1. We see the Sun during the day because we're on the part of the Earth that faces the Sun. There are lots of other stars behind the Sun, but they're light years away, so we don't see them. We might occasionally see the moon if it's in the "right spot" in its orbit of the Earth during our daytime.
2. We see other stars at night because the part of the Earth we're on no longer faces the Sun, and we can now see the other stars that are farther away. The stars we see change as the Earth orbits the Sun. In fact, we can predict what we see each year, because constellations reappear as we re-enter the same part of orbit each year. These stars appear so small compared to the Sun because they're really far away.
3. The moon appears sometimes during the night because it's lit up by the Sun and in our nighttime sky. What we see of the moon changes not because its size changes, but how much of it is illuminated by the Sun does.
This left us thinking...how does the Mystery Sun happen then?
Using the globe and our inflatable Sun, we tried to figure out how the Earth must be positioned and where it must be in its orbit of the Sun to ONLY get sunlight (and therefore never experience nighttime).
We set up some physical models that looked like this: