Analyse these repeating patterns. Decide on the conditions for a periodic pattern to occur and when the pattern extends to infinity.

A spiropath is a sequence of connected line segments end to end taking different directions. The same spiropath is iterated. When does it cycle and when does it go on indefinitely?

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

What is the sum of the angles of a triangle whose sides are circular arcs on a flat surface? What if the triangle is on the surface of a sphere?

Measure the two angles. What do you notice?

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

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

Use simple trigonometry to calculate the distance along the flight path from London to Sydney.

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?

A metal puzzle which led to some mathematical questions.

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?

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

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. . . .

P is a point inside a square ABCD such that PA= 1, PB = 2 and PC = 3. How big is angle APB ?

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