I’ve been meaning for some time to compile some items that I’ve developed over the years, since the courses I teach don’t lend themselves to using the straight up Modeling Instruction curriculum. I develop items mostly to fill gaps I see in understanding or content, depending on need. I wanted to be able to share this work with the larger physics community.

That said, I do need to issue a disclaimer; I think it is possible, likely even, that some of the materials I hope to post came out of conversations with others, or even after seeing someone else’s materials. If I’ve inadvertently stolen anyone’s work, please let me know and I will remove it immediately. To the best of my memory, however, the materials are original.

This first item arises out of a pretty standard physics problem; analyzing an elevator that is accelerating. I wanted a worksheet, however, that emphasized the similarity between speeding up while moving upward and slowing down while moving downward; likewise, slowing down while moving upward and speeding up while moving downward. I wanted to be able to point out that the common feature in these situations is the direction of the acceleration, and thus the direction of the net force. I also wanted to re-emphasize from our work with Constant Acceleration that a negative acceleration does not necessarily mean slowing down. Finally, I wanted to give students some easier situations before we moved on to ramps and angles.

I use this worksheet with my first year physics course that is similar to AP Physics 1 (it’s concurrent enrollment through the U of MN), and it’s the first thing we do after we build the model by pulling carts with spring scales (kinda like what Kelly does but with 1N spring scales). I also started last year doing it after model building with my ‘regular’ physics class, which is oddly between a standard HS physics course and a conceptual physics course.

I took the liberty of applying a creative commons license to the work, so it may be shared and/or adapted with attribution for noncommercial purposes under CC BY-NC 4.0. This is mostly so that it couldn’t be used for commercial purposes; feel free to use and change (and tell me about it)!

I’ve been looking for a better way to start circular motion for quite some time. Though many people use the spinning stopper lab, I found it difficult to get quality data or even a decent trend, even when I did the experiment. I tried using a pendulum setup, but I wasn’t happy with the hand-waving about non-uniform circular motion. I really like this particular treatment because it first focuses on the conceptual aspect of centripetal acceleration as toward the center, and then follows that up with lab quantifying the acceleration. It also gives some really nice opportunities to review and refine some lab techniques like uncertainty propagation and linearizing (both could be dropped or modified for your purposes) that my students will need as we progress through the year.

Day 1

We spend the first day investigating the accelerometers in Labquest 2s so that we understand the direction of the acceleration based on the graphs of x, y, and z acceleration versus time. (Note: certain this could fairly easily be adopted to use Arduinos or student phones; but note that asking students to risk their phones on a spinning apparatus in day 2 is tough). First the students turn on the built-in 3 axis accelerometer and observe that it reads approximately 9.8 m/s/s in the vertical direction, no matter the orientation. Then we watch the video below up to the two minute mark to explain why that is, and to help understand how the accelerometer is collecting data.

Next we do a series of short trials to confirm directions of positive and negative for each axis. We then turn off the z axis accelerometer as we will be working only with x and y.

Students are told they will be spinning in one place with arms outstretched, with very smooth, fast steps, as quickly as possible. They will be starting the data collection while spinning to eliminate the spin up process from data collection. I ask them to graph what they will see; most think it will be sinusoidal. Then students actually do the trial, and sketch what they see. It takes a lot of individual conversations here for them to see that the primary acceleration is in the negative y direction (based on how we hold the Labquests). Spinning faster and smoother helps this, and I have to point it out for a lot of groups, who then confirm with more trials (and more falling over from dizziness). It’s a good time.

Next we have a quick conversation about how the acceleration is negative y, so that means it’s….wait, what? Toward the center? “Hey everyone, go grab a bowling ball and a hammer.” I instruct them to make the ball go in a circle using small taps with the hammer. No, not spin…actually travel in a circle. Then I ask what direction they have to tap it in order for it to go in a circle. “Toward the center.” I go through a quick note about how applying linear taps speeds the ball up or slows it down, and that the net force from the taps is in the same direction as the acceleration. Thus for our circle, we apply a net force inward, and as a result the acceleration is inward. They don’t like this, not one bit.

