# Brainstorming a Special Relativity Activity

My AP Physics C: E&M class finished all the content we needed to get through with a few weeks to spare, so I decided (amidst their great excitement) to do a unit on relative motion and special relativity. Thankfully, the physics education twitter community is awesome, and I’ve been rescued from what would otherwise be certain doom with this unit, especially given that I had almost as much to learn about it as my students. Special thanks to @LCTTA and @kellyoshea for equipping me with a life preserver before I dove out into such choppy waters.

Through my various readings and video watchings, I came across this fantastic video by MinutePhysics. Take a moment to watch it before reading further; it’s only a 2 minute video, and it’ll save me a lot of typing. The inspiration for this activity came from what he call’s the switch-a-roo, in particular when “rotates” the slices vs. just sliding them (at about 1:03).

# Beginnings of an activity

My idea is really nothing more than making strips of paper that students can slide and rotate much like the “slices” shown in the MinutePhysics video. This will be given to them after a week of basic relative motion, so frames of reference and relative velocity are part of their vocabulary now. This will also go right after an introduction to the Michelson-Morley experiment and their “discovery” that the speed of light is constant in all frames of reference.

There will be two objects in motion relative to one another along with a photon. Both objects and the photon start at the same position. One object remains stationary, the other moves away with a constant velocity as does the photon in the same direction. Here’s how I’m imagining this might go:

Change the frame of reference such that the other object is at rest.

From this:To this:How does this violate the rules of special relativity? I would be looking for them to notice what’s in green above. I’m also curious as to what discussion the jaggedness of the photon’s line (look closely at the second picture) might generate.

How can you perform a switch-a-roo that doesn’t violate the rules of special relativity? I’m a little unsure how to guide them towards a condition for …doesn’t violate the rules of special relativity beyond that they have to keep the photon line “unbroken.” That makes it work out right, but I’d like it a little more grounded in physics. However, even if I can’t figure that out, it at least allows them to see the creation of a new time axis.

What do you think lines parallel to the bottom of each strip mean? It’s a new time axis! This will rely on their previous discussions about spacetime diagrams (which we’ve done for only non-relativistic scenarios in preparation for this part) about lines parallel to the time axis indicate constant position and vice-versa.

Does time in the new frame tick at the same rate as the first frame? Justify you claim with evidence and reasoning. This one’ll be a zinger! Time dilation!

Some other follow-up questions that I’ve yet to think of. Suggestions? Does the new frame have the same position axis as the old one? If not, what would a new one look like? See the next section on why I might ask this as I thought of it while typing it out. However, upon further reflection, I’m not even sure this can be done? Even if not through this particular example, it’s still a great discussion to have.

Some other follow-up questions that I’ve yet to think of. Suggestions?

# Ponderings

What I like: I think the most effective (and coolest) thing about this is that it gives students some kind of visual for how to think about time dilation and why the constancy of the speed of light necessitates such an effect. Linking the effects of special relativity to one of the key postulates in relativity, the speed of light in vacuum is constant in all frames of reference, was something that I never really understood until recently, and I want my students to appreciate that link. I think it also helps them appreciate the genius of the theory in that everything works out just fine if you just abandon the idea of absolute simultaneity. It sounds so simple when you say it like that, but it’s really rather profound!

It also sets them up for utilizing spacetime diagrams for relativistic scenarios, especially if we want to get into drawing the new time and position axes for a moving frame on top of those for a stationary frame, as shown here:

What I don’t like: I begin the activity by them seeing that applying a regular ‘ol switch-a-roo (i.e., the Galilean transformation) makes light travels a shorter distance in the same amount of time, thus violating special relativity. And that’s a great starting place, I think. But there’s not really a way to circle back to that to verify that the new switch-a-roo (i.e., the Lorentz transformation) succeeds where the old one fails. I see how that happens when the new position axis is drawn (like in the SparkNotes picture above), but I’m not sure my students will see that. Perhaps I should just ask them to discuss whether or not a new position axis is needed and, if so, what would it look like? Hmm..

I’ll be giving this to my students in a few days, and I’m hoping for the best!

# A New Found Appreciation

On my seemingly mountainous, yet steadily disintegrating, pile of Really Important Physics Things That I Never Learned in Undergrad (TM) was the importance of choosing a system when analyzing multiple objects in an interaction. Physics is, among other things, a study of how things interact, but when your analysis is only focused on one object, then deliberately choosing a system seems unnecessary. A rope pulling a crate is certainly “multiple objects interacting”, however in problems such as this, nobody actually cares about the rope. For much of the traditional kinematics/dynamics sequence, the choice of a system is largely invisible and often a single object. The car. The crate. The ball.

