I began my intro class on thursday by returning a mini assessment students had taken the day before.

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Like before, I asked for volunteers to go to different tables around the classroom and lead short 2 minute workshops on how to do the problems for students who didn’t get them. I’m really starting to love this teaching strategy.

We came back together just to go over the problem one more time, and there was a question about how to report the slope of the graph. Should it be:
$\frac{2}{3}$, $0.\bar{6}$,$0.63$, or simply $0.6$. I then asked what story to each of these numbers tell, and the students realized that really, only 0.6 tells the “I kinda eyeballed the graph to estimate the slope” story. Then I asked about why often students write out every decimal from their calculator, and asked them would they do this if pencil lead were really expensive, or kinda gruesomely, they had to write in blood. Some kid then said like Harry Potter with Delores Umbridge, and reminded me he was writing “I will not tell lies,” which led perfectly into a discussion about what the lie is that $0.\bar{6}$ is telling. Suppose another research wants to improve on my finding by buying every student an ipod touch and installing a sleep cycle, a cool sleep measuring app, but then they see you reported the slope as $0.\bar{6}$, they’ll decide they can’t do that well in the experiment and no ipod for you. 😦

Similarly we had a great discussion about which of the 3 following interpretations of slope is best

The slope is the amount by which your grade increases for sleep

• The slope says that your grade increased by 6.3 points for every 10 hours of sleep.
• The slope indicates that your grade increases by 0.63 points for every 1 additional hour of sleep.

Next, it was time to wrap up some of the pendulum unit. I asked the students to test their model, to see if $T=0.2\frac{\textrm{s}}{\sqrt{\textrm{cm}}}\sqrt{L}$ really does predict the period of the pendulum, and they all find it does. Next I have them apply the model to a meter stick with a hole drilled in one end, swinging as a pendulum does. Surprisingly, it doesn’t obey the model (if you think the length is 1). With very little prompting, many groups came to the idea that it acts like a much shorter pendulum, which makes sense, since all the mass of the meter stick isn’t at the bottom. Wow. The idea of what a model is, and its limited range of effectiveness is something I want to keep coming back to all year long.

Last, they put their model to the test by making a pendulum keep the beat to a song. This is an idea I think I got from Frank Noschese. This is a GREAT exercise. First the students have to figure out the beat frequency of a song. They try all sorts of stuff, and then someone googles “bpm counter” and they find this. Once they get that, I ask them how they can find how many seconds are in a beat. This is a real and interesting dimensional analysis problems (cf to “the sun is 8 light minutes away, find this distance in feet”). Pretty quickly they get to a count of beats per second, but then, it takes some real reasoning to figure out how many seconds per beat (great lead in question: if the bps is 3, do you expect it to take more or less than a second for the next beat to arrive? Why?). Pretty soon they were all figuring out why you wanted to prefer contry ballad over that hard driving techno mix. And amazingly, most kids got all this stuff done in 1 hour. This never would have happened in my previous life when I would have handed the kids a 6 page packet to trudge through.

1. September 4, 2010 1:57 pm

John,

The idea wasn’t mine originally, but I might have shared it w/ the modeling list. I checked my Gmail archive, but I cannot find the original source.

I love the meter stick pendulum as a model breaker. This kind of “when does the model apply” discussion is really important and, if time allowed, would make a great follow-up opened-ended investigation. I hardly do any of this, so its easy for kids to just apply the model we are currently studying.

Thank you so much for sharing!

• September 4, 2010 2:48 pm

I’m really thinking I need to emphasize modeling more, and in particular, the limitations of a model, so I’m trying to build a challenge problem or lab at the end of every unit to get kids to see what the model is. The crazy thing is that most of my students, despite the fact that CVPM is the only thing we’ve studied, don’t know what the CVPM is—it’s a forest for the trees issue. They’re too focused on the equations, and they somehow thing CVPM can be boiled down to one equation.