In this lesson, Mary and Claire modify the sine function to give the height above the ground of a student named April as she travels on a Ferris wheel.
Claire and Mary use a calculator to find April’s height above the ground when her angle of rotation is 1 radian, 2.5 radians, and when she is at the top of the Ferris wheel.
Mary and Claire explore how to modify the sine function to give April’s height above the ground at any angle of rotation as she travels around the Ferris wheel.
Mary and Claire reflect on how to modify a sine function to create a function that gives April’s height above the ground at any angle of rotation. They describe how the different parts of the function correspond to different features of the Ferris wheel.
Mathematics in this Lesson
Targeted Understandings
This lesson can help students:
Reason quantitatively about parameters in sinusoidal functions.
Interpret the output of the function y = a sin(x) + k as the height above the ground of an object traveling along the circumference of a circle whose radius is “a” units whose center is “k” units above the ground.
Understand how different parameters correspond to features of the context, like the radius and height above the ground.
Common Core Math Standards
CCSS.MATH.CONTENT.HSF.BF.B.3. Identify the effect on the graph of replacing f(x) by f(x) + k, k f(x), f(kx), and f(x + k) for specific values of k (both positive and negative); find the value of k given the graphs. Experiment with cases and illustrate an explanation of the effects on the graph using technology. Include recognizing even and odd functions from their graphs and algebraic expressions for them.
Claire and Mary explore a context involving a Ferris wheel, and they work to create a function that accurately models the height of a rider at any point during the Ferris wheel’s rotation. The students first explore how multiplying the sine function by a scalar affects the output. They also recognize that the expression 36 • sin(1) represents the height above the midline for an angle of rotation of 1 radian on a circle that has a radius of 36 feet. They try to adjust this to account for the height of the Ferris wheel by trying 2 • 36 • sin(1), but after some discussion, they realize that just represents the height above the midline for an angle of rotation of 1 radian on a circle that has a radius that is twice as large as a circle with a radius of 36 ft. The students also explore ways to account for the height of the Ferris wheel by adding onto the sine function, and eventually realize that to account for the height of a Ferris wheel that has a radius of 36 ft and sits 4 ft off the ground, they can add 40 feet to their function to get 36 • sin(n) + 40.
CCSS.MATH.CONTENT.HSF.BF.A.1.A. Determine an explicit expression, a recursive process, or steps for calculation from a context.
Mary and Claire determine how April’s height above the ground relates to her angle of rotation using their understanding of the sine function. They begin by exploring how multiplying the function affects the function output, and they make several important discoveries. For example, they initially multiply their function by 2 to account for the height of the Ferris wheel, but quickly they realize that they’ve modified the function to describe a Ferris Wheel that is twice as large. They also experiment with adding values to the sine function to account for the height of the Ferris wheel, and they discover that by adding to the function they can model the height of the Ferris wheel accurately. They modify their sine function to create a sinusoidal function and determine an explicit expression and steps for finding that height both in radii and other units.
This lesson features students trying to create a function that models the height of a rider on a Ferris wheel at points along the wheel’s journey. This proves to be challenging for Mary and Claire as they explore the effect of multiplying the sine function by scalars and of adding values to the function. In Episode 2, the students use given information and the sine function to find the height at certain points [e.g., the very top of the Ferris wheel; 6:14]. In Episode 3, they begin experimenting with the sine function using a systematic guess-and-check strategy. For example, they try 2x • sin(n) + 4 as a possible function [1:36], but after trying this function for known points [e.g., 2:34], they realize that multiplying by 2 effectively expresses the sine function for a circle that is twice as large as the Ferris wheel, rather than accounts for the height of the given Ferris wheel. They also explore the effect of adding to the function. For example, they work with a new possible function, 36 • sin(n) + 4, but after trying this function with a known value, they revise their work [Episode 3, 5:03]. The students finally figure out that x • sin(n) + h will describe the height for any Ferris wheel with a radius of x, an angle of rotation of n, and whose center point is h ft off the ground [Episode 4].
