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Teaching Math: A Video Library, 9-12

Ferris Wheel

Content Standards
Functions, Algebra, Geometry from an Algebraic Perspective, Trigonometry
Process Standards
Problem Solving, Communication, Reasoning, Connections
School: North Carolina School of Science and Mathematics, 550 students
Location: Durham, North Carolina
Teacher: Helen Compton
Years Teaching: 26
Students in Classroom: 16
Grade: 11
Ferris Wheel
Video Overview
The class reviews homework in which students wrote a function describing a rider's height on a single Ferris wheel. Then students work in groups, using graphing calculators to write a function for a rider who travels from the bottom to the top of a double Ferris wheel. At the end of the class, Ms. Compton reconvenes the students and presents that evening's homework on writing a function for the rider's horizontal position on a double Ferris wheel.

The Class Challenge
Students focus on the following problem:
A double Ferris wheel consists of two separate wheels, revolving at the end of a long bar which also turns. The rider experiences two circular motions. The bar revolves every 10 seconds. As it revolves, each wheel revolves so the rider is at the high point of the wheel every 7 seconds. The height from the ground to the center of the bar is 40 feet, each wheel's diameter is 20 feet, and the bar's length is 50 feet. Assume the rider begins timing at the lowest point and moves counterclockwise.

Develop a function to describe the rider's height above ground as a function of time (the rider starts at time, t = 0).
When will the rider be at the first high point? second high point?
How high is the highest possible point? When will the rider reach that point?

Course and Curriculum
The North Carolina School of Science and Mathematics, which includes only grades 11 – 12, is a state–supported residential magnet school that emphasizes mathematics and science. This course, Contemporary Precalculus through Applications, is required for all students. Prior to this course, students took Algebra 1, Algebra 2, and Geometry at their former high schools. In this course, students learn elementary functions with an emphasis on mathematical modeling, including such concepts as data analysis and matrices.
This lesson follows a six–week trigonometry unit in which the class used trigonometric functions to model phenomena, graphed parametric equations, solved trigonometric equations, used equations to model periodic phenomena, and discussed how to use parametric equations to place vertical and horizontal positions on the same graph. This lesson is the first time students added two trigonometric functions together. Through this lesson, Ms. Compton assessed students' understanding of initial conditions on trigonometric functions, such as amplitude and fundamental period. Students later wrote an explanation of the double Ferris wheel problem and included their assumptions, functions, and graphs.
Ms. Compton frequently uses group activities to introduce new concepts, although she occasionally presents new concepts in a teacher–led discussion. She emphasizes active discussions in both formats.

A Pre–Viewing Exploration for Teacher Workshops
Representing the Path of a Single Ferris Wheel
Objective: To determine and investigate different functions that represent the height of a person on a single Ferris wheel.
Materials: One graphing calculator with the capability to graph functions for each small group.
Activity
Solve this problem in small groups.

A Ferris wheel has a diameter of 50 feet and is suspended from a tower 40 feet above the ground. The wheel revolves every 10 seconds in a counterclockwise direction. Write a function that describes a rider's height above the ground as a function of the number of seconds since her start. Use the following questions to develop the function.

What is the radius of the circle? How does the radius affect the function?
How does the height of the tower affect the height of the rider?
How does the period of 10 seconds affect how the function is written?
What is the function?

Exploring Content
Use a calculator to find the answers.

Compare your functions with others and explain how you arrived at your result.
What determines the function used and the sign of that function? Graph and examine the following functions. Which functions represent the rider's height in the situations described in #4 – #7 below? (You can also use these functions to explore the rider's horizontal position.) y = 25 cos (2[pi]t/10) + 40 y = –25 cos (2[pi]t/10) + 40 y = 25 sin (2[pi]t/10) + 40 y = –25 sin (2[pi]t/10) + 40
The cycle starts on the horizontal with the rider at the right.
The cycle starts on the horizontal with the rider at the left.
The cycle starts with the rider at the top.
The cycle starts with the rider at the bottom.

Exploring Teaching Issues

What other real–world phenomena could be used to demonstrate relationships involving trigonometric functions?
How could a physical model of a Ferris wheel help students to explore properties in trigonometric functions?

Exploration Answers (Pre–Viewing)
Activity

25 feet. 25 feet is the amplitude and thus is the coefficient in the function.
40 must be added to the function, because this is the height of the tower, and the ground position is defined as y = 0.
2[pi](t/10). 2[pi] represents the period of the sine or cosine function, and it takes 10 seconds to complete a cycle; t represents the time in seconds.
y = 25 sin (2[pi]t/10) + 40, or y = 25 cos (2[pi]t/10) + 40.

