Ratio Tables and Unit Rates

Lesson 1 of 2: 6.RP.A.3

In this lesson:

• Build and use tables of equivalent ratios
• Plot ratio pairs and compare two ratios
• Solve unit-rate problems using constant speed and unit pricing

What You Will Learn Today

By the end of this lesson, you should be able to:

1. Build a table of equivalent ratios and find a missing value
2. Plot ratio pairs on the coordinate plane and explain collinearity
3. Compare two ratios using tables or graphs
4. Compute both unit rates and choose the correct direction

Can You Extend This Recipe?

A recipe calls for 3 cups of flour for every 4 cups of sugar.

• You want to use 6 cups of flour — how much sugar?
• What if you used 21 cups of flour?

What operation gets you from 3 to 6?

Build the Recipe Table Row by Row

Multiply BOTH quantities by the same factor.

Multiply Both — Never Just Add

Equivalent ratios: multiply both quantities by the same factor.

• ×2: (3, 4) → (6, 8) ✓
• ×7: (3, 4) → (21, 28) ✓
• +1 each time: (3, 4) → (4, 5) → (5, 6) → NOT equivalent ✗

Rule: Scale both quantities together, or the ratio changes.

Why the Graph Proves Equivalence

Multiplicative: all points on one line through (0, 0)

Additive: points miss the origin — different ratio every row

Scale Factor Strategy: Find Missing Values

Strategy: divide to find scale factor , then multiply.

21 cups flour → how much sugar?

32 cups sugar → how much flour?

Your Turn: Practice Scale Factors

A paint recipe uses 2 cups of blue for every 5 cups of yellow.

• You want to use 8 cups of blue. How many cups of yellow?
• You want 35 cups of yellow. How many cups of blue?

Write your scale factors before computing the answers.

Plotting Ratio Pairs on a Graph

Every pair lies on the line

Slope = unit rate: cups sugar per cup flour

Why Are These Points Always Collinear?

Every equivalent-ratio pair has the form :

• : — line passes through the origin
• :
• :

All satisfy — one line for all scale factors.

The slope equals the unit rate.

Compare Two Ratios Using a Table

Mix A: 2 blue : 3 yellow Mix B: 3 blue : 5 yellow

Scale to common yellow (15):

• Mix A ×5 → 10 blue, 15 yellow
• Mix B ×3 → 9 blue, 15 yellow

Same yellow, more blue → Mix A is bluer.

Tape Diagrams as a Third Representation

For ratio 2:3 (blue:yellow): tape has 5 equal parts total.

Example: A paint batch uses 25 cups total.

Check-In: Practice with a Tape Diagram

Paint mix 2:3 scaled to 20 cups total:

How many cups of blue? How many cups of yellow?

Draw the tape — label the parts, find one part's value, then scale.

Unit Rate Revisited from 6.RP.A.2

You already know: 75 dollars for 15 hamburgers.

Unit rate = divide until one quantity equals 1.

• 22 hamburgers cost dollars
• For 50 dollars: hamburgers

Every Ratio Has Two Unit Rates

7 hours to mow 4 lawns — both directions:

Both are valid. The question tells you which one you need.

Unit Cancellation Confirms the Right Direction

Unit cancellation is the structural check.

"How many lawns in 35 hours?" — answer needs lawns

Verify:

Constant Speed: Three Questions, One Rate

180 miles in 3 hours:

• Distance in 5 hr:
• Time for 240 mi:

Write units every step — they confirm the setup.

Guided Practice: Apply the Lawn Rate

At 1.75 hr/lawn, how long to mow 12 lawns?

Step 1: Rate is 1.75 hr/lawn → answer is in hr

Step 2: Set up:

Finish the computation. Lawns cancel; hours remain.

Comparative Unit Pricing: Which Is Cheaper?

Which cereal is the better deal per ounce?

• Box A: 12 oz / 3.60 dollars → dollars/oz
• Box B: 18 oz / 5.04 dollars → dollars/oz

Box B costs less per ounce — better deal.

Unscaffolded Transfer: Three Cyclist Questions

A cyclist rides 84 miles in 3.5 hours.

Show units at every step:

1. What is the speed in miles per hour?
2. How far in 6 hours at that speed?
3. How long to ride 120 miles?

No setup given — choose unit-rate direction each time.

Common Errors and How to Fix Them

Error 1 — Additive scaling: 2:3 → 3:4 → 4:5 ≠ equivalent
→ Multiply both quantities by the same factor

Error 2 — Wrong unit-rate direction: units don't cancel to the right answer
→ Check what units the answer needs; denominator unit must match the given

Error 3 — Skipping units: "20" instead of "20 lawns" or "20 hr"
→ Write units at every step — they confirm the setup is correct

Key Ideas from Today's Lesson

✓ Equivalent ratios multiply both quantities by the same factor
✓ Ratio pairs are collinear through the origin on a coordinate plane
✓ Every ratio has two unit rates — the question tells you which direction

Adding to both quantities does not preserve the ratio
Units that don't cancel → flip the rate

Coming Up in Lesson 2

In Lesson 2, the same scaling engine applies to:

• Percent — a ratio where the second quantity is always 100
• Unit conversion — a ratio that equals exactly 1 by definition

Same move. Two new surfaces.