How to Explain Multiplying Fractions to Fourth Graders

"How can you multiply by a fraction? Doesn't multiplying make things bigger?"

This question reveals a crucial misconception. In fourth grade, students discover that multiplication doesn't always mean "getting more"—and multiplying fractions by whole numbers is the perfect introduction to this idea.

Why This Concept Matters

Multiplying fractions by whole numbers bridges several important skills:

• Extends multiplication beyond whole numbers
• Prepares for fraction × fraction (fifth grade)
• Connects multiplication to repeated addition
• Builds foundation for ratios and proportions

This is where students begin to see that multiplication is about "groups of" rather than "getting bigger."

The Big Idea: Repeated Addition

3 × 2/5 means "3 groups of 2/5"

Just like 3 × 4 means "3 groups of 4":

• 3 × 4 = 4 + 4 + 4 = 12

3 × 2/5 means:

• 3 × 2/5 = 2/5 + 2/5 + 2/5 = 6/5

    2/5         2/5         2/5
[██|  |  |  |  ] + [██|  |  |  |  ] + [██|  |  |  |  ]

= [██|██|██|██|██|█]
        6/5 = 1 1/5

Two Ways to Think About It

Method 1: Whole Number × Fraction

4 × 3/8 = "4 groups of 3/8"

Group 1: 3/8 = [███|     ]
Group 2: 3/8 = [███|     ]
Group 3: 3/8 = [███|     ]
Group 4: 3/8 = [███|     ]

Total: 12/8 = 1 4/8 = 1 1/2

Method 2: Fraction × Whole Number

3/8 × 4 = "3/8 of 4"

Think: What is 3/8 of 4 wholes?

4 wholes: [████████][████████][████████][████████]

3/8 of 4:
- 1/8 of 4 = 4/8 = 1/2
- 3/8 of 4 = 3 × (1/2) = 3/2 = 1 1/2

Both methods give the same answer: 4 × 3/8 = 3/8 × 4 = 12/8 = 1 1/2

The Procedure: Multiply the Numerator

To multiply a fraction by a whole number:

1. Multiply the whole number by the numerator
2. Keep the denominator the same
3. Simplify if needed

Example 1: 5 × 2/3

5 × 2/3 = (5 × 2)/3 = 10/3

Convert to mixed number:
10 ÷ 3 = 3 R1

Answer: 5 × 2/3 = 10/3 = 3 1/3

Example 2: 4 × 3/4

4 × 3/4 = (4 × 3)/4 = 12/4 = 3

Answer: 4 × 3/4 = 3 (a whole number!)

Example 3: 6 × 1/2

6 × 1/2 = (6 × 1)/2 = 6/2 = 3

Answer: 6 × 1/2 = 3

This makes sense: Half of 6 is 3!

Why We Only Multiply the Numerator

Here's the key insight:

• The denominator tells you the SIZE of the pieces (fifths, eighths, etc.)
• The numerator tells you the NUMBER of pieces

When you multiply 3 × 2/5:

• The pieces are still fifths (size doesn't change)
• You just have 3 times as many of them (number changes)

So: 3 × 2/5 = (3 × 2)/5 = 6/5

Six fifths — more pieces, same size.

Visual Models

The Fraction Bar Model

Show 4 × 2/5:

1 group of 2/5:  [██|  |  |  |  ]
2 groups of 2/5: [██|██|  |  |  ]  (now 4/5)
3 groups of 2/5: [██|██|██|  |  ]  (now 6/5)
4 groups of 2/5: [██|██|██|██|  ]  (now 8/5)

8/5 = 1 3/5

The Number Line Model

Show 3 × 1/4 on a number line:

0    1/4   2/4   3/4   1
|-----|-----|-----|-----|

Jump 1/4 three times:
0 → 1/4 → 2/4 → 3/4

3 × 1/4 = 3/4

The Array Model

Show 3 × 2/5:

  2/5
┌─────┐
│     │  → 1 × 2/5 = 2/5
├─────┤
│     │  → 2 × 2/5 = 4/5
├─────┤
│     │  → 3 × 2/5 = 6/5
└─────┘

Real-World Contexts

Recipe Multiplication

"Each batch of cookies needs 3/4 cup of sugar. You're making 2 batches. How much sugar total?"

2 × 3/4 = 6/4 = 1 2/4 = 1 1/2 cups

Distance Problems

"You walk 2/5 mile to school. How far do you walk in 5 school days (one way)?"

5 × 2/5 = 10/5 = 2 miles

Pizza Problems

"Each person eats 1/8 of a pizza. How much pizza do 6 people eat?"

6 × 1/8 = 6/8 = 3/4 pizza

Time Problems

"Practice is 3/4 hour each day. How many hours of practice in 4 days?"

