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

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?

Can you see who the gold medal winner is? What about the silver medal winner and the bronze medal winner?

These rectangles have been torn. How many squares did each one have inside it before it was ripped?

You cannot choose a selection of ice cream flavours that includes totally what someone has already chosen. Have a go and find all the different ways in which seven children can have ice cream.

This problem focuses on Dienes' Logiblocs. What is the same and what is different about these pairs of shapes? Can you describe the shapes in the picture?

This article for teachers suggests activities based on pegboards, from pattern generation to finding all possible triangles, for example.

These practical challenges are all about making a 'tray' and covering it with paper.

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

In how many ways can you fit two of these yellow triangles together? Can you predict the number of ways two blue triangles can be fitted together?

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?

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

Can you help the children find the two triangles which have the lengths of two sides numerically equal to their areas?

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?

Can you order pictures of the development of a frog from frogspawn and of a bean seed growing into a plant?

If you split the square into these two pieces, it is possible to fit the pieces together again to make a new shape. How many new shapes can you make?

My local DIY shop calculates the price of its windows according to the area of glass and the length of frame used. Can you work out how they arrived at these prices?

When newspaper pages get separated at home we have to try to sort them out and get things in the correct order. How many ways can we arrange these pages so that the numbering may be different?

How many different triangles can you draw on the dotty grid which each have one dot in the middle?

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

Find all the different shapes that can be made by joining five equilateral triangles edge to edge.

How many triangles can you make using sticks that are 3cm, 4cm and 5cm long?

The challenge here is to find as many routes as you can for a fence to go so that this town is divided up into two halves, each with 8 blocks.

In this town, houses are built with one room for each person. There are some families of seven people living in the town. In how many different ways can they build their houses?

If we had 16 light bars which digital numbers could we make? How will you know you've found them all?

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

If you have three circular objects, you could arrange them so that they are separate, touching, overlapping or inside each other. Can you investigate all the different possibilities?

Ana and Ross looked in a trunk in the attic. They found old cloaks and gowns, hats and masks. How many possible costumes could they make?

How many different ways can you find to join three equilateral triangles together? Can you convince us that you have found them all?

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

Can you draw a square in which the perimeter is numerically equal to the area?

How many ways can you find of tiling the square patio, using square tiles of different sizes?

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?

This practical challenge invites you to investigate the different squares you can make on a square geoboard or pegboard.

If you had 36 cubes, what different cuboids could you make?

These activities lend themselves to systematic working in the sense that it helps to have an ordered approach.

Can you find all the different triangles on these peg boards, and find their angles?

What is the greatest number of counters you can place on the grid below without four of them lying at the corners of a square?

My briefcase has a three-number combination lock, but I have forgotten the combination. I remember that there's a 3, a 5 and an 8. How many possible combinations are there to try?

Start with three pairs of socks. Now mix them up so that no mismatched pair is the same as another mismatched pair. Is there more than one way to do it?

El Crico the cricket has to cross a square patio to get home. He can jump the length of one tile, two tiles and three tiles. Can you find a path that would get El Crico home in three jumps?

Lolla bought a balloon at the circus. She gave the clown six coins to pay for it. What could Lolla have paid for the balloon?

Kate has eight multilink cubes. She has two red ones, two yellow, two green and two blue. She wants to fit them together to make a cube so that each colour shows on each face just once.

Take a rectangle of paper and fold it in half, and half again, to make four smaller rectangles. How many different ways can you fold it up?

This task, written for the National Young Mathematicians' Award 2016, involves open-topped boxes made with interlocking cubes. Explore the number of units of paint that are needed to cover the boxes. . . .

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

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

Arrange 9 red cubes, 9 blue cubes and 9 yellow cubes into a large 3 by 3 cube. No row or column of cubes must contain two cubes of the same colour.

How many rectangles can you find in this shape? Which ones are differently sized and which are 'similar'?