Powers of 0 and Negative Powers - with a Twist

Most of you have probably done a lesson on non-positive exponents by looking at a pattern similar to this one. (Thanks to Desmos for the table). 
I typically start with showing the students something like this.  I ask them about the pattern of values on the right, focusing on the way the values change going up (most of the time students see that first) and the way the values change going down.  I also point out that there is a pattern on the left and we talk about that one.  We take some time to distinguish between the two patterns - one involving addition and subtraction, and the other involving multiplication and division.  And then I ask whether we could continue the patterns, and see what the students say. 

There are lots of directions to go from there, depending on what your students say, but it is always an interesting discussion, and fun to follow up with another exploration deriving the power rules for multiplication and division and tying that back to what you uncovered here.

The deeper symmetry that I appreciate in this is how the exponent function maps addition onto multiplication, and there are two features that I realized for the first time recently. 

- The pattern on the left ranges from positive infinity to negative infinity. The pattern on the right ranges from positive infinity to positive infinitesimals (infinitely small pieces).  That's cool how this joins the two forms of infinity, and how one side includes negatives and the other side excludes them.
- The two patterns arrive at the identity for their respective operations together.  0, the identity for addition/subtraction is matched with 1, the identity for multiplication/division.  In a way, they are equated in parallel universes.  I love that. 

There is a story that I tell my students about the Garden of Addition and the Garden of Multiplication that help them to see how addition/subtraction is really one operation, likewise multiplication/division, and the role of 0 and 1 as identities, and the unique (and frightening) role 0 plays in the Garden of Multiplication.  You could probably make up your own, but I will post it here in a little bit.