Download Shift from “Coverage” to “Understanding” with Algebra Tiles

Shift from
“Coverage” to
“Understanding”
with Algebra Tiles
Mícheál Marsh & Vivian Lezak
Culver City HS
Loyola Marymount University
Session #671
For more information about the CPM materials you find in this packet, contact:
Chris Mikles
(888) 808-4276
[email protected]
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CC1-6.2.2-Students will use variables to represent unknown lengths. They will find the area of algebra tiles using
variables and constants and will practice combining like terms in the context of finding the areas of
collections of algebra tiles.
CC1-6.2.3-Students will understand that combining like terms is a form of sorting. They will find the lengths of
the sides of algebra tiles and combine like terms as they find perimeters.
CC1-6.2.4-Students will continue combining like terms to generate equivalent expressions by finding the
perimeter of complex figures composed of algebra tiles.
CC1-6.2.5-Students will visually demonstrate that x can represent any number. They will continue to apply their
knowledge about creating equivalent expressions by combining like terms, and evaluating
expressions including expressions with exponents.
CC1-7.3.1-Students apply inverse operations to numbers and algebraic expressions in order to represent the steps
in a math “magic trick.”
CC1-7.3.2-Students will translate math steps into algebraic expressions and begin to explore the Distributive
Property to create equivalent pictures and expressions. They will write expressions that record
operations with numbers and letters.
CC1-7.3.3-Students will continue to translate math steps into the Distributive Property. They will use the
Distributive Property “in reverse.” Students use mathematics terminology from previous lessons to
identify parts of expressions.
CC2-4.3.3-The students will simplify algebraic expressions by combining like terms and using the Distributive
Property.
CC2-6.1.1-Students will build and simplify expressions on an expression mat to determine which of them is
greater. Students will also build an understanding of “legal moves” with the algebra tiles on the
Expression Comparison Mats.
CC2-6.1.2-Students will learn that sometimes it is not possible to determine whether one expression is greater
than another (or if they are equal). They will also learn two additional legal moves for simplification
of expressions.
CC2-6.2.1-Students will apply strategies for simplifying expressions to determine if one expression is greater and
will begin finding solutions when expressions are equal.
CC2-6.2.2-Students will develop an understanding of checking solutions while continuing to develop their
understanding of solving equations and the Distributive Property.
CC2-6.2.3-Students will use formal notation while simplifying expressions and solving equations. They will
compare arithmetic and algebraic methods for solving problems.
CC2-6.2.6-Students will solve equations that have infinite solutions and those with no solutions.
CC3-1.1.1-In the review of previous courses.
CC3-2.1.1-Students will be introduced to algebra tiles, which will lay the foundation for later work with
manipulating algebraic expressions and solving equations. Students will name each tile by its area
and will learn how to simplify an expression by combining like terms.
CCA-3.2.4-Students will continue to practice multiplying expressions and will begin to use generic rectangles to
simplify the process. Students will find missing dimensions of generic rectangles given pieces of area
and will find missing pieces of area given dimensions.
CCA-8.1.1-Students will review how to build rectangles with tiles and learn shortcuts for finding the dimensions
of a completed generic rectangle. Students will discover that the products of the terms in each
diagonal of a generic rectangle are equal.
CCA-8.1.2-Students will develop an algorithm to factor quadratic expressions without algebra tiles.
CCA-8.1.3-Students will continue to practice their factoring skills while learning about special cases: quadratic
expressions with missing terms, quadratics that are not in standard form, and quadratics with more
than one possible factored form.
CCA-8.1.4-Students will complete their focus on factoring by considering expressions that can be factored first
with a common factor and then again using the quadratic factoring method.
CCA-8.1.5-Students will learn a quick way to factor perfect square trinomials and quadratics that are a difference
of squares.
6.2.2
What is the area?
x2
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x
Area of a Rectangular Shape
In this lesson, you will continue to study variables in more depth by using them to
describe the dimensions and areas of different shapes. Then you will organize those
descriptions into algebraic expressions.
6-79.
Find the area of each rectangle below. Show your work. In part (b), each
square in the interior of the rectangle represents one square unit.
a.
b.
c.
0.75 m
15 cm
1.8 m
25 cm
d. Explain your method for finding the area of a rectangle.
6-80.
AREAS OF ALGEBRA TILES
Your teacher will provide your team with a set
of algebra tiles. Remove one of each shape
from the bag and put it on your desk. Trace
around each shape on your paper. Look at the
different sides of the shapes.
a.
With your team, discuss which shapes have
the same side lengths and which ones have
different side lengths. Be prepared to share
your ideas with the class. On your traced
drawings, color-code lengths that are the
same.
b. Each type of tile is named for its area. In this course, the smallest
square will have a side length of 1 unit, so its area is 1 square unit.
Thus, this tile will be called “one” or the “unit tile.” Can you use the
unit tile to find the side lengths of the other rectangles? Why or why
not?
c.
If the side lengths of a tile can be measured
exactly, then the area of the tile can be
calculated by multiplying these two lengths
together. The area is measured in square
units. For example, the tile at right
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2
measures 1 unit by 5 units, so it has an area
of 5 square units.
The next tile at right has one side length
that is exactly one unit long. If you cannot
give a numerical value to the other side
length, what can you call it?
1
?
d. If the unknown length is called “x,” label the side lengths of each of the
four algebra tiles you traced. Find each area and use it to name each
tile. Be sure to include the name of the type of units it represents.
6-81.
Jeremy and Josue each sketched three
x-tiles on their papers.
Jeremy labeled each tile with an x. “There
are three x-tiles, each with dimensions 1 by
x, so the total area is 3x square units,” he
said.
Josue labeled the dimensions (length and
width) of each tile. His sketch shows six
x-lengths.
Jeremy’s sketch
Josue’s sketch
1
1
x
x
x
x
x
1
a.
x
x
1
1
x
x
1
Why do the two sketches each show a different number of x labels on
the shapes?
b. When tiles are named by their areas, they are named in square units.
For example, the x-by-x tile (large square) is a shape that measures x 2
square units of area, so it is called an x 2 -tile.
What do the six x’s on Josue’s sketch measure? Are they measures of
square units?
6-82.
When a collection of algebra tiles is described with mathematical symbols,
it is called an algebraic expression. Take out the tiles shown in the picture
below and put them on your table.
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•
•
Use mathematical symbols (numbers, variables, and operations) to
record the area of this collection of tiles.
Write at least three different algebraic expressions that represent the
area of this tile collection.
x
x
x
x2
x2
x
6-83.
Take out the tiles pictured in each collection below, put them on your table,
and work with your team to find the area as you did in problem 6-82.
a.
x2
x
x2
x
x
x
b.
c.
x2
x
x
x
x
d. Is the area you found in part (a) the same or different from the area of
the collection in problem 6-82? Justify your answer using words,
pictures, or numbers.
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6.2.3
What is a variable?
x2
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x
Naming Perimeters of Algebra Tiles
How much homework do you have each night? Some nights you may have a lot, but
other nights you may have no homework at all. The amount of homework you have
varies from day to day.
In Lesson 6.2.2, you used variables to name lengths that could not be precisely
measured. Using variables allows you to work with lengths that you do not know
exactly. Today you will work with your team to write expressions to represent the
perimeters of different shapes using variables.
6-91.
TOOTHPICKS AND ALGEBRA TILES
In Chapter 1, you played the game “Toothpicks and Tiles.” Now you will
play it using algebra tiles!
Work with your team to find the area (“tiles”) and the perimeter
(“toothpicks”) for the following figures.
a.
b.
c.
x
x
x
d. What is different about the shape in part (c)?
e.
6-92.
Is the perimeter of the shape in part (c) greater or less than the
perimeter of the shape in part (a)? Explain your thinking.
The perimeter of each algebra tile can be written as an
expression using variables and numbers.
a.
Write at least two different expressions for the
perimeter of each tile shown at right.
b. Which way of writing the perimeter seems
clearest to you? What information can you get
from each expression?
c.
x
Lisa wrote the perimeter of the collection of
tiles at right as 2x +1 + 2x +1 units, but her
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x2
x
x
5
teammate Jody wrote it as 4x + 2 . How are
their expressions different?
d. Which expression represents the perimeter?
6-93.
For the shape at right, one way to write
the perimeter would be to include each
side length in the sum:
x + x + x + 1+ x + x + 1+ x .
a.
x2
How many x lengths are represented
in this expression? How many unit
lengths?
x
x
b. The expression above can be rearranged to x + x + x + x + x + x + 1+ 1 and
then be written as x(1+ 1+ 1+ 1+ 1+ 1) + 2 , which then equals 6x + 2 .
