Category Archives: Presentations

CAS Presentations at USACAS-9

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 y=ax^2+bx+c behaves when you vary a and c, but what happens when you change b?

I ‘blogged on this problem here and here.  In the session, we used TI-Nspire file QuadExplore.

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 r=cos \left( \frac{\theta}{k} \right).  Curiously, for k=3, 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 Ax^2+Bxy+Cy^2+Dx+Ey+F=0.  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 y=\frac{1}{x} and y=\frac{1}{x^2}, 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.

I used TI-Nspire Bending and Intro Polar files in the demonstration.  Here is my outline PowerPoint file for Bending Asymptotes.

Air Sketch iPad app

I’ve rarely been so jazzed by a piece of software that I felt compelled to write a review of it.  There’s plenty of folks doing that, so I figured there was no need for me to wander into that competitive field.  Then I encountered the iPad Air Sketch app (versions: free and $9.99 paid) last Monday and have been actively using in all of my classes since.

Here’s my synopsis of the benefits of Air Sketch after using it for one week:

–Rather than simply projecting my computer onto a single screen in the room, I had every student in my room tap into the local web page created by Air Sketch.  Projection was no longer just my machine showing on the wall; it was on every student machine in the room.  Working with some colleagues, we got the screen projections on iPhones, iPads, and computers.  I haven’t projected onto Windows machines, but can’t think of a reason in the world why that wouldn’t happen.

–In my last class Friday, I also figured out that I could project some math software using my computer while maintaining Air Sketch notes on my kids’ computers.  No more screen flipping or shrunken windows when I need to flip between my note-taking projection software and other software!

–When a student had a cool idea, I handed my iPad to her, and her work projected live onto every machine in the room.  About half of my students in some classes have now had an opportunity to drive class live.

–This is really cool:  One of my students was out of country this past week on an athletic trip, so he Skyped into class.  Air Sketch’s Web page is local, so he couldn’t see the notes directly, but his buddy got around that by sharing his computer screen within Skype.  The result:  my student half way around the globe got real-time audio and visual of my class.

–This works only in the paid version:  We reviewed a quiz much the way you would in Smart Notebook—opened a pdf in Air Sketch and marked it live—but with the advantage of me being able to zoom in as needed without altering the student views.

–Finally, because the kids can take screen shots whenever they want, they grabbed portions of the Air Sketch notes only when they needed them.  My students are using laptops with easily defined screen shot capture areas, but iPad users could easily use Skitch to edit down images.

–Admittedly, other apps give smoother writing, but none of them (that I know) project.   Air Sketch is absolutely good enough if you don’t rush.

By the way, the paid version is so much better than the free, allowing multiple colors, ability to erase and undo, saving work, and ability to ink pdfs.

Big down side:  When  you import a multi-page pdf, you can scroll multiple pages, but when creating notes, I’m restricted to a single page.  I give my students a 10-15 second warning when I’m about to clear a screen so that any who want cant take a screen shot.  It would be annoying to have to save multiple pages during a class and find a way to fuse all those pdfs into one document before posting.  The ad on the Air Sketch site was (TO ME) a bit misleading when it showed multiple pages being scrolled.  As far as I can tell, that happened on a pdf.  Perhaps it’s my bad, but I assumed that could happen when I was inking regular notes.  Give me this, and I’ll drop Smart Notebook forever.  Admittedly, SN has some features that Air Sketch doesn’t but I’m willing to work around those.

Overall, this is a GREAT app, and my students were raving about it last week.  I’ll certainly be using it all of my future presentations.

PreCalculus Summer Institute 2012

If you are in Atlanta, GA July 9-11, 2012, you might be interested in attending a workshop my co-author and I are offering at Westminster through the Center for Teaching.  A description of the workshop follows.  We hope to see some of you!

Title of Workshop:  PreCalculus Transformed
To register, click here.

PresentersChris Harrow & Nurfatimah Merchant from The Westminster Schools

Workshop Dates:  July 9-11, 2012 (Monday-Wednesday)

Workshop Description: PreCalculus Transformed highlights the under-explored role of non-standard transformations and function composition in learning algebra and precalculus concepts. Families of functions are identified first by immutable distinguishing characteristics and then modified through multiple representations & transformations. Participants will discover that many historically complicated precalculus problems and concepts are both richer and greatly simplified in this process. The course integrates computer algebra system (CAS) technology, but it is certainly possible to use and grasp its concepts without this technology.  Potential topics include expanded transformations, polynomials, rational functions, exponentials, logistics, and trigonometric functions.  Additional topics may be explored depending on time and participant needs or experience.  Textbook is included in the workshop price.

Target Audience:  Algebra II, Precalculus, and Calculus teachers at the high school or junior college level

Workshop Location:
The Center for Teaching at The Westminster Schools
1424 West Paces Ferry Road NW
Atlanta, GA 30327

Contact e-mail:

March 2011 Conference Presentations

T3 Regional Conference – Suwanee, GA – Saturday, March 19, 2011.

PreCalculus:  Transformed & Nspired

This workshop offers an innovative understanding of pre-calculus concepts through nonstandard transformations, allowing functions and concepts to be unified by a handful of underlying mathematical structures. It provides approaches that dramatically simplify many initially complicated-looking problems. CAS-enhanced ideas are presented.  (Co-presented with Nurfatimah Merchant)

Conics within Conics

This session presents the family of conic sections by connecting their algebraic and graphical representations, showing how each section can evolve from the others. The conclusion is a surprisingly elegant conic property and a 9th grader’s proof submitted for publication.