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?

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

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.

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

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

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?

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?

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.

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?

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. . . .

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

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.

Choose the size of your pegboard and the shapes you can make. Can you work out the strategies needed to block your opponent?

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 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?

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

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?

Use Excel to explore multiplication of fractions.

This article gives you a few ideas for understanding the Got It! game and how you might find a winning strategy.

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 relationship between the number of dots on the circle and the number of steps that will ensure that all points are hit?

A and B are two interlocking cogwheels having p teeth and q teeth respectively. One tooth on B is painted red. Find the values of p and q for which the red tooth on B contacts every gap on the. . . .

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

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?

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

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.

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

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

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?

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.

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.

A game for 2 players that can be played online. Players take it in turns to select a word from the 9 words given. The aim is to select all the occurrences of the same letter.

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.

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

An Excel spreadsheet with an investigation.

Use Excel to practise adding and subtracting fractions.

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

Choose 13 spots on the grid. Can you work out the scoring system? What is the maximum possible score?

Use an interactive Excel spreadsheet to investigate factors and multiples.

Use an Excel spreadsheet to explore long multiplication.

A simple file for the Interactive whiteboard or PC screen, demonstrating equivalent fractions.

Use an interactive Excel spreadsheet to explore number in this exciting game!

Use Excel to investigate the effect of translations around a number grid.

An interactive activity for one to experiment with a tricky tessellation