Can you show that you can share a square pizza equally between two people by cutting it four times using vertical, horizontal and diagonal cuts through any point inside the square?

What are the areas of these triangles? What do you notice? Can you generalise to other "families" of triangles?

Spotting patterns can be an important first step - explaining why it is appropriate to generalise is the next step, and often the most interesting and important.

Polygons drawn on square dotty paper have dots on their perimeter (p) and often internal (i) ones as well. Find a relationship between p, i and the area of the polygons.

Can you find a general rule for finding the areas of equilateral triangles drawn on an isometric grid?

Can you find sets of sloping lines that enclose a square?

Imagine starting with one yellow cube and covering it all over with a single layer of red cubes, and then covering that cube with a layer of blue cubes. How many red and blue cubes would you need?

Imagine a large cube made from small red cubes being dropped into a pot of yellow paint. How many of the small cubes will have yellow paint on their faces?

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. . . .

First of all, pick the number of times a week that you would like to eat chocolate. Multiply this number by 2...

ABC and DEF are equilateral triangles of side 3 and 4 respectively. Construct an equilateral triangle whose area is the sum of the area of ABC and DEF.

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

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

Take a look at the multiplication square. The first eleven triangle numbers have been identified. Can you see a pattern? Does the pattern continue?

Pick a square within a multiplication square and add the numbers on each diagonal. What do you notice?

Take any two positive numbers. Calculate the arithmetic and geometric means. Repeat the calculations to generate a sequence of arithmetic means and geometric means. Make a note of what happens to the. . . .

Imagine an infinitely large sheet of square dotty paper on which you can draw triangles of any size you wish (providing each vertex is on a dot). What areas is it/is it not possible to draw?

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?

Can you find the area of a parallelogram defined by two vectors?

Can you find an efficient method to work out how many handshakes there would be if hundreds of people met?

How could Penny, Tom and Matthew work out how many chocolates there are in different sized boxes?

Jo made a cube from some smaller cubes, painted some of the faces of the large cube, and then took it apart again. 45 small cubes had no paint on them at all. How many small cubes did Jo use?

Make some loops out of regular hexagons. What rules can you discover?

Some students have been working out the number of strands needed for different sizes of cable. Can you make sense of their solutions?

Explore the effect of reflecting in two intersecting mirror lines.

What is the volume of the solid formed by rotating this right angled triangle about the hypotenuse?

A game for 2 players. Set out 16 counters in rows of 1,3,5 and 7. Players take turns to remove any number of counters from a row. The player left with the last counter looses.

Rectangles are considered different if they vary in size or have different locations. How many different rectangles can be drawn on a chessboard?

If you can copy a network without lifting your pen off the paper and without drawing any line twice, then it is traversable. Decide which of these diagrams are traversable.

Triangle ABC is an equilateral triangle with three parallel lines going through the vertices. Calculate the length of the sides of the triangle if the perpendicular distances between the parallel. . . .

Do you notice anything about the solutions when you add and/or subtract consecutive negative numbers?

Charlie and Alison have been drawing patterns on coordinate grids. Can you picture where the patterns lead?

Sets of integers like 3, 4, 5 are called Pythagorean Triples, because they could be the lengths of the sides of a right-angled triangle. Can you find any more?

Can you find the values at the vertices when you know the values on the edges of these multiplication arithmagons?

Use the animation to help you work out how many lines are needed to draw mystic roses of different sizes.

Can you figure out how sequences of beach huts are generated?

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.

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

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

Choose a couple of the sequences. Try to picture how to make the next, and the next, and the next... Can you describe your reasoning?

An article for teachers and pupils that encourages you to look at the mathematical properties of similar games.

Dave Hewitt suggests that there might be more to mathematics than looking at numerical results, finding patterns and generalising.

How many moves does it take to swap over some red and blue frogs? Do you have a method?

Square numbers can be represented as the sum of consecutive odd numbers. What is the sum of 1 + 3 + ..... + 149 + 151 + 153?

Four bags contain a large number of 1s, 3s, 5s and 7s. Pick any ten numbers from the bags above so that their total is 37.

What is the ratio of the area of a square inscribed in a semicircle to the area of the square inscribed in the entire circle?

Alf Coles writes about how he tries to create 'spaces for exploration' for the students in his classrooms.

This task encourages you to investigate the number of edging pieces and panes in different sized windows.