Statway
TM
A statistics pathway for college students
Module 1:
Statistical Studies and Overview of the Data Analysis Process
Module 2:
Summarizing Data Graphically and Numerically
Module 3:
Reasoning About Bivariate Numerical Data—Linear
Relationships
Module 4:
Modeling Nonlinear Relationships
Module 5:
Reasoning About Bivariate Categorical Data and Introduction
to Probability
Module 6:
Formalizing Probability and Probability Distributions
Module 7:
Linking Probability to Statistical Inference
Module 8:
Inference for One Proportion
Module 9:
Inference for Two Proportions
Module 10:
Inference for Means
Module 11:
Chi-Squared Tests
Module 12:
Other Mathematical Content
Version 1.0
A resource from
The Charles A. Dana Center at
The University of Texas at Austin
July
2011
Frontmatter
Statway—Full Version 1.0, July 2011
Unless otherwise indicated, the materials found in this resource are
Copyright © 2011 by the Carnegie Foundation for the Advancement of Teaching and
the Charles A. Dana Center at The University of Texas at Austin
Outside the license described below, no part of this resource shall be reproduced, stored in a retrieval system, or
transmitted by any means—electronically, mechanically, or via photocopying, recording, or otherwise, including
via methods yet to be invented—without express written permission from the Foundation and the University.
The original version of this work was created by The Charles A. Dana Center at The University of Texas at
Austin under sponsorship of the Carnegie Foundation for the Advancement of Teaching.
STATWAYTM / StatwayTM is a trademark of the Carnegie Foundation for the Advancement of Teaching.
***
This copyright notice is intended to prohibit unlicensed commercial use of the Statway materials.
License for use
Statway Version 1.0, developed by the Charles A. Dana Center at The University of Texas at Austin under
sponsorship of the Carnegie Foundation for the Advancement of Teaching, is licensed under the Creative
Commons Attribution-Noncommercial-Share Alike 3.0 Unported license.
To view the details of this license, see creativecommons.org/licenses/by-nc-sa/3.0. In general, under this
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The original version of this work was developed by the Charles A. Dana Center at the University of
Texas at Austin under sponsorship of the Carnegie Foundation for the Advancement of Teaching.
This work is used (or adapted) under the Creative Commons Attribution-NonCommercialShareAlike 3.0 Unported (CC BY-NC-SA 3.0) license: creativecommons.org/licenses/by-nc-sa/3.0.
For more information about Carnegie’s work on Statway, see www.carnegiefoundation.org/statway;
for information on the Dana Center’s work on The New Mathways Project, see
www.utdanacenter.org/mathways.
Noncommercial—You may not use this work for commercial purposes.
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liability for any loss or damage resulting from the use of this resource. We have made extensive efforts to ensure
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otherwise comply with copyright law. If you find an error or you believe we have failed to provide proper
acknowledgment, please contact us at dana-txshop@utlists.utexas.edu.
The original versions of the Statway™ and Quantway™ courses were created by The Charles A. Dana Center at The University of Texas at Austin under
sponsorship of the Carnegie Foundation for the Advancement of Teaching, and are copyright © 2011 by the Carnegie Foundation for the Advancement of
Teaching and the Charles A. Dana Center at The University of Texas at Austin. STATWAY™/Statway™ and Quantway™ are trademarks of the Carnegie
Foundation for the Advancement of Teaching. The Dana Center’s frontmatter for Statway™ and Quantway™ is available at www.utdanacenter.org/mathways.
ii
Frontmatter
The Charles A. Dana Center
The University of Texas at Austin
1616 Guadalupe Street, Suite 3.206
Austin, TX 78701-1222
Fax: 512-232-1855
dana-txshop@utlists.utexas.edu
www.utdanacenter.org
Statway—Full Version 1.0, July 2011
The Carnegie Foundation for the Advancement of Teaching
51 Vista Lane
Stanford, California, 94305
Phone: 650-566-5110
pathways@carnegiefoundation.org
www.carnegiefoundation.org
About the development of this resource
The content for this full version of Statway was developed under a November 30, 2010, agreement by a team of
faculty authors and reviewers contracted and managed by the Charles A. Dana Center at the University of Texas
at Austin with funding from the Carnegie Foundation for the Advancement of Teaching.
This resource was produced in Microsoft Word 2008 and 2011 for the Mac. The content of these 12 modules
was developed and produced (that is, written, reviewed, edited, and laid out) by the Charles A. Dana Center at
The University of Texas at Austin and delivered by the Dana Center to the Carnegie Foundation for the
Advancement of Teaching on June 30, 2011.
Some issues to be aware of:
•
PDF files need to be viewed with Adobe Acrobat for full functionality. If viewed through Preview,
which is the default on some computers, the URLs may not be correct.
• The file names indicate the lesson number and whether the document is the instructor or student version
or the out-of-class experience.
The Dana Center is engaged in a process of revising and improving these materials to create the Dana Center
Statistics Pathway. We welcome feedback from the community as part of our course revision process. If you
would like to discuss these materials or learn more about the Dana Center’s plans for this course, contact us at
mathways@austin.utexas.edu.
About the Charles A. Dana Center at The University of Texas at Austin
The Dana Center collaborates with local and national entities to improve education systems so that they foster
opportunity for all students, particularly in mathematics and science. We are dedicated to nurturing students’
intellectual passions and ensuring that every student leaves school prepared for success in postsecondary
education and the contemporary workplace—and for active participation in our modern democracy.
The Center was founded in 1991 in the College of Natural Sciences at The University of Texas at Austin. Our
original purpose—which continues in our work today—was to raise student achievement in K–16 mathematics
and science, especially for historically underserved populations. We carry out our work by supporting high
standards and building system capacity; collaborating with key state and national organizations to address
emerging issues; creating and delivering professional supports for educators and education leaders; and writing
and publishing education resources, including student supports.
Our staff of more than 80 researchers and education professionals has worked intensively with dozens of school
systems in nearly 20 states and with 90 percent of Texas’s more than 1,000 school districts. As one of the
College’s largest research units, the Dana Center works to further the university’s mission of achieving
excellence in education, research, and public service. We are committed to ensuring that the accident of where a
student attends school does not limit the academic opportunities he or she can pursue.
For more information about the Dana Center and our programs and resources, see our homepage at
www.utdanacenter.org. To access our resources (many of them free) please see our products index at
www.utdanacenter.org/products. To learn about Dana Center professional development sessions, see our
professional development site at www.utdanacenter.org/pd.
The original versions of the Statway™ and Quantway™ courses were created by The Charles A. Dana Center at The University of Texas at Austin under
sponsorship of the Carnegie Foundation for the Advancement of Teaching, and are copyright © 2011 by the Carnegie Foundation for the Advancement of
Teaching and the Charles A. Dana Center at The University of Texas at Austin. STATWAY™/Statway™ and Quantway™ are trademarks of the Carnegie
Foundation for the Advancement of Teaching. The Dana Center’s frontmatter for Statway™ and Quantway™ is available at www.utdanacenter.org/mathways.
iii
Frontmatter
Statway—Full Version 1.0, July 2011
Acknowledgments
The original version of this work was created by The Charles A. Dana Center at The University of Texas at
Austin under sponsorship of the Carnegie Foundation for the Advancement of Teaching.
Carnegie Corporation of New York, The Bill & Melinda Gates Foundation, The William and Flora Hewlett
Foundation, Lumina Foundation, and The Kresge Foundation joined in partnership with the Carnegie
Foundation for the Advancement of Teaching in this work.
Leadership—Charles A. Dana Center at the University of Texas at Austin
Uri Treisman, director
Susan Hudson Hull, program director of mathematics national initiatives
Leadership—Carnegie Foundation for the Advancement of Teaching
Anthony S. Bryk, president
Bernadine Chuck Fong, senior managing partner
Louis Gomez, senior fellow
Paul LeMahieu, senior fellow
James Stigler, senior fellow
Uri Treisman, senior fellow
Guadalupe Valdés, senior fellow
Statway Project Leads
Kristen Bishop, former team lead for the New Mathways Project, the Charles A. Dana Center at the University
of Texas at Austin
Thomas J. Connolly, project lead, Statway, the Charles A. Dana Center at the University of Texas at Austin
Karon Klipple, director of Statway, the Carnegie Foundation for the Advancement of Teaching
Jane Muhich, director of Quantway, the Carnegie Foundation for the Advancement of Teaching
Project Staff—Charles A. Dana Center at the University of Texas at Austin
Richard Blount, advisor
Kathi Cook, project director, online services team
Jenna Cullinane, research associate
Steve Engler, lead editor and production editor
Amy Getz, team lead for the New Mathways Project
Susan Hudson Hull, program director of mathematics national initiatives
Joseph Hunt, graduate research assistant
Rachel Jenkins, consulting editor
Erica Moreno, program coordinator
Carol Robinson, administrative associate
Cathy Seeley, senior fellow
Rachele Seifert, administrative associate
Lilly Soto, senior administrative associate
Phil Swann, senior designer
Laura Torres, graduate research assistant
Thomas Wiegel, freelance formatter and proofreader
The original versions of the Statway™ and Quantway™ courses were created by The Charles A. Dana Center at The University of Texas at Austin under
sponsorship of the Carnegie Foundation for the Advancement of Teaching, and are copyright © 2011 by the Carnegie Foundation for the Advancement of
Teaching and the Charles A. Dana Center at The University of Texas at Austin. STATWAY™/Statway™ and Quantway™ are trademarks of the Carnegie
Foundation for the Advancement of Teaching. The Dana Center’s frontmatter for Statway™ and Quantway™ is available at www.utdanacenter.org/mathways.
iv
Frontmatter
Statway—Full Version 1.0, July 2011
Authors Contracted by the Dana Center
Roxy Peck, professor emerita of statistics, California Polytechnic State University, San Luis Obispo, California
Beth Chance, professor of statistics, California Polytechnic State University, San Luis Obispo, California
Robert C. delMas, associate professor of educational psychology, University of Minnesota, Minneapolis,
Minnesota
Scott Guth, professor of mathematics, Mt. San Antonio College, Walnut, California
Rebekah Isaak, graduate research student, University of Minnesota, Minneapolis, Minnesota
Leah McGuire, assistant professor, University of Minnesota, Minneapolis, Minnesota
Jiyoon Park, graduate research student, University of Minnesota, Minneapolis, Minnesota
Brian Kotz, associate professor of mathematics, Montgomery College, Germantown, Maryland
Chris Olsen, assistant professor of mathematics and statistics, Grinnell College, Grinnell, Iowa
Mary Parker, professor of mathematics, Austin Community College, Austin, Texas
Michael A. Posner, associate professor of statistics, Villanova University, Villanova, Pennsylvania
Thomas H. Short, professor, John Carroll University, University Heights, Ohio
Penny Smeltzer, teacher of statistics, Westwood High School, Austin, Texas
Myra Snell, professor of mathematics, Los Medanos College, Pittsburg, California
Laura Ziegler, graduate research student, University of Minnesota, Minneapolis, Minnesota
Reviewers Contracted by the Dana Center
Michelle Brock, American River College, Sacramento, California
Thomas J. Connolly, the Charles A. Dana Center at the University of Texas at Austin
Andre Freeman, Capital Community College, Hartford, Connecticut
Karon Klipple, the Carnegie Foundation for the Advancement of Teaching
Roxy Peck, professor emerita of statistics, California Polytechnic State University, San Luis Obispo, California
Jim Smart, Tallahassee Community College, Tallahassee, Florida
Myra Snell, Los Medanos College, Pittsburg, California
Committee for Statistics Learning Outcomes
Rose Asera, formerly of the Carnegie Foundation for the Advancement of Teaching
Kristen Bishop, formerly of the Charles A. Dana Center at the University of Texas at Austin
Richelle (Rikki) Blair, American Mathematical Association of Two-Year Colleges (AMATYC); Lakeland
Community College, Ohio
David Bressoud, Mathematical Association of America (MAA); Macalester College, Minnesota
John Climent, American Mathematical Association of Two-Year Colleges (AMATYC); Cecil College,
Maryland
Peg Crider, Lone Star College, Tomball, Texas
Jenna Cullinane, the Charles A. Dana Center at the University of Texas at Austin
Robert C. delMas, Consortium for the Advancement of Undergraduate Statistics Education (CAUSE);
University of Minnesota, Minneapolis, Minnesota
Bernadine Chuck Fong, the Carnegie Foundation for the Advancement of Teaching
Karen Givvin, the University of California, Los Angeles
Larry Gray, American Mathematical Society (AMS); University of Minnesota
Susan Hudson Hull, the Charles A. Dana Center at the University of Texas at Austin
Rob Kimball, American Mathematical Association of Two-Year Colleges (AMATYC); Wake Technical
Community College, North Carolina
Dennis Pearl, Consortium for the Advancement of Undergraduate Statistics Education (CAUSE); The Ohio
State University
Roxy Peck, American Statistical Association (ASA); Consortium for the Advancement of Undergraduate
Statistics Education (CAUSE); California Polytechnic State University, San Luis Obispo, California
The original versions of the Statway™ and Quantway™ courses were created by The Charles A. Dana Center at The University of Texas at Austin under
sponsorship of the Carnegie Foundation for the Advancement of Teaching, and are copyright © 2011 by the Carnegie Foundation for the Advancement of
Teaching and the Charles A. Dana Center at The University of Texas at Austin. STATWAY™/Statway™ and Quantway™ are trademarks of the Carnegie
Foundation for the Advancement of Teaching. The Dana Center’s frontmatter for Statway™ and Quantway™ is available at www.utdanacenter.org/mathways.
v
Frontmatter
Statway—Full Version 1.0, July 2011
Myra Snell, American Mathematical Association of Two-Year Colleges (AMATYC); Los Medanos College,
Pittsburg, California
Jim Stigler, the Carnegie Foundation for the Advancement of Teaching; the University of California, Los
Angeles
Daniel Teague, Mathematical Association of America (MAA); North Carolina School of Science and
Mathematics, Durham
Uri Treisman, the Carnegie Foundation for the Advancement of Teaching; the Charles A. Dana Center at the
University of Texas at Austin
Version 1.0 of Statway was developed in collaboration with faculty from the following colleges, the
“Collaboratory,” who advised on the development of the course. These Collaboratory colleges are:
Florida
Miami Dade College, Miami, Florida
Tallahassee Community College, Tallahassee,
Florida
Valencia Community College, Orlando,
Florida
California
American River College, Sacramento, California
Foothill College, Los Altos Hills, California
Mt. San Antonio College, Walnut, California
Pierce College, Woodland Hills, California
San Diego City College, San Diego, California
California State University System
Texas
CSU Northridge
Sacramento State University
San Jose State University
Austin Community College, Austin, Texas
El Paso Community College, El Paso, Texas
Houston Community College, Houston, Texas
Northwest Vista College, San Antonio, Texas
Richland College, Dallas, Texas
Connecticut
Washington
Capital Community College, Hartford, Connecticut
Gateway Community College, New Haven, Connecticut
Housatonic Community College, Bridgeport,
Connecticut
Naugatuck Valley Community College, Waterbury,
Connecticut
Seattle Central Community College, Seattle,
Washington
Tacoma Community College, Tacoma,
Washington
The original versions of the Statway™ and Quantway™ courses were created by The Charles A. Dana Center at The University of Texas at Austin under
sponsorship of the Carnegie Foundation for the Advancement of Teaching, and are copyright © 2011 by the Carnegie Foundation for the Advancement of
Teaching and the Charles A. Dana Center at The University of Texas at Austin. STATWAY™/Statway™ and Quantway™ are trademarks of the Carnegie
Foundation for the Advancement of Teaching. The Dana Center’s frontmatter for Statway™ and Quantway™ is available at www.utdanacenter.org/mathways.
vi
Frontmatter
Statway—Full Version 1.0, July 2011
Statway, Full Version 1.0, July 2011
Table of Contents
Module 1: Statistical Studies and Overview of the Data Analysis Process
Lesson 1.1.1: The Statistical Analysis Process
Lesson 1.1.2: Types of Statistical Studies and Scope of Conclusions
Lesson 1.2.1: Collecting Data by Sampling
Lesson 1.2.2: Random Sampling
Lesson 1.2.3: Other Sampling Strategies
Lesson 1.2.4: Sources of Bias in Sampling
Lesson 1.3.1: Collecting Data by Conducting an Experiment
Lesson 1.3.2: Other Design Considerations—Blinding, Control Groups, and Placebos
Lesson 1.4.1: Drawing Conclusions from Statistical Studies
Module 2: Summarizing Data Graphically and Numerically
Lesson 2.1.1: Dotplots, Histograms, and Distributions for Quantitative Data
Lesson 2.1.2: Constructing Histograms for Quantitative Data
Lesson 2.1.3: Comparing Distributions of Quantitative Data in Two Independent Samples
Lesson 2.2.1: Quantifying the Center of a Distribution—Sample Mean and Sample Median
Lesson 2.2.2: Constructing Histograms for Quantitative Data
Lesson 2.3.1: Quantifying Variability Relative to the Median
Lesson 2.4.1: Quantifying Variability Relative to the Mean
Lesson 2.4.2: The Sample Variance
Module 3: Reasoning About Bivariate Numerical Data—Linear Relationships
Lesson 3.1.1: Introduction to Scatterplots and Bivariate Relationships
Lesson 3.1.2: Developing an Intuitive Sense of Form, Direction, and Strength of the Relationship
Between Two Measurements
Lesson 3.1.3: Introduction to the Correlation Coefficient and Its Properties
Lesson 3.1.4: Correlation Formula
Lesson 3.1.5: Correlation Is Not Causation
Lesson 3.2.1: Using Lines to Make Predictions
Lesson 3.2.2: Least Squares Regression Line as Line of Best Fit
The original versions of the Statway™ and Quantway™ courses were created by The Charles A. Dana Center at The University of Texas at Austin under
sponsorship of the Carnegie Foundation for the Advancement of Teaching, and are copyright © 2011 by the Carnegie Foundation for the Advancement of
Teaching and the Charles A. Dana Center at The University of Texas at Austin. STATWAY™/Statway™ and Quantway™ are trademarks of the Carnegie
Foundation for the Advancement of Teaching. The Dana Center’s frontmatter for Statway™ and Quantway™ is available at www.utdanacenter.org/mathways.
