On a clock the three hands - the second, minute and hour hands - are on the same axis. How often in a 24 hour day will the second hand be parallel to either of the two other hands?

Join pentagons together edge to edge. Will they form a ring?

At the time of writing the hour and minute hands of my clock are at right angles. How long will it be before they are at right angles again?

What is the relationship between the angle at the centre and the angles at the circumference, for angles which stand on the same arc? Can you prove it?

Can you make a right-angled triangle on this peg-board by joining up three points round the edge?

Can you find triangles on a 9-point circle? Can you work out their angles?

Semi-regular tessellations combine two or more different regular polygons to fill the plane. Can you find all the semi-regular tessellations?

Consider a watch face which has identical hands and identical marks for the hours. It is opposite to a mirror. When is the time as read direct and in the mirror exactly the same between 6 and 7?

Draw some angles inside a rectangle. What do you notice? Can you prove it?

Construct two equilateral triangles on a straight line. There are two lengths that look the same - can you prove it?

Join the midpoints of a quadrilateral to get a new quadrilateral. What is special about it?

Pythagoras of Samos was a Greek philosopher who lived from about 580 BC to about 500 BC. Find out about the important developments he made in mathematics, astronomy, and the theory of music.

It is impossible to trisect an angle using only ruler and compasses but it can be done using a carpenter's square.

Shogi tiles can form interesting shapes and patterns... I wonder whether they fit together to make a ring?

ABCD is a square. P is the midpoint of AB and is joined to C. A line from D perpendicular to PC meets the line at the point Q. Prove AQ = AD.

The points P, Q, R and S are the midpoints of the edges of a non-convex quadrilateral.What do you notice about the quadrilateral PQRS and its area?

My train left London between 6 a.m. and 7 a.m. and arrived in Paris between 9 a.m. and 10 a.m. At the start and end of the journey the hands on my watch were in exactly the same positions but the. . . .

An equilateral triangle is constructed on BC. A line QD is drawn, where Q is the midpoint of AC. Prove that AB // QD.

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?

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

Have you ever noticed how mathematical ideas are often used in patterns that we see all around us? This article describes the life of Escher who was a passionate believer that maths and art can be. . . .

Can you use LOGO to create a systematic reproduction of a basic design? An introduction to variables in a familiar setting.

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

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

Can you use LOGO to create this star pattern made from squares. Only basic LOGO knowledge needed.

Explore patterns based on a rhombus. How can you enlarge the pattern - or explode it?

Where will the point stop after it has turned through 30 000 degrees? I took out my calculator and typed 30 000 ÷ 360. How did this help?

Jennifer Piggott and Charlie Gilderdale describe a free interactive circular geoboard environment that can lead learners to pose mathematical questions.

What angle is needed for a ball to do a circuit of the billiard table and then pass through its original position?

Make five different quadrilaterals on a nine-point pegboard, without using the centre peg. Work out the angles in each quadrilateral you make. Now, what other relationships you can see?

Suggestions for worthwhile mathematical activity on the subject of angle measurement for all pupils.