I had two presentations at last Saturday’s USACAS-9 conference at Hawken School in Cleveland, OH. Following are outline descriptions of the two sessions with links to the PowerPoint, pdf, and .tns files I used. I’m also adding all of this information to the Conference Presentations tab of this ‘blog.
Powerful Student Proofs
This session started with a brief introduction to a lab that first caught my eye at the first USACAS conference years ago.
You know how the graph of behaves when you vary a and c, but what happens when you change b?
Next, we explored briefly the same review of trigonometric and polar graphs not as static parent functions under static transformations, but as dynamic curves oscillating between their ceilings and floors. In the session, we used TI-Nspire file Intro Polar.
Having a complete grasp of polar graphs of limacons, cardioids, rose curves, and hybrids of these, I investigated what would happen for curves of the family . Curiously, for , I encountered a curve that looked like a horizontal translation of limacons–something that just shouldn’t happen within polar coordinates.
One of my former students, Sara, used a CAS to convert a polar curve to Cartesian, translate the curve, and convert back to polar. She then identified and solved a trig identity to confirm what the graph suggested. A complete description of Sara’s proof is below. I originally ‘blogged on Sara’s work here which was a much more elegant solution to the problem than my initial attempt. It’s always cool when a student’s work is better than her teacher’s! I used TI-Nspire file Polar Fractions in Saturday’s session.
The last example presented itself when I created a document to model the family of conic curves resulting from manipulating the coefficients of . After I created dynamic points for the foci of the conics, something unusual happened when the E parameter for horizontal ellipses and hyperbolas varied.
The foci for hyperbolas followed an ellipse, and the locus of elliptical foci appeared to be a hyperbola. Another former student, Lilly, proved this property to be true. A detailed explanation of Lilly’s proof is below. We were fortunate to have Lilly’s work published in the Mathematics Teacher in May, 2014.
To demonstrate this final part of the session, I used TI-Nspire file Hidden Conic Behavior.
Here is my PowerPoint file for Powerful Student Proofs. A more detailed sketch of the session and the student proofs is below.
Bending Asymptotes, Bouncing Off Infinity, and Going Beyond
The basic proposal was that adding the Reciprocal transformation to the palette of constant dilations and translations dramatically simplified understanding of the behavior of rational functions around even and odd vertical asymptotes (bouncing off and passing through infinity). Just like lead coefficients of polynomials determine their end behavior, so, too, do the lead coefficients of proper rational expressions define the end behavior of rational functions.
Extending the idea of reciprocating and transforming functions, you can quickly explain exponential decay from exponential growth, derive the graphs of and , and completely explain why logistic functions behave the way they do.
We finished with a quick exploration of trigonometric and polar graphs not as static parent functions under static transformations, but as dynamic curves oscillating between their ceilings and floors.