Chapter 2 · Lesson 3 of 9 · Grade 7 · The Number System · MAT-07-NS-003
Multiply Integers
Represent and solve multiply integers problems using precise mathematical notation, models, and reasoning.
Math V4 · Course chapter
Chapter 2: Rational Numbers and Operations
The Number System
Essential question: How do signed rational numbers extend arithmetic while preserving operation structure?
Where this lesson fits
Lesson 3 of 9. Each lesson builds on earlier chapter representations and ideas rather than resetting the topic.
Chapter destination
Create and defend a multi-step model using the central ideas of rational numbers and operations, including a representation, calculation, interpretation, and independent check.
Learning objectives
What you should be able to do
- Represent and solve multiply integers problems using precise mathematical notation, models, and reasoning.
- Explain why a method for multiply integers works, interpret the result in context, and verify it independently.
Prerequisite check
Make sure the foundation is ready.
Grade 7 uses prerequisite knowledge as active reasoning.
Learn
Build the idea from meaning, not memorization.
Chapter 2: Rational Numbers and Operations
The Number System. Lesson 3 of 9 extends earlier chapter reasoning.
Connection to the course
Use earlier Grade 7 chapters, Grade 6 prerequisite knowledge as active knowledge. Connect today's representation to at least one earlier chapter representation.
Mathematical meaning
Multiply Integers extends arithmetic to signed, fractional, decimal, or coordinate quantities. Operations must preserve both magnitude and direction.
Represent the relationship
The Math Box uses rational number line 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 with an inverse operation, equivalent representation, estimate, or second method so both the calculation and reasoning are auditable.
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.
Rational Number Line
Move through positive and negative halves and compare distance from zero.
- Model multiply integers in the Math Box.
- Change one input and predict the effect before testing it.
- Connect the visual, numerical, and symbolic representations.
- State what remains invariant and why.
Worked examples
Twenty different ways to see the concept work.
Compute (-2)(-3).
-2 × -3 = 6Same signs give a positive product; different signs give a negative product.
Represent before calculating. Use a number line, area/fraction model, place-value model, or operation equation for this task, then solve: Compute (3)(5).
3 × 5 = 15The representation should show the same mathematical relationship as the calculation. Same signs give a positive product; different signs give a negative product.
Explain why a valid method works, then solve: Compute (-4)(7).
-4 × 7 = -28A complete explanation names the relationship or property being preserved. Same signs give a positive product; different signs give a negative product.
Estimate or predict first, then calculate and decide whether the result is reasonable: Compute (5)(-9).
5 × -9 = -45The estimate is a reasonableness check, not a replacement for the exact result. Same signs give a positive product; different signs give a negative product.
Solve and verify the result with a second method, inverse operation, or equivalent representation: Compute (-6)(4).
-6 × 4 = -24Verification should independently support the result. Same signs give a positive product; different signs give a negative product.
Interpret the answer in context after solving. What does the result mean here? Compute (7)(6).
7 × 6 = 42State the result with its meaning, units, direction, or comparison—not only a number. Same signs give a positive product; different signs give a negative product.
Error analysis: a student says, "Applying sign rules without checking direction or magnitude." Explain the mistake, then solve this related task correctly: Compute (-8)(-8).
Use a number line, context, or inverse operation. Correct solution: -8 × -8 = 64Error analysis requires identifying the broken idea, not merely replacing the final answer. Same signs give a positive product; different signs give a negative product.
Compare two related cases and explain what changes and what stays mathematically consistent. Case A: Compute (9)(3). Case B: Compute (-2)(7).
Case A: 9 × 3 = 27 Case B: -2 × 7 = -14Compare the structure, representation, units, and result rather than only the numbers. Same signs give a positive product; different signs give a negative product.
Compute (-2)(5).
-2 × 5 = -10Same signs give a positive product; different signs give a negative product.
Represent before calculating. Use a number line, area/fraction model, place-value model, or operation equation for this task, then solve: Compute (3)(-7).
