The lesson I chose for this project was about
discontinuities in the domain of rational functions. The goal of the lesson was to have students
explore and gain an understanding of what causes the domain to be discontinuous
so that they would be able to identify functions where this type of behavior is
likely to occur as well as at what value(s) the function is discontinuous. I choose to use the TI-Nspire graphing
calculator in conjunction with this lesson.
The technology is not a necessity for students to understand the concept;
however, the goal of including this technology with the lesson is to help
students focus their attention on the stated goal as opposed to other
extraneous, albeit important in previous units, content.
I beta tested this lesson with my wife as my classes will
not be working on this for another week.
Just as I hoped she quickly identified that the graph had strange
vertical lines appearing in it, representing the discontinuities. When I directed her attention to the equation
I had her enter into the calculator she also quickly identified that the reason
that this was occurring was due to the fact that those x-values caused a
division by zero which as she noted “you can’t do”. I created a new equation for her to look at
and asked her to predict what the graph would look like. Again she picked out the values that would
cause division by zero. For our third
exercise I created an equation that was impossible to find the zeros of the
denominator by inspection. As it had
been years since she solved a quadratic equation, which is not the case for my
students, I showed her how to graph the denominator of the equation and asked
her what x values made it equal zero.
She estimated the two values from the graph and I showed her that she was
correct by demonstrating that the calculator would calculate them for her. Overall she maintained a focus on the reason
for the graphs strange behavior and was able to identify where it would occur,
precisely as the lesson had been designed to do.
The lesson did differ from the way I plan to teach to my
students in the sense that they have been working with solving quadratics, and
finding zeros with the calculator for several weeks. In this sense the technology should not
provide any sort of distraction from the content (i.e. students struggling to
use the technology) which has plagued some of my other attempts at integrating
technology into my lessons. As discussed
earlier the primary reason for using the calculator is to allow students to
focus on thinking about why the function is discontinuous so it is important
that one distraction is not replaced by another.
As I thought about this lesson I also discussed whether
students should be allowed to avoid having to use mathematics that they have
previously learned with other members in my department. At first, there was a strong sense that the
unit on rational functions provided a fantastic opportunity for students to
review concepts from previous units, which is certainly the case. However, when I noted that students seem to
get lost when the problems become so long (in the case of a quadratic
denominator in simplified form, roughly a half page of written calculations and
equation manipulation just to find two x-values) that they lose track of what
they are trying to do, several colleagues agreed that the review of old content
in this way may be distracting especially when our real goal is to focus on the
new content. I proposed that we may want
to consider asking them to do it without the technology at a later time, once
the new concept was more fully developed.
Another interesting idea that came out of these discussion
was that by letting students use the calculator to find the zeros of the
denominator they are freed up to explore denominators of higher degree (there
is no generalizable algorithm for solving polynomials higher than degree four
and the generalized solution to degree three and four polynomials is well
beyond the scope of an Algebra 1 class).
Hence, our students would be able to subvert these formulas and still be
able to make sense of what their solutions mean, even if it was a degree 50
polynomial and have it be almost no more work than if it were a first degree
polynomial.
Given the success this lesson was met with when I piloted it
with my wife I believe that the overwhelming majority of students will be able
to provide a detailed account of what causes discontinuities and how to
determine their values immediately after this lesson. The question that will be more difficult to
answer is regarding what the outcomes will be at the end of the unit. Students typically have a much more difficult
time being thoughtful about discontinuities when they are also responsible for
determining the x-intercepts, y- intercept and end behavior of a rational
function simultaneously. One other major
goal of this lesson is to make sure enough time is spent focusing on the “why discontinuities
exist question” that it is never entirely reduced to algorithm and therefore
not easily confused with other such algorithms.
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