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

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

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

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.

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

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

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?

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?

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

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?

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.

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

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?

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

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

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.

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?

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

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.

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

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

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?

What is the largest cuboid you can wrap in an A3 sheet of paper?

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

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

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

This activity asks you to collect information about the birds you see in the garden. Are there patterns in the data or do the birds seem to visit randomly?

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

How will you decide which way of flipping over and/or turning the grid will give you the highest total?

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?

Many natural systems appear to be in equilibrium until suddenly a critical point is reached, setting up a mudslide or an avalanche or an earthquake. In this project, students will use a simple. . . .

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

This challenge asks you to investigate the total number of cards that would be sent if four children send one to all three others. How many would be sent if there were five children? Six?

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?

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?

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?

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

In my local town there are three supermarkets which each has a special deal on some products. If you bought all your shopping in one shop, where would be the cheapest?

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!

A follow-up activity to Tiles in the Garden.

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

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

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?

Why does the tower look a different size in each of these pictures?

Bernard Bagnall looks at what 'problem solving' might really mean in the context of primary classrooms.

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