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?

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

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

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

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

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

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?

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

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

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?

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

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

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?

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?

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?

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

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

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. . . .

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?

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?

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

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

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?

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

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

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

Ask a friend to choose a number between 1 and 63. By identifying which of the six cards contains the number they are thinking of it is easy to tell them what the number is.

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

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?

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?

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?

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

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.

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

There are nine teddies in Teddy Town - three red, three blue and three yellow. There are also nine houses, three of each colour. Can you put them on the map of Teddy Town according to the rules?

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

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

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

Mr McGregor has a magic potting shed. Overnight, the number of plants in it doubles. He'd like to put the same number of plants in each of three gardens, planting one garden each day. Can he do it?

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

Week 2

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

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?

Arrange the four number cards on the grid, according to the rules, to make a diagonal, vertical or horizontal line.

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

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?

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.