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Statway—Full Version 1.0, July 2011
Lesson 3.2.3: Investigating the Meaning of Numbers in the Equation of a Line
Lesson 3.2.4: Special Properties of the Least Squares Regression Line
Lesson 3.3.1: Using Residuals to Determine If a Line Is a Good Fit
Lesson 3.3.2: Using Residuals to Determine If a Line Is an Appropriate Model
Module 4: Modeling Nonlinear Relationships
Lesson 4.1.1: Investigating Patterns in Data
Lesson 4.1.2: Exponential Models
Lesson 4.1.3: Assessing How Well a Model Fits the Data
Module 5: Reasoning About Bivariate Categorical Data and Introduction to Probability
Lesson 5.1.1: Reasoning About Risk and Chance
Lesson 5.1.2: Defining Risk
Lesson 5.1.3: Interpreting Risk
Lesson 5.1.4: Comparing Risks
Lesson 5.1.5: More on Conditional Risks
Module 6: Formalizing Probability and Probability Distributions
Lesson 6.1.1: Probability Lesson 6.1.2: Probability Rules
Lesson 6.1.3: Simulation, Discrete Random Variables, and Probability Distributions
Lesson 6.2.1: Probability Distributions of Continuous Random Variables
Lesson 6.2.2: Z-Scores and Normal Distributions
Lesson 6.2.3: Using Normal Distributions to Find Probabilities and Critical Values
Module 7: Linking Probability to Statistical Inference
Lesson 7.1.1: Predicting an Election—Statistics and Sampling Variability
Lesson 7.1.2: Sampling from a Population
Lesson 7.1.3: Testing Statistical Hypotheses
Lesson 7.2.1: Two Types of Inferential Procedures—Estimation and Hypothesis Testing
Lesson 7.2.2: Connecting Sampling Distributions and Confidence Intervals
Lesson 7.2.3: Connecting Sampling Distributions and Hypothesis Testing
The original versions of the Statway™ and Quantway™ courses were created by The Charles A. Dana Center at The University of Texas at Austin under
sponsorship of the Carnegie Foundation for the Advancement of Teaching, and are copyright © 2011 by the Carnegie Foundation for the Advancement of
Teaching and the Charles A. Dana Center at The University of Texas at Austin. STATWAY™/Statway™ and Quantway™ are trademarks of the Carnegie
Foundation for the Advancement of Teaching. The Dana Center’s frontmatter for Statway™ and Quantway™ is available at www.utdanacenter.org/mathways.
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Frontmatter
Statway—Full Version 1.0, July 2011
Module 8: Inference for One Proportion
Lesson 8.1.1: Sampling Distribution of One Proportion
Lesson 8.1.2: Sampling Distribution of One Proportion
Lesson 8.2.1: Estimation of One Proportion
Lesson 8.2.2: Estimation of One Proportion
Lesson 8.3.1: Estimation of One Proportion
Lesson 8.3.2: Hypothesis Testing for One Proportion
Module 9: Inference for Two Proportions
Lesson 9.1.1: Sampling Distribution of Differences of Two Proportions
Lesson 9.1.2: Using Technology to Explore the Sampling Distribution of the Differences in Two
Proportions
Lesson 9.2.1: Confidence Intervals for the Difference in Two Population Proportions
Lesson 9.2.2: Computing and Interpreting Confidence Intervals for the Difference in Two
Population Proportions
Lesson 9.3.1: A Statistical Test for the Difference in Two Population Proportions
Lesson 9.3.2: A Statistical Test for the Difference in Two Population Proportions
Lesson 9.3.3: Conducting a Statistical Test for the Difference in Two Population Proportions
Module 10: Inference for Means
Lesson 10.1.1: The Sampling Distribution of the Sample Mean
Lesson 10.1.2: Using an Applet to Explore the Sampling Distribution of the Mean with Focus on
Shape
Lesson 10.2.1: Estimating a Population Mean
Lesson 10.2.2: T-Statistics and T-Distributions
Lesson 10.2.3: The Confidence Interval for a Population Mean
Lesson 10.3.1: Testing Hypotheses About a Population Mean
Lesson 10.3.2: Test Statistic and P-Values, One-Sample T-Test
Lesson 10.4.1: Inferences About the Difference Between Two Population Means
Lesson 10.4.2: Inference for Paired Data
Lesson 10.4.3: Two-Sample T-Test
The original versions of the Statway™ and Quantway™ courses were created by The Charles A. Dana Center at The University of Texas at Austin under
sponsorship of the Carnegie Foundation for the Advancement of Teaching, and are copyright © 2011 by the Carnegie Foundation for the Advancement of
Teaching and the Charles A. Dana Center at The University of Texas at Austin. STATWAY™/Statway™ and Quantway™ are trademarks of the Carnegie
Foundation for the Advancement of Teaching. The Dana Center’s frontmatter for Statway™ and Quantway™ is available at www.utdanacenter.org/mathways.
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Frontmatter
Statway—Full Version 1.0, July 2011
Module 11: Chi-Squared Tests
Lesson 11.1.1: Introduction to Chi-Square Tests for One-Way Tables
Lesson 11.1.2: Executing the Chi-Square Test for One-Way Tables (Goodness-of-Fit)
Lesson 11.1.3: The Chi-Square Distribution and Degrees of Freedom
Lesson 11.2.1: Introduction to Chi-Square Tests for Two-Way Tables
Lesson 11.2.2: Executing the Chi-Square Test for Independence in Two-Way Tables
Lesson 11.2.3: Executing the Chi-Square Test for Homogeneity in Two-Way Tables
Module 12: Other Mathematical Content
Lesson 12.1.1: Statistical Linear Relationships and Mathematical Models of Linear Relationships
Lesson 12.1.2: Mathematical Linear Models
Lesson 12.1.3: Contrasting Mathematical and Statistical Linear Relationships
Lesson 12.1.4: Proportional Models
Lesson 12.2.1: Multiple Representations of Exponential Models
Lesson 12.2.2: Linear Models—Answering Various Types of Questions Algebraically
Lesson 12.2.3: Power Models
Lesson 12.2.4: Solving Inequalities
The original versions of the Statway™ and Quantway™ courses were created by The Charles A. Dana Center at The University of Texas at Austin under
sponsorship of the Carnegie Foundation for the Advancement of Teaching, and are copyright © 2011 by the Carnegie Foundation for the Advancement of
Teaching and the Charles A. Dana Center at The University of Texas at Austin. STATWAY™/Statway™ and Quantway™ are trademarks of the Carnegie
Foundation for the Advancement of Teaching. The Dana Center’s frontmatter for Statway™ and Quantway™ is available at www.utdanacenter.org/mathways.
x
Statway Instructor’s Notes April 23, 2012 (Full Version 1.0) Initiating Lesson 4.1.1: Investigating Patterns in Data
Estimated number of 50-­‐minute class sessions: 1 Learning Goals Students will understand that •
•
•
•
•
bivariate numerical data may not form a linear pattern. some types of physical phenomena give data that they can expect to have a particular type of underlying pattern. when they search for patterns to model data, usually they are looking for fairly smooth curves. while it is easier to identify patterns in almost non-­‐noisy data, it is also fairly easy to identify patterns in somewhat noisy data. (However, the noisier the data, the less confident students are in their assessment of the underlying pattern in the data.) when data fit a nonlinear pattern, it is usually true that over some interval of x-­‐values the pattern looks fairly close to linear. Students will be able to identify which pattern for fairly non-­‐noisy data best fits the data from a given repertoire of relationship types given by graphs. Students will begin to be able to think of some real-­‐world phenomena for several nonlinear patterns that might generate that type of pattern in the data. Part I, Rich Task Recommended Task Structure 2 minutes: Give a brief introduction about how radioactivity is dangerous and on these particular sites there had been a large amount of radioactivity that was decaying. The questions here focus on when the areas will be safe for people to live. There are two different levels that might be considered safe. Data are provided about the annual exposure to radioactivity for each site. 3 minutes: Have students look at the graphs and answer the questions about Area 2. 3 minutes: Conduct a wrap-­‐up and discussion of the Part I Rich Task. Introduction to the Context of Radioactivity Data Set [Student Handout] (Note: Provide background information [qualitative] to ensure that students are familiar with the concept of radioactive decay.) During the last half of the 20th century, several sites in Nevada were used for testing nuclear devices (i.e., dangerous nuclear devices were exploded on those sites). Nuclear explosions spread dangerous radioactive particles in the air, which settle on the ground; this is dangerous to living things in that area until the radioactive material decays. Testing ended around 1985, so no new radioactive sources were added after that time. The data in the table and graphs [on the next page] give the amount of radioactivity measured at various times at two locations. Radiation exposure is measured in milliroentgens (mR). Because radiation exposure is dangerous as it accumulates, these data are measured in mR/year, which is how many mR a person staying on that site for a full year would receive. The original versions of the Statway™ and Quantway™ courses were created by The Charles A. Dana Center at The University of Texas at Austin under sponsorship of the Carnegie Foundation for the Advancement of Teaching, and are copyright © 2011 by the Carnegie Foundation for the Advancement of Teaching and the Charles A. Dana Center at The University of Texas at Austin. STATWAY™/Statway™ and Quantway™ are trademarks of the Carnegie Foundation for the Advancement of Teaching. The Dana Center’s frontmatter for Statway™ and Quantway™ is available at www.utdanacenter.org/mathways. 1 Statway Instructor’s Notes April 23, 2012 (Full Version 1.0) Initiating Lesson 4.1.1: Investigating Patterns in Data
(Note: These data are from Position N-­‐8 in Area 2 and Position A-­‐9 in Area 4 of the Nevada Test Site of the U.S. Department of Energy.1) Year Area 2 mR/yr Area 4 mR/yr 1989 1,584 1,502 1990 1,453 1,420 1991 1,127 1,127 1992 1,273 1,110 1993 1,290 1,045 1994 1,094 849 1995 1,061 849 1996 914 784 1997 931 767 1998 849 735 1999 833 735 2000 914 751 2001 849 718 2002 800 686 2003 767 637 2004 751 604 2005 735 588 2006 702 571 2007 686 555 2008 653 522 (Note: Radioactive decay occurs in an exponential pattern. This means that how much radioactive material decays over a fixed time interval is not constant, but is a percentage of how much of the radioactive material is present. Facilitate discussion that allows students to arrive at the fact that the rate is not constant but instead a percentage. Then check for understanding.) When is each area safe? •
There is a certain amount of background radioactivity around. For a location to be safe for people, animals, and plants, the radioactivity level does not need to be zero. Natural 1
U.S. Department of Energy. (2009). Nevada test site environmental report 2008 (DOE/NV/25946—790). Washington D.C.: Author. Retrieved from http://tinyurl.com/3fcvuzd) The original versions of the Statway™ and Quantway™ courses were created by The Charles A. Dana Center at The University of Texas at Austin under sponsorship of the Carnegie Foundation for the Advancement of Teaching, and are copyright © 2011 by the Carnegie Foundation for the Advancement of Teaching and the Charles A. Dana Center at The University of Texas at Austin. STATWAY™/Statway™ and Quantway™ are trademarks of the Carnegie Foundation for the Advancement of Teaching. The Dana Center’s frontmatter for Statway™ and Quantway™ is available at www.utdanacenter.org/mathways. 2 Statway Instructor’s Notes April 23, 2012 (Full Version 1.0) Initiating Lesson 4.1.1: Investigating Patterns in Data
•
•
background radiation differs some in places, but 340 mR/year is a typical amount. A place is considered safe for people to live at up to about 100 mR/year above normal background radiation. The long-­‐term effects of low-­‐level radioactivity levels (up to 1,000 mR/year) are not clear. A one-­‐time radiation dose of 400,000 mR is typically fatal if untreated. Overall Question: When do you predict that Area 2 will be safe for people to live? First Analysis (1) Which of the following two questions is easier to answer? Why? (a) If you consider a level of 1,000 mR/year safe, when would you say that Area 2 was safe? (b) If you decide that the safety level should be no more than 440 mR/year, when will you consider Area 2 safe? (2) Answer the easier of Questions 1a and 1b, and explain how you found the answer. Discussion and Wrap-­‐Up of Part I Linear relationships are all students have seen at this point, so it is natural for them to summarize the relationship by a line. It is also sensible for some students to decide that it is not reasonable to use a line here because a line does not fit very well. At this point, it is not necessary to guide students much. Lead a class discussion if they do not quickly figure out that it is easier to find the year when radiation drops below 1,000 mR/year than the year it drops below 440 mR/year (because 1,000 is within the range observed and 440 is not). Then make sure that all students are working on the 1,000-­‐mR/year question for a few minutes. You could write their different predictions on the board in a table like the following: Area 2—when mR is <1,000 Method Here are some methods that students may use: •
•
•
•
doing it “by eye,” taking the year with the first value below 1,000, drawing a line with all the data, or drawing a line with just the part of the data in the first few years. In summary, help students notice that most of their predictions are pretty close, no matter which method they used. The original versions of the Statway™ and Quantway™ courses were created by The Charles A. Dana Center at The University of Texas at Austin under sponsorship of the Carnegie Foundation for the Advancement of Teaching, and are copyright © 2011 by the Carnegie Foundation for the Advancement of Teaching and the Charles A. Dana Center at The University of Texas at Austin. STATWAY™/Statway™ and Quantway™ are trademarks of the Carnegie Foundation for the Advancement of Teaching. The Dana Center’s frontmatter for Statway™ and Quantway™ is available at www.utdanacenter.org/mathways. 3 Statway Instructor’s Notes April 23, 2012 (Full Version 1.0) Initiating Lesson 4.1.1: Investigating Patterns in Data
Part II, Rich Task Recommended Task Structure 3 minutes: Have students do the activity for the Part II Rich Task. 3 minutes: Conduct a wrap-­‐up and discussion. Student Handout In this course, you have only worked with linear relationships. Let’s use the graph of a linear relationship to summarize the data and see how satisfactory it is to answer your questions. Area 2: mR/year (3) Use the line, extended as needed, to answer the following questions: (a) If you consider a level of 1,000 mR/year safe, when would you say that Area 2 was safe? (b) If you decide that the safety level should be no more than 440 mR/year, when will you consider Area 2 safe? (4) Based on your earlier ideas and what you see here, do you think this line is a good summary to use to answer these questions? Why or why not? Do you trust your answers to the questions in Part I? Wrap-­‐Up/Discussion of Part II Remind students that they could have used linear regression to find the line. Write this formula for the line on the board for students to see, telling them that this is one version of the formula: ŷ = –41.5(x – 1989) + 1357.5 The original versions of the Statway™ and Quantway™ courses were created by The Charles A. Dana Center at The University of Texas at Austin under sponsorship of the Carnegie Foundation for the Advancement of Teaching, and are copyright © 2011 by the Carnegie Foundation for the Advancement of Teaching and the Charles A. Dana Center at The University of Texas at Austin. STATWAY™/Statway™ and Quantway™ are trademarks of the Carnegie Foundation for the Advancement of Teaching. The Dana Center’s frontmatter for Statway™ and Quantway™ is available at www.utdanacenter.org/mathways. 4 Statway Instructor’s Notes April 23, 2012 (Full Version 1.0) Initiating Lesson 4.1.1: Investigating Patterns in Data
If students ask why x –1989 was used, show them how they can relabel the x-­‐values as years since 1989; it is then easy to interpret the intercept. If numbers 1989, 1990, and so on are used as the values of x, then the intercept is a really large number that is not meaningful in the context of this problem. Point out that this relabeling of x-­‐values is common when dealing with data where the interesting numbers are very far from x = 0. Tell students that they can use the formula to find that the value of y drops to 444 mR/year in 2011. However, they can also get a pretty good estimate of that by using a ruler to extend the line and approximating. This is a good place to talk about how they can often get answers in more than one way—symbolically or by estimating from a graph. Have students help you identify reasons why this line is not a good summary of the relationship. Highlight answers such as the following: The graph here shows that the data are somewhat curved. The y-­‐values are not decreasing as fast for the later years as they did in earlier years, so the graph is becoming flatter. The 2011 estimate is probably much too soon. Students may notice that if they extend the line and believe in this model, they predict that eventually there is a negative amount of radioactivity, which clearly cannot be true. Point out that in many math classes, teachers do not lead students through the wrong way to do something. That is what is done here. This wrong way is often used by people without their thinking about it being wrong. Just because you can get software to give you a linear regression line and you can make a prediction with it does not mean that is the right thing to do. Have students circle, highlight, or otherwise emphasize their answer to Question 4, which should say that this is not a good method and they would not trust the predictions. This is all motivation for learning a better way to make these predictions. Part III, Scaffolded Conceptual Task 1: Relating Graphs and Written Examples of Nonlinear Relationships Recommended Task Structure 8 minutes: Have students do Conceptual Task 1. 5 minutes: Conduct a wrap-­‐up of Conceptual Task 1. Introduction [Student Handout] In Module 3, you worked with linear relationships. Those are very useful, and so are some other types of relationships between variables. Following are some graphs and examples of nonlinear relationships frequently used in modeling. (Note: Provide an informal definition of modeling. For example, mathematical modeling consists of describing a physical phenomenon via the use of data and/or mathematical equations.) Activity [Student Handout] (Note: Have students work in small groups on this activity.) The original versions of the Statway™ and Quantway™ courses were created by The Charles A. Dana Center at The University of Texas at Austin under sponsorship of the Carnegie Foundation for the Advancement of Teaching, and are copyright © 2011 by the Carnegie Foundation for the Advancement of Teaching and the Charles A. Dana Center at The University of Texas at Austin. STATWAY™/Statway™ and Quantway™ are trademarks of the Carnegie Foundation for the Advancement of Teaching. The Dana Center’s frontmatter for Statway™ and Quantway™ is available at www.utdanacenter.org/mathways. 5 Statway Instructor’s Notes April 23, 2012 (Full Version 1.0) Initiating Lesson 4.1.1: Investigating Patterns in Data
(5) Notice the different types of patterns. Pick one pattern, and read its example(s) that follows all the graphs. Discuss whether the example’s description suggests the type of pattern seen in the graph. Exponential decay x y = 85 × 0.8
Quadratic 2 y = 50 + 42x + x
Exponential growth x y = 5 × 1.5
120
100
80
60
40
20
0
Constrained exponential (7 − x)
(logistic) y = 18/[1 + 2
] 0
10
20
30
40
Periodic (sinusoidal) y = 2 sin(1.57x) + 1 Cubic 3 2 y = x – 2x – x + 2
Exponential decay examples Radioactive decay: The amount of radioactivity emitted from an object (y) over time (x). Medicine: The amount of a medication in your system (y) over time in minutes (x) starting an hour after you took the dose. Quadratic example Height of an object: The height of an object thrown upward (y) over time in seconds (x). Exponential growth examples Population: The population of a country (y) over time in years (x). Money invested: The amount of money in an account (y) that is invested at a certain interest rate, compounded. The original versions of the Statway™ and Quantway™ courses were created by The Charles A. Dana Center at The University of Texas at Austin under sponsorship of the Carnegie Foundation for the Advancement of Teaching, and are copyright © 2011 by the Carnegie Foundation for the Advancement of Teaching and the Charles A. Dana Center at The University of Texas at Austin. STATWAY™/Statway™ and Quantway™ are trademarks of the Carnegie Foundation for the Advancement of Teaching. The Dana Center’s frontmatter for Statway™ and Quantway™ is available at www.utdanacenter.org/mathways. 6 Statway Instructor’s Notes April 23, 2012 (Full Version 1.0) Initiating Lesson 4.1.1: Investigating Patterns in Data
