 “digits” on our two hands. (The word digit literally means “finger” or “toe.”) Ten ones are replaced by 1 ten, 10 tens are replaced by 1 hundred, 10 hundreds are replaced by 1 thousand, and so on. Figure 2.25 shows how grouping is helpful when representing a collection of objects. The number of objects grouped together is called the base of the system; thus our Hindu–Arabic system is a base ten system.

NOTE: Recall that an element is listed only once in a set. Although all the dots in Figure 2.25 look the same, they are assumed to be unique, individual elements here and in all such subsequent figures.

Figure 2.25

The following two models are often used to represent multidigit numbers.

a. Bundles of sticks can be any kind of sticks banded together with rubber bands; 10 loose sticks are bound together with a rubber band to represent 10, then 10 bundles of 10 are bound together to represent 100, and so on (Figure 2.26).

Figure 2.26

b. Base ten pieces (also called Dienes blocks) consist of individual cubes, called “units,” “longs,” made up of 10 units, “flats,” made up of 10 longs, or 100 units and so on (Figure 2.27). Inexpensive two-dimensional sets of base ten pieces can be made using grid paper cutouts.

Figure 2.27

3. Place value (hence positional): Each of the various places in the numeral 6523, for example, has its own value.

Common Core – Grade 1 Understand that the two digits of a two-digit number represent amounts of tens and ones.

Reflection from Research Results from research suggest that base ten blocks should be used in the development of students’ place-value concepts (Fuson, 1990).

Common Core – Grade 2 Understand that the three digits of a three-digit number represent amounts of hundreds, tens, and ones (e.g., 706 equals 7 hun- dreds, 0 tens, and 6 ones).

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Section 2.3 The Hindu–Arabic System 69

thousand     hundred     ten     one

6 5 2 3

The 6 represents 6 thousands, the 5 represents 5 hundreds, the 2 represents 2 tens, and the 3 represents 3 ones due to the place-value attribute of the Hindu–Arabic system.

The following device is used to represent numbers written in place value. A chip abacus is a piece of paper or cardboard containing lines that form columns, on which chips or markers are used to represent unit values. Conceptually, this model is more abstract than the previous models because markers represent different values, depending on the columns in which the markers appear (Figure 2.28).

Figure 2.28

4. Additive and multiplicative: The value of a Hindu–Arabic numeral is found by multiplying each place value by its corresponding digit and then adding all the resulting products.

Place values: thousand hundred ten one Digits: 6 5 2 3 Numeral value: 6 1000 5 100 2 10 3 1× + × + × + × Numeral: 6523

Expressing a numeral as the sum of its digits times their respective place values is called the numeral’s expanded form or expanded notation. The expanded form of 83,507 is

8 10 000 3 1000 5 100 0 10 7 1× + × + × + × + ×, .

Because 7 1 7× = , we can simply write 7 in place of 7 1× when expressing 83,507 in expanded form.

NCTM Standard All students should use multiple models to develop initial under- standings of place value and the base ten number system.

Common Core – Grade 2 Read and write numbers to 1000 using base ten numerals, number names, and expanded form.

Express the following numbers in expanded form.

a. 437 b. 3001

S O L U T I O N

a. 437 4 100 3 10 7= + +( ) ( ) b. 3001 3 1000 0 100 0 10 1 3 1000 1= + + + +( ) ( ) ( ) ( ) or ■

Check for Understanding: Exercise/Problem Set A #1–3✔

Notice that our numeration system requires fewer symbols to represent numbers than did earlier systems. Also, the Hindu–Arabic system is far superior when performing computations. The computational aspects of the Hindu–Arabic system will be studied in Chapter 3.

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70 Chapter 2 Sets, Whole Numbers, and Numeration

Naming Hindu–Arabic Numerals Associated with each Hindu–Arabic numeral is a word name. Some of the English names are as follows:

0 zero 10 ten

1 one 11 eleven

2 two 12 twelve

3 three 13 thirteen (three plus ten)

4 four 14 fourteen (four plus ten)

5 five 21 twenty-one (2 tens plus one)

6 six 87 eighty-seven (8 tens plus seven)

7 seven 205 two hundred five (2 hundreds plus five)

8 eight 1,374 one thousand three hundred seventy-four

9 nine 23,100 twenty-three thousand one hundred

Here are a few observations about the naming procedure:

1. The numbers 0, 1, . . . , 12 all have unique names.

2. The numbers 13, 14, . . . , 19 are the “teens,” and are composed of a combination of earlier names, with the ones place named first. For example, “thirteen” is short for “three ten,” which means “ten plus three,” and so on.

3. The numbers 20, . . . , 99 are combinations of earlier names but reversed from the teens in that the tens place is named first. For example, 57 is “fifty-seven,” which means “5 tens plus seven,” and so on. The method of naming the numbers from 20 to 90 is better than the way we name the teens, due to the left-to-right agreement with the way the numerals are written.

4. The numbers 100, . . . , 999 are combinations of hundreds and previous names. For example, 538 is read “five hundred thirty-eight,” and so on.

5. In numerals containing more than three digits, groups of three digits are usually set off by commas. For example, the number

is read “one hundred twenty-three quadrillion four hundred fifty-six trillion seven hundred eighty-nine billion nine hundred eighty-seven million six hundred fifty-four thousand three hundred twenty-one.” (Internationally, the commas are omitted and single spaces are used instead. Also, in some countries, commas are used in place of decimal points.) Notice that the word and does not appear in any of these names: it is reserved to separate the decimal portion of a numeral from the whole-number portion.

Figure 2.29 graphically displays the three distinct ideas that children need to learn in order to understand the Hindu–Arabic numeration system.

Figure 2.29

Reflection from Research Children are able to recognize and read one- and two-digit numerals prior to being able to write them (Baroody, Gannon, Berent, & Ginsburg, 1983).

NCTM Standard All students should connect num- ber words and numerals to the quantities they represent, using various physical models and rep- resentations.

Check for Understanding: Exercise/Problem Set A #4–7✔

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From Lesson 1-3 “Greater Numbers” from Envision Math Common Core, by Randall I. Charles et al., Grade 3, copyright © by Pearson Education.

71

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72 Chapter 2 Sets, Whole Numbers, and Numeration

Nondecimal Numeration Systems Our Hindu–Arabic system is based on grouping by ten. To understand our system better and to experience some of the difficulties children have when learning our numeration system, it is instructive to study similar systems, but with different place values. For example, suppose that a Hindu–Arabic-like system utilized one hand (five digits) instead of two (ten digits). Then, grouping would be done in groups of five. If sticks were used, bundles would be made up of five each (Figure 2.30). Here seventeen objects are rep- resented by three bundles of five each with two left over. This can be expressed by the equation 17 32ten five= , which is read “seventeen base ten equals three two base five.” (Be careful not to read 32five as “thirty-two,” because thirty-two means “3 tens and two,” not “3 fives and two.”) The subscript words “ten” and “five” indicate that the grouping was done in tens and fives, respectively. For simplicity, the subscript “ten” will be omitted; hence 37 will always mean 37ten. (With this agreement, the numeral 24five could also be written 245 since the subscript “5” means the usual base ten 5.) The 10 digits 0, 1, 2, . . . , 9 are used in base ten; however, only the five digits 0, 1, 2, 3, 4 are necessary in base five. A few examples of base five numerals are illustrated in Figure 2.31.

Figure 2.30

Counting using base five names differs from counting in base ten. The first ten base five numerals appear in Figure 2.32. Interesting junctures in counting come after a number has a 4 in its ones column. For example, what is the next number (written in base five) after 24five? after 34five? after 44five ? after 444five ?

Reflection from Research Students experience more suc- cess counting objects that can be moved around than they do counting pictured objects that cannot be moved (Wang, Resnick, & Boozer, 1971).

The numbers from 0 to 26 are represented using only 3 symbols: ∼, Λ, and ⎡. Explain how the system works and list the next 10 numbers.

0 = ~ 9 = Λ ~~ 18 = ~~

1 = Λ 10 = Λ Λ~ 19 = ~Λ

2 = 11 = Λ~ 20 = ~

3 = Λ~ 12 = ΛΛ~ 21 = Λ~

4 = ΛΛ 13 = ΛΛΛ 22 = ΛΛ

5 = Λ 14 = ΛΛ 23 = Λ

6 = ~ 15 = Λ ~ 24 = ~

7 = Λ 16 = Λ Λ 25 = Λ

8 = 17 = Λ 26 =

How is this system related to the numeration systems we have discussed thus far?

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Section 2.3 The Hindu–Arabic System 73

Figure 2.31

Figure 2.32

Figure 2.33 shows how to find the number after 24five using multibase pieces.

Figure 2.33

Converting numerals from base five to base ten can be done using (1) multibase pieces or (2) place values and expanded notation.

Express 123five in base ten.

S O L U T I O N

a. Using Base Five Pieces: See Figure 2.34.

Figure 2.34

b. Using Place Value and Expanded Notation Place values

in base ten ⎯ →⎯ 123

25 5 1

1 2 3 1 25 2 5 3 1 38five = = + + =( ) ( ) ( ) ■

Converting from base ten to base five also utilizes place value.

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74 Chapter 2 Sets, Whole Numbers, and Numeration

When expressing place values in various bases, exponents can provide a convenient shorthand notation. The symbol am represents the product of m factors of a. Thus 5 5 5 5 7 7 7 3 3 3 3 33 2 4= = =¥ ¥ ¥ ¥ ¥ ¥, , , and so on. Using this exponential notation, the first several place values of base five, in reverse order, are 1, 5, 52, 53, 54. Although we have studied only base ten and base five thus far, these same place-value ideas can be used with any base greater than one. For example, in base two the place values, listed in reverse order, are 1, 2, 22, 23, 24, . . . ; in base three the place values are 1, 3, 32, 33, 34, . . . . The next two examples illustrate numbers expressed in bases other than five and their relationship to base ten.

Convert from base ten to base five.

a. 97 b. 341

S O L U T I O N

a. 97

25 5 1 =

← ⎯⎯ ? ? ?

Base five place values

expressed using base tenn numerals

Think: How many 25s are in 97? There are three since 3 25 75¥ = with 22 remain- ing. How many 5s in the remainder? There are four since 4 5 20¥ = . Finally, since 22 20 2− = , there are two 1s.

97 3 25 4 5 2 342= + + =( ) ( ) five

b. A more systematic method can be used to convert 341 to its base fi ve numeral. First, fi nd the highest power of 5 that will divide into 341; that is, which is the greatest among 1, 5, 25, 125, 625, 3125, and so on, that will divide into 341? The answer in this case is 125. The rest of that procedure uses long division, each time dividing the remainder by the next smaller place value.

Therefore, 341 2 125 3 25 3 5 1= + + +( ) ( ) ( ) , or 2331five. More simply, 341 2331= five, where 2, 3, and 3 are the quotients from left to right and 1 is the fi nal remainder. ■

Algebraic Reasoning The process described for con- verting 97 to a base five number is really solving the equation 97 25 5= + +x y z where x y, , and z are less than 5. The solution is x y= =3 4, , and z = 2. Although an equation was not used, the reasoning is still algebraic.

Express the following numbers in base ten.

a. 11011two b. 1234eight c. 1ETtwelve

(NOTE: Base twelve has twelve basic numerals: 0 through 9, T for ten, and E for eleven.)

S O L U T I O N

a. 11011 2 2 2 2 1

1 1 0 1 1 1 16 1 8 0 4 1 2 1 1 27

4 3 2

two = = + + + + =( ) ( ) ( ) ( ) ( )

b. 1234

8 8 8 1

1 2 3 4 1 8 2 8 3 8 4 1 512 128 24 4 66

3 2 3 2

eight = = + + + = + + + =( ) ( ) ( ) ( ) 88

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Section 2.3 The Hindu–Arabic System 75

c. 1 12 12 1

1 1 12 12 1 144 132 10 286

12 2ET

E T E Ttwelve = = + + = + + =( ) ( ) ( ) ■

Convert from base ten to the given base.

a. 53 to base two b. 1982 to base twelve

S O L U T I O N

a. 53 2 2 2 2 2 15 4 3 2

= ? ? ? ? ? ?

Think: What is the largest power of 2 contained in 53?

Answer: 2 325 = . Now we can fi nd the remaining digits by dividing by decreasing powers of 2.

) ) ) ) )32 53 1

32 21

16 21 1

16 5

8 5 0

0 5

4 5 1

4 1

2 1 0

0 1

1

Therefore, 53 110101= two.

b.  1982 12 1728 12 144 12 12 13 2 1

= = = =( ) ( ) ( ) ? ? ? ?

) ) ) )1728 1982 1

1728 254

144 254 1

144 110

12 110 9

108 2

1 2 2

2 0

Therefore, 1982 1192= twelve. ■

Check for Understanding: Exercise/Problem Set A #8–25✔

Consider the three cards shown here. Choose any number from 1 to 7 and note which cards your number is on. Then add the numbers in the upper right-hand corner of the cards containing your number. What did you find? This “magic” can be justified mathematically using the binary (base two) numeration system.

© R

on B

ag w

el l

Converting from base ten to other bases is accomplished by using grouping just as we did in base fi ve.

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76 Chapter 2 Sets, Whole Numbers, and Numeration

EXERCISES 1. Write each of the following numbers in expanded notation. a. 70        b. 300 c. 984        d. 60,006,060

2. Write each of the following expressions in standard place- value form.

a. 1 1000 2 100 7( ) ( )+ + b. 5 100 000 3 100( , ) ( )+ c. 8 10 7 10 6 10 5 16 4 2( ) ( ) ( ) ( )+ + + d. 2 10 3 10 3 10 4 109 4 3( ) ( ) ( ) ( )+ + +

3. State the place value of the digit 2 in each numeral. a. 6234       b. 5142       c. 2168

4. Words and their roots often suggest numbers. Using this idea, complete the following chart. (Hint: Look for a pattern.)

WORD LATIN ROOT MEANING OF ROOT POWER OF 10

Billion bi 2 9

Trillion tri 3 (a)

Quadrillion quater (b) 15

(c) quintus 5 18

Sextillion sex 6 21

(d) septem 7 (e)

Octillion octo 8 27

Nonillion novem (f) (g)

(h) decem 10 33

5. Write these numerals in words. a. 2,000,000,000 b. 87,000,000,000,000 c. 52,672,405,123,139

6. The following numbers are written in words. Rewrite each one using Hindu–Arabic numerals.

a. Seven million six hundred three thousand fifty-nine b. Two hundred six billion four hundred fifty-three

thousand

7. List three attributes of our Hindu–Arabic numeration system.

8. Write a base four numeral for the following set of base four pieces. Represent the blocks on the Chapter 2 eManipulative activity Multibase Blocks on our Web site and make all possible trades first.

9. Represent each of the following numerals with multi- base pieces. Use the Chapter 2 eManipulative activity Multibase Blocks on our Web site to assist you.

a. 134five b. 1011two c. 3211four

10. To express 69 with the fewest pieces of base three blocks, flats, longs, and units, you need _____ blocks, _____ flats, _____ longs, and _____ units. The Chapter 2 eManipula- tive activity Multibase Blocks on our Web site may help in the solution.

11. Represent each of the following with bundling sticks and chips on a chip abacus. (The Chapter 2 eManipulative Chip Abacus on our Web site may help in understanding how the chip abacus works.)

a. 24 b. 221five c. 167eight

12. a. Draw a sketch of 62 pennies and trade for nickels and quarters. Write the corresponding base five numeral.

b. Write the base five numeral for 93 and 2173.

13. How many different symbols would be necessary for a base twenty-three system?

14. What is wrong with the numeral 85eight ?

15. True or false? a. 7 7eight = b. 30 30four = c. 200 200three nine=

16. a. Write out the base five numerals in order from 1 to 100five.

b. Write out the base two numerals in order from 1 to 10000two.

c. Write out the base three numerals in order from 1 to 1000three.

d. In base six, write the next four numbers after 254six. e. What base four numeral follows 303four ?

17. Write each of the following base seven numerals in expanded notation.

a. 15seven b. 123seven c. 5046seven

18. a. What is the largest three-digit base four number? b. What are the five base four numbers that follow it? Give

your answers in base four numeration.

19. Use the Chapter 2 dynamic spreadsheet Base Converter on our Web site to convert the base ten numbers 2400 and 2402, which both have four digits, to a base seven number. What do you notice about the number of digits in the base seven representations of these numbers? Why is this?

20. Convert each base ten numeral into a numeral in the base requested.

a. 395 in base eight b. 748 in base four c. 54 in base two

EXERCISE/PROBLEM SET A

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Section 2.3 The Hindu–Arabic System 77

PROBLEMS 26. What bases make these equations true?

a. 32 44= _____ b. 57 10eight = _____ c. 31 11four = _____ d. 15 30x y=

27. The set of even whole numbers is the set { , , , , }0 2 4 6 . . . . What can be said about the ones digit of every even num- ber in the following bases?

a. 10 b. 4 c. 2 d. 5

28. Mike used 2989 digits to number the pages of a book. How many pages does the book have?

29. The sum of the digits in a two-digit number is 12. If the digits are reversed, the new number is 18 greater than the original number. What is the number?

30. To determine a friend’s birth date, ask him or her to perform the following calculations and tell you the result: Multiply the number of the month in which you were born by 4 and add 13 to the result. Then multiply that answer by 25 and subtract 200. Add your birth date (day of month) to that answer and then multiply by 2. Subtract 40 from the result and then mul- tiply by 50. Add the last two digits of your birth year to that answer and, finally, subtract 10,500.

a. Try this sequence of operations with your own birth date. How does place value allow you to determine a birth date from the final answer? Try the sequence again with a different birth date.

b. Use expanded notation to explain why this technique always works.

21. The base twelve numeration system has the following twelve symbols: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, T, E. Change each of the following numerals to base ten numerals. Use the Chapter 2 dynamic spreadsheet Base Converter on our Web site to check your answers.

a. 142twelve b. 234twelve c. 503twelve d. T twelve9 e. T Etwelve0 f. ETETtwelve

22. The hexadecimal numeration system, used in computer programming, is a base sixteen system that uses the symbols 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, E, and F. Change each of the following hexadecimal numerals to base ten numerals.

a. 213sixteen b. 1 2C Bsixteen

23. Write each of the following base ten numerals in base six- teen (hexadecimal) numerals.

a. 375 b. 2941 c. 9520 d. 24,274

24. Write each of the following numbers in base six and in base twelve.

a. 74         b. 128 c. 210        d. 2438

25. Find the missing base. a. 35 120= _____ b. 41 27six = _____ c. 52 34seven = _____

EXERCISES 1. Write each of the following numbers in expanded form. a. 409 b. 7094 c. 746 d. 840,001

2. Write each expression in place-value form. a. 3 1000 7 10 5( ) ( )+ + b. 7 10 000 6 100( , ) ( )+ c. 6 10 3 10 9 15 3( ) ( ) ( )+ + d. 6 10 9 107 5( ) ( )+

3. State the place value of the digit 0 in each numeral. a. 40,762          b. 9802          c. 0

4. The names of the months in the premodern calendar, which had a different number of months than we do now, was based on the Latin roots for bi, tri, quater, etc. in Set A Exercise 4. Consider the names of the months on the premodern calendar. These names suggest the number of the month. What was September’s number on the premodern calendar? What was October’s number? November’s? If December were the last month of the premodern calendar year, how many months made up a year?

5. Write these numerals in words. a. 32,090,047   b. 401,002,560,300 c. 98,000,000,000,000,000

6. The following numbers are written in words. Rewrite each one using Hindu–Arabic numerals.

a. Twenty-seven million sixty-nine thousand fourteen b. Twelve trillion seventy million three thousand five

7. Explain how the Hindu–Arabic numeration system is multi- plicative and additive.

8. Write a base three numeral for the following set of base three pieces. Represent the blocks on the Chapter 2 eMa- nipulative activity Multibase Blocks on our Web site and make all possible trades first.

EXERCISE/PROBLEM SET B

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78 Chapter 2 Sets, Whole Numbers, and Numeration

9. Represent each of the following numerals with multibase pieces. Use the Chapter 2 eManipulative Multibase Blocks on our Web site to assist you.

a. 221three b. 122four c. 112four d. List these three numbers from smallest to largest. e. Explain why you can’t just compare the first digit in

the different base representations to determine which is larger.

10. To express 651 with the smallest number of base eight pieces (blocks, flats, longs, and units), you need _____ blocks, _____ flats, _____ longs, and _____ units.

11. Represent each of the following with bundling sticks and chips on a chip abacus. (The Chapter 2 eManipulative Chip Abacus on our Web site may help in understanding how the chip abacus works.)

a. 38 b. 52six c. 1032five

12. Suppose that you have ten longs in a set of multibase pieces in each of the following bases. Make all possible exchanges and write the numeral the pieces represent in that base.

a. 10 longs in base eight b. 10 longs in base six c. 10 longs in base three

13. If all the letters of the alphabet were used as our single- digit numerals, what would be the name of our base sys- tem? If a represented zero, b represented one, and so on, what would be the base ten numeral for the “alphabet” numeral zz?

14. What is wrong with the numeral 24 three?

15. True or false? a. 8 8nine eleven= b. 30 30five six= c. 30 40eight six=

16. a. Write out the first 20 base four numerals. b. How many base four numerals precede 2000four ? c. Write out the base six numerals in order from 1 to 100six. d. What base nine numeral follows 888nine ?

17. Write each of the following base three numerals in expanded notation.

a. 22three b. 212three c. 12110three

18. a. What is the largest six-digit base two number? b. What are the next three base two numbers? Give your

answers in base two numeration.

19. Find two different numbers whose base ten representa- tions have a different number of digits but their base three representations each has 4 digits. The Chapter 2 dynamic spreadsheet Base Converter on our Web site can assist in solving this problem.

20. Convert each base ten numeral into its numeral in the base requested.

a. 142 in base twelve b. 72 in base two c. 231 in base eight

21. Find the base ten numerals for each of the following. Use the Chapter 2 dynamic spreadsheet Base Converter on our Web site to check your answers.

a. 342five b. TE twelve0 c. 101101two

22. Using the hexidecimal digits described in Set A, Exercise 22, rewrite the following hexidecimal numerals as base ten numerals.

a. A sixteen4 b. 420Esixteen

23. Convert these base two numerals into base eight numerals. Can you state a shortcut? [Hint: Look at part d.]

a. 1001two b. 110110two c. 10101010two d. 101111two

24. Convert the following base five numerals into base nine numerals.

a. 12five b. 204five c. 1322five

25. Find the missing base. a. 28 34= _____ b. 28 26= _____ c. 23 43twelve = _____

PROBLEMS 26. Under what conditions can this equation be true: a bb a= ?

Explain.

27. Propose new names for the numbers 11, 12, 13, . . . , 19 so that the naming scheme is consistent with the numbers 20 and above.

28. A certain number has four digits, the sum of which is 10. If you exchange the first and last digits, the new number will be 2997 larger. If you exchange the middle two digits of the original number, your new number will be 90 larger.

This last enlarged number plus the original number equals 2558. What is the original number?

29. What number is twice the product of its two digits?

30. As described in the Mathematical Morsel, the three cards shown here can be used to read minds. Have a person think of a number from 1 to 7 (say, 6) and tell you what card(s) it is on (cards A and B). You determine the per- son’s number by adding the numbers in the upper right- hand corner ( ).4 2 6+ =

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End of Chapter Material 79

a. How does this work? b. Prepare a set of four such magic cards for the numbers

1–15.

31. a. Assuming that you can put weights only on one pan, show that the gram weights 1, 2, 4, 8, and 16 are suf- ficient to weigh each whole-gram weight up to 31 grams using a pan balance. (Hint: Base two.)

b. If weights can be used on either pan, what would be the fewest number of gram weights required to weigh 31 grams, and what would they weigh?

32. What is a real-world use for number bases two and sixteen?

Analyzing Student Thinking

33. Jason asks why the class studied other numeration systems like the Roman system and base five. How should you respond?

34. Brandi asks why you are teaching the Hindu–Arabic numeration system using a chip abacus. How should you respond?

35. Jamie read the number 512 as “five hundred and twelve.” Is she correct? If not, why not?

36. The number 43 is read “forty-three” and means “4 tens plus 3 ones.” Since 13 is “1 ten and 3 ones,” Juan asks if he could call the number “onety-three.” How should you respond?

37. After studying base two, Gladys marvels at its simplicity and asks why we don’t use base two rather than base ten. How should you respond?

38. After asking Riann to name the numeral 1 000 0002, , , she says “one million.” Is she correct? Explain.

A survey was taken of 150 college freshmen. Forty of them were majoring in mathematics, 30 of them were majoring in English, 20 were majoring in science, 7 had a double major of mathematics and English, and none had a double (or triple) major with science. How many students had majors other than mathematics, English, or science?

Strategy: Draw a Diagram

A Venn diagram with three circles, as shown here, is useful in this problem. There are to be 150 students within the rectan- gle, 40 in the mathematics circle, 30 in the English circle, 20 in the science circle, and 7 in the intersection of the mathematics and English circles but outside the science circle.

There are 33 7 23 20+ + + , or 83, students accounted for, so there must be 150 83− , or 67, students outside the three circles. Those 67 students were the ones who did not major in math- ematics, English, or science.

Additional Problems Where the Strategy “Draw a Diagram” Is Useful

1. A car may be purchased with the following options: Radio: AM/FM, AM/FM Cassette, AM/FM Cassette/CD Sunroof: Pop-up, Sliding Transmission: Standard, Automatic How many different cars can a customer select among these options?

2. One morning a taxi driver travels the following routes: North: 5 blocks; West: 3 blocks; North: 2 blocks; East: 5 blocks; and South: 2 blocks. How far is she from where she started?

3. For every 50 cars that arrive at a highway intersection, 25 turn right to go to Allentown, 10 go straight ahead to Bos- ton, and the rest turn left to go to Canton. Half of the cars that arrive at Boston turn right to go to Denton, and one of every fi ve that arrive at Allentown turns left to go to Den- ton. If 10,000 cars arrive at Denton one day, how many cars arrived at Canton?

END OF CHAPTER MATERIAL

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80 Chapter 2 Sets, Whole Numbers, and Numeration

CHAPTER REVIEW

Review the following terms and problems to determine which require learning or relearning—page numbers are provided for easy reference.

Vocabulary/Notation Natural numbers  44 Counting numbers  44 Whole numbers  44 Set ({. . .})  45 Element (member)  45 Set-builder notation {. . . | . . .}  45 Is an element of ( )∈   45 Is not an element of ( )∉   45 Empty set ( )∅ or null set  45

Equal sets ( )=  45 Is not equal to (≠)  45 1-1 correspondence  45 Equivalent sets (~)  45 Matching sets  45 Subset (⊆)  46 Proper subset (⊂)  46 Venn diagram  46 Universal set ( )U   46

Finite set  46 Infinite set  46 Disjoint sets  47 Union (∪)  48 Intersection (∩)  48 Complement (A)  49 Set difference ( )−   49 Ordered pair  50 Cartesian product ( )×   50

Sets as a Basis for Whole Numbers

Exercises 1. Describe three different ways to define a set.

2. True or false? a. 1 1∈{ , , }a b b. a ∉{ , , }1 2 3

c. { , } { , , }x y x y z ⊂ d. { , } { , }a b a b ⊆ e. ∅ = {} f. { , } ~ { , }a b c d

David Hilbert (1862 1943)− David Hilbert attended the gymnasium in his home town of Königsberg and then went on to the University of Königsberg where he re- ceived his doctorate in 1885. He moved on and became a

professor of mathematics at the University of Gottin- gen, where, in 1895, he was appointed to the chair of mathematics. He spent the rest of his career at Gottingen. Hilbert’s work in geometry has been considered to have had the greatest infl uence in that area second only to Eu- clid. However, Hilbert is perhaps most famous for sur- veying the spectrum of unsolved problems in 1900 from which he selected 23 for special attention. He felt these 23 were crucial for progress in mathematics in the com- ing century. Hilbert’s vision was proved to be prophetic. Mathematicians took up the challenge, and a great deal of progress resulted from attempts to solve “Hilbert’s Problems.” Hilbert made important contributions to the foundations of mathematics and attempted to prove that mathematics was self-consistent. He became one of the most infl uential mathematicians of his time. Yet, when he read or heard new ideas in mathematics, they seemed “so diffi cult and practically impossible to understand,” until he worked the ideas through for himself.

Emmy Noether (1882 1935)− Emmy Noether was born and educated in Germany and grad- uated from the University of Erlangen, where her father, Max Noether, taught mathematics. There were few professional opportunities for a woman

mathematician, so Noether spent the next eight years doing research at home and teaching for her increas- ingly disabled father. Her work, which was in algebra, in particular, ring theory, attracted the attention of the mathematicians Hilbert and Klein, who invited her to the University of Gottingen. Initially, Noether’s lectures were announced under Hilbert’s name, because the uni- versity refused to admit a woman lecturer. Conditions improved, but in 18 years at Gottingen, she was rou- tinely denied the promotions that would have come to a male mathematician of her ability. When the Nazis came to power in 1933, she was dismissed from her position. She immigrated to the United States and spent the last two years of her life at Bryn Mawr College. Upon her death, Albert Einstein wrote in the New York Times that “In the judgment of the most competent living mathema- ticians, Fraulein Noether was the most signifi cant creative mathematical genius thus far produced since the higher education of women began.”

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Chapter Review 81

Vocabulary/Notation Number  57 Numeral  57 Cardinal number  57 Ordinal number  57 Identification number  57 Number of a set [ ( )]n A   58 Less than (<), greater than (>)  59 Less than or equal to (≤)  59

Greater than or equal to (≥)  59 Whole-number line  59 Tally numeration system  60 Grouping  60 Egyptian numeration system  60 Additive numeration system  61 Roman numeration system  61 Subtractive numeration system  61

Positional numeration system  62 Multiplicative numeration system  62 Babylonian numeration system  62 Place value  62 Placeholder  62 Mayan numeration system  63 Zero  63

Whole Numbers and Numeration

Exercises 1. Is the expression “house number” literally correct? Explain.

2. Give an example of a situation where each of the following is useful:

a. cardinal number b. ordinal number c. identification number

3. True or false? a. n a b c d({ , , , }) = 4 b. 7 7≤ c. 3 4≥ d. 5 50< e. |||| is three in the tally system

f. ∩ = ∩||| ||| in the Egyptian system g. IV VI= in the Roman system h. = in the Mayan system

4. Express each of the following in our system.

a. b. CXIV c.

5. Express 37 in each of the following systems: a. Egyptian b. Roman c. Mayan

6. Using examples, distinguish between a positional numera- tion system and a place-value numeration system.

Vocabulary/Notation Hindu–Arabic numeration system  68 Digits  68 Base  68

Bundles of sticks  68 Base ten pieces (Dienes blocks)  68 Chip abacus  69

Expanded form  69 Expanded notation  69 Exponents  74

The Hindu–Arabic System

Exercises 1. Explain how each of the following contributes to formulat-

ing the Hindu–Arabic numeration system. a. Digits b. Grouping by ten c. Place value d. Additive and multiplicative attributes

2. Explain how the names of 11, 12, . . . , 19 are inconsistent with the names of 21, 22, . . . , 29.

3. True or false? a. 100 212= five b. 172 146nine = c. 11111 2222two three= d. 18 11twelve nineteen=

4. What is the value of learning different number bases?

g.{ , } { , }a b c d = h.{ , , } { , , } { , , , }1 2 3 2 3 4 1 2 3 4 ∩ = i.{ , } { , , } { , , }2 3 1 3 4 2 3 4 ∪ = j.{ , }1 2 and { , }2 3 are disjoint sets k.{ , , , } { , , } { }4 3 5 2 2 3 4 5 − = l.{ , } { , } {( , ), ( , )}a b a b 1 2 1 2× =

3. What set is used to determine the number of elements in the set { , , , , , , }?a b c d e f g

4. Explain how you can distinguish between finite sets and infinite sets.

5. A poll at a party having 23 couples revealed that there were 25 people who liked both country-western and ballroom

dancing. 8 who liked only country-western dancing. 6 who liked only ballroom dancing.

How many did not like either type of dancing?

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82 Chapter 2 Sets, Whole Numbers, and Numeration

Knowledge 1. Identify each of the following as always true, sometimes

true, or never true. If it is sometimes true, give one true and one false example. (Note: If a statement is sometimes true, then it is considered to be mathematically false.)

a. If A B~ then A B= . b. If A B⊂ , then A B⊆ . c. A B A B∩ ⊆ ∪ . d. n a b x y z({ , } { , , }) . × = 6 e. If A B n A B n A∩ = ∅ − <, ( ) ( ). then . f. { , , , . . . , } { , , , . . .}2 4 6 2000000 4 8 12~ . g. The range of a function is a subset of the codomain of

the function. h. VI IV= in the Roman numeration system. i. ÷ represents one hundred six in the Mayan numeration

system. j. 123 321= in the Hindu–Arabic numeration system.

2. How many different symbols would be necessary for a base nineteen system?

3. Explain what it means for two sets to be disjoint.

Skill 4. For A a b c B b c d e C d e f g= = ={ , , }, { , , , }, { , , , },

D e f g= { , , }, find each of the following. a. A B∪ b. A C∩ c. A B∩ d. A D× e. C D− f. ( ) ( )B D A C∩ ∪ ∩

5. Write the equivalent Hindu–Arabic base ten numeral for each of the following numerals.

a. ∩ ∩ ∩ ||(Egyptian) b. CMXLIV (Roman)

c. .. . (Mayan)      d.

(Babylonian)

e. 10101two   f. ETtwelve

6. Express the following in expanded form. a. 759 b. 7002 c. 1001001two

7. Shade the region in the following Venn diagram to repre-

sent the set A B C− ∪( ).

8. Rewrite the base ten number 157 in each of the following number systems that were described in this chapter.

Babylonian Roman Egyptian Mayan

9. Write a base five numeral for the following set of base five blocks. (Make all possible trades first.)

10. Represent the shaded region using the appropriate set notation.

Understanding 11. Use the Roman and Hindu–Arabic systems to explain the

difference between a positional numeration system and a place-value numeration system.

12. Determine conditions, if any, on nonempty sets A and B so that the following equalities will be true.

a. A B B A∪ = ∪ b. A B B A∩ = ∩ c. A B B A− = − d. A B B A× = ×

13. If ( , )a b and ( , )c d are in A B× , name four elements in B A× .

14. Explain two distinctive features of the Mayan number system as compared to the other three non-Hindu–Arabic number systems described in this chapter.

15. Given the universal set of { , , , , }1 2 3 20. . . and sets A B, , and C as described, place all of the numbers from the universal set in the appropriate location on the following Venn diagram.

CHAPTER TEST

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Chapter Test 83

A = { , , , , , , }2 4 0 1 3 5 6 B = { , , , , , , , }10 11 12 1 2 7 6 14 C = { , , , , , , , }18 19 6 11 16 12 9 8

16. Let A x x= { | is a letter in the alphabet} and B = { , , , , }.10 11 12 40. . . Is it possible to have 1-1 correspondence between sets A and B? If so, describe the correspondence. If not, explain why not.

17. Represent the numeral 1212three with base three blocks and chips on a chip abacus.

Problem Solving/Application

18. If n A n B n A B( ) , ( ) , ( ) ,= = ∩ =71 53 27 what is n A B( )?∪

19. Find the smallest values for a and b so that 21 25b a= .

20. A number in some base b has two digits. The sum of the digits is 6 and the difference of the digits is 2. The number is equal to 20 in the base ten. What is the base, b, of the number?

21. What is the largest Mayan number that can be represented with exactly 4 symbols? Explain.

22. A third-grade class of 24 students counted the number of students in their class with the characteristics of brown eyes, brown hair, and curly hair. The students found the following:

12 students had brown eyes 19 students had brown hair 12 students had curly hair 5 students had brown eyes and curly hair 10 students had brown eyes and brown hair 7 students had brown, curly hair 3 students had brown eyes with brown, curly hair

To answer the following questions a–c, we will classify all eyes as either brown or blue. Similarly all hair will be either curly or straight. The color of the hair will be either brown or non-brown.

a. How many students have brown eyes with straight, non- brown hair?

b. How many students have curly brown hair and blue eyes?

c. How many students straight-haired, blue-eyed students are in the class?

23. To write the numbers 10, 11, 12, 13, 14, 15 in the Hindu– Arabic system requires 12 symbols. If the same numbers were written in the Babylonian, Mayan, or Roman sys- tems, which system would require the most symbols? Explain.

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84

T he Hindu–Arabic numeration system can be traced back to 250 B.C.E. However, it wasn’t until about 800 C.E. when a complete Hindu system was described in a book by the Persian mathematician al-Khwarizmi. Although the Hindu–Arabic numerals, as well as the Roman numeral system, were used to rep- resent numbers, they were not used for computations, mainly because of the lack of inexpensive, convenient writing equipment such as paper and pencil. In fact, the Romans used a sophisticated abacus or “sand tray” made of a board with small pebbles (calculi) that slid in grooves as their calculator. Another form of the abacus was a wooden frame with beads sliding on thin rods, much like those used by the Chinese as shown in the Focus On for Chapter 4.

From about 1100 to 1500 C.E. there was a great debate among Europeans regarding calculation. Those who advocated the use of Roman numerals along with the aba- cus were called the abacists. Those who advocated using the Hindu–Arabic numeration system together with writ- ten algorithms such as the ones we use today were called algorists. About 1500, the algorists won the argument and by the eighteenth century, there was no trace of the abacus in western Europe. However, parts of the world, notably, China, Japan, Russia, and some Arabian countries, con- tinued to use a form of the abacus.

It is interesting, though, that in the 1970s and 1980s, technology produced the inexpensive, handheld calculator,

which rendered many forms of written algorithms obso- lete. Yet the debate continues regarding what role the cal- culator should play in arithmetic. Could it be that a debate will be renewed between algorists and the modern-day abacists (or “calculatorists”)? Is it possible that we may someday return to being “abacists” by using our Hindu– Arabic system to record numbers while using calculators to perform all but simple mental calculations? Let’s hope that it does not take us 400 years to decide the appropriate balance between written and electronic calculations!

C H A P T E R

3 WHOLE NUMBERS: OPERATIONS AND PROPERTIES Calculation Devices Versus Written Algorithms: A Debate Through Time

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85

Problem-Solving Strategies 1. Guess and Test

2. Draw a Picture

3. Use a Variable

4. Look for a Pattern

5. Make a List

6. Solve a Simpler Problem

7. Draw a Diagram

8. Use Direct Reasoning

Use Direct Reasoning

The Use Direct Reasoning strategy is used virtually all the time in conjunction with other strategies when solving problems. Direct reasoning is used to reach a valid conclusion from a series of statements. Often, statements involving direct reasoning are of the form “If A then B.” Once this statement is shown to be true, statement B will hold whenever statement A does. (An expanded discussion of reasoning is contained in the Logic section near the end of the book.) In the following initial problem, no computations are required. That is, a solution can be obtained merely by using direct reasoning, and perhaps by drawing pictures.

Initial Problem In a group of nine coins, eight weigh the same and the ninth is either heavier or lighter. Assume that the coins are identical in appearance. Using a pan balance, what is the smallest number of balancings needed to identify the counterfiet coin?

Clues The Use Direct Reasoning strategy may be appropriate when

A proof is required.

A statement of the form “If . . . , then . . .” is involved.

You see a statement that you want to imply from a collection of known condi- tions.

A solution of this Initial Problem is on page 124.

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86

AUTHOR

WALK-THROUGH

I N T R O D U C T I O N The whole-number operations of addition, subtraction, multiplication, and division form the foundation of arithmetic. Because of their primary importance, this entire chapter is devoted to the study of these concepts independent of computational procedures. Each of the opera- tions is introduced independently but the connections between them are carefully highlighted. The properties of addition and multiplication are discussed as a way to assist in learning basic facts. Finally, exponents are introduced to simplify multiplication and to serve as a convenient notation for

representing large numbers.

Key Concepts from the NCTM Principles and Standards for School Mathematics

PRE-K-2–NUMBER AND OPERATIONS

Understand various meanings of addition and subtraction of whole numbers and the relationship between the two operations.

Understand situations that entail multiplication and division, such as equal groupings of objects and sharing equally. Develop and use strategies for whole-number computations, with a focus on addition and subtraction

PRE-K-2–ALGEBRA

Illustrate general principles and properties of operations, such as commutativity, using specific numbers. Model situations that involve the addition and subtraction of whole numbers, using objects, pictures, and symbols.

GRADES 3-5–NUMBER AND OPERATIONS

Understand various meanings of multiplication and division. Understand and use properties of operations, such as the distributivity of multiplication over addition. Develop fluency in adding, subtracting, multiplying, and dividing whole numbers.

GRADES 3-5–ALGEBRA

Identify such properties as commutativity, associativity, and distributivity and use them to compute with whole numbers.

Key Concepts from the NCTM Curriculum Focal Points

KINDERGARTEN: Representing, comparing, and ordering whole numbers and joining and separating sets. GRADE 1: Developing understandings of addition and subtraction and strategies for basic addition facts and

related subtraction facts. GRADE 2: Developing quick recall of addition facts and related subtraction facts and fluency with multidigit addi-

tion and subtraction. GRADE 3: Developing understandings of multiplication and division and strategies for basic multiplication facts

and related division facts. GRADE 4: Developing quick recall of multiplication facts and related division facts and fluency with whole-number

multiplication.

Key Concepts from the Common Core State Standards for Mathematics

KINDERGARTEN: Understand addition as putting together and adding to, and understand subtraction as taking apart and taking from.

GRADE 1: Represent and solve problems involving addition and subtraction. Understand and apply properties of operations and the relationship between addition and subtraction.

GRADE 2: Represent and solve problems involving addition and subtraction. GRADE 3: Represent and solve problems involving multiplication. Understand properties of multiplication and the

relationship between multiplication and division.

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Section 3.1 Addition and Subtraction 87

Addition and Its Properties Finding the sum of two whole numbers is one of the first mathematical ideas a child encounters after learning the counting chant “one, two, three, four, . . .” and the con- cept of number. In particular, the question “How many is 3 and 2?” can be answered using both a set model and a measurement model.

Set Model To find “3 2+ ” using a set model, one must represent two disjoint sets: one set, A, with three objects and another set, B, with 2 objects. Recall that n A( ) denotes the number of elements in set A. In Figure 3.1, n A( ) = 3 and n B( ) = 2.

Figure 3.1

With sets, addition can be viewed as combining the contents of the two sets to form the union and then counting the number of elements in the union, n A B( )∪ = 5. In this case, n A n B n A B( ) ( ) ( )+ = ∪ . The example in Figure 3.1 suggests the following general definition of addition.

NCTM Standard All students should model situ- ations that involve the addition and subtraction of whole num- bers using objects, pictures, and symbols.

ADDITION AND SUBTRACTION

Addition of Whole Numbers

Let a and b be any two whole numbers. If A and B are disjoint sets with a n A= ( ) and b n B= ( ), then a b n A B+ = ∪( ).

D E F I N I T I O N 3 . 1

The number a b+ , read “a plus b,” is called the sum of a and b, and a and b are called addends or summands of a b+ .

When using sets to discuss addition, care must be taken to use disjoint sets. In Figure 3.2, the sets A and B are not disjoint because n A B( )∩ = 1.

This nonempty intersection makes n A n B n A B( ) ( ) ( )+ ≠ ∪ and so it is not an example of addition using the set model. However, Figure 3.2 gives rise to a more general statement about the number of elements in the union of two sets. It is n A n B n A B n A B( ) ( ) ( ) ( )+ − ∩ = ∪ . The following example illustrates how to prop- erly use disjoint sets to model addition.Figure 3.2

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88 Chapter 3 Whole Numbers: Operations and Properties

Addition is called a binary operation because two (“bi”) numbers are combined to produce a unique (one and only one) number. Multiplication is another example of a binary operation with numbers. Intersection, union, and set difference are binary operations using sets.

Measurement Model Addition can also be represented on the whole-number line pictured in Figure 3.3. Even though we have drawn a solid arrow starting at zero and pointing to the right to indicate that the collection of whole numbers is unending, the whole numbers are represented by the equally spaced points labeled 0, 1, 2, 3, and so on. The magnitude of each number is represented by its distance from 0. In later chapters, the number line will be extended and filled in.

Figure 3.3

Addition of whole numbers is represented by directed arrows of whole-number lengths. The procedure used to find the sum 3 4+ using the number line is illustrated in Figure 3.4. Here the sum, 7, of 3 and 4 is found by placing arrows of lengths 3 and 4 end to end, starting at zero. Notice that the arrows for 3 and 4 are placed end to end and are disjoint, just as in the set model.

Figure 3.4

Next we examine some fundamental properties of addition of whole numbers that can be helpful in simplifying computations.

Use the definition of addition to compute 4 5+ .

S O L U T I O N   Let A a b c d= { , , , } and B e f g h i= { , , , , }. Then n A( ) = 4 and n B( ) = 5. Also, A and B have been chosen to be disjoint.

Therefore, ( )

({ , , , } { , , , , })

({ , , , ,

4 5+ = ∪ = ∪ =

n A B

n a b c d e f g h i

n a b c d e,, , , , })

.

f g h i

= 9 ■

Mentally compute the following and write down what you did. (Don’t convert to base ten to solve the problems.)

6 4seven seven+ 7 4 2nine nine nine+ +

By comparing your methods with those of your peers, determine if any similar methods were used.

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Section 3.1 Addition and Subtraction 89

Note that the root word of commutative is commute, which means “to inter- change.” Figure 3.5 illustrates this property for 3 2+ and 2 3+ .

Figure 3.5

Now suppose that a child knows all the addition facts through the fives but wants to find 6 3+ . A simple way to do this is to rewrite 6 3+ as 5 4+ by taking one from 6 and adding it to 3. Since the sum 5 4+ is known to be 9, the sum 6 3+ is 9. In sum- mary, this argument shows that 6 3+ can be thought of as 5 4+ by following this reasoning: 6 3 5 1 3 5 1 3 5 4+ = + + = + + = +( ) ( ) . The next property is most useful in simplifying computations in this way.

Problem-Solving Strategy Draw a Picture

Properties of Whole-Number Addition The fact that one always obtains a whole number when adding two whole numbers is summarized by the closure property.

Closure Property for Whole-Number Addition

The sum of any two whole numbers is a whole number.

P R O P E R T Y

In general, when an operation on a set satisfies a closure property, the set is said to be closed with respect to the given operation. Knowing that a set is closed under an operation is helpful when checking certain computations. For example, consider the set of all even whole numbers, { , , , }0 2 4 . . . , and the set of all odd whole numbers, { , , , }1 3 5 . . . . The set of even numbers is closed under addition since the sum of two even numbers is even. Therefore, if one adds a collection of even numbers and obtains an odd sum, an error has been made. The set of odd numbers is not closed under addi- tion because the sum 1 3+ is not an odd number.

