A lot of people used to think that the phases of the moon were caused by the shadow of the Earth on the moon; some still do. The problem with this explanation is that sometimes the "shadow" on the moon is concave and sometimes it is convex, and once every two weeks it is a straight line. The best way to understand moon phases is with a globe, a flashlight (the sun) and a grapefruit (the moon). In a darkened room, place the flashlight so that it shines on the globe. Move the grapefruit around the globe and note how, at different angles, people on Earth will see different parts of the moon, which is always half dark and half light--except during an eclipse.
Seen from Earth, the moon and the sun are almost exactly the same size. This makes for spectacular solar eclipses. If the moon appeared smaller, we would see a dark spot go across the sun. If the moon appeared larger, it would completely hide the sun. Because they appear to be the same size, we see spectacular blazes when the sun shines through the mountains on the moon. Lunar eclipses occur when the moon goes through Earth's shadow.
The time the moon rises (and sets) is coordinated with the phases. Kids can figure this out themselves if they look for the moon several days in a row. Ancient people always knew about this relationship because they spent so much more time outside. The new moon (very thin crescent) rises at dawn and sets at nightfall. The full moon rises at sundown and sets at sunrise, The half moon rises at noon and sets at midnight.
You can tell if the moon is growing or reducing by using the mnemonic "DOC." The relationship between the first two letters "D" and "O" show that when the curved part of the moon is to the right it is going to get fuller. The relationship between the last two letters shows what happens after the full moon: The moon gets smaller during this part of the cycle and the curved part is on the left. Given enough time, kids can figure out these relationships--although it is unlikely that they would also come up with the same mnemonic.