# Always New

A new winter quarter started this week at my community college and with it a new round of reflection on teaching and learning.  I teach math in the evening and this quarter I am participating, again, in a seven-week, multi-discipline exercise in blogging on all things academic.   There are no rules on what to write or how to write, but the blogs will hopefully be worth the time to read.  All participants are required to review the blogs of other participants so there is some hope for more than monologues.   Let the game begin.

Every time I prepare for a class I learn something new about what I already knew (don’t you love English and its homonyms).  For example, I have taught on exponential equations many times, but in reviewing for a review of the topic I marveled at how exponentials, well before the long run, magnificently outpace polynomials.   Nothing really profound—I was comparing a simple quadratic function and a simple exponential function, the two topics for the lecture.

I used a TI84 for demonstrations so I’ll include some screen shots.

 Here we have the two equations. The first graph and its window settings is shown below.  The leftmost curve is the quadratic.  The quadratic seems to handily outpacing the exponential.  The graph was created with the ZOOM 1ST QUADRANT command. Increasing the height of the viewing window, by about a factor of four, shows that the exponential is catching up to the quadratic (below). Increasing both Xmax and Ymax, but only doubling Xmax, it becomes very clear that the exponential leaves the quadratic forever behind starting at about 10 (below).

Admittedly, some of the effect of the graphs relies on the shallowness of the first two windows, but it still is exciting to see the power of an exponential growth.  Whenever I teach about exponentials, I try to emphasize to my students that any use of an exponential growth model must require the determination of a practical domain—going too far is going too far.

Since I’ve shown some calculator graphs, let me reflect on the use of a graphing calculator, such as the TI84, in a math class.  Why should an instructor require a graphing calculator when there are so many more visual, easier to use, graphing packages available for free on the internet?  Here are two reasons.

First, the TI84 is a readily-available, common-denominator tool.  Many students already own one, or can borrow one, and for those students who don’t own one, there are usually a pool of them available at the school library.  The calculator has sufficient capability to draw graphs of equations or create equations of scatter plots.  And it is not hard to master the basic commands.  As an alternative, requiring all students to bring a laptop, tablet or phone with appropriate app is not always possible.  And not every community college class can be equipped with student-available computers.

Second, a calculator is generally needed on math exams to help with at least arithmetic operations on decimals, fractions, or irrational numbers.  If the course is also teaching regression of data points to best fit curves, this level of calculator is also needed for exams.  Allowing students to use a laptop utility, or their phone, with substitute software, is just too much temptation for students to move beyond graphing/calculation software to symbolic manipulation software.  This is also a reason to not use higher-function, symbolic manipulation calculators instead of graphing calculators.

That’s my reflection for the week of January 9, 2017.