Grade 7 · Expressions & Equations · MAT-07-EE-002
Use the Distributive Property with Signed Numbers
Represent and solve use the distributive property with signed numbers problems using precise mathematical notation, models, and reasoning.
Learning objectives
What you should be able to do
- Represent and solve use the distributive property with signed numbers problems using precise mathematical notation, models, and reasoning.
- Explain why a method for use the distributive property with signed numbers works, interpret the result in context, and verify it independently.
Prerequisite check
Make sure the foundation is ready.
The new lesson depends on this prerequisite as active knowledge.
Learn
Build the idea from meaning, not memorization.
Mathematical meaning
Use the Distributive Property with Signed Numbers uses symbols to describe relationships. Equality and inequality are relationships that must be preserved on both sides.
Represent the relationship
The Math Box uses algebra tiles to expose the structure. Change one input, predict the result, then connect the visual change to an equation, table, graph, number line, or geometric model.
Calculate, interpret, and verify
Verify algebra by substituting the proposed solution into the original statement; equivalent expressions must agree for the same input.
Math Box · Interactive lesson
Touch the math. Change it. See what stays true.
Use the interactive model before and after the worked examples. Change the inputs, make a prediction, then use the model to test whether your reasoning holds.
Algebra Tiles Lab
Model a linear expression with x-tiles and unit tiles, then evaluate it.
- Model one example of use the distributive property with signed numbers in the Math Box.
- Change one input, predict the effect, and test the prediction.
- State the relationship or invariant that explains what stayed mathematically consistent.
Worked examples
Twenty different ways to see the concept work.
Expand -2(x + 3).
-2(x + 3) = -2x + -6Distribute the signed factor to every term, including the sign.
Expand 3(x + 5).
3(x + 5) = 3x + 15Distribute the signed factor to every term, including the sign.
Expand -4(x + 7).
-4(x + 7) = -4x + -28Distribute the signed factor to every term, including the sign.
Expand 5(x + 9).
5(x + 9) = 5x + 45Distribute the signed factor to every term, including the sign.
Expand -6(x + 3).
-6(x + 3) = -6x + -18Distribute the signed factor to every term, including the sign.
Expand 7(x + 5).
7(x + 5) = 7x + 35Distribute the signed factor to every term, including the sign.
Expand -8(x + 7).
-8(x + 7) = -8x + -56Distribute the signed factor to every term, including the sign.
Expand 2(x + 9).
2(x + 9) = 2x + 18Distribute the signed factor to every term, including the sign.
Expand -3(x + 3).
-3(x + 3) = -3x + -9Distribute the signed factor to every term, including the sign.
Expand 4(x + 5).
4(x + 5) = 4x + 20Distribute the signed factor to every term, including the sign.
Expand -5(x + 7).
-5(x + 7) = -5x + -35Distribute the signed factor to every term, including the sign.
Expand 6(x + 9).
6(x + 9) = 6x + 54Distribute the signed factor to every term, including the sign.
Expand -7(x + 3).
-7(x + 3) = -7x + -21Distribute the signed factor to every term, including the sign.
Expand 8(x + 5).
8(x + 5) = 8x + 40Distribute the signed factor to every term, including the sign.
Expand -2(x + 7).
-2(x + 7) = -2x + -14Distribute the signed factor to every term, including the sign.
Expand 3(x + 9).
3(x + 9) = 3x + 27Distribute the signed factor to every term, including the sign.
Expand -4(x + 3).
-4(x + 3) = -4x + -12Distribute the signed factor to every term, including the sign.
Expand 5(x + 5).
5(x + 5) = 5x + 25Distribute the signed factor to every term, including the sign.
Expand -6(x + 7).
-6(x + 7) = -6x + -42Distribute the signed factor to every term, including the sign.
Expand 7(x + 9).
7(x + 9) = 7x + 63Distribute the signed factor to every term, including the sign.
