Lesson 1.10 · Multiplying Fractions9 mins read

Multiplying Fractions: The 2D Area Model & Scaling

Why do we never need a common denominator to multiply? Discover how multiplying fractions slices pieces into a two-dimensional grid, how to cash in mixed numbers, and why multiplication doesn't always make numbers bigger!

➕ Addition: Slices must match (14 + 24)

✖️ Multiplication: Slices multiply directly into a 2D grid (34 × 25)

Part 1 of 6 · Notebook Page 1

🥖 1. Multiplication as Repeated Addition: 23 × 5

Multiplying a whole number by a fraction is just like multiplying ordinary numbers: it means taking repeated equal groups of that fraction!

Number of Groups:

5 ×23=23+23+23+23+23=103=313

5 separate groups of 23 (Total shaded pieces: 10)

Group 1

2 pieces of 13

Group 2

2 pieces of 13

Group 3

2 pieces of 13

Group 4

2 pieces of 13

Group 5

2 pieces of 13

The Whole Number Multiplication Rule: When multiplying a whole number by a fraction, multiply the whole number by the numerator, and keep the denominator unchanged:

a × bc = a × bc

Part 2 of 6 · Notebook Page 2

📦 2. Fraction of a Whole Number: What is 34 of 60?

In everyday life, we often ask: "What is 34 of \$60?" or "25 of 50 students?" The word "of" in mathematics directly translates to multiplication!

1 Full Collection = 60 items

60items in one single pile

Method A: Divide First, Then Multiply

(60 ÷ 4) × 3 = 15 × 3 = 45

Great for fast mental arithmetic!

Method B: Direct Fraction Multiplication

60 × 34 = 1804 = 45

The universal algebraic formula!

Part 3 of 6 · Notebook Pages 3 & 4

🪟 3. Fraction × Fraction: The 2D Grid / Area Slicing Model

What happens when you multiply a fraction by another fraction, like 34 × 25? You are finding a fraction of a fraction!

Notebook Examples:

↑ 5 Vertical Columns (2 shaded) ↑

Color Breakdown

Vertical columns: 25

Horizontal rows: 34

Overlapping Region = 620

The Calculation

34×25=3 × 24 × 5=620

Simplifies to: 310

What is 3/4 of 2/5? Slicing 2 fifths horizontally into 4 rows yields 6 overlapping pieces out of 20!

Part 4 of 6 · Notebook Page 5

⚡ 4. The Golden Algorithm: Multiply Straight Across!

Because the 2D grid splits each column into rows, the total number of unit pieces is always the product of the denominators (b × d), and the overlapping pieces is the product of numerators (a × c).

Universal Fraction Multiplication Rule

1. Multiply the numerators together → write as the new numerator.
2. Multiply the denominators together → write as the new denominator.
3. Simplify the resulting fraction if possible.

ab×cd=a × cb × d

Part 5 of 6 · Notebook Page 6

🪙 5. Mixed Numbers: "Cash In the Wholes First!"

WARNING: You cannot simply multiply the whole numbers together and fractions together! Doing 3 × 0 and 3/4 × 2/5 is completely wrong!

🔥 Golden Law: Convert ALL mixed numbers into improper fractions BEFORE multiplying!

Notebook Page 6 · Example 1: 334 × 25

Step 1: Cash In Wholes

334 → 3 × 4 + 34 = 154

3 wholes cash in for 12 fourths + 3 loose fourths = 15/4.

Step 2: Multiply Across

154 × 25 = 3020

15 × 2 = 30; 4 × 5 = 20.

Step 3: Simplify & Pack

3020 = 32 = 112

Divide top & bottom by 10 to get 1 1/2!

Example 2: Two Mixed Numbers

226 × 334

Cash in:146 × 154=21024=354=834

Part 6 of 6 · Notebook Pages 7 & 8

✂️ 6. Cross-Simplification: "One from the Top, One from the Bottom"

Why multiply giant numbers like 15 × 24 = 360 when you can simplify before multiplying? Cross-simplification saves enormous mental effort!

915

2024

Without cross-simplifying: 9 × 20 = 180 and 15 × 24 = 360. Simplifying 180/360 is painful!

Part 7 of 6 · Notebook Pages 9 & 10

⚖️ 7. Does Multiplication Always Make Numbers Bigger?

With whole numbers, multiplying always makes things bigger (6 × 2 = 12). But with fractions, multiplication acts as a scaling factor!

Original Length (Base = 6)Scaled Length

Original 6

12/5 = 2.4 (Smaller!)

Self-Check Quiz

🎯 Test Your Multiplication Mastery

Check your understanding of the core concepts before moving on to dividing fractions!

1What is the result of 23 × 34 in simplest form?

2How must you start multiplying 112 × 23?

3Which multiplication will result in a value SMALLER than 20?