Exploring Content

Answers will vary because equations will vary, depending upon start position. One possible answer is that if the cycle starts on the horizontal with the rider on the right, the function is y = 25 sin (2[pi]t/10) + 40. See Answers will vary because equations will vary, depending upon start position. One possible answer is that if the cycle starts on the horizontal with the rider on the right, the function is y = 25 sin (2[pi]t/10) + 40. See #4 – #7 for more possible answers.
Yes. It depends on where you begin the function. See #4 – #7 for possible answers.
Start position. Horizontal starts are sine functions and vertical starts are cosine functions. Starting at the right and the top are positive functions; starting at the left and the bottom are negative functions.
y = 25 sin (2[pi]t/10) + 40
x = 25 cos (2[pi]t/10)

y = –25 sin (2[pi]t/10) + 40
x = –25 cos (2[pi]t/10)
y = 25 cos (2[pi]t/10) + 40
x = –25 sin (2[pi]t/10)
y = –25 cos (2[pi]t/10) + 40
x = 25 sin (2[pi]t/10)
(y = vertical height)
(x = horizontal position)

A Post–Viewing Exploration for Teacher Workshops
Representing the Path of a Double Ferris Wheel
Objective: To apply what teachers learned in the pre-viewing exploration "Representing the Path of a Single Ferris Wheel" to investigate the function of a double Ferris wheel.
Materials: One graphing calculator with the capability to graph functions for each small group.
Activity
Solve this problem in small groups.
A double Ferris wheel consists of two separate wheels, each of which revolves at the end of a long bar that also turns. As a result, the rider experiences two circular motions. The bar revolves every 10 seconds. As the bar is revolving, each wheel revolves so that the rider is at the high point of the wheel every 7 seconds. The height of the tower is 40 feet, the diameter of each wheel is 20 feet, and the length of the bar is 50 feet. Assume the rider begins timing at the lowest point and moves in a counterclockwise direction. Develop a function to describe the rider's height above ground as a function of the number of seconds since her start.
Investigate the problem by looking at the functions that represent the height of the rider on the double Ferris wheel.

What function represents the motion of the 50–foot bar?
What function represents the smaller revolution of each individual wheel?
The combination function is the sum of these two functions. How would you write that function?

Exploring Content

What is the period of the graph?
Use a calculator to graph the function. When will the rider be at the first high point? the second high point? Discuss the symmetry.
What makes these high points different from each other?
What is the maximum height of the double Ferris wheel? Can the rider ever reach the maximum height? Explain your answer.
Can more than one function describe the phenomenon?

Exploring Teaching Issues

In your classroom, when could you use this application of using more than one function to describe this phenomenon?
How could you demonstrate to students who do not understand wave addition the reason for using the sum of the two functions?

Exploration Answers (Post–Viewing)
Activity

y = –25 cos (2[pi]t/10) + 40
y = –10 cos (2[pi]t/7) + 15
y = –10 cos (2[pi]t/ 7) + 40 + –25 cos (2[pi]t/10) (The 15 in answer #2 does not have to be included because it was accounted for in the position of the center.)

Exploring Content

70 seconds; 10 (the number of seconds in the revolution of the large bar) times 7 (the number of seconds in the revolution of the small ball) = 70.

After about 4 seconds; after about 16 seconds. (Graph on a calculator and zoom in on the points to find their exact value, or realize that the large bar will be at its maximum in 5 seconds, but the small bar will be at its maximum in 3.5 seconds. There is a 1.5 second difference in the time. After approximately half of that time, the rider is at a high point.)

The first point is higher than the second, because in the first maximum, both the large and small wheel are up, but in the second maximum, the small wheel is down. The position of the rider in the small wheel fluctuates. In these two cases, each time the bar reaches its maximum height, the rider is in a different position in the small wheel.

75 feet. No, because the two wheels do not align at the top at the same time. In order to reach the maximum height, the rider would have to start at the top.

Yes. You could use the sine function y = 25 sin (2[pi]t/10) + 10 sin (2[pi]t/7) + 40. However, this would represent the rider starting at a different position.

Topics for Discussion
Discuss the methods used by Ms. Compton to promote discourse and understanding.

How did Ms. Compton's interaction with students support their investigation and conjectures about making connections between their equations and graphs?
How did Ms. Compton use students' diagrams to assess student understanding?
How does this lesson address different learning styles? How can you make the content of your lessons accessible to different learning styles?
Cite examples of student responses and questions that revealed how the calculator helped them to understand the modeling problem.
How do you balance time presenting ideas with time having students apply those ideas?
How does learning mathematics in context enable students to monitor their own thinking?

How did this lesson relate mathematics to real–world applications? What other related applications could you make for this problem?
Given the geometric diagram of a double Ferris wheel, how did students determine the coordinates, the origin, x–axis, y–axis, and chair positions?
How did the students use trigonometric functions from the algebraic coordinates?
Give examples of real–world data you have used in your lessons. What criteria did you use to choose these real–world situations?
How do graphing calculators allow teachers to create lessons that help students to monitor their own thinking?
NCTM's Geometry from a Synthetic Perspective standard states that "Physical models and other real–world objects should be used to provide a strong base for the development of students' geometric intuition so that they can draw on these experiences in their work with abstract ideas." (Curriculum and Evaluation Standards for School Mathematics, p. 157) How is this idea demonstrated in the activity?
Why is it important to question assumptions in problem–solving situations?

How does Ms. Compton help students investigate their assumptions?
What affect would a different set of assumptions—such as a different starting point or the Ferris wheel moving clockwise—have on the way students approach the problem?
Describe how you have helped students identify their assumptions.
What are the advantages and disadvantages of establishing assumptions with the class before beginning work on a problem?