## Misusing commutativity in written division

When setting out short division, the amount being divided (the dividend) is usually larger than the number it is being divided by (the divisor).

635 ÷ 5 =

Thus, pupils can establish a process of writing the larger number in the ‘box’ or ‘bus stop’ and the smaller number before it, without considering the actual order of the numbers in a given calculation. As a consequence, when presented with a calculation where the dividend is smaller than the divisor, pupils will often switch the two numbers when setting it out. For example, when working out as instead of Pupils must remember that the first number in the calculation is the amount that is going to be divided up by the second number and is the number that always goes in the ‘box’. If this first number is smaller, then the first digit in their answer will be zero followed by a decimal point.

Similarly, individual digits can be switched within a calculation, when the digit being divided is smaller than the number it is being divided by. When calculating 2418 ÷ 6 a pupil may begin by saying ‘6 ÷ 2 = 3’ instead of ‘2 ÷ 6 = 0 rem 2’. instead of This is an easy mistake to make, given that we read the actual number sentence from left to right! Even more commonly, they will simply begin with the smaller of the two digits, saying: How many twos are in 6? The correct use of language needs to be reinforced to ensure consistent and correct application of the method.

Similar mistakes can be made mid-calculation:

Having correctly completed the initial part of the calculation, the pupil may then switch the next digits and say 6 ÷ 1 = 6 instead of 1 ÷ 6 = 0 rem 1. instead of 