Now is the right time to talk about turning in cars. I ask them to get in a car with me, and I slam on the gas. What way did the car accelerate? “Forward.” What way did you *feel* like you were moving? “Backward.” We do the same treatment of slamming on breaks, and talk about how really our bodies are just trying to keep on doing whatever they were doing, so we feel like we move the opposite direction as the actual acceleration. Ok, so now we are going to turn left. What direction does it *feel* like you are moving? “Right.” So what direction are you accelerating? “Huh. Left. Towards the center of the circle.”

And that’s enough for today.

Day 2

Now that we have the basic idea about centripetal acceleration it’s time to quantify it. We brainstorm; what factors affect how strongly you feel pushed to the outside of the car? (but are you really being pushed to the outside? No? Good) They come up with speed and radius pretty quickly. This part does have to go pretty fast, as data collection is tough to get done in our 48 minute periods. If there’s time we have a conversation about how investigating the radius and it’s affect on acceleration is tough because speed also depends on the radius. So we settle on changing the speed of rotation and measuring the resulting acceleration. On what you ask? Only the best equipment for my physics students.

To measure v; How far does an object go around this circle? “The circumference, .” Ok, so we’ll call the time it takes to go around once the period T, so the speed is .

Thus we measure the period to calculate the velocity (which we’ll do later). We use the statistics function on the Labquest to measure the mean of the acceleration, only while the acceleration is moderately constant, and use the standard deviation for the uncertainty. We collect data for 20 seconds since we now can’t avoid the spin up. Some very important student instructions;

Only use the linear section of the y acceleration graph

Each trial involves hitting play, starting to spin, maintaining that constant rate of rotation, then starting to count revolutions to measure the period.

It’s hard to actually spin the chair at a constant rate. I’ve seen a variety of techniques, but most groups either reach from above on the back, keeping their hand on the back the whole time (as discrete pushes show up as very obvious waves on the acceleration graph) or spin from below with quick, regular pushes.

This year I had some timing issues so we really only had time to collect data this day; in future years I think we’ll have time to go over the calculations of speed and uncertainty using Google Sheets here as well. I walk them through how to use the period data to automatically calculate speed using Sheets. We also have a conversation about uncertainty in the speed; it’s a propagation of the uncertainty in the radius and the period. So we estimate those uncertainties, then use sheets to calculate the maximum possible speed using the maximum radius and minimum period for any particular trial. It’s really nice to do this now, as we do an experiment later using photogates where we have to similarly propagate the uncertainty.

Day 3

Start today by graphing acceleration vs speed in linreg. In most cases their are two pieces of evidence that the trend isn’t linear; it looks a bit curved (though this depends a lot on the group), and the intercept is usually significantly negative.

As much as possible I have the conversation about these factors with each group, but as there gets to be more of them I toss it on the overhead and we hash it out there. We talk through why the intercept should be zero, and use the combined evidence to try linearizing. Below is a student spreadsheet with a wonderfully linearized graph.

Once the graphs all have linearized graphs, they whiteboard them. There will be a number of groups with data that makes no sense; I think they generally missed one or more of the “Important student instructions” bullets above. We talk about it, and I have them take a look at other groups’ data. The following discussion centers first on the quadratic nature of the data. Either someone does a unit analysis of the slope or I point out how nasty it is (), so we simplify it to 1/m. Eventually someone notices that the smallest radius has the largest slope and vice versa. I ask them to combine the evidence of the units of the slope with the radius–>slope information into a claim about the slope, and we end up with (note that facilitating this discussion is significant, but material for a different post).

I then emphasize the evidence that we’ve used to get to that point; the curve in the acceleration vs speed data and the negative intercept leading us to a quadratic relationship, and the units and radius comparisons leading us to an inverse relationship between acceleration and radius. We finally test it against our original musings; as we go faster around a curve, does it feel stronger? As we decrease the radius, does it feel stronger? It’s good that our equation matches our experiences.