Practice
Guided practice with hints
01Expand -8(x + 3).
Hint: Represent the relationship first, keep units visible, and identify a second verification method.
Answer: -8(x + 3) = -8x + -24
02Expand 2(x + 5).
Hint: Represent the relationship first, keep units visible, and identify a second verification method.
Answer: 2(x + 5) = 2x + 10
03Expand -3(x + 7).
Hint: Represent the relationship first, keep units visible, and identify a second verification method.
Answer: -3(x + 7) = -3x + -21
04Expand 4(x + 9).
Hint: Represent the relationship first, keep units visible, and identify a second verification method.
Answer: 4(x + 9) = 4x + 36
05Expand -5(x + 3).
Hint: Represent the relationship first, keep units visible, and identify a second verification method.
Answer: -5(x + 3) = -5x + -15
06Expand 6(x + 5).
Hint: Represent the relationship first, keep units visible, and identify a second verification method.
Answer: 6(x + 5) = 6x + 30
Practice
Independent practice
01Expand -7(x + 7).
Answer: -7(x + 7) = -7x + -49
02Expand 8(x + 9).
Answer: 8(x + 9) = 8x + 72
03Expand -2(x + 3).
Answer: -2(x + 3) = -2x + -6
04Expand 3(x + 5).
Answer: 3(x + 5) = 3x + 15
05Expand -4(x + 7).
Answer: -4(x + 7) = -4x + -28
06Expand 5(x + 9).
Answer: 5(x + 9) = 5x + 45
07Expand -6(x + 3).
Answer: -6(x + 3) = -6x + -18
08Expand 7(x + 5).
Answer: 7(x + 5) = 7x + 35
Common mistakes
Learn to catch the error, not just the answer.
Use equivalent operations on both sides.
Only terms with the same variable part are like terms.
Applications & challenge
Use the idea beyond a single exercise.
Applications
- Translate cost, distance, rate, and measurement situations into symbolic rules.
- Use equations and inequalities to determine unknown quantities, thresholds, and feasible ranges.
Challenge problems
Expand -8(x + 7).
-8(x + 7) = -8x + -56
Expand 2(x + 9).
2(x + 9) = 2x + 18
Expand -3(x + 3).
-3(x + 3) = -3x + -9
Expand 4(x + 5).
4(x + 5) = 4x + 20
Flashcards
Retrieve it from memory.
Flashcards
Card 1 of 4
Game mode
Grade 7 Reasoning Sprint
Solve use the distributive property with signed numbers problems accurately and justify each result with a valid check.
Practice
Mastery check
01Expand -5(x + 7).
Answer: -5(x + 7) = -5x + -35
02Expand 6(x + 9).
Answer: 6(x + 9) = 6x + 54
03Expand -7(x + 3).
Answer: -7(x + 3) = -7x + -21
04Expand 8(x + 5).
Answer: 8(x + 5) = 8x + 40
05Expand -2(x + 7).
Answer: -2(x + 7) = -2x + -14
06Expand 3(x + 9).
Answer: 3(x + 9) = 3x + 27
07Expand -4(x + 3).
Answer: -4(x + 3) = -4x + -12
08Expand 5(x + 5).
Answer: 5(x + 5) = 5x + 25
Terminology
Words to know
- variable
- A symbol representing a number that may vary or be unknown.
- coefficient
- A numerical factor multiplying a variable.
- expression
- A combination of numbers, variables, and operations.
- solution
- A value that makes an equation or inequality true.
Continue learning
Connected concepts
Curriculum references
Standards and instructional references
California Mathematics Framework (2023)
Middle-school instructional guidance, mathematical practices, modeling, and coherent progression.Common Core State Standards for Mathematics — Grade 7
Grade-level expectations for ratios, number systems, algebra, geometry, statistics, and probability.Principles to Actions: Ensuring Mathematical Success for All
Research-informed emphasis on reasoning, representations, discourse, productive struggle, and conceptual understanding.