Introduction

Rewriting Expressions to Solve Problems is an important Grade 7 math skill because students are moving from simple answers toward explaining how the math works.

In this lesson, students use models, real questions, worked examples, practice problems, and two online quizzes to build confidence with rewriting expressions to solve problems.

What Is Rewriting Expressions to Solve Problems?

Rewriting Expressions to Solve Problems means choosing a model, naming what each number means, and explaining the strategy.

The goal is not only to get the answer. Students should be able to show the idea, explain the strategy, and check whether the answer makes sense.

Understanding Rewriting Expressions to Solve Problems

Before solving, students should slow down and decide what each number, shape, unit, or label represents.

  • Read the question carefully and identify what is being asked.
  • Choose a model, equation, table, or diagram that matches the situation.
  • Solve one step at a time and keep units or labels attached.
  • Use the answer explanation to check that the result makes sense.

Visual Models

Visual Model 1

Question: Which expression correctly shows the relationship from the diagram?

Visual Model 1

  • A. \(0.9p = p - 0.1p\)
  • B. \(0.9p = p + 0.1p\)
  • C. \(0.9p = 1.1p\)
  • D. \(0.9p = 0.1p\)

Why it works: The diagram shows a discount reduces price from \(p\) to \(0.9p\). This means \(0.9p = p - 0.1p\), factoring as \(p(1-0.1)\).

Answer: \(0.9p = p - 0.1p\)

Visual Model 2

Question: Using the percent bar above, which expression represents the total cost (original price \(p\) plus tax)?

Visual Model 2

  • A. \(1.08p\)
  • B. \(0.92p\)
  • C. \(p + 0.08\)
  • D. \(p - 0.08p\)

Why it works: The percent bar shows \(100\% + 8\% = 108\%\) total. This is \(p + 0.08p = 1.08p\).

Answer: \(1.08p\)

Worked Examples

Example 1

Question: What expression goes in the box marked ??

Example 1

  • A. \(3 + 2x + 5\)
  • B. \(6x + 15\)
  • C. \(5x + 8\)
  • D. \(2x + 5 + 3\)
  1. Expanding \(3(2x+5)\) gives \(3 \cdot 2x + 3 \cdot 5 = 6x + 15\).

Answer: \(6x+15\)

Example 2

Question: Using the equivalence table, if both forms equal \($60\), what is \(x\)?

Example 2

  • A. \(x = 45\)
  • B. \(x = 75\)
  • C. \(x = 80\)
  • D. \(x = 100\)
  1. \(0.75x = 60\), so \(x = 60 \div 0.75 = 80\).
  2. Check: \(80-0.25(80) = 60\). \checkmark

Answer: \(x = 80\)

Example 3

Question: Which property explains why \(5x-10 = 5(x-2)\)?

Example 3

  • A. Associative property of multiplication
  • B. Distributive property (factoring)
  • C. Commutative property
  • D. Combining like terms
  1. Factoring uses the distributive property: \(5(x-2) = 5 \cdot x - 5 \cdot 2 = 5x - 10\).
  2. The property works both ways (distribute or factor).

Answer: Distributive property (factoring)

Real-World Word Problems

Problem 1

Question: A shirt costs \($60\). With a \(15\%\) discount, which expression represents the final price?

  • A. \(0.15 \times 60\)
  • B. \(0.85 \times 60\)
  • C. \(1.15 \times 60\)
  • D. \(60 - 15\)

Why it works: A \(15\%\) discount removes \(15\%\) from the original price. You pay the remaining \(100\%-15\%=85\%\). The multiplier \(0.85\) scales the price by this remaining percent: \(0.85 \times 60=51\).

Answer: \(0.85 \times 60\)

Problem 2

Question: A book costs \($25\). With sales tax of \(6\%\), what is the total cost?