Many children learn how to add by “counting on.” For example, to find 9 1+ , a child will count on 1 more from 9, namely, think “nine, then ten.” However, if asked to find 1 9+ , a child might say “1, then 2, 3, 4, 5, 6, 7, 8, 9, 10.” Not only is this inefficient, but also the child might lose track of counting on 9 more from 1. The fact that 1 9 9 1+ = + is useful in simplifying this computation and is an instance of the following property.

Common Core – Grade 1 Apply properties of operations as strategies to add and subtract (e.g., if 8 3 11+ = is known, then 3 8 11+ = is also known).

NCTM Standard All students should illustrate gen- eral principles and properties of operations, such as commutativ- ity, using specific numbers.

Commutative Property for Whole-Number Addition

Let a and b be any whole numbers. Then

a b b a+ = + .

P R O P E R T Y

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90 Chapter 3 Whole Numbers: Operations and Properties

The root word of associative is associate, which means “to unite” or, in this case, “reunite.” The example in Figure 3.6 illustrates this property.

Figure 3.6

Since the empty set has no elements, A A∪ ={ } . A numerical counterpart to this statement is one such as 7 0 7+ = . In general, adding zero to any number results in the same number. This concept is stated in generality in the next property.

Algebraic Reasoning When solving algebraic equa- tions, we often need to “combine like terms.” The commutative and associative properties make this possible. For example, (3 2 ) (4 5 )+ + +x x is simplified by changing the order and grouping of the terms to be (3 ) (2 5 )+ + +4 x x .

Children’s Literature www.wiley.com/college/musser See “The Mission of Addition” by Brian P. Cleary.

Because of this property, zero is called the additive identity or the identity for addition.

The previous properties can be applied to help simplify computations. They are especially useful in learning the basic addition facts (that is, all possible sums of the digits 0 through 9). Although drilling using flash cards or similar electronic devices is helpful for learning the facts, an introduction to learning the facts via the following thinking strategies will pay rich dividends later as students learn to perform multi- digit addition mentally.

Associative Property for Whole-Number Addition

Let a, b, and c be any whole numbers. Then

( ) ( ).a b c a b c+ + = + +

P R O P E R T YCommon Core – Grade 1 Apply properties of operations as strategies to add and subtract (e.g., to add 2 6 4+ + , the sec- ond two numbers can be added to make a ten, so 2 6 4+ + = 2 10 12+ = ).

Identity Property for Whole-Number Addition

There is a unique whole number, namely 0, such that for all whole numbers a,

a a a+ = = +0 0 .

P R O P E R T Y

Discuss how a 6-year-old would find the answer to the question “What is 7 2+ ?” If the 6-year-old were then asked “What is 2 7+ ?”, how would he find the answer to that ques-

tion? Is there a difference? Why or why not?

Thinking Strategies for Learning the Addition Facts The addition table in Figure 3.7 has 100 empty spaces to be filled. The sum of a b+ is placed in the intersection

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Section 3.1 Addition and Subtraction 91

of the row labeled a and the column labeled b. For example, since 4 1 5+ = , a 5 appears in the intersection of the row labeled 4 and the column labeled 1.

Figure 3.7 Figure 3.8 Figure 3.9

1. Commutativity: Because of commutativity and the symmetry of the table, a child will automatically know the facts in the shaded region of Figure 3.8 as soon as the child learns the remaining 55 facts. For example, notice that the sum 4 1+ is in the unshaded region, but its corresponding fact 1 4+ is in the shaded region.

2. Adding zero: The fact that a a+ =0 for all whole numbers fills in 10 of the remaining blank spaces in the “zero” column (Figure 3.9)—45 spaces to go.

3. Counting on by 1 and 2: Children find sums like 7 1+ , 6 2+ , 3 1+ , and 9 2+ by counting on. For example, to find 9 2+ , think 9, then 10, 11. This thinking strategy fills in 17 more spaces in the columns labeled 1 and 2 (Figure 3.10)—28 facts to go.

Figure 3.10 Figure 3.11

4. Combinations to ten: Combinations of the ten fingers can be used to find 7 3+ , 6 4+ , 5 5+ , and so on. Notice that now we begin to have some overlap. There are 25 facts left to learn (Figure 3.11).

5. Doubles: 1 1+ = 2, 2 2+ = 4, 3 3+ = 6, and so on. These sums, which appear on the main left-to-right downward diagonal, are easily learned as a consequence of counting by twos: namely, 2, 4, 6, 8, 10, . . . (Figure 3.12). Now there are 19 facts yet to be determined.

6. Adding ten: When using base ten pieces as a model, adding 10 amounts to laying down a “long” and saying the new name. For example, 3 10+ is 3 units and 1 long, or 13; 7 10+ is 17, and so on.

7. Associativity: The sum 9 5+ can be thought of as 10 4+ , or 14, because 9 5+ = 9 1 4 9 1 4+ + = + +( ) ( ) . Similarly, 8 7 10 5 15+ = + = , and so on. The rest of the addition table can be filled using associativity (sometimes called regrouping) com- bined with adding 10.

8. Doubles ±1 and ±2: This technique overlaps with the others. Many children use it effectively. For example, 7 8 7 7 1 14 1 15+ = + + = + = , or 8 7 8 8 1 15+ = + − = ; 5 7+ = 5 5 2 10 2 12+ + = + = , and so on.

By using thinking strategies 6, 7, and 8, the remaining basic addition facts needed to complete the table in Figure 3.12 can be determined.

Reflection from Research When students are taught strate- gies for thinking and working in mathematics, instead of just basic facts, their computational accu- racy, efficiency, and flexibility can be improved (Crespo, Kyriakides, & McGee, 2005).

Common Core – Grade 1 Relate counting to addition and subtraction (e.g., by counting on 2 to add 2).

Common Core – Kindergarten Decompose numbers less than or equal to 10 into pairs in more than one way (e.g., 5 2 3= + and 5 4 1= + ).

Figure 3.12

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92 Chapter 3 Whole Numbers: Operations and Properties

Thus far we have been adding single-digit numbers. However, thinking strate- gies can be applied to multidigit addition also. Figure 3.13 illustrates how multidigit addition is an extension of single-digit addition. The only difference is that instead of adding units each time, we might be adding longs, flats, and so on. Mentally combine similar pieces, and then exchange as necessary.

Figure 3.13

The next example illustrates how thinking strategies can be applied to multidigit numbers.

NCTM Standard All students should develop fluency with basic number combinations for addition and subtraction.

Use thinking strategies in three different ways to find the sum of 9 7+ .

S O L U T I O N 

a. 9 7 9 1 6 9 1 6 10 6 16+ = + + = + + = + =( ) ( ) b. 9 7 8 1 7 8 1 7 8 8 16+ = + + = + + = + =( ) ( ) c. 9 7 2 7 7 2 7 7 2 14 16+ = + + = + + = + =( ) ( ) ■

Common Core – Kindergarten Compose and decompose num- bers from 11 to 19 into 10 ones and some further ones such as 18 10 8= + .

Using thinking strategies, find the following sums.

a. 42 18+ b. 37 42 13+ +( ) c. 51 39+

S O L U T I O N 

a. 42 18 40 2 10 8 40 10 2 8

50 10 60

+ = + + + = + + + = + =

( ) ( ) ( ) ( )

Addition

Commutativity and associativity

Place value and combination to 10

Addition

b. 37 42 13 37 13 42 37 13 42

50 42 92

+ + = + + = + + = + =

( ) ( ) ( )

c. 51 39 50 1 39 50 1 39 50 40 90

+ = + + = + + = + =

( ) ( )

■

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Section 3.1 Addition and Subtraction 93

The use of other number bases can help you simulate how these think- ing strategies are experienced by students when they learn base ten arithmetic. Perhaps the two most powerful thinking strategies, especially when used together, are associativity and combinations to the base (base ten above). For ex ample, 7 6 7 2 4 7 2 4 10nine nine nine nine nine nine nine nine nine+ = + + = + + =( ) ( ) ++ =4 14nine nine (since the sum of 7nine and 2nine is one of the base in base nine), 4 5 3 1 5six six six six six+ = + + = 3 10 13six six six+ = (since 1 5six six+ is one of the base in base six), and so on.

Compute the following sums using thinking strategies.

a. 7 3eight eight+ b. 5 4seven seven+ c. 9 9twelve twelve+

S O L U T I O N 

a. 7 3 7 1 2 7 1 2eight eight eight eight eight eight eight eigh+ = + + = + +( ) ( ) tt eight eight eight= + =10 2 12

b. 5 4 5 2 2 5 2 2seven seven seven seven seven seven seven seve+ = + + = + +( ) ( ) nn seven seven seven

= +10 2 12

c. 9 9 9 3 6 9 3twelve twelve twelve twelve twelve twelve twelve+ = + + = +( ) ( )) + + =

6

10 6 16 twelve

twelve twelve twelve

Notice how associativity and combinations to the base are used. ■

Check for Understanding: Exercise/Problem Set A #1–7✔

Write a word problem that would have 7 – 2 as its solution. Compare your problem with those of several peers and determine how these problems are different and how they

are the same from a mathematical perspective. If possible categorize the problems according to the underlying math- ematical structure.

Subtraction The Take-Away Approach There are two distinct approaches to subtraction. The take-away approach is often used to introduce children to the concept of subtrac- tion. The problem “If you have 5 coins and spend 2, how many do you have left?” can be solved with a set model using the take-away approach. Also, the problem “If you walk 5 miles from home and turn back to walk 2 miles toward home, how many miles are you from home?” can be solved with a measurement model using the take- away approach (Figure 3.14).

Figure 3.14

This approach can be stated using sets.

Reflection from Research Students will benefit later in their mathematics if they are taught both the “take-away” and “determining the difference” approaches to subtraction, as well as the inverse relationship between addition and subtraction (Selter, Prediger, Nuhrenborger, Hussmann, 2012).

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94 Chapter 3 Whole Numbers: Operations and Properties

Subtraction of Whole Numbers: Take-Away Approach

Let a and b be any whole numbers and A and B be sets such that a n A= ( ), b n B= ( ), and B A⊆ . Then

a b n A B− = −( ).

D E F I N I T I O N 3 . 2

The number “a b- ” is called the difference and is read “a minus b,” where a is called the minuend and b the subtrahend. To fi nd 7 3− using sets, think of a set with seven elements, say { , , , , , , }a b c d e f g . Then, using set difference, take away a subset of three elements, say { , , }a b c . The result is the set { , , , }d e f g , so 7 3 4− = .

The Missing-Addend Approach The second method of subtraction, which is called the missing-addend approach, is often used when making change. For example, if an item costs 76 cents and 1 dollar is tendered, a clerk will often hand back the change by adding up and saying “76 plus four is 80, and twenty is a dollar” as four pennies and two dimes are returned. This method is illustrated in Figure 3.15.

Figure 3.15

Since 2 5+ =3 in each case in Figure 3.15, we know that 5 2− = 3.

NCTM Standard All students should understand various meanings of addition and subtraction of whole numbers and the relationship between the two operations.

Reflection from Research Children frequently have difficulty with missing-addend problems when they are not related to word problems. A common answer to 5 ? 8+ = is 13 (Kamii, 1985).

Common Core – Grade 1 Understand subtraction as an unknown-addend problem. Subtraction of Whole Numbers: Missing-Addend Approach

Let a and b be any whole numbers. Then a b c− = if and only if a b c= + for some whole number c.

A L T E R N A T I V E D E F I N I T I O N 3 . 3

In this alternative definition of subtraction, c is called the missing addend. The missing- addend approach to subtraction is very useful for learning subtraction facts because it shows how to relate them to the addition facts via four-fact families (Figure 3.16).

This alternative definition of subtraction does not guarantee that there is an answer for every whole-number subtraction problem. For example, there is no whole number c such that 3 4= + c, so the problem 3 4− has no whole-number answer. Another way of expressing this idea is to say that the set of whole numbers is not closed under subtraction.

Notice that we have two approaches to subtraction, (i) take away and (ii) missing addend and that each of these approaches can be modeled in two ways using (i) sets and (ii) measurement. The combination of these four methods can be visualized in a two-dimensional diagram (Figure 3.17).Figure 3.16

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Section 3.1 Addition and Subtraction 95

Figure 3.17

The reason for learning to add and subtract is to solve problems in the real world. For example, consider the next two problems.

1. Monica was 59" tall last year. She had a growth spurt and is now 66" tall. How much did she grow during this past year?

To solve this problem, we can use the lower right square in Figure 3.17 because we are dealing with her height (measurement) and we want to know how many more inches 66 is than 59 (missing addend).

2. Monica joined a basketball team this year. One of her teammates is 70" tall. How much taller is that teammate than Monica?

There is a new aspect to this problem. Instead of considering how much taller Monica is than she was last year, you are asked to compare her height with another player. More generally, you might want to compare her height to all the members of her team. This way of viewing subtraction adds a third dimension, comparison, to the 2-by-2 square in Figure 3.17. (Figure 3.18)

Figure 3.18

Children’s Literature www.wiley.com/college/musser See “Ten Sly Piranhas” by William Wise.

Common Core – Grade 1 Use addition and subtraction within 20 to solve word problems involving situations of adding to, taking from, putting together, taking apart, and comparing, with unknowns in all positions.

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96 Chapter 3 Whole Numbers: Operations and Properties

Benjamin Franklin was known for his role in politics and as an inven- tor. One of his mathematical discoveries was an 8-by-8 square made up of the counting numbers from 1 to 64. Check out the following properties:

1. All rows and columns total 260. 2. All half-rows and half-columns total 130. 3. The four corners total 130. 4. The sum of the corners in any 4-by-4 or 6-by-6 array is 130. 5. Every 2-by-2 array of four numbers totals 130.

(Note: There are 49 of these 2-by-2 arrays!) © R

on B

ag w

el l

Check for Understanding: Exercise/Problem Set A #8–14✔

To solve the problem of comparing Monica’s height with the heights of the rest of her teammates, we would use the smaller cube in the lower right back of the 2-by- 2-by-2 cube since it uses the measurement model and missing addend approach involv- ing the comparison with several teammates.

Following is another problem where comparison comes into play: If Larry has $7 and Judy has $3, how much more money does Larry have? Because Larry’s money and Judy’s money are two distinct sets, a comparison view would be used. To find the solution, we can mentally match up 3 of Larry’s dollars with 3 of Judy’s dollars and take those matched dollars away from Larry’s (see Figure 3.19).

Thus, this problem would correlate to the smaller cube in the top back left of Figure 3.18 because it is a set model using take-away approach involving the compari- son of sets of money. In subtraction situations where there is more than one set or one measurement situation involved as illustrated by the back row of the cube in Figure 3.18, the subtraction is commonly referred to as using the comparison approach.

Figure 3.19

EXERCISES 1. a. Draw a figure similar to Figure 3.1 to find 4 3+ . b. Find 3 5+ using a number line.

2. For which of the following pairs of sets is it true that n D n E n D E( ) ( ) ( )+ = ∪ ? When not true, explain why not.

a. D = { , , , }1 2 3 4 , E = { , , , }7 8 9 10 b. D = {}, E = { }1 c. D a b c d= { , , , }, E d c b a= { , , , }

3. Which of the following sets are closed under addition? Why or why not?

a. { , , , , }0 10 20 30 . . . b. { }0 c. { , , }0 1 2 d. { , }1 2 e. Whole numbers greater than 17

4. Identify the property or properties being illustrated. a. 1279 3847+ must be a whole number. b. 7 5 5 7+ = +

c. 53 47 50 50+ = + d. 1 0 1+ = e. 1 0 0 1+ = + f. ( )53 48 7 60 48+ + = +

5. Use the Chapter 3 eManipulative Number Bars on our Web site to model 7 2+ and 2 7+ on the same number line. Sketch what is represented on the computer and describe how the two problems are different. How are they similar?

6. What property or properties justify that you get the same answer to the following problem whether you add “up” (starting with 9 8+ ) or “down” (starting with 3 8+ )?

+

3

8

9

EXERCISE/PROBLEM SET A

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Section 3.1 Addition and Subtraction 97

7. Addition can be simplified using the associative property of addition. For example,

26 57 26 4 53 26 4 53

30 53 83

+ = + + = + + = + =

( ) ( )

. Complete the following statements. a. 39 68 40+ = + = b. 25 56 30+ = + = c. 47 23 50+ = + =

8. a. Complete the following addition table in base five. Remember to use the thinking strategies.

b. For each of the following subtraction problems in base five, rewrite the problem using the missing-addend ap- proach and find the answer in the table in part a.

i. 13 4five five− ii. 11 3five five− iii. 12 4five five− iv. 10 2five five−

9. Complete the following four-fact families in base five. a. 3 4 12five five five+ = b. ____________________________________ ____________________________________ 4 1 10five five five+ = ____________________________________ ____________________________________ ____________________________________ ____________________________________ c. ____________________________________ ____________________________________ 11 4 2five five five− = ____________________________________

10. In the following figure, centimeter strips are used to illus- trate 3 8 11+ = . What two subtraction problems are also being represented?

11. For the following figures, identify the problem being illus- trated, the model, and the conceptual approach being used.

a.

b.

c.

12. Using different-shaped boxes for variables provides a transition to alge.bra as well as a means of stating prob- lems. Try some whole numbers in the boxes to determine whether these properties hold.

a. Is subtraction closed?

b. Is subtraction commutative?

c. Is subtraction associative?

d. Is there an identity element for subtraction?

13. Each situation described next involves a subtraction problem. In each case, briefly name the small cube por- tion of Figure 3.18 that correctly classifies the problem. Typical answers may be set, take-away, no comparison or measurement, missing-addend, comparison. Finally, write an equation to fit the problem.

a. An elementary teacher started the year with a budget of $200 to be spent on manipulatives. By the end of December, $120 had been spent. How much money remained in the budget?

b. Doreen planted 24 tomato plants in her garden and Justin planted 18 tomato plants in his garden. How many more plants did Doreen plant?

c. Tami is saving money for a trip to Hawaii over spring break. The package tour she is interested in costs $1795. From her part-time job she has saved $1240 so far. How much more money must she save?

14. a. State a subtraction word problem involving 8 3− , the missing-addend approach, the set model, without comparison.

b. State a subtraction word problem involving 8 3− , the take-away approach, the measurement model, with comparison.

c. Sketch the set model representation of the situation described in part a.

d. Sketch the measurement model representation of the situation described in part b.

PROBLEMS 15. A given set contains the number 1. What other numbers

must also be in the set if it is closed under addition?

16. The number 100 can be expressed using the nine digits 1 2 9, , ,. . . with plus and minus signs as follows:

1 2 3 4 5 6 78 9 100+ + − + + + + =

Find a sum of 100 using each of the nine digits and only three plus or minus signs.

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98 Chapter 3 Whole Numbers: Operations and Properties

17. Complete the following magic square in which the sum of each row, each column, and each diagonal is the same. When completed, the magic square should contain each of the numbers 10 through 25 exactly once.

18. Magic squares are not the only magic figures. The following figure is a magic hexagon. What is “magic” about it?

19. A palindrome is any number that reads the same backward and forward. For example, 262 and 37673 are palindromes. In the accompanying example, the process of reversing the

digits and adding the two numbers has been repeated until a palindrome is obtained.

67

76

143

341

484

+

+

a. Try this method with the following numbers. i. 39 ii. 87 iii. 32 b. Find a number for which the procedure takes more than

three steps to obtain a palindrome.

20. Mr. Morgan has five daughters. They were all born the number of years apart as the youngest daughter is old. The oldest daughter is 16 years older than the youngest. What are the ages of Mr. Morgan’s daughters?

21. Use the Chapter 3 eManipulative activity, Number Puzzles exercise 2, on our Web site to arrange the numbers 1, 2, 3, 4, 5, 6, 7, 8, 9 in the circles below so the sum of the num- bers along each line of four is 23.

EXERCISES 1. Show that 2 6 8+ = using two different types of models.

2. For which of the following pairs of sets is it true that n D n E n D E( ) ( ) ( )+ = ∪ ? When not true, explain why not.

a. D a c e g= { , , , }, E b d f g= { , , , } b. D = { }, E = { } c. D = { , , , }1 3 5 7 , E = { , , }2 4 6

3. Which of the following sets are closed under addition? Why or why not?

a. { , , , , }0 3 6 9 . . . b. {1} c. { , , , ,1 5 9 13 . . . d. { , , , ,8 12 16 20 . . . e. Whole numbers less than 17

4. Each of the following is an example of one of the properties for addition of whole numbers. Fill in the blank to complete the statement, and identify the property.

a. 5 5+ = b. 7 5 7+ = + c. ( ) ( )4 3 6 4 6+ + = + +

d. ( ) ( )4 3 6 4 3+ + = + + e. ( ) ( )4 3 6 3 6+ + = + + f. 2 9+ is a ______ number.

5. Show that the commutative property of whole-number addition holds for the following examples in other bases by using a different number line for each base.

a. 3 4 4 3five five five five+ = +   b. 5 7 7 5nine nine nine nine+ = +

6. Without performing the addition, determine which sum (if either) is larger. Explain how this was accomplished and what properties were used.

3261

4287

5193+

4187

5291

3263+

7. Look for easy combinations of numbers to compute the following sums mentally. Show and identify the properties you used to make the groupings.

a. 94 27 6 13+ + + b. 5 13 25 31 47+ + + +

EXERCISE/PROBLEM SET B

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Section 3.1 Addition and Subtraction 99

8. a. Complete the following addition table in base six. Remember to use the thinking strategies.

b. For each of the following subtraction problems in base six, rewrite the problem using the missing-addend approach and find the answer in the table.

i. 13 5six six− ii. 5 4six six− iii. 12 4six six− iv. 10 2six six−

9. Using the addition table for base six given in Exercise 8, write the following four-fact families in base six.

a. 2 3 5six six six+ = b. 11 5 2six six six− =

10. Rewrite each of the following subtraction problems as an addition problem.

a. x − =156 279    b. 279 156− = x    c. 279 156− =x

11. For the following figures, identify the problem being illustrated, the model, and the conceptual approach being used.

a.

b.

c.

12. For each of the following, determine whole numbers x, y, and z that make the statement true.

a. x x x− = − =0 0 b. x y y x− = − c. ( ) ( )x y z x y z− − = − −

Which, if any, are true for all whole numbers x, y, and z?

13. Each situation described next involves a subtraction problem. In each case, briefly name the small cube por- tion of Figure 3.18 that correctly classifies the problem. Typical answers may be set, take-away, no comparison or measurement, missing-addend, comparison. Finally, write an equation to fit the problem.

a. Robby has accumulated a collection of 362 sports cards. Chris has a collection of 200 cards. How many more cards than Chris does Robby have?

b. Jack is driving from St. Louis to Kansas City for a meeting, a total distance of 250 miles. After 2 hours he notices that he has traveled 114 miles. How far is he from Kansas City at that time?

c. An elementary school library consists of 1095 books. As of May 8, 105 books were checked out of the library. How many books were still available for checkout on May 8?

14. a. State a subtraction word problem involving 9 5− , the missing-addend approach, the measurement model, without comparison.

b. State a subtraction word problem involving 9 5− , the take-away approach, the set model, with comparison.

c. Sketch the measurement model representation of the situation described in part a.

d. Sketch the set model representation of the situation described in part b.

PROBLEMS 15. Suppose that S is a set of whole numbers closed under

addition. S contains 3, 27, and 72. a. List six other elements in S . b. Why must 24 be in S?

16. A given set contains the number 5. What other numbers must also be in the set if it is closed under addition?

17. The next figure can provide practice in addition and sub- traction. The figure is completed by filling the upper two circles with the sums obtained by adding diagonally.

9 4 13 8 3 11+ = + =and

The circle at the lower right is filled in one of two ways:

i. Adding the numbers in the upper circles:

13 11 24+ =

ii. Adding across the rows, adding down the columns, and then adding the results in each case:

12 12 24 17 7 24+ = + =and

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100 Chapter 3 Whole Numbers: Operations and Properties

a. Fill in the missing numbers for the following figure.

b. Fill in the missing numbers for the following figure.

c. Use variables to show why the sum of the numbers in the upper two circles will always be the same as the sum of the two rows and the sum of the two columns.

18. Arrange numbers 1 to 10 around the outside of the circle shown so that the sum of any two adjacent numbers is the same as the sum of the two numbers on the other ends of the spokes. As an example, 6 and 9, 8 and 7 might be placed as shown, since 6 9 8 7+ = + .

19. Place the numbers 1 16− in the cells of the following magic square so that the sum of each row, column, and diagonal is the same.

20. Use the Chapter 3 eManipulative activity, Number Puzzles exercise 3, on our Web site to arrange the numbers 1, 2, 3, 4, 5, 6, 7, 8, 9 in the circles below so the sum of the num- bers along each line of three is 15.

21. Shown here is a magic triangle discovered by the mental calculator Marathe. What is its magic?

22. The property “If a c b c+ = + , then a b= ” is called the additive cancellation property. Is this property true for all whole numbers? If it is, how would you convince your students that it is always true? If not, give a counter example.

Analyzing Student Thinking

23. Kaitlyn doesn’t understand why the definition of Addition of Whole Numbers insists that sets A and B must be dis- joint. What can you say to help her?

24. Enrique says, “I think of closure as a bunch of numbers locked in a room and the operation, like addition, comes along and links any two of the numbers together. As long as the link is in the room, the room can be said to be closed with respect to addition. If the link is found outside of the room, then the set is not closed.” Is his reasoning math- ematically correct? Explain.

25. Monique prefers using rote memory to learn the addition facts rather than using thinking strategies. What case can be made to convince her that there is value in using think- ing strategies? Explain.

26. Theresa prefers using the number line for addition rather than the set model. Can she use the number line model to interpret the properties of whole-number addition? Explain.

27. Severino says, “When I added 27 and 36, I made the 27 a 30, then I added the 30 and the 36 and got 66, and then subtracted the 3 from the beginning and got 63.” Patricia says, “But when I added 27 and 36, I added the 20 and the 30 and got 50, then I knew 6 plus 6 was 12 so 6 plus 7 was 13, and then I added 50 and 13 and got 63.” Can you follow the students’ reasonings here? How would you describe some of their techniques?

28. Kayla claims that 7 3 10eight eight eight+ = since 7 3 10+ = . How should you respond?

29. Your classroom has 21 students and there are 9 boys. You ask two students to use these numbers to determine how many girls are in the classroom. Conner says, “9 plus 10 is 19, 19 plus 2 is 21, so there are 10 2 12+ = girls.” Chandler says that Conner is wrong because he did not use “take-away.” How should you respond?

30. Darren happens to see your copy of this book on your desk open to Figure 3.18. He says, “What is that used for?” How should you respond?

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Section 3.2 Multiplication and Division 101

Multiplication and Its Properties There are many ways to view multiplication.

Repeated-Addition Approach Consider the following problems: There are five children, and each has three silver dollars. How many silver dollars do they have altogether? The silver dollars are about 1 inch wide. If the silver dollars are laid in a single row with each dollar touching the next, what is the length of the row? These problems can be modeled using the set model and the measurement model (Figure 3.20).

Figure 3.20

These models look similar to the ones that we used for addition, since we are merely adding repeatedly. They show that 3 3 3 3 3 15+ + + + = , or that 5 3 15× = .

Children’s Literature www.wiley.com/college/musser See “Amanda Bean’s Amazing Dream: A Mathematical Story” by Cindy Neuschwander.

Common Core – Grade 3 Interpret products of whole numbers (e.g., 5 7× ) as the total number of objects in 5 groups of 7 objects each.

Reflection from Research When multiplication is repre- sented by repeated addition, students have a great deal of difficulty keeping track of the two sets of numbers. For instance, when considering how many sets of three there are in fifteen, students need to keep track of counting up the threes and how many sets of three they count (Steffe, 1988).

Write a word problem that would have 3 4× as its solution. Compare your problem with those of several peers and determine how these problems are different and how

they are the same from a mathematical perspective. If possible categorize the problems according to the underlying mathematical structure.

MULTIPLICATION AND DIVISION

Multiplication of Whole Numbers: Repeated-Addition Approach

Let a and b be any whole numbers where a ≠ 0. Then

ab b b b a

= + + ⋅ ⋅ ⋅ + addends

� ��� ���

If then also for alla ab b b b b= = = =1 1 0 0, ; .¥ ¥

D E F I N I T I O N 3 . 4 Children’s Literature www.wiley.com/college/musser See “One Hundred Hungry Ants” by Elinor Pinczes.

Since multiplication combines two numbers to form a single number, it is a binary operation. The number ab, read “a times b,” is called the product of a and b. The numbers a and b are called factors of ab. The product ab can also be written as “a b¥ ” and “a bë .” Notice that 0 0¥ b = for all b. That is, the product of zero and any whole number is zero.

Rectangular Array Approach If the silver dollars in the preceding problem are arranged in a rectangular array, multiplication can be viewed in a slightly different way (Figure 3.21).Figure 3.21

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102 Chapter 3 Whole Numbers: Operations and Properties

Cartesian Product Approach A third way of viewing multiplication is an abstraction of this array approach.

Multiplication of Whole Numbers: Rectangular Array Approach

Let a and b be any whole numbers. Then ab is the number of elements in a rectan- gular array having a rows and b columns.

Common Core – Grade 3 Use multiplication and division within 100 to solve word prob- lems in situations involving equal groups, arrays, and measurement quantities (e.g., by using draw- ings and equations with a symbol for the unknown number to rep- resent the problem).

A L T E R N A T I V E D E F I N I T I O N 3 . 5

Multiplication of Whole Numbers: Cartesian Product Approach

Let a and b be any whole numbers. If a n A= ( ) and b n B= ( ), then

ab n A B= ×( ).

A L T E R N A T I V E D E F I N I T I O N 3 . 6

For example, to compute 2 3¥ , let 2 = n a b({ , }) and 3 = n x y z({ , , }). Then 2 3¥ is the number of ordered pairs in { , } { , , }a b x y z× . Because { , } { , , } {( , ),a b x y z a x× = ( , ), ( , ), ( , ), ( , ), ( , )}a y a z b x b y b z has six ordered pairs, we conclude that 2 3 6¥ = . Actually, by arranging the pairs in an appropriate row and column configuration, this approach can also be viewed as the array approach (Figure 3.22).

Tree Diagram Approach Another way of modeling this approach is through the use of a tree diagram (Figure 3.23). Tree diagrams are especially useful in the field of probability, which we study in Chapter 11.

Properties of Whole-Number Multiplication You have probably observed that whenever you multiplied any two whole numbers, your product was always a whole number. This fact is summarized by the following property.

Figure 3.22

Figure 3.23

Reflection from Research Children can solve a variety of multiplicative problems long before being formally introduced to multiplication and division (Mulligan & Mitchelmore, 1995).

Closure Property for Multiplication of Whole Numbers

The product of two whole numbers is a whole number.

P R O P E R T Y

When two odd whole numbers are multiplied together, the product is odd; thus the set of odd numbers is closed under multiplication. Closure is a useful idea, since if we are multiplying two (or more) odd numbers and the product we calculate is even, we can conclude that our product is incorrect. The set { , , , , , }2 5 8 11 14 . . . is not closed under multiplication, since 2 5 10¥ = and 10 is not in the set.

The next property can be used to simplify learning the basic multiplication facts. For example, by the repeated-addition approach, 7 2× represents 2 2 2 2 2 2 2+ + + + + + , whereas 2 7× means 7 7+ . Since 7 7+ was learned as an addition fact, viewing 7 2× as 2 7× makes this computation easier.

Commutative Property for Whole-Number Multiplication

Let a and b be any whole numbers. Then

ab ba= .

P R O P E R T YCommon Core – Grade 3 Apply properties of operations as strategies to multiply and divide (e.g., if 6 4 24× = is known, then 4 6 24× = is also known).

c03.indd 102 7/30/2013 2:49:22 PM

103

From Lesson 4-1 “Multiplication as Repeated Addition” from Envision Math Common Core, by Randall I. Charles et al., Grade 3, copyright © by Pearson Education.

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104 Chapter 3 Whole Numbers: Operations and Properties

The example in Figure 3.24 should convince you that the commutative property for multiplication is true.

Figure 3.24

The product 5 2 13¥ ¥( ) is more easily found if it is viewed as ( )5 2 13¥ ¥ . Regrouping to put the 5 and 2 together can be done because of the next property.

Reflection from Research If multiplication is viewed as computing area, children can see the commutative property rela- tively easily, but if multiplication is viewed as computing the price of a number of items, the com- mutative property is not obvious (Vergnaud, 1981).

Problem-Solving Strategy Draw a Picture

Associative Property for Whole-Number Multiplication

Let a, b, and c be any whole numbers. Then

a bc ab c( ) ( ) .=

P R O P E R T Y

To illustrate the validity of the associative property for multiplication, consider the three-dimensional models in Figure 3.25.

Figure 3.25

The next property is an immediate consequence of each of our defi nitions of multiplication.

Common Core – Grade 3 Apply properties of operations as strategies to multiply and divide (e.g., 3 5 2× × can be found by 3 5 15× = , then 15 2 30× = or by 5 2 10× = , then 3 10 30× = ).

Problem-Solving Strategy Draw a Picture

Identity Property for Whole-Number Multiplication

The number 1 is the unique whole number such that for every whole number a,

a a a¥ ¥1 1= = .

P R O P E R T Y

Because of this property, the number 1 is called the multiplicative identity or the identity for multiplication.

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Section 3.2 Multiplication and Division 105

Let’s summarize the properties of whole-number addition and multiplication.

Distributive Property of Multiplication over Addition

Let a, b, and c be any whole numbers. Then

a b c ab ac( )+ = + .

NCTM Standard All students should understand and use properties of operations such as the distributivity of multi- plication over addition.

P R O P E R T Y

There is one other important property of the whole numbers. This property, distributivity, combines both multiplication and addition. Study the array model in Figure 3.26. This model shows that the product of a sum, 3 2 4( )+ , can be expressed as the sum of products, ( ) ( )3 2 3 4¥ ¥+ . This relationship holds in general.

Figure 3.26

Common Core – Grade 3 Apply properties of operations as strategies to multiply and divide (e.g., knowing that 8 5 40× = and 8 2 16× = , one can find 8 7× as 8 5 2 8 5 8 2 40 16 56

× + × × + = ( ) ( ) ( )

). = + =

Because of commutativity, we can also write ( )b c a ba ca+ = + . Notice that the distributive property “distributes” the a to the b and the c.

Algebraic Reasoning The distributive property is commonly used to “combine like terms” in problems like 3 2 3 2 5x x x x+ = + =( ) . Becoming comfortable with the distributive property with numbers develops a foundation for applying the property in algebra.

Rewrite each of the following expressions using the distributive property.

a. 3 4 5( )+ b. 5 7 5 3¥ + ¥ c. am an+ d. 31 76 29 76¥ ¥+ e. a b c d( )+ +

S O L U T I O N

a. 3 4 5 3 4 3 5( )+ = +¥ ¥ b. 5 7 5 3 5 7 3¥ ¥+ = +( ) c. am an a m n+ = +( ) d. 31 76 29 76 31 29 76¥ ¥+ = +( ) e. a b c d a b c d a b c ad ab ac ad( ) (( ) ) ( )+ + = + + = + + = + + ■

Whole-Number Properties

PROPERTY ADDITION MULTIPLICATION

Closure Yes Yes

Commutativity Yes Yes

Associativity Yes Yes

Identity Yes (zero) Yes (one)

Distributivity of multiplication over addition

Yes

P R O P E R T Y

In addition to these properties, we highlight the following property.

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106 Chapter 3 Whole Numbers: Operations and Properties

The following discussion shows how the properties are used to develop thinking strategies for learning the multiplication facts.

Thinking Strategies for Learning the Multiplication Facts The multipli- cation table in Figure 3.27 has 10 10 100× = unfilled spaces.

Figure 3.27 Figure 3.28 Figure 3.29

1. Commutativity: As in the addition table, because of commutativity, only 55 facts in the unshaded region in Figure 3.28 have to be found.

2. Multiplication by 0: a ¥ 0 0= for all whole numbers a. Thus the first column is all zeros (Figure 3.29).

3. Multiplication by 1: 1 1¥ ¥a a a= = . Thus the column labeled “1” is the same as the left-hand column outside the table (Figure 3.29).

Reflection from Research Rote memorization does not indicate mastery of facts. Instead, mastery of multiplication facts is indicated by a conceptual understanding and computational fluency (Wallace & Gurganus, 2005).

Using the missing-addend approach to subtraction, we will show that a b c( )− = ab ac− whenever b c− is a whole number. In words, multiplication distributes over subtraction.

Let

Then

Therefore

B

b c n

b c n

ab a c n

ab ac an

ab ac an

− = = + = + = +

− =

( )

,

uut b c n− =

Missing addend

Multipliction

Distributivity

Missing addend from the fi rst equation

So, substituting b c− for n, we have

ab ac a b c− = −( ).

Multiplication Property of Zero

For every whole number, a,

a a¥ ¥0 0 0= = .

P R O P E R T Y

Distributive Property of Multiplication over Subtraction

Let a, b, and c be any whole numbers where b c≥ . Then

a b c ab ac( ) .− = −

P R O P E R T Y

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Section 3.2 Multiplication and Division 107

4. Multiplication by 2: 2 ¥ a a a= + , which are the doubles from addition (Figure 3.29).

We have filled in 27 facts using thinking strategies 1, 2, 3, and 4. Therefore, 28 facts remain to be found out of our original 55.

5. Multiplication by 5: The counting chant by fives, namely 5, 10, 15, 20, and so on, can be used to learn these facts (see the column and/or row headed by a 5 in Figure 3.30).

6. Multiplication by 9: The multiples of 9 are 9, 18, 27, 36, 45, 54, 63, 72, and 81 (Figure 3.30). Notice how the tens digit is one less than the number we are multiply- ing by 9. For example, the tens digit of 3 9¥ is 2 (one less than 3). Also, the sum of the digits of the multiples of 9 is 9. Thus 3 9 27¥ = since 2 7 9+ = . The multiples of 5 and 9 eliminate 13 more facts, so 15 remain.

7. Associativity and distributivity: The remaining facts can be obtained using these two properties. For example, 8 4 8 2 2 8 2 2 16 2 32× = × × = × × = × =( ) ( ) or 8 4 8 2 2 8 2 8 2 16 16 32× = + = + = + =( ) ¥ ¥ .

In the next example we consider how knowledge of the basic facts and the proper- ties can be applied to multiplying a single-digit number by a multidigit number.

Figure 3.30

NCTM Standard All students should develop fluency with basic number com- binations for multiplication and division and use combinations to mentally compute related prob- lems, such as 30 50× .

Compute the following products using thinking strategies.

a. 2 34× b. 5 37 2( )¥ c. 7 25( )

S O L U T I O N 

a. 2 34 2 30 4 2 30 2 4 60 8 68× = + = + = + =( ) ¥ ¥ b. 5 37 2 5 2 37 5 2 37 370( ) ( ) ( )¥ ¥ ¥ ¥= = = c. 7 25 4 3 25 4 25 3 25 100 75 175( ) ( )= + = + = + =¥ ¥ ■

Check for Understanding: Exercise/Problem Set A #1–10✔

Write a word problem that would have 12 3÷ as its solution. There are some general types of division problems. Compare the underlying mathematical structure of your problem

with that of your peers and try to identify and describe some different types of division problems.

Division Just as with addition, subtraction, and multiplication, we can view division in differ- ent ways. Consider these two problems.

1. A class of 20 children is to be divided into four teams with the same number of children on each team. How many children are on each team?

2. A class of 20 children is to be divided into teams of four children each. How many teams are there?

Each of these problems is based on a different conceptual way of viewing division. A general description of the first problem is that you have a certain number of objects that you are dividing into or “sharing” among a specified number of groups and are asking how many objects are in each group. Because of its sharing nature, this type of division is referred to as sharing division. A general description of the second problem is that you have a certain number of objects and you are “measuring out” a specified number of objects to be in each group and asking how many groups there are. This type of division is called measurement division (Figure 3.31).

Reflection from Research Multiplication is usually intro- duced before division and sepa- rated from it, whereas children spontaneously relate them and do not necessarily find division more difficult than multiplication (Mulligan & Mitchelmore, 1997).

NCTM Standard All students should understand various meanings of multiplication and division.

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108 Chapter 3 Whole Numbers: Operations and Properties

Missing-Factor Approach Figure 3.32 shows that multiplication and division are related. This suggests the following definition of division.

Figure 3.32

When dealing with whole numbers, the difference between these two types of division may seem very subtle, but these differences become more apparent when considering the division of decimals or fractions.

Sharing

Figure 3.31

The following examples will help clarify the distinction between sharing and mea- surement division.

Common Core – Grade 3 Interpret whole-number quotients of whole numbers (e.g., 56 8÷ ) as the number of objects in each share when 56 objects are par- titioned equally into 8 shares or as a number of shares when 56 objects are partitioned into equal shares of 8 objects each.

Classify each of the following division problems as examples of either sharing or measurement division.

a. A certain airplane climbs at a rate of 300 feet per second. At this rate, how long will it take the plane to reach a cruising altitude of 27,000 feet?

b. A group of 15 friends pooled equal amounts of money to buy lottery tickets for a $1,987,005 jackpot. If they win, how much should each friend receive?

c. Shauna baked 54 cookies to give to her friends. She wants to give each friend a plate with 6 cookies on it. How many friends can she give cookies to?

S O L U T I O N 

a. Since every 300 feet can be viewed as a single group corresponding to 1 second, we are interested in fi nding out how many groups of 300 feet there are in 27,000 feet. Thus this is a measurement division problem.

b. In this case, each friend represents a group and we are interested in how much money goes to each group. Therefore, this is an example of a sharing division problem.

c. Since every group of cookies needs to be of size 6, we need to determine how many groups of size 6 there are in 54 cookies. This is an example of a measurement divi- sion problem. ■

Reflection from Research Allowing students to write their own division problems helps students to think about and apply their understanding of division to real-world contexts (Polly & Ruble, 2009).

Problem-Solving Strategy Draw a Picture

NCTM Standard All students should understand situations that entail multiplica- tion and division, such as equal groupings of objects and sharing equally.

Division of Whole Numbers: Missing-Factor Approach

If a and b are any whole numbers with b ≠ 0, then a b c÷ = if and only if a bc= for some whole number c.

D E F I N I T I O N 3 . 7

The symbol a ï b is read “a divided by b.” Also, a is called the dividend, b is called the divisor, and c is called the quotient or missing factor. The basic facts multiplication table can be used to learn division facts (Figure 3.32).

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Section 3.2 Multiplication and Division 109

The division problem in Example 3.8(d) can be generalized as follows and verified using the missing-factor approach.

Reflection from Research Fourth- and fifth-grade students most frequently defined division as undoing multiplication (Graeber & Tirosh, 1988).

Find the following quotients.

a. 24 8÷ b. 72 9÷ c. 52 4÷ d. 0 7÷

S O L U T I O N 

a. 24 8 3÷ = , since 24 8 3= × b. 72 9 8÷ = , since 72 9 8= × c. 52 4 13÷ = , since 52 13 4= × d. 0 7 0÷ = , since 0 7 0= × ■

Next, consider the situation of dividing by zero. Suppose that we extend the missing-factor approach of division to dividing by zero. Then we have the following two cases.

Case 1: a ÷ 0, where a ≠ 0. If a c÷ =0 , then a c= 0 ¥ , or a = 0. But a ≠ 0. Therefore, a ÷ 0 is undefi ned.

Case 2: 0 0÷ . If 0 0÷ = c, then 0 0= ¥ c. But any value can be selected for c, so there is no unique quotient c. Thus division by zero is said to be indeterminate, or unde- fi ned, here. These two cases are summarized by the following statement.

Algebraic Reasoning Notice how variables are used to represent any number in order to show that this explanation works for any number.

Division Property of Zero

If a ≠ 0, then 0 0÷ =a .

P R O P E R T Y

Division by 0 is undefi ned.

D I V I S I O N B Y Z E R O

If a and b are any whole numbers with b ≠ 0, then there exist unique whole num- bers q and r such that a bq r= + , where 0 ≤ <r b.

T H E D I V I S I O N A L G O R I T H M

Now consider the problem 37 4÷ . Although 37 4÷ does not have a whole-number answer, there are applications where it is of interest to know how many groups of 4 are in 37 with the possibility that there is something left over. For example, if there are 37 fruit slices to be divided among four children so that each child gets the same number of slices, how many would each child get? We can find as many as 9 fours in 37 and then have 1 remaining. Thus each child would get nine fruit slices with one left undistributed. This way of looking at division of whole numbers, with a remainder, is summarized next.

Reflection from Research Second- and third-grade students tend to use repeated addition to solve simple multiplication AND division problems. In a division problem, such as 15 5÷ , they will repeatedly add the divisor until they reach the quotient ( ; )5 5 10 10 5 15+ = + = , often using their fingers to keep track of the number of times they use 5 (Mulligan & Mitchelmore, 1995).

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110 Chapter 3 Whole Numbers: Operations and Properties

Find the quotient and remainder for these problems.

a. 57 9÷ b. 44 13÷ c. 96 8÷

S O L U T I O N 

a. 9 6 54× = , so 57 9 6 3= +¥ . The quotient is 6 and the remainder is 3 (Figure 3.33).

Figure 3.33

b. 13 3 39× = , so 44 13 3 5= +¥ . The quotient is 3 and the remainder is 5. c. 8 12 96× = , so 96 8 12 0= +¥ . The quotient is 12 and the remainder is 0. ■

Problem-Solving Strategy Draw a Picture

Repeated-Subtraction Approach Figure 3.34 suggests alternative ways of viewing division.

Figure 3.34

In Figure 3.34, 13 was subtracted from 44 three successive times until a number less than 13 was reached, namely 5. Thus 44 divided by 13 has a quotient of 3 and a remainder of 5. This example shows that division can be viewed as repeated subtrac- tion. In general, to find a b÷ using the repeated-subtraction approach, subtract b successively from a and from the resulting differences until a remainder r is reached, where r b< . The number of times b is subtracted is the quotient q.

Figure 3.35 provides a visual way to remember the main interconnections among the four basic whole-number operations. For example, multiplication of whole num- bers is defined by using the repeated-addition approach, subtraction is defined using the missing-addend approach, and so on. An important message in this diagram is that success in subtraction, multiplication, and division begins with a solid founda- tion in addition.

Reflection from Research The Dutch approach to written division calculations involves repeated subtraction using increasingly larger chunks. This approach, which has helped Dutch students to outperform students from other nations, builds progressively on intuitive strategies (Anghileri, Beishuizen, & Van Putten, 2002).

Figure 3.35

Check for Understanding: Exercise/Problem Set A #11–17✔

Here b is called the divisor, q is called the quotient, and r is the remainder. Notice that the remainder is always less than the divisor. Also, when the remainder is 0, this result coincides with the usual defi nition of whole-number division.

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Section 3.2 Multiplication and Division 111

The following note appeared in a newspaper.

“What is 241,573,142,393,627,673,576,957,439,048 times 45,994, 811,347,886,846,310,221,728,895,223,034,301,839? The answer is 71 consecutive 1s—one of the biggest numbers a computer has ever factored. The 71-digit number was factored in 9.5 hours of a Cray super-computer’s time at Los Alamos National Laboratory in New Mexico, besting the previous high—69 digits—by two.

Why bother? The feat might affect national security. Some com- puter systems are guarded by cryptographic codes once thought to be beyond factoring. The work at Los Alamos could help intelligence experts break codes.”

See whether you can find an error in the article and correct it.

© R

on B

ag w

el l

EXERCISES 1. What multiplication problems are suggested by the follow-

ing diagrams? a.

b. c.

d.

5

5 5 5 5 5 5

10 15 20 25 300

2. Illustrate 3 2× using the following combinations of models and approaches.

a. Set model; Cartesian product approach b. Set model; rectangular array approach c. Set model; repeated-addition approach d. Measurement model; rectangular array approach e. Measurement model; repeated-addition approach

3. Each situation described next involves a multiplication problem. In each case tell whether the problem situation is best represented by the repeated-addition approach, the rectangular array approach, or the Cartesian product approach, and why. Then write an appropriate equation to fit the situation.

a. A rectangular room has square tiles on the floor. Along one wall, Kurt counts 15 tiles and along an adjacent wall he counts 12 tiles. How many tiles cover the floor of the room?

b. Jack has three pairs of athletic shorts and eight different T-shirts. How many different combinations of shorts and T-shirts could he wear to play basketball?

c. A teacher provided three number-2 pencils to each stu- dent taking a standardized test. If a total of 36 students were taking the test, how many pencils did the teacher need to have available?

4. The repeated-addition approach can easily be illustrated using the calculator. For example, 4 3× can be found by pressing the following keys:

3 3 3 3 12+ + + =

or if the calculator has a constant key, by pressing

3 12

3 12

+ = = =

+ + = = =

or

Find the following products using one of these techniques. a. 3 12× b. 4 17× c. 7 93× d. 143 6× (Think!)

5. Which of the following sets are closed under multiplication? If the set is not closed, explain why not.

a. { , }2 4 b. { , , , , }0 2 4 6 . . . c. { , }0 3 d. { , }0 1 e. { }1 f. { }0 g. { , , , }5 7 9 . . . h. { , , , , }0 7 14 21 . . . i. { , , , , , , , }0 1 2 4 8 16 2. . . , . . .k

j. Odd whole numbers

6. Identify the property of whole numbers being illustrated. a. 4 5 5 4¥ ¥= b. 6 3 2 3 2 6( ) ( )+ = + c. 5 2 9 5 2 5 9( )+ = +¥ ¥ d. 1( )x y x y+ = +

EXERCISE/PROBLEM SET A

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112 Chapter 3 Whole Numbers: Operations and Properties

7. Rewrite each of the following expressions using the dis- tributive property for multiplication over addition or for multiplication over subtraction.

a. 4 60 37( )+ b. ( )21 35 6+ ¥ c. 37 60 22¥ ( )− d. 5 2x x+ e. 5 1 3 1( ) ( )a a+ − +

8. The distributive property of multiplication over addition can be used to perform some calculations mentally. For example, to find 13 î 12, you can think

13 12 13 10 2 13 10 13 2

130 26 156

( ) ( ) ( ) ( )

.

= + = + = + =

How could each of the following products be rewritten using the distributive property so that it is more easily computed mentally?

a. 45(11) b. 39(102) c. 23(21) d. 97(101)

9. a. Compute 374 12× without using the 4 key. Explain. b. Compute 374 12× without using the 2 key. Explain.

10. Compute using thinking strategies. Show your reasoning. a. 5 23 4( )× b. 12 25×

11. a. Complete the following multiplication table in base five.

b. Using the table in part a, complete the following four- fact families in base five.

i. 2 3 11five five five× = ii. ___________________ ___________________ ___________________ ___________________ 22 3 4five five five÷ = ___________________ ___________________

iii. ___________________ ___________________ ___________________ 13 2 4five five five÷ =

12. Identify each of the following problems as an example of either sharing or measurement division. Justify your answers.

a. Gabriel bought 15 pints of paint to redo all the doors in his house. If each door requires 3 pints of paint, how many doors can Gabriel paint?

b. Hideko cooked 12 tarts for her family of 4. If all of the family members receive the same amount, how many tarts will each person receive?

c. Ms. Ivanovich needs to give 3 straws to each student in her class for their art activity. If she uses 51 straws, how many students does she have in her class?

13. a. Write a sharing division problem that would have the equation 15 3÷ as part of its solution.

b. Write a measurement division problem that would have the equation 15 3÷ as part of its solution.

14. Rewrite each of the following division problems as a mul- tiplication problem.

a. 48 6 8÷ = b. 51 3÷ =x c. x ÷ =13 5

15. Find the quotient and remainder for the following. a. 53 8÷ b. 72 15÷ c. 137 6÷

16. Use the Chapter 3 eManipulative, Rectangular Division, on our Web site to investigate the three problems from Example 3.9. Explain why the remainder is always smaller than the divisor in terms of this rectangle division model.

17. In general, each of the following is false if x, y, and z are whole numbers. Give an example (other than dividing by zero) where each statement is false.

a. x y÷ is a whole number. b. x y y x÷ = ÷ c. ( ) ( )x y z x y z÷ ÷ = ÷ ÷ d. x y x y x y÷ = = ÷ for some e. x y z x y x z÷ + = ÷ + ÷( )

PROBLEMS 18. A square dancing contest has 213 teams of 4 pairs each.

How many dancers are participating in the contest?

19. A stamp machine dispenses twelve 32¢ stamps. What is the total cost of the twelve stamps?

20. If the American dollar is worth 121 Japanese yen, how many dollars can 300 yen buy?

21. An estate valued at $270,000 was left to be split equally among three heirs. How much did each one get (before taxes)?

22. Shirley meant to add 12349   +   29746 on her calculator. After entering 12349, she pushed the × button by mistake. What could she do next to keep from reentering 12349? What property are you using?

23. Suppose that A is a set of whole numbers closed under addition. Is A necessarily closed under multiplication? (If you think so, give reasons. If you think not, give a counter example, that is, a set A that is closed under addition but not multiplication.)

24. a. Use the numbers from 1 to 9 once each to complete this magic square. (The row, column, and diagonal sums are all equal.) (Hint: First determine the sum of each row.)

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Section 3.2 Multiplication and Division 113

b. Can you make a multiplicative magic square? (The row, column, and diagonal products are equal.) (Note: The numbers 1 through 9 will not work in this case.)

25. Predict the next three lines in this pattern, and check your work.

1

3 5

7 9 11

13 15 17 19

21 23 25 27 29

1

8

27

64

125

+ + +

+ + + + + + +

= = = = =

26. Using the digits 1 through 9 once each, fill in the boxes to make the equations true.

27. Take any number. Add 10, multiply by 2, add 100, divide by 2, and subtract the original number. The answer will be the number of minutes in an hour. Why?

28. Jason, Wendy, Kevin, and Michelle each entered a frog in an annual frog-jumping contest. Each of their frogs—Hippy, Hoppy, Bounce, and Pounce—placed first, second, or third in the contest and earned a blue, red, or white ribbon, respectively. Use the following clues to determine who entered which frog and the order in which the frogs placed.

a. Michelle’s frog finished ahead of both Bounce and Hoppy. b. Hippy and Hoppy tied for second place. c. Kevin and Wendy recaptured Hoppy when he escaped

from his owner. d. Kevin admired the blue ribbon Pounce received but was

quite happy with the red ribbon his frog received.

29. A café sold tea at 30 cents a cup and cakes at 50 cents each. Everyone in a group had the same number of cakes and the same number of cups of tea. (NOTE: This is not to say that the number of cakes is the same as the number of teas.) The bill came to $13.30. How many cups of tea did each have?

30. A creature from Mars lands on Earth. It reproduces itself by dividing into three new creatures each day. How many creatures will populate Earth on day 30 if there is one creature on the first day?

31. There are eight coins and a balance scale. The coins are alike in appearance, but one of them is counterfeit and lighter than the other seven. Find the counterfeit coin using two weighings on the balance scale.

32. Determine whether the property “If ac bc= , then a b= ” is true for all whole numbers. If not, give a counter-example. (Note: This property is called the multiplicative cancellation property when c ≠ 0.)

EXERCISES 1. What multiplication problems are suggested by the follow-

ing diagrams?

a. c.

b.

2. Illustrate 4 6× using the following combinations of models and approaches.

a. Set model; rectangular array approach b. Measurement model; rectangular array approach c. Set model; repeated-addition approach d. Measurement model; repeated-addition approach e. Set model; Cartesian product approach

3. Each situation described next involves a multiplication problem. In each case state whether the problem situation

is best represented by the repeated-addition approach, the rectangular array approach, or the Cartesian product approach, and why. Then write an appropriate equation to fit the situation.

a. At the student snack bar, three sizes of beverages are available: small, medium, and large. Five varieties of soft drinks are available: cola, diet cola, lemon-lime, root beer, and orange. How many different choices of soft drink does a student have, including the size that may be selected?

b. At graduation students file into the auditorium four abreast. A parent seated near the door counts 72 rows of students who pass him. How many students participated in the graduation exercise?

c. Kirsten was in charge of the food for an all-school picnic. At the grocery store she purchased 25 eight-packs of hot dog buns for 70 cents each. How much did she spend on the hot dog buns?

4. Use a calculator to find the following without using an X key. Explain your method.

a. 4 39× b. 231 3× c. 5 172× d. 6 843×

EXERCISE/PROBLEM SET B

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114 Chapter 3 Whole Numbers: Operations and Properties

5. a. Is the set of whole numbers with 3 removed i. closed under addition? Why? ii. closed under multiplication? Why? b. Answer the same questions for the set of whole numbers

with 7 removed.

6. Identify the property of whole-number multiplication being illustrated.

a. 3 5 2 3 5 3 2( )− = −¥ ¥ b. 6 7 2 6 7 2( ) ( )¥ ¥ ¥= c. ( )4 7 0 0+ =¥ d. ( )5 6 3 5 3 6 3+ = +¥ ¥ ¥

7. Rewrite each of the following expressions using the dis- tributive property for multiplication over addition or for multiplication over subtraction.

a. 3 29 30 6( )+ + b. 5 2( )x y− c. 3 6 4a a a+ − d. x x x( ) ( )+ + +2 3 2 e. 37 60 22( )−

8. The distributive property of multiplication over subtrac- tion can be used to perform some calculations mentally. For example, to find 7(99), you can think

7 99 7 100 1 7 100 7 1

700 7 693

( ) ( ) ( ) ( )

.

= − = − = − =

How could each of the following products be rewritten using the distributive property so that it is more easily computed mentally?

a. 14(19) b. 25(38) c. 35(98) d. 27(999)

9. a. Compute 463 17× on your calculator without using the 7 key.

b. Find another way to do it. c. Calculate 473 17× without using the 7 key.

10. Compute using thinking strategies. Show your reasoning. a. 8 85× b. 12(125)

11. a. Complete the following multiplication table in base eight. Remember to use the thinking strategies.

b. Rewrite each of the following division problems in base eight using the missing-factor approach, and find the answer in the table.

i. 61 7eight eight÷ ii. 17 3eight eight÷ iii. 30 6eight eight÷ iv. 16 2eight eight÷ v. 44 6eight eight÷ vi. 25 7eight eight÷

12. Identify each of the following problems as an example of either sharing or measurement division. Justify your answers.

a. For Amberly’s birthday, her mother brought 60 cupcakes to her mathematics class. There were 28 students in class that day. If she gives each student the same number of cupcakes, how many will each receive?

b. Tasha needs 2 cups of flour to make a batch of cookies. If she has 6 cups of flour, how many batches of cookies can she make?

c. Maria spent $45 on three shirts at the store. If the shirts all cost the same, how much did each shirt cost?

13. a. Write a measurement division problem that would have the equation 91 7÷ as part of its solution.

b. Write a sharing division problem that would have the equation 91 7÷ as part of its solution.

14. Rewrite each of the following division problems as a mul- tiplication problem.

a. 24 12÷ =x   b. x ÷ =3 27   c. a b x÷ =

15. Find the quotient and remainder for each problem. a. 7 3÷ b. 3 7÷ c. 7 1÷ d. 1 7÷ e. 15 5÷ f. 8 12÷

16. How many possible remainders (including zero) are there when dividing by the following numbers? How many pos- sible quotients are there?

a. 2 b. 12 c. 62 d. 23

17. Which of the following properties hold for division of whole numbers?

a. Closure b. Commutativity c. Associativity d. Identity

PROBLEMS 18. A school has 432 students and 9 grades. What is the aver-

age number of students per grade?

19. Twelve thousand six hundred people attended a golf tournament. If attendees paid $30 apiece and were dis- tributed equally among the 18 holes, how much revenue is collected per hole?

20. Compute mentally.

( , , ) ( , , )2348 7 653 214 7652 7 653 214× + ×

(Hint: Use distributivity.)

21. If a subset of the whole numbers is closed under multipli- cation, is it necessarily closed under addition? Discuss.

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Section 3.2 Multiplication and Division 115

22. Is there a subset of the whole numbers with more than one element that is closed under division? Discuss.

23. Complete the pattern and give a justification for your answers. If necessary, check your answers using your calculator.

12 345 679 9 111 111 111, , , ,× =

12 345 679 18 222 222 222, , , ,× =

12 345 679 27, , × = 12 345 679 63, , × = 12 345 679 81, , × =

24. Solve this problem posed by this Old English children’s rhyme.

As I was going to St. Ives I met a man with seven wives; Every wife had seven sacks; Every sack had seven cats; Every cat had seven kits. Kits, cats, sacks, and wives. How many were going to St. Ives?

How many wives, sacks, cats, and kits were met?

25. Write down your favorite three-digit number twice to form a six-digit number (e.g., 587,587). Is your six-digit number divisible by 7? How about 11? How about 13? Does this always work? Why? (Hint: Expanded form.)

26. Find a four-digit whole number equal to the cube of the sum of its digits.

27. Delete every third counting number starting with 3.

1 2 4 5 7 8 10 11 13 14 16 17, , , , , , , , , , ,

Write down the cumulative sums starting with 1.

1 3 7 12 19 27 37 48 61 75 91 108, , , , , , , , , , ,

Delete every second number from this last sequence, start- ing with 3. Then write down the sequence of cumulative sums. Describe the resulting sequence.

28. Write, side by side, the numeral 1 an even number of times. Take away from the number thus formed the num- ber obtained by writing, side by side, a series of 2s half the length of the first number. For example,

1111 22 1089 33 33− = = × .

Will you always get a perfect square? Why or why not?

29. Four men, one of whom committed a crime, said the following:

Bob: Charlie did it. Charlie: Eric did it. Dave: I didn’t do it. Eric: Charlie lied when he said I did it.

a. If only one of the statements is true, who was guilty? b. If only one of the statement is false, who was guilty?

30. Andrew and Bert met on the street and had the following conversation:

A: How old are your three children? B: The product of their ages is 36. A: That’s not enough information for me to know

their ages. B: The sum of their ages is your house number. A: That’s still not quite enough information. B: The oldest child plays the piano. A: Now I know!

Assume that the ages are whole numbers and that twins have the same age. How old are the children? (Hint: Make a list after Bert’s first answer.)

31. Three boxes contain black and white marbles. One box has all black marbles, one has all white marbles, and one has a mixture of black and white. All three boxes are mislabeled. By selecting only one marble, determine how you can correctly label the boxes. (Hint: Notice that “all black” and “all white” are the “same” in the sense that they are the same color.)

32. If a and b are whole numbers and ab = 0, what conclusion can you draw about a or b? Defend your conclusion with a convincing argument.

Analyzing Student Thinking

33. Pascuel claims that the rectangular array approach is the same as the Cartesian product approach. Is there a differ- ence? If so, what is it?

34. Phyllis claims that the tree diagram approach for multi- plication can only be used to model multiplying two num- bers, but not more than two. Is she correct? Explain.

35. Maurice rewrites 6 7 3( )¥ as ( ) ( )6 7 6 3¥ ¥× and says that he used distributivity. Is he correct in using the distribu- tive property this way? How should you respond to this student?

36. Before you can respond to Maurice in #35, Taylor shouts, “No, that is an application of the associative property for multiplication.” How should you respond to this student?

37. Breanne claims that the multiplication property of zero should be called the identity property of zero because when you multiply by zero you get zero. Is her claim reasonable? Explain.

38. Olga says she can’t see how distributivity can be used as a thinking strategy. How should you respond?

39. Ervin says that a a+ =0 and a a− =0 . Thus, since a × =0 0, it must be true that a ÷ =0 0. How could you help him better understand this situation?

40. A student asks you if “4 divided by 12” and “4 divided into 12” mean the same thing. How should you respond?

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116 Chapter 3 Whole Numbers: Operations and Properties

For example, 7 9< since 7 2 9+ = and 13 8> since 8 5 13+ = . The symbols “≤” and “≥” mean “less than or equal to” and “greater than or equal to,” respectively.

One useful property of “less than” is the transitive property.

The transitive property can be verifi ed using any of our defi nitions of “less than.” Consider the number line in Figure 3.36.

Figure 3.36

Since a b< , we have a to the left of b, and since b c< , we have b to the left of c. Hence a is to the left of c, or a c< .

The following is a more formal argument to verify the transitive property. It uses the definition of “less than” involving addition.

a b< means a n b+ = for some nonzero whole number n. b c< means b m c+ = for some nonzero whole number m. Adding m to a n+ and b, we obtain

a n m b m+ + = + . Thus a n m c b m c+ + = + =since . Therefore, a c< since a n m c+ + =( ) and n m+ is a nonzero

whole number.

NOTE: The transitive property of “less than” holds if “<” (and “≤”) are replaced with “greater than” for “>” (and “≥”) throughout.

There are two additional properties involving “less than.” The first involves addi- tion (or subtraction).

Problem-Solving Strategy Draw a Diagram

Ordering and Whole-Number Operations In Chapter 2, whole numbers were ordered in three different, though equivalent, ways using (1) the counting chant, (2) the whole-number line, and (3) a 1-1 correspondence. Now that we have defined whole-number addition, there is another, more useful way to define “less than.” Notice that 3 5< and 3 2 5 4 9+ = <, and 4 5 9+ = , and 2 11< and 2 9 11+ = . This idea is presented in the next definition of “less than.”

NCTM Standard All students should describe quantitative change, such as a student’s growing two inches in one year.

Common Core – Grade 1 Compare two 2-digit numbers based on meanings of the tens and ones digits, recording the results of comparisons with the symbols >, =, and <.

ORDERING AND EXPONENTS

“Less Than” for Whole Numbers

For any two whole numbers a and b, a b< (or b a> ) if and only if there is a nonzero whole number n such that a n b+ = .

D E F I N I T I O N 3 . 8

Transitive Property of “Less Than” for Whole Numbers

For all whole numbers a, b, and c, if a b< and b c< , then a c< .

P R O P E R T Y

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Section 3.3 Ordering and Exponents 117

As was the case with transitivity, this property can be verifi ed formally using the defi nition of “less than.” An informal justifi cation using the whole-number line fol- lows (Figure 3.37).

Figure 3.37

Notice that a b< , since a is to the left of b. Then the same distance, c, is added to each to obtain a c+ and b c+ , respectively. Since a c+ is left of b c+ , we have a c b c+ < + .

In the case of “less than” and multiplication, we have to assume that c ≠ 0. The proof of this property is left for the problem set.

Algebraic Reasoning When finding values of x that make the expression x − <3 6 true, this property can be used by adding 3 on both sides as follows: x − + < +3 3 6 3. Thus it can be seen that all values of x that satisfy x < 9 will make the original expression true.

Problem-Solving Strategy Draw a Diagram

Since c is a nonzero whole number, it follows that c > 0. In Chapter 8, where neg- ative numbers are fi rst discussed, this property will have to be modifi ed to include negatives.

Exponents Just as multiplication can be defined as repeated addition and division may be viewed as repeated subtraction, the concept of exponent can be used to simplify situations involving repeated multiplication.

Children’s Literature www.wiley.com/college/musser See “The King’s Chessboard” by David Birch.

Less Than and Addition Property for Whole Numbers

If a b< , then a c b c+ < + .

P R O P E R T Y

Less Than and Multiplication Property for Whole Numbers

If a b< and c ≠ 0, then ac bc< .

P R O P E R T Y

Check for Understanding: Exercise/Problem Set A #1–4✔

Whole-Number Exponent

Let a and m be any two whole numbers where m ≠ 0. Then

a a a am

m

= ⋅ … factors ��� �� .

D E F I N I T I O N 3 . 9

Given that we define whole-number exponents as a a a am

m

= ⋅ … factors

��� �� , discuss with a peer why each of the following is true.

a a am n m n¥ = + a b a bm m m¥ ¥= ( ) ( )a am n m n= ¥ a a am n m n÷ = −

The number m is called the exponent or power of a, and a is called the base. The number am is read “a to the power m” or “a to the mth power.” For example, 52, read

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118 Chapter 3 Whole Numbers: Operations and Properties

“5 to the second power” or “5 squared,” is 5 5 25¥ = ; 23, read “2 to the third power” or “2 cubed,” is 2 2 2 8¥ ¥ = ; and 3 3 3 3 3 814 = =¥ ¥ ¥ .

There are several properties of exponents that permit us to represent numbers and to do many calculations quickly.

The next example illustrates another way of rewriting products of numbers having the same exponent.

Rewrite each of the following expressions using a single exponent.

a. 2 23 4¥ b. 3 35 7¥

S O L U T I O N 

a. 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 23 4 7¥ ¥ ¥ ¥ ¥ ¥ ¥ ¥ ¥ ¥ ¥ ¥ ¥= = =( ) ( ) b. 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 35 7¥ ¥ ¥ ¥ ¥ ¥ ¥ ¥ ¥ ¥ ¥ ¥ ¥ ¥ ¥ ¥ ¥ ¥ ¥ ¥ ¥ ¥ ¥= = =( ) ( ) 3312 ■

Rewrite the following expressions using a single exponent.

a. 2 53 3¥   b. 3 7 112 2 2¥ ¥

S O L U T I O N 

a. 2 5 2 2 2 5 5 5 2 5 2 5 2 5 2 53 3 3¥ ¥ ¥ ¥ ¥ ¥ ¥ ¥ ¥= = =( )( ) ( ) ( ) ( ) ( ) b. 3 7 11 3 3 7 7 11 11 3 7 11 3 7 11 3 7 112 2 2 2¥ ¥ ¥ ¥ ¥ ¥ ¥ ¥ ¥ ¥ ¥= = =( )( ) ( ) ( )( ) ( ) ■

In Example 3.10(a) the exponents of the factors were 3 and 4, and the exponent of the product is 3 4 7+ = . Also, in (b) the exponents 5 and 7 yielded an exponent of 5 7 12+ = in the product.

The fact that exponents are added in this way can be shown to be valid in general. This result is stated next as a theorem. A theorem is a statement that can be proved based on known results.

Let a, m, and n be any whole numbers where m and n are nonzero. Then

a a am n m n¥ = + .

T H E O R E M 3 . 1

P R O O F

a a a a a a a a a am n

m n

¥ ¥= ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ = ⋅ ⋅ ⋅ ⋅( ) ( ) factors factors

��� �� ��� �� aa a m n

m n

+

+= factors

��� �� ■

The results in Example 3.11 suggest the following theorem.

Let a, b, and m be any whole numbers where m is nonzero. Then

a b abm m m¥ = ( ) .

T H E O R E M 3 . 2

P R O O F

a b a a a b b b ab abm m

m m

¥ ¥ ¥= ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ = ⋅ ⋅ factors factors

��� �� ��� �� ( )( ) ⋅⋅ =( ) ( )ab ab m

m

pairs of factors � ��� ��� ■

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Section 3.3 Ordering and Exponents 119

In general, we have the next theorem.

Rewrite the following expressions with a single exponent.

a. ( )53 2 b. ( )78 4

S O L U T I O N 

a. ( ) ( )5 5 5 5 5 53 2 3 3 3 3 6 3 2= = = =+¥ ¥

b. ( ) ( )7 7 7 7 7 7 78 4 8 8 8 8 32 8 4= = =¥ ¥ ¥ ¥ ■

The next example shows how to simplify expressions of the form ( )am n.

Let a, m, and n be any whole numbers where m and n are nonzero. Then

( ) .a am n mn=

T H E O R E M 3 . 3

The proof of this theorem is similar to the proofs of the previous two theorems. The previous three properties involved exponents and multiplication. However,

notice that ( )2 3 2 33 3 3+ ≠ + , so there is not a corresponding property involving sums or differences raised to powers.

The next example concerns the division of numbers involving exponents with the same base number.

Rewrite the following quotients with a single exponent.

a. 5 57 3÷      b. 7 78 5÷

S O L U T I O N 

a. 5 5 57 3 4÷ = , since 5 5 57 3 4= ¥ . Therefore, 5 5 57 3 7 3÷ = − . b. 7 7 78 5 3÷ = , since 7 7 78 5 3= ¥ . Therefore, 7 7 78 5 8 5÷ = − . ■

You are teaching a unit on exponents and a student asks you what 40 means. One student volunteers that it is 0 since exponents are a shortcut for multiplication. How do you

respond in a meaningful way?

In general, we have the following result.

Let a, m, and n be any whole numbers where m n> and a, m, and n are nonzero. Then

a a am n m n÷ = − .

T H E O R E M 3 . 4

P R O O F

If and only if a a cm n= ¥ . Since a a a an m n n m n m¥ − + −= =( ) , we have c am n= − . Therefore a a am n m n÷ = − . ■

a a cm n÷ =

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120 Chapter 3 Whole Numbers: Operations and Properties

Notice that 00 is not defi ned. To see why, consider the following two patterns.

PATTERN 1 PATTERN 2 3 10 = 0 03 = 2 10 = 0 02 = 1 10 = 0 01 = 0 ?0 = 00 = ?

Pattern 1 suggests that 00 should be 1 and pattern 2 suggests that 00 should be 0. Thus to avoid such an inconsistency, 00 is undefined.

Problem-Solving Strategy Look for a Pattern

Notice that we have not yet defined a0. Consider the following pattern

Extending this pattern, we see that the following definition is appropriate.

Problem-Solving Strategy Look for a Pattern

Zero as an Exponent

a0 1= for all whole numbers a ≠ 0.

D E F I N I T I O N 3 . 1 0

Check for Understanding: Exercise/Problem Set A #5–12✔

Order of Operations Now that the operations of addition, subtraction, multiplication, and division as well as exponents have been introduced, a point of confusion may occur when more than one operation is in the same expression. For example, does it matter which operation is performed first in an expression such as 3 4 5+ × ? If the 3 and 4 are added first, the result is 7 5 35× = , but if the 4 and 5 are multiplied first, the result is 3 20 23+ = .

Since the expression 3 5 5 5 5 23+ + + + =( ) can be written as 3 4 5+ × , it seems reasonable to do the multiplication, 4 5× , first. Similarly, since 4 5 5 5 500¥ ¥ ¥ ( )= can be rewritten as 4 53¥ , it seems reasonable to do exponents before multiplication. Also, to prevent confusion, parentheses are used to indicate that all operations within a pair of parentheses are done first. To eliminate any ambiguity, mathematicians have agreed that the proper order of operations shall be Parentheses, Exponents, Multiplication and Division, Addition and Subtraction (PEMDAS). Although mul- tiplication is listed before division, these operations are done left to right in order of appearance. Similarly, addition and subtraction are done left to right in order of appearance. The pneumonic device Please Excuse My Dear Aunt Sally is often used to remember this order.

Use the proper order of operations to simplify the following expressions.

a. 5 4 1 23 2− +¥ ( ) b. 11 4 2 5 3− ÷ +¥

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Section 3.3 Ordering and Exponents 121

S O L U T I O N 

a. 5 4 1 2 5 4 3

125 4 9

125 36

89

3 2 3 2− + = − = − = − =

¥ ¥ ¥

( )

b. 11 4 2 5 3 11 2 5 3

11 10 3

1 3

4

− ÷ + = − + = − + = + =

¥ ¥

■

© R

on B

ag w

el l

John von Neumann was a brilliant mathematician who made important contributions to several scientific fields, including the theory and appli- cation of high-speed computing machines. George Pólya of Stanford University admitted that “Johnny was the only student I was ever afraid of. If in the course of a lecture I stated an unsolved problem, the chances were he’d come to me as soon as the lecture was over, with the complete solu- tion in a few scribbles on a slip of paper.” At the age of 6, von Neumann could divide two 8-digit numbers in his head, and when he was 8 he had mastered the calculus. When he invented his first electronic computer, someone suggested that he race it. Given a problem like “What is the smallest power of 2 with the property that its decimal digit fourth from the right is a 7,” the machine and von Neumann started at the same time and von Neumann won!

Check for Understanding: Exercise/Problem Set A #13–14✔

EXERCISES 1. Find the nonzero whole number n in the definition of “less

than” that verifies the following statements. a. 12 31<   b. 53 37>

2. Using the definitions of < and > given in this section, write four inequality statements based on the fact that 2 8 10+ = .

3. The statement a x b< < is equivalent to writing a x< and x b< and is called a compound inequality. We often read a x b< < as “x is between a and b.” For the questions that follow, assume that a , x , and b are whole numbers.

If a x b< < and c is a nonzero whole number, is it always true that a c x c b c+ < + < + ? Try several examples to test your conjecture.

4. Does the transitive property hold for the following? Explain. a. = (equals)  b. ≠ (is not equal to)

5. Using exponents, rewrite the following expressions in a sim- pler form.

a. 3 3 3 3¥ ¥ ¥ b. 2 2 3 2 3 2¥ ¥ ¥ ¥ ¥ c. 6 7 6 7 6¥ ¥ ¥ ¥ d. x y x y y y¥ ¥ ¥ ¥ ¥ e. a b b a¥ ¥ ¥ f. 5 6 5 5 6 6¥ ¥ ¥ ¥ ¥

6. Evaluate each of the following without a calculator and order them from largest to smallest.

5 4 3 22 3 4 5, , ,

7. Write each of the following expressions in expanded form, without exponents.

a. 3 2 5x y z    b. 7 53¥    c. ( )7 5 3¥

8. Rewrite each with a single exponent. a. 5 53 4¥ b. 3 312 2÷ c. 2 57 7¥ d. 8 . 25

e. 25 53 2÷ f. 9 12 22 3¥ ¥

9. Express 514 in three different ways using the numbers 2, 5, 7 and exponents. (You may use a number more than once.)

10. Let a , b, n W∈ and n ≥ 1. Determine if ( )a b a bn n n+ = + is always true, sometimes true, or never true. Justify your conclusion.

11. Find x . a. 3 3 37 13¥ x = b. ( )3 34 20x = c. 3 2 6x x x¥ =

EXERCISE/PROBLEM SET A

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122 Chapter 3 Whole Numbers: Operations and Properties

12. Use the yx , xy , or ∧ key on your calculator to evaluate the following.

a. 68      b. ( )3 5 4¥       c. 3 54¥

13. Simplify each of the following expressions. a. 15 3 7 2− −( ) b. 2 52¥

c. 3 4 2 5 32 3¥ − −( ) d. 6 2 3 4 4 6 3 4

3 2 2 2

3 2

+ − + −

( ) ( )¥

14. For each of the following: i. Show that the expression is not equal to the numerical

value. ii. Insert parentheses in the expression so it will be equal to

the numerical value. a. 2 3 42¥ − ; 10 b. 4 3 4 22 − ÷¥ ; 26

PROBLEMS 15. Let a , m , and n be whole numbers where a is not zero.

Prove the following property: ( )a am n m n= ¥ .

16. Using properties of exponents, mentally determine the larger of the following pairs.

a. 610 and 320   b. 9 39 20and    c. 12 310 20and

17. The price of a certain candy bar doubled over a period of five years. Suppose that the price continued to double every five years and that the candy bar cost 25 cents in 2000.

a. What would be the price of the candy bar in the year 2015? b. What would be the price of the candy bar in the year 2040? c. Write an expression representing the price of the candy

bar after n five-year periods.

18. Verify the transitive property of “less than” using the 1-1 correspondence definition.

19. Observe the following pattern in the sums of consecutive whole numbers. Use your calculator to verify that the statements are true.

2 3 4 1 2

5 6 7 8 9 2 3

10 11 12 13 14 15 16 3 4

3 3

3 3

3 3

+ + = + + + + + = +

+ + + + + + = +

a. Write the next two lines in this sequence of sums. b. Express 9 103 3+ as a sum of consecutive whole numbers. c. Express 12 133 3+ as a sum of consecutive whole numbers. d. Express n n3 31+ +( ) as a sum of consecutive whole

numbers.

20. Pizzas come in four different sizes, each with or without a choice of four ingredients. How many ways are there to order a pizza?

21. The perfect square 49 is special in that each of the two nonzero digits forming the number is itself a perfect square.

a. Explain why there are no other two-digit squares with this property.

b. What three-digit perfect squares can you find that are made up of one 1-digit square and one 2-digit square?

c. What four-digit perfect squares can you find that are made up of two 1-digit squares and one 2-digit square?

d. What four-digit perfect squares can you find that are made up of one 3-digit square and one 1-digit square?

e. What four-digit perfect squares can you find that are made up of two 2-digit squares?

f. What four-digit perfect squares can you find that are made up of four 1-digit squares?

22. When asked to find four whole numbers such that the product of any two of them is one less than the square of a whole number, one mathematician said, “2, 4, 12, and 22.” A second mathematician said, “2, 12, 24, and 2380.” Which was correct?

23. a. Give a formal proof of the property of less than and addition. (Hint: See the proof in the paragraph follow- ing Figure 3.37.)

b. State and prove the corresponding property for sub- traction.

EXERCISES 1. Find the nonzero whole number n in the definition of “less

than” that verifies the following statements. a. 17 26<       b. 113 49>

2. Using the definitions of < and > given in this section, write four inequality statements based on the fact that 29 15 44+ = .

3. If a x b< < and c is a nonzero whole number, is it always true that ac xc bc< < ? Try several examples to test your conjecture.

4. Does the transitive property hold for the following? Explain. a. > (greater than)    b. ≤ (less than or equal to)

5. Using exponents, rewrite the following expressions in a simpler form.

a. 4 4 4 4 4 4 4¥ ¥ ¥ ¥ ¥ ¥ b. 4 3 3 4 4 3 3 3 3¥ ¥ ¥ ¥ ¥ ¥ ¥ ¥ c. y x y x x¥ ¥ ¥ ¥ ¥2 d. a b a b a b a b¥ ¥ ¥ ¥ ¥ ¥ ¥

6. Evaluate each of the following without a calculator and arrange them from smallest to largest.

5 6 3 23 2 5 7, , ,

7. Write each of the following expressions in expanded form, without exponents.

a. 10 4ab      b. ( )2 5x      c. 2 5x

EXERCISE/PROBLEM SET B

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Section 3.3 Ordering and Exponents 123

8. Write the following with only one exponent.

a. x x3 6¥ b. a a15 3÷ c. x y5 5 d. 2 163 ¥ e. 128 23÷ f. 3 9 277 5 2¥ ¥

9. Express 720 in three different ways using the numbers 7, 2, 5 and exponents. (You may use a number more than once.)

10. Let a , b, m, n W∈ and a is not zero. Determine if a an m¥ = an m¥ is always true, sometimes true, or never true. Justify your conclusion.

11. Find the value of x that makes each equation true. a. 2 2 25 7x ÷ = b. ( )6 362x x= c. 5 3 157 ¥ x x=

12. Use the yx , xy , or ∧ key on your calculator to evaluate the following.

a. 3 27 4+       b. 2 5 3 26 3¥ ¥−       c. 9 12 3

3 4

8

¥

13. Simplify each of the following expressions. a. 2 3 2 32 2¥ ¥−

b. 2 4 1 15 2 3 7 5

3

1 0

( ) ( )

− − +¥

c. 5 4 6 4 22 2 5 3− + − ÷( ) d. 4 2 6 2 33 4 3 2× − ÷ ( )¥

14. For each of the following: i. Show that the expression is not equal to the numerical value. ii. Insert parentheses in the expression so it will be equal to

the numerical value. a. 22 2 7 3− −¥ ; 137 b. 17 2 3 5 52+ + ÷¥ ; 8

PROBLEMS 15. a. How does the sum 1 2 3+ + compare to the sum

1 2 33 3 3+ + ? b. Try several more such examples and determine

the relationship between 1 2 3+ + ⋅ ⋅ ⋅ ++ n and 1 2 33 3 3 3+ + + ⋅ ⋅ ⋅ + n .

c. Use the pattern you observed to find the sum of the first 10 cubes, 1 2 3 103 3 3 3+ + + +… , without evaluating any of the cubes.

16. Order these numbers from smallest to largest using prop- erties of exponents and mental methods.

3 4 9 822 14 10 10, , ,

17. A pad of 200 sheets of paper is approximately 15 mm thick. Suppose that one piece of this paper were folded in half, then folded in half again, then folded again, and so on. If this folding process was continued until the piece of paper had been folded in half 30 times, how thick would the folded paper be?

18. Suppose that you can order a submarine sandwich with or without each of seven condiments.

a. How many ways are there to order a sandwich? b. How many ways can you order exactly two condiments? c. Exactly seven? d. Exactly one? e. Exactly six?

19. a. If n2 121= , what is n? b. If n2 1 234 321= , , , what is n? c. If n2 12 345 654 321= , , , , what is n? d. If n2 123 456 787 654 321= , , , , , what is n?

20. 12 is a factor of 10 22 2− , 27 is a factor of 20 72 2− , and 84 is a factor of 80 42 2− . Check to see whether these three statements are correct. Using a variable, prove why this works in general.

21. a. Verify that the property of less than and multiplication holds where a = 3 , b = 7 , and c = 5 .

b. Show that the property does not hold if c = 0.

c. Give a formal proof of the property. (Hint: Use the fact that the product of two nonzero whole numbers is nonzero.)

d. State the corresponding property for division.

22. Let a , m , and n be whole numbers where m n> and a is not zero. Prove the following property: a a am n m n÷ = − .

Analyzing Student Thinking

23. Brooke claims that because “less than and addition for whole numbers” is a property, so too should “less than and subtraction for whole numbers” be a property. How should you respond?

24. A student asks why there is not a property of “less than and division for whole numbers.” How should you respond?

25. Jarell was asked to simplify the expression 3 32 4¥ and wrote 3 3 92 4 6¥ = . How would you convince him that this answer is incorrect? That is, give the correct answer and explain why the student’s method is incorrect.

26. Viridiana observes 0 0 0 0+ = × and 2 2 2 2+ = × and assumes that addition and multiplication are the same. Is she correct? Explain.

27. Abbi was asked to simplify the expression 8 25 2÷ and wrote 8 2 45 2 3÷ = . How would you convince her that this answer is incorrect?

28. Cho rewrote ( )32 2 as 3 2 2( ) (a form of associativity, per-

haps). Was she correct or incorrect? She also rewrote ( )23 2 as 2 3

2( ) . Is this right or wrong? How should you respond?

29. Ayumi writes a b abm n m n¥ = +( ) . Is this statement correct or incorrect? Explain.

30. Gavin says that because of distributivity, the following equation is true: ( )a b a bm m m+ = + . What could you say to help him better understand the situation?

31. Bailey wrote the following using “Please Excuse My Dear Aunt Sally.” Explain where she may have misinterpreted this acronym.

( ) .5 3 17 5 3 17 125 27 17 1153 3 3− + = − + = − + =

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124 Chapter 3 Whole Numbers: Operations and Properties

In a group of nine coins, eight weigh the same and the ninth is either heavier or lighter. Assume that the coins are identical in appearance. Using a pan balance, what is the smallest number of balancings needed to identify the counterfeit coin?

Strategy: Use Direct Reasoning

Three balancings are sufficient. Separate the coins into three groups of three coins each.

A B C

Balance group A against group B. If they balance, we can deduce that the counterfeit coin is in group C. We then need to determine if the coin is heavier or lighter. Balance group C against group A. Based on whether group C goes up or down, we can deduce whether group C is light or heavy, respectively.

If the coins in group A do not balance the coins in group B, then we can conclude that either A is light or B is heavy (or vice versa). We now balance group A against group C.

If they balance, we know that group B has the counterfeit coin. If group A and C do not balance, then group A has the counterfeit coin. We also know whether group A or B is light or heavy depending on which one went down on the fi rst balancing.

In two balancings, we know which group of three coins contains the counterfeit one and whether the counterfeit coin is heavy or light. The third balancing will exactly identify the counterfeit coin.

From the group of three coins that has the counterfeit coin, select two coins and balance them against each other. This will determine which coin is the counterfeit one. Thus, the counter- feit coin can be found in three balancings.

Additional Problems Where the Strategy “Use Direct Reasoning” Is Useful

1. The sum of the digits of a three-digit palindrome is odd. De- termine whether the middle digit is odd or is even.

2. Jose’s room in a hotel was higher than Michael’s but lower than Ralph’s. Andre’s room is on a fl oor between Ralph’s and Jose’s. If it weren’t for Michael, Clyde’s room would be the lowest. List the rooms from lowest to highest.

3. Given an 8-liter jug of water and empty 3-liter and 5-liter jugs, pour the water so that two of the jugs have 4 liters.

END OF CHAPTER MATERIAL

Julia Bowman Robinson (1919–1985) Julia Bowman Robinson spent her early years in Ari- zona, near Phoenix. She said that one of her earliest memories was of arranging pebbles in the shadow of a

giant saguaro—“I’ve always had a basic liking for the natural numbers.” In 1948, Robinson earned her doc- torate in mathematics at Berkeley; she went on to con- tribute to the solution of “Hilbert’s tenth problem.” In 1975 she became the first woman mathematician elected to the prestigious National Academy of Sci- ences. Robinson also served as president of the Amer- ican Mathematical Society, the main professional organization for research mathematicians. “Rather than being remembered as the first woman this or that, I would prefer to be remembered simply for the theorems I have proved and the problems I have solved.”

John von Neumann (1903–1957) John von Neumann was one of the most remarkable math- ematicians of the twentieth century. His logical power was legendary. It is said that during and after World War II the U.S.

government reached many scientifi c decisions simply by asking von Neumann for his opinion. Paul Halmos, his one-time assistant, said, “The most spectacular thing about Johnny was not his power as a mathematician, which was great, but his rapidity; he was very, very fast. And like the modern computer, which doesn’t memorize logarithms, but computes them, Johnny didn’t bother to memorize things. He computed them.” Appropriately, von Neumann was one of the fi rst to realize how a general-purpose computing machine—a computer—should be designed. In the 1950s he invent- ed a “theory of automata,” the basis for subsequent work in artifi cial intelligence.

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Chapter Review 125

CHAPTER REVIEW

Review the following terms and exercises to determine which require learning or relearning—page numbers are provided for easy reference.

Vocabulary/Notation Plus  87 Sum  87 Addend  87 Summand  87 Binary operation  88 Closure for addition  89 Commutativity for addition  89

Associativity for addition  90 Identity for addition  90 Thinking strategies for addition

facts  90 Take-away approach  93 Difference  94 Minus  94

Minuend  94 Subtrahend  94 Missing-addend approach  94 Missing addend  94 Four-fact families  94 Comparison  95 Comparison approach  96

Addition and Subtraction

Exercises 1. Show how to find 5 4+ using a. a set model. b. a measurement model.

2. Name the property of addition that is used to justify each of the following equations.

a. 7 3 9 7 3 9+ + = + +( ) ( ) b. 9 0 9+ = c. 13 27 27 13+ = + d. 7 6+ is a whole number

3. Identify and use thinking strategies that can be used to find the following addition facts.

a. 5 6+ b. 7 9+

4. Illustrate the following using 7 3− . a. The take-away approach b. The missing-addend approach

5. Show how the addition table for the facts 1 through 9 can be used to solve subtraction problems.

6. Which of the following properties hold for whole-number subtraction?

a. Closure b. Commutative c. Associative d. Identity

Exercises 1. Illustrate 3 5× using each of the following approaches. a. Repeated addition with the set model b. Repeated addition with the measurement model c. Rectangular array with the set model d. Rectangular array with the measurement model

2. Name the property of multiplication that is used to justify each of the following equations.

a. 37 1 37× = b. 26 5 5 26× ×= c. 2 5 17 2 5 17× × × ×( ) ( )= d. 4 9× is a whole number

Vocabulary/Notation Repeated-addition approach  101 Times  101 Product  101 Factor  101 Rectangular array approach  101 Cartesian product approach  102 Tree diagram approach  102 Closure for multiplication  102 Commutativity for multiplication  102 Associativity for multiplication  104 Identity for multiplication  104 Multiplicative identity  104

Distributive property of multiplication over addition  105

Multiplication property of zero  106 Distributivity of multiplication over

subtraction  106 Thinking strategies for multiplication

facts  106 Sharing division  107 Measurement division  107 Missing-factor approach  108 Divided by  108 Dividend  108

Divisor  108 Quotient  108 Missing factor  108 Division property of zero  109 Division by zero  109 Division algorithm  109 Divisor  110 Quotient  110 Remainder  110 Repeated-subtraction approach  110

Multiplication and Division

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126 Chapter 3 Whole Numbers: Operations and Properties

Vocabulary/Notation Less than  116 Less than or equal to  116 Greater than or equal to  116 Transitive property of less

than  116

Less than and addition property  (subtraction)  117

Less than and multiplication property (division)  117

Exponent  117

Power  117 Base  117 Theorem  118 Zero as an exponent  120 Order of operations  120

Ordering and Exponents

Exercises 1. Describe how addition is used to define “less than” (“great-

er than”).

2. For whole numbers a , b, and c, if a b< , what can be said about

a. a c+ and b c+ ?  b. a c× and b c× ?

3. Rewrite 54 using the definition of exponent.

4. Rewrite the following using properties of exponents. a. ( )73 4      b. 3 75 5×       c. 5 57 3÷       d. 4 412 13×

5. Explain how to motivate the definition of zero as an exponent.

3. Show how the distributive property can be used to simplify these calculations.

a. 7 27 7 13× + × b. 8 17 8 7× ×−

4. Use and name the thinking strategies that can be used to find the following addition facts.

a. 6 7× b. 9 7×

5. Show how 17 3÷ can be found using a. Repeated subtraction with the set model b. Repeated subtraction with the measurement model

6. Show how the multiplication table for the facts 1 through 9 can be used to solve division problems.

7. Calculate the following if possible. If impossible, explain why. a. 7 0÷ b. 0 7÷ c. 0 0÷

8. Apply the division algorithm and apply it to the calcula- tion 39 7÷ .

9. Which of the following properties hold for whole-number division?

a. Closure b. Commutative c. Associative d. Identity

10. Label the following diagram and comment on its value.

CHAPTER TEST

Knowledge 1. True or false? a. n A B n A n B( ) ( ) ( )∪ = + for all finite sets A and B. b. If B A⊆ , then n A B n A n B( ) ( ) ( )− = − for all finite sets A

and B. c. Commutativity does not hold for subtraction of whole

numbers. d. Distributivity of multiplication over subtraction does not

hold in the set of whole numbers. e. The symbol ma, where m and a are nonzero whole num-

bers, represents the product of m factors of a . f. If a is the divisor, b is the dividend, and c is the quotient,

then ab c= . g. The statement “a b c+ = if and only if c b a− = ” is an

example of the take-away approach to subtraction. h. Factors are to multiplication as addends are to addition. i. If n ≠ 0 and b n a+ = , then a b< . j. If n A a( ) = and n B b( ) = , then A B× contains exactly ab

ordered pairs.

