Cooperative Learning
mrsishman
HW2D.MirroredTilesPlanWSampAnswers.pdf
1
Maria Lorelei Fernandez, Ph.D.
Florida International University
Adapted from M. L. Fernández & C. Anhalt, “Transition Toward Algebra,”
Mathematics Teaching in the Middle School, vol. 7 no. 4.
Mirrored Tiles Lesson Plan
Grade(s): 6-8 Algebra
Overarching (Long-term) Affective and Practice/Process Goal(s):
• Students will construct viable arguments using inductive and/or deductive reasoning to
discover and defend algebraic relationships. (CCSS.Math.Practice.MP3)
• Students will create and use representations to organize, record, and communicate
mathematical ideas.
• Students will apply and adapt a variety of appropriate strategies to make sense of and solve
a mathematical problem.
Florida (or other State) Standards for Mathematics and Lesson Objectives:
•MAFS.6.EE.3.9 Use variables to represent two quantities in a real-world problem that
change in relationship to one another; write an equation to express one quantity, thought of as
the dependent variable, in terms of the other quantity, thought of as the independent variable.
Analyze the relationship between the dependent and independent variables using graphs and
tables, and relate these to the equation.
or
•MAFS.7.EE.2.4 Use variables to represent quantities in a real-world or mathematical
problem, and construct simple equations and inequalities to solve problems by reasoning
about the quantities.
or
• MAFS.8.F.1.2 Compare properties of two functions each represented in a different way
(algebraically, graphically, numerically in tables, or by verbal descriptions).
Approx. Time: One block period (1hr 20 mins)
Materials
Colored Tiles, Mirrored Tiles Worksheet, graph paper, tools to randomly select group
members during whole class discussion (numbered tiles, dice, etc.)
Procedures
Motivation/Launch: Introduce mirrored tiles problem and ask participants to predict which
type of tile (2 beveled edges, 1 beveled edge, 0 beveled edges) M&M Designs should make
the most of. Record guesses on board.
1. Have participants get into groups of 3. Inform them that they are to work together and one
member will be selected at random to present group findings during the whole class
discussion. Assign group numbers and numbers for individuals in each group (i.e., 1, 2, or 3).
2. Distribute the Mirrored Tiles Worksheet. Role die to select one group member or ask
groups to send a member to get the materials for the problem solving. Ask groups to discuss
strategies for solving the problem for 5 minutes, helping each group member become
prepared to share ideas with class.
3. Randomly select participants, one at a time, to share one strategy for solving the problem.
Open up to class for additional strategies.
2
Maria Lorelei Fernandez, Ph.D.
Florida International University
Adapted from M. L. Fernández & C. Anhalt, “Transition Toward Algebra,”
Mathematics Teaching in the Middle School, vol. 7 no. 4.
4. Students will be asked to work in groups creating different sized mirrors, gathering data
about the numbers of different types of tiles in each mirror, developing tables for the data,
and looking for patterns in the data.
Teacher will walk around observing group interactions, responding to questions asked, and
asking questions or providing hints for groups having difficulty with gathering data.
Possible questions/hints: What is your group doing to solve the problem? How is it helping
you solve the problem? How is your group using the color tiles? How does making a table of
values help?
Possible student responses: We are creating the mirrors from the tiles, starting with a 3x3.
We are using different colored tiles for the inside the edges and the corners. We are making a
table for the numbers of different types of mirrored tiles.
5. Students working in groups will look for patterns in the data and/or in the mirrors created
with the tiles to construct algebraic formulas for the relationships representing number of tiles
with 2 beveled edges, 1 beveled edge, and 0 beveled edges (see possible answers attached).
Students will use inductive reasoning to construct relationships from the data and deductive
reasoning to explain and defend the algebraic formulas for the relationships.
Teacher continues walking around observing and interacting with groups as deemed needed.
Possible questions/hint: What relationships did your group construct? How did your group
use the table of values? How did your group use the tiled mirrors to create or defend the
relationships? How many sides on a square? How does number of sides relate to the algebraic
equation for tiles with 1 beveled edge? Etc.
Possible student Responses: The number of 2 beveled edge tiles for any mirror is always four
tiles, the corner tiles. The number of one beveled edge tiles is growing by 4 tiles as the size
of the square mirror grows by one each way. If we look at the drawing of the squares, for the
one beveled edge tiles, for any side length, subtract 2 from the side length and multiply by
four since a square has 4 sides.
