Mathematical Literacy: Making Sense of Negative Exponents

One of the unique characteristics of our math literacy class is the organization of content … in a given week, we usually have a blend of numeric and symbolic work mixing in some algebra within a set of concepts that have been building during the semester.

The last class had two primary tasks: Integer exponents in problems, and equations of lines.  In a typical algebra course, integer exponents are ‘covered’ as part of a concentrated sequence; in the Math Lit course, we have already been using basic exponent properties for several weeks (products, quotients, and simple powers).  The idea is to have the work make sense to students, as much as possible.  We had, in fact, done work with scientific notation including small numbers — before we dealt with negative exponents in general.

Since this was our first time doing negative exponents as a general idea, we started with a very basic problem:

Simplify x^2/x^4

We wrote out the powers of x and reduced; then we subtracted exponents.  Very typical stuff.  The difference was the next step:

The two answers are both correct (1/x^2 and x^-2).  What do you think negative exponents mean?

The first student response was ‘do the opposite’, meaning divide.  The second response was ‘turn it over’ meaning reciprocal.  Nonverbal clues indicated that most students understood one of these meanings, and several got both of them.  Only after this interaction did I say anything about what it meant — as we dealt with the idea of writing an expression with positive exponents.

It’s not that this was a magical moment.  Several students in class could not apply what they had just said, and some had a very incomplete understanding of exponents and coefficients.  I’m still looking for the magic path for this ‘sum versus product’ understanding.  However, less of the struggle was about negative exponents than I usually see.

Some progress is encouraging, and evidence of an idea making sense looks like progress.

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4 Comments

• 

  By schremmer, April 22, 2013 @ 10:26 am

  In my experience, exponents should be defined from the start as integral exponents.

  Actually, it is the Laurent monomial Ax^±n that should be defined—as a shorthand for the longhand “A multiplied/divided by n copies of x”. (Of course, when the coefficient A is 1, it can go without saying but only when the exponent is positive so I encourage the students always to write the coefficient, no matter what.)

  After the students have gotten used to go in and out of the shorthand, they should multiply/divide Laurent monomials for a while by expanding the shorthand into longhand, executing the longhand, and then writing the result in shorthand.

  After they get tired of it, one then points out that “oplussing” (adding signed numbers) the exponents takes care of multiplying Laurent monomials and “ominussing” the exponents takes care of dividing Laurent monomials.

  Regards
  –schremmer

• 

  By schremmer, April 22, 2013 @ 10:33 am

  Re “The last class had two primary tasks: Integer exponents in problems, and equations of lines.”

  What’s the connection?

  Regards
  –schremmer

• 

  By Jack Rotman, April 22, 2013 @ 11:35 am

  Only indirect … the course includes linear and exponential functions, so it is more a matter of ‘contrast’ than ‘connection’.

• 

  By schremmer, April 22, 2013 @ 1:27 pm

  Ah yes!, The (in)famous Hughes-Hallet, Andrew Gleason et al calculus book.

  Given that the connection is a bit … incidental, I have always wondered how much grief it ultimately causes.

  Regards
  –schremmer

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