Yesterday morning I had an epiphany and started on a new idea for a lesson which was rolled out exactly 43 minutes later. I've been trying to use GeoGebra more in my classes and I thought of an idea that would not only let me do that, but also help out with the actual concept of limits which some of my students were struggling with. While they can calculate limits pretty well and can throw back some of the things I've thrown at them(phrases like "what the function is approaching" and "what the point should be"), I've had this underlying feeling that there's still something missing in their understanding. We're also moving into derivatives, and I wanted a nice visual way to show them what exactly we are going to do. So I whipped up a few of these:
I split the class into a few small groups, had each group use their laptops to manipulate the GeoGebra files and gave them this worksheet which has them find various slopes at points on the curves, then had them explain their process and discuss what exactly that has to do with limits. The graphs increased in difficulty so the first answer was 0, which is intuitive just by moving the point around. The second answer was 1, which is also pretty intuitive, but a little more challenging. For questions 3 and 4 they had to get a little deeper into how exactly how to calculate the slope.
It was important to me that I didn't tell them *how* exactly how to find the slope, so that they could figure it out on their own. I did mention that if they put the points directly on top of each other, they wouldn't have a line anymore. For most of the students, I felt this activity went over really well.(A few students did struggle with not being told exactly how to do it and how exactly to discuss it with their group to make forward progress) For the most part they were able to figure out how to get the different slopes. A couple had an issue with the question about how this dealt with limits until the following exchanges:
"Okay so what exactly did you do to find it"
-"We just moved the point as close as we could to... Ooooh!"
or
"Hey wait a minute, you were assuming that point was (4, 6) but it's actually not exactly"
-"So as I move the point it's getting closer and closer to (4, 6) so.. Ooooh!!!"
That "Ooooh" moment was awesome, and exactly what I was looking to see. Through this activity, a lot of my students ended up with a better understanding of what limits are, and the stage was set much better this year for coming up with the limit definition of a derivative. I'm hoping this will lead to my students to a much better understanding of the relationship between limits and derivatives this year, but we'll have to wait and see.
I didn't end up with a ton of time to plan this activity, and though I'm happy with what came of it I would welcome any comments you have for improving it.
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In your GGB applet here, B is not actually "on" the curve (if you drag the curve the other pts move with it, but B doesn't).
ReplyDeleteAlso, if you click on the line and do Object Properties then the Algebra tab, you can tell it to display it in y = mx + b form. It is easier to see the slope that way, but developing the derivative formula will be harder than using the coefficients of x and y in standard form as you have it.
We are just finishing up derivatives in my AP classes, but they didn't quite get limits at the beginning of the year. This would be a good thing to use for review (if we can somehow get access to the computer lab and download GGB and all that mess in the next few weeks).