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?
Investigate how this pattern of squares continues. You could
measure lengths, areas and angles.
Can you work out the area of the inner square and give an
explanation of how you did it?
The first part of this problem was solved by Polly, West Flegg Middle School in Great Yarmouth; Joel, ACS in Singapore; and Luke, Dan, Nicholas and Luke, all from Clevedon Community School.
They all relized that the area of the triangles on either side of the squares are half that of the square (the base angle of triangle is 45°). Therefore, each square is half of the trapezium it is contained within, and the sum of all the squares will be half of the sum of all the trapezia that make up the triangle. The area covered by the squares is half the area covered by
the triangle for the right-angled isosceles triangle.
You will need to do some calculating to work out what happens then.