Lay three layers of paper towel on the plastic tray.
Spray the paper towel with the diluted plant food until saturated. This will supply some of the initial moisture to prevent the peels from drying out. In the normal sprouting process, whole seed potatoes are simply set in a box or crate whole, allowing the sprouts to feed from the nutrients and moisture in the un-cut potato.
Cover the paper towel with about 1/4 inch of potting soil.
Gently press the peels into the layer of soil. Give each peel a little wiggle for the soil to cover the edges without compacting the soil below the peel.
Spray a bit of diluted liquid fertilizer around the peels. Add a bit more of moisture to the soil, but do not spray directly onto the peels to prevent fungal growth on the peels and the buds.
Store the tray and peels. Set the tray in a warm place -- about 70 degrees Fahrenheit -- that will allow the peels moderate sun exposure.
Periodically moisten the soil around the peels. Check the peels about every three days, spraying an application of the diluted plant food onto the soil between the peels. The paper towels will absorb and redistribute the moisture. When the peels that grow buds reach about a half inch long, then those peels can be transplanted. Note: Not all peels will grow buds, because store-bought potatoes have been chemically treated with a sprout retardant.
Prepare the soil. Select a sunny place in the yard that gets a lot of drainage, a clean potting container, clean fish tank or other see-through container for educational projects. In the yard, dig and loosen the soil to a depth of about 18 inches. Mix in composted matter or potting soil if the ground itself is hard, light gray or light brown and powdery. When attempting to grow the potato plants in a container, use a new bag of potting soil, preferably with a compost additive to ensure the soil has sufficient plant nutrients.
Plant the sprouted peels about 10 to 14 inches apart. If a peel has more than one sprout, cut it apart, as each sprout equals one plant. Try to maintain a similar spacing in planters, if possible.
Lightly water the newly planted sprouts. Dampen but do not drench the soil with water or a dilution of liquid plant food. Water the soil every two to three days, depending upon environmental conditions, allowing the soil to dry between waterings. Plants in containers may require more frequent watering, but do not provide plant food more than once a week.
Hill the plants. When each plant reaches about 8 inches in height, pull a small mound of soil up around the stem -- equal to about half of the plant's height. This process is called hilling, which encourages the root system to grow deeper. Repeat the process every two to three weeks as the vine of the potato plant grows longer.