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

My cube has inky marks on each face. Can you find the route it has taken? What does each face look like?

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

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.

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?

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?

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 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?

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

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?

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.

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?

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?

Use the interactivity to find out how many quarter turns the man must rotate through to look like each of the pictures.

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

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

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

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

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

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.

Complete the squares - but be warned some are trickier than they look!

What does the overlap of these two shapes look like? Try picturing it in your head and then use the interactivity to test your prediction.

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?

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

This practical activity challenges you to create symmetrical designs by cutting a square into strips.

Can you sort these triangles into three different families and explain how you did it?

The computer has made a rectangle and will tell you the number of spots it uses in total. Can you find out where the rectangle is?

Eight children each had a cube made from modelling clay. They cut them into four pieces which were all exactly the same shape and size. Whose pieces are the same? Can you decide who made each set?

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?

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

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

How many balls of modelling clay and how many straws does it take to make these skeleton shapes?

A hundred square has been printed on both sides of a piece of paper. What is on the back of 100? 58? 23? 19?

Here are shadows of some 3D shapes. What shapes could have made them?

Create a pattern on the left-hand grid. How could you extend your pattern on the right-hand grid?

Sara and Will were sorting some pictures of shapes on cards. "I'll collect the circles," said Sara. "I'll take the red ones," answered Will. Can you see any cards they would both want?

Can you each work out what shape you have part of on your card? What will the rest of it look like?

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?