Identify which property is used to rewrite the expression with
parentheses.
c.
Like terms are terms that contain the same variable (as long as the
variable(s) are raised to the same power). Combining like terms is a
way of simplifying an expression. Rewriting the perimeter of the shape
above as 6x + 2 combines the separate x-terms to get 6x and combines
the units in the term to get 2.
If you have not already done so, combine like terms for the perimeter
of each of the different algebra tiles in problem 6-92.
6-94.
On your desk, use algebra tiles to make the shapes shown below. Trace
each shape and label the length of each side on your drawing. With your
team, find and record the total perimeter and area of each shape. If
possible, write the perimeter in more than one way.
a.
b.
x
x
x
x
6-95.
In problem 6-94, x is a variable that represents a number of units of length.
The value of x determines the size of the perimeter and area of the shape.
Using the shapes from parts (b) and (c) in problem 6-94, sketch and label
each shape with the new lengths given below. Rewrite the expressions with
numbers and simplify them to determine the perimeter and area of each
shape.
a.
x=6
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b.
x=2
c.
Compare your method for finding perimeter and area with the method
your teammates used. Is your method the same as your teammates’
methods? If so, is there a different way to find the perimeter and area?
Explain the different methods.
6.2.4
How can I rewrite it?
x2
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x
Combining Like Terms
In Lesson 6.2.3, you looked at different ways the perimeter of algebra tiles can be
written. You also created different expressions to describe the same perimeter.
Expressions that represent the same perimeter in different ways are called equivalent
expressions. Today, you will extend your work with writing and rewriting perimeters
to more complex shapes. You will rewrite expressions to determine whether two
perimeter expressions are equivalent or different.
6-101.
Build each of these shapes using algebra tiles and look
carefully at the lengths of the sides:
i.
ii.
x2
x
iii.
x2
x
x
x
a.
Sketch each figure on your paper and colorcode the lengths that are the same. Which
figures have side lengths that are different than
those you have measured before? How are they
different?
b. Label each length on your paper. Discuss with your team how to label
the lengths that are different than those you have measured before.
Explain your reasoning.
c . Find the perimeter of each figure. Write the perimeter in simplest form
by combining the like terms.
x
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x2
7
6-102.
In any expression, the number that tells you how
many of each variable or quantity you have is called a
coefficient.
For example, for the expression that
describes the collection at right, the
coefficient 3 shows that there are three
x 2 -tiles, and the coefficient 4 shows that
there are four x-tiles. The 6 is called the
constant term because it is a term that
does not contain variables and does not
change no matter what the value of x is.
Answer each question below for each of the
perimeters you found in problem 6-101.
6-103.
coefficients
constant
x
x2
x
x2
2
x
x
x
•
What is the coefficient in the expression for the perimeter?
•
How do you see the coefficient of x in the shape?
•
What is the constant term in the expression?
•
How do you see the constant term in the shape?
HOW MANY PERIMETERS?
Erik cannot keep his hands off the algebra tiles! He has made several
different shapes, each one using the same tiles. “Will every shape I create
with these tiles have the same perimeter?” he wonders.
Shares a
complete side
Help Erik investigate the question by making
different shapes with your team. Your shapes
must follow these rules:
x
• Shapes must use exactly three tiles: a
unit tile, an x-tile, and an x 2 -tile.
• Tiles must share a complete side.
Examples of tiles that do and do not
share complete sides are shown at
right.
a.
x
Does not share
a complete side
Rearrange the tiles until each teammate has a shape that follows the
rules and has a different perimeter. Discuss why the perimeters are
different. Trace each shape, color-code the sides, and label their
lengths. Write an expression for the perimeter of each shape and
simplify it by combining like terms.
b. Are other perimeters possible with the same pieces? As you find
others:
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6-102.
In any expression, the number that tells you how
many of each variable or quantity you have is called a
coefficient.
For example, for the expression that
describes the collection at right, the
coefficient 3 shows that there are
three x 2 -tiles, and the coefficient 4
shows that there are four x-tiles.
The 6 is called the constant term
because it is a term that does not
contain variables and does not
change no matter what the value of x
is.
coefficients
constant
x2
x
x
x2
2
x
x
x
Answer each question below for each of the
perimeters you found in problem 6-101.
•
What is the coefficient in the expression for the perimeter?
•
How do you see the coefficient of x in the shape?
•
What is the constant term in the expression?
•
How do you see the constant term in the shape?
6.2.5
What can a variable represent?
x2
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x
Evaluating Algebraic Expressions
In the previous lessons, you have learned how to find the perimeter and area of a
shape using algebra tiles. Today, you will challenge the class to find the perimeters
and areas of shapes that you create.
6-111.
Use the perimeter and area expressions you found in problem 6-106 to
answer the questions below.
x
x
x
x2
x2
x
a.
Determine each perimeter and area if x = 5 units.
b. Determine each perimeter and area if x = 2 12 units.
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c.
6-112.
Using a technology tool or graph paper as directed by your teacher,
carefully draw each shape with the specified length of x units.
SHAPE CHALLENGE
You and your team will choose four algebra tiles. Then
you will use them to build a shape to challenge your
classmates. You may choose whatever tiles you would
like to use as long as you use exactly four tiles.
As a team, decide on the shape you want to make.
Experiment with different shapes until you find one you
think will have a challenging perimeter and area for your
classmates to find. Then, to share your challenge with the
class:
• Build the shape with algebra tiles in the middle of your team so
everyone in your team can see it.
• Get an index card from your teacher. On one side, neatly draw the
shape and label each side.
• Write simplified expressions for the perimeter and the area on the
same side of the card. This will be the answer key. Show all of your
steps clearly.
• Turn the card face down so the answer is hidden. Then put the
names of your team members on the top of the card. Place the card
beside the shape you built with your algebra tiles.
Remember that your work needs to be clear enough for your classmates to
understand.
Follow your teacher’s directions to complete challenges created by other
teams. As you look at their shapes, sketch them on your paper. Work with
your team to label the sides and find the perimeter and area of each shape.
Be sure to combine like terms to make the expressions as simple as
possible.
6-113.
Choose two of the shapes from problem 6-112. Sketch each shape and
label it with its perimeter and area. Do not forget the correct units. It is not
necessary to draw the figures to scale. Rewrite each expression with the
values given below and then evaluate it.
a.
x = 1.5 units
b.
x = 3 43 units
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• Trace the shapes.
• Color-code and label the sides.
• Write the perimeter in simplest form.
Be prepared to share your list of perimeters with the class.
c.
6-104.
Are there different shapes that have the same perimeter? Why or why
not?
Additional Challenge: Build the shape at
right out of algebra tiles. Then, on graph
paper, draw the shape when x is equal to
each of the lengths below.
a.
7.3.1
x = 5 units
b.
x
x2
x = 3 units
Why does it work?
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Inverse Operations
Variables are useful tools for representing unknown numbers. In some situations, a
variable represents a specific number, such as the hop of a frog. In other situations, a
variable represents a collection of possible values, like the side lengths of Bonnie’s
picture frames. In previous chapters, you have also used variables to describe patterns
in scientific rules and to write lengths in perimeter expressions. In this section, you
will continue your work with variables and explore new ways to use them to represent
unknown quantities in word problems.
7-79.
THE MATHEMATICAL MAGIC TRICK
Have you ever seen a magician perform a seemingly impossible feat and
wondered how the trick works? Follow the steps below to participate in a
math magic trick.
Pick a number and write it down.
Add five to it.
Double the result.
Subtract four.
Divide by two.
Subtract your original number.
What did you get?
a.
Check with others in your study team and compare answers.
What was the result?
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11
b. Does this trick seem to work no matter what number you pick? Have
each member of your team test it with a different number. Consider
numbers that you think might lead to different answers, including zero,
fractions, and decimals. Keep track in the table below. For your
convenience, a copy of this table is on the Lesson 7.3.1 Resource Page.
Steps
1. Pick a number.
2. Add 5.
Trial 1
Trial 2
Trial 3
3. Double it.
4. Subtract 4.
5. Divide by 2.
6. Subtract the original
number.
c.
Which steps made the number you chose increase? When did the
number decrease? What connections do you see between the steps in
which the number increased and the steps in which the number
decreased?
d. Consider how this trick could be represented with math symbols. To
get started, think about different ways to represent just the first step,
“Pick a number.”
7-80.
Now you get to explore why the magic trick from problem 7-79 works.
