**The inverse of addition structure**

The key language to be developed in the inverse-of-addition structure of subtraction includes: What must be added? How many (much) more needed?

The inverse-of-addition structure refers to situations where we have to determine what must be added to a given quantity in order to reach some target. The phrase inverse-of-addition underlines the idea that subtraction and addition are inverse processes. This means, for example, that since 52 + 28 comes to 80, then 80 – 52 must be 28. The subtraction of 52 ‘undoes’ the effect of adding 52. Hence to solve a problem of the form ‘what must be added to x to give y

We subtract x from y. For example the entrance fee is 80c, but I have only 52c, how much more do I need? The calculation that would be entered on a calculator to solve this problem is the subtraction, 80 – 52. However how this subtraction structure might be interpreted as an action on a number line: starting at 52 we have to determine what must be added to get to 80. This is a particularly important structure to draw on when doing subtraction calculations by mental and informal strategies.

The subtraction structure is often the most difficult to recognise for primary pupils because of the language associated with it, such as ‘how much more is needed?’ and ‘what must be added?’, signals the idea of addition rather than subtraction e.g. if I scored 180 in darts, how much more do I need to reach 401?

The language used in problems with the inverse-of-addition structure often signals addition instead of subtraction, so that many pupils will automatically add the two numbers in the question. Such pupils will need targeted help to recognise the need for subtraction. Asking the question ‘What is the calculation to be entered on a calculator to solve this problem?’ helps to focus the pupils thinking on the underlying mathematical structure of the situation.

Children must be given opportunities to learn to apply their number skills to all of them. Being able to connect subtraction with the whole range of these situations and to switch freely from one to another is also the basis for being successful and efficient at mental and informal strategies for doing subtraction calculations. For example, a pupil may, to find out how much taller a girl of 167 cm is than a boy of 159 cm (comparison), recognise that this requires the subtraction ‘167 – 159’, but then do the actual calculation by interpreting it as ‘what must be added to 159 to get 167?’ (inverse addition).