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?

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

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

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

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

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

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?

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

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?

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

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

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.

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?

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

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

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

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

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?

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.

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

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?

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

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?

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

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?

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

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

Can you continue this pattern of triangles and begin to predict how many sticks are used for each new "layer"?

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?

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

When newspaper pages get separated at home we have to try to sort them out and get things in the correct order. How many ways can we arrange these pages so that the numbering may be different?

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

You cannot choose a selection of ice cream flavours that includes totally what someone has already chosen. Have a go and find all the different ways in which seven children can have ice cream.

Ana and Ross looked in a trunk in the attic. They found old cloaks and gowns, hats and masks. How many possible costumes could they make?

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.

If I use 12 green tiles to represent my lawn, how many different ways could I arrange them? How many border tiles would I need each time?

In this investigation we are going to count the number of 1s, 2s, 3s etc in numbers. Can you predict what will happen?

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?

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