How to Explain Multiplying Fractions to Fifth Graders

"Wait, why did the answer get smaller?"

When students first multiply 1/2 × 3/4 and get 3/8, they're often confused. Multiplication is supposed to make things bigger, right?

This moment of confusion is actually an opportunity. Let's explore how to make fraction multiplication make sense.

Why Fraction Multiplication Matters

Multiplying fractions appears everywhere:

• "I need half of a 3/4 cup measurement"
• "What's 2/3 of the pizza that's left?"
• "Find 3/4 of the distance"
• "Calculate 1/2 of 1/2"

Understanding fraction multiplication also prepares students for:

• Decimal multiplication
• Percent calculations
• Proportional reasoning
• Algebra

The Big Idea: "Of" Means "Times"

The word "of" in math usually means multiplication:

• 1/2 of 10 = 1/2 × 10 = 5
• 3/4 of 20 = 3/4 × 20 = 15
• 1/2 of 3/4 = 1/2 × 3/4 = ?

What Does 1/2 × 3/4 Mean?

It means: "What is half of three-fourths?"

If you have 3/4 of a pizza and eat half of that, how much did you eat?

Visual Proof: The Area Model

Example: 1/2 × 3/4

Start with a rectangle representing 1 whole:

+---------------+
|               |
|       1       |
|               |
+---------------+

Shade 3/4 horizontally:

+---+---+---+---+
| X | X | X |   |
| X | X | X |   |
| X | X | X |   |
+---+---+---+---+
     3/4

Now shade 1/2 vertically:

+---+---+---+---+
| X | X | X |   |  ← 1/2
| X | X | X |   |
+---+---+---+---+
| X | X | X |   |
| X | X | X |   |
+---+---+---+---+

The overlap (double-shaded) shows 1/2 × 3/4:

+---+---+---+---+
| # | # | # |   |  ← overlap
| # | # | # |   |
+---+---+---+---+
|   |   |   |   |
|   |   |   |   |
+---+---+---+---+

Count: 3 squares overlap out of 8 total squares.

Answer: 1/2 × 3/4 = 3/8

The Algorithm: Multiply Straight Across

The Rule

a/b × c/d = (a × c)/(b × d)

Multiply numerators. Multiply denominators.

Example: 2/3 × 4/5

2/3 × 4/5 = (2 × 4)/(3 × 5) = 8/15

That's it! No common denominators needed.

Example: 3/4 × 2/7

3/4 × 2/7 = (3 × 2)/(4 × 7) = 6/28 = 3/14

Simplifying: Before or After?

Simplifying After

Problem: 4/9 × 3/8

4/9 × 3/8 = 12/72

Now simplify: 12/72 = 1/6

Simplifying Before (Cross-Canceling)

Look for common factors diagonally:

4/9 × 3/8

4 and 8 share a factor of 4: 4÷4=1, 8÷4=2
9 and 3 share a factor of 3: 9÷3=3, 3÷3=1

1/3 × 1/2 = 1/6

Same answer, smaller numbers!

When to Simplify Before

Cross-canceling works well when:

• Numbers are large
• Common factors are obvious
• You want easier multiplication

But always check: the final answer should be the same either way!

Multiplying Whole Numbers by Fractions

Any Whole Number is a Fraction

5 = 5/1, so:

Problem: 5 × 3/4

5/1 × 3/4 = 15/4 = 3 3/4

The Shortcut

Multiply the whole number by the numerator, keep the denominator:

5 × 3/4 = (5 × 3)/4 = 15/4 = 3 3/4

Visual Understanding

5 × 3/4 means "five groups of 3/4":

Group 1: [===] (3/4)
Group 2: [===] (3/4)
Group 3: [===] (3/4)
Group 4: [===] (3/4)
Group 5: [===] (3/4)

Total: 15/4 = 3 3/4

Multiplying Mixed Numbers

Step 1: Convert to Improper Fractions

Problem: 2 1/3 × 1 1/2

Convert:

• 2 1/3 = 7/3
• 1 1/2 = 3/2

Step 2: Multiply

7/3 × 3/2 = 21/6

Step 3: Simplify and Convert Back

21/6 = 7/2 = 3 1/2

The Full Process

2 1/3 × 1 1/2
= 7/3 × 3/2
= 21/6
= 7/2
= 3 1/2

Why the Answer Gets Smaller (Sometimes)

The Pattern

• Multiply by a number greater than 1 → answer is larger
• Multiply by exactly 1 → answer is the same
• Multiply by a number less than 1 → answer is smaller

Examples

• 12 × 2 = 24 (bigger)
• 12 × 1 = 12 (same)
• 12 × 1/2 = 6 (smaller)
• 12 × 1/4 = 3 (even smaller)

The Intuition

1/2 × 3/4 asks "What is HALF of 3/4?"

