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

There are nine teddies in Teddy Town - three red, three blue and three yellow. There are also nine houses, three of each colour. Can you put them on the map of Teddy Town according to the rules?

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

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

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?

The letters of the word ABACUS have been arranged in the shape of a triangle. How many different ways can you find to read the word ABACUS from this triangular pattern?

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

Arrange eight of the numbers between 1 and 9 in the Polo Square below so that each side adds to the same total.

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

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

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!

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.

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?

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

How many models can you find which obey these rules?

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

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

It starts quite simple but great opportunities for number discoveries and patterns!

Let's suppose that you are going to have a magazine which has 16 pages of A5 size. Can you find some different ways to make these pages? Investigate the pattern for each if you number the pages.

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

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?

An investigation involving adding and subtracting sets of consecutive numbers. Lots to find out, lots to explore.

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.

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

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

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?

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?

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.

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

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

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?

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?

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?

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?

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

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

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

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

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

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?

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. . . .

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?

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

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