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'Tiles on a Patio' printed from http://nrich.maths.org/
Why do this problem?
Working on this investigation
provides many opportunities to speak with youngsters to discover how they are going about generating new ideas. The activity encourages growth in problem solving, spatial awareness and number awareness.
If you are doing this with a whole class, then it might be best to do one or two examples altogether and perhaps have some pretend tiles ready for use. When asking about the number of tiles that would be used if, instead of 1 by 1, they were 2 by 2 then very often, as you can imagine, children say that you would have to use 50 of them. Very gradually, pennies begin to drop, realisation of
"area increase" develops and they get to grips with the fact that you only need 25.
It is useful to put up a huge chart of squares from 1 to 100 for them to sign up when they have used that particular number of tiles in their shopping list for a patio. This, as a class activity, starts an interesting bit of challenge in itself as they try to get the ones that others have not. Arguments occur as some think that certain ones are impossible whilst others have solved them. When
" impossible" ones are noted it is interesting to get children to explain why they are impossible.
I have used this activity on a number of occasions on parents' evening when the school has tried to help parents understand what it is to do a mathematics investigation. The need for discussion and plenty of aids and time becomes apparent.
Tell me about this arrangement you've made.
Does this arrangement have MORE, LESS or THE SAME number of tiles than the others you've made?
By children looking at why you cannot use 2, 3, 98 or 99 tiles they may be encouraged to look at other sizes of patio and predict what numbers of tiles will not be possible. Other possibilities for extension are suggested at the end of the activity itself.
For the exceptionally mathematically able
These pupils could work towards comparing results with different sized patios and making statements about those arrangements that use the same number of tiles but different tiles. They could then look at My New Patio and before working on it, predict what the similarities and differences will be in the solutions of
I've found that not much support is needed but obviously using some materials, such as squared paper or different coloured sheets of paper cut into squares,can help.