But this becomes a problem when anything involving a conservation law comes into play. Whether or not some quantity is conserved is completely dependent upon the chosen system. Upon reflection, I imagine the unspoken, implicit choice of “system” becomes something murky like “all the objects mentioned in the problem” or even “the entire universe.” I also suspect that not explicitly defining a system causes issues with students applying conservation laws [citation to be added later, possibly?]. If students internalize “energy conservation” as “energy doesn’t change”, then I can see that causing issues with incorporating work/change in energy into their problem solving process. Same goes with the impulse-momentum theorem and momentum.

This new appreciation for the explicit choice of a system started with a twitter conversation earlier this year in which I expressed my dislike of the phrase “closed system.” Yet until recently, I wasn’t sure how to help my students develop the same appreciation and understanding of what it means to define a system in relation to conservation laws. Until now. I think.

# Introducing Momentum

My students were working through the paradigm lab for the momentum unit from the AMTA Modeling Curriculum. Two frictionless carts, one of which is spring-loaded, “explode” away from one another. Students are tasked with placing them on the track at such a location that the carts reach the edge of the track at the same time. Through varying the mass of one of the carts, students discover the inherent ratio nature of momentum and interactions and come up with the following equation (momentum is defined as mass*velocity after they come up with the top one):

Afterwards, they worked on using momentum bar charts to represent the experiment. I doctored the numbers a little to keep the focus on the concept of thinking proportionally while not being overshadowed by seemingly random decimals and fractions. I asked them to determine the initial (as defined before the carts exploded) total momentum and the final total momentum (as defined as after the carts stopped touching), to which their surprise was zero. Peculiar!

To be perfectly honest, I wasn’t planning what came next until the night before I did it, which is when I remembered the twitter conversation from earlier this year. As it turns out, asking students to sum up the momenta of the carts came in handy later on.

# Taking a Second Look

My general strategy for students discovering foundational laws and principles is to provide them both with an experiment and a way to view the results in such a way that whatever it is I’m wanting them to discover screams at them with a megaphone “HEY! LISTEN! HEY!” until they can’t stand it anymore. In this case, I’m wanting them to discover the Law of Conservation of Momentum while simultaneously approaching the Impulse-Momentum Theorem. Students were prompted at the beginning of the class to organize their notebooks into two columns with three rows each.

I started with analyzing the carts separately because that’s what they’re used to. We’d not formally discussed how to analyze a collection of objects, much less that such a thing was “allowed.” I allowed students to work primarily in their groups without assistance from me, and they required little help as all of the diagrams and analysis was nothing new to them. Once I was certain that all the groups had gotten everything, I put my analysis on the whiteboard.

The diagrams with all the circles on the left are system schemas, which were introduced months before when students were first learning how to draw free-body diagrams. Emphasis was placed on determining whether the momentum of each “system” (i.e., cart) was equal or not equal to zero, along with the final change in momentum from before, to during, to after.

Next up was for students to re-do the analysis, but treating both carts as a single system.

# What’s in the booooxxxx?

What I was most worried about was the level of abstraction required for students to truly understand what it meant to analyze both carts as a single system. We hadn’t discussed center-of-mass, nor had we discussed what it would mean for the center of mass of an object or collection of objects to be located outside of those objects. While center-of-mass would be the most accurate way to articulate the effects due to conservation of momentum, it would require more new concepts and abstraction, which I was sure to be too much for my students to assimilate at once. Instead, I placed a box over both carts.

I emphasized that this sort of analysis is something that we’d done many times before. The carts themselves are made of individual atoms all interacting with each other, yet we never worried about it because it was unnecessary. Treating the carts as a “single” object was really no different, and the visual of the box really seemed to drive the point home.

During the times in which students were moving through each of the before, during, and after steps, I would raise the box, set the carts accordingly, and put it back down. I emphasized that we were only concerned with what happened to the box from the outside. Just like we didn’t care what individual atoms were doing before, we don’t really care about what happens inside the box. Without much help from me, students produced the following analysis in their notebooks. Again, emphasis is placed on determining the total change in momentum from before, during, to after the interaction.

