Last week we had some Year 10 students visiting NRICH as part of a summer school. The theme for the week was proof, and over the course of the week we worked on problems designed to expose the students to opportunities for convincing arguments, reasoning and proof.

I led a session about how to build up simple proofs. We started with these two problems (you might want to have a go for yourself before reading on):

Find three numbers whose squares add up to a multiple of 4.What do you notice about the numbers?Will it always be the case?Can you prove it?Find a pair of numbers whose squares add up to a multiple of 3.What do you notice about the numbers?Will it always be the case?Can you prove it?

We discussed the first problem all together, starting by finding some examples and collecting them together on the board. Then we commented on what we noticed, and considered how to represent odd and even numbers algebraically to see if we could prove our conjecture that the three numbers had to be even. This led us into discussions of remainders when you divide by four, so we were effectively working on modular arithmetic although we weren’t using the notation. I pointed the students in the direction of the introductions to modular arithmetic on the NRICH site if they wanted to follow this up. Then they used what they’d learned from the first example to have a go at the second, and I was delighted that most of them quickly began to consider the remainders when they divided by three as being important.

I then threw in a quick one-liner to provoke some more thought:

Is 6n-1 a prime number for all positive integer values of n?

They weren’t having any of it – they found counter-examples extremely quickly, and this led to a couple of interesting discussions – that we need to have a body of watertight logic to prove something, but we only need one counter-example to disprove it. We also discussed the consequences for world mathematics if we HAD managed to find a formula that generated prime numbers…

We finished off with a couple of number patterns that lead to some nice algebraic proofs:

1x2x3 + 2 = 82x3x4 + 3 = 273x4x5 + 4 = 64Will the answer always be a cube number?1x2x3x4 + 1 = 252x3x4x5 +1 = 1213x4x5x6 +1 = 361Will the answer always be a square number?

This is the third study school these students have attended with us, and it’s a testament to how much they’ve progressed with their mathematical thinking that some of them dispatched the first problem almost immediately by representing the middle number as n and expanding some brackets. They found the second example a bit harder, until they were prompted to guess a form for the right hand side and expand left and right sides to see if they were equal.

This was just one session in a week devoted to thinking about what it means to prove something, and what level of justification is necessary – throughout the week I was delighted that they rose to every challenge we gave them, even when we asked them some pretty challenging questions. One session turned into a discussion of complex numbers, another touched on summing infinite series – ideas that Year 10 students might not normally be exposed to, but that they eagerly had a go at when it was presented to them. I hope they will go away from this study school with a better grasp of proof and a belief in themselves as young mathematicians who can persevere with hard problems.

Advertisements

August 4, 2010 at 12:14 |

It’s really exciting reading this account, Alison. I have to say that a large part of the success of the week will be down to the way you have led the sessions. I imagine that you will have been making it clear (sometimes explicitly and sometimes implicitly) what kind of ‘behaviours’ you value in your maths classroom and this has obviously had a huge impact on the way the students are working.

You should at least take some of the credit!

August 4, 2010 at 12:18 |

Of course you’re right Liz that the classroom atmosphere and the action of the teacher are very important in developing a culture of conjecture and proof, so I will take a little credit for how well they did! I hope teachers will find our September material on NRICH useful as the theme is closely linked with getting started on using rich tasks at the beginning of a new term – I’m sure I’ll be writing more about this next month.