Final answer:
A moon that looks half full and is on its way to becoming a full moon is called a waxing gibbous.
Step-by-step explanation:
A moon that looks half full and is on its way to becoming a full moon is called a waxing gibbous. During the week after the first quarter phase, the Moon goes through the waxing gibbous phase, where we see more and more of its illuminated hemisphere. Eventually, the Moon reaches the full phase, where it and the Sun are opposite each other in the sky.