In addition to the reasons stated at the beginning of this post, I love that the kids have a blast doing the lab. Playing with spinning chairs is fun for people of all ages.

As I start my 3rd year of Modeling Instruction, I’m happy to be in a place where I can start tweaking rather than making sweeping changes to my courses. My primary goal this year is to give more help and attention to students who struggle, and one of the ways I plan to do this is to pointedly seek methods for engaging them more during class. My first plan of attach on this concerns board meetings.

If you are not familiar, a “board meeting” is loosely defined as having students form a large circle so that they can observe each group’s whiteboards. I typically use this method of whiteboarding to have groups compare data from the same lab in order to induce aspects of a particular model.

Despite having 25 students in a class, I noticed last year that board meetings tended to be dominated by less than 5 people. I want to try to get all the students involved; I want them all contributing and wrestling with the data. This year I’m going to have board meetings start by giving students 1-2 minutes to simply look around and make at least one observation. I want them to do this silently, individually. I think that sometimes there are students (like me) who are comfortable word-vomiting immediately about what they see, which then overwhelms students who prefer to sit back, take in info, and process before speaking. I want to give that second group time to process. After this time period, I’m going to have them turn to share their observation with the person next to them. Again, I want every single student in the room interacting about the data. After that I think I’ll have them go around the circle to share with the whole group. I thought about letting groups volunteer or cold-calling on groups, but by going around the circle I can step out and simply record their thoughts with minimal guidance and intervention. As groups report in, I think I’ll stick with my observations/claims approach to help students organize the information reported out.

I think throughout the year I will slowly remove the scaffolds like turning to partner or going around the circle in favor of more organic approaches, but I’m thinking I’d keep the 1-2 minutes of process time. I really want to help the processors engage before the vomiters get in their way.

I know this isn’t new in general (yeah, yeah, it’s basically ‘think pair share’), but I think applying the idea specifically to a board meeting has some merit. I’ll report back with how it goes. I’ll also hopefully be posting with other possibilities for getting *all* students engaging in the various aspects of a modeling classroom.

UPDATE: I did this will all my classes and I believe it was very successful. In addition, we had finished collecting data in one class period but didn’t have time to whiteboard it, so I had them put it in their lab notebooks (sketch a graph, record the equation in words, write the slope and intercept with units and uncertainties), and then to write a couple of sentences summarizing what the results meant. When they came back the next day, I had them take 2 minutes to discuss their paragraphs with each other. I like that this both helped them think about the data first, and then also incorporated some writing, which I hope to do more. After discussing their summaries, I had them gather in a circle and do what I described above. I really believe that this process helped get more students directly involved in wrestling with the data than only doing a standard board meeting.

I want to thank Patrick Briggs, who keynoted for our all-district kickoff yesterday, for explicitly pointing out that many students need time to think and prepare before they are willing/able to have an academic conversation.

I only have one standard for CVPM, as I didn’t want to get bogged down with a super granular standard list.

CVPM.1: I can represent a constant velocity problems graphically and algebraically and solve problems using both numeric and algebraic methods.

I start day one of my essentially honors level, first year physics course with the Buggy Lab. (If you’re not familiar with the Buggy Lab, or even if you are, read Kelly’s post about it). This takes 2 full days, sometimes 2.5, with 45 minute periods.

From there I use Practice 1 stolen from Kelly, found in my CVPM Packet, which takes me about a day and a half (of 45 minute periods). Here’s a post about the board meeting to discuss the data.

Next is Practice 2, also stolen from Kelly, though I add that we walk them with motion detectors, 1 day ish. (Update: Whiteboarding took the whole period and I decided that that was more worthwhile than actually walking them with motion detectors, we’ll do more of that in CAPM)

The last worksheet is Practice 3, which I developed to help develop more algebraic problem solving. This is because my class is actually a U of MN class taught at the HS level, and the U emphasizes algebraic problem solving. 2 days. This worksheet went very well, and here are some notes about starting the whiteboarding process with it as well as the ensuing conversation.