At this point, students are asked to make a claim, and support it with evidence and reasoning. Most groups’ claims were of the following form:

If the forces on a system are balanced, then it’s momentum doesn’t change. If the forces are unbalanced, then it’s momentum does change.

Sounds a lot like a conservation law to me, which is typically stated like this:

In a closed system, the momentum of that system doesn’t change.

I’m with Frank; ditch the phrasing of “open” and “closed” system all together.

# Closing Thoughts

There’s no reason to wait until momentum to introduce the idea of analyzing systems of single and multiple objects. Next year, I certainly won’t wait so long. I imagine the same type of demonstration can be performed and analyzed to get the point across. This will be particularly useful once we get into energy, where the entire concepts of work and power are defined by what’s happening by or on the system and, of course, the link between the system definition and the law of conservation of energy.

This has also helped me realize something about the kind of physics teacher that I am. I am continually fascinated by all the connections I’m discovering that I never found on the first, second, or third time around with all of this content. Now, more than ever, I see physics as an intricately constructed puzzle-that’s-also-a-tower, built from the ground up from a few simple principles and definitions. I want to help my students construct this tower for themselves and see all the beauty in the details that I do. More practically, I think that students being aware of these details helps them achieve a more deep understanding of physics.

# Motivation

Students struggle with both a conceptual understanding and the mathematical application of Kirchoff’s Voltage Law (KVL). The highly abstract nature of potential and electricity creates difficulties in students differentiating between and simply understanding such topics as charge, current, potential difference, and power (1), (2). I have developed a diagram that builds conceptual understanding of potential and potential difference and also aids in generating a system of equations adhering to KVL.

While I’ve seen similar qualitative potential vs. position graphs in several common introductory physics textbooks, none of them incorporated multiple loops within the circuit nor were they explicitly discussed as a tool to use in problem solving. My goal here is to address both of those points and have these diagrams to be as integral to circuit analysis as free-body diagrams are to dynamics.

I’ll be presenting this during a poster session at the 2015 Physics Education Research Conference in College Park, MD. Come and find me during then if you’ll be there!

# The KVL Diagram

At it’s core, the diagram is a qualitative potential vs. position graph that incorporates all loops within the circuit. Electric potential at any point is represented by the height of the line on the diagram. Different loops within the circuit are represented by different branches on the diagram. To generate a diagram, the following guidelines should be followed.

• Draw a horizontal dashed line to represent the ground potential, 0 V.
• Pick a point on the circuit to begin. The battery or other source is best.
• Upward slanted lines represent the rise in potential provided by a source.
• Flat lines represent the wire between elements in the circuit.
• Downward slanted lines represent the drop in potential that charges experience when passing through that element.
• Choose a single loop to follow and complete each loop one at a time.
• After the final element in a circuit is reached in a loop, make sure the line drops to the 0 V dashed line.
• For additional loops, simply extend a horizontal flat line from the first loop at the point at which the paths diverge.

I’ve taken three simple circuits and diagrammed them below to give an idea of how they look. The points (A, B, C, etc) aren’t part of the diagram, but are presented here to help you see how the schematic “translates” to the KVL diagram.

# Limitations

The diagram’s primary advantage is in generating equations according to KVL without knowing anything quantitative about the circuit. The diagram is meant to be a stepping stone towards such an analysis. Given this, without any quantitative information, the relationship between the magnitudes of different potential differences cannot be known. However, since the height of a line clearly represents absolute potential, students could mistakenly conclude that one potential drop is more, less, or equal to another. In the first example, the drop in height of the line is shown to be the same, but without any additional information, it cannot be known that ΔV1 is equal to ΔV2. Therefore, this limitation must be explicitly discussed with students. Additionally, the slope of slanted lines do not yet provide any information about the circuit.

# Student Work Sample

The following picture shows an example of student work using the KVL diagram. This was taken from an AP Physics 1 class during the Spring 2015 semester.

Students worked on the problem below in groups of 2-3. The problem was taken from TIPERs: Sensemaking Tasks for Introductory Physics by  C. J. Hieggelke, Steve Kanim, D. P. Maloney, and  T. L. O’Kuma. Students were asked to rank the magnitude of the potential difference between points M and N. All of the bulbs and batteries in each circuit are identical.

