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?
Can you make sense of these three proofs of Pythagoras' Theorem?
L triominoes can fit together to make larger versions of themselves. Is every size possible to make in this way?
The opposite vertices of a square have coordinates (a,b) and (c,d). What are the coordinates of the other vertices?
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 tool for generating random integers.
Join pentagons together edge to edge. Will they form a ring?
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.
Semi-regular tessellations combine two or more different regular polygons to fill the plane. Can you find all the semi-regular tessellations?
How good are you at estimating angles?
Draw some angles inside a rectangle. What do you notice? Can you prove it?
Explore the area of families of parallelograms and triangles. Can you find rules to work out the areas?
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 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?
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
Kyle and his teacher disagree about his test score - who is right?
There are lots of ideas to explore in these sequences of ordered fractions.
Can you work out how to produce different shades of pink paint?
Triangular numbers can be represented by a triangular array of squares. What do you notice about the sum of identical triangle numbers?
Can you find an efficent way to mix paints in any ratio?
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?
Experiment with the interactivity of "rolling" regular polygons, and explore how the different positions of the red dot affects its speed at each stage.
The farmers want to redraw their field boundary but keep the area the same. Can you advise them?
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...
Is there a relationship between the coordinates of the endpoints of a line and the number of grid squares it crosses?
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?
Move the point P to see how P' moves. Then use your insights to calculate a missing length.
Medieval stonemasons used a method to construct octagons using ruler and compasses... Is the octagon regular? Proof please.
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?
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?
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 decide whether two lines are perpendicular or not? Can you do this without drawing them?
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
Some of the numbers have fallen off Becky's number line. Can you figure out what they were?
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?
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?
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.
Use these four dominoes to make a square that has the same number of dots on each side.
In how many ways can you fit all three pieces together to make shapes with line symmetry?
Can you devise a system for making sense of complex multiplication?
Under what circumstances can you rearrange a big square to make three smaller squares?
Take any triangle, and construct squares on each of its sides. What do you notice about the areas of the new triangles formed?
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.