Advanced Mathematical Decision Making:

Advanced Mathematical Decision Making: Unit I: Analyzing Numerical Data AMDM is a project of The Texas Association of Supervisors of Mathematics and The Charles A. Dana Center at The University of Texas With support from the Greater Texas Foundation 2010

Copyright 2010, Charles A. Dana Center at The University of Texas This 2010 implementation edition is copyrighted by the Charles A. Dana Center at The University of Texas. Unless otherwise indicated, the materials found in this resource are the copyrighted property of the Charles A. Dana Center at The University of Texas (the University). 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 University, except under the following conditions: 1) The following guidelines address only the materials owned by the Charles A. Dana Center at the University of Texas. a) The people of Texas are granted by the University a nonexclusive license in perpetuity subject to the terms and conditions listed below (item 2) to use the 2010 edition of the AMDM instructional materials (reflecting improvements based on the 2009 2010 pilot) in Texas classrooms and homes for the education of our children. b) Teachers and administrators from outside Texas who participate in the Dana Center s AMDM professional development in summer 2010 (sign-up information here: www.utdanacenter.org/pd) may use the 2010 edition of the AMDM instructional materials in their classrooms and schools, subject to the terms and conditions listed below (item 2). c) Organizations or individuals other than those listed above (items a and b) must obtain prior written permission from the Charles A. Dana Center at The University of Texas at Austin for the use of the 2010 AMDM instructional materials, the terms of which may be set forth in a copyright license agreement, and which may include the payment of a licensing fee, or royalties, or both. 2) Terms and conditions: Any portion reproduced must retain the original copyright notice, as written above. We use all funds generated through use of our materials to further our nonprofit educational mission. Please send your permission requests or questions to us at this address: Charles A. Dana Center Fax: 512-232-1855 The University of Texas dana-txshop@utlists.utexas.edu 1616 Guadalupe Street, Suite 3.206 www.utdanacenter.org Austin, TX 78701-1222 The Charles A. Dana Center and The University of Texas, as well as the authors and editors, assume no liability for any loss or damage resulting from the use of this resource. We have made extensive efforts to ensure the accuracy of the information in this resource, to provide proper acknowledgement of original sources, and to 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. Charles A. Dana Center at The University of Texas ii

About the Charles A. Dana Center at The University of Texas The Dana Center works to raise student achievement in K 16 mathematics and science, especially for historically underserved populations. We do so by providing direct service to school districts and institutions of higher education; to local, state, and national education leaders; and to agencies, nonprofits, and professional organizations concerned with strengthening American mathematics and science education. The Center was founded in 1991 in the College of Natural Sciences at The University of Texas. Our original purpose which continues in our work today was to increase the diversity of students who successfully pursue careers in science, technology, engineering, and mathematics (STEM) fields. We carry out our work by supporting high standards and building system capacity; developing collaborations 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 child 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. And for updates and background on the Advanced Mathematical Decision Making project, see www.utdanacenter.org/amdm. About the Texas Association of Supervisors of Mathematics The mission of the Texas Association of Supervisors of Mathematics is to assist in promoting effectiveness in the supervision, coordination, and teaching of mathematics, especially in the elementary and secondary fields. TASM accomplishes this by holding meetings for the presentation and discussion of papers; by conducting public discussion groups, forums, panels, lectures, or other similar programs; by conducting or promoting investigations for the purpose of improving the teaching of mathematics; and by the publication of papers, journals, books, and reports, thus vitalizing and coordinating the work of mathematics supervisors across Texas and bringing the interests of mathematics to the attention and consideration of the larger education community in Texas. For more information about TASM, visit its website at www.tasmonline.net. Charles A. Dana Center at The University of Texas iii