3 × -7 = -21The representation should show the same mathematical relationship as the calculation. Same signs give a positive product; different signs give a negative product.
Explain why a valid method works, then solve: Compute (-4)(9).
-4 × 9 = -36A complete explanation names the relationship or property being preserved. Same signs give a positive product; different signs give a negative product.
Estimate or predict first, then calculate and decide whether the result is reasonable: Compute (5)(4).
5 × 4 = 20The estimate is a reasonableness check, not a replacement for the exact result. Same signs give a positive product; different signs give a negative product.
Solve and verify the result with a second method, inverse operation, or equivalent representation: Compute (-6)(-6).
-6 × -6 = 36Verification should independently support the result. Same signs give a positive product; different signs give a negative product.
Interpret the answer in context after solving. What does the result mean here? Compute (7)(8).
7 × 8 = 56State the result with its meaning, units, direction, or comparison—not only a number. Same signs give a positive product; different signs give a negative product.
Error analysis: a student says, "Applying sign rules without checking direction or magnitude." Explain the mistake, then solve this related task correctly: Compute (-8)(3).
Use a number line, context, or inverse operation. Correct solution: -8 × 3 = -24Error analysis requires identifying the broken idea, not merely replacing the final answer. Same signs give a positive product; different signs give a negative product.
Compare two related cases and explain what changes and what stays mathematically consistent. Case A: Compute (9)(-5). Case B: Compute (-2)(-9).
Case A: 9 × -5 = -45 Case B: -2 × -9 = 18Compare the structure, representation, units, and result rather than only the numbers. Same signs give a positive product; different signs give a negative product.
Compute (-2)(7).
-2 × 7 = -14Same signs give a positive product; different signs give a negative product.
Represent before calculating. Use a number line, area/fraction model, place-value model, or operation equation for this task, then solve: Compute (3)(9).
3 × 9 = 27The representation should show the same mathematical relationship as the calculation. Same signs give a positive product; different signs give a negative product.
Explain why a valid method works, then solve: Compute (-4)(-4).
-4 × -4 = 16A complete explanation names the relationship or property being preserved. Same signs give a positive product; different signs give a negative product.
Estimate or predict first, then calculate and decide whether the result is reasonable: Compute (5)(6).
5 × 6 = 30The estimate is a reasonableness check, not a replacement for the exact result. Same signs give a positive product; different signs give a negative product.
Practice
Guided practice with hints
01Solve and verify the result with a second method, inverse operation, or equivalent representation: Compute (-6)(8).
Hint: Choose a representation before computing.
Answer: -6 × 8 = -48
02Interpret the answer in context after solving. What does the result mean here? Compute (7)(-3).
Hint: Explain why the method works and name a verification strategy.
Answer: 7 × -3 = -21
03Error analysis: a student says, "Applying sign rules without checking direction or magnitude." Explain the mistake, then solve this related task correctly: Compute (-8)(5).
Hint: Choose a representation before computing.
Answer: Use a number line, context, or inverse operation. Correct solution: -8 × 5 = -40
04Compare two related cases and explain what changes and what stays mathematically consistent. Case A: Compute (9)(7). Case B: Compute (-2)(4).
Hint: Explain why the method works and name a verification strategy.
Answer: Case A: 9 × 7 = 63 Case B: -2 × 4 = -8
05Compute (-2)(-9).
Hint: Choose a representation before computing.
Answer: -2 × -9 = 18
06Represent before calculating. Use a number line, area/fraction model, place-value model, or operation equation for this task, then solve: Compute (3)(4).
Hint: Explain why the method works and name a verification strategy.
Answer: 3 × 4 = 12
Practice
Independent practice
01Explain why a valid method works, then solve: Compute (-4)(6).
Answer: -4 × 6 = -24
02Estimate or predict first, then calculate and decide whether the result is reasonable: Compute (5)(-8).
Answer: 5 × -8 = -40
03Solve and verify the result with a second method, inverse operation, or equivalent representation: Compute (-6)(3).
