The sums of the squares of three related numbers is also a perfect
square - can you explain why?
If a sum invested gains 10% each year how long before it has
doubled its value?
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?
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.
Choose four consecutive whole numbers. Multiply the first and last numbers together. Multiply the middle pair together. What do you notice?
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?
Many numbers can be expressed as the difference of two perfect squares. What do you notice about the numbers you CANNOT make?
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?
Can you find the area of a parallelogram defined by two vectors?
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...
Show that if you add 1 to the product of four consecutive numbers
the answer is ALWAYS a perfect square.
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?
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.
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 explain the surprising results Jo found when she calculated
the difference between square 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?
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?
Think of two whole numbers under 10, and follow the steps. I can work out both your numbers very quickly. How?
Given an equilateral triangle inside an isosceles triangle, can you find a relationship between the angles?
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 ?
Is there a relationship between the coordinates of the endpoints of a line and the number of grid squares it crosses?
What angle is needed for a ball to do a circuit of the billiard
table and then pass through its original position?
Explore the effect of combining enlargements.
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?
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. . . .
Square numbers can be represented as the sum of consecutive odd
numbers. What is the sum of 1 + 3 + ..... + 149 + 151 + 153?
Investigate how you can work out what day of the week your birthday will be on next year, and the year after...
Can you find an efficient method to work out how many handshakes
there would be if hundreds of people met?
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?
Can you describe this route to infinity? Where will the arrows take you next?
There are four children in a family, two girls, Kate and Sally, and
two boys, Tom and Ben. How old are the children?
Powers of numbers behave in surprising ways. Take a look at some of these and try to explain why they are true.
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.
What is the smallest number with exactly 14 divisors?
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 you guarantee that, for any three numbers you choose, the
product of their differences will always be an even number?
Sissa cleverly asked the King for a reward that sounded quite modest but turned out to be rather large...
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?
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?
Do you notice anything about the solutions when you add and/or
subtract consecutive negative numbers?
Can all unit fractions be written as the sum of two unit fractions?
What does this number mean ? Which order of 1, 2, 3 and 4 makes the
highest value ? Which makes the lowest ?
Which set of numbers that add to 10 have the largest product?
A jigsaw where pieces only go together if the fractions are
Mathematicians are always looking for efficient methods for solving problems. How efficient can you be?
The Egyptians expressed all fractions as the sum of different unit
fractions. The Greedy Algorithm might provide us with an efficient
way of doing this.