Creativity + Desmos = A Rewarding Math Project

A couple of weeks ago, I assigned a project to my Honors Precalculus students that made use of the fantastic online calculator, Desmos :

Honors Precalculus Desmos Project

In this project, you get to combine your mathematical knowledge with your artistic creativity.

Use the Desmos online graphing calculator ( to plot a set of functions that create a picture. You must use at least 25 functions. You may use any type of function we’ve learned so far this year: polynomial, rational, piece-wise, trigonometric. You can add shading by using inequalities.

Save your finished project (include your name in the title), and submit it to me by using the “share graph” button on the top right:

 This project is worth 40 points.

Your masterpiece is due at the beginning of class Monday, November 25. We’ll view everyone’s submissions, and vote on the “Best in Show”. The winner will get a special prize!

The results far exceeded my expectations. The students threw themselves into the task with amazing enthusiasm. They learned all about restricting domains of functions, using inequalities for shading, and transformations. One student even researched how to rotate conic sections, and shared her new knowledge with her classmates.

If you are concerned about spending a lot of time learning a new program, fear not: Desmos is one of the easiest and most intuitive graphers I’ve ever worked with. They provide a brief but excellent user guide that can be downloaded here, as well as lots of video tutorials.

The gallery below contains all of my students’ final submissions, but I have to spotlight a couple students’ masterpieces. In the Beauty and the Beast one, the student used 406 equations to create it, and it is simply spectacular!


And here is a magnificent rendering of the Taj Mahal by another gifted mathematical artist:


Here are the rest of their creations. Clicking on a thumbnail brings up the full-size image. Enjoy!

Update: Desmos featured one of my student’s work on Twitter!


Vectors and Dancing – A Dangerous Mix

I came across this activity from Jim Noble at the other day and tried it out with my precalculus class. We defined four different dance steps with vectors, and combined them into a routine.


We then tried out the routine while listening to Donna Summers’ “Hot Stuff”. As you can see, we could have spent a little more time practicing!

I liked this activity, because it got girls up and moving. We had a good discussion of the various ways to combine the steps to reach the same destination, and they mapped them out using the vector definitions. Jim Noble has made available some excellent worksheets that go along with this project.

If you’re trying to teach simple vector addition and scalar multiplication give this activity a try!

A Simple Way To Teach Polar Coordinates

The courtyard outside my building is laid in a circular pattern, which makes it a perfect setting to teach polar coordinates: Sr Patio   I took my Precalculus class outside and stood on the polar axis. I assigned an r and theta-value to each student. They then had to start at the pole and figure out where their given polar coordinates would position them. It was a great way to incorporate physical movement with math. They even understood what happens when is negative! To wrap things up, we worked out what the equations theta = pi/4 and r = 6 would look like, and we compared them to their rectangular counterparts. Then, back inside to apply our new knowledge to some problems!

The Wheels On The Bus Go Round and Round…

A quick post today about a lesson in precalculus that went a lot better than I expected (and isn’t it nice when that happens!). The objective was to understand linear velocity and angular velocity, and the differences between them.

I like to reduce as much as possible the number of formulas students memorize, so when we discuss this topic, I try to get them to use unit cancellation to arrive at the desired result. We talked a little about how to convert rpm to rad/sec, and mph to rpm. Then I sent them out in teams of four to the parking lot, armed with rulers, to figure out the rpms necessary for the wheels of cars traveling at 35 mph.

I didn’t tell them what to use the rulers for, so they had to figure out for themselves that they needed to calculate their car’s wheel circumference. When they returned to the classroom, they got right to work, and wrestled with the proper setup for their expressions. Eventually, every team got a good answer, and they didn’t use the same process to arrive at their result (which is great!). Here’s one team’s work:


Comparing the teams’ different results led to a nice discussion on how wheel size affects the rpms needed to roll at 35 mph, and why cars need differential gears. Merely getting my students out of the classroom and moving for about ten minutes really energized them, and made them interested in figuring out the answer to an admittedly simple problem. Whenever possible, I need to incorporate movement and outside activity into my lessons, even if it’s only for a few minutes.