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?

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

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

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

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.

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

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?

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?

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

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?

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?

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

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

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

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

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?

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.

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?

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

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

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

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?

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

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

Investigate these hexagons drawn from different sized equilateral triangles.

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

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

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

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

The challenge here is to find as many routes as you can for a fence to go so that this town is divided up into two halves, each with 8 blocks.

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

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?

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

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?

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

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

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

A follow-up activity to Tiles in the Garden.

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?

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?