Pre-Calculus and the Witch of Agnesi

This year, a colleague announced that he needed to take a short leave from school. This is the second teacher in our department who has needed to take time from work this year. Due to an unfortunate shortage of qualified substitutes, teachers in our department have been asked to take on another teaching section. Some teachers are eager to do it, thanks to the bump in pay (after all, who couldn't use a few extra bucks in the bank?). However, some teachers are reluctant to teach an additional period, citing an already busy personal and/or professional schedule.

So when I was asked to teach one of my colleague's classes, I was hesitant. But wanting to help out (and with a son looking at tuition payments next year), I agreed. I knew the leave would only be a few weeks and most of it was during the month of November, a month with days off for Election Day, Veterans' Day, and the Thanksgiving break.

So when I was asked to cover his Honors Pre-Calculus class, that combination of words instantly gave me sweaty palms and fear of sleepless nights. You see, I haven't done anything related to calculus since my college days. I have been teaching for 24 years, so that was ancient history. Unfortunately, we do not have a policy of moving teachers around in our department and I have not taught anything higher than Intermediate Algebra 2. So facing honors students in a course that I have no expertise in, was cause for concern.

The teacher I was helping assured me that he had pushed ahead, knowing he was going to be out. Also, for the first two weeks, we worked on SAT review packets. But in the third week, we started getting into derivative rules and implicit differentiation. Yikes! Lucky for me, he was able to record flipped videos that the students and I could watch. This way, he was able to keep the instruction moving forward. The students would watch the videos, take notes, and work on the homework problems. I did too.

As I worked on the problems, I found myself turning to Desmos for support. It became a valuable resource that allowed me to check my work. I would graph the original equation, plot a point on the curve, find the derivative (aka the slope of the tangent line), then graph the tangent line. Here is a sample:

I found the Desmos graphs to be a way to reassure and lift my confidence levels as I reviewed these challenging problems with the students. But I also found that is provided powerful visual models of these problems.

One of the problems that was assigned involved finding the slope of the tangent line to a curve titled the "Witch of Agnesi":

I worked through the problem, got a solution, and checked it with Desmos. But as we were reviewing the problem, a student asked about the name of the graph. I didn't know, so I encouraged her to Google it. The back story on the naming of the graph is an interesting one, and I encourage you to read more about it.

When we looked at the page from Wolfram, we were fascinated by the GIF showing how the curve is generated. After studying it for a bit, I was able to recreate it using GeoGebra:

I showed the students the next day and they thought it was neat that I took the time to play with the math and put that little applet together. So we moved on... except for one student.

After a period of students watching the next flipped video, working on homework, and getting help from me and their classmates, I was working the room and asking the students to sign out and return their Chromebooks. But one student, who is a little on the shy side, said "Hey Mr. K, check this out. I'm trying to make the Witch of Agnesi in Desmos." 

Now the only exposure that they have had to dynamic math technology such as Desmos & GeoGebra
has been what I have shown them in just a few short weeks as their substitute. But this young man had begun to create the workings of a dynamic Witch of Agnesi app in Desmos.

Needless to say, I was thrilled and he and I spent the last few minutes and an additional half-hour after school putting together his app. We worked through some intense algebra and coding to get it to work:

It has been a wonderful experience working with some challenging mathematics and with some very impressive students. Although I am looking forward to my colleague's return to school in a few days, I will miss the experiences that have resulted from this time with the Honors Pre-Calc students.

These were magical moments for me. Thank you for letting me share them with you.


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