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

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

This article for teachers suggests ideas for activities built around 10 and 2010.

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?

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.

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?

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

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?

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

How many different ways can you find of fitting five hexagons together? How will you know you have found all the ways?

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

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.

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.

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.

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.

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?

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

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?

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

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?

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

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?

Can you continue this pattern of triangles and begin to predict how many sticks are used for each new "layer"?

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

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

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

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?

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?

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?

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?

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?

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.

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!

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?

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?

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

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

How many models can you find which obey these 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.

It starts quite simple but great opportunities for number discoveries and patterns!

A follow-up activity to Tiles in the Garden.

In how many ways can you stack these rods, following the rules?

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?

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

How many shapes can you build from three red and two green cubes? Can you use what you've found out to predict the number for four red and two green?

An investigation that gives you the opportunity to make and justify predictions.