# How I wish I'd always taught... addition and subtraction.

Maths is full of symbols. Some of the more familiar are that addition and subtraction symbols. The equals symbol is a relative newcomer having been introduced in 1557 by Welsh mathematician Robert Recorde.

__Addition__

When we add, we sum together two (or more) quantities. This results in an increase in quantity (__unless we are working with negative numbers, of course__). Numbers being added together have useful names which can aid children to reason clearly. Let's take 9 + 1 = 10. 9 is the **augend**: the number in which is to be added to. 1 is the **addend**: the number too add to the **augend**. The total is 10 and so the **sum** is 10. As children gain fluency is adding 1, they begin to make links and connections known as **number bonds** which make patterns and can expose the structure of the number system.

This may be the first time children look for structures and make generalisations in maths. They are useful to know as their recall aids fluency with arithmetic and understanding of the number system which is built around sets of 10 (including the decimal system).

__Laws__

The three fundamental laws of addition unlock the structure of maths: **associative** law, **commutative** law and the **distributive** law. Understanding how the laws work leads to a deep understanding of how mathematics works.

__Associative law.__

This law states it does not matter how we group numbers with addition for the **sum** will always be the same. This law allows children to make calculations more efficient. For example:

9 + 12 + 8: it is more efficient to do 9 + (12 + 8) than (9 + 12) + 8. **Number bonds** to 20 help here too.

__Commutative law__

The commutative law states the order of addition makes no difference to the **sum**. For example: 1 + 2 = 2 + 1.

This is useful as children may encounter problems such as: 9 + 12 + 8 + 1. It would be much more efficient to look for number bonds such as 9 + 1 + 12 + 8 and achieving the same **sum**. We can express this algebraically: *a + b = b + a*.

__Distributive law__

This law requires numbers to be 'broken down' in small components. This is an essential skill for children to master to look for numbers *within* numbers. Essentially, looking for **number bonds** within a set of numbers. Let's look at this example:

27 + 14 + 11.

Here, we want to encourage children to apply the **distributive law** to break the problem down. It may look something like this:

(20 + 7) + (10 + 4) + (10 + 1)

(20 + 10+ 10) + (7 + 4 + 1)

(40 + 12)

= 52