## Zero with a remainder

When the answer to a division question is zero with a remainder, pupils can find this very difficult to understand and work out correctly. Confidence and correct understanding are vital for future work with short and long division written methods.

When presented with a calculation where the initial number is smaller than the number it is divided by, it is common for pupils to automatically switch the digits around, incorrectly using commutativity . So in the example above, the pupil has actually calculated 5 ÷ 4.

If a pupil doesn’t switch the digits, they may instead say, ‘You can’t do 4 ÷ 5’. They need prompting to explain why (because 4 isn’t big enough) and then encouraging to see that because 4 isn’t enough to make a group of 5, they will be able to make zero groups. Pupils don’t always realise that zero is a valid answer , so this point may need some reinforcement.

The next step is to calculate the remainder. Here, the pupil has recalled that this involves finding the difference (by counting on or subtracting), and has just mistakenly found the difference between the two digits in the calculation. They knew that 5 – 4 = 1, so wrote this as the remainder.

In this calculation, the pupil knew that the answer was zero, and that its remainder would be the same as one of the numbers in the calculation – but they didn’t understand which number or why.

Using objects to physically demonstrate this division will remind pupils that they are finding the difference between the number they started with and how many they used to make groups.