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

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?

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.

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.

Explore the area of families of parallelograms and triangles. Can you find rules to work out the areas?

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.

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

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

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. . . .

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

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?

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

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.

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

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.

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?

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

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

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.

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?

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

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.

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?

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

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?

Pick the number of times a week that you eat chocolate. This number must be more than one but less than ten. Multiply this number by 2. Add 5 (for Sunday). Multiply by 50... Can you explain why it. . . .

What is the total number of squares that can be made on a 5 by 5 geoboard?

A little bit of algebra explains this 'magic'. Ask a friend to pick 3 consecutive numbers and to tell you a multiple of 3. Then ask them to add the four numbers and multiply by 67, and to tell you. . . .

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?

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

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

If you have a large supply of 3kg and 8kg weights, how many of each would you need for the average (mean) of the weights to be 6kg?

Can you see how to build a harmonic triangle? Can you work out the next two rows?

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

Charlie has moved between countries and the average income of both has increased. How can this be so?

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

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

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

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?

It would be nice to have a strategy for disentangling any tangled ropes...

Start with two numbers and generate a sequence where the next number is the mean of the last two numbers...

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?

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?”

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.

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.