Last time we were talking about subtraction, among other things. Let’s continue with this, and then discuss addition and multiplication. We’ll save division for another day – young people, unless they’re very well trained, find sharing difficult!

I’ve suggested that subtraction may well involve partitioning the number which is being subtracted. That’s OK, but I’m not sure that people would partition 69 when subtracting it from 97.

I think that most people would count up from 69: 69, 70 (1), 80, 90 (21), 97 (28). And this is important – *a lot of subtraction is done by adding not taking away*. In other words, subtraction is the operation but it is very often done by adding not taking away.

These are very obvious, but very important, ideas, and they illustrate a crucial principle for mathematics teachers. Mostly, it doesn’t matter what method is used for these calculations – if a student can find a method which works for them, then mostly it should be encouraged. And we mustn’t be hide-bound in using methods which we think are ‘right’. ‘Right’ is right for them.

**Task 1: Find the dates
**

**1**Today is Monday 13

^{th}June. Find the date in three weeks and 4 days.

**Strategy
**Add the correct number of days to 13.

If the answer is more than 30, subtract 30 (the number of days in June).

If the answer is more than 61, subtract 61 (the number of days in June and July).

**Thought process
**3 × 7 = 21 → 21 + 4 = 25, 13 + 25 = 38 → 38 – 30 = 8

The answer is 8

^{th}July.

**2** Today is Monday 13^{th} June. Find the date in 9 weeks and 1 day.

**Thought process
**9 × 7 = 63 → 63 + 1 + 13 = 77 → 77 – 61 = 16

The answer is 16

^{th}August.

There are lots of good things going on here. Practice with 7 times table, adding and subtracting, keeping information (number of days in June and July) in one’s head and available when required.

Keep doing this every week, start simply – they’ll get better at it surprisingly quickly. Plus it will have a knock-on effect in their written solutions to problems.

**Task 2: Number bases
**Explore number bases using only mental methods – which are, of course, how they learnt about base 10 in the first place.

**Base 10
**Go around the room in order – or by pointing at random to individuals – and get them to count from 1 to 30.

Repeat but this time, they’re only allowed to say digits.

When they get to 10 they have to say 1-0, 1-1 for 11, and so on.

Explain that this is how it’s written, because in base 10 there is no digit for 10; only digits 0, 1, 2 … 9.

**Extend to base 8
**This means that there is no digit for 8, only digits 0, 1, 2 … 7.

Go round again, counting: 1, 2, 3 … 7, 1-0, 1-1 …

This time keep going… 1-6, 1-7, 2-0 … 7-5, 7-6, 7-7, 1-0-0 …

This is hard. But nothing like as hard as base 2 – ‘you can’t say 2’.

1, 1-0, 1-1, 1-0-0, 1-0-1, 1-1-0, 1-1-1, 1-0-0-0 …

The fantastic thing is that a class of middle ability student in years 8, 9 or 10 (one-zero?) will be able to do this after a few weeks. Only then should you talk about the ideas of place value; that is, in base 8, the number 123 means 3 ones, 2 eights and 1 eight squared (64). This is the point where, arguably, they understand place value properly for the first time ever.

And they’ll understand that great mathematics joke: there are only 10 types of people in the world: those who understand binary and those who don’t. But they probably won’t laugh.

**Task 3: Extend to algebra
**Students should understand that they first learnt to count without any written work – the order in which they developed their understanding of number was: mental methods followed by written consolidation of those ideas.

Finally, extend this to other branches of mathematics, algebra in particular.

**Traditional, written methods
**Solve 2

*x*+ 1 = 11

Take 1 from both sides, giving 2

*x*= 11 – 1, etc.

When they’re fluent with those ideas (it may be terms, or a year after), get them to understand that the ‘grown up’ way to do algebra is to take things to the other side of the equals sign, when they become their inverse operation, with the numbers furthest from *x* being dealt with first.

**Solving mentally
**Write on the board: Solve 2

*x*+ 1 = 11

They have to say the following:

Two *x* plus one equals 11

Two *x* equals 11 minus 1

Two *x* equals 10

*x* equals 10 divided by 2

*x* equals 5.

Exactly that – first the original equation, then every line exactly as above. They’ll get the hang of it quickly and soon be able to solve equations mentally more easily than by writing.

Good luck,

John Rayneau

*John Rayneau , BSc MSc RGN CertEd, is the lead practitioner in maths at a school in Oxford. John’s articles draw from his experiences in mathematics education and nursing – as well as being inspired by the absurdities of school life.*