The Phases of the Moon

As the Moon moves in its orbit around the Sun, our view of the side illuminated by the Sun changes.

Let us work through the cycle, starting with the full Moon.

At full Moon, the moon rises at sunset and sets at sunrise.

Excuse the misspelling in the figure; someday we will fix it!:
At 3rd quarter, the moon rises at midnight and sets at noon. Then we see only a crescent.
At new, the moon rises at sunrise and sets at sunset, and we don't see any of the illuminated side! Then we see only a crescent, but compare this to the waning crescent: what's different? Why?
At first quarter, the moon rises at noon and sets at midnight, Then it continues to grow until it is full.