**Using multiples of 10 and 100 as stepping stones **

Remember to take every opportunity to explain mental strategies with reference to hundred squares and empty number lines, in order to provide pupils with mental images that will underpin their manipulation of numbers.

Notice what happens when we add 5 to 57 on the hundred square. We have to break the 5 down into two bits, 3 and 2. The 3 gets us to the next multiple of 10 (60) and then we have 2 more to count on. This process of using a multiple of ten (60) as a stepping stone is an important mental strategy for addition and subtraction. (*Note*: multiples of 10 are 10, 20, 30, 40, 50, 60, …and so on.

Here is how we might use this idea of a stepping stone for calculating, say, 57 + 28. First, we could count on in 10s, to deal with adding the 20: 57 …67, 77. Then break the 8 up into 3 and 5, to enable us to use 80 as a stepping stone: 77 + 8 = 77 + 3 + 5 = 80 + 5 = 85. A number-line diagram is a useful image for supporting this kind of reasoning.

Pupils can be taught to use an ‘empty number line’, which is simply a line on which they can put whatever numbers they like, not worrying about the scale, just ensuring that numbers are in the right order relative to each other. The diagram shows an empty number-line representation of the calculation of 57 + 28, using 80 as a stepping stone.

*Using multiples of 100 as stepping stones in a subtraction*

Figure 2 above, shows how we might use multiples of 100 as stepping stones for calculating 542 -275, using an empty number line. Using the inverse-of-addition structure, the sub- traction can be interpreted as, ‘What do you add to 275 to get 542?’ This is done in three steps (25 + 200 + 42), with 300 and 500 as convenient stepping stones lying between the 275 and the 542.