The gradient of ax + by = c is -a/b.
This is what Year 10 wish I had told them lessons ago.
They've been looking at parallel and perpendicular lines lately, which involves finding lots and lots of gradients. They like it when the equation of a line is in gradient-intercept form. The gradient is m from y = mx + b, and everyone is happy.
As an aside, why is m used for gradient/slope? Does anyone know? I've had kids ask me that so many times, and my honest answer has been "I haven't got a clue."
I was annoyed at the lack of images in this post. So here's some parallel and perpendicular lines, just because.
Anyway, they aren't so happy with standard form. They know how to rearrange between the forms, so they've been changing them into gradient-intercept form to find the gradient. But they've been getting really fed up with it. "This takes too long, Mr Carter! There has to be a quicker way."
As I mentioned at the start of this post, there is a quicker way. But if I'd just given them that shortcut from the start, they would've been able to find gradients a whole lot faster. But they wouldn't have understood why.
So I hinted that there is a quicker method. I wrote an equation on the board (I don't remember what, I didn't save my notes for some silly reason), and asked them to tell what to do to find the gradient. Which they did.
"So what's the gradient of this?" I asked as I wrote ax + by = c on the board. There was a moment of quiet, before someone called out, "minus a over b!" Someone else added their agreement. Someone else asked "How did you get that?" Someone else disagreed, saying it can't that simple. "Where did the minus come from?" asked another. And though they wanted me to confirm the answer, I just waited quietly for the class to come to their own agreement about what they'd just discovered.
They weren't all that impressed with me when they realised I'd known this rule all along...
I feel like I should point out that this is not how the majority of my classes go. This is just an example of one time when everything went really right, and the class responded the way I wanted. My main motivation for writing this post is to remind myself that this type of moment, where I bait the class into their own discussions which lead to their own discoveries, is what I want to happen in my classroom. All the time.
Something I'm realising more and more is that mathematical knowledge has much more value to students when they earn it themselves. Would it have been more efficient to give the class the rule straight away. Well, yes, if by efficient you really mean "students were able to get more questions done". But I don't think that's what efficient learning in class is. Rather than tell my students one specific rule that than can make use of in one specific scenario (and probably completely forget in a week), I want to teach my students to think mathematically and discover mathematics for themselves.