Multiplication Properties
Multiplication properties are fundamental mathematical rules that govern how numbers behave when multiplied together. These properties include the commutative property (order doesn't matter), associative property (grouping doesn't matter), identity property (multiplying by 1), and distributive property (spreading multiplication over addition). Understanding these properties forms the foundation for mental arithmetic and algebraic manipulation in Key Stage 2 and beyond.
Why it matters
These properties appear throughout mathematics education and everyday calculations. In Year 4 SATs, pupils use the commutative property to check multiplication facts like 6 × 7 = 42 by calculating 7 × 6. The distributive property becomes essential for GCSE algebra when expanding brackets such as 3(x + 5) = 3x + 15. Shopkeepers use these properties mentally when calculating prices: finding the cost of 8 items at £3 each can be worked out as 8 × 3 = 24 or broken down using the distributive property as 8 × (2 + 1) = 16 + 8 = 24. Engineers and scientists rely on these properties for complex calculations, whilst computer programmers use them to optimise code efficiency. The associative property helps when calculating compound interest or working with multiple discounts in retail environments.
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 7 × 3 the same as 3 × 7?
Answer: Yes (21)
- Calculate the first side → 7 × 3 = 21 — Think of 7 rows with 3 in each row. That is 21 altogether.
- Calculate the second side → 3 × 7 = 21 — Now flip the array: 3 rows with 7 in each row. Still 21!
- 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 3 × 0?
Answer: 0
- Think about what × 0 means → 3 × 0 = 0 groups of 3 — Multiplying by 0 means you have 0 groups. If you have zero bags of sweets, you have no sweets at all!
- Name the property → Zero property — The zero property says any number multiplied by 0 is always 0.
- Write the answer → 3 × 0 = 0 — No matter how big the number is, 3 × 0 = 0.
(5 × 3) × 2 = 5 × (3 × 2) = ?
Answer: 30
- Calculate left grouping first → (5 × 3) × 2 = 15 × 2 = 30 — First multiply 5 × 3 = 15, then 15 × 2 = 30.
- Calculate right grouping → 5 × (3 × 2) = 5 × 6 = 30 — First multiply 3 × 2 = 6, then 5 × 6 = 30.
- Name the property → Associative property: both = 30 — 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 is commutative, writing 12 ÷ 3 = 3 ÷ 12, which gives 4 = 0.25 instead of recognising that 12 ÷ 3 = 4 but 3 ÷ 12 = 0.25.
- Confusing the identity property with the zero property, believing that 7 × 0 = 7 instead of 7 × 0 = 0, or thinking 7 × 1 = 0 instead of 7 × 1 = 7.
- Misapplying the distributive property by writing 3 × (4 + 5) = 3 × 4 + 5 = 17 instead of 3 × 4 + 3 × 5 = 27.
- Incorrectly grouping in associative calculations, computing (2 × 3) × 4 as 2 × (3 + 4) = 14 instead of 2 × (3 × 4) = 24.