All types of mathematical problems serve a useful purpose in mathematics teaching, but different types of problem will achieve different learning objectives. In generalmore open-ended problems have. . . .

We think this 3x3 version of the game is often harder than the 5x5 version. Do you agree? If so, why do you think that might be?

In this investigation, we look at Pascal's Triangle in a slightly different way - rotated and with the top line of ones taken off.

A follow-up activity to Tiles in the Garden.

Numbers arranged in a square but some exceptional spatial awareness probably needed.

There are three tables in a room with blocks of chocolate on each. Where would be the best place for each child in the class to sit if they came in one at a time?

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

Place four pebbles on the sand in the form of a square. Keep adding as few pebbles as necessary to double the area. How many extra pebbles are added each time?

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

How will you decide which way of flipping over and/or turning the grid will give you the highest total?

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?

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

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

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?

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

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

These caterpillars have 16 parts. What different shapes do they make if each part lies in the small squares of a 4 by 4 square?

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

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?

48 is called an abundant number because it is less than the sum of its factors (without itself). Can you find some more abundant numbers?

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?

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

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?

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

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

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

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?

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!

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?

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

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?

What is the largest cuboid you can wrap in an A3 sheet of paper?

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

In this investigation, you are challenged to make mobile phone numbers which are easy to remember. What happens if you make a sequence adding 2 each time?

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

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

Complete these two jigsaws then put one on top of the other. What happens when you add the 'touching' numbers? What happens when you change the position of the jigsaws?

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?

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

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.

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.