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

Investigate the different shaped bracelets you could make from 18 different spherical beads. How do they compare if you use 24 beads?

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

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?

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?

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

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.

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

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

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

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

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

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

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?

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

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?

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

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

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

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?

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

The ancient Egyptians were said to make right-angled triangles using a rope with twelve equal sections divided by knots. What other triangles could you make if you had a rope like this?

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

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?

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

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

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?

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

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

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?

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?

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

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?

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

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

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?

A follow-up activity to Tiles in the Garden.

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

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?

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?

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

How many models can you find which obey these rules?