Shakar decided to represent the steps with algebra tiles. Since he could
start the trick with any number, he let an x-tile represent the “Pick a
number” step. With your team, analyze his work with the tiles. Then
answer the questions below.
Steps
Trial 1
Trial 2
Trial 3
Algebra Tile Picture
1. Pick a number.
x
2. Add 5.
x
3. Double it.
x
x
4. Subtract 4.
x
x
5. Divide by 2.
x
6. Subtract the
original number.
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a.
For the second step, “Add 5,” what did Shakar do with the tiles?
b. What did Shakar do with his tiles to
“Double it”? Explain why that works.
c.
7-81.
How can you tell from Shakar’s table
that this trick will always end with 3?
Explain why the original number does
not matter.
The table below has the steps for a new “magic trick.” Use the Lesson
7.3.1 Resource Page to complete parts (a) through (d) that follow.
Steps
Trial 1
Trial 2
Trial 3
Algebra Tile Picture
1. Pick a number.
2. Add 2.
3. Multiply by 3.
4. Subtract 3.
5. Divide by 3.
6. Subtract the
original number.
a.
Pick a number and place it in the top row of the “Trial 1” column.
Then follow each of the steps for that number. What was the end
result?
b. Now repeat this process for two new numbers in the “Trial 2” and
“Trial 3” columns. Remember to consider trying fractions, decimals,
and zero. What do you notice about the end result?
c.
Use algebra tiles to see why your observation from part (b) works. Let
an x-tile represent the number chosen in Step 1 (just as Shakar did in
problem 7-80). Then follow the instructions with the tiles. Be sure to
draw diagrams on your resource page to show how you built each step.
d. Explain how the algebra tiles help show that your conclusion in part (b)
will always be true no matter what number you originally select.
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7-82.
Now reverse your thinking to figure out a new “magic trick.” Locate the
table below on the Lesson 7.3.1 Resource Page and complete parts (a)
through (c) that follow.
Steps
Trial 1
Trial 2
Trial 3
1. Pick a number.
x
2.
x
4.
x
x
x
x
5.
x
6.
x
3.
a.
Algebra Tile Picture
Use words to fill in the steps of the trick like those in the previous
tables.
b. Use your own numbers in the trials, again considering fractions,
decimals, and zero. What do you notice about the result?
c.
7-83.
Why does this result occur? Use the algebra tiles to help explain this
result.
In the previous math “magic tricks,” did you notice how multiplication by a
number was later followed by division by the same number? These are
known as inverse operations (operations that “undo” each other).
a.
What is the inverse operation for addition?
b. What is the inverse operation for multiplication?
c.
What is the inverse operation for “Divide by 2”?
d. What is the inverse operation for “Subtract 9”?
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7-84.
Now you get to explore one more magic trick. Locate the table below on
the Lesson 7.3.1 Resource Page. For this trick:
• Complete three trials using different numbers. Use at least one
fraction or decimal.
• Use algebra tiles to help you analyze the trick, as you did in problem
7-81. Draw the tiles in the table on the resource page.
• Find at least two pairs of inverse operations in the process that are
“undoing” each other.
Steps
Trial 1
Trial 2
Trial 3
Algebra Tile Picture
1. Pick a number.
2. Double it.
3. Add 4.
4. Multiply by 2.
5. Divide by 4.
6. Subtract the
original number.
7.3.2
How can I write it?
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Distributive Property
In Lesson 7.3.1, you looked at how mathematical “magic tricks” work by using
inverse operations. In this lesson, you will connect algebra tile pictures to algebraic
expressions. An algebraic expression is another way to represent a mathematical
situation.
7-91.
Today you will consider a more complex math magic trick. The table you
use to record your steps will have only two trials, but it will add a new
column to represent the algebra tiles with an algebraic expression. To
begin this activity, get a Lesson 7.3.2 Resource Page from your teacher.
Then:
• Work with your team to choose different numbers for the trials.
• Decide how to write algebraic expressions that represent what is
happening in each step.
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Steps
Trial 1 Trial 2
Algebraic
Expression
Algebra Tile Picture
1. Pick a number.
x
2. Add 7.
x
3. Triple the result.
x
x
x
4. Add 9.
x
x
x
5. Divide by 3.
x
6. Subtract the
original number.
7-92.
For this number trick, the steps and trials are left for you to complete by
using the algebraic expressions. To start, copy the table below on your
paper and build each step with algebra tiles.
Steps
a.
Trial 1
Trial 2
Algebraic Expression
1.
x
2.
x+4
3.
2(x + 4)
4.
2x + 20
5.
x +10
6.
10
Describe Steps 1, 2, and 3 in words.
b. Look at the algebra tiles you used to build Step 3. Write a different
expression to represent those tiles.
c.
What tiles do you have to add to build Step 4? Complete Steps 4, 5,
and 6 in the chart.
d. Complete two trials and record them in the chart.
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16
7-93.
In Step 3 of the last magic trick (problem 7-92), you rewrote the expression
2(x + 4) as 2x + 8 . Can all expressions like 2(x + 4) be rewritten without
parentheses? For example, can 3(x + 5) be rewritten without parentheses?
Build 3(x + 5) with tiles and write another expression to represent it. Does
this seem to work for all expressions?
7-94.
Diana, Sam, and Elliot were working on two different mathematical magic
tricks shown below. Compare the steps in their magic tricks. You may
want to build the steps with algebra tiles.
Magic Trick A
Magic Trick B
1. Pick a number.
1. Pick a number.
2. Add 3.
2. Multiply by 2.
3. Multiply by 2.
7-95.
3. Add 3.
a.
Each student had completed one of
the tricks. After the third step,
Diana had written 2x + 6 , Sam had
written 2(x + 3) , and Elliot had
written 2x + 3 . Which
expression(s) are valid for Magic
Trick A? Which one(s) are valid
for Magic Trick B? How do you
know? Use tiles, sketches,
numbers, and reasons to explain
your thinking.
b.
How are the steps and results of the two magic tricks different? How
can this difference be seen in the expression used to represent each
trick?
Parentheses allow us to consider the number of groups of tiles that are
present. For example, when the group of tiles x + 3 in problem 7-94 is
doubled in Magic Trick A, the result can be written 2(x + 3) . However,
sometimes it is more efficient to write the result as 2x + 6 instead of
2(x + 3) . You may remember this as an application of the Distributive
Property that you first learned about in Chapter 2, only now with variables
instead of just numbers.
a.
Show at least two ways to write the result of these steps:
1. Pick a number.
2. Add 5.
3. Multiply by 3.
© CPM Educational Program 2013, all rights reserved CC1 and CC2
17
b.
Write three steps that will result in 4(x + 2) . How can the result be
written so that there are no parentheses?
c.
Build the following steps with tiles. Write the result in two ways.
1. Pick a number.
2. Triple it.
3. Add 6.
4. Multiply by 2.
7.3.3
How can I talk about it?
•••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••
Distributive Property and Expressions Vocabulary
Today you will continue your work writing algebraic expressions using the
Distributive Property. You will label parts of an algebraic expression with their
mathematical names.
7-101.
At right is an algebra tile drawing that shows the result
of the first three steps of a number trick.
a.
What are three possible steps that led to this
drawing?
x
x
x
x
b. Use a variable to write at least two expressions that represent the tiles
in this problem. Write your expressions so that one of them contains
parentheses.
c.
7-102.
If the next step in the trick is “Divide by 2,” what should the simplified
drawing and two algebraic expressions look like?
Recall that parentheses allow us to consider the number of groups of tiles
that are present.
a.
Below are four steps of a math magic trick. Write the result of the
steps in two different ways. Build it with tiles if it helps you.
1. Pick a number.
2. Triple it.
3. Add 1.
4. Multiply by 2.
b.
Write 4(2x + 3) in another way.
© CPM Educational Program 2013, all rights reserved CC1 and CC2
18
7-103.
c.
Build 9x + 3 with tiles. How many groups can you divide the tiles
into evenly? Write the expression two ways, one with parentheses and
one without.
d.
Build 15x +10 with tiles and write the expression another way.
You have been writing expressions in different ways to mean the same
thing. The way you write an expression depends on whether you see tiles
grouped by rows (like four sets of x + 3 in problem 7-101) or whether you
see separate groups (like 4x and 12 in that problem). The Distributive
Property is the formal name for linking these two equivalent expressions.
Write each of the following descriptions in another way. For example,
4(x + 3) can also be written 4x +12 . (Hint: Divide each expression into as
many equal groups as possible.)
7-104.
a.
6(8 + x)
b.
12x + 4
c.
21x +14
d.
18 +12x
e.
