How to Explain Ratios to Sixth Graders

Ratios are one of the most important concepts students encounter in sixth grade—and one that causes frequent confusion. When taught well, ratio thinking becomes a powerful tool that students use throughout their math education and daily lives. This guide will help you explain ratios in ways that make sense and stick.

Why Ratios Matter for Sixth Graders

Ratios appear everywhere in the real world:

• Cooking: A recipe calls for 2 cups flour to 1 cup sugar
• Sports: A basketball player makes 3 out of every 4 free throws
• Maps: 1 inch represents 10 miles
• Money: Exchange rates between currencies
• Science: Mixing solutions in specific proportions

More importantly, ratio reasoning is the foundation for:

• Proportional relationships (7th grade)
• Percentages and percent change
• Slope and linear equations
• Probability and statistics
• Scale drawings and similarity

When students truly understand ratios, they're prepared for years of connected mathematical concepts.

Key Concepts Broken Down Simply

What Is a Ratio?

A ratio compares two quantities. It tells us "for every ___ of this, there are ___ of that."

Simple definition: A ratio is a way to compare two amounts using division or the word "to."

Three Ways to Write Ratios

The same ratio can be written three different ways:

Using "to":     3 to 4
Using a colon:  3:4
As a fraction:  3/4

All three mean the same thing: for every 3 of one thing, there are 4 of another.

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

This distinction confuses many students. Let's clarify with an example:

Scenario: A class has 12 boys and 18 girls.

Part-to-Part Ratios:
┌─────────────────────────────────────────┐
│  Boys to Girls = 12:18 (simplifies to 2:3)  │
│  Girls to Boys = 18:12 (simplifies to 3:2)  │
└─────────────────────────────────────────┘

Part-to-Whole Ratios:
┌─────────────────────────────────────────┐
│  Boys to Total = 12:30 (simplifies to 2:5)  │
│  Girls to Total = 18:30 (simplifies to 3:5) │
└─────────────────────────────────────────┘

Key insight: The order matters! 2:3 is NOT the same as 3:2.

Equivalent Ratios

Just like equivalent fractions, ratios can be written in different forms that represent the same relationship.

Original ratio: 2:3

Equivalent ratios (multiply both parts by the same number):
  2:3  =  4:6  =  6:9  =  8:12  =  10:15

    ×2     ×3     ×4      ×5
   ↓      ↓      ↓       ↓
  2:3 → 4:6 → 6:9 → 8:12 → 10:15

The rule: Multiply or divide BOTH parts of a ratio by the same number to find equivalent ratios.

Simplifying Ratios

To simplify a ratio, divide both parts by their greatest common factor (GCF).

Example: Simplify 24:36

Step 1: Find GCF of 24 and 36
        Factors of 24: 1, 2, 3, 4, 6, 8, 12, 24
        Factors of 36: 1, 2, 3, 4, 6, 9, 12, 18, 36
        GCF = 12

Step 2: Divide both parts by 12
        24 ÷ 12 = 2
        36 ÷ 12 = 3

Answer: 24:36 = 2:3

Visual Examples and Diagrams

Tape Diagrams

Tape diagrams (also called bar models) are powerful tools for visualizing ratios.

Problem: The ratio of cats to dogs at a shelter is 3:5. If there are 15 cats, how many dogs are there?

Cats:   [===][===][===]           3 parts
Dogs:   [===][===][===][===][===] 5 parts

Each part = 15 ÷ 3 = 5 animals

Cats:   [ 5 ][ 5 ][ 5 ]           = 15 cats ✓
Dogs:   [ 5 ][ 5 ][ 5 ][ 5 ][ 5 ] = 25 dogs

Ratio Tables

Ratio tables help students see patterns and find equivalent ratios.

Example: Lemonade recipe uses 2 cups lemon juice to 5 cups water.

┌─────────────┬─────────────┐
│ Lemon Juice │   Water     │
├─────────────┼─────────────┤
│      2      │      5      │
│      4      │     10      │
│      6      │     15      │
│      8      │     20      │
│     10      │     25      │
└─────────────┴─────────────┘
     ×2         ×2
     ×3         ×3
     ...        ...

Double Number Lines

Double number lines show the relationship between two quantities visually.

Example: 3 apples cost $2

Apples: 0────3────6────9────12───15
        |    |    |    |    |    |
Dollars:0────2────4────6────8────10

This shows that 6 apples cost $4, 9 apples cost $6, and so on.

Hands-On Activities

Activity 1: Recipe Scaling

Materials: A simple recipe, measuring cups, ingredients

Instructions:

1. Start with a basic recipe (like trail mix: 2 cups cereal, 1 cup raisins, 1 cup nuts)
2. Have students double the recipe (maintaining ratios)
3. Then make a batch that's 1.5 times the original
4. Discuss: What stayed the same? What changed?

