See if you can anticipate successive 'generations' of the two animals shown here.

Three circles have a maximum of six intersections with each other. What is the maximum number of intersections that a hundred circles could have?

It is possible to dissect any square into smaller squares. What is the minimum number of squares a 13 by 13 square can be dissected into?

In a right angled triangular field, three animals are tethered to posts at the midpoint of each side. Each rope is just long enough to allow the animal to reach two adjacent vertices. Only one animal. . . .

How can you make an angle of 60 degrees by folding a sheet of paper twice?

Can you recreate these designs? What are the basic units? What movement is required between each unit? Some elegant use of procedures will help - variables not essential.

These are pictures of the sea defences at New Brighton. Can you work out what a basic shape might be in both images of the sea wall and work out a way they might fit together?

Seven small rectangular pictures have one inch wide frames. The frames are removed and the pictures are fitted together like a jigsaw to make a rectangle of length 12 inches. Find the dimensions of. . . .

Use a single sheet of A4 paper and make a cylinder having the greatest possible volume. The cylinder must be closed off by a circle at each end.

A circle rolls around the outside edge of a square so that its circumference always touches the edge of the square. Can you describe the locus of the centre of the circle?

You can move the 4 pieces of the jigsaw and fit them into both outlines. Explain what has happened to the missing one unit of area.

Points P, Q, R and S each divide the sides AB, BC, CD and DA respectively in the ratio of 2 : 1. Join the points. What is the area of the parallelogram PQRS in relation to the original rectangle?

Is it possible to remove ten unit cubes from a 3 by 3 by 3 cube made from 27 unit cubes so that the surface area of the remaining solid is the same as the surface area of the original 3 by 3 by 3. . . .

Use the interactivity to listen to the bells ringing a pattern. Now it's your turn! Play one of the bells yourself. How do you know when it is your turn to ring?

Looking at the picture of this Jomista Mat, can you decribe what you see? Why not try and make one yourself?

Can you use small coloured cubes to make a 3 by 3 by 3 cube so that each face of the bigger cube contains one of each colour?

What happens to the perimeter of triangle ABC as the two smaller circles change size and roll around inside the bigger circle?

What shape is the overlap when you slide one of these shapes half way across another? Can you picture it in your head? Use the interactivity to check your visualisation.

Use the interactivity to play two of the bells in a pattern. How do you know when it is your turn to ring, and how do you know which bell to ring?

Choose a box and work out the smallest rectangle of paper needed to wrap it so that it is completely covered.

Imagine you are suspending a cube from one vertex (corner) and allowing it to hang freely. Now imagine you are lowering it into water until it is exactly half submerged. What shape does the surface. . . .

ABCDEFGH is a 3 by 3 by 3 cube. Point P is 1/3 along AB (that is AP : PB = 1 : 2), point Q is 1/3 along GH and point R is 1/3 along ED. What is the area of the triangle PQR?

Here is a solitaire type environment for you to experiment with. Which targets can you reach?

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?

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?

What is the shape of wrapping paper that you would need to completely wrap this model?

What shape has Harry drawn on this clock face? Can you find its area? What is the largest number of square tiles that could cover this area?

What is the total area of the four outside triangles which are outlined in red in this arrangement of squares inside each other?

A cheap and simple toy with lots of mathematics. Can you interpret the images that are produced? Can you predict the pattern that will be produced using different wheels?

This article introduces the idea of generic proof for younger children and illustrates how one example can offer a proof of a general result through unpacking its underlying structure.

The triangle ABC is equilateral. The arc AB has centre C, the arc BC has centre A and the arc CA has centre B. Explain how and why this shape can roll along between two parallel tracks.

Can you fit the tangram pieces into the outlines of the workmen?

This article for teachers discusses examples of problems in which there is no obvious method but in which children can be encouraged to think deeply about the context and extend their ability to. . . .

If you move the tiles around, can you make squares with different coloured edges?

Can you fit the tangram pieces into the outlines of the candle and sundial?

Here's a simple way to make a Tangram without any measuring or ruling lines.

A game for 2 players. Given a board of dots in a grid pattern, players take turns drawing a line by connecting 2 adjacent dots. Your goal is to complete more squares than your opponent.

Players take it in turns to choose a dot on the grid. The winner is the first to have four dots that can be joined to form a square.

Charlie and Alison have been drawing patterns on coordinate grids. Can you picture where the patterns lead?

Lyndon Baker describes how the Mobius strip and Euler's law can introduce pupils to the idea of topology.

A bus route has a total duration of 40 minutes. Every 10 minutes, two buses set out, one from each end. How many buses will one bus meet on its way from one end to the other end?

Can you describe this route to infinity? Where will the arrows take you next?

Choose a couple of the sequences. Try to picture how to make the next, and the next, and the next... Can you describe your reasoning?

Can you fit the tangram pieces into the outlines of Mai Ling and Chi Wing?

Slide the pieces to move Khun Phaen past all the guards into the position on the right from which he can escape to freedom.

The aim of the game is to slide the green square from the top right hand corner to the bottom left hand corner in the least number of moves.

This article for teachers describes how modelling number properties involving multiplication using an array of objects not only allows children to represent their thinking with concrete materials,. . . .

Can you fit the tangram pieces into the outline of Little Ming and Little Fung dancing?

A shape and space game for 2,3 or 4 players. Be the last person to be able to place a pentomino piece on the playing board. Play with card, or on the computer.