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

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

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

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

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

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. . . .

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

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?

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?

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?

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

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

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?

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

Triangular numbers can be represented by a triangular array of squares. What do you notice about the sum of identical triangle numbers?

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

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?

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

An environment that enables you to investigate tessellations of regular polygons

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

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

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

We can show that (x + 1)² = x² + 2x + 1 by considering the area of an (x + 1) by (x + 1) square. Show in a similar way that (x + 2)² = x² + 4x + 4

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?

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?

Draw some isosceles triangles with an area of $9$cm$^2$ and a vertex at (20,20). If all the vertices must have whole number coordinates, how many is it possible to draw?

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

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.

The number of plants in Mr McGregor's magic potting shed increases overnight. He'd like to put the same number of plants in each of his gardens, planting one garden each day. How can he do it?

There are lots of ideas to explore in these sequences of ordered fractions.

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

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

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.

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.

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

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

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

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

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.

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

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

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

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

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?

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

How much of the square is coloured blue? How will the pattern continue?

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