I had a flash of insight yesterday while teaching exponents to my 8th graders.

As any math teachers are probably aware, it is difficult to get students to really understand why

I usually go through the process of showing them what happens every time you lower the exponent by 1, and show the pattern, which is also a nice way to motivate negative exponents. For example:



Here the explanation is that when the exponent goes down by 1, the value is divided by 2. The same pattern continues to 0 and negative exponents.

They are usually able to guess the negative values from the pattern (when encouraged to “follow the same pattern”). But in general they REALLY want the value to be zero when the exponent is zero. “You have zero 2’s, so it should be zero!”
So I started talking about the invisible 1 that is always there in multiplication and causes so many problems to students by being invisible all the time (I am finding more and more places as a teacher that it is useful to make this invisible 1 visible).


So I came up with this great idea (yeah, I know, not very modest, but it was just sooo effective) for explaining the zero exponent which got a full class of “oh, I see”. I now explain that exponents really work like this instead:



Now the zero exponent follow exactly the same rule and the answer follows completely naturally. This will also help them with cancelling later (ever tried to teach canceling as a shortcut and have students put a zero instead of a 1 when they cancel everything on the top or bottom of a fraction?).