Introduction
Expanding Expressions with the Distributive Property 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 expanding expressions with the distributive property.
What Is Expanding Expressions with the Distributive Property?
Expanding Expressions with the Distributive Property 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 Expanding Expressions with the Distributive Property
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: The rectangle below is divided into two regions to model \(5(x+2)\). What is the total area?
- A. \(5x+10\)
- B. \(5x+2\)
- C. \(7x\)
- D. \(x+7\)
Why it works: The rectangle shows two parts: \(5x\) and \(10\). Total: \(5x+10\).
Answer: \(5x+10\)
Visual Model 2
Question: The rectangle below models \(4(a+2)\). What is the total area expressed as a sum?
- A. \(6a\)
- B. \(4a+2\)
- C. \(4a+8\)
- D. \(a+8\)
Why it works: The area model shows \(4a\) (left) and \(8\) (right). Sum: \(4a+8\).
Answer: \(4a+8\)
Worked Examples
Example 1
Question: A rectangular room has length \((2w+3)\) meters and width \(4\) meters. The diagram shows the area split into parts. What is the total area in expanded form?
- A. \(8w+12\)
- B. \(8w+3\)
- C. \(2w+12\)
- D. \(6w+7\)
- Area \(= 4(2w+3)=8w+12\).
- The model shows \(4 \times 2w=8w\) and \(4 \times 3=12\).
Answer: \(8w+12\)
Example 2
Question: A student expanded \(-2(x-3)\) as \(-2x-6\). The diagram below shows the correct distribution. What is the correct answer?
- A. \(-2x+6\)
- B. \(-2x-6\)
- C. \(2x-6\)
- D. \(2x+6\)
- Correct: \(-2(x)-2(-3)=-2x+6\).
- Student forgot that negative times negative is positive.
Answer: \(-2x+6\)
Example 3
Question: Which expression is equivalent to \(2(3x+4)\)? The diagram shows the split:
- A. \(6x+4\)
- B. \(6x+8\)
- C. \(5x+6\)
- D. \(3x+8\)
- Distribute \(2\): \(2(3x)+2(4)=6x+8\).
- The area model shows the two parts: \(6x\) and \(8\).
Answer: \(6x+8\)
Real-World Word Problems
Problem 1
Question: A store marks up the cost \(c\) by \($5\), then multiplies by \(3\) for sale price. Which expression shows the final price?
- A. \(3c+5\)
- B. \(3c+15\)
- C. \(3c-5\)
- D. \(c+15\)
Why it works: First add the markup: \((c+5)\). Then multiply by \(3\): \(3(c+5)=3c+15\). Distribute to both terms.
Answer: \(3c+15\)
Problem 2
Question: A student expanded \(3(2x+5)\) and got \(6x+5\). What error did the student make?
- A. Forgot to distribute to the constant term.
- B. Distributed incorrectly to the variable term.
- C. Used addition instead of multiplication.
- D. The answer is correct.
Why it works: The student skipped distributing \(3\) to the \(5\), a common error.
Answer: Correct expansion: \(3(2x)+3(5)=6x+15\). Student only distributed to the first term.
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
Expand: \(4(2x-3)\)
- A. \(6x-12\)
- B. \(8x-12\)
- C. \(8x-3\)
- D. \(6x-7\)
Question 2
Expand: \(3(x+5)\)
- A. \(3x+5\)
- B. \(3x+8\)
- C. \(3x+15\)
- D. \(x+15\)
Question 3
Expand: \(-2(x+4)\)
- A. \(-2x-8\)
- B. \(-2x+8\)
- C. \(2x-8\)
- D. \(-2x+4\)
Question 4
Expand: \(5(3y-2)\)
- A. \(15y-2\)
- B. \(8y-7\)
- C. \(15y-10\)
- D. \(3y-10\)
Question 5
Expand: \(-3(2a-5)\)
- A. \(-6a-15\)
- B. \(-6a+15\)
- C. \(6a-15\)
- D. \(-6a-5\)
Question 6
Expand: \(\frac{1}{2}(4x+6)\)
- A. \(2x+3\)
- B. \(4x+6\)
- C. \(2x+6\)
- D. \(x+3\)
Full Answer Explanations Click to show all answers and explanations
Question 1
Answer: \(8x-12\)
Distribute \(4\) to both terms: \(4(2x)-4(3)=8x-12\). (Positive times negative gives negative.)
Question 2
Answer: \(3x+15\)
Distribute \(3\) to both terms: \(3(x)+3(5)=3x+15\).
Question 3
Answer: \(-2x-8\)
Distribute \(-2\): \(-2(x)-2(4)=-2x-8\). Negative times positive is negative.
Question 4
Answer: \(15y-10\)
Distribute \(5\): \(5(3y)-5(2)=15y-10\).
Question 5
Answer: \(-6a+15\)
Distribute \(-3\) to both terms: \(-3(2a)=-6a\) and \(-3(-5)=+15\) (negative times negative is positive). Result: \(-6a+15\).
Question 6
Answer: \(2x+3\)
Distribute \(\frac{1}{2}\): \(\frac{1}{2}(4x)+\frac{1}{2}(6)=2x+3\).
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
Expanding Expressions with the Distributive Property 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.
