Good day, student of introductory astronomy!

Since you were interested enough to inquire, I have decided to pass
along the following to you. This is in regards to the easterly shift of
the Moon in the sky, along the ecliptic, from one day to the next,
say if you were to observe the moon at the same clock time over a
period of a week or so.

If you were to figure how much the moon shifts along the ecliptic
relative to the background stars then:

shift = 360 degrees / 27.32 days per revolution (with respect to the
stars) = 13.2 degrees/day, and so the Moon would lie 13.2 degrees
to the east relative to the stars (using the stars along the ecliptic as
reference points).

On the other hand, if you were to figure out how much the Moon
shifts relative to a direction along the horizon, that is with respect to
a moving earth, then this is slightly different. Let's say that at 10 pm
the moon lay on a line connecting your zenith with due south along
your horizon from your location, and you checked again at 10 pm
the next night, then that shift would be:

shift = 360 / 29.53 days per lunar month = 12.2 degrees per day, and
so the moon would lie 12.2 degrees to the east of due south.

The difference in these shifts is of course due to the fact that the
Earth revolves around the Sun, about a degree per day, so that the
stars shift westward by about 1 degree per day.

If you really want to split hairs, there are other complicating factors
involved as well, but 12 or 13 degrees eastward per day are good
round numbers and are all anybody need understand in this class.

thanks for your interest,

Professor Korista