Constrained exponential (logistic) example Population: The population (y) over time in years (x) in a situation where there are constraints, such as the amount of room or food. It starts from x = 0 with an exponential growth pattern, but there is a time where it is not growing as fast, and the rate of growth tapers off. Periodic (sinusoidal) examples Ferris wheel: The height of a particular Ferris wheel car (y) over time in seconds (x). If you look at that over several revolutions of the Ferris wheel, you see a periodic pattern. Temperature: For a particular location, the average daily temperature for each month (y) over time measured in months (x). If you look at that over several years, you see a periodic pattern. Tide: For a particular location on the beach, the height of the tide (y) at a given time in hours (x) after midnight. Cubic example U.S. natural gas consumption (y) over the years 1960 to 2000 (x). Consumption increased from 1960 to 1969, decreased from 1970 to 1989, and then increased from 1990 to 2000. A function that can change directions twice is needed here, and a cubic function does that. (Retrieved from the U.S. Energy Information Administration at www.eia.doe.gov/dnav/ng/hist/n9140us2a.htm) Discussion and Wrap-­‐Up of Scaffolding Conceptual Task 1 Reassure students that they are not expected at this point to remember these examples or to pay attention to the formulas. They will see several of these again later in the course. The point here is to remember several of these types of nonlinear graphs and recognize them in a set of noisy data. Invite several groups to further explain an example they found interesting. If a group has a different example than was on the list, invite the group to describe the example to the class and tell what about the graph reminded it of that example. Part IV, Scaffolding Conceptual Task 2: Recognizing/Summarizing a Nonlinear Pattern in a Data Set Recommended Task Structure 3 minutes: Have students do the graph activity. 5 minutes: Conduct a discussion and wrap-­‐up of the graph activity. Student Handout (6) Do the following for each of the six graphs: (a) Consider the underlying relationship in the two variables as you look through the noise in the data. Visualize a smooth curve of the same shape as one of those from the previous task that summarizes the data. (b) Sketch a smooth curve of that form on the data-­‐only graph that goes through the middle of the data. The original versions of the Statway™ and Quantway™ courses were created by The Charles A. Dana Center at The University of Texas at Austin under sponsorship of the Carnegie Foundation for the Advancement of Teaching, and are copyright © 2011 by the Carnegie Foundation for the Advancement of Teaching and the Charles A. Dana Center at The University of Texas at Austin. STATWAY™/Statway™ and Quantway™ are trademarks of the Carnegie Foundation for the Advancement of Teaching. The Dana Center’s frontmatter for Statway™ and Quantway™ is available at www.utdanacenter.org/mathways. 7 Statway Instructor’s Notes April 23, 2012 (Full Version 1.0) Initiating Lesson 4.1.1: Investigating Patterns in Data
(c) Write the name of that type of relationship above the graph. Graph A Graph B
Graph D Graph C
Graph E
Graph F Discussion and Wrap-­‐Up of Scaffolding Conceptual Task 2 Have students compare papers with other classmates. •
•
Did they agree on the names of each relationship? How similar did their smooth curves look for each graph? Did students have trouble seeing through the noise to see a smooth curve summarizing the data? Were some graphs more difficult to do this with? Discuss each graph individually and ask students if there was anything that seemed difficult in that particular graph as they drew a smooth summary curve. The original versions of the Statway™ and Quantway™ courses were created by The Charles A. Dana Center at The University of Texas at Austin under sponsorship of the Carnegie Foundation for the Advancement of Teaching, and are copyright © 2011 by the Carnegie Foundation for the Advancement of Teaching and the Charles A. Dana Center at The University of Texas at Austin. STATWAY™/Statway™ and Quantway™ are trademarks of the Carnegie Foundation for the Advancement of Teaching. The Dana Center’s frontmatter for Statway™ and Quantway™ is available at www.utdanacenter.org/mathways. 8 Statway Instructor’s Notes April 23, 2012 (Full Version 1.0) Initiating Lesson 4.1.1: Investigating Patterns in Data
If students have difficulty drawing smooth curves through data, the following comments may be helpful. However, do not repeat too many of these because that could be overwhelming to students—watch their reactions. If you overexplain, they might think it is hard to draw a smooth curve through the data. •
•
•
•
In Graph A, the smooth curve does not necessarily go through the highest point as in Graph B. The smooth curve should not go through the lowest point near x = 2.5. In Graph E, the point with the x-­‐value of about 6 is clearly somewhat outside the pattern. If your curve drops down to include that point, you are not using a smooth curve of one of our types. Modelers generally do not modify their idea of the overall pattern by one point like this in the middle. On the other hand, in Graph C, the point with the largest x-­‐value should have a strong influence on how curved the smooth curve is. Here, that point fits the overall pattern, unlike that low point in Graph E. In Graphs D and F, these data are pretty noisy, so it is harder to draw a good smooth curve. That is okay—some real data are that noisy. In real problems, you obtain your graph and formula data with nonlinear relationships from similar types of regression techniques as those used for linear models. However, you must know the name of the relationship type you want to use to model the data in order to choose from a menu in the software to find the formula for the model that summarizes the data. Part V: Return to the Rich Task Recommended Task Structure 7 minutes: Have students answer questions about Area 2. 5 minutes: Conduct a discussion and wrap-­‐up of these answers. Student Handout (7) Do the following for the Area 2 data: (a) Consider the underlying relationship in the two variables as you look through the noise in the data. Visualize a smooth curve of the same shape as one of those you have seen in this lesson. (b) Sketch a smooth curve of that form on the data-­‐only graph (see the next page) that goes through the middle of the data. (c) What is the name of this type of relationship? (d) Notice the second graph, which includes a smooth curve and extended time. Using that, estimate when the amount of radioactivity will decrease to a safe level for people to live in the area (440 mR/year or less). (e) Using that smooth curve, estimate when the amount of radioactivity decreased to 1,000 mR/year. (f) Look back at Part I of this lesson, where you observed the data graph that was provided without any summary line or curve and estimated the time when the amount of The original versions of the Statway™ and Quantway™ courses were created by The Charles A. Dana Center at The University of Texas at Austin under sponsorship of the Carnegie Foundation for the Advancement of Teaching, and are copyright © 2011 by the Carnegie Foundation for the Advancement of Teaching and the Charles A. Dana Center at The University of Texas at Austin. STATWAY™/Statway™ and Quantway™ are trademarks of the Carnegie Foundation for the Advancement of Teaching. The Dana Center’s frontmatter for Statway™ and Quantway™ is available at www.utdanacenter.org/mathways. 9 Statway Instructor’s Notes April 23, 2012 (Full Version 1.0) Initiating Lesson 4.1.1: Investigating Patterns in Data
radioactivity decreased to 1,000 mR/year. What year did you find and what was your method for finding it? (g) How close was that first estimate of the time when it decreased to 1,000 mR/year to your estimate from this smooth exponential curve? Was it pretty good or not so good? (h) If you did not have software to provide you with a graph of the best exponential smooth curve to fit the data, could you sketch something to make the predictions? (i) Compare this model to the linear model and note the differences, particularly how much the linear model is off (error). Area 2—mR/year, with exponential model and with extended time The original versions of the Statway™ and Quantway™ courses were created by The Charles A. Dana Center at The University of Texas at Austin under sponsorship of the Carnegie Foundation for the Advancement of Teaching, and are copyright © 2011 by the Carnegie Foundation for the Advancement of Teaching and the Charles A. Dana Center at The University of Texas at Austin. STATWAY™/Statway™ and Quantway™ are trademarks of the Carnegie Foundation for the Advancement of Teaching. The Dana Center’s frontmatter for Statway™ and Quantway™ is available at www.utdanacenter.org/mathways. 10 Statway Instructor’s Notes April 23, 2012 (Full Version 1.0) Initiating Lesson 4.1.1: Investigating Patterns in Data
Discussion/Wrap-­‐Up of Rich Task Area 2 Questions The homework for this lesson is answering the same eight questions for the Area 4 data. Quickly check with students to see whether they had difficulty with any of these questions. You will probably want to write some answers on the board. Emphasize that the numerical answers are just estimates, so they are not supposed to all get the same exact number. For the exponential model, the estimate of when the radioactivity falls below 1,000 mR/year is about 1996 or 1997, and a good estimate of when the radioactivity falls below 440 mR/year is about 2022 or 2023. Point out that there is a formula for this curve, and, just as for a linear relationship, they could use the formula to solve for x for a given y. The problem is that solving an equation of this exponential type requires skills usually not learned until College Algebra. However, it is just as easy to estimate a solution from the graph here as it is to estimate a solution from a linear graph. Extra Information for Teachers The exponential model for the Area 2 data on radioactivity is ŷ = 1132(0.93)(x – 1989) + 340 The exponential model fitted here goes down to a baseline of 340 mR/year instead of 0 in order to better match reality. This was fitted by minimizing the sum of squared deviations. Not all software packages use that criterion for exponential fitting—sometimes they use minimizing the sum of the squared relative deviations. At this point, you are not asking students to use technology to find the formula. That comes in later lessons. For the Area 4 data, the models with the optimal parameter values are the following: Linear: ŷ = –0.427(x – 1989) + 1233
Exponential decay model: ŷ = 1071.6(0.90339)(x – 1989) The original versions of the Statway™ and Quantway™ courses were created by The Charles A. Dana Center at The University of Texas at Austin under sponsorship of the Carnegie Foundation for the Advancement of Teaching, and are copyright © 2011 by the Carnegie Foundation for the Advancement of Teaching and the Charles A. Dana Center at The University of Texas at Austin. STATWAY™/Statway™ and Quantway™ are trademarks of the Carnegie Foundation for the Advancement of Teaching. The Dana Center’s frontmatter for Statway™ and Quantway™ is available at www.utdanacenter.org/mathways. 11 Statway Instructor’s Notes April 23, 2012 (Full Version 1.0) Initiating Lesson 4.1.1: Investigating Patterns in Data
Homework [Student Handout] (1) Summarize what you learned today. Questions 2–4 use the Area 4 data, for which various graphs are given at the end of the homework assignment. The following are the basic questions: •
•
If you consider a level of 1,000 mR/year safe, when would you say that Area 4 was safe? If you decide that the safety level should be no more than 440 mR/year, when will you consider Area 4 safe? (2) Use the data-­‐only graph to estimate your answers to the two questions about when the area will be safe. (Answer: After 1993 for the 1,000-­‐mR/year level, and roughly sometime after 2010 for the 440-­‐
mR/year level. It is hard to estimate the latter since there are no data near it.) (3) Is it a good idea to use the linear model graph (with the data and the summary straight line) to answer these questions? Explain your reasoning. If it is a good idea, do it. (Answer: No. The summary line does not fit the pattern of the data very well. In the early 1990s, the data show a more rapid decrease from year to year than the line shows. After 2010, the data show a much less rapid decrease from year to year than the line shows.) (4) Is it a good idea to use the exponential model graph (with the data and the summary curved line) to answer these questions? Explain your reasoning. If it is a good idea, do it. (Answer: This is a good way to answer these questions because the exponential pattern seems to be as good a smooth curve as you would expect to find to fit the data. In addition, the description for exponential decay gave radioactive decay as a typical example, so you expect radioactive decay to have an exponential pattern. The radioactivity is at 1,000 mR/year or less after about 1993 or 1994. The radioactivity is at 440 mR/year or less after about 2012 or 2013.) The original versions of the Statway™ and Quantway™ courses were created by The Charles A. Dana Center at The University of Texas at Austin under sponsorship of the Carnegie Foundation for the Advancement of Teaching, and are copyright © 2011 by the Carnegie Foundation for the Advancement of Teaching and the Charles A. Dana Center at The University of Texas at Austin. STATWAY™/Statway™ and Quantway™ are trademarks of the Carnegie Foundation for the Advancement of Teaching. The Dana Center’s frontmatter for Statway™ and Quantway™ is available at www.utdanacenter.org/mathways. 12 Statway Instructor’s Notes April 23, 2012 (Full Version 1.0) Initiating Lesson 4.1.1: Investigating Patterns in Data
Area 4—mR/year with linear model and extended time Area 4—mR/year with exponential model and extended time The original versions of the Statway™ and Quantway™ courses were created by The Charles A. Dana Center at The University of Texas at Austin under sponsorship of the Carnegie Foundation for the Advancement of Teaching, and are copyright © 2011 by the Carnegie Foundation for the Advancement of Teaching and the Charles A. Dana Center at The University of Texas at Austin. STATWAY™/Statway™ and Quantway™ are trademarks of the Carnegie Foundation for the Advancement of Teaching. The Dana Center’s frontmatter for Statway™ and Quantway™ is available at www.utdanacenter.org/mathways. 13 Statway Student Handout April 23, 2012 (Full Version 1.0) Initiating Lesson 4.1.1: Investigating Patterns in Data
Part I During the last half of the 20th century, several sites in Nevada were used for testing nuclear devices (i.e., dangerous nuclear devices were exploded on those sites). Nuclear explosions spread dangerous radioactive particles in the air, which settle on the ground; this is dangerous to living things in that area until the radioactive material decays. Testing ended around 1985, so no new radioactive sources were added after that time. The data in the table and graphs give the amount of radioactivity measured at various times at two locations. Radiation exposure is measured in milliroentgens (mR). Because radiation exposure is dangerous as it accumulates, these data are measured in mR/year, which is how many mR a person staying on that site for a full year would receive. Year Area 2 mR/yr Area 4 mR/yr 1989 1,584 1,502 1990 1,453 1,420 1991 1,127 1,127 1992 1,273 1,110 1993 1,290 1,045 1994 1,094 849 1995 1,061 849 1996 914 784 1997 931 767 1998 849 735 1999 833 735 2000 914 751 2001 849 718 2002 800 686 2003 767 637 2004 751 604 2005 735 588 2006 702 571 2007 686 555 2008 653 522 The original versions of the Statway™ and Quantway™ courses were created by The Charles A. Dana Center at The University of Texas at Austin under sponsorship of the Carnegie Foundation for the Advancement of Teaching, and are copyright © 2011 by the Carnegie Foundation for the Advancement of Teaching and the Charles A. Dana Center at The University of Texas at Austin. STATWAY™/Statway™ and Quantway™ are trademarks of the Carnegie Foundation for the Advancement of Teaching. The Dana Center’s frontmatter for Statway™ and Quantway™ is available at www.utdanacenter.org/mathways. 1 Statway Student Handout April 23, 2012 (Full Version 1.0) Initiating Lesson 4.1.1: Investigating Patterns in Data
When is each area safe? •
•
•
There is a certain amount of background radioactivity around. For a location to be safe for people, animals, and plants, the radioactivity level does not need to be zero. Natural background radiation differs some in places, but 340 mR/year is a typical amount. A place is considered safe for people to live at up to about 100 mR/year above normal background radiation. The long-­‐term effects of low-­‐level radioactivity levels (up to 1,000 mR/year) are not clear. A one-­‐time radiation dose of 400,000 mR is typically fatal if untreated. Overall Question: When do you predict that Area 2 will be safe for people to live? (1) Which of the following two questions is easier to answer? Why? (a) If you consider a level of 1,000 mR/year safe, when would you say that Area 2 was safe? (b) If you decide that the safety level should be no more than 440 mR/year, when will you consider Area 2 safe? (2) Answer the easier of Questions 1a and 1b, and explain how you found the answer. The original versions of the Statway™ and Quantway™ courses were created by The Charles A. Dana Center at The University of Texas at Austin under sponsorship of the Carnegie Foundation for the Advancement of Teaching, and are copyright © 2011 by the Carnegie Foundation for the Advancement of Teaching and the Charles A. Dana Center at The University of Texas at Austin. STATWAY™/Statway™ and Quantway™ are trademarks of the Carnegie Foundation for the Advancement of Teaching. The Dana Center’s frontmatter for Statway™ and Quantway™ is available at www.utdanacenter.org/mathways. 2 Statway Student Handout April 23, 2012 (Full Version 1.0) Initiating Lesson 4.1.1: Investigating Patterns in Data
Part II In this course, you have only worked with linear relationships. Let’s use the graph of a linear relationship to summarize the data and see how satisfactory it is to answer your questions. Area 2: mR/year (3) Use the line, extended as needed, to answer the following questions: (a) If you consider a level of 1,000 mR/year safe, when would you say that Area 2 was safe? (b) If you decide that the safety level should be no more than 440 mR/year, when will you consider Area 2 safe? (4) Based on your earlier ideas and what you see here, do you think this line is a good summary to use to answer these questions? Why or why not? Do you trust your answers to the questions in Part I? The original versions of the Statway™ and Quantway™ courses were created by The Charles A. Dana Center at The University of Texas at Austin under sponsorship of the Carnegie Foundation for the Advancement of Teaching, and are copyright © 2011 by the Carnegie Foundation for the Advancement of Teaching and the Charles A. Dana Center at The University of Texas at Austin. STATWAY™/Statway™ and Quantway™ are trademarks of the Carnegie Foundation for the Advancement of Teaching. The Dana Center’s frontmatter for Statway™ and Quantway™ is available at www.utdanacenter.org/mathways. 3 Statway Student Handout April 23, 2012 (Full Version 1.0) Initiating Lesson 4.1.1: Investigating Patterns in Data
Part III: Relating Graphs and Written Examples of Nonlinear Relationships In Module 3, you worked with linear relationships. Those are very useful, and so are some other types of relationships between variables. Following are some graphs and examples of nonlinear relationships frequently used in modeling. (5) Notice the different types of patterns. Pick one pattern, and read its example(s) that follows all the graphs. Discuss whether the example’s description suggests the type of pattern seen in the graph. Exponential decay x y = 85 × 0.8
Quadratic 2 y = 50 + 42x + x
Exponential growth x y = 5 × 1.5
120
100
80
60
40
20
0
Constrained exponential (7 − x)
(logistic) y = 18/[1 + 2
] 0
10
20
30
40
Periodic (sinusoidal) y = 2 sin(1.57x) + 1 Cubic 3 2 y = x – 2x – x + 2
Exponential decay examples Radioactive decay: The amount of radioactivity emitted from an object (y) over time (x). Medicine: The amount of a medication in your system (y) over time in minutes (x) starting an hour after you took the dose. Quadratic example Height of an object: The height of an object thrown upward (y) over time in seconds (x). The original versions of the Statway™ and Quantway™ courses were created by The Charles A. Dana Center at The University of Texas at Austin under sponsorship of the Carnegie Foundation for the Advancement of Teaching, and are copyright © 2011 by the Carnegie Foundation for the Advancement of Teaching and the Charles A. Dana Center at The University of Texas at Austin. STATWAY™/Statway™ and Quantway™ are trademarks of the Carnegie Foundation for the Advancement of Teaching. The Dana Center’s frontmatter for Statway™ and Quantway™ is available at www.utdanacenter.org/mathways. 4 Statway Student Handout April 23, 2012 (Full Version 1.0) Initiating Lesson 4.1.1: Investigating Patterns in Data
Exponential growth examples Population: The population of a country (y) over time in years (x). Money invested: The amount of money in an account (y) that is invested at a certain interest rate, compounded. Constrained exponential (logistic) example Population: The population (y) over time in years (x) in a situation where there are constraints, such as the amount of room or food. It starts from x = 0 with an exponential growth pattern, but there is a time where it is not growing as fast, and the rate of growth tapers off. Periodic (sinusoidal) examples Ferris wheel: The height of a particular Ferris wheel car (y) over time in seconds (x). If you look at that over several revolutions of the Ferris wheel, you see a periodic pattern. Temperature: For a particular location, the average daily temperature for each month (y) over time measured in months (x). If you look at that over several years, you see a periodic pattern. Tide: For a particular location on the beach, the height of the tide (y) at a given time in hours (x) after midnight. Cubic example U.S. natural gas consumption (y) over the years 1960 to 2000 (x). Consumption increased from 1960 to 1969, decreased from 1970 to 1989, and then increased from 1990 to 2000. A function that can change directions twice is needed here, and a cubic function does that. (Retrieved from the U.S. Energy Information Administration at www.eia.doe.gov/dnav/ng/hist/n9140us2a.htm) The original versions of the Statway™ and Quantway™ courses were created by The Charles A. Dana Center at The University of Texas at Austin under sponsorship of the Carnegie Foundation for the Advancement of Teaching, and are copyright © 2011 by the Carnegie Foundation for the Advancement of Teaching and the Charles A. Dana Center at The University of Texas at Austin. STATWAY™/Statway™ and Quantway™ are trademarks of the Carnegie Foundation for the Advancement of Teaching. The Dana Center’s frontmatter for Statway™ and Quantway™ is available at www.utdanacenter.org/mathways. 5 Statway Student Handout April 23, 2012 (Full Version 1.0) Initiating Lesson 4.1.1: Investigating Patterns in Data
