Skip to main content
### Number and algebra

### Geometry and measure

### Probability and statistics

### Working mathematically

### For younger learners

### Advanced mathematics

# Twice as Big?

## Twice as Big?

**Why do this problem?**

### Possible approach

### Key questions

### Possible extension

### Possible support

## You may also like

### Cutting Corners

### Cut and Make

### Square to L

Links to the University of Cambridge website
Links to the NRICH website Home page

Nurturing young mathematicians: teacher webinars

30 April (Primary), 1 May (Secondary)

30 April (Primary), 1 May (Secondary)

Or search by topic

Age 7 to 11

Challenge Level

- Problem
- Getting Started
- Student Solutions
- Teachers' Resources

If we double each side of a small square we get a new enlarged square:

The new enlarged square is the size of four of the smaller squares.

This also happens when we enlarge other shapes. Some, like the squares, can be filled with the same smaller shape.

Look at these:

Can you work out how the four shapes fit to make the enlarged shape each time?

You need to rotate or reflect the smaller shapes to fit them in. (This means that if you make them from squared paper you will need to turn them round or turn them over.)

Please send us pictures of your completed shapes.

In this interactivity the rotation and the reflection of the shapes has been done for you.

If you enjoyed working on this problem, you might like to investigate some more shapes. Have a look at Two Squared or print out this sheet which contains some other examples as well as the shapes above.

This problem encourages children to use visualisation and will help to improve their spatial awareness.

Although, as it stands, the problem focuses on fitting the shapes into their enlarged version, it makes a good stepping stone to discussing what "bigger" means. Ask the class to investigate the difference in perimeter and area of the small and large version of the shapes. What do they notice?

You may like to talk to the group about some good ways to approach the problem - ask for their ideas and model some behaviours. You could place two shapes into the larger one leaving a small space which could not be filled to draw their attention to something to avoid!

If you are not using the interactivity, you may like to print off this sheet and cut out the shapes for the children. (The sheet also contains some shapes based on triangles as well as squares.) You may find that if they are working from "concrete" examples, the class will need reminding that they can flip
and rotate the shapes.

Can you think of a good way to start on this?

Why don't you try putting one shape in at a time?

Are you being careful not to leave any gaps which you won't be able to fill?

Have you remembered you can flip and rotate the shapes?

Learners could do the examples on the second page of this sheet or try Two Squared.

Suggest using the interactivity workingon one shape at a time from the top.

Can you make the most extraordinary, the most amazing, the most unusual patterns/designs from these triangles which are made in a special way?

Cut a square of paper into three pieces as shown. Now,can you use the 3 pieces to make a large triangle, a parallelogram and the square again?

Find a way to cut a 4 by 4 square into only two pieces, then rejoin the two pieces to make an L shape 6 units high.