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.

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).

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).

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).