About the development of this resource The development and production of the AMDM student expectations excerpted in this resource, as well as the AMDM instructional materials that constitute Units I through VII of this resource, were supported by the Greater Texas Foundation, based in Bryan, Texas. Any opinions, findings, conclusions, or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of the Greater Texas Foundation or The University of Texas. The AMDM student expectations excerpted in this resource were developed by a writing group of 20 people, consisting primarily of Texas mathematics educators and mathematicians at the secondary and postsecondary level, with additional assistance from mathematicians outside Texas. These AMDM student expectations reflect many months of development and feedback from across the state and from expert reviewers outside the state as described in updates presented to the Texas State Board of Education in November 2007, June 2008, and March 2009. These or any other AMDM student expectations, if adopted by the state of Texas as Texas Essential Knowledge and Skills, will become part of state law and thus available to all Texans at no charge. The Dana Center will not claim copyright or charge usage fees for the use of any AMDM TEKS. The AMDM instructional materials (Units I VII) were developed by mathematics teachers and faculty from Texas and beyond, with support from the Texas Association of Supervisors of Mathematics and others around the state who have donated their time or served as consultants for the project. See the acknowledgments section for a complete listing of the authors, reviewers, advisory team, and staff supporting this work. As noted in the copyright language above, the people of Texas are granted by the University a nonexclusive license in perpetuity to use the 2010 edition of the AMDM instructional materials (reflecting improvements based on the 2009 2010 pilot) in Texas classrooms and homes for the education of our children. This resource was produced in Microsoft Word 2004 for Mac. This version constitutes the 2010 implementation edition and reflects enhancements from the pilot edition (2009). Released June 2010. Art credits p. I-12, istockphoto; p. I-13, Dana Center staff; p. I-27, Dana Center; p. I-30, Dana Center staff; p. I-32, Phil Swann; p. I-34, Dana Center staff; p. I-38, Dana Center staff; p. I-40, istockphoto; p. I-41, istockphoto and Dana Center staff; p. I-46, Dana Center staff; p. I-86, Dana Center staff; p. I-91, Dana Center staff Charles A. Dana Center at The University of Texas iv

Acknowledgments Here we list alphabetically the authors and reviewers for AMDM Units I through VII. Affiliations are listed to clarify individuals credentials and locations. When affiliations other than the Charles A. Dana Center at The University of Texas are listed, it should be understood that in most cases the individual was acting as an independent consultant and not necessarily as a representative of his or her organization. Project director Cathy Seeley, senior fellow and AMDM project director, Charles A. Dana Center at the University of Texas Director of materials development Pam Walker, senior program coordinator, Charles A. Dana Center at the University of Texas Pilot coordinator Mary E. Molly Ewing, graduate research assistant, Charles A. Dana Center at the University of Texas Authors Stuart Boersma, professor of mathematics, Central Washington University, Ellensburg, Washington Tom Butts, professor, Department of Science/Mathematics Education, The University of Texas at Dallas, Dallas, Texas Paul Gray, director of mathematics and science, Pearland Independent School District, Pearland, Texas Susan May, senior program coordinator, Charles A. Dana Center at the University of Texas Michael Ota, graduate research assistant, Charles A. Dana Center at the University of Texas Connie Richardson, instructor of mathematics, Midwestern State University, Wichita Falls, Texas Carmen Whitman, independent consultant, Mathematics for All Consulting Reviewers Mark L. Daniels, clinical associate professor, UTeach Science Program, The University of Texas Mary E. Molly Ewing, graduate research assistant, Charles A. Dana Center at the University of Texas Kelly Flickinger, mathematics teacher, Bowie High School, Austin Independent School District, Austin, Texas Stephen Maurer, professor of mathematics and chair, Department of Mathematics and Statistics, Swarthmore College, Swarthmore, Pennsylvania Margaret E. Maggie Myers, lecturer, Department of Computer Sciences, The University of Texas Cathy Seeley, senior fellow and AMDM project director, Charles A. Dana Center at the University of Texas Rita Tellez, mathematics instructional specialist, Ysleta Independent School District, El Paso, Texas Charles A. Dana Center at The University of Texas v

