Here is a chance to create some attractive images by rotating shapes through multiples of 90 degrees, or 30 degrees, or 72 degrees or...

Have a go at creating these images based on circles. What do you notice about the areas of the different sections?

Using a ruler, pencil and compasses only, it is possible to construct a square inside any triangle so that all four vertices touch the sides of the triangle.

You are only given the three midpoints of the sides of a triangle. How can you construct the original triangle?

Design and construct a prototype intercooler which will satisfy agreed quality control constraints.

What shape and size of drinks mat is best for flipping and catching?

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

Investigate constructible images which contain rational areas.

How can you represent the curvature of a cylinder on a flat piece of paper?

Drawing a triangle is not always as easy as you might think!

What fractions can you divide the diagonal of a square into by simple folding?

Jenny Murray describes the mathematical processes behind making patchwork in this article for students.

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

Construct a line parallel to one side of a triangle so that the triangle is divided into two equal areas.

Can you reproduce the Yin Yang symbol using a pair of compasses?

Explain how to construct a regular pentagon accurately using a straight edge and compass.

Draw a line (considered endless in both directions), put a point somewhere on each side of the line. Label these points A and B. Use a geometric construction to locate a point, P, on the line,. . . .

Draw a square and an arc of a circle and construct the Golden rectangle. Find the value of the Golden Ratio.

Describe how to construct three circles which have areas in the ratio 1:2:3.

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

The challenge is to produce elegant solutions. Elegance here implies simplicity. The focus is on rhombi, in particular those formed by jointing two equilateral triangles along an edge.

The diagonal of a square intersects the line joining one of the unused corners to the midpoint of the opposite side. What do you notice about the line segments produced?

Medieval stonemasons used a method to construct octagons using ruler and compasses... Is the octagon regular? Proof please.

Prove that, given any three parallel lines, an equilateral triangle always exists with one vertex on each of the three lines.

Without using a calculator, computer or tables find the exact values of cos36cos72 and also cos36 - cos72.

Two perpendicular lines are tangential to two identical circles that touch. What is the largest circle that can be placed in between the two lines and the two circles and how would you construct it?