# 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.

## Problem

Image

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.

Image

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.

## Getting Started

It might help to have some large sheets of paper for this activity.

## Student Solutions

We had just three contributions sent in, which is sometimes the case with these more visual challenges. However it is good to receive anything you have to say about your work having had a go at the challenge. Imogen-Rose from Sutton Bonington wrote;

You start with a square, then all you do is tilt the square into a diamond and then back to a normal square and so on.

That's interesting - I wonder what a 'normal' square is? Does a square change shape if it is tilted?

A pupil from Pownhall Hall School wrote;

It got small really quickly at first and then it got smaller slowly. The shapes got more regular as it went smaller. The measurements got harder as the shape got smaller.

Finally Chris from Northowram Primary School wrote;

To start with I started with a rectangle and marked halfway on
all four vertices with a dot then used a ruler to join the
dots up to make a tilted parallelogram. I did the same thing with
the parallelogram and then the shape was a rectangle. I did the
same again but with a trapezium, then an isosceles triangle, then a
hexagon, then finally an octagon.

Well Chris, it is good to see that you went further in investigating this challenge. So, thank you for these contributions and do feel encouraged to send in ideas from any challenge even though you may not have 'completed' it.

## Teachers' Resources

### Why do this problem?

This activity enables pupils with a wide range of attainment level to work on the same challenge to improve their concepts of shape and space. It also offers opportunities for further investigation and for pupils to create their own challenges.

### Possible approach

This might need to vary according to your learners' experiences. For those who have good pencil, ruler and measuring skills, some of the examples here (Word document or pdf) could be presented. You could ask children to describe what they see, with a partner first and then open it out to the whole group. Discussion could follow that would allow pupils to decide what their own explorations would be. Some may want to create their own whereas those who struggle with motor control may wish to explore the
given ones.

It could be that you ask children to feed back about their discoveries orally, or you may wish them to create a poster of some kind. Encourage them to explain their observations. This activity could lend itself to being investigated over an extended period of time (a 'simmering activity') and it would be useful to dedicate a space on the wall for learners to contribute their work
during that time.

### Key questions

What have you been exploring?

Tell me what you have found.