Chapter 2 · Lesson 2 of 9 · Grade 7 · The Number System · MAT-07-NS-002
Subtract Integers
Represent and solve subtract 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 2 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 subtract integers problems using precise mathematical notation, models, and reasoning.
- Explain why a method for subtract 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 2 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
Subtract 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 subtract 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 -8 − (-5).
-8 − (-5) = -3Subtracting a number is equivalent to adding its opposite.
Represent before calculating. Use a number line, area/fraction model, place-value model, or operation equation for this task, then solve: Compute -7 − (-4).
-7 − (-4) = -3The representation should show the same mathematical relationship as the calculation. Subtracting a number is equivalent to adding its opposite.
Explain why a valid method works, then solve: Compute -6 − (-3).
-6 − (-3) = -3A complete explanation names the relationship or property being preserved. Subtracting a number is equivalent to adding its opposite.
Estimate or predict first, then calculate and decide whether the result is reasonable: Compute -5 − (-2).
-5 − (-2) = -3The estimate is a reasonableness check, not a replacement for the exact result. Subtracting a number is equivalent to adding its opposite.
Solve and verify the result with a second method, inverse operation, or equivalent representation: Compute -4 − (-1).
-4 − (-1) = -3Verification should independently support the result. Subtracting a number is equivalent to adding its opposite.
Interpret the answer in context after solving. What does the result mean here? Compute -3 − (0).
-3 − (0) = -3State the result with its meaning, units, direction, or comparison—not only a number. Subtracting a number is equivalent to adding its opposite.
Error analysis: a student says, "Applying sign rules without checking direction or magnitude." Explain the mistake, then solve this related task correctly: Compute -2 − (1).
Use a number line, context, or inverse operation. Correct solution: -2 − (1) = -3Error analysis requires identifying the broken idea, not merely replacing the final answer. Subtracting a number is equivalent to adding its opposite.
Compare two related cases and explain what changes and what stays mathematically consistent. Case A: Compute -1 − (2). Case B: Compute 8 − (0).
Case A: -1 − (2) = -3 Case B: 8 − (0) = 8Compare the structure, representation, units, and result rather than only the numbers. Subtracting a number is equivalent to adding its opposite.
Compute 0 − (3).
0 − (3) = -3Subtracting a number is equivalent to adding its opposite.
Represent before calculating. Use a number line, area/fraction model, place-value model, or operation equation for this task, then solve: Compute 1 − (4).
1 − (4) = -3The representation should show the same mathematical relationship as the calculation. Subtracting a number is equivalent to adding its opposite.
Explain why a valid method works, then solve: Compute 2 − (5).
2 − (5) = -3A complete explanation names the relationship or property being preserved. Subtracting a number is equivalent to adding its opposite.
Estimate or predict first, then calculate and decide whether the result is reasonable: Compute 3 − (-5).
3 − (-5) = 8The estimate is a reasonableness check, not a replacement for the exact result. Subtracting a number is equivalent to adding its opposite.
Solve and verify the result with a second method, inverse operation, or equivalent representation: Compute 4 − (-4).
4 − (-4) = 8Verification should independently support the result. Subtracting a number is equivalent to adding its opposite.
Interpret the answer in context after solving. What does the result mean here? Compute 5 − (-3).
5 − (-3) = 8State the result with its meaning, units, direction, or comparison—not only a number. Subtracting a number is equivalent to adding its opposite.
Error analysis: a student says, "Applying sign rules without checking direction or magnitude." Explain the mistake, then solve this related task correctly: Compute 6 − (-2).
Use a number line, context, or inverse operation. Correct solution: 6 − (-2) = 8Error analysis requires identifying the broken idea, not merely replacing the final answer. Subtracting a number is equivalent to adding its opposite.
Compare two related cases and explain what changes and what stays mathematically consistent. Case A: Compute 7 − (-1). Case B: Compute 16 − (-3).
Case A: 7 − (-1) = 8 Case B: 16 − (-3) = 19Compare the structure, representation, units, and result rather than only the numbers. Subtracting a number is equivalent to adding its opposite.
Compute 8 − (0).
8 − (0) = 8Subtracting a number is equivalent to adding its opposite.
Represent before calculating. Use a number line, area/fraction model, place-value model, or operation equation for this task, then solve: Compute 9 − (1).
9 − (1) = 8The representation should show the same mathematical relationship as the calculation. Subtracting a number is equivalent to adding its opposite.
Explain why a valid method works, then solve: Compute 10 − (2).
