Multiplication

What is Multiplication?

Multiplication is a fast way of doing repeated addition — adding the same number multiple times.

4 × 3 means 4 added 3 times: 4 + 4 + 4 = 12

Or 3 added 4 times: 3 + 3 + 3 + 3 = 12

The symbol for multiplication is × and the result is called the product.

Key Terms

• Factor — a number being multiplied (in 4 × 3, both 4 and 3 are factors)
• Product — the result of multiplication (4 × 3 = 12; product is 12)
• Multiple — the product of a number and any integer (multiples of 4: 4, 8, 12, 16…)
• Array — a rectangular arrangement of objects that models multiplication

Times Tables

Fluency with times tables (up to 12 × 12) is essential for speed and accuracy in all areas of maths.

×	6	7	8	9
6	36	42	48	54
7	42	49	56	63
8	48	56	64	72
9	54	63	72	81

The most commonly forgotten facts are 6×7, 7×8, and 8×9. A trick: 5,6,7,8 → 56 = 7 × 8.

Properties of Multiplication

Commutative Property: the order of factors does not change the product. a × b = b × a

• 6 × 7 = 7 × 6 = 42

Associative Property: the grouping of factors does not matter. (a × b) × c = a × (b × c)

• (2 × 3) × 4 = 6 × 4 = 24; also 2 × (3 × 4) = 2 × 12 = 24

Distributive Property: multiply a sum by splitting it into parts. a × (b + c) = a×b + a×c

• 6 × 14 = 6 × 10 + 6 × 4 = 60 + 24 = 84

Identity Property: any number multiplied by 1 stays the same. a × 1 = a

Zero Property: any number multiplied by 0 equals 0. a × 0 = 0

Multiplying by Powers of 10

When multiplying by 10, 100, or 1,000, move digits to the left (add zeros):

• 7 × 10 = 70
• 7 × 100 = 700
• 300 × 40 = 3 × 4 × 100 × 10 = 12 × 1,000 = 12,000

Short Multiplication

Multiply a multi-digit number by a single digit, working right to left:

Worked Example: 34 × 6

• 4 × 6 = 24, write 4 carry 2
• 3 × 6 = 18, plus 2 = 20
• Result: 204

Worked Example: 147 × 8

• 7 × 8 = 56, write 6 carry 5
• 4 × 8 = 32, plus 5 = 37, write 7 carry 3
• 1 × 8 = 8, plus 3 = 11
• Result: 1,176

Long Multiplication

For multiplying two multi-digit numbers, split into partial products:

    245
  ×  12
  -----
    490   (245 × 2)
  2450    (245 × 10)
  -----
  2940

Worked Example: 245 × 12

• 245 × 2 = 490
• 245 × 10 = 2,450
• 490 + 2,450 = 2,940

Grid Method (Alternative to Long Multiplication)

The grid method makes long multiplication visual:

For 34 × 26:

×	30	4
20	600	80
6	180	24

Total = 600 + 80 + 180 + 24 = 884

Word Problem Clues

Multiplication is used in problems involving:

• Equal groups: "6 bags with 9 apples each" → 6 × 9
• Total cost: "8 tickets at £3.50 each" → 8 × 3.50
• Area: "room 12 m × 8 m" → 12 × 8
• Scaling: "3 times as many"
• Key words: each, per, times, product, groups of, total, area, altogether

Worked Example: A box holds 24 chocolates. There are 15 boxes. How many chocolates in total? 24 × 15 = 24 × 10 + 24 × 5 = 240 + 120 = 360 chocolates

Common Mistakes

• Mistake: forgetting to carry when multiplying. Fix: write the carry digit small above the next column and add it on.
• Mistake: in long multiplication, forgetting to add a zero when multiplying by the tens digit. Fix: write a placeholder zero (or indent the partial product) before multiplying.
• Mistake: confusing commutative and associative properties. Fix: commutative changes order (a×b = b×a); associative changes grouping ((a×b)×c = a×(b×c)).
• Mistake: 300 × 40 = 1,200 (forgetting that 3 × 4 × 10 × 10 × 10 = 12 × 1,000). Fix: count the total zeros separately.

Tips and Tricks

• To multiply by 9: multiply by 10 and subtract the number. 9 × 7 = 70 - 7 = 63.
• To multiply by 5: multiply by 10 and halve. 5 × 14 = 140 ÷ 2 = 70.
• Check your answer using the inverse: if 245 × 12 = 2,940, then 2,940 ÷ 12 should equal 245.
• Grid method and long multiplication give the same answer — use whichever feels clearer to you.