Grade 7 Workbook — Chapter 9: Equivalent Algebraic Expressions

SOL 7.PFA.2 (b–d) · Companion to Textbook Chapter 9

Each page below is one Canva page. Headings are sized for direct paste: page title as H1, section labels as H2. Figures referenced by filename live in ../figures/. Every numbered item is the same problem as in the textbook, so one answer key serves both. Item numbers run continuously from 1 to 120.

PAGE 1 — Chapter opener

Chapter 9 · Equivalent Algebraic Expressions

Standard 7.PFA.2 (b–d)

In this chapter you will:

Build algebraic expressions out of algebra tiles and colored chips
Use zero pairs to show that two expressions are equivalent
Find terms, coefficients, constants, and like terms
Combine like terms, including negative and fractional coefficients
Use the distributive property to rewrite expressions
Evaluate expressions for given replacement values

Words to know: algebraic expression · variable · equivalent · algebra tile · $x$ -tile · unit tile · colored chip · zero pair · term · coefficient · constant · like terms · distributive property · evaluate · replacement value

Calculator note: simplifying algebraic expressions is tested without a calculator. Every number on pages 2 through 17 is mental-math sized.

PAGE 2 — The pieces and what they are worth

9.1 Modeling with Tiles and Chips

FIGURE: fig1-tile-and-chip-key.png (full width)

Fill in the blanks.

A long rectangle is an ________-tile. It is worth ________.

A small square is a ________ tile. It is worth ________.

Blue pieces are ____________ and red pieces are ____________.

We never say how many unit tiles fit inside an $x$ -tile, because $x$ is ____________.

FIGURE: fig2-model-3x-plus-2.png (full width)

A model shows three $x$ -tiles and two positive unit tiles. Write the expression.

Expression: _______________

Draw the tiles. Use a long rectangle for an $x$ -tile and a small square for a unit tile. Shade negative pieces.

$2x + 5$

TILE DRAWING SPACE: 1.5 in tall, full width
$-x + 4$

TILE DRAWING SPACE: 1.5 in tall, full width

PAGE 3 — Reading and writing models

From Tiles to Symbols and Back

A model shows four $x$ -tiles and three negative unit tiles.

Expression: _______________
What is the value of one zero pair? ______

Why does removing one leave the total unchanged?
A model shows three $x$ -tiles and two negative $x$ -tiles. Cross out the zero pairs above, then write what remains.

TILE DRAWING SPACE: 1.2 in tall, full width — pre-drawn 3 positive x-tiles and 2 negative x-tiles

What remains: _______________
Write the expression modeled by each set of tiles.

Model	Expression
a) two $x$ -tiles and six negative unit tiles
b) three negative $x$ -tiles and one positive unit tile
c) one $x$ -tile and five positive unit tiles

Describe a tile model for $-2x - 3$ .

PAGE 4 — Chips and zero pairs

Chips, Zero Pairs, and Equivalence

FIGURE: fig3-zero-pairs.png (full width)

FIGURE: fig4-chip-value.png (full width)

Seven positive chips and four negative chips. Value: ______
Five negative chips and two positive chips. Value: ______
Model $4x + (-4x)$ with tiles. What is left?

TILE DRAWING SPACE: 1.2 in tall, full width

Explain why the expression is equivalent to $0$ .
Model A: three $x$ -tiles and two negative unit tiles. Model B: one $x$ -tile, five negative unit tiles, two $x$ -tiles, and three positive unit tiles.

Model A simplifies to: _______________

Model B simplifies to: _______________

Equivalent? ______ How do you know?
Apply it. A concert ticket costs $x$ dollars and the site adds a flat $2 fee to the order. Write an expression for three tickets plus the fee, then describe the model.

Expression: _______________ Model: _______________________________________________
Find the error. To model $5 - 2x$ , a student lays out five negative unit tiles and two positive $x$ -tiles.

What went wrong? _______________________________________________

Correct model: _______________________________________________

PAGE 5 — Exit ticket 9.1

Exit Ticket · Lesson 9.1

Name: ________________________ Date: ____________

Write the expression modeled by two $x$ -tiles and four negative unit tiles. _______________
Describe a tile model for $-3x + 1$ .
A model shows five $x$ -tiles and three negative $x$ -tiles. What remains after all zero pairs are removed? ______
Why is one positive chip together with one negative chip worth zero?

