Introduction
The Distributive Property with Common Factors is an important Grade 6 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 the distributive property with common factors.
What Is The Distributive Property with Common Factors?
The Distributive Property with Common Factors 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 The Distributive Property with Common Factors
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 area model below shows a rectangle divided into two parts. Which expression does it represent?
- A. \(3(2+5)\)
- B. \(2(3+5)\)
- C. \(5(3+2)\)
- D. \((3+2)(3+5)\)
Why it works: The rectangle has height \(3\) and is divided horizontally into two widths: \(2\) and \(5\). The total area is \(3 \times 2 + 3 \times 5 = 3(2+5)\).
Answer: \(3(2+5)\)
Visual Model 2
Question: The table below shows pairs of expressions. Which pair represents the same value?
| Expression 1 | Expression 2 | |
|---|---|---|
| A. | \(4(5+7)\) | \(20+27\) |
| B. | \(6(3+7)\) | \(18+42\) |
| C. | \(3(9+5)\) | \(27+10\) |
| D. | \(2(11+6)\) | \(22+11\) |
- A. Choice A
- B. Choice B
- C. Choice C
- D. Choice D
Why it works: A adds to \(47\) vs \(48\) on the expanded side---close call but not equal. B matches term by term (\(18+42\)). C wrongly lists \(27+10\) instead of \(27+15\), and D shortchanges \(2\times 6\).
Answer: Choice B: \(6(3+7) = 18+42\)
Worked Examples
Example 1
Question: The area model below represents a factored sum. What is the factored form?
- A. \(5(4+6) = 50\)
- B. \(4(5+6) = 44\)
- C. \((4+6)(5) = 50\)
- D. \(5 \times 10 = 50\)
- The rectangle shows a common factor of \(5\) with widths \(4\) and \(6\).
- Factor first: \(5(4+6)\) totals \(50\).
- Choice B flips roles (height \(5\) splits differently), while D skips the inner sum.
Answer: \(5(4+6) = 50\)
Example 2
Question: An area model is shown below. Which factored expression does it represent?
- A. \(6(3+8) = 66\)
- B. \(3(6+8) = 42\)
- C. \((6+3)(6+8) = 154\)
- D. \(6(3 \times 8) = 144\)
- The rectangle shows a common factor of \(6\) with addends \(3\) and \(8\).
- The area is \(6 \times 3 + 6 \times 8 = 18 + 48 = 66 = 6(3+8)\).
- Choice A is correct.
Answer: \(6(3+8) = 66\)
Example 3
Question: Which sum matches the factored form \(6(7+4)\)? Look at the area model.
- A. \(42+24\)
- B. \(42+4\)
- C. \(6+7+6+4\)
- D. \(40+26\)
- \(6(7+4) = 6 \cdot 7 + 6 \cdot 4 = 42 + 24\).
- The area model shows the two parts: height \(6\) with widths \(7\) and \(4\).
- Only Choice A shows the correct expansion.
Answer: \(42+24\)
Real-World Word Problems
Problem 1
Question: A soccer coach is buying uniforms. She buys \(12\) red shirts and \(18\) blue shirts for the team. If she wants to divide them equally into piles with no shirts left over, what is the greatest number of piles she can make?
- A. \(2\)
- B. \(3\)
- C. \(6\)
- D. \(9\)
Why it works: Find GCF\((12, 18)\). Factors of \(12\): \(1, 2, 3, 4, 6, 12\). Factors of \(18\): \(1, 2, 3, 6, 9, 18\). GCF is \(6\). She can make \(6\) piles with \(2\) red shirts and \(3\) blue shirts each.
Answer: \(6\)
Problem 2
Question: A student tries to use the area model below to find \(5(3+8)\) but makes an error. The student writes: \(5(3+8) = 15 + 8 = 23\). What is the error?
- A. \(5\) was only distributed to the first term
- B. The sum \(3+8\) should equal \(11\), not both terms
- C. \(5(3+8)\) should simplify to \(55\), not \(23\)
- D. Choices A and C are both correct
Why it works: The student only multiplied \(5 \times 3=15\) instead of distributing to both terms. The correct solution is \(5(3+8) = 5 \cdot 3 + 5 \cdot 8 = 15 + 40 = 55\). The area model shows both parts, confirming both must be distributed to.
Answer: Choices A and C are both correct
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 \(18+24\) using the distributive property?
- A. \(6(3+4)\)
- B. \(6(3+24)\)
- C. \(2(9+24)\)
- D. \(3(6+8)\)
Question 2
Factor \(12+20\) using the distributive property.
- A. \(4(3+5)\)
- B. \(2(6+10)\)
- C. \(3(4+20)\)
- D. \(5(2+4)\)
Question 3
Which shows \(15+10\) factored using the GCF?
- A. \(3(5+10)\)
- B. \(5(3+2)\)
- C. \(2(8+5)\)
- D. \(10(2+1)\)
Question 4
What is the GCF of \(21\) and \(28\)?
- A. \(2\)
- B. \(4\)
- C. \(7\)
- D. \(14\)
Question 5
Which expression is equivalent to \(30+45\)?
- A. \(5(6+9)\)
- B. \(15(2+3)\)
- C. \(3(10+15)\)
- D. \(6(5+7.5)\)
Question 6
Expand \(7(4+6)\).
- A. \(28+42\)
- B. \(28+6\)
- C. \(4+42\)
- D. \(11 \times 7\)
Full Answer Explanations Click to show all answers and explanations
Question 1
Answer: \(6(3+4)\)
Peek for the biggest factor both sums share evenly: that's \(6\). Pull it out softly: \(18+24=6\cdot3+6\cdot4=6(3+4)\)---the distributive property read backward.
Question 2
Answer: \(4(3+5)\)
The GCF of \(12\) and \(20\) is \(4\). So \(12+20 = 4(3+5)\). Choice B uses GCF \(2\), which is correct but not the greatest common factor. Choice A uses the greatest common factor.
Question 3
Answer: \(5(3+2)\)
GCF of \(15\) and \(10\) is \(5\). Thus \(15+10=5(3+2)\). Choice A uses factor \(3\) (not the greatest); Choice C uses \(2\) (not a common factor of both); Choice D uses \(10\) but yields \(30\), not \(25\).
Question 4
Answer: \(7\)
Factors of \(21\): \(1, 3, 7, 21\). Factors of \(28\): \(1, 2, 4, 7, 14, 28\). Common factors: \(1, 7\). The GCF is \(7\).
Question 5
Answer: \(15(2+3)\)
GCF of \(30\) and \(45\) is \(15\). So \(30+45 = 15(2+3)\). Choice A uses GCF \(5\), which works but is not the greatest. Choice B is correct.
Question 6
Answer: \(28+42\)
Using the distributive property: \(7(4+6) = 7 \cdot 4 + 7 \cdot 6 = 28 + 42 = 70\).
Connection to Standards
This lesson supports Grade 6 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
The Distributive Property with Common Factors 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.
