Distributive Property Factoring

(−8x)÷4 = −2x; (−12)÷4 = −3

4(−2x) + 4(−3) = −8x − 12 ✓

Equals −8x − 12? [ ✓ ] Yes

Factored Form B (Negative Multiplier — GCF) Multiplier: −4

−4(2x + 3)

Work & Distribution Check:

(−8x)÷(−4) = 2x; (−12)÷(−4) = 3

(−4)(2x) + (−4)(3) = −8x − 12 ✓

Equals −8x − 12? [ ✓ ] Yes

*Also accept: \(2(-4x - 6)\) or \(-2(4x + 6)\) as valid alternate factored forms.

Expression 2: 15x − 25 Factors of 15 & 25: 1, 5

Factored Form A (GCF Multiplier) Multiplier: 5

5(3x − 5)

Work & Distribution Check:

(15x)÷5 = 3x; (−25)÷5 = −5

5(3x) + 5(−5) = 15x − 25 ✓

Equals 15x − 25? [ ✓ ] Yes

Factored Form B (Negative Multiplier) Multiplier: −5

−5(−3x + 5)

Work & Distribution Check:

(15x)÷(−5) = −3x; (−25)÷(−5) = 5

(−5)(−3x) + (−5)(5) = 15x − 25 ✓

Equals 15x − 25? [ ✓ ] Yes

*Also accept: \(1(15x - 25)\) or \(-1(-15x + 25)\) as valid alternate forms.

Part C: Think Deeply — Exemplar Reasoning & Rubric Exemplar Explanations

1. How can two different factored expressions represent the exact same expanded expression?

Exemplar Response: Two different factored expressions are equivalent because distributing the outside multiplier into each term produces the exact same expanded expression. Factoring out different common factors simply groups the terms into different equal-sized multiples, but the overall value never changes (e.g., \(2(6x+9) = 6(2x+3) = 12x+18\)).

2. Which of your factored expressions in Part A and Part B used the greatest common factor (GCF)? Explain how you know.

Exemplar Response: In Part A, 6(2x + 3) used the GCF because 6 is the largest integer that divides both 12 and 18, leaving terms inside \((2x + 3)\) whose coefficients share no common factor other than 1. In Part B, −4(2x + 3) (or \(4(-2x-3)\)) and 5(3x − 5) used the GCF because 4 and 5 are the greatest common divisors of the terms.

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Teacher Answer Key 7th Grade Math

Lesson 33 Exemplar & Facilitation Guide

Task 3: Factor and Expand Suggested Time: 18 Minutes

Learning Goal: I can factor an expression using the greatest common factor and verify my answer using the distributive property.

★ Two Essential Requirements for Fully Factored Expressions:

Requirement 1: The multiplier outside must have the greatest possible absolute value (GCF).

Requirement 2: The first term inside the parentheses must be positive (choose multiplier sign accordingly).

Part A: Guided Example — Factoring with Negatives (Completed) Target: −18x + 24

Steps 1 & 2: GCF [Modeled]

Coefficients: −18 and +24

Absolute values: \(|-18| = 18\) and \(|24| = 24\)

\(\text{GCF}(18, 24) = 6\)

Steps 3 & 4: Sign & Divide [Key]

Sign Rule: First term is −18x (negative), so choose −6 as multiplier.

Completed Division:

(−18x) ÷ (−6) = 3x

(+24) ÷ (−6) = −4

Steps 5 & 6: Factor & Verify [Key] Factored Form:

−6(3x − 4)

Distribute to verify:

(−6)(3x) + (−6)(−4)
= −18x + 24 ✓

Part B: Independent Practice — Factoring with GCF Exemplar Solutions

1. 12x + 20

Both terms positive → positive multiplier

|

\(\text{GCF}(|12|, |20|) =\) 4

Multiplier = +4

|

4(3x + 5)

First term positive inside? [ ✓ ] Yes

|

= 4(3x) + 4(5)

