Can you lay out the pictures of the drinks in the way described by the clue cards?

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

A group of children are discussing the height of a tall tree. How would you go about finding out its height?

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.

We went to the cinema and decided to buy some bags of popcorn so we asked about the prices. Investigate how much popcorn each bag holds so find out which we might have bought.

Here's a simple way to make a Tangram without any measuring or ruling lines.

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

Can you make the birds from the egg tangram?

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

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

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?

This problem invites you to build 3D shapes using two different triangles. Can you make the shapes from the pictures?

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

Exploring and predicting folding, cutting and punching holes and making spirals.

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?

Use the lines on this figure to show how the square can be divided into 2 halves, 3 thirds, 6 sixths and 9 ninths.

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?

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

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?

How many different cuboids can you make when you use four CDs or DVDs? How about using five, then six?

Paint a stripe on a cardboard roll. Can you predict what will happen when it is rolled across a sheet of paper?

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

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

What happens to the area of a square if you double the length of the sides? Try the same thing with rectangles, diamonds and other shapes. How do the four smaller ones fit into the larger one?

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?

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?

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?

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

Looking at the picture of this Jomista Mat, can you decribe what you see? Why not try and make one yourself?

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

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?

For this task, you'll need an A4 sheet and two A5 transparent sheets. Decide on a way of arranging the A5 sheets on top of the A4 sheet and explore ...

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?

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?

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

Make new patterns from simple turning instructions. You can have a go using pencil and paper or with a floor robot.

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

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

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

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?

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

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

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?

What are the next three numbers in this sequence? Can you explain why are they called pyramid numbers?