Chapter 1 · Lesson 4 of 8 · Grade 6 · Ratios & Proportional Reasoning · MAT-06-RP-004
Compare Unit Rates
Represent and solve compare unit rates problems using precise mathematical notation, models, and reasoning.
Math V4 · Course chapter
Chapter 1: Ratios, Rates, and Percent
Comparing quantities multiplicatively
Essential question: How can one multiplicative relationship be represented, scaled, interpreted, and used to make decisions?
Where this lesson fits
Lesson 4 of 8. Each lesson builds on earlier chapter representations and ideas rather than resetting the topic.
Chapter destination
Plan a small event using recipes, prices, unit rates, percentages, and unit conversions. Defend every comparison with a table, diagram, or equation.
Learning objectives
What you should be able to do
- Represent and solve compare unit rates problems using precise mathematical notation, models, and reasoning.
- Explain why a method for compare unit rates 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.
Learn
Build the idea from meaning, not memorization.
Chapter 1: Ratios, Rates, and Percent
Comparing quantities multiplicatively. This is Lesson 4 of 8 in Chapter 1. It builds on earlier chapter ideas instead of restarting the topic from scratch.
Connection to the course
Cumulative knowledge used here includes Grade 5 fractions, decimal multiplication and division, measurement conversions. As you work, connect today's idea to earlier lessons in the chapter and keep those earlier representations available for comparison.
Mathematical meaning
Compare Unit Rates 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 one example of compare unit rates in the Math Box.
- Change one input, predict the effect, and test the prediction.
- Connect the model to another representation used earlier in this chapter.
- State the relationship or invariant that explains what stayed mathematically consistent.
Worked examples
Twenty different ways to see the concept work.
Plan A gives 3 units for 4 dollars; Plan B gives 4 units for 7 dollars. Which has the greater units-per-dollar rate?
A: 0.75; B: 0.571; A is greaterConvert both comparisons to the same unit rate before deciding.
Represent before calculating. Use a ratio table, tape diagram, double number line, or equation for this task, then solve: Plan A gives 4 units for 5 dollars; Plan B gives 6 units for 8 dollars. Which has the greater units-per-dollar rate?
A: 0.8; B: 0.75; A is greaterThe representation should show the same mathematical relationship as the calculation. Convert both comparisons to the same unit rate before deciding.
Explain why a valid method works, then solve: Plan A gives 5 units for 6 dollars; Plan B gives 8 units for 9 dollars. Which has the greater units-per-dollar rate?
A: 0.833; B: 0.889; B is greaterA complete explanation names the relationship or property being preserved. Convert both comparisons to the same unit rate before deciding.
Estimate or predict first, then calculate and decide whether the result is reasonable: Plan A gives 6 units for 7 dollars; Plan B gives 10 units for 10 dollars. Which has the greater units-per-dollar rate?
A: 0.857; B: 1; B is greaterThe estimate is a reasonableness check, not a replacement for the exact result. Convert both comparisons to the same unit rate before deciding.
Solve and verify the result with a second method, inverse operation, or equivalent representation: Plan A gives 7 units for 8 dollars; Plan B gives 12 units for 11 dollars. Which has the greater units-per-dollar rate?
A: 0.875; B: 1.091; B is greaterVerification should independently support the result. Convert both comparisons to the same unit rate before deciding.
Interpret the answer in context after solving. What does the result mean here? Plan A gives 8 units for 9 dollars; Plan B gives 5 units for 5 dollars. Which has the greater units-per-dollar rate?
A: 0.889; B: 1; B is greaterState the result with its meaning, units, direction, or comparison—not only a number. Convert both comparisons to the same unit rate before deciding.
Error analysis: a student says, "Treating a ratio as additive." Explain the mistake, then solve this related task correctly: Plan A gives 9 units for 10 dollars; Plan B gives 7 units for 6 dollars. Which has the greater units-per-dollar rate?
Check for a common scale factor or constant unit rate. Correct solution: A: 0.9; B: 1.167; B is greaterError analysis requires identifying the broken idea, not merely replacing the final answer. Convert both comparisons to the same unit rate before deciding.
Compare two related cases and explain what changes and what stays mathematically consistent. Case A: Plan A gives 10 units for 4 dollars; Plan B gives 9 units for 7 dollars. Which has the greater units-per-dollar rate? Case B: Plan A gives 10 units for 6 dollars; Plan B gives 9 units for 9 dollars. Which has the greater units-per-dollar rate?
