This is Our Theorem – College Algebra

“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.

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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:

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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:

“Class. We have found that all polynomials blah blah blah…” [while writing the statement of the theorem on the board.]  In mathematics, when we have an important finding like this, and when all mathematicians have agreed the finding is true, it gets a name.  Sometimes it is named for a person, such as ‘Fermat’s Last Theorem’; sometimes it is named for what it says, as in ‘The Triangle Inequality’.  But that name makes it possible to refer to it going forward. It helps us to remember and to use the thing we figured out. So we need to name our theorem. Who has a name they’d like to suggest?”

Alas, the excitement of naming the theorem will have to wait until tomorrow.

A Speedy Makeover for the Intermediate Value Theorem

As a college algebra teacher, I was not satisfied with the way I presented the intermediate value theorem last trimester.  I felt the lesson was somewhat isolated from other concepts we had studied and definitely was disconnected from the real world.  My approach lacked a hook and was laddened with procedure.   Committed to teaching the concept better this trimester, I recorded the following video while (someone else) was driving:

I know, not a high quality masterpiece, but I think I captured what I needed to illustrate the theorem.

I ask the students to draw a graph of the speed of the car with respect to time.  After playing the video a number of times, I had them share their graphs with their seat partner.  As I circulated the room, I noticed their results fell into one of these three categories:

Graph A

Graph A

 

Graph B

Graph B

 

Graph C

Graph C

After examining the options, I had them choose which graph they felt represented the situation most accurately.  Spoiler Alert:  The overwhelming majority of them chose Graph B.  Their reasoning:  it’s unclear what happened to the speed between seconds 10 and 15 therefore, there should be a space in the graph.  Those vying for Graph C cleverly argued that there was no audible “revving of the engine,” indicating that the car continued to slow.  Others supporting C claimed that even though we could not see the speed, they know how a speedometer works and can make a reasonable assumption about what happened in that time frame.

Enter this student’s graph and the Intermediate Value Theorem (trumpets):

IVT graph

I liked this students “shading” through unknown speed region, so I projected it for everyone to discuss.  They were able to determine the value of the function at ten seconds, f(10), was approximately 45 miles per hour and the value of the function at fifteen seconds, f(15), was approximately 35 miles per hour.  They also knew that the car must have reached 40 miles per hour sometime in between 10 and 15 seconds.  “How do you know that?” I pryed.  Gem response of the day:  “Well, speed is continuous and I can’t go from 45 mph to 35 mph without going through 44, 43, 42, 41, 40 mph, and so on.”  Bingo.  Intermediate Value Theorem.  No boring procedural explanation necessary.

We applied this “new” knowledge to a polynomial function so that they could get a handle on some of the algebra and notation used.   And as a bonus, they also seemed to grasp that this theorem does not only apply to crossing the x-axis, a common misconception students had last trimester.

Moving forward, I’ll definitely work on creating a better video!