Math Worksheet

Homework for section 4.3

1. Exercise 4.3 #15 Explain your reasoning and show your work. Make connections to both math and diagrams.

2. Exercise 4.3 #17 Explain your reasoning and show your work. Make connections to

both math and diagrams.

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3. Name two fractions between ! " and "

# . Explain your reasoning, use a diagram like a

number line show all your work. Make connections to both math and diagrams.

4. Exercise 4.3 #26 Explain your reasoning, draw a diagram, and show all your work. Make connections to both math and diagrams. Remember that multiplication is always stated as

“the number of groups” x “ the quantity in each group”

So ½ x ¼ would be ½ group times ¼ of an object in the group.

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b. A proper fraction times a whole number

c. A proper fraction times a mixed number

d. A mixed number times a nrixed number

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e. A proper fraction divided by a proper fraction

f. A mixed number divided by a mixed number

94 CHAPTER 4 Extending the Number System

EXPLORATION 4.13'~ Ordering Fractions

This kind of exploration is used in many elementary school curricula to help students develop fraction sense by ordering fractions. This is done before they have developed algorithms for equivalent fractions or converting fractions into decimals. While children will begin with manip- ulatives and pictorial representations, at some point we want to move beyond them, as we saw in Chapter 3 with base ten blocks. For example, demonstrating that t > t cannot be done easily with a diagram, but it can be done by noting that both fractions are one piece away from 1, and since t > j, ~ is missing a bigger piece, and thus t < t.

There are many tools that you do have, including resorting to the meaning of numerator and denominator (as used in the example above), benchmarks (t, ·h t, etc.), and other ways of look- ing at fractions, for example, as the ratio of the numerator and denominator.

1. Predict the relative value of the pairs of fractions on page 95 by inserting the symbol <, =, or > into the space between the fractions without referring to pictures, finding the lowest common denominator of both fractions, or converting them to decimals. Briefly justify your choice.

2. Order each set of fractions, again without referring to pictures, finding the lowest common denominator of both fractions, or converting them to decimals. Briefly justify your choice.

1 ~ i a. 2 7 9

2 3 11 b. 5 TO T9

c. ~ li 1!. 3 17 80

1 1 .1.. i 7 d. 2 4 10 6 8

e. i ~ i ~ ~ 6 5 11 9 9

f. 1 ~ i i 3 3 8 5 8 6 49 56

n

n

SECTION 4.3 • Exploring Operations with Fractions 95

Table for EXPLORATION 4.13: Ordering fractions

u >, =, Fraction or< Fraction Justification

a. l l 5 8

b. 5 7 6 8

c. l J.. .5 12

d. l .!1_ 2 31

e. l ~ 8 9

f. f. l 7 8

g. l 2 4 9

h. _2_ l 11 9

i. J. ~ 8 5

j. .1_ _2_ 10 23

k. 2 .1_ 9 10

u

Mathematics for ~· ..... ;;. Elementary · . .l "·;

U School Teachers · p.212

v

SECTION 4.3 • Exploring Operations with Fractions 97

EXPLORATION \4~1:4.' Adding Fractions

While adding fractions is conceptually simpler than multiplying fractions, the algorithm for adding fractions is actually more challenging, and many children stumble with this algorithm.

1. For the moment, I would like you to suspend your knowledge of needing a common denom- inator to add fractions. Imagine that you don't have any algorithms and thus ~ave to find the answers using the manipulatives and your knowledge of what fraction means. Using your manipulatives, add the following fractions and briefly explain your reasoning.

1 1 a. 2 + 3

2 3 d. 13 + 4

2. On the basis of your work above, explain why we have to find a common denominator in order to add fractions. Some students find this alternative version of the question preferable: Why can't we just add the top numbers and add the bottom numbers? For example, why isn't t + t = rn?

98 CHAPTER 4 Extending the Number System

EXPLORATION :4i1-5·:' Making Sense of Wholes and Units

As you have discovered in previous explorations, it is easy to get lost in fractions in tenns of dis- tinguishing between wholes and units. For example, when fmding the sum oft and t using a diagram, children will often give an answer. oft. They have confused the whole and the unit. The following problems are designed to help deepen your understanding.

PART 1 : Using area models

1. a. If the rectangle has a value oft, show I.

b~ If the rectangle has a value oft, show I.

c. If the rectangle has a value of lt, show I.

2. The following questions are related to Pattern Blocks. These can be found on the National Library of Virtual Manipulatives website, there are also printable templates available online, or you or your instructor may have the physical blocks.

a. If the hexagon has a value oft, show I.

b. If the trapezoid has a value oft, show 1.

c. If two hexagons have a value of It, show I.

d. If the trapezoid has a value oft, what is the value of the rhombus?

PART 2: Using discrete models

1. Draw a diagram that has the specified value. Justify your solutions.

a. If the given diagram has a value oft, show I.

b. If the given diagram has a value of 1, show t.

c. If the given diagram has a value of2t, show 1.

d. If the given diagram has a value oft, showt.

2. What fraction of the circles in the diagram below are black? Your answer needs to have a denominator other than 12. Justify your answer.

D D D

• •• ••• •••• ·:·

• ••• •••• • ••• •••• ••

• • •••• eeoo eeoo

100 CHAPTER 4 Extending the Number System

Mathematics far Elementary Schaal Teachers p.213

a EXPLORATION 4.16 Multiplying Fractions

The goal of this exploration is to understand why the procedures for multiplying fractions work. Thus, as we have done in the past, we w ill ask that you assume you do not know the algorithm so you can immerse yourself in the why; you already know the how!

1. Below is a representation of t X 21-. We can see this as the area of a rectangle w ith these dimensions. We can also see it as t 21- times.

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a. In order to transfom1 the area into a number, we need to partition the shaded area into equal-sized pieces.

First, state how many equal-sized pieces we have. This is the numerator of our answer.

b. Then, using your understanding of what the denominator means, determine the denomi- nator.

c. Di scuss your solution and justification with your partners and then the w hole class.

d. What did you learn from this exploration?

2. Let us now look at the problem 31- X 2t . a. Draw a diagram to represent this problem.

b. D etermine the answer from your diagram.

c. Di scuss your solution and justification with your partners and then the whole class.

d. Draw the same diagram to represent the problem once again.

e. Th is time partition the diagram into equa l-s ized pieces if your orig ina l solution did not involve partitioning the diagram into equal-sized pieces. Determine the number of pieces and determine the answer as an improper fraction.

f. Now solve the problem using the algorithm.

g. Compare the d iagram in part (e) to the a lgorithm. Describe the connections you see between the diagram and the algorithm.

Only do 1a-b and 2a-b