Proportion

$\frac{-3}{6}=\frac{a}{18}\quad$

$\frac{a}{18}=-\frac{1}{2}$

$:\quad a=-9$

-Vinculum

Since $\alpha$ is inversely proportional to $\beta$, $\alpha = \dfrac k\beta$ for some nonzero real number $k$, which we call the proportionality constant.

We find k by substituting a pair of values of $\alpha$ and $\beta$ into the equation.

Substituting $\alpha = -3$ and $\beta = -6$, $-3 = \dfrac k{-6}$. Solving gives $k = 18$.

Now, substitute $\beta = 18$ and $k = 18$ to get the corresponding value of $\alpha$ when $\beta = 18$.

$\alpha = \dfrac{18}{18} = \boxed{1}$.