At the end of the first post in this series I lamented that starting energy empirically meant that I couldn’t include changes in thermal energy like starting this modeling unit more traditionally does. I shouldn’t have worried. Turns out that emphasizing that changing the energy of a system through working, heating, or radiating helps them overall with energy conservation despite that thermal energy in particular isn’t address. But I’m getting ahead of myself.
Days 1-4 ish are outlined in the first post of this series. I’m now picking up at around Day 5.
We started this unit by finding that the area under the force vs. postion graphs for two different springs, when made equal, yielded equal velocities when launching carts. I emphasized at this time (and over and over again as we went through the unit) that the area under graphs, if it has a physical meaning, means a change in something. In this case it’s a change in energy, though we hadn’t gotten that far yet. I just emphasized it’s a change in something. So in the first activity the change in something predicted velocities. In the second it correlated with a change in height. At that point we coined the term gravitational interaction energy, and we looked at how the final gravitational interaction energy was the same as the initial plus the change in energy (as found from the area under the F vs. x graph) The third, starting now, looks at the correlation of that change with velocity. They now know that this has something to do with kinetic energy, since we had the energy=pain talk, but not exactly how.
There are many variations of this lab, most using springs. I found that if you attach a force detector to a cart (which we did for the area vs. change in height experiment previously), you can just pull the cart with a rope and get pretty good data for area vs. v^2 even though the force isn’t constant. Which I think is extra cool. Basic setup for this experiment is below. Note the horizontal track.
I learned one pretty neat trick when I performed the lab myself. For each trial, it doesn’t really matter where the end point is, as long as you find the area for some displacement and then record the final velocity that corresponds to the end point for that displacement (assuming you start from rest, which I did). So I had students graph force vs. position to find the area (change in energy) that we were interested in, and then plot velocity vs. position so that they easily find the corresponding ending velocity. This way they can set the integral (area) section to be the same for each trial, then quickly use the examine function in logger pro to find the ending velocity at that same endpoint for each trial. Slick.
Plotting change in energy vs. v looks like this. Note that since I took this data I actually called the area work, since that is the means by which the energy is changing in this case. I did not instruct them to do that, however.
It actually looks fairly linear, especially to kids who are looking for things to be linear. However, typically data was non-linear enough, and we linearized a quadratic doing central force, so most groups linearized using v^2 on the x axis.
When the data is linearized, it looks like this.
Certainly that looks more linear! Student data actually turned out good as well. Always nice when that happens.
The board meeting for this went amazingly fast. In the first class a student commented almost right away about the units of the slope. They started trying to figure out what the units should be, and I wrote on the board. With a little prodding we finally figured this out;
Whoa. All that simplifies to kg? Cool.
The classes did this in different orders, but essentially within 10 minutes they had figured out that the intercept was zero (both empirically from their data as well as logically by thinking through why it should be zero), that the slope was half the mass, and that the slope relating to the mass made sense because the units of the slope simplify to kg.
From here we went on to be explicit about the names of everything. The area represented a change in energy. In the first case (pulling carts up ramps), it’s a change in gravitational interaction energy. In this case, it’s a change in kinetic energy.
This is more or less where day 5 ended. No, seriously, at this point they (keep in mind this is a college level class taught at the high school, so essentially top 20% kids) took data, whiteboarded it, and figured out meaning in a 45 minute class period.
Day 6 ish: Lab wrap up and transition to Energy Bar Charts
I started the day by teaching energy bar charts (LOLs). (Need a primer on energy bar charts? Kelly comes through again). We then went through the labs drawing the LOL for each one. This did two things; first, and most importantly, it emphasized that the area under the force vs. position graph found a value that measured how energy changed from the first snapshot to the second snapshot. Secondly, it was a way to show students how to draw LOLs. After drawing the LOLs for our two experiments, we had a conversation about how energy changes. The modeling instruction teacher notes lists that there are three ways energy changes; working, heating, and radiating. (Side note: I strongly prefer starting energy from a First Law of Thermodynamics perspective (strict conservation of energy) rather than from a Work-KE theorem perspective. More on that in a later post on partial truths) They brought up convection and conduction, and I talked about how these are just two different ways for heat to transfer. We briefly talked about molecular interactions and KE transfer here, but I kept it quick. The point here was to plant the seed that what we are doing generalizes beyond work performing the energy transfers in and out of the system, but that for now we are going to focus on work (rather than heating or radiating) as a mechanism to transfer energy.
This took an entire day, as I have them draw the LOLs first, then we have a conversation about them. After today I assigned a worksheet on drawing LOLs and writing the qualitative energy conservation equations. This is a modified version of worksheet 3 in the standard modeling curriculum, modified by myself, Kelly O’Shea, and Marc Schrober (in reverse order?).
I’m hoping to write more about the development process, but overall I found, very anecdotally, that starting energy this way helped students see conservation on a system basis, and they have no problems with the idea that energy can enter or leave a system through working, heating, or radiating. It took a while to differentiate between energy stored in the system as thermal energy versus energy leaving the system through work done by friction, air resistance, or normal force (bouncing ball or other examples), but that’s to be expected no matter how this is done. My regular physics students certainly had trouble with that distinction despite starting ETM ‘traditionally.’ Both classes saw this demonstration (video here) to show that kinetic energy certainly does, often, transfer to thermal energy. The difficultly generally is tracking that energy; is it stored as a change in E_therm in the system, or does it leave via work? It took a while to work through that (pun intended).
I’m going to leave you with this. When I first started learning about Modeling Instruction, I assumed it was all about the labs, such as those outlined so far in this series. I have since learned, however, that though the labs provide a foundation for the concepts being learned, working through those concepts through whiteboarding is as much as important as the paradigm labs. Whiteboarding is where students flesh out the differences between what they think and what science demonstrates as a better truth, and where they hopefully cement their beliefs as those that align with science. Don’t underestimate the full framework of Modeling Instruction as a complete system for helping students through the process of learning like scientists.