Actually I work in the hospital. We use supine and prone a lot. They are medical terms. Since I have been in US for 9 years, I have never heard them outside of the hospital. You are using them with your children. I think using lay on your back or lay on your stomach would be ok.
1. to place oneself or be in a horizontal position in order to rest
2. to yield to: never take any attack on your candidate lying down
1. to place on the ground or a surface
2. to sacrifice: willing to lay down their lives for the truth
3. to formulate (a rule or principle)
4. to record (plans) on paper
5. to store or stock: the speed with which we lay down extra, unwanted fat
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lie down, to assume a horizontal or prostrate position, as for the purpose of resting. lay down,
a. to give up; yield: to lay down one's arms.
b. to assert firmly; state authoritatively: to lay down rigid rules of conduct.
c. to stock; store: to lay down wine.
d. Shipbuilding. to draw at full size (the lines of a hull), as on the floor of a mold loft; lay off; loft.