Now, write the following number trick as two different expressions.
1. Pick a number.
2. Multiply by 4.
3. Add 7.
4. Multiply by 3.
MATH TALK
Read the Math Notes box at the end of this lesson as a review of the math
words you have already been exposed to in previous chapters.
a.
Compare the expression without parentheses in parts (b), (c), and (d)
of problem 7-103. Which has the largest coefficient? Which
expression has the largest constant term?
b.
What are the two factors in part (a)? What are the two factors in part
(b) when it is written with parentheses?
c.
Write an expression with a variable of y, a coefficient of 18, and a
constant term of 9. Rewrite your expression as two factors.
d.
Use the words coefficient, constant term, term, expression, variable,
and factor to describe 12x 2 +19y !1 .
e.
Use the words factor, product, quotient, and sum to describe the parts
+5.
of 3(a + b) + 12!a
b
© CPM Educational Program 2013, all rights reserved CC1 and CC2
19
7-105.
Additional Challenge: Mrs. Baker demonstrated an interesting math magic
trick to her class. She said:
“Think of a two-digit number and write it down without showing
me.
Add the ‘magic number’ of 90 to your number.
Take the digit that is now in the hundreds place, cross it
out, and add it to the ones place. Now tell me the result.”
As each student told Mrs. Baker his or her result, she quickly told each
student his or her original number. Why does this trick work? Consider
the following questions as you unravel the trick.
a.
Could you represent the original number with a variable? What is the
algebraic expression after adding 90?
b. What is the largest number the students could have now? The
smallest?
c.
When Mrs. Baker says to cross out the digit in the hundreds place, the
students cross off the 1. Was any other number possible? What have
the students subtracted from the expression? What is the new
expression?
d. When the students add the 1 to the ones place, what is the simplified
version of the expression they have created? What does Mrs. Baker
mentally add to their result to reveal their original number?
© CPM Educational Program 2013, all rights reserved CC1 and CC2
20
4.3.3
How can I simplify with negatives?
x
2
•••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••• x
Simplifying With Zero
In the previous lessons, you simplified and rewrote algebraic expressions. In this
lesson, you will continue to explore various ways to make expressions simpler by
finding parts of them that make zero.
Zero is a relative newcomer to the number system. Its first appearance
was as a placeholder around 400 B.C. in Babylon. The Ancient Greeks
philosophized about whether zero was even a number: “How can
nothing be something?” East Indian mathematicians are generally
recognized as the first people to represent the quantity zero as a
numeral and number in its own right about 600 A.D.
Zero now holds an important place in mathematics both as a numeral representing
the absence of quantity and as a placeholder. Did you know there is no year 0 in the
Gregorian calendar system (our current calendar system of 365 days in a year)?
Until the creation of zero, number systems began at one.
4-103.
CONCEPTS OF ZERO
Zero is a special and unusual number. As you read above, it has an
interesting history. What do you know about zero mathematically? The
questions below
will test your knowledge of zero.
a.
If two quantities are added and the sum is zero, what do you know
about the quantities?
b. If you add zero to a number, how does the number change?
c.
If you multiply a number by zero, what do you know about the
product?
d. What is the opposite of zero?
4-104.
e.
If three numbers have a product of zero, what do you know about at
least one of the numbers?
f.
Is zero even or odd?
When you use algebra tiles, +1 is
represented with algebra tiles as a
shaded small square and is always a
positive unit. The opposite of 1,
written –1, is an open small square and
© CPM Educational Program 2013, all rights reserved CC1 and CC2
=+1
=–1
x
1 unit
any
Can be any length
21
is always negative. Let’s explore the
variable x-tiles.
a.
The variable x-tile is shaded, but is the number
represented by a variable such as x always
positive? Why or why not?
x
b. The opposite of the variable x, written –x, looks
like it might be negative, but since the value of a
variable can be any number ( the opposite of –2 is
2 ), what can you say about the opposite of the
variable x?
c.
x
Is it possible to determine which is greater, x or –x? Explain.
d. What is true about 6 + (!6) ? What is true about x + (!x) (the sum of a
variable and its opposite)?
4-105.
Get a Lesson 4.3.3 Resource Page from your teacher,
which is called “Expression Mat.” The mat will help
you so you can tell the difference between the
expression you are working on and everything else on
your desk.
Expression Mat
From your work in problem 4-104, you can say that situations like 6 + (!6)
and x + (!x) “create zeros.” That is, when you add an equal number of tiles
and their opposites, the result is zero. The pairs of unit tiles and x-tiles
shown in that problem are examples of “zero pairs” of tiles.
Build each collection of tiles represented below on the mat. Name the
collection using a simpler algebraic expression (one that has fewer terms).
You can do this by finding and removing zero pairs and combining like
terms. Note: A zero pair is two of the same kind of tile (for example, unit
tiles), one of them positive and the other negative.
a.
2 + 2x + x + (!3) + (!3x)
b.
!2 + 2x + 1! x + (!5) + 2x
© CPM Educational Program 2013, all rights reserved CC1 and CC2
22
4-106.
An equivalent expression refers to the same amount with a different name.
Build the expression mats shown in the pictures below. Write the
expression shown on the expression mat, then write its simplified
equivalent expression by making zeros (zero pairs) and combining like
terms.
a.
4-107.
x
x
x
x
b.
x2
x2
x
x
x
x
On your Expression Mat, build what is described below. Then write two
different equivalent expressions to describe what is represented. One of the
two representations should include parentheses.
a.
The area of a rectangle with a width of 3 units and a length of x + 5 .
b. Two equal groups of 3x ! 2 .
c.
Four rows of 2x + 1 .
d. A number increased by one, then tripled.
6.1.1
Word
How do these compare?
•••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••
<
Graph
!
! Symbol
Comparing Expressions
In Chapter 4, you worked with writing and simplifying expressions. As you wrote
expressions, you learned that it was helpful to simplify them by combining like terms
and removing zeros. In this lesson, you and your teammates will use a tool for
comparing expressions. The tool will allow you to determine whether one expression
is greater than the other or if they are equivalent ways of writing the same thing (that
is, if they are equal).
Remember that to represent expressions with algebra tiles, you will need to
be very careful about how positives and negatives are distinguished. To
help you understand the diagrams in the text, the legend at right will be
placed on every page containing a mat. It shows the shading for +1 and –1.
This model also represents a zero pair.
© CPM Educational Program 2013, all rights reserved CC1 and CC2
= +1
= –1
23
6-1.
COMPARING EXPRESSIONS
Mat A
Ignacio and Oliver were playing a game.
Each of them grabbed a handful of
algebra tiles. They wanted to see whose
expression had the greater value.
Two expressions can be compared by
dividing the expression mat in half to
change it into an Expression
Comparison Mat. Then the two
expressions can be built side by side and
compared to see which one is greater.
•
Oliver put his tiles on Mat A in the
picture above and described it as
5 + (!3) .
•
Ignacio put his tiles on Mat B
and said it was (!5) + 2 .
Mat B
?
5 + (!3)
(!5) + 2
Mat A
Mat B
With your team, find two different methods
to simplify the two expressions so you can
compare them. Which side of the mat is
larger?
6-2.
6-3.
Using your Expression Comparison Mat, build
the two expressions at right. Find a way to
determine which side is greater, if possible.
Show your work by sketching it on the Lesson
6.1.1B Resource Page. Be ready to share your
conclusion and your justification.
x2
x2
?
x
x
x
x
MORE COMPARING EXPRESSIONS – Is one expression greater?
Consider how you were able to compare the expressions in the previous
problems. When is it possible to remove tiles to compare the expressions
on the mats? In this problem, you will work with your team to identify two
different “legal moves” for simplifying expressions.
Build the mat below using tiles and simplify the expressions. Record your
work by drawing circles around the zeros or the balanced sets of tiles that
you remove in each step on the Lesson 6.1.1B
Mat A
Mat B
Resource Page. Which expression is greater?
= +1
= –1
© CPM Educational Program 2013, all rights reserved CC1 and CC2
x
?
x
24
6-4.
There are two kinds of moves you
could use in problem 6-3 to simplify
expressions with algebra tiles. First,
you could remove zeros. Second, you
could remove matching (or balanced)
sets of tiles from both sides of the mat.
Both moves are shown in the figures
below. Justify why each of these
moves can be used to simplify
expressions.
Removing Balanced Sets
Mat A
x
6-5.
Mat B
?
Removing Zeros
Mat A
Mat B
?
x
WHICH SIDE IS GREATER?
For each of the problems below, use the Lesson 6.1.1 Resource Page
and:
a.