2. Complete the following table for the operations on the set of whole numbers. Write True in the box if the indicated property holds for the indicated operation on whole num- bers and write False if the indicated property does not hold for the indicated operation on whole numbers.

ADD SUBTRACT MULTIPLY DIVIDE

Closure Commutative Associative Identity

3. Identify the property of whole numbers being illustrated. a. a b c a c b¥ ¥ ¥ ¥( ) ( )= b. ( )3 2 1 3 2+ = +¥ c. ( ) ( )4 7 8 4 7 8+ + = + + d. ( ) ( )2 3 5 2 3 5¥ ¥ ¥ ¥= e. ( )6 5 2 6 2 5 2− = −¥ ¥ ¥ f. m n p m p n+ + = + +( ) ( )

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Chapter Test 127

Skill 4. Find the following sums and products using thinking strate-

gies. Show your work. a. 39 12+ b. 47 87+ c. 5 73 2( )¥ d. 12 33×

5. Find the quotient and remainder when 321 is divided by 5.

6. Rewrite the following using a single exponent in each case. a. 3 37 12¥ b. 5 531 7÷ c. ( )73 5 d. 45 ÷ 28

e. 7 2 1412 12 3¥ ¥ f. ( ) ( )12 38 4 6 2÷125 ¥

7. Perform the following calculations by applying appropriate properties. Which properties are you using?

a. 13 97 13 3¥ ¥+ b. ( )194 86 6+ + c. 7 23 23 3¥ ¥+ d. 25 123 8( )¥

8. Classify each of the following division problems as exam- ples of either sharing or measurement division.

a. Martina has 12 cookies to share between her three friends and herself. How many cookies will each person receive?

b. Coach Massey had 56 boys sign up to play intramural basketball. If he puts 7 boys on each team, how many teams will he have?

c. Eduardo is planning to tile around his bathtub. If he wants to tile 48 inches up the wall and the individual tiles are 4 inches wide, how many rows of tile will he need?

9. Each of the following situations involves a subtraction problem. In each case, tell whether the problem is best represented by the take-away approach or the missing- addend approach, and why. Then write an equation that could be used to answer the question.

a. Ovais has 137 basketball cards and Quinn has 163 bas- ketball cards. How many more cards does Quinn have?

b. Regina set a goal of saving money for a $1500 down payment on a car. Since she started, she has been able to save $973. How much more money does she need to save in order to meet her goal?

c. Riley was given $5 for his allowance. After he spent $1.43 on candy at the store, how much did he have left to put into savings for a new bike?

Understanding 10. Using the following table, find A B− .

+ A B C A C A B

B A B C

C B C A

11. a. Using the definition of an exponent, provide an expla- nation to show that ( )7 73 4 12= .

b. Use the fact that a a am n m n¥ = + to explain why ( )7 73 4 12= .

12. Show why 3 0÷ is undefined.

13. Explain in detail why a b a bm m m¥ ¥= ( ) .

14. Explain why ( )a b a bc c¥ ¥≠ for all a , b, c W, ∈ .

15. If we make up another operation, say Ω, and we give it a table using the letters A, B, C, D as the set on which this symbol is operating, determine whether this operation is commutative.

Ω A B C D A B B B B

B B C D A

C B A C D

D B D A C

Then create your own 4 4× table for a new operation @, operating on the same four numbers, A, B, C, D. Make sure your operation is commutative. Can you see some- thing that would always be true in the table of a commuta- tive operation?

Problem Solving/Application 16. Which is the smallest set of whole numbers that

contains 2 and 3 and is closed under addition and multiplication?

17. If the product of two numbers is even and their sum is odd, what can you say about the two numbers?

18. Illustrate the following approaches for 4 3× using the measurement model. Please include a written description to clarify the illustration.

a. Repeated-addition approach b. Rectangular array approach

19. Use a set model to illustrate that the associative property for whole-number addition holds.

20. Illustrate the missing-addend approach for the subtraction problem 8 5− by using

a. The set model b. The measurement model

21. Use the rectangular array approach to illustrate that the commutative property for whole-number multiplication holds.

22. Find a whole number less than 100 that is both a perfect square and a perfect cube.

23. Find two examples where a , b W∈ and a b a b¥ = + .

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128

T he abacus was one of the earliest computational devic-es. The Chinese abacus, or suan-pan, was composed of a frame together with beads on fixed rods.

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In 1617, Napier invented lattice rods, called Napier’s bones. To multiply, appropriate rods were selected, laid side by side, and then appropriate “columns” were added—thus only addition was needed to multiply.

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About 1594, Napier invented logarithms. Using loga- rithms, multiplication can be performed by adding respec- tive logarithms. The slide rule, used extensively by engi- neers and scientists through the 1960s, was designed to use properties of logarithms.

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In 1642, Pascal invented the first mechanical adding machine.

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In 1671, Leibniz developed his “reckoning machine,” which could also multiply and divide.

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In 1812, Babbage built his Difference Engine, which was the first “computer.”

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By 1946, ENIAC (Electronic Numerical Integrator and Computer) was developed. It filled a room, weighed over 30 tons, and had nearly 20,000 vacuum tubes. Chips permitted the manufacture of microcomputers, such as the Apple and DOS-based computers in the late 1970s, the Apple Macintosh and Windows-based versions in the 1980s, to laptop computers in the 1990s.

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Chip manufacture also allowed calculators to become more powerful to the point that the dividing line between calculator and computer has become blurred.

C H A P T E R

4 WHOLE-NUMBERCOMPUTATION—MENTAL, ELECTRONIC, AND WRITTEN Computational Devices from the Abacus to the Computer

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129

Occasionally, in mathematics, there are problems that are not easily solved using direct reasoning. In such cases, indirect reasoning may be the best way to solve the problem. A simple way of viewing indirect reasoning is to consider an empty room with only two entrances, say A and B. If you want to use direct reasoning to prove that someone entered the room through A, you would watch entrance A. However, you could also prove that someone entered through A by watching entrance B. If a person got into the room and did not go through B, the person had to go through entrance A. In mathematics, to prove that a condition, say “A,” is true, one assumes that the condition “not A” is true and shows the latter con- dition to be impossible.

Initial Problem The whole numbers 1 through 9 can be used once, each being arranged in a 3 3× square array so that the sum of the numbers in each of the rows, columns, and diago- nals is 15. Show that 1 cannot be in one of the corners.

Clues The Use Indirect Reasoning strategy may be appropriate when

Direct reasoning seems too complex or does not lead to a solution.

Assuming the negation of what you are trying to prove narrows the scope of the problem.

A proof is required.

A solution of this Initial Problem is on page 169.

Use Indirect ReasoningProblem-Solving Strategies

1. Guess and Test

2. Draw a Picture

3. Use a Variable

4. Look for a Pattern

5. Make a List

6. Solve a Simpler Problem

7. Draw a Diagram

8. Use Direct Reasoning

9. Use Indirect Reasoning

c04.indd 129 7/30/2013 2:52:17 PM

130

AUTHOR

WALK-THROUGH

I N T R O D U C T I O N In the past, much of elementary school mathematics was devoted to learning written methods for doing addition, subtraction, multiplication, and division. Due to the availability of calculators and computers, less emphasis is being placed on doing written calculations involving numbers with many digits. Instead, an emphasis is being placed on developing skills in the use of all three types of com- putations: mental, written, and electronic. Then, depending on the size of the numbers involved, the accuracy desired in the answer, and the time available, the appropriate model(s) of calculation will be

selected and employed. In this chapter, you will study all three forms of computations: mental, electronic, and written.

Key Concepts from the NCTM Principles and Standards for School Mathematics

PRE-K-2–NUMBER AND OPERATIONS

Develop and use strategies for whole-number computations, with a focus on addition and subtraction. Use a variety of methods and tools to compute, including objects, mental computation, estimation, paper and pencil,

and calculators.

GRADES 3-5–NUMBER AND OPERATIONS

Develop fluency with basic number combinations for multiplication and division and use these combinations to men- tally compute related problems, such as 30 50× .

Develop fluency in adding, subtracting, multiplying, and dividing whole numbers. Develop and use strategies to estimate the results of whole-number computations and to judge the reasonableness of

such results. Select appropriate methods and tools for computing with whole numbers from among mental computation, estima-

tion, calculators, and paper and pencil according to the context and nature of the computation and use the selected method or tool.

Key Concepts from the NCTM Curriculum Focal Points

GRADE 1: Developing an understanding of whole-number relationships, including grouping in tens and ones. GRADE 2: Developing quick recall of addition facts and related subtraction facts and fluency with multidigit addi-

tion and subtraction. GRADE 4: Developing quick recall of multiplication facts and related division facts and fluency with whole number

multiplication. GRADE 5: Developing an understanding of and fluency with division of whole numbers. GRADES 4 AND 5: Students select appropriate methods and apply them accurately to estimate products and cal-

culate them mentally, depending on the context and numbers involved.

Key Concepts from the Common Core State Standards for Mathematics

GRADE 1: Use place value understanding and properties of operations to add and subtract two-digit numbers. GRADE 2: Represent and solve problems involving addition and subtraction. Use place value understanding and

properties of operations to add and subtract three-digit numbers. Relate addition and subtraction to length. GRADE 3: Represent and solve problems involving multiplication and division within 100. Understand properties

of multiplication and the relationship between multiplication and division. Use place value understanding and properties of operations to perform multidigit arithmetic.

GRADE 4: Use the four operations with whole numbers to solve problems. Use place value understanding and properties of operations to perform multidigit arithmetic. Multiply four-digit whole numbers by one-digit whole numbers and multiply two two-digit whole numbers. Divide four-digit whole numbers by one-digit divisors.

GRADE 5: Perform operations with multidigit whole numbers. Specifically, divide four-digit whole numbers by two-digit divisors.

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Section 4.1 Mental Math, Estimation, and Calculators 131

Mental Math The availability and widespread use of calculators and computers have permanently changed the way we compute. Consequently, there is an increasing need to develop students’ skills in estimating answers when checking the reasonableness of results obtained electronically. Computational estimation, in turn, requires a good working knowledge of mental math. Thus this section begins with several techniques for doing calculations mentally.

In Chapter 3 we saw how the thinking strategies for learning the basic arithmetic facts could be extended to multidigit numbers, as illustrated next.

NCTM Standard All students should select appro- priate methods and tools for computing with whole numbers from among mental computation, estimation, calculators, and paper and pencil according to the con- text and nature of the computa- tion and use the selected method or tool.

Children’s Literature www.wiley.com/college/musser See “Great Estimations’’ by Bruce Goldstone.

Compute each of the following mentally and write a sentence for each problem describing your thought processes. After a discussion with classmates, determine some

common strategies. 32 26¥ ¥− × × + +

+ − × ÷ 23 32 16 9 25 25 39 105

49 27 152 87 46 99 252 12

( ) ( )

MENTAL MATH, ESTIMATION, AND CALCULATORS

Calculate the following mentally.

a. 15 27 25+ +( ) b. 21 17 13 21¥ ¥− c. ( )8 7 25× × d. 98 59+ e. 87 29+ f. 168 3÷

S O L U T I O N 

a. 15 27 25 27 25 15 27 25 15 27 40 67+ + = + + = + + = + =( ) ( ) ( ) . Notice how com- mutativity and associativity play a key role here.

b. 21 1 7 13 21 21 17 21 13 21 17 13 21 4 84¥ ¥ ¥ ¥ ¥− = − = − = =( ) . Observe how commuta- tivity and distributivity are useful here.

c. ( ) ( ) ( )8 7 25 7 8 25 7 8 25 7 200 1400× × = × × = × × = × = . Here commutativity is used fi rst; then associativity is used to group the 8 and 25 since their product is 200.

d. 98 59 98 2 57 98 2 57 157+ = + + = + + =( ) ( ) . Associativity is used here to form 100. e. 87 29 80 20 7 9 100 16 116+ = + + + = + = using associativity and commutativity. f. 168 3 150 3 18 3 50 6 56÷ = ÷ + ÷ = + =( ) ( ) . ■

Reflection from Research Flexibility in mental calculation cannot be taught as a “process skill” or holistic “strategy.” Instead, flexibility can be devel- oped by using the solutions stu- dents find in mental calculations to show how numbers can be dealt with, be taken apart and put back together, be rounded and adjusted, etc. (Threlfall, 2002).

Observe that part (f ) makes use of right distributivity of division over addition; that is, whenever the three quotients are whole numbers, ( ) ( ) ( )a b c a c b c+ ÷ = ÷ + ÷ . Right distributivity of division over subtraction also holds.

The calculations in Example 4.1 illustrate the following important mental tech- niques.

Properties Commutativity, associativity, and distributivity play an important role in simplifying calculations so that they can be performed mentally. Notice how useful these properties were in parts (a), (b), (c), (d), and (e) of Example 4.1. Also, the solution in part (f ) uses right distributivity.

Common Core – Grade 3 Multiply one-digit whole numbers by multiples of 10 in the range 10–90 (e.g., 9 80× , 5 60× ) using strategies based on place value and properties of operations.

When Joelle, a fourth grader, was asked to solve 15 8,− she said, “Well 8 5− is 3 and 10 3− is 7 so 15 8− is 7.” Discuss with a peer what you think Joelle’s mental math methods are

and whether or not you think they are valid.

Compatible Numbers Compatible numbers are numbers whose sums, differ- ences, products, or quotients are easy to calculate mentally. Examples of compatible

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132 Chapter 4 Whole-Number Computation—Mental, Electronic, and Written

numbers are 86 and 14 under addition (since 86 14 100+ = ), 25 and 8 under multi- plication (since 25 8 200× = ), and 600 and 30 under division (since 600 30 20÷ = ). In part (a) of Example 4.1, adding 15 to 25 produces a number, namely 40, that is easy to add to 27. Notice that numbers are compatible with respect to an operation. For example, 86 and 14 are compatible with respect to addition but not with respect to multiplication.

Calculate the following mentally using properties and/or com- patible numbers.

a. ( )4 13 25× × b.1710 9÷          c. 86 15×

S O L U T I O N

a. ( ) ( )4 13 25 13 4 25 1300× × = × × = b.1710 9 1800 90 9 1800 9 90 9 200 10 190÷ = − ÷ = ÷ − ÷ = − =( ) ( ) ( ) c. 86 15 86 10 86 5 860 430 1290× = × + × = + =( ) ( ) (Notice that 86 5× is half of

86 10× .) ■

Compensation The sum 43 38 17+ +( ) can be viewed as 38 60 98+ = using commutativity, associativity, and the fact that 43 and 17 are compatible numbers. Finding the answer to 43 36 19+ +( ) is not as easy. However, by reformulating the sum 36 19+ mentally as 37 18+ , we obtain the sum ( )43 37 18 80 18 98+ + = + = . This process of reformulating a sum, difference, product, or quotient to one that is more readily obtained mentally is called compensation. Some specific techniques using compensation are introduced next.

In the computations of Example 4.1(d), 98 was increased by 2 to 100 and then 59 was decreased by 2 to 57 (a compensation was made) to maintain the same sum. This technique, additive compensation, is an application of associativity. Similarly, additive compensation is used when 98 59+ is rewritten as 97 60+ or 100 57+ . The problem 47 29− can be thought of as 48 30 18− =( ). This use of compensation in subtraction is called the equal additions method since the same number (here 1) is added to both 47 and 29 to maintain the same difference. This compensation is performed to make the subtraction easier by subtracting 30 from 48. The product 48 5× can be found using multiplicative compensation as follows: 48 5 24 10 240× = × = . Here, again, associativity can be used to justify this method.

Left-to-Right Methods To add 342 and 136, first add the hundreds ( )300 100+ , then the tens ( )40 30+ , and then the ones ( )2 6+ , to obtain 478. To add 158 and 279, one can think as follows: 100 200 300+ = , 300 50 70 420+ + = , 420 8 9 437+ + = . Alternatively, 158 279+ can be found as follows: 158 200 358+ = , 358 70 428+ = , 428 9 437+ = . Subtraction from left to right can be done in a similar manner. Research has found that people who are excellent mental calculators utilize this left-to-right method to reduce memory load, instead of mentally picturing the usual right-to-left written method. The multiplication problem 3 123× can be thought of mentally as 3 100 3 20 3 3× + × + × using distributivity. Also, 4 253× can be thought of mentally as 800 200 12 1012+ + = or as 4 250 4 3 1000 12 1012× + × = + = .

Multiplying Powers of 10 These special numbers can be multiplied mentally in either standard or exponential form. For example, 100 1000 100 000× = , ,10 10 104 5 9× = , 20 300 6000× = , and 12 000 110 000 12 11 10 1 320 000 0007, , , , ,× = × × = .

Multiplying by Special Factors Numbers such as 5, 25, and 99 are regarded as special factors because they are convenient to use mentally. For example, since 5 10 2= ÷ , we have 38 5 38 10 2 380 2 190× = × ÷ = ÷ = . Also, since 25 100 4= ÷ , 36 25 3600 4 900× = ÷ = . The product 46 99× can be thought of as 46 100 1 4600 46 4554( )− = − = . Also, dividing by 5 can be viewed as dividing by 10, then multiplying by 2. Thus 460 5 460 10 2 46 2 92÷ = ÷ × = × =( ) .

Reflection from Research Recent research shows that com- putational fluency and number sense are intimately related. These tend to develop together and facilitate the learning of the other (Griffin, 2003).

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133

From Chapter 3, Lesson 1 “Take Apart Tens to Add” from My Math, Volume 1 Common Core State Standards, Grade 2, copyright © 2013 by McGraw-Hill Education,

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134 Chapter 4 Whole-Number Computation—Mental, Electronic, and Written

Computational Estimation The process of estimation takes on various forms. The number of beans in a jar may be estimated using no mathematics, simply a “guesstimate.” Also, one may estimate how long a trip will be, based simply on experience. Computational estimation is the process of finding an approximate answer (an estimate) to a computation, often using mental math. With the use of calculators becoming more commonplace, computa- tional estimation is an essential skill. Next we consider various types of computa- tional estimation.

Front-End Estimation Three types of front-end estimation will be demonstrated.

Range Estimation Often it is sufficient to know an interval or range—that is, a low value and a high value—that will contain an answer. The following example shows how ranges can be obtained in addition and multiplication.

NCTM Standard All students should develop and use strategies to estimate the results of whole-number compu- tations and to judge the reason- ableness of such results.

Reflection from Research Students can develop a bet- ter understanding of number through activities involving esti- mation (Leutzinger, Rathmell, & Urbatsch, 1986).

Calculate mentally using the indicated method.

a.197 248+ using additive compensation b.125 44× using multiplicative compensation c. 273 139− using the equal additions method d. 321 437+ using a left-to-right method e. 3 432× using a left-to-right method f. 456 25× using the multiplying by a special factor method

S O L U T I O N

a.197 248 197 3 245 200 245 445+ = + + = + = b.125 44 125 4 11 500 11 5500× = × × = × = c. 273 139 274 140 134− = − = d. 321 437 758+ = [Think: ( ) ( ) ( )300 400 20 30 1 7+ + + + + ] e. 3 432 1296× = [Think: ( ) ( ) ( )3 400 3 30 3 2× + × + × ] f. 456 25 114 100 11 400× = × = , (Think: 25 4 100× = . Thus 456 25 114 4 25× = × ×

= × =114 100 11 400, .) ■

Check for Understanding: Exercise/Problem Set A #1–7✔

Use estimation to answer the following two questions: 1) How much blood does the average heart pump in a day? 2) Is $10 enough to buy items with the following prices? $1.49, $0.99, $2.14, $3.23, $1.33, $0.25.

How is the type of estimation used to solve these two problems different?

Find a range for answers to these computations by using only the leading digits.

a. 257

576+          b. 294

53×

S O L U T I O N

a. Sum Low Estimate High Estimate

257

576+ 200

500 700

+ 300

600 900

+

Children’s Literature www.wiley.com/college/musser See “How Much, How Many, How Far, How Heavy, How Long, How Tall Is 1000?’’ by Helen Nolan.

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Section 4.1 Mental Math, Estimation, and Calculators 135

One-Column/Two-Column Front-End We can estimate the sum 498 251+ using the one-column front-end estimation method as follows: To estimate 498 251+ , think 400 200 600+ = (the estimate). Notice that this is simply the low end of the range estimate. The one-column front-end estimate always provides low estimates in addition problems as well as in multiplication problems. In the case of 376 53 417+ + , the one-column estimate is 300 400 700+ = , since there are no hundreds in 53. The two-column front-end estimate also provides a low estimate for sums and products. However, this estimate is closer to the exact answer than one obtained from using only one column. For example, in the case of 376 53 417+ + , the two-column front- end estimation method yields 370 50 410 830+ + = , which is closer to the exact answer 842 than the 700 obtained using the one-column method.

Front-End with Adjustment This method enhances the one-column front-end esti- mation method. For example, to find 498 251+ , think 400 200 600+ = and 98 51+ is about 150. Thus the estimate is 600 150 750+ = . Unlike one-column or two-column front-end estimates, this technique may produce either a low estimate or a high esti- mate, as in this example.

Keep in mind that one estimates to obtain a “rough” answer, so all of the preced- ing forms of front-end estimation belong in one’s estimation repertoire.

Algebraic Reasoning Estimation is a valuable tool of algebraic reasoning. It can be used to determine if a solution to an equation seems reasonable for the situation.

Thus a range for the answer is from 700 to 900. Notice that you have to look at only the digits having the largest place values (2 5 7+ = , or 700) to arrive at the low estimate, and these digits each increased by one (3 6 9+ = , or 900) to fi nd the high estimate.

b. Product Low Estimate High Estimate

294

53×

200

50 10 000

× ,

300

60 18000

× ,

Due to the nature of multiplication, this method gives a wide range, here 10,000 to 18,000. Even so, this method will catch many errors. ■

Estimate using the method indicated.

a. 503 813× using one-column front-end b. 1200 35× using range estimation c. 4376 1889− using two-column front-end d. 3257 874+ using front-end adjustment

S O L U T I O N

a. To estimate 503 813× using the one-column front-end method, think 500 800× = 400 000, . Using words, think “5 hundreds times 8 hundreds is 400,000.”

b. To estimate a range for 1200 35× , think 1200 30 36 000× = , and 1200 40 48000× = , . Thus a range for the answer is from 36,000 to 48,000. One also could use 1000 30 30 000× = , and 2000 40 80 000× = , . However, this yields a wider range.

c. To estimate 4376 1889− using the two-column front-end method, think 4300 − 1800 2500= . You also can think 43 18 25− = and then append two zeros after the 25 to obtain 2500.

d. To estimate 3257 874+ using front-end with adjustment, think 3000, but since 257 874+ is about 1000, adjust to 4000. ■

Rounding Rounding is perhaps the best-known computational estimation tech- nique. The purpose of rounding is to replace complicated numbers with simpler numbers. Here, again, since the objective is to obtain an estimate, any of several rounding techniques may be used. However, some may be more appropriate than

Reflection from Research To be confident and successful estimators, children need numer- ous opportunities to practice estimation and to learn from their experiences. Thus, the skills required by students to develop a comfort level with holistic, or range-based, estimation need to be included throughout a young child’s education (Onslow, Adams, Edmunds, Waters, Chapple, Healey, & Eady, 2005).

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136 Chapter 4 Whole-Number Computation—Mental, Electronic, and Written

others, depending on the word problem situation. For example, if you are estimating how much money to take along on a trip, you would round up to be sure that you had enough. When calculating the amount of gas needed for a trip, one would round the miles per gallon estimate down to ensure that there would be enough money for gas. Unlike the previous estimation techniques, rounding is often applied to an answer as well as to the individual numbers before a computation is performed.

Several different methods of rounding are illustrated next. What is common, how- ever, is that each method rounds to a particular place. You are asked to formulate rules for the following methods in the problem set.

Round Up (Down) The number 473 rounded up to the nearest tens place is 480 since 473 is between 470 and 480 and 480 is above 473 (Figure 4.1). The number 473 rounded down to the nearest tens place is 470. Rounding down is also called truncat- ing (truncate means “to cut off”). The number 1276 truncated to the hundreds place is 1200.

Round a 5 Up The most common rounding technique used in schools is the round a 5 up method. This method can be motivated using a number line. Suppose that we wish to round 475 to the nearest ten (Figure 4.2).

Figure 4.2 Since 475 is midway between 470 and 480, we have to make an agreement

concerning whether we round 475 to 470 or to 480. The round a 5 up method always rounds such numbers up, so 475 rounds to 480. In the case of the numbers 471 to 474, since they are all nearer 470 than 480, they are rounded to 470 when rounding to the nearest ten. The numbers 476 to 479 are rounded to 480.

One disadvantage of this method is that estimates obtained when several 5s are involved tend to be on the high side. For example, the “round a 5 up” to the nearest ten estimate applied to the addends of the sum 35 45 55 65+ + + yields 40 50 60 70 220+ + + = , which is 20 more than the exact sum 200. Round to the Nearest Even Rounding to the nearest even can be used to avoid errors of accumulation in rounding. For example, if 475 545 1020+ =( ) is estimated by rounding up to the tens place or rounding a 5 up, the answer is 480 550 1030+ = . By rounding down, the estimate is 470 540 1010+ = . Since 475 is between 480 and 470 and the 8 in the tens place is even, and 545 is between 550 and 540 and the 4 is even, round to the nearest even method yields 480 540 1020+ = .

Figure 4.1

Estimate using the indicated method. (The symbol “≈” means “is approximately.”)

a. Estimate 2173 4359+ by rounding down to the nearest hundreds place. b. Estimate 3250 1850− by rounding to the nearest even hundreds place. c. Estimate 575 398− by rounding a 5 up to the nearest tens place.

S O L U T I O N

a. 2173 4359 2100 4300 6400+ ≈ + = b. 3250 1850 3200 1800 1400− ≈ − = c. 575 398 580 400 180− ≈ − = ■

Round to Compatible Numbers Another rounding technique can be applied to esti- mate products such as 26 37× . A reasonable estimate of 26 37× is 25 40 1000× = . The numbers 25 and 40 were selected since they are estimates of 26 and 37, respectively, and are compatible with respect to multiplication. (Notice that the rounding up tech- nique would have yielded the considerably higher estimate of 30 40 1200× = , whereas the exact answer is 962.) This round to compatible numbers technique allows one to round either up or down to compatible numbers to simplify calculations, rather

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Section 4.1 Mental Math, Estimation, and Calculators 137

than rounding to specified places. For example, a reasonable estimate of 57 98× is 57 100 5700× =( ). Here, only the 98 needed to be rounded to obtain an estimate mentally. The division problem 2716 75÷ can be estimated mentally by considering 2800 70 40÷ =( ). Here 2716 was rounded up to 2800 and 75 was rounded down to 70 because 2800 70÷ easily leads to a quotient since 2800 and 70 are compatible numbers with respect to division.

Estimate by rounding to compatible numbers in two different ways.

a. 43 21× b. 256 33÷

S O L U T I O N

a. 43 21 40 21 840× ≈ × = b. 256 33 240 30 8÷ ≈ ÷ = 43 21 43 20 860× ≈ × = 256 33 280 40 7÷ ≈ ÷ =

(The exact answer is 903.) (The exact answer is 7 with remainder 25.) ■

Reflection from Research Good estimators use three com- putational processes. They make the number easier to manage (possibly by rounding), change the structure of the problem itself to make it easier to carry out, and compensate by making adjust- ments in their estimation after the problem is carried out (Reys, Rybolt, Bestgen, & Wyatt, 1982).

Rounding is a most useful and flexible technique. It is important to realize that the main reasons to round are (1) to simplify calculations while obtaining reasonable answers and (2) to report numerical results that can be easily understood. Any of the methods illustrated here may be used to estimate.

The ideas involving mental math and estimation in this section were observed in children who were facile in working with numbers. The following suggestions should help develop number sense in all children:

1. Learn the basic facts using thinking strategies, and extend the strategies to multi- digit numbers.

2. Master the concept of place value.

3. Master the basic addition and multiplication properties of whole numbers.

4. Develop a habit of using the front-end and left-to-right methods.

5. Practice mental calculations often, daily if possible.

6. Accept approximate answers when exact answers are not needed.

7. Estimate prior to doing exact computations.

8. Be flexible by using a variety of mental math and estimation techniques.

Check for Understanding: Exercise/Problem Set A #8–15✔

Using a Calculator Although a basic calculator that costs less than $10 is sufficient for most elementary school students, there are features on $15 to $30 calculators that simplify many complicated calculations. Also, most smart phones have basic as well as scientific calculators. When we use the word calculator throughout the book, it may encom- pass many of these types of calculators. The TI-34 MultiView, manufactured by Texas Instruments, is shown in Figure 4.3. The TI-34 MultiView, which is designed especially for elementary and middle schools, performs fraction as well as the usual decimal calculations, can perform long division with remainders directly, and has the functions of a scientific calculator. One nice feature of the TI-34 MultiView is that it has mutiple lines of display, which allows the student to see the input and output at the same time.

The on key turns the calculator on. The delete key is an abbreviation for “delete” and allows the user to delete one character at a time from the right if the cursor is at the end of an expression or delete the character under the cursor. Pressing the clear key will clear the current entry. The previous entry can be retrieved by pressing the m on the upper right key.

Two types of logic are available on common calculators: arithmetic and algebraic.

V in

ce nt

L aR

us sa

/J oh

n W

ile y

an d

S on

s

Figure 4.3

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138 Chapter 4 Whole-Number Computation—Mental, Electronic, and Written

(NOTE: For ease of reading, we will write numerals without the usual squares around them to indicate that they are keys.)

Arithmetic Logic In arithmetic logic, the calculator performs operations in the order they are entered. For example, if 3 4 5+ × = is entered, the calculations are performed as follows: ( )3 4 5 7 5 35+ × = × = . That is, the operations are performed from left to right as they are entered.

Algebraic Logic If your calculator has algebraic logic and the expression 3 + 4 5× = is entered, the result is different; here the calculator evaluates expressions according to the usual mathematical convention for order of operations.

If a calculator has parentheses, they can be inserted to be sure that the desired operation is performed first. In a calculator using algebraic logic, the calculation 13 5 4 2 7− × ÷ + will result in 13 10 7 10− + = . If one wishes to calculate 13 5− first, parentheses must be inserted. Thus ( )13 5 4 2 7 23− × ÷ + = .

Reflection from Research Students in classrooms where calculators are used tend to have more positive attitudes about mathematics than students in classrooms where calculators are not used (Reys & Reys, 1987).

Common Core – Mathematical Practices Use appropriate tools strategically. Proficient students are sufficiently familiar with tools appropriate for their grade or course to make sound decisions about when each of these tools might be helpful, recognizing both the insight to be gained and their limitations.

Use the order of operations to mentally calculate the following and then enter them into a calculator to compare results.

a. ( )4 2 5 7 3+ × ÷ + b. 8 2 3 22 2÷ + × c. 17 4 5 2− −( ) d. 40 5 2 2 33÷ × − ×

S O L U T I O N

a. ( ) ( )

( )

4 2 5 7 3 4 10 7 3

14 7 3

2 3 5

+ × ÷ + = + ÷ + = ÷ + = + =

b. 8 2 3 2 8 4 3 4

2 12 14

2 2÷ + × = ÷ + × = + =

( ) ( )

c. 17 4 5 2 17 4 3

17 12 5

− − = − × = − =

( ) ( )

d. 40 5 2 2 3 40 5 2 2 3

8 8 2 3

64 6 58

3 3÷ × − × = ÷ × − × = × − × = − =

[( ) ] ( )

■

Now let’s consider some features that make a calculator helpful both as a compu- tational and a pedagogical device. Several keystroke sequences will be displayed to simulate the variety of calculator operating systems available.

Parentheses As mentioned earlier when we were discussing algebraic logic, one must always be attentive to the order of operations when several operations are pres- ent. For example, the product 2 3 4× +( ) can be found in two ways. First, by using commutativity, the following keystrokes will yield the correct answer:

3 4 2 14+ = × = .

Alternatively, the parentheses keys may be used as follows:

2 3 4 14× + =( ) .

Parentheses are needed, since pressing the keys 2 3 4× + = on a calculator with algebraic logic will result in the answer 10. Distributivity may be used to simplify calculations. For example, 753 8 753 9¥ ¥+ can be found using 753 8 9× + =( ) instead of 753 8 753 9× + × = . The enter key on the TI-34 MultiView is often used in place of an equals key.

Reflection from Research Young people in the workplace believe that menial tasks, such as calculations, are low-order tasks that should be undertaken by technology and that their role is to identify problems and solve them using technology to sup- port that solution. Thus, math- ematics education may need to reshift its focus from accuracy and precision relying on arduous calculations to one that will bet- ter fit the contemporary work- place and life beyond schools (Zevenbergen, 2004).

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Section 4.1 Mental Math, Estimation, and Calculators 139

Constant Functions In Chapter 3, multiplication was viewed as repeated addi- tion; in particular, 5 3 3 3 3 3 3 15× = + + + + = . Repeated operations are carried out in different ways depending on the model of calculator. For example, the following keystroke sequence is used to calculate 5 3× on one calculator that has a built-in constant function:

3 15+ = = = = .

Numbers raised to a whole-number power can be found using a similar technique. For example, 35 can be calculated by replacing the + with a × in the preceding examples. A constant function can also be used to do repeated subtraction to find a quotient and a remainder in a division problem. For example, the following sequence can be used to find 35 8÷ :

35 8 3− = = = = .

The remainder (3 here) is the first number displayed that is less than the divisor (8 here), and the number of times the equal sign was pressed is the quotient (4 here).

Because of the multiple lines of display, the TI-34 MultiView can handle most of these examples by typing in the entire expression. For example, representing 5 3× as repeated addition is entered into the TI-34 MultiView as

3 3 3 3 3+ + + + enter

The entire expression of 3 3 3 3 3+ + + + appears on the first line of display and the result 15 appears on the end of the same line of display when mode is NORM.

Exponent Keys There are three common types of exponent keys: x y2 2, , and ∧ . The x2 key is used to find squares in one of two ways:

3 9 3 92 2x x= or .

The yx and ∧ keys are used to find more general powers and have similar key- strokes. For example, 73 may be found as follows:

7 3 343 7 3 343yx = ∧ or .

Memory Functions Many calculators have a memory function designated by the keys M+ , M − , MR , or STO , RCL , SUM . Your calculator’s display will probably show an “M” to remind you that there is a nonzero number in the memory. The problem 5 9 7 8× + × may be found as follows using the memory keys:

5 9 7 8 101× = + × = +M M MR or

5 9 7 8 101× = × =SUM SUM RCL .

It is a good practice to clear the memory for each new problem using the all clear key. The TI-34 MultiView has seven memory locations—x, y, z, t, a, b, c—which can be

accessed by pressing the stoc key and then pressing the x yztabc key multiple times to select the desired variable. The following keystrokes are used to evaluate the expres- sion above.

5 9 7 8× ×sto and stoc cx yztabc x yzt abc x

yzt abcenter enter

The above keys will store the value 45 in the memory location x and the value 56 in the memory location y. The values x and y can then be added together as follows.

2 2nd enter nd enter enterrecall + recall .

This will show 45 56+ on the left of the calculator display and 101 on the right of the display.

Additional special keys will be introduced throughout the book as the need arises.

Scientific Notation Input and output of a calculator are limited by the number of places in the display (generally 8, 10, or 12). Two basic responses are given when

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140 Chapter 4 Whole-Number Computation—Mental, Electronic, and Written

a number is too large to fit in the display. Simple calculators either provide a par- tial answer with an “E” (for “error”), or the word “ERROR” is displayed. Many scientific calculators automatically express the answer in scientific notation (that is, as the product of a decimal number greater than or equal to 1 but less than 10, and the appropriate power of 10). For example, on the TI-34 MultiView, the product of 123,456,789 and 987 is displayed 1 218518507 1011. × (NOTE: Scientific notation is discussed in more detail in Chapter 9 after decimals and negative numbers have been studied.) If an exact answer is needed, the use of the calculator can be com- bined with paper and pencil and distributivity as follows:

123 456 789 987 123 456 789 900 123 456 789

80 123 456 789

, , , , , ,

, ,

× = × + × + ××

= + +

7

111 111 110 100 9 876 543 120

864 197 523

, , , , , ,

, , .

Now we can obtain the product by adding:

111 111 110 100

9 876 543 120

864 197 523

121 851 850 7

, , ,

, , ,

, ,

, , ,

+

443

Calculations with numbers having three or more digits will probably be performed on a calculator (or computer) to save time and increase accuracy. Even so, it is pru- dent to estimate your answer when using your calculator.

Check for Understanding: Exercise/Problem Set A #16–24✔

Shakuntala Devi (1929-2013), an Indian mathematical wizard, was dubbed ‘the human computer’ because she could calculate swiftly mentally. At six, she performed at circuses and other road shows and became the source of support for her family. In 1977, she found the 23rd root of a 201-digit number in 50 seconds while a Univac computer required 62 seconds. In 1982, she earned a place in Guinness Book of World Records for mentally multiplying two 13 digit numbers in 28 seconds. She also had careers in astrology and as a cookbook author and a novelist.

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EXERCISES 1. Calculate mentally using properties. a. ( )37 25 43+ +        b. 47 15 47 85¥ ¥+ c. ( )4 13 25× ×           d. 26 24 21 24¥ ¥−

2. Find each of these differences mentally using equal additions. Write out the steps that you thought through.

a. 43 17−                   b. 62 39− c. 132 96−                 d. 250 167−

3. Calculate mentally left to right. a. 123 456+ b. 342 561+ c. 587 372− d. 467 134−

4. Calculate mentally using the indicated method. a. 198 387+ (additive compensation) b. 84 5× (multiplicative compensation) c. 99 53× (special factor) d. 4125 25÷ (special factor)

EXERCISE/PROBLEM SET A

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Section 4.1 Mental Math, Estimation, and Calculators 141

5. Calculate mentally. a. 58000 5 000 000, , ,× b. 7 10 210005× × , c. 13000 7 000 000, , ,× d. 4 10 3 10 7 105 6 3× × × × × e. 5 10 7 10 4 103 7 5× × × × × f. 17 000 000 6 000 000 000, , , , ,×

6. The sum 26 38 55+ + can be found mentally as follows: 26 30 56+ = , 56 8 64+ = , 64 50 114+ = , 114 5 119+ = . Find the following sums mentally using this technique.

a. 32 29 56+ + b. 54 28 67+ + c. 19 66 49+ + d. 62 84 27 81+ + +

7. In division you can sometimes simplify the problem by multiplying or dividing both the divisor and dividend by the same number. This is called division compensation. For example,

72 12 72 2 12 2 36 6 6÷ = ÷ ÷ ÷ = ÷ =( ) ( ) and 145 5 145 2 5 2 290 10 29÷ = × ÷ × = ÷ =( ) ( )

Calculate the following mentally using this technique. a. 84 14÷       b. 234 26÷       c. 120 15÷       d. 168 14÷

8. Before granting an operating license, a scientist has to estimate the amount of pollutants that should be allowed to be discharged from an industrial chimney. Should she overestimate or underestimate? Explain.

9. Estimate each of the following using the four front-end methods: (i) range, (ii) one-column, (iii) two-column, and (iv) with adjustment.

a. 3741 1252+

b. 1591 346

589

163+

c. 2347 58

192

5783+

10. Find a range estimate for these products. a. 37 24×         b. 157 231×         c. 491 8×

11. Estimate using compatible number estimation. a. 63 97× b. 51 212× c. 3112 62÷ d. 103 87× e. 62 58× f. 4254 68÷

12. Round as specified. a. 373 to the nearest tens place b. 650 using round a 5 up method to the hundreds place c. 1123 up to the tens place d. 457 to the nearest tens place e. 3457 to the nearest thousands place

13. Cluster estimation is used to estimate sums and products when several numbers cluster near one number. For example, the addends in 789 810 792+ + cluster around 800. Thus 3 800 2400× = is a good estimate of the sum. Estimate the following using cluster estimation.

a. 347 362 354 336+ + + b. 61 62 58× × c. 489 475 523 498× × × d. 782 791 834 812 777+ + + +

14. Here are four ways to estimate 26 12× :

26 10 260 30 12 360× = × = 25 12 300 30 10 300× = × =

Estimate the following in four ways. a. 31 23× b. 35 46× c. 48 27× d. 76 12×

15. Estimate the following values and check with a calculator. a. 656 74× is between _____ 000 and _____ 000. b. 491 3172× is between _____ 00000 and _____ 00000. c. 1432 is between _____ 0000 and _____ 0000.

16. Guess what whole numbers can be used to fill in the blanks. Use your calculator to check.

a. _____ 6 = 4096 b. _____ 4 = 28,561

17. Guess which is larger. Check with your calculator. a. 54 or 45 ? b. 73 or 37 ?

c. 74 or 47 ? d. 63 or 36 ?

18. Some products can be found most easily using a combination of mental math and a calculator. For example, the product 20 47 139 5× × × can be found by calculating 47 139× on a calculator and then multiplying your result by 100 mentally ( )20 5 100× = . Calculate the following using a combination of mental math and a calculator.

a. 17 25 817 4× × × b. 98 2 673 5× × × c. 674 50 889 4× × × d. 783 8 79 125× × ×

19. Compute the quotient and remainder (a whole number) for the following problems on a calculator without using repeated subtraction. Describe the procedure you used.

a. )8 103 b. )17 543 c. )123 849 d. )894 107 214, 20. 1233 12 332 2= + and 8833 88 332 2= + . How about 10,100

and 5,882,353? (Hint: Think 588 2353.)

21. Determine whether the following equation is true for n = 1 2, , or 3.

1 6 8 2 4 9n n n n n n+ + = + +

22. Notice that 153 1 5 33 3 3= + + . Determine which of the fol- lowing numbers have the same property.

a. 370 b. 371 c. 407

23. Determine whether the following equation is true when n is 1, 2, or 3.

1 4 5 5 6 9

2 3 3 7 7 8

n n n n n n

n n n n n n

+ + + + + = + + + + +

24. Simplify the following expressions using a calculator. a. 135 7 48 33− −( ) b. 32 512¥ c. 13 34 3 45 372 3¥ − −( )

d. 26 4 31 172 2 401 26 31 172

3 2 2

3 2

+ − + −

( ) ( )

¥ ¥

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142 Chapter 4 Whole-Number Computation—Mental, Electronic, and Written

25. Five of the following six numbers were rounded to the nearest thousand, then added to produce an estimated sum of 87,000. Which number was not included?

5228 14 286 7782 19 628 9168 39 228, , ,

26. True or false?

493 827 156 246 913 578 987 654 3122, , , , , ,= ×

27. Place a multiplication sign or signs so that the product in each problem is correct; for example, in 1 2 3 4 5 6 = 41,472, the multiplication sign should be between the 2 and the 3 since 12 3456 41 472× = , .

a. 1 3 5 7 9 0 = 122,130 b. 6 6 6 6 6 6 = 439,956 c. 7 8 9 3 4 5 6 = 3,307,824 d. 1 2 3 4 5 6 7 = 370,845

28. Develop a method for finding a range for subtraction problems for three-digit numbers.

29. Find 13 333 3332, , .

30. Calculate 99 36¥ and 99 23¥ and look for a pattern. Then predict 99 57¥ and 99 63¥ mentally and check your predic- tions with a calculator.

31. a. Calculate 25 35 452 2 2, , , and 552 and look for a pattern. Then find 65 752 2, , and 952 mentally and check your answers.

b. Using a variable, prove that your result holds for squar- ing numbers that have a 5 as their ones digit.

32. George Bidder was a calculating prodigy in England dur- ing the nineteenth century. As a nine-year-old, he was asked: If the moon were 123,256 miles from the Earth and sound traveled at the rate of 4 miles a minute, how long would it be before inhabitants of the moon could hear the battle of Waterloo? His answer—21 days, 9 hours, 34 minutes—was given in 1 minute. Was he correct? Try to do this calculation in less than 1 minute using a calculator. (NOTE: The moon is about 240,000 miles from Earth and sound travels about 12.5 miles per second.)

33. Found in a newspaper article: What is

241, 573, 142, 393, 627, 673, 576, 957, 439, 048 × 45, 994, 811, 347, 886, 846, 310, 221, 728, 895, 223, 034, 301, 839?

The answer is 71 consecutive 1s—one of the biggest num- bers a computer has ever factored. This factorization bested the previous high, the factorization of a 69-digit number. Find a mistake here, and suggest a correction.

34. Some mental calculators use the following fact: ( )( )a b a b a b+ − = −2 2. For example, 43 37× = ( )( )40 3 40 3 40 3 1600 9 15912 2+ − = − = − = . Apply this technique to find the following products mentally.

a. 54 46× b. 81 79× c. 122 118× d. 1210 1190×

35. Fermat claimed that

100 895 598 169 898 423 112 303, , , , , .= ×

Check this on your calculator.

36. Show how to find 439 268 6852, × using a calculator that displays only eight digits.

37. Insert parentheses (if necessary) to obtain the following results.

a. 76 54 97 11 476× + = , b. 4 13 27042× = c. 13 59 47 163 6202+ × = , d. 79 43 2 17 3072− ÷ + =

38. a. Find a shortcut.

24 26 624× = 62 68 4216× = 73 77 5621× = 41 49 2009× = 86 84 7224× = 57 53× =

b. Prove that your result works in general. c. How is this problem related to Problem 31?

39. Develop a set of rules for the round a 5 up method.

40. There are eight consecutive odd numbers that when multi- plied together yield 34,459,425. What are they?

41. Jill goes to get some water. She has a 5-liter pail and a 3-liter pail and is supposed to bring exactly 1 liter back. How can she do this?

