Some numbers are definitely more important than others, says Colin Foster – and grasping why can transform students’ mathematical understanding…
Why Teach This?
Our world is full of numbers, but not all of them are created equal. Some – the prime numbers – play a critical role in enabling us to generate all the other whole numbers. Learning about the prime numbers and how they work is a crucial part of understanding about number. Sometimes KS3 students are aware of prime numbers as those whose only factors are “one and itself”. This definition can lead to ambiguity about whether 1 itself should or should not be counted as prime, but another problem is that it can seem a very arbitrary definition. Why should numbers with exactly two factors be worth giving a name to and learning about? Students may learn about factorising numbers into primes without realising why this is of any more significance than factorising numbers into, say, square numbers or even numbers or any other kind of numbers. This lesson focuses on what it is about the prime numbers that makes them special – the idea that they are the building blocks of all the whole numbers greater than 1. So breaking a number down into its prime factors is a bit like breaking down a chemical molecule into is constituent elements – you really discover its structure. Since every whole number can be factorised into a unique product of primes, the prime numbers really form the basis for generating all the whole numbers that there are!
Q. Do you think that some numbers are more important than other numbers or are all numbers equally important? Why? Students might think about this for a minute in pairs before sharing their ideas. There is no single right answer; the aim of this initial question is just for students to think about the possibility of some numbers being more important or useful than others. They might say, for example, that 12 is the most important number because it is their age, or because it has a lot of factors. They might mention telephone numbers or numbers that you see a lot in the world as being particularly important. Q. I want to make all the whole numbers from 1 to 20 by multiplying numbers together. What would be a good set of starting numbers to use? As students suggest possibilities you can introduce the idea of the smallest possible set of starting numbers that will do it. Again, this could be worth thinking about in pairs before sharing. Students might say that 2s are very useful because you can make all the even numbers, but that involves adding, and here we are restricted to multiplying only. So if all you have is a box of 2s then you can only make the powers of 2, not all the even numbers. So you need 2s and 3s, for instance, but you don’t need 4s or 6s, because you can make those out of 2s and 3s. Likewise, students may think that 1s are useful, whereas in fact 1 is useful only for making 1 and doesn’t contribute to making any other number, because multiplying by 1 has no effect. The smallest set of numbers that will do it are: 1, 2, 3, 5, 7, 11, 13, 17 and 19; the first 8 prime numbers, together with 1. Students may use the language of prime numbers or they might not at this stage. Show the image below of the periodic table. Q. Has anyone seen this before? Where? Do you know what it is? If no one has any idea, you could suggest that they might have seen it around the science classrooms. Q. What is special about the substances on this table? Hydrogen, helium, lithium, etc. These are the chemical elements and every substance in the universe is made up of arrangements of just these 100 or so elements, just like every English word is made up of combinations of the 26 letters of the alphabet. Q. Do you think there are numbers that are the building blocks of all the whole numbers? How could we find them? These are called the prime numbers. Working systematically to find them involves excluding all the numbers you can make using the primes that you have so far. A nice approach, called the Sieve of Eratosthenes, is to begin with a table of numbers (see below) and:
- Colour in 1, because 1 is not a prime number.
- Leave the next empty cell, but colour in all its higher multiples.
- Keep going until you have coloured all the non-primes. (How will you know?)
A task sheet containing this table is available at teachsecondary.com/downloads/maths-resources Students may find following these instructions difficult, so you could begin together on the board and then students could continue individually or in pairs, using a calculator where necessary. Students are often confused about whether 1 is a prime number of not. If we think of the prime numbers as the building blocks of the integers, like the chemical elements, then 1 is not of much use (except to generate itself), so that is one way to think about why we don’t include it as a prime number – multiplying by 1 doesn’t do anything.
Q. What does your table look like? Which numbers are not coloured in? Why? Can you really make all the other numbers (except 1) out of just these ones? How can you be sure? Q. What patterns do you notice in the table? There are lots of things that students can point out. Students could be asked to make a poster to illustrate what they find. For example, some columns never contain any primes after the top row. Why should this be? Students might think that the 3rd column, for instance, has no primes because it consists of the multiples of 3, but that is not quite right – in general, the rth column contains the numbers 12n + r, where n is the row number (beginning with row zero on the top), so if r has a common factor with 12 that is greater than 1 then there will be no primes in that column. So we have no primes (after the zeroth row) in columns 2, 3, 4, 6, 8, 9, 10 and 12. So after the zeroth row all of the primes come in columns 1, 5, 7 and 11 only. Q. Do you think that the prime numbers go on for ever, or do you think they eventually stop, so there is a last prime number? This can be a nice opportunity for a simple proof by contradiction, although you don’t necessarily need to use that language. Suppose there are exactly 1000 primes and no more, and you write them down on a piece of paper. Imagine multiplying them all together and adding 1. It would make a very big number, but would that number be prime? None of your 1000 primes would go into it, because they all leave a remainder of 1. So either this big number is a prime or it has a prime factor that is bigger than any of your 1000 primes. Either way, the 1000 primes on your list can’t be all of the primes. This argument would work no matter how many primes you had written down on your list. So the primes must go on forever. Students could begin working out which primes they need to multiply together to make each number; e.g., 18 = 2 × 3 × 3 or 2 × 32. They will find that each whole number greater than 1 has exactly one way of being expressed as a product of primes (ignoring the order in which you write the product). This is called unique factorisation.
About the author
Colin Foster is an Assistant Professor in mathematics education in the School of Education at the University of Nottingham. He has written many books and articles for mathematics teachers (see www.foster77.co.uk).
Stretch them further
Students could pose their own questions about the completed grid. For example:
- What is the longest sequence of consecutive composite (non-prime) numbers?
- Do the primes get more spread out as you go on?
Twin primes are prime numbers that are 2 apart (e.g., 5 and 7). How many twin primes can you find? (No one knows how many twin primes there are, or whether there are infinitely many, even though lots of people have tried very hard to work it out!)
For a huge selection of free maths lesson plans for KS3 and KS4 click here.