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 today, you should be able to:

1. Build a table of equivalent ratios; find a missing value
2. Plot ratio pairs; explain collinearity (lying on one line)
3. Compare two ratios using tables or graphs
4. Compute both unit rates and choose the right 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 collinear (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?

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

All satisfy — 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

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

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 hours per lawn, how long to mow 12 lawns?

Step 1: Rate is 1.75 hours per lawn → answer is in hours

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 ounces, 3.60 dollars → dollars per ounce
• Box B: 18 ounces, 5.04 dollars → dollars per ounce

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 ≠ equivalent
→ Multiply both quantities by the same factor

Error 2 — Wrong direction: units don't cancel
→ Choose the rate whose denominator matches the given

Error 3 — No units: write "20 lawns" not "20"
→ Units confirm the setup

Key Ideas from Today's Lesson

✓ Equivalent ratios: multiply both quantities by the same factor
✓ Ratio pairs lie on a line through the origin
✓ Every ratio has two unit rates — question picks 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.