If a sum invested gains 10% each year how long before it has
doubled its value?
The sums of the squares of three related numbers is also a perfect
square - can you explain why?
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?
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 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?
Choose four consecutive whole numbers. Multiply the first and last numbers together. Multiply the middle pair together. What do you notice?
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?
Start with two numbers and generate a sequence where the next number is the mean of the last two numbers...
Can you explain the surprising results Jo found when she calculated
the difference between square numbers?
Many numbers can be expressed as the difference of two perfect squares. What do you notice about the numbers you CANNOT make?
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...
What angle is needed for a ball to do a circuit of the billiard
table and then pass through its original position?
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.
Show that if you add 1 to the product of four consecutive numbers
the answer is ALWAYS a perfect square.
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?
Given an equilateral triangle inside an isosceles triangle, can you find a relationship between the angles?
Think of two whole numbers under 10, and follow the steps. I can work out both your numbers very quickly. How?
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.
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. . . .
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.
Can you find an efficient method to work out how many handshakes
there would be if hundreds of people met?
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?
What is the greatest volume you can get for a rectangular (cuboid)
parcel if the maximum combined length and girth are 2 metres?
An aluminium can contains 330 ml of cola. If the can's diameter is 6 cm what is the can's height?
Manufacturers need to minimise the amount of material used to make
their product. What is the best cross-section for a gutter?
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?
Is there a relationship between the coordinates of the endpoints of a line and the number of grid squares it crosses?
How many winning lines can you make in a three-dimensional version of noughts and crosses?
Which set of numbers that add to 10 have the largest product?
Explore the effect of combining enlargements.
Explore the effect of reflecting in two parallel mirror lines.
Can you describe this route to infinity? Where will the arrows take you next?
Square numbers can be represented as the sum of consecutive odd
numbers. What is the sum of 1 + 3 + ..... + 149 + 151 + 153?
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?
Can all unit fractions be written as the sum of two unit fractions?
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
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.
Can you arrange these numbers into 7 subsets, each of three
numbers, so that when the numbers in each are added together, they
make seven consecutive numbers?
Take any four digit number. Move the first digit to the 'back of
the queue' and move the rest along. Now add your two numbers. What
properties do your answers always have?
Here is a chance to create some attractive images by rotating
shapes through multiples of 90 degrees, or 30 degrees, or 72
If you move the tiles around, can you make squares with different coloured edges?
On the graph there are 28 marked points. These points all mark the
vertices (corners) of eight hidden squares. Can you find the eight
What does this number mean ? Which order of 1, 2, 3 and 4 makes the
highest value ? Which makes the lowest ?
A jigsaw where pieces only go together if the fractions are
Sissa cleverly asked the King for a reward that sounded quite modest but turned out to be rather large...