A fragment is a small piece that’s come off a larger whole, and to fragment is to break. If your teacher writes "frag" on your paper, you've got an incomplete sentence.
Fragment, meaning "a tiny, brittle shard," first appeared as a noun and later as a verb. That afternoon you hacked away at the fireplace in your parents’ living room in search of secret treasure as a child? Those dusty chunks of brick you scattered all over their shag carpet were fragments of a once-intact wall, and a happier time before you were grounded. Not only did you cause physical damage that day, but you also fragmented their trust in you.