Case A: A: 2.5; B: 1.286; A is greater Case B: A: 1.667; B: 1; A is greaterCompare the structure, representation, units, and result rather than only the numbers. Convert both comparisons to the same unit rate before deciding.
Plan A gives 11 units for 5 dollars; Plan B gives 11 units for 8 dollars. Which has the greater units-per-dollar rate?
A: 2.2; B: 1.375; A is greaterConvert both comparisons to the same unit rate before deciding.
Represent before calculating. Use a ratio table, tape diagram, double number line, or equation for this task, then solve: Plan A gives 3 units for 6 dollars; Plan B gives 4 units for 9 dollars. Which has the greater units-per-dollar rate?
A: 0.5; B: 0.444; A is greaterThe representation should show the same mathematical relationship as the calculation. Convert both comparisons to the same unit rate before deciding.
Explain why a valid method works, then solve: Plan A gives 4 units for 7 dollars; Plan B gives 6 units for 10 dollars. Which has the greater units-per-dollar rate?
A: 0.571; B: 0.6; B is greaterA complete explanation names the relationship or property being preserved. Convert both comparisons to the same unit rate before deciding.
Estimate or predict first, then calculate and decide whether the result is reasonable: Plan A gives 5 units for 8 dollars; Plan B gives 8 units for 11 dollars. Which has the greater units-per-dollar rate?
A: 0.625; B: 0.727; B is greaterThe estimate is a reasonableness check, not a replacement for the exact result. Convert both comparisons to the same unit rate before deciding.
Solve and verify the result with a second method, inverse operation, or equivalent representation: Plan A gives 6 units for 9 dollars; Plan B gives 10 units for 5 dollars. Which has the greater units-per-dollar rate?
A: 0.667; B: 2; B is greaterVerification should independently support the result. Convert both comparisons to the same unit rate before deciding.
Interpret the answer in context after solving. What does the result mean here? Plan A gives 7 units for 10 dollars; Plan B gives 12 units for 6 dollars. Which has the greater units-per-dollar rate?
A: 0.7; B: 2; B is greaterState the result with its meaning, units, direction, or comparison—not only a number. Convert both comparisons to the same unit rate before deciding.
Error analysis: a student says, "Treating a ratio as additive." Explain the mistake, then solve this related task correctly: Plan A gives 8 units for 4 dollars; Plan B gives 5 units for 7 dollars. Which has the greater units-per-dollar rate?
Check for a common scale factor or constant unit rate. Correct solution: A: 2; B: 0.714; A is greaterError analysis requires identifying the broken idea, not merely replacing the final answer. Convert both comparisons to the same unit rate before deciding.
Compare two related cases and explain what changes and what stays mathematically consistent. Case A: Plan A gives 9 units for 5 dollars; Plan B gives 7 units for 8 dollars. Which has the greater units-per-dollar rate? Case B: Plan A gives 9 units for 7 dollars; Plan B gives 7 units for 10 dollars. Which has the greater units-per-dollar rate?
Case A: A: 1.8; B: 0.875; A is greater Case B: A: 1.286; B: 0.7; A is greaterCompare the structure, representation, units, and result rather than only the numbers. Convert both comparisons to the same unit rate before deciding.
Plan A gives 10 units for 6 dollars; Plan B gives 9 units for 9 dollars. Which has the greater units-per-dollar rate?
A: 1.667; B: 1; A is greaterConvert both comparisons to the same unit rate before deciding.
Represent before calculating. Use a ratio table, tape diagram, double number line, or equation for this task, then solve: Plan A gives 11 units for 7 dollars; Plan B gives 11 units for 10 dollars. Which has the greater units-per-dollar rate?
A: 1.571; B: 1.1; A is greaterThe representation should show the same mathematical relationship as the calculation. Convert both comparisons to the same unit rate before deciding.
Explain why a valid method works, then solve: Plan A gives 3 units for 8 dollars; Plan B gives 4 units for 11 dollars. Which has the greater units-per-dollar rate?
A: 0.375; B: 0.364; A is greaterA complete explanation names the relationship or property being preserved. Convert both comparisons to the same unit rate before deciding.
