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?
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.
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?
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.
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?
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.
Can you make sense of these three proofs of Pythagoras' Theorem?
Manufacturers need to minimise the amount of material used to make their product. What is the best cross-section for a gutter?
Medieval stonemasons used a method to construct octagons using ruler and compasses... Is the octagon regular? Proof please.
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?
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. . . .
If the sides of the triangle in the diagram are 3, 4 and 5, what is the area of the shaded square?
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?
Can you find rectangles where the value of the area is the same as the value of the perimeter?
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?
Given an equilateral triangle inside an isosceles triangle, can you find a relationship between the angles?
Pick a square within a multiplication square and add the numbers on each diagonal. What do you notice?
Think of a number and follow the machine's instructions... I know what your number is! Can you explain how I know?
Can you use the diagram to prove the AM-GM inequality?
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. . . .
Show that if you add 1 to the product of four consecutive numbers the answer is ALWAYS a perfect square.
Create some shapes by combining two or more rectangles. What can you say about the areas and perimeters of the shapes you can make?
When number pyramids have a sequence on the bottom layer, some interesting patterns emerge...
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.
Can you find a rule which connects consecutive triangular numbers?
Think of a number and follow my instructions. Tell me your answer, and I'll tell you what you started with! Can you explain how I know?
An algebra task which depends on members of the group noticing the needs of others and responding.
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?
How many winning lines can you make in a three-dimensional version of noughts and crosses?
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?
What is special about the difference between squares of numbers adjacent to multiples of three?
Robert noticed some interesting patterns when he highlighted square numbers in a spreadsheet. Can you prove that the patterns will continue?
There are unexpected discoveries to be made about square numbers...
Jo made a cube from some smaller cubes, painted some of the faces of the large cube, and then took it apart again. 45 small cubes had no paint on them at all. How many small cubes did Jo use?
The opposite vertices of a square have coordinates (a,b) and (c,d). What are the coordinates of the other vertices?
How could Penny, Tom and Matthew work out how many chocolates there are in different sized boxes?
A task which depends on members of the group noticing the needs of others and responding.
How to build your own magic squares.
How good are you at finding the formula for a number pattern ?
Surprising numerical patterns can be explained using algebra and diagrams...
Imagine starting with one yellow cube and covering it all over with a single layer of red cubes, and then covering that cube with a layer of blue cubes. How many red and blue cubes would you need?
A moveable screen slides along a mirrored corridor towards a centrally placed light source. A ray of light from that source is directed towards a wall of the corridor, which it strikes at 45 degrees. . . .
Show that all pentagonal numbers are one third of a triangular number.
Can you find the area of a parallelogram defined by two vectors?
If you know the perimeter of a right angled triangle, what can you say about the area?
A cyclist and a runner start off simultaneously around a race track each going at a constant speed. The cyclist goes all the way around and then catches up with the runner. He then instantly turns. . . .
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. . . .
Can you explain how this card trick works?
What is the total number of squares that can be made on a 5 by 5 geoboard?