Chapter 1 · Lesson 9 of 9 · Grade 7 · Ratios & Proportional Reasoning · MAT-07-RP-009
Use Proportions in Multi-Step Contexts
Represent and solve use proportions in multi-step contexts problems using precise mathematical notation, models, and reasoning.
Math V4 · Course chapter
Chapter 1: Proportional Relationships, Percent, and Financial Reasoning
Ratios & Proportional Reasoning
Essential question: How can proportional relationships describe change, comparison, percent, and real decisions?
Where this lesson fits
Lesson 9 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 proportional relationships, percent, and financial reasoning, including a representation, calculation, interpretation, and independent check.
Learning objectives
What you should be able to do
- Represent and solve use proportions in multi-step contexts problems using precise mathematical notation, models, and reasoning.
- Explain why a method for use proportions in multi-step contexts 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 1: Proportional Relationships, Percent, and Financial Reasoning
Ratios & Proportional Reasoning. Lesson 9 of 9 extends earlier chapter reasoning.
Connection to the course
Use Grade 6 ratios, rates, fractions, and decimals as active knowledge. Synthesize the chapter through this project: Create and defend a multi-step model using the central ideas of proportional relationships, percent, and financial reasoning, including a representation, calculation, interpretation, and independent check.
Mathematical meaning
Use Proportions in Multi-Step Contexts is about multiplicative comparison. Equivalent representations preserve the same scale relationship, not merely the same difference.
Represent the relationship
The Math Box uses double 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.
Double Number Line
Track two linked quantities with a constant unit rate.
- Model use proportions in multi step contexts 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.
A machine makes 3 parts every 2 minutes. How many parts in 4 minutes at the same rate?
4 ÷ 2 = 2; 3 × 2 = 6 partsEquivalent ratios use the same scale factor on both quantities.
Represent before calculating. Use a ratio table, tape diagram, double number line, or equation for this task, then solve: A machine makes 4 parts every 3 minutes. How many parts in 9 minutes at the same rate?
9 ÷ 3 = 3; 4 × 3 = 12 partsThe representation should show the same mathematical relationship as the calculation. Equivalent ratios use the same scale factor on both quantities.
Explain why a valid method works, then solve: A machine makes 5 parts every 4 minutes. How many parts in 16 minutes at the same rate?
16 ÷ 4 = 4; 5 × 4 = 20 partsA complete explanation names the relationship or property being preserved. Equivalent ratios use the same scale factor on both quantities.
Estimate or predict first, then calculate and decide whether the result is reasonable: A machine makes 6 parts every 5 minutes. How many parts in 25 minutes at the same rate?
25 ÷ 5 = 5; 6 × 5 = 30 partsThe estimate is a reasonableness check, not a replacement for the exact result. Equivalent ratios use the same scale factor on both quantities.
Solve and verify the result with a second method, inverse operation, or equivalent representation: A machine makes 7 parts every 6 minutes. How many parts in 36 minutes at the same rate?
36 ÷ 6 = 6; 7 × 6 = 42 partsVerification should independently support the result. Equivalent ratios use the same scale factor on both quantities.
Interpret the answer in context after solving. What does the result mean here? A machine makes 8 parts every 7 minutes. How many parts in 14 minutes at the same rate?
14 ÷ 7 = 2; 8 × 2 = 16 partsState the result with its meaning, units, direction, or comparison—not only a number. Equivalent ratios use the same scale factor on both quantities.
Error analysis: a student says, "Treating a ratio as additive." Explain the mistake, then solve this related task correctly: A machine makes 9 parts every 2 minutes. How many parts in 6 minutes at the same rate?
Check for a common scale factor or constant unit rate. Correct solution: 6 ÷ 2 = 3; 9 × 3 = 27 partsError analysis requires identifying the broken idea, not merely replacing the final answer. Equivalent ratios use the same scale factor on both quantities.
Compare two related cases and explain what changes and what stays mathematically consistent. Case A: A machine makes 10 parts every 3 minutes. How many parts in 12 minutes at the same rate? Case B: A machine makes 3 parts every 6 minutes. How many parts in 18 minutes at the same rate?
