Mental strategies for addition and subraction

Like the commutative law (a + b = b + a) the associative law is a fundamental property of addition. Written formally, as a generalization, it is the assertion that for any numbers a, b and c:

a + (b + c) = (a + b) + c

Using a particular example, this might be: 7 + (13 + 18) = (7 + 13) + 18

The brackets indicate which addition should be done first. In simple terms, the associative law says that if you have three numbers to add together you get the same answer whether you start by adding the second and third or start by adding the first and second. In the example above, it’s probably easier to start by adding the 7 and 13, but you get the same answer if you start with 13 + 18.

One way to remember the associative law is by thinking of it as a picture of three political parties: sometimes the party in the centre associates with the right and sometimes it associates with the left, but it doesn’t make any difference!

This law allows us to write down 7 + 13 + 18, without any brackets to indicate which two numbers should be added first. We can choose whichever we prefer. The commutative and associative laws combined give us the freedom to add a string of numbers together in any order we like.

For example, 7 + (13 + 18) could be changed as follows:

7 + (13 + 18) = 7 + (18 + 13) (using the commutative law)

= (7 + 18) + 13 (using the associative law)

= (18 + 7) + 13 (using the commutative law)

= 18 + (7 + 13) (using the associative law)

= 18 + (13 + 7) (using the commutative law)

= (18 + 13) + 7 (using the associative law)

= (13 + 18) + 7 (using the commutative law).