### Maths Help for Age 10-12: Multiples, Factors, Equations

Here's a neat way to revise multiples, practise factors and begin equations which I stumbled on when helping my son. His times tables and mental maths are very good and he's currently working at Level 5. Best not to use a calculator.

What are Multiples?
The multiples of 2 are 2 4 6 8 10 12 14 16 18 etc
multiples of 5 are 5 10 15 20 25 30 35 40 45 etc

What are Factors?
My son had a dim memory of studying Factors. So I explained, in a sense, factors are the opposite of multiples.
e.g. factors of 12 are1x12 = 2x6 = 3x4
1, 12, 2, 6, 3, 4 > Factors of 12: 1, 2, 3, 4, 6, 12

What we did
We made factor spiders, like this one for 36:

Then I asked my son to make factor spiders for:
a. 24 (easy) 1, 2, 3, 4, 6, 8, 12, 24
b. 48 (harder) 1, 2, 3, 4, 6, 8, 12, 16, 24, 48
c. 144 (v. hard)  1, 2, 3, 4, 6, 8, 9, 12, 16, 18, 24, 36, 48, 72, 144
I had to encourage him to be methodical, working from 1x?=144 2x?=144 3x?=144 4x?=144 5x?=144 ruling out any that didn't work (like 5x?=144).
We figured out that if there was a 12x?=144 then there would also be a 3x?=144 and a 4x?=144 because 3 and 4 are factors of 12. We also worked out that 144 couldn't possibly have 5, 10, 15 or 20 as a factor because all multiples of these end in 5 or 0, and 144 ends in 4.

How did we move onto equations?
While we were trying to work out factors, I started to write (n) instead of ? in 2x?=144 3x?=144 4x?=144 so we had 2 x n = 144, 3 x n = 144, 4 x n = 144, etc. I explained it like this: '(n) is 'the mystery number' (aka 'the answer').
Then we figured out, when we were doing factors of 144:
x 144 = 2 72 = 4 36 = 8 18

We noticed a pattern that helped us work out the answer. As one side doubles, the other halves:
72 = 4 36
36 = 8 18

So then we set up more puzzles (equations), to work out more 'mystery numbers':
16 = 6 n    answer: n = 32
x 20 = 10 n     answer: n = 10
2 50 = 4 n     answer n = 25
Last, but not least, I showed him that we write n as 6n (six lots of n), 10 n as 10n and n as 2n.
so  16 = 6n,  x 20 = 10n and 2 50 = 4n

After that, for some reason he wrote down n, so I asked him if he knew that we can write 2 x 2 as 2². (He said yes.) So then I said we write n x n as n².

Obviously all this was spontaneous, tracking his interest and understanding (we made it up as we went along), which may be why it worked. He drew a 'factor tree', which he was really pleased with, and he asked me to post it here (photo on its way!).