PAGE 6 — Terms, coefficients, constants

9.2 Like Terms

Rewrite every subtraction as adding the opposite first. The sign in front of a term BELONGS to that term.

Complete the table.

Expression	Terms	Coefficient of $x$	Constant
$7x - 4$
$-x + 9$
$8 - 3x$
$\tfrac{2}{3}x$			none

In $7x - 4$ : coefficient ______ constant ______
How many terms are in $3x + 5 - 2x$ ? ______
In $-x + 9$ , the coefficient of $x$ is ______
Are $4x$ and $4$ like terms? ______ Why?
Like terms in $6x - 3 + 2x + 8$ : ______ and ______ ; ______ and ______
Are $0.5x$ and $-\tfrac{3}{4}x$ like terms? ______

PAGE 7 — Sorting practice

Sorting Terms

List the terms of $5x - 7 + x$ : ______ , ______ , ______
Name the coefficient.

a) $9x$	b) $-x$	c) $\tfrac{2}{3}x$	d) $-2.5x$

Circle YES or NO for each pair of like terms.

a) $3x$ and $-8x$	YES / NO
b) $5$ and $-2$	YES / NO
c) $7x$ and $7$	YES / NO
d) $-\tfrac{1}{2}x$ and $4x$	YES / NO

Sort the terms of $-2x + 6 + 5x - 11$ .

$x$ -terms: _______________ constants: _______________
Explain. Using algebra tiles, why are $3x$ and $3$ not like terms?

PAGE 8 — Testing and correcting

Test It, Then Fix It

Are $x + x + x$ and $3x$ equivalent?

$x$	$x + x + x$	$3x$
$4$
$-2$

What do the models look like? _______________________________________________

Apply it. A rectangle has length $x$ cm and width $3$ cm. Its perimeter is $x + 3 + x + 3$ .

Like terms: ______ and ______ ; ______ and ______
Find the error. A student says the coefficient of $x$ in $8 - 3x$ is $3$ .

What went wrong? _______________________________________________

Correct coefficient: ______

PAGE 9 — Exit ticket 9.2

Exit Ticket · Lesson 9.2

Name: ________________________ Date: ____________

In $-6x + 2.5$ : coefficient ______ constant ______
Are $-\tfrac{1}{3}x$ and $7x$ like terms? ______ Why?
How many terms are in $4x - 9 + x$ ? ______
What makes two terms like terms?

PAGE 10 — Combining like terms

9.3 Combining Like Terms

FIGURE: fig5-combining-like-terms.png (full width)

Why it works — fill in the frame.

$5x + 3x = (\underline{\ \ \ \ } + \underline{\ \ \ \ })x = \underline{\ \ \ \ }x$

Pulling the common factor $x$ out front is the ______________ property read backwards.

Simplify. Use the step frame for items 41 and 42.

$4x + 3x =$ ______
$9x - 2x =$ ______
$5x + 2 + 3x =$ ______
$-3x + 7x =$ ______
$6 - 4x - 9$

As a sum: _______________ Combine: _______________
$2x - 5 - 7x + 1$

As a sum: _______________ $x$ -terms: _______________ constants: _______________

Simplified: _______________
$0.5x + 1.5x =$ ______
$\tfrac{1}{4}x + \tfrac{1}{2}x =$ ______

PAGE 11 — More practice

Practice · Combining Like Terms

Simplify.

a) $8x - 12x$	b) $-x + 6x$	c) $3.2x - 1.2x$	d) $-\tfrac{2}{3}x - \tfrac{1}{3}x$

$7x + 4 - 2x - 9 =$ _______________
$-5 + 3x + 8 - 10x =$ _______________
$\tfrac{1}{2}x + 3 - 2x + \tfrac{1}{2} =$ _______________
$-2.5x - 4 + 2.5x + 4 =$ _______________
$6x - x + 3 - 3 =$ _______________

PAGE 12 — Reasoning and application

Why It Works

Explain. Use the distributive property to show why $5x + 3x = 8x$ .

Why does the same reasoning NOT turn $5x + 3$ into $8x$ ?
Show $4x - x + 2$ and $3x + 2$ are equivalent.

Simplify $4x - x + 2$ : _______________

$x$	$4x - x + 2$	$3x + 2$
$5$
$-2$

Apply it. A triangle has sides $x$ , $x + 4$ , and $2x - 1$ inches.