Estimate or predict first, then calculate and decide whether the result is reasonable: Plan A gives 4 units for 9 dollars; Plan B gives 6 units for 5 dollars. Which has the greater units-per-dollar rate?
A: 0.444; B: 1.2; B is greaterThe estimate is a reasonableness check, not a replacement for the exact result. Convert both comparisons to the same unit rate before deciding.
Practice
Guided practice with hints
01Solve and verify the result with a second method, inverse operation, or equivalent representation: Plan A gives 5 units for 10 dollars; Plan B gives 8 units for 6 dollars. Which has the greater units-per-dollar rate?
Hint: Choose a representation before computing.
Answer: A: 0.5; B: 1.333; B is greater
02Interpret the answer in context after solving. What does the result mean here? Plan A gives 6 units for 4 dollars; Plan B gives 10 units for 7 dollars. Which has the greater units-per-dollar rate?
Hint: Explain what relationship or property makes your method valid.
Answer: A: 1.5; B: 1.429; A is greater
03Error analysis: a student says, "Treating a ratio as additive." Explain the mistake, then solve this related task correctly: Plan A gives 7 units for 5 dollars; Plan B gives 12 units for 8 dollars. Which has the greater units-per-dollar rate?
Hint: Choose a representation before computing.
Answer: Check for a common scale factor or constant unit rate. Correct solution: A: 1.4; B: 1.5; B is greater
04Compare two related cases and explain what changes and what stays mathematically consistent. Case A: Plan A gives 8 units for 6 dollars; Plan B gives 5 units for 9 dollars. Which has the greater units-per-dollar rate? Case B: Plan A gives 8 units for 8 dollars; Plan B gives 5 units for 11 dollars. Which has the greater units-per-dollar rate?
Hint: Explain what relationship or property makes your method valid.
Answer: Case A: A: 1.333; B: 0.556; A is greater Case B: A: 1; B: 0.455; A is greater
05Plan A gives 9 units for 7 dollars; Plan B gives 7 units for 10 dollars. Which has the greater units-per-dollar rate?
Hint: Choose a representation before computing.
Answer: A: 1.286; B: 0.7; A is greater
06Represent before calculating. Use a ratio table, tape diagram, double number line, or equation for this task, then solve: Plan A gives 10 units for 8 dollars; Plan B gives 9 units for 11 dollars. Which has the greater units-per-dollar rate?
Hint: Explain what relationship or property makes your method valid.
Answer: A: 1.25; B: 0.818; A is greater
Practice
Independent practice
01Explain why a valid method works, then solve: Plan A gives 11 units for 9 dollars; Plan B gives 11 units for 5 dollars. Which has the greater units-per-dollar rate?
Answer: A: 1.222; B: 2.2; B is greater
02Estimate or predict first, then calculate and decide whether the result is reasonable: Plan A gives 3 units for 10 dollars; Plan B gives 4 units for 6 dollars. Which has the greater units-per-dollar rate?
Answer: A: 0.3; B: 0.667; B is greater
03Solve and verify the result with a second method, inverse operation, or equivalent representation: Plan A gives 4 units for 4 dollars; Plan B gives 6 units for 7 dollars. Which has the greater units-per-dollar rate?
Answer: A: 1; B: 0.857; A is greater
04Interpret the answer in context after solving. What does the result mean here? Plan A gives 5 units for 5 dollars; Plan B gives 8 units for 8 dollars. Which has the greater units-per-dollar rate?
Answer: A: 1; B: 1; B is greater
05Error analysis: a student says, "Treating a ratio as additive." Explain the mistake, then solve this related task correctly: Plan A gives 6 units for 6 dollars; Plan B gives 10 units for 9 dollars. Which has the greater units-per-dollar rate?
Answer: Check for a common scale factor or constant unit rate. Correct solution: A: 1; B: 1.111; B is greater
06Compare two related cases and explain what changes and what stays mathematically consistent. Case A: Plan A gives 7 units for 7 dollars; Plan B gives 12 units for 10 dollars. Which has the greater units-per-dollar rate? Case B: Plan A gives 7 units for 9 dollars; Plan B gives 12 units for 5 dollars. Which has the greater units-per-dollar rate?
Answer: Case A: A: 1; B: 1.2; B is greater Case B: A: 0.778; B: 2.4; B is greater
07Plan A gives 8 units for 8 dollars; Plan B gives 5 units for 11 dollars. Which has the greater units-per-dollar rate?
