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

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?

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?

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?

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.

This practical problem challenges you to create shapes and patterns with two different types of triangle. You could even try overlapping them.

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

The ancient Egyptians were said to make right-angled triangles using a rope with twelve equal sections divided by knots. What other triangles could you make if you had a rope like this?

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

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

This practical investigation invites you to make tessellating shapes in a similar way to the artist Escher.

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.

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

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 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?

Can you make the most extraordinary, the most amazing, the most unusual patterns/designs from these triangles which are made in a special way?

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?

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

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

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?

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

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.

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

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

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.

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

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

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

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?

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

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?

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

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?

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?

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.

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?

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

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

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.

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

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

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

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. . . .