Unit 5: Measuring and Mapping the Sky

Observation and recording what we see in an accurate way is the foundation of all scientific knowledge. Map making is one of the oldest scientific activities, it certainly predates written language and recorded history by many millennia. The oldest known drawings of constellations are on clay tablets more than 15,000 years old; maps of the lunar phases date back more than 30,000 years. Even though map making is a very ancient activity, it is not a natural one. Map making is an acquired skill that requires practice, but with the use of simple tools even very young students can do a remarkable job of it.

Maps are also great teaching tools. Keep in mind that younger students are very visual learners. Young students who possess only basic literary and logical skills often find it difficult to follow ideas or arguments that are presented through language – this is also a fundamental problem for the ESL student.

Maps put information in an easy to understand visual format, as well as putting information into context which helps the student build a mental framework. Helping students to integrate new knowledge in with what they already know can be a daunting challenge. Map making helps make this process easier, and more effective.

Activity 9: Altitude and Azimuth – Your Place in the Sky

The focus of this activity is to teach students to use some simple tools, a compass and a protractor. The compass will be used to measure bearing or azimuth of a distant object such as a tree or telephone pole. The protractor will be used to measure the angle between the horizon and the distant object, this is also called the altitude. The protractor is not the plastic half-circle model you may be thinking of – instead we will use a human arm and a common classroom ruler to measure angles! It turns out that if you hold a ruler at arm’s length, one centimeter measures an angle of one degree [1].

This activity is also best conducted in the daytime, and can even be done indoors although it works best out in the school yard or playground. After your students learn to use these tools properly, the Being an Astronomer section will give them an activity they can use to try their new skills out after dark at home in their own back yards.

Academic Standards

Science and Engineering Practices

  • Planning and carrying out investigations.

  • Analyzing and interpreting data.

  • Using mathematics.

Crosscutting Concepts

  • Scale, proportion, and quantity.

Next Generation Science Standards

  • Engineering and design (K-5, 6-8, 9-12).

For the Educator

Facts you need to know

  1. A small magnetic compass can tell us which way we are pointing; this direction or compass bearing is also called azimuth. In this system, north is 0o, east is 90o, south is 180o, and west is 270o
  2. How high something is off the horizon is called altitude. We record this angle between the horizon and any object in degrees and measure it with a simple classroom ruler.
  3. By measuring altitude and azimuth together, we can precisely locate any object in the sky!
  4. Measuring angles is typically done with a protractor. We can make a simple device using two rulers bolted together to reproduce angles and record them accurately, allowing us measure them later in the classroom. This will be very helpful in mapping constellations!

Teaching and Pedagogy

Unlike our previous activities, this one is about learning to use tools to measure things. You may be thinking: ‘But I already teach my students how to use a ruler and a protractor to measure things!’ This activity is fundamentally different.

With this new activity, students can learn to measure things that are too big, or too far away to measure in any conventional way. Learning how to measure distant things like the Moon, the Sun, and other planets and stars is a problem that astronomers have been dealing with for many thousands of years – and we are still working on it today!

Once your students have mastered using the compass and ruler to measure altitude and azimuth, students can apply these skills to actually map the position of the Moon in the sky! The important thing with this activity is to make sure the students hold the ruler at arm’s length. Holding the ruler at arm’s length insures that the distance between the eye and the ruler is the same every time. If your students do not do this, their results will not be consistent!

Student Outcomes

What will the student discover?

  1. Using a magnetic compass and a ruler to measure altitude and azimuth will allow your students to accurately observe and record the position of any object in the sky whether near or far!
  2. Using two classroom rulers fastened together, your students will learn to methodically produce accurate maps of any constellation in the sky, reproducing size and shape accurately.
  3. Map making is a valuable scientific skill that requires good observing skills and patience! Accurate maps of constellations help us understand the relative size and shape of constellations – even if they are in very different parts of the sky!

What will your students learn about science?

  1. Many students confuse observing with looking. Observing is a useful and practical skill that is essential to the scientist and astronomer. These exercises will help develop this valuable skill in your students, regardless of age.
  2. Mapping, recording the position and size of an object relative to the things around it is another way to make a scientific model. In this case, the model is put down on paper instead of being made of objects, but the principle and usefulness is precisely the same!

Conducting the Activity

Materials

  1. Small (at least 1-inch, larger 2 or 3-inch sizes will be easier to use) magnetic compass. If your students have smart phones, there are many compass apps available for free.
  2. A Ruler marked in centimeters
  3. Sidewalk chalk

Building the Altitude-Azimuth Measuring Device

This activity requires no construction – we are simply learning to use a ruler and compass in a new way!

Exploring and Measuring Altitude and Azimuth

  1. [Teacher] Take sidewalk chalk out to the play yard and mark an X to identify 10 or so places for students to stand while taking measurements. You may also wish to number these spots and write the name of the target next to the X. A simple worksheet which asks students to record the altitude and azimuth and then describe or even draw the object they are measuring is useful.
  2. Have the students stand on a fixed place (X marks the spot!) and hold the compass flat and level in their hands. Now turn toward the target (a distant tree or any other object) and adjust the compass so the N lines up with the compass needle; the direction you are looking toward the object shows you the bearing or azimuth direction. Using the compass properly will take some practice. This is often best done in the classroom where everyone can turn to each of the walls and corners of the room and measure azimuth bearings together to be sure everyone is doing this correctly and getting the same results[2].
  3. Once everyone has become familiar with the compass and taking azimuth bearings, it is now time to try measuring altitude. Once again, this can be practiced indoors or out. Have students stand on the mark and look toward the object they wish to measure. Hold the ruler at arm’s length and count how many centimeters ‘tall’ the object is. It is sometimes useful for students to work in pairs. One student holds the ruler and sights the object, while the other runs their finger slowly up the ruler. When the finger reaches the top of the object, the observer calls “Stop!” and the measurement is read off the ruler. Record the measurement on the worksheet.

