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'Shaping It' printed from http://nrich.maths.org/
These pictures were made quite simply by starting with a square,
finding the half-way point on each side and joining those points
up. This creates a smaller shape (which also happens to be a
square) inside the original. The half-way points of this new
shape are then joined up to make a third shape. This way of making
new shapes is continued until it gets too small to do
You can, of course, start with any straight-lined shape.
Here's one where I've coloured each new halving line to help to see
what has happened more clearly.
So, it's your turn to have a go.
It's probably good to start with a fairly large shape since it's
going to get smaller and smaller each time.
Here are some challenges for you to
- Having made a design like one above,
cut out the triangles and the smallest inner shape and rearrange
the pieces to form a new shape/design.
- Talk about and record the things you
notice as you have drawn more and more halving lines.
- What is happening to the enclosed
area each time the sides are halved? (Try
investigating a regular shape first.)
This problem is based on an idea
suggested by Ian Short.