Students were not given any of the typical “rules of thumb” for circuits, such as the potential drops across parallel elements are equal. Through working problems like these, this was done to allow students to discover such relationships on their own as well as to be a soft test of the diagram’s effectiveness in developing conceptual understanding. Anecdotally, students found the diagrams useful, especially in differentiating between current and potential difference

# Invitation to collaborate

This is only the beginning for me with this diagram. Over the next several years, I will be doing some kind of research on the effectiveness of these diagrams. I’m not yet sure what my research questions will be or what such studies will look like, but I know I’ll be doing something! If you’ve got any feedback on the diagrams or want to participate in research with me, then let me know! Find me on twitter (@TRegPhysics) or shoot me an email!

# References

(1) Cohen, R., Eylon, B., and Ganiel, U. (1983), “Potential Difference and Current in Simple Electric Circuits: A Study of Students’ Concepts,” Am. J. Phys. 51, 407.

(2) McDermott, L. C., and Shaffer, P. (1992), (a) “Research as a Guide for Curriculum Development: An Example from Introductory Electricity, Part I: Investigation of Student Understanding,” Am. J. Phys. 60(11), 994.

# Crisis Is the Best Motivation

I was stumped. Absolutely stumped. I’m teaching rotation for the first time and had just finished blazing through rotational kinematics. I wasn’t thinking more than a day or two ahead (still am… thankfully, spring break is next week). So Friday happened, and I had no idea where to go next.

Sunday evening rolled around, and still nothing. I was at the “google iterations of different phrases” stage of desperation. And at some point in my google hole, I was looking at a cook-book lab for torque. And then… huzzah! An idea.

I wasn’t sure at the time if it was going to work, but I didn’t have anything else to work with…

# It’s All About the Ratios

Students quickly notice that ratio nature of this lab. I stressed in the beginning to pick 2-3 combinations of masses that were even multiples of one another: “Picking a few data points where the ratio of masses is easy to see will help you see the pattern. Then, apply that pattern to your data points that aren’t even multiples to see if it still holds up.”

I wasn’t sure about giving them that tip when I did, but I knew that we didn’t have 2-3 days to spend on this. As it turns out, it was the right thing to do. Sure, it takes a little bit of the discovery away, but not in a way that lessens the benefit of an activity that essentially has them inventing torque.

Once most groups had noticed the pattern, I asked each group to write an equation for the pattern they’d discovered that included the force of both hanging masses and the distance of each from the center. Each group was instructed to include that on their results. All of the groups came up with a ratio equation, which later I asked them to “get all the R’s on one side and all the L’s on the other.”

At this point, everyone’s back in their desks ready to take notes. I introduce the concept of torque and talk for a few minutes about it being an analogue to force for rotational dynamics. I also include some snippets from the rant below.

What fascinates me about this lab is that it allows students to go through an incredibly authentic scientific experience because they essentially invent the torque quantity. This activity answers the question Why is torque = force*distance, which is something that I never understood. Because of this whole ratios thing! Nature has decided that if the force*distance on one side of a reference point is the same as the force*distance on the other side of that point, then the object’s angular velocity doesn’t change. The “invented quantity” aspect of this activity lays the entire foundation for rotational dynamics as everything else depends on knowing what a “torque” is.

All in all, this lab is simple and straightforward. No tricks, twists, or surprises. And I’m sure it’s not a new or unique approach in the grand ‘ol world of physics education, though it’s very new to me. But I think that’s a big part of why I like it. If you’ve got suggestions on how to improve it, I’d love to hear ’em!

# No rotation equipment? No problem!

Like many AP Physics 1 teachers, I’ve had to figure out how to teach rotation this year. As I’m still new to the profession, the amount of lead time I have on my lesson plans has shrunk to about 2-3 days by this time of year. Here I was with a few days to plan an entire unit that would ideally start off with a lab that I didn’t have equipment for. Thankfully, there exists a thing called Direct Measurement Videos that have got me covered. I wrote about why they’re awesome earlier in the school year. Later in this post, I’ll talk about how I integrated this with Desmos to get a match made in heaven.

# Introducing Rotational Motion

I first started with a simple video of a rotating disk with some markers on it. I talk briefly about the differences between translational and rotational motion and give them their first task: come up with a way to to measure the position of the dots in as simple a way as possible. Essentially, I asked them to invent the concept of angular position. A few groups proposed using the arc length that the dot traces out, bu once they saw how complicated the math would be they took a different approach.