•
Build the two expressions on your mat.
•
Write an expression for each side below the mats for parts (a)
through (d) OR draw the tiles in the space given on the resource
page for parts (e) and (f).
•
Use legal moves to determine which mat is greater, if
possible. Record your work by drawing circles around
the zeros or the balanced (matching) sets of tiles that you
remove in each problem.
Mat A
x x
Mat B
?
x
x
Mat B
Mat A
b.
x
© CPM Educational Program 2013, all rights reserved CC1 and CC2
x
x
x
= +1
= –1
?
x x
25
c.
Mat A
Mat B
x
6.1.2
x
x
x
?
x
?
x
Mat B
Mat A
d.
Word
What if I cannot tell?
<
•••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••
Graph
!
! Symbol
Comparing Quantities with Variables
= +1
Have you ever tried to make a decision when the information you have is uncertain? = –1
Perhaps you have tried to make plans on a summer day only to learn that it might rain.
In that case, your decision might have been based on the weather, such as, “I will go
swimming if it does not rain, or stay home and play video games if it does rain.”
Sometimes in mathematics, solutions might depend on something you do not know,
like the value of the variable. Today you will study this kind of situation.
6-12.
For each of the problems below, build the given expressions
on your Expression Comparison Mat. Then use the
simplification strategies of removing zeros and simplifying by
removing matching pairs of tiles to determine which side is
greater, if possible. Record your steps on the Lesson 6.1.2
Resource Page.
a.
Mat A
x
x
?
x x
c.
Mat A
x x
x
x
x
b.
Mat B
x
x
x
Mat A: 2(x + 3) ! 4
Mat B: 3x + (!1) ! x + 4
x
d.
Mat B
x
Mat A
Mat B
x
x
x
?
x
x x x
x
© CPM Educational Program 2013, all rights reserved CC1 and CC2
?
x
26
6-13.
WHAT HAPPENED?
When Ignacio and Oliver compared the expressions in part
(d) of problem 6-12, they could not figure out which side was
greater.
a.
Is it always possible to determine
which side of the Expression
Comparison Mat is greater (has the
greater value)? Why or why not? Be
prepared to share your reasoning.
Mat A
Mat B
x
x x x
b. How is it possible for Mat A to have the
greater value?
c.
= +1
= –1
?
x
How is it possible for Mat B to have the
greater value?
d. In what other way can Mat A and B be related? Explain.
6-14.
Ignacio and Oliver are playing another game
with the algebra tiles. After they simplify two
new expressions, they are left with the
expressions on their mats shown at right.
They could not tell which part of the mat is
greater just by looking.
a.
Mat B
x
?
One way to compare the mats is to
separate the x-tiles and the unit tiles on
different sides of the mat. Work with
your team to find a way to have only
x-tiles on Mat A. Make sure that you are
able to justify that your moves are legal.
b. Using the same reasoning from part (a),
what would you do to have only the
variable on Mat B in the Expression
Comparison Mat at right?
c.
Mat A
Write a short note to Ignacio and Oliver
explaining this new strategy. Feel free
to give it a name so it is easier for them
to remember.
© CPM Educational Program 2013, all rights reserved CC1 and CC2
Mat A
Mat B
?
x
27
6-15.
Ignacio and Oliver are trying to decide if there are other
ways to change expressions on the Expression Comparison
Mat without affecting which side is greater. They have
invented some new strategies and described them below.
Your Task: For each of the moves below:
•
Build the Expression Comparison Mats on your paper.
•
Follow each set of directions for the mat shown in each
strategy below.
•
Determine if the move in the strategy is valid for maintaining the
relationship between the two expressions. Be prepared to justify
your response.
Strategy #1
Strategy #2
“If you have a mat like the one
drawn below, you can add the
same number of tiles to both
sides. In this case, I added 3
negative tiles to both sides.”
Mat A
x
Mat B
?
“On a mat like the one below, I
added +3 to Mat A and added –3 to
Mat B.”
= +1
= –1
Mat A
x
?
x
Strategy #3
“To simplify, I removed a positive
x-tile from one side and a
negative x-tile from the other
side.”
Mat A
x
Mat B
?
Mat B
x
Strategy #4
“On a mat like the one below, I
would add three zero pairs to Mat
B.”
Mat A
x
x
© CPM Educational Program 2013, all rights reserved CC1 and CC2
Mat B
?
x
28
6.2.1
What values make expressions equal?
••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••
Solving Equations
x
In the last section, you figured out how to determine what values of x make one
expression greater than another. In this lesson, you will study what can be learned
about x when two expressions are equal.
6-48.
CHOOSING A PRICE PLAN
Sandeep works at a bowling alley
that currently charges players $3
to rent shoes and $4 per game.
However, his boss is thinking
about charging $11 to rent shoes
and $2 per game.
a.
If a customer rents shoes and plays two games, will he or she pay more
with the current price plan or the new price plan? Show how you
know.
b. If the customer bowls 7 games, which price plan is cheaper?
6-49.
WILL THEY EVER BE EQUAL?
= +1
= –1
Mat A
Mat B
x
x
Sandeep decided to represent the two price plans
from problem 6-48 with the expressions below,
where x represents the number of games bowled.
Then he placed them on the Expression Comparison
Mat shown at right.
x
x
x
?
x
Original price: 4x + 3 New price 2x + 11
a.
Are his expressions correct? Find both the original and new prices
when x = 2 and then again when x = 7 games. Did you get the same
prices as you found in problem 6-46?
Mat A
b. Sandeep then simplified the expressions on
the mat. What steps did Sandeep take to
simplify the mat to this point?
c.
Sandeep noticed that for a certain number of
x
games, customers would pay the same
amount no matter which price plan his boss
used. That is, he found a value of x that
will make 4x + 3 = 2x + 11 . How many games would that
customer bowl? What was the price he paid? Explain.
© CPM Educational Program 2013, all rights reserved CC1 and CC2
Mat B
x
?
= +1
= –1
29
d. The value of x you found in part (c) is called a solution to the
equation 4x + 3 = 2x + 11 because it makes the equation true. That is, it
makes both expressions have the same value.
Is x = 6 also a solution? How can you tell?
6-50.
SOLVING FOR X
When the expressions on each side of the comparison mat are equal, they
can be represented on a mat called an Equation Mat. Obtain a Lesson
6.2.1 Resource Page and algebra tiles from your teacher. Now the “=”
symbol on the central line indicates that the expressions on each side of the
mat are equal.
a.
Build the equation represented by the
Equation Mat at right on your own mat
using algebra tiles.
x
x
x
b. On your paper, record the original
equation represented on your Equation
Mat.
c.
6-51.
Simplify the tiles on the mat as much as possible. Record what is on
the mat after each legal move as you simplify each expression. What
value
of x will make the expressions equal?
Amelia wants to solve the equation shown on
the Equation Mat at right. After she
simplified each expression as much as
possible, she was confused by the tiles that
were left on the mat.
a.
x
x
x
x
x
x
What was Amelia’s original
equation?
b. Remove any zero pairs that you find on each side of the Equation Mat.
What happens?
c.
6-52.
What is the solution to this equation? That is, what value of x makes
this equation true? Explain your reasoning.
Amelia now wants to solve the equation 2x + 2 + (!3) = 5x + 8 . Help her
find the value of x that makes these expressions equal. Be sure to:
• Build the expressions using algebra tiles on your Equation Mat.
• Draw the mat on your paper.
• Simplify the mat to help you figure out what value of x makes this
equation true. Be sure to record your work in symbols on your
paper.
© CPM Educational Program 2013, all rights reserved CC1 and CC2
30
6.2.2
How do I know that it is correct?
•••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••
Checking Solutions and the Distributive Property
x
Sometimes a lot can depend on the solution of a problem. For example, when
businesses calculate the cost of packaging and shipping a product, they need to come
up with an accurate value. If they miscalculate by only $0.01 per package but ship
one million packages per year, this small miscalculation could be costly.
Solving a problem is one challenge. However, once it is solved, it is important to
have ways to know whether the solution you found is correct. In this lesson, you will
be solving equations and finding ways to determine whether your solution makes the
equation true.
6-60.
Chen’s sister made this riddle for him to solve: “I am
thinking of a number. If you add two to the number
then triple it, you get nine.”
a.
Build the equation on an Equation Mat. What are
two ways that Chen could write this equation?
b. Solve the equation and show your work by
writing the equation on your paper after each
legal move.
c.
6-61.
6-62.
When Chen told his sister the mystery number in the riddle, she said he
was wrong. Chen was sure that he had figured out the correct number.
