Why do this problem?
can be a rich source of exploration both in calculation and shape and space.
It's good to make it as practical as possibe for many pupils and working together through a few simple examples before getting them to explore on their own is helpful.
Which tiles have you used in this one?
How many tiles altogether have you used?
Tell me about the way you are deciding which tiles to use.
Is there a pattern in the numbers for the smallest number of tiles?
Can you explain any patterns you've found?
- go much further with bigger starting squares
- allow square tiles to be cut into half
- have the space left in a different position and discuss whether this makes any difference to the tiles used
- investigate whether there is anything in the number of each size of square tiles that are used in each successive sized area [i.e. area 4 x 4 uses ? of 1 x 1's; ? of 2 x 2's; ? of 3 x 3's]. How does that compare with the tiles that are used in the 5 x 5 area?
- investigate whether there are different tiles that can be used to produce the same smallest number of tiles for any given area
- try to find a system for filling up the area
- make predictions based on their findings
Some pupils will benefit by going up to areas that exceed 20 x 20. It is certainly worthwhile looking at the implications of where the bush is in relation to the patio. I found it interesting when exploring the odd numbered sided patios with the bush in the centre position: factors, and even/odd numbers, and fractions of the lengths of sides, and addition and subtraction. What else could you
For more extension work
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 of different sizes. There is also the idea of looking at more squares that are not tiled because of Garden trees, ornaments etc. even a stream.
As much practical equipment as possible will help many children.