Case A: 12 ÷ 3 = 4; 10 × 4 = 40 parts Case B: 18 ÷ 6 = 3; 3 × 3 = 9 partsCompare the structure, representation, units, and result rather than only the numbers. Equivalent ratios use the same scale factor on both quantities.
A machine makes 3 parts every 4 minutes. How many parts in 20 minutes at the same rate?
20 ÷ 4 = 5; 3 × 5 = 15 partsEquivalent ratios use the same scale factor on both quantities.
Represent before calculating. Use a ratio table, tape diagram, double number line, or equation for this task, then solve: A machine makes 4 parts every 5 minutes. How many parts in 30 minutes at the same rate?
30 ÷ 5 = 6; 4 × 6 = 24 partsThe representation should show the same mathematical relationship as the calculation. Equivalent ratios use the same scale factor on both quantities.
Explain why a valid method works, then solve: A machine makes 5 parts every 6 minutes. How many parts in 12 minutes at the same rate?
12 ÷ 6 = 2; 5 × 2 = 10 partsA complete explanation names the relationship or property being preserved. Equivalent ratios use the same scale factor on both quantities.
Estimate or predict first, then calculate and decide whether the result is reasonable: A machine makes 6 parts every 7 minutes. How many parts in 21 minutes at the same rate?
21 ÷ 7 = 3; 6 × 3 = 18 partsThe estimate is a reasonableness check, not a replacement for the exact result. Equivalent ratios use the same scale factor on both quantities.
Solve and verify the result with a second method, inverse operation, or equivalent representation: A machine makes 7 parts every 2 minutes. How many parts in 8 minutes at the same rate?
8 ÷ 2 = 4; 7 × 4 = 28 partsVerification should independently support the result. Equivalent ratios use the same scale factor on both quantities.
Interpret the answer in context after solving. What does the result mean here? A machine makes 8 parts every 3 minutes. How many parts in 15 minutes at the same rate?
15 ÷ 3 = 5; 8 × 5 = 40 partsState the result with its meaning, units, direction, or comparison—not only a number. Equivalent ratios use the same scale factor on both quantities.
Error analysis: a student says, "Treating a ratio as additive." Explain the mistake, then solve this related task correctly: A machine makes 9 parts every 4 minutes. How many parts in 24 minutes at the same rate?
Check for a common scale factor or constant unit rate. Correct solution: 24 ÷ 4 = 6; 9 × 6 = 54 partsError analysis requires identifying the broken idea, not merely replacing the final answer. Equivalent ratios use the same scale factor on both quantities.
Compare two related cases and explain what changes and what stays mathematically consistent. Case A: A machine makes 10 parts every 5 minutes. How many parts in 10 minutes at the same rate? Case B: A machine makes 3 parts every 2 minutes. How many parts in 12 minutes at the same rate?
Case A: 10 ÷ 5 = 2; 10 × 2 = 20 parts Case B: 12 ÷ 2 = 6; 3 × 6 = 18 partsCompare the structure, representation, units, and result rather than only the numbers. Equivalent ratios use the same scale factor on both quantities.
A machine makes 3 parts every 6 minutes. How many parts in 18 minutes at the same rate?
18 ÷ 6 = 3; 3 × 3 = 9 partsEquivalent ratios use the same scale factor on both quantities.
Represent before calculating. Use a ratio table, tape diagram, double number line, or equation for this task, then solve: A machine makes 4 parts every 7 minutes. How many parts in 28 minutes at the same rate?
28 ÷ 7 = 4; 4 × 4 = 16 partsThe representation should show the same mathematical relationship as the calculation. Equivalent ratios use the same scale factor on both quantities.
Explain why a valid method works, then solve: A machine makes 5 parts every 2 minutes. How many parts in 10 minutes at the same rate?
10 ÷ 2 = 5; 5 × 5 = 25 partsA complete explanation names the relationship or property being preserved. Equivalent ratios use the same scale factor on both quantities.
Estimate or predict first, then calculate and decide whether the result is reasonable: A machine makes 6 parts every 3 minutes. How many parts in 18 minutes at the same rate?
