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

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

I took the graph y=4x+7 and performed four transformations. Can you find the order in which I could have carried out the transformations?

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?

What angle is needed for a ball to do a circuit of the billiard table and then pass through its original position?

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?

Charlie likes tablecloths that use as many colours as possible, but insists that his tablecloths have some symmetry. Can you work out how many colours he needs for different tablecloth designs?

Can you draw the shape that is being described by these cards?

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.

Explore the meaning behind the algebra and geometry of matrices with these 10 individual problems.

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?

A fire-fighter needs to fill a bucket of water from the river and take it to a fire. What is the best point on the river bank for the fire-fighter to fill the bucket ?.

See how 4 dimensional quaternions involve vectors in 3-space and how the quaternion function F(v) = nvn gives a simple algebraic method of working with reflections in planes in 3-space.

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?

I noticed this about streamers that have rotation symmetry : if there was one centre of rotation there always seems to be a second centre that also worked. Can you find a design that has only. . . .

When a strip has vertical symmetry there always seems to be a second place where a mirror line could go. Perhaps you can find a design that has only one mirror line across it. Or, if you thought that. . . .

A design is repeated endlessly along a line - rather like a stream of paper coming off a roll. Make a strip that matches itself after rotation, or after reflection

A moveable screen slides along a mirrored corridor towards a centrally placed light source. A ray of light from that source is directed towards a wall of the corridor, which it strikes at 45 degrees. . . .

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?

Investigate the transfomations of the plane given by the 2 by 2 matrices with entries taking all combinations of values 0. -1 and +1.

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

Given that ABCD is a square, M is the mid point of AD and CP is perpendicular to MB with P on MB, prove DP = DC.

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

Follow hints to investigate the matrix which gives a reflection of the plane in the line y=tanx. Show that the combination of two reflections in intersecting lines is a rotation.

Follow hints using a little coordinate geometry, plane geometry and trig to see how matrices are used to work on transformations of the plane.

Choose some complex numbers and mark them by points on a graph. Multiply your numbers by i once, twice, three times, four times, ..., n times? What happens?

In a snooker game the brown ball was on the lip of the pocket but it could not be hit directly as the black ball was in the way. How could it be potted by playing the white ball off a cushion?

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?