Lesson 18

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.

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.

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).

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.

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.

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.

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).

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. 

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.