# Scaffolding the Uniform Acceleration Paradigm Lab

Since being told about the Hudl Technique slow motion video app (free in Android/iOS), an entire world of new labs opened up for my students. Hand-controlled timers are just too inaccurate for most experiments, but a slow motion video that also shows real time to the hundredth of a second is more than adaquate for almost any experiment in an introductory physics class. Shown below is one group setting up their experiment to be recorded by the app.

## Past Failures

My college-prep physics class (non-AP, trig-based, but more conceptual than not) began this lab today. Since the goal of this lab is to have them model the relationship between velocity and time, the follow up to generating the position vs. time graph for the tennis ball is then calculating the velocity of that ball at different times.

I’ve tried having them do this in several different ways over the years, all of which that have been some shade of failure. Most of my attempts have revolved around having them draw tangent lines at different points of the curve (a curve which they hand-drew), calculate the slopes of those lines, and then using that to generate the velocity data. The problem there is that having them hand-draw a curve then hand-draw tangent lines makes the uncertainty in the velocity data skyrocket thus destroying any real linearity with the data.

Additionally, there was just too much for my students to grasp all at once. Even the students that got “good” data still didn’t really understand what they were doing.

I’ve also tried having them use a motion detector to generate velocity data, which they then pick out and graph. But I didn’t like that either as it felt too much like magic hand-waving.

So with this lab approaching, I was dreading how all of it would go again. A lot of confusion, poor data, and blank stares.

## Using Desmos

One of the primary issues I’ve discovered with the tangent line method (accuracy issues aside) is that it skips a step. A tangent line is the point at which two points on a curve converge (i.e., calculus), and understanding that the definition for instantaneous velocity comes from squeezing two points infinitely close is essential for understanding how the operational definition of velocity (slope of a position vs. time graph) is generalized to non-linear position functions.

Once I realized this, my goal became to figure out a way to take the above into account without a mountain of cognitive overload. Usually this means turning to Desmos and, as expected, Desmos continues to be one of my favorite tools for the physics classroom. Here’s the tool I made.

Students first enter their position vs. time data in the table to the left…

…which then desmos uses to generate the function that best fits the data (all of which is in the WIZARDRY folder):

The back-end of this tool generates a fit for any function $ax^n$.

## Understanding Tangent Lines

Once students have entered their data, I then have them talk in their groups about how they can calculate the velocity of the ball. When the topic of “the” slope comes up, I then point out that there is no “the” slope because the line gets steeper and steeper… which makes sense because the ball’s velocity gets greater and greater.

But then how do we calculate the velocity of the ball at any particular point? What points do we choose?

After a few more minutes of them discussing this amongst themselves, I allow groups to throw out suggestions to the class. Often times they won’t come to the “choose two points very close together” conclusion on their own, but one group will say “choose a point before” and “choose a point after.” From there, I lead that into “but how close? should the points be?” Or, an even better question: “But why do they need to be close?”

At this point I put Desmos up on the projector and show them two things:

Note: the tangent line is turned off by default as I think it can confuse things when they’re making their calculations later on. To turn it on, go into the Wizardry folder and click the empty circle next to the very last equation

I point out that the orange and black lines deviate quite a bit for almost all of the graph. However, if you zoom in…

…they overlap perfectly. This is the justifcation for “pick two points really close together.” To wrap up, I ask them to write a few sentences that explain and justify the process of calculating the velocity at a particular point in time.

This also just-so-miraculously-happens (read: completely and totally on purpose) sets them up for calculus later on should they take it.

## Generating Velocity Data

At this point, students are ready to start generating their velocity data. I tell students to drag the blue dot (the red follows along) such that they center each of their data points. See the two screenshots above for what that looks like for data points not on/close to the curve. The slope of the line between the red and blue points is the velocity at he x-coordinate of that data point (1 s for the point above). Students generate around 10 velocity-time data points, and then move to graphing their new data.

Because the velocity data is generated from a perfect curve, the velocity data is almost perfectly linear, thus making the further analysis much easier.

If you have any questions about how all of this works or suggestions on how to improve, please let me know!

# Replacing Textbook Problems with Lab Experiences

This was published in the Talkin’ Physics column of The Physics Teacher’s October 2017 issue.

