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

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

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?

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

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

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

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

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.

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

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?

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?

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

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

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

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?

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?

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?

How can you arrange the 5 cubes so that you need the smallest number of Brush Loads of paint to cover them? Try with other numbers of cubes as well.

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

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

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

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

Investigate the area of 'slices' cut off this cube of cheese. What would happen if you had different-sized block of cheese to start with?

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

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?

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

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

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

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?

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

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

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?

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.

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?

How many models can you find which obey these rules?

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

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

How many faces can you see when you arrange these three cubes in different ways?

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

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?

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?

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?

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?

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