**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**are: 1x12 = 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.

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 2

*x*?=144 3*x*?=144 4*x*?=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**n**umber' (aka 'the answer').
Then we figured out, when we were doing factors of 144:

1

*x*144 = 2*x*72 = 4*x*36 = 8*x*18
We noticed a pattern that helped us work out the answer. As one side doubles, the other halves:

2

*x*72 = 4*x*36
4

*x*36 = 8*x*18
So then we set up more puzzles (equations), to work out more 'mystery

**n**umbers':
3

*x*16 = 6*x*n answer: n = 32
5

*x*20 = 10*x*n answer: n = 10
2

Last, but not least, I showed him that we write 6

so 3

After that, for some reason he wrote down n

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!).

*x*50 = 4*x*n answer n = 25Last, but not least, I showed him that we write 6

*x*n as 6n (six lots of n), 10*x*n as 10n and 2*x*n as 2n.so 3

*x*16 = 6n, 5*x*20 = 10n and 2*x*50 = 4nAfter that, for some reason he wrote down n

*x*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!).