Discussion Questions

  1. If everyone measured the same things, why did we get so many different answers? Shouldn’t there be one correct answer?
    • Answer: The idea that there can be more than one correct answer can be disconcerting to some! In this case, apart from natural errors in measurement, some children have shorter or longer arms, some may not have stood in exactly the same place when they took their measurements. For nearby objects like buildings and flagpoles, the errors can be significant! Remind the students that this activity is about learning to use tools correctly, not necessarily about getting the right answers!
  2. If everyone measures a building or a flagpole so differently, how can we expect to measure the Moon and get a good answer?
    • Answer: When we measure things that are nearby such as a building or a streetlight, they are so close to us that moving our position just a little can cause a big change in the measurement. When we measure very distant things like the Moon however, it is so far away that the little distance between one person and another – even across town – will make no change in our measurement.

Supplemental Materials

Going Deeper:

Altitude-Azimuth is only one way of measuring the sky. This measuring system is centered on the point where the student stands. If two students were measuring the altitude and azimuth of Mars in the night sky, their measurements would depend not only on where they were standing, but the exact time when the measurements were taken.

The other principal measurement system for astronomers is called the Right Ascension – Declination system, or RA-Dec. This system borrows from the latitude-longitude system we use to measure our position on the Earth. Unlike the Altitude-Azimuth system, the RA-Dec system does not depend upon the observer at all.

See if you can find a map of the night sky using the RA-Dec system. What similarities do you see between this and the latitude-longitude system we use on Earth? What advantages would this system have for astronomers?

Being an Astronomer:

Now that your students have learned to measure altitude and azimuth, let’s apply these skills to measure and plot the path of the Moon! There are two ways to do this, the one-nighter activity that measures the Moon’s path through the sky over a single evening; and the multi-night activity that measures the Moon’s orbital motion over several days. Let’s look at each activity separately.

Being a Scientist:

Many coordinate systems have something in common – they can use the Pythagorean Theorem to determine distances. Take a look at a star map with lines of right ascension and declination on it. Each hour of right ascension = 15 degrees.

Find two stars or constellations and measure the distance between them in both the right ascension direction and the declination direction. Treat these measurements like two sides of a triangle and use Pythagoras’ equation to find the distance.

Distance=RA2+Dec2

Following Up:

Ancient cultures used many different ways to measure and mark the positions of objects in the sky. Pyramids, henges, and Sun-circles are just a few. See if you can find out how the Pyramids of Giza in Egypt or the Stonehenge in England were used for astronomy.

Activity 10: Measuring the Nightly Path of the Moon

There is a misconception that ‘doing real astronomy’ is difficult and expensive, only highly trained and generously funded people can do it; this book is designed to show that both of these ideas are false. Measuring the Moon’s orbital path through the sky is simple enough that a seven year old can do it in their own back yard with a little parental help.

This activity is simple enough in concept, and can be conducted any night the Moon is visible for several hours in the sky; practically speaking, this works best in the week between first quarter moon and full moon. Students will be taking an altitude and azimuth measurement of the Moon every hour for 4-5 hours. At least four separate measurements are needed for best results. The Moon’s diurnal motion will be plotted on a simple graph after the measurements are taken.

Academic Standards

Science and Engineering Practices

  • Asking questions and defining problems.
  • Planning and carrying out investigations.
  • Analyzing and interpreting data.
  • Using mathematics.
  • Obtain, evaluate, and communicate information.

Crosscutting Concepts

  • Patterns in nature.

  • Scale, proportion, and quantity.

  • Stability and change.

Next Generation Science Standards

  • Space systems (K-5, 6-8, 9-12).
  • The Earth-Moon system (6-8, 9-12).
  • Gravitation and orbits (6-8, 9-12).

For the Educator

Facts you need to know

  1. The Moon’s nightly path across the sky is apparent motion. This movement is actually an illusion caused by the rotation of the Earth.
  2. We see moonrise and moonset primarily because the Earth spins on its axis once every 24 hours.
  3. When we measure the Moon’s nightly motion, we are actually measuring the rotational motion of the Earth.

Teaching and Pedagogy

This activity is certainly about applying the measuring skills that we learned in Activity #9, but it does more than that. This activity allows students to take real measurements and then plot them out on a graph to help them understand what is actually happening in the sky as they watch the Moon sink toward the western horizon.

All too often, graphing is put forward with data that is detached from reality – this activity puts the activity of graphing solidly in the child’s realm of experience and allows them to see that mathematics and graphing have a concrete benefit in real-world situations.

Even if you think that the graphing activity is a bit too much for your younger students, you can still take these measurements and plot them on the board together. This activity makes a wonderful introduction to graphing and its power to reveal mathematical truths in an appealing, visual format.

Student Outcomes

What will the student discover?

