Perfect numbers can be fun for bright primary mathematicians
As a retired comprehensive school maths teacher now volunteering in my local primary school, one thing I am asked to do is to provide some support for the very brightest in the school. What follows are some examples of material I have used with these students.
One thing I use regularly with students of all abilities is Codes. Using the simple code A=1, B=2 to Y=25 and Z=26 you can create work for all levels of ability. You can start with, for example, just the first five letters and create words like BAD (2,1,4), BED (2,5,4), ADD (1 4,4) and CAB (3,1,2). You can keep it simple (and 3 letters) by using the whole alphabet and creating rhyming words, like a line with SAY, MAY and DAY converted to code, which I used just today as I write this and the students were amused by it. Even if you are only using the first five letters, you can create simple problems for each letter. Hence BAD could be (5-3), (6-5), (2×2).
At the most difficult end, you can create “thinking” codes. I don’t actually teach things like powers and roots – I can just put them into code sheets and students just ask if they don’t know what something means. Mathematics does, after all, contain a language of its own. The answer to the question “Why use something like 45?” is, to me, quite simple. It is so much quicker than writing 4×4×4×4×4!
A simple example of “thinking” comes when you use a cube root. Recently, a Year 6 student had to deal with 3√1728. This was his thinking.
“Firstly, it has to be a whole number from 1 to 26. What single digit cubed ends in an 8? There’s only 23. Now 103 is 1000 and 203 is 8000, so the only possible answer between those two is 12. It’s L” he said.
It turns out that all the cubes of the whole numbers from 0 to 9 end in a different number so, given in this case it must be a whole number answer, you can immediately identify the letter – assuming, of course, you have thought it through AND your arithmetic can cope! A similar “thinking” question is √625 or √225. Of course, the only square to end in 5 must have a 5 on the end to begin with! Given that 10² is 100 and 20² is 400, then √625 can ONLY be 25 and √225 can ONLY be 15.
Another piece of mathematical language bright students always seem to enjoy is the factorial. 4! means 4×3×2×1 – again, a simple use of a sign to save time and space. This is a very important mathematical sign because, for example, the number of ways of arranging five different letters is 5! In modern day life, one of the ways this appears each week is that the number of ways of picking 6 numbers from 49 is 49C6 or 49!÷(43!×6!). This comes to about 14 million, so 1 in 14 million is your chance of a jackpot in the Lottery! I’ve never played the Lottery – I’m a mathematician!!
While in this area, a little note about something I did after the SATs with a group of perhaps half of Year 6. I have bought for the school a set of Scientific Calculators as teaching aids. I asked this class what would happen if I borrowed £1 and had to repay 2% each day if I didn’t pay it back. On reflection, a more realistic way of putting it might be that I borrowed £21 and paid back £20 quickly, but forgot I still owed £1!
I asked the students how much they thought I would owe after 1 year? One answer came back as £8.30, interesting as it was £1 plus 365 lots of 2p. I used that answer to try to clarify compound rather than simple interest. Other answers varied from £30 to, perhaps, £100. I then explained how to use the power button on the calculator to get the answer. The result was, I think, the most amazing collection of sounds I’ve heard in a class ! The amount of £1377.41 was universally regarded as staggering! It prompted a discussion about what was on a TV screen and easy to read – and what was perhaps deliberately NOT easy to read.
One way of moving on from the simple code is to divide each number by, say, 24. This gives you a fraction code going from 1/24 to 12/24. When I use it, I cancel all fractions, so Z is 11/12. I have then written a first sheet with straightforward codes on it, then a second where I have forgotten to cancel some fractions! Thus 4/8 needs to be cancelled to ½ before students can find it in the codes I put at the top of the sheet. For the brightest students, you can include fraction questions to be solved to find the letter. So, if I’ve got it right, 9/5 × 10/27 would result in 2/3, leading to the letter P.
There is a similar approach for decimals. Instead of A=1, B=2, you can use A=0.1, B=0.2 or even A=0.01, B=0.02 and create questions using decimals or even, of course, using fractions which the students then convert to decimals.
Another area which you can expand on is that of Factors. There is a simple extension of exercises in which you ask students to find the factors of certain numbers. Whole numbers bigger than 1 divide into 3 groups. These groups are called Abundant, Deficient and Perfect numbers. For example, 12 is Abundant because when you add 1+2+3+4+6 (ie the factors other than the number itself) you get more than 12. 10 is Deficient because 1+2+5 comes to less than 10. 6 is Perfect because 1+2+3 is exactly 6. Believe it or not, I have read a number of articles in daily newspapers when a new Perfect number has been found!
Perfect numbers are all generated by what are known as Mersenne primes. Mersenne primes are numbers one below powers of 2 that are prime. Algebraically, they are of the form 2p-1, where p stands for a prime number – though not all prime numbers create Mersenne primes. One of my students last year went away and decided to find the 5th Mersenne prime – which led to a perfect number in the tens of millions! As an example of finding a perfect number: 7 is a Mersenne prime, being one below 2×2×2. To find the perfect number, just multiply it by 2×2, giving 28. Feel free to check out that the factors of 28 – other than 28 itself – add up to 28!
Clearly, all prime numbers are Deficient - more interesting is finding the first odd number that is NOT Deficient. You have to get to 945 to find it – which, it turns out, is 3×5×7×9. Any multiples of Perfect and Abundant numbers will always be Abundant. All of this can lead to outcomes that might appear strange. For example, look at multiples of 3. Until you get to 945, all ODD multiples are Deficient. 6 is Perfect, so all EVEN multiples of 3 from 12 on are Abundant.
Another area that I use with my students is divisibility tests. One intriguing example is to take the number 91 and put the same single digit at the front and the back – like 3913. The number you finish up with ALWAYS divides by 91! The same is true for the number 77. Why those two – and how many more numbers does this work for and why?
A more commonly recognised test is how you can tell very quickly (and without dividing !) that the number 837 divides by 9? Well, 8+3+7 comes to 18 and 1+8=9. That’s called the digital root. If the digital root is 9, then the original number divides by 9, no matter how big the original number is!
What if a number doesn’t divide by 9 ? Take the number 698. To find the digital root we add 6+9+8, which is 23 and 2+3 = 5. This tells us that 698 doesn’t divide exactly by 9 – but it also tells that if we divide by 9, then the remainder is 5!
There is some sort of divisibility test for all numbers from 2 to 16 – come to think of it if, in a 4-figure number, the second pair is twice the first pair (like 2958) then that’s a test for being divisible by 17!