Chapter 1 · Lesson 2 of 9 · Grade 7 · Ratios & Proportional Reasoning · MAT-07-RP-002
Find the Constant of Proportionality
Represent and solve find the constant of proportionality 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 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 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 find the constant of proportionality problems using precise mathematical notation, models, and reasoning.
- Explain why a method for find the constant of proportionality 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 1: Proportional Relationships, Percent, and Financial Reasoning
Ratios & Proportional Reasoning. Lesson 2 of 9 extends earlier chapter reasoning.
Connection to the course
Use Grade 6 ratios, rates, fractions, and decimals as active knowledge. Connect today's representation to at least one earlier chapter representation.
Mathematical meaning
Find the Constant of Proportionality is about multiplicative comparison. Equivalent representations preserve the same scale relationship, not merely the same difference.
Represent the relationship
The Math Box uses ratio table 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.
Ratio Table Lab
Scale both quantities by the same factor and watch equivalent ratios stay aligned.
- Model find the constant of proportionality 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.
The point (3, 6) lies on y = kx. Find k.
k = 6/3 = 2The constant of proportionality is the unit rate y divided by x.
Represent before calculating. Use a ratio table, tape diagram, double number line, or equation for this task, then solve: The point (4, 12) lies on y = kx. Find k.
k = 12/4 = 3The representation should show the same mathematical relationship as the calculation. The constant of proportionality is the unit rate y divided by x.
Explain why a valid method works, then solve: The point (5, 20) lies on y = kx. Find k.
k = 20/5 = 4A complete explanation names the relationship or property being preserved. The constant of proportionality is the unit rate y divided by x.
Estimate or predict first, then calculate and decide whether the result is reasonable: The point (6, 30) lies on y = kx. Find k.
k = 30/6 = 5The estimate is a reasonableness check, not a replacement for the exact result. The constant of proportionality is the unit rate y divided by x.
Solve and verify the result with a second method, inverse operation, or equivalent representation: The point (7, 42) lies on y = kx. Find k.
k = 42/7 = 6Verification should independently support the result. The constant of proportionality is the unit rate y divided by x.
Interpret the answer in context after solving. What does the result mean here? The point (8, 56) lies on y = kx. Find k.
k = 56/8 = 7State the result with its meaning, units, direction, or comparison—not only a number. The constant of proportionality is the unit rate y divided by x.
Error analysis: a student says, "Treating a ratio as additive." Explain the mistake, then solve this related task correctly: The point (9, 72) lies on y = kx. Find k.
Check for a common scale factor or constant unit rate. Correct solution: k = 72/9 = 8Error analysis requires identifying the broken idea, not merely replacing the final answer. The constant of proportionality is the unit rate y divided by x.
Compare two related cases and explain what changes and what stays mathematically consistent. Case A: The point (10, 90) lies on y = kx. Find k. Case B: The point (3, 27) lies on y = kx. Find k.
Case A: k = 90/10 = 9 Case B: k = 27/3 = 9Compare the structure, representation, units, and result rather than only the numbers. The constant of proportionality is the unit rate y divided by x.
The point (3, 30) lies on y = kx. Find k.
k = 30/3 = 10The constant of proportionality is the unit rate y divided by x.
Represent before calculating. Use a ratio table, tape diagram, double number line, or equation for this task, then solve: The point (4, 8) lies on y = kx. Find k.
k = 8/4 = 2The representation should show the same mathematical relationship as the calculation. The constant of proportionality is the unit rate y divided by x.
Explain why a valid method works, then solve: The point (5, 15) lies on y = kx. Find k.
k = 15/5 = 3A complete explanation names the relationship or property being preserved. The constant of proportionality is the unit rate y divided by x.
Estimate or predict first, then calculate and decide whether the result is reasonable: The point (6, 24) lies on y = kx. Find k.
k = 24/6 = 4The estimate is a reasonableness check, not a replacement for the exact result. The constant of proportionality is the unit rate y divided by x.
Solve and verify the result with a second method, inverse operation, or equivalent representation: The point (7, 35) lies on y = kx. Find k.
k = 35/7 = 5Verification should independently support the result. The constant of proportionality is the unit rate y divided by x.
Interpret the answer in context after solving. What does the result mean here? The point (8, 48) lies on y = kx. Find k.
k = 48/8 = 6State the result with its meaning, units, direction, or comparison—not only a number. The constant of proportionality is the unit rate y divided by x.
Error analysis: a student says, "Treating a ratio as additive." Explain the mistake, then solve this related task correctly: The point (9, 63) lies on y = kx. Find k.
Check for a common scale factor or constant unit rate. Correct solution: k = 63/9 = 7Error analysis requires identifying the broken idea, not merely replacing the final answer. The constant of proportionality is the unit rate y divided by x.
Compare two related cases and explain what changes and what stays mathematically consistent. Case A: The point (10, 80) lies on y = kx. Find k. Case B: The point (3, 24) lies on y = kx. Find k.
Case A: k = 80/10 = 8 Case B: k = 24/3 = 8Compare the structure, representation, units, and result rather than only the numbers. The constant of proportionality is the unit rate y divided by x.
The point (3, 27) lies on y = kx. Find k.
k = 27/3 = 9The constant of proportionality is the unit rate y divided by x.
Represent before calculating. Use a ratio table, tape diagram, double number line, or equation for this task, then solve: The point (4, 40) lies on y = kx. Find k.
k = 40/4 = 10The representation should show the same mathematical relationship as the calculation. The constant of proportionality is the unit rate y divided by x.
