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Answer: (long answer) Concepts like phases of the moon and seasons are difficult for high school students and adults to learn! And when you put them together, it turns out to be an even greater challenge. OK, here goes...
The simple, approximate answer is that the moon would be in your sky about half of the time. But it would be up in the sky for about 2 weeks at a time. Keep in mind that by approximately, we mean within a couple of days. To understand this concept, you might want to use a scale model and a computer model to work through the motions, angles, and situations (see below for more information on the models).
In the United States (and anywhere from 45?N to 45?S really), the moon is in our sky half of the time due to the Earth's rotation, and you can see the full range of phases throughout each month. The time of day or night that you can see it in the sky just depends on the phase of the moon. You'll notice on our Daytime Moon Calendar that when the moon is waxing (past New Moon to Full Moon) it's best to look for the moon in the afternoon or evening. And when the moon is waning (past Full Moon to New Moon), it's best to look for the moon in the morning or late at night.
North and South Poles:
At the north or south poles, the moon would still be in the sky about half of the time, but it would be due to the moon's orbit around the Earth. At the beginning of Summer, when the sun is highest in the sky (and daytime all of the time), the moon would be in the sky for (approximately) 2 weeks; the 2 weeks closest to New Moon. (Granted, we can't see the moon when it is close to the sun in the sky, but you could observe it through the wider crescent phases.) Then the moon would be below the horizon for 2 weeks. At the beginning of Fall, when the sun is at sunset all of the time, the moon would be up in your sky for the 2 weeks closest to Last Quarter (waning), and then below the horizon for the next 2 weeks. At the beginning of Winter, when it's nighttime all of the time, the moon would be in the sky for the 2 weeks closest to Full Moon, and then below the horizon for the next 2 weeks. And at the beginning of Spring (click on the graphic), when the sun is at sunrise all of the time, the moon would be up in the sky for the 2 weeks closest to First Quarter (waxing), and then below the horizon for the next 2 weeks.
In astronomy, we often use models to explore concepts like this. I'd recommend playing around with two types of models; an actual 3D scale model, and a computer model. Both models have advantages and limitations, but they compliment each other nicely.
Use a computer program like Celestia (free) or Starry Night (easier) to simulate your observations. These programs simulate the view from any location, date, and time like a planetarium; and any program which will allow you to do this will work fine. Set your location for one of the poles, and explore various dates and times to look for the patterns described above. You may want to consider turning on the line that shows the moon's orbit, center and lock your view on the moon, and set your horizon to a flat horizon. Have fun!
(NOTE: the illustration to the right was not drawn to scale. Credit: Earth and moon images made from Starry Night.) [Click on illustration for larger image and explanation.] Set up a scale model of the Earth-Moon system, and a light to represent the sun. To make a scale model of the Earth-Moon system, see if you can find a small globe. Using a typical 12 inch (30 cm) diameter globe would require that the moon be about 30 feet (9 m) away. You can use something the size of a tennis ball, and carefully (safely) insert some type of stick to represent the north and south poles. Whatever you use for the Earth, the moon's diameter is a quarter of the Earth's diameter. In the example of using a tennis ball, the moon would be a ball the size of a penny (.75 in or 2 cm). To calculate the distance between the Earth and the moon, the distance would be about 10 times the circumference of the Earth. Using a tennis ball, the moon would be about 80 inches (2 m) away. You can either calculate it (circumference = 2 x pie x radius), or simply wrap a string around the Earth ten times. If you use a string to measure the circumference 10 times, you can also use the string to maintain the proper distance between the Earth and moon in your model.
Don't worry about trying to incorporate the scaled size and distance for the sun: it's too big and too far. For the sun, simply place a bare, lighted light bulb in the center of the room, and figure out where the Earth's winter (north pole facing away from the sun), spring, summer (north pole facing toward the sun), and fall positions would be.
Remember to tilt the Earth about 23.5 degrees, and be careful to always have the north pole pointed in the same direction, as the Earth orbits the sun counter-clockwise as seen from above the north pole. The moon orbits the Earth in a counter-clockwise direction as seen from above the North Pole. The moon's orbit around the Earth is tilted 5 degrees to the line between the Earth and the sun, but don't worry about that, just use that fact to eliminate solar and lunar eclipses.
Start with the Earth in the winter position, figure out the moon's position around the Earth for each of the phases, and imagine being at the north pole. What would you see? Then, move the Earth to the summer position, and do the same. And if you want the additional challenge, try the spring and fall positions as well.
Good luck, and have fun!