# 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:

- The frame count starts at 0 when the ribbon is cut. This makes converting frames to time values much easier for students.
- 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.
- 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 *t *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.

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.