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?

Sort the houses in my street into different groups. Can you do it in any other ways?

Use the interactivity to investigate what kinds of triangles can be drawn on peg boards with different numbers of pegs.

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

Use your mouse to move the red and green parts of this disc. Can you make images which show the turnings described?

Place the 16 different combinations of cup/saucer in this 4 by 4 arrangement so that no row or column contains more than one cup or saucer of the same colour.

Investigate the number of paths you can take from one vertex to another in these 3D shapes. Is it possible to take an odd number and an even number of paths to the same vertex?

Explore the different tunes you can make with these five gourds. What are the similarities and differences between the two tunes you are given?

There are nine teddies in Teddy Town - three red, three blue and three yellow. There are also nine houses, three of each colour. Can you put them on the map of Teddy Town according to the rules?

This challenging activity involves finding different ways to distribute fifteen items among four sets, when the sets must include three, four, five and six items.

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

This challenge extends the Plants investigation so now four or more children are involved.

How many models can you find which obey these rules?

This challenge involves eight three-cube models made from interlocking cubes. Investigate different ways of putting the models together then compare your constructions.

A challenging activity focusing on finding all possible ways of stacking rods.

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

Investigate the number of faces you can see when you arrange three cubes in different ways.

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

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?

Make new patterns from simple turning instructions. You can have a go using pencil and paper or with a floor robot.

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

We went to the cinema and decided to buy some bags of popcorn so we asked about the prices. Investigate how much popcorn each bag holds so find out which we might have bought.

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

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

How many different cuboids can you make when you use four CDs or DVDs? How about using five, then six?

This challenge is to design different step arrangements, which must go along a distance of 6 on the steps and must end up at 6 high.

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?

Let's suppose that you are going to have a magazine which has 16 pages of A5 size. Can you find some different ways to make these pages? Investigate the pattern for each if you number the pages.

Write the numbers up to 64 in an interesting way so that the shape they make at the end is interesting, different, more exciting ... than just a square.

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

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

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?

Three children are going to buy some plants for their birthdays. They will plant them within circular paths. How could they do this?

We think this 3x3 version of the game is often harder than the 5x5 version. Do you agree? If so, why do you think that might be?

In this investigation, you must try to make houses using cubes. If the base must not spill over 4 squares and you have 7 cubes which stand for 7 rooms, what different designs can you come up with?

An activity making various patterns with 2 x 1 rectangular tiles.

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?

Vincent and Tara are making triangles with the class construction set. They have a pile of strips of different lengths. How many different triangles can they make?

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

What is the smallest cuboid that you can put in this box so that you cannot fit another that's the same into it?

This problem is based on the story of the Pied Piper of Hamelin. Investigate the different numbers of people and rats there could have been if you know how many legs there are altogether!

Let's say you can only use two different lengths - 2 units and 4 units. Using just these 2 lengths as the edges how many different cuboids can you make?

Investigate the different ways you could split up these rooms so that you have double the number.

Take 5 cubes of one colour and 2 of another colour. How many different ways can you join them if the 5 must touch the table and the 2 must not touch the table?

I like to walk along the cracks of the paving stones, but not the outside edge of the path itself. How many different routes can you find for me to take?

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

How many different shaped boxes can you design for 36 sweets in one layer? Can you arrange the sweets so that no sweets of the same colour are next to each other in any direction?

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?