This practical challenge invites you to investigate the different
squares you can make on a square geoboard or pegboard.
How many ways can you find of tiling the square patio, using square
tiles of different sizes?
Place four pebbles on the sand in the form of a square. Keep adding
as few pebbles as necessary to double the area. How many extra
pebbles are added each time?
I met up with some friends yesterday for lunch. On the table was a good big block of cheese. It looked rather like a cube. As the meal went on we started cutting off slices, but these got smaller and smaller! It got me thinking ...
What if the cheese cube was $5$ by $5$ by $5$ and each slice was always $1$ thick?
It wouldn't be fair on everyone else's lunch if I cut up the real cheese so I made a model out of multilink cubes:
You can see that it's a $5$ by $5$ by $5$ because of the individual cubes, so the slices will have to be $1$ cube thick.
So let's take a slice off the right hand side, I've coloured it in so you can see which bit I'm talking about:
The next slice will be from the left hand side (shown in a different colour again):
So the next cut is from the top. Hard to cut this so I would have put it on its side!
I do three more cuts to get to the $3$ by $3$ by $3$ and these leave the block like this:
If we keep all the slices and the last little cube, we will have pieces that look like (seen from above):
Investigate sharing these thirteen pieces out so that everyone gets an equal share.
I guess that once you've explored the pattern of numbers you'll be able to extend it as if you had started with a $10$ by $10$ by $10$ cube of cheese.