Explain why a valid method works, then solve: The point (5, 10) lies on y = kx. Find k.
k = 10/5 = 2A complete explanation names the relationship or property being preserved. The constant of proportionality is the unit rate y divided by x.
Estimate or predict first, then calculate and decide whether the result is reasonable: The point (6, 18) lies on y = kx. Find k.
k = 18/6 = 3The estimate is a reasonableness check, not a replacement for the exact result. The constant of proportionality is the unit rate y divided by x.
Practice
Guided practice with hints
01Solve and verify the result with a second method, inverse operation, or equivalent representation: The point (7, 28) lies on y = kx. Find k.
Hint: Choose a representation before computing.
Answer: k = 28/7 = 4
02Interpret the answer in context after solving. What does the result mean here? The point (8, 40) lies on y = kx. Find k.
Hint: Explain why the method works and name a verification strategy.
Answer: k = 40/8 = 5
03Error analysis: a student says, "Treating a ratio as additive." Explain the mistake, then solve this related task correctly: The point (9, 54) lies on y = kx. Find k.
Hint: Choose a representation before computing.
Answer: Check for a common scale factor or constant unit rate. Correct solution: k = 54/9 = 6
04Compare two related cases and explain what changes and what stays mathematically consistent. Case A: The point (10, 70) lies on y = kx. Find k. Case B: The point (3, 21) lies on y = kx. Find k.
Hint: Explain why the method works and name a verification strategy.
Answer: Case A: k = 70/10 = 7 Case B: k = 21/3 = 7
05The point (3, 24) lies on y = kx. Find k.
Hint: Choose a representation before computing.
Answer: k = 24/3 = 8
06Represent before calculating. Use a ratio table, tape diagram, double number line, or equation for this task, then solve: The point (4, 36) lies on y = kx. Find k.
Hint: Explain why the method works and name a verification strategy.
Answer: k = 36/4 = 9
Practice
Independent practice
01Explain why a valid method works, then solve: The point (5, 50) lies on y = kx. Find k.
Answer: k = 50/5 = 10
02Estimate or predict first, then calculate and decide whether the result is reasonable: The point (6, 12) lies on y = kx. Find k.
Answer: k = 12/6 = 2
03Solve and verify the result with a second method, inverse operation, or equivalent representation: The point (7, 21) lies on y = kx. Find k.
Answer: k = 21/7 = 3
04Interpret the answer in context after solving. What does the result mean here? The point (8, 32) lies on y = kx. Find k.
Answer: k = 32/8 = 4
05Error analysis: a student says, "Treating a ratio as additive." Explain the mistake, then solve this related task correctly: The point (9, 45) lies on y = kx. Find k.
Answer: Check for a common scale factor or constant unit rate. Correct solution: k = 45/9 = 5
06Compare two related cases and explain what changes and what stays mathematically consistent. Case A: The point (10, 60) lies on y = kx. Find k. Case B: The point (3, 18) lies on y = kx. Find k.
Answer: Case A: k = 60/10 = 6 Case B: k = 18/3 = 6
07The point (3, 21) lies on y = kx. Find k.
Answer: k = 21/3 = 7
08Represent before calculating. Use a ratio table, tape diagram, double number line, or equation for this task, then solve: The point (4, 32) lies on y = kx. Find k.
Answer: k = 32/4 = 8
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: The point (5, 45) lies on y = kx. Find k.
k = 45/5 = 9
Estimate or predict first, then calculate and decide whether the result is reasonable: The point (6, 60) lies on y = kx. Find k.
k = 60/6 = 10
Solve and verify the result with a second method, inverse operation, or equivalent representation: The point (7, 14) lies on y = kx. Find k.
k = 14/7 = 2
Interpret the answer in context after solving. What does the result mean here? The point (8, 24) lies on y = kx. Find k.
k = 24/8 = 3
Flashcards
Retrieve it from memory.
Flashcards
Card 1 of 4
Game mode
Grade 7 Chapter 1 Reasoning Lab
Solve find the constant of proportionality 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: The point (9, 36) lies on y = kx. Find k.
Answer: Check for a common scale factor or constant unit rate. Correct solution: k = 36/9 = 4
02Compare two related cases and explain what changes and what stays mathematically consistent. Case A: The point (10, 50) lies on y = kx. Find k. Case B: The point (3, 15) lies on y = kx. Find k.
Answer: Case A: k = 50/10 = 5 Case B: k = 15/3 = 5
03The point (3, 18) lies on y = kx. Find k.
Answer: k = 18/3 = 6
04Represent before calculating. Use a ratio table, tape diagram, double number line, or equation for this task, then solve: The point (4, 28) lies on y = kx. Find k.
Answer: k = 28/4 = 7
05Explain why a valid method works, then solve: The point (5, 40) lies on y = kx. Find k.
Answer: k = 40/5 = 8
06Estimate or predict first, then calculate and decide whether the result is reasonable: The point (6, 54) lies on y = kx. Find k.
Answer: k = 54/6 = 9
07Solve and verify the result with a second method, inverse operation, or equivalent representation: The point (7, 70) lies on y = kx. Find k.
Answer: k = 70/7 = 10
08Interpret the answer in context after solving. What does the result mean here? The point (8, 16) lies on y = kx. Find k.
Answer: k = 16/8 = 2
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.