Explore the Rebound Height of A Ball

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Objective

SWBAT recognize the bouncing ball activity as an exponential decay function from a table in the form of y=ab^x, where 0<b<1 and y=a(1-r)^t.

Big Idea

To compare an exponential decay function to an exponential growth function in the previous day's lesson from a table using the equation y=ab^x.

Warm up

10 minutes

 

I intend today's Warm-up to take about 10 minutes to complete. The goal is for me to introduce the lesson.  I expect my students will approach the task by building on the previous day's lesson. The previous lesson was an introduction to an exponential growth function. This lesson is an introduction to an exponential decay problem.

During the warm up, students will attempt to identify patterns by inspecting the values in Table A and in Table B.  Table A is an example of an exponential growth function, similar to yesterday's lesson.  Table B is an example of an exponential decay function. The warm up is meant to expose students to the difference between an exponential growth function and an exponential decay function.  

I discuss the implementation of the Warm-up in greater detail in the video below:

 

 

Teaching NoteFor this lesson, I will prepare copies of the following before class:

  1. Warm up- one for each student
  2. Bouncing ball activity- 1 copy per pair of students
  3. Exit slip- one for each student

Rebounding Ball Activity

15 minutes

 

I introduce the context for today's rebounding ball activity to my students with this short 6 second video.  I state, "Today we will be comparing the number of bounces of a tennis ball to the height of each bounce."  I have students work on this problem with their assigned elbow partner. In this lesson, I provide the students with the problem and with a data table.  

Problem-Bouncing tennis ball :  A tennis ball is dropped from a height of 100 inches. The data below shows the height of the ball after each bounce, which is called the rebound height.  Analyze the pattern of the output of the table, determine the type of function, and write the equation of the function.

The independent variable in this activity is the number of bounces and the dependent variable is the rebound height of the bouncing ball after each bounce.  The introductory video provides a visual anchor for students. It helps them to arrive at the conjecture that the rebound height reduces after each bounce.  

As they analyze the data, I want my students to recognize this function as an example of exponential decay, like Table B in the warm up.  Some mathematical facts and observations that I want to cover with my students are:

  • The parameter, a, in the equation y=ab^x is the starting value or the dropping height which is 100.  This is also the y intercept.  
  • The (.55) is the geometric ratio for this data, which is b in the equation found by dividing each output by the previous output in the table.  
  • The value of parameter a changes according to the dropping height.
  • The value of parameter b in the equation changes according to the elasticity of the ball or the rebound height. For some materials, this can be a constant, rather than a variable.
  • Changing from a tennis ball to a basketball, for example, will change the b value, which is the common ratio, and the elasticity of the ball. 

 

Frayer Model on Exponential Decay

15 minutes

 

After a period of sharing student responses for the bouncing ball activity in class, I will ask my students to take notes on a Frayer Model graphic organizer.  I begin the class notes section by requesting that the students take out their notes from yesterday's exponential growth lesson. I want to continue to strand of comparative thinking.  I choose to use the Frayer Model because it helps my students better understand the concepts, particulary when the Frayer structure allows for comparisons. My goal for this section is to build a solid foundation so that students will soon be able to write the equation for exponential functions in both forms: y=ab^x and y=(1-r)^t.  

 

I present and describe a completed Frayer Model in the video below:

 

 

Exit Slip

10 minutes

 

I use today's Exit slip as a formative assessment of student understanding with respect to writing an exponential equation to model a given situation.  

During the class notes presentation for this lesson, I emphasized the differences between exponential growth and exponential decay. At this point, I want my students to be able to identify the percent of increase or decrease in an exponential function, and, the common ratio being multiplied between intervals.

Some of the results that I am looking for in student responses are as follows:

  • the function is decreasing by a factor of .88, 88/100, or 22/25 each year
  • the function is decreasing by a percentage of 12% each year
  • the equation y=18,000(.88)^t models the function
  • the equation y=18,000(1-.12)^t models the function 

The price of the car after 8 years is $6473.42.