Commutative and Associative Laws
The commutative law lets you swap the order of numbers, like 3 + 5 = 5 + 3. The associative law lets you regroup them, like (2 × 5) × 4 = 2 × (5 × 4). Both keep the answer the same when you add or multiply.
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What is the Commutative Property?
- The commutative property means you can swap the order of numbers
- It works for addition and multiplication, but not for subtraction or division
Commutative Property: Examples
- In addition, changing the order gives the same total
- In multiplication, changing the order gives the same product
What is the Associative Property?
- The associative property means you can change how numbers are grouped
- It works for addition and multiplication, but not for subtraction or division
Associative Property: Examples
- In addition, regrouping numbers does not change the total
- In multiplication, regrouping numbers does not change the product
Using Both Properties Together
- You can reorder and regroup numbers to make calculations easier
- This helps you with mental maths and simplifying expressions
Practice Questions
Test your understanding
The commutative law says you can swap the order of the two numbers without changing the answer. Which pair of calculations does it apply to?
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The commutative law holds for addition and multiplication: and , so the order makes no difference.
It does not hold for subtraction or division. but ; but ; and but .
?
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The associative law lets you group the numbers in whichever order is easiest.
Group the two that make a round number: .
Then .
?
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Multiplication can be done in any order and in any grouping, so pair the numbers that make a round number first.
, then .
?
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Addition can be done in any order, so pair the numbers that make a round total first.
, then .
?
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Addition can be done in any order, so pair the numbers that make a round total first.
, then .
?
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Multiplication can be reordered and regrouped, so pair the numbers that make round values first.
and , so .
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Interactive Activity
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Frequently asked questions
The questions students bump into most on this topic
Commute means to move around. The commutative law, or commutative property, says you can swap the order of numbers when you add or multiply and the answer stays the same: 4 + 6 + 3 is 13 whichever way round you add the fruit. It does not work for subtraction or division, where changing the order changes the answer.
Associate means to join together. The associative law says you can change how numbers are grouped with brackets when you add or multiply, and the answer stays the same: (48 + 52) + 127 gives the same 227 as 48 + (52 + 127). Like the commutative law, it does not work for subtraction or division.
The commutative law swaps the order of the numbers; the associative law changes the grouping with brackets. To tell them apart, look at what moved: if the numbers changed places, it is commutative; if only the brackets moved, it is associative. Both keep the answer the same, and both work for addition and multiplication only.
You can add numbers in any order and get the same total. Adding 77 and 246 is awkward, but 77 + 23 is exactly 100, so swap the numbers in 77 + 246 + 23 to make 77 + 23 + 246, which is 100 + 246 = 346. Looking for a pair that makes a round number is the whole trick.
You can change which numbers are multiplied together first and still get the same product. In 128 × 5 × 2, working out 128 × 5 is hard, but moving the brackets to 128 × (5 × 2) gives 128 × 10 = 1,280 in one step. Group the pair that makes 10 or 100 and the rest is easy.
No. Neither law works for subtraction or division, because swapping or regrouping the numbers changes the answer: 10 − 6 is 4 but 6 − 10 is −4, and 12 ÷ 3 is 4 but 3 ÷ 12 is a quarter. Use the commutative and associative shortcuts for addition and multiplication only.
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