Chapter 1 · Lesson 2 of 8 · Grade 6 · Ratios & Proportional Reasoning · MAT-06-RP-002
Represent Ratios with Tables and Diagrams
Represent and solve represent ratios with tables and diagrams 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 2 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 represent ratios with tables and diagrams problems using precise mathematical notation, models, and reasoning.
- Explain why a method for represent ratios with tables and diagrams 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 2 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
Represent Ratios with Tables and Diagrams 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 represent ratios with tables and diagrams 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.
A recipe uses 2 cups of A for 3 cups of B. Scale both quantities by 2.
2:3 = 4:6Multiplying both quantities by the same positive scale factor preserves the ratio.
Represent before calculating. Use a ratio table, tape diagram, double number line, or equation for this task, then solve: A recipe uses 3 cups of A for 5 cups of B. Scale both quantities by 3.
3:5 = 9:15The representation should show the same mathematical relationship as the calculation. Multiplying both quantities by the same positive scale factor preserves the ratio.
Explain why a valid method works, then solve: A recipe uses 4 cups of A for 7 cups of B. Scale both quantities by 4.
4:7 = 16:28A complete explanation names the relationship or property being preserved. Multiplying both quantities by the same positive scale factor preserves the ratio.
Estimate or predict first, then calculate and decide whether the result is reasonable: A recipe uses 5 cups of A for 9 cups of B. Scale both quantities by 5.
5:9 = 25:45The estimate is a reasonableness check, not a replacement for the exact result. Multiplying both quantities by the same positive scale factor preserves the ratio.
Solve and verify the result with a second method, inverse operation, or equivalent representation: A recipe uses 6 cups of A for 11 cups of B. Scale both quantities by 6.
6:11 = 36:66Verification should independently support the result. Multiplying both quantities by the same positive scale factor preserves the ratio.
Interpret the answer in context after solving. What does the result mean here? A recipe uses 7 cups of A for 3 cups of B. Scale both quantities by 7.
7:3 = 49:21State the result with its meaning, units, direction, or comparison—not only a number. Multiplying both quantities by the same positive scale factor preserves the ratio.
Error analysis: a student says, "Treating a ratio as additive." Explain the mistake, then solve this related task correctly: A recipe uses 8 cups of A for 5 cups of B. Scale both quantities by 2.
Check for a common scale factor or constant unit rate. Correct solution: 8:5 = 16:10Error analysis requires identifying the broken idea, not merely replacing the final answer. Multiplying both quantities by the same positive scale factor preserves the ratio.
Compare two related cases and explain what changes and what stays mathematically consistent. Case A: A recipe uses 9 cups of A for 7 cups of B. Scale both quantities by 3. Case B: A recipe uses 9 cups of A for 5 cups of B. Scale both quantities by 6.
Case A: 9:7 = 27:21 Case B: 9:5 = 54:30Compare the structure, representation, units, and result rather than only the numbers. Multiplying both quantities by the same positive scale factor preserves the ratio.
A recipe uses 10 cups of A for 9 cups of B. Scale both quantities by 4.
10:9 = 40:36Multiplying both quantities by the same positive scale factor preserves the ratio.
Represent before calculating. Use a ratio table, tape diagram, double number line, or equation for this task, then solve: A recipe uses 2 cups of A for 11 cups of B. Scale both quantities by 5.
2:11 = 10:55The representation should show the same mathematical relationship as the calculation. Multiplying both quantities by the same positive scale factor preserves the ratio.
Explain why a valid method works, then solve: A recipe uses 3 cups of A for 3 cups of B. Scale both quantities by 6.
3:3 = 18:18A complete explanation names the relationship or property being preserved. Multiplying both quantities by the same positive scale factor preserves the ratio.
Estimate or predict first, then calculate and decide whether the result is reasonable: A recipe uses 4 cups of A for 5 cups of B. Scale both quantities by 7.
4:5 = 28:35The estimate is a reasonableness check, not a replacement for the exact result. Multiplying both quantities by the same positive scale factor preserves the ratio.
