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?

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?

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?

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

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

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?

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

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

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

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

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?

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

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

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

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.

Week 2

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

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

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

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

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?

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?

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

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

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

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

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

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

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

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

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 sense of these three proofs of Pythagoras' Theorem?

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?

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

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?

Can you arrange the digits 1, 1, 2, 2, 3 and 3 to make a Number Sandwich?

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

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 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?

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 the distance it travels at each stage.

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

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?

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