Lesson 18

Modeling Circular Motion

18.1: Comparing Bikes (5 minutes)

Warm-up

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 where 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.

Student Facing

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

Graph of 2 functions on a coordinate grid, origin O.
  1. Which bike has larger wheels? Explain how you know.
  2. Which bike’s wheels are spinning faster? Explain how you know.

Student Response

Student responses to this activity are available at one of our IM Certified Partners

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 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 (\(\frac{1}{2.5}\) seconds for Graph A and \(\frac{1}{3.5}\) seconds for Graph B).

18.2: Around a Carousel (15 minutes)

Activity

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 the position of a point on the carousel with trigonometric functions and 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, providing 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.

Launch

Reading: MLR6 Three Reads. Use this routine to support reading comprehension of this problem. Ask students to keep their books or devices closed and display only the image and the task statement without revealing the questions that follow. Use the first read to orient students to the situation. After a shared reading, ask students, “What is this situation about?” (Jada, Noah, and Elena are riding on a carousel, which moves counterclockwise and are all at different positions). After the second read, students list any quantities that can be counted or measured, without focusing on specific values (the position each person is at and the different angle measures). During the third read, the question or prompt is revealed. Invite students to discuss possible strategies, referencing the relevant quantities named after the second read. This helps students connect the language in the word problem and the reasoning needed to solve the problem while keeping the cognitive demand of the task.
Design Principle: Support sense-making

Student Facing

A photograph of a large carousel 

Jada, Noah, and Elena are riding a carousel. Here is a view, from above, of the carousel.

Circle that represents a carousel with center O.

The carousel moves in a counterclockwise direction. When the ride begins, Jada is at position \(J\), Noah is at position \(N\), and Elena is at position \(E\). The measure of angle \(JON\) is \(\frac{\pi}{2}\) and the measure of angle \(NOE\) is \(\frac{2\pi}{3}\).

  1. The radius of the carousel is 20 feet. How far does Jada travel to reach Noah’s starting position? What about Elena’s starting position? Explain or show how you know.
  2. The carousel makes 1 complete rotation every 10 seconds. At which times will Jada be at her starting position? At which times will she be at Noah's starting position? Explain or show how you know.
  3. The carousel ride lasts for 3.25 minutes. Where will Elena be when the ride ends? How far will she have traveled? Explain or show how you know.

Student Response

Student responses to this activity are available at one of our IM Certified Partners

Anticipated Misconceptions

If students struggle to visualize the situation, have them cut out a circle to use as a physical representation on which they can mark the locations of Jada, Noah, and Elena. Using a pencil or pen on the center of the circle, students can rotate the paper around the center.

Activity Synthesis

Emphasize that radian angle measure is ideally suited for measuring distances when something is moving in a circle. With the carousel context, here are some questions for discussion:

  • “How far does Elena travel when the carousel rotates through an angle of \(\frac{\pi}{3}\) radians?” (\(\frac{20\pi}{3}\) feet, because \(\frac{\pi}{3} \boldcdot 20 = \frac{20\pi}{3}\).)
  • “How far does Elena travel when the carousel rotates through an angle of \(5\pi\) radians?” (\(100\pi\) feet, because \(5\pi \boldcdot 20 = 100\pi\).)
  • “How far does Elena travel when the carousel rotates through an angle of \(a\) radians?” (\(20a\) feet, because \(a \boldcdot 20 = 20a\).)

Tell students that, in general, if \(C\) is a circle of radius \(r\) units and \(P\) a point on \(C\) then when the circle rotates by an angle of \(\theta\) radians, the point \(P\) travels a distance of \(r\theta \) units.

Representation: Internalize Comprehension. Use color and annotations to illustrate student thinking. As students share their reasoning about Jada, Noah, and Elena’s position on the carousel, scribe their thinking on a visible display. Keep the display visible for the following activity.
Supports accessibility for: Visual-spatial processing; Conceptual processing

18.3: Modeling the Carousel Motion (15 minutes)

Activity

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, the model requires a horizontal translation if sine is used in addition to parameters representing the amplitude and the period.

Monitor for students who use different variables for location and time 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. Chunk this task into manageable parts for students who benefit from support with organizational skills in problem solving. Check in with students after the first 2–3 minutes of work time. Invite 1–2 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

Student Facing

Jada begins the carousel ride at point \(J\) and Noah begins the ride at point \(N\). The radius of the carousel is 20 feet and it rotates in a counterclockwise direction, making one complete rotation every 10 seconds.

