A sentence fragment is a group of words that looks like a sentence, but isn't. To be a sentence, a group of words must have at least one independent clause ( a group of words that contains a subject and a verb and can stand on its own as a complete thought). Taking this into account, the answers to your questions would be the following ones:
Children can learn to help out by completing their chores.
Explanation:
"Their" specifies that it is plural which is also what "Children" specifies. If it were to be singular, you could put "A child can learn to help out by completing his or her chores."