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Key Stage 3
National Curriculum - Calculating
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# 1. How confident are you that you understand:

## Example

Any operation ● which has the property that a ● b = b ● a for all members a and b of a given set is called commutative.

For the set of real numbers:

Addition is commutative, e.g. 2 + 3 = 3 + 2.

Multiplication is commutative, e.g. 2 × 3 = 3 × 2.

Subtraction and division are not commutative because, as counter-examples, 2 – 3 ≠ 3 – 2 and 2 ÷ 3 ≠ 3 ÷ 2.

## What this might look like in the classroom

Problem:
Take 20 counters and make a rectangular array. What multiplication facts does this tell you?

Solution:
A 4 by 5 array demonstrates that 4 × 5 = 5 × 4 because it doesn't matter what orientation the rectangle is in, the total number of counters is the same. Similarly, solutions could demonstrate that 2 × 10 = 10 × 2 or 20 × 1 = 1 × 20. The most significant conclusion from this is that the number of multiplication facts that need to be learnt in order to 'know your tables' is halved.

## Taking this mathematics further

The laws of arithmetic also apply within algebra, for example:

• a + b = b + a
• a × b = b × a (= ab)
• a × a × b = a × b × a = b × a × a (= a2b)
• 4 × y = y × 4 = 4y

Note: The convention of algebra is to write expressions in alphabetical order (e.g. ab) and numbers before letters (e.g .4y)

## Making connections

Learners are likely to have no problem understanding that addition is commutative because 'it's obvious!' The danger is that because it is seen as obvious, the commutative law is incorrectly applied to subtraction and division − particularly when it comes to writing the mathematics down. Use an incorrect written response such as 5 − 7 = 2 or 3 ÷ 18 = 6 to challenge misconceptions and discuss commutativity.

The laws of arithmetic also apply within algebra, but while 4x is technically the same as x4, the convention is that only 4x is used.

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