18 ÷ 3 = 6; 6 × 6 = 36 partsThe estimate is a reasonableness check, not a replacement for the exact result. Equivalent ratios use the same scale factor on both quantities.
Practice
Guided practice with hints
01Solve and verify the result with a second method, inverse operation, or equivalent representation: A machine makes 7 parts every 4 minutes. How many parts in 8 minutes at the same rate?
Hint: Choose a representation before computing.
Answer: 8 ÷ 4 = 2; 7 × 2 = 14 parts
02Interpret the answer in context after solving. What does the result mean here? A machine makes 8 parts every 5 minutes. How many parts in 15 minutes at the same rate?
Hint: Explain why the method works and name a verification strategy.
Answer: 15 ÷ 5 = 3; 8 × 3 = 24 parts
03Error analysis: a student says, "Treating a ratio as additive." Explain the mistake, then solve this related task correctly: A machine makes 9 parts every 6 minutes. How many parts in 24 minutes at the same rate?
Hint: Choose a representation before computing.
Answer: Check for a common scale factor or constant unit rate. Correct solution: 24 ÷ 6 = 4; 9 × 4 = 36 parts
04Compare two related cases and explain what changes and what stays mathematically consistent. Case A: A machine makes 10 parts every 7 minutes. How many parts in 35 minutes at the same rate? Case B: A machine makes 3 parts every 4 minutes. How many parts in 16 minutes at the same rate?
Hint: Explain why the method works and name a verification strategy.
Answer: Case A: 35 ÷ 7 = 5; 10 × 5 = 50 parts Case B: 16 ÷ 4 = 4; 3 × 4 = 12 parts
05A machine makes 3 parts every 2 minutes. How many parts in 12 minutes at the same rate?
Hint: Choose a representation before computing.
Answer: 12 ÷ 2 = 6; 3 × 6 = 18 parts
06Represent before calculating. Use a ratio table, tape diagram, double number line, or equation for this task, then solve: A machine makes 4 parts every 3 minutes. How many parts in 6 minutes at the same rate?
Hint: Explain why the method works and name a verification strategy.
Answer: 6 ÷ 3 = 2; 4 × 2 = 8 parts
Practice
Independent practice
01Explain why a valid method works, then solve: A machine makes 5 parts every 4 minutes. How many parts in 12 minutes at the same rate?
Answer: 12 ÷ 4 = 3; 5 × 3 = 15 parts
02Estimate or predict first, then calculate and decide whether the result is reasonable: A machine makes 6 parts every 5 minutes. How many parts in 20 minutes at the same rate?
Answer: 20 ÷ 5 = 4; 6 × 4 = 24 parts
03Solve and verify the result with a second method, inverse operation, or equivalent representation: A machine makes 7 parts every 6 minutes. How many parts in 30 minutes at the same rate?
Answer: 30 ÷ 6 = 5; 7 × 5 = 35 parts
04Interpret the answer in context after solving. What does the result mean here? A machine makes 8 parts every 7 minutes. How many parts in 42 minutes at the same rate?
Answer: 42 ÷ 7 = 6; 8 × 6 = 48 parts
05Error analysis: a student says, "Treating a ratio as additive." Explain the mistake, then solve this related task correctly: A machine makes 9 parts every 2 minutes. How many parts in 4 minutes at the same rate?
Answer: Check for a common scale factor or constant unit rate. Correct solution: 4 ÷ 2 = 2; 9 × 2 = 18 parts
06Compare two related cases and explain what changes and what stays mathematically consistent. Case A: A machine makes 10 parts every 3 minutes. How many parts in 9 minutes at the same rate? Case B: A machine makes 3 parts every 6 minutes. How many parts in 12 minutes at the same rate?
Answer: Case A: 9 ÷ 3 = 3; 10 × 3 = 30 parts Case B: 12 ÷ 6 = 2; 3 × 2 = 6 parts
07A machine makes 3 parts every 4 minutes. How many parts in 16 minutes at the same rate?
Answer: 16 ÷ 4 = 4; 3 × 4 = 12 parts
08Represent before calculating. Use a ratio table, tape diagram, double number line, or equation for this task, then solve: A machine makes 4 parts every 5 minutes. How many parts in 25 minutes at the same rate?
