Choose four consecutive whole numbers. Multiply the first and last numbers together. Multiply the middle pair together. What do you notice?
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?
Many numbers can be expressed as the difference of two perfect squares. What do you notice about the numbers you CANNOT make?
A car's milometer reads 4631 miles and the trip meter has 173.3 on
it. How many more miles must the car travel before the two numbers
contain the same digits in the same order?
Can you describe this route to infinity? Where will the arrows take you next?
Can you find the area of a parallelogram defined by two vectors?
Is there a relationship between the coordinates of the endpoints of a line and the number of grid squares it crosses?
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 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?
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?
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...
If a sum invested gains 10% each year how long before it has
doubled its value?
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?
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?
The sums of the squares of three related numbers is also a perfect
square - can you explain why?
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?
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 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?
Explore the effect of combining enlargements.
Explore the effect of reflecting in two parallel mirror lines.
In a three-dimensional version of noughts and crosses, how many winning lines can you make?
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?
Square numbers can be represented as the sum of consecutive odd
numbers. What is the sum of 1 + 3 + ..... + 149 + 151 + 153?
Can you find an efficient method to work out how many handshakes
there would be if hundreds of people met?
Can all unit fractions be written as the sum of two unit fractions?
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.
There are four children in a family, two girls, Kate and Sally, and
two boys, Tom and Ben. How old are the children?
Sissa cleverly asked the King for a reward that sounded quite modest but turned out to be rather large...
Many numbers can be expressed as the sum of two or more consecutive integers. For example, 15=7+8 and 10=1+2+3+4. Can you say which numbers can be expressed in this way?
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
Do you notice anything about the solutions when you add and/or
subtract consecutive negative numbers?
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?
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.
Manufacturers need to minimise the amount of material used to make
their product. What is the best cross-section for a gutter?
The number 2.525252525252.... can be written as a fraction. What is
the sum of the denominator and numerator?
Given an equilateral triangle inside an isosceles triangle, can you find a relationship between the angles?
Different combinations of the weights available allow you to make different totals. Which totals can you make?
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 ?
A 2 by 3 rectangle contains 8 squares and a 3 by 4 rectangle
contains 20 squares. What size rectangle(s) contain(s) exactly 100
squares? Can you find them all?
Show that is it impossible to have a tetrahedron whose six edges
have lengths 10, 20, 30, 40, 50 and 60 units...
What angle is needed for a ball to do a circuit of the billiard
table and then pass through its original position?
Have a go at creating these images based on circles. What do you notice about the areas of the different sections?
Powers of numbers behave in surprising ways. Take a look at some of these and try to explain why they are true.
A country has decided to have just two different coins, 3z and 5z
coins. Which totals can be made? Is there a largest total that
cannot be made? How do you know?
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. . . .
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?