How would you explain to a 9th grader the negative exponent rule?

Solution 1:

You could say that the minus sign means an opposite; for instance, subtraction is the opposite of addition, and it uses the - symbol. Similarly division is the opposite of multiplication, and the division symbol has a minus sign in it (this isn't where it comes from, but is only good for memorizing purposes).

So a minus in the exponent is the opposite of multiplying over and over again, namely dividing over and over again.

Solution 2:

How about motivate them by using the familiar example of scientific number? For example, $1.23\times 10^{2}=123$ where the dot is moved to right by 2 place, and so it make sense that the other direction apply too, that is $1.23\times 10^{-2}=0.0123$. And the only way that is true is if $10^{-2}=\frac{1}{100}$. And once you got that worked out for 10 power it's easy to just generalize it.

Solution 3:

Of course it's confusing to them.

"They have just been introduced to the generalization: $a^{-x} = \frac{1}{a^x}$ from the pattern method: $2^2 = 4, 2^1 = 2, 2^0 = 1, 2^{-1} = \frac{1}{2}$ etc"

In other words, you've made them memorize this thing that doesn't make any sense to them. If I didn't already know about exponents, I'd be confused too. A pattern is only useful if they can anticipate it. How is a normal kid supposed to go from positive exponents to a zero exponent, let alone a negative exponent, when you haven't even explained the basic rule of exponents?

Solution 4:

You could use the rules that they already know.

Since $-n=0-n$, $$ x^{-n}=x^{0-n}=\frac{x^0}{x^n}=\frac{1}{x^n}. $$

Solution 5:

You could make a table like this:

$$ \begin{array}{c | c} n & 2^n \\ \hline \cdot & \cdot \\ \cdot & \cdot \\ \cdot & \cdot \\ 1 & 2 \\ 2 & 4 \\ 3 & 8 \\ 4 & 16 \end{array} $$

on a blackboard and ask how the "natural" continuation upwards of each column looks like. So if you read from the bottom $4,3,2,1,\ldots$ what numbers follow? And in $16,8,4,2,\ldots$ what numbers come next in that sequence (the "rule" is to half each term, clearly).