  1. The sky is always changing! The idea that things in the sky are constant and unchanging is a common misconception. By observing the sky over just a few hours, students will see that the objects in the sky move, changing position in a regular way.
  2. Math helps us describe the change we see in a clear and precise way. Students often ask: “What do we need this for?” By adding numbers into our lessons in a natural way, we show our students that math is good for something, it isn’t just a puzzle to solve and struggle over!
  3. Things that look the same are not always identical. The idea of observing the Moon for a few hours one night – and then doing the same observation at the same time over several nights – might seem nonsensical. But there is power in observation, on one night, we see the Earth spinning. Over several nights, we see the Moon moving around the Earth in orbit!

What will your students learn about science?

  1. These activities bring home to students that there is no such thing as ‘just looking’ or ‘just measuring’. Just like playing the violin or dribbling a soccer ball, observing and then carefully measuring and recording what you see are skills that require patience and discipline to master.
  2. Some students may feel frustrated at first when they try these activities, especially if they do not get the quick and easy results they had been expecting. To be quite frank, some teachers delving into STEM science activities in the classroom for the first time often feel the same!
  3. Remind your students (and yourself!) that simple isn’t always easy! This elementary fact is a stumbling block for students of every age and academic level. The corollary idea that diligent practice brings results is also worth teaching – and remembering! As you and your students practice these activities together, your results and consistency will improve over time!
  4. Science often does not proceed smoothly. Often there are bumps and missteps along the way. As we have seen with Aristotle’s Earth-centered model of the solar system, sometimes these wrong ideas can persist for a very long time! It is good for our students to understand that science is a practical skill, not unlike playing a sport or a musical instrument; it requires some talent, (and lots of practice!) to excel at it.

Conducting the Activity

Materials

  1. A ruler marked in centimeters
  2. A yardstick, tape measure, or a ruler marked in inches will work equally well – the measurements just need to be converted before plotting them on a graph.
  3. A compass for measuring direction.
  4. If the student doesn’t have a compass, the parent’s phone will suffice. Most smart phones already have a compass app on them – if not, there are many free apps of this type readily available.

Measuring the Moon’s Nightly Path Across the Sky

  1. Begin at sunset by measuring the altitude of the Moon with a ruler – this is the Moon’s apparent distance above the horizon. Hold the ruler at arm’s length and measure the distance from the horizon to the center of the Moon’s disk.
    • If the Moon is too high off the horizon to measure with a simple ruler, try stretching a piece of string from the horizon to the Moon’s altitude, tie a knot to mark the length and then measure the string later.
    • If your ruler does not show centimeters, that’s okay! Just take the altitude in inches and multiply by 2.5 to get centimeters – and degrees!
    • Example: The string measures as 18 inches. 18 x 2.5 = 45 cm = 45 degrees altitude!
  2. Measure the azimuth of the Moon with a compass. The easiest way to do this is with a compass app on a smartphone. Point the smartphone at the Moon and read the azimuth angle off the display. If you use a conventional compass, keep the needle aligned with north, then look in the direction of the Moon and find the azimuth bearing. Use the instructions that come with the compass to help you.
  3. Repeat the exercise, measuring the altitude and azimuth position 4-5 times. Measurements should be taken at least 45 minutes apart to insure that the Moon has moved measurably. Record your measurements: time, altitude, and azimuth neatly each time so you can graph them later.
  4. The next day in class, plot out your Moon position data on a graph as shown below. You can use color-dot stickers to plot the Moon’s position and color in the phase if you like!
image

Discussion Questions

  1. How do the ideas of altitude and azimuth fit into this activity?
    • Answer: With any graph, we need two measurements to locate a point. In math, we normally label one axis x and the other axis y. In this activity, the vertical axis is altitude (the distance off the horizon) and the horizontal axis is azimuth (the compass direction).
  2. Graphs in math usually show locations (points) or equations (lines), what does this graph show?
    • Answer: The diurnal (daily) motion of the Moon across the sky.
  3. What is causing the motion of the Moon that we see in a single evening as it sinks toward the horizon?
    • Answer: The rotation of the Earth (The Moon actually moves from west to east!)

Supplemental Materials

Going Deeper

This time, our Going Deeper activity asks our students to change the time scale of the activity. Instead of observing the Moon for a few hours over the course of one evening, this activity asks them to observe consistently for 5-7 successive nights. While this may seem like a small change, the requirement to continue an investigation in a focused way over a longer period of time is excellent exercise for the gifted child, it teaches persistence and resilience as well as scientific facts. You will find precise instructions for this in Activity #11.

It is also useful to know that this activity, although superficially the same, really measures something quite different! Observing the Moon’s motion for a few hours over a single evening shows us the Moon’s east to west motion which is due to the Earth’s rotation every 24 hours on its own axis.

However, when we observe the Moon at the same time over a period of days, we are now recording something very different. We are now measuring the Moon’s orbital motion as it travels around the Earth each month!

This difference will become apparent when your students plot their data on the graph. Instead of seeing the points move from left to right (east to west) across the paper, the new graph shows the points moving the other direction – west to east! This is because the Moon in orbit actually moves eastward across the sky as it circles the Earth in space.

Being a Scientist

Part of the power of science is when we add careful numerical measurements to our observations, wonderful mathematical patterns emerge that help us understand, and predict Nature.

When we see anything moving, one natural question to ask is: “How fast is it going?” There are many ways to answer such a question; it is common to measure speed in either miles (or kilometers) per hour.

This is not the only way to measure speed! When something moves in a circle like the Moon circling the Earth, we don’t measure its change in distance because the Moon is always about the same distance away from the Earth. Instead, we measure degrees instead of miles or kilometers.

Your activity is already doing this; when students record the compass direction of the Moon in degrees, they are measuring the Moon’s position. By adding the time of day to each observation, they will have everything they need to measure the Moon’s angular velocity in degrees per minute.

