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Unit 6, Lesson 20

Beyond Circles

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Integrated Math 3

20.1

Warm-up

5 mins

Examining Data

Standards Alignment

Building On

Addressing

Building Toward

Instructional Routines

None

Materials

None

Activity Narrative

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).

Launch

Display the table. Tell students, “The headings have been removed from these tables of data. What do you think the data could be referring to?”

Give students 1 minute of quiet think time, and then 1 minute to discuss with a partner, followed by a whole-class discussion.

Activity

None

Student Task Statement

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

input	output
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

input	output
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

input	output
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

Activity Synthesis

Ask students to share their ideas about what the data could be referring to. Ask them what patterns they noticed that helped them come up with their ideas.

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.

20.2

Activity

30 mins

Watching the Evening Sky

Instructional Routines

Aspects of Mathematical Modeling MLR1: Stronger and Clearer Each Time

Materials

To Gather

Graphing technology

Activity Narrative

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 (MP4).

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.

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.

to access Cool-down.

Student Lesson Summary

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.

The midline of the graph appears to be around 4.5 feet. Notice that each day (or each 24-hour period) there are two high tides, a small one where the water goes up by a little less than 3 feet and then a bigger one when the tide goes up by a little more than 3 feet. Since there are two high tides per day, the period for this graph is about 12 hours. The data begins about 1 hour before the tide is at the 4.5 midline value. Since , this would make the sine function a good choice for modeling the tide.

Putting together all of our information gives the model , where measures hours since midnight on September 1.

<p>Graph of two sine functions, on a coordinate plane with grid, origin <span class="math">$O$</span>.</p>

Notice that:

Adding 4.5 gives a midline of , which means that the average water height is about 4.5 feet.
The coefficient of 3 is the amplitude, which averages out the difference in the high tides. In general, the water rises by about 3 feet for each high tide and lowers by about 3 feet for each low tide.
makes the period 12 hours.
-5 translates the sine graph to the right by 5 hours, so it has a value of 4.5 at about 5 hours after midnight.

Launch

Graphing technology is needed for every student.

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, use a flashlight and a sphere to demonstrate for students what causes different amounts of the moon to be visible. 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.

Instruct students to record “day” as the header of the input column of the data tables and to record “amount visible” in the output column of the data tables.

MLR1 Stronger and Clearer Each Time. Before the whole-class discussion, give students time to meet with 2–3 partners to share and get feedback on their first draft response to the fourth question about explaining each parameter of the function to model the moon data. Invite listeners to ask questions and give feedback that will help their partner clarify and strengthen their ideas and writing. Give students 3–5 minutes to revise their first draft based on the feedback they receive.
Advances: Writing, Speaking, Listening

Representation: Develop Language and Symbols. Create a display of important terms and vocabulary. Invite students to suggest language or diagrams to include that will support their understanding of modeling the visible amount of the moon. This may include a graph and equation of a transformed periodic function with these terms labeled: “midline,” “amplitude,” “translation,” and “period.” If possible, display a diagram showing different phases of the moon and how they change from new moon to full moon and back again.
Supports accessibility for: Conceptual Processing, Language

Activity

None

Student Task Statement

The data 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 the illuminated portion of the moon's face is visible. A value of 0.25 means that one-fourth of the illuminated portion of the moon's face is visible.

What is an appropriate midline for modeling the moon data? What about the amplitude? Explain your reasoning.
What is an appropriate period for modeling the moon data? Explain your reasoning.
Choose a sine or cosine function to model the data. What is the horizontal translation for your choice of function?
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.
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?
Use your model to predict when the next two full moons will be in 2018. Are your predictions accurate?
How much of the moon do you expect to be visible on your birthday? Explain your reasoning.

Student Response

to access Student Response.

Building on Student Thinking

If students struggle with the horizontal translation in their function that models the amount of the moon that is visible, consider saying:

“Tell me more about the amplitude, midline, and period for your model.”
“Try writing and graphing your model without the horizontal translation. How does the maximum of your model compare to where you want the maximum to occur?"

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Lesson 20

Beyond Circles

Warm-up

Examining Data

Standards Alignment

Building On

Addressing

Building Toward

HSF-TF.B

Instructional Routines

Materials

Activity Narrative

Launch

Activity

Student Task Statement

Activity Synthesis

Activity

Watching the Evening Sky

Instructional Routines

Materials

To Gather

Activity Narrative

Lesson Synthesis

Student Lesson Summary

Student Response

Building on Student Thinking

Are You Ready for More?

Launch

Activity

Student Task Statement

Student Response

Building on Student Thinking

Are You Ready for More?

Activity Synthesis

Standards Alignment

Building On

Addressing

HSF-IF.C.7.e

HSF-TF.B.5

Building Toward