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

How many models can you find which obey these rules?

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

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?

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?

How can you arrange the 5 cubes so that you need the smallest number of Brush Loads of paint to cover them? Try with other numbers of cubes as well.

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?

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

Suppose there is a train with 24 carriages which are going to be put together to make up some new trains. Can you find all the ways that this can be done?

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

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

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

This challenge is to design different step arrangements, which must go along a distance of 6 on the steps and must end up at 6 high.

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

What happens if you join every second point on this circle? How about every third point? Try with different steps and see if you can predict what will happen.

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?

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"?

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?

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

In a Magic Square all the rows, columns and diagonals add to the 'Magic Constant'. How would you change the magic constant of this square?

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

This problem is based on the story of the Pied Piper of Hamelin. Investigate the different numbers of people and rats there could have been if you know how many legs there are altogether!

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

Suppose we allow ourselves to use three numbers less than 10 and multiply them together. How many different products can you find? How do you know you've got them all?

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

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.

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?

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?

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

How many different shaped boxes can you design for 36 sweets in one layer? Can you arrange the sweets so that no sweets of the same colour are next to each other in any direction?

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?

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

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?

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

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.

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

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

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

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

A follow-up activity to Tiles in the Garden.

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

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

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