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?
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?
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 ?
Choose four consecutive whole numbers. Multiply the first and last numbers together. Multiply the middle pair together. What do you notice?
Manufacturers need to minimise the amount of material used to make their product. What is the best cross-section for a gutter?
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?
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.
What angle is needed for a ball to do a circuit of the billiard table and then pass through its original position?
Show that if you add 1 to the product of four consecutive numbers the answer is ALWAYS a perfect square.
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.
What is the same and what is different about these circle questions? What connections can you make?
Show that is it impossible to have a tetrahedron whose six edges have lengths 10, 20, 30, 40, 50 and 60 units...
If a sum invested gains 10% each year how long before it has doubled its value?
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 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?
Many numbers can be expressed as the difference of two perfect squares. What do you notice about the numbers you CANNOT make?
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. . . .
Which has the greatest area, a circle or a square inscribed in an isosceles, right angle triangle?
Can you explain the surprising results Jo found when she calculated the difference between square numbers?
A napkin is folded so that a corner coincides with the midpoint of an opposite edge . Investigate the three triangles formed .
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?
The sums of the squares of three related numbers is also a perfect square - can you explain why?
Is there a relationship between the coordinates of the endpoints of a line and the number of grid squares it crosses?
Can you find rectangles where the value of the area is the same as the value of the perimeter?
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?
Have a go at creating these images based on circles. What do you notice about the areas of the different sections?
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?
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?
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.
A hexagon, with sides alternately a and b units in length, is inscribed in a circle. How big is the radius of the circle?
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?
Square numbers can be represented as the sum of consecutive odd numbers. What is the sum of 1 + 3 + ..... + 149 + 151 + 153?
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?
There are four children in a family, two girls, Kate and Sally, and two boys, Tom and Ben. How old are the children?
How many pairs of numbers can you find that add up to a multiple of 11? Do you notice anything interesting about your results?
How many winning lines can you make in a three-dimensional version of noughts and crosses?
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?
Explore the effect of reflecting in two parallel mirror lines.
If you have only 40 metres of fencing available, what is the maximum area of land you can fence off?
Can you maximise the area available to a grazing goat?
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?
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?
Explore the effect of combining enlargements.
Why does this fold create an angle of sixty degrees?
Can you find an efficient method to work out how many handshakes there would be if hundreds of people met?
Investigate how you can work out what day of the week your birthday will be on next year, and the year after...
Can you find the area of a parallelogram defined by two vectors?
The diagonals of a trapezium divide it into four parts. Can you create a trapezium where three of those parts are equal in area?