Look at the example data chart below. The student records time and compass position of the Moon in the first two columns. To get degrees moved, start with position #2 and subtract the value above – here we subtract 202 – 185 = 17 degrees. Time is treated in the same way – here we get 62 minutes from 6:18 to 7:18.

imageThe Velocity is calculated by dividing degrees moved by time change – here we divide 17 / 62 = 0.27 degrees per minute. Finding similar values in the last column every time gives us confidence that we have made good measurements.

Remember: if you chart data taken over a single evening, you are measuring the speed at which the Earth spins. If you chart data taken over several nights, plotting the position of the Moon at the same time each night, then you are measuring the orbital speed of the Moon!

Following Up

Whatever the age level or math level of your students, every one of them can observe the Moon moving in the sky. Watching the Moon sink slowly into the west on a clear night a few days after the new moon can be very gratifying. Students will notice that not only does the Moon move westward, but so do bright stars in the sky. This is observing the rotation of the Earth.

When students later observe the Moon several nights in succession, looking at the same time each night, they will notice something different. Unlike the stars which start out in roughly the same position each night, the Moon begins in a different position each night! When we observe this, we are seeing the Moon moving in orbit around the Earth.

Activity 11: Measuring the Moon’s Orbital Motion

Try activity 9 again, but this time measure the Moon’s position in altitude and azimuth at the same time for several days beginning shortly after new moon, you will find that the graph is similar except that the points move eastward showing orbital motion and that the phase will change over several days! In order for this activity to be successful, students must remember to take their measurements at approximately the same time every day.

Academic Standards

Science and Engineering Practices

  • Asking questions and defining problems.
  • Planning and carrying out investigations.
  • Analyzing and interpreting data.
  • Using mathematics.
  • Argument from evidence.

Crosscutting Concepts

  • Patterns in nature.

  • Systems and system models.

  • Stability and change.

Next Generation Science Standards

  • Space systems (K-5, 6-8, 9-12).
  • The Earth-Moon system (6-8, 9-12).
  • Gravitation and orbits (6-8, 9-12).

For the Educator

Facts you need to know

  1. The Moon’s orbit around the Earth takes approximately 28 days.
  2. Because the Moon takes 4 weeks to orbit the Earth once – it takes about two weeks for the Moon to move from new moon (on the western horizon) to full moon (on the eastern horizon.)
  3. You will see that the Moon’s orbital motion moves west to east – this is in the opposite direction from its apparent east to west nightly motion.

Teaching and Pedagogy

As we have discussed in Activity #10 above, this activity is very similar. The process of measuring the Moon’s position in the sky (Altitude and Azimuth) are identical; the recording of the data will be made on an identical graph.

There are differences in the two activities, and these need to be emphasized for your students. Activity #10 is a one night event, all the data needed is gathered on one night, preferably just a few nights after the new moon. For Activity #10, students observe the Moon multiple times on the same night.

Activity #11 is different. This activity requires the student to observe the Moon of multiple successive nights – making a single observation at the same time each evening.

This sort of activity requires patience and persistence. There is no way to speed up the process, and neglecting the observations will spoil the data. Each observation takes only a few minutes, but the requirement for the observation to be taken at the same time means that parent support is needed for this activity.

Looking at it another way, this activity is an excellent way to improve parent involvement! You might wish to present this activity at a Back to School event, and get the parents involved in your school’s STEM program.

Student Outcomes

What will the student discover?

  1. Observations that look similar don’t always yield the same results. Sometimes paying attention to subtle details can yield ingenious discoveries.
  2. It is possible to track the Moon’s movement around the Earth. The Moon in orbit seems to represent the unreachable in Nature; it passes above us in the skies, but we cannot touch or influence it. Science gives us the ability to track, measure, and understand things that we cannot reach or touch.
  3. The Moon actually moves eastward in orbit around the Earth. Everything we observe in the skies moves westward, rising in the east and setting in the west. It is astonishing to many people to learn that the Moon travels in the opposite direction as it orbits the Earth.

What will your students learn about science?

  1. Science rewards the persistent. It is not easy to make observations over several nights, but the reward is the discovery of something astonishing – the Moon travels eastward – unlike most other objects in the sky.
  2. Planning and foresight are essential in any scientific activity. These skills pay many dividends in everyday life as well.

Conducting the Activity

Materials

  1. A ruler marked in centimeters
  2. A yardstick, tape measure, or a ruler marked in inches will work equally well – the measurements just need to be converted before plotting them on a graph.
  3. A compass for measuring direction.
  4. If the student doesn’t have a compass, the parent’s phone will suffice. Most smart phones already have a compass app on them – if not, there are many free apps of this type readily available.

