  We investigate a large sample of massive galaxies at $z\sim1$ with combined
$HST$ broad-band and grism observations to constrain the star-formation
histories of these systems as they transition from a star-forming state to
quiescence. Among our sample of massive $(M_*>10^{10}~{\rm M_\odot})$ galaxies
at $0.7<z<1.2$, dust-corrected H$\alpha$ and UV star-formation indicators agree
with a small dispersion ($\sim0.2$~dex) for galaxies on the main sequence, but
diverge and exhibit substantial scatter ($\sim0.7$~dex) once they drop
significantly below the star-forming main sequence. Significant H$\alpha$
emission is present in galaxies with low dust-corrected UV SFR values as well
as galaxies classified as quiescent using the $UVJ$ diagram. We compare the
observed H$\alpha$ flux distribution to the expected distribution assuming
bursty or smooth star-formation histories, and find that massive galaxies at
$z\sim1$ are most consistent with a quick, bursty quenching process. This
suggests that mechanisms such as feedback, stochastic gas flows, and minor
mergers continue to induce low-level bursty star formation in massive galaxies
at moderate redshift, even as they quench.
