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

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

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

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

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?

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.

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

Explore the effect of combining enlargements.

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.

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

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. . . .

Two ladders are propped up against facing walls. The end of the first ladder is 10 metres above the foot of the first wall. The end of the second ladder is 5 metres above the foot of the second. . . .

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?

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?

The Egyptians expressed all fractions as the sum of different unit fractions. The Greedy Algorithm might provide us with an efficient way of doing this.

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

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

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

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?

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

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

What is the greatest volume you can get for a rectangular (cuboid) parcel if the maximum combined length and girth are 2 metres?

Manufacturers need to minimise the amount of material used to make their product. What is the best cross-section for a gutter?

An aluminium can contains 330 ml of cola. If the can's diameter is 6 cm what is the can's height?

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

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

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

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

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?

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

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

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

There are lots of different methods to find out what the shapes are worth - how many can you find?

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

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?

A hexagon, with sides alternately a and b units in length, is inscribed in a circle. How big is the radius of the circle?

If you are given the mean, median and mode of five positive whole numbers, can you find the numbers?

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. . . .

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?

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?

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

Each of the following shapes is made from arcs of a circle of radius r. What is the perimeter of a shape with 3, 4, 5 and n "nodes".

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

Here is a chance to create some attractive images by rotating shapes through multiples of 90 degrees, or 30 degrees, or 72 degrees or...

Chris is enjoying a swim but needs to get back for lunch. If she can swim at 3 m/s and run at 7m/sec, how far along the bank should she land in order to get back as quickly as possible?

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.

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?