It all started when I tried to get an interesting variation on graphs of inverse trigonometric functions. Tiring of constant scale changes and translations of inverse trig graphs, I tried , thinking that this product of odd functions leading to an even function would be a nice, but minor, extension for my students.
I reasoned that because the magnitude of arctangent approached as , the graph of must approach . As shown and to my surprise, seemed to parallel the anticipated absolute value function instead of approaching it. Hmmmm…..
If this is actually true, then the gap between and must be constant. I suspected that this was probably beyond the abilities of my precalculus students, but with my CAS in hand, I (and they) could compute that limit anyway.
Now that was just too pretty to leave alone. Because the values of x are positive for the limit, this becomes .
So, four things my students should see here (with guidance, if necessary) are
- actually approaches ,
- the limit can be expressed as a product,
- each of the terms in the product describes what is happening to the individual terms of the factors of as x approaches infinity, and
- (disturbingly) this limit seems to approach . A less-obvious recognition is that as , must behave exactly like because its product with x becomes 1,
But what do I do with this for my precalculus students?
NOTE: As a calculus teacher, I immediately recognized the product as a precursor to L’Hopital’s rule.
and this form permits L’Hopital’s rule
OK, that proves what the graph suggests and the CAS computes. Rather than leaving students frustrated with a point in a problem that they couldn’t get past (determining the gap between the suspected and actual limits), the CAS kept the problem within reach. Satisfying enough for some, I suspect, but I’d love suggestions on how to make this particular limit more attainable for students without invoking calculus. Ideas, anyone?
I recently was notified that a former student’s idea is expected to be published in the Mathematics Teacher in early Fall, 2011. Ian discovered a way to use a single quadratic function with a variable y-intercept to capture a limited number of values of an increasingly long sum of absolute value functions. I’ll link a copy of the brief article once it is published.
The coolest part of the development of this idea is that Ian’s access to a CAS and the encouragement of his class to explore their ideas made this problem possible. When Ian needed to learn additional mathematics (induction) to prove his theorem, he was receptive and confident.
Every student should have access to a CAS (Computer Algebra System) in a handheld and/or computer-based format at least as early as he or she begins learning algebraic concepts.
Used properly, a CAS creates a dynamic laboratory environment for a student in which he or she can explore algebraic relationships, receive instantaneous confirmation of the validity of algebraic manipulations, and scaffolding for deeper exploration and understanding of mathematics. In short, a CAS enables a student to have a mathematical solving expert available at all times in all places. Most importantly, students get the opportunity to explore mathematics without needing
Of course, to use a CAS, one needs to learn how to ask questions and how to interpret the solutions. A CAS will always provide an answer to the question asked. Users must know precisely what is being asked so that they can interpret their results.
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.