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?
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?
Show that if you add 1 to the product of four consecutive numbers
the answer is ALWAYS a perfect square.
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?
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?
Start with two numbers. This is the start of a sequence. The next
number is the average of the last two numbers. Continue the
sequence. What will happen if you carry on for ever?
Sissa cleverly asked the King for a reward that sounded quite modest but turned out to be rather large...
Can you find the area of a parallelogram defined by two vectors?
Do you notice anything about the solutions when you add and/or
subtract consecutive negative numbers?
Can you describe this route to infinity? Where will the arrows take you next?
Is there a relationship between the coordinates of the endpoints of a line and the number of grid squares it crosses?
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?
Explore the effect of reflecting in two parallel mirror lines.
Explore the effect of combining enlargements.
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?
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?
The sums of the squares of three related numbers is also a perfect
square - can you explain why?
Think of two whole numbers under 10, and follow the steps. I can work out both your numbers very quickly. How?
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.
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 ?
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?
Square numbers can be represented as the sum of consecutive odd
numbers. What is the sum of 1 + 3 + ..... + 149 + 151 + 153?
How many pairs of numbers can you find that add up to a multiple of 11? Do you notice anything interesting about your results?
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?
Can you find an efficient method to work out how many handshakes
there would be if hundreds of people met?
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 are four children in a family, two girls, Kate and Sally, and
two boys, Tom and Ben. How old are the children?
Have a go at creating these images based on circles. What do you notice about the areas of the different sections?
Can you explain the surprising results Jo found when she calculated
the difference between square numbers?
What does this number mean ? Which order of 1, 2, 3 and 4 makes the
highest value ? Which makes the lowest ?
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 is the area of the quadrilateral APOQ? Working on the building
blocks will give you some insights that may help you to work it
What is the greatest volume you can get for a rectangular (cuboid)
parcel if the maximum combined length and girth are 2 metres?
If a sum invested gains 10% each year how long before it has
doubled its value?
Do you know a quick way to check if a number is a multiple of two? How about three, four or six?
What is the smallest number with exactly 14 divisors?
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.
The number 2.525252525252.... can be written as a fraction. What is
the sum of the denominator and numerator?
Can all unit fractions be written as the sum of two unit fractions?
What size square corners should be cut from a square piece of paper to make a box with the largest possible volume?
Can you find rectangles where the value of the area is the same as the value of the perimeter?
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?
A circle is inscribed in a triangle which has side lengths of 8, 15
and 17 cm. What is the radius of the circle?
Mathematicians are always looking for efficient methods for solving problems. How efficient can you be?
Manufacturers need to minimise the amount of material used to make
their product. What is the best cross-section for a gutter?