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

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

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

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?

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

Learn about Pen Up and Pen Down in Logo

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

Turn through bigger angles and draw stars with Logo.

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

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

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.

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 practical problem challenges you to make quadrilaterals with a loop of string. You'll need some friends to help!

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

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

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

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

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

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

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

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

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

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.

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

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

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?

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.

Can you make the birds from the egg tangram?

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

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?

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?