10 − (2) = 8A complete explanation names the relationship or property being preserved. Subtracting a number is equivalent to adding its opposite.
Estimate or predict first, then calculate and decide whether the result is reasonable: Compute 11 − (3).
11 − (3) = 8The estimate is a reasonableness check, not a replacement for the exact result. Subtracting a number is equivalent to adding its opposite.
Practice
Guided practice with hints
01Solve and verify the result with a second method, inverse operation, or equivalent representation: Compute 12 − (4).
Hint: Choose a representation before computing.
Answer: 12 − (4) = 8
02Interpret the answer in context after solving. What does the result mean here? Compute 13 − (5).
Hint: Explain why the method works and name a verification strategy.
Answer: 13 − (5) = 8
03Error analysis: a student says, "Applying sign rules without checking direction or magnitude." Explain the mistake, then solve this related task correctly: Compute 14 − (-5).
Hint: Choose a representation before computing.
Answer: Use a number line, context, or inverse operation. Correct solution: 14 − (-5) = 19
04Compare two related cases and explain what changes and what stays mathematically consistent. Case A: Compute 15 − (-4). Case B: Compute 24 − (5).
Hint: Explain why the method works and name a verification strategy.
Answer: Case A: 15 − (-4) = 19 Case B: 24 − (5) = 19
05Compute 16 − (-3).
Hint: Choose a representation before computing.
Answer: 16 − (-3) = 19
06Represent before calculating. Use a number line, area/fraction model, place-value model, or operation equation for this task, then solve: Compute 17 − (-2).
Hint: Explain why the method works and name a verification strategy.
Answer: 17 − (-2) = 19
Practice
Independent practice
01Explain why a valid method works, then solve: Compute 18 − (-1).
Answer: 18 − (-1) = 19
02Estimate or predict first, then calculate and decide whether the result is reasonable: Compute 19 − (0).
Answer: 19 − (0) = 19
03Solve and verify the result with a second method, inverse operation, or equivalent representation: Compute 20 − (1).
Answer: 20 − (1) = 19
04Interpret the answer in context after solving. What does the result mean here? Compute 21 − (2).
Answer: 21 − (2) = 19
05Error analysis: a student says, "Applying sign rules without checking direction or magnitude." Explain the mistake, then solve this related task correctly: Compute 22 − (3).
Answer: Use a number line, context, or inverse operation. Correct solution: 22 − (3) = 19
06Compare two related cases and explain what changes and what stays mathematically consistent. Case A: Compute 23 − (4). Case B: Compute 32 − (2).
Answer: Case A: 23 − (4) = 19 Case B: 32 − (2) = 30
07Compute 24 − (5).
Answer: 24 − (5) = 19
08Represent before calculating. Use a number line, area/fraction model, place-value model, or operation equation for this task, then solve: Compute 25 − (-5).
Answer: 25 − (-5) = 30
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 26 − (-4).
26 − (-4) = 30
Estimate or predict first, then calculate and decide whether the result is reasonable: Compute 27 − (-3).
27 − (-3) = 30
Solve and verify the result with a second method, inverse operation, or equivalent representation: Compute 28 − (-2).
28 − (-2) = 30
Interpret the answer in context after solving. What does the result mean here? Compute 29 − (-1).
29 − (-1) = 30
Flashcards
Retrieve it from memory.
Flashcards
Card 1 of 4
Game mode
Grade 7 Chapter 2 Reasoning Lab
Solve subtract 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 30 − (0).
Answer: Use a number line, context, or inverse operation. Correct solution: 30 − (0) = 30
02Compare two related cases and explain what changes and what stays mathematically consistent. Case A: Compute 31 − (1). Case B: Compute 40 − (-1).
Answer: Case A: 31 − (1) = 30 Case B: 40 − (-1) = 41
03Compute 32 − (2).
Answer: 32 − (2) = 30
04Represent before calculating. Use a number line, area/fraction model, place-value model, or operation equation for this task, then solve: Compute 33 − (3).
Answer: 33 − (3) = 30
05Explain why a valid method works, then solve: Compute 34 − (4).
Answer: 34 − (4) = 30
06Estimate or predict first, then calculate and decide whether the result is reasonable: Compute 35 − (5).
Answer: 35 − (5) = 30
07Solve and verify the result with a second method, inverse operation, or equivalent representation: Compute 36 − (-5).
Answer: 36 − (-5) = 41
08Interpret the answer in context after solving. What does the result mean here? Compute 37 − (-4).
Answer: 37 − (-4) = 41
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.