PROBLEMS

EXERCISES 1. Calculate mentally using properties. a. 52 14 52 4¥ ¥− b. ( )5 37 20× × c. ( )56 37 44+ + d. 23 4 23 5 7 9¥ ¥ ¥+ +

2. Find each of these differences mentally using equal additions. Write out the steps that you thought through.

a. 56 29− b. 83 37− c. 214 86− d. 542 279−

3. Calculate mentally using the left-to-right method. a. 246 352+ b. 49 252+ c. 842 521− d. 751 647−

4. Calculate mentally using the method indicated. a. 359 596+ (additive compensation) b. 76 25× (multiplicative compensation) c. 37 98× (special factor) d. 1240 5÷ (special factor)

EXERCISE/PROBLEM SET B

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Section 4.1 Mental Math, Estimation, and Calculators 143

PROBLEMS

5. Calculate mentally. a. 32 000 400, × b. 6000 12 000× , c. 4000 5000 70× × d. 5 10 30 104 5× × × e. 12 000 4 107, × × f. 23000 000 5 000 000, , , ,×

6. Often subtraction can be done more easily in steps. For example, 43 37− can be found as follows: 43 37 43 30 7 13 7 6− = − − = − =( ) . Find the following differences using this technique.

a. 52 35− b. 173 96− c. 241 159− d. 83 55−

7. The halving and doubling method can be used to multi- ply two numbers when one factor is a power of 2. For example, to find 8 17× , find 4 34× or 2 68 136× = . Find the following products using this method.

a. 16 21×          b. 4 72×         c. 8 123×         d. 16 211×

8. In determining an evacuation zone, a scientist must esti- mate the distance that lava from an erupting volcano will flow. Should she overestimate or underestimate? Explain.

9. Estimate each of the following using the four front-end methods: (i) one-column, (ii) range, (iii) two-column, and (iv) with adjustment.

a. 4652 8134+

b. 2659 3752

79

143+

c. 15923 672

2341

251+

10. Find a range estimate for these products. a. 57 1924× b. 1349 45× c. 547 73 951× ,

11. Estimate using compatible number estimation. a. 84 49× b. 5527 82÷ c. 2315 59÷ d. 78 81× e. 207 73× f. 6401 93÷

12. Round as specified. a. 257 down to the nearest tens place b. 650 to the nearest even hundreds place c. 593 to the nearest tens place d. 4157 to the nearest hundreds place e. 7126 to the nearest thousands place

13. Estimate using cluster estimation. a. 547 562 554 556+ + +           b. 31 32 35 28× × × c. 189 175 193 173+ + +            d. 562 591 634× ×

14. Estimate the following products in two different ways and explain each method.

a. 52 39× b. 17 74× c. 88 11× d. 26 42×

15. Estimate the following values and check with a calculator. a. 324 56× is between _____ 000 and _____ 000.

b. 5714 13× is between _____ 0000 and _____ 0000. c. 2563 is between _____ 000000 and _____ 000000.

16. Guess what whole numbers can be used to fill in the blanks. Use your calculator to check.

a. _____ 4 = 6561 b. _____ 5 = 16,807

17. Guess which is larger. Check with your calculator. a. 64 or 55 ? b. 38 or 46 ? c. 54 or 93 ? d. 85 or 66 ?

18. Compute the following products using a combination of mental math and a calculator. Explain your method.

a. 20 14 39 5× × × b. 40 27 25 23× × × c. 647 50 200 89× × × d. 25 91 2 173 2× × × ×

19. Find the quotient and remainder using a calculator. Check your answers.

a. 18 114 37, ÷ b. 381 271 147, ÷ c. 9 346 870 1349, , ÷ d. 817 293 749, ÷

20. Check to see that 1634 1 6 3 44 4 4 4= + + + . Then determine which of the following four numbers satisfy the same property.

a. 8208 b. 9474 c. 1138 d. 2178

21. It is easy to show that 3 4 52 2 2+ = , and 5 12 132 2 2+ = . However, in 1966 two mathematicians claimed the following:

27 84 110 133 1445 5 5 5 5+ + + = .

True or false?

22. Verify the following patterns.

3 4 5

10 11 12 13 14

21 22 23 24 25 26 27

2 2 2

2 2 2 2 2

2 2 2 2 2 2 2

+ = + + = +

+ + + = + +

23. For which of the values n = 1 2 3 4, , , is the following true?

1 5 8 12 18 19

2 3 9 13 16 20

n n n n n n

n n n n n n

+ + + + + = + + + + +

24. Simplify each of the following expressions using a calculator.

a. 42 63 52 232 2¥ ¥−

b. 72 43 31 162 2 3 27 115

3

2 0

( ) ( ) −

− +¥

c. 35 24 26 14 22 2 5 3− + − ÷( )

d. 14 23 36 12 33 4 3 2× − ÷ ¥

25. Notice how by starting with 55 and continuing to raise the dig- its to the third power and adding, 55 reoccurs in three steps.

55 5 5 250 2 5

133 1 3 3 55

3 3 3 3

3 3 3

⎯ →⎯ + = ⎯ →⎯ +

= ⎯ →⎯ + + =

Check to see whether this phenomenon is also true for these three numbers:

a. 136 b. 160 c. 919

26. What is interesting about the quotient obtained by divid- ing 987,654,312 by 8? (Do this mentally.)

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144 Chapter 4 Whole-Number Computation—Mental, Electronic, and Written

27. Using distributivity, show that ( ) .a b a ab b− = − +2 2 22 How can this idea be used to compute the following squares mentally? (Hint: 99 100 1= − .)

a. 992 b. 9992 c. 99992

28. Fill in the empty squares to produce true equations.

29. Discuss the similarities/differences of using (i) special fac- tors and (ii) multiplicative compensation when calculating 36 5× mentally.

30. Find 166 666 6662, , .

31. Megan tried to multiply 712,000 by 864,000 on her calculator and got an error message. Explain how she can use her calculator (and a little thought) to find the exact product.

32. What is the product of 777,777,777 and 999,999,999?

33. Explain how you could calculate 342 143× even if the 3 and 4 keys did not work.

34. Find the missing products by completing the pattern. Check your answers with a calculator.

11 11 121

111 111 12321

1111 1111

11111 1

× = × =

× = ×

11111

111111 111111

= × =

35. Use your calculator to find the following products.

12 11 24 11 35 11× × ×

Look at the middle digit of each product and at the first and last digits. Write a rule that you can use to multiply by 11. Now try these problems using your rule and check your answers with your calculator.

54 11 62 11 36 11× × ×

Adapt your rule to handle

37 11 59 11 76 11× × × .

36. When asked to multiply 987,654,321 by 123,456,789, one mental calculator replied, “I saw in a flash that 987 654 321 81 80 000 000 001, , , , , ,× = so I multiplied 123,456,789 by 80,000,000,001 and divided by 81.” Determine whether his reasoning was correct. If it was, see whether you can find the answer using your calculator.

37. a. Find a pattern for multiplying the following pairs. 32 72 2304 43 63 2709

73 33 2409

× = × = × =

Try finding these products mentally.

17 97 56 56 42 62× × ×

b. Prove why your method works.

38. Have you always wanted to be a calculating genius? Amaze yourself with the following problems.

a. To multiply 4,109,589,041,096 by 83, simply put the 3 in front of it and the 8 at the end of it. Now check your answer.

b. After you have patted yourself on the back, see whether you can find a fast way to multiply

7 894 736 842 105 263 158 86, , , , , , . by

(NOTE: This works only in special cases.)

39. Develop a set of rules for the round to the nearest even method.

40. Find the ones digits. a. 210     b. 43210     c. 36     d. 2936

41. A magician had his subject hide an odd number of coins in one hand and an even number of coins in another. He then told the subject to multiply the number of coins in the right hand by 2 and the number of coins in the left hand by 3 and to announce the sum of these two numbers. The magician immediately then correctly stated which hand had the odd number of coins. How did he do it?

Analyzing Student Thinking

42. Jessica calculated 84 28− as 84 30 54− = and 54 2 56+ = . Is her method valid? Explain.

43. Kalil tells you that multiplying by 5 is a lot like dividing by 2. For example, 48 5 240× = , but it is easier just to go 48 2 24÷ = and then affix a zero at the end. Will his meth- od always work? Explain.

44. Kalil also says that dividing by 5 is the same as mul- tiplying by 2 and “dropping” a zero. Does this work? Explain.

45. To work the problem 125 5÷ , Paula did the following: ( ) ( ) .100 5 25 5 20 5 25÷ + ÷ = + = She said that she used a form of distributivity. Do you agree? Why or why not?

46. Alyssa was asked to find a reasonable range for the sum 158 547+ . She said 600 to 700. How should you respond?

47. Jared was asked which method would give the best esti- mate for addition of numbers with at least 3 digits: one- column front-end or two-column front-end. He said that it didn’t matter because an estimate is an estimate. Is he correct? Explain.

48. Nicole multiplied 3472 and 259 on her calculator and she wrote down the answer of 89,248. How can you see imme- diately using estimation that she made an error? Can you see how she made the error?

49. Kysha was asked to calculate the product of 987,654,321 and 123 on her 10 digit calculator. She said that it was impossible. How should you respond?

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Section 4.2 Written Algorithms for Whole-Number Operations 145

Section 4.1 was devoted to mental and calculator computation. This section presents the common written algorithms as well as some alternative ones that have historical interest and can be used to help students better understand how their algorithms work.

WRITTEN ALGORITHMS FOR WHOLE-NUMBER OPERATIONS

When a class of students was given the problem 48 35,+ the following three responses were typical of what the students did.

I combined 40 and

30 to get 70 8 + 5 = 13. 70 and 13 is 83

Nick

48

35 70

13 83

+ 8 + 5 is 13

Carry the 1 write down 3

4 + 3 + 1 = 8 write down 8

Trevor

48

35 83

1

+

48 plus 30 is 78.

Now I add the 5 and get 83.

Courtney

48

+ 35 83

Which of these methods demonstrates the best (least) understanding of place value? Which method is the best? Justify.

Algorithms for the Addition of Whole Numbers An algorithm is a systematic, step-by-step procedure used to find an answer, usu- ally to a computation. The common written algorithm for addition involves two main procedures: (1) adding single digits (thus using the basic facts) and (2) carrying (regrouping or exchanging).

A development of our standard addition algorithm is used in Figure 4.4 to find the sum 134 325+ .

+ +

Figure 4.4

Observe how the left-to-right sequence in Figure 4.4 becomes progressively more abstract. When one views the base ten pieces (1), the hundreds, tens, and ones are distinguishable due to their sizes and the number of each type of piece. In the chip abacus (2), the chips all look the same. However, representations are distinguished by the number of chips in each column and by the column containing the chips (i.e., place value). In the place-value representation (3), the numbers are distinguished by the digits and the place values of their respective columns. Representation (4) is the common “add in columns” algorithm. The place-value method can be justified using expanded form and properties of whole-number addition as follows.

Common Core – Grade 4 Fluently add and subtract mul- tidigit whole numbers using the standard algorithm.

Reflection from Research Students in upper grades can develop shortcut strategies for addition and subtraction, but younger students need careful instruction to support the discovery of such shortcut strategies (Torbeyns, De Smedt, Ghesquiere, & Vershaffel, 2009).

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146 Chapter 4 Whole-Number Computation—Mental, Electronic, and Written

134 325 1 10 3 10 4 3 10 2 10 5

1 10 3 10 3 10

2 2

2 2

+ = + + + + + = + + +

( ) ( )

( ) (

¥ ¥ ¥ ¥ ¥ ¥ ¥ 22 10 4 5

1 3 10 3 2 10 4 5

4 10 5 10 9

459

2

2

¥

¥ ¥

)

( )

( ) ( ) ( )

+ + = + + + + + = + + =

Expandeed form

Associativity and commutativity

Distributivity

Additionn

Simplified form

Note that 134 325+ can be found working from left to right (add the hundreds first, etc.) or from right to left (add the ones first, etc.).

An addition problem when regrouping is required is illustrated in Figure 4.5 to find the sum 37 46+ . Notice that grouping 10 units together and exchanging them for a long with the base ten pieces is not as abstract as exchanging 10 ones for 1 ten in the chip abacus. The abstraction occurs because the physical size of the 10 units is main- tained with the 1 long of the base ten pieces but is not maintained when we exchange 10 dots in the ones column for only 1 dot in the tens column of the chip abacus.

Figure 4.5

The procedure illustrated in the place-value representation can be refined in a series of steps to lead to our standard carrying algorithm for addition. Intermediate algorithms that build on the physical models of base ten pieces and place-value repre- sentations lead to our standard addition algorithm and are illustrated next.

(a) INTERMEDIATE (b) INTERMEDIATE (c) STANDARD ALGORITHM 1 ALGORITHM 2 ALGORITHM

568

394

12

150

800

962

+

sum of ones

sum of tens

sum of hundreds

final sum

568

394

12

15

8

962

+ 568

394 962

1 1

+

The preceding intermediate algorithms are easier to understand than the stan- dard algorithm. However, they are less efficient and generally require more time and space.

Throughout history many other algorithms have been used for addition. One of these, the lattice method for addition, is illustrated next.

Common Core – Grade 1 Understand that in adding two- digit numbers, one adds tens and tens, ones and ones; and some- times it is necessary to compose a ten.

Reflection from Research Students who initially used invented strategies for addition and subtraction demonstrated knowledge of base ten number concepts before students who relied primarily on standard algorithms (Carpenter, Franke, Jacobs, Fennema, & Empson, 1997).

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Section 4.2 Written Algorithms for Whole-Number Operations 147

Notice how the lattice method is very much like intermediate algorithm 2. Other interesting algorithms are contained in the problem sets.

Check for Understanding: Exercise/Problem Set A #1–11✔

Use base ten pieces to solve the problem 146 – 25. On a piece of paper, sketch the actions used with the pieces to solve the problem. How do the actions with the blocks relate to the

subtraction algorithms?

Algorithms for the Subtraction of Whole Numbers The common algorithm for subtraction involves two main procedures: (1) subtract- ing numbers that are determined by the addition facts table and (2) exchanging or regrouping (the reverse of the carrying process for addition). Although this exchang- ing procedure is commonly called “borrowing,” we choose to avoid this term because the numbers that are borrowed are not paid back. Hence the word borrow does not represent to children the actual underlying process of exchanging.

A development of our standard subtraction algorithm is used in Figure 4.6 to find the difference 357 123− .

Figure 4.6 

Notice that in the example in Figure 4.6, the answer will be the same whether we subtract from left to right or from right to left. In either case, the base ten blocks are used by first representing 357 and then taking away 123. Since there are 7 units from which 3 can be taken and there are 5 longs from which 2 can be removed, the use of the blocks is straightforward. The problem 423 157− is done differently because we cannot take 7 units away from 3 units directly. In this problem, a long is broken into 10 units to create 13 units and a flat is exchanged from 10 longs and combined with the remaining long to create 11 longs (see Figure 4.7). Once these exchanges have been made, 157 can be taken away to leave 266.

NCTM Standard All students should develop and use strategies for whole-number computations, with a focus on addition and subtraction.

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148 Chapter 4 Whole-Number Computation—Mental, Electronic, and Written

In Figure 4.7, the representations of the base ten blocks, chip abacus, and place- value model become more and more abstract. The place-value procedure is finally shortened to produce our standard subtraction algorithm. Even though the stan- dard algorithm is abstract, a connection with the base ten blocks can be seen when exchanging 1 long for 10 units because this action is equivalent to a “borrow.”

423

157

4 2 3

1 5 7

2 6 6

3 11 13

− +⎯→

make exchanges

One nontraditional algorithm that is especially effective in any base is called the subtract-from-the-base algorithm. This algorithm is illustrated in Figure 4.8 using base ten pieces to find 323 64− .

Figure 4.7 

Figure 4.8 

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Section 4.2 Written Algorithms for Whole-Number Operations 149

In (1), observe that the 4 is subtracted from the 10 (instead of finding 13 4− , as in the standard algorithm). The difference, 10 4 6− = , is then combined with the 3 units in (2) to form 9 units. Then the 6 longs are subtracted from 1 flat (instead of find- ing 11 6− ). The difference, 10 6 4− = longs, is then combined with the 1 long in (3) to obtain 5 longs (or 50). Thus, since 2 flats remain, 323 64 259− = . The following illustrates this process symbolically.

The advantage of this algorithm is that we only need to know the addition facts and differences from 10 (as opposed to differences from all the teens). As you will see later in this chapter, this method can be used in any base (hence the name subtract- from-the-base). After a little practice, you may find this algorithm to be easier and faster than our usual algorithm for subtraction.

Reflection from Research Having students create their own computational algorithms for large number addition and sub- traction is a worthwhile activity (Cobb, Yackel, & Wood, 1988).

Check for Understanding: Exercise/Problem Set A #12–19✔

1. How could the base ten pieces be used to model 13 24× ? 2. What might an intermediate algorithm for 13 24× look like?

Discuss how the work with the blocks in number 1 is related to the intermediate algorithm in number 2.

Algorithms for the Multiplication of Whole Numbers The standard multiplication algorithm involves the multiplication facts, distributivity, and a thorough understanding of place value. A development of our standard multipli- cation algorithm is used in Figure 4.9 to find the product 3 213× .

Place-Value Representations

Figure 4.9 

Next, the product of a two-digit number times a two-digit number is found. Calculate 34 12× .

34 12 34 10 2

34 10 34 2

30 4 10 30 4 2

30 10 4 10 30 2

× = + = + = + + + = + +

( )

( ) ( )

¥ ¥

¥ ¥ ¥ ++ = + + + =

4 2

300 40 60 8

408

¥

Expanded form

Distributivity

Expanded formm

Distributivity

Multiplication

Addition

The product 34 12× also can be represented pictorially (Figure 4.10).

Common Core – Grade 5 Fluently multiply multidigit whole numbers using the standard algo- rithm.

Reflection from Research Teachers should help students to develop a conceptual under- standing of how multidigit mul- tiplication and division relates to and builds upon place value and basic multiplication combinations (Fuson, 2003).

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150 Chapter 4 Whole-Number Computation—Mental, Electronic, and Written

Figure 4.10

It is worthwhile to note how the Intermediate Algorithm 1 closely connects to the base block representation of 34 12× . The numbers in the algorithm have been color coded with the regions in the rectangular array in Figure 4.10 to make the connection more apparent. The intermediate algorithms assist in the transition from the concrete blocks to the abstract standard algorithm.

(A) INTERMEDIATE (B) INTERMEDIATE (C) STANDARD ALGORITHM 1 ALGORITHM 2 ALGORITHM

34

12

8

60

40

300

408

× 34

12

68

340

408

2 34

10 34

×

×

×

Think

Think

One final complexity in the standard multiplication algorithm is illustrated next. Calculate 857 9× .

(A) INTERMEDIATE ALGORITHM (B) STANDARD ALGORITHM

Notice the complexity involved in explaining the standard algorithm! The lattice method for multiplication is an example of an extremely simple multipli-

cation algorithm that is no longer used, perhaps because we do not use paper with lattice markings on it and it is too time-consuming to draw such lines.

To calculate 35 4967× , begin with a blank lattice and find products of the digits in intersecting rows and columns. The 18 in the completed lattice was obtained by multiplying its row value, 3, by its column value, 6 (Figure 4.11). The other values are filled in similarly. Then the numbers are added down the diagonals as in lattice addi- tion. The numbers in the “6’’ diagonal add to 12, but 1 (actually 10) is carried from the previous “7’’ diagonal that yields 13. Then the 1 (actually 10) from 13 is carried to the “9’’ diagonal to yield 2 2 2 1 7+ + + = . The answer, read counterclockwise from left to right, is 173,845.

Algebraic Reasoning In Figure 4.10, the model of a rectangular array can be used to multiply expressions such as x + 2 and x + 3. Tiles that look similar to base ten blocks are placed with an x + 2 down the side and x + 3 across the top. When the rectangle is filled in, the product of ( )x + 2 ( )x + 3 is seen to be x x2 5 6+ + .

Figure 4.11

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Section 4.2 Written Algorithms for Whole-Number Operations 151

Check for Understanding: Exercise/Problem Set A #20–28✔

Use base ten pieces to solve the problem 432 3÷ . On a piece of paper, sketch the actions used with the pieces to solve the problem. Next solve the problem using the standard

long division algorithm. There are several actions with the blocks that relate to specific steps in the algorithm. Explain those connections.

Algorithms for the Division of Whole Numbers The long-division algorithm is the most complicated procedure in the elementary mathematics curriculum. Because of calculators, the importance of written long divi- sion using multidigit numbers has greatly diminished. However, the long-division algorithm involving one- and perhaps two-digit divisors continues to have common applications. The main idea behind the long-division algorithm is the division algo- rithm, which was given in Section 3.2. It states that if a and b are any whole numbers with b ≠ 0, there exist unique whole numbers q and r such that a bq r= + , where 0 ≤ <r b. For example, if a = 17 and b = 5, then q = 3 and r = 2 since 17 5 3 2= +¥ . The purpose of the long division algorithm is to find the quotient, q, and remainder, r, for any given divisor, b, and dividend, a.

To gain an understanding of the division algorithm, we will use base ten blocks and a fundamental definition of division. Find the quotient and remainder of 461 divided by 3. This can be thought of as 461 divided into groups of size 3. As you read this example, notice how the manipulation of the base ten blocks parallels the written algorithm. The following illustrates long division using base ten blocks.

Reflection from Research A weak understanding of the place-value system and a procedure-oriented performance of algorithms cause students to struggle to understand the traditional long division algorithm (Lee, 2007).

Children’s Literature www.wiley.com/college/musser See “The Doorbell Rang” by Pat Hutchins.

Common Core – Grade 6 Fluently divide multidigit numbers using the standard algorithm.

Thought One

Think: One group of 3 flats, which is 100 groups of 3, leaves 1 flat left over.

Thought Two

Think: Convert the 1 leftover flat to 10 longs and add it to the existing 6 longs to make 16 longs.

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152 Chapter 4 Whole-Number Computation—Mental, Electronic, and Written

Thought Three

Think: Five groups of 3 longs, which is 50 groups of 3, which leaves 1 long left over.

Thought Four

Think: Convert the 1 leftover long into 10 units and add it to the existing 1 unit to make 11 units.

Thought Five

Think: Three groups of 3 units each leaves 2 units left over. Since there is one group of flats (hundreds), five groups of longs (tens), and three groups of units (ones) with 2 left over, the quotient is 153 with a remainder of 2.

We will arrive at the final form of the long-division algorithm by working through various levels of complexity to illustrate how one can gain an understanding of the algorithm by progressing in small steps.

Find the quotient and remainder for 5739 31÷ . The scaffold method is a good one to use first.

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Section 4.2 Written Algorithms for Whole-Number Operations 153

    )31 5739 3100

2639

1550

1089

930

159

155

4

100 31

50 31

30 31

5 3

−

−

−

+

( )

( )

( )

( 11

185 31

100

)

( )

How many 31s in 5739?

How many 31s in 2

Guess:

6639?

How many 31s in 1089?

How many 31s i

Guess:

Guess:

50

30

nn 159?

Since 4 we stop and add 100

The

Guess: 5

31 50 30 5< + + +, . rrefore, the quotient is 185 and remainder is 4.

Check: 31 185 4 5735 4 5739¥ + = + = .

As just shown, various multiples of 31 are subtracted successively from 5739 (or the resulting difference) until a remainder less than 31 is found. The key to this method is how well one can estimate the appropriate multiples of 31. In the scaffold method, it is better to estimate too low rather than too high, as in the case of the 50. However, 80 would have been the optimal guess at that point. Thus, although the quotient and remainder can be obtained using this method, it can be an inefficient application of the Guess and Test strategy.

The next example illustrates how division by a single digit can be done more efficiently. Find the quotient and remainder for 3159 7÷ .

INTERMEDIATE ALGORITHM

Start here: )

451

50

7 3159 400

2800 359 350

9 7 2

1

−

−

−

How many 7s in 310Think: 00? 400

How many 7s in 350? 50

How many 7s in

Think:

Think: 99? 1

Therefore, the quotient is the sum 400 50 1+ + , or 451, and the remainder is 2. Check: 7 451 2 3157 2 3159¥ + = + = .

Now consider division by a two-digit divisor. Find the quotient and remainder for 1976 32÷ .

INTERMEDIATE ALGORITHM

)32 1976 60

1

61

1920 56 32 24

How many 32s in 1976? 60 −

−

Think:

Thhink: How many 32s in 56? 1

Therefore, the quotient is 61 and the remainder is 24. Check: 32 61 24 1952 24 1976¥ + = + = .

Reflection from Research When solving arithmetic prob- lems, students who move directly from using concrete strategies to algorithms, without being allowed to generate their own abstract strategies, are less likely to develop a conceptual under- standing of multidigit numbers (Ambrose, 2002).

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154 Chapter 4 Whole-Number Computation—Mental, Electronic, and Written

Next we will employ rounding to help estimate the appropriate quotients. Find the quotient and remainder of 4238 56÷ .

)56 4238 70

5

3920

318 280

38

How many 60s in 42 −

−

Think: 000? 70

70

How many 60s in 310? 5

5

× =

× =

56 3920

56 280

Think:

Therefore, the quotient is 70 5 75+ = and the remainder is 38. Check: 56 75 38 4200 38 4238¥ + = + = .

Observe how we rounded the divisor up to 60 and the dividend down to 4200 in the first step. This up/down rounding assures us that the quotient at each step will not be too large.

This algorithm can be simplified further to our standard algorithm for division by reducing the “think” steps to divisions with single-digit divisors. For example, in place of “How many 60s in 4200?” one could ask equivalently, “How many 6s in 420?” Even easier, “How many 6s in 42?” Notice also that in general, the divisor should be rounded up and the dividend should be rounded down.

Algebraic Reasoning Solving equations like x x x3 22 3 6 0− − + = can be done by factoring. One way to factor is a type of algebraic long division. The standard division algorithm provides a foundation for this type of algebraic reasoning.

Check for Understanding: Exercise/Problem Set A #29–34✔

Find the quotient and remainder for 4238 56÷ .

S O L U T I O N 

)56 4238 75

392

How many 6s in 42? 7

Put the 7 above th

Think:

ee 3 since we are actually finding

4230 56.

The 392 is 7

÷ ¥ 56.

⎧⎧

⎨ ⎪⎪

⎩ ⎪ ⎪

318

280

38

− Think: How many 6s in 31? 5

Put the 5 above the 8 ssince we are finding 318 56.

The 280 is 5 56.

÷ ⎧ ⎨ ⎪

⎩⎪ ¥ ■

The quotient and remainder for 4238 56÷ can be found using a calculator. The TI-34 MultiView does long division with remainder directly using 4238 2nd int÷ 56 enter . The result for this problem is shown in Figure 4.12 as it appears on the second line of the calculator’s display. To find the quotient and remainder using a standard calculator, press these keys: 4238 ÷ 56 = . Your display should read 75.678571 (perhaps with fewer or more decimal places). Thus the whole-number quotient is 75. From the relationship a bq r= + , we see that r a bq= − is the remainder. In this case r = −4238 56 75¥ , or 38.

As the preceding calculator example illustrates, calculators virtually eliminate the need for becoming skilled in performing involved long divisions and other tedious calculations.

In this section, each of the four basic operations was illustrated by different algo- rithms. The problem set contains examples of many other algorithms. Even today, other countries use algorithms different from our “standard” algorithms. Also, students may even invent “new” algorithms. Since computational algorithms are aids for simpli- fying calculations, it is important to understand that all correct algorithms are accept- able. Clearly, computation in the future will rely less and less on written calculations and more and more on mental and electronic calculations.

75r38

Figure 4.12

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Section 4.2 Written Algorithms for Whole-Number Operations 155

Have someone write down a three-digit number using three different digits hidden from your view. Then have the person form all of the other five 3-digit numbers that can be obtained by rearranging his or her three digits. Add these six numbers together with a seventh number, which is any other one of the six. The person tells you the sum, and then you, in turn, tell him or her the seventh number.

Here is how. Add the thousands digit of the sum to the remaining three-digit number (if the sum was 3635, you form 635 3 638+ = ). Then take the remainder upon division of this new number by nine (638 9÷ leaves a remainder of 8). Multiply the remainder by 111 8 111 888( )× = and add this to the previous number ( )638 888 1526+ = . Finally, add the thousands digit (if there is one) to the remaining number (1526 yields 526 1 527+ = , the seventh number!).

© R

on B

ag w

el l

EXERCISES 1. Using the Chapter 4 eManipulative activity Base Blocks—

Addition on our Web site, model the following addition problems using base ten blocks. Sketch how the base ten blocks would be used.

a. 327 61+ b. 347 86+

2. The physical models of base ten blocks and the chip abacus have been used to demonstrate addition. Bundling sticks can also be used. Sketch 15 32+ using the following models.

a. Chip abacus b. Bundling sticks

3. Give a reason or reasons for each of the following steps to justify the addition process.

17 21 4 10 7 2 10 1

1 10 2 10 7 1

3 10 8

38

+ = + + + = + + + = + =

( ) ( )

( ) ( )

¥ ¥ ¥ ¥ ¥

4. There are many ways of providing intermediate steps between the models for computing sums (base ten blocks, chip abacus, etc.) and the algorithm for addition. One of these is to represent numbers in their expanded forms. Consider the following examples:

hundreds tens

hundreds tens

246 2 4 6

352 3 5 2

= + + + = + +

hundreds tens

= + + = = +

5 9 8

598

547 5 10 42( ) (( )

( ) ( )

( ) ( )

10 7

296 2 10 9 10 6

7 10 13 10 13

2

2

+ + = + +

+ + ←

⎯⎯⎯⎯⎯

← ⎯⎯ + +

+ +

⎫

⎬ ⎪⎪

⎭ ⎪ ⎪

7 10 14 10 3

8 10 4 10 3

2

2

( ) ( )

( ) ( )

regroupping

843=

Use this expanded form of the addition algorithm to com- pute the following sums.

a. 351 635+

b. 564 345+

5. Use the Intermediate Algorithm 1 to compute the following sums.

a. 598 396+ b. 322 799 572+ +

6. An alternative algorithm for addition, called scratch addi- tion, is shown next. Using it, students can do more compli- cated additions by doing a series of single-digit additions. This method is sometimes more effective with students hav- ing trouble with the standard algorithm. For example, to compute 78 56 38+ + :

7 8

5 6

3 84+

Add the number in the units place start- ing at the top. When the sum is ten or more, scratch a line through the last num- ber added and write down the unit. The scratch represents 10.

7 8

5 6

3 8

2

2

4 +

Continue adding units (adding 4 and 8). Write the last number of units below the line. Count the number of scratches, and write above the second column.

7 8

5 6

3 8

1 7 2

2

4 4 +

Repeat the procedure for each column.

Compute the following additions using the scratch algorithm.

a. 734 468

27+

b. 1364 7257

4813+

section 4.2 EXERCISE/PROBLEM SET A

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156 Chapter 4 Whole-Number Computation—Mental, Electronic, and Written

7. Compute the following sums using the lattice method.

a. 482 269+

b. 567 765+

8. Give an advantage and a disadvantage of each of the following methods for addition.

a. Intermediate algorithm b. Lattice

9. Add the following numbers. Then turn this page upside down and add it again. What did you find? What feature of the numerals 1, 6, 8, 9 accounts for this?

986 818

969

989

696

616

10. Without performing the addition, tell which sum, if either, is greater.

23 456

23 400

23 000

20 002

20 002

32

432

65 432

,

,

,

,

,

,+ +

11. Without calculating the actual sums, select the smallest sum, the middle sum, and the largest sum in each group. Mark them A, B, and C, respectively. Use your estimating powers.

a. _____ 283 109+ _____ 161 369+ _____ 403 277+ b. _____ 629 677+ _____ 723 239+ _____ 275 631+

Now check your estimate with a calculator.

12. Using the Chapter 4 eManipulative activity Base Blocks— Subtraction on our Web site, model the following subtraction problems using base ten blocks. Sketch how the base ten blocks would be used.

a. 87 35−

b. 483 57−

13. Sketch solutions to the following problems, using bundling sticks and a chip abacus

a. 57 37−

b. 34 29−

14. 9342 is usually thought of as 9 thousands, 3 hundreds, 4 tens, and 2 ones, but in subtracting 6457 from 9342 using the customary algorithm, we regroup and think of 9342 as

_____ thousands, _____ hundreds, _____ tens, and _____ ones.

15. Order these computations from easiest to hardest.

a. 809 306−

b. 8 3− c. 82

67−

16. To perform some subtractions, it is necessary to reverse the rename and regroup process of addition. Perform the following subtraction in expanded form and follow regrouping steps.

732

378

700 30 2

300 70 8− + +

− + +( ) or

700 20 12

300 70 8

+ + − + +( )

or

600 120 12

300 70 8

300 50 4 354

+ + − + +

+ + =

( )

Use expanded form with regrouping as necessary to  perform the following subtractions.

a. 652 175−        b. 923 147−      c. 8257 6439−

17. The cashier’s algorithm for subtraction is closely related to the missing-addend approach to subtraction; that is, a b c− = if and only if a b c= + . For example, you buy $23 of school supplies and give the cashier a $50 bill. While handing you the change, the cashier would say “$23, $24, $25, $30, $40, $50.” How much change did you receive?

What cashier said: $23, $24, $25, $30, $40, $50 Money received: 0, $1, $1, $5, $10, $10

Now you are the cashier. The customer owes you $62 and gives you a $100 bill. What will you say to the customer, and how much will you give back?

18. Subtraction by adding the complement goes as follows:

619 476−

619 523

1142 1

+

+

143

The complement of 476 is 523 since 4476 The sum in each place is 9.)

Cross out the le

+ =523 999. (

aading digit. Add 1. Answer = 143

Find the following differences using this algorithm.

a. 537 − 179

b. 86 124 38 759

,

,−

19. A subtraction algorithm, popular in the past, is called the equal-additions algorithm. Consider this example.

436

282−           4 10 6

2 10 2

2

2

¥ ¥ ¥ ¥

+ + − + +

3 10 8 10

( )

4 10 6 8 10 2

1 10 5 10 4 154

2

2

¥ ¥ ¥ ¥ ¥ ¥

+ + − + +

+ + =

13 10 3 102

( )

     4 3 6

15 4

13

− 2 82 3

To subtract 8 10¥ from 3 10¥ , add 10 tens to the minuend and 1 102¥ (or 10 tens) to the subtrahend. The problem is changed, but the answer is the same. Use the equal-additions algorithm to find the following differences.

a. 421 286−         b. 92 863 75 387, ,−

20. Consider the product of 27 33× .

a. Sketch how base ten blocks can be used in a rectangular array to model this product.

b. Find the product using the Intermediate Algorithm 1. c. Describe the relationship between the solutions in parts

a and b.

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Section 4.2 Written Algorithms for Whole-Number Operations 157

21. Show how to find 28 34× on this grid paper.

22. The pictorial representation of multiplication can be adapt- ed as follows to perform 23 16× .

Use this method to find the following products. a. 15 36× b. 62 35×

23. Justify each step in the following proof that 72 10 720× = .

72 10 70 2 10

70 10 2 10

7 10 10 2 10

7 10 10 2 10

7

× = + × = × + × = × × + × = × × + × =

( )

( )

( )

×× + × = + =

100 2 10

700 20

720

24. Solve the following problems using (i) the lattice method for multiplication and (ii) an intermediate algorithm.

a. 23 62×             b. 17 45×

25. Study the pattern in the following left-to-right multiplication.

731

238

1462

2193

5848

173978

2 731

3 731

8 731

× ( )

( )

( )

¥ ¥ ¥

Use this algorithm to do the following computations. a. 75 47×             b. 364 421×

26. The Russian peasant algorithm for multiplying 27 51× is illustrated as follows:

Notice that the numbers in the first column are halved (disregarding any remainder) and that the numbers in the second column are doubled. When 1 is reached in the halving column, the process is stopped. Next, each row with an even number in the halving column is crossed out and the remaining numbers in the doubling column are added. Thus

27 51 51 102 408 816 1377× = + + + = .

Use the Russian peasant algorithm to compute the following products.

a. 68 35×            b. 38 62×

27. The use of finger numbers and systems of finger com- putation has been widespread through the years. One such system for multiplication uses the finger positions shown for computing the products of numbers from 6 through 10.

The two numbers to be multiplied are each represented on a different hand. The sum of the raised fingers is the number of tens, and the product of the closed fingers is the number of ones. For example, 1 3 4+ = fingers raised, and 4 2 8× = fingers down.

Use this method to compute the following products. a. 7 8×            b. 6 7×           c. 6 10×

28. The duplication algorithm for multiplication combines a succession of doubling operations, followed by addition. This algorithm depends on the fact that any number can be written as the sum of numbers that are powers of 2. To compute 28 36× , the 36 is repeatedly doubled as shown.

1 36 36

2 36 72

4 36 144

8 36 288

16 36 576

× = × =

→ × = → × = → × =

↑

powers off 2

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158 Chapter 4 Whole-Number Computation—Mental, Electronic, and Written

This process stops when the next power of 2 in the list is greater than the number by which you are multiplying. Here we want 28 of the 36s, and since 28 16 8 4= + +( ), the product of 28 36 16 8 4 36× = + +( ) .¥ From the last column, you add 144 288 576 1008+ + = .

Use the duplication algorithm to compute the following products.

a. 25 62×         b. 35 58×         c. 73 104×

29. Use the scaffold method in the Chapter 4 dynamic spreadsheet Scaffold Division on our Web site to find the following quotients. Write down how the scaffold method was used.

a. 899 13÷           b. 5697 23÷

30. A third-grade teacher prepared her students for division this way:

20 4 20

4

16

4

12

4

8

4

4

4

0

20 4 5

÷ −

−

−

−

−

÷ =

✔

✔

✔

✔

✔

How would her students find 42 6÷ ?

31. Without using the divide key, find the quotient for each of the following problems.

a. 24 4÷            b. 56 7÷ c. Describe how the calculator was used to find the quo-

tients.

32. When asked to find the quotient and remainder of 431 17÷ , one student did the following with a calculator:

Display

431 17 25 35294÷ = −.

Display

25 17 6 0000000= × = .

a. Try this method on the following pairs. (i) 1379 87÷ (ii) 69 431 139, ÷ (iii) 1 111 111 333, , ÷ b. Does this method always work? Explain.

33. Sketch how base ten blocks can be used to model 762 5÷ .

34. When performing the division problem 2137 14÷ using the standard algorithm, the first few steps look like what is shown here. The next step is to “bring down” the 3. Explain how that process of “bringing down” the 3 is modeled using base ten blocks.

)14 2137 1

14 7

PROBLEMS 35. Larry, Curly, and Moe each add incorrectly as follows.

Larry

: 29

83

1012

+      Curly

: 29

83

121

2

+      Moe

: 29

83

102

+

How would you explain their mistakes to each of them?

36. Use the digits 1 to 9 to make an addition problem and answer. Use each digit only once.

37. Place the digits 2, 3, 4, 6, 7, 8 in the boxes to obtain the following sums.

a. The greatest sum     b. The least sum

38. Arrange the digits 1, 2, 3, 4, 5, 6, 7 such that they add up to 100. (For example, 12 34 56 7 109+ + + = .)

39. Given next is an addition problem. a. Replace seven digits with 0s so that the sum of the num-

bers is 1111.

999

777

555

333

111+

b. Do the same problem by replacing (i) eight digits with 0s, (ii) nine digits with 0s, (iii) ten digits with 0s.

40. Consider the following array:

Compare the pair 26, 37 with the pair 36, 27.

a. Add: 26 37+ = _____. 36 27+ = _____. What do you notice about the answers?

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Section 4.2 Written Algorithms for Whole-Number Operations 159

b. Subtract: 37 26− =_____. 36 27− =_____. What do you notice about the answers?

c. Are your findings true for any two such pairs? d. What similar patterns can you find?

41. The x’s in half of each figure can be counted in two ways.

a. Draw a similar figure for 1 2 3 4+ + + . b. Express that sum in a similar way. Is the sum correct? c. Use this idea to find the sum of whole numbers from 1

to 50 and from 1 to 75.

42. In the following problems, each letter represents a differ- ent digit and any of the digits 0 through 9 can be used. However, both additions have the same result. What are the problems?

ZZZ

KKK

LLL

RSTU

+      

ZZZ

PPP

QQQ

RSTU

+

43. Following are some problems worked out by students. Each student has a particular error pattern. Find that error, and tell what answer that student will get for the last problem.

What instructional procedures might you use to help each of these students?

44. The following is an example of the German low-stress algorithm. Find 5314 79× using this method and explain how it works.

4967 35

21

1835

2730

1245

20

1738

×

445

45. Select any four-digit number. Arrange the digits to form the largest possible number and the smallest possible number. Subtract the smallest number from the largest number. Use the digits in the difference and start the process over again. Keep repeating the process. What do you discover?

EXERCISES 1. Sketch how base ten blocks could be used to solve the fol-

lowing problems. You may may find the Chapter 4 eMa- nipulative activity Base Blocks—Addition on our Web site to be useful in this process.

a. 258 149+

        b. 627 485+

2. Sketch the solution to 46 55+ using the following models. a. Bundling sticks     b. Chip abacus

3. Give a reason for each of the following steps to justify the addition process.

38 56 3 10 8 5 10 6

3 10 5 10 8 6

3 10 5 10 14

+ = + + + = + + + = + + =

( ) ( )

( ) ( )

( )

(

¥ ¥ ¥ ¥ ¥ ¥

33 10 5 10 1 10 4¥ ¥ ¥+ + +)

3 10 5 10 1 10 4

3 5 1 1 10 4

9 10

¥ ¥ ¥ ¥ ¥ ¥

= + + + = + + + = +

( )

( )

44 94=

4. Use the expanded form of the addition algorithm (see Set A, Exercise 4) to compute the following sums.

a. 478 269+

         b. 1965 857+

5. Use the Intermediate Algorithm 2 to compute the following sums.

a. 347 679+          b. 3538 784+ c. Describe the advantages and disadvantages of

Inter-mediate Algorithm 2 versus Intermediate Algorithm 1.

EXERCISE/PROBLEM SET B

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160 Chapter 4 Whole-Number Computation—Mental, Electronic, and Written

6. Another scratch method involves adding from left to right, as shown in the following example.

987

356

12

+

9 8 7

3 5 6

1 2 3 3

+     

9 8 7

3 5 6

1 2 3 3 3 4

+     

987

356

1343

+

First the hundreds column was added. Then the tens column was added and, because of carrying, the 2 was scratched out and replaced by a 3. The process continued until the sum was complete. Apply this method to com- pute the following sums.

a. 475 381+

        b. 856 907+

        c. 179 356+

7. Compute the following sums using the lattice method. a. 982

659+         b. 4698

5487+

8. Give an advantage and a disadvantage of each of the following methods for addition.

a. Expanded form b. Standard algorithm

9. In Set A, Exercise 9, numbers were listed that add to the same sum whether they were read right side up or upside down. Find another 6 three-digit numbers that have the same sum whether read right side up or upside down.

10. Gerald added 39,642 and 43,728 on his calculator and got 44,020 as the answer. How could he tell, mentally, that his sum is wrong?

11. Without calculating the actual sums, select the smallest sum, the middle sum, and the largest sum in each group. Mark them A B, , and C, respectively.

a. _____ 284 625+ _____ 593 237+ _____ 304 980+ b. _____ 427 424+ _____ 748 611+ _____ 272 505+

12. Sketch how base ten blocks could be used to solve the fol- lowing problems. You may find the Chapter 4 eManipulative activity Base Blocks—Subtraction on our Web site to be useful in this process.

a. 536 54− b. 625 138−

13. Sketch the solution to the following problems using bundling sticks and a chip abacus.

a. 42 27− b. 324 68−

14. To subtract 999 from 1111, regroup and think of 1111 as _____ hundreds, _____ tens, and _____ ones.

15. Order these computations from easiest to hardest. a. 81

36−          b. 80

30−         c. 8819

3604−

16. Use the expanded form with regrouping (see Set A, Exercise 16) to perform the following subtractions.

a. 455 278− b. 503 147− c. 3426 652−

17. Use the cashiers algorithm (see Set A, Exercise 17) to describe what the cashier would say to the customer in

each of the following cases. How much change does the customer receive?

a. Customer owes: $28; Cashier receives: $40 b. Customer owes: $33; Cashier receives: $100

18. Use the adding the complement algorithm described in Set A, Exercise 18 to find the differences in parts a and b.

a. 3479 2175−

b. 6 002 005 4 187 269

, ,

, ,− c. Explain how the method works with three-digit numbers.

19. Use the equal-additions algorithm described in Set A, Exercise 19 to find the differences in parts a and b.

a. 3476 558− b. 50 004 36 289, ,− c. Will this algorithm work in general? Why or why not?

20. Consider the product of 42 24× . a. Sketch how base ten blocks can be used in a rectangular

array to model this product. b. Find the product using the Intermediate Algorithm 1. c. Describe the relationship between the solutions in parts

a and b.

21. Show how to find 17 26× on this grid paper.

22. Use the pictorial representation shown in Set A, Exercise 22 to find the products in parts a and b.

a. 23 48× b. 34 52× c. How do the numbers within the grid compare with the

steps of Intermediate Algorithm 1?

23. Justify each step in the following proof that shows 573 100 57 300× = , .

573 100 500 70 3 100

500 100 70 100 3 100

5 100 100 7

× = + + × = × + × + × = × × +

( )

( ) ( ×× × + ×

= × × + × × + ×

= × + ×

10 100

3 100

5 100 100 7 10 100

3 100

5 10 000 7 10

)

( ) ( )

, 000 300

50 000 7000 300

57 300

+ = + + =

,

,

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Section 4.2 Written Algorithms for Whole-Number Operations 161

24. Solve the following problems using (i) the lattice method for multiplication and (ii) an intermediate algorithm.

a. 237 48× b. 617 896×

25. Use the left-to-right multiplication algorithm shown in Set A, Exercise 25 to find the following products.

a. 276 43× b. 768 891×

26. Use the Russian peasant algorithm for multiplication shown in Set A, Exercise 26 to find the following products.

a. 44 83× b. 31 54×

27. Use the finger multiplication method described in Set A, Exercise 27 to find the following products. Explain how the algorithm was used in each case.

a. 9 6×              b. 7 9×              c. 8 8×

28. Use the duplication algorithm for multiplication shown in Set A, Exercise 28 to find the products in parts a, b, and c.

a. 14 43× b. 21 67× c. 43 73× d. Which property justifies the algorithm?

29. Use the scaffold method to find the following quotients. Show how the scaffold method was used.

a. 749 22÷ b. 3251 14÷

30. a. Use the method described in Set A, Exercise 30 to find the quotient 63 9÷ and describe how the method was used.

b. Which approach to division does this method illustrate?

31. Without using the divide key, use a calculator to find the quotient and remainder for each of the following problems.

a. )3 39          b. )8 89        c. )6 75 d. Describe how the calculator was used to find the re-

mainders.

32. Find the quotient and remainder to the problems in parts a, b, and c using a calculator and the method illustrated in Set A, Exercise 32. Describe how the calculator was used.

a. 18 114 37, ÷ b. 381 271 47, ÷ c. 9 346 870 349, , ÷ d. Does this method always work? Explain.

