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?

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

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

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?

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?

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

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

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

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

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

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

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?

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

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

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

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.

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

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?

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

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

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

There are to be 6 homes built on a new development site. They could be semi-detached, detached or terraced houses. How many different combinations of these can you find?

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

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

How can you arrange these 10 matches in four piles so that when you move one match from three of the piles into the fourth, you end up with the same arrangement?

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?

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

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?

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

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

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

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?

Roll two red dice and a green dice. Add the two numbers on the red dice and take away the number on the green. What are all the different possible answers?

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

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?

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.

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?

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

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

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

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

Investigate these hexagons drawn from different sized equilateral 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?

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?