Substitute -1, -2 or -3, into an algebraic expression and you'll
get three results. Is it possible to tell in advance which of those
three will be the largest ?
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 ?
Show that is it impossible to have a tetrahedron whose six edges
have lengths 10, 20, 30, 40, 50 and 60 units...
If you have only 40 metres of fencing available, what is the maximum area of land you can fence off?
If a sum invested gains 10% each year how long before it has
doubled its value?
An investigation involving adding and subtracting sets of consecutive numbers. Lots to find out, lots to explore.
Is there a relationship between the coordinates of the endpoints of a line and the number of grid squares it crosses?
Given an equilateral triangle inside an isosceles triangle, can you find a relationship between the angles?
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
Different combinations of the weights available allow you to make different totals. Which totals can you make?
Your school has been left a million pounds in the will of an ex-
pupil. What model of investment and spending would you use in order
to ensure the best return on the money?
The sums of the squares of three related numbers is also a perfect
square - can you explain why?
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.
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?
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.
Which set of numbers that add to 10 have the largest product?
Show that if you add 1 to the product of four consecutive numbers
the answer is ALWAYS a perfect square.
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. . . .
Powers of numbers behave in surprising ways. Take a look at some of these and try to explain why they are true.
Start with two numbers and generate a sequence where the next number is the mean of the last two numbers...
An aluminium can contains 330 ml of cola. If the can's diameter is
6 cm what is the can's height?
What is the greatest volume you can get for a rectangular (cuboid)
parcel if the maximum combined length and girth are 2 metres?
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 plastic funnel is used to pour liquids through narrow apertures.
What shape funnel would use the least amount of plastic to
manufacture for any specific volume ?
Do you notice anything about the solutions when you add and/or
subtract consecutive negative numbers?
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 does this number mean ? Which order of 1, 2, 3 and 4 makes the
highest value ? Which makes the lowest ?
Can you find the area of a parallelogram defined by two vectors?
Can you find an efficient method to work out how many handshakes
there would be if hundreds of people met?
A game for 2 or more people, based on the traditional card game
Rummy. Players aim to make two `tricks', where each trick has to
consist of a picture of a shape, a name that describes that shape,
and. . . .
Chris is enjoying a swim but needs to get back for lunch. If she can swim at 3 m/s and run at 7m/sec, how far along the bank should she land in order to get back as quickly as possible?
Which has the greatest area, a circle or a square inscribed in an
isosceles, right angle triangle?
On the graph there are 28 marked points. These points all mark the
vertices (corners) of eight hidden squares. Can you find the eight
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?
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?
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?
The diagonals of a trapezium divide it into four parts. Can you
create a trapezium where three of those parts are equal in area?
A napkin is folded so that a corner coincides with the midpoint of
an opposite edge . Investigate the three triangles formed .
In a three-dimensional version of noughts and crosses, how many winning lines can you make?
Two ladders are propped up against facing walls. The end of the
first ladder is 10 metres above the foot of the first wall. The end
of the second ladder is 5 metres above the foot of the second. . . .
Chris and Jo put two red and four blue ribbons in a box. They each
pick a ribbon from the box without looking. Jo wins if the two
ribbons are the same colour. Is the game fair?
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 hexagon, with sides alternately a and b units in length, is inscribed in a circle. How big is the radius of the circle?
Have a go at creating these images based on circles. What do you notice about the areas of the different sections?
If you are given the mean, median and mode of five positive whole numbers, can you find the numbers?
Can you maximise the area available to a grazing goat?
Can you find rectangles where the value of the area is the same as the value of the perimeter?
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. . . .
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?
Each of the following shapes is made from arcs of a circle of
radius r. What is the perimeter of a shape with 3, 4, 5 and n