Find the missing angle between the two secants to the circle when the two angles at the centre subtended by the arcs created by the intersections of the secants and the circle are 50 and 120 degrees.
Semicircles are drawn on the sides of a rectangle. Prove that the sum of the areas of the four crescents is equal to the area of the rectangle.
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?
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?
If you know the perimeter of a right angled triangle, what can you say about the area?
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?
How good are you at finding the formula for a number pattern ?
Arrange the numbers 1 to 16 into a 4 by 4 array. Choose a number. Cross out the numbers on the same row and column. Repeat this process. Add up you four numbers. Why do they always add up to 34?
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?
A job needs three men but in fact six people do it. When it is finished they are all paid the same. How much was paid in total, and much does each man get if the money is shared as Fred suggests?
Here are three 'tricks' to amaze your friends. But the really clever trick is explaining to them why these 'tricks' are maths not magic. Like all good magicians, you should practice by trying. . . .
Think of a number and follow the machine's instructions... I know what your number is! Can you explain how I know?
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?
Write down a three-digit number Change the order of the digits to get a different number Find the difference between the two three digit numbers Follow the rest of the instructions then try. . . .
A task which depends on members of the group noticing the needs of others and responding.
Can you explain how this card trick works?
The sums of the squares of three related numbers is also a perfect square - can you explain why?
Two semi-circles (each of radius 1/2) touch each other, and a semi-circle of radius 1 touches both of them. Find the radius of the circle which touches all three semi-circles.
If the sides of the triangle in the diagram are 3, 4 and 5, what is the area of the shaded square?
Can you use the diagram to prove the AM-GM inequality?
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.
Pick a square within a multiplication square and add the numbers on each diagonal. What do you notice?
How to build your own magic squares.
Surprising numerical patterns can be explained using algebra and diagrams...
What is special about the difference between squares of numbers adjacent to multiples of three?
Can you explain why a sequence of operations always gives you perfect squares?
Robert noticed some interesting patterns when he highlighted square numbers in a spreadsheet. Can you prove that the patterns will continue?
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. . . .
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. . . .
What is the total number of squares that can be made on a 5 by 5 geoboard?
There are unexpected discoveries to be made about square numbers...
Account of an investigation which starts from the area of an annulus and leads to the formula for the difference of two squares.
Can you explain what is going on in these puzzling number tricks?
Can you make sense of these three proofs of Pythagoras' Theorem?
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?
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. . . .
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. . . .
The Number Jumbler can always work out your chosen symbol. Can you work out how?
Choose four consecutive whole numbers. Multiply the first and last numbers together. Multiply the middle pair together. What do you notice?
How many winning lines can you make in a three-dimensional version of noughts and crosses?
Use the numbers in the box below to make the base of a top-heavy pyramid whose top number is 200.
What would you get if you continued this sequence of fraction sums? 1/2 + 2/1 = 2/3 + 3/2 = 3/4 + 4/3 =
Think of a two digit number, reverse the digits, and add the numbers together. Something special happens...
Given an equilateral triangle inside an isosceles triangle, can you find a relationship between the angles?
Some students have been working out the number of strands needed for different sizes of cable. Can you make sense of their solutions?
Find some examples of pairs of numbers such that their sum is a factor of their product. eg. 4 + 12 = 16 and 4 × 12 = 48 and 16 is a factor of 48.
Crosses can be drawn on number grids of various sizes. What do you notice when you add opposite ends?
Try entering different sets of numbers in the number pyramids. How does the total at the top change?
When number pyramids have a sequence on the bottom layer, some interesting patterns emerge...
The heptathlon is an athletics competition consisting of 7 events. Can you make sense of the scoring system in order to advise a heptathlete on the best way to reach her target?