some of my favorite Desmos projects from this semester

Important note: You can click on any graph in this post to go to an interactive version. The interactivity is kind of the whole point, so please do take a few moments to experiment with some of these.

At the start of last summer, I announced that the Desmos graphing calculator had sold me on its usefulness “after just a few minutes of playing around”. Since then, the Desmos team has added a lot more features, without ever sacrificing user-friendliness (which, for those of us using Desmos to teach, is paramount).

During the fall semester, I used Desmos extensively in my calculus classes at Smith College. I made “worksheets” that allowed students to interact with mathematical ideas in an incredibly direct way; I also had fun creating them. Eventually I figured out that Desmos and Google docs could be used together to make more fully developed worksheets. (A brief word about my teaching situation: at Smith, all students have a Google account for their email, and thus all have a school-related Google drive by which documents could be shared. On days we used worksheets, about half of the students would bring in laptops, and they would work in groups of 2–3. At the end of class, or afterwards if they had sections to finish, they would share their work with me so that I could review it.) I’ve shared some of these worksheets over time, but I thought it would be nice to have some of my favorites collected in a single place. For simplicity, I’ve removed a bunch of the “worksheet” structure to these, so that they have become more like demonstrations others can use as they wish. Not all of these were used in class, as sometimes I just had to play around with some ideas.

First, some play. That you could not only define variables but also define functions in a Desmos graph and use them elsewhere came as a revelation to me, as did the fact that you could create sums with a variable number of terms. I first learned this while adding up sine functions à la Fourier sine series, which I wrote about here. After that, I made the blancmange curve, a classic example of a continuous but nowhere-differentiable function:

The ever-responsive and ever-creative Desmos team turned the blancmange curve into a mountain range, with a setting sun and moving train (you’ll definitely want to play with this one):

Onto the calculus demonstrations. Teaching calculus in the fall almost always leads to introducing derivatives near the equinox, around the time that days are getter shorter at their fastest rate. I have in the past just mentioned this as an illustration of the derivative. This fall, however, I had students explore how the changing amount of daylight is affected by time of year, latitude, axial tilt, etc. Here, for instance, is a graph depicting the amount of daylight on each of the year at latitude 35°N:

Here is the amount of daylight each day at latitude 50°S: And here’s what the amount of daylight would be like just a few degrees away from the equator, if Earth had the same axial tilt as Uranus (about 82 degrees):

Optimization takes up a chunk of time after derivatives have been introduced. Several classic problems deal with boxes whose surface area must be minimized, or whose volume must be maximized, under various constraints. I’ve always suspected students have trouble imagining what it means, for instance, to require a box have a square base and a fixed volume. What do the various shapes of such boxes look like? So I made a simple model of an open-topped box whose volume and base side length could be manipulated:

Then integration rolls around, with the requisite Riemann sums. Between the introduction of sigma notation, Δx‘s, and a host of other notation, it’s easy for students to feel like they have no idea what is going on. A picture can clear things up, because the idea is quite simple, but drawing enough pictures to show what it means for Riemann sums to converge can take an incredibly long time. Isn’t it nice that we can just show this now?

(I adapted this from another Riemann sums demonstration, made by Evan R.)

In discussing differential equations, we took a day to look at the logistic model of population growth. I asked on Twitter if anyone had a suggestion for real-world data to base a project on. Lia Santilli came up with a great idea I would never have considered: the number of Starbucks locations open t years after the company started. I had the students create a table using the data available here, then try to match the data as nearly as possible with a logistic curve. Here was my attempt:

What I like about this example is that you can see how difficult it is to distinguish between exponential and logistic growth early on. Right up until the inflection point of the logistic curve, the growth seems exponential, and so naturally the company continues that growth trend for another few years. But as the market reaches saturation, it becomes clear that they’ve overshot the mark, and one year (2009) they actually have to close more locations than they open. After that, the growth is more restrained. I don’t know if the 20,000 locations I built into my model is actually the largest sustainable number for this “population”, but I like the challenges for management highlighted by this analysis.

I also used Desmos a few times in my probability class for illustrations. I wrote about one example here. By using the built-in floor functions and combinations, it’s easy to show what various probability distributions look like, and how they change with the various parameters. For example, here is the probability distribution for the number of times a coin comes up heads in 20 tosses, if it is weighted so that it comes up heads 70% of the time (the vertical dotted line indicates the expected value of 14):

The ability to change parameters also makes it possible to nicely illustrate the Central Limit Theorem. Here, for instance, is a graph showing the standard normal distribution (black), the distribution of an exponential random variable (blue), the sum of ten such random variables (green, heading off the right side of the graph), and a normalization of the sum to have mean 0 and variance 1 (red): You can see the convergence of the sum to a normal distribution beginning.

Finally, I also found Desmos useful for illustrating parts of my research. One of the dynamical systems I’m studying is related to the three-cusped hypocycloid, or “deltoid”, which is traced out by a point marked on the circumference of a circle rolling around the inside of a circle three times the size:

Each point inside the deltoid lies on three tangent segments: Perhaps most exciting for me was when I discovered that all the pedal curves of the deltoid could be easily seen and manipulated. A pedal curve is determined by the orthogonal projections of a fixed point onto the tangent lines of the deltoid:
The pedal curves in the above examples were drawn using explicit parametrizations. They can also be defined implicitly by fourth-degree polynomial equations. Since Desmos recently added the ability to plot implicit curves where both variables appear with degree greater than 2, I thought I’d share another, simpler graph that illustrates this functionality:

So that’s an assortment of things I’ve done with Desmos over the past few months, some big, some small. For teachers planning to use Desmos with their students, I would make the following suggestions:

1. Draw them in with something interactive and manipulable. Teach them early to recognize that different shapes can be given by the same formula simply by changing a few parameters, and to explore the effects that the parameters have.
2. Get them to create their own graphs. In the past, we had to do all the work to create the worksheets and the models, but now students can be enabled to build their own; when they do, they will benefit from creating, not just responding.
3. Give them questions that require thoughtful use of the technology they have; simply having access is not a panacea. For example, real-world problems often have models that call for very different scales on the vertical and horizontal axes. Students can be tempted just to use the zoom buttons, causing them to miss important details. Make sure they know they have to think about the graphs they’re creating, not just rely on the computer to show them everything, because it won’t.

For everyone, I encourage widespread use of Desmos and similar tools for education, illustration, research, and entertainment. The Desmos development team deserves an immense amount of thanks for providing us with such graphing and computational power.

Added 12/17– Check out these other graphs for calculus, made by Patrick Honner: http://mrhonner.com/desmos