The Cycle of Phases in a Picture
Phases and Angles
As most of us are probably aware, the moon’s phases proceed in a repeating sequence. The moon grows from ‘new’ up to ‘full’, then shrinks back down to ‘new’, and then it starts over again. Many of us may not have noticed that this pattern is also connected to the position of the moon in the sky, and to the time of day when you see it. The ‘waxing crescent’, for example, is always a beautiful sliver over the sunset. With my students, I call it the ‘sunset moon’. Conversely, the ‘waning crescent’ is always visible over the sunrise — it's the ‘sunrise moon’. The full moon is the ‘opposite moon’ — it is always the opposite of the sun, on the other side of the sky, and on the other side of us, from the sun. It rises in the east as the sun sets in the west, and it can never be in the sky at the same time as the sun. Similarly, the ‘gibbous’ phases are always far from the sun in the sky, and the ‘quarter’ phases are always a quarter-turn from the sun in the sky.
Some time early in the school year, when the moon is a waxing crescent, I make sure to have the students watch what happens to this ‘sunset moon’ for at least a few days, so they can notice it getting higher over the sunset each day, and growing larger as it goes. And I try to take at least one trip outdoors during the daytime to point out the moon in the daytime sky. (Trying to find a thin crescent moon in the daytime can be a fun challenge.) Then later in the year when it is time to put the pieces together, I give them a handout with two pictures of a horizon, stretching from the eastern horizon on the left to the western horizon on the right, and representing the view of a person facing south and observing the sky. On one of these pictures, we draw the sun setting in the west, and we draw a series of moon phases, growing from the ‘sunset moon’ through first quarter to full, as the moon creeps westward away from the sun, and ending with the full moon rising in the east as the sun sets in the west. On the other horizon, we switch our attention to morning, we draw the sun rising in the east, and draw the reverse sequence of shrinking moon phases as the moon creeps closer to the sun across the morning sky. (I have included a blank and a filled-in version of the handout in the downloads section so you can see what I mean. This is perhaps not my finest work of craftsmanship, but I think it gets the point across.)
This handout helps make clear that there is a direct connection between the phase of the moon, and how far away it is from the sun in the sky. This isn’t strictly necessary, but I find it helpful for older students to review a little geometry, while simultaneously reinforcing the names of the phases: The moon is always crescent when it is at an acute angle from the sun in the sky, it is always a quarter when it is a quarter-turn or a right angle from the sun in the sky, and it is always gibbous when it is an obtuse angle from the sun in the sky. By extension, we can also say that the new moon is at a ‘zero angle’, or nearly so, and the full moon is at a ‘straight angle’, i.e. 180°, or nearly so. (If the moon were exactly in the same place in the sky as the sun, we would see a solar eclipse, and if it were exactly 180° from the sun, we would see a lunar eclipse — but that’s a discussion for another time...)
A Visual Summary of the Phase Cycle
What if we wanted a single picture, a cartoon of sorts, depicting the entire cycle, rather than two halves? It takes very few hints, if any at all, to get a student to suggest turning one of the pictures upside-down, and uniting the two pictures. Doing so produces a circular picture something like this:
This represents the entire repeating sequence of phases in a single ring, with all of the phases shown at their respective angles from the sun in the sky. What can you do with such a graphical model of the phase cycle? For one thing, you can figure out (approximately) what phase the moon will be at any point in the future, or at any point in the past, simply by counting phases counterclockwise or clockwise, one quarter-turn for each week, and one full turn for each month.
Representing the Horizon and the Sky
Another thing you could do with such a dial is add a horizon cover. Assuming you live in the northern hemisphere, you can cover the bottom half of the dial, and the upper half will represent the path across the sky along which the sun and moon travel, as seen by someone facing south, from rising in the east to high in the south to setting in the west. Furthermore, you can make it represent any time of day, simply by turning the dial so that the sun is in the appropriate part of the sky. For example, if you turn the dial so that the sun is touching the horizon on the right (west), representing sunset, the dial will look just like the ‘evening moon’ figure we drew earlier, and it will show the positions in the sky of all the moon phases visible at the time of sunset. If you turn the dial so that the sun is on the left, it will represent sunrise, it should look like the ‘morning moon’ figure we drew earlier, and will show the positions in the sky of all the moon phases visible in the morning. Similarly, you can turn the dial to represent noon by placing the sun at the top, or midnight by placing the sun at the bottom, and the dial will then show all the moon phases visible at those respective times. With such a dial, we have a ‘phase clock’ or ‘phase calendar’ with which we can see where the moon will be in the sky at any time of day for any given phase.
