Why I Write of Life on the Asymptote: On Not Being Well-Behaved

First of all, let me answer the burning question: what the heck is an asymptote? Basically, an asymptote is a “line that a curve approaches.”

Although asymptotes can take a variety of forms—horizontal, vertical and oblique—possibly the most common encountered by a high school math student is the vertical asymptote, which occurs where a function has a non-removable discontinuity.

For example, consider this function: y = 1 / x – 2.

I might ask my students, “what are the possible values for x and y?”  Students might then say, “Well, if x is 3, then y is 1” and give other points on the graph. This continues until one brave soul might dare to ask, “What about when x is 2?”

Well, when x is 2, chaos ensues!  Not really, but something weird does happen.  You probably have heard things like “Only Chuck Norris can divide by 0” and “The math police arrest you when you try to divide by 0.”  It is true that the graph doesn’t “work” when x is 2. This means there is a discontinuity at x = 2.  Here is what the graph of the function looks like, where the function is blue and the asymptote red.

Graph of asymptote

Interestingly, the word “asymptote” comes from the Greek word “asumptotos” which means “not falling together.” The graph shows that even though both blue graphs are parts of the function, they do not fall together.

In math, functions can be classified as well-behaved or non well-behaved.  A function that has an asymptote is definitely not well-behaved.

One of the most annoying (endearing?) side effects of being a math teacher is that I tend to relate just about every experience to something in a math textbook.  And my journey as a teacher leader is no exception!

Like the dotted red line, I don’t “fit” exactly in any category that currently exists in the education system. Being a teacherpreneur means that I am a part-time classroom teacher and part-time education leader.  I participate in policy discussions, yet I am not a politician or official decision maker.  I lead education efforts and help teachers implement standards, yet I am not an administrator.

Nor am I a full-time teacher.

In addition, like the asymptote, my leadership role is something that traditionalists would describe as “not well-behaved.” After all, I challenge convention.  I attend policy meetings and also have in the forefront of my mind the students that I taught yesterday. I am able to “lead without leaving,” and that perspective is invaluable—if unconventional.

Teacher leaders, what are your metaphors? (Please tell me I’m not the only one who thinks in the language of the subject I teach…)

Note: This blog post is brought to you by CCSS F-IF.7.d Graph rational functions, identifying zeros and asymptotes when suitable factorizations are available, and showing end behavior.

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  • SandyMerz

    My Metaphors

    I think a lot about extreme values, vanishingly small probabilities, thresholds,  attractors, tolerances, specifications, the cost of a wrong decision (as in, “let’s try it and see, the cost of a wrong decision isn’t high.)  Oh, I love to say, “Orders of Magnitude.” 


    • Yale Landsberg

      An example of where 999… is not well-behaved? Maybe.

      As b->1, each associated list of b^n begins at b and accounts for an infinite number of numbers. Each entry on each infinite list is different from all of the other number on that list.

      On the other hand, for b=1, all numbers of 1*n’s infinite list are the same one number, 1.

      In terms of two representations of the same number having the same behavior, what do the above different behaviors of .999… versus 1 indicate about .999…=1? 

      Warmest regards, YaleLandsberg@gmail.com



  • AnneJolly


    Hmmmm – interesting exercise, Ali.  In borrowing a familiar science term, I think of teacher leaders as catalysts. A catalyst is a substance that causes or accelerates a chemical reaction without being affected itself.  Teacher leaders can cause and accelerate change within the faculty, within the school, and even outside the school.  Often these changes are small, but they can grow exponentially. 


    • AliCrowley

      Great metaphor!

      Hi Anne,

      Thank you for accepting the challenge!  That is a very inspirational metaphor.  I wonder if many teachers realize the impact that they have- not just on the their students- but also on stakeholders. I love the idea of a catalyst.  I also am thinking- what fuels a teacher leader’s ability to be a catalyst?  Passion for the profession?  Student achievement?  A desire to improve teaching and learning overall?  

      Thanks for keeping my wheels spinning, Ali 🙂

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