Answer: -6 × 3 = -18
04Interpret the answer in context after solving. What does the result mean here? Compute (7)(5).
Answer: 7 × 5 = 35
05Error analysis: a student says, "Applying sign rules without checking direction or magnitude." Explain the mistake, then solve this related task correctly: Compute (-8)(-7).
Answer: Use a number line, context, or inverse operation. Correct solution: -8 × -7 = 56
06Compare two related cases and explain what changes and what stays mathematically consistent. Case A: Compute (9)(9). Case B: Compute (-2)(6).
Answer: Case A: 9 × 9 = 81 Case B: -2 × 6 = -12
07Compute (-2)(4).
Answer: -2 × 4 = -8
08Represent before calculating. Use a number line, area/fraction model, place-value model, or operation equation for this task, then solve: Compute (3)(-6).
Answer: 3 × -6 = -18
Common mistakes
Learn to catch the error, not just the answer.
Use a number line, context, or inverse operation.
For negatives, values closer to zero are greater.
Applications & challenge
Use the idea beyond a single exercise.
Applications
- Model temperature, elevation, debt, gains and losses, coordinates, and measurements beyond whole numbers.
- Compute accurately with fractions, decimals, and signed quantities while checking magnitude and sign.
Challenge problems
Explain why a valid method works, then solve: Compute (-4)(8).
-4 × 8 = -32
Estimate or predict first, then calculate and decide whether the result is reasonable: Compute (5)(3).
5 × 3 = 15
Solve and verify the result with a second method, inverse operation, or equivalent representation: Compute (-6)(-5).
-6 × -5 = 30
Interpret the answer in context after solving. What does the result mean here? Compute (7)(7).
7 × 7 = 49
Flashcards
Retrieve it from memory.
Flashcards
Card 1 of 4
Game mode
Grade 7 Chapter 2 Reasoning Lab
Solve multiply integers tasks using calculation, representation, explanation, interpretation, verification, error analysis, and comparison.
Practice
Mastery check
01Error analysis: a student says, "Applying sign rules without checking direction or magnitude." Explain the mistake, then solve this related task correctly: Compute (-8)(9).
Answer: Use a number line, context, or inverse operation. Correct solution: -8 × 9 = -72
02Compare two related cases and explain what changes and what stays mathematically consistent. Case A: Compute (9)(-4). Case B: Compute (-2)(-8).
Answer: Case A: 9 × -4 = -36 Case B: -2 × -8 = 16
03Compute (-2)(6).
Answer: -2 × 6 = -12
04Represent before calculating. Use a number line, area/fraction model, place-value model, or operation equation for this task, then solve: Compute (3)(8).
Answer: 3 × 8 = 24
05Explain why a valid method works, then solve: Compute (-4)(-3).
Answer: -4 × -3 = 12
06Estimate or predict first, then calculate and decide whether the result is reasonable: Compute (5)(5).
Answer: 5 × 5 = 25
07Solve and verify the result with a second method, inverse operation, or equivalent representation: Compute (-6)(7).
Answer: -6 × 7 = -42
08Interpret the answer in context after solving. What does the result mean here? Compute (7)(-9).
Answer: 7 × -9 = -63
Terminology
Words to know
- rational number
- A number expressible as a ratio of two integers with nonzero denominator.
- absolute value
- Distance from zero on a number line.
- opposite
- A number the same distance from zero on the other side.
- coordinate
- A number locating a point relative to an axis.
Continue learning
Connected concepts
Curriculum references
Standards and instructional references
Grade 7 scope, sequence, and public task structure
Coherent sequencing, representations, and reasoning-rich tasks.Eureka Math² Grade 7 program structure
Module coherence, cumulative practice, and fluency.enVision Mathematics Grade 7 instructional model
Problem-based entry points, visual learning, modeling, and application.Common Core State Standards for Mathematics — Grade 7
Grade-level content and mathematical-practice expectations.