Primary Mathematics/Dividing numbers

Dividing numbers
When first teaching young children to divide, you may like to try getting a small group together and sharing out a number of objects evenly between them. This will ensure that the students get a clear understanding of the concept of division before moving on to the written process of short and long division.

The process of division should be taught as the reverse of multiplication, again the times tables are needed. The links between addition and subtraction and multiplication and division become important later when the 'opposite' or inverse operation will be needed to solve algebraic problems. Primary teachers can help with later learning by emphasising that division is the undoing of multiplication, as well as pointing children in the right direction for actually doing division.

The simplest division is where the multiplication is known:

$$\frac{10}{2} = 5$$

(Note that I will write division as fractions - the formating is easier - the two are the same thing, children need to know this; I have met GCSE students, 15 and 16 years old, who were confused about this. The usual line with dots above and below (÷) is normally used for these simple problems.)

Harder are problems where the inverse is not known:

$$\frac{98}{2}$$

As 98 is greater than 10×2, children are unlikely to know the answer. This is where we introduce long division, which we write like this: __ 2/98

The first step is then to divide 9 by 2, however 2 does not go into 9. We find the highest number less than 9 into which 2 does go (8) and divide it by 2. The remainder (9-8=1) we add to the units as a ten (we have taken 10 away from 90 to get 80, we then add the 10 to the 8 to get 18). Having done this we divide 18 by 2 to get 9. This is often set out like this (more often it is set out as a crossed out 9 with a little 1 squished in by the 8, however this setting out is clearer and much better for harder problems):

49 2/98  - 80    18  - 18     0

The 8 under the 9 is 4×2 and is the lowest multiple of 2 less than 9. As the 9 is in the 10s column it is 90 and the 8 is 80. We then subtract the 80 from the original 98 and divide the resulting 18 by 2.

In a longer example we do the same thing more than once:

78487 2/156974  - 140000  (2 does not go into 1 or 15, but 2×7 is 14) 16974    - 16000  (2 does go into 16, 8 times) 974    - 800  (2 does not go into 9 but it does 8) 174    - 160  (not 17 but 16) 14     - 14  (and finally 7×2=14) 0

This method extends directly to division by larger numbers. For numbers larger than 12 writing out the multiples of the number to the side can be useful - only go up to what is required, calculating more as you need them. It also extends to decimals - just continue after the decimal point, allthough if the number you are dividing by is a decimal you need to multiply both it and the number you are dividing it by by 10 or 100 or 1000 etc. in order to make it an integer (whole number).