A description of some experiments in which you can make discoveries about 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.

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

Investigate these hexagons drawn from different sized equilateral triangles.

These pictures were made by starting with a square, finding the half-way point on each side and joining those points up. You could investigate your own starting shape.

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

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?

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

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?

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

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?

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.

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?

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

Investigate the different shaped bracelets you could make from 18 different spherical beads. How do they compare if you use 24 beads?

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

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

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

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

This challenge involves calculating the number of candles needed on birthday cakes. It is an opportunity to explore numbers and discover new things.

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

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

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?

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"?

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

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.

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

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!

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?

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?

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

How many models can you find which obey these rules?

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

What shapes should Elly cut out to make a witch's hat? How can she make a taller hat?

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. . . .

A follow-up activity to Tiles in the Garden.

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

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

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

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