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.