When you place a sweet potato into a mason jar full of water, new stems and roots will grow from the potato. It may take a while, so be patient. I poke three toothpicks into the potato to hold it upright in the jar. Keep the jar filled with more water as it is used by the tuber and it evaporates.Once you start to see little white "nubs" on the side and bottom of the potato, you'll know that roots will be growing out soon.
I took cuttings, some rooted and some not, from my two sweet potatoes that have been sitting in these jars full of water for the last few months. It is obvious where you should make your cut with the stems that have roots on them, but a non rooting stems should be cut below the leaf (where you see little root "nubs" ever so slightly protruding from the stem (just below each leaf).
I cut anywhere from 1/4 to 1/2 inch below the leaf junction so that when I place these cuttings into another jar of water, I have a few nubs ready to grow more roots.
When these non rooting cuttings have some root growth on them, I'll plant them out into the garden bed. It may take a few more weeks to get some roots on them, so a little more patience is needed. Once rooted, plant the stems 2-3" inches deep and in rows about 18-24" apart in soil amended with compost.