Grade 8 Workbook — Chapter 4: Equivalent Algebraic Expressions

SOL 8.PFA.1 · Companion to Textbook Chapter 4

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 126.

PAGE 1 — Chapter opener

Chapter 4 · Equivalent Algebraic Expressions

Standard 8.PFA.1

In this chapter you will:

Build algebraic expressions out of algebra tiles, and read an expression off a model
Arrange tiles in a rectangle to see the distributive property
Name the property of real numbers behind every move you make
Combine like terms when the coefficients are fractions or decimals
Expand with the distributive property, including fractional and decimal factors
Generate several expressions equivalent to a given one, including a factored form

Words to know: algebraic expression · variable · equivalent · algebra tile · $x$ -tile · unit tile · zero pair · term · coefficient · numeric term · like terms · commutative property · associative property · identity property · inverse property · distributive property · expand · factor · generate

Calculator note: equivalence is a reasoning question, not an arithmetic one — there is nothing here for a calculator to compute. Every fraction and decimal on these pages is mental-math sized.

PAGE 2 — The pieces and what they are worth

4.1 Representing Expressions with Tiles

FIGURE: fig1-tile-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 ____________.

Two expressions are equivalent when they give the same value for ____________ replacement value of the variable.

FIGURE: fig2-model-3x-minus-4.png (full width)

A model shows two $x$ -tiles and five negative unit tiles.

Expression: _______________

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

$4x + 3$

TILE DRAWING SPACE: 1.5 in tall, full width
$-3x + 2$

TILE DRAWING SPACE: 1.5 in tall, full width

PAGE 3 — Zero pairs

Zero Pairs and Equivalence

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

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

Expression: _______________
What is one zero pair worth? ______

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

TILE DRAWING SPACE: 1.2 in tall, full width — pre-drawn 5 positive x-tiles and 3 negative x-tiles

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

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

Describe a tile model for $6 - 2x$ .

PAGE 4 — Rectangles and the distributive property

Tiles in a Rectangle

FIGURE: fig4-tile-array-distributive.png (full width)

Read the rectangle two ways. By rows: ______ rows of $(x + 2)$ , which is ____________. By counting the whole rectangle: ______ $x$ -tiles and ______ unit tiles, which is ____________.

$2(x + 4)$

TILE DRAWING SPACE: 1.6 in tall, full width — 2 rows

Expanded: _______________
$3(2x + 1)$

TILE DRAWING SPACE: 2.0 in tall, full width — 3 rows

Expanded: _______________
Explain. Model $\tfrac{1}{2}(6x + 4)$ by splitting a pile of tiles into equal groups.

How many groups? ______ What is in one group? _______________

Result: _______________

PAGE 5 — Practice and error hunt

Reading Models Carefully

A model shows three $x$ -tiles, two negative $x$ -tiles, five positive unit tiles, and eight negative unit tiles.

$x$ -tiles left: ______ unit tiles left: ______ Simplified: _______________
Apply it. Five identical goody bags each hold $x$ pencils and $3$ stickers.

With parentheses: _______________ Expanded: _______________

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

What went wrong? _______________________________________________

Correct model: _______________________________________________

PAGE 6 — Exit ticket 4.1

Exit Ticket · Lesson 4.1

Name: ________________________ Date: ____________

Four $x$ -tiles and two negative unit tiles. Expression: _______________
Describe a tile model for $-x + 5$ .
Describe a rectangular arrangement for $2(3x + 2)$ . Expanded: _______________
How does a rectangular arrangement show that the outside factor multiplies every term inside?

PAGE 7 — The properties of real numbers

4.2 Properties of Real Numbers

Complete the table.

