I cut this square into two different shapes. What can you say about the relationship between them?

Investigate how this pattern of squares continues. You could measure lengths, areas and angles.

Try continuing these patterns made from triangles. Can you create your own repeating pattern?

What is the smallest number of tiles needed to tile this patio? Can you investigate patios of different sizes?

A thoughtful shepherd used bales of straw to protect the area around his lambs. Explore how you can arrange the bales.

What do these two triangles have in common? How are they related?

Investigate all the different squares you can make on this 5 by 5 grid by making your starting side go from the bottom left hand point. Can you find out the areas of all these squares?

What happens to the area of a square if you double the length of the sides? Try the same thing with rectangles, diamonds and other shapes. How do the four smaller ones fit into the larger one?

How many ways can you find of tiling the square patio, using square tiles of different sizes?

A group of children are discussing the height of a tall tree. How would you go about finding out its height?

Explore the triangles that can be made with seven sticks of the same length.

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.

A follow-up activity to Tiles in the Garden.

These pictures show squares split into halves. Can you find other ways?

If I use 12 green tiles to represent my lawn, how many different ways could I arrange them? How many border tiles would I need each time?

Using different numbers of sticks, how many different triangles are you able to make? Can you make any rules about the numbers of sticks that make the most triangles?

Arrange eight of the numbers between 1 and 9 in the Polo Square below so that each side adds to the same total.

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

In this challenge, you will work in a group to investigate circular fences enclosing trees that are planted in square or triangular arrangements.

Use the interactivity to find all the different right-angled triangles you can make by just moving one corner of the starting triangle.

Is there a best way to stack cans? What do different supermarkets do? How high can you safely stack the cans?

We need to wrap up this cube-shaped present, remembering that we can have no overlaps. What shapes can you find to use?

This practical investigation invites you to make tessellating shapes in a similar way to the artist Escher.

If we had 16 light bars which digital numbers could we make? How will you know you've found them all?

How could you put eight beanbags in the hoops so that there are four in the blue hoop, five in the red and six in the yellow? Can you find all the ways of doing this?

The red ring is inside the blue ring in this picture. Can you rearrange the rings in different ways? Perhaps you can overlap them or put one outside another?

Can you find out how the 6-triangle shape is transformed in these tessellations? Will the tessellations go on for ever? Why or why not?

Cut differently-sized square corners from a square piece of paper to make boxes without lids. Do they all have the same volume?

Arrange your fences to make the largest rectangular space you can. Try with four fences, then five, then six etc.

Can you make these equilateral triangles fit together to cover the paper without any gaps between them? Can you tessellate isosceles triangles?

Suppose we allow ourselves to use three numbers less than 10 and multiply them together. How many different products can you find? How do you know you've got them all?

Explore ways of colouring this set of triangles. Can you make symmetrical patterns?

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?

When newspaper pages get separated at home we have to try to sort them out and get things in the correct order. How many ways can we arrange these pages so that the numbering may be different?

This practical problem challenges you to create shapes and patterns with two different types of triangle. You could even try overlapping them.

This tricky challenge asks you to find ways of going across rectangles, going through exactly ten squares.

The challenge here is to find as many routes as you can for a fence to go so that this town is divided up into two halves, each with 8 blocks.

Suppose there is a train with 24 carriages which are going to be put together to make up some new trains. Can you find all the ways that this can be done?

This problem is intended to get children to look really hard at something they will see many times in the next few months.

You cannot choose a selection of ice cream flavours that includes totally what someone has already chosen. Have a go and find all the different ways in which seven children can have ice cream.

Ana and Ross looked in a trunk in the attic. They found old cloaks and gowns, hats and masks. How many possible costumes could they make?

While we were sorting some papers we found 3 strange sheets which seemed to come from small books but there were page numbers at the foot of each page. Did the pages come from the same book?

In this investigation we are going to count the number of 1s, 2s, 3s etc in numbers. Can you predict what will happen?

If you have three circular objects, you could arrange them so that they are separate, touching, overlapping or inside each other. Can you investigate all the different possibilities?

There are to be 6 homes built on a new development site. They could be semi-detached, detached or terraced houses. How many different combinations of these can you find?

Investigate these hexagons drawn from different sized equilateral triangles.

Polygonal numbers are those that are arranged in shapes as they enlarge. Explore the polygonal numbers drawn here.