What are the commutative, associative and distributive property?

What are the commutative, associative and distributive property?
The commutative, associative and distributive properties or laws underpin algebra and are first introduced to children, in very broad terms, in the primary-school years. We explain how your child will start to understand the basics of higher maths in our guide for parents.
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What are the commutative, associative and distributive property?

The commutative, associative and distributive property are used in algebra to help us solve number problems.

The commutative property explained for parents

The word 'commutative' comes from 'commute' or 'move around', so the commutative property refers to being able to move numbers around within number sentences

For example: 2 + 3 gives the answer 5, and if we move the numbers around to make 3 + 2, we still get the answer 5. Equally, with multiplication: 6 x 4 = 24, just as 4 x 6 = 24.

In algebra we would write the commutative property as:

  • a + b  =  b + a
  • a × b  =  b × a

The commutative property applies to addition and multiplication, but not to subtraction or division.

Primary-school children are extremely unlikely to hear the words 'commutative property', but children in Year 2 will be taught that addition of numbers can be done in any order, but subtraction of numbers cannot.

The associative property explained for parents

The associative property says that when we add or multiply numbers it doesn't matter how we group them.

This rule applies to numbers that are grouped within brackets, for example: 2 + (3 + 4) or 5 x (2 x 3). 

We put calculations in brackets when we want someone to do that particular calculation first.  However, when it comes to a group of numbers that need to be added, it does not matter where the brackets go, the answer will still be the same. Equally, with a group of numbers that need to be multiplied, the answer will be the same regardless of how the numbers are grouped.

In algebra we write the associative property as:

  • (a + b) + c  =  a + (b + c)
  • (a × b) × c  =  a × (b × c)

The associative property applies to addition and multiplication, but not to subtraction or division.

It is unlikely that children will hear the term 'associative property' at primary school, but they may be asked to solve a problem, such as the following, where the rule is applied:

Three friends have collected pine cones and are collecting them in a container. Mary and Paul each collect 20 pine cones. John adds another 15 to the collection. How many are there altogether?

When solving the word problem above, (20 + 20) + 15 = 20 + (20 + 15)

The distributive property explained for parents

This is the rule that says that multiplication distributes over addition.

For example: if we want to carry out the following multiplication 2(5 +3), it means that we are working out what 2 lots of 5 are, plus 2 lots of 3, so the 2x can be 'distributed' across the 5 + 3, into 2 x 5 and 2 x 3. This can be written as the following formula: 

  • a × (b + c)  =  a × b  +  a × c

Again, the term 'distributive property' will not be explicitly referred to at primary school, however, children may be asked to solve a problem similar to the following:

There are 25 children in Birch Class and 28 children in Chestnut Class. Each child in both classes needs to bring a pair of gloves to school on Monday. How many individual gloves will be brought in altogether?

Here, the child would need to realise that they can either work this out as:

  • 25 + 28 = 53
  • 2 x 53 = 106

Or they could do:

  • 2 x 25 = 50
  • 2 x 28 = 56
  • 50 + 56 = 106

 

How algebra is taught in KS2

Children will start to learn algebra in Year 6, where they will learn about how to use simple formulae and find missing numbers in equations.

It is not stated in the national curriculum that KS2 children need to learn about the use of brackets in calculations. It is possible, however, that more able Year 6 children will be introduced to the use of brackets; the rules about the order in which to complete calculations are taught as BODMAS (Brackets, Other / Indices, Division, Multiplication, Addition and Subtraction).