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

Make an equilateral triangle by folding paper and use it to make patterns of your own.

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.

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?

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.

In this article for teachers, Bernard uses some problems to suggest that once a numerical pattern has been spotted from a practical starting point, going back to the practical can help explain. . . .

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

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

Make a clinometer and use it to help you estimate the heights of tall objects.

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

Write a Logo program, putting in variables, and see the effect when you change the variables.

Learn about Pen Up and Pen Down in Logo

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

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

Can you puzzle out what sequences these Logo programs will give? Then write your own Logo programs to generate sequences.

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

What happens when a procedure calls itself?

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

Turn through bigger angles and draw stars with Logo.

Learn how to draw circles using Logo. Wait a minute! Are they really circles? If not what are they?

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

This part introduces the use of Logo for number work. Learn how to use Logo to generate sequences of numbers.

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

How many differently shaped rectangles can you build using these equilateral and isosceles triangles? Can you make a square?

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

Where can you put the mirror across the square so that you can still "see" the whole square? How many different positions are possible?

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

Learn to write procedures and build them into Logo programs. Learn to use variables.

More Logo for beginners. Learn to calculate exterior angles and draw regular polygons using procedures and variables.

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

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?

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

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.

The triangle ABC is equilateral. The arc AB has centre C, the arc BC has centre A and the arc CA has centre B. Explain how and why this shape can roll along between two parallel tracks.

Here is a solitaire type environment for you to experiment with. Which targets can you reach?

I start with a red, a green and a blue marble. I can trade any of my marbles for two others, one of each colour. Can I end up with five more blue marbles than red after a number of such trades?

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

These are pictures of the sea defences at New Brighton. Can you work out what a basic shape might be in both images of the sea wall and work out a way they might fit together?

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

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

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

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

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

Imagine you have an unlimited number of four types of triangle. How many different tetrahedra can you make?

Factors and Multiples game for an adult and child. How can you make sure you win this game?

Can you make the birds from the egg tangram?