Equivalent Expressions: Three Transformations

Lesson 3 of 5: Expressions and Equations

In this lesson:

• Expand expressions using the distributive property
• Factor expressions by finding the GCF
• Combine like terms to simplify

What You Will Learn Today

By the end of this lesson, you should be able to:

1. Expand expressions like using the distributive property
2. Factor expressions by identifying and pulling out the GCF
3. Combine like terms by adding or subtracting their coefficients

Same Cost, Two Different Forms

You buy 3 items, each priced at dollars.

• Option A: — factor times a sum
• Option B: — expanded terms

Try : does Option A equal Option B?

Pick a form and compute before advancing.

What Makes Two Expressions Equivalent

Equivalent expressions give the same value for every variable value.

Same output every time — these expressions are equivalent.

Area Model: Why Distribution Works

Total area = — the same as

Arrow Annotation: Count Every Term

• One arrow per inside term — one product per arrow
• Two terms inside → two arrows → two products
• Draw arrows before computing — every time

Expanding: Two and Three Terms

:

Verify at : ; ✓

— three terms, three arrows:

Your Turn: Expand and Verify

Expand .

1. Draw one arrow from 4 to each term inside
2. Write the product each arrow represents
3. Write the expanded expression
4. Verify at : does equal your answer?

Draw your arrows before writing any numbers.

From Expanded Back to Factored

You expanded:

Now the question flips:

Starting from — what was the factor outside?

The same operation runs in reverse.

Factoring Means Finding the GCF

Factoring reverses distribution — pull out the GCF of all terms.

Factor :

• Factors of 24 and 18 share: 1, 2, 3, 6
• GCF = 6

Inside-check: do 4x and 3y share a factor? No — done.

Watch Out: Use the Full GCF

, student pulls out 2:

Inside-check: and share factor 2 — not done!

3 and 2 share no factor — complete.

Your Turn: Factor and Check

Factor .

1. List factors of 15 and factors of 10
2. Identify the GCF
3. Write the factored form
4. Inside-check: do the terms inside share any factor?

Verify by expanding back — does it give ?

Like Terms Share the Same Variable

Like terms share the same variable and exponent.

• and — like (same variable )
• and — like (same variable )
• and — unlike (different variables)

Add the coefficients — the variable part stays unchanged.

Addition vs. Multiplication of Variables

• — addition counts copies → coefficient
• — multiplication stacks copies → exponent

Ask first: addition or multiplication?

Combining Terms with Two Variable Groups

Combine :

• -terms:
• -terms:
• Result:

Can we combine and ? No — different variables.

Apples and oranges can't be added together.

Three Groups: Variables and Constants

Combine :

• -terms:
• Constants:
• Result:

Constants are like terms with each other — numbers without variables form their own group.

Different Variables Cannot Be Combined

You see . A student writes . Is that right?

• has variable ; the has no variable
• Unlike terms — they cannot combine
• is already fully simplified

Name each term's variable before combining — stop if they differ.

Expand, Then Combine: Two Steps

Simplify :

Step 1 — Expand (Chunk 1):

Step 2 — Combine like terms (Chunk 3):

Two operations from today, applied in sequence.

Your Turn: Expand Then Combine Terms

Simplify .

1. Step 1: Expand — draw your arrows first
2. Step 2: Identify the like terms in the full expression
3. Step 3: Combine them

Name each step as you go: "I'm expanding" then "I'm combining."

Three Tools for Equivalent Expressions

✓ Expand: outside factor reaches every inside term

✓ Factor: pull out GCF; inside-check confirms done

✓ Combine: add coefficients of like terms only

Every term needs its own arrow

Inside-check — non-GCF leaves work unfinished

, not

Next Lesson: Identifying Equivalent Expressions

You can now produce equivalent expressions — you know how to get from one form to another.

The next lesson asks: given two expressions someone else wrote, how do you identify whether they're equivalent?

The substitution check you practiced today is your starting tool.