Math Misconceptions: zero and negative exponents

Are you looking for a lesson to explan negative exponents to students? This post explains what a negative exponent means and also zero as an exponent.

What math misconceptions have you seen every year in your classroom? There seems to be a common thread with these misconceptions and the same ones also seem to come up over and over again. We've seen everything from students thinking that they are bad at math to misreading the height of a triangle. Oh and fractions! Yes, fractions. And have you heard, "Just tell me the answer!"? We have too! What other misconceptions have you seen?

For me, one of the things I like best about teaching algebra 2 are the misconceptions that come up about algebra 1. This sounds crazy, right? Maybe I secretly miss teaching algebra 1.

Wait, no I don't. Freshmen!  Eeeeeeek! :) 

But seriously, I love algebra misconceptions. One of my very favorite has to do with zero and negative exponents. My students always seem to think at a zero exponent makes zero and a negative exponent makes a negative number. I mean, this seems to make sense, right?

"A negative exponent becomes positive in the denominator"

Um, what? I always try this one out first to see just how much they remember from algebra 1. My hope is that I can jog their memory, but it hardly ever works. I'll get a lot of blank stares or, "Oh right"s that I know are just to make me feel better (and for them to save face). This is when I stop what we're doing, bust out the dry erase markers and show them WHY exponents do what they do.

negative and zero exponents

But first

Before x, we look at 2 raised to the 4th, 3rd, 2nd, and 1st. Everyone follows along. We look at the pattern (divided by 2 each time). It's then an easy jump to see that 20 is 1 and 2-1 is 1/2. It just follows the pattern they already see happening in the table. 

And finally x
It's then an easy transition to see how these exponent rules translate to variables. Now my students can see how any number raised to the zero power is 1 and that an x-2 is 1/x2.  After this, they are completely bought into the idea that a negative exponent in the denominator will become positive in the numerator. 

And an activity
Here is a fun activity to summarize a unit on exponent rules. Since we always learned about exponents in the Spring, I decorated this pennant with flowers for students to color. I remember freshmen loving to color.

And a new exponent rules digital math escape room:

Exponent Rules (with evaluating) Digital Math Escape Room

OK, maybe I do miss freshmen... a little. :)

Other common Math misconceptions
Here are some great Math misconception blog posts (with solutions!) from some of my math teacher friends....


  1. Such a confusing topic for students... Great post!

  2. That chart makes perfect sense! And the pennants are adorable. :)

    1. Thank you Amber! Maybe we'll meet at the Science Museum again! :)