After Practice 3 I have two days of difficult problem solving practice. The first is the standard lab practicum where students must cause two buggies of different speeds to head-on crash at a particular location. Here’s a post describing the practicum. The second is a difficult, context rich problem that students work on in groups.

All in all the unit takes me 13-14 days, including the quiz at the end and a day to FCI pretest.

In my college level physics class we study Energy right before momentum. I really like this, particularly because we can begin our study of momentum as driven by the fact that a pattern emerges from data that is not explainable by Energy.

On the first day of my momentum unit I typically do a fun car crash activity to help students start thinking about how force and time are related in collisions. The next day we start building the momentum transfer model. (We’ll come back to force-time relationship at the end of this paradigm series) Last year, not having experience with Modeling Instruction, I just dove right in (chronicled starting with day 1 here). This year I wanted to utilize the discover, build, break cycle that Frank Noschese talked about in his TEDx talk. One of the tenants of modeling is that models are useful for certain cases and not for others. Thus I used an inelastic collision to springboard into momentum based on the fact that an energy analysis is not particularly useful for this situation.

When students walked in I showed them a scenario where a moving cart (A) collides with a stationary cart (B) of equal mass. I asked them to use the Energy Transfer Model (ETM) to predict the final velocity of the carts. A typical analysis looks something like this;

Assuming there is no conversion of energy to thermal energy, the kinetic energy of the first cart should end up as combined kinetic energy for both carts after the collision;

Noting that for this case and , the whole thing simplifies to

Solving for the final velocity of the two carts together in terms of the initial velocity of the first one,

Once we got to here I simply said “Go test it,” and they got to work in the lab.

Before I go on I want to comment on the lack of thermal energy in the above derivation. Many of my students correctly tried to include E_therm in their analysis. This is great, but I pointed out that today was a lab day and thus we need to be able to measure things. Me: “Can we easily measure E_therm?” Student:”Ummmm…no.” “Right, so let’s ignore it and see if the data upholds that assumption.” They almost always (correctly) want to include E_therm in every energy analysis, but we have done a couple situations in the lab where stored gravitational interaction energy transfers to kinetic energy for dynamics carts where assuming no changes in E_therm yielded good data. Thus students were primed for me to suggest that we could ignore E_therm. However, this is tempered with the fact that I do a demonstration showing that kinetic energy transfers to thermal energy in collisions (a couple weeks prior) and that they are used to me guiding towards ‘wrong’ answers. So I believe students went into lab cautiously optimistic that our the lab evidence would support the derived equation.

It doesn’t.

It only takes students 5-10 minutes to realize that the final velocities are closer to half the initial rather than the initial divided by the square root of two. Some of them try to justify the data (well, it seems kind of close to root two…), but after conferring with their classmates they give up and go with two. At that point I pulled them back up to the front of the room.

Me: So, did our equation work? Students: Nope M: But was their a pattern? S: Yep. Final velocity is half the initial. M: Wait, you mean that energy doesn’t predict the final velocity, but something else does? S: Um…..

We had a quick discussion about how something must be going on that is different from energy. We also talked about how it makes sense that energy wouldn’t work; we expect some of the initial kinetic energy to convert to E_therm after the collision.

From here I continued day 1 in pretty much the same way as last year. I found after a 45 minute period students were just about ready to talk about a relationship, just slightly behind where day 1 ended before adding the energy piece. My students are much more used to the idea of paradigm labs this year and are getting pretty good at looking for meaning in lab data, so I am not surprised that this addition didn’t significantly change the day one timeframe. Tomorrow we start with presenting the student derived relationships.

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.

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.

Thus

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).

Concluding Thoughts

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.

I’ve been thinking a lot about the Energy Transfer Model (ETM). The Modeling Instruction curriculum seems to start this model by jumping right into the concept of Energy Transfer without much empirical model building, contrary to many of the earlier models. I really like the way Kelly starts energy, showing students how previous models don’t work to predict the desired outcome. Still, I was unsatisfied in that I felt like I would just be telling students what energy is and how it transfers without letting them get a feel for it for themselves. So I set out to design my own version of the beginning of ETM. I used this version of ETM in my college physics class after starting ETM the standard way in regular physics.