  • A. \(25 + 6\)
  • B. \(25 \times 0.06\)
  • C. \(25 \times 1.06\)
  • D. \(25 - 0.06\)

Why it works: With a \(6\%\) tax, you pay \(100\%+6\%=106\%\) of the original price: \(25 \times 1.06\).

Answer: \(25 \times 1.06\)

Common Mistakes

  • Rushing before identifying what the numbers represent.
  • Choosing an operation that does not match the situation.
  • Dropping labels, units, or context from the answer.
  • Skipping the estimate or reasonableness check.

Strategy Tips

  • Underline the question being asked.
  • Use a model before jumping to computation.
  • Write an equation that matches the story or picture.
  • Explain the final answer in a sentence.

Practice Questions

Question 1

Which expression is equivalent to \(0.95p\)?

  • A. \(p-0.95p\)
  • B. \(p-0.05p\)
  • C. \(p+0.05p\)
  • D. \(0.95+p\)

Question 2

Which expression is equivalent to \(1.08m\)?

  • A. \(m+0.08m\)
  • B. \(m-0.08m\)
  • C. \(1.08+m\)
  • D. \(0.92m\)

Question 3

Use the distributive property to simplify \(3(x+5)-2x\).

  • A. \(3x+15-2x=x+15\)
  • B. \(x+5\)
  • C. \(3x+5-2x\)
  • D. \(3x+15\)

Question 4

Which expression is equivalent to \(2(4y-1)+3y\)?

  • A. \(8y-2+3y=11y-2\)
  • B. \(8y+1\)
  • C. \(11y-1\)
  • D. \(4y+2\)

Question 5

A pair of jeans costs \($80\). After a \(20\%\) discount, express the final price in factored form.

  • A. \(80(1-0.2)\)
  • B. \(80(0.2)\)
  • C. \(0.8(80)\)
  • D. \(80 \times 20\)

Question 6

Simplify: \(5(2a+3)-(a+6)\).

  • A. \(10a+15-a-6=9a+9\)
  • B. \(10a+9\)
  • C. \(11a-6\)
  • D. \(10a+3\)
Full Answer Explanations Click to show all answers and explanations

Question 1

Answer: \(p-0.05p\)

The decimal \(0.95\) means keeping \(95\%\) of the price (removing \(5\%\)). This is the same as \(p-0.05p=(1-0.05)p=0.95p\). Both forms show a \(5\%\) discount: one expanded (subtract the discount), one simplified (multiply by the remaining scale factor).

Question 2

Answer: \(m+0.08m\)

\(m+0.08m=(1+0.08)m=1.08m\). An \(8\%\) tax adds \(8\%\) to the original price.

Question 3

Answer: \(x+15\)

Distribute: \(3(x+5)-2x=3x+15-2x\). Combine like terms: \(3x-2x+15=x+15\).

Question 4

Answer: \(11y-2\)

Distribute \(2\): \(2(4y)-2(1)+3y=8y-2+3y\). Combine: \(8y+3y-2=11y-2\).

Question 5

Answer: \(80(1-0.2)\)

A \(20\%\) discount means you keep \(100\%-20\%=80\%\) of the price. The factored form \(80(1-0.2)\) emphasizes the scale factor \((1-0.2)\), which equals \(0.8\). This form reveals the structure of the discount.

Question 6

Answer: \(9a+9\)

Distribute: \(5(2a)+5(3)-(a+6)=10a+15-a-6\). Combine: \((10a-a)+(15-6)=9a+9\).

Timed practice quizzes

Related Quizzes

After studying Rewriting Expressions to Solve Problems, open these two timed quizzes in a new tab for focused extra practice.

Connection to Standards

This lesson supports Grade 7 math expectations for reasoning, modeling, problem solving, and explaining answers clearly. It connects classroom skills to the kind of questions students see on state math assessments.

Summary

Rewriting Expressions to Solve Problems becomes easier when students connect the question to a model, use clear steps, and explain why the answer fits.

GOLDEN RULE

Understand the model before choosing the operation.