Chapter 2 · Lesson 8 of 9 · Grade 7 · The Number System · MAT-07-NS-008
Solve Multi-Step Rational-Number Problems
Represent and solve solve multi-step rational-number problems 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 8 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 solve multi-step rational-number problems problems using precise mathematical notation, models, and reasoning.
- Explain why a method for solve multi-step rational-number problems 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.
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 8 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
Solve Multi-Step Rational-Number Problems 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 solve multi step rational number problems 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.
Evaluate -10 + 3 × 2.
3 × 2 = 6; -10 + 6 = -4Use multiplication before addition.
Represent before calculating. Use a number line, area/fraction model, place-value model, or operation equation for this task, then solve: Evaluate -9 + 4 × 3.
4 × 3 = 12; -9 + 12 = 3The representation should show the same mathematical relationship as the calculation. Use multiplication before addition.
Explain why a valid method works, then solve: Evaluate -8 + 5 × 4.
5 × 4 = 20; -8 + 20 = 12A complete explanation names the relationship or property being preserved. Use multiplication before addition.
Estimate or predict first, then calculate and decide whether the result is reasonable: Evaluate -7 + 6 × 5.
6 × 5 = 30; -7 + 30 = 23The estimate is a reasonableness check, not a replacement for the exact result. Use multiplication before addition.
Solve and verify the result with a second method, inverse operation, or equivalent representation: Evaluate -6 + 7 × 6.
7 × 6 = 42; -6 + 42 = 36Verification should independently support the result. Use multiplication before addition.
Interpret the answer in context after solving. What does the result mean here? Evaluate -5 + 8 × 2.
8 × 2 = 16; -5 + 16 = 11State the result with its meaning, units, direction, or comparison—not only a number. Use multiplication before addition.
Error analysis: a student says, "Applying sign rules without checking direction or magnitude." Explain the mistake, then solve this related task correctly: Evaluate -4 + 9 × 3.
Use a number line, context, or inverse operation. Correct solution: 9 × 3 = 27; -4 + 27 = 23Error analysis requires identifying the broken idea, not merely replacing the final answer. Use multiplication before addition.
Compare two related cases and explain what changes and what stays mathematically consistent. Case A: Evaluate -3 + 3 × 4. Case B: Evaluate 6 + 5 × 3.
Case A: 3 × 4 = 12; -3 + 12 = 9 Case B: 5 × 3 = 15; 6 + 15 = 21Compare the structure, representation, units, and result rather than only the numbers. Use multiplication before addition.
Evaluate -2 + 4 × 5.
4 × 5 = 20; -2 + 20 = 18Use multiplication before addition.
Represent before calculating. Use a number line, area/fraction model, place-value model, or operation equation for this task, then solve: Evaluate -1 + 5 × 6.
5 × 6 = 30; -1 + 30 = 29The representation should show the same mathematical relationship as the calculation. Use multiplication before addition.
Explain why a valid method works, then solve: Evaluate 0 + 6 × 2.
6 × 2 = 12; 0 + 12 = 12A complete explanation names the relationship or property being preserved. Use multiplication before addition.
Estimate or predict first, then calculate and decide whether the result is reasonable: Evaluate 1 + 7 × 3.
7 × 3 = 21; 1 + 21 = 22The estimate is a reasonableness check, not a replacement for the exact result. Use multiplication before addition.
Solve and verify the result with a second method, inverse operation, or equivalent representation: Evaluate 2 + 8 × 4.
8 × 4 = 32; 2 + 32 = 34Verification should independently support the result. Use multiplication before addition.
Interpret the answer in context after solving. What does the result mean here? Evaluate 3 + 9 × 5.
9 × 5 = 45; 3 + 45 = 48State the result with its meaning, units, direction, or comparison—not only a number. Use multiplication before addition.
Error analysis: a student says, "Applying sign rules without checking direction or magnitude." Explain the mistake, then solve this related task correctly: Evaluate 4 + 3 × 6.
Use a number line, context, or inverse operation. Correct solution: 3 × 6 = 18; 4 + 18 = 22Error analysis requires identifying the broken idea, not merely replacing the final answer. Use multiplication before addition.
Compare two related cases and explain what changes and what stays mathematically consistent. Case A: Evaluate 5 + 4 × 2. Case B: Evaluate 14 + 6 × 6.
Case A: 4 × 2 = 8; 5 + 8 = 13 Case B: 6 × 6 = 36; 14 + 36 = 50Compare the structure, representation, units, and result rather than only the numbers. Use multiplication before addition.
Evaluate 6 + 5 × 3.
5 × 3 = 15; 6 + 15 = 21Use multiplication before addition.
Represent before calculating. Use a number line, area/fraction model, place-value model, or operation equation for this task, then solve: Evaluate 7 + 6 × 4.
6 × 4 = 24; 7 + 24 = 31The representation should show the same mathematical relationship as the calculation. Use multiplication before addition.
Explain why a valid method works, then solve: Evaluate 8 + 7 × 5.
7 × 5 = 35; 8 + 35 = 43A complete explanation names the relationship or property being preserved. Use multiplication before addition.
Estimate or predict first, then calculate and decide whether the result is reasonable: Evaluate 9 + 8 × 6.
8 × 6 = 48; 9 + 48 = 57The estimate is a reasonableness check, not a replacement for the exact result. Use multiplication before addition.
