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?

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

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?

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

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

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

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 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.

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

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

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?

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

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?

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?

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?

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

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

Investigate these hexagons drawn from different sized equilateral triangles.

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

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

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.

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

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

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

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

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

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

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

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

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

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

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?

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.

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?

A follow-up activity to Tiles in the Garden.

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?

A thoughtful shepherd used bales of straw to protect the area around his lambs. Explore how you can arrange the bales.

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

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

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?