Measuring the Moon’s Orbital Motion

  1. Begin at sunset by measuring the altitude of the Moon with a ruler – this is the Moon’s apparent distance above the horizon. Hold the ruler at arm’s length and measure the distance from the horizon to the center of the Moon’s disk. If the Moon is too high off the horizon to measure with a simple ruler, try stretching a piece of string from the horizon to the Moon’s altitude, tie a knot to mark the length and then measure the string later. If your ruler does not show centimeters, that’s okay! Just take the altitude in inches and multiply by 2.5 to get centimeters – and degrees!
    • Example: The string measures as 18 inches. 18 x 2.5 = 45 cm = 45 degrees altitude!
  2. Measure the azimuth of the Moon with a compass. The easiest way to do this is with a compass app on a smartphone. Point the smartphone at the Moon and read the azimuth angle off the display. If you use a conventional compass, keep the needle aligned with north, then look in the direction of the Moon and find the azimuth bearing. Use the instructions that come with the compass to help you.
  3. Repeat the exercise for 3-5 nights in a row, measuring the altitude and azimuth position at the same time each night. Measurements should be taken as close to the same time as possible each night. Record your measurements: time, date, altitude, and azimuth neatly each time so you can graph them later.
  4. The next day in class, plot out your Moon position data on a graph as shown below. You can use color-dot stickers to plot the Moon’s position and color in the phase if you like!
image

Discussion Questions

  1. How does Activity #11 differ from Activity #10?
    • Answer: Activity #10 was a one-night activity that we used to measure the Moon’s daily motion. Activity #11 requires several nights to measure the Moon’s movement in orbit around the Earth.
  2. Why must we observe the Moon for several days to see its orbital motion?
    • Answer: The Moon takes 28 days to circle the Earth once – it moves too little in a single night to measure this change easily.
  3. What is causing the Moon to appear to move eastward over several days?
    • Answer: This is the Moon’s actual orbital motion around the Earth.

Supplemental Materials

Going Deeper

It is often valuable in science to repeat an activity a number of times to see if you get the same answer. Getting a repeatable answer is considered to be an indication that the experiment was done correctly and that the conclusions drawn from the results are reasonable.

For this activity, it turns out that once again, things are not as simple as they seem. If you run the activity the first time in the fall semester, it will be instructive to run the activity again late in the spring semester. You will find the results to be quite different!

In the fall and winter, the Moon travels high above the horizon, taking a longer path through the night skies. While in the late spring and summer, the Moon travels a lower path much closer to the southern horizon [3].

The reason for this is the tilt of the Earth’s axis. We shall examine this idea later in the book.

Being an Astronomer

Ancient astronomers paid great attention to the constellations of the zodiac. These 13 constellations lie along the path of the Moon, Sun, and planets as they move across the sky.

Many smartphones have apps available that allow you to point the phone at the sky and see a map of constellations. These applications help people identify constellations, planets, and find the names of stars in the sky.

Try using one of these applications and identify which constellation the Moon lies in as you observe it for several nights. The fact that the Moon lies in different constellations as you observe it over several days is additional confirmation that the Moon is really moving in orbit around the Earth. Add this constellation data to your graph of the Moon’s orbital motion!

Want more challenge? Leave the smartphone alone and try to identify constellations from the patterns of the stars and a star map. Excellent monthly star maps are available on line for free.

Being a Scientist

Once again, we ask the question: “How fast the Moon moving in orbit?” This time we will not be measuring degrees per minute, but rather how many degrees per day does the Moon move in orbit?

Look at the example data chart below. The student records the day and compass position of the Moon in the first two columns. To get degrees moved, start with position #2 and subtract the value above – here we subtract 272 – 285 = -13 degrees (the negative value indicates eastward motion.) Time is always 1 day because we observe the Moon at the same time each evening.

The Velocity is calculated by dividing degrees moved by time change, here the time change is always 1, so the degrees moved is the same as the velocity. Finding similar values in the last column every time gives us confidence that we have made good measurements.

imageFor our last step, determine how long it would take the Moon to orbit the Earth at this speed. To do this, divide 360 degrees by the average velocity. Here: 360 / 13.3 = 27 days per orbit. Any value between 25 and 30 days per orbit is a reasonably good match to the true value of 28.3 days.

Following Up

The two moons of Mars, Phobos and Deimos, are an interesting comparison to Earth’s moon. These two moons move around Mars at very different speeds from each other – and much faster than Earth’s moon.

What can you find out about the orbital period of Phobos and Deimos? Why are they so different from each other? What controls the speed of a satellite in orbit?

Activity 12: Measuring the Earth with Eratosthenes

An ancient Greek astronomer named Eratosthenes was the first man to measure the size of the Earth accurately. His method was very simple: he measured the angle made by a shadow cast from a vertical stick in two different cities on the same day and time. With the help of another teacher, you can recreate Eratosthenes’ experiment and your students can measure the size of the Earth for themselves! All you will need is two yardsticks, a protractor, a magnetic compass, and a bit of string.

Academic Standards

Science and Engineering Practices

  • Asking questions and defining problems.
  • Planning and carrying out investigations.
  • Analyzing and interpreting data.
  • Using mathematics.
  • Constructing explanations.
  • Argument from evidence.
  • Obtain, evaluate, and communicate information.

Crosscutting Concepts

  • Scale, proportion, and quantity.
  • Systems and system models.

Next Generation Science Standards

  • Engineering and design (K-5, 6-8, 9-12).
  • The Earth-Moon system (6-8, 9-12).

For the Educator

Facts you need to know

  1. The Earth’s circumference was first accurately measured more than 2,200 years ago by a Greek astronomer named Eratosthenes.
  2. Eratosthenes method was very simple; he measured the length of a shadow from a vertical stick of a known height in two cities on the same day. The ratio between the north-south distance between the two cities and the angles measured gave a ratio which allowed Eratosthenes to calculate the size of the Earth.

Teaching and Pedagogy

This is a wonderful example of practical geometry and a powerful introduction into ancient cultures; the activity is not just STEM, but cross-curricular as well. It is a common misconception that just because cultures were ancient, they must have been primitive or simplistic. We often confuse technological sophistication for learning and knowledge. The activity where students actually work together with children from another school is living proof that this is not so.

