Save those broken leaves/petals, don't throw them away! Just place in the shade and let them sit for a month or so. You can also gently pull away outer/lower leaves to do the same. Some succulents won't propagate this way, but many will. Experiment, you can even lay them on top of soil or gently place an end into soil. Key is that they are completely dried/calloused/scabbed on the end, otherwise they can get infected and rot on you.