Rose of Sharon can be propagated from softwood or hardwood cuttings. This refers to the stage of growth. Softwood is the stage just after the new growth has fully expanded but not turned woody. This usually happens in late spring. Hardwood is the stage after the new growth has fully lignified, or turned woody. In either case, the stem cuttings should be dipped in a rooting hormone and stuck in a well-drained rooting medium such as a mixture of peat and perlite. Keep the root zone moist but not saturated and maintain high humidity around the cuttings by covering them with plastic.
Rose of Sharon can also be started from seed, but the resulting seedlings will vary from the parent plant.