Month: January 2008

Paper folding and exponential functions

I am teaching exponential functions in my Algebra II classes this week. And I just came back from this teaching conference, where one of the sessions included a few handouts of the types of problems that this one charter school uses. And lucky for me, one was on exponential growth and decay.

I wholesale retyped this activity-based lesson up and gave it to my students. I can’t say it was the “most awesome thing ever,” but I can say that it got students to think for themselves instead of being spoon fed everything. What it had students do is to:

Fold a piece of paper in half and record: (1) the number of folds made, (2) the number of regions the paper is divided into, and (3) the fractional area of each region. Then fold the paper again and record those numbers again. (So after 1 fold, there are 2 regions, each with fractional area 1/2; after 2 folds, there are 4 regions, each with fractional area 1/4; …).

What students discovered was exponential growth and decay. What was interesting i&s that when I had them try to come up with a function relating fold number to the number of regions (y=2^x), many of them couldn’t do it. They would try thinks like y=2x, or y=x^2, but it wasn’t until I reminded them that the number of regions (2, 4, 8, 16, 32, …) could be re-written as (2^1, 2^2, 2^3, 2^4, 2^5, …) that the majority of them could figure it out.

In any case, it took a good 20 – 30 minutes for them to finish the activity (which included some plotting, and some discussion of independent and dependent variables), but overall, I’d like to think they got more out of it than me simply explaining in words what an exponential function is.

Not that I have the time to come up with a bunch more of these, nor the classtime to implement them, but I think having one or two per chapter up my sleeve would be perfect.

"Someone told me…"

I told my calculus class, in the last 15 minutes of class on Monday, that:

“Somebody told me something, and I don’t know if it’s true. They said that if there’s a cubic equation that hits the x-axis three times, then there’s a point of inflection, and it will be the average of these three x-intercepts. I don’t know if I believe them. It sounds plausible, but I’m skeptical. Because why would the zeros be related to the inflection point? And why would they be related to it in such an elegant way? Crazy talk, I said.”

Of course, they called me on it, saying that of course I knew if it was true or not. So I chuckled, and said that they got me, and of course I know if it’s true or not. But I wanted them to figure it out. So I asked them to figure out what the problem is, and guess how we would solve it.

By the end of the class, we together (but letting them lead the discussion) determined that our general equation for the cubic would be f(x)=k(x-a)(x-b)(x-c), where the x-intercepts are at a, b, and c.

Then for homework I let them loose on figuring out if what I was told was true.

The next day (today), I asked them how far they got. One person solved it, and a few had the right idea, but got frustrated with the algebra. No one “checked” to see if the inflection point truly was an inflection point (if at that point, the function switched curvature from being concave up to concave down or vice versa). But going over the solution together was awesome because:

1. I got to reinforce that a, b, and c in this equation are constants, not variables (a few were confused about that)
2. I got to show them a quick way to “foil” out (x-a)(x-b)(x-c)
3. I got to remind them how to prove something is an inflection point
4. I got to show them what a formal proof looks like

and most importantly,

I got a few of them to see how cool it was. I basically told them why I loved problems like this… because even though the algebra can get hairy, even though you might make a wrong turn somewhere along the line, we were able to show something that is totally not intuitive. To use the words I used in class, that “the payoff is worth so much more than the work.” And even though only one person solved it on their own, I think a few of my students felt that ownership as we solved it together in class.

In theory and practice it was 30 minutes of class well spent. I should do more of these sorts of problems. Hard things we do together in class.

presenting on presenting

For this week in the faculty periodic professional development meeting, I was asked to present a problem, observation, case study (basically anything) to a set of 10 new teachers. (We switch off presenters each time we meet.) I chose to do mine on the relationship between teaching, presenting, and smartboard — and the evolution of my understanding of this relationship since I’ve started teaching.

Unlike animal and plant evolution which can take hundreds of years for permanent changes to become stable (I guess bacterial evolution can happen in a matter of hours), I’ve evolved a lot in my teaching style and I wanted to share that with others. So instead of bringing up problem and talking it to death without any resolution or actionable steps — which is the norm — I decided that my group would learn something.

And so I presented. I only had 1 hour to create the ten minute presentation, which is why it is on the bland side, but I think I inspired a few of the teachers. Especially one of the music teachers, for whom I see this to be a huge timesaver.

One of the lower school teachers who had heard about my presentation randomly stopped me and said that we should talk, because she too has been using smartboard a lot and we could trade observations and ideas.

But honestly, I’m not here to change anyone’s mind about anything. I’m here to do my stuff in the classroom. Whatever else comes out of it, great.

I’m second place! Woo hoo!

I’m snug in a bug in a rug now. It’s cold. So cold, in fact, that I’ve plugged in my electric blanket and I’m tucking myself in my big fluffy down comforter. And better than that, I found out today that I came in second place in the Annual Report design contest I wrote about earlier.

I was so elated that I sent an email with the good news to my school’s “faculty folderol” course conference (basically, a place where faculty can post about anything from an apartment they are subletting to finding good restaurant recommendations)!

You can view the winners and read the judges analyses, the other entrants, and the initial competition description.

The two best things about this competition is that (1) I got a chance to reflect on the year, on the information we save and the information we discard and the information that we discarded that we wished we had saved, and on how my life has differed from January to December. (2) I learned to use a program like photoshop (but without 99% of the bells and whistles). It’s an opensource program called Seashore; you have to do your work on “layers.” A lot more work, but very easy to play around and edit in. The learning curve is steep. I made one slide a night; the first night’s work took about 3 hours, but the next three nights’ work took about 1 hour each.

So congrats to me. I’m proud.

Annual Report 2007

I did get to enter dy/dan’s Annual Report contest. (It involved a lot of time management to spew these out — I did one slide a day.) I admit that there are some very easy ways to improve my images (my astute sister made some helpful suggestions), but since I have already submitted the four images, I’m going to leave them be.

The hard thing with design, I’ve learned in this short time, is that you are almost never going to be satisfied, because there is always one or two things that you realize could be made better (constantly tweaking and adjusting), or you come up with an entirely new idea that would take a huge amount of time to execute.

[Update: I am now officially on dy/dan’s page! Post here, and my contest entry here. Lookin’ good!]

A bouquet of thanks

This morning I gave a presentation to the 10th grade about a blog I set up for them to archive their community service week (which is two weeks from now). It went well. I left knowing I did a good (darn good) job with my presentation — I joked and got my point across.

The dean who was there said that after the presentation, he wanted to hug me (because it really got the kids to realize what the whole community service thing was about).

I had done a lot of work to set up this blog — but I did it because I believe in it, because I wanted to.

I got back to my office in the middle of the day, and on my desk was a bouquet of flowers. From the 10th grade advisers (homeroom teachers), in appreciation of all the work I’ve done on this. My heart glowed; it’s nice to have hard work is acknowledged.