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?

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

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.

Take any prime number greater than 3 , square it and subtract one. Working on the building blocks will help you to explain what is special about your results.

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

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

The sums of the squares of three related numbers is also a perfect square - can you explain why?

Powers of numbers behave in surprising ways. Take a look at some of these and try to explain why they are true.

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

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

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

If a sum invested gains 10% each year how long before it has doubled its value?

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

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?

The diagram illustrates the formula: 1 + 3 + 5 + ... + (2n - 1) = n² Use the diagram to show that any odd number is the difference of two squares.

Given an equilateral triangle inside an isosceles triangle, can you find a relationship between the angles?

All CD Heaven stores were given the same number of a popular CD to sell for £24. In their two week sale each store reduces the price of the CD by 25% ... How many CDs did the store sell at. . . .

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

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?

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 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.

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

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

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

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

A 2-Digit number is squared. When this 2-digit number is reversed and squared, the difference between the squares is also a square. What is the 2-digit number?

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

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

A car's milometer reads 4631 miles and the trip meter has 173.3 on it. How many more miles must the car travel before the two numbers contain the same digits in the same order?

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?

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 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?

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

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

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 guarantee that, for any three numbers you choose, the product of their differences will always be an even number?

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

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 ?

Find the decimal equivalents of the fractions one ninth, one ninety ninth, one nine hundred and ninety ninth etc. Explain the pattern you get and generalise.

A mother wants to share a sum of money by giving each of her children in turn a lump sum plus a fraction of the remainder. How can she do this in order to share the money out equally?

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?

Substitute -1, -2 or -3, into an algebraic expression and you'll get three results. Is it possible to tell in advance which of those three will be the largest ?

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

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

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

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?

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. . . .

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?