Wrong! Earth's shadow has nothing to do with Moon phases, only lunar eclipses. And your half-moon is really a quarter-moon! Take a large and a small ball and a single light source from a distance. Set them side by side and imagine your point of view as some point on the big ball and as you do this remember that the moon's orbit about the Earth is highly incline, i.e. it does not follow around the equator but it is like a hat brim tilted back on your head. Very seldom does the Moon ever cast its shadow on the earth or vice versa.

As you move the moon ( the small ball ) in a circle about the big ball (Earth) you will see that half the ball is always in light but that lit portion is not always fully visible from Earth. When it is on the farther point away from the light source ( the Sun) we see a full Moon. In seven days the Moon is at a right angle from the Earth/Sun line. That's when we see the "half-moon" which is the third quarter: three quarters of the way around the Earth. When the Moon is more or less between the Earth and Sun (7 days later) it disappears and this is the "New Moon". Seven more days it is at a right angle to the left again we see a "half-moon" and this is the first quarter which is on the PM side of things. The Moon transverses about 13 degrees of sky a day and makes an orbit in about 29 days. Approximations because there are other considerations which show up as variables.

If the Earth/Moon/Sun system actually did lie in the same plane as on a floor then the would be a lunar eclipse at every full moon and a solar eclipse at every new moon. But such is not the case. Again, the orbit is inclined so such encounters are relatively rare events. I think I remember reading that in 360 years nearly every spot on Earth will have experienced a solar eclipse, but I'm not waiting! 2012 I hope to be in Australia for such an event!

In interesting phenomenon can be seen clearly around the crescent phases and that is Earthshine falling on the otherwise darkened portion of the Moon. Sometimes a faint reddish glow is seen caused by the crud in Earth's atmosphere. I hope this has helped.

As the Earth and the Moon plow around the Sun in their annual journey they weave an interesting dance. The Earth is little moved by the Moon but the Moon traces a wavy line, in and out, back and forth across the Earth's track. It's a fairly easy thing to plot this out, either in whole or in portion, on a piece of paper and then connect the dots.

Tom