Solve and verify the result with a second method, inverse operation, or equivalent representation: A recipe uses 5 cups of A for 7 cups of B. Scale both quantities by 2.
5:7 = 10:14Verification should independently support the result. Multiplying both quantities by the same positive scale factor preserves the ratio.
Interpret the answer in context after solving. What does the result mean here? A recipe uses 6 cups of A for 9 cups of B. Scale both quantities by 3.
6:9 = 18:27State the result with its meaning, units, direction, or comparison—not only a number. Multiplying both quantities by the same positive scale factor preserves the ratio.
Error analysis: a student says, "Treating a ratio as additive." Explain the mistake, then solve this related task correctly: A recipe uses 7 cups of A for 11 cups of B. Scale both quantities by 4.
Check for a common scale factor or constant unit rate. Correct solution: 7:11 = 28:44Error analysis requires identifying the broken idea, not merely replacing the final answer. Multiplying both quantities by the same positive scale factor preserves the ratio.
Compare two related cases and explain what changes and what stays mathematically consistent. Case A: A recipe uses 8 cups of A for 3 cups of B. Scale both quantities by 5. Case B: A recipe uses 8 cups of A for 11 cups of B. Scale both quantities by 2.
Case A: 8:3 = 40:15 Case B: 8:11 = 16:22Compare the structure, representation, units, and result rather than only the numbers. Multiplying both quantities by the same positive scale factor preserves the ratio.
A recipe uses 9 cups of A for 5 cups of B. Scale both quantities by 6.
9:5 = 54:30Multiplying both quantities by the same positive scale factor preserves the ratio.
Represent before calculating. Use a ratio table, tape diagram, double number line, or equation for this task, then solve: A recipe uses 10 cups of A for 7 cups of B. Scale both quantities by 7.
10:7 = 70:49The representation should show the same mathematical relationship as the calculation. Multiplying both quantities by the same positive scale factor preserves the ratio.
Explain why a valid method works, then solve: A recipe uses 2 cups of A for 9 cups of B. Scale both quantities by 2.
2:9 = 4:18A complete explanation names the relationship or property being preserved. Multiplying both quantities by the same positive scale factor preserves the ratio.
Estimate or predict first, then calculate and decide whether the result is reasonable: A recipe uses 3 cups of A for 11 cups of B. Scale both quantities by 3.
3:11 = 9:33The estimate is a reasonableness check, not a replacement for the exact result. Multiplying both quantities by the same positive scale factor preserves the ratio.
Practice
Guided practice with hints
01Solve and verify the result with a second method, inverse operation, or equivalent representation: A recipe uses 4 cups of A for 3 cups of B. Scale both quantities by 4.
Hint: Choose a representation before computing.
Answer: 4:3 = 16:12
02Interpret the answer in context after solving. What does the result mean here? A recipe uses 5 cups of A for 5 cups of B. Scale both quantities by 5.
Hint: Explain what relationship or property makes your method valid.
Answer: 5:5 = 25:25
03Error analysis: a student says, "Treating a ratio as additive." Explain the mistake, then solve this related task correctly: A recipe uses 6 cups of A for 7 cups of B. Scale both quantities by 6.
Hint: Choose a representation before computing.
Answer: Check for a common scale factor or constant unit rate. Correct solution: 6:7 = 36:42
04Compare two related cases and explain what changes and what stays mathematically consistent. Case A: A recipe uses 7 cups of A for 9 cups of B. Scale both quantities by 7. Case B: A recipe uses 7 cups of A for 7 cups of B. Scale both quantities by 4.
Hint: Explain what relationship or property makes your method valid.
Answer: Case A: 7:9 = 49:63 Case B: 7:7 = 28:28
05A recipe uses 8 cups of A for 11 cups of B. Scale both quantities by 2.
Hint: Choose a representation before computing.
Answer: 8:11 = 16:22
06Represent before calculating. Use a ratio table, tape diagram, double number line, or equation for this task, then solve: A recipe uses 9 cups of A for 3 cups of B. Scale both quantities by 3.
Hint: Explain what relationship or property makes your method valid.
