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?
Back in August $1998$ we had a challenge called Pebbles and this investigation could have grown out of it.
We are looking at making squares from several points. So you might like to think of these dots or points:-
Whichever way you care to think about them, we are going to make some squares.
The squares are made by first drawing just one side, always starting that one side from the bottom left-hand square as shown below:
... and draw the rest of the square in by making sure that the sides are at right angles and of the same length - things that you know about squares!
That would give us:
We can of course draw other starting lines, such as:
Your challenge is to make more and more squares by using your starting side (roughly in the lower left-hand part) to other points marked in the $5$ by $5$ arrangement.
The investigation is about ways of finding out the areas of all these squares. You do not need any special knowledge but you may need lots of squared paper and a pair of scissors. You may be wanting to use a piece of cut-out card. I guess you'll need a chance to discuss this with friends.