6. Bring class together for whole class discussion. Use students’ suggestions to set-up a table
on the board (or overhead) to be filled as a whole class to determine the general formula
(table column headers may include: side length (dimension of square), 2 beveled edges, 1
beveled edge, 0 beveled edges).
7. Randomly select participants, one at a time, to come to board (or overhead) and fill one
table column (2 beveled edges, 1 beveled edge or 0 beveled edges) and explain patterns
observed and relationships/formulas for their column. Select one individual to fill 2 beveled
edges, another from a different group to fill 1 beveled edge, and another for 0 beveled edges.
(see possible relationships/formulas attached). After each column is filled, open discussion
for other groups to add their algebraic formulas for the relationships or other observations.
Teacher orchestrates discussion of data gathered and algebraic formulas discovered.
8. Randomly select students, one at a time, to verify the algebraic formulas for the
relationships deductively through the use of the tiles for a partly visual argument. (See the
handout attached for possible explanations.) Open discussion for other groups to discuss how
they verified their algebraic formulas.
3
Maria Lorelei Fernandez, Ph.D.
Florida International University
Adapted from M. L. Fernández & C. Anhalt, “Transition Toward Algebra,”
Mathematics Teaching in the Middle School, vol. 7 no. 4.
Teacher orchestrates discussion to verify algebraic formulas through deductive reasoning
using the tiles for a partly visual argument. Possible questions: How many 2 beveled edge
tiles are needed for each square mirror? How did you use the concrete tiles to defend the
relationships? How many sides on a square? How does the number of sides relate to the
algebraic equation for tiles with 1 beveled edge or 0 beveled edges? Etc.
9. Ask the students to explain which amount of each tile is growing the fastest as the size of
the square mirror is increasing. How do they know?
Possible student response: The no beveled edge tile is growing the fastest because they are in
the middle of the mirror so the bigger the mirror, the more space in the middle.
10. Engage the students in graphing the relationships on graph paper and/or graphing
calculators. Discuss observations about the graphs of the different types of tiles. How are the
graphs related to the tables of values? Discuss the change in values for the constant (2
beveled edges), linear (1 beveled edge), and quadratic (0 beveled edges) relationships. What
do the points of intersection represent?
11. Bring class together for whole class discussion. Randomly select students to explain their
graphs and observations about the graphs. Ask students to use the graphs to discuss which
amount of tile (2, 1, or 0 beveled edges) is growing the fastest.
Teacher orchestrates discussion of graphs. Possible questions: What observations can be
made about all of the graphs and the pattern of growth for the amounts of each type of tile?
For which type of tile is the amount growing fastest as square mirrors increase in size? When
do the graphs of the relationships intersect? What do the intersections represent? What
observations can be made about those intersections?
Possible student observations: The number of 2-beveled edge tiles is constant, always 4
(implies linear relationship with slope of 0); the number of 1-beveled edge tiles is growing at
a constant rate (implies linear relationship with positive slope); and the number of 0-beveled
edge tiles is growing at a rate that is increasing by a constant amount (implies quadratic
relationship). Students can observe and discuss the graphical shapes of these different
relationships. Students can discuss the intersection of the number of 1-beveled edge tiles and
the number of 0-beveled edge tiles occurring when the mirror is 6x6. Students can discuss
which tiles M&M Designs should make the most of. Have students defend their positions
using the mathematics they have explored. Their arguments might include issues of what
size mirrors customers may tend to make most often.
Extensions
Ask participants to create other patterns that can be explored with the colored tiles and
develop relationships for these patterns.
Assessment
Informal observations of participants working in groups and observations of responses made
by random group members. Collect written work (e.g., graphs of relationships with
interpretations about the graphs and explanation of which types of tiles the M&M Design
company should make most of).
Accommodations
4
Maria Lorelei Fernandez, Ph.D.
Florida International University
Adapted from M. L. Fernández & C. Anhalt, “Transition Toward Algebra,”
Mathematics Teaching in the Middle School, vol. 7 no. 4.
The use of small groups will provide support for students with special needs such as second
language learners and special education students. Place students in groups as needed to
benefit the learning of all students. The use of multiple representations also provides support
for learners with different needs and approaches to developing their understanding of the
mathematics.
Handout
Mirror Tiles Problem handout (attached)
Bibliography
Created by Maria Lorelei Fernandez and presented in M. L. Fernández & C. Anhalt,
“Transition Toward Algebra,” Mathematics Teaching in the Middle School, vol. 7 no. 4.
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Maria Lorelei Fernandez, Ph.D.
Florida International University
Adapted from M. L. Fernández & C. Anhalt, “Transition Toward Algebra,”
Mathematics Teaching in the Middle School, vol. 7 no. 4.
