jmc

Since octagon $ABCDEFGH$ is a regular octagon, it is split into $8$ equal parts, such as triangles $△ABO,△BCO,△CDO$, etc. These parts, since they are all equal, are $\frac{1}{8}$ of the octagon each. The shaded region consists of $3$ of these equal parts plus half of another, so the fraction of the octagon that is shaded is $\frac{1}{8}+\frac{1}{8}+\frac{1}{8}+\frac{1}{16}=\boxed{\frac{7}{16}}.$