Ratio Tools and Unit Rate Problems

Lesson 1 of 2: Building and Using the Toolkit

In this lesson:

• Build equivalent ratio tables and find missing values
• Plot ratio pairs on the coordinate plane
• Solve unit rate problems forward and backward

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 missing values
2. Plot ratio pairs on a coordinate plane and describe the pattern
3. Solve unit rate problems — forward and backward — across contexts

Four Tools for Ratio Problem-Solving

Applying All Four Tools to One Ratio

A lemonade recipe: 2 cups lemon juice for every 5 cups water

• Table: each row is an equivalent ratio
• Tape diagram: two bars that scale by the same factor
• Double number line: two parallel lines, values aligned
• Coordinate plane: ratio pairs form a straight line

Tool 1 — Equivalent Ratio Table

Lemonade: 2 cups lemon for every 5 cups water

• Multiply both values by the same factor each row
• Unit rate: cups water per cup lemon
• Check: every row divides to give the same unit rate

Tool 2 — Tape Diagram

Lemonade: 2 cups lemon for every 5 cups water

• Lemon bar: 2 equal segments; water bar: 5 equal segments
• If each segment = 3 cups → 6 cups lemon, 15 cups water
• Both bars scale together — the ratio stays constant

Tool 3 — Double Number Line

Lemonade: 2 cups lemon for every 5 cups water

One cup of lemon corresponds to 2.5 cups of water — that's the unit rate, read directly from the line.

Tool 4 — Coordinate Plane

Plot each ratio pair: (lemon, water)

• Points: (2,5), (4,10), (6,15), (8,20)
• They form a straight line through the origin
• Steepness of the line = the unit rate (2.5 cups water per cup lemon)

Using the Unit Rate to Find Missing Values

How much water for 7 cups of lemon?

Step 1 — divide: cups water per cup lemon

Step 2 — multiply: cups water

Check: ✓

Using Tables to Compare Two Ratios

Which recipe makes stronger lemonade?

• Recipe A: 2:5 → cups water per cup lemon
• Recipe B: 3:7 → cups water per cup lemon

Recipe B is stronger — less water per cup of lemon

Quick Check — Build a Ratio Table

New ratio: 3 cups orange juice for every 8 cups water

1. Build a ratio table with at least 4 rows
2. Find the unit rate
3. How much water for 9 cups of orange juice?

Pause and try before the next slide.

Every Unit Rate Problem Has Three Questions

1. Find the rate: divide to get the unit rate
2. Forward: rate × given quantity = answer
3. Backward: given amount ÷ rate = answer

Lawn-Mowing Example — All Three Questions

7 hours to mow 4 lawns

Q1 — Rate:

Q2 — Forward: lawns in 35 hours?

Q3: same rate — lawn/hr or hr/lawn

Unit Pricing — Compare and Calculate

Store A: 3 lb for $2.49 · Store B: 5 lb for $3.90

• Store A: /lb
• Store B: /lb → Better deal

For 8 pounds at Store B:

Worked Example — Constant Speed

A cyclist travels at 12 miles per hour.

Forward — how far in 2.5 hours?

Backward — how long to travel 42 miles?

Always check: do the units work out?

Unit Direction Error — Wrong vs. Right

Wrong direction (units don't work out):

Right direction (hours cancel):

Ask first: "What unit do I want in my answer?"

Practice — Apply the Three-Question Template

1 (unit pricing): 16-oz peanut butter $3.84; 28-oz $6.44.
Better deal? Cost for 40 oz at the better price?

2 (constant speed): Train at 75 mph.
Distance in 3.5 hr? Time for 300 miles?

Show unit rate; carry units through.

Practice Answers — Unit Rate Problems

1: /oz vs /oz → 28-oz jar wins
40 × $0.23 = $9.20

2: $75 \times 3.5 = $ 262.5 mi · $300 \div 75 = $ 4 hr

Units: mi/hr × hr = mi ✓ | mi ÷ mi/hr = hr ✓

Key Takeaways — Lesson 1

✓ Four ratio tools represent the same relationship in different ways

✓ To find a missing value in a ratio table: compute the unit rate, then multiply

✓ Plotting ratio pairs gives a straight line through the origin

✓ Unit rate problems: find the rate → use forward (×) or backward (÷)

Ratio tables grow by multiplication, not addition — (2,5), (4,10) not (3,6)

Check your units — the wrong direction gives a nonsense unit like hr²/lawn

Next Lesson — Percent and Unit Conversion

• Percent as a special rate — "per 100"
• Two percent problem types: find the part, find the whole
• Unit conversion using the cancellation method

The same ratio/rate thinking you practiced today applies directly.