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?
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.
Show that if you add 1 to the product of four consecutive numbers the answer is ALWAYS a perfect square.
Can you find the area of a parallelogram defined by two vectors?
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.
The sums of the squares of three related numbers is also a perfect square - can you explain why?
Choose four consecutive whole numbers. Multiply the first and last numbers together. Multiply the middle pair together. What do you notice?
Have a go at creating these images based on circles. What do you notice about the areas of the different sections?
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?
Given an equilateral triangle inside an isosceles triangle, can you find a relationship between the angles?
Can you explain the surprising results Jo found when she calculated the difference between square numbers?
Can you maximise the area available to a grazing goat?
Is there a relationship between the coordinates of the endpoints of a line and the number of grid squares it crosses?
Many numbers can be expressed as the difference of two perfect squares. What do you notice about the numbers you CANNOT make?
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?
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?
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.
Can you see how to build a harmonic triangle? Can you work out the next two rows?
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?
Investigate how you can work out what day of the week your birthday will be on next year, and the year after...
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?
What angle is needed for a ball to do a circuit of the billiard table and then pass through its original position?
Can you find rectangles where the value of the area is the same as the value of the perimeter?
Manufacturers need to minimise the amount of material used to make their product. What is the best cross-section for a gutter?
Show that is it impossible to have a tetrahedron whose six edges have lengths 10, 20, 30, 40, 50 and 60 units...
Which has the greatest area, a circle or a square inscribed in an isosceles, right angle triangle?
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 ?
What is the same and what is different about these circle questions? What connections can you make?
If a sum invested gains 10% each year how long before it has doubled its value?
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. . . .
There are four children in a family, two girls, Kate and Sally, and two boys, Tom and Ben. How old are the children?
Can you guarantee that, for any three numbers you choose, the product of their differences will always be an even 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?
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?
Find the decimal equivalents of the fractions one ninth, one ninety ninth, one nine hundred and ninety ninth etc. Explain the pattern you get and generalise.
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.
How many winning lines can you make in a three-dimensional version of noughts and crosses?
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 napkin is folded so that a corner coincides with the midpoint of an opposite edge . Investigate the three triangles formed .
There are lots of different methods to find out what the shapes are worth - how many can you find?
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?
Explore when it is possible to construct a circle which just touches all four sides of a quadrilateral.
Can you find an efficient method to work out how many handshakes there would be if hundreds of people met?
Explore the effect of reflecting in two parallel mirror lines.
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?
A country has decided to have just two different coins, 3z and 5z coins. Which totals can be made? Is there a largest total that cannot be made? How do you know?
Start with two numbers and generate a sequence where the next number is the mean of the last two numbers...
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?
Do you notice anything about the solutions when you add and/or subtract consecutive negative numbers?