Rectangle PQRS has X and Y on the edges. Triangles PQY, YRX and XSP have equal areas. Prove X and Y divide the sides of PQRS in the golden ratio.
Find the sum of this series of surds.
What is the same and what is different about these circle questions? What connections can you make?
What does this number mean ? Which order of 1, 2, 3 and 4 makes the highest value ? Which makes the lowest ?
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?
Why does this fold create an angle of sixty degrees?
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?
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.
A napkin is folded so that a corner coincides with the midpoint of an opposite edge . Investigate the three triangles formed .
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?
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?
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. . . .
Given an equilateral triangle inside an isosceles triangle, can you find a relationship between the angles?
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?
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. . . .
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?
Show that if you add 1 to the product of four consecutive numbers the answer is ALWAYS a perfect square.
The sums of the squares of three related numbers is also a perfect square - can you explain why?
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. . . .
Can you find the area of a parallelogram defined by two vectors?
Investigate how you can work out what day of the week your birthday will be on next year, and the year after...
The diagonals of a trapezium divide it into four parts. Can you create a trapezium where three of those parts are equal in area?
Square numbers can be represented as the sum of consecutive odd numbers. What is the sum of 1 + 3 + ..... + 149 + 151 + 153?
How many pairs of numbers can you find that add up to a multiple of 11? Do you notice anything interesting about your results?
Five children went into the sweet shop after school. There were choco bars, chews, mini eggs and lollypops, all costing under 50p. Suggest a way in which Nathan could spend all his money.
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 ?
Can you find an efficient method to work out how many handshakes there would be if hundreds of people met?
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?
Have a go at creating these images based on circles. What do you notice about the areas of the different sections?
Can you maximise the area available to a grazing goat?
If you move the tiles around, can you make squares with different coloured edges?
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?
Manufacturers need to minimise the amount of material used to make their product. What is the best cross-section for a gutter?
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?
Start with two numbers and generate a sequence where the next number is the mean of the last two numbers...
Is there a relationship between the coordinates of the endpoints of a line and the number of grid squares it crosses?
Can you describe this route to infinity? Where will the arrows take you next?
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 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?
Show that is it impossible to have a tetrahedron whose six edges have lengths 10, 20, 30, 40, 50 and 60 units...
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?
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?
Explore the effect of reflecting in two parallel mirror lines.
Explore the effect of combining enlargements.
A hexagon, with sides alternately a and b units in length, is inscribed in a circle. How big is the radius of the circle?
How many winning lines can you make in a three-dimensional version of noughts and crosses?
There are four children in a family, two girls, Kate and Sally, and two boys, Tom and Ben. How old are the children?
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.