Do any of your students get in a mess with “double borrowing”? There’s a very simple solution which builds on their existing knowledge of subtraction with borrowing….
Callum buys a packet of crisps for 62p. He pays with a £2 coin. How much change will he get?
The correct answer is £1.38 but a proportion of children will answer £1.48. The logic goes like this:
- Count up from 8p to 10p, that’s 2p
- Count up from 60p to 100p (or £1) that’s 40p
- Count up from £1 to £2, that’s £1.
The tiny error that has crept in is not noticing that, the counting up from 8p to 10p actually means we start from 70p… and if you’re still following this, then you are clearly good at understanding abstract verbal ways of looking at Maths. My hunch is most people who see themselves as “good at Maths” may be Auditory learners. Most of the people I end up teaching 1-1 don’t favour this learning style. They prefer visual approaches, or Kinaesthetic.
Auditory/Visual learners may confidently get the right answer to this question of giving change, using the standard Numeracy Strategy approach of “counting along the number line”. They may sketch the line and, in due course, be able to “see it in their head”, or just recite the “jumps” up to £2 (That’s visual and Auditory respectively.)
Kinaesthetic learners may struggle to jot down the number line with enough confidence to be able to get the right answer. They may not be clear what “lies between” 62p and £2. (or, worse, between £3.27 and £10.00).
The best standard resource in the classroom for these learners is a 100-square but you will need 2 squares for this problem. Click the picture for a black and white printable pdf (Two Pounds Square) of the £2 hundred-squares. Or, if you want a coloured one, print this one (Two Pound Square colour version)
To use this printable to work out the change, ask the student to find the square containing 62p. Then count to the end of the row, that’s 8p. Now count up to £1. That’s 30p. Now jump to £2. That’s £1. So the “jumps” total £1.38.
You will need the printable cards Click for the printable pdf
Have some cards marked 10p, 20p, 30p, 12p, 11p, 31p. Take turns to draw a card, and move your counter up, by that amount of money. First person to go over £2 wins (and it’s a nice touch to ask “where would you be now, if the next card was printed?”. That will tell you if they are able to extrapolate the ideas on beyond £2). This game will get the players used to “counting up” on the board, and difference between a move of 1p and a move of 10p.
You will need the rest of the cards from the set you just printed (the ones in normal type). Place face down on the table and shuffle. Take turns to look at a card, and calculate the change from £2. If you can turn over the correct card as your second one, keep them both. Otherwise, turn them back face down again. The winner is the one who has the most cards. (Note, Kinaesthetic learners seem to be really good at “pairs”, because they can remember where the cards are. Since I don’t have this strength, I lose, pretty well every time…)
Over to you
Did you try this idea? Or adapt the resource to your own game or lesson? Leave a reply below to let me know how it went!
There are some issues with Maths that pop up across all agegroups, and the upside down subtraction bug is one of them. If you now have a mental picture of some sort of 6-legged brightly coloured ladybird creature hanging under a branch, then that’s not quite what I mean. I’m talking about the commonest error people make in traditional “Column Subtraction”.
The one where they say that 573 – 254 = 321.
It goes like this: 500 – 200 = 300, then 70 – 50 = 20, then lastly 3 – 4 = 1.
Most kids, when you point out their error, say “Oh yes”, and try again, doing the borrowing correctly and getting the correct answer of 319. But I just have a sinking feeling that the bug will reappear the very next time they subtract… nothing is solved. And I hate not solving a problem….
Working as a maths tutor in a primary school gives me a new perspective on how maths is learned and taught. The language that is used is pretty consistent – the teachers all sing pretty much from the same hymnsheet, and a lot of the kids make really good progress. And this is how this sum “should” be done….
3 – 4 YOU CAN’T so go next door. 7 becomes 6 and 3 becomes 13. 13-4 is…. 13, 12, 11, 10, 9. NINE. 6-5=1. 5-2=3. The answer is 319.
This makes me uncomfortable, it’s simply not true that 3 – 4 is impossible, and it set me wondering if kids’ later dislike of negative numbers partly stems from this rather strange approach to subtraction. What if the poor things really believe that you can’t do 3-4? I remember being 6 and if an adult told me I couldn’t do something, then it was TRUE.
So that’s one reason that traditional subtraction bothers me, but the other reason is that it bears no relationship with the mental method that is taught in primary, the whole concept of counting on. Using this method, 573-254 becomes:
+6 +40 +100 +100 +70 +3
254 —–> 260 ——> 300 ——> 400 —–> 500 —-> 570 ——> 573
the answer is 100+100+70+40+6+3 = 219 and that is actually VERY difficult. So it’s only really suitable if the 2 numbers are a small number of steps apart.
I like written methods that flow seamlessly into mental methods, because kids feel SOO good if they can do maths mentally, but a lot of them do need to jot as a first step. So the other day, I tried out this with Year 6:
573-254 goes like this:
500 – 200 = 300
70 – 50 = 20
3 – 4 = -1
300 + 20 – 1 = 319
Several of them experienced genuine panic and distress – why were we working left to right? Surely this was wrong? 3-4 I REALLY can’t. They also believed that they needed to learn a whole load of new subtraction facts like 4-9=-5. It took a while for some of them to learn to cheat (ie do 9-4=5, so 4-9=-5) and yet cheating is pretty much the upside down subtraction bug, used properly…
What was surprising, after I had done this with 4 groups over the morning, was that the kids who had the biggest panics were the ones who ended up fastest and most confident with the method.
Like so many alternative methods, I don’t expect it will suit them all… but it opened their minds a bit, and empowered one or two of them to do some wonderful, fast, mental work.