Prove Pythagoras' Theorem using enlargements and scale factors.

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?

Show that for any triangle it is always possible to construct 3 touching circles with centres at the vertices. Is it possible to construct touching circles centred at the vertices of any polygon?

The computer has made a rectangle and will tell you the number of spots it uses in total. Can you find out where the rectangle is?

It's easy to work out the areas of most squares that we meet, but what if they were tilted?

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?

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.

An environment that enables you to investigate tessellations of regular polygons

Do you know how to find the area of a triangle? You can count the squares. What happens if we turn the triangle on end? Press the button and see. Try counting the number of units in the triangle now. . . .

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?

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?

Some treasure has been hidden in a three-dimensional grid! Can you work out a strategy to find it as efficiently as possible?

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?

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

An environment that simulates a protractor carrying a right- angled triangle of unit hypotenuse.

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

How good are you at finding the formula for a number pattern ?

Players take it in turns to choose a dot on the grid. The winner is the first to have four dots that can be joined to form a square.

When number pyramids have a sequence on the bottom layer, some interesting patterns emerge...

A java applet that takes you through the steps needed to solve a Diophantine equation of the form Px+Qy=1 using Euclid's algorithm.

Triangle ABC has equilateral triangles drawn on its edges. Points P, Q and R are the centres of the equilateral triangles. What can you prove about the triangle PQR?

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

Can you find a way to turn a rectangle into a square?

A game for two people, or play online. Given a target number, say 23, and a range of numbers to choose from, say 1-4, players take it in turns to add to the running total to hit their target.

To avoid losing think of another very well known game where the patterns of play are similar.

A counter is placed in the bottom right hand corner of a grid. You toss a coin and move the star according to the following rules: ... What is the probability that you end up in the top left-hand. . . .

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?

Charlie likes tablecloths that use as many colours as possible, but insists that his tablecloths have some symmetry. Can you work out how many colours he needs for different tablecloth designs?

This problem is about investigating whether it is possible to start at one vertex of a platonic solid and visit every other vertex once only returning to the vertex you started at.

Use Excel to explore multiplication of fractions.

This is an interactivity in which you have to sort the steps in the completion of the square into the correct order to prove the formula for the solutions of quadratic equations.

Which dilutions can you make using only 10ml pipettes?

Can you explain the strategy for winning this game with any target?

This game challenges you to locate hidden triangles in The White Box by firing rays and observing where the rays exit the Box.

Here is a chance to play a fractions version of the classic Countdown Game.

Here is a chance to play a version of the classic Countdown Game.

Can you fill in the mixed up numbers in this dilution calculation?

Meg and Mo need to hang their marbles so that they balance. Use the interactivity to experiment and find out what they need to do.

The aim of the game is to slide the green square from the top right hand corner to the bottom left hand corner in the least number of moves.

Can you beat the computer in the challenging strategy game?

It is possible to identify a particular card out of a pack of 15 with the use of some mathematical reasoning. What is this reasoning and can it be applied to other numbers of cards?

Try entering different sets of numbers in the number pyramids. How does the total at the top change?

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

Place a red counter in the top left corner of a 4x4 array, which is covered by 14 other smaller counters, leaving a gap in the bottom right hand corner (HOME). What is the smallest number of moves. . . .

Use an Excel to investigate division. Explore the relationships between the process elements using an interactive spreadsheet.

Euler discussed whether or not it was possible to stroll around Koenigsberg crossing each of its seven bridges exactly once. Experiment with different numbers of islands and bridges.