Subtraction by Adding On

This is also sometimes called “subtraction using a blank numberline” and I’ve even heard it called “that nonsensical modern method”. This latter is a real misnomer since it is neither nonsensical nor modern. In fact it’s a method that dates back to before I was born, in days before we had calculators and electronic tills. It’s also a really useful method involving counting forwards, which is always easier than counting backwards – even for maths geniuses!

Let’s return to those old-fashioned little shops. I buy some sweets for 24p and hand over a £5. To work out my change, the shopkeeper needs to calculate £5 – 24p. Now she could count £4.99, £4.98, £4.97 until she had subtracted 24p, but what you actually would have heard is this:

24 and 6 makes 30, and 20 makes 50 and another 50 makes £1. 2, 3, 4, £5. And while doing this they counted the change (£4.76) into your hand.

This is the method that schools have returned to. To begin with, Children use a “blank numberline” – that is, a line that they can write the numbers on themselves. They then write the lower number at one end, and see what they need to add to make the higher number. Here’s an example:

96-38

numberline

The children first of all use their knowledge of number bonds to add to the next 10 (38 + 2 = 40).
They then use their ability to add multiples of 10. In the example above the child has done 40+10 = 50 and then 50 + 40 = 90. They may have been able to see straight away 40 + 50 = 90 and done this as one step, or they may have needed to do 40 + 10 = 50, 50 + 10 = 60, 60 + 10 = 70 and so on up to 90. The method isn’t about having to complete it in a certain number of steps, it’s about each child breaking it down into the smallest number of steps that they can manage.
When they reach the multiple of 10 before the higher number, they add on whatever units are needed to make the higher number, in this case it was +6 to make 96.
Finally, they add up all the numbers they added on to find the answer: 2+10+40+6 = 58 so 96 – 38 = 58.

When they are confident with this, they move on to jotting down only the numbers they are adding on, and they keep the tally in their head, until eventually they develop their working memory enough to hold all of the numbers in their head and write down just the answer.

Related posts: Teaching the Times Tables, Teaching Sequencing and Column Addition

Teaching sequencing and column addition to a child with dyscalculia

Recently a colleague asked me for some suggestions to help one of his pupils with her maths. She was having various problems, such as

  • Difficulty with sequencing numbers
  • Getting confused as to which way to move on a number-line to add or subtract single digits
  • Getting confused as to which way to move on a number-square to add or subtract multiples of 10
  • Not understanding whether an answer she got when performing a calculation was “reasonable” or way off
  • Confusing the Hundreds Tens and Units columns and so not always starting in the correct place when performing calculations.

He suspected that the child in question might have dyslexia and / or dyscalculia, and if that is the case then I can understand why they might have trouble with column addition/subtraction. They’ll be concentrating really hard on left to right, left to right for their writing, and then suddenly column calculations go right to left – no wonder they get confused!

My advice was to make the learning experience completely multi-sensory, even if it meant taking the learning outside.   These were some of my suggestions:

  • Make a physical number line on the floor/front driveway/back garden/anywhere with plenty of space. Place one object with the label “1”, then two objects labelled “2”, three objects and a label 3 and so on to help her equate the number 3 with the value 3
  • Chalk the numbers outside, and get her to walk along it counting forward, and then walk the other way counting backwards.  Get her to jump along it landing on every other number counting forwards in twos and then backwards in twos.
  • Move on to a number square in chalk so that they can change direction to add on/take away 10. This should also help with “reasonable” answers because, for example, she would come to understand that she had to walk further to add on 49 than to add on 12.
    • Always make the number square start with 1 at the bottom, rather than at the top like most number squares – then the higher numbers are at the top of the square and the lower numbers are at the bottom which also helps with understanding the value of numbers.
    • Use an abacus for additions/subtractions instead of written methods. I’ve done this with Y6 children who had no concept of place value and it made a huge difference!
  • When moving on to column addition and subtraction, colour-code the numbers in each column with a known sequence of colours (eg Red White and Blue so they do red units first, then white tens, then blue hundreds). Put the numbers either on coloured card – or even better use painted wooden numbers so she can pick them up and feel the shape of each number
  • People with dyslexia tend to think in pictures, so when finally moving onto pen and paper calculations, try putting pictures of Strictly Come Dancing / X-Factor judges at the top of each column. The judges always sit in the same order on the shows, so it’s easy to picture them sitting in a row – then you know that you always have to add Bruno Tonioli’s numbers first!

One final tip I picked up at a session on dyscalculia to help children with sequencing numbers was to give them something associated with each number so that they have something to relate that number to – seeing how the number matches the object and handling the objects while they count makes it more of a multi-sensory experience.  For example if you want them to count in 6s, rather than giving them something generic like pictures of 6 spots or sets of 6 cubes, give them egg-boxes.