Find a way to justify that you have the correct solution in part (b).
Now solve the equation 4(x + 3) = 8 . Remember to:
•
Build the equation on your Equation Mat with algebra tiles.
•
Simplify the equation using your legal moves.
•
Record your work on your paper.
•
Solve for x. That is, find the value of x that makes the equation
true.
CHECKING YOUR SOLUTION
When you solve an equation that has one solution, you get a value for the
variable. But how do you know that you have done the steps
correctly and that your answer “works”?
a.
Look at your answer for problem 6-61. How could you
verify that your solution is correct and convince someone
else? Discuss your ideas with your team.
© CPM Educational Program 2013, all rights reserved CC1 and CC2
31
b. When Kelly and Madison compared their solutions for
the equation 2x ! 7 = !2x + 1 , Kelly got a solution of
x = 2 and Madison got a solution of x = !1 . To decide
whether the solutions were correct, the girls decided to
check their answers to see if they made the expressions
equal.
Finish their work below to determine whether either girl
has the correct solution.
Kelly’s Work
?
2x ! 7 = ! 2x + 1
?
2(2) ! 7 = ! 2(2) + 1
c.
Madison’s Work
?
2x ! 7 = ! 2x + 1
?
2(!1) ! 7 = ! 2(!1) + 1
When checking, Kelly ended up with !3 = !3 . Does this mean that her
answer is correct or incorrect? If it is correct, does this mean the
solution is x = !3 or x = 2 ? Explain.
d. Go back to problem 6-61 and show how to check your solution for that
problem.
6-63.
Kelly solved the equation 4(x + 3) = 8 from problem
6-61. Her work is shown at right.
a.
If 4(x + 3) = 8 , does x + 3 have to equal 2?
Why?
4(x + 3)
x +3
x + 3 + (!3)
x
=
=
=
=
8
2
2 + (!3)
!1
b. What did Kelly do to remove the 3 unit tiles from the left side of the
equation? Does this move affect the equality?
c.
6.2.3
If Kelly were solving the equation 3(x ! 5) = 9 , what might her first
step be? What would she have after that step? You may want to build
this equation on an Equation Mat to help make sense of her strategy.
How can I record it?
•••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••
Solving Equations and Recording Work
x
In this lesson, you will continue to improve your skills of simplifying and solving
more complex equations. You will develop ways to record your solving strategies so
that another student can understand your steps without seeing your Equation Mat.
© CPM Educational Program 2013, all rights reserved CC1 and CC2
32
6-71.
Gene and Aidan were using algebra tiles to solve equations. Aidan was
called away.
Help Gene finish by completing the table shown below and on the Lesson
6.2.3 Resource Page.
Mat A
Mat B
2x + 2(2x + 1) + (!3x) + (!6)
4x + 3 + (!3) + x + 8
Steps taken
Original Equation
1. Use the Distributive
Property.
3x + (!4)
5x + 8
2.
3. Subtract 3x from both
sides.
!12
2x
4.
5. Divide both sides by 2.
6-72.
Aidan was frustrated that he needed to write so much when solving an
equation. He decided to come up with a shortcut for recording his work to
solve a new equation.
As you look at Aidan’s recording below of how he solved 2x + 4 = !12 ,
visualize an Equation Mat with algebra tiles. Then answer the questions for
each step below.
a.
What legal move does writing – 4 twice
represent?
b. What legal move does circling the + 4 and the – 4
represent?
c.
What does the box around the
2
2
2 x + 4 = !12
!4 = !4
2x
2
represent?
=
!16
2
x = !8
d. Why did Aidan divide both sides by 2?
e.
Check Aidan’s solution in the original equation. Is his solution
correct?
© CPM Educational Program 2013, all rights reserved CC1 and CC2
33
6-73.
The method of recording the steps in the solution of an equation is useful
only if you understand what operations are being used and how they relate
to the legal moves on your Equation Mat.
Find the work shown at right on your resource
page for this lesson.
a.
For each step in the solution, add the
missing work below each line that shows
what legal moves were used. You may
want to build the equation on an Equation
Mat.
x + (!4) + 6 x = 3 x ! 1 + 5
!4 + 7 x = 3 x + 4
b. Check that the solution is correct.
6-74.
7x = 3x + 8
4x = 8
x =2
For each equation below, solve for x. You may want to build the equation
on your Equation Mat. Record your work in symbols using Aidan’s
method from problem 6-72. Remember to check your solution.
a.
!2x + 5 + 2x ! 5 = !1+ (!1) + 6x + 2
b.
3(4 + x) = x + 6
6.2.6
Is there always a solution?
•••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••
Cases With Infinite or No Solutions
x
Are all equations solvable? Are all solutions a single number? Think about this:
Annika was born first, and her brother William was born 4 years later. How old will
William be when Annika is twice his age? How old will William be when Annika is
exactly the same as his age?
In this lesson, you will continue to practice your strategies of combining like terms,
removing zeros, and balancing to simplify and compare two expressions. You will
also encounter unusual situations where the solution may be unexpected.
6-116.
Many students believe that every equation has only one
solution. However, in the introduction to this lesson you
might have noticed that if Annika was four years older
than her brother, William, they could never be the same
age. Some situations have one solution, others have no
solution, and still others have all numbers as solutions.
© CPM Educational Program 2013, all rights reserved CC1 and CC2
34
For each of the following equations, reason with your team to decide if the
answer would be “One solution,” “No solution,” or “All numbers are
solutions.” If there is a single number solution, write it down. If you are
not sure how many solutions there are, have each member of your team try
a number to see if you can find a value that makes the equation work.
6-117.
a.
x=x
b.
x +1 = x
c.
x = 2x
d.
x+ x = 2+ x
e.
x+x= x!x
f.
x + x = 2x
g.
x ! x = x2
h.
x !1 = x
Use the 5-D Process to write an equation for the problem below. Then
answer the question.
Kelly is 6 years younger than her twin brothers Bailey and Larry. How old
will Kelly be when the sum of her brothers ages is 12 more than twice
Kelly’s?
6-118.
SPECIAL CASES, Part One
Use the equation 8 + x + (!5) = (!4) + x + 7 to complete parts (a) through (c).
a.
Build the equation on your Equation Mat and simplify it as much as
possible. Record your steps and what you see when you have
simplified the equation fully. Draw a picture of your final mat.
b. Have each member of your team test a different value for x in the
original equation, such as x = 0 , x = 1 , x = !5 , x = 10 , etc. What
happens in each case?
c.
6-119.
Are there any solutions to this equation? If so, how many?
SPECIAL CASES, Part Two
Use the equation x + x + 2 = 2x to complete parts (a) through (c).
a.
Build the equation on your Equation Mat and simplify it as much as
possible. Record your steps and what you see when you have
simplified the equation fully. Draw a picture of your final mat.
b. Have each member of your team test a different value for x in the
equation, such as x = 0 , x = 1 , x = !5 , x = 10 , etc. What happens? Is
there a pattern to the results you get from the equation?
c.
Did you find any values for x that satisfied the equation in part (a)?
When there is an imbalance of units left on the mat (such as 2 = 0),
what does this mean? Is x = 0 a solution to the equation?
© CPM Educational Program 2013, all rights reserved CC1 and CC2
35
1-4.
DIAMOND PROBLEMS
Finding and using a pattern is an important problem-solving skill you will use in algebra.
The patterns in Diamond Problems will be used later in the course to solve other types of
algebraic problems.
Look for a pattern in the first three diamonds below. For the fourth diamond, explain
how you could find the missing numbers (?) if you know the two numbers (#).
10
5
6
2
2
7
4
3
–1
5
?
–4
#
#
–5
?
Copy the Diamond Problems below onto your paper. Then use the pattern you
discovered to complete each one.
a.
b.
c.
12
3
4
7
2-1.
d.
6
–2
e.
12
–3
3
–5
4
4
8
7
8 12
1
2
8
–2
–4
–6
Your teacher will distribute a set of algebra tiles for your team to use during this
course. As you explore the tiles, address the following questions with your
team. Be prepared to share your responses with the class.
• How many different shapes are there? What are all
of the different shapes?
• How are the shapes different? How are they the same?
• How are the shapes related? Which fit together and which do not?
2-4.
Build each collection of tiles represented below. Then name the collection using a
simpler algebraic expression, if possible. If it is not possible to simplify the
expression, explain why not.
a.
3x + 5 + x 2 + y + 3x 2 + 2
b.
c.
