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

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

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?

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

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

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

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

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

Can you find ways of joining cubes together so that 28 faces are visible?

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?

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

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

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

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

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.

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?

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?

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.

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

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

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

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

A follow-up activity to Tiles in the Garden.

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

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

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?

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?

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

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

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!

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

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

In a Magic Square all the rows, columns and diagonals add to the 'Magic Constant'. How would you change the magic constant of this square?

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

Place four pebbles on the sand in the form of a square. Keep adding as few pebbles as necessary to double the area. How many extra pebbles are added each time?

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

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?

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

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

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 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.

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.

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