Lesson 19

Beyond Circles

19.1: Notice and Wonder: Examining Data (5 minutes)


The goal of this warm-up is for students to preview and start making sense of the data they will study in this lesson (MP1). The data is for the amount of the Moon visible from a particular location on Earth each night over the course of one month. The context is not provided so students can focus on patterns that they identify in the raw data before attempting to model it with a trigonometric function.

The data is taken for January 2018 in the mountain time zone at midnight (the very beginning of the day). 


Tell students that their job is to think of at least one thing they notice and at least one thing they wonder. Display the table for all to see. Give students 1 minute of quiet think time, and then 1 minute to discuss the things they notice with their partner, followed by a whole-class discussion.

Student Facing

Here is some data that we will study in today’s lesson.

day amount
1 0.99
2 1.00
3 0.98
4 0.93
5 0.86
6 0.77
7 0.67
8 0.57
9 0.46
10 0.37
day amount
11 0.28
12 0.19
13 0.13
14 0.07
15 0.03
16 0.01
17 0.00
18 0.01
19 0.04
20 0.09
day amount
21 0.16
22 0.24
23 0.33
24 0.43
25 0.54
26 0.65
27 0.76
28 0.85
29 0.92
30 0.98
31 1.00

What do you notice? What do you wonder? 

Student Response

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Activity Synthesis

Ask students to share the things they noticed and wondered. Record and display their responses for all to see. If possible, record the relevant reasoning on or near the tables. After all responses have been recorded without commentary or editing, ask students, “Is there anything on this list that you are wondering about now?” Encourage students to respectfully disagree, ask for clarification, or point out contradicting information.

If the overall pattern  (the numbers decrease and then increase) does not come up during conversation, ask students to discuss how this is like other situations they have examined in this unit.

19.2: Watching the Evening Sky (30 minutes)


The goal of this activity is to model data with a trigonometric function in a context that is not directly about circular motion. The periodic behavior of the amount of the Moon visible is governed by the Moon’s orbit around Earth and Earth’s orbit around the Sun. These orbits can be modeled by ellipses rather than circles though they are both pretty close to being circular. In this activity, students analyze the data from the warm-up, making predictions about the amplitude, midline, period, and horizontal translation for a trigonometric model.

Next students check how well their model fits the actual data and use technology to modify the parameters in their model as needed. Finally, students use their models to make predictions about the Moon for later dates in 2018. This prediction work is a very important part of the task, not only because making predictions is one of the main reasons for creating models but also because this can bring out deficiencies that are otherwise not apparent. For example, a 30 day period fits the January data well but it predicts a full moon on December 28, 2018 when only 62% of the Moon is visible. A 29 day period does not fit the January data as well and predicts a full moon on December 16 which is too soon. A 29.5 day period turns out to fit the overall data better than 29 or 30 days, even though it does not seem to fit the January data as well as 30. To the nearest hundredth, the orbit of the Moon takes 29.53 days.

For reference as students work on the problems, the next two full moons in 2018 in the mountain time zone (the source of the data) are March 2, 2018 and March 31, 2018. There are a variety of online sites where more data about the moon on specific dates can be retrieved in order for students to check their birthday predictions.

If students do not have access to Desmos, they can experiment with the different parameters in their trigonometric model using other graphing technology. Some students may also use technology to propose a trigonometric model for the data.


Tell students that the data from the warm-up is the amount of the Moon that is visible from a particular location on Earth at midnight for each day in January 2018. If possible, demonstrate for students what causes different amounts of the Moon to be visible with a flashlight and a sphere. At any given moment in time, half of the Moon is illuminated by the Sun (the half “facing” the Sun) and half is dark. When we see a full moon, that means the part of the Moon illuminated by the Sun is facing our location on Earth, and when we see little or none of the Moon, that means the dark half of the Moon is facing us. As the Moon orbits Earth and Earth orbits the Sun, the relative positions of the three bodies change, and this produces the Moon cycle investigated in this activity.

Writing, Conversing: MLR1 Stronger and Clearer Each Time. Use this routine to help students improve their writing, by providing them with multiple opportunities to clarify their explanations through conversation. Give students time to meet with 2–3 peers to share and get feedback on their explanation of each of the parameters of their function to model the Moon data. Invite students to press for additional details referring to specific features of the function or graph such as the midline, amplitude, and estimated period. Display feedback prompts such as, “What is your input and how does it relate to the context?”, “What is your output and how does it relate to the context?”, and “A detail (or word) you could add is _____, because . . . .” Give students 3–4 minutes to revise their initial draft based on feedback from their peers. This will help students create a trigonometric equation and discuss features in the context of visibility of the moon.
Design Principle(s): Support sense-making; Cultivate conversation
Representation: Develop Language and Symbols. Create a display of important terms and vocabulary. During the launch take time to review terms that students will need to access for this activity. Invite students to suggest language or diagrams to include that will support their understanding of: midline, amplitude, translation, and period.
Supports accessibility for: Conceptual processing; Language

Student Facing

The data from the warm-up is the amount of the Moon that is visible from a particular location on Earth at midnight for each day in January 2018. A value of 1 represents a full moon in which all of illuminated portion of the moon's face is visible. A value of 0.25 means one fourth of the illuminated portion of the moon's face is visible. 

