Have a go at creating these images based on circles. What do you notice about the areas of the different sections?

If you move the tiles around, can you make squares with different coloured edges?

On the graph there are 28 marked points. These points all mark the vertices (corners) of eight hidden squares. Can you find the eight hidden squares?

Which has the greatest area, a circle or a square inscribed in an isosceles, right angle triangle?

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?

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

A napkin is folded so that a corner coincides with the midpoint of an opposite edge . Investigate the three triangles formed .

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

This shape comprises four semi-circles. What is the relationship between the area of the shaded region and the area of the circle on AB as diameter?

If the hypotenuse (base) length is 100cm and if an extra line splits the base into 36cm and 64cm parts, what were the side lengths for the original right-angled triangle?

Explore the effect of reflecting in two parallel mirror lines.

How many winning lines can you make in a three-dimensional version of noughts and crosses?

A 2 by 3 rectangle contains 8 squares and a 3 by 4 rectangle contains 20 squares. What size rectangle(s) contain(s) exactly 100 squares? Can you find them all?

What is the same and what is different about these circle questions? What connections can you make?

A spider is sitting in the middle of one of the smallest walls in a room and a fly is resting beside the window. What is the shortest distance the spider would have to crawl to catch the fly?

If you have only 40 metres of fencing available, what is the maximum area of land you can fence off?

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

The area of a square inscribed in a circle with a unit radius is, satisfyingly, 2. What is the area of a regular hexagon inscribed in a circle with a unit radius?

A square of area 40 square cms is inscribed in a semicircle. Find the area of the square that could be inscribed in a circle of the same radius.

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.

Imagine you have a large supply of 3kg and 8kg weights. How many of each weight would you need for the average (mean) of the weights to be 6kg? What other averages could you have?

There are four children in a family, two girls, Kate and Sally, and two boys, Tom and Ben. How old are the children?

Can you find rectangles where the value of the area is the same as the value of the perimeter?

How many different symmetrical shapes can you make by shading triangles or squares?

Liam's house has a staircase with 12 steps. He can go down the steps one at a time or two at time. In how many different ways can Liam go down the 12 steps?

Many numbers can be expressed as the difference of two perfect squares. What do you notice about the numbers you CANNOT make?

Can all unit fractions be written as the sum of two unit fractions?

Which set of numbers that add to 10 have the largest product?

Investigate how you can work out what day of the week your birthday will be on next year, and the year after...

A decorator can buy pink paint from two manufacturers. What is the least number he would need of each type in order to produce different shades of pink.

What size square corners should be cut from a square piece of paper to make a box with the largest possible volume?

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

Is it always possible to combine two paints made up in the ratios 1:x and 1:y and turn them into paint made up in the ratio a:b ? Can you find an efficent way of doing this?

Explore when it is possible to construct a circle which just touches all four sides of a quadrilateral.

Explore the effect of combining enlargements.

Two motorboats travelling up and down a lake at constant speeds leave opposite ends A and B at the same instant, passing each other, for the first time 600 metres from A, and on their return, 400. . . .

What angle is needed for a ball to do a circuit of the billiard table and then pass through its original position?

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

A circle of radius r touches two sides of a right angled triangle, sides x and y, and has its centre on the hypotenuse. Can you prove the formula linking x, y and r?

A 1 metre cube has one face on the ground and one face against a wall. A 4 metre ladder leans against the wall and just touches the cube. How high is the top of the ladder above the ground?

Different combinations of the weights available allow you to make different totals. Which totals can you make?

Five children went into the sweet shop after school. There were choco bars, chews, mini eggs and lollypops, all costing under 50p. Suggest a way in which Nathan could spend all his money.

The diagonals of a trapezium divide it into four parts. Can you create a trapezium where three of those parts are equal in area?

There is a particular value of x, and a value of y to go with it, which make all five expressions equal in value, can you find that x, y pair ?

Show that is it impossible to have a tetrahedron whose six edges have lengths 10, 20, 30, 40, 50 and 60 units...

How many pairs of numbers can you find that add up to a multiple of 11? Do you notice anything interesting about your results?

A game for 2 or more people, based on the traditional card game Rummy. Players aim to make two `tricks', where each trick has to consist of a picture of a shape, a name that describes that shape, and. . . .