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?

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

What is the greatest number of squares you can make by overlapping three squares?

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

Use the interactivity to investigate what kinds of triangles can be drawn on peg boards with different numbers of pegs.

Investigate how the four L-shapes fit together to make an enlarged L-shape. You could explore this idea with other shapes too.

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.

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.

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?

What shape is the overlap when you slide one of these shapes half way across another? Can you picture it in your head? Use the interactivity to check your visualisation.

Seeing Squares game for an adult and child. Can you come up with a way of always winning this game?

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

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?

Can you find a reliable strategy for choosing coordinates that will locate the treasure in the minimum number of guesses?

How many different triangles can you make on a circular pegboard that has nine pegs?

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 find all the different triangles on these peg boards, and find their angles?

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.

Investigate the smallest number of moves it takes to turn these mats upside-down if you can only turn exactly three at a time.

An environment that enables you to investigate tessellations of regular polygons

You have 4 red and 5 blue counters. How many ways can they be placed on a 3 by 3 grid so that all the rows columns and diagonals have an even number of red counters?

Use the interactivity to find all the different right-angled triangles you can make by just moving one corner of the starting triangle.

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?

Cut four triangles from a square as shown in the picture. How many different shapes can you make by fitting the four triangles back together?

A game in which players take it in turns to choose a number. Can you block your opponent?

A game for 1 or 2 people. Use the interactive version, or play with friends. Try to round up as many counters as possible.

A game for two people that can be played with pencils and paper. Combine your knowledge of coordinates with some strategic thinking.

A game for 1 person. Can you work out how the dice must be rolled from the start position to the finish? Play on line.

Imagine a wheel with different markings painted on it at regular intervals. Can you predict the colour of the 18th mark? The 100th mark?

What shaped overlaps can you make with two circles which are the same size? What shapes are 'left over'? What shapes can you make when the circles are different sizes?

Watch this film carefully. Can you find a general rule for explaining when the dot will be this same distance from the horizontal axis?

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?

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

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

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?

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

Three beads are threaded on a circular wire and are coloured either red or blue. Can you find all four different combinations?

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

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

An interactive game to be played on your own or with friends. Imagine you are having a party. Each person takes it in turns to stand behind the chair where they will get the most chocolate.

Two engines, at opposite ends of a single track railway line, set off towards one another just as a fly, sitting on the front of one of the engines, sets off flying along the railway line...

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.

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

Try to stop your opponent from being able to split the piles of counters into unequal numbers. Can you find a strategy?

Use Excel to explore multiplication of fractions.

Can you beat the computer in the challenging strategy game?

Each light in this interactivity turns on according to a rule. What happens when you enter different numbers? Can you find the smallest number that lights up all four lights?

Train game for an adult and child. Who will be the first to make the train?