Two circles of equal radius touch at P. One circle is fixed whilst the other moves, rolling without slipping, all the way round. How many times does the moving coin revolve before returning to P?

Find out how we can describe the "symmetries" of this triangle and investigate some combinations of rotating and flipping it.

How many different ways can you find of fitting five hexagons together? How will you know you have found all the ways?

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.

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

What happens to these capital letters when they are rotated through one half turn, or flipped sideways and from top to bottom?

Sort the frieze patterns into seven pairs according to the way in which the motif is repeated.

Can you recreate this Indian screen pattern? Can you make up similar patterns of your own?

Can you picture where this letter "F" will be on the grid if you flip it in these different ways?

How many different symmetrical shapes can you make by shading triangles or squares?

What are the coordinates of this shape after it has been transformed in the ways described? Compare these with the original coordinates. What do you notice about the numbers?

Can you work out what kind of rotation produced this pattern of pegs in our pegboard?

What is the same and what is different about these tiling patterns and how do they contribute to the floor as a whole?

This article describes the scope for practical exploration of tessellations both in and out of the classroom. It seems a golden opportunity to link art with maths, allowing the creative side of your. . . .

A triangle ABC resting on a horizontal line is "rolled" along the line. Describe the paths of each of the vertices and the relationships between them and the original triangle.

Does changing the order of transformations always/sometimes/never produce the same transformation?

Why not challenge a friend to play this transformation game?

The diagram shows a very heavy kitchen cabinet. It cannot be lifted but it can be pivoted around a corner. The task is to move it, without sliding, in a series of turns about the corners so that it. . . .

See the effects of some combined transformations on a shape. Can you describe what the individual transformations do?

How many different transformations can you find made up from combinations of R, S and their inverses? Can you be sure that you have found them all?

Explore the effect of reflecting in two intersecting mirror lines.

What is the relationship between these first two shapes? Which shape relates to the third one in the same way? Can you explain why?

Overlaying pentominoes can produce some effective patterns. Why not use LOGO to try out some of the ideas suggested here?

This problem explores the shapes and symmetries in some national flags.

Look carefully at the video of a tangle and explain what's happening.

This article for teachers suggests ideas for activities built around 10 and 2010.

A brief video looking at how you can sometimes use symmetry to distinguish knots. Can you use this idea to investigate the differences between the granny knot and the reef knot?

What mathematical words can be used to describe this floor covering? How many different shapes can you see inside this photograph?

This article describes a practical approach to enhance the teaching and learning of coordinates.

A train leaves on time. After it has gone 8 miles (at 33mph) the driver looks at his watch and sees that the hour hand is exactly over the minute hand. When did the train leave the station?

A red square and a blue square overlap so that the corner of the red square rests on the centre of the blue square. Show that, whatever the orientation of the red square, it covers a quarter of the. . . .

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.

Patterns that repeat in a line are strangely interesting. How many types are there and how do you tell one type from another?

Here is a chance to create some attractive images by rotating shapes through multiples of 90 degrees, or 30 degrees, or 72 degrees or...

A gallery of beautiful photos of cast ironwork friezes in Australia with a mathematical discussion of the classification of frieze patterns.

Proofs that there are only seven frieze patterns involve complicated group theory. The symmetries of a cylinder provide an easier approach.

A security camera, taking pictures each half a second, films a cyclist going by. In the film, the cyclist appears to go forward while the wheels appear to go backwards. Why?

This investigation explores using different shapes as the hands of the clock. What things occur as the the hands move.