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:



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

How I passed the QTS maths test – part 1

Like a lot of people, I was always scared of maths. I hated it at school – somehow those numbers never made as much sense to me as they did to my peers. But because claiming to be bad at maths is seen as something to be proud of in this country – it’s up there with not being able to speak another language – I never really worried about it.

I’d somehow managed to scrape through O level, and somewhere along the way I learnt how to work out a gross profit margin, which was all I needed to do my job, so everything was fine. Until I decided I wanted to retrain as a teacher.

Suddenly I had the prospect of the QTS skills test looming over me. I wasn’t worried about the English and ICT ones, but the maths one filled me with fear. I tried the online practice test and ended up a weeping, soggy mess on my desk. So how did I get from there to where I am now, which is a qualified teacher who

  • passed the skills test first time
  • has the confidence to teach maths up to Y6
  • is able to tutor pupils in years 7, 8 and 9 in maths
  • tutors trainee teachers to help them pass the same test

The short answer is practice! The longer answer is more practice and a lot of help, and I began by dividing the test into its two parts: the mental arithmetic section and the traditional pen and paper maths section. I tackled each part separately, and in the next two posts I will explain how.

Related posts: Passing the QTS maths skills test – Part 2, Passing the QTS maths skills test – Part 3

Finding a square of a 2-digit number in your head

Recently we looked at how to square two-digit numbers ending in 5 in your head.  What about the other square numbers?  Well, there’s a quick way to do that in your head as well.

Let’s look at 372.
First of all round the 37 up or down to the nearest 10.
You added 3 to get 40, so you also need to subtract 3 to get 34 (37-3=34)
Multiply these to numbers together 40×34 = 4 x 34 x 10 = 1360 (4×30=120,  4×4=16, 120 +16 = 136, 136×10=1360)
Finally, add on the square of the number you added and subtracted (here it was 3 and 3×3=9 so add 9) 1360+9=1369.
So 372 = 1369

Here’s another example: 622
Round this down to 60, and then because you subtracted 2, you also have to add 2 so you get 60×64 (6 x 64 x 10). 3 x 60 = 360, 6 x 4 = 24, 360 + 24 = 384, 384 x 10 = 3840
Remember to add the square of the number you rounded down by. 22 = 4 so add 4.
3840 + 4 = 3844
So 622 = 3844

If you’ve enjoyed these tricks then you might enjoy the Secrets of Mental Maths course from The Great Courses. It’s fascinating!

Related post: How to multiply any 2-digit number by 11 (in your head!)

How to square 2 digit numbers ending in 5 (in your head)

This is another trick that my family already knew, but I only learnt it recently so I’m sharing it for anybody out there who doesn’t already know.

Squaring 2 digit numbers is easy if they end in 5.

5×5 = 25, so the last 2 digits of the answer will always be 25.

To find the first digits of the answer, take the first digit of the number you are squaring and multiply it by the next ten up.

Eg 65 x 65

The first digit is 6. The next ten up is 7, so to find the first digits of the answer you multiply 6×7.

6×7 = 42

The last two digits will always be 25 so 652 = 4225

352 = 1225  (3×4 = 12 and the last 2 digits are 25)

852 – 7225 (8×9 = 72 and the last 2 digits are 25)

It’s especially useful for squaring something-and-a-half eg 7.5 x 7.5. 7 x 8 = 56. The last two digits will be 2 and 5, but here you are using ½ not a whole number, so it won’t be 25 but .25 so the answer is 56.25.  3.52 is 12.25 and so on.

If you enjoyed this trick, you might also enjoy this post on how to multiply 2 digit numbers by 11 in your head.

Multiplying 2 digit numbers by 11

Earlier this year I discovered a trick for multiplying 2 digit numbers by 11.  I taught this to some of the children I tutor to help them work out 11×11 and 11×12 which they had been struggling with but then as it works for all 2 digit numbers they have enjoyed baffling their family and friends with their ability to perform calculations such as 54 x 11 in their head.

Apparently it’s not as big a secret as I had thought (well, everyone in my family already knew how), but I’m going to share it anyway just in case some of you don’t already know how…

First of all you split the digits of the two digit number, and these become the first and last digits of the answer. Eg with 54 x 11, the answer is going to start with 5 and end with 4. To get the number in the middle you just add the two digits together. 5 + 4 = 9, so put 9 in the middle. Your answer is 594.

35 x 11 = 3 [3+5] 5   that is 385
42 x 11 = 4 [4+2] 2   that is 462.

That’s all very well, but what if the two digits add up to more than 9?  That’s easy too. You partition that number, the units become the middle digit and the tens get added to the first digit.

86 x 11 = 8 [8+6] 68 [14] 6 → uh oh!  It can’t be 8146. That’s ok – split the 14. The 4 becomes the middle digit, and the ten (1) gets added to the first digit → 8+1=9, so the answer is 946.

Happy calculating!

Related posts: finger tables, how do you know if a number is divisble by…?How to square a 2 digit number ending in 5 in your head