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.
What does this number mean ? Which order of 1, 2, 3 and 4 makes the
highest value ? Which makes the lowest ?
Find the sum of the series.
Powers of numbers behave in surprising ways. Take a look at some of these and try to explain why they are true.
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.
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?
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?
Sissa cleverly asked the King for a reward that sounded quite modest but turned out to be rather large...
A jigsaw where pieces only go together if the fractions are
Show that if you add 1 to the product of four consecutive numbers
the answer is ALWAYS a perfect square.
Can you explain the surprising results Jo found when she calculated
the difference between square numbers?
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?
Can you find the area of a parallelogram defined by two vectors?
Choose four consecutive whole numbers. Multiply the first and last numbers together. Multiply the middle pair together. What do you notice?
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?
Have a go at creating these images based on circles. What do you notice about the areas of the different sections?
The sums of the squares of three related numbers is also a perfect
square - can you explain why?
Can you find an efficient method to work out how many handshakes
there would be if hundreds of people met?
How many pairs of numbers can you find that add up to a multiple of 11? Do you notice anything interesting about your results?
Investigate how you can work out what day of the week your birthday will be on next year, and the year after...
Given an equilateral triangle inside an isosceles triangle, can you find a relationship between the angles?
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
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.
Which has the greatest area, a circle or a square inscribed in an
isosceles, right angle triangle?
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. . . .
Square numbers can be represented as the sum of consecutive odd
numbers. What is the sum of 1 + 3 + ..... + 149 + 151 + 153?
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?
On the graph there are 28 marked points. These points all mark the
vertices (corners) of eight hidden squares. Can you find the eight
Can you maximise the area available to a grazing goat?
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. . . .
If you move the tiles around, can you make squares with different coloured edges?
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?
Start with two numbers and generate a sequence where the next number is the mean of the last two numbers...
Manufacturers need to minimise the amount of material used to make
their product. What is the best cross-section for a gutter?
What size square corners should be cut from a square piece of paper to make a box with the largest possible volume?
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?
Can you describe this route to infinity? Where will the arrows take you next?
Explore the effect of reflecting in two parallel mirror lines.
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?
Show that is it impossible to have a tetrahedron whose six edges
have lengths 10, 20, 30, 40, 50 and 60 units...
How many winning lines can you make in a three-dimensional version of noughts and crosses?
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?
Why does this fold create an angle of sixty degrees?
Explore the effect of combining enlargements.
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?
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?