We had a look at this one in year 2 today. We started off by doing the 'number trick'.
The children then practised this in their books to make sure it happened every time.
After we'd established that this rule was always correct e.g. you always end up with the number you started with, I asked the children to discuss in small groups why they thought this was the case. I listened to different children and this is what they said:
Mutahara:'I did it 1,2,3,4 in order, and the answers went 10,11,12,13...in order'
Yusuf said:
'9 is one less than 10, and 1 is one less than 2' (We were looking at 2+9 as an example)
'I notice that the 1st ones number is bigger and the 2nd ones number. It's 1 bigger, and 9 is one smaller than ten.
'You're just adding 10s to the number and when you partition you just ignore the tens.' I asked him what he meant by 'ignore' and he wasn't able to explain'.
Francis said:
'If you add ten it will be ten more so one less must be the same... adding 10 is just adding a 1 on the front (to the tens column) and if you then -1 it will = one less. Were adding 9, which is one less than ten.'
We then agreed as a class that adding 9 was the same as +10 and -1, and tested this hypothesis using different numbers. Once we'd all agreed this was true, we resumed our small group discussions:
Ryu said: 'Becuase there's a one in the tens, if you break down the 1-s clumn, and pretend there's not such thing as a tens column, you're really just adding one.'
We re-convened and collated what we knew. I asked the children to think about why we always end up in the tens column. In his explanation, Amaan said '9 is next to ten', and used the word 'inverse'. I asked him to elaborate. 'Plus is the opposite of minus,' he said.
Milo responded to this excitedly with the equation '1+1-1=1'. We explored this concept with different numbers, e.g. 6+1-1=6, and decided that since we were taking a number, +10 then -1, then recombining the digits, that was essentially the same as +1-1, therefore we always ended up with the number we started with.