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
Show that if you add 1 to the product of four consecutive numbers
the answer is ALWAYS a perfect square.
Have a go at creating these images based on circles. What do you notice about the areas of the different sections?
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?
Is there a relationship between the coordinates of the endpoints of a line and the number of grid squares it crosses?
Can you maximise the area available to a grazing goat?
The sums of the squares of three related numbers is also a perfect
square - can you explain why?
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.
Choose four consecutive whole numbers. Multiply the first and last numbers together. Multiply the middle pair together. What do you notice?
Many numbers can be expressed as the difference of two perfect squares. What do you notice about the numbers you CANNOT make?
Can you explain the surprising results Jo found when she calculated
the difference between square numbers?
Which has the greatest area, a circle or a square inscribed in an
isosceles, right angle triangle?
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
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...
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 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?
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.
If a sum invested gains 10% each year how long before it has
doubled its value?
Show that is it impossible to have a tetrahedron whose six edges
have lengths 10, 20, 30, 40, 50 and 60 units...
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?
Can you guarantee that, for any three numbers you choose, the
product of their differences will always be an even number?
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?
What angle is needed for a ball to do a circuit of the billiard
table and then pass through its original position?
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 are lots of different methods to find out what the shapes are worth - how many can you find?
Given an equilateral triangle inside an isosceles triangle, can you find a relationship between the angles?
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?
If you have only 40 metres of fencing available, what is the maximum area of land you can fence off?
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?
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?
Square numbers can be represented as the sum of consecutive odd
numbers. What is the sum of 1 + 3 + ..... + 149 + 151 + 153?
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...
What size square corners should be cut from a square piece of paper to make a box with the largest possible volume?
How many pairs of numbers can you find that add up to a multiple of 11? Do you notice anything interesting about your results?
A napkin is folded so that a corner coincides with the midpoint of
an opposite edge . Investigate the three triangles formed .
The diagonals of a trapezium divide it into four parts. Can you
create a trapezium where three of those parts are equal in area?
How many winning lines can you make in a three-dimensional version of noughts and crosses?
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?
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 combining enlargements.
Explore the effect of reflecting in two parallel mirror lines.
Explore when it is possible to construct a circle which just
touches all four sides of a quadrilateral.
On the graph there are 28 marked points. These points all mark the
vertices (corners) of eight hidden squares. Can you find the eight
If you move the tiles around, can you make squares with different coloured edges?