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?

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

Investigate the number of faces you can see when you arrange three cubes in different ways.

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

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

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

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

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

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

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

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

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

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

Take 5 cubes of one colour and 2 of another colour. How many different ways can you join them if the 5 must touch the table and the 2 must not touch the table?

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?

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?

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

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?

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.

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

An investigation that gives you the opportunity to make and justify predictions.

How many models can you find which obey these rules?

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

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

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

A follow-up activity to Tiles in the Garden.

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

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

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

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

Explore the different tunes you can make with these five gourds. What are the similarities and differences between the two tunes you are given?

In this article for teachers, Bernard gives an example of taking an initial activity and getting questions going that lead to other explorations.

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

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

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?

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

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

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

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

While we were sorting some papers we found 3 strange sheets which seemed to come from small books but there were page numbers at the foot of each page. Did the pages come from the same book?

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

Place the 16 different combinations of cup/saucer in this 4 by 4 arrangement so that no row or column contains more than one cup or saucer of the same colour.

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.

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?