Incidentally, is the moon up more in the daytime, or is it up more at night? Or is that a trick question? We normally think of the sun as the ‘ruler of the day’, and the moon as the ‘ruler of the night’, but is that really accurate? Well, yes and no. The moon is actually above the horizon in the daytime just as often as it is up at night. But there’s a reason we think of the moon as being a nighttime thing. Turn the dial to represent noon, and notice which phases are visible at that time. Do the same for midnight. Do you see a reason why we might think of the moon as a nighttime thing, even though it is up in the daytime just as often? For one thing, the FULL moon is in fact a purely nighttime object. The full moon can never share the sky with the sun. For another thing, all the phases that can be up during the day are the thinner, dimmer ones, and are much harder to see in the daytime sky. All of the larger, brighter phases are up mostly at night.
The Two Motions, and the Moon’s Orbit
In the version of ‘phase dial’ that I present here, I have also included two arrows on the dial indicating the two directions of motion in the sky, clockwise and counterclockwise. "Which way does the moon go?" can be a bit of a trick question, and I think it is important to spend some time on clarifying which way things go in the sky.
If you watch the sun or the moon for a few hours and compare their positions in the sky at different times of day, you will observe that they both go clockwise (if you live in the northern hemisphere, and face south). In other words, they both rise in the east, pass from east to west across the sky, and set in the west, along roughly the same path, going once around each day. It is as if they are racing each other from east to west across the sky each day. With the phase dial (and assuming you reside in the northern hemisphere), this corresponds to facing south and turning the dial clockwise. As you turn the dial slowly clockwise, it will show the sun and moon rising on the left (i.e. in the east), passing in an arc across the top (i.e. high in the southern sky), and setting on the right (i.e. in the west). Clockwise motion represents daily motion across the sky.
But if you notice the beautiful crescent moon over the sunset one day, and you observe it at the same time each day for many days, you will notice that, compared to the sun, the moon goes counterclockwise. If the sun and moon are racing east to west across the sky, then the moon is gradually falling behind the sun a little bit more each day, and thus creeping gradually from west to east as the days go by. It is falling behind in the race, and its relative motion is west to east. With the phase dial, this corresponds to shifting your attention counterclockwise through the phases. To follow the moon through its phases, you have to go counterclockwise around the dial.
With sufficiently mature students, maybe 3rd or 4th grade and above, depending on circumstances, I might spend a little time exploring the causes for these visible motions. Everything in the sky rotates from east to west about once a day. One explanation for this might be that everything in outer space is more or less standing still, and we are spinning west to east as we watch it, like watching the world go by as we turn an a merry-go-round. But if this is so … if the apparent motion in the sky is really due to the earth spinning from west to east, and everything only looks like it's moving east to west … then what about the much slower creeping motion of the moon? That slow west-to-east motion of the moon compared to the sun, once around each month — that's a real circular motion of the moon around us. That's the orbit of the moon. When you watch the crescent moon rise higher over the sunset day by day, you are watching the moon orbit slowly west-to-east around us.
Making a Lunar Phase Dial
Printing and Filling in the Template
Start by downloading the phase dial, and printing it onto card stock. Normal paper works almost as well, but card stock will be more durable and have more heft. Shade in the moon phases, and label the sun and the phases. When shading, remember that the side of the moon facing the sun is always bright, and the side facing away is always dark. As far as I can tell, ‘third quarter’ and ‘last quarter’ are equally common names, but I chose to go with ‘third quarter’. The names ‘waxing crescent’ and ‘waning crescent’ are a little more standard than ‘new crescent’ and ‘old crescent’, but also less picturesque. I have omitted arrowheads on the arrows, so that we can spend some time clarifying "which way things go", and then the students can draw in their own arrowheads.
