Get a good-sized water glass or tumbler--clear sides will make it easier to see progress, and entertain kids of all ages! Stick three toothpicks into the sweet potato around the middle. Place the sweet potato in the glass, with the toothpicks resting on the rim of the glass, and one end hanging down in. Pour enough water into the glass to cover the lower third of the tuber. If your tap water is heavily chlorinated, you might want to let the water sit in an open pitcher for 24 hours, to let the chlorine evaporate. If your water is very hard, you might want to use purified water, instead. If you have access to unused aquarium charcoal, or horticultural charcoal, put a few chips into the water to help keep it pure. Gently change the water every 2 weeks. Put the glass in a warm, sunny window.
It will be at least 3 days, and may take up to 2 weeks for roots to show, and about another week for shoots to develop. Fertilizer will be unecessary, unless you plan to keep it as a permanent house plant.
You may have to start two or three tubers to get one to sprout--sometimes they are treated to prevent sprouting before being shipped to the grocery store.