Building intuitions for non-positive exponents

In my Algebra II classes I often find that students aren’t really comfortable with the idea of zero or negative exponents, even though they are part of the Algebra I curriculum. So I need to teach negative and zero exponents to them. Here are some approaches that I’ve found to building intuitions about these non-positive exponents.

(Some of these approaches were discussed in this comment thread on a post about a common exponent mistake.)

  1. If you teach exponential functions before handling exponents, then you can use the continuity of the graph to explore zero and negative exponents. Like, what seems to be happening with the negative exponents? What’s the y-value? What seems to be going on?
  2. Use students’ pattern observation skills to build a sense for what zero and negative exponents should be. This means asking students to evaluate 3^3, then to evaluate 3^2,  then 3^1, and asking students what happens each time you reduce the power by 1. Then what should happen with 3^0? And so on.
  3. You could use something that CME calls the “duck principle” i.e. if it quacks like a duck, smells like a duck, tastes(?) like a duck, then it’s a duck. So they’ll ask students to figure out what 3^0 means by reference to a rule that they’re comfortable with, such as x^a * x^b = x^{a+b}. So what does 3^0*3^5 evaluate to?
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