33. Sketch how to use base ten blocks to model the operation 673 4÷ .

34. When finding the quotient for 527 3÷ using the standard division algorithm, the first few steps are shown here.

)3 527 1

3 2

−

Explain how the process of subtracting 3 from 5 is modeled with base ten blocks.

PROBLEMS 35. Peter, Jeff, and John each perform subtraction incorrectly

as follows:

Peter:  503

269

366

−     Jeff:  5 0 3

2 6 9

2 4 4

4 10 13

−

    John:  5 0 3 2 6 9

1 3 4

4 3

10 9

13

−

How would you explain their mistakes to each of them?

36. Let A B C, ,, and D represent four consecutive whole numbers. Find the values for A B C, , , and D if the four boxes are replaced with A B C, , , and D in an unknown order.

37. Place the digits 3, 5, 6, 2, 4, 8 in the boxes to obtain the following differences.

a. The greatest difference b. The least difference

38. a. A college student, short of funds and in desperate need, writes the following note to his father:

SEND

MORE

MONEY

+

If each letter in this message represents a different digit, how much MONEY (in cents) is he asking for?

b. The father, considering the request, decides to send some money along with some important advice.

SAVE

MORE

MONEY

+

However, the father had misplaced the request and could not recall the amount. If he sent the largest amount of MONEY (in cents) represented by this sum, how much did the college student receive?

39. Consider the sums

1 11

1 11 111

1 11 111 1111

+ = + + =

+ + + = a. What is the pattern? b. How many addends are there the first time the pattern

no longer works?

40. Select any three-digit number whose first and third digits are different. Reverse the digits and find the difference between the two numbers. By knowing only the hundreds digit in this difference, it is possible to deter- mine the other two digits. How? Explain how the trick works.

41. Choose any four-digit number, reverse its digits, and add the two numbers. Is the sum divisible by 11? Will this always be true?

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162 Chapter 4 Whole-Number Computation—Mental, Electronic, and Written

42. Select any number larger than 100 and multiply it by 9. Select one of the digits of this result as the “missing digit.” Find the sum of the remaining digits. Continue adding digits in resulting sums until you have a one-digit number. Subtract that number from 9. Is that your miss- ing digit? Try it again with another number. Determine which missing digits this procedure will find.

43. Following are some division exercises done by students.

Determine the error pattern and tell what each student will get for the last problem.

What instructional procedures might you use to help each of these students?

44. Three businesswomen traveling together stopped at a motel to get a room for the night. They were charged $30 for the room and agreed to split the cost equally. The manager later realized that he had overcharged them by $5. He gave the refund to his son to deliver to the women. This smart son, realizing it would be difficult to split $5 equally, gave the women $3 and kept $2 for himself. Thus, it cost each woman $9 for the room. Therefore, they spent $27 for the room plus the $2 “tip.” What happened to the other dollar?

45. Show that a perfect square is obtained by adding 1 to the product of two whole numbers that differ by 2. For exam- ple, 8 10 1 92( ) ,+ = 11 13 1 122× + = , etc.

Analyzing Student Thinking

46. A student’s father was curious about why his daughter was learning addition using base ten pieces when he had not learned it that way. How should you respond?

47. Is it okay for a student to use the lattice method for addi- tion or must all of your students eventually use the stan- dard algorithm? Explain.

48. When finding the difference 123 67− , why might the sub- tract-from-the-base algorithm be easier for some students?

49. What is one advantage and one disadvantage of having your students use base ten blocks to model the standard multiplication algorithm? Explain.

50. After two subtraction algorithms were shared in class, Nicholas asked why they needed two different methods and which of them he was supposed to learn. How should you respond?

51. A student found 4005 37− as follows:

4005

37

2078

−

Diagnose what the student was doing wrong.

52. Natasha asked what the advantages of the lattice method are as compared to the standard multiplication algorithm. How would you respond?

53. Andres found the product of 46 and 53 horizontally as follows: ( )( )40 6 50 3 40 50 40 3 6 50+ + = × + × + × + 6 3 2000 120 300 18 2438× = + + + = . Will this method always work? Explain.

54. When using the standard division algorithm to solve 621 3÷ , Dylan found the quotient to be 27 instead of 207. How could base ten pieces be used to clarify this misun- derstanding?

ALGORITHMS IN OTHER BASES

All of the algorithms you have learned can be used in any base. In this section we apply the algorithms in base five and then let you try them in other bases in the problem set. The purpose for doing this is to help you see where, how, and why your future students might have difficulties in learning the standard algorithms.

Use any of the algorithms learned in Section 4.2 and your knowledge of base five numbersto solve the following problems. Compare and contrast your strategies with

those of your peers.

132

433 five

five+

241

143 five

five−

34

24 five

five× )3 433five five

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Section 4.3 Algorithms in Other Bases 163

Operations in Base Five Addition Addition in base five is facilitated using the thinking strategies in base five. These thinking strategies may be aided by referring to the base five number line shown in Figure 4.13, in which 2 4 11five five five+ = is illustrated. This number line also provides a representation of counting in base five. (All numerals on the number line are written in base five with the subscripts omitted.)

Figure 4.13

Check for Understanding: Exercise/Problem Set A #1–6✔

Subtraction There are two ways to apply a subtraction algorithm successfully. One is to know the addition facts table forward and backward. The other is to use the missing-addend approach repeatedly. For example, to find 12 4five five,− think “What number plus 4five is 12five?” To answer this, one could count up “10 11 12five five five, , .” Thus 12 4 3five five five− = (one for each of 10 11five five , , and 12five). A base five number line can also illustrate 12 4 3five five five− = . The missing-addend approach is illustrated in Figure 4.14 and the take-away approach is illustrated in Figure 4.15.

Figure 4.14

Figure 4.15

Find 342 134five five+ using the following methods.

a. Lattice method      b. Intermediate algorithm      c. Standard algorithm

S O L U T I O N a. Lattice Method          b. Intermediate      c. Standard Algorithm         Algorithm

■

Be sure to use thinking strategies when adding in base five. It is helpful to find sums to five first; for example, think of 4 3five five+ as 4 1 2 4 1five five five five five( ) ( )+ + = + + 2 12five five= and so on.

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164 Chapter 4 Whole-Number Computation—Mental, Electronic, and Written

A copy of the addition table for base five is included in Figure 4.16 to assist you in working through Example 4.11. (All numerals in Figure 4.16 are written in base five with the subscripts omitted.)

Figure 4.16

Calculate 412 143five five− using the following methods.

S O L U T I O N

a. Standard Algorithm b. Subtract-from-the-Base Think: Think:

■

Notice that to do subtraction in base five using the subtract-from-the-base algorithm, you only need to know two addition combinations to five, namely 1five + 4 10five five= and 2 3 10five five five+ = . These two, in turn, lead to the four subtraction facts you need to know, namely 10 4 1five five five− = , 10 1 4 10 3 2five five five five five five − = − =, , and 10 2 3five five five− = .

Check for Understanding: Exercise/Problem Set A #7–11✔

Multiplication To perform multiplication efficiently, one must know the multipli- cation facts. The multiplication facts for base five are displayed in Figure 4.17. The entries in this multiplication table can be visualized by referring to the number line shown in Figure 4.18.

Figure 4.17

The number line includes a representation of 4 3 22five five five× = using the repeated- addition approach. (All numerals in the multiplication table and number line are written in base five with subscripts omitted.)

Algebraic Reasoning To find 3 4¥ in base five, one must solve 12 5= +a b where a b, .< 5 Although this is typically done mentally, algebraic reason- ing is required to complete this base five multiplication table.

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Section 4.3 Algorithms in Other Bases 165

Notice how efficient the lattice method is. Also, instead of using the multiplica- tion table, you could find single-digit products using repeated addition and thinking strategies. For example, you could find 4 2five five× as follows.

4 2 2 4 4 4 4 1 3

4 1 five five five five five five five five

f

× = × = + = + + = +

( )

( ) iive five five+ =3 13

Although this may look like it would take a lot of time, it would go quickly mentally, especially if you imagine base five pieces.

Check for Understanding: Exercise/Problem Set A #12–14✔

Calculate 43 123five five× using the following methods.

S O L U T I O N

a. Lattice Method                  b. Intermediate        c. Standard Algorithm Algorithm

■

Division As shown in Section 3.2, division can be displayed using a number line. Figure 4.19 displays 23 4five five÷ using the repeated subtraction approach. (All numerals below the number line without subscripts are assumed to be base five.)

Figure 4.19

Figure 4.19 shows that 23 4five five÷ has a quotient of 3five with a remainder of 1five. Doing long division in other bases points out the difficulties of learning this algo-

rithm and, especially, the need to become proficient at approximating multiples of numbers.

Figure 4.18

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166 Chapter 4 Whole-Number Computation—Mental, Electronic, and Written

In summary, doing computations in other number bases can provide insights into how computational difficulties arise, whereas our own familiarity and competence with our algorithms in base ten tend to mask the trouble spots that children face.

Find the quotient and remainder for 1443 34five five÷ using the following methods.

S O L U T I O N

a. Scaffold Method           b. Standard Long-Division Algorithm

)34 1443 20

five five

five 1230

213

212

1

−

− 33 23

five

five

)34 1443 23

five five

five

123

213

212

1

−

−

five five

five five

( )

( )

2 34

3 34

×

×

Quotient: 23five Remainder: 1five

In the scaffold method, the fi rst estimate, namely 20five, was selected because 2 3five five× is 11five, which is less than 14five. ■

Check for Understanding: Exercise/Problem Set A #15–18✔

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George Parker Bidder (1806–1878), who lived in Devonshire, England, was blessed with an incredible memory as well as being a calculating prodigy. When he was 10, he was read a number backward and he immediately gave the number back in its correct form. An hour later, he repeated the original number, which was

2,563,721,987,653,461,598,746,231,905,607,541,127,975,231.

Furthermore, his brother memorized the entire Bible and could give the chapter and verse of any quoted text. Also, one of Bidder’s sons could multiply 15-digit numbers in his head.

EXERCISES 1. Create a number line to illustrate the following operations. a. 12 3four four+              b. 4 15six six+

2. Use multibase blocks to illustrate each of the following base four addition problems. The Chapter 4 eManipulative activity Multibase Blocks on our Web site may be helpful in the solution process.

a. 1 2four four+              b. 11 23four four+ c. 212 113four four+ d. 2023 3330four four+

3. Use bundling sticks or chip abacus for the appropriate base to illustrate the following sums.

a. 41 33six six+              b. 555 66seven seven+ c. 3030 322four four+

4. Write out a base six addition table. Use your table and the intermediate algorithm to compute the following sums.

a. 32 23six six+                b. 45 34six six+ c. 145 541six six+                d. 355 211six six+

5. Use the lattice method to compute the following sums. a. 46 13seven seven+     b. 13 23four four+

6. Use the standard algorithm to compute the following sums. a. 213 433five five+      b. 716 657eight eight+

7. Create a number line to illustrate the following operations.

a. 12 3four four−                   b. 14 5six six−

EXERCISE/PROBLEM SET A

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Section 4.3 Algorithms in Other Bases 167

8. Use multibase blocks to illustrate each of the following base four subtraction problems. The Chapter 4 eManip- ulative activity Multibase Blocks on our Web site may be helpful in the solution process.

a. 31 12four four−    b. 123 32four four− c. 1102 333four four−

9. Use bundling sticks or chip abacus for the appropriate base to illustrate the following differences.

a. 41 33six six−           b. 555 66seven seven− c. 3030 102four four−

10. Solve the following problems using both the standard algorithm and the subtract-from-the-base algorithm.

a. 36 17eight eight−         b. 1010 101two two− c. 32 13four four−

11. Find 10201 2122three three− using “adding the complement.” What is the complement of a base three number?

12. Create a number line to illustrate the following operations. a. 2 13four four×        b. 3 5six six×

13. Sketch the rectangular array of multibase pieces and find the products of the following problems.

a. 23 3four four× b. 23 12five five×

14. Solve the following problems using the lattice method, an intermediate algorithm, and the standard algorithm.

a. 31four four2×

          b. 43 3

five

five ×         c. 312

3 four

four ×

15. Create a number line to illustrate the following operations.

a. 22 3four four÷        b. 24 5six six÷

16. Use the scaffold method of division to compute the following numbers. (Hint: Write out a multiplication table in the appropriate base to help you out.)

a. 22 2six six÷ b. 4044 51seven seven÷ c. 13002 33four four÷

17. Solve the following problems using the missing-factor defi- nition of division. (Hint: Use a multiplication table for the appropriate base.)

a. 21 3four four÷         b. 23 3six six÷ c. 24 5eight eight÷

18. Sketch how to use base seven blocks to illustrate the oper- ation 534 4seven seven÷ .

PROBLEMS 19. 345 _____ + 122 _____ = 511 _____ is an addition problem

done in base _____.

20. Jane has $10 more than Bill, Bill has $17 more than Tricia, and Tricia has $21 more than Steve. If the total amount of all their money is $115, how much money does each have?

21. Without using a calculator, determine which of the five numbers is a perfect square. There is exactly one.

39 037 066 087

39 037 066 084

39 037 066 082

38 336 073 623

38

, , ,

, , ,

, , ,

, , ,

,, , ,414 432 028

22. What single number can be added separately to 100 and 164 to make them both perfect square numbers?

23. What is wrong with the following problem in base four? Explain.

1022four

four 413+

24. Write out the steps you would go through to mentally subtract 234five from 421five using the standard subtraction algorithm.

EXERCISES 1. Create a number line to illustrate the following operations. a. 12 6seven seven+              b. 4 16eight eight+

2. Use multibase blocks to illustrate each of the following base five addition problems. The Chapter 4 eManipulative activ- ity Multibase Blocks on our Web site may be helpful in the solution process.

a. 12 20five five+         b. 24 13five five+ c. 342 13five five+         d. 2134 1330five five+

3. Use bundling sticks or chip abacus for the appropriate base to illustrate the following problems.

a. 32 33four four+           b. 54 55eight eight+ c. 265 566nine nine+

4. Use an intermediate algorithm to compute the following sums.

a. 78 65nine nine+              b. TE EEtwelve twelve+

EXERCISE/PROBLEM SET B

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168 Chapter 4 Whole-Number Computation—Mental, Electronic, and Written

5. Use the lattice method to compute the following sums. a. 54 34six six+ b. 1 8 499T eleven eleven+

6. Use the standard algorithm to compute the following sums. a. 79 85twelve twelve+ b. T eleven eleven1 99+

7. Create a number line to illustrate the following operations. a. 12 3seven seven−          b. 14 5eight eight−

8. Use multibase blocks to illustrate each of the following base five subtraction problems. The Chapter 4 eManipulative activity Multibase Blocks on our Web site may be helpful in the solution process.

a. 32 14five five−   b. 214 32five five− c. 301 243five five−

9. Use bundling sticks or chip abacus for the appropriate base to illustrate the following problems.

a. 123 24five five−             b. 253 76eight eight− c. 1001 110two two−

10. Solve the following problems using both the standard algorithm and the subtract-from-the-base algorithm.

a. 45 36seven seven−           b. 99 7twelve twelveT− c. 100 77eight eight−

11. Find 1001010 111001two two− by “adding the complement.”

12. Create a number line to illustrate the following operations. a. 2 13seven seven×             b. 3 5eight eight×

13. Sketch the rectangular array of multibase pieces and find the products of the following problems.

a. 32 13six six×               b. 34 21seven seven×

14. Solve the following problems using the lattice method, an intermediate algorithm, and the standard algorithm.

a. 11011two two1101×

                b. 43twelve twelve23×

             c. 66seven seven66×

15. Create a number line to illustrate the following operations. a. 22 3seven seven÷             b. 23 5eight eight÷

16. Use the scaffold method of division to compute the fol- lowing numbers. (Hint: Write out a multiplication table in the appropriate base to help you out.)

a. 14 3five five÷                     b. 2134 14six six÷ c. 61245 354seven seven÷

17. Solve the following problems using the missing-factor defi- nition of division. (Hint: Use a multiplication table for the appropriate base.)

a. 42 5seven seven÷   b. 62 7nine nine÷ c. 92twelve twelveE÷

18. Sketch how to use base four blocks to illustrate the opera- tion 3021 11four four.÷

PROBLEMS 19. 320 _____ − 42 _____ = 256 _____ is a correct subtraction

problem in what base?

20. Betty has three times as much money as her brother Tom. If each of them spends $1.50 to see a movie, Betty will have nine times as much money left over as Tom. How much money does each have before going to the movie?

21. To stimulate his son in the pursuit of mathematics, a math professor offered to pay his son $8 for every equation cor- rectly solved and to fine him $5 for every incorrect solu- tion. At the end of 26 problems, neither owed any money to the other. How many did the boy solve correctly?

22. Prove: If n is a whole number and n2 is odd, then n is odd. (Hint: Use indirect reasoning. Either n is even or it is odd. Assume that n is even and reach a contradiction.)

Analyzing Student Thinking

23. Antolina asks why they have to be doing algorithms in other bases. How should you respond?

24. Ralph asks if it is okay to use the base five addition table when subtracting in base five or does he have to memorize the table first?

25. You ask a student to make up a subtraction problem in base four. The student writes the following:

1022four

four 413−

Is this a reasonable base four subtraction problem? Explain.

26. When doing a multiplication problem in base five, Veronica uses the intermediate algorithm. Geoff doesn’t want to use it because it has too many steps. How should you respond in this situation?

27. One of your students says that she is having difficulty doing division in base five using the standard algorithm. Suggest how you might help the student.

28. When Sherry did the subtraction problem 342 135six six ,− she “borrowed” from the 4 to get 12 in the ones columns as shown.

3 4 21 3

six

six 1 3 5−

She got confused when she subtracted 12 5− and got 7 because the digit “7” is not in base six. However, when she converted 7 to 11six she didn’t know what to do with both ones. Where is Sherry’s misunderstanding and how would you help her clarify her thinking?

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End of Chapter Material 169

The whole numbers 1 through 9 can be used once, each being arranged in a 3 3× square array so that the sum of the numbers in each of the rows, columns, and diagonals is 15. Show that 1 cannot be in one of the corners.

Strategy: Use Indirect Reasoning

Suppose that 1 could be in a corner as shown in the following figure. Each row, column, and diagonal containing 1 must have a sum of 15. This means that there must be three pairs of numbers among 2 through 9 whose sum is 14. Hence the sum of all three pairs is 42. However, the largest six numbers—9, 8, 7, 6, 5, and 4—have a sum of 39, so that it is impossible to find three pairs whose sum is 14. Therefore, it is impossible to have 1 in a corner.

Additional Problems Where the Strategy “Use Indirect Rea- soning” Is Useful

1. If x represents a whole number and x x2 2 1+ + is even, prove that x cannot be even.

2. If n is a whole number and n4 is even, then prove that n is even.

3. For whole numbers x and y, if x y2 2+ is a square, then prove that x and y cannot both be odd.

END OF CHAPTER MATERIAL

John Kemeny (1926–1992) John Kemeny was born in Budapest, Hungary. His family immigrated to the United States, and he at- tended high school in New York City. He entered Princeton where he stud-

ied mathematics and philosophy. As an undergraduate, he took a year off to work on the Manhattan Project. While he and Tom Kurtz were teaching in the mathemat- ics department at Dartmouth in the mid-1960s, they cre- ated BASIC, an elementary computer language. Keme- ny made Dartmouth a leader in the educational uses of computers in his era. Kemeny’s fi rst faculty position was in philosophy. “The only good job offer I got was from the Princeton philosophy department,” he said, explain- ing that his degree was in logic, and he studied philoso- phy as a hobby in college. Kemeny had the distinction of having served as Einstein’s mathematical assistant while still a young graduate student at Princeton. Einstein’s as- sistants were always mathematicians. Contrary to popu- lar belief, Einstein did need help in mathematics. He was very good at it, but he was not an up-to-date research mathematician.

Grace Brewster Murray Hopper (1906–1992) Grace Brewster Murray Hop- per recalled that as a young girl, she disassembled one of her family’s alarm clocks. When she was unable to reas- semble the parts, she disman-

tled another clock to see how those parts fi t together. This process continued until she had seven clocks in pieces. This attitude of exploration foreshadowed her innovations in computer programming. Trained in mathematics, Hopper worked with some of the fi rst computers and invented the business language CO- BOL. After World War II she was active in the Navy, where her experience in computing and programming began. In 1985 she was promoted to the rank of rear admiral. Known for her common sense and spirit of invention, she kept a clock on her desk that ran (and kept time) counterclockwise. Whenever someone ar- gued that a job must be done in the traditional man- ner, she just pointed to the clock. Also, to encourage risk taking, she once said, “I do have a maxim—I teach it to all youngsters: A ship in port is safe, but that’s not what ships are built for.”

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170 Chapter 4 Whole-Number Computation—Mental, Electronic, and Written

CHAPTER REVIEW

Review the following terms and exercises to determine which require learning or relearning—page numbers are provided for easy reference.

Vocabulary/Notation Right distributivity  131 Compatible numbers  131 Compensation  132 Additive compensation  132 Equal additions method  132 Multiplicative compensation  132 Left-to-right methods  132 Powers of 10  132 Special factors  132 Computational estimation  134

Front-end estimation  134 Range estimation  134 Range  134 One-column front-end estimation

method  135 Two-column front-end estimation

method  135 Front-end with adjustment  135 Round up/down  136 Truncate  136

Round a 5 up  136 Round to the nearest even  136 Round to compatible numbers  136 Arithmetic logic  138 Algebraic logic  138 Parentheses  138 Constant function  139 Exponent keys  139 Memory functions  139 Scientific notation  139

Mental Math, Estimation, and Calculators

Exercises 1. Calculate the following mentally, and name the

property(ies) you used. a. 97 78+ b. 267 3÷ c. ( )16 7 25× × d. 16 9 6 9× − × e. 92 15× f. 17 99× g. 720 5÷ h. 81 39−

2. Estimate using the techniques given. a. Range: 157 371+ b. One-column front-end: 847 989× c. Front-end with adjustment: 753 639+ d. Compatible numbers: 23 56×

3. Round as indicated. a. Up to the nearest 100: 47,943 b. To the nearest 10: 4751 c. Down to the nearest 10: 576

4. Insert parentheses (wherever necessary) to produce the indi- cated results.

a. 3 7 5 38+ × = b. 7 5 2 3 24× − + = c. 15 48 3 4 19+ ÷ × =

5. Fill in the following without using a calculator.

a. 5 2 3+ × =

b. 8 6 3− ÷ =

c. 2 3 2× + =( )

d. 2 3yx =

e. 3 7STO RCL + =

f. 3 5 8 2× = ÷ = + =STO RCL

Exercises 1. Find 837 145+ using a. the lattice method. b. an intermediate algorithm. c. the standard algorithm.

2. Find 451 279− using a. the standard algorithm. b. a nonstandard algorithm.

3. Find 72 43× using a. an intermediate algorithm. b. the standard algorithm. c. the lattice method.

4. Find 253 27÷ using a. the scaffold method. b. an intermediate algorithm. c. the standard algorithm. d. a calculator.

Vocabulary/Notation Algorithm  145 Standard addition algorithm  145 Lattice method for addition  146 Standard subtraction algorithm  147

Subtract-from-the-base algorithm  148 Standard multiplication

algorithm  149 Lattice method for multiplication  150

Long division using base ten blocks  151

Scaffold method for division  152 Standard algorithm for division  154

Written Algorithms for Whole-Number Operations

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Chapter Test 171

Exercises 1. Find 413 254six six+ using a. the lattice method. b. an intermediate algorithm. c. the standard algorithm.

2. Find 234 65seven seven− using a. the standard algorithm. b. a nonstandard algorithm.

3. Find 21 32four four× using a. an intermediate algorithm. b. the standard algorithm. c. the lattice method.

4. Find 213 41five five÷ using a. repeated subtraction b. the scaffold method. c. the standard algorithm.

Vocabulary/Notation Operations in base five  163

Algorithms in Other Bases

Knowledge 1. True or false? a. An algorithm is a technique that is used exclusively for

doing algebra. b. Intermediate algorithms are helpful because they require

less writing than their corresponding standard algo- rithms.

c. There is only one computational algorithm for each of the four operations: addition, subtraction, multiplication, and division.

d. Approximating answers to computations by rounding is useful because it increases the speed of computation.

Skill 2. Compute each of the following using an intermediate algo-

rithm. a. 376 594+ b. 56 73×

3. Compute the following using the lattice method. a. 568 493+ b. 37 196×

4. Compute the following mentally. Then, explain how you did it.

a. 54 93 16 47+ + + b. 9223 1998− c. 3497 1362− d. 25 52×

5. Find 7496 32÷ using the standard division algorithm, and check your results using a calculator.

6. Estimate the following using (i) one-column front-end, (ii) range estimation, (iii) front-end with adjustment, and (iv) rounding to the nearest 100.

a. 546 971 837 320+ + + b. 731 589×

Understanding 7. Compute 32 21× using expanded form; that is, continue the

following.

32 21 30 2 20 1× = +( ) +( ) = . . .

8. In the standard multiplication algorithm, why do we “shift over one to the left” as illustrated in the 642 in the follow- ing problem?

321

23

963

642

×

+

9. To check an addition problem where one has “added down the column,” one can “add up the column.” Which prop- erties guarantee that the sum should be the same in both directions?

10. Sketch how a chip abacus could be used to perform the following operation.

374 267+

11. Use an appropriate intermediate algorithm to compute the following quotient and remainder.

7261 43÷

12. Sketch how base ten blocks could be used to find the quo- tient and remainder of the following.

538 4÷

13. Sketch how base four blocks could be used to find the fol- lowing difference.

32 13four four−

14. Sketch how a chip abacus could be used to find the fol- lowing sum.

278 37nine nine+

15. Sketch how base ten blocks could be used to model the following operations and explain how the manipulations of the blocks relate to the standard algorithm.

a. 357 46+      b. 253 68−      c. 789 5÷

CHAPTER TEST

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172 Chapter 4 Whole-Number Computation—Mental, Electronic, and Written

16. Compute 492 37× using an intermediate algorithm and a standard algorithm. Explain how the distributive property is used in each of these algorithms.

17. State some of the advantages and disadvantages of the standard algorithm versus the lattice method for multipli- cation.

18. State some of the advantages and disadvantages of the standard subtraction algorithm versus the subtract-from the-base algorithm.

19. Show how to find 17 23× on the following grid paper and explain how the solution on the grid paper can be related to the intermediate algorithm for multiplication.

Problem Solving/Application 20. If each different letter represents a different digit, find

the number “HE” such that (HE) SHE.2 = (Note: “HE” means 10 ¥H E+ due to place value.)

21. Find values for a b, , and c in the lattice multiplication problem shown. Also find the product.

22. Find digits represented by A, B, C, and D so that the fol- lowing operation is correct.

ABA

BAB

CDDC

+

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c04.indd 173 7/30/2013 2:54:27 PM

174

N umber theory provides a rich source of intrigu-ing problems. Interestingly, many problems in number theory are easily understood, but still have never been solved. Most of these problems are statements or conjectures that have never been proven right or wrong. The most famous “unsolved” problem, known as Fermat’s Last Theorem, is named after Pierre de Fermat who is pictured below. It states “There are no nonzero whole numbers a b c, , , where a b cn n n+ = , for n a whole number greater than two.”

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Fermat left a note in the margin of a book saying that he did not have room to write up a proof of what is now called Fermat’s Last Theorem. However, it remained an unsolved problem for over 350 years because mathematicians were unable to prove it. In 1993, Andrew Wiles, an English math- ematician on the Princeton faculty, presented a “proof ” at a conference at Cambridge University. However, there was a hole in his proof. Happily, Wiles and Richard Taylor produced a valid proof in 1995, which followed from work done by Serre, Mazur, and Ribet beginning in 1985.

The following list contains several such problems that are still unsolved. If you can solve any of them, you will surely become famous, at least among mathematicians.

1. Goldbach’s conjecture. Every even number greater than 4 can be expressed as the sum of two odd primes. For example, 6 3 3 8 3 5 10 5 5 12 5 7= + = + = + = +, , , , and so on. It is interesting to note that if Goldbach’s conjec- ture is true, then every odd number greater than 7 can be written as the sum of three odd primes.

2. Twin prime conjecture. There is an infinite number of pairs of primes whose difference is two. For example, (3, 5), (5, 7), and (11, 13) are such prime pairs. Notice that 3, 5, and 7 are three prime numbers where 5 3 2− = and 7 5 2− = . It can easily be shown that this is the only such triple of primes.

3. Odd perfect number conjecture. There is no odd perfect number; that is, there is no odd number that is the sum of its proper factors (factors less than the number). For example, 6 1 2 3= + + ; hence 6 is a perfect number. It has been shown that the even perfect numbers are all of the form 2 2 11p p− −( ), where 2 1p − is a prime.

4. Ulam’s conjecture. If a nonzero whole number is even, divide it by 2. If a nonzero whole number is odd, multi- ply it by 3 and add 1. If this process is applied repeat- edly to each answer, eventually you will arrive at 1. For example, the number 7 yields this sequence of numbers: 7, 22, 11, 34, 17, 52, 26, 13, 40, 20, 10, 5, 16, 8, 4, 2, 1. Interestingly, there is a whole number less than 30 that requires at least 100 steps before it arrives at 1. It can be seen that 2n requires n steps to arrive at 1. Hence one can find numbers with as many steps (finitely many) as one wishes.

C H A P T E R

5 NUMBER THEORY Famous Unsolved Problems

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175

Understanding the intrinsic nature of numbers is often helpful in solving problems. For example, knowing that the sum of two even numbers is even and that an odd number squared is odd may simplify checking some computations. The solution of the initial problem will seem to be impossible to a naive problem solver who attempts to solve it using, say, the Guess and Test strategy. On the other hand, the solution is immediate for one who understands the concept of divisibility of numbers.

Initial Problem A major fast-food chain held a contest to promote sales. With each purchase a cus- tomer was given a card with a whole number less than 100 on it. A $100 prize was given to any person who presented cards whose numbers totaled 100. The following are several typical cards. Can you find a winning combination?

3 9 12 15 18 27 51 72 84

Can you suggest how the contest could be structured so that there would be at most 1000 winners throughout the country? (Hint: What whole number divides evenly in each sum?)

Clues The Use Properties of Numbers strategy may be appropriate when

Special types of numbers, such as odds, evens, primes, and so on, are involved.

A problem can be simplifi ed by using certain properties.

A problem involves lots of computation.

A solution of this Initial Problem is on page 202.

Use Properties of NumbersProblem-Solving Strategies 1. Guess and Test

2. Draw a Picture

3. Use a Variable

4. Look for a Pattern

5. Make a List

6. Solve a Simpler Problem

7. Draw a Diagram

8. Use Direct Reasoning

9. Use Indirect Reasoning

10. Use Properties of Numbers

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176

AUTHOR

WALK-THROUGH

I N T R O D U C T I O N Number theory is a branch of mathematics that is devoted primarily to the study of the set of counting numbers. The aspects of the counting numbers central to the elementary curriculum that are covered in this chapter include primes, composites, and divisibility tests as well as the notions of greatest com- mon factor and least common multiple. Many of these topics are useful in other areas of mathematics such as fractions (Chapter 6) and algebra. Being able to find common factors and greatest common factors between pairs of numbers is useful when simplifying fractions. Understanding least common

multiples is useful for finding a common denominator when adding and subtracting fractions. The principles of factor- ing numbers studied in this chapter generalize into similar situations in algebra when factoring expressions.

Key Concepts from the NCTM Principles and Standards for School Mathematics

PRE-K-2–NUMBER AND OPERATIONS

Develop a sense of whole numbers and represent and use them in flexible ways, including relating, composing, and decomposing numbers.

GRADES 3-5–NUMBER AND OPERATIONS

Recognize equivalent representations for the same number and generate them by decomposing and composing num- bers.

Describe classes of numbers according to characteristics such as the nature of their factors.

GRADES 6-8–NUMBER AND OPERATIONS

Use factors, multiples, prime factorization, and relatively prime numbers to solve problems.

Key Concepts from the NCTM Curriculum Focal Points

GRADE 3: Developing understandings of multiplication and division and strategies for basic multiplication facts and related division facts.

GRADE 4: Developing quick recall of multiplication facts and related division facts and fluency with whole-number multiplication.

Key Concepts from the Common Core State Standards for Mathematics

GRADE 4: Gain familiarity with factors and multiples. GRADE 6: Find common factors and multiples. Apply and extend previous understandings of arithmetic to alge-

braic expressions.

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Section 5.1 Primes, Composites, and Tests for Divisibility 177

For example, 2, 3, 5, 7, 11 are primes, since they have only themselves and 1 as fac- tors; 4, 6, 8, 9, 10 are composites, since they each have more than two factors; 1 is neither prime nor composite, since 1 is its only factor.

An algorithm used to find primes is called the Sieve of Eratosthenes (Figure 5.1).

Figure 5.1

The directions for using this procedure are as follows: Skip the number 1. Circle 2 and cross out every second number after 2. Circle 3 and cross out every third number after 3 (even if it had been crossed out before). Continue this procedure with 5, 7, and each succeeding number that is not crossed out. The circled numbers will be the primes and the crossed-out numbers will be the composites, since prime factors cause them to be crossed out. Again, notice that 1 is neither prime nor composite.

Composite numbers have more than two factors and can be expressed as the prod- uct of two smaller numbers. Figure 5.2 shows how a composite can be expressed as the product of smaller numbers using factor trees.

Figure 5.2

NCTM Standard All students should use factors, multiples, prime factorization, and relatively prime numbers to solve problems.

Reflection from Research Talking, drawing, and writing about types of numbers such as factors, primes, and composites, allows students to explore and explain their ideas about general- izations and patterns dealing with these types of numbers (Whitin & Whitin, 2002).

On a piece of paper, sketch all of the possible rectangles that can be made up of exactly 10 squares. An example of a rectangle consisting of

6 squares is shown at the right.

Repeat these sketches for 11 squares and 12 squares. How are the dimensions of the rectangles related to the number of squares? For each number of squares, the number of rectangles that can be con- structed varies. Why?

PRIMES, COMPOSITES, AND TESTS FOR DIVISIBILITY

Prime and Composite Numbers

A counting number with exactly two different factors (itself and 1) is called a prime number, or a prime. A counting number with more than two factors is called a composite number, or a composite.

Common Core – Grade 4 Determine whether a given whole number in the range 1–100 is prime or composite.

D E F I N I T I O N 5 . 1

Primes and Composites Prime numbers are building blocks for the counting numbers 1 2 3 4, , , , . . . .

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178 Chapter 5 Number Theory

Next, we will study shortcuts that will help us find prime factors. When division yields a zero remainder, as in the case of 15 3÷ , for example, we say that 15 is divisible by 3, 3 is a divisor of 15, or 3 divides 15. In general, we have the following definition.

Children’s Literature www.wiley.com/college/musser See “A Remainder of One” by Elinor Pinczes.

Notice that 60 was expressed as the product of two factors in several different ways. However, when we kept factoring until we reached primes, each method led us to the same prime factorization, namely 60 2 2 3 5= ¥ ¥ ¥ . This example illustrates the following important result.

Algebraic Reasoning To factor an expression such as x x2 5 24− − it is important to be able to factor 24 into pairs of factors.

Fundamental Theorem of Arithmetic

Each composite number can be expressed as the product of primes in exactly one way (except for the order of the factors).

T H E O R E M 5 . 1

Express each number as the product of primes.

a. 84     b. 180     c. 324

S O L U T I O N

a. 84 4 21 2 2 3 7 2 3 72= × = =¥ ¥ ¥ ¥ ¥

b.180 10 18 2 5 2 3 3 2 3 52 2= × = =¥ ¥ ¥ ¥ ¥ ¥

c. 324 4 81 2 2 3 3 3 3 2 32 4= × = =¥ ¥ ¥ ¥ ¥ ¥ ■

Divides

Let a and b be any whole numbers with a ≠ 0. We say that a divides b, and write a b| , if and only if there is a whole number x such that ax b= . The symbol a b� means that a does not divide b.

D E F I N I T I O N 5 . 2

In words, a divides b if and only if a is a factor of b. When a divides b, we can also say that a is a divisor of b a, is a factor of b b, is a multiple of a, and b is divisible by a.

We can also say that a b| if b objects can be arranged in a rectangular array with a rows. For example, 4 12| because 12 dots can be placed in a rectangular array with 4 rows, as shown in Figure 5.3(a). On the other hand, 5 12� because if 12 dots are placed in an array with 5 rows, a rectangular array cannot be formed [Figure 5.3(b)].

Figure 5.3

Common Core – Grade 4 Recognize that a whole number is a multiple of each of its fac- tors. Determine whether a given whole number in the range 1–100 is a multiple of a given one-digit number.

Algebraic Reasoning Each time we ask ourselves if a b| , we are solving the equation ax b= . For example, we use algebraic reasoning when we decide if a statement like 7 91| is true because we are looking for a whole number x that will make 7 91x = true.

Determine whether the following are true or false. Explain.

a. 3 12| . b. 8 is a divisor of 96. c. 216 is a multiple of 6. d. 51 is divisible by 17. e. 7 divides 34. f. ( ) | ( ).2 3 2 3 52 3 2¥ ¥ ¥

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179

From Lesson 1 “Prime Factorization” from My Math, Volume 1 Common Core State Standards, Grade 5, copyright © 2013 by McGraw-Hill Education.

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180 Chapter 5 Number Theory

Tests for Divisibility Some simple tests can be employed to help determine the factors of numbers. For example, which of the numbers 27, 45, 38, 70, and 111,110 are divisible by 2, 5, or 10? If your answers were found simply by looking at the ones digits, you were applying tests for divisibility. The tests for divisibility by 2, 5, and 10 are stated next.

Reflection from Research When students are allowed to use calculators to generate data and are encouraged to examine the data for patterns, they often discover divisibility rules on their own (Bezuszka, 1985).

Check for Understanding: Exercise/Problem Set A #1–6✔

S O L U T I O N a. True. 3 12| , since 3 4 12¥ = . b. True. 8 is a divisor of 96, since 8 12 96¥ = . c. True. 216 is a multiple of 6, since 6 36 216¥ = . d. True. 51 is divisible by 17, since 17 3 51¥ = . e. False. 7 34� , since there is no whole number x such that 7 34x = . f. True. ( ) | ( ),2 3 2 3 52 3 2⋅ ⋅ ⋅ since ( )( ) ( )2 3 2 3 5 2 3 52 3 2¥ ¥ ¥ ¥ ¥= . ■

Tests for Divisibility by 2, 5, and 10

A number is divisible by 2 if and only if its ones digit is 0, 2, 4, 6, or 8. A number is divisible by 5 if and only if its ones digit is 0 or 5. A number is divisible by 10 if and only if its ones digit is 0.

T H E O R E M 5 . 2

Now notice that 3 27| and 3 9| . It is also true that 3 27 9| ( )+ and that 3 27 9| ( ).− This is an instance of the following theorem.

Children’s Literature www.wiley.com/college/musser See “The Great Divide” by Dayle Ann Dodds.

Let a m n, , , and k be whole numbers where a ≠ 0.

a. If a m| and a n| , then a m n| ( ).+ b. If a m| and a n| , then a m n| ( )− for m n≥ . c. If a m| , then a km| .

T H E O R E M 5 . 3

P R O O F

a. If a m| , then ax m= for some whole number x. If a n| , then ay n= for some whole number y. Therefore, adding the respective sides of the two equations, we have ax ay+ = m n+ , or

a x y m n( ) .+ = +

Since x y+ is a whole number, this last equation implies that a m n| ( ).+ Part (b) can be proved simply by replacing the plus signs with minus signs in this discussion. The proof of (c) follows from the definition of divides. ■

Part (a) in the preceding theorem can also be illustrated using the rectangular array description of divides. In Figure 5 4 3 9. , | is represented by a rectangle with 3 rows of 3 blue dots. 3 12| is represented by a rectangle of 3 rows of 4 black dots. By placing

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Section 5.1 Primes, Composites, and Tests for Divisibility 181

the 9 blue dots and the 12 black dots together, there are ( )9 12+ dots arranged in 3 rows, so 3 9 12| ( ).+

Figure 5.4

Using this result, we can verify the tests for divisibility by 2, 5, and 10. The main idea of the proof of the test for 2 is now given for an arbitrary three-digit number (the same idea holds for any number of digits).

Let r a b c= + +¥ ¥10 102 be any three-digit number. Observe that a b a b¥ ¥ ¥10 10 10 102 + = +( ). Since 2 10| , it follows that 2 10 10| ( )a b¥ + or 2 10 102| ( )a b⋅ + ⋅ for any digits a

and b.

Thus if 2 | c (where c is the ones digit), then 2 10 10| [ ( ) ].a b c⋅ + + Thus 2 10 10 22| ( ), | .a b c r¥ ¥+ + or Conversely, let 2 10 102| ( ).a b c⋅ + ⋅ + Since 2 10 102| ( ),a b¥ ¥+ it follows that

2 10 10 10 102 2| [( ) ( )]a b c a b¥ ¥ ¥ ¥+ + − + or 2 | c.

Therefore, we have shown that 2 divides a number if and only if 2 divides the num- ber’s ones digit. One can apply similar reasoning to see why the tests for divisibility for 5 and 10 hold.

The next two tests for divisibility can be verified using arguments similar to the test for 2. Their verifications are left for the problem set.

Algebraic Reasoning To the right, notice how the vari- ables a, b, and c represent the digits 0,1, 2, . . . , 9. In this way, all three-digit numbers may be represented (where a ≠ 0). Then, rearranging sums using proper- ties and techniques of algebra shows that the digit c is the one that determines whether a num- ber is even or not.

Tests for Divisibility by 4 and 8

A number is divisible by 4 if and only if the number represented by its last two digits is divisible by 4.

A number is divisible by 8 if and only if the number represented by its last three digits is divisible by 8.

T H E O R E M 5 . 4

Determine whether the following are true or false. Explain.

a. 4 1432| b. 8 4204| c. 4 2 345 678| , , d. 8 98 765 432| , ,

S O L U T I O N a. True. 4 1432| , since 4 32| . b. False. 8 4204� , since 8 204� . c. False. 4 2 345 678� , , , since 4 78� . d. True. 8 98 765 432| , , , since 8 432| . ■

Notice that the test for 4 involves two digits and 2 42 = . Also, the test for 8 requires that one consider the last three digits and 2 83 = .

The next two tests for divisibility provide a simple way to test for factors of 3 or 9.

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182 Chapter 5 Number Theory

The following justification of the test for divisibility by 3 in the case of a three- digit number can be extended to prove that this test holds for any whole number.

Let r a b c= + +¥ ¥10 102 be any three-digit number. We will show that if 3 | ( ),a b c+ + then 3 | r. Rewrite r as follows:

r a b c

a a b b c

a b a b c

a

= +( ) + +( ) + = + + + + = + + + + =

¥ ¥ ¥ ¥ ¥ ¥ ¥ ¥ ¥

99 1 9 1

99 1 9 1

99 9

111 9+( ) + + +b a b c.

Since 3 9| , it follows that 3 11 9| ( ) .a b⋅ + Thus if 3 | ( ),a b c+ + where a b c+ + is the sum of the digits of r, then 3 | r since 3 11 9| [( ) ( )].a b a b c¥ + + + + On the other hand, if 3 | r, then 3 | ( ),a b c+ + since 3 11 9| [ ( ) ]r a b− +¥ and r a b a b c− + = + +( ) .¥11 9

The test for divisibility by 9 can be justified in a similar manner. The following is a test for divisibility by 11.

Determine whether the following are true or false. Explain.

a. 3 12 345| , b. 9 12 345| ,           c. 9 6543|

S O L U T I O N a. True. 3 12 345| , , since 1 2 3 4 5 15+ + + + = and 3 15| . b. False. 9 12 345� , , since 1 2 3 4 5 15+ + + + = and 9 15� . c. True. 9 6543| , since 9 6 5 4 3| ( ).+ + + ■

Test for Divisibility by 11

A number is divisible by 11 if and only if 11 divides the difference of the sum of the digits whose place values are odd powers of 10 and the sum of the digits whose place values are even powers of 10.

T H E O R E M 5 . 6

When using the divisibility test for 11, first compute the sums of the two different sets of digits, and then subtract the smaller sum from the larger sum.

Determine whether the following are true or false. Explain.

a. 11 5346| b. 11 909 381| ,           c. 11 76 543| ,

S O L U T I O N a. True. 11 5346| , since 5 4 9 3 6 9 9 9 0+ = + = − =, , , and 11 0| . b. True. 11 909 381| , , since 0 3 1 4 9 9 8 26 26 4 22+ + = + + = − =, , , and 11 22| . c. False. 11 76 543� , , since 6 4 10 7 5 3 15 15 10 5+ = + + = − =, , , and 11 5� . ■

Tests for Divisibility by 3 and 9

A number is divisible by 3 if and only if the sum of its digits is divisible by 3. A number is divisible by 9 if and only if the sum of its digits is divisible by 9.

T H E O R E M 5 . 5

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Section 5.1 Primes, Composites, and Tests for Divisibility 183

This technique of applying two tests simultaneously can be used in other cases also. For example, the test for 10 can be thought of as applying the tests for 2 and 5 simultaneously. By the test for 2, the ones digit must be 0, 2, 4, 6, or 8, and by the test for 5 the ones digit must be 0 or 5. Thus a number is divisible by 10 if and only if its ones digit is zero. Testing for divisibility by applying two tests can be done in general.

Test for Divisibility by 6

A number is divisible by 6 if and only if both of the tests for divisibility by 2 and 3 hold.

T H E O R E M 5 . 7

A number is divisible by the product, ab, of two nonzero whole numbers a and b if it is divisible by both a and b, and a and b have only the number 1 as a common factor.

T H E O R E M 5 . 8

The justification of this test for divisibility by 11 is left for Problem 44 in the Exercise/Problem Set A. Also, a test for divisibility by 7 is given in Exercise 10 in the Exercise/Problem Set A.

One can test for divisibility by 6 by applying the tests for 2 and 3.

According to this theorem, a test for divisibility by 36 would be to test for 4 and test for 9, since 4 and 9 both divide 36 and 4 and 9 have only 1 as a common factor. However, the test “a number is divisible by 24 if and only if it is divisible by 4 and 6” is not valid, since 4 and 6 have a common factor of 2. For example, 4 36| and 6 36| , but 24 36� . The next example shows how to use tests for divisibility to find the prime factorization of a number.

Find the prime factorization of 5148.

S O L U T I O N   First, since the sum of the digits of 5148 is 18 (which is a mul- tiple of 9), we know that 5148 9 572= ¥ . Next, since 4 72| , we know that 4 572| . Thus 5148 9 572 9 4 143 3 2 1432 2= = =¥ ¥ ¥ ¥ ¥ . Finally, since in 143, 1 3 4 0+ − = is divisible by 11, the number 143 is divisible by 11, so 5148 2 3 11 132 2= ¥ ¥ ¥ . ■

Determine a process that could be used to determine if a number such as 211 is prime. Write a brief description of that process that could easily be generalized to determine if

any given number is prime.

