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).

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  radians?” ( feet, because  .)
• “How far does Elena travel when the carousel rotates through an angle of  radians?” ( feet, because .)
• “How far does Elena travel when the carousel rotates through an angle of  radians?” ( feet, because .)

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

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  for Noah's vertical location, the translation by for Noah represents the fact that he did not start in the “” position but instead in the "" 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.