Mirrored Tiles Problem
M&M Designs is making mirrored tiles that can be used to tile walls (like tiles on floors).
There are 3 types of square one foot tiles: plain (0 beveled edges), one beveled edge, and 2
beveled edges for corners. The outer tiles of any tiling will have beveled edges for safety.
Sizes for square mirrors include 2’x2’, 3’x3’, 4’x4’, 5’x5’ and so on. Your task is to
determine the number of mirrors for each of the 3 types needed to make any square mirror
out of the mirrored tiles.
A. How many tiles of each type do you need for a 10’x10‘ square mirror (side length 10
feet)? Explain.
B. Develop a general formula for the number of each type of tiles needed to make any square
mirror of whole length sides if you know the length of a side of the mirror. Defend your
formulas.
C. Which type of tile should M&M Designs make most of? Explain your reasoning.
D. Graph all of your information on the same coordinate axes. What observations can you
make about the relationships? Use this information to discuss part C.
2 Beveled Edges
Corner Tile
1 Beveled Edge
Outer Tile
0 Beveled Edges
Inner Tile
\
6
Maria Lorelei Fernandez, Ph.D.
Florida International University
Adapted from M. L. Fernández & C. Anhalt, “Transition Toward Algebra,”
Mathematics Teaching in the Middle School, vol. 7 no. 4.
Mirrored Tiles Problem Possible Answers
The Many Mirrors Design Company is making mirrored tiles that can be used to tile walls
(like tiles on floors). There are 3 types of square one foot tiles: plain (0 beveled edges), one
beveled edge, and 2 beveled edges for corners. The outer edge of any square tiling will have
beveled edges for safety. Sizes for square mirrors include 2’x2’, 3’x3’, 4’x4’, 5’x5’ and so
on. Your task is to determine the number of mirrors for each of the 3 types needed to make a
square mirror of any size out of the mirrored tiles.
2 Beveled Edges
Corner Tile
1 Beveled Edge
Outer Tile
0 Beveled Edges
Inner Tile
\
Examples:
2’ x 2’ 3’ x 3’
Randomly select participants (e.g., draw numbers from cup or roll dice), one at a time, to
come to board (or overhead) and fill one table column (2 beveled edges, 1 beveled edge or 0
beveled edges) and explain any observed patterns, relationships and formulas for their
column. Select one individual to fill 2 beveled edges, another from a different group to fill 1
Beveled edge, and another for 0 Beveled edges. After each column is filled, open discussion
for other groups to add their observations and formulas for the relationships.
Possible observations and inductive reasoning to generate formulas: For 2 beveled edges, always 4.
For 1 beveled edge, growing by 4 tiles each time (recursive formula); for any side length,
subtract 2 from the side length and multiply by four (explicit formula); explicit formulas in
algebraic symbols, #Tiles = 4(x-2) or #Tiles = 4x-8.
For 3 beveled edges, growing by ascending odd numbers (1, 3, 5, 7, etc.) (recursive
formula); for any side length, subtract 2 and square the value (e.g., for side length of 5, (5 –
2) squared) (explicit formula).
Call on participants to verify their formulas or algebraic relationships deductively through
the use of the tiles for a partly visual argument. Select one individual to discuss their
justifications for 2 beveled edges, another from a different group to fill 1 beveled edge, and
another for 0 beveled edges. Open discussion for other groups to discuss how they verified
their algebraic relationships.
7
Maria Lorelei Fernandez, Ph.D.
Florida International University
Adapted from M. L. Fernández & C. Anhalt, “Transition Toward Algebra,”
Mathematics Teaching in the Middle School, vol. 7 no. 4.
Possible deductive explanations (partly visual argument):
2 Beveled Edges 4: One on each corner of any square mirror, so always has 4 corners.
1 Beveled Edge 4(n – 2): n – 2 on each edge of the square, which has 4 edges; n – 2 on each edge because we
need to subtract the two corner tiles along each edge. In the 3 × 3 example below, the number
of one-beveled-edge tiles is 3 – 2 corner tiles along each edge, so the number of one-beveled-
edge tiles is 4(1) = 4.
0 Beveled Edges (n – 2)
2 : (n – 2)
2 because we need to subtract two rows of tiles along opposing edges to
determine the length of each dimension of the square with zero beveled edges found in the
center of each mirror. In the 4 × 4 example below, the dimension of the inner square mirror is
(4 – 2) along each dimension of the square; thus, the number of tiles with zero beveled edges
is 2 × 2.
