Powers of numbers behave in surprising ways. Take a look at some of these and try to explain why they are true.
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?
Ben passed a third of his counters to Jack, Jack passed a quarter of his counters to Emma and Emma passed a fifth of her counters to Ben. After this they all had the same number of counters.
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.
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?
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?
How many pairs of numbers can you find that add up to a multiple of 11? Do you notice anything interesting about your results?
Can you guarantee that, for any three numbers you choose, the product of their differences will always be an even number?
Find a cuboid (with edges of integer values) that has a surface area of exactly 100 square units. Is there more than one? Can you find them all?
Investigate how you can work out what day of the week your birthday will be on next year, and the year after...
Start with two numbers and generate a sequence where the next number is the mean of the last two numbers...
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?
Square numbers can be represented as the sum of consecutive odd numbers. What is the sum of 1 + 3 + ..... + 149 + 151 + 153?
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?
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?
Can you explain the surprising results Jo found when she calculated the difference between square numbers?
Mathematicians are always looking for efficient methods for solving problems. How efficient can you be?
Which set of numbers that add to 10 have the largest product?
Many numbers can be expressed as the difference of two perfect squares. What do you notice about the numbers you CANNOT make?
Choose four consecutive whole numbers. Multiply the first and last numbers together. Multiply the middle pair together. What do you notice?
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.
Show that if you add 1 to the product of four consecutive numbers the answer is ALWAYS a perfect square.
How many winning lines can you make in a three-dimensional version of noughts and crosses?
Can you maximise the area available to a grazing goat?
If you move the tiles around, can you make squares with different coloured edges?
Have a go at creating these images based on circles. What do you notice about the areas of the different sections?
If you are given the mean, median and mode of five positive whole numbers, can you find the 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?
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 the area of a parallelogram defined by two vectors?
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?
Explore the effect of combining enlargements.
Explore the effect of reflecting in two parallel mirror lines.
Can you find an efficient method to work out how many handshakes there would be if hundreds of people met?
There are four children in a family, two girls, Kate and Sally, and two boys, Tom and Ben. How old are the children?
What is the largest number which, when divided into 1905, 2587, 3951, 7020 and 8725 in turn, leaves the same remainder each time?
Play the divisibility game to create numbers in which the first two digits make a number divisible by 2, the first three digits make a number divisible by 3...
Is it always possible to combine two paints made up in the ratios 1:x and 1:y and turn them into paint made up in the ratio a:b ? Can you find an efficent way of doing this?
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.
Sissa cleverly asked the King for a reward that sounded quite modest but turned out to be rather large...
What does this number mean ? Which order of 1, 2, 3 and 4 makes the highest value ? Which makes the lowest ?
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 out.
A decorator can buy pink paint from two manufacturers. What is the least number he would need of each type in order to produce different shades of pink.
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 clues for this Sudoku are the product of the numbers in adjacent squares.
Some 4 digit numbers can be written as the product of a 3 digit number and a 2 digit number using the digits 1 to 9 each once and only once. The number 4396 can be written as just such a product. Can. . . .
Do you notice anything about the solutions when you add and/or subtract consecutive negative numbers?
Do you know a quick way to check if a number is a multiple of two? How about three, four or six?
Is there an efficient way to work out how many factors a large number has?