This activity is also another example of the practical application of mathematics. Math needn’t be complex or totally divorced from reality; children actually respond and learn better when mathematics are presented in a real-world concept. I can think of no more dramatic answer to the perennial question: “What are we gonna use this math junk for anyway?” than to say: “We’re going to measure the size of the Earth today!”

Student Outcomes

What will the student discover?

  1. This is a lovely project for many reasons; as with Activity #10 and #11, students are able to use simple methods to do amazing things, in this case to measure the entire Earth.
  2. Eratosthenes measured the Earth to within 2% of the modern measured value. Using a stick, protractor, and a piece of string you students can easily do as well.

What will your students learn about science?

  1. Science is a cooperative venture. Without the help of student scientists at another school, this activity is not possible. Even though the activity itself is extremely simple (measure one angle at a specific time of day,) without cooperation nothing is gained.

Conducting the Activity

Materials

  1. A meter stick
  2. String or twine
  3. An accurate protractor

Measuring the Earth with Eratosthenes

  1. The first step is to contact another teacher at your same grade level who lives at least 100 miles directly north or south of you – farther apart is better for this experiment. A direct north-south line between the cities is also important for this, you will need to know as exactly as possible how many miles north or south of you the other school is as opposed to the direct mileage between the cities. Look a map and select a likely city, research their schools on the internet and reach out to someone by email and send them an invitation to join your class in this exciting project. It may take one or two tries, but I bet you can find a partner without too much difficulty!
  2. When the big day arrives, send an email in the morning to be sure you have sunny weather in both cities. A few minutes before noon, set up the yard sticks in the playground area. One stick should be held vertically, (use a small carpenter’s level for this). Use the compass to lay out the second yardstick flat on the ground so that it points directly north. You have now made a simple sundial! Watch as the shadow moves clockwise; when the shadow lies directly along the flat yardstick, measure and record the position where the tip of the shadow falls. Depending on your location and the time of year, the shadow may extend past the end of the flat yardstick – that’s okay, just mark its position with some sidewalk chalk.
  3. Now that you’ve marked the tip of the shadow, stretch a piece of string from the top of the vertical yardstick down to where the tip of the shadow touched the ground. Measure the angle between the vertical stick and the string with a protractor as accurately as you can and record it. Email this information to each other – it will be the difference between the angles that will be important for this activity!
  4. Eratosthenes believed that the Earth was round, and so the angle of the Sun in the sky would be different depending on how far north you were from the equator – and he was right! By setting up a simple ratio and proportion between the difference in the two angles and the distance between the cities, he was able to accurately measure the circumference of the Earth for the first time about 2,300 years ago. Eratosthenes’ calculation for the size of the Earth was accurate to within about 2% of our modern value, how close can your students get? Set up your calculation as shown below!

 

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5. The actual circumference of the Earth is 24,900 miles. The example above was done by my own students several years ago and shows a value within 4% of the true size of the Earth – pretty good for kids using some string and a protractor! How close will your students get!

Discussion Questions

  1. Eratosthenes obviously didn’t have a telephone or the internet, how do you think he managed to do this activity in ancient Egypt? (Egypt was then part of the Greek/Macedonian empire.)
    • Answer: Eratosthenes did not take both measurements on the same day! The astronomer took a measure of the solar angle in the town of Syene in southern Egypt on the summer solstice. He then walked to the town of Alexandria in northern Egypt and carefully measured the distance along the way and measured the solar angle again on the summer solstice in the following year.
  2. We sometimes think of ancient peoples as ‘primitive’ or even ‘ignorant’. What do you think of the ancient Greek culture of Eratosthenes now that you know that people in this era were able to measure the size of the Earth and Moon, and even measure the distance between them accurately?
    • Answer: The ancient cultures were not all ignorant or primitive! Many cultures have had ‘dark ages’ where learning was not advanced, but ancient cultures were in many ways remarkably advanced!

Supplemental Materials

Going Deeper

Understanding what is happening when we measure the solar angle at two different locations, and how this helps us measure the Earth, is a masterpiece of scientific thinking. Sometimes the power of a simple experiment or argument are difficult to grasp.

One of the ways to comprehend the thinking of Eratosthenes is to draw the Earth and Sun, showing the angles between the Earth’s core and the lines representing the rays of the Sun. See if you can understand Eratosthenes ideas this way!

There are many drawings of Eratosthenes ideas on the internet to help you!

Being an Astronomer

Measuring the solar angle with a stick, string, and protractor is another exercise that can show how the sky changes through the seasons. If your students can measure the solar angle once a week and keep a running record of the results, you will find that the solar angle changes measurably through the seasons.

Can you find a relation between the solar angle and the season?

Being a Scientist:

Climatic change is a hot topic in research and political debate these days, but climate doesn’t just change slowly over centuries. The climatic change of the seasonal weather caused by the change in the solar angle is both powerful and measureable.

If your students keep a running record of both the solar angle and the average high temperature for each week, and interesting relationship will be revealed.

Create two graphs, one showing the solar angle over time, the other showing the weekly average high temperature over time. Compare the two graphs; what do you find?

The Sun is the most powerful factor in our climatic change. By comparing solar angle to temperature fluctuations, we can find a powerful link between how much sunlight we receive and our local temperatures.

Following Up

Ancient scientists like Eratosthenes, Pythagoras, Aristotle, and many others contributed to our modern scientific knowledge. Look into some of the ideas and discoveries of these ancient masters and see what you can find!