  1. What is an appropriate midline for modeling the Moon data? What about the amplitude? Explain your reasoning.
  2. What is an appropriate period for modeling the Moon data? Explain your reasoning.
  3. Choose a sine or cosine function to model the data. What is the horizontal translation for your choice of function?
  4. Propose a function to model the Moon data. Explain the meaning of each parameter in your model and specify units for the input and output of your function.
  5. Plot the data using graphing technology and check your choice of parameters (midline, amplitude, period, horizontal translation). What changes did you make to your model?
  6. Use your model to predict when the next two full moons will be in 2018. Are your predictions accurate?
  7. How much of the Moon do you expect to be visible on your birthday? Explain your reasoning.

Student Response

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Anticipated Misconceptions

Students may struggle with the horizontal translation in their function modeling the amount of the Moon visible. Start by asking what they chose for the amplitude, midline, and period for their model. If they have chosen \(0.5\), \(0.5\), 29 days and a cosine model, ask them first to write a cosine model without a horizontal translation. This will give a function like \(f(x) = 0.5\cos\left(\frac{2\pi d}{29}\right) + 0.5\). Next ask them when this function takes its maximum value (when \(d = 0\)). For which value of \(d\) is the Moon full? (\(d = 1\)) How can you adjust the expression \(\frac{2\pi d}{29}\) so that it takes the value 0 when \(d = 1\)? (subtract \(\frac{2\pi}{29}\))

Activity Synthesis

Highlight these features of the Moon data:

  • It ranges from 0 (none of the Moon visible) to 1 (full moon). This means that the midline is \(0.5\) since it is the average of the minimum and maximum values. This also gives an amplitude of \(0.5\).
  • The amount of time between full moons (the period) is about 4 weeks. There are 29 days between the two full moons but there are several days when the Moon is “almost full” so it is hard to tell what the exact period is.

Ask students how they used the midline, amplitude, and estimated period in their equation. For the equation \(f(d) = 0.5\cos\left(\frac{2\pi d}{29}-\frac{2\pi}{29}\right) + 0.5\), the added constant \(0.5\) is the midline while the coefficient of \(0.5\) is the amplitude. This means that the graph will be horizontally centered at \(0.5\) (half of the Moon visible) with a maximum of \(0.5 + 0.5\) (all of the Moon visible) and a minimum of \(0.5 - 0.5\) (none of the Moon visible). The denominator of 29 in the expression \(\frac{2\pi d}{29}\)  gives the function a period of 29. The term \(\frac{2\pi}{29}\) is a horizontal translation to the right by 1 so that the maximum (full moon) is predicted for January 2, one day after January 1.

Discuss how well the models worked for predicting the fraction of the Moon visible for dates further out in the year. The models should work well for January and the next couple months but the further out you go, the period begins to be off (for a choice of 29 or 30 days). This makes sense because a small error in the period becomes more pronounced with each cycle through. Ask students what they might do to better estimate the period of the Moon’s orbit around Earth. Looking at the table shows that the time between full moons consistently alternates between 29 and 30 days, making 29.5 days an appropriate estimate for the period.

Lesson Synthesis

Lesson Synthesis

If time allows, ask students:

  • “Which parts of the moon model were the most challenging to find?” (The period is 29 days in January but this does not always work well to estimate the amount of the Moon visible for other months later in the year.)
  • “Was your prediction of how much of the Moon will be visible on your birthday accurate?” (responses will vary, but in general the later in the year the birthday is, the less accurate the model will be.)
  • “What could you do to improve the accuracy of the model?” (Look at more data to better estimate the period.)

A key point to highlight here is that with a given set of data it is not always possible to produce an accurate model.

19.3: Cool-down - A Titanic Orbit (5 minutes)


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Student Lesson Summary

Student Facing

Sometimes a phenomenon can be periodic even though it is not connected to motion in a circle. For example, here is a graph of the water level in Bridgeport, Connecticut, over a 50 hour period in 2018.

Graph of a function, origin O.

Notice that each day (or each 24 hour period) there are two tides, a small one where the water goes up to a little less than 3 feet and then a bigger one when the tide goes up to a little more than 3 feet. Since there are two tides per day, the period for this graph is about 12 hours. The data begins about 1 hour before the tide is at the 0 value. Since \(\sin(0) = 0\), this would make the sine a good choice for modeling the tide.

Putting together all of our information gives model \(f(h) = 3\sin\left(\frac{2\pi}{12}(h -1)\right)\), where \(h\) measures hours since midnight on September 1.

Graph of two sine functions, on a coordinate plane with grid, origin \(O\).

Notice that:

  • The coefficient of 3 is the amplitude, which averages out the bigger and smaller tides.
  • \(\frac{2\pi}{12}\) makes the period 12 hours.
  • -1 translates the sine graph to the right by 1 hour to so it has a value of 0 at about 1 hour after midnight.