Ratio as a Relationship

Lesson 1 of 2: Introducing Ratios

In this lesson:

• Understand what a ratio is — and how it differs from a difference
• Use ratio language: "for every," "to," "per," "for each"
• Write ratios in all three notations and classify them as part-to-part or part-to-whole

Learning Objectives

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

1. Define a ratio as a relationship between two quantities — not a single number
2. Use ratio language — "for every," "to," "for each," "per" — to describe a ratio
3. Write a ratio in all three notations: a to b, a:b, and a/b
4. Identify whether a ratio is part-to-part or part-to-whole
5. Explain why a ratio is not the same as a fraction

The Candy Jar — What Do You Notice?

What do you notice about the two colors of candy?

Two Ways to Describe: Additive vs. Multiplicative

Additive description: "There are 4 more purple candies than yellow."

→ Tells you the difference

Multiplicative description: "For every 2 yellow candies, there are 3 purple."

→ Tells you the relationship — this is a ratio

Why ratios matter: If you doubled the jar (16 yellow, 24 purple), the difference becomes 8 — but the ratio is still 2 to 3. The relationship stays the same.

What Is a Ratio?

A ratio describes how two quantities are related to each other.
It is not a single number — it is a relationship.

Ratio language:

• "For every 2 yellow, there are 3 purple" → ratio of yellow to purple is 2 to 3
• "For each beak, there are 2 wings" → ratio of beaks to wings is 1 to 2
• "3 cups of fruit to 1 cup of yogurt" → ratio of fruit to yogurt is 3 to 1
• "$12 per hour" → ratio of dollars to hours is 12 to 1

Standard Examples: Wings, Beaks, and Votes

Bird example:

• 2 wings for every 1 beak
• Ratio of wings to beaks: 2 to 1
• "For every 2 wings, there is 1 beak" ✓
• "For every 1 beak, there are 2 wings" ✓ (same relationship, reversed order)

Voting example:

• For every vote Candidate A received, Candidate C received nearly 3 votes
• Ratio of A's votes to C's votes: 1 to 3 (approximately)

Order matters: "Wings to beaks is 2:1" is different from "beaks to wings is 1:2"

Your Turn: Write Ratio Statements

Use "for every" language and write the ratio for each:

1. A recipe uses 4 cups of flour and 2 cups of butter.

For every ___ cups of flour, there are ___ cups of butter.
Ratio of flour to butter: ___ to ___

2. A parking lot has 9 cars and 3 trucks.

For every ___ cars, there is/are ___ truck(s).
Ratio of cars to trucks: ___ to ___

Name the quantities in the order given — match your numbers to the right quantity.

Quick Check

Look at this statement: "The ratio of students to teachers is 28 to 2."

Which description is correct?

• A: There are 26 more students than teachers
• B: For every 28 students, there are 2 teachers
• C: There are 28 students and 2 teachers in the whole school

Think: what does a ratio tell you? What is it NOT telling you?

Three Ways to Write a Ratio

All three notations mean the same thing — choose based on context.

Notation Practice

Write each ratio in all three forms:

1. The ratio of girls to boys is "7 to 3"

2. A trail mix uses nuts and raisins in a 5:2 ratio

3. Write the ratio of seconds to minutes: 60 seconds for every 1 minute

Part-to-Part vs. Part-to-Whole

The same collection generates multiple ratio descriptions.

Worked Example: Baseball and Tennis Balls

Bag: 5 baseballs, 3 tennis balls (8 total)

Also: 8:12 from the candy jar and 2:3 are both correct — neither needs to be "simplified."

Ratio vs. Fraction — Not the Same

Same bag: 5 baseballs, 3 tennis balls, 8 total

Part-to-part ratio: 5:3 → written as 5/3

• Reads as "5 to 3" — NOT "five-thirds"
• Does NOT mean 5/3 of the balls are baseballs (that would be more than all the balls!)

Part-to-whole ratio: 5:8 → written as 5/8

• Reads as "5 to 8" — but this one DOES tell us the fraction of baseballs: 5 out of 8

Fraction: 5/8 names a single number (0.625) — "five-eighths of the balls are baseballs"

Only a part-to-whole ratio matches a fraction of the whole.

Your Turn: Classify and Write

A fruit basket has 6 apples and 4 oranges (10 total).

Complete the table:

Bonus: Which of the three ratios above matches the fraction "6 out of 10"?

Key Takeaways

✓ A ratio describes a multiplicative relationship — not a difference or a single number
✓ Ratio language: "for every," "for each," "to," "per" all signal a ratio
✓ Three notations: a to b = a:b = a/b — all mean the same thing
✓ Part-to-part: compares two parts — Part-to-whole: compares one part to the total
✓ Only a part-to-whole ratio can be interpreted as a fraction of the whole
✓ Both 8:12 and 2:3 are valid — ratios do not have to be simplified

Watch Out! Common Errors

"4 more purple" is NOT a ratio — that is additive comparison. Ratios use "for every."

Order matters. "Wings to beaks is 2:1" ≠ "beaks to wings is 2:1" — those mean different things.

5/3 does NOT mean 5/3 of the balls are baseballs. Only a part-to-whole ratio can be read as a fraction of the whole.

8:12 is NOT wrong just because it can be simplified. Unsimplified ratios are completely valid.

Coming Up: Lesson 2 of 2

In the next lesson, you will:

• Apply ratio language and all three notations to real-world scenarios
• Practice the full four-step ratio process: identify → describe → notate → classify
• Work through smoothie recipes, classroom preferences, and nature contexts

All five skills from today come together in Lesson 2.