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

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. . . .

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

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.

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?

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?

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.

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. . . .

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?

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?

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. . . .

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. . . .

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

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?

I found these clocks in the Arts Centre at the University of Warwick intriguing - do they really need four clocks and what times would be ambiguous with only two or three of them?

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?

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

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

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.

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

Starting with four different triangles, imagine you have an unlimited number of each type. How many different tetrahedra can you make? Convince us you have found them all.

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

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?

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?

You have 27 small cubes, 3 each of nine colours. Use the small cubes to make a 3 by 3 by 3 cube so that each face of the bigger cube contains one of every colour.

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

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

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?

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?

ABCD is a regular tetrahedron and the points P, Q, R and S are the midpoints of the edges AB, BD, CD and CA. Prove that PQRS is a square.

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?

Can you make sense of the charts and diagrams that are created and used by sports competitors, trainers and statisticians?

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.

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

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,. . . .

What happens when you turn these cogs? Investigate the differences between turning two cogs of different sizes and two cogs which are the same.

This article looks at levels of geometric thinking and the types of activities required to develop this thinking.

Can you fit the tangram pieces into the outline of this sports car?

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

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

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.

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.

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

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

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

Billy's class had a robot called Fred who could draw with chalk held underneath him. What shapes did the pupils make Fred draw?

Can you fit the tangram pieces into the outline of this goat and giraffe?

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