Galileo, a famous inventor who lived about 400 years ago, came up with an idea similar to this for making a time measuring instrument. Can you turn your pendulum into an accurate minute timer?

Can you put these shapes in order of size? Start with the smallest.

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?

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

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?

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?

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

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

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

Can you order pictures of the development of a frog from frogspawn and of a bean seed growing into a plant?

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

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

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

Investigate the smallest number of moves it takes to turn these mats upside-down if you can only turn exactly three at a time.

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.

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

Are all the possible combinations of two shapes included in this set of 27 cards? How do you know?

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?

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

Make a cube out of straws and have a go at this practical challenge.

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

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

An activity making various patterns with 2 x 1 rectangular tiles.

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

Learn about Pen Up and Pen Down in Logo

If you'd like to know more about Primary Maths Masterclasses, this is the package to read! Find out about current groups in your region or how to set up your own.

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?

Can you fit the tangram pieces into the outline of this junk?

Can you fit the tangram pieces into the outline of this telephone?

Can you fit the tangram pieces into the outline of Little Ming playing the board game?

Can you fit the tangram pieces into the outline of this brazier for roasting chestnuts?

Can you fit the tangram pieces into the outline of Little Fung at the table?

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

Have you ever noticed the patterns in car wheel trims? These questions will make you look at car wheels in a different way!

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?

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?

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.

Let's say you can only use two different lengths - 2 units and 4 units. Using just these 2 lengths as the edges how many different cuboids can you make?

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

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?

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?

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

The class were playing a maths game using interlocking cubes. Can you help them record what happened?

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?

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