2 + x 2 + 3x + y 2 + 4y + xy
d. 3y + 2 + 2xy + 4x + y 2 + 4y + 1
© CPM Educational Program 2013, all rights reserved
2x 2 + 1 + xy + x 2 + 2xy + 5
2
3-65.
Write the area as a product and as a sum
for the composite rectangle shown at
right.
x
x
x
3-66.
x
x
x
Now examine the following diagram. How is it similar to the set of tiles in problem 522? How is it different? Talk with your teammates and write down all of your
observations.
3-68.
3
12x
15
2x
8x 2
10x
4x
5
Use a generic rectangle to multiply the following expressions. Write each solution both
as a sum and as a product.
a. (2x + 5)(x + 6)
b.
(m ! 3)(3m + 5)
c. (12x + 1)(x ! 5)
d.
(3 ! 5y)(2 + y)
8-2.
The process of changing a sum to a product is called factoring. Can every expression be
factored? That is, does every sum have a product that can be represented with tiles?
Investigate this question by building rectangles with algebra tiles for the following
expressions. For each one, write the area as a sum and as a product. If you cannot
build a rectangle, be prepared to convince the class that no rectangle exists (and thus
the expression cannot be factored
a. 2x 2 + 7x + 6
b.
6x 2 + 7x + 2
c. x 2 + 4x + 1
d.
2xy + 6x + y 2 + 3y
8-3.
Work with your team to find the sum and the product for the following generic
rectangles. Are there any special strategies you discovered that can help you
determine the dimensions of the rectangle? Be sure to share these strategies with your
teammates.
a.
b.
c.
2x
5
!2 y
!6
!9 x
!12
6x 2
15x
5xy
15x
12x 2
16x
© CPM Educational Program 2013, all rights reserved
3
8-4.
8-14.
While working on problem 8-3, Casey noticed a pattern with the
diagonals of each generic rectangle. However, just before she
shared her pattern with the rest of her team, she was called out
of class! The drawing on her paper looked like the diagram at
right. Can you figure out what the two diagonals have in
common?
2x
5
6x 2
15x
FACTORING QUADRATIC EXPRESSIONS
To develop a method for factoring without algebra tiles, first study how to factor with
algebra tiles, and then look for connections within a generic rectangle.
a. Using algebra tiles, factor 2x 2 + 5x + 3 ; that is, use the tiles to build a rectangle, and then
write its area as a product.
b. To factor with tiles (like you did in part (a)), you need to determine how the tiles need to
be arranged to form a rectangle. Using a generic rectangle to factor requires a
different process.
Miguel wants to use a generic rectangle to factor 3x 2 + 10x + 8 .
He knows that 3x 2 and 8 go into the rectangle in the
locations shown at right. Finish the rectangle by deciding
how to place the ten x-terms. Then write the area as a
product.
c. Kelly wants to find a shortcut to factor 2x 2 + 7x + 6 . She
knows that 2x 2 and 6 go into the rectangle in the
locations shown at right. She also remembers Casey’s
pattern for diagonals. Without actually factoring yet,
what do you know about the missing two parts of the
generic rectangle?
8
3x 2
?
6
2x 2
?
d. To complete Kelly’s generic rectangle, you need two xterms that have a sum of 7x and a product of 12x 2 . Create and
solve a Diamond Problem that represents this situation.
product
e. Use your results from the Diamond Problem to complete the
generic rectangle for 2x 2 + 7x + 6 , and then write the area as a
product of factors.
sum
© CPM Educational Program 2013, all rights reserved
4
8-15.
Factoring with a generic rectangle is especially convenient when algebra tiles are not
available or when the number of necessary tiles becomes too large to manage. Using
a Diamond Problem helps avoid guessing and checking, which can at times be
challenging. Use the process from problem 8-13 to factor 6x 2 + 17x + 12 . The
questions below will guide your process.
a. When given a trinomial, such as 6x 2 + 17x + 12 , what two parts of a generic rectangle
can you quickly complete?
b. How can you set up a Diamond Problem to help factor a trinomial
such as 6x 2 + 17x + 12 ? What goes on the top? What goes on
the bottom
c. Solve the Diamond Problem for 6x 2 + 17x + 12 and complete its
generic rectangle.
product
sum
d. Write the area of the rectangle as a product.
8-16.
8-24.
8-25.
Use the process you developed in problem 8-14 to factor the following quadratics,
if possible. If a quadratic cannot be factored, justify your conclusion.
a. x 2 + 9x + 18
b.
4x 2 + 17x ! 15
c. 4x 2 ! 8x + 3
d.
3x 2 + 5x ! 3
Factor each quadratic expression below, if possible. Use a Diamond Problem and
generic rectangle for each one.
a. x 2 + 6x + 9
b.
2x 2 + 5x + 3
c. x 2 + 5x ! 7
d.
3m 2 + m ! 14
SPECIAL CASES
Most quadratic expressions are written in the form
ax 2 + bx + c . But what if a term is missing?
Or what if the terms are in a different order?
Consider these questions while you factor the
expressions below. Share your ideas with your
teammates and be prepared to demonstrate your
process for the class.
a.
9x 2 ! 4
b.
12x 2 ! 16x
c.
3 + 8k 2 ! 10k
d.
40 ! 100m
© CPM Educational Program 2013, all rights reserved
5
8-26.
Now turn your attention to the quadratic expression below. Use a generic
rectangle and Diamond Problem to factor this expression. Compare your
answer with your teammates’ answers. Is there more than one possible
answer?
4x 2 ! 10x ! 6
8-36.
8-37.
c. Without factoring, predict which quadratic expressions below may have more than
one factored form. Be prepared to defend your choice to the rest of the class.
i. 12t 2 ! 10t + 2
ii. 5 p 2 ! 23p ! 10
iii. 10x 2 + 25x ! 15
iv. 3k 2 + 7k ! 6
FACTORING COMPLETELY
In part (c) of problem 8-36, you should have noticed that each term in
12t 2 ! 10t + 2 is divisible by 2. That is, it has a common factor of 2.
a. An expression is considered completely factored if none of the factors can be
factored any more. Often it is easiest to remove common factors first, before
factoring with a generic rectangle. Rewrite this expression 10x 2 + 25x ! 15 with
the common factor factored out.
b. Your result in part (a) is not completely factored if either factor can be factored.
Factor 10x 2 + 25x ! 15 completely.
8-100.
COMPLETING THE SQUARE
Jessica was at home struggling with her Algebra homework. She had missed class
and did not understand the new method called completing the square. She
was supposed to use it to change y = x 2 + 8x + 10 to graphing form. Then her
precocious younger sister, who was playing with algebra tiles, said, “Hey, I bet
I know what they mean.” Anita’s Algebra class had been using tiles to multiply
and factor binomials.
Anita explained: “ x 2 + 8x + 10 would look like this;”
x
x
2
x
x
x
x
x
1
1
1
x
1
x
1
1
1
1
1
1
“Yes,” said Jessica, “I’m taking Algebra too, remember?”
Anita continued, “And you need to make it into a square!”
© CPM Educational Program 2013, all rights reserved
6
“OK,” said Jessica, and
she arranged her tiles on
an equation mat as
shown at right.
+
x
y
+
4
x
x2
x x x x
+
x
x
x
x
1 1 1 1
1 1 1 1
1 1
4
“Oh,” said Jessica. “I
need 16 small unit tiles
to fill in the corner!”
+
“But you only have 10,”
Anita reminded her.
_
_
“Right, I only have ten,”
Jessica replied. She
drew the outline of the
whole square and said:
“Oh, I get it! To complete the square, I need to add six tiles to each side of the
equation:”
+
x +
y
1 1
1 1
1 1
x
+
4
_
x
+
4
x x x x
2
x
x
x
x
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
_
“Oh, I see,” said Anita. “You started with y = x 2 + 8x + 10 , but now you can
rewrite it as y + 6 = (x + 4)2 .”
“Thank you so much, Anita! Now I can easily write the function in graphing form,
y = (x + 4)2 ! 6 .”
How can you use your graphing calculator to verify that y = x 2 + 8x + 10 and
y = (x + 4)2 ! 6 are equivalent functions?
© CPM Educational Program 2013, all rights reserved
7
8-101.
Write each function in graphing form, then state the vertex and y-intercept of each
parabola.
a. f (x) = x 2 + 6x + 7
8-102.
f (x) = x 2 + 4x + 11
b.
Help Jessica with a new problem. She needs to complete the square to write
y = x 2 + 4x + 9 in graphing form.
a. Draw tiles to help her figure out how to make this expression into a square. Does
she have too few or too many unit squares this time? Write her function in
graphing form.
b. Find the vertex and the x-intercepts. What happened? What does that mean?
c. Algebraically find the y-intercept. Sketch the graph.
