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. . . .

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

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A game to make and play based on the number line.

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

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

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

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Learn about Pen Up and Pen Down in Logo

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Turn through bigger angles and draw stars with Logo.

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Use the tangram pieces to make our pictures, or to design some of your own!

Draw whirling squares and see how Fibonacci sequences and golden rectangles are connected.

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

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

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

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

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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?

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

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

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

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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.

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

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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.

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

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What happens when a procedure calls itself?

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A jigsaw where pieces only go together if the fractions are equivalent.

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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?

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Make some celtic knot patterns using tiling techniques

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You can use a clinometer to measure the height of tall things that you can't possibly reach to the top of, Make a clinometer and use it to help you estimate the heights of tall objects.

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This is the second in a twelve part introduction to Logo for beginners. In this part you learn to draw polygons.

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Delight your friends with this cunning trick! Can you explain how it works?

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I start with a red, a blue, a green and a yellow marble. I can trade any of my marbles for three others, one of each colour. Can I end up with exactly two marbles of each colour?

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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?

A description of how to make the five Platonic solids out of paper.

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Generate three random numbers to determine the side lengths of a triangle. What triangles can you draw?

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Interior angles can help us to work out which polygons will tessellate. Can we use similar ideas to predict which polygons combine to create semi-regular solids?

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Move your counters through this snake of cards and see how far you can go. Are you surprised by where you end up?

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Here is a chance to create some Celtic knots and explore the mathematics behind them.

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The Tower of Hanoi is an ancient mathematical challenge. Working on the building blocks may help you to explain the patterns you notice.

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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?

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Can you use small coloured cubes to make a 3 by 3 by 3 cube so that each face of the bigger cube contains one of each colour?

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A game in which players take it in turns to choose a number. Can you block your opponent?

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

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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.

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Time for a little mathemagic! Choose any five cards from a pack and show four of them to your partner. How can they work out the fifth?

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

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

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How can you make an angle of 60 degrees by folding a sheet of paper twice?

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