Rewriting Expressions in Equivalent Forms

Grade 7 Mathematics — 7.EE.A.2

In this lesson:

• Rewrite expressions in different but equal forms
• Decode what each form reveals about a situation
• Connect percent language to multiplication expressions

Learning Objectives for This Lesson

By the end, you should be able to:

1. Rewrite an expression in an equivalent form
2. Explain what each form reveals about a situation
3. Connect percent language to expressions like
4. Recognize that equivalent forms are exactly equal
5. Choose the most useful form for the task

The Shirt and Tax Problem Setup

A shirt costs $a. The store adds 5% sales tax.

What expression represents the total price you pay?

Think for a moment before the next slide...

Two Forms of the Same Expression

Both expressions give the same total for any value of .

What Each Form Reveals to Us

Expanded form:

• Shows the two-part structure: original + tax
• Makes the reason for the calculation visible

Factored form:

• Shows the single multiplier: multiply by 1.05
• One operation — faster for computation

Verifying Both Expressions Give Same Value

Let (a $40 shirt):

Note: 5% of means , not — the percent becomes a decimal coefficient.

Equivalent Expressions Are Exactly Equal

Definition: Two expressions are equivalent if they give the same value for every value of the variable.

• for every value of ✓
• Not an approximation — an algebraic identity
• Simplifying does not round off or lose precision

Quick Check: Rewrite as Single-Multiplier Expression

Write as a single-multiplier expression.

Hint: Use the same reasoning as

What does your answer tell you in one sentence?

From One Example to a Pattern

The shirt-and-tax example is just one case of a general pattern.

Every time a value changes by a percent of itself:

We can build a reference table to capture all the cases — increases and decreases.

Row 1: Sales Tax at 8%

Situation: Price with 8% sales tax added

Multiplier greater than 1 → result is bigger than original ✓

Row 2: Discount at 20%

Situation: Price with 20% discount removed

Decrease → multiplier is less than 1 (0.80, not 1.20)

Row 3: Tip at 15%

Situation: Bill with 15% tip added

A 15% tip means the total is 115% of the original bill.

Row 4: Value Decrease at 6%

Situation: Value decreases by 6%

A 6% decrease means keeping 94% of the original value.

Complete Pattern Table for Reference

Pattern: Increase → multiplier = | Decrease → multiplier =

Quick Check: Reading the Coefficient

A coat is priced at after a markdown.

1. What percent markdown is this?
2. Write the expanded form that shows the reduction explicitly.

The multiplier tells you everything — decode it.

Form Choice: Context Determines Utility

Same value — different forms — different uses.

Neither form is always best — the task determines the choice.

Worked Comparison: Two Methods Solve Same Problem

A coat costs $120 with a 15% markdown. What is the sale price?

Method A (Expanded):

Method B (Factored):

Both give $102. Method B is faster for computation.

Your Turn: Choose Your Form

Write an expression and state which form you prefer:

1. Laptop: $800 with a 10% sale discount.
2. Phone bill: plus 8% taxes and fees. Total?
3. Population: 5,000 decreasing by 3% per year. Year-1 total?

For each, write one sentence explaining your form choice.

Form Choice Answers Reviewed and Explained

Problem 1: — factored; one step, mental-math friendly

Problem 2: — both valid; expanded shows structure, factored computes total

Problem 3: — factored; 0.97 means keeping 97% each year

Reversing Direction: From Expanding to Factoring

So far: factored → expanded (distributing).

Now the reverse: expanded → factored (factoring).

You already know:

So backward: $5x + 10 = $ ___?

Factoring as the Reverse of Distribution

• Distributing:
• Factoring:

Both directions are equally valid — choose the direction that serves your goal.

Factoring Example: Worked Step by Step

Factor :

Step 1: GCF of 6 and 9 → 3

Step 2: Divide each term: ;

Step 3:

Verify: ✓

Factoring Back to Percent Expressions

Connect back to percent context:

The GCF is — factor it out, then simplify inside the parentheses.

Factoring out (or ) is exactly what we've been doing all lesson.

Your Turn: Factor Five Expressions

Factor each expression. State the GCF and write the factored form.

Quick Check: Factor and Verify

Factor .

Then verify your answer by distributing back.

What is the GCF? What is the factored form?

Advance to the next slide to check your work.

Common Factoring Error: Analyze It

Factor 2 was applied to only — not to the 9.

The GCF must divide every term inside.

Key Takeaways from Today's Lesson

✓ Equivalent expressions are exactly equal — not approximations

✓ Expanded form shows structure; factored form speeds computation

✓ Increase → multiplier above 1; decrease → multiplier below 1

✓ Factoring is the reverse of distributing — verify by distributing back

5% of is — not
20% decrease → , not
The GCF must divide every term — not just the first

What's Coming Next in 7.EE

Next: solving equations with rational coefficients.

• Use equivalent forms to simplify before solving
• Choose the form that makes solving easiest
• Apply the same "both directions" thinking to equations

Moving between forms — today's skill — is your main tool.