Fit these shapes

What is the largest number of circles we can fit into the frame without them overlapping? How do you know? What will happen if you try the other shapes?

Age

5 to 11

Challenge level

Exploring and noticing Working systematically Conjecturing and generalising Visualising and representing Reasoning, convincing and proving

Being curious Being resourceful Being resilient Being collaborative

Problem

If we take sixteen of the blue squares we can fill the frame below.

What if we used the green circles? What is the largest number of circles we can fit into the frame without them overlapping?

Then there are the other shapes. (They each just fit into the size of the blue square.)

What is the largest number of each of these you can fit into the frame?

You may like to make some predictions before trying out your ideas. You can print the frame and the shapes here.

Teachers' Resources

Fit These Shapes

If we take sixteen of the blue squares we can fill the frame below.

What if we used the green circles? What is the largest number of circles we can fit into the frame without them overlapping?

Then there are the other shapes. (They each just fit into the size of the blue square.)

What is the largest number of each of these you can fit into the frame?

You may like to make some predictions before trying out your ideas. You can print the frame and the shapes here.

Why do this problem?

This activity challenges the exploration skills of the pupils. They will need to try ideas, and after testing them, will probably need to make adjustments before being satisfied. During these processes there will be opportunities for plenty of mathematical discussion and perhaps for new exploration ideas to develop.

Possible approach

You could introduce the challenge using a large version of the frame and cut-out shapes, or you could make an interactive whiteboard version from the template. Invite children to make predictions about the number of each shape they will be able to fit in the frame, encouraging them to articulate their reasoning.

Print and cut out sets of the shapes and the frame on thick paper/card, enough for each pair or small group of children. Encourage them to test out their ideas, trying to find the largest number of each shape that will fit in the frame. Allow the children to approach the task in whatever way they see fit. Some might draw around one shape, some might want to have multiple copies of each shape, some might do some measuring and calculating ...

You could invite two pairs or two groups to join together to compare findings. Encourage learners to explain to each other how they went about the task and invite them to challenge each other's findings, where appropriate.

Key questions

Tell me about your ideas for getting an answer.

I see you've ... can you tell me about that?

Any ideas for what you could try now?

Possible extension

What about trying combinations of shapes? What would make a good pair to try?

For the highest-attaining

Some children might be keen to explore the area of the spaces left when using each shape.

Possible support

Some learners will need something which is handled more easily so bigger plastic/foam shapes and an appropriately-sized frame would be fantastic.

Or search by topic

Number and algebra

Geometry and measure

Probability and statistics

Working mathematically

Advanced mathematics

For younger learners