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

Can you make the most extraordinary, the most amazing, the most unusual patterns/designs from these triangles which are made in a special way?

Explore the triangles that can be made with seven sticks of the same length.

Using different numbers of sticks, how many different triangles are you able to make? Can you make any rules about the numbers of sticks that make the most triangles?

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

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

Use the three triangles to fill these outline shapes. Perhaps you can create some of your own shapes for a friend to fill?

In how many ways can you fit two of these yellow triangles together? Can you predict the number of ways two blue triangles can be fitted together?

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

Ideas for practical ways of representing data such as Venn and Carroll diagrams.

Can you work out what shape is made by folding in this way? Why not create some patterns using this shape but in different sizes?

What do these two triangles have in common? How are they related?

Can you see which tile is the odd one out in this design? Using the basic tile, can you make a repeating pattern to decorate our wall?

We can cut a small triangle off the corner of a square and then fit the two pieces together. Can you work out how these shapes are made from the two pieces?

Kaia is sure that her father has worn a particular tie twice a week in at least five of the last ten weeks, but her father disagrees. Who do you think is right?

NRICH December 2006 advent calendar - a new tangram for each day in the run-up to Christmas.

Follow the diagrams to make this patchwork piece, based on an octagon in a square.

Surprise your friends with this magic square trick.

Have you noticed that triangles are used in manmade structures? Perhaps there is a good reason for this? 'Test a Triangle' and see how rigid triangles are.

Make a mobius band and investigate its properties.

Did you know mazes tell stories? Find out more about mazes and make one of your own.

Have a go at drawing these stars which use six points drawn around a circle. Perhaps you can create your own designs?

Watch this "Notes on a Triangle" film. Can you recreate parts of the film using cut-out triangles?

This problem focuses on Dienes' Logiblocs. What is the same and what is different about these pairs of shapes? Can you describe the shapes in the picture?

Take 5 cubes of one colour and 2 of another colour. How many different ways can you join them if the 5 must touch the table and the 2 must not touch the table?

Make a flower design using the same shape made out of different sizes of paper.

Have a go at making a few of these shapes from paper in different sizes. What patterns can you create?

Arrange your fences to make the largest rectangular space you can. Try with four fences, then five, then six etc.

Watch the video to see how to fold a square of paper to create a flower. What fraction of the piece of paper is the small triangle?

What is the largest number of circles we can fit into the frame without them overlapping? How do you know? What will happen if you try the other shapes?

For this activity which explores capacity, you will need to collect some bottles and jars.

In this activity focusing on capacity, you will need a collection of different jars and bottles.

Can you describe a piece of paper clearly enough for your partner to know which piece it is?

These practical challenges are all about making a 'tray' and covering it with paper.

You will need a long strip of paper for this task. Cut it into different lengths. How could you find out how long each piece is?

This is a simple paper-folding activity that gives an intriguing result which you can then investigate further.

How can you make a curve from straight strips of paper?

These pictures show squares split into halves. Can you find other ways?

Make a chair and table out of interlocking cubes, making sure that the chair fits under the table!

Can you split each of the shapes below in half so that the two parts are exactly the same?

Follow these instructions to make a three-piece and/or seven-piece tangram.

You have a set of the digits from 0 – 9. Can you arrange these in the five boxes to make two-digit numbers as close to the targets as possible?

The ancient Egyptians were said to make right-angled triangles using a rope with twelve equal sections divided by knots. What other triangles could you make if you had a rope like this?

This challenge invites you to create your own picture using just straight lines. Can you identify shapes with the same number of sides and decorate them in the same way?

In this challenge, you will work in a group to investigate circular fences enclosing trees that are planted in square or triangular arrangements.

Take a rectangle of paper and fold it in half, and half again, to make four smaller rectangles. How many different ways can you fold it up?

Our 2008 Advent Calendar has a 'Making Maths' activity for every day in the run-up to Christmas.