A circle drawn on an x y axis with the x axis labeled h and the y axis labeled v.
    1. Write an equation describing the horizontal coordinate of Jada’s location as a function of time relative to the center of the carousel. Make sure to indicate the units of your variables.
    2. Sketch a graph of your function.
      Blank coordinate plane. Horizontal axis has 6 tick marks. Vertical axis has 5 tick marks above and 5 ticks below the horizontal axis.
    3. What does the graph tell you about Jada’s location during the carousel ride?
    1. Write an equation describing the vertical coordinate of Noah’s location as a function of time relative to the center of the carousel. Make sure to indicate the units of your variables.
    2. Sketch a graph of your function.
      Blank coordinate plane. Horizontal axis has 6 tick marks. Vertical axis has 5 tick marks above and 5 ticks below the horizontal axis.
    3. What does the graph tell you about Noah’s location during the carousel ride?

Student Response

Student responses to this activity are available at one of our IM Certified Partners

Student Facing

Are you ready for more?

Diego rides a different carousel and begins at position \(D\). An equation describing the horizontal coordinate, \(d(t)\), of his location in feet relative to the center of the carousel as a function of time is \(d(t)=15\cos(\frac{\pi t}{15}-\pi)\) where \(t\) is the number of seconds since the carousel started to move.

  1. What is the radius of the carousel?
  2. How long does it take the carousel to make a complete rotation?
  3. Where did Diego start?

Student Response

Student responses to this activity are available at one of our IM Certified Partners

Activity Synthesis

Select previously identified students to share their equations and graphs for Jada and Noah. Discuss why it is important to indicate the units for the input and output of the functions. (These equations all represent the same situation, but different letters were used to represent the inputs and outputs of the functions.)

Discuss the meaning of the amplitude, horizontal translation, and period of the functions. Highlight:

  • The amplitude is the distance to the center of the carousel.
  • In the equation \(n(t) = 20\sin\left(\frac{\pi}{2} + \frac{2\pi t}{10}\right)\) for Noah's vertical location, the translation by \(\frac{\pi}{2}\) for Noah represents the fact that he did not start in the “\((1,0)\)” position but instead the "\((0,1)\)" position.
  • The period for one complete revolution is 10 seconds.

Conclude the discussion by asking students why it makes sense that the two graphs look the same. (The horizontal displacement for Jada and the vertical displacement for Noah are identical. They start at 1, the largest possible value, decreasing to -1, before increasing again. Or, geometrically, rotating a quarter turn counterclockwise takes the horizontal displacement from the center to the vertical displacement from the center). After some quiet think time, invite students to share their thinking. An important takeaway here is that while these two graphs look the same, they are describing different physical motions.

Speaking, Representing: MLR8 Discussion Supports. Use this routine to support whole-class discussion. For each response, that is shared, ask students to restate what they heard using precise mathematical language. Consider providing students time to restate what they heard to a partner before selecting one or two students to share with the class. Ask the original speaker if their peer was accurately able to restate their thinking. Call students' attention to any words or phrases that helped to clarify the original statement. This provides more students with an opportunity to produce language as they interpret the reasoning of others.
Design Principle(s): Support sense-making; Maximize meta-awareness

Lesson Synthesis

Lesson Synthesis

Tell students that the horizontal position of Lin in feet relative to the center of a carousel is given by \(h = 20\cos\left(\frac{2\pi t}{25} +\frac{\pi}{4}\right)\) where \(t\) 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 \(\frac{\pi}{4}\) 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 of 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.

18.4: Cool-down - A different Carousel (5 minutes)

Cool-Down

Cool-downs for this lesson are available at one of our IM Certified Partners

Student Lesson Summary

Student Facing

Here is a point \(P\) on a Ferris wheel:

A ferris wheel with point P at the top of the wheel.

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 \(P\), in feet, after \(t\) 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 \(\frac{\pi}{2}\) (\(P\) starts at the angle \(\frac{\pi}{2}\) 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 \(P\) is modeled by the function \(h = 50\sin\left(\frac{\pi}{2} + \frac{2\pi t}{5}\right) + 60\). Here is a graph of the function:

Graph of function f.