Part IV: Recognizing/Summarizing a Nonlinear Pattern in a Data Set (6) Do the following for each of the six graphs: (a) Consider the underlying relationship in the two variables as you look through the noise in the data. Visualize a smooth curve of the same shape as one of those from the previous task that summarizes the data. (b) Sketch a smooth curve of that form on the data-­‐only graph that goes through the middle of the data. (c) Write the name of that type of relationship above the graph. Graph A Graph B
Graph D Graph C
Graph E
Graph F The original versions of the Statway™ and Quantway™ courses were created by The Charles A. Dana Center at The University of Texas at Austin under sponsorship of the Carnegie Foundation for the Advancement of Teaching, and are copyright © 2011 by the Carnegie Foundation for the Advancement of Teaching and the Charles A. Dana Center at The University of Texas at Austin. STATWAY™/Statway™ and Quantway™ are trademarks of the Carnegie Foundation for the Advancement of Teaching. The Dana Center’s frontmatter for Statway™ and Quantway™ is available at www.utdanacenter.org/mathways. 6 Statway Student Handout April 23, 2012 (Full Version 1.0) Initiating Lesson 4.1.1: Investigating Patterns in Data
Part V (7) Do the following for the Area 2 data: (a) Consider the underlying relationship in the two variables as you look through the noise in the data. Visualize a smooth curve of the same shape as one of those you have seen in this lesson. (b) Sketch a smooth curve of that form on the data-­‐only graph (see the next page) that goes through the middle of the data. (c) What is the name of this type of relationship? (d) Notice the second graph, which includes a smooth curve and extended time. Using that, estimate when the amount of radioactivity will decrease to a safe level for people to live in the area (440 mR/year or less). (e) Using that smooth curve, estimate when the amount of radioactivity decreased to 1,000 mR/year. (f) Look back at Part I of this lesson, where you observed the data graph that was provided without any summary line or curve and estimated the time when the amount of radioactivity decreased to 1,000 mR/year. What year did you find and what was your method for finding it? (g) How close was that first estimate of the time when it decreased to 1,000 mR/year to your estimate from this smooth exponential curve? Was it pretty good or not so good? The original versions of the Statway™ and Quantway™ courses were created by The Charles A. Dana Center at The University of Texas at Austin under sponsorship of the Carnegie Foundation for the Advancement of Teaching, and are copyright © 2011 by the Carnegie Foundation for the Advancement of Teaching and the Charles A. Dana Center at The University of Texas at Austin. STATWAY™/Statway™ and Quantway™ are trademarks of the Carnegie Foundation for the Advancement of Teaching. The Dana Center’s frontmatter for Statway™ and Quantway™ is available at www.utdanacenter.org/mathways. 7 Statway Student Handout April 23, 2012 (Full Version 1.0) Initiating Lesson 4.1.1: Investigating Patterns in Data
(h) If you did not have software to provide you with a graph of the best exponential smooth curve to fit the data, could you sketch something to make the predictions? (i) Compare this model to the linear model and note the differences, particularly how much the linear model is off (error). Area 2—mR/year, with exponential model and with extended time The original versions of the Statway™ and Quantway™ courses were created by The Charles A. Dana Center at The University of Texas at Austin under sponsorship of the Carnegie Foundation for the Advancement of Teaching, and are copyright © 2011 by the Carnegie Foundation for the Advancement of Teaching and the Charles A. Dana Center at The University of Texas at Austin. STATWAY™/Statway™ and Quantway™ are trademarks of the Carnegie Foundation for the Advancement of Teaching. The Dana Center’s frontmatter for Statway™ and Quantway™ is available at www.utdanacenter.org/mathways. 8 Statway Student Handout April 23, 2012 (Full Version 1.0) Initiating Lesson 4.1.1: Investigating Patterns in Data
Homework (1) Summarize what you learned today. Questions 2–4 use the Area 4 data, for which various graphs are given at the end of the homework assignment. The following are the basic questions: •
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If you consider a level of 1,000 mR/year safe, when would you say that Area 4 was safe? If you decide that the safety level should be no more than 440 mR/year, when will you consider Area 4 safe? (2) Use the data-­‐only graph to estimate your answers to the two questions about when the area will be safe. (3) Is it a good idea to use the linear model graph (with the data and the summary straight line) to answer these questions? Explain your reasoning. If it is a good idea, do it. (4) Is it a good idea to use the exponential model graph (with the data and the summary curved line) to answer these questions? Explain your reasoning. If it is a good idea, do it. The original versions of the Statway™ and Quantway™ courses were created by The Charles A. Dana Center at The University of Texas at Austin under sponsorship of the Carnegie Foundation for the Advancement of Teaching, and are copyright © 2011 by the Carnegie Foundation for the Advancement of Teaching and the Charles A. Dana Center at The University of Texas at Austin. STATWAY™/Statway™ and Quantway™ are trademarks of the Carnegie Foundation for the Advancement of Teaching. The Dana Center’s frontmatter for Statway™ and Quantway™ is available at www.utdanacenter.org/mathways. 9 Statway Student Handout April 23, 2012 (Full Version 1.0) Initiating Lesson 4.1.1: Investigating Patterns in Data
Area 4—mR/year with linear model and extended time Area 4—mR/year with exponential model and extended time The original versions of the Statway™ and Quantway™ courses were created by The Charles A. Dana Center at The University of Texas at Austin under sponsorship of the Carnegie Foundation for the Advancement of Teaching, and are copyright © 2011 by the Carnegie Foundation for the Advancement of Teaching and the Charles A. Dana Center at The University of Texas at Austin. STATWAY™/Statway™ and Quantway™ are trademarks of the Carnegie Foundation for the Advancement of Teaching. The Dana Center’s frontmatter for Statway™ and Quantway™ is available at www.utdanacenter.org/mathways. 10 Statway Instructor’s Notes April 23, 2012 (Full Version 1.0) Supporting Lesson 4.1.2: Exponential Models Estimated number of 50-­‐minute class sessions: 2 If your class period is longer than 50 minutes, you may go on to Lesson 4.1.3 after Part I of this lesson and then finish Part II of this lesson in the next class period. Learning Goals Students will understand •
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that there is a difference between exact bivariate relationships, which can be expressed exactly by mathematical functions—called mathematical models in this context—and bivariate data, which may have a relationship that you can model as data = model + error. In this lesson, students are investigating exact exponential bivariate relationships. that exponential relationships are curved and not linear. how to interpret the parameters in an exponential function (given with a growth factor parameter) and the limitations on the possible values for each parameter. the relationship between the growth factor parameter and the growth rate of the process. that for a given exponential process, for the purpose of writing a function, the starting point where x = 0 is sometimes arbitrary and the implications of that for the parameter values. that if they have x-­‐values that increase by exactly 1 each time, then the common ratio of the y-­‐values is the growth factor; this is easy to find in the table of values. However, when students have data where the x-­‐values do not increase by exactly 1 going down the rows in the table, they will not see the common ratios in the table, even if the data are well modeled by an exponential model. Students will be able to •
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identify from a graph, function, table of values, or verbal explanation of an exponential relationship whether it is growth or decay. evaluate function values from an exponential formula. from an appropriate graph or table, estimate the initial value parameter. This includes, as necessary, redefining the x-­‐variable to have x = 0 in the data set or at least not far from the x-­‐values in the data. use a graph or table of values to find an approximate solution to the question, “For what values of x is y less than (greater than, equal to) k, where k is some constant.” (Note: This same method could be used for a linear relationship to answer the same questions, but it is not usually necessary since students can easily solve it algebraically.) Part I [estimated time: 50 minutes] Introduction [Student Handout; estimated time: 6 minutes] You have an opportunity to invest $2,000 at either 3% or 6% interest, compounded annually, for 10 years. There are various differences in the accounts (e.g., withdrawal flexibility), but you need to focus on the different amounts of money you will earn in each. The original versions of the Statway™ and Quantway™ courses were created by The Charles A. Dana Center at The University of Texas at Austin under sponsorship of the Carnegie Foundation for the Advancement of Teaching, and are copyright © 2011 by the Carnegie Foundation for the Advancement of Teaching and the Charles A. Dana Center at The University of Texas at Austin. STATWAY™/Statway™ and Quantway™ are trademarks of the Carnegie Foundation for the Advancement of Teaching. The Dana Center’s frontmatter for Statway™ and Quantway™ is available at www.utdanacenter.org/mathways. 1 Statway Instructor’s Notes April 23, 2012 (Full Version 1.0) Supporting Lesson 4.1.2: Exponential Models (1) Before doing any computation, choose the answer that best expresses your expectation: (a) I expect to earn about twice as much in the 6% account than the 3% account. (b) I expect to earn significantly less than twice as much in the 6% account than the 3% account. (c) I expect to earn significantly more than twice as much in the 6% account than the 3% account. Now, let’s look at graphs of the amount of money in each account over a 10-­‐year period. Account Earning 3% Annual Interest Account Earning 6% Annual Interest (2) Put a check by each statement that is correct and an X by each statement that is incorrect. (a) Both graphs go up to the same height, so both accounts earn the same amounts. (b) In 10 years, one account earns about $2,700 and the other account about $3,600. (c) In 10 years, one account earns about $700 and the other account about $1,600. (d) I stand by my answer in Question 1. Lecture/Discussion of How to Find the Formula for the 3% Account [Student Handout; estimated time: 12 minutes, including helping students with their calculators to use the exponent key] (Note: There is too much here for students to quickly read on their own, and it is important for students who are not strong in algebra to go fairly quickly through this discussion of using the distributive law to develop the pattern they need to go from a growth rate statement to a growth factor formula. The printed discussion goes slowly because students may want to look at some parts in more detail. However, your discussion should go more quickly to keep them from getting bogged down. You really want students to see the transition from the iterative idea of adding 3% of the amount each time to the formula, so that they believe in the equivalence of adding on a certain percentage each time and the amount growing by a common growth factor. The original versions of the Statway™ and Quantway™ courses were created by The Charles A. Dana Center at The University of Texas at Austin under sponsorship of the Carnegie Foundation for the Advancement of Teaching, and are copyright © 2011 by the Carnegie Foundation for the Advancement of Teaching and the Charles A. Dana Center at The University of Texas at Austin. STATWAY™/Statway™ and Quantway™ are trademarks of the Carnegie Foundation for the Advancement of Teaching. The Dana Center’s frontmatter for Statway™ and Quantway™ is available at www.utdanacenter.org/mathways. 2 Statway Instructor’s Notes April 23, 2012 (Full Version 1.0) Supporting Lesson 4.1.2: Exponential Models Pay careful attention to how students are computing the formula values. Many will not know how to use an exponent key on their calculators. Be sure that they learn to do that, explaining the various different ways it can look. It is not acceptable here for students to just punch in the growth factor numerous times to avoid using the exponent key. [This indicates correct understanding and that is good, but it is much too easy to make a mistake in how many times you punch it in!]) How do you compute the amounts in an account at the end of each year to make that graph? [Answer: Assume $2,000 was deposited on January 1, 2008, in a 3% interest account. On January 1, 2009, the amount in the account is $2,000 + 0.03($2,000) = $2,000 + $60 = $2,060 Recall that 3% = 3/100 = 0.03 and that the decimal version of this number is used for computation. On January 1, 2010, the amount in the account grows further to become $2,060 + 0.03($2,060) = $2,060 + $61.80 = $2,121.80.] Working in groups, fill in the following table: Years Since Amount at 3% Amount at 6% January 1, 2008 Date January 1, 2008 0 $2,000.00 $2,000.00 January 1, 2009 1 $2,060.00 January 1, 2010 2 $2,121.80 January 1, 2011 3 January 1, 2012 4 The method you used here is an iterative approach. You find the amount at the end of one year, and then based on that, you find the amount for the next year and continue until you have computed the amounts for as many years as needed. However, finding the amount at the end of 10 years is tedious. It is even more tedious to find the amount at the end of 20 years! Surely someone has discovered a better way. Yes, you can find a pattern here and, using that pattern, develop a formula. Let’s see how to do this. First, use some arithmetic to find a pattern. Start by noticing the pattern of how you computed the amount at Year 1 of the 3% investment: new amount = 2,000 + 0.03(2,000) Now, from arithmetic class, remember the distributive property: 2,000 + 0.03(2,000) = 2,000(1 + 0.03) = 2,000(1.03) Therefore, the distributive property says that for every time you add 3% of its value to a number, you can also do that by multiplying the number by 1.03. The original versions of the Statway™ and Quantway™ courses were created by The Charles A. Dana Center at The University of Texas at Austin under sponsorship of the Carnegie Foundation for the Advancement of Teaching, and are copyright © 2011 by the Carnegie Foundation for the Advancement of Teaching and the Charles A. Dana Center at The University of Texas at Austin. STATWAY™/Statway™ and Quantway™ are trademarks of the Carnegie Foundation for the Advancement of Teaching. The Dana Center’s frontmatter for Statway™ and Quantway™ is available at www.utdanacenter.org/mathways. 3 Statway Instructor’s Notes April 23, 2012 (Full Version 1.0) Supporting Lesson 4.1.2: Exponential Models Notice that you did that for Year 2, since 2,060 + 0.03(2,060) = 2,060(1.03). You also did it when computing the other years in the table for the 3% account. So, here is the shortcut to compute the amount in the account at the end of each year: after 1 year, 2,000(1.03) after 2 years, [2,000(1.03)](1.03) = 2,000(1.03)2 after 3 years, {[2,000(1.03)](1.03)}(1.03) = 2,000(1.03)3 and so on. Therefore, the pattern is, “Each year, multiply the amount in the account by 1.03 to get the new amount.” This means that the amount for Year 4 is 2,000(1.03)(1.03)(1.03)(1.03), which can be written as 2,000(1.03)4. Compute the amounts at the end of each year using this pattern. Date Years Since January 1, 2008 Amount at 3% Amount at 6% January 1, 2008 0 2,000.00 2,000.00 January 1, 2009 1 2,000(1.03) = 2,060.00 January 1, 2010 2 2
2,000(1.03) = 2,121.80 2,000(1.06) = January 1, 2011 3 2,000(1.03) = 3
2,000(1.06) = January 1, 2012 4 2,000(1.03) = 4
2,000(1.06) = 2,000(1.06) = 2,120.00 2
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After you have computed some values in this table, check to see that they agree with the amounts found when you used the iterative method (the first method). Now that you have found a pattern, let’s write it in a more general way, as a formula. It can also be called a function. Consider the 3% account. Let t = the years since January 1, 2008, and A = the amount of money in the bank at the end of t years. Restating part of the above information, you have At t = 1, then A = 2,000(1.03) = 2,060.00 At t = 2, then A = 2,000(1.03)2 = 2,121.80 At t = 3, then A = 2,000(1.03)3 = At t = 4, then A = 2,000(1.03)4 = So, a general formula for the amount (A) in the 3% account at the tth year is A = 2,000(1.03)t. This process has demonstrated the power of algebra. You had a rule that showed how to do one amount at a time to build up a table of values, you analyzed it to see an underlying pattern, and then you summarized that pattern with a formula (also called a function.) The original versions of the Statway™ and Quantway™ courses were created by The Charles A. Dana Center at The University of Texas at Austin under sponsorship of the Carnegie Foundation for the Advancement of Teaching, and are copyright © 2011 by the Carnegie Foundation for the Advancement of Teaching and the Charles A. Dana Center at The University of Texas at Austin. STATWAY™/Statway™ and Quantway™ are trademarks of the Carnegie Foundation for the Advancement of Teaching. The Dana Center’s frontmatter for Statway™ and Quantway™ is available at www.utdanacenter.org/mathways. 4 Statway Instructor’s Notes April 23, 2012 (Full Version 1.0) Supporting Lesson 4.1.2: Exponential Models Tasks [Student Handout; estimated time: 10 minutes] In your group, do the following: (3) Develop the formula for the amount of money in the 6% account at the tth year. (Note: It might be easier to use a different label for this amount. Instead of A, use V for value. Or use A for both formulas, and employ words and ideas to keep track of the formula that is needed rather than the letter used for the variable.) (4) Use your formulas to find the amount in the two accounts at the end of 10 years. (5) Go back to the graphs in Question 1 and plot the points representing what you found. Do these points agree with the values given by the graphs? (6) Calculate the values in the following table and make graphs of each by hand. These will be similar to the graphs given in Question 1, but the pattern of growth will be even clearer because of the longer time. Date Years since January 1, 2008 Amount at 3% Amount at 6% January 1, 2008 0 2,000.00 2,000.00 January 1, 2013 5 10 10 January 1, 2018 10 2,000(1.03) =
2,000(1.06) = January 1, 2023 15 January 1, 2028 20 Conclusion [Student Handout; estimated time: 5 minutes] Return to Question 2. Discuss each statement with your group. Can your group agree on which statements are correct or incorrect? Write a summary of your conclusions. Wrap-­‐Up [estimated time: 12 minutes] Discuss each answer for Question 2. Bring out the following points: Question 2a Even though the graphs end at a place that looks the same in the pictures, notice the scales. It is very important to read the scales of graphs when you are interpreting them. These two graphs do not indicate that the final amounts are the same even though they are at the same height. Question 2b The total amount in the account is not the same as the amount earned, which is the amount over $2,000. Question 2c This is correct. Question 2d Go back and talk about the various options in Question 1. It is likely that many students thought that since 6% is twice 3%, then the amount earned in the 6% account at the end of 10 years would be twice the amount earned by the 3% account. This is proportional reasoning, which is an important type of reasoning. The original versions of the Statway™ and Quantway™ courses were created by The Charles A. Dana Center at The University of Texas at Austin under sponsorship of the Carnegie Foundation for the Advancement of Teaching, and are copyright © 2011 by the Carnegie Foundation for the Advancement of Teaching and the Charles A. Dana Center at The University of Texas at Austin. STATWAY™/Statway™ and Quantway™ are trademarks of the Carnegie Foundation for the Advancement of Teaching. The Dana Center’s frontmatter for Statway™ and Quantway™ is available at www.utdanacenter.org/mathways. 5 Statway Instructor’s Notes April 23, 2012 (Full Version 1.0) Supporting Lesson 4.1.2: Exponential Models However, many students have not learned when it can be applied and when it cannot. This topic is explored more in Module 12, but it is important to discuss it here. Point out that proportional reasoning is a linear process, with linear growth, and the example here is a different type of growth. They can look at the graph and see a curve in it; this is not linear. Wrap up the entire activity by discussing the following points: •
