What do we do with the remains?
I am a classroom assistant and I don't know what happens with remainders when you do division. I hated it at school and I hate it now. But I have been asked to work with a small group of pupils and I don't like to let them know that I really don't understand how to help with division.
I was once told a story about someone observing a teacher who was teaching division who did not understand what happened with a remainder. They taught the children that when 13 is divided by four the answer is 3.1, as they incorrectly thought the remainder should be written as a decimal. After the lesson the observer tried to explain the correct form of division to the teacher. But the observer found this task very difficult. When I was told this story I first thought how the teacher must have felt and how they probably found it difficult to take anything on board having realised that they had taught the whole lesson incorrectly. I wondered how I might work with someone who has a misconception about remainders. My suggestion is to take 13 pieces of A5 card and ask the person to work out 13 V 4. They will either arrange the cards into three groups of four or four groups of three, but will still have the answer 3 with one card remaining.
Ask them about this card, what should be done with it? It should also be divided by four, giving the fraction 14. Then talk about how this could be written as a decimal: 0.25. This might not be clear but could be demonstrated using the card. First divide the card into 10 equal rectangles. Then shade a quarter (to work out a quarter fold in half vertically and then horizontally). As in Figure 2, this leaves one rectangle half shaded, so it is 0.05 of the whole. Now count up the shaded parts: 0.1 + 0.1 + 0.05 = 0.25. So the answer to 13 V 4 is 3.25.
Now we return to our original problem and write the division in a more formal way as: In the first stage we get the 13 V 4 = 3, with the remainder of one (Figure 3). In writing the sum (Figure 4), point out that there is a decimal point after the 13 and that there are zeros after that (one zero shown initially), and that a decimal point is also written after the three above the line. We now put the one (the remainder) beside the zero, to make 10 which we read as 10 V 4, giving 2 with a remainder of 2. Write the 2 (tenths) above the line after the decimal point. The 2 (the new remainder) is then carried to before the next zero, which we have just written although it was there all the time. The next sum becomes 20 V 4, with an answer of 5 (five hundredths). This is written after the 2 to give an answer of 3.25. The same as with the cards.
Try several examples that you've made up, check them with a calculator.
When you feel confident, then work with the pupils you have been allocated.
Remember to use lots of different ways to ask division. My poem below brings these words together!
The quotient is where
You divide your share
To split the group.
Divide implied By the words?
When you divide take a side
When you share, have a care
When you split, you must quit
Owning a quotient is quite potent!
Wendy Fortescue-Hubbard is a teacher and game inventor. She has been awarded a three-year fellowship by the National Endowment for Science, Technology and the Arts (NESTA) to spread maths to the masses. Email your questions to Mathagony Aunt at email@example.comOr write to TES Teacher, Admiral House, 66-68 East Smithfield, London E1W 1BX