I was surprised that it only took about 15 minutes for each group to independently settle on using the angle from a chosen reference point to measure position. I did this because it’s critical for students to have a strong conceptual understanding of the fundamentals of rotational motion. If they don’t understand the basic idea that position is measured relative to a reference point, then everything else becomes much more difficult. I wanted to pair that idea with their idea to measure the angle relative to the reference point. Since they’ve been in physics for a semester-and-a-half by now, it went fairly smoothly. Nonetheless, it’s a critical step, and the 30 total minutes they spent on the activity saves them (and me!) a bunch of time later on.

# Integrating Desmos

The next day, I use a different video, a disk accelerated by a falling mass, to be their first foray into exploring rotational kinematics relationships. This video is a good choice for several reasons:

1. The frame count starts at 0 when the ribbon is cut. This makes converting frames to time values much easier for students.
2. It has 3 different points marked on the disk. I had groups investigating different points. They’ll see at the end that the angular acceleration was the same for all the dots, which is surprising to them. It’s also sets a good time to differentiate between angular and translational quantities.
3. The disk only goes through about a 1/4 turn. It helps with the data turning out very clean as frictional, air resistance, etc. effects are minimal.

Students generate an angular position vs. time graph from their data. Now, normally, I’d have them open up a blank instance of Desmos, input their data, and then find the parent function that fits their data. This time, however, I used a template that I adapted from one of Desmos’ stock graphs, Calculus: Tangent Line. What I wanted was for students to first find the equation for angular position, and then use the slopes of the tangent lines to generate an angular velocity vs. time data table from which they could make another graph. This can be done by hand, of course, but we definitely don’t have time for that. I slightly modified the stock graph by adding a data table, changing some variable names, and adding some instructions specific to the activity. Once they’ve figured out the parent function (SPOILER: it’s quadratic.), they can use the slider to pick a point on the parabola through which a tangent line is drawn. The t point gives the time value, and the slope of that line gives the angular velocity at that time.

This screenshot doesn’t show any actual data points, just the parent function and tangent line.

Once they have an angular velocity vs. time graph, which I have them do on a new blank Desmos graph since they don’t need the template for a linear relationship, they find the equation of the line, find the slope, and start figuring out what it all means.

What I really like about the template (and Desmos in general) is that it allows students to play around with the different sliders and explore how the slope of the tangent line changes as it moves back-and-forth on the parabola. It gives a very convincing visual showing why this specific curve yields an increasing velocity; they can see the tangent line getting steeper and steeper right there in front of them! It also saves a lot of time. I had them do this by hand when we did translational kinematics at the beginning of the year, and it took two full 50 minute class periods. And, even then, I’m certain that many of my students weren’t fully understanding exactly what they were doing as they were so focused on carefully drawing lines, estimating points to calculate slopes, and all the other mechanics of doing this manually.

Desmos takes all that away and lets them only have to mess with the important stuff. For this activity, I had students using a set of iPads that my school has. This can just as easily be done on desktop computers, laptops, or even students’ phones. Here’s the template that I gave to my students. If you create an account and save it, you can modify it for your own purposes.

# Thoughts on Desmos

Ever since I found Desmos at the end of last school year, I’ve been finding more and more ways to integrate it into my classes. For example, at the beginning of our simple harmonic motion unit, students discovered that the position of a mass-on-a-spring follows the cos(x) function. After having been introduced to some vocabulary, I gave them this warm up the next day: I’ve also noticed throughout this year that students think that anything (and I mean anything) that’s not linear is automatically “exponential”. The joke ends up being on them, however, as nothing we graph ever ends up being truly exponential. This wasn’t a misconception I thought needed to be tackled, but we’ve knocked that one out nonetheless! When trying to find functions that fit the data, being able to effortlessly zoom in and out as well as change the range of the axes takes care of another problem I’d noticed when doing this by hand at the beginning of the year.

I’ve battled and battled, mostly unsuccessfully, to have students choose axis ranges that spread the data out as much as possible on graph paper. Because if they crowd the data in one corner, then it’s probably going to look linear. Not a problem in Desmos once I show them how to rescale and zoom. I had an additional idea while writing this post. I’ll make it happen eventually, or maybe you will and you can save us all the work! I’m imagining a “worksheet” utilizing the sliding tangent line idea. I could give them a set of premade position vs. time graphs of accelerated motion. They could slide the tangent line, observe the changes in slope, and use that to translate the position graph to a velocity vs. time graph. I could even use curly braces to piece together a section of motion that’s accelerated and then a section that’s constant velocity like this: y=x^2 {0 < x < 4} and y = 16 { x > 16}.

