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.
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?
Show that if you add 1 to the product of four consecutive numbers the answer is ALWAYS a perfect square.
Choose four consecutive whole numbers. Multiply the first and last numbers together. Multiply the middle pair together. What do you notice?
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.
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?
The sums of the squares of three related numbers is also a perfect square - can you explain why?
Can you see how to build a harmonic triangle? Can you work out the next two rows?
Show that is it impossible to have a tetrahedron whose six edges have lengths 10, 20, 30, 40, 50 and 60 units...
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. . . .
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.
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 ?
How many more miles must the car travel before the numbers on the milometer and the trip meter contain the same digits in the same order?
If a sum invested gains 10% each year how long before it has doubled its value?
Investigate how you can work out what day of the week your birthday will be on next year, and the year after...
Square numbers can be represented as the sum of consecutive odd numbers. What is the sum of 1 + 3 + ..... + 149 + 151 + 153?
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?
Can all unit fractions be written as the sum of two unit fractions?
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.
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?
Many numbers can be expressed as the difference of two perfect squares. What do you notice about the numbers you CANNOT make?
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 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?
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?
Given an equilateral triangle inside an isosceles triangle, can you find a relationship between the angles?
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?
There are lots of different methods to find out what the shapes are worth - how many can you find?
What angle is needed for a ball to do a circuit of the billiard table and then pass through its original position?
Can you explain the surprising results Jo found when she calculated the difference between square numbers?
Water freezes at 0°Celsius (32°Fahrenheit) and boils at 100°C (212°Fahrenheit). Is there a temperature at which Celsius and Fahrenheit readings are the same?
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 ?
What is the same and what is different about these circle questions? What connections can you make?
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?
How many winning lines can you make in a three-dimensional version of noughts and crosses?
Can you maximise the area available to a grazing goat?
If you move the tiles around, can you make squares with different coloured edges?
Can you describe this route to infinity? Where will the arrows take you next?
Have a go at creating these images based on circles. What do you notice about the areas of the different sections?
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?
Which of these games would you play to give yourself the best possible chance of winning a prize?
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...
Explore the effect of reflecting in two parallel mirror lines.
Explore the effect of combining enlargements.
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. . . .
Which has the greatest area, a circle or a square inscribed in an isosceles, right angle triangle?
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?