Throughout the day, I’ve seen and been a part of a great discussion about how to do the Modeling Buggy Lab. For those not familiar, think classic algebra problem with 2 trains. After screaming, “I hated that problem!” Ask yourself what did you dislike about it? Probably the fact that it had no context. This is where modeling steps in.

Here’s the set up, you show the class a motorized car that moves at constant speed. And, through socratic questioning, lead them to realize that the position and elapsed time are related. You then ask them to determine the “graphical and mathematical” relationship between position and elapsed time (or clock reading). You leave it up to the students to figure out how to find those relationships.

If you try this, your students might come up with one of these two solutions, (A) set up fixed distances and measure the time to travel those distances, or (B) place marks at distances for fixed values of time (place a piece of tape where the car is at 1 sec, 2 sec, …)

This is where are tweebate begins. Since the “convention” is to plot the independent variable on the horizontal axis, the two different means of collecting data would produce two different plots. Option A, since the student set up fixed distances, that would be the independent variable. Thus elapsed time (clock reading) would be on the vertical axis and distance traveled (or more specifically position) on the horizontal. Option B would yield the opposite.

Our debate was whether or not we should let this happen. Also discussed was the above was ok, but to just tell them to all plot time on the horizontal axis.

So here’s my two cents:

What I love about the modeling curriculum is that we (teachers) are trying to foster discussion, which in the end should induce critical thinking skills. So to me, let the students measure it how they think they should. Let them graph it how they think they should. As they all come together in the “Board Meeting,” where they share their results, as the teacher try to help them draw out the important conclusion. Did the two methods produce different results? Did both methods produce a straight line? What does that say about the relationship between position and time for the buggy? Assuming the buggies go at the same speed, how do the slopes compare? If some have a slope that has units of m/s and some have s/m, lead them to generate the algebraic equation for the line. Have one group rearrange their equation

$y=mx+b$

$\large x=\frac{y-b}{m}$

Discussing the “10% rule” which helps them figure out if the y-intercept is significant, can also come into play here. If $b$ is 0, then:

$\large x=\left(\frac{1}{m}\right)y$

Thus, how do the slopes compare. Again if the cars all travel at the same approximate speed, won’t this be a great aha moment? To me it’s worth the time, rather than the teacher merely saying, “Don’t worry about the convention you learned in middle school, just plot it this way.” In no way am I trying to demean those that do this, I just think it’s worth the 10 minutes to let the discussion play out.

If you have the time, you can also bring in the discussion of slope recommended by Arons, the “bible” of physics teachers (personally, I think it’s an amazing book, but it’s still no Bible, sorry Arons). He recommends discussing the meaning of slope. What does the slope of position vs time mean? Most students will probably respond “change in y over change in x,” which would confirm what Arons suggests. He writes that most students don’t understand slope, they just know how to calculate it. This is where you can take another 10 minutes and lead the students to realize that option A will produce a graph the measures how fast the buggy is moving while group B will produce a graph that measures how slow it is moving. This is now a great time to lead them to see why the convention of independent variables on the horizontal and dependent on the vertical is not a convention that is always used. Rather think ahead of the relationship you want to show, then set up the axes to show that relationship.

I know I’m new at this, so I would love to hear where, other than possibly time, we wouldn’t want to have these discussions.

PS – for those wondering where the train problem comes in – take away the first buggy and now give them a second buggy (which travels at a different constant speed). While the students turn in that buggy, have them graph (and find equation for the slope) the data for the second buggy. Now, without the buggies in hand, have them predict where the collision will occur if you start them at opposite ends (meter stick, room, table, etc). Viola, the train problem will come to life. I did this with my AP kids and they were so excited when they say that their calculation was “right” (read within error predicted by a monte carlo analysis, which I say a big thank you to Andy and the rest of the crew in the Global Physics Dept).

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