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?

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

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

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

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

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

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?

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.

Bernard Bagnall describes how to get more out of some favourite NRICH investigations.

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

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

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

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?

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

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.

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

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 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?

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

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?

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

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

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

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

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?

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

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

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

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

A follow-up activity to Tiles in the Garden.

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?

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

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

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

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?

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

Cut differently-sized square corners from a square piece of paper to make boxes without lids. Do they all have the same volume?

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

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?

Here is your chance to investigate the number 28 using shapes, cubes ... in fact anything at all.

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