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

Investigate what happens to the equations of different lines when you reflect them in one of the axes. Try to predict what will happen. Explain your findings.

Can you explain why it is impossible to construct this triangle?

Explore the effect of reflecting in two intersecting mirror lines.

Explore the effect of reflecting in two parallel mirror lines.

Why not challenge a friend to play this transformation game?

Where can you put the mirror across the square so that you can still "see" the whole square? How many different positions are possible?

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

In how many ways can you fit all three pieces together to make shapes with line symmetry?

These clocks have been reflected in a mirror. What times do they say?

Numbers arranged in a square but some exceptional spatial awareness probably needed.

A challenging activity focusing on finding all possible ways of stacking rods.

In how many ways can you stack these rods, following the rules?

How will you decide which way of flipping over and/or turning the grid will give you the highest total?

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

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

Can you place the blocks so that you see the relection in the picture?

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

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

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?

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

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

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

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. . . .

What is the missing symbol? Can you decode this in a similar way?

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

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

Some local pupils lost a geometric opportunity recently as they surveyed the cars in the car park. Did you know that car tyres, and the wheels that they on, are a rich source of geometry?

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

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

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.

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?

Which times on a digital clock have a line of symmetry? Which look the same upside-down? You might like to try this investigation and find out!