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Teacher Guide: Understanding Ratios

Learn what ratios are and how to use them to compare quantities in everyday situations.

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10 questions on Ratios. Students join with a name, you see everyone's score.

For Teachers

Learning Objectives
  • Define a ratio as a comparison of two quantities
  • Write ratios in three different forms (colon, word, fraction)
  • Simplify ratios using greatest common factor
  • Distinguish between part-to-part and part-to-whole ratios
  • Apply ratios to solve real-world problems
Prerequisites
  • • Understanding of multiplication and division
  • • Knowledge of fractions
  • • Ability to find greatest common factor (GCF)
Discussion Starters
  • 1. Where have you seen ratios used in your daily life?
  • 2. If a recipe calls for a 2:12:1 ratio of flour to sugar, what happens if you reverse it to 1:21:2?
  • 3. A class has 15 boys and 10 girls. How would you describe this using a ratio?
  • 4. Why is it important that ratios can be simplified? When might you want the original numbers?
Common Misconceptions

Thinking 3:53:5 is the same as 5:35:3

Remediation: Use concrete examples: '3 cats to 5 dogs' is very different from '5 cats to 3 dogs'. Have students draw pictures to see the difference.

Adding instead of multiplying to find equivalent ratios

Remediation: Show that 2:32:3 plus 2 to each gives 4:54:5, which is NOT equivalent. Demonstrate with tape diagrams that multiplying preserves the relationship.

Differentiation Ideas

For Struggling Students:

  • • Use physical manipulatives (colored blocks) to build ratios
  • • Start with simple ratios like 1:21:2 or 2:12:1
  • • Focus on part-to-part ratios before introducing part-to-whole

For On-Level Students:

  • • Practice writing ratios in all three forms
  • • Solve word problems involving finding missing values
  • • Compare ratios by finding common terms

For Advanced Students:

  • • Work with three-part ratios (e.g., 2:3:52:3:5)
  • • Explore the relationship between ratios and percentages
  • • Create and solve their own ratio word problems
Standards Alignment
  • 6.RP.A.1 (CCSS.MATH.CONTENT.6.RP.A.1)

    Understand the concept of a ratio and use ratio language to describe a ratio relationship between two quantities

  • 6.RP.A.2 (CCSS.MATH.CONTENT.6.RP.A.2)

    Understand the concept of a unit rate a/b associated with a ratio a:b

  • 6.RP.A.3 (CCSS.MATH.CONTENT.6.RP.A.3)

    Use ratio and rate reasoning to solve real-world and mathematical problems

Lesson Resources
  • visualInteractive Tape Diagram

    Students explore ratios by adjusting tape lengths

  • activityRecipe Scaling Challenge

    Scale a recipe up and down using ratios

  • gameRatio Match

    Match equivalent ratios in different forms

Lesson Content

Everything students see: definition, examples, common mistakes, applications. Tap to open.

Definition

A ratio is a comparison of two quantities. It tells us how much of one thing there is compared to another.
We can write ratios in three ways:
  • Using a colon: 2:32:3
  • Using the word "to": 2 to 3
  • As a fraction: 23\frac{2}{3}
For example, if a classroom has 12 boys and 18 girls, the ratio of boys to girls is:
12:18or12 to 18or121812:18 \quad \text{or} \quad 12 \text{ to } 18 \quad \text{or} \quad \frac{12}{18}
This ratio can be simplified to 2:32:3 by dividing both numbers by 6.

Worked Examples

A fruit bowl has 4 apples and 6 oranges. Write the ratio of apples to oranges.
1

Identify the quantities

Apples = 4, Oranges = 6 → Two quantities to compare
2

Write in order mentioned

Apples TO oranges means apples first → 4:64:6
3

Simplify if possible

Both 4 and 6 divide by 2: 42=2\frac{4}{2} = 2, 62=3\frac{6}{2} = 3 → 2:32:3

Common Mistakes

Writing the ratio in the wrong order

Why it's wrong: The order matters! Apples to oranges (4:64:6) is different from oranges to apples (6:46:4).

