Regardless of the time of the month, half of the moon is always illuminated by sunlight. The lunar phases depend on what portion of that half we see from Earth. If we see all of that illuminated half, then we observe a full Moon. If we see half of that illuminated half, we see a quarter phase (either 1st or 3rd).
Another thing your can teach your kids or learn with your kids is that specific lunar phases are up in the sky at similar times of the day. First quarter Moon phases are always up at the same time, while a Full Moon is up at a different time. Over the course of a couple of months notice the change in phase versus time and determine if together you can find any patterns. I bet you can!