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MEASURES OF RESTITUTION
by Chris Hiatt
University of Idaho

ITV SERIES LIFE BY NUMBERS #102 THE NUMBERS GAME

GRADES 7 - 9

OVERVIEW

An important part of mathematics is data gathering and analysis. Students should be able to gather data with accuracy and precision. They should also be able to analyze the data for its mean. This lesson will introduce several problem-solving skills used to analyze data. To give students practice with data gathering and analysis, this lesson will have students, in groups of three, gather data on the coefficient of restitution of several types of balls. Then they will compare their data with the other groups to analyze the accuracy and precision of the data. Precision in data collection for this lesson will be difficult to achieve, so it will be a good introduction to data collection, because it will force them to be very careful in their measuring.


LEARNING OBJECTIVES

Students should be able to:

  • Gather data in a scientific manner
  • Compute means from data
  • Use analysis skills to interpret data
  • Know the difference between precision and accuracy in data collection
  • Write a letter to a university engineering program for research.

 

MATERIALS

  • Pencils and paper
  • Calculator
  • Several types of balls (golf balls, tennis balls, racket balls, elastic rubber balls, etc...)
  • Yardsticks for measuring
  • Word Processor and Internet accessibility to write letter to university engineering department
  • Group measuring worksheet

 


PREVIEWING ACTIVITIES

Review the idea of mean. Ask students to demonstrate how means are calculated. Have several students go to the board and calculate means of data that you give them. Also ask students what the difference is between accuracy and precision. Have them give examples of both. Show them archery examples of accuracy and precision.

 

FOCUS FOR VIEWING

To give students a specific responsibility while viewing have students look for and write down the mathematical formula for the coefficient of restitution that is presented in the video. There are several spots in the video for students to participate in discussion. Students will need to have paper and pencil during viewing.


VIEWING ACTIVITIES

BEGIN the video when the words A Measure of Power come onto the screen. PAUSE the video when Rom Ramnath says that his is the "only independent testing facility in the country." Ask the students why it is important to have an independent tester. Ask what would happen if competing companies were testing each other's rackets.

RESUME the video and PAUSE the video again when Ramnath gives the example of the coefficient of restitution being 0.8. Ask the students what other equipment could be tested with this formula. Possible answers would be baseball bats, racket ball rackets, ping pong paddles, etc... Ask how other equipment could be tested such as golf clubs, any kind of ball, etc...

RESUME the video and PAUSE the video again when Danny Glover says "and defines it as the rackets sweet spot." Ask the students what is good about the sweet spot on a racket. Answer: it gives more power to the ball. Ask what other equipment has a sweet spot. Answers could be baseball bats, golf clubs, etc...

RESUME the video and PAUSE it again right after Ramnath says "to very complex issues" and walks up the steps of the large building with columns.

 

ACTION PLAN

Ask students what substance they think has the largest coefficient of restitution. Have the students finalize the project by having each group write to different universities engineering department to ask them what material has the largest coefficient of restitution.

Have them explain their project and why they want to know. Have them go onto to the Internet to find the address for the university. Possible university engineering departments and their web addresses:

University of Idaho
http://www.uidaho.edu/engr/cedept/

University of Southern California
http://www.usc.edu/dept/engineering/

University of California Berkley
http://www-cive.stanford.edu/

geninfo.html

Massachusetts Institute of Technology
http://esd.mit.edu/

University of Washington
http://www.engr.washington.edu/

 

POST VIEWING ACTIVITIES

Introduce the experiment of testing the coefficient of restitution of different types of balls. Explain that the coefficient of restitution of the ball is the amount of energy it converts into motion when it hits a solid object. Explain that it is the same as the coefficient of restitution for the rackets, except to measure it we will drop a ball and then measure how high it bounces back after it hits the ground. The coefficient of restitution of the ball would then be the height that the ball bounces back divided by the height that it was dropped from. Give example that if you dropped a ball from two feet and it bounced one foot in the air after it was dropped then its coefficient of restitution would be 0.5.

Show students the different balls that will be tested and then have them write down a guess of what the coefficient of restitution of each ball is. Break students up into groups of three and give each group a set of balls. Each group should test each type of ball. For example if you are testing golf balls, basketballs, and tennis balls. Each group should test each of these three types of balls. Give each group a yardstick also to make the measurements. Explain the rules of the test. Each group should split up into a dropper, catcher, and a recorder.

Each ball will be tested 10 times. It must be dropped such that the bottom of the ball is even with the top of the yardstick. The catcher should then catch the ball when it is at the peak of its bounce after it has hit the ground. Then the recorder should record the height it bounced back. The recorder needs to make sure that the measurement was done from the bottom of the ball so that they remain accurate in their testing. Have the recorder put the data into the worksheet included with this lesson. Then, have the students find the average bounce height of each ball. To find the coefficient of restitution of each ball, have them divide the average bounce height by the drop height: which is one yard.

Explain how the units must be the same. If they measured the bounce height in inches, the drop height must be in inches as well. Have each group put their coefficients of restitution on the board. Then ask the students why all of their coefficients of restitution aren't the same for the same type of ball. Answers should be differences in recording the bounce heights, slight differences in the balls, difference in the precision of the measurements, etc..

Have students compute the average coefficients of restitution from all of their average coefficients of restitution for each ball. Then have them compare these final coefficients of restitution to their guesses at the beginning of the lesson. Ask them how close they were, and why they were or weren't close.

EXTENSIONS

Ask the students to figure out how high each of their balls would bounce if they were dropped off the Empire State Building. Ask them what else could affect the height that the ball bounced besides the ball's coefficient of restitution. Possible answers could be wind, surface that the ball hits, the ball breaking, etc... Have the students search the Internet for other tall buildings or structures, and then have them figure out how high each ball would bounce from these structures. Possible search engine to use would be http://www.askjeeves.com/

Bring in a part of the video Flubber. Ask the students if it is possible to make a substance like Flubber (a substance with a coefficient of restitution greater than one). Ask them why or why not. Possible answer could be that it goes against the first law of thermodynamics (energy is neither created not destroyed).

Ask the students how data can be manipulated to better fit a bias. Have the students manipulate their own data by removing two measures from their data. Have each group be biased toward a certain type of ball, and manipulate their data to make their ball look better. Have them present the manipulated data to show how much of a difference bias makes in data collection.

 


For additional lesson plans and ideas relating to this topic and many others try TeacherSource at PBS Online! You will find activities, lesson plans, teacher guides and links to other great educational web sites! Search the database by keyword, grade level or subject area! Mathline and Scienceline are also great resources for teachers seeking teaching tips, lesson plans, assessment methods, professional development, and much more!


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