The Distributive Law

Starter: Mental Maths

Calculate these:

A Concrete Problem

Problem: Find the area of this rectangle

┌─────────────┬─────┐
│     20      │  3  │
│             │     │  ← height = 7
│             │     │
└─────────────┴─────┘

What’s the total width? 20 + 3 = 23

So the area is: 7 × 23 = ?

Method 1: Direct Calculation

┌─────────────────────┐
│   width = 23        │
│                     │  ← height = 7
│                     │
└─────────────────────┘

Area = 7 × 23 = 161

But what if 23 is hard to multiply?

Method 2: Split the Rectangle

┌─────────────┬─────┐
│             │     │
│     20      │  3  │  ← height = 7
│             │     │
└─────────────┴─────┘

Left rectangle: 7 × 20 = 140

Right rectangle: 7 × 3 = 21

Total area: 140 + 21 = 161 ✓

The Key Insight

The Distributive Law

When we multiply a number by a sum, we can multiply the number by each part separately, then add the results.

In symbols: × (□ + △) = × □ + × △

(We avoid letters for now!)

Visual Proof with Areas

For any rectangle split into two parts:

┌─────────┬─────┐
│    A    │  B  │  ← height = h
└─────────┴─────┘

Total area = h × (A + B)

Also equals = h × A + h × B

Therefore: h × (A + B) = h × A + h × B

Example 1: 6 × 47

Method 1 (Hard): 6 × 47 = ?

┌─────────┬───┐
│   40    │ 7 │  ← height = 6
└─────────┴───┘

Example 2: 8 × 35

Let’s split 35 = 30 + 5

┌─────────┬───┐
│   30    │ 5 │  ← height = 8
└─────────┴───┘

8 × (30 + 5) = 8 × 30 + 8 × 5 = 240 + 40 = 280

Check: 8 × 35 = 280 ✓

What About Subtraction?

Problem: 9 × 48

Think: 48 is close to 50…

┌─────────────┬──┐
│     50      │-2│  ← height = 9
└─────────────┴──┘

9 × 48 = 9 × (50 - 2) = 9 × 50 - 9 × 2 = 450 - 18 = 432

Distributive Law (Complete)

For Addition: × (□ + △) = × □ + × △

For Subtraction:
× (□ - △) = × □ - × △

Your Turn: Practice

Calculate using the distributive law:

1. 7 × 26 (Hint: 26 = 20 + 6)
2. 5 × 83 (Hint: 83 = 80 + 3)
   
3. 4 × 29 (Hint: 29 = 30 - 1)
4. 6 × 198 (Hint: 198 = 200 - 2)

Solutions

1. 7 × 26 = 7 × (20 + 6) = 7 × 20 + 7 × 6 = 140 + 42 = 182

2. 5 × 83 = 5 × (80 + 3) = 5 × 80 + 5 × 3 = 400 + 15 = 415

3. 4 × 29 = 4 × (30 - 1) = 4 × 30 - 4 × 1 = 120 - 4 = 116

4. 6 × 198 = 6 × (200 - 2) = 6 × 200 - 6 × 2 = 1200 - 12 = 1188

Going Backwards: Factorising

Problem: 15 × 8 + 15 × 2 = ?

┌─────────┬───┐
│    8    │ 2 │  ← height = 15
└─────────┴───┘

Factorising Examples

Look for the common factor:

1. 12 × 5 + 12 × 3 = 12 × (5 + 3) = 12 × 8 = 96

2. 9 × 7 - 9 × 2 = 9 × (7 - 2) = 9 × 5 = 45

3. 20 × 6 + 20 × 4 = 20 × (6 + 4) = 20 × 10 = 200

Real-World Application

Problem: A school has two courtyards: - Court A: 12m × 8m
- Court B: 12m × 5m

Find the total area.

Court A          Court B
┌─────────┐     ┌─────┐
│    8    │     │  5  │  ← height = 12
└─────────┘     └─────┘

Method 1: 12 × 8 + 12 × 5 = 96 + 60 = 156 m² Method 2: 12 × (8 + 5) = 12 × 13 = 156 m² ✓

Key Benefits

Why use the distributive law?

Challenge Questions

1. True or False: 5 × (8 + 3) = 5 × 8 + 3

What’s Next?

Next lesson: We’ll use the same distributive law but with letters instead of numbers!

• Instead of: 3 × (4 + 5) = 3 × 4 + 3 × 5
• We’ll see: 3 × (x + y) = 3x + 3y

The pattern stays the same!

Summary

The Distributive Law helps us:

• Split hard multiplications into easier ones
• Use area models to visualise
• Factor out common numbers
• Prepare for algebra

Remember: × (□ + △) = × □ + × △

Next time: Same idea, but with pronumerals!