8-103.
How could you complete the square to change f (x) = x 2 + 5x + 2 into graphing
form? How would you split the five x-tiles into two equal parts?
Jessica decided to use force! She cut one tile in half, as shown below. Then she
added her two small unit tiles.
x + 2.5
x
x2
+
x
x
2.5
1x
2
x x
1x
2
Figure A
x + 2.5
x
x2
x x
+
x
x
1
2.5
1x
2
1
1x
2
Figure B
a.
How many small unit tiles are missing from Jessica’s square?
b.
Write the graphing form of the function, name the vertex and y-intercept, and
sketch the graph.
© CPM Educational Program 2013, all rights reserved
8
Lesson 6.2.2 Resource Page
© CPM Educational Program 2013, all rights reserved CC1 and CC2
Page 1 of 2
36
Lesson 6.2.2 Resource Page
© CPM Educational Program 2013, all rights reserved CC1 and CC2
Page 2 of 2
37
Lesson 7.3.1 Resource Page
Page 1 of 2
Mathematical Magic Tricks
7-79.
Steps
Trial 1
Trial 2
Trial 3
1. Pick a number.
2. Add 5.
3. Double it.
4. Subtract 4.
5. Divide by 2.
6. Subtract the
original number.
7-81.
Steps
Trial 1
Trial 2
Trial 3
Algebra Tile Picture
1. Pick a number.
2. Add 2.
3. Multiply by 3.
4. Subtract 3.
5. Divide by 3.
6. Subtract the
original number.
© CPM Educational Program 2013, all rights reserved CC1 and CC2
38
Lesson 7.3.1 Resource Page
Page 2 of 2
Mathematical Magic Tricks
7-82.
Steps
Trial 1
Trial 2
Trial 3
Algebra Tile Picture
1. Pick a number.
2.
3.
4.
5.
6.
7-84.
Steps
Trial 1
Trial 2
Trial 3
Algebra Tile Picture
1. Pick a number.
2. Double it.
3. Add 4.
4. Multiply by 2.
5. Divide by 4.
6. Subtract the
original number.
© CPM Educational Program 2013, all rights reserved CC1 and CC2
39
Lesson 7.3.2 Resource Page
7-91.
Steps
Trial 1
Trial 2
Algebra Tile Picture
1. Pick a number.
x
2. Add 7.
x
3. Triple the result.
x
x
x
4. Add 9.
x
x
x
5. Divide by 3.
x
Algebraic
Expression
6. Subtract the
original number.
© CPM Educational Program 2013, all rights reserved CC1 and CC2
40
Lesson 6.1.1A Resource Page
Expression Comparison Mat
© CPM Educational Program 2013, all rights reserved CC1 and CC2
41
Lesson 6.1.1B Resource Page
= +1
= –1
Comparing Expressions
6-2.
6-3.
Mat A
Mat B
Mat A
Mat B
x2
x2
?
x
x
x
x
Expressions:
x
?
x
Expressions:
6-5.
a.
Mat A
x x
Mat B
?
x
x
x
Expressions:
c.
x
x
x
Mat B
?
x x
Expressions:
Mat A
x
x
Mat B
?
Mat A
d.
x
x
x
Expressions:
e.
Mat A
b.
Mat B
?
x
Expressions:
Mat A
Mat B
Mat A
f.
?
?
3x ! 4 ! 2 x + 3 ! 5 + 2x
Lesson 6.1.2 Resource Page
Mat B
5 + (!3x) + 5x
© CPM Educational Program 2013, all rights reserved CC1 and CC2
x 2 + 2x + 1! x 2
42
Simplify and Compare
6-12.
a.
b.
Mat A
x
x
?
x x
Mat B
x
x
x
Mat A
Mat B
?
x
Expressions: 2(x + 3) ! 4
Expressions:
c.
3x + (!1) ! x + 4
d.
Mat A
x x
Mat B
x
Mat A
Mat B
x
x
x
x
x
x
?
x
x x x
x
?
x
Expressions:
Expressions:
6-14.
a.
b.
Mat A
Mat B
Mat A
Mat B
x
?
Expressions:
?
x
Expressions:
© CPM Educational Program 2013, all rights reserved CC1 and CC2
43
Lesson 6.2.1 Resource Page
Equation Mat
© CPM Educational Program 2013, all rights reserved CC1 and CC2
44
Lesson 6.2.3 Resource Page
Recording Work
Problem 6-71.
Mat A
Mat B
2x + 2(2x + 1) + (!3x) + (!6)
4x + 3 + (!3) + x + 8
Steps taken
Original Equation
1. Use the Distributive
Property.
3x + (!4)
5x + 8
2.
3. Subtract 3x from
both sides.
!12
4.
2x
5. Divide both sides
by 2.
Problem 6-73.
x + (!4) + 6 x = 3 x ! 1 + 5
!4 + 7 x = 3 x + 4
7x = 3x + 8
4x = 8
x =2
© CPM Educational Program 2013, all rights reserved CC1 and CC2
45
Printed on the Envelopes
•
•
•
•
FORTUNE COOKIE
The first person takes out a fortune and reads it to the group. They respond to it in 30
seconds.
The first person passes the fortune to the second person who also responds to it and passes it
to the next person. This continues until everyone has responded.
The team then decides upon a team response to the fortune.
Now the second person takes out a fortune and reads it to the group. The process continues
as described above.
2015 M. Marsh & V. Lezak
Fortune Cookies
How would the use of algebra tiles to improve student learning in your
classroom?
How can you make sure the students connect the algebra tiles to the
mathematics, and are not just playing with tiles?
What did your team say to help you in your learning as you worked on the
problem?
What mathematical concepts do algebra tiles help students learn? Both
presented so far and ideas you have?
What would a good management system for algebra tiles look like in your
classroom?
What behaviors, on your part or the team’s part, hindered your learning?
When students are struggling while using algebra tiles, what are some
questions you can ask to help them move forward?
Fortune Cookies adapted from CPM Educational Program
2015 M. Marsh & V. Lezak
Mathematics | Standards for Mathematical Practice
The Standards for Mathematical Practice describe varieties of expertise that mathematics educators at all
levels should seek to develop in their students. These practices rest on important “processes and
proficiencies” with longstanding importance in mathematics education. The first of these are the NCTM
process standards of problem solving, reasoning and proof, communication, representation, and connections.
The second are the strands of mathematical proficiency specified in the National Research Council’s report
Adding It Up: adaptive reasoning, strategic competence, conceptual understanding (comprehension of
mathematical concepts, operations and relations), procedural fluency (skill in carrying out procedures flexibly,
accurately, efficiently and appropriately), and productive disposition (habitual inclination to see mathematics
as sensible, useful, and worthwhile, coupled with a belief in diligence and one’s own efficacy).
1
•
•
•
•
•
•
Make sense of problems and persevere in solving them.
Find meaning in problems
Look for entry points
Analyze, conjecture and plan solution pathways
Monitor and adjust
Verify answers
Ask themselves the question: “Does this make sense?”
2
•
•
•
Reason abstractly and quantitatively.
Make sense of quantities and their relationships in problems
Learn to contextualize and decontextualize
Create coherent representations of problems
3
•
•
•
•
Construct viable arguments and critique the reasoning of others.
Understand and use information to construct arguments
Make and explore the truth of conjectures
Recognize and use counterexamples
Justify conclusions and respond to arguments of others
4
•
•
•
•
Model with mathematics.
Apply mathematics to problems in everyday life
Make assumptions and approximations to simplify a complicated situation
Identify quantities in a practical situation
Interpret results in the context of the situation and reflect on whether the results make
sense
5
•
•
Use appropriate tools strategically.
Consider the available tools when solving problems
Are familiar with tools appropriate for their grade or course ( pencil and paper, concrete
models, ruler, protractor, calculator, spreadsheet, computer programs, digital content
located on a website, and other technological tools)
Make sound decisions of which of these tools might be helpful
•
6
•
•
•
Attend to precision.
Communicate precisely to others
Use clear definitions, state the meaning of symbols and are careful about specifying units
of measure and labeling axes
Calculate accurately and efficiently
7
•
•
•
Look for and make use of structure.
Discern patterns and structures
Can step back for an overview and shift perspective
See complicated things as single objects or as being composed of several objects
8
•
•
•
Look for and express regularity in repeated reasoning.
Notice if calculations are repeated and look both for general methods and shortcuts
In solving problems, maintain oversight of the process while attending to detail
Evaluate the reasonableness of their immediate results
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46