The formula can be thought of in two different ways—as a growth factor formula or a growth rate formula. Growth factor formula: Growth rate formula: A = 2,000(1.03)t A = 2,000(1.00 + 0.03)t Growth rate = 0.03 = 3% Growth factor: 1.03 •
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The idea of the growth in y being by a certain percentage for each increase in x instead of a certain amount for each increase in x is crucial. This distinguishes this from linear relationships. This type of growth is called exponential growth and gives a curved graph. The smaller the growth rate, the shallower the curve. The numbers students have computed here can be thought of as data, even though there is no noise. These numbers are the data you could gather from a bank if you asked about the amount of money that would be in an account at each of these times. They have no noise in them because there is no variability. The bank has promised to pay the money according to a completely predictable rule, so it can be given by a formula. So, the relationship of these two variables (amount and time) is an exact relationship, not a statistical relationship. When studying statistics, you typically encounter noisy data, where they do not follow an exact mathematical relationship. However, when you are discussing the nature of the underlying model, it is more convenient to start with exact relationships so that you can see the structure clearly. Many statistical relationships (ones with noise) have an underlying exponential structure; you will discuss statistical relationships again in Lesson 4.1.3. Homework for Part I Preparing Students to Work on Homework [10 minutes] The following set of practice exercises should be assigned as homework to be worked on before the next class. Tell students the following: Some of these exercises are more challenging than just a simple application of what you have just done. Let’s look at some of them now. What is different about Question 2? Do you want to let t = 1950 and t = 1951? Would you want to plug 1950 into your formula? When your original problem today was set up, did you use x = January 1, 2008? No. Could you have plugged that into a formula? No, that would have been confusing. So, The original versions of the Statway™ and Quantway™ courses were created by The Charles A. Dana Center at The University of Texas at Austin under sponsorship of the Carnegie Foundation for the Advancement of Teaching, and are copyright © 2011 by the Carnegie Foundation for the Advancement of Teaching and the Charles A. Dana Center at The University of Texas at Austin. STATWAY™/Statway™ and Quantway™ are trademarks of the Carnegie Foundation for the Advancement of Teaching. The Dana Center’s frontmatter for Statway™ and Quantway™ is available at www.utdanacenter.org/mathways. 6 Statway Instructor’s Notes April 23, 2012 (Full Version 1.0) Supporting Lesson 4.1.2: Exponential Models here you are doing the same kind of thing. You are picking a starting point and calling it x = 0, or here, t = 0. What is different about Question 3? Here the population is declining, not growing. However, it is doing that in the same kind of pattern you were just studying: It changes by the same percentage each time instead of the same amount. So, you can start your table for this in the same way that you started the table for your “money in the bank” example. What are you expected to do with these exercises by the next class period? •
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Read all of the problems. Start all of them, each on a separate sheet of paper. When you get to something that is so confusing you cannot go forward on that part, stop and write a question on your homework paper. Then go to the next part. If this is also confusing for the same reason, that may be as far as you can go with that problem. So go to the next problem. Student Handout Most examples of exponential growth and decay come from areas where you expect the relationships not to be exact, but statistical (with error or noise.) For the purposes of this lesson, consider them as exact relationships. (1) In a biology lab, a scientist has a population of fruit flies that has 120 flies on Day 0, and they increase by 8% per day. (a) Make a table for the number of fruit flies (N) on each day (d) for Days 0 to 6. d N 0 120 1 130 2 140 3 151 4 163 5 176 6 190 The original versions of the Statway™ and Quantway™ courses were created by The Charles A. Dana Center at The University of Texas at Austin under sponsorship of the Carnegie Foundation for the Advancement of Teaching, and are copyright © 2011 by the Carnegie Foundation for the Advancement of Teaching and the Charles A. Dana Center at The University of Texas at Austin. STATWAY™/Statway™ and Quantway™ are trademarks of the Carnegie Foundation for the Advancement of Teaching. The Dana Center’s frontmatter for Statway™ and Quantway™ is available at www.utdanacenter.org/mathways. 7 Statway Instructor’s Notes April 23, 2012 (Full Version 1.0) Supporting Lesson 4.1.2: Exponential Models (b) Graph the data in the table. N = number of fruit flies, d = days (c) Did the population of fruit flies get as high as 180 flies in this time period? If so, approximately when? (Answer: Yes, the population got as high as 180 flies, about Day 5.) (d) Write a formula for the number of fruit flies. [Answer: N = 120(1.08)d ] (e) In your formula, identify the initial amount, growth factor, and growth rate. (Answer: Initial amount: 120; growth factor: 1.08; growth rate: 0.08, or 8%) (2) The population of a certain country was 5.83 million in 1950, and grew by 2.5% per year for the next 20 years. (a) Make a table of the population of that country in 1950, 1951, and 1952. Year Years Since 1950 (t) Population, in millions (P) 1950 0 5.83 1951 1 5.98 1952 2 6.13 1953 3 6.28 (b) Let t = year – 1950. Write a formula for the population of that country (P). [Answer: P = 5.83(1.025)t] (c) In your formula, identify the initial amount, growth factor, and growth rate. (Answer: Initial amount: 5.83 million; growth factor: 1.025; growth rate: 0.025, or 2.5%) The original versions of the Statway™ and Quantway™ courses were created by The Charles A. Dana Center at The University of Texas at Austin under sponsorship of the Carnegie Foundation for the Advancement of Teaching, and are copyright © 2011 by the Carnegie Foundation for the Advancement of Teaching and the Charles A. Dana Center at The University of Texas at Austin. STATWAY™/Statway™ and Quantway™ are trademarks of the Carnegie Foundation for the Advancement of Teaching. The Dana Center’s frontmatter for Statway™ and Quantway™ is available at www.utdanacenter.org/mathways. 8 Statway Instructor’s Notes April 23, 2012 (Full Version 1.0) Supporting Lesson 4.1.2: Exponential Models (3) A certain country was losing population over a 20-­‐year period. The country had 11 million people in 1990 and lost 1% of its population each year during the next 20 years. (a) Make a table of the population for that country in 1990, 1991, 1992, and 1993. Year Years Since 1990 (t) Population, in millions (P) 1990 0 11.00 1991 1 10.89 1992 2 10.78 1993 3 10.67 1995 5 10.46 2000 10 9.95 2005 15 9.46 2010 20 9.00 (b) In this case, the factor is (1.00 – 0.01) = 0.99. Since this is not growth (the amount is declining), it is called a decay factor, and this is exponential decay instead of exponential growth. So, the formula is P = 11(0.99)t, where t = year – 1990. Use this formula to determine the population in 2010. (Answer: See the table for Question 3a.) (c) In this formula, identify the initial amount, decay factor, and decay rate. (Answer: Initial amount: 11 million; decay factor: 0.99; decay rate: 0.01, or 1%) (d) Use this formula to check your values for the population of that country in 1990, 1991, 1992, and 1993. (e) Notice that the description of this scenario is clearly exponential decay, yet the graph looks linear. Why? (Answer: This graph looks quite linear. Apparently a decay rate of 1% is so small that just looking at a graph does not help students see that the underlying pattern is exponential instead of linear.1) 1
The next lesson focuses on using residual plots and provides a much easier method to check how closely the data fit a given model. The original versions of the Statway™ and Quantway™ courses were created by The Charles A. Dana Center at The University of Texas at Austin under sponsorship of the Carnegie Foundation for the Advancement of Teaching, and are copyright © 2011 by the Carnegie Foundation for the Advancement of Teaching and the Charles A. Dana Center at The University of Texas at Austin. STATWAY™/Statway™ and Quantway™ are trademarks of the Carnegie Foundation for the Advancement of Teaching. The Dana Center’s frontmatter for Statway™ and Quantway™ is available at www.utdanacenter.org/mathways. 9 Statway Instructor’s Notes April 23, 2012 (Full Version 1.0) Supporting Lesson 4.1.2: Exponential Models P = population, t = years since 1990 (4) Water is treated with chlorine to kill germs to make it safe for drinking, swimming, and so on. Some gardeners, however, want water without chlorine because the chemical is somewhat harmful to plants. Chlorine dissipates from water when the water is exposed to air and sunlight. A typical dissipation rate for chlorine in water is about 15% per hour. Suppose you have a tub of water with an initial chlorine concentration of 2.00 milligrams per liter that is exposed to air and sunlight for 10 hours and the dissipation rate for that time is 15% per hour. (a) Make a table showing the chlorine concentration (C) at the end of each hour (h) for Hours 0, 1, and 2. Hours (h) Amount of Chlorine (C) 0 2.00 1 1.70 2 1.45 5 0.89 7 0.64 10 0.39 (b) Write a formula for the chlorine concentration (C) at the end of h hours. [Answer: C = 2.00(0.85)h] (c) In your formula, identify the initial amount, growth/decay factor, and growth/decay rate. (Answer: Initial amount: 2.00; decay factor: 0.85; decay rate: 0.15, or 15%) (d) Use the formula to check your values of C for Hours 0, 1, and 2. Then compute C for Hours 5, 7, and 10. (Answer: See the table for Question 4a.) The original versions of the Statway™ and Quantway™ courses were created by The Charles A. Dana Center at The University of Texas at Austin under sponsorship of the Carnegie Foundation for the Advancement of Teaching, and are copyright © 2011 by the Carnegie Foundation for the Advancement of Teaching and the Charles A. Dana Center at The University of Texas at Austin. STATWAY™/Statway™ and Quantway™ are trademarks of the Carnegie Foundation for the Advancement of Teaching. The Dana Center’s frontmatter for Statway™ and Quantway™ is available at www.utdanacenter.org/mathways. 10 Statway Instructor’s Notes April 23, 2012 (Full Version 1.0) Supporting Lesson 4.1.2: Exponential Models (e) Use your values to make a graph of chlorine concentration. C = amount of chorine, h = hours in air and sunlight (f) The gardener is happy to use the water for his plants when the chlorine concentration drops below 1.00 milligram per liter. Will it fall that low within the 10-­‐hour period? If so, approximately when? Calculate some additional values, if needed, to get a pretty good estimate of when. (Answer: It will fall below 1 milligram per liter within the 10-­‐hour time. From the graph, the time is at about 5 hours or so.) (5) Radioactive decay has this same exponential property that we have been discussing. That is, a radioactive substance loses a certain percentage of its radioactivity each year. Below is a statement about Cobalt-­‐60 that might be found in a technical article. Cobalt-­‐60 has a half-­‐life of 5.27 years, which means that 12.3% of its atoms decay each year. The gamma rays from radioactive cobalt are used with X-­‐ray film to inspect large welds. If an industrial gamma ray source is manufactured with 45 curies of cobalt-­‐60 (1 curie of a radioactive material results in 37 billion disintegrations per second), what will its strength measured in curies be at the end of each of the next 10 years? From this statement, do the following: (a) Find the initial amount and growth/decay rate. (Answer: Initial amount: 2.00 curies: growth/decay rate: –0.123 or –12.3%.) (b) Write a formula for the relationship. [Answer: R = 45(1.000 – 0.123)t = 45(0.877)t (c) In your formula, identify the initial amount, growth/decay factor, and growth/decay rate. The original versions of the Statway™ and Quantway™ courses were created by The Charles A. Dana Center at The University of Texas at Austin under sponsorship of the Carnegie Foundation for the Advancement of Teaching, and are copyright © 2011 by the Carnegie Foundation for the Advancement of Teaching and the Charles A. Dana Center at The University of Texas at Austin. STATWAY™/Statway™ and Quantway™ are trademarks of the Carnegie Foundation for the Advancement of Teaching. The Dana Center’s frontmatter for Statway™ and Quantway™ is available at www.utdanacenter.org/mathways. 11 Statway Instructor’s Notes April 23, 2012 (Full Version 1.0) Supporting Lesson 4.1.2: Exponential Models (Answer: Initial amount: 2.00 curies; growth/decay factor: 0.877; growth/decay rate: –0.123 or –12.3%.) (d) Use the formula to determine the amount of radioactive material left at the end of each of the next 10 years. Years (t) Amount of Radioactivity (R) 0 45.00 1 39.47 2 34.61 3 30.35 4 26.62 5 23.35 6 20.47 7 17.96 8 15.75 9 13.81 10 12.11 (e) For this type of inspection, the strength of the gamma ray source must be at least 20 curies. How long will this source be adequate for this type of inspection? R = amount of radioactivity, t = years The original versions of the Statway™ and Quantway™ courses were created by The Charles A. Dana Center at The University of Texas at Austin under sponsorship of the Carnegie Foundation for the Advancement of Teaching, and are copyright © 2011 by the Carnegie Foundation for the Advancement of Teaching and the Charles A. Dana Center at The University of Texas at Austin. STATWAY™/Statway™ and Quantway™ are trademarks of the Carnegie Foundation for the Advancement of Teaching. The Dana Center’s frontmatter for Statway™ and Quantway™ is available at www.utdanacenter.org/mathways. 12 Statway Instructor’s Notes April 23, 2012 (Full Version 1.0) Supporting Lesson 4.1.2: Exponential Models Part II This part of the lesson is to be done after students have had time out of class to work on the five exercises in Part I’s homework. Much of Part II should be done in class without the full student handout, or perhaps without any handout at all. A student handout is provided to distribute at the end of this lesson. Discussion of Homework Choose two problems to focus on in each of these discussions. They can be the same two problems or different ones. Definitely include one exponential decay problem in each. 10 minutes: In groups, students share and discuss their work on two of the problems. Questions 2 and 4 would be good. 10 minutes: In groups, students look at the formula, table, and graph for the two problems and discuss how they would answer the exercises after they have produced the formula, table, and graph. 5-­‐minute wrap-­‐up: Lecture/discuss how to do the two problems chosen. Lecture/Discussion How much should you round the response variable values that you compute from the model? In several of the problems, the y-­‐values in any data observed would be whole numbers. This is true for population data, because a population of fruit flies cannot have 137.273 flies in it. However, when you are finding a good model for the data, it will be produced from a mathematical formula; so, response variable values may not be whole numbers, but instead numbers like 137.23 flies. This is fine. As in Module 3, predictions are for the average number of flies you would expect to see at that given time. Limitations on the Parameter Values Consider exponential growth. You had the amount in the bank and the population of fruit flies. Is there any limitation on the value of the initial value? What about the growth rate? What about the growth factor? Discuss. (Answers: positive initial amount, positive growth rate, and growth factor larger than 1.) Consider exponential decay. You had the declining population, chlorine dissipation, and radioactive decay. Is there any limitation on the value of the initial value? What about the decay factor rate? What about the decay rate? Discuss. (Answers: positive initial amount; decay factor must be less than 1; the decay rate is less than 100%, so the growth factor must be between 0 and 1.) The original versions of the Statway™ and Quantway™ courses were created by The Charles A. Dana Center at The University of Texas at Austin under sponsorship of the Carnegie Foundation for the Advancement of Teaching, and are copyright © 2011 by the Carnegie Foundation for the Advancement of Teaching and the Charles A. Dana Center at The University of Texas at Austin. STATWAY™/Statway™ and Quantway™ are trademarks of the Carnegie Foundation for the Advancement of Teaching. The Dana Center’s frontmatter for Statway™ and Quantway™ is available at www.utdanacenter.org/mathways. 13 Statway Instructor’s Notes April 23, 2012 (Full Version 1.0) Supporting Lesson 4.1.2: Exponential Models Conclusions Growth rates can be larger than 100%, but decay rates cannot. Growth factors are larger than 1. Decay factors are between 0 and 1. Initial amounts were positive in all of their applications, and it is hard to think of how they could be negative.2 Starting Point Is Somewhat Arbitrary In Questions 2 and 3, the problem was stated by giving the population in a particular year, so it is natural to use that year as a starting point. However, it might easily have been true that this pattern persisted over more years than were claimed; the pattern in Question 2 possibly started as far back as 1940. If that were true, it would be reasonable to choose as a starting point any of these years. Certainly 1940 would be reasonable, but if you really only care about predicting it in the future, it is okay to start with 1940 or 1950 or any other year within the range where the model fits. Of course, if you change the starting year, you must determine the population in that year to use as your initial amount. So, get the class involved here. If you want to use 1955 as the starting point, what would the initial value have to be? Write the formula for that. Remember the limitations: The claim was only that the model is good to 2010. So now the limitation on x must stop at 15. Identifying Exponential Growth/Decay from a Table of Values If you have non-­‐noisy data where the x-­‐values are evenly spaced, it is often possible to tell whether the relationship is exponential, linear, or neither. •
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Recall that a linear relationship has a slope, which is the constant amount that the response variable y changes whenever the explanatory variable x increases by 1 unit. Recall that an exponential relationship has a growth/decay factor, which comes from the fact that the response variable y changes by the same percentage whenever the explanatory variable x increases by 1 unit. Student Handout [Includes some summary] Ideas from Part I Homework •
How much should you round your response variable values from the model and why? Even if the data must have output as whole numbers, the predictions come from a mathematical model, which is an exact relationship. It is acceptable to have one or more decimal places in a predicted value. If you predict 572.23 people in a given year, everybody would recognize that you mean “close to 572.23.” 2
The positive initial amount is not absolutely required; it just makes sense. However, in fact, there are applications of exponential decay where the initial amount is modeled as a negative number, such as the depth of a point in water. That is probably much too complex to say to the class. It is mentioned here just as a caution that the initial amount being positive is not an intrinsic limitation of the model, as are the restrictions on the growth/decay factors and the decay rate. The original versions of the Statway™ and Quantway™ courses were created by The Charles A. Dana Center at The University of Texas at Austin under sponsorship of the Carnegie Foundation for the Advancement of Teaching, and are copyright © 2011 by the Carnegie Foundation for the Advancement of Teaching and the Charles A. Dana Center at The University of Texas at Austin. STATWAY™/Statway™ and Quantway™ are trademarks of the Carnegie Foundation for the Advancement of Teaching. The Dana Center’s frontmatter for Statway™ and Quantway™ is available at www.utdanacenter.org/mathways. 14 Statway Instructor’s Notes April 23, 2012 (Full Version 1.0) Supporting Lesson 4.1.2: Exponential Models •
What are the limitations on the parameter values in an exponential model? If it is exponential decay, the decay rate must be between 0% and 100%. You cannot have a process where more than 100% of the amount is disappearing at any point. If it is exponential growth, the growth rate must be positive, but it can be as large as needed, including more than 100%. A start-­‐up business might easily increase its sales by more than 200% per month in the first few months of starting. Another way of summing up these two statements is that the growth factor must be greater than 0. In most cases, the initial amount is greater than 0, but that is not mathematically necessary. •