If this is your first time seeing Desmos, and your reaction is anything like mine… THE POSSIBILITIES ARE ENDLESS…! then my suggestion is to not look around for premade activities to suit your needs, but to instead just spend some time playing around with it. Look for every excuse you can to do something in Desmos. Once you’ve figured the basics out, look at the premade ones for more ideas. Eventually, ideas will just start coming to you! Above all else, Desmos is a sandbox, and the more you understand all the nuts and bolts, the better suited you are for bending it to suit your needs.

# Setting the Stage

I introduce the impulse-momentum theorem immediately after a lab introducing momentum. At this point, students have a general idea on what momentum is, including that it’s defined as the product of mass and velocity. The introduction lab helped them see the ratio-centric nature of momentum and interactions. All that said, the general understanding of momentum is still fairly basic and developing.

I start with a simple question that essentially has students state the impulse-momentum theorem conceptually.

I immediately get responses such as change the mass! and change the velocity!. It only takes a little digging to get them to expand upon change the velocity to include that an unbalanced force must be present on the object. Bam, impulse-momentum theorem:

To change the momentum of an object, the forces on that object must be unbalanced.

Then, we move to deriving it. I’ve been including more derivations lately, but I’ve been careful to only do it when I feel that the students can do it themselves (with a little poking in the right direction by myself). My only direction here was this:

So, we said above that a net force is required to change something’s momentum. But the only equation we have that includes net force says nothing about momentum. That doesn’t sit well with me. Let’s see if we can get momentum in there. Be sure to start from  $F_{net} = ma$

The “ideal” derivation I’d imagined looks something like this:

As it turns out, only a couple of students actually did it along the lines of the way I did it. Which is actually fine! There’s no single “right” way to go through stuff like this.What’s important is that most kids were able to figure out how to get a in there somewhere. What most students did was this:

And I was quite satisfied! My response fort when students asked the oft repeated physics class refrain Mr. Register, is this right?, was Almost! Watch me go through it and you’ll see… When I went through my version of the derivation, they seem to see the difference. What’s important is that they’ve gone through a pretty important physics derivation (mostly) on their own.

Ok, let’s move that delta-t over to the left side… and we have the impulse-momentum theorem!

# Enter the Mythbusters

Huzzah! We have an equation! It’s got a fancy sounding name! Let’s start plugging in numbers!

Nope.

I’m trying to put more focus on understanding equations conceptually, hence the full-stop on jumping straight to plug-n-chug. To accomplish that along with really grabbing my students’ attention, I use this clip from a Mythbusters episode.

I stop it before they actually crash the cars together and poll the room, which instantly explodes in conversation and wild theorizing. I have a few students share their thoughts, throw out a few Hmm…‘s and What do you mean by that?‘s, a few inquisitive looks here and there. A head tilt or two. At this point, students are biting at the bit to know the answer.

But finishing the video would, in my opinion, flop the entire lesson. While about half my students guessed the right answer, none of them had a fully well thought out explanation supported by even (in my opinion) a reasonable amount of evidence to support their claims. Essentially, they were guessing. Which is a fine starting point, but a poor ending one (see The Most Important Thing (TM) on this post for more of my thoughts on this). I’m also trying to develop my students’ scientific argument writing skills a la’ the Claim-Evidence-Reasoning framework.

I want my students to be able to explain, in detail, exactly why the answer is what it is.

# Analysis

The analysis I have my students do is aided greatly by momentum bar-charts, which they were briefly exposed to during the lab and notes introducing momentum. I also provide some data.

I chose to use tons and not pounds or kilograms because of the numbers involved. Using either of those gives momenta in the thousands, especially if the mi/hr is converted to m/s. Using tons and mi/hr keeps the numbers low. Students work through all the bars, and I go through with them to make sure they have a solid set of data from which they can later draw their conclusions. I give them the additional piece of information:

The time for every collision is the same.

The goal is for students to see that the change in momentum for the cars hitting each other is the same as the 50 mi/hr car hitting a wall. If the change in momentum is the same and the collision time, delta-t, is the same, then the net force on each car must be the same as the 50 mi/hr car hitting a wall.

One common thing I noticed is that some groups wanted to add the impulses from each of the 50 mi/hr cars hitting each other together to get 150. Once I pointed out that the impulse for each car was +75 and -75, which added to zero, they’d go back to the drawing board. Others ignored me completely and stuck to their guns. As it turns out, the total momentum being equal to zero is important, but that’s for another day.