= 12x + 20 ✓

Check: = 12x + 20 [ ✓ ]

| |

2. −15x + 25

First term negative → negative multiplier!

|

\(\text{GCF}(|-15|, |25|) =\) 5

Multiplier = −5

|

−5(3x − 5)

First term positive inside? [ ✓ ] Yes

|

= (−5)(3x) + (−5)(−5)

= −15x + 25 ✓

Check: = −15x + 25 [ ✓ ]

|

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Teacher Answer Key Lesson 33: Distributive Property

Task 3 (Continued) Exemplar

Part B: Independent Practice (Continued) — Solutions Exemplar Solutions

3. −24x − 36

Both negative → negative multiplier!

|

\(\text{GCF}(|-24|, |-36|) =\) 12

Multiplier = −12

|

−12(2x + 3)

First term positive inside? [ ✓ ] Yes

|

= (−12)(2x) + (−12)(3)

= −24x − 36 ✓

Check: = −24x − 36 [ ✓ ]

| |

4. 18x − 30

First term positive → positive multiplier

|

\(\text{GCF}(|18|, |-30|) =\) 6

Multiplier = +6

|

6(3x − 5)

First term positive inside? [ ✓ ] Yes

|

= 6(3x) + 6(−5)

= 18x − 30 ✓

Check: = 18x − 30 [ ✓ ]

|

Independent Challenge — Exemplar Solutions Factor Completely

Exemplar Notes: Ensure students selected the full GCF for each problem (7 and 8) and applied the negative multiplier rule on problem 6.

5. 21x + 35 GCF: 7 | Multiplier: +7

Factored Expression 7(3x + 5)

Distribution & Verification Work:

= 7(3x) + 7(5)

= 21x + 35 ✓

Note: Inside terms (3x & 5) share no factors > 1.

Equals 21x + 35? [ ✓ ] Yes

6. −16x + 40 GCF: 8 | Multiplier: −8

Factored Expression −8(2x − 5)

Distribution & Verification Work:

= (−8)(2x) + (−8)(−5)

= −16x + 40 ✓

First inside term is +2x (positive as required).

Equals −16x + 40? [ ✓ ] Yes

Teacher Answer Key • Lesson 33 Page 4 “The only way to learn mathematics is to do mathematics.”

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Teacher Answer Key Lesson 33: Distributive Property

Task 3 Error Analysis Exemplar

Part C: Error Analysis — Exemplar & Misconception Guide Teacher Exemplar

A student was asked to factor −12x + 18 using the greatest possible absolute value for the multiplier while keeping the first term inside the parentheses positive.

Student's Incorrect Form 6(−2x + 3)

1. What mistake did the student make? Did they fail Requirement 1, Requirement 2, or both? Explain your reasoning.

Exemplar Critique: The student failed Requirement 2.

The student correctly identified the GCF of 12 and 18 as 6, so they satisfied Requirement 1 (\(|6| = 6\) is the greatest possible absolute value). However, they chose a positive multiplier (\(+6\)), leaving the first term inside as −2x, which is negative.

To satisfy Requirement 2 (making the first inside term positive), the student should have chosen a negative multiplier (−6) because \((\minus 12x) \div (\minus 6) = +2x\).

2. Determine the correct factored expression that follows BOTH requirements:

\(\text{GCF}(|-12|, |18|) =\) 6

Multiplier sign: Negative (−6)

Correct Factored Form −6(2x − 3)

3. Justify and verify your expression:

Verification (Show Distribution):

= (−6)(2x) + (−6)(−3)

= −12x + 18 ✓

Equals −12x + 18? [ ✓ ] Yes

Why it meets both requirements:

Requirement 1 Met: The multiplier has \(|-6| = 6\), the GCF of 12 and 18. Inside terms (2x and 3) share no common factor > 1.

Requirement 2 Met: The leading inside term is +2x, which is positive.

Teacher Answer Key • Lesson 33 Page 5 “The only way to learn mathematics is to do mathematics.”