Teaching Number Bonds

Number bonds. We all use them in our adult life without even realising it. When adding up items in our basket at the supermarket, we know that 3p and 7p is 10p, and that 2x 50p is £1. When we buy something for £5.60 at the market and hand over a £10 note, we know that £5.60 + 40p is £6 and another £4 makes £10, so we know to expect £4.40 change. Knowing our number bonds is extremely useful, but a lot of children struggle to learn them.

Over the years I have successfully taught many children how to remember their number bonds. As with times tables, the key is to find multi-sensory ways to teach, and to make practising fun.

Some children respond very well to visual clues, and to help these I use colour strips. These are strips marked out in 10 sections and coloured in contrasting colours, so that children can see clearly that 2 red squares plus 8 green squares equals 10 squares altogether, and that 8 green squares plus 2 red squares also equals 10 squares altogether. They are small enough to hold in the hand, and I tend to use them in conjunction with other methods. The children I tutor find them really useful to refer to during games.

Snap and pelmanism are always popular games, and I have made two sets of cards for this. The first set is colour-coded, so when the children turn the cards over there is a visual clue as to whether the two cards add up to 10. When they turn over the first card, I encourage them to work out what number they need to find to make 10. When the children are a little more confident I switch to the black and white ones to remove the visual clue, but we still play the same games to keep some familiarity.

Another card game I play is Imprison the Villain, which is played in the same way as Old Maid/Donkey/Chase the Ace. I use these lovely monster cards, which I downloaded from Primary Resources. I coloured mine in for added attraction and laminated them for durability.

Bengali - number bonds to 10I have one last set of number bonds cards which I made especially for one boy. He was struggling with number bonds, and as his home language was Bengali not English, as an experiment I found the numbers in Bengali on the internet and made him a set of dual language numbers cards. They have the numbers in words and figures in Bengali and English. They worked! When he saw them his eyes lit up and he pointed excitedly saying “I know these numbers!” The cards really increased his motivation and it didn’t take him long to learn them.

There are also a couple of fun number bonds games on Sue Kerrigan’s Let Me Learn website. I have Number Bombs and the similar but seasonal Elf Splat. There are two game boards for each of these games, one for addition and one for subtraction, which are both versatile enough to use for number bonds to 10 and 20. I haven’t tried this game with any girls yet, but the boys love it, and now I get greeted with “Are we going to play Number Bombs today?” My answer is always “We don’t need to now – you already know your number bonds to ten!”  However because it is so popular we do sometimes play it at the end of a session as a reward for good work!

Another game that children seem to love is ping pong. I can’t remember where I first heard about this game, but it’s played like a game of table tennis except that you bounce numbers backwards and forwards instead of a ball. I begin by establishing a rhythm – I say ‘ping’ and the child replies ‘pong’. Then I start calling numbers from 1-10 and the child gives me the corresponding number that adds to 10.  As we speak we swing an imaginary table tennis paddle to hit the numbers. For extra fun and an exercise involving whole body movement, stand opposite sides of the table with real paddles.

dominoesFor progression to 20, I have found a dominoes game which you will find if you follow the link to Primary Resources and type dominoes in the search box.  It takes a little practise because many children don’t know how to play dominoes, but once they get the hang of it, it proves quite a popular game.

For moving children on and helping them to see that when they know their number bonds to ten, and understand the pattern for making twenty, it’s quite easy to use this knowledge to find bonds for any multiples of ten, Sue is developing some resources based around football which highlight the patterns for making 30. I have been lucky enough to trial these resources. There is a write-on wipe-off card which explains how to use your knowledge of number bonds to make 30, and then a booklet to practise in.   There is also a game to play – similar to Number Bombs, but with the added excitement that if you land on certain squares you can get ‘sent off’ and you have to remove one of your counters. I have only recently starting trialing this game, but it is proving popular so far. Finally, the set includes a match report card, where you can tick off each goal as you achieve them: working out number bonds to 10 on fingers, knowing number bonds to 10 from memory etc.

Games are fun, but sometimes children need other ways to make those numbers stick, and word association is often helpful. I have found a few number bonds rhymes on the internet, but I find it works better when the children write their own. I have helped children to write their own rhymes, such as “5 plus 5 like to go for a drive” or “7 plus 3 go to Devon for their tea”. They then illustrate each of the rhymes with pictures such as two number 5s in an open top car, or the numbers 7 and 3 having a picnic under a sign saying Devon.

One Year 6 girl I worked with found it really hard to remember which numbers added to ten, and she was really down-hearted at being so far behind her classmates. We used this word association method and she found that she could remember her own poem and the pictures really easily. To begin with she said her rhyme every time before deciding which numbers went together. Eventually the numbers became so well embedded that she was able to dispense with the rhyme.

By the time she moved on to secondary school, she was still behind her classmates, but she now knew that she could achieve in maths which really increased her confidence. And that’s why I really love my job.

Related posts: Teaching the Times Tables    A Multisensory Approach to Reading