After a successful proffessional development morning back in February, I invited Don Steward back to Huddersfield University this month to run another session for trainees, NQTs and local (and some not so local!) maths teachers as part of our mini Maths Teacher Conference. The event included other speakers and workshops too, which I’ll write a little about later in the week. (Check out http://www.missblilley.co.uk/ to read about one of them, which Don attended and described as ‘incredible’ yesterday).
Anyway, moving on. Don has kindly let me write up his session, and I think the materials will be available on his blog as well, which if you haven’t seen it, is here)
The session was very focused around proofs, how to introduce them, and how to derive them.
He opened with this:
This is intended for Year 7 students. Before students do anything, the question “will we get the same answer if we go in either direction?” is asked. A discussion can be had, then students can place a low number in the top left circle, and work right, then down first, then start again but go down and right. Was the answer the same? Nothing too taxing so far. Students then try again with a second number, and then again with a third. Is there a pattern emerging (spoiler alert – yes).
Students are then encouraged to try a big number (in fact, it was suggested that younger students often choose a big number in a bid to try and show it works for all numbers). This time however, we cannot fit say, a million in that small circle, so we’re going to call it ‘m’ (genius!!).
In this way we’re essentially moving discretely into algebra, whilst keeping it all very approachable for students. Running through creates algebraic expressions now.
Then you can get a student to think of a number, but not say it to anyone. If that student is called Beth, we’ll use ‘b’ for ‘Beth’s number’. We run through again, deriving the same expressions as before, which highlight the pattern, but this time, even though b and m are both algebraic representations, crucially for students, they do not know what ‘b’ is, yet they did for ‘m’. They now have a proof. Beth’s number could be anything, and the pattern still stands. There are variations on this idea too. You could give students the end numbers and work backwards for example.
Here is a more visual proof of the pattern:
The second proof exercise looked at the cases where 8n + 1 generates a square number (see below)
Is there a pattern emerging?
You may be able to see that the ‘n’ values that create a square number are in fact triangle numbers. I should point out that this is where a lot of the greatness of the presentation is lost in blog format, as that thinking time and discovery was very much handed over to us, the audience, rather than being given the answers instantly as I’m doing here.
Again, a few different methods of proving this were mooted and considered, and a nice visual proof is shown below (n = 10):
This proof was derived nicely by a trainee in the audience too.
A third proof activity is shown below. This time, looking at the answers generated for a general formula (n+2)^2 – n^2 . No focus is initally given to algebra at all, it’s about playing around with real examples, looking at the answers, and trying to spot the pattern. In this case, the answers are 4(n+1) each time. Which would initially be desribed without algebra before diving into proofs (ie first we spot that each answer is a multiple of 4, then we look at ‘what’ is multiplied by 4 to get that particular answer, then we discover it is in fact the number between 7 and 5, 3 and 1, 10 and 8 etc).
The eventual visual proof for this result is shown below:
It’s a rearrangement of a more intuitive pair of diagrams:
It turns out the diagram is used as a visual proof for several theorems
I recognised it from a Pythagorean proof:
But there are some other nifty proofs too (see the PowerPoint at the end).
Further investigations involved the relationships between area and perimeter:
Using angle bisectors to investigate the (infinitely) more interesting inscribed circles:
and taking a very mathematical method to approach magic squares (which will merit a post of its own soon!)
A very thought provoking, and inspiring presentation once again. Many thanks Don.
Here’s the slideshow