Perimeter expression: _______________ Simplified: _______________
Find the error. A student simplifies $9 - 4x$ to $5x$ .

What went wrong? _______________________________________________

Correct answer: _______________

PAGE 13 — Exit ticket 9.3

Exit Ticket · Lesson 9.3

Name: ________________________ Date: ____________

$3x + 8x =$ ______
$-6x + 2 + x - 9 =$ _______________
$\tfrac{3}{4}x - \tfrac{1}{4}x =$ ______
Why can $7x - 7$ not be simplified any further?

PAGE 14 — The distributive property

9.4 Distributing with Variables

FIGURE: fig6-area-model-distribution.png (full width)

$a(b + c) = ab + ac$

The outside factor multiplies EVERY term inside.

Watch the signs. $-3(x - 6) = (-3)(x) + (-3)(-6) = -3x + \underline{\ \ \ \ }$

Expand. Use the arrows to show both products.

$3(x + 4) =$ _______________
$2(x - 5) =$ _______________
$-4(x + 2) =$ _______________
$5(2x - 3) =$ _______________
$-(x - 7) =$ _______________
$\tfrac{1}{2}(6x + 8) =$ _______________
$3(x + 2) + 4x$

Distribute: _______________ Combine: _______________
$2(x + 3) - (x + 1)$

Distribute: _______________ Combine: _______________

PAGE 15 — Expand and simplify

Practice · Distribute, Then Combine

Expand.

a) $6(x + 1)$	b) $-2(3x - 5)$

c) $0.5(4x + 2)$	d) $-(2x + 9)$

$4(x - 2) + 3(x + 5) =$ _______________
$5(2x + 1) - 3(x - 4) =$ _______________
$-3(x - 6) - 2x =$ _______________
$\tfrac{2}{3}(3x - 9) + 4 =$ _______________

PAGE 16 — Reasoning, application, error hunt

Show What You Know

Explain. Use the area model to explain why $3(x + 4) = 3x + 12$ .

The left region represents _______________

The right region represents _______________

Together: _______________________________________________
Apply it. Four identical party bags each hold $x$ stickers and $3$ erasers.

With parentheses: _______________ Expanded: _______________
Find the error. A student writes $-3(x - 2) = -3x - 6$ .

What went wrong? _______________________________________________

Correct answer: _______________

PAGE 17 — Exit ticket 9.4

Exit Ticket · Lesson 9.4

Name: ________________________ Date: ____________

$7(x - 3) =$ _______________
$2(x + 4) + 3x =$ _______________
$-(x - 5) + 2x =$ _______________
Why must the outside factor multiply every term inside?

PAGE 18 — Substituting a value

9.5 Evaluating Expressions

Always substitute inside parentheses. $3x + 5$ at $x = -2$ becomes $3(\underline{\ \ \ \ }) + 5 = \underline{\ \ \ \ }$

Evaluate. Show the substitution step.

$3x + 5$ at $x = 4$ : $3(\underline{\ \ \ \ }) + 5 =$ ______
$3x + 5$ at $x = -2$ : $3(\underline{\ \ \ \ }) + 5 =$ ______
$-2x + 7$ at $x = 3$ : _______________ $=$ ______
$4x - 9$ at $x = -\tfrac{1}{2}$ : _______________ $=$ ______
$x + 2^3$ at $x = -5$ : _______________ $=$ ______
$5x - \sqrt{36}$ at $x = 2$ : _______________ $=$ ______
$|x| + 4$ at $x = -7$ : _______________ $=$ ______
$2[x + 3] - 1$ at $x = -4$ : _______________ $=$ ______

PAGE 19 — Evaluation practice

Practice · Evaluating

$6x - 4$

$x = 3$	$x = -3$	$x = 0.5$

$-x + 10$

$x = 12$	$x = -6$	$x = \tfrac{1}{3}$

$3[2x - 5] + \sqrt{49}$ at $x = 4$ : _______________ $=$ ______
$2^4 - 3x$ at $x = -2$ : _______________ $=$ ______
$|2x| - 5$ at $x = -3$ : _______________ $=$ ______

PAGE 20 — Equivalence and error hunt

Simplify First, Then Check

FIGURE: fig7-equivalence-substitution.png (full width)

Simplify $4(x - 1) + 2x$ : _______________

$x$	$4(x - 1) + 2x$	simplified form
$3$
$-\tfrac{1}{2}$

What do the results show? _______________________________________________

Apply it. A kayak rental costs $12 per hour plus a $5 launch fee, so the total is $12h + 5$ dollars.

