Normally, I don’t have trouble teaching implicit differentiation. However, I’m never satisfied with what I do. I’m fairly certain that I have taught it four different ways in the past four years. But what’s common is that we do a lot of algebra. By the end, they can find for a relation like . Or something like that. But we lose the meaning of what we’re doing.
I realized we can do all this algebra, but it’s all procedure. And so there’s no real depth.
So today, after introducing implicit differentiation (including some visual motivation), I assigned 5 basic problems from the textbook. Each of the problems has an equation like and students are asked to find . My kids are going to go home today and struggle with it. We’ll spend about 20 or 25 minutes in our next class going over their solutions, talking about things, whatever.
And then… then… I’m going to hand out this sheet I wrote today.
View this document on Scribd
[if you’re wondering, the graphs were made by the fabulous winplot which I adore… it can do implicit plotting!]
My kids found for homework. Now in class, my kids are going to interrogate what that means.
I am not sure yet how I’m going to structure the class. I think I might have us all work together on the first problem (#9), and then assign pairs to work on two of the remaining problems. And then I’ll pick one problem for each pair to present to the class. But what I’m truly happy about is that each problem gets kids to relate implicit differentiation to a graphical understanding of the derivative. It forces my kids to look at the derivative equation, and make connections to the original graph.
Although I’m proud of it, I’m honestly just not sure if this investigation is beyond the scope of my kids’s abilities. It pulls together a lot of concepts. I think it’ll work for them. This year I have a really really strong crew so I have faith. However, it’s an activity I’m going to have to give my kids time to do, and room to struggle. I know me, and I’m going to want to rush it, and I’m going to want to help them in ways that aren’t good for them. The struggle is where they’re going to learn in this, so I have to give it time and stay out.
I am in the middle of a hellish week, but if I have time, I’ll try to report back how it goes after we do it in class.
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