The operation $\otimes$ is defined for all nonzero numbers by $a\otimes b =\frac{a^{2}}{b}$. Determine $[(1\otimes 2)\otimes 3]-[1\otimes (2\otimes 3)]$.
$\text{(A)}\ -\frac{2}{3}\qquad\text{(B)}\ -\frac{1}{4}\qquad\text{(C)}\ 0\qquad\text{(D)}\ \frac{1}{4}\qquad\text{(E)}\ \frac{2}{3}$
Follow PE(MD)(AS), doing the innermost parentheses first.
$[(1\otimes 2)\otimes 3]-[1\otimes (2\otimes 3)]$
$[\frac{1^2}{2}\otimes 3]-[1\otimes \frac{2^2}{3}]$
$[\frac{1}{2}\otimes 3]-[1\otimes \frac{4}{3}]$
$[\frac{(\frac{1}{2})^{2}}{3}]-[\frac{1^2}{(\frac{4}{3})}]$
$[\frac{1}{4} \cdot \frac{1}{3}]-[\frac{3}{4}]$
$\frac{1}{12} - \frac{3}{4}$
$\frac{1}{12} - \frac{9}{12}$
$\frac{-8}{12}$
$-\frac{2}{3}$, which is answer $\boxed{A}$