Systematically explore the range of symmetric designs that can be created by shading parts of the motif below. Use normal square lattice paper to record your results.

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

This activity investigates how you might make squares and pentominoes from Polydron.

This practical problem challenges you to create shapes and patterns with two different types of triangle. You could even try overlapping them.

This practical problem challenges you to make quadrilaterals with a loop of string. You'll need some friends to help!

Can you deduce the pattern that has been used to lay out these bottle tops?

Follow these instructions to make a five-pointed snowflake from a square of paper.

Mathematics is the study of patterns. Studying pattern is an opportunity to observe, hypothesise, experiment, discover and create.

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

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

It's hard to make a snowflake with six perfect lines of symmetry, but it's fun to try!

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

When dice land edge-up, we usually roll again. But what if we didn't...?

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

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

Use the information on these cards to draw the shape that is being described.

Someone at the top of a hill sends a message in semaphore to a friend in the valley. A person in the valley behind also sees the same message. What is it?

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

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?

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

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. . . .

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

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

Are these statements always true, sometimes true or never true?

Here is a chance to create some Celtic knots and explore the mathematics behind them.

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

These images are taken from the Topkapi Palace in Istanbul, Turkey. Can you work out the basic unit that makes up each pattern? Can you continue the pattern? Can you see any similarities and. . . .

Each of these solids is made up with 3 squares and a triangle around each vertex. Each has a total of 18 square faces and 8 faces that are equilateral triangles. How many faces, edges and vertices. . . .

Place the numbers 1, 2, 3,..., 9 one on each square of a 3 by 3 grid so that all the rows and columns add up to a prime number. How many different solutions can you find?

Toni Beardon has chosen this article introducing a rich area for practical exploration and discovery in 3D geometry

Find the missing coordinates which will form these eight quadrilaterals. These coordinates themselves will then form a shape with rotational and line symmetry.

Using the 8 dominoes make a square where each of the columns and rows adds up to 8

Can all but one square of an 8 by 8 Chessboard be covered by Trominoes?

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

Create a symmetrical fabric design based on a flower motif - and realise it in Logo.

Take a look at the photos of tiles at a school in Gibraltar. What questions can you ask about them?

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

Scheduling games is a little more challenging than one might desire. Here are some tournament formats that sport schedulers use.

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

An article for students and teachers on symmetry and square dancing. What do the symmetries of the square have to do with a dos-e-dos or a swing? Find out more?