Another thing I noticed and also found fascinating is that several of my students across different classes mentioned something their Driver’s Ed classes say. Apparently, they’re told that two cars getting hit head on at 50 is like hitting a wall at 100. I responded by asking the students if they thought their driver’s ed teachers were physicists or knew anything about physics.

Yeah, I guess?.

So, it’s true because somebody you perceive to be an authority told you so?

Yes. Uhh, no. Maybe?.

That was usually enough to get them to keep digging with the data they had.

Up until now, students were doing things in their notebooks. This was enough to fill a 50 minute class period. Day 2 consisted of them putting their arguments on whiteboards and debating with each other in small groups.

# Whiteboarding

I had students form pairs and each put their arguments in Claim-Evidence-Reasoning format. Then, I shuffled all the groups around the room so that each pair was presenting to another pair that doesn’t normally sit at their table. Each group was to present their argument and (heavily) encouraged to modify their own boards if they changed their mind as a result of seeing another group’s presentation. Or add more evidence to support their own arguments if both groups agreed.

This here was the highlight of my week.

Students explaining and defending their ideas to each other is so beyond more effective than me “sage on the stage”-ing it. And way more fun for them. Many groups got (playfully) passionate about their arguments. They were already biting at the bit for the answer at the end of the previous day. They were downright starving for it today.

Once I was satisfied that enough discussion had happened, I played the rest of the video. I then I went through the explanation myself.

I will absolutely be doing more stuff like this again. One thing I need to figure out is how to get groups to more readily consider the evidence and arguments of other groups. While the discussions were great, very few groups actually made any changes to their boards, especially if the two groups disagreed. While I enjoyed watching them defend their arguments, nobody seemed ready to budge. I’m not sure if this is even a problem, just unexpected.

This activity also ended up being a textbook example of confirmation bias. Students would make a claim, and then do whatever they could to justify that claim. Some would ignore the +75 and -75 and add them to 150 anyway because the 100 mi/hr car hitting a wall had a change in momentum of 150. They’d do this even after I pointed it out. When I asked them to justify adding the numbers in the first place, the response was simply that the cars were hitting each other. Which, as we know, isn’t wrong per se, but it doesn’t necessarily help answer the question. When I’d ask how net force fits into that, I couldn’t get a satisfying answer… but students would continue to dig in their heels. This is basic human nature, of course, but it was interesting to be confronted with it so blatantly. I gave a speech emphasizing to draw conclusions from data, not draw conclusion and then hunt to find a way to justify it. But that alone isn’t going to be nearly enough.

One thing I’ll do differently next time is allow one group that got it right to explain what happened instead of me. I was just pressed for time this time, but that shouldn’t be an issue next time as I’ve got a much better idea of how to pace things now.

# A Revelation!

Just before the holiday break I had a bit of a revelation. I’d heard of this “Claim, Evidence, Reasoning” thing (resources on this at the end of this post) as a framework for writing scientific arguments, and I was hooked from the start. This was it! This is how I get my students to stop staring at their papers blankly when I say “justify your answer.” Or flail about including as many fancy-sounding vocabulary words as possible in a desperate attempt to get something, anything right. I dug through a ton of resources, read a few papers, and I was set!

Before you go further, if you’re not at all familiar with what I’m talking about, take some time to read up. A quick Google search will turn up lots of great stuff. This example is an excellent start.

My original vision for using the CER framework was in writing good conclusions to lab reports. And it’s a fantastic tool for that! But that’s only one way to use it. It was especially problematic for me because I wanted to offer my students opportunities to practice, but that would require doing a lab and having them write a paragraph-length conclusion. And then I’d have to grade them. I did this for one lab. Some kids got it, most didn’t, and I didn’t have the energy to try again. Eventually, it dawned on me what I was doing wrong.

Additionally, I think that CER is the way to go about preparing AP Physics 1 and 2 students for the new writing-focused section of the Free-Response questions.

CER isn’t just for elaborate conclusion writing. It’s for anything that ends with “justify your answer” or “explain your reasoning” or some other clever iteration of the same thing we always want students to do: tell me why.

# Introducing CER

Disclaimer: the idea presented below was shamelessly ripped from Eric Brunsell‘s Edutopia article Designing Science Inquiry: Claim + Evidence + Reasoning = Explanation. You can find the slides I used here, and here is the graphic organizer that I use. The organizer is a PowerPoint slide, but it prints like a regular sheet of paper.

