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

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?

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?

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

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?

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

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

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

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

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

Show that if you add 1 to the product of four consecutive numbers the answer is ALWAYS a perfect square.

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

In 15 years' time my age will be the square of my age 15 years ago. Can you work out my age, and when I had other special birthdays?

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.

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

A circle is inscribed in a triangle which has side lengths of 8, 15 and 17 cm. What is the radius of the circle?

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

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

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?

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

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?

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

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?

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?

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

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.

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

Start with two numbers. This is the start of a sequence. The next number is the average of the last two numbers. Continue the sequence. What will happen if you carry on for ever?

Find a cuboid (with edges of integer values) that has a surface area of exactly 100 square units. Is there more than one? Can you find them all?

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

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

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?

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

Here are four tiles. They can be arranged in a 2 by 2 square so that this large square has a green edge. If the tiles are moved around, we can make a 2 by 2 square with a blue edge... Now try to. . . .

Explore the effect of reflecting in two parallel mirror lines.

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

Explore the effect of combining enlargements.

What is the area of the quadrilateral APOQ? Working on the building blocks will give you some insights that may help you to work it out.

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 ?

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

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

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 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?

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

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

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. . . .

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

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