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

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

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?

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.

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

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.

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.

Week 2

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

Under what circumstances can you rearrange a big square to make three smaller squares?

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

Kyle and his teacher disagree about his test score - who is right?

Some of the numbers have fallen off Becky's number line. Can you figure out what they were?

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

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

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?

A red square and a blue square overlap so that the corner of the red square rests on the centre of the blue square. Show that, whatever the orientation of the red square, it covers a quarter of the. . . .

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

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

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

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

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?

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?

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

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 devise a system for making sense of complex multiplication?

What happens when we add together two complex numbers?

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?

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

Use these four dominoes to make a square that has the same number of dots on each side.

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.

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

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.

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

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?

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

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

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?

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?

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

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

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