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.
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.
Powers of numbers behave in surprising ways. Take a look at some of these and try to explain why they are true.
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?
Find the sum of the series.
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?
Show that if you add 1 to the product of four consecutive numbers
the answer is ALWAYS a perfect square.
Choose four consecutive whole numbers. Multiply the first and last numbers together. Multiply the middle pair together. What do you notice?
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?
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?
Can you describe this route to infinity? Where will the arrows take you next?
Is there a relationship between the coordinates of the endpoints of a line and the number of grid squares it crosses?
Have a go at creating these images based on circles. What do you notice about the areas of the different sections?
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
A napkin is folded so that a corner coincides with the midpoint of
an opposite edge . Investigate the three triangles formed .
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
Which has the greatest area, a circle or a square inscribed in an
isosceles, right angle triangle?
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 lots of different methods to find out what the shapes are worth - how many can you find?
Do you notice anything about the solutions when you add and/or
subtract consecutive negative numbers?
What is the largest number which, when divided into 1905, 2587,
3951, 7020 and 8725 in turn, leaves the same remainder each time?
Sissa cleverly asked the King for a reward that sounded quite modest but turned out to be rather large...
What does this number mean ? Which order of 1, 2, 3 and 4 makes the
highest value ? Which makes the lowest ?
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?
Why does this fold create an angle of sixty degrees?
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?
On the graph there are 28 marked points. These points all mark the
vertices (corners) of eight hidden squares. Can you find the eight
Show that is it impossible to have a tetrahedron whose six edges
have lengths 10, 20, 30, 40, 50 and 60 units...
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?
In a three-dimensional version of noughts and crosses, how many winning lines can you make?
Investigate how you can work out what day of the week your birthday will be on next year, and the year after...
Explore when it is possible to construct a circle which just
touches all four sides of a quadrilateral.
How many pairs of numbers can you find that add up to a multiple of
11? Do you notice anything interesting about your results?
Can you find an efficient method to work out how many handshakes
there would be if hundreds of people met?
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.
Use the differences to find the solution to this Sudoku.
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 is inscribed in a triangle which has side lengths of 8, 15
and 17 cm. What is the radius of the circle?
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?
A hexagon, with sides alternately a and b units in length, is
inscribed in a circle. How big is the radius of the circle?
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?
Find a cuboid (with edges of integer values) that has a surface
area of exactly 100 square units. Is there more than one? Can you
find them all?
An aluminium can contains 330 ml of cola. If the can's diameter is
6 cm what is the can's height?
Which of these games would you play to give yourself the best possible chance of winning a prize?
Some people offer advice on how to win at games of chance, or how
to influence probability in your favour. Can you decide whether
advice is good or not?