What shape would fit your pens and pencils best? How can you make it?

Build a scaffold out of drinking-straws to support a cup of water

This article for students gives some instructions about how to make some different braids.

This article for pupils gives an introduction to Celtic knotwork patterns and a feel for how you can draw them.

Can Jo make a gym bag for her trainers from the piece of fabric she has?

How does the time of dawn and dusk vary? What about the Moon, how does that change from night to night? Is the Sun always the same? Gather data to help you explore these questions.

Design and construct a prototype intercooler which will satisfy agreed quality control constraints.

As part of Liverpool08 European Capital of Culture there were a huge number of events and displays. One of the art installations was called "Turning the Place Over". Can you find our how it works?

What shape and size of drinks mat is best for flipping and catching?

What shapes should Elly cut out to make a witch's hat? How can she make a taller hat?

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?

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?

Take a counter and surround it by a ring of other counters that MUST touch two others. How many are needed?

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

You could use just coloured pencils and paper to create this design, but it will be more eye-catching if you can get hold of hammer, nails and string.

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

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.

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?

Make some celtic knot patterns using tiling techniques

Make your own double-sided magic square. But can you complete both sides once you've made the pieces?

Here is a version of the game 'Happy Families' for you to make and play.

It might seem impossible but it is possible. How can you cut a playing card to make a hole big enough to walk through?

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

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

This package contains hands-on code breaking activities based on the Enigma Schools Project. Suitable for Stages 2, 3 and 4.

What is the greatest number of counters you can place on the grid below without four of them lying at the corners of a square?

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?

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?

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?

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

How many models can you find which obey these rules?

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

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

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

If these balls are put on a line with each ball touching the one in front and the one behind, which arrangement makes the shortest line of balls?

Can you each work out the number on your card? What do you notice? How could you sort the cards?

These models have appeared around the Centre for Mathematical Sciences. Perhaps you would like to try to make some similar models of your own.

How can you put five cereal packets together to make different shapes if you must put them face-to-face?

More Logo for beginners. Now learn more about the REPEAT command.

Logo helps us to understand gradients of lines and why Muggles Magic is not magic but mathematics. See the problem Muggles magic.

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?

Kate has eight multilink cubes. She has two red ones, two yellow, two green and two blue. She wants to fit them together to make a cube so that each colour shows on each face just once.

Exploring balance and centres of mass can be great fun. The resulting structures can seem impossible. Here are some images to encourage you to experiment with non-breakable objects of your own.

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.

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?