Activity 13: Mapping the Constellations

One of the most fundamental activities of science – and exploration – is to record what we see. Map making is perhaps the oldest expression of this human need to record what we know and share it with others; it long predates other scientific activities and even predates written language.

When we want to make a map of a place where we live, such as our school neighborhood, or even make a map of a place we have been to, such as a summer vacation spot, that may be one thing. How do make a map of a place so far away we can never possibly go there? How do we make a map of the stars? Fortunately, this is not as hard as it sounds! Once again, science extends our reach and allows our minds to go where our bodies could not possibly follow.

The device that we will build is called a pantograph. This device is based upon an old-fashioned drawing tool that allowed the user to copy down drawings and make them different sizes without distortions. We will use our pantograph to accurately copy the constellation patterns that we see in the sky. All we need to do is measure distances between points with a ruler, and copy down angles!

Academic Standards

Science and Engineering Practices

  • Planning and carrying out investigations.
  • Analyzing and interpreting data.
  • Using mathematics.
  • Obtain, evaluate, and communicate information.
Crosscutting Concepts
  • Patterns in nature.
  • Stability and change.

Next Generation Science Standards

  • Earth’s place in the Universe.
  • Motion and Stability.

For the Educator:

Facts you need to know

  1. Seeing patterns of stars (constellations) and naming them is an ancient activity. We have evidence recorded in clay tablets over 15,000 years old documenting and naming star patterns. Almost every ancient and modern culture has done this.
  2. The sky has 88 modern constellations that cover the entire visible sky the way states or countries cover a map – there is no space between them.
  3. From the continental United States and most of Europe, we can see about 65 constellations – those constellations that lie closer to the southern celestial pole are visible only to those who live in the southern hemisphere.

Teaching and Pedagogy

While very young students may have difficulty with the manual dexterity needed for this activity, older children between grades 3-6 should be able to handle it easily. Once again we see that simple methods can produce beautiful and accurate results. This lab activity will also underscore the idea that observing and recording what you see in an accurate way is a definite skill. It is not always easy to determine which students in your class will be the most skillful at this sort of work, the results may surprise you!

For your students, the idea that they can make beautiful and accurate maps of constellations without a camera or a telescope may amaze them. This method is actually an example of 16th century technology that was used by Danish astronomer Tycho Brahe (Tee’‑kō Bra’-hey).

Tycho is considered by many to have been the greatest observer in history, without the use of a telescope or camera, he mapped the positions of the stars and planets so accurately that their positions were known to an accuracy of 1/5000th of a degree! These measurements were used years later by his assistant, German astronomer Johannes Kepler to prove that the planets orbited the Sun in elliptical paths instead of circular ones.

Student Outcomes:

What will the student discover?

  1. Human beings are very good at recognizing patterns in Nature, for millions of years our survival depended upon it. Humans are so good at finding patterns, that we tend to see them even when they are not there; anyone who has played the “What does that cloud look like?” game has seen this pattern recognition ability in action.
  2. Constellations are patterns we see in the stars. Different cultures recognize different objects when looking at the same stars. The constellation pattern we call the ‘Big Dipper’ in America is called ‘The Plough’ in Britain, and ‘The Ax’ by some Native American cultures.
  3. Part of discovery is the naming process. A biologist who discovers a new species of beetle, an astronomer who discovers a new asteroid, all discoverers are granted the privilege of naming their discoveries. When children discover and record a new pattern of stars, they can name their discovery, too.

What will your students learn about science?

  1. The first task of any scientist is to observe accurately and record what they see. Accurately recording the positions of things you see relative to one another creates a map – perhaps the oldest and most fundamental type of scientific model! Astronomers from many cultures around the world have been making maps of constellations to help them create calendars and predict the changing of the seasons for many thousands of years.
  2. We have evidence of constellation maps recorded in clay tablets from ancient Persia that are more than 15,000 years old. Many scientists believe that structures such as Stonehenge were actually maps and calendar measuring devices made of wood and stone that helped pre-historic mankind mark the constellations and measure the changing of the seasons.
  3. Scientific models, whether we build them physically, create them on paper, or record them in the language of mathematics all serve to help us understand the world we live in. Learning to create these scientific models in any form can be a valuable job skill – and an exciting career!

Conducting the Activity

Materials

  1. Two flat, wooden classroom rulers for each student, marked in centimeters
  2. One 3/16 x ½ bolt and a lock washer and wingnut to match for each student (The local home improvement center can easily help you with this!)
  3. Electric drill with 3/16 – ¼ inch drill bit
  4. 6 pieces of large butcher paper or craft paper, each appx. 30” x 48”
  5. Construction paper, pencils, markers

Building the Pantograph:

  1. [Teacher] Use the butcher paper and draw six constellations on paper and label them. This works well if you use well-known and recognizable constellations such as Ursa Major (The Big Dipper), Orion (The Hunter), Gemini (The Twins), etc. Draw in the brightest stars (make the dots large – 1” or better) and connect them with clear lines drawn in with a heavy marker. Place these constellation diagrams around the room well up on your classroom walls where they can be easily seen. If you haven’t much wall space in your classroom, these often work well when posted in the hallways or even outside the classroom on the building wall.
  2. image[Teacher] Now you must attach two rulers together using the bolt and wingnut. Some rulers come with holes near one end, if yours do not have this you will have to drill the holes. Rubber band two rulers together and drill a hole about ¾ inch from one end – be sure you have a block of wood behind the rulers as you drill to keep from marring your classroom tables! If you are really on a budget, try using cheap yardsticks from the paint department at the home improvement store – each one can be cut into three inexpensive, 12-inch rulers!
  3. With the hole drilled, slip the bolt through the hole and secure the rulers together using the wingnut. This needn’t be over tight, students must be able to slide the rulers apart to form an angle. If the rulers slide too easily, try putting a piece of stick-on felt between the rulers for added friction. These two rulers form a simple pantograph, a device for copying shapes and angles precisely.

