How many possible necklaces can you find? And how do you know you've found them all?

This task depends on groups working collaboratively, discussing and reasoning to agree a final product.

Are these statements relating to calculation and properties of shapes always true, sometimes true or never true?

Use the clues about the symmetrical properties of these letters to place them on the grid.

Systematically explore the range of symmetric designs that can be created by shading parts of the motif below. Use normal square lattice paper to record your results.

Here are four cubes joined together. How many other arrangements of four cubes can you find? Can you draw them on dotty paper?

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?

Sally and Ben were drawing shapes in chalk on the school playground. Can you work out what shapes each of them drew using the clues?

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?

In how many different ways can you break up a stick of 7 interlocking cubes? Now try with a stick of 8 cubes and a stick of 6 cubes.

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?

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?

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?

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?

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?

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

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

Use the interactivity to help get a feel for this problem and to find out all the possible ways the balls could land.

How many DIFFERENT quadrilaterals can be made by joining the dots on the 8-point circle?

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

Arrange the shapes in a line so that you change either colour or shape in the next piece along. Can you find several ways to start with a blue triangle and end with a red circle?

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

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

This ladybird is taking a walk round a triangle. Can you see how much he has turned when he gets back to where he started?

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

For this task, you'll need an A4 sheet and two A5 transparent sheets. Decide on a way of arranging the A5 sheets on top of the A4 sheet and explore ...

Investigate the number of faces you can see when you arrange three cubes in different ways.

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?

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

Can you dissect an equilateral triangle into 6 smaller ones? What number of smaller equilateral triangles is it NOT possible to dissect a larger equilateral triangle into?

This cube has ink on each face which leaves marks on paper as it is rolled. Can you work out what is on each face and the route it has taken?

Use your mouse to move the red and green parts of this disc. Can you make images which show the turnings described?

We have a box of cubes, triangular prisms, cones, cuboids, cylinders and tetrahedrons. Which of the buildings would fall down if we tried to make them?

This challenge invites you to create your own picture using just straight lines. Can you identify shapes with the same number of sides and decorate them in the same way?

Shapes are added to other shapes. Can you see what is happening? What is the rule?

Can you work out what shape is made when this piece of paper is folded up using the crease pattern shown?

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

These points all mark the vertices (corners) of ten hidden squares. Can you find the 10 hidden squares?

Here are some pictures of 3D shapes made from cubes. Can you make these shapes yourself?

Here are the six faces of a cube - in no particular order. Here are three views of the cube. Can you deduce where the faces are in relation to each other and record them on the net of this cube?

Here are some arrangements of circles. How many circles would I need to make the next size up for each? Can you create your own arrangement and investigate the number of circles it needs?

How many different triangles can you make on a circular pegboard that has nine pegs?

Can you describe a piece of paper clearly enough for your partner to know which piece it is?

Have a go at making a few of these shapes from paper in different sizes. What patterns can you create?

You have been given three shapes made out of sponge: a sphere, a cylinder and a cone. Your challenge is to find out how to cut them to make different shapes for printing.

Use the isometric grid paper to find the different polygons.

What are the coordinates of the coloured dots that mark out the tangram? Try changing the position of the origin. What happens to the coordinates now?