Why do this problem?
As a number activity
I have found this to be one of the very best for both engaging pupils in thoughtful work and for getting them to push forward their own understanding of number.
Working at the front on some board to display the pupils' ideas I have started by vaguely grouping their suggestions into the separate four rules of number: addition, subtraction, multiplication and division. After a while I asked the youngsters if there are any more, but just wrote 'etc' to indicate that there were some more if they thought so.
Then I pointed out that they had used two numbers for each suggestion; so, could they use three or four numbers and, by adding, get to 8? After more examples I asked if they could start with a biggish number and then take some away and then have to take some more away in order to end up with only 8?
If and when slip-ups occur (suppose they have suggested 15 - 6 - 6), I would ask what has to happen to the answer so far, so that the answer can get to 8. I then talked with the children about the fact they they can use any mathematics that they understand as long as the answer is 8.
(Not assuming that you follow their recording - which may have to be a bit unorthodox - it's good to pose questions that help you to know how the child was thinking.)
Tell me what you are doing here.
How have you got these?
Could you find any more like that?
1) You might find that, for example, a pupil continues with loads and loads of subtractions raising the starting number by just one each time. My own feeling is that I'll allow that to happen for the first two or three lessons in which I use this starting point. If they carry on in the next lesson I would encourage them to venture further. Usually there is no need, they have already changed
things. Maybe the pupil just had to work at something they felt very confident with, or maybe they just liked the patterns that came from the work.
2) Sometimes when children have written something very confidently you can 'dangle a carrot' in front of them and ask them if they know anything about halves or quarters, and if so they could use them also. Very often pupils have done so when they have received no formal teaching of that subject yet.
For the exceptionally mathematically able
The pupil in this category will presumably have many more arithmetic and geometric skills and knowledge of more sophisticated processes. Then the pupil can be expected to obtain the number $8$ using their knowledge and experience.
If a pupil has eight objects then they can access this activity by just putting the eight into a number of groups and say "this, plus this, plus this, makes 8". In this way, the eight objects can be put together and set out into different groups.