Shaping it

These pictures were made by starting with a square, finding the half-way point on each side and joining those points up. You could investigate your own starting shape.
Age
5 to 11
Challenge level
filled star empty star empty star
Exploring and noticing Working systematically Conjecturing and generalising Visualising and representing Reasoning, convincing and proving
Being curious Being resourceful Being resilient Being collaborative

Problem



Shaping It
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 properly. 

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.

 

 
Shaping It


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 pursue:
 
  • 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.