I start off with some definitions that I pieced together through a few minutes of Googling. I’m not sure how useful copying definitions is for the students, but it feels like the right thing to start with. And it probably doesn’t hurt, so why not?

I spend a few minutes talking about each one, mostly to fill the silence while they dutifully copy. I answer “what’s olfactory?” a few times, too. Next, the video:

First off, the little girl is adorable. And the commercial is rather amusing, which helps with engaging students. Most importantly, as odd as it may sound, it’s very straightforward. She states that she has evidence and then lists off what that evidence is. Very little room for interpretation here. This is nice because it allows the discussion to focus on what I feel is actually the most difficult part of CER: the reasoning. And this is where student creativity can shine.

# Here Comes the Hard Part

Why is the reasoning difficult? Because it is generated solely by the student. It requires the student to connect the dots from evidence to claim. The best example I saw of this creativity was from this piece of evidence that the little girl provides:

He says he’s from Albuquerque? I’m not buying it.

My handout only has spaces for 4 pieces of evidence, but she presents 5. The Albuquerque one was the one that was left off the most often. To be perfectly honest, I didn’t consider it a viable piece of evidence until my students chimed in. I love it when I’m wrong! Here are some responses:

• It’s a very strange sounding name for a city. She thought he made it up.
• She’s never heard of this place before, so it couldn’t exist.
• He slipped and spoke some of his “weird language.”

My favorite bit of reasoning was in response to the He speaks a weird language piece of evidence: no other human language requires water to be spoken.

Making a claim is easy. Students do this all the time. Finding evidence to support it can also be easy. Just throw some numbers and vocab words out there and see what sticks. The true test of understanding is if the student can link it together with reasoning. Another reason why this video is a great start to the topic is because coming up with reasoning for the evidence presented is still fairly simple. It illustrates the idea to the students without imposing too much of a cognitive demand.

The reasoning is, in my opinion, the most critical part of the CER structure. It’s the glue that holds it all together. And, despite the relative ease of the introductory example, I maintain that it’s the most difficult part.

# And Now, a Physics Example

I formally introduced CER in the middle of my energy unit, so I used this as a follow-up example.

In both my AP and College-Prep classes, students quickly and easily identified that the car slowed down (claim) because it lost kinetic energy (evidence). However, I was awash with blank faces when I asked “So, class, who wants to share their reasoning?”

*crickets chirping*

It’s actually quite difficult for a student in an intro physics class to answer the seemingly simple question of “Why does an object slow down if it loses kinetic energy?”

Many students said “well, it’s going uphill (a claim unto itself) because it’s losing gravitational potential energy (evidence).” To which my response was “How do you know the driver doesn’t have their foot on the gas pedal? You can speed up and go up a hill at the same time, ya know…”

I was satisfied with one of two responses:

• Kinetic energy is the energy of motion. If kinetic energy decreases, then the motion decreases. – I was less a fan of this because of how vague “motion decreases” is.
• The equation for kinetic energy is (1/2)mv^2. If you plug in a bigger number for v, then the kinetic energy increases. If you plug in a smaller number for v, then kinetic energy goes down. – I like this response much better because it references an equation in a conceptual way.

To help students see the connection to the equation, I wrote it up on the board and asked a few leading questions.

• Which of these variables are constant? Which changes?
• Think about how each variable “interacts” mathematically. Are they added, multiplied, divided by, exponentiated, something else?
• Imagine you are making up numbers to plug into each of these variables. Now imagine plugging in a larger number, think about what that means physically, and figure out how that affects the kinetic energy. Do the same for a smaller number.

This was also the first time that I’d asked students to think about equations conceptually in this way (which I need to do more of!), so they struggled for a bit. When I asked them to do the same for question 2, it went much more smoothly.

# What’s Next?

While the introduction to CER went exceptionally well, my students still need lots of practice. I’m going to spend the next week or so doing warm ups that are exclusively this. One way to scaffold this is to simply ask them to write C: E: R: on three lines on their paper, and to fill in the blanks.

Another great candidate for using CER is the nTIPERS book. It’s swimming in problems begging to have CER slapped on them.

This also ties into a new framework that my colleague came up with for writing learning goals. I’m dividing them up by qualitative, quantitative, graph/data interpretation, and lab.

Who knows if this will work out at all as I haven’t tried any of it yet, but it’s nice that the CER thing seems to work well with it.

# Update

I’m loving how this is turning out. My students are rockin’ it so far!