Activity 2: Color Mixing Lab

Materials: Red, blue, and yellow paint; paper; mixing containers

Instructions:

1. Create a "purple" using 2 parts red to 3 parts blue
2. Make larger batches keeping the same ratio
3. Explore: What happens if you change the ratio?
4. Record findings in a ratio table

Activity 3: Sports Statistics Analysis

Materials: Sports statistics from newspapers or websites

Instructions:

1. Find a player's shooting statistics (made/attempted)
2. Write as a ratio and simplify
3. Compare players using ratios
4. Predict future performance using the ratio

Activity 4: Map Scale Investigation

Materials: Local map, ruler, calculator

Instructions:

1. Find the map scale (e.g., 1 inch = 5 miles)
2. Measure distances between local landmarks on the map
3. Calculate actual distances using the ratio
4. Verify with online mapping tools

Common Mistakes and How to Fix Them

Mistake 1: Order Confusion

Wrong: Writing "boys to girls" as 18:12 when there are 12 boys and 18 girls

Why it happens: Students don't pay attention to the order specified

Fix: Always ask "What comes first in the comparison?" Circle the first quantity mentioned.

"boys to girls" → boys first → 12:18
"girls to boys" → girls first → 18:12

Mistake 2: Adding Instead of Multiplying

Wrong: To get from 2:3 to an equivalent ratio, adding 2 to get 4:5

Why it happens: Students apply wrong operation

Fix: Use visual models to show why this doesn't work:

Original:   [==][===]     = 2:3
Wrong way:  [====][=====] = 4:5 (different shape!)
Right way:  [==][==][===][===] = 4:6 (same proportions)

Mistake 3: Confusing Ratios with Fractions

Wrong: Saying a ratio of 2:3 means 2/3 of the total

Why it happens: Similar notation causes confusion

Fix: Emphasize the difference:

• Ratio 2:3 compares parts (2 of this TO 3 of that)
• Fraction 2/3 is part OF a whole (2 out of 3 total)

In a 2:3 ratio, there are 5 total parts, so:

• First quantity is 2/5 of total
• Second quantity is 3/5 of total

Mistake 4: Not Simplifying Consistently

Wrong: Simplifying 12:8 to 6:4 and stopping

Why it happens: Not finding the GCF

Fix: Always ask "Can both numbers still be divided by something?"

12:8 → 6:4 → 3:2 ✓ (fully simplified)

Mistake 5: Unit Ratio Confusion

Wrong: Thinking 1:4 and 4:1 are the same

Fix: Use concrete examples:

• 1:4 = 1 teacher for every 4 students (few teachers)
• 4:1 = 4 teachers for every 1 student (many teachers!)

Practice Ideas for Home

Everyday Ratio Spotting

Challenge your child to find ratios during daily activities:

• Grocery shopping: Compare prices (price per ounce)
• Cooking: Read recipes as ratios
• Driving: Miles per hour, miles per gallon
• Screen time: Ratio of homework time to game time

Kitchen Math

1. Double or halve recipes together
2. Make different batch sizes of the same recipe
3. Compare nutritional labels using ratios

Sports Connection

If your child enjoys sports:

1. Calculate batting averages or shooting percentages
2. Compare two players' statistics
3. Predict outcomes based on historical ratios

Building and Creating

1. LEGO ratios: Build structures using specific color ratios
2. Drawing to scale: Create scale drawings of their room
3. Music: Explore musical intervals as ratios

Ratio Games

1. Ratio War: Like card war, but players must find equivalent ratios
2. Ratio Bingo: Call out ratios, mark equivalent ones on cards
3. Ratio Scavenger Hunt: Find items in specific ratios around the house

Connection to Future Math Concepts

Understanding ratios now prepares students for:

7th Grade: Proportional Relationships

If 3:4 = 6:x, solve for x
(Cross multiplication: 3x = 24, so x = 8)

7th-8th Grade: Percentages

Percent is a ratio with 100 as the second term
75% = 75:100 = 3:4

8th Grade: Slope

Slope = rise:run = change in y : change in x
A slope of 2/3 means "up 2, right 3"

High School: Similar Figures

Corresponding sides of similar triangles form equal ratios
If triangles are similar: side₁/side₂ = 3/6 = 4/8 = 5/10

Real-World Applications

• Chemistry: Mixing solutions in proper ratios
• Finance: Understanding interest rates and returns
• Engineering: Scale models and blueprints
• Medicine: Dosage calculations

Tips for Teaching Success

1. Start concrete: Use physical objects before numbers
2. Use student interests: Sports, cooking, gaming—ratios are everywhere
3. Emphasize vocabulary: "For every," "to," "per," "out of"
4. Practice multiple representations: Words, symbols, tables, diagrams
5. Connect to prior knowledge: Build on fraction understanding
6. Highlight order: Consistently emphasize that order matters

Quick Reference

┌────────────────────────────────────────────────────┐
│              RATIO QUICK REFERENCE                 │
├────────────────────────────────────────────────────┤
│ Ways to write: 3 to 4  |  3:4  |  3/4             │
│                                                    │
│ Part-to-Part: compares parts to each other        │
│ Part-to-Whole: compares one part to total         │
│                                                    │
│ Equivalent ratios: multiply/divide both by same # │
│ Simplify: divide both by GCF                      │
│                                                    │
│ ORDER MATTERS!  2:3 ≠ 3:2                         │
└────────────────────────────────────────────────────┘

Ratios are a gateway to proportional thinking—a skill that extends far beyond the math classroom. With patience, practice, and real-world connections, your sixth grader will master this essential concept and be well-prepared for the mathematical challenges ahead.