The starting point is somewhat arbitrary. In any linear growth relationship or exponential growth relationship, you can redefine the x-­‐variable (years since 1700) to move the starting point (y-­‐intercept for linear, initial amount for exponential) close to the data you are actually observing. You can describe exactly the same relationship using this as you did before. The description differs somewhat, and a parameter value differs from the value you found for the model with nonadjusted starting point. This makes the model easier to interpret. •
Identifying exponential growth/decay from a table of values. For example, consider the population of Asia at 50-­‐year intervals from 1700 to 1950. For this example, you do not have the actual data, but have fitted a good model to the data, so that you have an exact relationship to work with here. In addition, 1 unit of the x-­‐values is 50 years, or “half-­‐centuries after 1700.” Year Half-­‐Centuries After 1700 (x) Asia Predicted Population, in Millions (y) 1700 0 386 1750 1 495 1800 2 636 1850 3 816 1900 4 1,047 1950 5 1,343 The original versions of the Statway™ and Quantway™ courses were created by The Charles A. Dana Center at The University of Texas at Austin under sponsorship of the Carnegie Foundation for the Advancement of Teaching, and are copyright © 2011 by the Carnegie Foundation for the Advancement of Teaching and the Charles A. Dana Center at The University of Texas at Austin. STATWAY™/Statway™ and Quantway™ are trademarks of the Carnegie Foundation for the Advancement of Teaching. The Dana Center’s frontmatter for Statway™ and Quantway™ is available at www.utdanacenter.org/mathways. 15 Statway Instructor’s Notes April 23, 2012 (Full Version 1.0) Supporting Lesson 4.1.2: Exponential Models Asia 1500 1000 500 0 1700 1750 1800 1850 1900 1950 When you only look at the graph, it is not completely clear whether the relationship is close enough to linear to call it linear. To determine whether it is linear, look at some differences. Fill in the rest of the differences. Are the differences in population constant for each half-­‐century? x y Differences between successive y’s 0 386 1 495 495 – 386 = 109 2 636 636 – 495 = 141 3 816 4 1,047 5 1,343 No, the differences in population are not constant for each half-­‐century. So this is not linear growth. To determine whether it is exponential, look at the ratios, which give the growth factor. Fill in the rest of the ratios. Are the ratios in population constant for each half-­‐century? x y Ratios of successive y’s 0 386 1 495 495/386 = 1.28 2 636 636/495 = 1.28 3 816 4 1047 5 1343 Yes. The ratios of successive y’s are 1.28. This means that, for each time x increases by 1, the predicted model value is 1.28 times the previous model value. The original versions of the Statway™ and Quantway™ courses were created by The Charles A. Dana Center at The University of Texas at Austin under sponsorship of the Carnegie Foundation for the Advancement of Teaching, and are copyright © 2011 by the Carnegie Foundation for the Advancement of Teaching and the Charles A. Dana Center at The University of Texas at Austin. STATWAY™/Statway™ and Quantway™ are trademarks of the Carnegie Foundation for the Advancement of Teaching. The Dana Center’s frontmatter for Statway™ and Quantway™ is available at www.utdanacenter.org/mathways. 16 Statway Instructor’s Notes April 23, 2012 (Full Version 1.0) Supporting Lesson 4.1.2: Exponential Models Exercises/Assignments If you did not completely work all five of the exercises from the Part I of the lesson correctly, correct or rework those now and submit them along with these exercises. For the descriptions of a relationship between two variables in Questions 1–4, identify each as one of the following: •
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•
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a linear relationship, an exponential decay relationship, an exponential growth relationship, a relationship that is none of the above, or cannot tell whether it is one of these options. If it is exponential, identify the initial amount, growth/decay factor, and growth/decay rate. If it is linear, give the slope. (1) A patient starts with 3 million bacteria in her body. The treatment prescribed by her doctor will reduce the number of bacteria in her body by 40% every day. (Answer: Exponential decay, with an initial amount of 3 million, a growth/decay factor of 0.60, and a decay rate of 0.40, or 40%. [Or call it a growth rate of –0.40.]) (2) Mr. Smith’s salary has risen by $100 per month for the last two years. (Answer: Linear growth, with a slope of $100 per month) (3) A particular town’s population is best modeled by P = –140n + 13,000 over the past eight years, where n is the number of years. (Answer: Linear decrease, with a slope of –140 people per year) (4) In Glen Rose, Texas, in 2008, there were 672 registered cats, and the number of registered cats has increased at the rate of 11% per year. (Answer: Exponential growth, with an initial amount of 6.72, a growth factor of 1.11, and a growth rate of 0.11, or 11%) The relationship is summarized in the given table for Questions 5–7: (5) t P 0 9,500 1 9,690 2 9,884 3 10,081 4 10,283 5 10,489 6 10,699 7 10,913 8 11,131 The original versions of the Statway™ and Quantway™ courses were created by The Charles A. Dana Center at The University of Texas at Austin under sponsorship of the Carnegie Foundation for the Advancement of Teaching, and are copyright © 2011 by the Carnegie Foundation for the Advancement of Teaching and the Charles A. Dana Center at The University of Texas at Austin. STATWAY™/Statway™ and Quantway™ are trademarks of the Carnegie Foundation for the Advancement of Teaching. The Dana Center’s frontmatter for Statway™ and Quantway™ is available at www.utdanacenter.org/mathways. 17 Statway Instructor’s Notes April 23, 2012 (Full Version 1.0) Supporting Lesson 4.1.2: Exponential Models (6)
(Answer: Exponential growth because it has a common ratio of 1.02. The initial amount is 9,500, the growth factor 1.02, and the growth rate 0.02, or 2%.) x y 0 1,200.0 1 1,203.5 2 1,207.0 3 1,210.5 4 1,214.0 5 1,217.5 (Answer: Linear growth, because it has a constant amount of increase of 3.5. The slope is 3.5.) (7) t A 1 1,248 3 2,496 7 3,744 8 4,992 9 6,240 11 7,488 13 8,736 17 9,984 (Answer: Neither. Although the table has common differences, since the x-­‐values are not evenly spaced, the common differences are not the slope of a linear relationship.) The original versions of the Statway™ and Quantway™ courses were created by The Charles A. Dana Center at The University of Texas at Austin under sponsorship of the Carnegie Foundation for the Advancement of Teaching, and are copyright © 2011 by the Carnegie Foundation for the Advancement of Teaching and the Charles A. Dana Center at The University of Texas at Austin. STATWAY™/Statway™ and Quantway™ are trademarks of the Carnegie Foundation for the Advancement of Teaching. The Dana Center’s frontmatter for Statway™ and Quantway™ is available at www.utdanacenter.org/mathways. 18 Statway Student Handout April 23, 2012 (Full Version 1.0) Supporting Lesson 4.1.2: Exponential Models Part I You have an opportunity to invest $2,000 at either 3% or 6% interest, compounded annually, for 10 years. There are various differences in the accounts (e.g., withdrawal flexibility), but you need to focus on the different amounts of money you will earn in each. (1) Before doing any computation, choose the answer that best expresses your expectation: (a) I expect to earn about twice as much in the 6% account than the 3% account. (b) I expect to earn significantly less than twice as much in the 6% account than the 3% account. (c) I expect to earn significantly more than twice as much in the 6% account than the 3% account. Now, let’s look at graphs of the amount of money in each account over a 10-­‐year period. Account Earning 3% Annual Interest Account Earning 6% Annual Interest (2) Put a check by each statement that is correct and an X by each statement that is incorrect. (a) Both graphs go up to the same height, so both accounts earn the same amounts. (b) In 10 years, one account earns about $2,700 and the other account about $3,600. (c) In 10 years, one account earns about $700 and the other account about $1,600. (d) I stand by my answer in Question 1. The original versions of the Statway™ and Quantway™ courses were created by The Charles A. Dana Center at The University of Texas at Austin under sponsorship of the Carnegie Foundation for the Advancement of Teaching, and are copyright © 2011 by the Carnegie Foundation for the Advancement of Teaching and the Charles A. Dana Center at The University of Texas at Austin. STATWAY™/Statway™ and Quantway™ are trademarks of the Carnegie Foundation for the Advancement of Teaching. The Dana Center’s frontmatter for Statway™ and Quantway™ is available at www.utdanacenter.org/mathways. 1 Statway Student Handout April 23, 2012 (Full Version 1.0) Supporting Lesson 4.1.2: Exponential Models How to Find the Formula for the 3% Account How do you compute the amounts in an account at the end of each year to make that graph? Working in groups, fill in the following table: Years Since Amount at 3% Amount at 6% January 1, 2008 Date January 1, 2008 0 $2,000.00 $2,000.00 January 1, 2009 1 $2,060.00 January 1, 2010 2 $2,121.80 January 1, 2011 3 January 1, 2012 4 The method you used here is an iterative approach. You find the amount at the end of one year, and then based on that, you find the amount for the next year and continue until you have computed the amounts for as many years as needed. However, finding the amount at the end of 10 years is tedious. It is even more tedious to find the amount at the end of 20 years! Surely someone has discovered a better way. Yes, you can find a pattern here and, using that pattern, develop a formula. Let’s see how to do this. First, use some arithmetic to find a pattern. Start by noticing the pattern of how you computed the amount at Year 1 of the 3% investment: new amount = 2,000 + 0.03(2,000) Now, from arithmetic class, remember the distributive property: 2,000 + 0.03(2,000) = 2,000(1 + 0.03) = 2,000(1.03) Therefore, the distributive property says that for every time you add 3% of its value to a number, you can also do that by multiplying the number by 1.03. Notice that you did that for Year 2, since 2,060 + 0.03(2,060) = 2,060(1.03). You also did it when computing the other years in the table for the 3% account. So, here is the shortcut to compute the amount in the account at the end of each year: after 1 year, 2,000(1.03) after 2 years, [2,000(1.03)](1.03) = 2,000(1.03)2 after 3 years, {[2,000(1.03)](1.03)}(1.03) = 2,000(1.03)3 and so on. Therefore, the pattern is, “Each year, multiply the amount in the account by 1.03 to get the new amount.” This means that the amount for Year 4 is 2,000(1.03)(1.03)(1.03)(1.03), which can be written as 2,000(1.03)4. The original versions of the Statway™ and Quantway™ courses were created by The Charles A. Dana Center at The University of Texas at Austin under sponsorship of the Carnegie Foundation for the Advancement of Teaching, and are copyright © 2011 by the Carnegie Foundation for the Advancement of Teaching and the Charles A. Dana Center at The University of Texas at Austin. STATWAY™/Statway™ and Quantway™ are trademarks of the Carnegie Foundation for the Advancement of Teaching. The Dana Center’s frontmatter for Statway™ and Quantway™ is available at www.utdanacenter.org/mathways. 2 Statway Student Handout April 23, 2012 (Full Version 1.0) Supporting Lesson 4.1.2: Exponential Models Compute the amounts at the end of each year using this pattern. Date Years Since January 1, 2008 Amount at 3% Amount at 6% January 1, 2008 0 2,000.00 2,000.00 January 1, 2009 1 2,000(1.03) = 2,060.00 January 1, 2010 2 2,000(1.03) = 2,121.80 2
2,000(1.06) = January 1, 2011 3 2,000(1.03) = 3
2,000(1.06) = January 1, 2012 4 2,000(1.03) = 4
2,000(1.06) = 2,000(1.06) = 2,120.00 2
3
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After you have computed some values in this table, check to see that they agree with the amounts found when you used the iterative method (the first method). Now that you have found a pattern, let’s write it in a more general way, as a formula. It can also be called a function. Consider the 3% account. Let t = the years since January 1, 2008, and A = the amount of money in the bank at the end of t years. Restating part of the above information, you have At t = 1, then A = 2,000(1.03) = 2,060.00 At t = 2, then A = 2,000(1.03)2 = 2,121.80 At t = 3, then A = 2,000(1.03)3 = At t = 4, then A = 2,000(1.03)4 = So, a general formula for the amount (A) in the 3% account at the tth year is A = 2,000(1.03)t. This process has demonstrated the power of algebra. You had a rule that showed how to do one amount at a time to build up a table of values, you analyzed it to see an underlying pattern, and then you summarized that pattern with a formula (also called a function.) Tasks In your group, do the following: (3) Develop the formula for the amount of money in the 6% account at the tth year. (4) Use your formulas to find the amount in the two accounts at the end of 10 years. The original versions of the Statway™ and Quantway™ courses were created by The Charles A. Dana Center at The University of Texas at Austin under sponsorship of the Carnegie Foundation for the Advancement of Teaching, and are copyright © 2011 by the Carnegie Foundation for the Advancement of Teaching and the Charles A. Dana Center at The University of Texas at Austin. STATWAY™/Statway™ and Quantway™ are trademarks of the Carnegie Foundation for the Advancement of Teaching. The Dana Center’s frontmatter for Statway™ and Quantway™ is available at www.utdanacenter.org/mathways. 3 Statway Student Handout April 23, 2012 (Full Version 1.0) Supporting Lesson 4.1.2: Exponential Models (5) Go back to the graphs in Question 1 and plot the points representing what you found. Do these points agree with the values given by the graphs? (6) Calculate the values in the following table and make graphs of each by hand. These will be similar to the graphs given in Question 1, but the pattern of growth will be even clearer because of the longer time. Date Years since January 1, 2008 Amount at 3% Amount at 6% January 1, 2008 0 2,000.00 2,000.00 January 1, 2013 5 10 10 January 1, 2018 10 2,000(1.03) =
2,000(1.06) = January 1, 2023 15 January 1, 2028 20 Conclusion Return to Question 2. Discuss each statement with your group. Can your group agree on which statements are correct or incorrect? Write a summary of your conclusions. The original versions of the Statway™ and Quantway™ courses were created by The Charles A. Dana Center at The University of Texas at Austin under sponsorship of the Carnegie Foundation for the Advancement of Teaching, and are copyright © 2011 by the Carnegie Foundation for the Advancement of Teaching and the Charles A. Dana Center at The University of Texas at Austin. STATWAY™/Statway™ and Quantway™ are trademarks of the Carnegie Foundation for the Advancement of Teaching. The Dana Center’s frontmatter for Statway™ and Quantway™ is available at www.utdanacenter.org/mathways. 4 Statway Student Handout April 23, 2012 (Full Version 1.0) Supporting Lesson 4.1.2: Exponential Models Homework for Part I Most examples of exponential growth and decay come from areas where you expect the relationships not to be exact, but statistical (with error or noise.) For the purposes of this lesson, consider them as exact relationships. (1) In a biology lab, a scientist has a population of fruit flies that has 120 flies on Day 0, and they increase by 8% per day. (a) Make a table for the number of fruit flies (N) on each day (d) for Days 0 to 6. (b) Graph the data in the table. (c) Did the population of fruit flies get as high as 180 flies in this time period? If so, approximately when? (d) Write a formula for the number of fruit flies. (e) In your formula, identify the initial amount, growth factor, and growth rate. The original versions of the Statway™ and Quantway™ courses were created by The Charles A. Dana Center at The University of Texas at Austin under sponsorship of the Carnegie Foundation for the Advancement of Teaching, and are copyright © 2011 by the Carnegie Foundation for the Advancement of Teaching and the Charles A. Dana Center at The University of Texas at Austin. STATWAY™/Statway™ and Quantway™ are trademarks of the Carnegie Foundation for the Advancement of Teaching. The Dana Center’s frontmatter for Statway™ and Quantway™ is available at www.utdanacenter.org/mathways. 5 Statway Student Handout April 23, 2012 (Full Version 1.0) Supporting Lesson 4.1.2: Exponential Models (2) The population of a certain country was 5.83 million in 1950, and grew by 2.5% per year for the next 20 years. (a) Make a table of the population of that country in 1950, 1951, and 1952. (b) Let t = year – 1950. Write a formula for the population of that country (P). (c) In your formula, identify the initial amount, growth factor, and growth rate. (3) A certain country was losing population over a 20-­‐year period. The country had 11 million people in 1990 and lost 1% of its population each year during the next 20 years. (a) Make a table of the population for that country in 1990, 1991, 1992, and 1993. (b) In this case, the factor is (1.00 – 0.01) = 0.99. Since this is not growth (the amount is declining), it is called a decay factor, and this is exponential decay instead of exponential growth. So, the formula is P = 11(0.99)t, where t = year – 1990. Use this formula to determine the population in 2010. The original versions of the Statway™ and Quantway™ courses were created by The Charles A. Dana Center at The University of Texas at Austin under sponsorship of the Carnegie Foundation for the Advancement of Teaching, and are copyright © 2011 by the Carnegie Foundation for the Advancement of Teaching and the Charles A. Dana Center at The University of Texas at Austin. STATWAY™/Statway™ and Quantway™ are trademarks of the Carnegie Foundation for the Advancement of Teaching. The Dana Center’s frontmatter for Statway™ and Quantway™ is available at www.utdanacenter.org/mathways. 6 Statway Student Handout April 23, 2012 (Full Version 1.0) Supporting Lesson 4.1.2: Exponential Models (c) In this formula, identify the initial amount, decay factor, and decay rate. (d) Use this formula to check your values for the population of that country in 1990, 1991, 1992, and 1993. (e) Notice that the description of this scenario is clearly exponential decay, yet the graph looks linear. Why? P = population, t = years since 1990 The original versions of the Statway™ and Quantway™ courses were created by The Charles A. Dana Center at The University of Texas at Austin under sponsorship of the Carnegie Foundation for the Advancement of Teaching, and are copyright © 2011 by the Carnegie Foundation for the Advancement of Teaching and the Charles A. Dana Center at The University of Texas at Austin. STATWAY™/Statway™ and Quantway™ are trademarks of the Carnegie Foundation for the Advancement of Teaching. The Dana Center’s frontmatter for Statway™ and Quantway™ is available at www.utdanacenter.org/mathways. 7 Statway Student Handout April 23, 2012 (Full Version 1.0) Supporting Lesson 4.1.2: Exponential Models (4) Water is treated with chlorine to kill germs to make it safe for drinking, swimming, and so on. Some gardeners, however, want water without chlorine because the chemical is somewhat harmful to plants. Chlorine dissipates from water when the water is exposed to air and sunlight. A typical dissipation rate for chlorine in water is about 15% per hour. Suppose you have a tub of water with an initial chlorine concentration of 2.00 milligrams per liter that is exposed to air and sunlight for 10 hours and the dissipation rate for that time is 15% per hour. (a) Make a table showing the chlorine concentration (C) at the end of each hour (h) for Hours 0, 1, and 2. (b) Write a formula for the chlorine concentration (C) at the end of h hours. (c) In your formula, identify the initial amount, growth/decay factor, and growth/decay rate. (d) Use the formula to check your values of C for Hours 0, 1, and 2. Then compute C for Hours 5, 7, and 10. (e) Use your values to make a graph of chlorine concentration. The original versions of the Statway™ and Quantway™ courses were created by The Charles A. Dana Center at The University of Texas at Austin under sponsorship of the Carnegie Foundation for the Advancement of Teaching, and are copyright © 2011 by the Carnegie Foundation for the Advancement of Teaching and the Charles A. Dana Center at The University of Texas at Austin. STATWAY™/Statway™ and Quantway™ are trademarks of the Carnegie Foundation for the Advancement of Teaching. The Dana Center’s frontmatter for Statway™ and Quantway™ is available at www.utdanacenter.org/mathways. 8 Statway Student Handout April 23, 2012 (Full Version 1.0) Supporting Lesson 4.1.2: Exponential Models (f) The gardener is happy to use the water for his plants when the chlorine concentration drops below 1.00 milligram per liter. Will it fall that low within the 10-­‐hour period? If so, approximately when? Calculate some additional values, if needed, to get a pretty good estimate of when. (5) Radioactive decay has this same exponential property that we have been discussing. That is, a radioactive substance loses a certain percentage of its radioactivity each year. Below is a statement about Cobalt-­‐60 that might be found in a technical article. Cobalt-­‐60 has a half-­‐life of 5.27 years, which means that 12.3% of its atoms decay each year. The gamma rays from radioactive cobalt are used with X-­‐ray film to inspect large welds. If an industrial gamma ray source is manufactured with 45 curies of cobalt-­‐60 (1 curie of a radioactive material results in 37 billion disintegrations per second), what will its strength measured in curies be at the end of each of the next 10 years? From this statement, do the following: (a) Find the initial amount and growth/decay rate. (b) Write a formula for the relationship. (c) In your formula, identify the initial amount, growth/decay factor, and growth/decay rate. The original versions of the Statway™ and Quantway™ courses were created by The Charles A. Dana Center at The University of Texas at Austin under sponsorship of the Carnegie Foundation for the Advancement of Teaching, and are copyright © 2011 by the Carnegie Foundation for the Advancement of Teaching and the Charles A. Dana Center at The University of Texas at Austin. STATWAY™/Statway™ and Quantway™ are trademarks of the Carnegie Foundation for the Advancement of Teaching. The Dana Center’s frontmatter for Statway™ and Quantway™ is available at www.utdanacenter.org/mathways. 9 Statway Student Handout April 23, 2012 (Full Version 1.0) Supporting Lesson 4.1.2: Exponential Models (d) Use the formula to determine the amount of radioactive material left at the end of each of the next 10 years. Years (t) Amount of Radioactivity (R) 0 1 2 3 4 5 6 7 8 9 10 (e) For this type of inspection, the strength of the gamma ray source must be at least 20 curies. How long will this source be adequate for this type of inspection? The original versions of the Statway™ and Quantway™ courses were created by The Charles A. Dana Center at The University of Texas at Austin under sponsorship of the Carnegie Foundation for the Advancement of Teaching, and are copyright © 2011 by the Carnegie Foundation for the Advancement of Teaching and the Charles A. Dana Center at The University of Texas at Austin. STATWAY™/Statway™ and Quantway™ are trademarks of the Carnegie Foundation for the Advancement of Teaching. The Dana Center’s frontmatter for Statway™ and Quantway™ is available at www.utdanacenter.org/mathways. 10 Statway Student Handout April 23, 2012 (Full Version 1.0) Supporting Lesson 4.1.2: Exponential Models Part II Ideas from Part I Homework •