Answer: 9:3 = 27:9
Practice
Independent practice
01Explain why a valid method works, then solve: A recipe uses 10 cups of A for 5 cups of B. Scale both quantities by 4.
Answer: 10:5 = 40:20
02Estimate or predict first, then calculate and decide whether the result is reasonable: A recipe uses 2 cups of A for 7 cups of B. Scale both quantities by 5.
Answer: 2:7 = 10:35
03Solve and verify the result with a second method, inverse operation, or equivalent representation: A recipe uses 3 cups of A for 9 cups of B. Scale both quantities by 6.
Answer: 3:9 = 18:54
04Interpret the answer in context after solving. What does the result mean here? A recipe uses 4 cups of A for 11 cups of B. Scale both quantities by 7.
Answer: 4:11 = 28:77
05Error analysis: a student says, "Treating a ratio as additive." Explain the mistake, then solve this related task correctly: A recipe uses 5 cups of A for 3 cups of B. Scale both quantities by 2.
Answer: Check for a common scale factor or constant unit rate. Correct solution: 5:3 = 10:6
06Compare two related cases and explain what changes and what stays mathematically consistent. Case A: A recipe uses 6 cups of A for 5 cups of B. Scale both quantities by 3. Case B: A recipe uses 6 cups of A for 3 cups of B. Scale both quantities by 6.
Answer: Case A: 6:5 = 18:15 Case B: 6:3 = 36:18
07A recipe uses 7 cups of A for 7 cups of B. Scale both quantities by 4.
Answer: 7:7 = 28:28
08Represent before calculating. Use a ratio table, tape diagram, double number line, or equation for this task, then solve: A recipe uses 8 cups of A for 9 cups of B. Scale both quantities by 5.
Answer: 8:9 = 40:45
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 recipe uses 9 cups of A for 11 cups of B. Scale both quantities by 6.
9:11 = 54:66
Estimate or predict first, then calculate and decide whether the result is reasonable: A recipe uses 10 cups of A for 3 cups of B. Scale both quantities by 7.
10:3 = 70:21
Solve and verify the result with a second method, inverse operation, or equivalent representation: A recipe uses 2 cups of A for 5 cups of B. Scale both quantities by 2.
2:5 = 4:10
Interpret the answer in context after solving. What does the result mean here? A recipe uses 3 cups of A for 7 cups of B. Scale both quantities by 3.
3:7 = 9:21
Flashcards
Retrieve it from memory.
Flashcards
Card 1 of 4
Game mode
Grade 6 Chapter 1 Reasoning Lab
Solve represent ratios with tables and diagrams 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: A recipe uses 4 cups of A for 9 cups of B. Scale both quantities by 4.
Answer: Check for a common scale factor or constant unit rate. Correct solution: 4:9 = 16:36
02Compare two related cases and explain what changes and what stays mathematically consistent. Case A: A recipe uses 5 cups of A for 11 cups of B. Scale both quantities by 5. Case B: A recipe uses 5 cups of A for 9 cups of B. Scale both quantities by 2.
Answer: Case A: 5:11 = 25:55 Case B: 5:9 = 10:18
03A recipe uses 6 cups of A for 3 cups of B. Scale both quantities by 6.
Answer: 6:3 = 36:18
04Represent before calculating. Use a ratio table, tape diagram, double number line, or equation for this task, then solve: A recipe uses 7 cups of A for 5 cups of B. Scale both quantities by 7.
Answer: 7:5 = 49:35
05Explain why a valid method works, then solve: A recipe uses 8 cups of A for 7 cups of B. Scale both quantities by 2.
Answer: 8:7 = 16:14
06Estimate or predict first, then calculate and decide whether the result is reasonable: A recipe uses 9 cups of A for 9 cups of B. Scale both quantities by 3.
Answer: 9:9 = 27:27
07Solve and verify the result with a second method, inverse operation, or equivalent representation: A recipe uses 10 cups of A for 11 cups of B. Scale both quantities by 4.
Answer: 10:11 = 40:44
08Interpret the answer in context after solving. What does the result mean here? A recipe uses 2 cups of A for 3 cups of B. Scale both quantities by 5.
Answer: 2:3 = 10:15
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.