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?

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

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

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

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

An environment that enables you to investigate tessellations of regular polygons

Draw some quadrilaterals on a 9-point circle and work out the angles. Is there a theorem?

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

Show how this pentagonal tile can be used to tile the plane and describe the transformations which map this pentagon to its images in the tiling.

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

Never used GeoGebra before? This article for complete beginners will help you to get started with this free dynamic geometry software.

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

A dot starts at the point (1,0) and turns anticlockwise. Can you estimate the height of the dot after it has turned through 45 degrees? Can you calculate its height?

A circle rolls around the outside edge of a square so that its circumference always touches the edge of the square. Can you describe the locus of the centre of the circle?

Move the point P to see how P' moves. Then use your insights to calculate a missing length.

Can you find an efficent way to mix paints in any ratio?

Can you make sense of these three proofs of Pythagoras' Theorem?

L triominoes can fit together to make larger versions of themselves. Is every size possible to make in this way?

Complex numbers can be represented graphically using an Argand diagram. This problem explains more...

Can you work out how to produce different shades of pink paint?

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

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?

Take any triangle, and construct squares on each of its sides. What do you notice about the areas of the new triangles formed?

Can you put the 25 coloured tiles into the 5 x 5 square so that no column, no row and no diagonal line have tiles of the same colour in them?

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.

A right-angled isosceles triangle is rotated about the centre point of a square. What can you say about the area of the part of the square covered by the triangle as it rotates?

How can visual patterns be used to prove sums of series?

If you continue the pattern, can you predict what each of the following areas will be? Try to explain your prediction.

What happens when we multiply a complex number by a real or an imaginary number?

What happens when we add together two complex numbers?

Let's go further and see what happens when we multiply two complex numbers together!

Can you devise a system for making sense of complex multiplication?

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?

It would be nice to have a strategy for disentangling any tangled ropes...

Is there a relationship between the coordinates of the endpoints of a line and the number of grid squares it crosses?

Move the corner of the rectangle. Can you work out what the purple number represents?

Can you decide whether two lines are perpendicular or not? Can you do this without drawing them?

Experiment with the interactivity of "rolling" regular polygons, and explore how the different positions of the red dot affects its speed at each stage.

Week 2

How well can you estimate angles? Playing this game could improve your skills.

A tilted square is a square with no horizontal sides. Can you devise a general instruction for the construction of a square when you are given just one of its sides?

In how many ways can you fit all three pieces together to make shapes with line symmetry?

Investigate what happens to the equations of different lines when you reflect them in one of the axes. Try to predict what will happen. Explain your findings.

Investigate what happens to the equation of different lines when you translate them. Try to predict what will happen. Explain your findings.

P is a point on the circumference of a circle radius r which rolls, without slipping, inside a circle of radius 2r. What is the locus of P?

Use the applet to make some squares. What patterns do you notice in the coordinates?