How much should you round your response variable values from the model and why? Even if the data must have output as whole numbers, the predictions come from a mathematical model, which is an exact relationship. It is acceptable to have one or more decimal places in a predicted value. If you predict 572.23 people in a given year, everybody would recognize that you mean “close to 572.23.” •
What are the limitations on the parameter values in an exponential model? If it is exponential decay, the decay rate must be between 0% and 100%. You cannot have a process where more than 100% of the amount is disappearing at any point. If it is exponential growth, the growth rate must be positive, but it can be as large as needed, including more than 100%. A start-­‐up business might easily increase its sales by more than 200% per month in the first few months of starting. Another way of summing up these two statements is that the growth factor must be greater than 0. In most cases, the initial amount is greater than 0, but that is not mathematically necessary. •
The starting point is somewhat arbitrary. In any linear growth relationship or exponential growth relationship, you can redefine the x-­‐variable (years since 1700) to move the starting point (y-­‐intercept for linear, initial amount for exponential) close to the data you are actually observing. You can describe exactly the same relationship using this as you did before. The description differs somewhat, and a parameter value differs from the value you found for the model with nonadjusted starting point. This makes the model easier to interpret. •
Identifying exponential growth/decay from a table of values. For example, consider the population of Asia at 50-­‐year intervals from 1700 to 1950. For this example, you do not have the actual data, but have fitted a good model to the data, so that you have an exact relationship to work with here. In addition, 1 unit of the x-­‐values is 50 years, or “half-­‐
centuries after 1700.” Year Half-­‐Centuries After 1700 (x) Asia Predicted Population, in Millions (y) 1700 0 386 1750 1 495 1800 2 636 1850 3 816 1900 4 1,047 1950 5 1,343 The original versions of the Statway™ and Quantway™ courses were created by The Charles A. Dana Center at The University of Texas at Austin under sponsorship of the Carnegie Foundation for the Advancement of Teaching, and are copyright © 2011 by the Carnegie Foundation for the Advancement of Teaching and the Charles A. Dana Center at The University of Texas at Austin. STATWAY™/Statway™ and Quantway™ are trademarks of the Carnegie Foundation for the Advancement of Teaching. The Dana Center’s frontmatter for Statway™ and Quantway™ is available at www.utdanacenter.org/mathways. 11 Statway Student Handout April 23, 2012 (Full Version 1.0) Supporting Lesson 4.1.2: Exponential Models Asia 1500 1000 500 0 1700 1750 1800 1850 1900 1950 When you only look at the graph, it is not completely clear whether the relationship is close enough to linear to call it linear. To determine whether it is linear, look at some differences. Fill in the rest of the differences. Are the differences in population constant for each half-­‐century? x y Differences between successive y’s 0 386 1 495 495 – 386 = 109 2 636 636 – 495 = 141 3 816 4 1,047 5 1,343 No, the differences in population are not constant for each half-­‐century. So this is not linear growth. To determine whether it is exponential, look at the ratios, which give the growth factor. Fill in the rest of the ratios. Are the ratios in population constant for each half-­‐century? x y Ratios of successive y’s 0 386 1 495 495/386 = 1.28 2 636 636/495 = 1.28 3 816 4 1047 5 1343 Yes. The ratios of successive y’s are 1.28. This means that, for each time x increases by 1, the predicted model value is 1.28 times the previous model value. The original versions of the Statway™ and Quantway™ courses were created by The Charles A. Dana Center at The University of Texas at Austin under sponsorship of the Carnegie Foundation for the Advancement of Teaching, and are copyright © 2011 by the Carnegie Foundation for the Advancement of Teaching and the Charles A. Dana Center at The University of Texas at Austin. STATWAY™/Statway™ and Quantway™ are trademarks of the Carnegie Foundation for the Advancement of Teaching. The Dana Center’s frontmatter for Statway™ and Quantway™ is available at www.utdanacenter.org/mathways. 12 Statway Student Handout April 23, 2012 (Full Version 1.0) Supporting Lesson 4.1.2: Exponential Models Exercises/Assignments If you did not completely work all five of the exercises from the Part I of the lesson correctly, correct or rework those now and submit them along with these exercises. For the descriptions of a relationship between two variables in Questions 1–4, identify each as one of the following: •
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•
•
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a linear relationship, an exponential decay relationship, an exponential growth relationship, a relationship that is none of the above, or cannot tell whether it is one of these options. If it is exponential, identify the initial amount, growth/decay factor, and growth/decay rate. If it is linear, give the slope. (1) A patient starts with 3 million bacteria in her body. The treatment prescribed by her doctor will reduce the number of bacteria in her body by 40% every day. (2) Mr. Smith’s salary has risen by $100 per month for the last two years. (3) A particular town’s population is best modeled by P = –140n + 13,000 over the past eight years, where n is the number of years. (4) In Glen Rose, Texas, in 2008, there were 672 registered cats, and the number of registered cats has increased at the rate of 11% per year. The original versions of the Statway™ and Quantway™ courses were created by The Charles A. Dana Center at The University of Texas at Austin under sponsorship of the Carnegie Foundation for the Advancement of Teaching, and are copyright © 2011 by the Carnegie Foundation for the Advancement of Teaching and the Charles A. Dana Center at The University of Texas at Austin. STATWAY™/Statway™ and Quantway™ are trademarks of the Carnegie Foundation for the Advancement of Teaching. The Dana Center’s frontmatter for Statway™ and Quantway™ is available at www.utdanacenter.org/mathways. 13 Statway Student Handout April 23, 2012 (Full Version 1.0) Supporting Lesson 4.1.2: Exponential Models The relationship is summarized in the given table for Questions 5–7: (5) t P 0 9,500 1 9,690 2 9,884 3 10,081 4 10,283 5 10,489 6 10,699 7 10,913 8 11,131 (6) x y 0 1,200.0 1 1,203.5 2 1,207.0 3 1,210.5 4 1,214.0 5 1,217.5 (7) t A 1 1,248 3 2,496 7 3,744 8 4,992 9 6,240 11 7,488 13 8,736 17 9,984 The original versions of the Statway™ and Quantway™ courses were created by The Charles A. Dana Center at The University of Texas at Austin under sponsorship of the Carnegie Foundation for the Advancement of Teaching, and are copyright © 2011 by the Carnegie Foundation for the Advancement of Teaching and the Charles A. Dana Center at The University of Texas at Austin. STATWAY™/Statway™ and Quantway™ are trademarks of the Carnegie Foundation for the Advancement of Teaching. The Dana Center’s frontmatter for Statway™ and Quantway™ is available at www.utdanacenter.org/mathways. 14 Statway Instructor’s Notes April 23, 2012 (Full Version 1.0) Supporting Lesson 4.1.3: Assessing How Well a Model Fits the Data Estimated number of 50-­‐minute class sessions: 0.5 to 1 (depending on whether students use technology to produce the models and graphs) Learning Goals Students will understand that •
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•
•
the residual plot is a picture of the error in the structure: data = model + error. it is important to make a residual plot when they have fitted a model so that it is easier to see a pattern in the data that is not explained by the chosen model. when two models look as if they are a close fit to the data shown, both may give reasonable predictions within the range of the observed data, but often they give quite different predictions when outside the data range provided. Thus, if students want to make predictions outside the range of the observed data, it is particularly important that they have found a model that encompasses the entire pattern in the data. they can compute the coefficient of determination for fitting nonlinear models to data, and the definition is the same as in the linear case. However, the shortcut to computing it (squaring the correlation coefficient) does not work when fitting nonlinear relationships. Students will be able to •
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fit a linear model to a set of data and prepare the residual plot. fit an exponential model to a set of data and prepare the residual plot. When the true model for the data is exponential with a small growth rate (so that a scatterplot suggests that a linear model is adequate), identify the characteristics of the two residual plots that make clear that the exponential model is a better fit. Activity: Take the Population of Europe from 1700 to 19991 [Student Handout] Years Since 1700 (t) Population in millions (P) 0 125 50 163 100 203 150 276 200 408 250 547 299 729 (Note: Check for understanding around the data. For example, t = 250 refers to what year?) 1
United Nations. (1999). The world at six billion (ESA/P/WP.154). New York: Author. Retrieved from www.un.org/esa/population/publications/sixbillion/sixbilcover.pdf. The original versions of the Statway™ and Quantway™ courses were created by The Charles A. Dana Center at The University of Texas at Austin under sponsorship of the Carnegie Foundation for the Advancement of Teaching, and are copyright © 2011 by the Carnegie Foundation for the Advancement of Teaching and the Charles A. Dana Center at The University of Texas at Austin. STATWAY™/Statway™ and Quantway™ are trademarks of the Carnegie Foundation for the Advancement of Teaching. The Dana Center’s frontmatter for Statway™ and Quantway™ is available at www.utdanacenter.org/mathways. 1 Statway Instructor’s Notes April 23, 2012 (Full Version 1.0) Supporting Lesson 4.1.3: Assessing How Well a Model Fits the Data (1) Graph the data in the table. Graph of the Population Data Versus the Years Since 1700 t = years since 1700, P = population in millions (2) Does the graph give a compelling view that the relationship is straight or curved? Or is it not so clear? (Answer: The graph looks slightly curved.) (3) Fit a linear model to these data, show the linear model on the scatterplot of the data, and make the residual plot of the residuals versus the explanatory variable (years since 1700). (Note: Teachers must have a handout available for classes in which student technology is not available.) P = 1.99t + 51.75 Year Population of Europe 0 125 115.9698 50 163 157.7131 100 203 214.4819 150 276 291.6847 200 408 396.6766 250 547 539.4603 300 729 733.639 Prediction The original versions of the Statway™ and Quantway™ courses were created by The Charles A. Dana Center at The University of Texas at Austin under sponsorship of the Carnegie Foundation for the Advancement of Teaching, and are copyright © 2011 by the Carnegie Foundation for the Advancement of Teaching and the Charles A. Dana Center at The University of Texas at Austin. STATWAY™/Statway™ and Quantway™ are trademarks of the Carnegie Foundation for the Advancement of Teaching. The Dana Center’s frontmatter for Statway™ and Quantway™ is available at www.utdanacenter.org/mathways. 2 Statway Instructor’s Notes April 23, 2012 (Full Version 1.0) Supporting Lesson 4.1.3: Assessing How Well a Model Fits the Data Year Residuals 0 9.030245 50 5.286929 100 –11.48189 150 –15.68465 200 11.32345 250 7.539716 299 –4.639023 100 80 60 40 20 0 -­‐20 0 -­‐40 -­‐60 -­‐80 -­‐100 Residuals 50 100 150 200 250 300 350 (4) Interpret the residual plot to determine whether a linear model fits these data well. Discuss what you see in the residual plot and how it relates to the graph of the data with the model. (Answer: The linear model does not fit these data well. In the scatterplot, you can see that the predicted population for years in the middle of this range is systematically too small, and those at the ends of the range are systematically too large. The residual plot uses a different y-­‐axis scale that allows you to see the pattern more clearly.) (5) Fit an exponential model to these data, show the exponential model on the scatterplot of the data, and make the residual plot of the residuals versus the explanatory variable (years since 1700). Discuss what you see in the residual plot and how it relates to the graph of the data with the model. P = 115.97(1.0062)t Year Population Prediction of Europe 0 125 115.9698 50 163 157.7131 100 203 214.4819 150 276 291.6847 200 408 396.6766 250 547 539.4603 299 729 733.639 800 700 600 500 400 pop'n of Europe 300 Predicaon 200 100 0 0 100 200 300 400 The original versions of the Statway™ and Quantway™ courses were created by The Charles A. Dana Center at The University of Texas at Austin under sponsorship of the Carnegie Foundation for the Advancement of Teaching, and are copyright © 2011 by the Carnegie Foundation for the Advancement of Teaching and the Charles A. Dana Center at The University of Texas at Austin. STATWAY™/Statway™ and Quantway™ are trademarks of the Carnegie Foundation for the Advancement of Teaching. The Dana Center’s frontmatter for Statway™ and Quantway™ is available at www.utdanacenter.org/mathways. 3 Statway Instructor’s Notes April 23, 2012 (Full Version 1.0) Supporting Lesson 4.1.3: Assessing How Well a Model Fits the Data Year Residuals Residuals 0 9.030245 50 5.286929 15 100 –11.48189 10 150 –15.68465 200 11.32345 250 7.539716 -­‐10 299 –4.639023 -­‐15 5 0 -­‐5 0 100 200 300 400 Residuals -­‐20 (6) Interpret the residual plot to determine whether an exponential model seems to fit the data fairly well. (Answer: The exponential model seems to fit the data very well. The residual plot does not exactly look random, but there is not a clear pattern here, so it is not clear what formula fits it better.) (7) Use both models to predict the population of Europe in 1785. How different are those predictions? (Answer: The linear model predicts 220.84 and the exponential model 196.) (8) Use both models to predict the population of Europe in 2020. How different are those predictions? (Answer: The linear model predicts 688.32 and the exponential model 830.) (9) Do your graphs show why the predictions are closer for the two models in 1785 than in 2020? Explain your reasoning. (Answer: The models are closer to each other in 1785, so it is not surprising that they are providing similar predictions.) (10) Extrapolation means to make a prediction outside the range of data collected. Interpolation means to make a prediction inside the range of data collected. For Questions 7 and 8, characterize each as extrapolation or interpolation. (Answer: The prediction for 1785 is interpolation and for 2020 extrapolation.) (11) If you are using the best-­‐fitting linear model for data that actually have exponential growth, which type of prediction is more likely to be very wrong—interpolation or extrapolation? Justify your answer by explaining something you see on the graph. (Answer: Extrapolation is more risky because the models are further apart when you get outside the domain in which they are very close to each other. If you need to extrapolate, it is crucial to obtain the correct form for the model—linear or exponential here. The original versions of the Statway™ and Quantway™ courses were created by The Charles A. Dana Center at The University of Texas at Austin under sponsorship of the Carnegie Foundation for the Advancement of Teaching, and are copyright © 2011 by the Carnegie Foundation for the Advancement of Teaching and the Charles A. Dana Center at The University of Texas at Austin. STATWAY™/Statway™ and Quantway™ are trademarks of the Carnegie Foundation for the Advancement of Teaching. The Dana Center’s frontmatter for Statway™ and Quantway™ is available at www.utdanacenter.org/mathways. 4 Statway Instructor’s Notes April 23, 2012 (Full Version 1.0) Supporting Lesson 4.1.3: Assessing How Well a Model Fits the Data Lecture/Discussion If you do not expect students to do exponential regression using technology, distribute the student handout with the appropriate parts of the answer key for this activity and the homework. If you expect students to use technology outside of class to do these, lead them through the process of producing the formulas and graphs during the classroom presentation. If students have individual or group access to appropriate technology in the classroom to do linear and exponential regression, allow enough time for them to construct the formulas and graphs. Eleven questions is an ambitious task for students to work through on their own or even in groups. This task is intended to be a learning experience to increase their patience with and skill in carrying out multistep analyses. Such skills are important in professional-­‐level work. In the discussion, mention the following points to students: •
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In Module 3, they learned that a good residual plot has points randomly scattered around the middle, which is a zero residual. If there is a pattern instead of random scatter, that indicates a pattern in the data that they have not yet found in the model. For the correspondence of the way the data do not fit the linear model with the pattern in the residual plot for the linear model, point to where the model is below the data in the original scatterplot and where it is above the data. Show how that corresponds to the curve on the residual plot. It is important to remember the difference between extrapolation (think of extra at the beginning of the word) and interpolation (think of inter at the beginning of the word) and why extrapolation in particular can lead students far astray if they are using a model that does not really fit the shape of the data. What Proportion of the Variability in y Can Be Explained by the Regression Model? [Student Handout] When you were studying linear relationships, you computed the coefficient of determination ! ! = 1 – !!"#$%&'()
!!*+*()
as a numerical measure of the proportion of the variability in y that can be explained by the regression model. That definition is not limited to linear regression; it can be used with nonlinear regression as well. However, the shortcut to computing it for a linear model (squaring the correlation coefficient) does not work for nonlinear models. This shortcut is only for the linear model. To find the coefficient of determination for fitting another model, you must use the definition and not the shortcut. In practice, this means that the computer program computes it for you. Usually, it is given with an uppercase R so not to suggest the same relationship to the correlation coefficient that is true for the coefficient of determination of the linear relationship. !! = 1 – !!"#$%&'()
!!*+*()
