Powers of numbers behave in surprising ways. Take a look at some of these and try to explain why they are true.
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?
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
Show that if you add 1 to the product of four consecutive numbers
the answer is ALWAYS a perfect square.
Can you guarantee that, for any three numbers you choose, the
product of their differences will always be an even number?
Which set of numbers that add to 10 have the largest product?
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. . . .
Show that is it impossible to have a tetrahedron whose six edges
have lengths 10, 20, 30, 40, 50 and 60 units...
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?
Can you explain the surprising results Jo found when she calculated
the difference between square numbers?
Find the sum of the series.
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 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?
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?
What is the smallest number with exactly 14 divisors?
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.
Investigate how you can work out what day of the week your birthday will be on next year, and the year after...
What is the largest number which, when divided into 1905, 2587,
3951, 7020 and 8725 in turn, leaves the same remainder each time?
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.
Given an equilateral triangle inside an isosceles triangle, can you find a relationship between the angles?
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 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 ?
Mathematicians are always looking for efficient methods for solving problems. How efficient can you be?
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?
Can you find the area of a parallelogram defined by two vectors?
If a sum invested gains 10% each year how long before it has
doubled its value?
How many pairs of numbers can you find that add up to a multiple of 11? Do you notice anything interesting about your results?
Choose four consecutive whole numbers. Multiply the first and last numbers together. Multiply the middle pair together. What do you notice?
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 describe this route to infinity? Where will the arrows take you next?
Sissa cleverly asked the King for a reward that sounded quite modest but turned out to be rather large...
Is there a relationship between the coordinates of the endpoints of a line and the number of grid squares it crosses?
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?
What does this number mean ? Which order of 1, 2, 3 and 4 makes the
highest value ? Which makes the lowest ?
Manufacturers need to minimise the amount of material used to make
their product. What is the best cross-section for a gutter?
Do you know a quick way to check if a number is a multiple of two? How about three, four or six?
What angle is needed for a ball to do a circuit of the billiard
table and then pass through its original position?
Start with two numbers and generate a sequence where the next number is the mean of the last two numbers...
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?
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?
Square numbers can be represented as the sum of consecutive odd
numbers. What is the sum of 1 + 3 + ..... + 149 + 151 + 153?
What size square corners should be cut from a square piece of paper to make a box with the largest possible volume?
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 ?
What is the same and what is different about these circle
questions? What connections can you make?
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 maximise the area available to a grazing goat?
Explore the effect of reflecting in two parallel mirror lines.