Multiplying Fractions and Whole Numbers

In this lesson:

• Multiply a fraction by a whole number using partition-and-count
• Multiply a fraction by a fraction using area models
• Tile rectangles with unit fraction squares
• Create story contexts for fraction multiplication

Learning Objectives for This Lesson

By the end of this lesson, you will:

1. Interpret as partitioning and taking
2. Use visual models for fraction-times-whole-number problems
3. Use area models for fraction-times-fraction problems
4. Tile rectangles with unit fraction squares
5. Create story contexts for fraction multiplication

What Does "Two-Thirds of Four" Mean?

• From our last lesson: multiplying by a fraction means taking a part
• means "4 groups of one-third"
• Today's question: what does mean?
• Read it as: "two-thirds of four"

Partition and Count: Two-Thirds of Four

: partition 4 into 3 equal parts, take 2 of them

Connecting the Visual to Computation

• We found
• Notice the pattern: gives the numerator
• The denominator stays 3
• General rule:
• Convert:

Multiplication Can Make Things Smaller

• Is more or less than 4? Less!
• We took only two-thirds of 4 — a part, not the whole
• When you multiply by a fraction less than 1, the result is smaller
• Think: "two-thirds of a pizza is less than the whole pizza"

Quick Check: Predict Before Computing

Will be more or less than 8?

• , so the product is less than 8
• Partition 8 into 4 parts, take 3:

Your Turn: Fraction Times Whole Number

Compute and write a story for it.

• Partition 8 into ____ equal parts
• Each part is ____
• Take ____ of those parts
• Result: ____

Story starter: "There are 8 ___. Someone takes 3/4 of them..."

Try it, then advance for one possible story.

Now: A Fraction of a Fraction

• We can find a fraction of a whole number
• Next question: what is a fraction of a fraction?
• Example: means "two-thirds of four-fifths"
• We will use an area model to see why the answer is

Area Model: Two-Thirds of Four-Fifths

Shade horizontally, then take vertically

Counting the Pieces: Why the Rule Works

• Whole square: equal pieces
• Overlap: pieces
• Result:
• Numerator ; Denominator

The General Multiplication Rule for Fractions

• Multiply the numerators to get the new numerator
• Multiply the denominators to get the new denominator
• The area model shows why: numerators count shaded pieces, denominators count total pieces

Second Example: One-Half Times Three-Fourths

• Shade horizontally (3 of 4 rows)
• Take vertically (1 of 2 columns)
• Total pieces:
• Overlap pieces:

Size Check: Products Are Smaller

• is less than — because
• is less than — because
• Both times, we multiplied by a fraction less than 1
• Pattern: multiplying by a fraction < 1 always shrinks the result

Quick Check: Apply the Rule

Compute

• Multiply numerators:
• Multiply denominators:

Is less than ? Yes —

Connecting Area Models to Actual Area

• The area model for fractions is about area
• A rectangle with sides and has area
• We can tile it with tiny unit fraction squares
• The tile count confirms the multiplication

Tiling a Fractional Rectangle with Unit Squares

Tiles are wide and tall — each tile has area

Counting Tiles Confirms the Product

• Tiles across: columns
• Tiles down: rows
• Total tiles:
• Each tile has area
• Area

Second Tiling: Three-Fourths by One-Half

• Tiles: wide and tall → each tile is
• Columns: 3, Rows: 1 → total tiles: 3
• Area:
• Verify: ✓

Your Turn: Tile a Rectangle

A rectangle has sides and .

• What size tiles fit exactly? by
• How many tiles across? How many tiles down?
• What is each tile's area?
• What is the total area?

Try it, then check with the multiplication rule.

Quick Check: What Tiles Fit?

A rectangle has sides and .

• Tile width: (because the side is fifths)
• Tile height: (because the side is fourths)
• Each tile: of the unit square
• Tiles: → Area:

Three Models Lead to One Rule

• Partition-and-count: fraction × whole number
• Area model: fraction × fraction
• Tiling: fractional rectangle area
• All three models lead to the same rule:

Practice: Apply the Multiplication Rule

Compute each product. Simplify if possible.

For each: is the answer less than the second factor?

Practice Answers with Size Check

1. — less than 10 ✓
2. — less than ✓
3. — less than ✓

Each product is smaller because each first factor is less than 1.

Writing Stories for Fraction Multiplication

For : need 10 of something, take

Story: "A carton has 10 eggs. Dani used for breakfast. How many eggs?"

Check: eggs ✓

Your Turn: Write a Story

Write a word problem for .

• Start with of something, then take
• The answer should be

Example: "A sidewalk is mile. Snow covers of it. How much?"

Exit Ticket: Compute and Create

Compute:

Then write a story that matches this expression.

Check: Is your answer less than ?

Key Takeaways from Fraction Multiplication

✓ Partition into parts, take
✓
✓ Tiling confirms area = length × width
✓ Fraction < 1 as multiplier shrinks the result

Multiply both denominators, not just one
Denominators multiply — never add
"Of" means multiply, not divide

What Comes Next in Fraction Multiplication

• Next lesson: Multiplying fractions as scaling (5.NF.B.5)
• Explore: when does multiplication make things bigger vs. smaller?
• Key connection: today's "less than" pattern is part of a bigger picture