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.
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.
The new lesson depends on this prerequisite as active knowledge.
The new lesson depends on this prerequisite as active knowledge.
Learn
Build the idea from meaning, not memorization.
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 one example of use proportions in multi step contexts in the Math Box.
- Change one input, predict the effect, and test the prediction.
- State the relationship or invariant that explains what stayed mathematically consistent.
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.
A machine makes 4 parts every 3 minutes. How many parts in 9 minutes at the same rate?
9 ÷ 3 = 3; 4 × 3 = 12 partsEquivalent ratios use the same scale factor on both quantities.
A machine makes 5 parts every 4 minutes. How many parts in 16 minutes at the same rate?
16 ÷ 4 = 4; 5 × 4 = 20 partsEquivalent ratios use the same scale factor on both quantities.
A machine makes 6 parts every 5 minutes. How many parts in 25 minutes at the same rate?
25 ÷ 5 = 5; 6 × 5 = 30 partsEquivalent ratios use the same scale factor on both quantities.
A machine makes 7 parts every 6 minutes. How many parts in 36 minutes at the same rate?
36 ÷ 6 = 6; 7 × 6 = 42 partsEquivalent ratios use the same scale factor on both quantities.
A machine makes 8 parts every 7 minutes. How many parts in 14 minutes at the same rate?
14 ÷ 7 = 2; 8 × 2 = 16 partsEquivalent ratios use the same scale factor on both quantities.
A machine makes 9 parts every 2 minutes. How many parts in 6 minutes at the same rate?
6 ÷ 2 = 3; 9 × 3 = 27 partsEquivalent ratios use the same scale factor on both quantities.
A machine makes 10 parts every 3 minutes. How many parts in 12 minutes at the same rate?
12 ÷ 3 = 4; 10 × 4 = 40 partsEquivalent 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.
A machine makes 4 parts every 5 minutes. How many parts in 30 minutes at the same rate?
30 ÷ 5 = 6; 4 × 6 = 24 partsEquivalent ratios use the same scale factor on both quantities.
A machine makes 5 parts every 6 minutes. How many parts in 12 minutes at the same rate?
12 ÷ 6 = 2; 5 × 2 = 10 partsEquivalent ratios use the same scale factor on both quantities.
A machine makes 6 parts every 7 minutes. How many parts in 21 minutes at the same rate?
21 ÷ 7 = 3; 6 × 3 = 18 partsEquivalent ratios use the same scale factor on both quantities.
A machine makes 7 parts every 2 minutes. How many parts in 8 minutes at the same rate?
8 ÷ 2 = 4; 7 × 4 = 28 partsEquivalent ratios use the same scale factor on both quantities.
A machine makes 8 parts every 3 minutes. How many parts in 15 minutes at the same rate?
15 ÷ 3 = 5; 8 × 5 = 40 partsEquivalent ratios use the same scale factor on both quantities.
A machine makes 9 parts every 4 minutes. How many parts in 24 minutes at the same rate?
24 ÷ 4 = 6; 9 × 6 = 54 partsEquivalent ratios use the same scale factor on both quantities.
A machine makes 10 parts every 5 minutes. How many parts in 10 minutes at the same rate?
10 ÷ 5 = 2; 10 × 2 = 20 partsEquivalent 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.
A machine makes 4 parts every 7 minutes. How many parts in 28 minutes at the same rate?
28 ÷ 7 = 4; 4 × 4 = 16 partsEquivalent ratios use the same scale factor on both quantities.
A machine makes 5 parts every 2 minutes. How many parts in 10 minutes at the same rate?
10 ÷ 2 = 5; 5 × 5 = 25 partsEquivalent ratios use the same scale factor on both quantities.
A machine makes 6 parts every 3 minutes. How many parts in 18 minutes at the same rate?
18 ÷ 3 = 6; 6 × 6 = 36 partsEquivalent ratios use the same scale factor on both quantities.
Practice
Guided practice with hints
01A machine makes 7 parts every 4 minutes. How many parts in 8 minutes at the same rate?
Hint: Represent the relationship first, keep units visible, and identify a second verification method.
Answer: 8 ÷ 4 = 2; 7 × 2 = 14 parts
02A machine makes 8 parts every 5 minutes. How many parts in 15 minutes at the same rate?
Hint: Represent the relationship first, keep units visible, and identify a second verification method.
Answer: 15 ÷ 5 = 3; 8 × 3 = 24 parts
03A machine makes 9 parts every 6 minutes. How many parts in 24 minutes at the same rate?
Hint: Represent the relationship first, keep units visible, and identify a second verification method.
Answer: 24 ÷ 6 = 4; 9 × 4 = 36 parts
04A machine makes 10 parts every 7 minutes. How many parts in 35 minutes at the same rate?
Hint: Represent the relationship first, keep units visible, and identify a second verification method.
Answer: 35 ÷ 7 = 5; 10 × 5 = 50 parts
05A machine makes 3 parts every 2 minutes. How many parts in 12 minutes at the same rate?
Hint: Represent the relationship first, keep units visible, and identify a second verification method.
Answer: 12 ÷ 2 = 6; 3 × 6 = 18 parts
06A machine makes 4 parts every 3 minutes. How many parts in 6 minutes at the same rate?
Hint: Represent the relationship first, keep units visible, and identify a second verification method.
Answer: 6 ÷ 3 = 2; 4 × 2 = 8 parts
Practice
Independent practice
01A 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
02A 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
03A 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
04A 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
05A machine makes 9 parts every 2 minutes. How many parts in 4 minutes at the same rate?
Answer: 4 ÷ 2 = 2; 9 × 2 = 18 parts
06A machine makes 10 parts every 3 minutes. How many parts in 9 minutes at the same rate?
Answer: 9 ÷ 3 = 3; 10 × 3 = 30 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
08A 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
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
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
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
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 Reasoning Sprint
Solve use proportions in multi-step contexts problems accurately and justify each result with a valid check.
Practice
Mastery check
01A machine makes 9 parts every 4 minutes. How many parts in 20 minutes at the same rate?
Answer: 20 ÷ 4 = 5; 9 × 5 = 45 parts
02A machine makes 10 parts every 5 minutes. How many parts in 30 minutes at the same rate?
Answer: 30 ÷ 5 = 6; 10 × 6 = 60 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
04A 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
05A 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
06A 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
07A 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
08A 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
California Mathematics Framework (2023)
Middle-school instructional guidance, mathematical practices, modeling, and coherent progression.Common Core State Standards for Mathematics — Grade 7
Grade-level expectations for ratios, number systems, algebra, geometry, statistics, and probability.Principles to Actions: Ensuring Mathematical Success for All
Research-informed emphasis on reasoning, representations, discourse, productive struggle, and conceptual understanding.