Tag Archives: geogebra

Trig Functions – a w00t and a help request

First off, a “hey this went well,” and then a “help me out.” :) These are both about my trig class.

Hey, This Went Well
I started today talking about amplitude and period of sine functions. They’d already understood that the period of sin(x) is 2pi, and they got pretty easily that the amount it goes above and below the x-axis is 1; I just had to add the word “amplitude” to their vocabulary. Check.

So then I put a GeoGebra window up on the projector and entered f(x)=sin(x), and I asked them what I should change about the equation in order to make the amplitude different. They weren’t quite sure what I meant, so I said, “Should I multiply sine x by something? Should I multiply the x by something before I take the sine? Should I divide instead? Should I add something at the end? Should I add something to the x before I take the sine? Should I subtract instead?”

Then they started conjecturing. I made them explain why they thought their particular change would effect the desired transformation, made them convince each other before I would type it in so we could see what it actually did. After we saw each change, I went back to f(x)=sin(x) before making another change, so we were always comparing our changes with the plain ol’ sine function. They were really pleased with being able to see the changes immediately.

It was also cool because we’d looked at what happened when we used f(x)=2sin(x) and f(x)=0.5sin(x), and then somebody said, “What if we used a hundred?” So I changed the coefficient to 100, and a couple of them said, “Whoa! It’s just straight lines!” Others said, “No, it just looks like that because it goes so far up and down.” The magic of the zoom feature allowed us to see that it was, in fact, still curving like we expected it to, just with really skinny humps. :)

So I think that using GeoGebra like that was a good way of looking at these transformations – they can make a guess, and we can see whether it’s right immediately. We can also use crazy big (or crazy small) numbers if we want to.

Help Me Out
I’ve got an idea, but it’s only partially formed right now. I don’t want it to flop like the last one (which involved kids spinning in circles and then releasing a stuffed animal – it was supposed to be about linear velocity, but it was a Fail, though they did enjoy going outside). So…here goes.

You know how when you walk, your body moves up and down? Walk across the room now. When your legs are side by side, you’re at your full height, but when one leg is in front of the other, the distance from the ground to the top of your head is less than your height. If you hold an arm straight out with a marker held to a long piece of paper, and you walk normally, the mark you make on the paper will be sinusoidal. (I remember my instructor showing us this in the health class I took in college…I don’t have a clue what his point was, but I thought it was cool.)

So I want to let my students do this. I want them to ask questions about what will be different for the curves formed by the tallest person versus the shortest person; what will be different between a girl and a guy; what will be different if a person is walking versus running; what changes if a person is on crutches (okay, I’m not sure if we could pull that one off or not, but it’s an interesting question, right?); what will be the same in each of these cases as well.

But I don’t want to get out there (I have an idea for where we can do this) and then just be wasting time with questions we won’t be able to answer. So I need to figure out how to structure this thing. Suggestions?

Historical Intro to the Tangent Line

I mentioned when I started this blog that this is my first year teaching math, but that I spent the last seven years teaching history. So it’s probably not a surprise that I find the history of mathematics interesting, though I’ll admit I have a LOT to learn in that area. I’ve pulled a few lessons from this CD-ROM, called Historical Modules for the Teaching and Learning of Mathematics, which I highly recommend. And then I developed a lesson for my calculus students as an introduction to The Derivative.

For full disclosure, you guys are actually getting the edited version. What I gave the students was overly ambitious and overlooked the fact that ellipses and hyperbolas have more complicated derivatives than parabolas, and I was really just going for something simple to ease them into things. I made some on-the-spot changes during the presentation of the lesson, and now I’ve come back to the lesson and changed it to what I really ended up doing.

All that said, here’s the lesson.

Some things to note about the execution of it…

  • I didn’t actually say the word “derivative” until they got to the very end. Their textbook doesn’t give them the word until the next chapter, but hey, they found it! Why not tell them that’s what they were doing?
  • Your students need to be familiar with GeoGebra in order to make this thing work. Mine ended up having to use the Van Schooten worksheet to find their tangent line slopes because they couldn’t actually do what I’d asked them to – this was the first time I’d told them to do something other than graph a function on GeoGebra, and it was over their heads. At the same time, they did learn a lot about what it can do and what doesn’t work! :) I’m hoping to have next year’s group prepared by working with GeoGebra a good bit while I have them in trig/analyt this year.
  • Don’t tell your students you just ordered a book on the history of calculus unless you want them to look at you like you’re insane.
  • My students didn’t know what “analytic” meant or how it was different from “geometric.” My own understanding of geometric solutions to problems has been greatly increased as I’ve done my thesis (on Descartes’ La Geometrie), so hopefully I was able to explain it to them well. I am much more of an analytic thinker than a geometric one.

My goal in using this was to have the historical setting provide a context for studying the mathematical concept. I think it worked pretty well. In any case, staring at the board where they’d done a lot of the work together and where I’d circled an expression and written “derivative,” one student said,

You know, I really do think I get this. It looks complicated, but it’s really not that hard – it’s just slope.