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?
Show that if you add 1 to the product of four consecutive numbers
the answer is ALWAYS a perfect square.
Can you find the area of a parallelogram defined by two vectors?
Sissa cleverly asked the King for a reward that sounded quite modest but turned out to be rather large...
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?
If a sum invested gains 10% each year how long before it has
doubled its value?
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?
Many numbers can be expressed as the difference of two perfect squares. What do you notice about the numbers you CANNOT make?
Start with two numbers and generate a sequence where the next number is the mean of the last two 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?
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 ?
Can you see how to build a harmonic triangle? Can you work out the next two rows?
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.
Square numbers can be represented as the sum of consecutive odd
numbers. What is the sum of 1 + 3 + ..... + 149 + 151 + 153?
Think of two whole numbers under 10, and follow the steps. I can work out both your numbers very quickly. How?
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?
Investigate how you can work out what day of the week your birthday will be on next year, and the year after...
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?
Can you find an efficient method to work out how many handshakes
there would be if hundreds of people met?
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?
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 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 reflecting in two parallel mirror lines.
Explore the effect of combining enlargements.
How many pairs of numbers can you find that add up to a multiple of 11? Do you notice anything interesting about your results?
There are four children in a family, two girls, Kate and Sally, and
two boys, Tom and Ben. How old are the children?
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?
The number 2.525252525252.... can be written as a fraction. What is
the sum of the denominator and numerator?
Can you guarantee that, for any three numbers you choose, the
product of their differences will always be an even number?
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 area of the quadrilateral APOQ? Working on the building
blocks will give you some insights that may help you to work it
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.
Is there an efficient way to work out how many factors a large number has?
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?
Do you know a quick way to check if a number is a multiple of two? How about three, four or six?
Do you notice anything about the solutions when you add and/or
subtract consecutive negative numbers?
What does this number mean ? Which order of 1, 2, 3 and 4 makes the
highest value ? Which makes the lowest ?
Can all unit fractions be written as the sum of two unit fractions?
What angle is needed for a ball to do a circuit of the billiard
table and then pass through its original position?
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?