The original versions of the Statway™ and Quantway™ courses were created by The Charles A. Dana Center at The University of Texas at Austin under sponsorship of the Carnegie Foundation for the Advancement of Teaching, and are copyright © 2011 by the Carnegie Foundation for the Advancement of Teaching and the Charles A. Dana Center at The University of Texas at Austin. STATWAY™/Statway™ and Quantway™ are trademarks of the Carnegie Foundation for the Advancement of Teaching. The Dana Center’s frontmatter for Statway™ and Quantway™ is available at www.utdanacenter.org/mathways. 5 Statway Instructor’s Notes April 23, 2012 (Full Version 1.0) Supporting Lesson 4.1.3: Assessing How Well a Model Fits the Data Homework [Student Handout] Recall the data set from Lesson 4.1.1 about the radioactivity of Area 2 in Nevada for many years after nuclear testing had ended. In that lesson, you learned that there is a certain amount of background radiation present, so a simple exponential decay model where the amount eventually decreases to zero is not appropriate. There are various ways that can be handled, but the easiest is to restate the response variable as the amount of radioactivity in excess of the expected background level of 340 milliroentgens (mR)/year. As you learned in the Lesson 4.1.2, it is often convenient to revise the explanatory variable from the date to “years since ...”. Following is the revised data from Area 2 in Lesson 4.1.1, where t = years since 1989 and A = amount of radioactivity per year in excess of 340 mR. t A t A 0 1,244 10 493 1 1,113 11 574 2 787 12 509 3 933 13 460 4 950 14 427 5 754 15 411 6 721 16 395 7 574 17 362 8 591 18 346 9 509 19 313 (1) Graph the data from the table. A = amt of excess radioacNvity 1400 1200 1000 800 600 400 200 0 0 5 10 15 20 t = years since 1989, A = amount of radioactivity per year in excess of 340 mR The original versions of the Statway™ and Quantway™ courses were created by The Charles A. Dana Center at The University of Texas at Austin under sponsorship of the Carnegie Foundation for the Advancement of Teaching, and are copyright © 2011 by the Carnegie Foundation for the Advancement of Teaching and the Charles A. Dana Center at The University of Texas at Austin. STATWAY™/Statway™ and Quantway™ are trademarks of the Carnegie Foundation for the Advancement of Teaching. The Dana Center’s frontmatter for Statway™ and Quantway™ is available at www.utdanacenter.org/mathways. 6 Statway Instructor’s Notes April 23, 2012 (Full Version 1.0) Supporting Lesson 4.1.3: Assessing How Well a Model Fits the Data (2) Does the graph give a compelling view that the relationship is straight or curved? Or is it not so clear? (Answer: The graph looks slightly curved.) (3) Fit a linear model to these data, show the linear model on the scatterplot of the data, and make the residual plot of the residuals versus the explanatory variable (year). A = –41.5t + 1,357.5 t A t A 0 1,244 10 493 1 1,113 11 574 2 787 12 509 3 933 13 460 4 950 14 427 5 754 15 411 6 721 16 395 7 574 17 362 8 591 18 346 9 509 19 313 1800 1600 1400 1200 1000 A2: mR/yr 800 Predicaon 600 400 200 0 0 5 10 15 20 The original versions of the Statway™ and Quantway™ courses were created by The Charles A. Dana Center at The University of Texas at Austin under sponsorship of the Carnegie Foundation for the Advancement of Teaching, and are copyright © 2011 by the Carnegie Foundation for the Advancement of Teaching and the Charles A. Dana Center at The University of Texas at Austin. STATWAY™/Statway™ and Quantway™ are trademarks of the Carnegie Foundation for the Advancement of Teaching. The Dana Center’s frontmatter for Statway™ and Quantway™ is available at www.utdanacenter.org/mathways. 7 Statway Instructor’s Notes April 23, 2012 (Full Version 1.0) Supporting Lesson 4.1.3: Assessing How Well a Model Fits the Data Year Residuals Year Residuals 0 226.5 10 –109.6 1 137.0 11 12.9 2 –147.5 12 –10.6 3 40.0 13 –18.1 4 98.5 14 –9.6 5 –56.0 15 15.9 6 –47.5 16 41.4 7 –153.0 17 49.9 8 –94.5 18 75.4 9 –135.0 19 83.9 Residuals 250.0 200.0 150.0 100.0 50.0 0.0 -­‐50.0 0 5 10 15 20 -­‐100.0 -­‐150.0 -­‐200.0 (4) Interpret the residual plot to determine whether a linear model fits these data well. Discuss what you see in the residual plot and how it relates to the graph of the data with the model. (Answer: The linear model does not fit these data well. In the scatterplot, you can see that the predicted radioactivity for years in the middle of this range is fairly systematically too small, and those that the ends of the range are systematically too large. The residual plot uses a different y-­‐axis scale that allows you to see the pattern more clearly.) The original versions of the Statway™ and Quantway™ courses were created by The Charles A. Dana Center at The University of Texas at Austin under sponsorship of the Carnegie Foundation for the Advancement of Teaching, and are copyright © 2011 by the Carnegie Foundation for the Advancement of Teaching and the Charles A. Dana Center at The University of Texas at Austin. STATWAY™/Statway™ and Quantway™ are trademarks of the Carnegie Foundation for the Advancement of Teaching. The Dana Center’s frontmatter for Statway™ and Quantway™ is available at www.utdanacenter.org/mathways. 8 Statway Instructor’s Notes April 23, 2012 (Full Version 1.0) Supporting Lesson 4.1.3: Assessing How Well a Model Fits the Data (5) Fit a exponential model to these data, show the exponential model on the scatterplot of the data, and make the residual plot of the residuals versus the explanatory variable (year). Discuss what you see in the residual plot and how it relates to the graph of the data with the model. A = 1,131.95(0.93)t t A t A 0 1,244 10 493 1 1,113 11 574 2 787 12 509 3 933 13 460 4 950 14 427 5 754 15 411 6 721 16 395 7 574 17 362 8 591 18 346 9 509 19 313 1800 1600 1400 1200 1000 A2: mR/yr 800 Predicaon 600 400 200 0 0 5 10 15 20 The original versions of the Statway™ and Quantway™ courses were created by The Charles A. Dana Center at The University of Texas at Austin under sponsorship of the Carnegie Foundation for the Advancement of Teaching, and are copyright © 2011 by the Carnegie Foundation for the Advancement of Teaching and the Charles A. Dana Center at The University of Texas at Austin. STATWAY™/Statway™ and Quantway™ are trademarks of the Carnegie Foundation for the Advancement of Teaching. The Dana Center’s frontmatter for Statway™ and Quantway™ is available at www.utdanacenter.org/mathways. 9 Statway Instructor’s Notes April 23, 2012 (Full Version 1.0) Supporting Lesson 4.1.3: Assessing How Well a Model Fits the Data Year Residuals Year Residuals 0 112.1 10 –57.2 1 59.8 11 62.1 2 –192.9 12 32.7 3 21.3 13 16.9 4 101.8 14 14.7 5 –35.2 15 27.4 6 –13.2 16 38.1 7 –109.1 17 30.0 8 –44.6 18 37.1 9 –82.3 19 25.6 (6) Interpret the residual plot to determine whether an exponential model seems to fit the data fairly well. (Answer: The exponential model seems to fit the data much better than the linear model. The residual plot does not exactly look random, but there is not a clear pattern here, so it is not clear what formula fits it better.) (7) Use both models to predict the amount of radioactivity in 1993, which is Year 4. How different are those predictions? (Answer: The linear model predicts 1,192 and the exponential model 1,188.) The original versions of the Statway™ and Quantway™ courses were created by The Charles A. Dana Center at The University of Texas at Austin under sponsorship of the Carnegie Foundation for the Advancement of Teaching, and are copyright © 2011 by the Carnegie Foundation for the Advancement of Teaching and the Charles A. Dana Center at The University of Texas at Austin. STATWAY™/Statway™ and Quantway™ are trademarks of the Carnegie Foundation for the Advancement of Teaching. The Dana Center’s frontmatter for Statway™ and Quantway™ is available at www.utdanacenter.org/mathways. 10 Statway Instructor’s Notes April 23, 2012 (Full Version 1.0) Supporting Lesson 4.1.3: Assessing How Well a Model Fits the Data (8) Use both models to predict the amount of radioactivity in 2012, which is Year 23. How different are those predictions? (Answer: The linear model predicts 403 and the exponential model 555.) (9) Do your graphs show why the predictions are closer for the two models in 1993 than in 2012? Explain your reasoning. (Answer: The models are closer to each other in 1993, so it is not surprising that they are providing similar predictions.) (10) Extrapolation means to make a prediction outside the range of data collected. Interpolation means to make a prediction inside the range of data collected. For Questions 7 and 8, characterize each as extrapolation or interpolation. (Answer: The prediction for 1993 is interpolation and for 2012 extrapolation.) (11) When you use technology to fit a linear model to data, you are getting the best-­‐fitting model of this type. So, the technology is only finding the best parameter values for the model you chose—
not finding the best type of model. If you are using the best-­‐fitting linear model for data that actually have exponential growth, which type of prediction is more likely to be very wrong—
interpolation or extrapolation? Justify your answer by explaining something you see on the graph. (Answer: Extrapolation is more risky because the models are further apart when you get outside the domain in which they are very close to each other. If you need to extrapolate, it is crucial to obtain the correct form for the model—linear or exponential here.) The original versions of the Statway™ and Quantway™ courses were created by The Charles A. Dana Center at The University of Texas at Austin under sponsorship of the Carnegie Foundation for the Advancement of Teaching, and are copyright © 2011 by the Carnegie Foundation for the Advancement of Teaching and the Charles A. Dana Center at The University of Texas at Austin. STATWAY™/Statway™ and Quantway™ are trademarks of the Carnegie Foundation for the Advancement of Teaching. The Dana Center’s frontmatter for Statway™ and Quantway™ is available at www.utdanacenter.org/mathways. 11 Statway Student Handout April 23, 2012 (Full Version 1.0) Supporting Lesson 4.1.3: Assessing How Well a Model Fits the Data Take the Population of Europe from 1700 to 19991 Years Since 1700 (t) Population in millions (P) 0 125 50 163 100 203 150 276 200 408 250 547 299 729 (1) Graph the data in the table. (2) Does the graph give a compelling view that the relationship is straight or curved? Or is it not so clear? 1
United Nations. (1999). The world at six billion (ESA/P/WP.154). New York: Author. Retrieved from www.un.org/esa/population/publications/sixbillion/sixbilcover.pdf. The original versions of the Statway™ and Quantway™ courses were created by The Charles A. Dana Center at The University of Texas at Austin under sponsorship of the Carnegie Foundation for the Advancement of Teaching, and are copyright © 2011 by the Carnegie Foundation for the Advancement of Teaching and the Charles A. Dana Center at The University of Texas at Austin. STATWAY™/Statway™ and Quantway™ are trademarks of the Carnegie Foundation for the Advancement of Teaching. The Dana Center’s frontmatter for Statway™ and Quantway™ is available at www.utdanacenter.org/mathways. 1 Statway Student Handout April 23, 2012 (Full Version 1.0) Supporting Lesson 4.1.3: Assessing How Well a Model Fits the Data (3) Fit a linear model to these data, show the linear model on the scatterplot of the data, and make the residual plot of the residuals versus the explanatory variable (years since 1700).
(4) Interpret the residual plot to determine whether a linear model fits these data well. Discuss what you see in the residual plot and how it relates to the graph of the data with the model. The original versions of the Statway™ and Quantway™ courses were created by The Charles A. Dana Center at The University of Texas at Austin under sponsorship of the Carnegie Foundation for the Advancement of Teaching, and are copyright © 2011 by the Carnegie Foundation for the Advancement of Teaching and the Charles A. Dana Center at The University of Texas at Austin. STATWAY™/Statway™ and Quantway™ are trademarks of the Carnegie Foundation for the Advancement of Teaching. The Dana Center’s frontmatter for Statway™ and Quantway™ is available at www.utdanacenter.org/mathways. 2 Statway Student Handout April 23, 2012 (Full Version 1.0) Supporting Lesson 4.1.3: Assessing How Well a Model Fits the Data (5) Fit an exponential model to these data, show the exponential model on the scatterplot of the data, and make the residual plot of the residuals versus the explanatory variable (years since 1700). Discuss what you see in the residual plot and how it relates to the graph of the data with the model.
(6) Interpret the residual plot to determine whether an exponential model seems to fit the data fairly well. The original versions of the Statway™ and Quantway™ courses were created by The Charles A. Dana Center at The University of Texas at Austin under sponsorship of the Carnegie Foundation for the Advancement of Teaching, and are copyright © 2011 by the Carnegie Foundation for the Advancement of Teaching and the Charles A. Dana Center at The University of Texas at Austin. STATWAY™/Statway™ and Quantway™ are trademarks of the Carnegie Foundation for the Advancement of Teaching. The Dana Center’s frontmatter for Statway™ and Quantway™ is available at www.utdanacenter.org/mathways. 3 Statway Student Handout April 23, 2012 (Full Version 1.0) Supporting Lesson 4.1.3: Assessing How Well a Model Fits the Data (7) Use both models to predict the population of Europe in 1785. How different are those predictions? (8) Use both models to predict the population of Europe in 2020. How different are those predictions? (9) Do your graphs show why the predictions are closer for the two models in 1785 than in 2020? Explain your reasoning. (10) Extrapolation means to make a prediction outside the range of data collected. Interpolation means to make a prediction inside the range of data collected. For Questions 7 and 8, characterize each as extrapolation or interpolation. (11) If you are using the best-­‐fitting linear model for data that actually have exponential growth, which type of prediction is more likely to be very wrong—interpolation or extrapolation? Justify your answer by explaining something you see on the graph. The original versions of the Statway™ and Quantway™ courses were created by The Charles A. Dana Center at The University of Texas at Austin under sponsorship of the Carnegie Foundation for the Advancement of Teaching, and are copyright © 2011 by the Carnegie Foundation for the Advancement of Teaching and the Charles A. Dana Center at The University of Texas at Austin. STATWAY™/Statway™ and Quantway™ are trademarks of the Carnegie Foundation for the Advancement of Teaching. The Dana Center’s frontmatter for Statway™ and Quantway™ is available at www.utdanacenter.org/mathways. 4 Statway Student Handout April 23, 2012 (Full Version 1.0) Supporting Lesson 4.1.3: Assessing How Well a Model Fits the Data What Proportion of the Variability in y Can Be Explained by the Regression Model? When you were studying linear relationships, you computed the coefficient of determination ! ! = 1 – !!"#$%&'()
!!*+*()
as a numerical measure of the proportion of the variability in y that can be explained by the regression model. That definition is not limited to linear regression; it can be used with nonlinear regression as well. However, the shortcut to computing it for a linear model (squaring the correlation coefficient) does not work for nonlinear models. This shortcut is only for the linear model. To find the coefficient of determination for fitting another model, you must use the definition and not the shortcut. In practice, this means that the computer program computes it for you. Usually, it is given with an uppercase R so not to suggest the same relationship to the correlation coefficient that is true for the coefficient of determination of the linear relationship. !! = 1 – !!"#$%&'()
!!*+*()
The original versions of the Statway™ and Quantway™ courses were created by The Charles A. Dana Center at The University of Texas at Austin under sponsorship of the Carnegie Foundation for the Advancement of Teaching, and are copyright © 2011 by the Carnegie Foundation for the Advancement of Teaching and the Charles A. Dana Center at The University of Texas at Austin. STATWAY™/Statway™ and Quantway™ are trademarks of the Carnegie Foundation for the Advancement of Teaching. The Dana Center’s frontmatter for Statway™ and Quantway™ is available at www.utdanacenter.org/mathways. 5 Statway Student Handout April 23, 2012 (Full Version 1.0) Supporting Lesson 4.1.3: Assessing How Well a Model Fits the Data Homework Recall the data set from Lesson 4.1.1 about the radioactivity of Area 2 in Nevada for many years after nuclear testing had ended. In that lesson, you learned that there is a certain amount of background radiation present, so a simple exponential decay model where the amount eventually decreases to zero is not appropriate. There are various ways that can be handled, but the easiest is to restate the response variable as the amount of radioactivity in excess of the expected background level of 340 milliroentgens (mR)/year. As you learned in the Lesson 4.1.2, it is often convenient to revise the explanatory variable from the date to “years since ...”. Following is the revised data from Area 2 in Lesson 4.1.1, where t = years since 1989 and A = amount of radioactivity per year in excess of 340 mR. t A t A 0 1,244 10 493 1 1,113 11 574 2 787 12 509 3 933 13 460 4 950 14 427 5 754 15 411 6 721 16 395 7 574 17 362 8 591 18 346 9 509 19 313 (1) Graph the data from the table. (2) Does the graph give a compelling view that the relationship is straight or curved? Or is it not so clear? The original versions of the Statway™ and Quantway™ courses were created by The Charles A. Dana Center at The University of Texas at Austin under sponsorship of the Carnegie Foundation for the Advancement of Teaching, and are copyright © 2011 by the Carnegie Foundation for the Advancement of Teaching and the Charles A. Dana Center at The University of Texas at Austin. STATWAY™/Statway™ and Quantway™ are trademarks of the Carnegie Foundation for the Advancement of Teaching. The Dana Center’s frontmatter for Statway™ and Quantway™ is available at www.utdanacenter.org/mathways. 6 Statway Student Handout April 23, 2012 (Full Version 1.0) Supporting Lesson 4.1.3: Assessing How Well a Model Fits the Data (3) Fit a linear model to these data, show the linear model on the scatterplot of the data, and make the residual plot of the residuals versus the explanatory variable (year). (4) Interpret the residual plot to determine whether a linear model fits these data well. Discuss what you see in the residual plot and how it relates to the graph of the data with the model. The original versions of the Statway™ and Quantway™ courses were created by The Charles A. Dana Center at The University of Texas at Austin under sponsorship of the Carnegie Foundation for the Advancement of Teaching, and are copyright © 2011 by the Carnegie Foundation for the Advancement of Teaching and the Charles A. Dana Center at The University of Texas at Austin. STATWAY™/Statway™ and Quantway™ are trademarks of the Carnegie Foundation for the Advancement of Teaching. The Dana Center’s frontmatter for Statway™ and Quantway™ is available at www.utdanacenter.org/mathways. 7 Statway Student Handout April 23, 2012 (Full Version 1.0) Supporting Lesson 4.1.3: Assessing How Well a Model Fits the Data (5) Fit a exponential model to these data, show the exponential model on the scatterplot of the data, and make the residual plot of the residuals versus the explanatory variable (year). Discuss what you see in the residual plot and how it relates to the graph of the data with the model. (6) Interpret the residual plot to determine whether an exponential model seems to fit the data fairly well. The original versions of the Statway™ and Quantway™ courses were created by The Charles A. Dana Center at The University of Texas at Austin under sponsorship of the Carnegie Foundation for the Advancement of Teaching, and are copyright © 2011 by the Carnegie Foundation for the Advancement of Teaching and the Charles A. Dana Center at The University of Texas at Austin. STATWAY™/Statway™ and Quantway™ are trademarks of the Carnegie Foundation for the Advancement of Teaching. The Dana Center’s frontmatter for Statway™ and Quantway™ is available at www.utdanacenter.org/mathways. 8 Statway Student Handout April 23, 2012 (Full Version 1.0) Supporting Lesson 4.1.3: Assessing How Well a Model Fits the Data (7) Use both models to predict the amount of radioactivity in 1993, which is Year 4. How different are those predictions? (8) Use both models to predict the amount of radioactivity in 2012, which is Year 23. How different are those predictions? (9) Do your graphs show why the predictions are closer for the two models in 1993 than in 2012? Explain your reasoning. (10) Extrapolation means to make a prediction outside the range of data collected. Interpolation means to make a prediction inside the range of data collected. For Questions 7 and 8, characterize each as extrapolation or interpolation. (11) When you use technology to fit a linear model to data, you are getting the best-­‐fitting model of this type. So, the technology is only finding the best parameter values for the model you chose—
not finding the best type of model. If you are using the best-­‐fitting linear model for data that actually have exponential growth, which type of prediction is more likely to be very wrong—
interpolation or extrapolation? Justify your answer by explaining something you see on the graph. The original versions of the Statway™ and Quantway™ courses were created by The Charles A. Dana Center at The University of Texas at Austin under sponsorship of the Carnegie Foundation for the Advancement of Teaching, and are copyright © 2011 by the Carnegie Foundation for the Advancement of Teaching and the Charles A. Dana Center at The University of Texas at Austin. STATWAY™/Statway™ and Quantway™ are trademarks of the Carnegie Foundation for the Advancement of Teaching. The Dana Center’s frontmatter for Statway™ and Quantway™ is available at www.utdanacenter.org/mathways. 9