Property	Addition form	Multiplication form
Commutative	$a + b =$ ________	$ab =$ ________
Associative	$(a + b) + c =$ ________	$(ab)c =$ ________
Identity	$a +$ ______ $= a$	$a \cdot$ ______ $= a$
Inverse	$a +$ ________ $= 0$	$a \cdot$ ______ $= 1$
Distributive	—	$a(b + c) =$ ____________

Careful. Subtraction is NOT commutative: $4 - 9 =$ ______ but $9 - 4 =$ ______.

Name the property.

$5 + x = x + 5$ : ____________________
$3(4x) = (3 \cdot 4)x$ : ____________________
$7(x + 2) = 7x + 14$ : ____________________
$x + 0 = x$ : ____________________
$6x + (-6x) = 0$ : ____________________
$0.5(8) + 3 =$ ______

PAGE 8 — Naming properties

Which Property?

Name the property illustrated.

a) $-\tfrac{2}{3}x + \tfrac{2}{3}x = 0$
b) $(x + 4) + 6 = x + (4 + 6)$
c) $\tfrac{1}{4} \cdot 4 = 1$
d) $1 \cdot 9x = 9x$

Name the property justifying each step of the simplification of $3 + 2(x + 5)$ .

a) $3 + 2(x + 5) = 3 + (2x + 10)$
b) $3 + (2x + 10) = 3 + (10 + 2x)$
c) $3 + (10 + 2x) = (3 + 10) + 2x$

$\tfrac{1}{2}(10) + 4 \cdot 3 - 6 =$ ______
$2.5(4) - 3 + 0.5 =$ ______

PAGE 9 — Reasoning with properties

Why the Rules Are Rules

Explain. Is subtraction commutative? $4 - 9 =$ ______ and $9 - 4 =$ ______

Why does rewriting subtraction as adding the opposite let you use the commutative property?
Explain. Use the distributive property read backwards to show why $6x + 4$ is not $10x$ .
Apply it. A rectangle has length $x + 2.5$ cm and width $1.5$ cm, so the perimeter is $2(x + 2.5) + 2(1.5)$ .

Simplified: _______________ Properties used: ____________________
Find the error. A student writes $3 + 4x = 7x$ and cites the commutative property.

What went wrong? _______________________________________________

Correct simplification: _______________

PAGE 10 — Exit ticket 4.2

Exit Ticket · Lesson 4.2

Name: ________________________ Date: ____________

$\tfrac{3}{5}x \cdot 1 = \tfrac{3}{5}x$ : ____________________
$8\left(x - \tfrac{1}{2}\right) = 8x - 4$ : ____________________
$6 + 2 \cdot 9 - \tfrac{1}{3}(12) =$ ______
Which property rewrites $-7 + 2x$ as $2x - 7$ ? ____________________ Why?

PAGE 11 — Combining like terms

4.3 Combining Like Terms

FIGURE: fig6-combining-like-terms.png (full width)

Why it works — fill in the frame.

$5x + 8x = (\underline{\hspace{1cm}} + \underline{\hspace{1cm}})x = \underline{\hspace{1cm}}x$

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

Fractions need a common denominator first.

$\tfrac{1}{2}x + \tfrac{1}{4}x = \left(\tfrac{2}{4} + \tfrac{1}{4}\right)x = \underline{\hspace{1.5cm}}$

Simplify. Use the step frame for items 40 and 42.

$5x + 8x =$ ______
$4.5x - 1.5x =$ ______
$\tfrac{2}{5}x + \tfrac{1}{5}x =$ ______
$3x - 7 - 8x + 2$

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

Simplified: _______________
$\tfrac{1}{2}x + 4 + \tfrac{1}{4}x =$ _______________
$0.75x - 2.5 - 0.25x + 1$

As a sum: _______________ Simplified: _______________
$-\tfrac{2}{3}x + 5 + \tfrac{1}{6}x =$ _______________
$6 - 1.2x - 9 + 0.2x =$ _______________

PAGE 12 — More practice

Practice · Rational Coefficients

Simplify.