We can also use divisibility tests to help decide whether a particular counting number is prime. For example, we can determine whether 137 is prime or composite by determining if it has any prime factors less than 137. None of the numbers 2, 3, or 5 divides 137. How about 7? 11? 13? How many prime factors must be considered before we know whether 137 is a prime? Consider the following example.

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184 Chapter 5 Number Theory

Example 5.7 suggests that to determine whether a number n is prime, we need only search for prime factors p, where p n2 ≤ . Recall that y x= (read “the square root of x”) means that y x2 = where y ≥ 0. For example, 25 5= since 5 252 = . Not all whole numbers have whole-number square roots. For example, using a calculator, 27 5 196≈ . , since 5 196 272. .≈ (A more complete discussion of the square root is contained in Chapter 9.) Thus the search for prime factors  of a number n by considering only those primes p where p n2 ≤ can be simplified even further by using the x key on a calculator and checking only those primes p where p n≤ .

Determine whether 137 is a prime.

S O L U T I O N   First, by the tests for divisibility, none of the numbers 2, 3, or 5 is a factor of 137. Next try 7, 11, 13, and so on.

Notice that the numbers in column 1 form an increasing list of primes and the numbers in column 2 are decreasing. Also, the numbers in the two columns “cross over” between 11 and 13. Thus, if there is a prime factor of 137, it will appear in col- umn 1 fi rst and reappear later as a factor of a number in column 2. Thus, as soon as the crossover is reached, there is no need to look any further for prime factors. Since the crossover point was passed in testing 137 and no prime factor of 137 was found, we conclude that 137 is prime. ■

Prime Factor Test

To test for prime factors of a number n, one need only search for prime factors p of n, where p n2 ≤ (or p n≤ ).

T H E O R E M 5 . 9

Determine whether the following numbers are prime or composite.

a. 299          b. 401

S O L U T I O N a. Only the prime factors 2 through 17 need to be checked, since 17 299 192 2< <

(check this on your calculator). None of the numbers 2, 3, 5, 7, or 11 is a factor, but since 299 13 23= ¥ , the number 299 is composite.

b. Only primes 2 through 19 need to be checked, since 401 ≈ 20. Since none of the primes 2 through 19 are factors of 401, we know that 401 is a prime. (The tests for divisibility show that 2, 3, 5, and 11 are not factors of 401. A calculator, tests for divisibility, or long division can be used to check 7, 13, 17, and 19.) ■

Check for Understanding: Exercise/Problem Set A #7–14✔

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Section 5.1 Primes, Composites, and Tests for Divisibility 185

Finding large primes is a favorite pastime of some mathematicians. Before the advent of calculators and computers, this was certainly a time-consuming endeavor. Three anecdotes about large primes follow.

Euler once announced that 1,000,009 was prime. However, he later found that it was the product of 293 and 3413. At the time of this discovery, Euler was 70 and blind.

Fermat was once asked whether 100,895,598,169 was prime. He replied shortly that it had two factors, 898,423 and 112,303.

For more than 200 years the Mersenne number 2 167 − was thought to be prime. In 1903, Frank Nelson Cole, in a speech to the American Mathematical Society, went to the blackboard and without uttering a word, raised 2 to the power 67 (by hand, using our usual multiplication algorithm!) and subtracted 1. He then multiplied 193,707,721 by 761,838,257,287 (also by hand). The two numbers agreed! When asked how long it took him to crack the number, he said, “Three years of Sundays.”

© R

on B

ag w

el l

EXERCISES 1. Using the Chapter 5 eManipulative activity Sieve of

Eratosthenes on our Web site, find all primes less than 100.

2. Find a factor tree for each of the following numbers. a. 36         b. 54         c. 102         d. 1000

3. A factor tree is not the only way to find the prime factor- ization of a composite number. Another method is to divide the number first by 2 if possible, then divide the quotient by 2 if possible, then by 3, then by 5, and so on, until all possible divisions by prime numbers have been performed. For example, to find the prime factorization of 108, you might organize your work as follows to conclude that 108 2 32 3= × .

3 3 9

3 27 2 54

2 108

Use this method to find the prime factorization of the following numbers.

a. 216     b. 2940     c. 825     d. 198,198

4. Determine which of the following are true. If true, illustrate it with a rectangular array. If false, explain.

a. 3 9| b. 12 6| c. 3 is a divisor of 21. d. 6 is a factor of 3. e. 4 is a factor of 16. f. 0 5| g. 11 11| h. 48 is a multiple of 16.

5. If 21 divides m, what else must divide m?

6. If the variables represent counting numbers, determine whether each of the following is true or false.

a. If x y� and x z� , then x y z� ( ).+ b. If 2 | a and 3 | a, then 6 | a.

7. a. Show that 8 123 152| , using the test for divisibility by 8. b. Show that 8 123 152| , by finding x such that 8 123 152x = , . c. Is the x that you found in part b a divisor of 123,152?

Prove it.

8. Which of the following are multiples of 3? of 4? of 9? a. 123,452        b. 1,114,500

9. Use the test for divisibility by 11 to determine which of the following numbers are divisible by 11.

a. 2838         b. 71,992         c. 172,425

10. A test for divisibility by 7 is illustrated as follows. Does 7 divide 17,276?

Test: 17276

12 1715 10

161

− ×

− ×

Subtract 2 6 from 1727

Subtract 2 5 frrom 171

14 Subtract 2 1 from 16− ×2

Since 7 14| , we also have 7 17 276| , . Use this test to see whether the following numbers are divisible by 7.

a. 8659     b. 46,187     c. 864,197,523

EXERCISE/PROBLEM SET A

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186 Chapter 5 Number Theory

11. True or false? Explain. a. If a counting number is divisible by 9, it must be divis-

ible by 3. b. If a counting number is divisible by 3 and 11, it must be

divisible by 33.

12. Decide whether the following are true or false using only divisibility ideas given in this section (do not use long division or a calculator). Give a reason for your answers.

a. 6 80| b. 15 10 000| , c. 4 15 000| , d. 12 32 304| ,

13. Which of the following numbers are composite? Why? a. 12     b. 123     c. 1234     d. 12,345

14. To determine if 467 is prime, we must check to see if it is divisible by any numbers other than 1 and itself. List all of the numbers that must be checked as possible factors to see if 467 is prime.

PROBLEMS 15. a. Write 36 in prime factorization form. b. List the divisors of 36. c. Write each divisor of 36 in prime factorization

form. d. What relationship exists between your answer to part a

and each of your answers to part c? e. Let n = ×13 292 5. If m divides n, what can you conclude

about the prime factorization of m?

16. Justify the tests for divisibility by 5 and 10 for any three- digit number by reasoning by analogy from the test for divisibility by 2.

17. The symbol 4! is called four factorial and means 4 3 2 1× × × ; thus 4! = 24. Which of the following state- ments are true?

a. 6 6| ! b. 5 6| ! c. 11 6| ! d. 30 30| ! e. 40 30| ! f. 30 30 1| ( ! )+

(Do not multiply out parts d to f.)

18. a. Does 8 | 7!?     b. Does 7 | 6!? c. For what counting numbers n will n divide ( )!?n − 1

19. There is one composite number in this set: 331, 3331, 33,331, 333,331, 3,333,331, 33,333,331, 333,333,331. Which one is it? (Hint: It has a factor less than 30.)

20. Show that the formula p n n n( ) = + +2 17 yields primes for n = 0 1 2, , , and 3. Find the smallest whole number n for which p n n n( ) = + +2 17 is not a prime.

21. a. Compute n n2 41+ + , where n = 0 1 2 10, , , . . . , , and deter- mine which of these numbers are prime.

b. On another piece of paper, continue the following spiral pattern until you reach 151. What do you notice about the main upper left to lower right diagonal?

etc. 53 52 51 50

43 42 49 44 41 48 45 46 47

22. In his book The Canterbury Puzzles (1907), Dudeney men- tioned that 11 was the only number consisting entirely of ones that was known to be prime. In 1918, Oscar Hoppe proved that the number 1,111,111,111,111,111,111 (19 ones) was prime. Later it was shown that the number made up of 23 ones was also prime. Now see how many of these “repunit” numbers up to 19 ones you can factor.

23. Which of the following numbers can be written as the sum of two primes, and why?

7 17 27 37 47, , , , , . . .

24. One of Fermat’s theorems states that every prime of the form 4 1x + is the sum of two square numbers in one and only one way. For example, 13 4 3 1= +( ) , and 13 4 9= + , where 4 and 9 are square numbers.

a. List the primes less than 100 that are of the form 4 1x + , where x is a whole number.

b. Express each of these primes as the sum of two square numbers.

25. The primes 2 and 3 are consecutive whole numbers. Is there another such pair of consecutive primes? Justify your answer.

26. Two primes that differ by 2 are called twin primes. For example, 5 and 7, 11 and 13, 29 and 31 are twin primes. Using the Chapter 5 eManipulative activity Sieve of Eratosthenes on our Web site to display all primes less than 200, find all twin primes less than 200.

27. One result that mathematicians have been unable to prove true or false is called Goldbach’s conjecture. It claims that each even number greater than 2 can be expressed as the sum of two primes. For example.

4 2 2 6 3 3 8 3 5

10 5 5 12 5 7

= + = + = + = + = + , , ,

, .

a. Verify that Goldbach’s conjecture holds for even num- bers through 40.

b. Assuming that Goldbach’s conjecture is true, show how each odd whole number greater than 6 is the sum of three primes.

28. For the numbers greater than 5 and less than 50, are there at least two primes between every number and its double? If not, for which number does this not hold?

29. Find two whole numbers with the smallest possible dif- ference between them that when multiplied together will produce 1,234,567,890.

30. Find the largest counting number that divides every num- ber in the following sets.

a. { , , , . . .}1 2 3 2 3 4 3 4 5¥ ¥ ¥ ¥ ¥ ¥ b. { , , , . . .}1 3 5 2 4 6 3 5 7¥ ¥ ¥ ¥ ¥ ¥

Can you explain your answer in each case?

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Section 5.1 Primes, Composites, and Tests for Divisibility 187

31. Find the smallest counting number that is divisible by the numbers 2 through 10.

32. What is the smallest counting number divisible by 2, 4, 5, 6, and 12?

33. Fill in the blank. The sum of three consecutive counting numbers always has a divisor (other than 1) of _____. Prove.

34. Choose any two numbers, say 5 and 7. Form a sequence of numbers as follows: 5, 7, 12, 19, and so on, where each new term is the sum of the two preceding numbers until you have 10 numbers. Add the 10 numbers. Is the seventh num- ber a factor of the sum? Repeat several times, starting with a different pair of numbers each time. What do you notice? Prove that your observation is always true.

35. a. 5! = 5 · 4 · 3 · 2 · 1 is divisible by 2, 3, 4, and 5. Prove that 5! + 2, 5! + 3, 5! + 4, and 5! + 5 are all composite.

b. Find 1000 consecutive numbers that are composite.

36. The customer said to the cashier, “I have 5 apples at 27 cents each and 2 pounds of potatoes at 78 cents per pound. I also have 3 cantaloupes and 6 lemons, but I don’t remem- ber the price for each.” The cashier said, “That will be $3.52.” The customer said, “You must have made a mis- take.” The cashier checked and the customer was correct. How did the customer catch the mistake?

37. There is a three-digit number with the following property: If you subtract 7 from it, the difference is divisible by 7; if you subtract 8 from it, the difference is divisible by 8; and if you subtract 9 from it, the difference is divisible by 9. What is the number?

38. Paula and Ricardo are serving cupcakes at a school party. If they arrange the cupcakes in groups of 2, 3, 4, 5, or 6, they always have exactly one cupcake left over. What is the smallest number of cupcakes they could have?

39. Prove that all six-place numbers of the form abcabc (e.g., 416,416) are divisible by 13. What other two numbers are always factors of a number of this form?

40. a. Prove that all four-digit palindromes are divisible by 11. b. Is this also true for every palindrome with an even num-

ber of digits? Prove or disprove.

41. The annual sales for certain calculators were $2567 one year and $4267 the next. Assuming that the price of the calculators was the same each of the two years, how many calculators were sold in each of the two years?

42. Observe that 7 divides 2149. Also check to see that 7 divides 149,002. Try this pattern on another four-digit number using 7. If it works again, try a third. If that one also works, formulate a conjecture based on your three examples and prove it. (Hint: 7 1001| .)

43. How long does this list continue to yield primes?

17 2 19

19 4 23

23 6 29

29 8 37

+ = + = + = + =

44. Justify the test for divisibility by 11 for four-digit numbers by completing the following: Let a b c d¥ ¥ ¥10 10 103 2+ + + be any four-digit number. Then

a b c d

a b c

¥ ¥ ¥10 10 10 1001 1 99 1 11 1

3 2+ + + = −( ) + +( ) + −( ) + dd

= ⋅ ⋅ ⋅ ⋅

45. If p is a prime greater than 5, then the number 111 1. . . , consisting of p − 1 ones, is divisible by p. For example, 7 111 111| , , since 7 15 873 111 111× =, , . Verify the initial sen- tence for the next three primes.

46. a. Find the largest n such that 3 24n | !. b. Find the smallest n such that 36 | n!. c. Find the largest n such that 12 24n | !.

47. Do Problem 20 using a spreadsheet to create a table of values, n and p n( ), for n = 1 2 20, , . . . . Once the smallest value of n is found, evaluate p n( )+ 1 and p n( ).+ 2 Are they prime or composite?

48. In Exercise 10, 2 times the ones digit was subtracted from the remaining number repeatedly as part of the divisibility test for 7. Use this same technique to develop a test for divisibility by 11. (Hint: Use something other than 2.)

EXERCISES 1. An efficient way to find all the primes up to 100 is to

arrange the numbers from 1 to 100 in six columns. As with the Sieve of Eratosthenes, cross out the multiples of 2, 3, 5, and 7. What pattern do you notice? (Hint: Look at the col- umns and diagonals.)

1 2 3  4 5 6 7 8 9 10 11 12

13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 3 6 37 38 39 40 41 42

43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100

2. Find a factor tree for each of the following numbers. a. 192     b. 380     c. 1593     d. 3741

EXERCISE/PROBLEM SET B

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188 Chapter 5 Number Theory

3. Factor each of the following numbers into primes. a. 39 b. 1131 c. 55 d. 935 e. 3289 f. 5889

4. Use the definition of divides to show that each of the following is true. (Hint: Find x that satisfies the definition of divides.)

a. 7 49| b. 21 210| c. 3 9 18| ( )× d. 2 2 5 72| ( )× × e. 6 2 3 7 134 2 3 5| ( )× × × f. 100,000 | (2 3 5 17 )7 9 11 8× × × g. 6000 2 3 5 2921 17 89 37| ( )× × × h. 22 121 4| ( )× i. p q r p q r s t3 5 5 13 7 2 27| ( ) j. 7 5 21 14| ( )× +

5. If 24 divides b, what else must divide b?

6. If the variables represent counting numbers, determine whether each of the following is true or false.

a. If 2 | a and 6 | a, then 12 | a. b. 6 | xy, then 6 | x or 6 | y.

7. a. Prove in two different ways that 2 divides 114. b. Prove in two different ways that 3 336| .

8. Which of the following are multiples of 3? of 4? of 9? a. 2,199,456     b. 31,020,417

9. Use the test for divisibility by 11 to determine which of the following numbers are divisible by 11.

a. 945,142     b. 6,247,251     c. 385,627

10. There is a test for divisibility by 17 similar to the one for 7 as shown in Set A, Exercise 10. Rather than using 2 times the ones digit, however, one must use 5 in place of 2 before subtracting. Determine which of the following numbers are divisible by 17 using that test.

a. 2963     b. 37,142     c. 95,757,923

11. True or false? Explain. a. If a counting number is divisible by 6 and 8, it must be

divisible by 48. b. If a counting number is divisible by 4, it must be divis-

ible by 8.

12. Decide whether the following are true or false using only divisibility ideas given in this section (do not use long divi- sion or a calculator). Give a reason for your answers.

a. 24 325 608| , b. 45 13 075| , c. 40 1 732 800| , , d. 36 677 916| ,

13. Which of the following numbers are composite? Why? a. 123,456 b. 1,234,567 c. 123,456,789

14. To determine if 769 is prime, we must check to see if it is divisible by any numbers other than 1 and itself. List all of the numbers that must be checked as possible factors to see if 769 is prime.

PROBLEMS 15. A calculator may be used to test for divisibility of one

number by another, where n and d represent counting numbers.

a. If n d÷ gives the answer 176, is it necessarily true that d n| ?

b. If n d÷ gives the answer 56.3, is it possible that d n| ?

16. Justify the test for divisibility by 9 for any four-digit number. (Hint: Reason by analogy from the test for divisibility by 3.)

17. Justify the tests for divisibility by 4 and 8.

18. Find the first composite number in this list.

3 2 1 5

4 3 2 1 19

5 4 3 2 1 101

! ! !

! ! ! !

! ! ! ! !

− + = − + − = − + − + =

Prime

Prime

Primme

Continue this pattern. (Hint: The first composite comes within the first 10 such numbers.)

19. In 1845, the French mathematician Bertrand made the following conjecture: Between any whole number greater than 1 and its double there is at least one prime. In 1911, the Russian mathematician Tchebyshev proved the con- jecture true. Using the Chapter 5 eManipulative activity Sieve of Eratosthenes on our Web site to display all primes less than 200, find three primes between each of the fol- lowing numbers and its double.

a. 30     b. 50     c. 100

20. The numbers 2, 3, 5, 7, 11, and 13 are not factors of 211. Can we conclude that 211 is prime without checking for more factors? Why or why not?

21. It is claimed that the formula n n2 41− + yields a prime for all whole-number values for n. Decide whether this state- ment is true or false.

22. In 1644, the French mathematician Mersenne asserted that 2 1n − was prime only when n = 2 3 5 7 13 17 19 31 67 127, , , , , , , , , , and 257. As it turned out, when n = 67 and n = 257, 2 1n − was a composite, and 2 1n − was also prime when n = 89 and n = 107. Show that Mersenne’s assertion was correct concerning n = 3 5 7, , , and 13.

23. It is claimed that every prime greater than 3 is either one more or one less than a multiple of 6. Investigate. If it seems true, prove it. If it does not, find a counterexample.

24. Is it possible for the sum of two odd prime numbers to be a prime number? Why or why not?

25. Mathematician D. H. Lehmer found that there are 209 consecutive composites between 20,831,323 and 20,831,533. Pick two numbers at random between 20,831,323 and 20,831,533 and prove that they are com- posite.

26. Prime triples are consecutive primes whose difference is 2. One such triple is 3, 5, 7. Find more or prove that there cannot be any more.

27. A seventh-grade student named Arthur Hamann made the following conjecture: Every even number is the difference of two primes. Express the following even numbers as the difference of two primes.

a. 12    b. 20    c. 28

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Section 5.1 Primes, Composites, and Tests for Divisibility 189

28. The numbers 1, 7, 13, 31, 37, 43, 61, 67, and 73 form a 3 3× additive magic square. (An additive magic square has the same sum in all three rows, three columns, and two main diagonals.) Find it.

29. Can you find whole numbers a and b such that 3 5a b= ? Why or why not?

30. I’m a two-digit number less than 40. I’m divisible by only one prime number. The sum of my digits is a prime, and the difference between my digits is another prime. What number could I be?

31. What is the smallest counting number divisible by 2, 4, 6, 8, 10, 12, and 14?

32. What is the smallest counting number divisible by the numbers 1 2 3 4 24 25, , , , . . . , ? (Hint: Give your answer in prime factorization form.)

33. The sum of five consecutive counting numbers has a divi- sor (other than 1) of _____. Prove.

34. Take any number with an even number of digits. Reverse the digits and add the two numbers. Does 11 divide your result? If yes, try to explain why.

35. Take a number. Reverse its digits and subtract the smaller of the two numbers from the larger. Determine what num- ber always divides such differences for the following types of numbers.

a. A two-digit number     b. A three-digit number c. A four-digit number

36. Choose any three digits. Arrange them three ways to form three numbers. Claim: The sum of your three numbers has a factor of 3. True or false?

Example:

371

137

713

1221

+

and 1221 3 407= ×

37. Someone spilled ink on a bill for 36 sweatshirts. If only the first and last digits were covered and the other three digits were, in order, 8, 3, 9 as in ?83.9?, how much did each cost?

38. Determine how many zeros are at the end of the numerals for the following numbers in base ten.

a. 10!        b. 100!        c. 1000!

39. Find the smallest number n with the following property: If n is divided by 3, the quotient is the number obtained by moving the last digit (ones digit) of n to become the first digit. All of the remaining digits are shifted one to the right.

40. A man and his grandson have the same birthday. If for six consecutive birthdays the man is a whole number of times as old as his grandson, how old is each at the sixth birthday?

41. Let m be any odd whole number. Then m is a divisor of the sum of any m consecutive whole numbers. True or false? If true, prove. If false, provide a counter- example.

42. A merchant marked down some pads of paper from $2 and sold the entire lot. If the gross received from the sale was $603.77, how many pads did she sell?

43. How many prime numbers less than 100 can be written using the digits 1, 2, 3, 4, 5 if

a. no digit is used more than once? b. a digit may be used twice?

44. Which of the numbers in the set

{ , , , , . . .}9 99 999 9999

are divisible by 7?

45. Two digits of this number were erased: 273*49*5. However, we know that 9 and 11 divide the number. What is it?

46. This problem appeared on a Russian mathematics exam: Show that all the numbers in the sequence 100001, 10000100001 1000010000100001, , . . . . are composite. Show that 11 divides the first, third, fifth numbers in this sequence, and so on, and that 111 divides the second. An American engineer claimed that the fourth number was the product of

21401 4672725574038601 and .

Was he correct?

47. The Fibonacci sequence, 1 1 2 3 5 8 13, , , , , , , . . . , is formed by adding any two consecutive numbers to find the next term. Prove or disprove: The sum of any ten consecutive Fibonacci numbers is a multiple of 11.

48. Do Set A, Problem 21 (a) using a spreadsheet to create a table of values, n and p n( ), for n = 1 2 20, , . . . .

49. In Exercise 10 of Set A, 2 times the ones digit was sub- tracted from the remaining number repeatedly as part of the divisibility test for 7. Use this same technique to develop a test for divisibility by 13. (Hint: Use something other than 2.).

Analyzing Student Thinking

50. Courtney asserts that if the Sieve of Eratosthenes can be used to find all the primes, then it should also be able to find all the composite numbers. Is she correct? Explain.

51. One of the theorems in this section states “If a m| and a n| , then a m n| ( )+ when a is nonzero.” A student suggests that “If a m n| ( )+ , then a m| or a n| .” Is the student cor- rect? Explain.

52. Christa asserts that the test for divisibility by 9 is similar to the test for divisibility by 3 because every power of 10 is one more than a multiple of 9. Is she correct? Explain.

53. Brooklyn says to you that to find all of the prime factors of 113, she only has to check to see if any of 2, 3, 5, 7 are a factors of 113. How do you respond?

54. A student says that she is confused by the difference between a b| , which means “a divides b,” and a b/ , which is read “a divided by b.” How would you help her resolve her confusion?

55. Kelby notices that if 2 36| and 9 36| , then 18 36| since 18 2 9= × . He also notices that 4 36| and 6 36| but 24 36� . How could you help him understand why one case works and the other does not?

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190 Chapter 5 Number Theory

Counting Factors In addition to finding prime factors, it is sometimes useful to be able to find how many factors (not just prime factors) a number has. The fundamental theorem of arithmetic is helpful in this regard. For example, to find all the factors of 12, consider its prime factorization 12 2 32 1= ¥ . All factors of 12 must be made up of products of at most 2 twos and 1 three. All such combinations are contained in the table. Therefore, 12 has six factors, namely, 1, 2, 3, 4, 6, and 12.

EXPONENT OF 2 EXPONENT OF 3 FACTOR

0 0 2 3 10 0¥ = 1 0 2 3 21 0¥ = 2 0 2 3 42 0¥ = 0 1 2 3 30 1¥ = 1 1 2 3 61 1¥ = 2 1 2 3 122 1¥ =

The technique used in this table can be used with any whole number that is expressed as the product of primes with their respective exponents. To find the num- ber of factors of 2 53 2¥ , a similar list could be constructed. The exponents of 2 would range from 0 to 3 (four possibilities), and the exponents of 5 would range from 0 to 2 (three possibilities). In all there would be 4 3¥ combinations, or 12 factors of 2 53 2¥ , as shown in the following table.

EXPONENTS OF 5

2 0 1 2

0 2050 = 1 2051 = 5 2052 = 25 1 2150 = 2 2151 = 10 2152 = 50 2 2250 = 4 2251 = 20 2252 = 100 3 2350 = 8 2351 = 40 2352 = 200

This method for finding the number of factors of any number can be summarized as follows.

Problem-Solving Strategy Look for a Pattern

Common Core – Grade 4 Find all factor pairs for a whole number in the range 1–100.

Children’s Literature www.wiley.com/college/ musser See “You Can Count on Monsters” by Richard Evan Schwartz.

56. Lorena claims that since a number is divisible by 5 if its ones digit is a 5, then the number 357 is divisible by 7 because the last digit is a 7. How would you respond?

57. Suppose a student said that the sum of the digits of the number 354 is 12 and therefore 354 is divisible by any number that divides into 12, like 2, 3, 4, and 6. Would you agree with the student? Explain.

Following recess, the 1000 students of Wilson School lined up for the following activity: The first student opened all of the 1000 lockers in the school. The second student closed

all lockers with even numbers. The third student “changed” all lockers that were numbered with multiples of 3 by closing those that were open and opening those that were closed. The fourth student changed each locker whose number was a multiple of 4, and so on. After all 1000 students had completed the activity, which lockers were open? Why?

COUNTING FACTORS, GREATEST COMMON FACTOR, AND LEAST COMMON MULTIPLE

Suppose that a counting number n is expressed as a product of distinct primes with their respective exponents, say n p p pn n m

nm= ⋅ ⋅ ⋅( )( ) ( ).1 21 2 Then the number of factors of n is the product ( ) ( ) ( ).n n nm1 21 1 1+ + ⋅ ⋅ ⋅ +¥

T H E O R E M 5 . 1 0

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Section 5.2 Counting Factors, Greatest Common Factor, and Least Common Multiple 191

Greatest Common Factor The concept of greatest common factor is useful when simplifying fractions.

Common Core – Grade 6 Find the greatest common factor of two whole numbers less than or equal to 100.

Find the number of factors.

a. 144     b. 2 5 73 7 4¥ ¥      c. 9 115 2¥

S O L U T I O N

a.144 2 34 2= ¥ . So, the number of factors of 144 is ( )( ) .4 1 2 1 15+ + = b. 2 5 73 7 4¥ ¥ has ( )( ) ( )3 1 7 1 4 1 160+ + + = factors. c. 9 11 3 115 2 10 2¥ ¥= has ( )( )10 1 2 1 33+ + = factors. (NOTE: 95 had to be rewritten as 310,

since 9 was not prime.) ■

Check for Understanding: Exercise/Problem Set A #1–2✔

Notice that the number of factors does not depend on the prime factors, but rather on their respective exponents.

Greatest Common Factor

The greatest common factor (GCF) of two (or more) nonzero whole numbers is the largest whole number that is a factor of both (all) of the numbers. The GCF of a and b is written GCF(a, b).

D E F I N I T I O N 5 . 3

Use your understanding of the greatest common factor to find GCF(27, 45). Write a detailed description of the method used to find it.

By definition, GCF( ,a a a) = . There are two elementary ways to find the greatest common factor of two num-

bers: the set intersection method and the prime factorization method. The GCF(24, 36) is found next using these two methods.

Set Intersection Method

Step 1

Find all factors of 24 and 36. Since 24 2 33= ¥ , there are 4 2 8¥ = factors of 24, and since 36 2 32 2= ¥ , there are 3 3 9¥ = factors of 36. The set of factors of 24 is {1, 2, 3, 4, 6, 8, 12, 24}, and the set of factors of 36 is {1, 2, 3, 4, 6, 9, 12, 18, 36}.

Step 2

Find all common factors of 24 and 36 by taking the intersection of the two sets in step 1.

{ , , , , , , , } { , , , , , , , , }1 2 3 4 6 8 12 24 1 2 3 4 6 9 12 18 36∩ = {{ , , , , , }1 2 3 4 6 12

Step 3

Find the largest number in the set of common factors in step 2. The largest number in {1, 2, 3, 4, 6, 12} is 12. Therefore, 12 is the GCF of 24 and 36. (NOTE: The set intersection method can also be used to find the GCF of more than two numbers in a similar manner.)

Algebraic Reasoning The concept of greatest common factor (applied to 6, 15, and 12 in 6 15 12x y− + ) is useful when factoring or simplifying such expressions.

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192 Chapter 5 Number Theory

While the set intersection method can be cumbersome and at times less efficient than other methods, it is conceptually a natural way to think of the greatest common factor. The “common” part of the greatest common factor is the “intersection” part of the set intersection method. Once students have a good understanding of what the greatest common factor is by using the set intersection method, the prime factoriza- tion method, which is often more efficient, can be introduced.

Prime Factorization Method

Step 1

Express the numbers 24 and 36 in their prime factor exponential form: 24 2 33= ¥ and 36 2 32 2= ¥ .

Step 2

The GCF will be the number 2 3m n where m is the smaller of the exponents of the 2s and n is the smaller of the exponents of the 3s. For 2 33 ¥ and 2 32 2¥ , m is the smaller of 3 and 2, and n is the smaller of 1 and 2. Therefore, the GCF of 2 33 1¥ and 2 32 2¥ is 2 3 122 1¥ = . Review this method so that you see why it always yields the largest number that is a factor of both of the given numbers.

Find GCF(42, 24) in two ways.

S O L U T I O N

Set Intersection Method 42 2 3 7= ¥ ¥ , so 42 has 2 2 2 8¥ ¥ = factors. 24 2 33= ¥ , so 24 has 4 2 8¥ = factors. Factors of 42: {1, 2, 3, 6, 7, 14, 21, 42}. Factors of 24: {1, 2, 3, 4, 6, 8, 12, 24}. Common factors: {1, 2, 3, 6}. GCF(42, 24) = 6.

Prime Factorization Method 42 2 3 7= ¥ ¥ and 24 2 33= ¥ . GCF ( , ) .42 24 2 3 6= =¥ Notice that only the common primes (2 and 3) are used, since the exponent on the 7 is zero in the prime factorization of 24. ■

Earlier in this chapter we obtained the following result: If a m| , a n| , and m n≥ , then a m n| ( ).− In words, if a number divides each of two numbers, then it divides their difference. Hence, if c is a common factor of a and b, where a b≥ , then c is also a common factor of b and a b− . Since every common factor of a and b is also a common factor of b and a b− , the pairs (a, b) and ( , )a b b− have the same common factors. So GCF(a, b) and GCF( , )a b b− must also be the same.

If a and b are whole numbers, with a b≥ , then

GCF( ) GCF( )a b a b b, , .= −

T H E O R E M 5 . 1 1

The usefulness of this result is illustrated in the next example.

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Section 5.2 Counting Factors, Greatest Common Factor, and Least Common Multiple 193

Using a calculator, we can find the GCF(546, 390) as follows:

546 − 390 = 156

390 − 156 = 234

234 − 156 = 78

156 − 78  = 78

Therefore, since the last two numbers in the last line are equal, then the GCF(546, 390) = 78. Notice that this procedure may be shortened by storing 156 in the calcula- tor’s memory.

This calculator method can be refined for very large numbers or in exceptional cases. For example, to find GCF(1417, 26), 26 must be subtracted many times to produce a number that is less than (or equal to) 26. Since division can be viewed as repeated subtraction, long division can be used to shorten this process as follows:

)26 1417 54 13

R

Here 26 was “subtracted” from 1417 a total of 54 times to produce a remainder of 13. Thus GCF(1417, 26) = GCF(13, 26). Next, divide 13 into 26.

)13 26 2 0

R

Thus GCF(13, 26) = 13, so GCF(1417, 26) = 13. Each step of this method can be justified by the following theorem.

Find the GCF(546, 390).

S O L U T I O N

GCF( ) GCF( )

GCF( )

GCF(

546 390 546 390 390

156 390

390 156 15

, ,

,

,

= − = = − 66

234 156

156 156

156

78 7

)

GCF( )

GCF(234 )

GCF(78 )

GCF(156

= = − = = −

,

,

,

, 88

78 78

78

)

GCF( )= =

,

■

If a and b are whole numbers with a b≥ and a bq r= + , where r b< , then

GCF( ) GCF( )a b r b, , .=

T H E O R E M 5 . 1 2

Thus, to find the GCF of any two numbers, this theorem can be applied repeat- edly until a remainder of zero is obtained. The final divisor that leads to the zero remainder is the GCF of the two numbers. This method is called the Euclidean algorithm.

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194 Chapter 5 Number Theory

A calculator also can be used to calculate the GCF(3432, 840) using the Euclidean algorithm.

3432 840 4 085714286÷ = .

3432 4 840 72− × = .

840 72 11 66666667÷ = .

840 11 72 48− × = .

72 48 1 5÷ = .

72 1 48 24− × = .

48 24 2÷ = .

Therefore, 24 is the GCF(3432, 840). Notice how this method parallels the one in Example 5.12.

Find the GCF(840, 3432).

S O L U T I O N

)840 3432 4 72

R

)72 840 11 48

R

)48 72 1 24 R

)24 48 2 0

R

Therefore, GCF(840, 3432) = 24. ■

Check for Understanding: Exercise/Problem Set A #3–8✔

Least Common Multiple The least common multiple is useful when adding or subtracting fractions.

Refl ection from Research Possibly because students often confuse factors and multiples, the greatest common factor and the least common multiple are dif- ficult topics for students to grasp (Graviss & Greaver, 1992).

Common Core – Grade 6 Find the least common multiple of two whole numbers less than or equal to 12.

Least Common Multiple

The least common multiple (LCM) of two (or more) nonzero whole numbers is the smallest nonzero whole number that is a multiple of each (all) of the numbers. The LCM of a and b is written LCM(a, b).

D E F I N I T I O N 5 . 4

Use your understanding of the least common multiple to find LCM(18, 24). Write a detailed description of the method used to find it.

By definition, LCM(a a a, ) = .

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Section 5.2 Counting Factors, Greatest Common Factor, and Least Common Multiple 195

For the LCM, there are three elementary ways to find the least common multiple of two numbers: the set intersection method, the prime factorization method, and the build-up method. The LCM(24, 36) is found next using these three methods.

Set Intersection Method

Step 1

List the first several nonzero multiples of 24 and 36. The set of nonzero mul- tiples of 24 is { , , , , , , . . .},24 48 72 96 120 144 and the set of nonzero multiples of 36 is { , , , , . . .}.36 72 108 144 (NOTE: The set of multiples of any nonzero whole number is an infinite set.)

Step 2

Find the first several common multiples of 24 and 36 by taking the intersection of the two sets in step 1:

{ , , , , , , . . .} { , , , , . . .} { , , . . .}24 48 72 96 120 144 36 72 108 144 72 144∩ = ..

Step 3

Find the smallest number in the set of common multiples in step 2. The smallest number in { , , . . .}72 144 is 72. Therefore, 72 is the LCM of 24 and 36 (Figure 5.5).

Similar to the methods for finding the greatest common factor, the “intersection” part of the set intersection method illustrates the “common” part of the least com- mon multiple. Thus, the set intersection method is a natural way to introduce the least common multiple.

Prime Factorization Method

Step 1

Express the numbers 24 and 36 in their prime factor exponential form: 24 2 33= ¥ and 36 2 32 2= ¥ .

Step 2

The LCM will be the number 2 3r s, where r is the larger of the exponents of the 2s and s is the larger of the exponents of the 3s. For 2 33 1¥ and 2 32 2¥ , r is the larger of 3 and 2 and s is the larger of 1 and 2. That is, the LCM of 2 33 1¥ and 2 32 2¥ is 2 33 2¥ , or 72. Review this procedure to see why it always yields the smallest number that is a multiple of both of the given numbers.

Build-up Method

Step 1

As in the prime factorization method, express the numbers 24 and 36 in their prime factor exponential form: 24 2 33= ¥ and 36 2 32 2= ¥ .

Step 2

Select the prime factorization of one of the numbers and build the LCM from that as follows. Beginning with 24 2 33= ⋅ , compare it to the prime factorization of 36 2 32 2= ⋅ . Because 2 32 2⋅ has more factors of three than 2 33 1⋅ , build up the 2 33 1⋅ to have the same number of factors of three as 2 32 2¥ , making the LCM 2 33 2⋅ . If there are more than two numbers for which the LCM is to be found, continue to compare and build with each subsequent number.

Algebraic Reasoning Finding a Least Common Multiple is really solving a system of equa- tions. When searching for the LCM(24, 36), we are searching for whole numbers x , y , and z that will make the equations 24x z= and 36y z= true.

Figure 5.5

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196 Chapter 5 Number Theory

Extending the Concepts of GCF and LCM These methods can also be applied to find the GCF and LCM of several numbers.

Find the LCM(42, 24) in three ways.

S O L U T I O N

Set Intersection Method Multiples of 42: { , , , , . . . .}.42 84 126 168 Multiples of 24: { , , , , , , , . . . .}.24 48 72 96 120 144 168 Common multiples: { , . . .}.168 LCM(42, 24) = 168.

Prime Factorization Method 42 2 3 7= ¥ ¥ and 24 2 33= ¥ . LCM ( , ) .42 24 2 3 7 1683= =¥ ¥

Build-up Method 42 2 3 7= ¥ ¥ and 24 2 33= ¥ . Beginning with 24 2 33= ¥ , compare to 2 3 7¥ ¥ and build 2 33 ¥ up to 2 3 73 ¥ ¥ . LCM( , ) .42 24 2 3 7 1683= =¥ ¥

Notice that all primes from either number are used when forming the least common multiple. ■

Check for Understanding: Exercise/Problem Set A #9–11✔

In the diagram at the right, the circle labeled F48 contains the prime factors of 48 and the circle labeled F72 contains

the prime factors of 72. Place all the prime factors of 48 and 72 in the appropriate location in the Venn diagram at the right.

How are the GCF(48, 72) and the LCM(48, 72) related to the placement of the numbers in the diagram?

F48 F72

Find the (a) GCF and (b) LCM of the three numbers 2 3 5 2 3 5 75 2 7 4 4 3¥ ¥ ¥ ¥ ¥, , and 2 3 5 136 4 2¥ ¥ ¥ .

S O L U T I O N a. The GCF is 2 3 51 2 3¥ ¥ (use the common primes and the smallest respective

exponents). b. Using the build-up method, begin with 2 3 55 2 7¥ ¥ . Then compare it to 2 3 5 74 4 3¥ ¥ ¥

and build up the LCM to 2 3 5 75 4 7¥ ¥ ¥ . Now compare with 2 3 5 136 4 2¥ ¥ ¥ and build up 2 3 5 7 2 3 5 7 135 4 7 5 6 7 2¥ ¥ ¥ ¥ ¥ ¥ ¥ to . ■

If you are trying to find the GCF of several numbers that are not in prime-factored exponential form, as in Example 5.14, you may want to use a computer program. By considering examples in exponential notation, one can observe that the GCF of a, b, and c can be found by finding GCF(a, b) first and then GCF(GCF(a, b), c). This idea can be extended to as many numbers as you wish. Thus one can use the Euclidean algorithm by finding GCFs of numbers, two at a time. For example, to find GCF(24, 36, 160), find GCF(24, 36), which is 12, and then find GCF(12, 160), which is 4.

Finally, there is a very useful connection between the GCF and LCM of two num- bers, as illustrated in the next example.

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Section 5.2 Counting Factors, Greatest Common Factor, and Least Common Multiple 197

Example 5.15 illustrates that all of the prime factors and their exponents from the original number are accounted for in the GCF and LCM. This relationship is stated next.

Find the GCF and LCM of a and b, for the numbers a = 2 3 5 75 7 2¥ ¥ ¥ and b = 2 3 5 113 2 6¥ ¥ ¥ .

S O L U T I O N   Notice in the following solution that the products of the factors of a and b, which are in bold type, make up the GCF, and the products of the remaining factors, which are circled, make up the LCM.

Hence

GCF( ) LCM( )a b a b, , ( )( )

( ) (

× =

=

2 3 5 2 3 5 7 11

2 3 5 7

3 2 2 5 7 6

5 7 2

¥ ¥ ¥ ¥ ¥ ¥ ¥ ¥ ¥ ¥ 22 3 5 113 2 6¥ ¥ ¥ )

.= ×a b ■

Let a and b be any two whole numbers. Then

GCF( ) LCM( )a b a b ab, , .× =

T H E O R E M 5 . 1 3

Also, LCM( )a b ab

a b ,

( , ) =

GCF is a consequence of this theorem. So if the GCF of two

numbers is known, the LCM can be found using the GCF.

Find the LCM(36, 56).

S O L U T I O N   GCF(36, 56) = 4. Therefore, LCM = = =36 56 4

9 56 504 ¥ ¥ . ■

This technique applies only to the case of finding the GCF and LCM of two num- bers.

We end this chapter with an important result regarding the primes by proving that there is an infinite number of primes.

P R O O F

Either there is a finite number of primes or there is an infinite number of primes. We will use indirect reasoning. Let us assume that there is only a finite number of primes, say 2 3 5 7 11, , , , , . . . , ,p where p is the greatest prime. Let N p= +( . . . )2 3 5 7 11 1¥ ¥ ¥ ¥ . This number, N , must be 1, prime, or composite. Clearly, N is greater than 1. Also, N is greater than any prime. But then, if N is composite, it must have a prime factor. Yet whenever N is divided by a prime, the remainder is always 1 (think about this)! Therefore, N is neither 1, nor a prime, nor a composite. But that is impossible. Using indirect reasoning, we conclude that there must be an infinite number of primes. ■

Problem-Solving Strategy Use Indirect Reasoning

There is an infinite number of primes.

T H E O R E M 5 . 1 4

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198 Chapter 5 Number Theory

There are also infinitely many composite numbers (for example, the even numbers greater than 2).

Check for Understanding: Exercise/Problem Set A #12–17✔

On January 25, 2013, a new large Mersenne prime was found. Mersenne primes are special primes that take on a certain form, 2n– 1. This largest known prime is 257885161 – 1 and has 17,425,170 digits. The second largest known prime was found on August 23, 2008 the prime 243112609 – 1 was found and verified to have 12,978,189 digits. Typically each new prime that is found is larger than the previous ones. However, between 2008 and 2013, two other large primes were found but neither was as large as the prime found in August 2008. If the largest prime, found in 2013, were written out in a typical newsprint size, it would fill about 900 pages of a newspaper. Why search for such large primes? One reason is that it requires trillions of calculations and hence can be used to test computer speed and reliability. Also, it is important in writing messages in code. Besides, as a computer expert put it, “it's like Mount Everest; why do people climb mountains?” To keep up on the ongoing race to find a big- ger prime, visit the Web site www.mersenne.org.

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EXERCISES 1. How many factors do each of the following numbers have? a. 2 32 × b. 3 53 2× c. 5 7 112 3 4× ×

2. a. Factor 36 into primes. b. Factor each divisor of 36 into primes. c. What relationship exists between the prime factors

in part b and the prime factors in part a? d. Let x = ×7 174 2. If n is a divisor of x, what can you say

about the prime factorization of n? e. How many factors does x = 7 174 2¥ have? List them.

3. Use the set intersection method to find the GCFs. a. GCF(12, 18) b. GCF(42, 28) c. GCF(60, 84)

4. Use the prime factorization method to find the GCFs. a. GCF(8, 18) b. GCF(36, 42) c. GCF(24, 66)

5. Using a calculator method, find the following. a. GCF(138, 102)              b. GCF(484, 363) c. GCF(297, 204)              d. GCF(222, 2222)

6. Using the Euclidean algorithm, find the following GCFs. a. GCF(24, 54) b. GCF(39, 91) c. GCF(72, 160) d. GCF(5291, 11951)

7. Use any method to find the following GCFs. a. GCF(51, 85) b. GCF(45, 75) c. GCF(42, 385) d. GCF(117, 195)

8. Two counting numbers are relatively prime if the greatest common factor of the two numbers is 1. Which of the fol- lowing pairs of numbers are relatively prime?

a. 4 and 9 b. 24 and 123 c. 12 and 45

9. Use the set intersection method to find the following LCMs.

a. LCM(24, 30) b. LCM(42, 28) c. LCM(12, 14)

10. Use the (i) prime factorization method and the (ii) build- up method to find the following LCMs.

a. LCM(6, 8) b. LCM(4, 10) c. LCM(7, 9) d. LCM(8, 10)

11. Find the following LCMs using any method. a. LCM(60, 72) b. LCM(35, 110) c. LCM(45, 27)

12. Use any method to find the following GCFs. a. GCF(12, 60, 90) b. GCF(55, 75, 245) c. GCF(1105, 1729, 3289) d. GCF(1421, 1827, 2523) e. GCF ( , , , )3 5 11 3 5 11 3 5 11 3 5 113 5 9 4 3 7 7 7 7 11 6 5¥ ¥ ¥ ¥ ¥ ¥ ¥ ¥ f. GCF ( , , )2 3 11 13 2 3 7 13 2 3 5 133 4 2 2 6 2 4 5 3¥ ¥ ¥ ¥ ¥ ¥ ¥ ¥ ¥

13. Use any method to find the following LCMs. a. LCM(2, 3, 5) b. LCM(8, 12, 18) c. LCM ( , , )3 5 11 3 5 11 2 3 5 133 5 9 11 6 5 4 5 3¥ ¥ ¥ ¥ ¥ ¥ ¥ d. LCM ( , , )2 3 11 13 2 3 7 13 3 5 113 4 2 2 6 2 7 7 7¥ ¥ ¥ ¥ ¥ ¥ ¥ ¥

EXERCISE/PROBLEM SET A

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Section 5.2 Counting Factors, Greatest Common Factor, and Least Common Multiple 199

14. Another method of finding the LCM of two or more num- bers is shown. Find the LCM(27, 36, 45, 60). Use this method to find the following LCMs.

2 27 36 45 60 2 27 18 45 30 3 27 9 45 15 3 9 3 15 5 3 3 1 5 5 5 1 1 5 5

1 1 1 1

Divide all even numbers by 2. If not divisible by 2, bring down. Repeat

until none are divisible by 2. Proceed to the next prime number and repeat the process.

Continue until the last row is all ones. LCM = 22 · 33 · 5 (see the left column)

a. LCM(21, 24, 63, 70) b. LCM(20, 36, 42, 33) c. LCM(15, 35, 42, 80)

15. a. For a = 91, b = 39 find GCF(a , b) and a b¥ . Use these two values to find LCM(a , b).

b. For a b= =3 7 11 2 3 76 3 5 3 4 5¥ ¥ ¥ ¥, find LCM(a, b) and a b¥ . Use these values to find GCF(a , b).

