How would you explain to a 9th grader the negative exponent rule? Let us assume that the students haven't been exposed to these two rules: $a^{x+y} = a^{x}a^{y}$ and $\frac{a^x}{a^y} = a^{x-y}$. 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. However, some students confuse $2^{-3}$ to be $(-2)(-2)(-2)$ since they are familiar with $2^{3} = 2 \cdot 2 \cdot 2$. This is a low-income urban school and most kids in this algebra class struggle with math dealing with exponents, fractions and decimals. What would be the best approach to reach all 32 students? 


some students confuse $2^{-3}$ to be $(-2)(-2)(-2)$

Simple, $(-2)(-2)(-2)$ means ${-2}^{3}$, that is the first thing you have to show when they make that mistake.
Edit: That is the number minus 2, not a 2 combined with the unarry (-) and a exponent. Or it probably is, you might want to write ${(-2)}^{3}$ if it confuses you, i'm not even sure which bind stronger.
Next, show the relation between $2^3$ and $2^2$. One equals $2\cdot2\cdot2$ and the other $2\cdot2$. So $2^3$ = $2^2\cdot2$ and $\frac{2^3}{2} = 2^2$. Also, ${2\cdot2\cdot2\over2}=2\cdot2$
We can then move forwards (well, backwards),
$\frac{2^2}{2} = 2^1$,
$\frac{2^1}{2} = 2^0$,
$\frac{2^0}{2} = 2^{-1}$
etc.
