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?

An environment that enables you to investigate tessellations of regular polygons

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?

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

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?

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

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?

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

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.

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

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?

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

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

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

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

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

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?

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

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.

Explore the area of families of parallelograms and triangles. Can you find rules to work out the areas?

Can you work out the fraction of the original triangle that is covered by the inner triangle?

The farmers want to redraw their field boundary but keep the area the same. Can you advise them?

The length AM can be calculated using trigonometry in two different ways. Create this pair of equivalent calculations for different peg boards, notice a general result, and account for it.

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

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

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

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

Week 2

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

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?

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

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?

Use the applet to explore the area of a parallelogram and how it relates to vectors.

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

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

Two circles of equal radius touch at P. One circle is fixed whilst the other moves, rolling without slipping, all the way round. How many times does the moving coin revolve before returning to P?

The opposite vertices of a square have coordinates (a,b) and (c,d). What are the coordinates of the other vertices?

The sine of an angle is equal to the cosine of its complement. Can you explain why and does this rule extend beyond angles of 90 degrees?

Draw a square. A second square of the same size slides around the first always maintaining contact and keeping the same orientation. How far does the dot travel?

Can you explain what is happening and account for the values being displayed?

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

Position the lines so that they are perpendicular to each other. What can you say about the equations of perpendicular lines?

How does the position of the line affect the equation of the line? What can you say about the equations of parallel lines?

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

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

What happens when we add together two complex numbers?

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