4 × 3/4 = 12/4 = 3 hours

When the Answer is Greater Than 1

Many problems give answers greater than 1. Convert improper fractions to mixed numbers:

6 × 2/3 = 12/3 = 4

5 × 3/4 = 15/4 = 3 3/4

8 × 5/6 = 40/6 = 6 4/6 = 6 2/3

The Conversion Process

To convert 15/4 to a mixed number:

1. Divide numerator by denominator: 15 ÷ 4 = 3 R3
2. Quotient is the whole number: 3
3. Remainder is the new numerator: 3
4. Keep the same denominator: 4

15/4 = 3 3/4

Hands-On Activities

Fraction Strips Multiplication

Use paper fraction strips:

• Start with 2/5 strips
• Lay out 3 of them end to end
• Count how much you have: 6/5 = 1 1/5

Build and Count

Use LEGO or blocks:

• If 5 blocks = 1 whole
• Then 2 blocks = 2/5
• Build 4 groups of 2 blocks = 8 blocks = 8/5 = 1 3/5

Recipe Scaling

Take real recipes and scale them:

• "Double this recipe that uses 2/3 cup flour"
• "Triple this recipe that uses 1/4 teaspoon salt"

Measurement Practice

Use actual measuring cups:

• Fill 1/4 cup, pour it 3 times into a larger container
• Verify you now have 3/4 cup

Common Mistakes and How to Fix Them

Mistake 1: Multiplying Both Numerator and Denominator

Wrong: 3 × 2/5 = 6/15

Fix: The denominator is the SIZE of the pieces. The size doesn't change—we just get more pieces. Only multiply the numerator:
3 × 2/5 = 6/5

Mistake 2: Forgetting to Simplify

Wrong: Leaving 6 × 1/2 = 6/2

Fix: Always simplify when possible:
6/2 = 3

Mistake 3: Confusion with Adding Fractions

Wrong: Thinking 3 × 2/5 = 3/5 + 2/5 = 5/5

Fix: Multiplication means repeated addition of THE SAME fraction:
3 × 2/5 = 2/5 + 2/5 + 2/5 = 6/5

Mistake 4: Not Converting Improper Fractions

Oversight: Leaving answer as 10/3 when a mixed number is clearer

Fix: Convert improper fractions to mixed numbers for final answers:
10/3 = 3 1/3

Connecting to Multiplication Properties

Commutative Property

3 × 2/5 = 2/5 × 3 = 6/5

Order doesn't matter—both mean "3 groups of 2/5" or "2/5 of 3."

Associative Property (Preview)

3 × (2 × 1/5) = (3 × 2) × 1/5 = 6 × 1/5 = 6/5

Grouping doesn't change the product.

Identity Property

1 × 4/7 = 4/7

Multiplying by 1 doesn't change the fraction.

Mental Math Strategies

Unit Fraction Shortcut

For fractions with 1 as numerator:

5 × 1/3 = 5/3 (just put the whole number over the denominator)

8 × 1/4 = 8/4 = 2

Recognizing When Answers are Whole Numbers

If the whole number × numerator equals a multiple of the denominator:

4 × 3/4 = 12/4 = 3 (since 12 is divisible by 4)

6 × 2/3 = 12/3 = 4 (since 12 is divisible by 3)

Building Toward Future Concepts

This prepares students for:

Fraction × Fraction (Fifth Grade)

2/3 × 4/5 = 8/15

Mixed Number Multiplication

2 1/2 × 3 (convert to improper first)

Ratios and Proportions

"If 1/4 of the class is absent, and there are 28 students, how many are absent?"

Percent Problems

"What is 3/4 of 80?" (same as 3/4 × 80)

Practice Ideas for Home

Cooking Together

Scale recipes up or down:

• "We need 2/3 cup, but we're making triple. How much?"

Daily Fraction Questions

• "If you practice piano for 1/2 hour each day, how many hours in 6 days?"
• "Each cookie uses 1/8 cup of chocolate chips. How much for 4 cookies?"

Visual Drawing Practice

Draw fraction bars showing:

• 3 × 1/4
• 2 × 3/5
• 4 × 2/3

Estimation First

Before calculating, estimate:

• "Is 5 × 2/3 more or less than 5?" (Less—we're taking 2/3 of it each time... wait, we're multiplying! It's more than 2/3 but the answer is 10/3 ≈ 3.3)

Actually, think: "2/3 is less than 1, so 5 times 2/3 is less than 5."

The Bottom Line

Multiplying fractions by whole numbers is really just repeated addition in disguise. When your fourth grader can see that 3 × 2/5 means "three groups of two-fifths," the procedure makes sense.

The key insight is that multiplication changes how MANY pieces you have (numerator), not what SIZE the pieces are (denominator). This understanding will carry them through fraction multiplication, division, and into algebra.

Don't rush to the procedure. Build understanding through visual models and real-world problems first. When students understand why we multiply only the numerator, they're not just following rules—they're reasoning mathematically.