# Round and Round We Go

## Relationship Between the Unit Circle and the Sine Curve

K20 Center, Cacey Wells, Kate Raymond, Nicole Shobert | Published: July 13th, 2022 by K20 Center

• Subject Mathematics
• Course Precalculus
• Time Frame 140 minutes
• Duration 2-4 class periods

### Summary

Students will explore the rotation of a Ferris wheel and make conjectures about the function represented by the motion. Students will find and plot points in the motion and use technology to find the line of best fit. Students will also manipulate the equation for their function to discover what the variables and constants mean in relation to the Ferris wheel and other sine functions.

### Essential Question(s)

What can trigonometric functions tell us about real world situations?

### Snapshot

Engage

Students watch a variety of video clips and create graphs that match each situation.

Explore

Students record the height of the rider through several rotations of the ride using a K'Nex Ferris Wheel or video clips.

Explain

Students participate in a Gallery Walk to see their peers' work and to clarify misconceptions.

Extend

Students use Desmos to find the approximate equation of their Ferris Wheel's motion. Students manipulate and explain the various parts of the function and their relationship with the Ferris Wheel itself as well as the data collected during the ride (e.g., height, time, amplitude, period). Students explore the different variables of the function y = A·sin[B(x – C)] + D.

Evaluate

Evaluation primarily occurs throughout the lesson through informal formative assessment; however, students conclude the lesson by completing an Exit Ticket.

### Materials

• Lesson Slides (attached)

• Paper Airplanes Graphs handout (attached; one per student)

• Ferris Wheel Mathematics and Desmos handout (attached; one per student)

• K'Nex Ferris Wheels (ideally, one per groups of 3-4 students) or Video of the Ferris Wheel

• Timer or stopwatch (students can also use a timer or stopwatch on their smartphone, assuming the teacher is comfortable with this)

• Large poster paper

• Copy paper (for data collection; one per group of 3-4 students)

• Markers

• Rulers (one per group of 3-4 students)

• Yardsticks (one per group of 3-4 students)

• Computer, Chromebook, or iPad with internet access

• Small sticky notes

• Graph paper (1 sheet per group of 3-4 students)

• Optional: Flip camera, iPod, cell phone camera, etc. (if using the K'Nex Ferris Wheel)

### Engage

20 Minute(s)

Introduce the lesson using the attached Lesson Slides. Display slide 3 and discuss the lesson's Essential Question: What can trigonometric functions tell us about real world situations? Display slide 4 and identify the lesson's learning objectives. Review each of these with your class to the extent you feel necessary.

Go to slide 5. Pass out the attached Paper Airplanes Graphs handout. Have students watch the "Graphing Stories - Elevation of a Plane" video about the elevation of a paper airplane.

While the students are watching the video, have them graph the elevation of the paper airplane on the Paper Airplane Graphs. Watch the video as many times as needed for the students to get the information they need to make an accurate prediction. After graphing, ask students to defend their graph to their peers using the appropriate mathematical language.

Go to slide 6 and have students watch the "Ferris Wheel - The Giant Wheel Ride At Yazoo Park Virar" video.

Using the Think-Pair-Share strategy, have students first hypothesize how they would graph the elevation of a rider on the Ferris wheel. After they individually come up with a solution, have them discuss it with a partner. Lastly, as a pair, have them share their solution with the class.

### Explore

40 Minute(s)

Divide students into groups (depending on the number of K'Nex Ferris Wheels you have). Make sure each group has a Ferris Wheel, yardstick, timer, graph paper, and data collection sheet. If you do not have access to Ferris Wheels that students can use in class, feel free to show a clip of a Ferris wheel or ask students to pull up a clip of a Ferris wheel on their smartphone or other device.

Go to slide 7. Before collecting data, ask groups to devise a plan for how they will collect their data points. Have students record the height of the rider at least three times per rotation for at least five rotations, making that 15 data points total. If multiple students are having difficulty looking at the Ferris wheel at the same time, record the rotations of the Ferris wheel so everyone can be collecting data at the same time.

Once students have all their measurements, they will create a graph of their findings. If possible, use the large sticky note graph paper. If not, have students make as large a graph as possible on the paper provided.

### Explain

30 Minute(s)

Go to slide 8. To clarify misconceptions and to get a better feel for what other groups have accomplished, have the students participate in a Gallery Walk. Have students display their group's graph on the wall and stand by it. Be sure to have groups spread out throughout the room.

All groups will rotate clockwise and look at the graphs of their peers. With sticky notes, have each group leave at least two comments on each of the other graphs:

1. What is one thing your graphs have in common?

2. What is one thing you have a question about?

After the gallery walk, have students return to their poster and read what the other groups have commented. As a class, take some time for groups to ask their questions in order to clarify misconceptions.

### Extend

40 Minute(s)

Go to slide 9. Have students reunite with their group at their desks and pick up one laptop or Chromebook to complete this section. If there are not enough laptops for each student, students can share.

Pass out the attached Ferris Wheel Mathematics and Desmos handout and have students work in their groups to complete the handout. Students should go to this link: https://www.desmos.com/calculator/pcy0aa4ztq.

Once students complete the handout, check in with individual groups to help clarify misconceptions they may have. Ask students to explain their reasoning.

### Evaluate

10 Minute(s)

Go to slide 10. To evaluate understanding, check the Ferris Wheel Mathematics and Desmos handout for accuracy.

Just before students leave class, have students use the Exit Ticket strategy. Ask them to get a sticky note to do the following:

• Have students draw a picture that depicts how they are feeling about the material covered in class.

• Have students write down one question or concern they have about moving forward.

Have students stick their sticky note to the whiteboard as they exit.