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?

Learning Targets and Brainstorming

Status on Learning Targets…

• Calculus: done
  
• Statistics: not all the way through the text, but I’ve gotten through everything we’ve done in the class I’m taking, and I can add to it later
  
• Trigonometry & Analytic Geometry: done, unless I get through things and have time for an extra chapter, at which point I’ll add in those LTs
  
• Geometry: just opened the book (I’ll spare you a link to an empty document ;))

If anyone feels like giving me feedback, please do so. This has been a bigger task than I anticipated!

Now for some brainstorming…I have an idea for how I want to structure my units. For each unit, I want to put one or more real problems (of the WCYDWT variety) up on the bulletin board. The idea is that I will choose these Big Problems (don’t I come up with creative names?) as things that can be found using what they will learn in the unit. We’ll keep looking at them as we go through the unit, evaluating how our new knowledge can help us reach new understandings about the Big Problems. (I totally need a better name than Big Problems.)

What I want my kids to recognize is that math can be useful. I think it will also push them in the right direction for completing each semester’s final assessment, assuming I stick with that idea.

So once I get my Learning Targets finished, I think I will tackle the first couple of units for each class and figure out…

1. Big Problems
   
2. Assessments
   
3. Instruction
   
4. Homework/practice sets

I think doing that will help me feel ready to start the year, though the more units I can prepare ahead of time, the better. Oh, and I also need to work on my Policies & Procedures sheet (to explain things like my crazy new grading idea).

Reading too much into the question

More thoughts on my statistics class from today. We were going over some review questions, and I said that one of them didn’t have enough information for us to determine an answer. The instructor responded,

You’re reading way too much into the question.

Well, no, I really wasn’t. It was a poor question. I’m really not sure why she didn’t just acknowledge that it’s a poor question – it’s not even a pride thing, because these questions are from the textbook publisher, not from her. And even when I am the one who wrote questions, I’ve told kids who challenged them, “You know what? You’re right. I could be asking this question more precisely.” And I make a note to improve the question for the next time I come to that material.

So the first part of my point here is, it’s okay to recognize that a question/assignment/whatever is not the best and could be improved. I think my students like knowing that I will receive their critiques (delivered respectfully, of course) and consider their feedback. I know that I didn’t like feeling like I wasn’t being listened to, being told to just write down the “right answer” and move on.

But the second thing I wanted to say in this post is a question I thought of as a result of this experience. Is there a point at which it’s okay to tell a student “you’re reading too much into the question”? Obviously we want to stop kids from saying,

“Well, Johnny won’t have ANY apples left after he gives 3 to Suzie, because right at that moment Suzie gets turned into a ZOMBIE, and she doesn’t care about the apples anymore but just wants to eat Johnny’s BRAAAAAAINS, so Johnny figures life is more important than apples and he drops the apples so he can run away. Will he survive Zombie Suzie’s attack? Just wait until I turn in my next homework!”

Because that would just be silly.

But. There’s a lot of good thinking that students can do in between “just give the answer you know I’m looking for” and the zombie scenario. I’ve read several bloggers talking about WCYDWT (What Can You Do With This), which is all about posing new problems, digging deeper into what’s right in front of you instead of just relying on the (possibly/probably poorly written) questions in the textbook.

There probably are good occasions for telling a kid he’s reading too much into a problem. But I think we might jump there a little more quickly than we ought to sometimes.