a) $\tfrac{3}{4}x + \tfrac{1}{4}x$	b) $-2.4x + 0.4x$	c) $\tfrac{5}{6}x - \tfrac{1}{2}x$	d) $-\tfrac{3}{8}x - \tfrac{1}{8}x$

$7x + \tfrac{1}{2} - 3x - \tfrac{3}{2} =$ _______________
$-0.5x + 6 + 2.5x - 10 =$ _______________
$\tfrac{1}{3}x - 4 + \tfrac{1}{3}x + 4 =$ _______________
$9 - \tfrac{3}{5}x - 9 + \tfrac{3}{5}x =$ _______________
$2.5x + 1.5 - 2.5x + 3.5 =$ _______________

PAGE 13 — Reasoning and application

Show Why

Explain. Use the distributive property to show why $\tfrac{1}{2}x + \tfrac{1}{4}x = \tfrac{3}{4}x$ .

Why does the same reasoning NOT combine $\tfrac{1}{2}x + \tfrac{1}{4}$ ?
Show $1.5x + 2 - 0.5x$ and $x + 2$ are equivalent.

Simplify $1.5x + 2 - 0.5x$ : _______________

$x$	$1.5x + 2 - 0.5x$	$x + 2$
$4$
$-2$

Apply it. A triangle has sides $\tfrac{1}{2}x$ , $\tfrac{1}{2}x + 3$ , and $2x - 1.5$ inches.

Perimeter expression: _______________ Simplified: _______________
Find the error. A student simplifies $\tfrac{1}{2}x + \tfrac{1}{3}x$ to $\tfrac{2}{5}x$ .

What went wrong? _______________________________________________

Correct answer: _______________

PAGE 14 — Exit ticket 4.3

Exit Ticket · Lesson 4.3

Name: ________________________ Date: ____________

$\tfrac{7}{8}x - \tfrac{3}{8}x =$ ______
$-1.4x + 3 + 0.4x - 5 =$ _______________
$4 - \tfrac{2}{3}x + \tfrac{1}{6}x =$ _______________
Why can $0.6x - 0.6$ not be simplified any further?

PAGE 15 — The distributive property

4.4 Expanding with the Distributive Property

FIGURE: fig5-area-model-rational-factor.png (full width)

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

The outside factor multiplies EVERY term inside — including a fractional factor.

Watch the signs. $-3(x - 1.5) = (-3)(x) + (-3)(-1.5) = -3x + \underline{\hspace{1.5cm}}$

Expand. Draw arrows to show both products.

$4(x + 2.5) =$ _______________
$\tfrac{1}{2}(4x + 6) =$ _______________
$-3(x - 1.5) =$ _______________
$\tfrac{2}{3}(9x - 6) =$ _______________
$-(x - 2.5) =$ _______________
$0.2(10x + 5) =$ _______________
$5(x + 1.4) - 2x$

Distribute: _______________ Combine: _______________
$\tfrac{3}{4}(8x - 4) + 3$

Distribute: _______________ Combine: _______________

PAGE 16 — Expand and simplify

Practice · Distribute, Then Combine

Expand.

a) $6\left(x + \tfrac{1}{3}\right)$	b) $-4(2x - 0.25)$

c) $\tfrac{1}{5}(10x - 15)$	d) $-(0.5x + 7)$

$3(x - 1.5) + 2(x + 4) =$ _______________
$\tfrac{1}{2}(6x + 8) - \tfrac{1}{3}(9x - 3) =$ _______________
$-2(x + 2.5) - 4(x - 1) =$ _______________
$8\left(\tfrac{1}{4}x - \tfrac{1}{2}\right) + 3x =$ _______________
$0.5(4x - 6) - (x - 5) =$ _______________

PAGE 17 — Reasoning, application, error hunt

Show What You Know

Explain. Use the area model to explain why $\tfrac{1}{2}(4x + 6) = 2x + 3$ .

The left region represents _______________

The right region represents _______________

Together: _______________________________________________
Apply it. Six identical smoothies each cost $x$ dollars, and each gets a $1.50 topping.

With parentheses: _______________ Expanded: _______________
Find the error. A student writes $\tfrac{1}{3}(6x + 9) = 2x + 9$ .

What went wrong? _______________________________________________

Correct answer: _______________
Find the error. A student writes $-5(x - 0.4) = -5x - 2$ .

What went wrong? _______________________________________________

Correct answer: _______________

PAGE 18 — Exit ticket 4.4

Exit Ticket · Lesson 4.4

Name: ________________________ Date: ____________

$\tfrac{2}{5}(10x - 5) =$ _______________
$3(x + 0.5) - (x - 1.5) =$ _______________
$-6\left(\tfrac{1}{2}x - \tfrac{1}{3}\right) =$ _______________
Why does a fractional factor still have to multiply every term inside?

PAGE 19 — Generating equivalent expressions

4.5 Generating Equivalent Expressions

Many forms, one value. Complete the table for $4x + 10$ .

Equivalent form	Property
$10 + 4x$
$4x + 6 + 4$
$2(2x + 5)$

To factor, pull out a common factor. $6x + 15 = 3(\underline{\hspace{1cm}}) + 3(\underline{\hspace{1cm}}) = 3(\underline{\hspace{2cm}})$

Nested grouping: work from the inside out. $4[x + 2(x - 1)] = 4[\underline{\hspace{2cm}}] = \underline{\hspace{2cm}}$

Write two expressions equivalent to $4x + 10$ . At least one must use parentheses.

Expression 1: _______________ Expression 2: _______________
Factor $6x + 15$ : _______________
Factor $0.5x + 1.5$ : _______________
$2(3x - 1) - 4(x + 0.5) =$ _______________
$\tfrac{1}{4}(8x + 12) - 3 =$ _______________
Are $3(x - 2) + 5$ and $3x - 1$ equivalent?

Simplify $3(x - 2) + 5$ : _______________ Check at $x = 2$ : ______ and ______ Equivalent? ______

PAGE 20 — Factoring and simplifying

Practice · Factor and Simplify

Factor.

a) $8x + 12$	b) $-6x + 9$	c) $\tfrac{1}{2}x + \tfrac{3}{2}$	d) $0.3x - 1.2$

Write three expressions equivalent to $2x + 6$ , at least one with parentheses, and name the property for each.

Expression	Property

$5 - 2(x - 3.5) =$ _______________
$\tfrac{2}{3}(6x - 9) + \tfrac{1}{2}(4x + 2) =$ _______________
$4[x + 2(x - 1)] =$ _______________
$-0.5(6x - 4) + 3(x - 1) =$ _______________

PAGE 21 — Equivalence and error hunt

Is It Really Equivalent?

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

Is $2(x + 4.5)$ equivalent to $2x + 4.5$ ? ______

Why? _______________________________________________

$2(x + 4.5)$ is equivalent to: _______________
Explain. Why are $6x + 12$ and $6(x + 2)$ equivalent for every value of $x$ ?

Why would one test value not have been enough? Use $x + 3$ and $4x$ .
Apply it. A phone plan costs $x$ dollars per month plus $4.50 per month in fees, for four months.

With parentheses: _______________ Expanded: _______________ Another equivalent form: _______________
Find the error. A student simplifies $10 - 3(x + 2)$ to $7(x + 2) = 7x + 14$ .

What went wrong? _______________________________________________

Rule broken: ____________________ Correct answer: _______________

PAGE 22 — Exit ticket 4.5

Exit Ticket · Lesson 4.5

Name: ________________________ Date: ____________

Factor $0.4x + 1.6$ : _______________
$3[2x - (x + 1)] =$ _______________
Write an expression equivalent to $\tfrac{1}{2}x + 5$ that uses parentheses: _______________
Why does simplifying never change the value of an expression?

