Conic, Parametric and Polar Equations. Oh, my!

At ISB, we let students skip ahead in math courses, trying to meet ability rather than age.  However, we want them to be with their peers in the last two years of high school.  In part because the IB is changing their mathematics courses for the DP (grade 11/12 courses) so we did not want them to be learning old content but also because the IB does not allow students to take Higher Level exams early, so finishing IBHL early is of no benefit.  So we needed a “holding” course for our grade 10 students who are in the “normal” progression ready to enter IBDP math courses.  I was tasked with creating a class that included math content they haven’t seen before and that they won’t necessarily see in their IBDP classes.  Fun, right?  I wanted the course to be fully project based, but even with projects there still needs to be some content taught.

So, I began with locus of points leading into conic sections. Students had been briefly introduced to conic sections along side quadratic functions, but we don’t get into the development of conics as a locus of points. Teaching it from this perspective allowed us to develop equations for “tilted” conic sections, such as ellipses that have a major axis which has a non-zero/infinite slope. We also had the chance to do constructions and paper folding!  They wrote instructions and taught the principal how to do it.

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Teaching the principal to construct an ellipse.

From this, I segued into polar graphs and finally parametric equations.  Along the way I had students using Flip Grid to give me reflections on their learning.  Mostly 2 minutes videos about how the were making connections between prior learning and new learning.

Last month, I started introducing them to the idea of writing math papers. Since we are not a full IB school, most students had never been exposed to the idea of writing a paper in math.  So I started them out where we were all working on the same thing. First I had them answer the questions from Underground Mathematics Conic Sections in Real Life. From there, I asked them to write a maximum 4 page paper summing up their findings on how mathematics can be used to explain GPS location determination methods and gave them an outline for a math paper.

Here’s an excerpt from an introduction:

In our current math class, I have been learning the conic section for a few months. We defined the maths behind it not only by its functional equations, but also by the locus of points it satisfies. Moreover, by studying the concept of how certain locus of points form the shape of the conics, we were able to understand the characteristics behind it; such as, ellipse is a locus of points where the sum of distance between that point and other two fixed point is constant. Now, I am trying to imply my learning to real life situation and I found out that the satellites use the conics to identify the location of the individuals using the GPS.

Another student wrote and created diagrams:

Screen Shot 2018-11-26 at 4.58.47 PM

While students were writing, I was providing comments via Google Classroom and Google Docs.  The papers were good, but we all know, reading many of the essentially same paper can be boring.  So next up, I had students choose their own topics to write about.  We started with a brainstorm:

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Brainstorming uses of conic and parametric equations.

Next, students picked their topics.  This class is small, only 8 students.  I had them start by writing to me about what math the would be using to see if their ideas were feasible.  To do this, I highlighted the key parts of the outline for a math paper.  In the end, here are the eight topics I approved:

  • Sundials
  • Path of a Roller Coaster
  • Path of a paper airplane
  • Light refraction in telescopes
  • Whisper Galleries
  • Hyperbolic Navigation
  • Planetary Orbits
  • Trajectory of a Baseball

Way more fun to read. The student who wrote about the whisper galleries spent a full class up at the white board trying to figure out how to model his elliptical shape using the speed of sound.  When he got it, well, it was magical.

The student who was looking at the path of a roller coaster designed a series of parametric conic sections and then drove a roller coaster over them on Desmos!

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