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?

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. . . .

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

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?

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

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

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.

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?

Can you explain the surprising results Jo found when she calculated the difference between square numbers?

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.

An account of some magic squares and their properties and and how to construct them for yourself.

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

Is there a relationship between the coordinates of the endpoints of a line and the number of grid squares it crosses?

With one cut a piece of card 16 cm by 9 cm can be made into two pieces which can be rearranged to form a square 12 cm by 12 cm. Explain how this can be done.

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. . . .

Start with any number of counters in any number of piles. 2 players take it in turns to remove any number of counters from a single pile. The loser is the player who takes the last counter.

You can work out the number someone else is thinking of as follows. Ask a friend to think of any natural number less than 100. Then ask them to tell you the remainders when this number is divided by. . . .

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

It starts quite simple but great opportunities for number discoveries and patterns!

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

Jo has three numbers which she adds together in pairs. When she does this she has three different totals: 11, 17 and 22 What are the three numbers Jo had to start with?”

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?

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.

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

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.

Caroline and James pick sets of five numbers. Charlie chooses three of them that add together to make a multiple of three. Can they stop him?

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?

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.

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

Imagine we have four bags containing numbers from a sequence. What numbers can we make now?

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

Got It game for an adult and child. How can you play so that you know you will always win?

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

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?

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

Choose four consecutive whole numbers. Multiply the first and last numbers together. Multiply the middle pair together. What do you notice?

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

Nim-7 game for an adult and child. Who will be the one to take the last counter?

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

Can you describe this route to infinity? Where will the arrows take you next?

Can you find the values at the vertices when you know the values on the edges?

What's the largest volume of box you can make from a square of paper?

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

Imagine we have four bags containing a large number of 1s, 4s, 7s and 10s. What numbers can we make?

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?

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

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