Cost of a 3-hour rental: $______
Find the error. Asked to evaluate $5 - 2x$ at $x = -3$ , a student writes $5 - 2 \cdot 3 = -1$ .

What went wrong? _______________________________________________

Correct value: ______

PAGE 21 — Exit ticket 9.5

Exit Ticket · Lesson 9.5

Name: ________________________ Date: ____________

$7x - 2$ at $x = -1$ : ______
$x + 3^2$ at $x = -4$ : ______
$|x| - 6$ at $x = -10$ : ______
Why should you write parentheses around a negative replacement value? Use $5 - 2x$ at $x = -3$ .

PAGE 22 — Chapter 9 review, part 1

Chapter 9 Review

Part A · Modeling with tiles and chips

Three $x$ -tiles and five negative unit tiles. Expression: _______________
Describe a tile model for $-2x + 4$ : _______________________________________________
Six $x$ -tiles and four negative $x$ -tiles. After removing zero pairs: _______________
Eight positive chips and eleven negative chips. Value: ______
Explain how zero pairs show $5x + (-5x) = 0$ .

PAGE 23 — Chapter 9 review, part 2

Chapter 9 Review (continued)

Part B · Simplifying and generating equivalent expressions

Simplify.

a) $9x + 5x$	b) $-7x + 3x$	c) $2.5x - 4.5x$	d) $\tfrac{5}{6}x - \tfrac{1}{3}x$

$8x - 3 - 5x + 11 =$ _______________
$5(x - 2) + 3x =$ _______________
$-2(4x + 1) + 6x =$ _______________
$\tfrac{3}{4}(8x - 4) =$ _______________
$3(x + 5) - (2x - 1) =$ _______________
Write two expressions equivalent to $6x + 12$ . At least one must use parentheses.

Expression 1: _______________ Expression 2: _______________

Why is each equivalent? _______________________________________________

PAGE 24 — Chapter 9 review, part 3

Chapter 9 Review (continued)

Part C · Evaluating expressions

$4x - 7$

$x = 5$	$x = -2$	$x = \tfrac{1}{4}$

$-3x + 2^3$ at $x = -1$ : ______
$|x| + \sqrt{25}$ at $x = -6$ : ______
$2[x - 4] + 9$ at $x = 1.5$ : ______
$10 - 5x$ at $x = -0.2$ : ______

PAGE 25 — Chapter 9 review, part 4

Chapter 9 Review (continued)

Part D · Application and reasoning

Simplify $3(2x - 1) + 4$ : _______________

$x$	$3(2x - 1) + 4$	$6x + 1$
$2$
$-3$

Apply it. A ride costs $3 plus $2 per mile, so one rider pays $2m + 3$ dollars. Two friends each take their own ride of the same length.

With parentheses: _______________ Simplified: _______________

Total for a 7-mile trip: $______
Find the error. A student simplifies $4 - 2(x + 3)$ to $-2x + 10$ .

What went wrong? _______________________________________________

Correct answer: _______________
Explain. Why is checking one replacement value not enough to prove two expressions are equivalent? Use $x + 2$ and $3x$ .
Use algebra tiles to justify that $2(x + 3)$ and $2x + 6$ are equivalent.

TILE DRAWING SPACE: 2 in tall, full width — two groups of (1 x-tile + 3 unit tiles), then one combined row

Explain: _______________________________________________

Canva production notes

Page size: 8.5 × 11 in, 0.75 in margins
Type: headings 24–28 pt, body 12–14 pt, answer blanks 14 pt with 1.5 line spacing
Tile drawing spaces: plain white boxes with a hairline border; students draw their own rectangles and squares. Keep the height called out on each page so a full model fits on one line.
Color convention: blue = positive, red = negative, matching fig1-tile-and-chip-key.png. If pages are printed in grayscale, add the instruction "shade negative pieces" wherever tiles are drawn.
Figure widths: every figure in this chapter is full width. fig5-combining-like-terms.png is the tallest; give it its own block with no text wrapping beside it.
Answer blanks: keep every blank on the same line as its prompt so the Canva text box does not reflow.