The fragment part of A is the first part. While this may look like a sentence and may also transmit the same ideas as would a properly formed sentence, this is a fragment.
A sentences fragment is not a sentence because it is not an independent clause containing a subject and a predicate. In order to be a sentence, the fragment would need to be changed to something like the following:
He was a child who grew up in the New Jersey suburbs in the fifties.