Day 1: Area of Force vs. Position graphs

Day 1 started just as Kelly’s post details above, though she has modified it since posting to use Pasco’s spring cart launcher instead of regular springs. The idea is simple. How can I make the final velocity of these carts the same if they are launched by two different springs? We spent 10 minutes playing with the carts, and I showed them at maximum compression, both at about 8 cm, the carts launch at different speeds. Predictably, the spring with the highest spring constant launches fastest. So how can we make them go the same speed using their Force vs Position graphs?

We (my colleague Ben, with whom I teach the regular class, and I) tested the springs and their constants fell very close to those stated in the documentation, so we used that to make expected F vs. x graphs rather than take real data. It worked just fine.

In all classes I did this (three different sections, one regular and two college), the first guess was to make the force equal for each spring. So we did that. My regular class just looked at the graph, saw that if we wanted a force just over 4 N we could use about 5 cm for the red spring and 3 cm for the blue one. For the college classes I asked them to choose an arbitrary compression for the red spring, then find the blue compression to give the same force.

Either way, it failed miserably.

Turns out that if two different springs are compressed to the same force value, they do in fact have the same average force, and thus the same average acceleration. However, the weaker spring has to be compressed further to get that same force value, and thus the same acceleration happens over a larger distance. The weaker spring actually gives a faster speed when the force each exerts is the same!

They get this. I asked them what would happen if you had two cars that had the same acceleration, but one accelerates for 10 meters and one for 20 meters. The 20 meter one ends up at a faster speed. Yep, that happens here too. The red spring car goes faster because it has the same acceleration on average as the blue spring but for a longer distance.

So anyway, what now? I had to guide them to check area. I did not do as awesome of a job as I would like using the area under velocity vs. time graphs to find displacement, and as a result area of graphs is not a formost thought for them. However, all classes jumped on the idea once I led them there (by referring back to kinematics graphs and the parts of those graphs that do in fact have physical meaning). Most students needed help with the idea that they should pick an arbitrary compression of the weak spring. Once there, however, we worked through the math and found the compression of the blue spring such that its area equaled that of the red spring with our arbitrary compression.

The launch was perfect. In all 3 sections.

Kids really like it when things work, and boy, does this work. It took about one 45 minute class period to get this done, but they definitely got the idea that the area under the F vs x graph meant something. I emphasized, over and over, that area under graphs, if it has a physical meaning, means a change in something. We don’t know, however, what that something is yet.

This is where the classes diverged. The regular class went into a lecture day on types of energy and energy pie charts. But that’s not what I want to write about.

To continue empirically, I wanted them to see that the area under the F vs. x graph (a change in something, as I kept calling it) was meaningful in other situations as well. So next we looked at ramps.

Day 2: Ramps and the Area of F vs. x graphs

On day 2 I told them we were going to again look at the Area of F vs. x graphs, but this time in a different situation. We started with a cart at rest at point A, arbitrary but constant. We wanted to end with the cart at rest at point B up the ramp, also arbitrary and constant. I had them pull carts from A to B in any way they wanted and to find the area under the F vs. x graph. Here’s a sample trial.

I learned some things. First of all, most of them didn’t end the cart at rest at point B at first. But we did, however, use that to establish that the faster the cart was going at B, the larger the area seemed to be. We will go back and quantify this later (part 3 or 4 of this series, I believe). So we went back and got some data for starting and ending at the same points each time, starting and ending at rest, but getting from A to B in different ways. Here’s some sample data.

In discussion it became evident that outliers appeared in one of two general cases; when the cart was difficult to actually stop at point B, and when the cart moved backward at some point. On the whole, it was pretty easy to convince them that the area was the same no matter how you got from A to B as long as the cart didn’t move backward and the cart was at rest again at B. Pretty awesome.

