Multiplication Properties
Year 4 pupils often struggle when Oliver calculates 7 × 8 = 56 but Amelia writes 8 × 7 and gets confused about whether the answer changes. Understanding multiplication properties helps students recognise patterns that make mental maths faster and builds confidence for algebraic thinking in Key Stage 3.
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Why it matters
Multiplication properties form the foundation for efficient mental calculations and algebraic reasoning throughout the National Curriculum. When pupils grasp that 25 × 4 equals 4 × 25 (commutative property), they can choose the easier calculation. The distributive property becomes essential for GCSE topics like expanding brackets: 3(x + 5) = 3x + 15. In everyday situations, these properties help students calculate costs efficiently—working out 8 × £12.50 by thinking (8 × £12) + (8 × 50p) = £96 + £4 = £100. Primary teachers use these properties to introduce factor pairs and times table patterns, whilst secondary teachers rely on them for polynomial multiplication and factorisation. Students who master these properties in Key Stage 2 show stronger performance in algebraic manipulation throughout Key Stages 3 and 4.
How to solve multiplication properties
Multiplication & Division Properties
- Commutative: a × b = b × a.
- Associative: (a × b) × c = a × (b × c).
- Identity: a × 1 = a (multiplying by 1 changes nothing).
- Distributive: a × (b + c) = a × b + a × c.
- Division is NOT commutative or associative.
Example: 5 × (2 + 3) = 5 × 2 + 5 × 3 = 10 + 15 = 25.
Worked examples
Is 6 × 9 the same as 9 × 6?
Answer: Yes (54)
- Calculate the first side → 6 × 9 = 54 — Think of 6 rows with 9 in each row. That is 54 altogether.
- Calculate the second side → 9 × 6 = 54 — Now flip the array: 9 rows with 6 in each row. Still 54!
- Name the property → Commutative property — The commutative property of multiplication says you can swap the numbers around and still get the same answer. It works because an array of 3 rows of 4 has the same number of squares as 4 rows of 3.
What is 6 × 1?
Answer: 6
- Think about what × 1 means → 6 × 1 = 1 group of 6 — Multiplying by 1 means you have exactly 1 group. One bag with 6 apples inside — you still have 6 apples.
- Name the property → Identity property — The identity property says any number multiplied by 1 stays the same.
- Write the answer → 6 × 1 = 6 — 1 is called the multiplicative identity because it does not change the number.
(5 × 4) × 3 = 5 × (4 × 3) = ?
Answer: 60
- Calculate left grouping first → (5 × 4) × 3 = 20 × 3 = 60 — First multiply 5 × 4 = 20, then 20 × 3 = 60.
- Calculate right grouping → 5 × (4 × 3) = 5 × 12 = 60 — First multiply 4 × 3 = 12, then 5 × 12 = 60.
- Name the property → Associative property: both = 60 — The associative property says you can regroup the numbers when multiplying and get the same answer. This is useful because sometimes one grouping is easier to calculate in your head.
Common mistakes
- Assuming division follows the commutative property, writing 12 ÷ 3 = 3 ÷ 12, giving 4 instead of the correct answers 4 and 1/4 respectively.
- Misapplying the distributive property to multiplication over multiplication, calculating 3 × (4 × 5) as (3 × 4) × (3 × 5) = 12 × 15 = 180 instead of 3 × 20 = 60.
- Confusing the identity property with adding zero, stating that 7 × 0 = 7 instead of 7 × 0 = 0, mixing up multiplicative and additive identities.
- Incorrectly grouping in associative problems, writing (6 × 2) × 3 as 6 × (2 + 3) = 6 × 5 = 30 instead of 6 × (2 × 3) = 6 × 6 = 36.