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Unit 6, Lesson 19

Modeling Circular Motion

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Integrated Math 3

19.1

Warm-up

Standards Alignment

Building On

Addressing

Building Toward

Instructional Routines

None

Materials

None

Activity Narrative

In this Warm-up, students interpret graphs of periodic functions in context. Students use features of the graph like midline, amplitude, and period to give a qualitative description of spinning wheels. This prepares students for upcoming activities in which they come up with functions to model other situations involving circular motion.

Launch

Arrange students in groups of 2. Give students quiet work time and then time to share their work with a partner. Follow with a whole-class discussion.

Activity

None

Each graph shows the vertical position, , in inches, of a point on the outside of a bike wheel, seconds after the wheel begins to spin.

<p>Graph of 2 functions on a coordinate grid, origin O.</p>

Which bike has larger wheels? Explain how you know.
Which bike’s wheels are spinning faster? Explain how you know.

Student Response

to access Student Response.

Building on Student Thinking

Activity Synthesis

Invite partners to share their responses and explain their reasoning. As students share, encourage them to refine their language and use appropriate terminology, such as ”amplitude,” “period,” or “midline,” and to explain how those words are related to the context of the spinning bike wheels. Highlight:

The midline tells where the center of the wheel is with respect to the ground.
The amplitude is the radius of the wheel.
The period is how long it takes the wheel to make a complete revolution ( second for Graph A and second for Graph B).

19.2

Activity

Instructional Routines

MLR6: Three Reads

Materials

To Gather

Scientific calculators

Activity Narrative

The goal of this activity is to introduce the circular motion of a carousel and study the motion using radian angle measure. In the next activity, students will model, with trigonometric functions, the position of a point on the carousel and then graph those functions, relating the graphs to the position of different points on the carousel. In this situation, a trigonometric function models the situation essentially perfectly. This provides students an opportunity to interpret the parameters of a trigonometric function in context before being asked to make more difficult modeling decisions to fit real-world data in the next lesson, as they reason abstractly and quantitatively (MP2).

Launch

Provide access to scientific calculators.

MLR6 Three Reads. Keep books or devices closed. Display only the image and the problem statement, without revealing the questions. “We are going to read this statement 3 times.”

After the 1st read: “Tell your partner what this situation is about.”
After the 2nd read: “List the quantities. What can be counted or measured?”
For the 3rd read: Reveal and read the questions. Ask, “What are some ways we might get started on this?”

Advances: Reading, Representing

19.3

Activity

Instructional Routines

Graph It MLR8: Discussion Supports

Materials

None

Activity Narrative

This activity is a continuation of the previous. Here, students model the positions of Jada and Noah with equations and use those equations to examine what is happening as Jada and Noah go around the carousel. While the midline for these trigonometric functions is 0, if the sine function is used, the model requires a horizontal translation in addition to parameters representing the amplitude and the period. Students use the structure of the functions to explain why the graphs look the same, even though one is a sine function and one is a cosine function (MP7).

Monitor for students who use different variables for location and time. Invite them to share during the discussion.

Making graphing technology available gives students an opportunity to choose appropriate tools strategically (MP5).

Launch

Tell students that they are going to continue to work with the carousel information from the first activity. They will write equations to describe the locations of Jada and Noah on the carousel as a function of time.

Action and Expression: Internalize Executive Functions. To support development of organizational skills in problem-solving, chunk this task into more manageable parts. For example, check in with students after the first 2–3 minutes of work time. Invite 1 or 2 students to share their equation and graph for the first question. Record their thinking for all to see, and keep the work visible as students continue to work on the second question.
Supports accessibility for: Organization, Attention

Lesson Synthesis

Tell students that the horizontal position of Lin in feet relative to the center of a carousel is given by , where is the number of seconds since the carousel started moving. Ask students:

“What is the radius of the carousel?” (20 feet, the amplitude of the function)
“How long does it take for the carousel to make one complete revolution?” (25 seconds, the period of the function)
“Where is Lin on the carousel circle when the ride begins?” (at the location, in the middle of the first quadrant)

Conclude the thinking about this function by asking students to sketch its graph without using technology. Once students have made their sketches, display the actual graph of the function for students to check their work. If time permits, invite students who made accurate sketches to share their strategies, such as by marking the midline, minimum, and maximum to help with accuracy. While it is not an expectation of these materials that students be able to sketch trigonometric functions by hand, it is useful practice for students to make connections between equation parameters and graphical features, particularly for equations representing situations.

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Student Lesson Summary

Here is a point on a Ferris wheel:

<p>A ferris wheel with point P at the top of the wheel.</p>

This Ferris wheel has a diameter of 100 feet and its center is 60 feet off the ground. The Ferris wheel makes one revolution every 5 minutes. We can use this information to write a function that describes the vertical position of , in feet, after minutes. We know

The amplitude is 50 (the diameter of the Ferris wheel is 100 feet).
The midline is at 60 (the center of the Ferris wheel is 60 feet high).
The horizontal translation is ( starts at the angle on the circle).
The period is 5 (every 5 minutes the Ferris wheel makes one complete revolution).

Since we want the vertical position, let's use the sine function. Putting all of this information together, the height of is modeled by the function . Here is a graph of the function:

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Lesson 19

Modeling Circular Motion

Warm-up

Standards Alignment

Building On

Addressing

HSF-IF.B.4

Building Toward

Instructional Routines

Materials

Activity Narrative

Launch

Activity

Student Response

Building on Student Thinking

Activity Synthesis

Activity

Instructional Routines

Materials

To Gather

Activity Narrative

Launch

Activity

Instructional Routines

Materials

Activity Narrative

Launch

Lesson Synthesis

Student Lesson Summary

Activity

Student Response

Building on Student Thinking

Activity Synthesis

Activity

Student Response

Building on Student Thinking

Activity Synthesis

Standards Alignment

Building On

HSG-C.B.5

Addressing

HSF-TF.A.1

Building Toward

HSF-TF.B.5

Standards Alignment

Building On

HSF-BF.A.1

Addressing

HSF-IF.C.7.e

HSF-TF.B.5

HSN-Q.A.1

Building Toward