Using the Pantograph to Record Constellations:

  1. To copy and map a constellation, we need only look at three stars at a time. Any three stars will form an angle, with the center star at the vertex of the angle. Let’s take the Big Dipper as an example, see figure below.
image
Constellation diagram of Ursa Major

 

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2. Have your student stand 8-10 feet back from your Big Dipper poster – have them hold their rulers at arm’s length, if the line between stars #1 and #2 appears to be 4-6 cm long, they are at the right distance. (You can try this yourself to help them!)

3. Now adjust the two rulers so that star #2 is at the center, and they can measure the distance to the other two stars simultaneously. Now without adjusting the angle between the rulers, transfer the measurements to a piece of construction paper.

image4. Next measure the angle and distance between stars #2 – #3 – #4. Transfer this angle and distance to your paper, which adds star #4 to your map. Continue to proceed along the diagram until you have measured and mapped all seven stars in the constellation.

Discussion Questions

  1. We have some constellation maps that are over 15,000 years old! How do you think those ancient people made these constellation maps?
    • Answer: The most ancient constellation maps were etched into clay slabs and then fired to make a permanent record. These astronomers probably sketched what they saw as an artist would. By the 1300’s, astronomers were using methods very much like those you just used! Modern astronomers use photographs from telescopes and satellites and even computer software to help them make even more accurate maps!
  2. What else could you use this mapping method for?
    • Answer: Interestingly, this method is based upon the pantograph – a device that allows an artist or illustrator to copy a drawing and even change its size in a precise way. Your star-mapping device can be used to map any object where there are distinctive points. Try mapping a school building! Just remember to number the points and measure them in ordered sequence one after the other!

Supplemental Materials

Going Deeper

Most constellations have a connection to mythology, the constellations of the Zodiac are good examples of this. Look at a star map and pick a constellation that interest your students. Try doing an internet search or looking in a book of myths and legends to see if you can find more information about what the constellation is supposed to represent.

Being an Astronomer:

If your students have had good success with mapping constellations on the walls of your classroom, it is now time for them to try the same activity at night with a real constellation. It will not matter which constellation they choose, and in fact, students often have trouble picking out the constellations unless there is someone knowledgeable there to help them!

Lack of constellation knowledge won’t matter a bit – constellations are just arbitrary patterns chosen by people anyway. Any group of bright stars your students choose can easily be measured and recorded accurately on a piece of paper from their back yard. Have your students bring their results back to the classroom. If the children do not know the name of the constellation – have them make one up and tell everyone what these constellation represent to them. Hang these masterworks on your classroom walls for everyone to enjoy!

Being a Scientist:

When you use your pantograph to map a constellation, you should be holding the device at arm’s length to make sure the constellation will fit on a single sheet of drawing or construction paper.

After you have drawn the constellation, use a ruler to measure the distance between the stars in centimeters – write the length down on the lines defining your constellation. The reason that we do this is simple, at arm’s length (57.2 cm to be precise) one centimeter of length also measures one degree of arc. This is called angular distance, and it is the way that we measure size of, and distance between, objects in the sky.

How large are your constellations? If you measure the distance from the center of one constellation to the next with a ruler at night, how far apart are they? Keep in mind that the sky is 180 degrees wide from horizon to horizon; this will give you a better idea of how large the constellations are compared to each other, and compared to the size of the entire sky.

Following Up:

We assume the stars are unchanging, but in fact they are not. The constellations that we see in the sky start each night about 1 degree farther west. The result is that over a period of months, we see different constellations when we go outdoors after sunset.

Programs such as Stellarium (free star-mapping software from www.stellarium.org) show us star maps for the sky any night of the year, and for any location on Earth. You can use such software to see how the constellations change from month to month.

Set up the Stellarium software (or any night sky mapping software) to show the evening sky for today’s date. Then use the calendar function to advance the date one month at a time. You will notice that the April sky in springtime looks little like the summer sky in July, or the autumn sky in October.

Keep an eye on your own sky at night and watch the constellations change from month to month. You will say goodbye to old friends as they sink in the west and welcome new constellations as they rise out of the east as the seasons go by!


  1. To be mathematically precise, holding a ruler 57.2 cm away from your eye will make 1 cm subtend an angle of exactly 1o – and this corresponds nicely with the length of the average adult human arm. Children’s arms are significantly shorter, so the angle measure will be inaccurate in an absolute scientific sense. It is the concept of measuring angles and the technique of using a ruler to measure them that we are interested in here however, not whether or not a 2nd grader is taking technically precise scientific data for an experiment!
  2. Keep in mind that metal distorts a compass! Compasses often will not work properly on a desktop because the metal supports beneath the desk will interfere and throw off the reading – this is why we hold the compass in our hands when using it to take a bearing
  3. This book is written for teachers and students in the northern hemisphere. If you are teaching in the southern hemisphere, the situation will be reversed. The summer moon rides very high above the northern horizon, while the winter Moon stays closer to the northern horizon as it crosses the sky.

License

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Astronomy for Educators by Daniel E. Barth, PhD is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License, except where otherwise noted.

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