Practice
Guided practice with hints
01Solve and verify the result with a second method, inverse operation, or equivalent representation: Evaluate 10 + 9 × 2.
Hint: Choose a representation before computing.
Answer: 9 × 2 = 18; 10 + 18 = 28
02Interpret the answer in context after solving. What does the result mean here? Evaluate 11 + 3 × 3.
Hint: Explain why the method works and name a verification strategy.
Answer: 3 × 3 = 9; 11 + 9 = 20
03Error analysis: a student says, "Applying sign rules without checking direction or magnitude." Explain the mistake, then solve this related task correctly: Evaluate 12 + 4 × 4.
Hint: Choose a representation before computing.
Answer: Use a number line, context, or inverse operation. Correct solution: 4 × 4 = 16; 12 + 16 = 28
04Compare two related cases and explain what changes and what stays mathematically consistent. Case A: Evaluate 13 + 5 × 5. Case B: Evaluate 22 + 7 × 4.
Hint: Explain why the method works and name a verification strategy.
Answer: Case A: 5 × 5 = 25; 13 + 25 = 38 Case B: 7 × 4 = 28; 22 + 28 = 50
05Evaluate 14 + 6 × 6.
Hint: Choose a representation before computing.
Answer: 6 × 6 = 36; 14 + 36 = 50
06Represent before calculating. Use a number line, area/fraction model, place-value model, or operation equation for this task, then solve: Evaluate 15 + 7 × 2.
Hint: Explain why the method works and name a verification strategy.
Answer: 7 × 2 = 14; 15 + 14 = 29
Practice
Independent practice
01Explain why a valid method works, then solve: Evaluate 16 + 8 × 3.
Answer: 8 × 3 = 24; 16 + 24 = 40
02Estimate or predict first, then calculate and decide whether the result is reasonable: Evaluate 17 + 9 × 4.
Answer: 9 × 4 = 36; 17 + 36 = 53
03Solve and verify the result with a second method, inverse operation, or equivalent representation: Evaluate 18 + 3 × 5.
Answer: 3 × 5 = 15; 18 + 15 = 33
04Interpret the answer in context after solving. What does the result mean here? Evaluate 19 + 4 × 6.
Answer: 4 × 6 = 24; 19 + 24 = 43
05Error analysis: a student says, "Applying sign rules without checking direction or magnitude." Explain the mistake, then solve this related task correctly: Evaluate 20 + 5 × 2.
Answer: Use a number line, context, or inverse operation. Correct solution: 5 × 2 = 10; 20 + 10 = 30
06Compare two related cases and explain what changes and what stays mathematically consistent. Case A: Evaluate 21 + 6 × 3. Case B: Evaluate 30 + 8 × 2.
Answer: Case A: 6 × 3 = 18; 21 + 18 = 39 Case B: 8 × 2 = 16; 30 + 16 = 46
07Evaluate 22 + 7 × 4.
Answer: 7 × 4 = 28; 22 + 28 = 50
08Represent before calculating. Use a number line, area/fraction model, place-value model, or operation equation for this task, then solve: Evaluate 23 + 8 × 5.
Answer: 8 × 5 = 40; 23 + 40 = 63
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: Evaluate 24 + 9 × 6.
9 × 6 = 54; 24 + 54 = 78
Estimate or predict first, then calculate and decide whether the result is reasonable: Evaluate 25 + 3 × 2.
3 × 2 = 6; 25 + 6 = 31
Solve and verify the result with a second method, inverse operation, or equivalent representation: Evaluate 26 + 4 × 3.
4 × 3 = 12; 26 + 12 = 38
Interpret the answer in context after solving. What does the result mean here? Evaluate 27 + 5 × 4.
5 × 4 = 20; 27 + 20 = 47
Flashcards
Retrieve it from memory.
Flashcards
Card 1 of 4
Game mode
Grade 7 Chapter 2 Reasoning Lab
Solve solve multi-step rational-number problems 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: Evaluate 28 + 6 × 5.
Answer: Use a number line, context, or inverse operation. Correct solution: 6 × 5 = 30; 28 + 30 = 58
02Compare two related cases and explain what changes and what stays mathematically consistent. Case A: Evaluate 29 + 7 × 6. Case B: Evaluate 38 + 9 × 5.
Answer: Case A: 7 × 6 = 42; 29 + 42 = 71 Case B: 9 × 5 = 45; 38 + 45 = 83
03Evaluate 30 + 8 × 2.
Answer: 8 × 2 = 16; 30 + 16 = 46
04Represent before calculating. Use a number line, area/fraction model, place-value model, or operation equation for this task, then solve: Evaluate 31 + 9 × 3.
Answer: 9 × 3 = 27; 31 + 27 = 58
05Explain why a valid method works, then solve: Evaluate 32 + 3 × 4.
Answer: 3 × 4 = 12; 32 + 12 = 44
06Estimate or predict first, then calculate and decide whether the result is reasonable: Evaluate 33 + 4 × 5.
Answer: 4 × 5 = 20; 33 + 20 = 53
07Solve and verify the result with a second method, inverse operation, or equivalent representation: Evaluate 34 + 5 × 6.
Answer: 5 × 6 = 30; 34 + 30 = 64
08Interpret the answer in context after solving. What does the result mean here? Evaluate 35 + 6 × 2.
Answer: 6 × 2 = 12; 35 + 12 = 47
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.