PAGE 23 — Chapter 4 review, part 1

Chapter 4 Review

Part A · Representing expressions with manipulatives and pictures

Five $x$ -tiles and three negative unit tiles. Expression: _______________
Describe a tile model for $-4x + 2$ : _______________________________________________
Describe a rectangular arrangement for $4(x + 3)$ : _______________________________

Expanded: _______________
Four $x$ -tiles, one negative $x$ -tile, two positive unit tiles, six negative unit tiles.

Simplified: _______________
Explain with zero pairs why $3x + 5 - 3x - 5 = 0$ . Property: ____________________
Model $\tfrac{1}{3}(6x + 9)$ by splitting into equal groups.

Groups: ______ One group holds: _______________ Result: _______________
Two identical rows, each with three $x$ -tiles and one unit tile.

Factored: _______________ Expanded: _______________
Why can $x$ -tiles and unit tiles never be combined into one count?

PAGE 24 — Chapter 4 review, part 2

Chapter 4 Review (continued)

Part B · Simplifying and generating equivalent expressions

Simplify.

a) $\tfrac{5}{8}x + \tfrac{1}{8}x$	b) $-3.6x + 1.6x$	c) $\tfrac{7}{10}x - \tfrac{1}{5}x$	d) $0.9x - 0.9x$

$6x - 2.5 - 4x + 4 =$ _______________
$\tfrac{1}{3}x + 7 - \tfrac{5}{6}x - 2 =$ _______________
$5(x - 1.2) + 2x =$ _______________
$-\tfrac{3}{4}(8x - 12) =$ _______________
$0.4(5x + 10) - (x - 3) =$ _______________
$2[3x - (x - 4)] =$ _______________
$\tfrac{2}{3}(9x + 6) - \tfrac{1}{2}(6x - 8) =$ _______________

PAGE 25 — Chapter 4 review, part 3

Chapter 4 Review (continued)

Factor.

a) $10x + 25$	b) $-8x + 4$	c) $1.5x + 4.5$

Write two expressions equivalent to $5x - 20$ , at least one factored.

Expression 1: _______________ Expression 2: _______________

Why is each equivalent? _______________________________________________
Name the property for each step.

a) $\tfrac{1}{2}(4x + 6) = 2x + 3$
b) $2x + 3 = 3 + 2x$
c) $(3 + 2x) + 0 = 3 + 2x$

$7 - 2(3x - 1.5) =$ _______________

PAGE 26 — Chapter 4 review, part 4

Chapter 4 Review (continued)

Part C · Application and reasoning

Simplify $4(x - 0.75) + 3$ : _______________

$x$	$4(x - 0.75) + 3$	$4x$
$2$
$-1$

Apply it. Three identical crates each hold $x$ kg of apples, and each crate weighs $2.5$ kg.

With parentheses: _______________ Expanded: _______________
Find the error. A student writes $-\tfrac{1}{2}(6x - 4) = -3x - 2$ .

What went wrong? _______________________________________________

Correct answer: _______________
Explain. Why is checking one replacement value not enough to prove equivalence? Use $x + 3$ and $4x$ .
Use a rectangular tile arrangement to justify that $3(x + 2) = 3x + 6$ .

TILE DRAWING SPACE: 2 in tall, full width — 3 rows of (1 x-tile + 2 unit tiles)

Reading by rows: _______________ Reading the whole rectangle: _______________
Write an expression with a fractional coefficient that simplifies to $2x - 1$ .

Expression: _______________ Simplification: _______________________________

Properties used: ____________________

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. Pages 4 and 26 need extra height because the arrangements are rectangles, not rows.
Color convention: blue = positive, red = negative, matching fig1-tile-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. fig6-combining-like-terms.png is the tallest and widest; give it its own block with no text wrapping beside it.
Fractions: set every fraction as a real stacked fraction, not as 1/2, so students read the coefficient correctly at a glance.
Answer blanks: keep every blank on the same line as its prompt so the Canva text box does not reflow.