“We came up with a theorem once at my old school. The teacher has it in a frame behind his desk.”
This statement from one of my college algebra students made me both elated and sad at the same time. Thrilled because this is the type of mathematics I believe all students should have the chance to engage in on a regular basis. Disappointed because this type of discovery happens so infrequently in American mathematics classrooms that the incident warranted a sacred place on the wall of this teacher’s room.
In College Algebra, part of today’s learning objective was to define a polynomial function and determine some key features. I have the awesome types of students that if I were to write down the surly definition and features of a polynomial function onto the whiteboard, each would follow in lock-step and write it in their notebooks solidifying it’s place among mathematical obscurity.
Today, we were going to break that cycle with something different.
But I needed to know where they were at, so I had them write down what they knew about a polynomial function.
After some discussion and leading questions, we were sure that linear, quadratic, cubic, quartic, x^5, x^6, and so on were all polynomial functions. Awesome. We weren’t, however, as sure about functions including negative exponents, roots, sin/cos, or algebraic fractions.
What makes this group we are sure about special? Last week, we spent a considerable amount of time on features of functions including domains, end behavior, intercepts, intervals, symmetry, and turning points. In their groups, I had them examine the graphs of these alleged “polynomials” through the lens of the features of functions.
Two similarities emerged as significant:
Questions: Was this true of all polynomial functions? And if both conditions were not met, could we exclude it from our known polynomial functions? Hiding my initial excitement, I then had them look at our list of “questionable” functions. For example, did “y = 9 + 1/x” meet each of these two criteria?
Christopher Danielson suggested that my class give this new theorem a name, so we could refer back to it with ease: