A focused Grade 6 learning session for The Distributive Property with Common Factors using visual models, worked examples, strategy guidance, and a short mastery check.
Student goal: I can solve problems involving 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.
Question: The area model below shows a rectangle divided into two parts. Which expression does it represent?
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: \(3(2+5)\)
Question: The area model below represents a factored sum. What is the factored form?
Answer: \(5(4+6) = 50\)
Question: An area model is shown below. Which factored expression does it represent?
Answer: \(6(3+8) = 66\)
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?
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: \(6\)
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.
Understand the model before choosing the operation.
Use scratch paper. The explanation matters more than simply seeing whether the choice is correct.
Which expression is equivalent to \(18+24\) using the distributive property?
Answer A: \(6(3+4)\)
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.
Factor \(12+20\) using the distributive property.
Answer A: \(4(3+5)\)
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.
Which shows \(15+10\) factored using the GCF?
Answer B: \(5(3+2)\)
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 \(25\).
What is the GCF of \(21\) and \(28\)?
Answer C: \(7\)
Factors of \(21\): \(1, 3, 7, 21\). Factors of \(28\): \(1, 2, 4, 7, 14, 28\). Common factors: \(1, 7\). The GCF is \(7\).
Which expression is equivalent to \(30+45\)?
Answer B: \(15(2+3)\)
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.
Expand \(7(4+6)\).
Answer A: \(28+42\)
Using the distributive property: \(7(4+6) = 7 \cdot 4 + 7 \cdot 6 = 28 + 42 = 70\).
Which two expressions are equivalent?
Answer A: \(8(5+3)\) and \(40+24\)
\(8(5+3) = 8 \cdot 5 + 8 \cdot 3 = 40 + 24\). Choice B gives \(8+8=16\), so it is not equivalent.
Factor \(16+32\) using the GCF.
Answer D: \(16(1+2)\)
The greatest common factor of \(16\) and \(32\) is \(16\). Factoring out \(16\) gives \(16+32=16(1+2)\).
Expand \(6(2+9)\).
Answer A: \(12+54\)
Distributive property: \(6(2+9) = 6 \cdot 2 + 6 \cdot 9 = 12 + 54 = 66\). Choice A shows the expanded form before simplifying.
Which expression is NOT equivalent to \(24+36\)?
Answer D: \(3(7+12)\)
\(24+36 = 60\). Choice A: \(6(4+6)=60\) ✓. Choice B: \(12(2+3)=60\) ✓. Choice C: \(4(6+9)=60\) ✓. Choice D: \(3(7+12)=57\) ✗. Only Choice D is not equivalent to \(60\).
The area model below shows a rectangle divided into two parts. Which expression does it represent?
Answer A: \(3(2+5)\)
The rectangle has height \(3\) and is divided horizontally into two widths: \(2\) and \(5\). The total area is \(3 \times 2 + 3 \times 5 = 3(2+5)\).
