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?
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?
Many numbers can be expressed as the difference of two perfect squares. What do you notice about the numbers you CANNOT make?
Show that if you add 1 to the product of four consecutive numbers
the answer is ALWAYS a perfect square.
Start with two numbers and generate a sequence where the next number is the mean of the last two numbers...
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?
Can you find the area of a parallelogram defined by two vectors?
What is the smallest number with exactly 14 divisors?
Do you notice anything about the solutions when you add and/or
subtract consecutive negative numbers?
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?
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.
The sums of the squares of three related numbers is also a perfect
square - can you explain why?
Can you explain the surprising results Jo found when she calculated
the difference between square numbers?
My two digit number is special because adding the sum of its digits to the product of its digits gives me my original number. What could my number be?
Can you see how to build a harmonic triangle? Can you work out the next two rows?
Think of two whole numbers under 10, and follow the steps. I can work out both your numbers very quickly. How?
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?
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?
Have a go at creating these images based on circles. What do you notice about the areas of the different sections?
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?
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?
What is the greatest volume you can get for a rectangular (cuboid)
parcel if the maximum combined length and girth are 2 metres?
What size square corners should be cut from a square piece of paper to make a box with the largest possible volume?
Is there a relationship between the coordinates of the endpoints of a line and the number of grid squares it crosses?
Can you describe this route to infinity? Where will the arrows take you next?
How many pairs of numbers can you find that add up to a multiple of 11? Do you notice anything interesting about your results?
Square numbers can be represented as the sum of consecutive odd
numbers. What is the sum of 1 + 3 + ..... + 149 + 151 + 153?
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?
Explore the effect of combining enlargements.
Explore the effect of reflecting in two parallel mirror lines.
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 ?
There are four children in a family, two girls, Kate and Sally, and
two boys, Tom and Ben. How old are the children?
Can all unit fractions be written as the sum of two unit fractions?
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?
Can you guarantee that, for any three numbers you choose, the
product of their differences will always be an even number?
The number 2.525252525252.... can be written as a fraction. What is
the sum of the denominator and numerator?
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.
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.
Do you know a quick way to check if a number is a multiple of two? How about three, four or six?
If a sum invested gains 10% each year how long before it has
doubled its value?
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
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?
What angle is needed for a ball to do a circuit of the billiard
table and then pass through its original position?
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?
Investigate how you can work out what day of the week your birthday will be on next year, and the year after...
A jigsaw where pieces only go together if the fractions are
What does this number mean ? Which order of 1, 2, 3 and 4 makes the
highest value ? Which makes the lowest ?