Correct: Always write quantities in the order they are mentioned in the problem.

Forgetting to simplify ratios

Why it's wrong: While 12:1812:18 is correct, 2:32:3 is the simplified form and easier to work with.

Correct: Always check if both numbers share a common factor and simplify when possible.

Confusing ratios with fractions in all contexts

Why it's wrong: A ratio 3:53:5 compares two quantities, while 35\frac{3}{5} often means 3 out of 5 (part of a whole).

Correct: Understand whether you're comparing part-to-part (3:53:5) or part-to-whole (3:83:8, because the whole is 3+5=83 + 5 = 8).

Why It Matters

Ratios are everywhere in daily life:
  • Cooking: A recipe might call for 2 cups of flour for every 1 cup of sugar (ratio 2:12:1)
  • Maps: A map scale of 1:1000001:100000 means 1 cm on the map equals 100,000 cm in real life
  • Sports: A team's win-loss ratio of 5:35:3 means they won 5 games for every 3 they lost
  • Mixing: Paint colors are mixed using ratios like 3 parts blue to 1 part white
Understanding ratios helps you compare quantities, scale recipes, read maps, and solve many practical problems!

Real World Applications

Cooking and Recipes

Chefs use ratios to scale recipes up or down while keeping the same taste.

Example:

A lemonade recipe uses 2 cups of lemon juice for every 8 cups of water (ratio 2:82:8 or 1:41:4). To make twice as much, use 4 cups of juice and 16 cups of water.

1Try It Yourself

A pancake recipe uses flour and milk in a ratio of 3:23:2. You have 9 cups of flour.

How many cups of milk do you need?

Step 1: Write the mathematical expression

If 3 parts flour needs 2 parts milk, and you have 9 cups of flour:

Sports Statistics

Sports analysts use ratios to compare player and team performance.

Example:

A basketball player made 24 shots out of 30 attempts. The ratio of made shots to attempts is 24:3024:30 or simplified 4:54:5.

2Try It Yourself

A soccer team has a win-to-loss ratio of 5:25:2. They played 21 games total (only wins and losses, no ties).

How many games did they win?

Step 1: Write the mathematical expression

Total parts = 5+2=75 + 2 = 7. Write an expression for the number of wins:

Key Takeaways

  • 1A ratio compares two quantities and can be written as a:ba:b, "a to b", or ab\frac{a}{b}
  • 2The order of numbers in a ratio matters: 2:32:3 is different from 3:23:2
  • 3Ratios can be simplified by dividing both numbers by their greatest common factor
  • 4Part-to-part ratios compare parts (2:32:3), while part-to-whole ratios compare a part to the total
  • 5Equivalent ratios are found by multiplying or dividing both parts by the same number

Frequently Asked Questions

What's the difference between a ratio and a fraction?

A ratio compares any two quantities (like boys to girls: 3:23:2). A fraction usually shows part of a whole (like 35\frac{3}{5} of the pizza). However, ratios can be written as fractions when comparing part-to-whole.

Do I always need to simplify a ratio?

Not always, but simplified ratios are easier to understand and compare. 2:32:3 is clearer than 14:2114:21, even though they're equivalent.

Can a ratio have more than two numbers?

Yes! Extended ratios compare three or more quantities. For example, mixing paint in a ratio of red:blue:yellow = 2:3:12:3:1.

Glossary

Ratio
A comparison of two or more quantities using division
Equivalent ratios
Ratios that represent the same comparison (e.g., 2:42:4 and 1:21:2)
Part-to-part ratio
Compares one part of a group to another part (e.g., boys to girls)
Part-to-whole ratio
Compares one part to the total (e.g., boys to all students)
Simplify
Reduce a ratio to its smallest whole numbers by dividing by the GCF

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