End-of-the-chapter textbook problems are often the bread-and-butter of any traditional physics classroom.  However, research strongly suggests that students be given the opportunity to apply their knowledge in multiple contexts as well as be provided with opportunities to do the process of science through laboratory experiences (Mestre, 2001). Little correlation has been shown linking the number of textbook problems solved with conceptual understanding of topics in mechanics (Kim & Pak, 2002). Furthermore, textbook problems as the primary source of practice for students robs them of the joy and productive struggle of learning how to think like an experimental physicist. Methods such as Modeling Instruction tackle this problem head-on by starting each instructional unit with an inquiry-based lab aimed at establishing the important concepts and equations for the unit, and this article will discuss ideas and experiences for how to carry that philosophy throughout a unit.

# Practicums, practicums, practicums!

Designing the right kind of lab experience is more than simply having students make calculations based on making real measurements. What makes a lab experience a practicum[1] is that the calculation students make is verifiable. An experiment should be able to be carried out that undeniably shows, either through visual confirmation or an additional measurement, that their prediction is accurate or not. This allows nature to be the arbiter of the quality and accuracy of their work instead of the teacher, thus placing students in an environment far more representative of how real science actually works.

Through lab practicums, students also get to see the importance of measurement uncertainty and significant figures. If in a multi-step calculation a student rounds their results in between each calculation, their final prediction is likely to greatly deviate from reality regardless of the quality of their solution or initial measurements. The same applies for poor measurement techniques in that an accurate analytic solution is worthless for predictions if the measurements are taken poorly. Practicums also emphasize that equations are not black boxes that take in numbers and magically eject answers, but instead are mathematical models that provide predictions (accurate or otherwise) for how the world works.

Lab practicums can be as simple or as complicated as they need to be for the desired level of rigor or available time. After students have begun tackling the concepts of constant velocity, instead of assigning them problems of the A car drives at a constant velocity of…variety, why not instead give them a constant-velocity tumble buggy and have them make predictions about its motion ? Have them calculate the velocity and then place a piece of tape on the floor where they predict the buggy will be after some amount of time determined by the teacher. If more pressed for time, calculate the velocity beforehand and provide that to students. For an added challenge, provide students two buggies with different velocities and ask them to place tape on the floor where the two will collide.

For calculus-based classes, have students use a large rubber band to launch a wooden block across the floor and predict where on the floor the block will skid to a stop. They will need to model the non-linear force-position function for the rubber band to figure out the initial amount of elastic potential energy that is then transferred to kinetic energy in the block and thermal energy due to the friction between the block and the floor.

Accurate time measurements for experiments like this are often difficult to achieve. However, a free mobile app called Hudl Technique, on both Android and iOS, takes not only slow-motion video, but also provides accurate time measurements down to the hundredth of a second. This opens the door for experiments for virtually all topics within mechanics at whatever level of challenge desired.

For statics problems hang objects from multiple spring scales all pulling on an object from different angles. Give students the mass of the object and have them predict a reading on a scale or vice-versa.

For energy, use two spring scales with different k-values that are tied to low-friction carts along a string. Task students with figuring out how far to stretch each spring such that the carts have the same velocity (O’Shea, 2012).

Once thinking has been shifted towards lab practicums, an entire new world of practicing physics in the literal sense is opened up. But even in the face of time and equipment limitations, there are Direct Measurement Videos[2] (DMV). The Science Education Resource Center (SERC) at Carlton College has a series of high-quality slow motion videos of different experiments relevant for a variety of physics topics, primarily mechanics. What sets these apart is that students can use the DMV web player to advance movies frame-by-frame as well as use screen overlays to take time and position measurements directly from the videos. They work seamlessly on laptops, desktops, and mobile devices.

What ties all of this together is the attempt to provide students with as authentic a scientific experience as possible. Real scientists solve new problems every day without an authority figure, other than nature itself, to appeal to when things get tough. They must deal with all kinds of problems associated with accurate data collection, reliable experimental setups, and finicky equipment. The end product of an experiment is rarely what it started out as, and allowing students a variety of opportunities to experience that process, to do science, will provide them with the kind of problem solving skills that physics teachers claim as the benefits of studying physics.

# References

Kim, E., & Pak, S.-J. (2002). Students do not overcome conceptual difficulties after solving 1000 traditional problems. American Journal of Physics, 70(6), 759.

Mestre, J. P. (2001). Implications of research on learning for the education of prospective science and physics teachers. Physics Education, 36(1), 44.

O’Shea, K. (2012, February 5). Building the Energy Transfer Model. Retrieved from Physics! Blog!: https://kellyoshea.blog/2012/02/05/building-the-energy-transfer-model/

[1] Practicums for Physics Teachers: http://www.physicsforce.com/class-practicums/

[2] Direct Measurement Videos http://serc.carleton.edu/dmvideos/index.html

# 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.”

# Comments on Torque

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.