That same day I asked them what measurement would always correlate with the area. Horizontal distance up the ramp? Angle? Height? We were able to quickly show that though distance correlated with area, it didn’t work well if we kept the same distance and changed the angle (we got different areas then). Thus distance is not a universal predictor of the area. How about angle? Similar problem; for one angle you could get infinite areas. How about height? We spent the last minutes of this period showing that if we had an equal change in height, even for two different ramps (same cart of course), that the area was approximately the same. Cool.

Day 3 and 4: Finding the Correlation with Height and the Entrance of Energy

Day 3 was short classes, only 30 minutes because of a pep fest, and I think data collection and whiteboarding could probably be done in one class period. However, the conversation we had about types of energy at the end of day 4 fit really well and it was nice to have that there. But I’m getting ahead of myself.

Day 3, 30 minutes, was spent collecting area vs. change in height data. Some students changed the height just by pulling the cart further up the ramp, and some by changing the angle of the ramp, or a combination of the two. Part of the awesomeness of this lab is that it doesn’t matter; no matter how they change the height, if they collect data consistently and correctly, the results turn out well. (Students won’t, by the way, take data consistently and correctly; I had at least 2 groups in each class with non-sensical data. They don’t set the endpoints of the integral in Loggerpro correctly, or they don’t change the endpoints (thus making the change in height the same for all trials), or they measure change in distance rather than height, or they do one of I’m sure many other things that yield poor results. It’s a learning experience though, and the conversations that come from ‘bad’ data are often just as useful as those that come from ‘good.’)

In any case, the graphs were decently linear. Through a board meeting (circle sharing) groups quickly noticed that the intercept was zero, and that that made sense as if we don’t have any change in height, we shouldn’t have gone anywhere, so the area of F vs x would also be zero. They then noticed that some groups (conveniently with carts of different masses, *cough cough*) had different slopes. At some point someone notices that the slope appears to be approximately 10 times the mass. Hmmm, isn’t g really close to 10? Then we look at units. The slope must have units of Newtons, as y axis has units N*m and the x has units of meters. If the slope was mass times g, then the units would be in Newtons. Hmm. Note: In all this, I try to at ask questions with a couple of words max and let the conversation take its course.

This was convincing enough for my students that the slope should be mg. It was, pretty close, for the groups that had decent data. I then asked them to write a general equation to model our data. Most were able to get here;

where A is the area under the Force vs Position graph, in N*m.

I pointed out that even though this was a different situation than day 1, the area still gave us something meaningful. But seemingly unrelated to speed! We’re getting there. Let’s rearrange the above equation a bit.

which leads to

Here is where I finally defined that the quantity mgh is called Gravitational Interaction (or Potential) Energy. I took a side trip for a bit on energy as pain, as described very well (better than I could) in Kelly’s aforementioned post on building the ETM.

Thus what we have found is that the initial gravitational interaction energy plus a the Area under F vs x (which recall we had emphasized as a change something) gave us the final gravitational interaction energy. So I guess the area is a change in Energy, huh?

Starting with Day 5 we are going to look at how the area correlates with speed, and use that to figure out Kinetic Energy. We will then use that to transition in to Energy Bar Charts and the rest of the energy unit. More on that in later posts (I think 1731 words is enough for now, huh?)

Concluding thoughts, for now.

I really like that this method strongly emphasizes that the energy is changing due to the Work done (though we haven’t used that word yet), and I plan to use it to strengthen both their methods of using graphs and multiple representations to solve problems as well as to help with the idea of Work itself, which when taught traditionally has really only served to confuse my students. I don’t like, however, that for now I am ignoring changes in thermal energy, which the typical intro to ETM in Modeling Instruction emphasizes from the get go. I used to teach energy where we would ignore friction for weeks, then finally add it in and start all over, and didn’t like that. I think, however, that the idea that the F vs x graph influences the transfer of energy will transfer (hehe) to friction as well. We’ll see, and I’ll keep you updated.