Answer: 25 ÷ 5 = 5; 4 × 5 = 20 parts
Common mistakes
Learn to catch the error, not just the answer.
Check for a common scale factor or constant unit rate.
Convert both to a common unit rate first.
Applications & challenge
Use the idea beyond a single exercise.
Applications
- Compare prices, speeds, recipes, maps, taxes, discounts, tips, and rates using common units.
- Scale designs and predict related quantities while preserving multiplicative relationships.
Challenge problems
Explain why a valid method works, then solve: A machine makes 5 parts every 6 minutes. How many parts in 36 minutes at the same rate?
36 ÷ 6 = 6; 5 × 6 = 30 parts
Estimate or predict first, then calculate and decide whether the result is reasonable: A machine makes 6 parts every 7 minutes. How many parts in 14 minutes at the same rate?
14 ÷ 7 = 2; 6 × 2 = 12 parts
Solve and verify the result with a second method, inverse operation, or equivalent representation: A machine makes 7 parts every 2 minutes. How many parts in 6 minutes at the same rate?
6 ÷ 2 = 3; 7 × 3 = 21 parts
Interpret the answer in context after solving. What does the result mean here? A machine makes 8 parts every 3 minutes. How many parts in 12 minutes at the same rate?
12 ÷ 3 = 4; 8 × 4 = 32 parts
Flashcards
Retrieve it from memory.
Flashcards
Card 1 of 4
Game mode
Grade 7 Chapter 1 Reasoning Lab
Solve use proportions in multi-step contexts tasks using calculation, representation, explanation, interpretation, verification, error analysis, and comparison.
Practice
Mastery check
01Error analysis: a student says, "Treating a ratio as additive." Explain the mistake, then solve this related task correctly: A machine makes 9 parts every 4 minutes. How many parts in 20 minutes at the same rate?
Answer: Check for a common scale factor or constant unit rate. Correct solution: 20 ÷ 4 = 5; 9 × 5 = 45 parts
02Compare two related cases and explain what changes and what stays mathematically consistent. Case A: A machine makes 10 parts every 5 minutes. How many parts in 30 minutes at the same rate? Case B: A machine makes 3 parts every 2 minutes. How many parts in 10 minutes at the same rate?
Answer: Case A: 30 ÷ 5 = 6; 10 × 6 = 60 parts Case B: 10 ÷ 2 = 5; 3 × 5 = 15 parts
03A machine makes 3 parts every 6 minutes. How many parts in 12 minutes at the same rate?
Answer: 12 ÷ 6 = 2; 3 × 2 = 6 parts
04Represent before calculating. Use a ratio table, tape diagram, double number line, or equation for this task, then solve: A machine makes 4 parts every 7 minutes. How many parts in 21 minutes at the same rate?
Answer: 21 ÷ 7 = 3; 4 × 3 = 12 parts
05Explain why a valid method works, then solve: A machine makes 5 parts every 2 minutes. How many parts in 8 minutes at the same rate?
Answer: 8 ÷ 2 = 4; 5 × 4 = 20 parts
06Estimate or predict first, then calculate and decide whether the result is reasonable: A machine makes 6 parts every 3 minutes. How many parts in 15 minutes at the same rate?
Answer: 15 ÷ 3 = 5; 6 × 5 = 30 parts
07Solve and verify the result with a second method, inverse operation, or equivalent representation: A machine makes 7 parts every 4 minutes. How many parts in 24 minutes at the same rate?
Answer: 24 ÷ 4 = 6; 7 × 6 = 42 parts
08Interpret the answer in context after solving. What does the result mean here? A machine makes 8 parts every 5 minutes. How many parts in 10 minutes at the same rate?
Answer: 10 ÷ 5 = 2; 8 × 2 = 16 parts
Terminology
Words to know
- ratio
- A multiplicative comparison between two quantities.
- unit rate
- A rate expressed per one unit of the second quantity.
- equivalent ratios
- Ratios made by scaling both quantities by the same nonzero factor.
- constant of proportionality
- The unit rate k in a proportional equation y = kx.
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.