16. For each of the pairs of numbers in parts a c− below i. sketch a Venn diagram with the prime factors of a and b

in the appropriate locations.

ii. find GCF(a, b) and LCM(a, b).

Use the Chapter 5 eManipulative Factor Tree on our Web site to better understand the use of the Venn diagram.

a. a = 63, b = 90 b. a = 48, b = 40 c. a = 16, b = 49

17. Which is larger, GCF(12, 18) or LCM(12, 18)?

PROBLEMS 18. The factors of a number that are less than the number

itself are called proper factors. The Pythagoreans classified numbers as deficient, abundant, or perfect, depending on the sum of their proper factors.

a. A number is deficient if the sum of its proper factors is less than the number. For example, the proper factors of 4 are 1 and 2. Since 1 2 3 4 4+ = < , is a deficient number. What other numbers less than 25 are deficient?

b. A number is abundant if the sum of its proper factors is greater than the number. Which numbers less than 25 are abundant?

c. A number is perfect if the sum of its proper factors is equal to the number. Which number less than 25 is perfect?

19. A pair of whole numbers is called amicable if each is the sum of the proper divisors of the other. For example, 284 and 220 are amicable, since the proper divisors of 220 are 1, 2, 4, 5, 10, 11, 20, 22, 44, 55, 110, which sum to 284, whose proper divisors are 1, 2, 4, 71, 142, which sum to 220.

Determine which of the following pairs are amicable. a. 1184 and 1210 b. 1254 and 1832 c. 5020 and 5564

20. Two numbers are said to be betrothed if the sum of all proper factors greater than 1 of one number equals the other, and vice versa. For example, 48 and 75 are betrothed, since

48 3 5 15 25= + + + ,

proper factors of 75 except for 1, and

75 2 3 4 6 8 12 16 24= + + + + + + + ,

proper factors of 48 except for 1.

Determine which of the following pairs are betrothed. a. (140, 195) b. (1575, 1648) c. (2024, 2295)

21. In the following problems, you are given three pieces of information. Use them to answer the question.

a. GCF( , ) ,a b = ×2 3 LCM( ) 2a b, ,= × ×2 3 53

b = × ×2 3 52 . What is a? b. GCF( , ) ,a b = × ×2 7 112 LCM( , ) ,a b = × × × ×2 3 5 7 115 2 3 2

b = × × × ×2 3 5 7 115 2 . What is a?

22. What is the smallest whole number having exactly the following number of divisors?

a. 1 b. 2 c. 3 d. 4 e. 5 f. 6 g. 7 h. 8

23. Find six examples of whole numbers that have the follow- ing number of factors. Then try to characterize the set of numbers you found in each case.

a. 2 b. 3 c. 4 d. 5

24. Euclid (300 B.C.E.) proved that 2 2 11n n− −( ) produced a perfect number [see Exercise 13(c)] whenever 2 1n − is prime, where n = 1 2 3, , , . . . . Find the first four such perfect num- bers. (Note: Some 2000 years later, Euler proved that this formula produces all even perfect numbers.)

25. Find all whole numbers x such that GCF(24, x) = 1 and 1 24≤ ≤x .

26. George made enough money by selling candy bars at 15 cents each to buy several cans of pop at 48 cents each. If he had no money left over, what is the fewest number of candy bars he could have sold?

27. Three chickens and one duck sold for as much as two geese, whereas one chicken, two ducks, and three geese

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200 Chapter 5 Number Theory

were sold together for $25. What was the price of each bird in an exact number of dollars?

28. Which, if any, of the numbers in the set { , , , , , . . .}10 20 40 80 160 is a perfect square?

29. What is the largest three-digit prime all of whose digits are prime?

30. Take any four-digit palindrome whose digits are all non- zero and not all the same. Form a new palindrome by interchanging the unlike digits. Add these two numbers.

Example:   8 448 4 884

13 332

,

,

,

+

a. Find a whole number greater than 1 that divides every such sum.

b. Find the largest such whole number.

31. Fill in the following 4 4× additive magic square, which is comprised entirely of primes.

3 61 19 37

43 31 5 —

— — — 29

— — 23 —

32. What is the least number of cards that could satisfy the following three conditions?

If all the cards are put in two equal piles, there is one card left over.

If all the cards are put in three equal piles, there is one card left over.

If all the cards are put in five equal piles, there is one card left over.

33. Show that the number 343 is divisible by 7. Then prove or disprove: Any three-digit number of the form 100 10a b a+ + , where a b+ = 7, is divisible by 7.

34. In the set {18, 96, 54, 27, 42}, find the pair(s) of numbers with the greatest GCF and the pair(s) with the smallest LCM.

35. Using the Chapter 5 eManipulative Fill ’n Pour on our Web site, determine how to use container A and container B to measure the described target amount.

a. Container A = 8 ounces Container B = 12 ounces Target = 4 ounces

b. Container A = 7 ounces Container B = 11 ounces Target = 1 ounce

EXERCISES 1. How many factors do each of the following numbers have? a. 2 32 2× b. 7 113 3× c. 7 19 7911 6 23× × d. 124

e. How many factors does x = 11 135 3¥ have? List them.

2. a. Factor 120 into primes. b. Factor each divisor of 120 into primes. c. What relationship exists between the prime factors in

part b and the prime factors in part a? d. Let x = ×11 135 3. If n is a divisor of x, what can you say

about the prime factorization of n?

3. Use the set intersection method to find the GCFs. a. GCF(24, 16)        b. GCF(48, 64)          c. GCF(54, 72)

4. Use the prime factorization method to find the following GCFs.

a. GCF(36, 54) b. GCF(16, 51) c. GCF(136, 153)

5. Using a calculator method, find the following. a. GCF(276, 54) b. GCF(111, 111111) c. GCF(399, 102) d. GCF(12345, 54323)

6. Using the Euclidean algorithm and your calculator, find the following GCFs.

a. GCF(2244, 418) b. GCF(963, 657) c. GCF(7286, 1684)

7. Use any method to find the following GCFs. a. GCF(38, 68) b. GCF(60, 126) c. GCF(56, 120) d. GCF(338, 507)

8. a. Show that 83,154,367 and 4 are relatively prime. b. Show that 165,342,985 and 13 are relatively prime. c. Show that 165,342,985 and 33 are relatively prime.

9. Use the set intersection method to find the following LCMs.

a. LCM(15, 18) b. LCM(26, 39) c. LCM(36, 45)

10. Use the (i) prime factorization method and the (ii) build- up method to find the following LCMs.

a. LCM(15, 21) b. LCM(14, 35) c. LCM(75, 100) d. LCM(130, 182)

11. Find the following LCMs using any method. a. LCM(21, 51) b. LCM(111, 39) c. LCM(125, 225)

12. Use any method to find the following GCFs. a. GCF(15, 35, 42) b. GCF(28, 98, 154) c. GCF(1449, 1311, 1587) d. GCF(2737, 3553, 3757) e. GCF( , , , )5 7 13 5 7 11 13 5 7 13 5 7 134 3 6 6 5 2 4 4 7 11 3 6 8¥ ¥ ¥ ¥ ¥ ¥ ¥ ¥ ¥ f. GCF(3 5 7 11 , 2 3 7 13 , 3 7 11 13 )4 4 2 2 3 7 3 5 2 2¥ ¥ ¥ ¥ ¥ ¥ ¥ ¥ ¥

EXERCISE/PROBLEM SET B

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Section 5.2 Counting Factors, Greatest Common Factor, and Least Common Multiple 201

13. Use any method to find the following LCMs. a. LCM(4, 5, 6) b. LCM(9, 15, 25) c. LCM(3 5 7 , 5 7 11 , 3 7 11 )3 5 4 2 6 4 4 3 3¥ ¥ ¥ ¥ ¥ d. LCM ( , , )2 5 11 3 7 13 2 5 7 133 4 3 8 2 7 4 2 5 3¥ ¥ ¥ ¥ ¥ ¥ ¥

14. Use the method described in Set A, Exercise 14 to find the following LCMs.

a. LCM(12, 14, 45, 35) b. LCM(54, 40, 44, 50) c. LCM(39, 36, 77, 28)

15. a. For a = 49, b = 84 find GCF(a b, ) and a b¥ . Use these two values to find LCM(a b, ).

b. Given LCM ( , )a b = 2 5 7 117 4 5 4¥ ¥ ¥ and a b¥ ¥ ¥ ¥= 2 5 7 117 7 9 4, find GCF(a , b).

c. Find two different candidates for a and b in part b.

16. For each of the pairs of numbers in parts a c− below i. sketch a Venn diagram with the prime factors of a and b

in the appropriate locations.

ii. find GCF(a, b) and LCM(a, b).

Use the Chapter 5 eManipulative Factor Tree on our Web site to better understand the use of the Venn diagram.

a. a = 30, b = 24 b. a = 4, b = 27 c. a = 18, b = 45

17. Let a and b represent two nonzero whole numbers. Which is larger, GCF(a, b) or LCM(a, b)?

PROBLEMS 18. Identify the following numbers as deficient, abundant, or

perfect. [See Set A, Problem 18(a)] a. 36          b. 28 c. 60          d. 51

19. Determine if the following pairs of numbers are amicable. (See Set A, Problem 19)

a. 1648, 1576 b. 2620, 2924 c. If 17,296 is one of a pair of amicable numbers, what is

the other one? Be sure to check your work.

20. Determine if the following pairs of numbers are betrothed. (See Set A, Problem 20).

a. (248, 231) b. (1050, 1925) c. (1575, 1648)

21. a. Complete the following table by listing the factors for the given numbers. Include 1 and the number itself as factors.

b. What kind of numbers have only two factors? c. What kind of numbers have an odd number of factors?

NUMBER FACTORS NUMBER OF FACTORS

1 1 1 2 1, 2 2 3 4 5 6 7 8 9

10 11 12 13 14 15 16

22. Let the letters p, q , and r represent different primes. Then p qr2 3 has 24 divisors. So would p23. Use p, q , and r to describe all whole numbers having exactly the following number of divisors.

a. 2     b. 3     c. 4     d. 5     e. 6     f. 12

23. Let a and b represent whole numbers. State the conditions on a and b that make the following statements true.

a. GCF(a b a, ) = b. LCM(a b a, ) = c. GCF ( , )a b a b= × d. LCM( , )a b a b= ×

24. If GCF(x y, ) = 1, what is GCF ( , )?x y2 2 Justify your answer.

25. It is claimed that every perfect number greater than 6 is the sum of consecutive odd cubes beginning with 1. For example, 28 1 33 3= + . Determine whether the preceding statement is true for the perfect numbers 496 and 8128.

26. Plato supposedly guessed (and may have proved) that there are only four relatively prime whole numbers that satisfy both of the following equations simultaneously.

x y z x y z w2 2 2 3 3 3 3+ = + + =and

If x = 3 and y = 4 are two of the numbers, what are z and w?

27. Tilda’s car gets 34 miles per gallon and Naomi’s gets 8 miles per gallon. When traveling from Washington, D.C., to Philadelphia, they both used a whole number of gallons of gasoline. How far is it from Philadelphia to Washington, D.C.?

28. Three neighborhood dogs barked consistently last night. Spot, Patches, and Lady began with a simultaneous bark at 11 P.M. Then Spot barked every 4 minutes, Patches every 3 minutes, and Lady every 5 minutes. Why did Mr. Jones suddenly awaken at midnight?

29. The numbers 2, 5, and 9 are factors of my locker number and there are 12 factors in all. What is my locker number, and why?

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202 Chapter 5 Number Theory

30. Which number less than 70 has the greatest number of factors?

31. The theory of biorhythm states that there are three “cycles” to your life:

The physical cycle: 23 days long The emotional cycle: 28 days long The intellectual cycle: 33 days long

If your cycles are together one day, in how many days will they be together again?

32. Show that the number 494 is divisible by 13. Then prove or disprove: Any three-digit number of the form 100 10a b a+ + , where a b+ = 13, is divisible by 13.

33. A Smith number is a counting number the sum of whose digits is equal to the sum of all the digits of its prime factors. Prove that 4,937,775 (which was discovered by Harold Smith) is a Smith number.

34. a. Draw a 2 3× rectangular array of squares. If one diago- nal is drawn in, how many squares will the diagonal go through?

b. Repeat for a 4 6× rectangular array. c. Generalize this problem to an m n× array of

squares.

35. Ramanujan observed that 1729 was the smallest number that was the sum of two cubes in two ways. Express 1729 as the sum of two cubes in two ways.

36. The Euclidean algorithm is an iterative process of find- ing the remainder over and over again in order to find the GCF. Use the dynamic spreadsheet Euclidean on our Web site to identify two numbers that will require the

Euclidean algorithm of at least 10 steps to find the GCF. (Hint: The numbers are not necessarily large, but they can be found by thinking of doing the algorithm backward.)

Analyzing Student Thinking 37. Colby used the ideas in this section to claim that 8 93 4¥ has

( )( )3 1 4 1 20+ + = factors. Is he correct? Explain.

38. Eva is confused by the terms LCM and GCF. She says that the LCM of two numbers is greater than the GCF of two numbers. How can you help her understand this apparent contradiction?

39. A student noticed the terms LCD and GCD in another math book. What do you think these abbreviations stand for?

40. Brett knows that all primes have exactly two factors, but does not think that there is anything special about whole numbers that have exactly three factors. How should you respond?

41. Vivian says that her older brother told her that the GCF and LCM are useful when studying fractions. Is the older brother correct in both cases? Explain.

42. Cecilia says that she prefers to use the Set Intersection method to find the GCF of two numbers, but Mandy prefers the Prime Factorization Method. How can you help the students see the value of learning both methods?

43. In the theorem proving that there is an infinite number of primes, it considers ( ) .2 3 5 7 11 1¥ ¥ ¥ ¥ ¥ ¥⋅ ⋅ ⋅ +p One of your students says that 2 3 5 7 11 13 1¥ ¥ ¥ ¥ ¥ + is not a prime because it is equal to 59 509.¥ Is the student correct? Does this invalidate the theorem? Explain.

A major fast-food chain held a contest to promote sales. With each purchase a customer was given a card with a whole num- ber less than 100 on it. A $100 prize was given to any person who presented cards whose numbers totaled 100. The following are several typical cards. Can you find a winning combination?

3 9 12 15 18 27 51 72 84

Can you suggest how the contest could be structured so that there would be at most 1000 winners throughout the country?

Strategy: Use Properties of Numbers

Perhaps you noticed something interesting about the num- bers that were on sample cards—they are all multiples of 3. From work in this chapter, we know that the sum of two (hence any number of) multiples of 3 is a multiple of 3. Therefore, any combination of the given numbers will produce a sum that is a multiple of 3. Since 100 is not a multiple of 3, it is impossible to win with the given numbers. Although there are several ways to

control the number of winners, a simple way is to include only 1000 cards with the number 1 on them.

Additional Problems Where the Strategy “Use Properties of Numbers” Is Useful

1. How old is Mary?

She is younger than 75 years old.

Her age is an odd number.

The sum of the digits of her age is 8.

Her age is a prime number.

She has three great-grandchildren.

2. A folding machine folds letters at a rate of 45 per minute and a stamping machine stamps folded letters at a rate of 60 per minute. What is the fewest number of each machine required so that all machines are kept busy?

3. Find infi nitely many natural numbers each of which has exactly 91 factors.

END OF CHAPTER MATERIAL

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Chapter Review 203

Constance Bowman Reid (1918–2010) Constance Bowman Reid, Julia Bowman Robinson’s older sister, became a high school English and journalism teacher. She gave up

teaching after marriage to become a freelance writer. She coauthored Slacks and Calluses, a book about two young teachers who, in 1943, decided to help the war effort dur- ing their summer vacation by working on a B-24 bomber production line. Reid became fascinated with number theory because of discussions with her mathematician sister. Her fi rst popular exposition of mathematics in a 1952 Scientifi c American article was about fi nding per- fect numbers using a computer. One of the readers com- plained that Scientifi c American articles should be written by recognized authorities, not by housewives! Reid has written the popular books From Zero to Infi nity, A Long Way from Euclid, and Introduction to Higher Mathemat- ics as well as biographies of mathematicians. She also wrote Julie: A Life of Mathematics, which contains an autobiography of Julia Robinson and three articles about Julia’s work by outstanding mathematical colleagues.

Srinivasa Ramanujan (1887–1920) Srinivasa Ramanujan devel- oped a passion for mathemat- ics when he was a young man in India. Working from nu- merical examples, he arrived at astounding results in num-

ber theory. Yet he had little formal training beyond high school and had only vague notions of the principles of mathematical proof. In 1913 he sent some of his results to the English mathematician George Hardy, who was astounded at the raw genius of the work. Hardy ar- ranged for the poverty-stricken young man to come to England. Hardy became his mentor and teacher but later remarked, “I learned from him much more than he learned from me.” After several years in England, Ramanujan’s health declined. He went home to India in 1919 and died of tuberculosis the following year. On one occasion, Ramanujan was ill in bed. Hardy went to visit, arriving in taxicab number 1729. He remarked to Ramanujan that the number seemed rather dull, and he hoped it wasn’t a bad omen. “No,” said Ramanujan, “it is a very interesting number; it is the smallest number expressible as a sum of cubes in two different ways.”

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Exercises 1. Find the prime factorization of 17,017.

2. Find all the composite numbers between 90 and 100 inclusive.

3. True or false?` a. 51 is a prime number. b. 101 is a composite number. c. 7 91| . d. 24 is a divisor of 36.

e. 21 is a factor of 63. f. 123 is a multiple of 3. g. 81 is divisible by 27. h. 169 13= .

4. Using tests for divisibility, determine whether 2, 3, 4, 5, 6, 8, 9, 10, or 11 are factors of 3,963,960.

5. Invent a test for divisibility by 25.

CHAPTER REVIEW

Review the following terms and exercises to determine which require learning or relearning—page numbers are provided for easy reference.

Vocabulary/Notation Prime number  177 Composite number  177 Sieve of Eratosthenes  177 Factor tree  177 Prime factorization  178

Fundamental Theorem of Arithmetic  178

Divides ( | )a b   178 Does not divide ( )a b�   178 Divisor  178

Factor  178 Multiple  178 Is divisible by  178 Square root  184 Prime Factor Test  184

Primes, Composites, and Tests for Divisibility

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204 Chapter 5 Number Theory

Exercises

1. How many factors does 3 75 3¥ have?

2. Find GCF(144, 108) using the prime factorization method.

3. Find GCF(54, 189) using the Euclidean algorithm.

4. Find LCM(144, 108).

5. Given that there is an infinite number of primes, show that there is an infinite number of composite numbers.

6. Explain how the four numbers 81, 135, GCF(81, 135), and LCM(81, 135) are related.

Greatest common factor [GCF (a, b)]  191

Euclidean algorithm  193 Least common multiple [LCM (a, b)]  194

Vocabulary/Notation

Counting Factors, Greatest Common Factor, and Least Common Multiple

Knowledge 1. True or false? a. Every prime number is odd. b. The Sieve of Eratosthenes is used to find primes. c. A number is divisible by 6 if it is divisible by 2 and 3. d. A number is divisible by 8 if it is divisible by 4 and 2. e. If a b≠ , then GCF( LCM(a b a b, ) , )< . f. The number of factors of n can be determined by the

exponents in its prime factorization. g. The prime factorization of a and b can be used to find

the GCF and LCM of a and b. h. The larger a number, the more prime factors it has. i. The number 12 is a multiple of 36. j. Every counting number has more multiples than factors.

2. Write a complete sentence that conveys the meaning of and correctly uses each of the following terms or phrases.

a. divided into     b. divided by     c. divides

Skill 3. Find the prime factorization of each of the following numbers. a. 120     b. 10,800     c. 819

4. Test the following for divisibility by 2, 3, 4, 5, 6, 8, 9, 10, and 11.

a. 11,223,344     b. 6,543,210     c. 2 3 53 4 6¥ ¥

5. Determine the number of factors for each of the following numbers.

a. 360     b. 216     c. 900

6. Find the GCF and LCM of each of the following pairs of numbers.

a. 144, 120 b. 147, 70 c. 2 3 5 2 3 53 5 7 7 4 3¥ ¥ ¥ ¥, d. 2419, 2173 e. 45, 175, 42, 60

7. Use the Euclidean algorithm to find GCF(6273, 1025).

8. Find the LCM(18, 24) by using the (i) set intersection method, (ii) prime factorization method, and (iii) build-up method.

Understanding 9. Explain how the Sieve of Eratosthenes can be used to find

composite numbers.

10. Is it possible to find nonzero whole numbers x and y such that 7 11 ?x = y Why or why not?

11. Show that the sum of any four consecutive counting num- bers must have a factor of 2.

12. a. Show why the following statement is not true. “If 4 | m and 6 | m then 24 | m.”

b. Devise a divisibility test for 18.

13. Given that a b¥ = 270 and GCF(a b, ) = 3, find LCM(a, b).

14. Use rectangular arrays to illustrate why 4 8| but 3 8� .

15. If n = 2 3 2 7 2 3¥ ¥ ¥ ¥ ¥ and m = 2 2 3 72 2¥ ¥ ¥ , how are m and n related? Justify your answer.

Problem Solving/Application 16. Find the smallest number that has factors of 2, 3, 4, 5, 6,

7, 8, 9, and 10.

17. The primes 2 and 5 differ by 3. Prove that this is the only pair of primes whose difference is 3.

18. If a = 2 32 3¥ and the LCM(a, b) is 1080, what is the (a) smallest and (b) the largest that b can be?

19. Find the longest string of consecutive composite numbers between 1 and 50. What are those numbers?

20. Identify all numbers between 1 and 20 that have an odd number of divisors. How are these numbers related?

21. What is the maximum value of n that makes the statement 2 15 14 13 12 3 2 1n | ¥ ¥ ¥ ¥ ¥⋅ ⋅ ⋅ true?

22. Find two pairs of numbers a and b such that GCF(a b, ) = 15 and LCM(a b, ) = 180.

23. When Alpesh sorts his marbles, he notices that if he puts them into groups of 5, he has 1 left over. When he puts them in groups of 7, he also has 1 left over, but in groups of 6, he has none left over. What is the smallest number of marbles that he could have?

CHAPTER TEST

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c05.indd 205 7/30/2013 2:58:06 PM

206

T he first extensive treatment of fractions known to us appears in the Ahmes (or Rhind) Mathematical Papyrus (1600 B.C.E.), which contains the work of Egyptian mathematicians.

A lb

um /O

ro no

z/ N

ew sC

om

The Egyptians expressed fractions as unit fractions (that is, fractions in which the numerator is 1). Thus, if they wanted to describe how much fish each person would get if they were dividing 5 fish among 8 people, they wouldn’t write it as 58 but would express it as

1 2

1 8+ .

Of course, the Egyptians used hieroglyphics to represent these unit fractions.

Although the symbol looks similar to the

zero, , in the Mayan number system, the Egyptians

used it to denote the unit fraction with the denominator of the fraction written below the .

Because of its common use, the Egyptians did not write 2 3 as the sum

1 2

1 6+ , but rather wrote it as a unit fraction

with a denominator of 32 . Therefore, 2 3 was written as the

reciprocal of 32 with the special symbol shown here.

Our present way of expressing fractions is probably due to the Hindus. Brahmagupta (circa 630 C.E.) wrote the sym- bol 23 (with no bar) to represent “two-thirds.” The Arabs introduced the “bar” to separate the two parts of a frac- tion, but this first attempt did not catch on. Later, due to typesetting constraints, the bar was omitted and, at times, the fraction “two-thirds” was written as 2/3.

The name fraction comes from the Latin word fractus, which means “to break.” The term numerator comes from the Latin word numerare, which means “to num- ber” or to count and the name denominator comes from nomen or name. Therefore, in the fraction two-thirds

2 3( ), the denominator tells the reader the name of the

objects (thirds) and the numerator tells the number of the objects (two).

C H A P T E R

6 FRACTIONS Fractions—A Historical Sketch

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207

One’s point of view or interpretation of a problem can often change a seemingly difficult problem into one that is easily solvable. One way to solve the next problem is by drawing a picture or, perhaps, by actually finding some representative blocks to try various combinations. On the other hand, another approach is to see whether the problem can be restated in an equivalent form, say, using numbers. Then if the equivalent problem can be solved, the solution can be interpreted to yield an answer to the original problem.

Initial Problem A child has a set of 10 cubical blocks. The lengths of the edges are 1 cm, 2 cm, 3 cm, . . . , 10 cm. Using all the cubes, can the child build two towers of the same height by stacking one cube upon another? Why or why not?

Clues The Solve an Equivalent Problem strategy may be appropriate when

You can find an equivalent problem that is easier to solve.

A problem is related to another problem you have solved previously.

A problem can be represented in a more familiar setting.

A geometric problem can be represented algebraically, or vice versa.

Physical problems can easily be represented with numbers or symbols.

A solution of this Initial Problem is on page 247.

Solve an Equivalent ProblemProblem-Solving Strategies

1. Guess and Test

2. Draw a Picture

3. Use a Variable

4. Look for a Pattern

5. Make a List

6. Solve a Simpler Problem

7. Draw a Diagram

8. Use Direct Reasoning

9. Use Indirect Reasoning

10. Use Properties of Numbers

11. Solve an Equivalent Problem

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208

AUTHOR

WALK-THROUGH

I N T R O D U C T I O N Chapters 2 to 5 have been devoted to the study of the system of whole numbers. Understanding the system of whole numbers is necessary to ensure success in mathematics later. This chapter is devoted to the study of fractions. Fractions were invented because it was not convenient to describe many problem situations using only whole numbers. As you study this chapter, note the importance that the whole numbers play in helping to make fraction concepts easy to understand.

Key Concepts from the NCTM Principles and Standards for School Mathematics

PRE-K-2–NUMBER AND OPERATIONS Understand and represent commonly used fractions, such as 1/4, 1/3, and 1/2.

GRADES 3-5–NUMBER AND OPERATIONS Develop understanding of fractions as parts of unit wholes, as parts of a collection, as locations on number lines, and

as divisions of whole numbers. Use models, benchmarks, and equivalent forms to judge the size of fractions.

GRADES 6-8–NUMBER AND OPERATIONS

Understand the meaning and effects of arithmetic operations with fractions. Develop and analyze algorithms for computing with fractions and develop fluency in their use.

Key Concepts from the NCTM Curriculum Focal Points

GRADE 3: Developing an understanding of fractions and fraction equivalence. GRADE 5: Developing an understanding of and fluency with addition and subtraction of fractions and decimals. GRADE 6: Developing an understanding of and fluency with multiplication and division of fractions and decimals.

Key Concepts from the Common Core State Standards for Mathematics

GRADE 3: Develop understanding of fractions as numbers. Specifically, explain equivalence of fractions and com- pare fractions with the same numerator or same denominator.

GRADE 4: Extend understanding of fraction equivalence and ordering by comparing two fractions with different numerators and different denominators. Add and subtract fractions with like denominators. Multiply fractions by whole numbers.

GRADE 5: Use equivalent fractions as a strategy to add and subtract fractions with unlike denominators. Apply and extend previous understandings of multiplication and division to multiply fractions by fractions and divide whole numbers by a unit fraction or a unit fraction by a nonzero whole number.

GRADE 6 : Apply and extend previous understandings of multiplication and division to divide fractions by fractions.

6.1

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Section 6.1 The Set of Fractions 209

The Concept of a Fraction There are many times when whole numbers do not fully describe a mathematical situ- ation. For example, using whole numbers, try to answer the following questions that refer to Figure 6.1: (1) How much pizza is left? (2) How much of the stick is shaded? (3) How much paint is left in the can?

Figure 6.1

Although it is not easy to provide whole-number answers to the preceding questions, the situations in Figure 6.1 can be conveniently described using fractions. Reconsider the preceding questions in light of the subdivisions added in Figure 6.2. Typical answers to these questions are (1) “Three-fourths of the pizza is left,” (2) “Four-tenths of the stick is shaded,” (3) “The paint can is three-fifths full.”

Figure 6.2

The term fraction is used in two distinct ways in elementary mathematics. Initially, fractions are used as numerals to indicate the number of parts of a whole to be consid- ered. In Figure 6.2, the pizza was cut into 4 equivalent pieces, and 3 remain. In this case we use the fraction 34 to represent the 3 out of 4 equivalent pieces (i.e., equivalent in size). The use of a fraction as a numeral in this way is commonly called the “part-to-whole”

model. Succinctly, if a and b are whole numbers, where b ≠ 0, then the fraction a b

or a b/ ,

represents a of b equivalent parts; a is called the numerator and b is called the denominator.

Children’s Literature www.wiley.com/college/musser See “Give me Half” by Stuart J. Murphy.

Algebraic Reasoning While the equation x − =6 2 can be solved using whole numbers, an equation such as 3 2x = has a fraction as a solution.

Reflection from Research Students need experiences that build on their informal frac- tion knowledge before they are introduced to fraction symbols (Sowder & Schappelle, 1994).

Reflection from Research When using area models to represent fractions, it should be noted that students often confuse length and area (Kouba, Brown, Carpenter, Lindquist, Silver, & Swafford, 1988).

For each of the following visual representations of fractions, there is a corresponding incorrect symbolic expression. Discuss what aspects of the visual representation might

lead a student to the incorrect expression.

THE SET OF FRACTIONS

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210 Chapter 6 Fractions

The word denominator comes from the word nomenclature which means “to name.” Thus, the denominator tells you the name of the parts: fourths, tenths, fifths, etc. The word numerator means “to number” so the numerator is the number of parts.

The term equivalent parts means equivalent in some attribute, such as length, area, volume, number, or weight, depending on the composition of the whole and appro- priate parts. In Figure 6.2, since 4 of 10 equivalent parts of the stick are shaded, the fraction 410 describes the shaded part when it is compared to the whole stick. Also, the fraction 35 describes the filled portion of the paint can in Figure 6.2.

As with whole numbers, a fraction also has an abstract meaning as a number. What do you think of when you look at the relative amounts represented by the shaded regions in Figure 6.3?

Figure 6.3

Although the various diagrams are different in size and shape, they share a com- mon attribute—namely, that 5 of 8 equivalent parts are shaded. That is, the same relative amount is shaded. This attribute can be represented by the fraction. 58 . Thus, in addition to representing parts of a whole, a fraction is viewed as a number repre- senting a relative amount. Hence we make the following definition.

Reflection from Research Students often think of a fraction as two numbers separated by a line rather than as representing a single number (Bana, Farrell, & McIntosh, 1995).

NCTM Standard All students should develop understanding of fractions as parts of unit wholes, as a part of a collection, as locations on number lines, and as divisions of whole numbers.

Reflection from Research Students need to develop not only a conceptual understanding of fractions, but also an under- standing of the proper notation of fractions (Brizuela, 2006).

Fractions

A fraction is a number that can be represented by an ordered pair of whole num- bers

a b

(or a b/ ), where b ≠ 0. In set notation, the set of fractions is

F a b

a b b= ≠⎧⎨ ⎩

⎫ ⎬ ⎭

| .and are whole numbers, 0

D E F I N I T I O N 6 . 1

On a piece of paper explain, using pictures, why 34 -and 6 8 are equivalent. Compare and

contrast your explanation with that of your peers.

Fractions can also be represented on a number line. The fraction 1 b

is located on a

number line by defining the interval from 0 to 1 as the whole and partitioning it into b

equal parts. Each part is of length 1 b

and the number 1 b

is located at the right endpoint

of the part with its left endpoint at 0 as shown in Figure 6.4.

Figure 6.4

Common Core – Grade 3 Understand a fraction 1/b as the quantity formed by 1 part when a whole is partitioned into b equal parts; understand a fraction a/b as the quantity formed by a parts of size 1/b.

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Section 6.1 The Set of Fractions 211

The fraction a b

is located on the number line by starting at 0 and marking off a

copies of the length determined by the fraction 1 b

end to end as shown in Figure 6.5.

The right endpoint of the ath copy is the location of the fraction a b

.

0 1

Figure 6.5

Before proceeding with the computational aspects of fractions as numbers, it is instructive to comment further on the complexity of this topic—namely, viewing fractions as numerals and as numbers. Recall that the whole number three was the attribute common to all sets that match the set { , , }a b c . Thus if a child who under- stands the concept of a whole number is shown a set of three objects, the child will answer the question “How many objects?” with the word “three” regardless of the size, shape, color, and so on of the objects themselves. That is, it is the “numerous- ness” of the set on which the child is focusing.

With fractions, there are two attributes that the child must observe. First, when consid- ering a fraction as a number, the focus is on relative amount. For example, in Figure 6.6 the relative amount represented by the various shaded regions is described by the fraction 1 4 (which is considered as a number). Notice that

1 4 describes the relative amount shaded

without regard to size, shape, arrangement, orientation, number of equivalent parts, and so on; thus it is the “numerousness” of a fraction on which we are focusing.

Figure 6.6

Second, when considering a fraction as a numeral representing a part-to-whole relationship, many numerals can be used for the relationship. For example, the three diagrams in Figure 6.6 can be labeled differently (Figure 6.7).

Figure 6.7

In Figures 6.7(b) and (c), the shaded regions have been renamed using the fractions 2 8 and

3 12 , respectively, to call attention to the different subdivisions. The notion of

Common Core – Grade 3 Understand a fraction as a number on the number line; represent fractions on a number line diagram.

Reflection from Research The complex nature of fractions requires students to develop more than a part-to-whole under- standing. Students also need to develop an understanding based

on the quotient 3 4 34÷ =( ) and measurement (location on a number line) interpretations of fractions (Flores, Samson, & Yanik, 2006).

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212 Chapter 6 Fractions

fraction as a numeral displaying a part-to-whole relationship can be merged with the concept of fraction as a number. That is, the fraction (number) 14 can also be thought of and represented by any of the fractions 28 ,

3 12 ,

4 16 , and so on. Figure 6.8 brings this

into sharper focus. On a more general note, a model such as those in Figure 6.8, which uses geometric

shapes or “regions” to represent fractions, is called a region model.

Figure 6.8

In each of the pairs of diagrams in Figure 6.8, the same relative amount is shaded, although the subdivisions into equivalent parts and the sizes of the diagrams are dif- ferent. As suggested by the shaded regions, the fractions 416 ,

2 8 , and

3 12 all represent the

same relative amount as 14 . Two fractions that represent the same relative amount are said to be equivalent

fractions. The diagrams in Figure 6.8 were different shapes and sizes. Fraction strips, which

can be constructed out of paper or cardboard, can be used to visualize fractional parts (Figure 6.9). The advantage of this model is that the unit strips are the same size—only the shading and number of subdivisions vary.

Figure 6.9

Equivalent fractions are also located at the same point on a number line. The equivalence of 34 and

6 8 can be seen by their common location on the number line in

Figure 6.10.

Figure 6.10

It is useful to have a simple test to determine whether fractions such as 28 , 3

12 , and 4

16 represent the same relative amount without having to draw a picture of each represen-

tation. Two approaches can be taken. First, observe that 28 1 2 4 2

3 12

1 3 4 3= =

¥ ¥

¥ ¥, , and

4 16

1 4 4 4= ¥ ¥ .

These equations illustrate the fact that 28 can be obtained from 1 4 by equally subdivid-

ing each portion of a representation of 14 by 2 [Figure 6.8(b)]. A similar argument can

be applied to the equations 312 1 3 4 3= ¥ ¥ [Figure 6.8(c)] and

4 16

1 4 4 4= ¥ ¥ [Figure 6.8(a)]. Thus it

appears that any fraction of the form 1 4 ¥ ¥ n n

, where n is a counting number, represents

NCTM Standard All students should understand and represent commonly used fractions, such as 1

4 1 3, , and

1 2 .

Reflection from Research A child’s understanding of frac- tions needs to coordinate the knowledge of notation and the knowledge of some part-whole relations. However, these two types of knowledge develop independently (Saxe, Taylor, McIntosh, & Gearhart, 2005).

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213

From Lesson 10-5 “Finding Equivalent Fractions” from Envision Math Common Core, by Randall I.Charles et al., Grade 3, copyright © by Pearson Education.

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214 Chapter 6 Fractions

the same relative amount as 14 , or that an bn

a b

= , in general. When an bn

is replaced with a b

,

where n ≠ 1, we say that an bn

has been simplified.

To determine whether 312 and 4

16 are equal, we can simplify each of them: 3

12 1 4=

and 416 1 4= . Since they both equal

1 4 , we can write

3 12

4 16= . Alternatively, we can view

3 12 and

4 16 as

3 16 12 16 ¥ ¥ and

4 12 16 12 ¥ ¥ instead. Since the numerators 3 16 48¥ = and 4 12 48¥ = are

equal and the denominators are the same, namely 12 16¥ , the two fractions 312 and 4

16 must be equal. As the next diagram suggests, the numbers 3 16¥ and 4 12¥ are called the cross-products of the fractions 312 and

4 16 .

The technique, which can be used for any pair of fractions, leads to the following definition of fraction equality.

Fraction Equality

Let a b

and c d

be any fractions. Then a b

c d

= if and only if ad bc= .

D E F I N I T I O N 6 . 2

In words, two fractions are equal fractions if and only if their cross-products, that is, products ad and bc obtained by cross-multiplication, are equal. The first method described for determining whether two fractions are equal is an immediate conse- quence of this definition, since a bn b an( ) ( )= by associativity and commutativity. This is summarized next.

Let a b

be any fraction and n a nonzero whole number. Then

a b

an bn

na nb

= = .

T H E O R E M 6 . 1

It is important to note that this theorem can be used in two ways: (1) to replace the

fraction a b

with an bn

and (2) to replace the fraction an bn

with a b

. Occasionally, the term

reducing is used to describe the process in (2). However, the term reducing can be mis- leading, since fractions are not reduced in size (the relative amount they represent) but only in complexity (the numerators and denominators are smaller).

Common Core – Grade 3 Explain equivalence of fractions in special cases, and understand two fractions as equivalent (equal) if they are the same size, or the same point on a number line.

Algebraic Reasoning Although established on the prin- ciples of equivalent fractions, the process of cross-multiplication is commonly used when solving proportions algebraically.

Common Core – Grade 4 Explain why a fraction a b/ is equivalent to a fraction ( ) )n a n b× ×/( by using visual fraction models, with attention to how the number and size of the parts differ even though the two fractions themselves are the same size. Use this principle to recognize and generate equivalent fractions.

Verify the following equations using the definition of fraction equality or the preceding theorem.

a. 5 6

25 30

= b. 27 36

54 72

=      c. 16 48

1 3

=

S O L U T I O N

a. 5 6

5 5 6 5

25 30

= =¥ ¥

by the preceding theorem.

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Section 6.1 The Set of Fractions 215

b. 54 72

27 2 36 2

27 36

= =¥ ¥

by simplifying. Alternatively, 27 36

3 9 4 9

3 4

= =¥ ¥

and

54 72

3 18 4 18

3 4

= =¥ ¥

, so 27 36

54 72

= .

c. 16 48

1 3

= , since their cross-products, 16 3¥ and 48 1¥ , are equal. ■

Fraction equality can readily be checked on a calculator using an alternative version

of cross-multiplication—namely, a b

c d

= if and only if ad b

c= . Thus the equality 2036 30 54

can be checked by pressing 20 54 36 30× ÷ = . Since 30 obtained in this way is equal to the numerator of 3054 , the two fractions are equal.

Since a b

an bn

= for n = 1 2 3, , , . . . , every fraction has an infinite number of representa-

tions (numerals). For example,

1 2

2 4

3 6

4 8

5 10

6 12

= = = = = = ⋅ ⋅ ⋅

are different numerals for the number 12. In fact, another way to view a fraction as a number is to think of the idea that is common to all of its various representations. That is, the fraction 23 , as a number, is the idea that one associates with the set of all fractions, as numerals, that are equivalent to 23 , namely

2 3 ,

4 6 ,

6 9 ,

8 12 , . . . . Notice that the

term equal refers to fractions as numbers, while the term equivalent refers to fractions as numerals.

Every whole number is a fraction and hence has an infinite number of fraction representations. For example,

1 1 1

2 2

3 3

2 2 1

4 2

6 3

0 0 1

0 2

= = = = ⋅ ⋅ ⋅ = = = = ⋅ ⋅ ⋅ = = = ⋅ ⋅ ⋅, , ,

and so on. Thus the set of fractions extends the set of whole numbers. A fraction is written in its simplest form or lowest terms when its numerator and

denominator have no common prime factors.

Find the simplest form of the following fractions.

a. 12 18

b. 36 56

     c. 9 31

     d. ( ) )2 3 5 3 73 5 7 4¥ ¥ ¥ ¥/(26

S O L U T I O N

a. 12 18

2 6 3 6

2 3

= =¥ ¥

b. 36 56

18 2 28 2

18 28

9 2 14 2

9 14

= = = =¥ ¥

¥ ¥

    or     36 56

2 2 3 3 2 2 2 7

3 3 2 7

9 14

= = =¥ ¥ ¥ ¥ ¥ ¥

¥ ¥

c. 9 31

is in simplest form, since 9 and 31 have only 1 as a common factor.

d. To write ( ) )2 3 5 3 73 5 7 4¥ ¥ ¥ ¥/(26 in simplest form, fi rst fi nd the GCF of 2 3 53 5 7¥ ¥ and 2 3 76 4¥ ¥ :

GCF(23 ¥ ¥ ¥ ¥ ¥3 5 2 3 7 2 35 7 6 4 3 4, ) .=

Then

23 ¥ ¥ ¥ ¥

¥ ¥ ¥ ¥

¥ ¥

3 5 2 3 7

3 5 2 3 2 7 2 3

3 5 2 7

5 7

6 4

7 3 4

3 3 4

7

3= = ( )( ) ( )( )

. ■

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216 Chapter 6 Fractions

Fractions with numerators greater than or equal to their denominators fall into two categories. The fractions 11 ,

2 2 ,

3 3 ,

4 4 , . . . , in which the numerators and denomina-

tors are equal, represent the whole number 1. Fractions where the numerators are greater than the denominators are called improper fractions. For example, 72 ,

8 5 , and

117 35 are improper fractions. The fraction

7 2 would mean that an object was divided into

2 equivalent parts and 7 such parts are designated. Figure 6.11 illustrates a model for 7 2 because seven halves are shaded where one shaded circle represents one unit. The diagram in Figure 6.11 illustrates that 72 can also be viewed as 3 wholes plus

1 2 , or 3

1 2 .

A combination of a whole number with a fraction juxtaposed to its right is called a mixed number. Mixed numbers will be studied in Section 6.2.

Figure 6.11

The TI-34 MultiView calculator can be used to convert improper fractions to mixed numbers. For example, to convert 1234378 to a mixed number in its simplest form press

1234 nd 378 enter which yields 3 100 378 . Pressing csimp enter yields 3

50 189 , which is in

simplest form. For calculators without the fraction function, 1234 378 3 2645502÷ = . shows the whole-number part, namely 3, together with a decimal fraction. The calcula-

tion 1234 3 378 100− × = gives the numerator of the fraction part; thus 1234378 100 3783= ,

which is 3 50189 in simplest form.

Reflection from Research When dealing with improper frac- tions, like 1211 , students struggle to focus on the whole, 1111, and not the number of parts, 12 (Tzur, 1999).

Check for Understanding: Exercise/Problem Set A #1–11✔

Order the following fractions from smallest to largest without converting them to deci- mals. Justify your method of ordering.

12 22

, 8 17

, 10 21

Ordering Fractions The concepts of less than and greater than in fractions are extensions of the respective whole-number concepts. Fraction strips can be used to order fractions (Figure 6.12).

Next consider the three pairs of fractions on the fraction number line in Figure 6.13.

Figure 6.13

As it was in the case of whole numbers, the smaller of two fractions is to the left of the larger fraction on the fraction number line. Also, the three examples in Figure 6.13 suggest the following definition, where fractions having common denominators can be compared simply by comparing their numerators (which are whole numbers).

Figure 6.12

Reflection from Research Young children have a very diffi- cult time believing that a fraction such as one-eighth is smaller than one-fifth because the number eight is larger than five (Post, Wachsmuth, Lesh, & Behr, 1985).

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Section 6.1 The Set of Fractions 217

NOTE: Although the definition is stated for “less than,” a corresponding state- ment holds for “greater than.” Similar statements hold for “less than or equal to” and “greater than or equal to.”

For example, 37 5 7< , since 3 5< ;

4 13

10 13< , since 4 10< ; and so on. The numbers

2 7 and

4 13 can be compared by getting a common denominator.

2 7

2 13 7 13

26 91

4 13

4 7 13 7

28 91

= = = =¥ ¥

¥ ¥

and

Since 2691 28 91< , we conclude that

2 7

4 13< .

This last example suggests a convenient shortcut for comparing any two frac- tions. To compare 27 and

4 13 , we compared

26 91 and

28 91 and, eventually, 26 and 28. But

26 2 13= ¥ and 28 7 4= ¥ . In general, this example suggests the following theorem.

Children’s Literature www.wiley.com/college/musser See “Fraction Action” by Loreen Leedy.

Less Than for Fractions

Let a c

and b c

be any fractions. Then a c

b c

< if and only if a b< .

D E F I N I T I O N 6 . 3Common Core – Grade 3 Compare two fractions with the same numerator or the same denominator by reasoning about their size. Recognize that com- parisons are valid only when the two fractions refer to the same whole. Record the results of com- parisons with the symbols >, =, or <, and justify the conclusions (e.g., by using a visual fraction model).

Cross-Multiplication of Fraction Inequality

Let a b

and c d

be any fractions. Then a b

c d

< if and only if ad bc< .

T H E O R E M 6 . 2

Notice that this theorem reduces the ordering of fractions to the ordering of whole

numbers. Also, since a b

c d

< if and only if c d

a b

> , we can observe that c d

a b

> if and only

if bc ad> .

Arrange in order.

a. 78 and 9

11     b. 17 32 and

19 40

S O L U T I O N 

a. 78 9

11< if and only if 7 11 8 9¥ ¥< . But 77 72> ; therefore, 7 8

9 11> .

b.17 40 680¥ = and 32 19 608¥ = . Since 32 19 17 40¥ ¥< , we have 1940 17 32< . ■

Often fractions can be ordered mentally using your “fraction sense.” For example, fractions like 45 ,

7 8 ,

11 12 , and so on are close to 1, fractions like

1 15 ,

1 14 ,

1 10 , and so on are

close to 0, and fractions like 614 , 8

15 , 11 20 , and so on are close to

1 2 . Thus

4 7

12 13< , since

4 7

1 2≈

and 1213 1≈ . Also, 1 11

7 13< , since

1 11 0≈ and

7 13

1 2≈ .

Keep in mind that this procedure is just a shortcut for finding common denomina- tors and comparing the numerators.

Cross-multiplication of fraction inequality can also be adapted to a T1-34 MultiView cal-

culator as follows: a b

c d

< if and only if ad b

c< (or a b

c d