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

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

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

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

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

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

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

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

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

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?

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?

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?

Investigate the area of 'slices' cut off this cube of cheese. What would happen if you had different-sized block of cheese to start with?

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?

Here are many ideas for you to investigate - all linked with the number 2000.

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

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

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.

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

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

I like to walk along the cracks of the paving stones, but not the outside edge of the path itself. How many different routes can you find for me to take?

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?

Let's say you can only use two different lengths - 2 units and 4 units. Using just these 2 lengths as the edges how many different cuboids can you make?

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?

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

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

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

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?

Investigate the different ways you could split up these rooms so that you have double the number.

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

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

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

Take a look at these data collected by children in 1986 as part of the Domesday Project. What do they tell you? What do you think about the way they are presented?

This challenging activity involves finding different ways to distribute fifteen items among four sets, when the sets must include three, four, five and six items.

This challenge extends the Plants investigation so now four or more children are involved.

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

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

Many natural systems appear to be in equilibrium until suddenly a critical point is reached, setting up a mudslide or an avalanche or an earthquake. In this project, students will use a simple. . . .

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 challenging activity focusing on finding all possible ways of stacking rods.

In how many ways can you stack these rods, following the rules?

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

How many models can you find which obey these rules?

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?

Follow the directions for circling numbers in the matrix. Add all the circled numbers together. Note your answer. Try again with a different starting number. What do you notice?

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

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

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

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