Assembling the Dial
To assemble the dial, cut out both pieces around their outlines. Place the cover over the disk portion with the centers aligned, and press a 2″ brass fastener through the center of both pieces. Fold the legs flat against the back of the dial and flatten them as much as possible. I find that if you turn the dial right-side-up and press against the brass knob with your thumb, or lay a weight on top of the dial, it helps to make the brass legs on the back side very smooth and flat. Finally, tape the legs of the brass fastener against the back of the dial. The assembled large version looks like this:
If you have followed the instructions to this point, you will now have an assembled dial, but the dial will be stuck and not turn. If I had made the hump in the middle of the dial a little smaller, it would have been possible to reach the center mark with a hand-held hole punch, and to punch a hole in which the brass fastener could turn. As it is, if you gently turn the dial by hand until it spins freely, it will ream out its own hole. That’s not the most elegant way to do things, perhaps, but it works perfectly well.
Problem-Solving With the Dial
I generally take a couple of class periods with my students when making these dials to practice using them to solve problems, and I have included a couple sets of example problems in the downloads section.
We normally stop after the labeling step, but before cutting out the parts, and practice using just the dial part to look ahead or look back and deduce what the moon phase used to be, and what it will be. Taking each quarter-turn around the dial as lasting approximately a week, and a full phase cycle as approximately four weeks, which is approximately a month (or a ‘moonth’ as I say to my kids), then even small children can easily get the knack of saying what the phase will be or what it was any number of weeks in the past or the future. Note that two weeks forward or back always brings you to the ‘opposite’ phase from where you started, and four weeks forward or back always brings you back to where you started. (If you want to get more precise with your students, you can tell them that the ‘synodic period’ of the moon, i.e. the time to complete one phase cycle, is actually 29½ days. For the purposes of these estimations, however, we can just take it to be four weeks or a month.)
Can you tell approximately when the moon will rise or when it will set? If the moon is currently a third quarter, and you go outdoors to look for it after dinner tonight, will you be able to find it? After assembling the dial, we move on to questions of time of day, and when the moon rises or sets. I have also included a set of example ‘moon time’ problems in the downloads section. In these problems, you are given a moon phase, and the question is to find out what time it rises or sets or is in any particular position in the sky. To solve them, find the intended moon phase, and turn the dial until the moon is in to the desired place in the sky (i.e. at the left for ‘rising’, at the top for ‘high in the south’, and so on). The time of day will then be determined by where the sun is (i.e. ‘sunrise’ if the sun is at the left, ‘noon’ if the sun is at the top, and so on). For ‘sunrise’, ‘sunset’, ‘noon’, and ‘midnight’, the corresponding word should also show in the window. The ‘challenge’ problems are not so much challenging to solve as they are challenging to express the answer in words. In all previous cases, the sun is either at the left, right, top, or bottom, giving times of sunrise, sunset, noon, or midnight. In the ‘challenge’ problems, the sun will be at one of the four ‘corners’, i.e. top left, top right, bottom left, or bottom right. I find it helpful to point this out to the students, and we discuss the best way to express the time of day at these times. Sometimes we give an approximate clock time (‘around 9 O'Clock in the morning’), but I usually try to get them to agree on ‘late morning’, ‘early afternoon’, ‘late evening’, and ‘very early morning’ or ‘the wee hours’.
Given a moon phase and a time of day, one could also deduce where in the sky the moon will be, but I find it harder to keep the student's interest with these sorts of problems. With the moon time problems, some children might actually have an interest in knowing when during their day to look for the moon. With the ability to solve direction problems, one would have another way to tell direction if you were lost in the wilderness at night, and some children might think that would be a good ability to have. But after all the other problems we've solved, solving more problems risks getting a little monotonous. Sometimes I give direction problems as an optional challenge, and there are usually a few children who are eager to try. For these problems, you would turn your dial to the specified time of day, find the current moon phase, and note the position in the sky (or note that it is ‘below the horizon’) at that time.
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