Can you see how to build a harmonic triangle? Can you work out the next two rows?
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
Can all unit fractions be written as the sum of two unit fractions?
Show that if you add 1 to the product of four consecutive numbers
the answer is ALWAYS a perfect square.
What angle is needed for a ball to do a circuit of the billiard
table and then pass through its original position?
Many numbers can be expressed as the difference of two perfect squares. What do you notice about the numbers you CANNOT make?
Think of two whole numbers under 10, and follow the steps. I can work out both your numbers very quickly. How?
Can you explain the surprising results Jo found when she calculated
the difference between square numbers?
How many more miles must the car travel before the numbers on the milometer and the trip meter contain the same digits in the same order?
If a sum invested gains 10% each year how long before it has
doubled its value?
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?
Is there a relationship between the coordinates of the endpoints of a line and the number of grid squares it crosses?
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.
Choose four consecutive whole numbers. Multiply the first and last numbers together. Multiply the middle pair together. What do you notice?
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.
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 the area of a parallelogram defined by two vectors?
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?
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 ?
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?
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.
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?
The sums of the squares of three related numbers is also a perfect
square - can you explain why?
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?
There are four children in a family, two girls, Kate and Sally, and
two boys, Tom and Ben. How old are the children?
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?
A jigsaw where pieces only go together if the fractions are
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?
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?
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. . . .
Can you describe this route to infinity? Where will the arrows take you next?
Given an equilateral triangle inside an isosceles triangle, can you find a relationship between the angles?
Water freezes at 0°Celsius (32°Fahrenheit) and boils at 100°C (212°Fahrenheit). Is there a temperature at which Celsius and Fahrenheit readings are the same?
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?
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. . . .
If you move the tiles around, can you make squares with different coloured edges?
Explore the effect of reflecting in two parallel mirror lines.
Square numbers can be represented as the sum of consecutive odd
numbers. What is the sum of 1 + 3 + ..... + 149 + 151 + 153?
On the graph there are 28 marked points. These points all mark the
vertices (corners) of eight hidden squares. Can you find the eight
How many winning lines can you make in a three-dimensional version of noughts and crosses?
Show that is it impossible to have a tetrahedron whose six edges
have lengths 10, 20, 30, 40, 50 and 60 units...
Can you find an efficient method to work out how many handshakes
there would be if hundreds of people met?
Can you guarantee that, for any three numbers you choose, the
product of their differences will always be an even number?
Can you maximise the area available to a grazing goat?
How many pairs of numbers can you find that add up to a multiple of 11? Do you notice anything interesting about your results?
Powers of numbers behave in surprising ways. Take a look at some of these and try to explain why they are true.
Explore the effect of combining enlargements.
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.
Have a go at creating these images based on circles. What do you notice about the areas of the different sections?
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?