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?

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

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

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

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

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

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.

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

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?

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

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

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

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

How many models can you find which obey these rules?

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.

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

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

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

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?

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

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?

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

Compare the numbers of particular tiles in one or all of these three designs, inspired by the floor tiles of a church in Cambridge.

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

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

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

I cut this square into two different shapes. What can you say about the relationship between them?

How many shapes can you build from three red and two green cubes? Can you use what you've found out to predict the number for four red and two green?

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?

This activity asks you to collect information about the birds you see in the garden. Are there patterns in the data or do the birds seem to visit randomly?

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?

Start with four numbers at the corners of a square and put the total of two corners in the middle of that side. Keep going... Can you estimate what the size of the last four numbers will be?

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

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

This challenge encourages you to explore dividing a three-digit number by a single-digit number.

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

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

Investigate what happens when you add house numbers along a street in different ways.

In this investigation, you must try to make houses using cubes. If the base must not spill over 4 squares and you have 7 cubes which stand for 7 rooms, what different designs can you come up with?

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