Half of anything is less than what you started with!

Hands-On Activities

Paper Folding

1. Take a square piece of paper (this is 1 whole)
2. Fold it in half. Unfold and shade half
3. Now fold the shaded part in thirds
4. What fraction of the original is one section?

This shows: 1/2 × 1/3 = 1/6

Fraction Tile Multiplication

Use fraction tiles or strips:

• Start with 3/4 of a whole
• Find 1/2 of that
• Compare to the whole

Recipe Scaling

Take a recipe and make 1/2 or 2/3 of it:

• Original: 3/4 cup sugar
• Half batch: 1/2 × 3/4 = 3/8 cup sugar

Area Models with Graph Paper

Draw rectangles on graph paper:

1. Make a 3×4 rectangle (12 squares)
2. Shade 2/3 of the rows
3. Shade 3/4 of the columns
4. Count the overlap
5. Compare to total

Common Mistakes and How to Fix Them

Mistake 1: Adding Instead of Multiplying

Wrong: 1/2 × 3/4 = 4/6 (added numerators and denominators)

Fix: Remind: "Multiply means multiply straight across."

• Numerators: 1 × 3 = 3
• Denominators: 2 × 4 = 8
• Answer: 3/8

Mistake 2: Finding Common Denominators

Wrong: Converting to common denominators before multiplying.

Fix: "That's for addition! For multiplication, just multiply across."

Mistake 3: Forgetting to Simplify

Problem: 2/4 × 4/6 = 8/24

Better: 8/24 = 1/3

Fix: Always check if the answer can be simplified. Or cross-cancel before multiplying.

Mistake 4: Mixed Number Errors

Wrong: 2 1/2 × 3 = 6 1/2 (only multiplied the whole number)

Fix: "The whole thing gets multiplied." Convert to improper fraction first:

• 2 1/2 = 5/2
• 5/2 × 3 = 15/2 = 7 1/2

Mistake 5: Expecting a Larger Answer

Confusion: "I multiplied but the answer got smaller!"

Fix: "You took a PART of something. Half of 3/4 is smaller than 3/4, just like half of your allowance is less than your whole allowance."

Practice Ideas for Home

"What's Fraction Of?"

Quiz each other:

• "What's 1/2 of 8?"
• "What's 3/4 of 20?"
• "What's 2/3 of 1/2?"

Cooking Together

Make a half batch or double batch of a recipe:

• "The recipe calls for 2/3 cup. We're making half. What's 1/2 of 2/3?"

Estimate First

Before calculating, estimate:

• "Is 3/4 × 4/5 more or less than 1/2?"
• "About how much is 2/3 × 9?"

The Shrinking Challenge

Start with 1 whole. Multiply by 1/2 five times:

• 1 × 1/2 = 1/2
• 1/2 × 1/2 = 1/4
• 1/4 × 1/2 = 1/8
• 1/8 × 1/2 = 1/16
• 1/16 × 1/2 = 1/32

Watch numbers shrink!

Connecting to Future Concepts

Decimal Multiplication

0.5 × 0.75 uses the same concept:

• Half of 0.75 is 0.375
• Same as 1/2 × 3/4 = 3/8 = 0.375

Percent Problems

"What is 25% of 80?" is really:

• 25% = 1/4
• 1/4 × 80 = 20

Probability

"What's the probability of flipping heads twice?"

• P(heads) = 1/2
• P(heads twice) = 1/2 × 1/2 = 1/4

Algebra

The distributive property with fractions:

• 2/3 × (6 + 9) = 2/3 × 6 + 2/3 × 9

The Bottom Line

Multiplying fractions is algorithmically simple—just multiply across. But the real learning happens when students understand what they're calculating.

When your fifth grader sees 2/3 × 3/4 and thinks "I'm finding two-thirds OF three-fourths," they've moved beyond procedure to genuine mathematical understanding.

And that understanding—that multiplying by a fraction less than 1 makes things smaller, that "of" means "times," that the answer represents a fraction of a fraction—that's the foundation for proportional reasoning, percentages, and so much more.