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 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.

One of my very favorite math misconceptions 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"

Wait a second. 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 memories, but it rarely works. This is when I stop what we're doing, bust out the dry erase markers and show them WHY exponents do what they do.

But first...

Before x, we look at 2 raised to the 4th, 3rd, 2nd, and 1st. Can you see a pattern?

zero exponent pattern - what will 2 to the 0 be?

Everyone follows along. We look at the pattern (divided by 2 each time). 

Here we can see the dividing by 2 pattern, showing that 2 to the 0 is 1

If 20 is 1 then what is 2-1

What will 2 to the -1 be?

It follows the exact same pattern of dividing by 2.

If we continue the pattern, we see that negative exponents create fractions

If we continue the pattern, we can see that negative exponents follow the exact same pattern. Instead of making negative numbers, negative exponents make fractions. 

Here's a quick video showing why negative exponents create fractions and why numbers raised to the zero power equal 1:

Moving to variables with exponents:
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. 

Here we can see what happens when we raise variable x to exponents

I decorated this exponent rules math pennant with flowery clipart for students to color. I remember freshmen loving to color. The final math pennants make a colorful, student-created classroom d├ęcor. 

Exponent rules math pennant activity
Exponent rules math pennant activity

It's been a bunch of years since I originally wrote this post, so while I'm here updating with new photos, I also wanted to link some more exponents activities to help with those misconceptions.

Exponent Rules (with evaluating) Digital Math Escape Room puzzle #2
Exponent rules digital escape room (with evaluating)

Students use their knowledge of exponent rules and then evaluate their final expressions in this exponent rules (with evaluating) digital math escape room. Above is puzzle #2 of the escape room. Students find their answers in the grid, then type their code to unlock the puzzle. The entire escape room is one answer-validated Google Form to make it super simple to assign.

Exponent Rules Digital Math Escape Room gif showing the first puzzle to unlock
Exponent rules digital escape room

If you'd rather your students work with variables, this exponent rules digital math escape room has students working with variables instead of numbers, allowing students to generalize the rules. 

More math misconceptions:

Below are links to more answers to common math misconceptions (with solutions!) written by some math friends on finding the base and height of triangles, solving equations, repairing the "I'm bad at math" attitude, the 1/2 fraction, dividing fractions and the importance of showing work.

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  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! :)