Chapter 2 · Lesson 1 of 9 · Grade 7 · The Number System · MAT-07-NS-001
Add Integers
Represent and solve add 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 1 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 add integers problems using precise mathematical notation, models, and reasoning.
- Explain why a method for add 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. This lesson opens the chapter with: How do signed rational numbers extend arithmetic while preserving operation structure?
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
Add 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 add 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 -12 + 7.
-12 + 7 = -5Add signed quantities by combining direction and magnitude.
Represent before calculating. Use a number line, area/fraction model, place-value model, or operation equation for this task, then solve: Compute -11 + 6.
-11 + 6 = -5The representation should show the same mathematical relationship as the calculation. Add signed quantities by combining direction and magnitude.
Explain why a valid method works, then solve: Compute -10 + 5.
-10 + 5 = -5A complete explanation names the relationship or property being preserved. Add signed quantities by combining direction and magnitude.
Estimate or predict first, then calculate and decide whether the result is reasonable: Compute -9 + 4.
-9 + 4 = -5The estimate is a reasonableness check, not a replacement for the exact result. Add signed quantities by combining direction and magnitude.
Solve and verify the result with a second method, inverse operation, or equivalent representation: Compute -8 + 3.
-8 + 3 = -5Verification should independently support the result. Add signed quantities by combining direction and magnitude.
Interpret the answer in context after solving. What does the result mean here? Compute -7 + 2.
-7 + 2 = -5State the result with its meaning, units, direction, or comparison—not only a number. Add signed quantities by combining direction and magnitude.
Error analysis: a student says, "Applying sign rules without checking direction or magnitude." Explain the mistake, then solve this related task correctly: Compute -6 + 1.
Use a number line, context, or inverse operation. Correct solution: -6 + 1 = -5Error analysis requires identifying the broken idea, not merely replacing the final answer. Add signed quantities by combining direction and magnitude.
Compare two related cases and explain what changes and what stays mathematically consistent. Case A: Compute -5 + 0. Case B: Compute 4 + 6.
Case A: -5 + 0 = -5 Case B: 4 + 6 = 10Compare the structure, representation, units, and result rather than only the numbers. Add signed quantities by combining direction and magnitude.
Compute -4 + -1.
-4 + -1 = -5Add signed quantities by combining direction and magnitude.
Represent before calculating. Use a number line, area/fraction model, place-value model, or operation equation for this task, then solve: Compute -3 + -2.
-3 + -2 = -5The representation should show the same mathematical relationship as the calculation. Add signed quantities by combining direction and magnitude.
Explain why a valid method works, then solve: Compute -2 + -3.
-2 + -3 = -5A complete explanation names the relationship or property being preserved. Add signed quantities by combining direction and magnitude.
Estimate or predict first, then calculate and decide whether the result is reasonable: Compute -1 + -4.
-1 + -4 = -5The estimate is a reasonableness check, not a replacement for the exact result. Add signed quantities by combining direction and magnitude.
Solve and verify the result with a second method, inverse operation, or equivalent representation: Compute 0 + -5.
0 + -5 = -5Verification should independently support the result. Add signed quantities by combining direction and magnitude.
Interpret the answer in context after solving. What does the result mean here? Compute 1 + -6.
1 + -6 = -5State the result with its meaning, units, direction, or comparison—not only a number. Add signed quantities by combining direction and magnitude.
Error analysis: a student says, "Applying sign rules without checking direction or magnitude." Explain the mistake, then solve this related task correctly: Compute 2 + -7.
Use a number line, context, or inverse operation. Correct solution: 2 + -7 = -5Error analysis requires identifying the broken idea, not merely replacing the final answer. Add signed quantities by combining direction and magnitude.
Compare two related cases and explain what changes and what stays mathematically consistent. Case A: Compute 3 + 7. Case B: Compute 12 + -2.
Case A: 3 + 7 = 10 Case B: 12 + -2 = 10Compare the structure, representation, units, and result rather than only the numbers. Add signed quantities by combining direction and magnitude.
Compute 4 + 6.
4 + 6 = 10Add signed quantities by combining direction and magnitude.
Represent before calculating. Use a number line, area/fraction model, place-value model, or operation equation for this task, then solve: Compute 5 + 5.
5 + 5 = 10The representation should show the same mathematical relationship as the calculation. Add signed quantities by combining direction and magnitude.
Explain why a valid method works, then solve: Compute 6 + 4.
