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?

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

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?

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

Can you explain why it is impossible to construct this triangle?

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

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

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

What is the total area of the first two triangles as a fraction of the original A4 rectangle? What is the total area of the first three triangles as a fraction of the original A4 rectangle? If. . . .

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

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

Investigate constructible images which contain rational areas.

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?

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.

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

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.

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,. . . .

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

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