Answer: A: 1; B: 0.455; A is greater
08Represent before calculating. Use a ratio table, tape diagram, double number line, or equation for this task, then solve: Plan A gives 9 units for 9 dollars; Plan B gives 7 units for 5 dollars. Which has the greater units-per-dollar rate?
Answer: A: 1; B: 1.4; B is greater
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: Plan A gives 10 units for 10 dollars; Plan B gives 9 units for 6 dollars. Which has the greater units-per-dollar rate?
A: 1; B: 1.5; B is greater
Estimate or predict first, then calculate and decide whether the result is reasonable: Plan A gives 11 units for 4 dollars; Plan B gives 11 units for 7 dollars. Which has the greater units-per-dollar rate?
A: 2.75; B: 1.571; A is greater
Solve and verify the result with a second method, inverse operation, or equivalent representation: Plan A gives 3 units for 5 dollars; Plan B gives 4 units for 8 dollars. Which has the greater units-per-dollar rate?
A: 0.6; B: 0.5; A is greater
Interpret the answer in context after solving. What does the result mean here? Plan A gives 4 units for 6 dollars; Plan B gives 6 units for 9 dollars. Which has the greater units-per-dollar rate?
A: 0.667; B: 0.667; B is greater
Flashcards
Retrieve it from memory.
Flashcards
Card 1 of 4
Game mode
Grade 6 Chapter 1 Reasoning Lab
Solve compare unit rates problems accurately, choose an appropriate representation, and justify each result with a check.
Practice
Mastery check
01Error analysis: a student says, "Treating a ratio as additive." Explain the mistake, then solve this related task correctly: Plan A gives 5 units for 7 dollars; Plan B gives 8 units for 10 dollars. Which has the greater units-per-dollar rate?
Answer: Check for a common scale factor or constant unit rate. Correct solution: A: 0.714; B: 0.8; B is greater
02Compare two related cases and explain what changes and what stays mathematically consistent. Case A: Plan A gives 6 units for 8 dollars; Plan B gives 10 units for 11 dollars. Which has the greater units-per-dollar rate? Case B: Plan A gives 6 units for 10 dollars; Plan B gives 10 units for 6 dollars. Which has the greater units-per-dollar rate?
Answer: Case A: A: 0.75; B: 0.909; B is greater Case B: A: 0.6; B: 1.667; B is greater
03Plan A gives 7 units for 9 dollars; Plan B gives 12 units for 5 dollars. Which has the greater units-per-dollar rate?
Answer: A: 0.778; B: 2.4; B is greater
04Represent before calculating. Use a ratio table, tape diagram, double number line, or equation for this task, then solve: Plan A gives 8 units for 10 dollars; Plan B gives 5 units for 6 dollars. Which has the greater units-per-dollar rate?
Answer: A: 0.8; B: 0.833; B is greater
05Explain why a valid method works, then solve: Plan A gives 9 units for 4 dollars; Plan B gives 7 units for 7 dollars. Which has the greater units-per-dollar rate?
Answer: A: 2.25; B: 1; A is greater
06Estimate or predict first, then calculate and decide whether the result is reasonable: Plan A gives 10 units for 5 dollars; Plan B gives 9 units for 8 dollars. Which has the greater units-per-dollar rate?
Answer: A: 2; B: 1.125; A is greater
07Solve and verify the result with a second method, inverse operation, or equivalent representation: Plan A gives 11 units for 6 dollars; Plan B gives 11 units for 9 dollars. Which has the greater units-per-dollar rate?
Answer: A: 1.833; B: 1.222; A is greater
08Interpret the answer in context after solving. What does the result mean here? Plan A gives 3 units for 7 dollars; Plan B gives 4 units for 10 dollars. Which has the greater units-per-dollar rate?
Answer: A: 0.429; B: 0.4; A is greater
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 6 scope, sequence, standards, and public task structure
Reference for coherent sequencing, representations, dependency-aware progression, and reasoning-rich task types.Eureka Math² Grade 6 program structure
Reference for module/topic coherence, recap, mixed-practice, and cumulative-learning patterns.enVision Mathematics Grade 6 instructional model
Reference for problem-based entry points, visual learning, modeling, and application patterns.Common Core State Standards for Mathematics — Grade 6
Grade-level content and mathematical-practice expectations.