Choose four consecutive whole numbers. Multiply the first and last numbers together. Multiply the middle pair together. What do you notice?
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?
Given an equilateral triangle inside an isosceles triangle, can you find a relationship between the angles?
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?
Show that if you add 1 to the product of four consecutive numbers the answer is ALWAYS a perfect square.
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.
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?
Can all unit fractions be written as the sum of two unit fractions?
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?
Many numbers can be expressed as the difference of two perfect squares. What do you notice about the numbers you CANNOT make?
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?
Square numbers can be represented as the sum of consecutive odd numbers. What is the sum of 1 + 3 + ..... + 149 + 151 + 153?
Can you describe this route to infinity? Where will the arrows take you next?
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?
Start with two numbers and generate a sequence where the next number is the mean of the last two numbers...
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?
Explore the effect of reflecting in two parallel mirror lines.
Explore the effect of combining enlargements.
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 an efficient method to work out how many handshakes there would be if hundreds of people met?
Can you find the area of a parallelogram defined by two vectors?
How many winning lines can you make in a three-dimensional version of noughts and crosses?
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?
There are four children in a family, two girls, Kate and Sally, and two boys, Tom and Ben. How old are the children?
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.
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?
How many pairs of numbers can you find that add up to a multiple of 11? Do you notice anything interesting about your results?
Is there a relationship between the coordinates of the endpoints of a line and the number of grid squares it crosses?
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. . . .
Do you know a quick way to check if a number is a multiple of two? How about three, four or six?
If you move the tiles around, can you make squares with different coloured edges?
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 ?
Do you notice anything about the solutions when you add and/or subtract consecutive negative numbers?
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?
Can you maximise the area available to a grazing goat?
Different combinations of the weights available allow you to make different totals. Which totals can you make?
An investigation involving adding and subtracting sets of consecutive numbers. Lots to find out, lots to explore.
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?
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?
On the graph there are 28 marked points. These points all mark the vertices (corners) of eight hidden squares. Can you find the eight hidden squares?
An aluminium can contains 330 ml of cola. If the can's diameter is 6 cm what is the can's height?
What size square corners should be cut from a square piece of paper to make a box with the largest possible volume?
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?
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?
Is there an efficient way to work out how many factors a large number has?
Manufacturers need to minimise the amount of material used to make their product. What is the best cross-section for a gutter?
Powers of numbers behave in surprising ways. Take a look at some of these and try to explain why they are true.
Can you guarantee that, for any three numbers you choose, the product of their differences will always be an even number?