6 + 4 = 10A complete explanation names the relationship or property being preserved. Add signed quantities by combining direction and magnitude.
Estimate or predict first, then calculate and decide whether the result is reasonable: Compute 7 + 3.
7 + 3 = 10The estimate is a reasonableness check, not a replacement for the exact result. Add signed quantities by combining direction and magnitude.
Practice
Guided practice with hints
01Solve and verify the result with a second method, inverse operation, or equivalent representation: Compute 8 + 2.
Hint: Choose a representation before computing.
Answer: 8 + 2 = 10
02Interpret the answer in context after solving. What does the result mean here? Compute 9 + 1.
Hint: Explain why the method works and name a verification strategy.
Answer: 9 + 1 = 10
03Error analysis: a student says, "Applying sign rules without checking direction or magnitude." Explain the mistake, then solve this related task correctly: Compute 10 + 0.
Hint: Choose a representation before computing.
Answer: Use a number line, context, or inverse operation. Correct solution: 10 + 0 = 10
04Compare two related cases and explain what changes and what stays mathematically consistent. Case A: Compute 11 + -1. Case B: Compute 20 + 5.
Hint: Explain why the method works and name a verification strategy.
Answer: Case A: 11 + -1 = 10 Case B: 20 + 5 = 25
05Compute 12 + -2.
Hint: Choose a representation before computing.
Answer: 12 + -2 = 10
06Represent before calculating. Use a number line, area/fraction model, place-value model, or operation equation for this task, then solve: Compute 13 + -3.
Hint: Explain why the method works and name a verification strategy.
Answer: 13 + -3 = 10
Practice
Independent practice
01Explain why a valid method works, then solve: Compute 14 + -4.
Answer: 14 + -4 = 10
02Estimate or predict first, then calculate and decide whether the result is reasonable: Compute 15 + -5.
Answer: 15 + -5 = 10
03Solve and verify the result with a second method, inverse operation, or equivalent representation: Compute 16 + -6.
Answer: 16 + -6 = 10
04Interpret the answer in context after solving. What does the result mean here? Compute 17 + -7.
Answer: 17 + -7 = 10
05Error analysis: a student says, "Applying sign rules without checking direction or magnitude." Explain the mistake, then solve this related task correctly: Compute 18 + 7.
Answer: Use a number line, context, or inverse operation. Correct solution: 18 + 7 = 25
06Compare two related cases and explain what changes and what stays mathematically consistent. Case A: Compute 19 + 6. Case B: Compute 28 + -3.
Answer: Case A: 19 + 6 = 25 Case B: 28 + -3 = 25
07Compute 20 + 5.
Answer: 20 + 5 = 25
08Represent before calculating. Use a number line, area/fraction model, place-value model, or operation equation for this task, then solve: Compute 21 + 4.
Answer: 21 + 4 = 25
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 22 + 3.
22 + 3 = 25
Estimate or predict first, then calculate and decide whether the result is reasonable: Compute 23 + 2.
23 + 2 = 25
Solve and verify the result with a second method, inverse operation, or equivalent representation: Compute 24 + 1.
24 + 1 = 25
Interpret the answer in context after solving. What does the result mean here? Compute 25 + 0.
25 + 0 = 25
Flashcards
Retrieve it from memory.
Flashcards
Card 1 of 4
Game mode
Grade 7 Chapter 2 Reasoning Lab
Solve add 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 26 + -1.
Answer: Use a number line, context, or inverse operation. Correct solution: 26 + -1 = 25
02Compare two related cases and explain what changes and what stays mathematically consistent. Case A: Compute 27 + -2. Case B: Compute 36 + 4.
Answer: Case A: 27 + -2 = 25 Case B: 36 + 4 = 40
03Compute 28 + -3.
Answer: 28 + -3 = 25
04Represent before calculating. Use a number line, area/fraction model, place-value model, or operation equation for this task, then solve: Compute 29 + -4.
Answer: 29 + -4 = 25
05Explain why a valid method works, then solve: Compute 30 + -5.
Answer: 30 + -5 = 25
06Estimate or predict first, then calculate and decide whether the result is reasonable: Compute 31 + -6.
Answer: 31 + -6 = 25
07Solve and verify the result with a second method, inverse operation, or equivalent representation: Compute 32 + -7.
Answer: 32 + -7 = 25
08Interpret the answer in context after solving. What does the result mean here? Compute 33 + 7.
Answer: 33 + 7 = 40
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.