Professional development team Kelly Flickinger, mathematics teacher, Bowie High School, Austin Independent School District, Austin, Texas Susan May, senior program coordinator, Charles A. Dana Center at the University of Texas at Austin Bonnie McNemar, independent mathematics consultant, Georgetown, Texas, and former director of TEXTEAMS Michael Ota, graduate research assistant, Charles A. Dana Center at the University of Texas Tricia Rothenberg, mathematics curriculum coordinator, Round Rock Independent School District, Round Rock, Texas Pam Walker, senior program coordinator, Charles A. Dana Center at the University of Texas Jim Wohlgehagen, mathematics coordinator, secondary curriculum and instruction, Plano Independent School District, Plano, Texas Professional development delivery team Roger Burnham, mathematics teacher, McAllen Memorial High School, McAllen Independent School District, McAllen, Texas Joyce Collett, mathematics coordinator 6 12, Clear Creek Independent School District, League City, Texas Kelly Flickinger, mathematics teacher, Bowie High School, Austin Independent School District, Austin, Texas Jennifer Harper, mathematics teacher, Hightower High School, Fort Bend Independent School District, Sugar Land, Texas William Bill Jones, mathematics teacher, Clark High School, Northside Independent School District, San Antonio, Texas Frakeetta Marshall-Yancey, mathematics teacher, Hightower High School, Fort Bend Independent School District, Sugar Land, Texas Bonnie McNemar, independent mathematics consultant, Georgetown, Texas, and former director of TEXTEAMS Jonathan Newman, mathematics teacher, Billy Ryan High School, Denton Independent School District, Denton, Texas Michael Ota, graduate research assistant, Charles A. Dana Center at the University of Texas David Pulsipher, mathematics teacher, Plano East Senior High School, Plano Independent School District, Plano, Texas Connie Richardson, instructor of mathematics, Midwestern State University, Wichita Falls, Texas Tricia Rothenberg, mathematics curriculum coordinator, Round Rock Independent School District, Round Rock, Texas Miguel Salazar, mathematics teacher and department head, Ralls High School, Ralls Independent School District, Ralls, Texas Pam Walker, senior program coordinator, Charles A. Dana Center at the University of Texas Advisory team Julie Acosta, secondary mathematics coordinator, McAllen Independent School District, McAllen, Texas Sandra Browning, assistant professor of mathematics education, University of Houston Clear Lake, Clear Lake, Texas Charles A. Dana Center at The University of Texas vi

Tom Butts, professor, Department of Science/Mathematics Education, The University of Texas at Dallas, Dallas, Texas Joyce Collett, mathematics coordinator 6 12, Clear Creek Independent School District, League City, Texas Wade Ellis, (retired) instructor of mathematics, West Valley Community College, Saratoga, California James A. Mendoza Epperson, associate professor of mathematics, The University of Texas at Arlington, Arlington, Texas Gregory Foley, Robert L. Morton professor of mathematics education, Ohio University, Athens, Ohio Linda Gann, instructional specialist, secondary mathematics, Northside Independent School District, San Antonio, Texas Sarah Sallie Kay Janes, academic dean, San Jacinto College, North Campus, Houston, Texas Stephen Maurer, professor of mathematics and chair, Department of Mathematics and Statistics, Swarthmore College, Swarthmore, Pennsylvania Susan May, senior program coordinator, Charles A. Dana Center at the University of Texas Bonnie McNemar, independent mathematics consultant, Georgetown, Texas, and former director of TEXTEAMS Michael Ota, graduate research assistant, Charles A. Dana Center at the University of Texas Rita Tellez, mathematics instructional specialist, Ysleta Independent School District, El Paso, Texas Sherri Waddey, former consultant teacher preparation and certification, Region 10 Education Service Center, Richardson, Texas Pam Walker, senior program coordinator, Charles A. Dana Center at the University of Texas Jim Wohlgehagen, mathematics coordinator, secondary curriculum and instruction, Plano Independent School District, Plano, Texas Linda Zientek, assistant professor of mathematics, Department of Mathematics and Statistics, Sam Houston State University, Huntsville, Texas Lana Zimmer, high school mathematics teacher, Clear Creek Independent School District, League City, Texas Research and evaluation team Lesley Leach, research scientist associate III, Charles A. Dana Center at the University of Texas Cynthia Schneider, director of research and evaluation, Charles A. Dana Center at the University of Texas Editorial and production team Amy Dolejs, webmaster and copyeditor, Charles A. Dana Center at the University of Texas Steve Engler, lead editor and production editor, Charles A. Dana Center at the University of Texas Rachel Jenkins, consulting editor, Charles A. Dana Center at the University of Texas Sarah Searcy, copyeditor, Charles A. Dana Center at the University of Texas Phil Swann, senior designer, Charles A. Dana Center at the University of Texas With Special Thanks... To the teachers participating in the 2009 2010 pilot and sharing their ideas for improving this course. Charles A. Dana Center at The University of Texas vii

Table of Contents This unit is one of seven that make up the 2010 implementation edition of the Advanced Mathematical Decision Making instructional materials. The copyright and acknowledgments for this unit apply to all the AMDM units of this 2010 edition. Unit I: Analyzing Numerical Data Unit II: Probability Unit III: Statistical Studies Unit IV: Using Recursion in Models and Decision Making Unit V: Using Functions in Models and Decision Making Unit VI: Decision Making in Finance Unit VII: Networks and Graphs Advanced Mathematical Decision Making Unit I: Analyzing Numerical Data Teacher Materials Unit I: Analyzing Numerical Data: Overview...I-1 Section I.A: Estimating Large Numbers: Section Planning...I-3 I.A.1: Estimating Crowds...I-6 I.A.2: Filling Your Classroom with Tennis Balls...I-8 I.A.3: Not Enough Numbers... I-10 I.A Student Activity Sheet 1 Teacher Version... I-12 I.A Student Activity Sheet 2 Teacher Version... I-15 I.A Student Activity Sheet 3 Teacher Version... I-18 Section I.B: Using Ratios: Section Planning... I-22 I.B.4: Ratios in the Media... I-25 I.B.5: Changing Tires... I-28 I.B Student Activity Sheet 4 Teacher Version... I-30 I.B Student Activity Sheet 5 Teacher Version... I-40 Section I.C: Indices Using Weighted Sums and Averages: Section Planning... I-47 I.C.6: Final Grade Averages... I-50 I.C.7: Slugging Averages... I-51 I.C.8: Quarterback Ratings... I-52 I.C.9: Fan Cost Index... I-53 I.C.10: Readability Indices... I-55 I.C.11: Simpson s Paradox... I-57 Charles A. Dana Center at The University of Texas viii

I.C Student Activity Sheet 6 Teacher Version... I-58 I.C Student Activity Sheet 7 Teacher Version... I-61 I.C Student Activity Sheet 8 Teacher Version... I-64 I.C Student Activity Sheet 9 Teacher Version... I-67 I.C Student Activity Sheet 10 Teacher Version... I-73 I.C Student Activity Sheet 11 Teacher Version... I-79 Section I.D: Validating Identification Numbers: Section Planning... I-80 I.D.12: Universal Product Codes... I-82 I.D.13: Credit Card Numbers... I-84 I.D Student Activity Sheet 12 Teacher Version... I-86 I.D Student Activity Sheet 13 Teacher Version... I-91 Advanced Mathematical Decision Making Unit I: Analyzing Numerical Data Student Materials I.A Student Activity Sheet 1: Estimating Crowds... 1 I.A Student Activity Sheet 2: Filling Your Classroom with Tennis Balls... 4 I.A Student Activity Sheet 3: Not Enough Numbers... 7 I.B Student Activity Sheet 4: Ratios in the Media... 11 I.B Student Activity Sheet 5: Changing Tires... 21 I.C Student Activity Sheet 6: Final Grade Averages... 28 I.C Student Activity Sheet 7: Slugging Averages... 31 I.C Student Activity Sheet 8: Quarterback Ratings... 34 I.C Student Activity Sheet 9: Fan Cost Index... 37 I.C Student Activity Sheet 10: Readability Indices... 42 I.C Student Activity Sheet 11: Simpson s Paradox... 48 I.D Student Activity Sheet 12: Universal Product Codes... 49 I.D Student Activity Sheet 13: Credit Card Numbers... 54 Charles A. Dana Center at The University of Texas ix