The Moon doesn't actually change shape, and it never really goes away. Half of the Moon is always lit by the Sun — the half facing the Sun. What changes is how much of that lit half we can see from Earth, and that depends on where the Moon is in its trip around us.
Watch the "from space" view: the white half of the Moon always points at the sunlight, no matter where it is. Now look at the dashed sight line. When we're looking at the Moon edge-on to that lit half, we get a half moon. When we're looking straight at the lit half, we get a full moon. The Moon never changed — our viewing angle did.
This is the part almost nobody is taught, and the "from the ground" view makes it obvious. The phase decides what time the Moon is up. Drag the month slider and watch the moonrise time change:
So "I saw the Moon this afternoon" isn't strange at all — it just means the Moon was somewhere in the first half of its month.
Seeing the Moon low in the east just before sunset is one thing. Seeing it high up in a bright blue sky, with the Sun also out, surprises people more. The key is that the Sun and the Moon travel the same path across the sky every day, east to west, one behind the other. How far apart they are on that path is the phase.
You can still catch a full-looking Moon in a blue sky, just low. For a day or two on either side of full it looks full to the eye. The day before, it comes up in the east an hour or so before sunset. The day after, it's still hanging in the west after sunrise. Summer evenings at our latitude stay bright long after sunset, which makes this easy to see.
So the Moon is highest in daylight around the quarters. From a thick crescent up to first quarter it's high in the afternoon, trailing behind the Sun. From last quarter back down to a thick crescent it's high in the morning, leading ahead of the Sun.
Set the month slider just before First qtr and the time to about 4 PM. The "from the ground" view shows the Sun sinking toward the west while the Moon sits near the top of the sky. In the "from space" view the sight line to the Moon is at a right angle to the sunlight. That right angle is what puts the Moon high up in the middle of the day.
Why a blue sky doesn't hide it: the blue is sunlight bouncing off air, and it's fairly dim. The Moon's surface is lit by direct sunlight, so it's much brighter than the blue around it. Stars are too faint to beat the blue, but the Moon is bright enough.
A bonus for right now: the height also depends on the season. On autumn mornings the last-quarter Moon sits where the Sun sits at midsummer, so it rides about as high as the June noon Sun, roughly 66° up at the Pass. On spring afternoons the first-quarter Moon does the same. That's why a high morning Moon in September or October looks so striking.
You just made every moon phase there is. Your turning is the Moon going around the Earth — one full spin is one month.
Tip: the hardest part is keeping the ball out of your head's shadow. Hold it a little high. If the ball goes dark all at once, that's an eclipse — a different thing, explained below.
This is the big one, and it's worth a whole poster board. The dark part of the Moon isn't Earth's shadow falling on it — it's just the half of the Moon that's facing away from the Sun, the same way the back of your head is dark when you face a lamp. You can prove it with the flashlight: at the moment you see a half-lit ball, your own shadow is pointing somewhere completely different.
When Earth's shadow does land on the Moon, that's a lunar eclipse, and it's rare and obvious — the Moon turns a dim coppery red for an hour or two.
A good question to anticipate from a judge. The Moon's orbit is tilted about 5° compared to Earth's path around the Sun, so most months the Moon passes a little above or below the shadow instead of through it. Hold the ball slightly high or slightly low during your demo and you've just shown exactly that.
One more for the board: the Moon keeps the same face pointed at us all the time, so there's no permanent "dark side." Every part of the Moon gets two weeks of daylight and two weeks of night — we just never see the far half from here.
The phase shape, the percentage lit and the "jump to right now" button are computed properly and match the real sky. The two diagrams are teaching pictures, though, and they round off some corners: