Thinking with Mathematical Models Linear Functions and Bivariate Data 4 CONNECTED MATHEMATICS® Student Edition SAMPLE
SAMPLE
CONNECTED MATHEMATICS 4 ® Student Edition Thinking with Mathematical Models Linear Functions and Bivariate Data SAMPLE
© Copyright 2025 by Michigan State University. Published by Lab-Aids, Inc., Ronkonkoma, New York, 11779. All rights reserved. This publication is protected by copyright, and permission should be obtained from the publisher prior to any prohibited reproduction, storage in a retrieval system, or transmission in any form or by any means electronic, mechanical, photocopying, recording, or likewise. For information regarding permission(s) or other customer service topics, please contact Lab-Aids, Inc. Connected Mathematics® is used under license from Michigan State University. Acknowledgments: Connected Mathematics® was developed at Michigan State University with financial support from the Michigan State University Office of the Provost and the College of Natural Science. This material is based upon work supported by the National Science Foundation under Grant No. 9150217 and Grant No. ESI 9986372. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of the National Science Foundation. Thinking with Mathematical Models | Student Edition | Connected Mathematics® 4 ISBN-13: 979-8-89101-126-7 eISBN: 979-8-89101-178-6 v1 Part ID: CMP8-1-4SE Print number: 01 Print year: 2024 Printed in the United States of America. Photo credits: Page xviii, (top): © Oliver Förstner/Alamy Stock Photo, (middle): © George Ostertag/Alamy Stock Photo; page xix, (left): © Arina Habich/Alamy Stock Photo, (right): © Ilja Enger-Tsizikov/Alamy Stock Photo; page 2, (left): © Guoqiang Xue/Alamy Stock Photo, (right): © Björn Wylezich/Alamy Stock Photo; page 8, © Feng Yu/Alamy Stock Photo; page 9, Courtesy of the American Galvanizers Association; page 12, © vast natalia/Alamy Stock Photo; page 24, © rphstock/Alamy Stock Photo; page 28, ARphotography/Alamy Stock Photo; page 72, © Alamy Stock Photo; page 73, (left): © PitK/Alamy Stock Photo, (right): © Georgi Stoyanov/ Alamy Stock Photo; page 85, Alamy Stock Photos; page 94, © Henk Wallays/Alamy Stock Photo Cover image: Alex Potemkin/E+ via Getty Images (Image 1303779896) Developed at East Lansing, MI 48824 connectedmath.msu.edu Published by 17 Colt Court Ronkonkoma, NY 11779 lab-aids.com SAMPLE
v ABOUT THE AUTHORS Elizabeth Difanis Phillips, a former high school teacher, is a Senior Academic Specialist in the Program in Mathematics Education (PRIME) at Michigan State University. She is interested in the teaching and learning of mathematics, with a special interest in the teaching and learning of algebra across the grades. She is the author of numerous articles and book chapters and a speaker at national and international conferences. Recently, in recognition of her scholarly work, she received an Honorary Doctor of Science degree in 2022 from Michigan State University, the 2023 National Council of Teachers of Mathematics (NCTM) Lifetime Achievement Award, and the 2023 Ross Taylor/Glenn Gilbert National Leadership Award from the National Council of Supervisors of Mathematics (NCSM). Currently, she is the principal investigator for several National Science Foundation grants that are developing a collaborative digital platform for students and teachers of Connected Mathematics® 4. Glenda Lappan is a University Distinguished Professor Emeritus in the Program in Mathematics Education (PRIME) at Michigan State University. Her research and development interests are in the connected areas of students’ learning, mathematics teachers’ professional growth, and change related to the development and enactment of K–12 curriculum materials. She served as president of the National Council of Teachers of Mathematics (NCTM) from 1998 to 2000 and played a major role in NCTM’s 1989 Standards for Curriculum and Evaluation Standards for School Mathematics, and 2000 Principals and Standards for School Mathematics. In addition to authoring numerous articles and book chapters and speaking at national and international conferences, she has won numerous awards, including the 2004 NCTM Lifetime Achievement Award and the 2007 Ross Taylor/Glenn Gilbert National Leadership Award from NCSM. James T. Fey is a Professor Emeritus at the University of Maryland. His consistent professional interests have been development and research focused on curriculum materials that engage middle and high school students in problem-based collaborative investigations of mathematical ideas and their applications. He won the 2005 NCTM Lifetime Achievement Award. Susan N. Friel is a Professor Emeritus of Mathematics Education in the School of Education at the University of North Carolina at Chapel Hill. Her research interests focus on statistics education for middle-grade students and, more broadly, on teachers’ professional development and growth in teaching mathematics K–8. Yvonne Slanger-Grant is an Academic Specialist in the Program in Mathematics Education (PRIME) at Michigan State University. Her professional interests focus on SAMPLE
vi About the Authors helping teachers develop understanding and agency in the teaching of mathematics. She has held various roles in education, including middle school mathematics teacher, elementary teacher, instructional coach, professional development consultant, and mentor to teachers, school leaders, and graduate assistants. Many of these responsibilities have involved the Connected Mathematics Project since its beginning in 1991. Alden J. Edson is a Research Assistant Professor in the Program in Mathematics Education (PRIME) at Michigan State University. His research and development interests center on improving the teaching and learning of mathematics through innovations in curriculum and technology. Specifically, his research studies the enactment of problem-based, inquiry-oriented mathematics curriculum by students and their teacher in a digital world. He also studies the affordances of innovative mathematics curriculum materials as a context for teacher learning. Since joining the Connected Mathematics Project in 2014, AJ has been writing and carrying out research and development grants. He also teaches mathematics education courses, advises doctoral students, and facilitates professional learning with teachers of mathematics. With . . . Kathy Dole and Jacqueline Stewart. Kathy is a recently retired middle school teacher of mathematics at Portland Middle School in Portland, Michigan. Jacqueline is a retired high school teacher of mathematics at Okemos High School, Okemos, Michigan. Both Kathy and Jacqueline have worked on a variety of activities related to the development, professional learning, and implementation of the Connected Mathematics® curriculum since its beginning in 1991. In memory of . . . William Fitzgerald (Deceased) Bill through his making “good trouble” made substantial contributions to conceptualizing and creating Connected Mathematics®. SAMPLE
vii ACKNOWLEDGMENTS CONNECTED MATHEMATICS ® 4 DEVELOPMENT TEAM Elizabeth Difanis Phillips, Senior Academic Specialist, Michigan State University Glenda Lappan, University Distinguished Professor Emeritus, Michigan State University James T. Fey, Professor Emeritus, University of Maryland Susan N. Friel, Professor Emeritus, University of North Carolina at Chapel Hill Yvonne Slanger-Grant, Academic Outreach Specialist, Michigan State University Alden J. Edson, Research Assistant Professor, Michigan State University With . . . Kathy Dole, Middle School Mathematics Teacher (Retired), Portland Middle School, Portland, MI Jacqueline Stewart, High School Mathematics Teacher (Retired), Okemos Public Schools, Okemos, MI In Memory of . . . William M. Fitzgerald, Professor (Deceased), Michigan State University CONNECTED MATHEMATICS PROJECT STAFF Elizabeth Difanis Phillips, Senior Academic Specialist, Michigan State University, East Lansing, MI Alden J. Edson, Research Assistant Professor, Michigan State University, East Lansing, MI Taren Going, Postdoctoral Research Associate, Michigan State University, East Lansing, MI Elizabeth Lozen, Consortium Coordinator, Michigan State University, East Lansing, MI Sunyoung Park, Postdoctoral Research Associate, Michigan State University, East Lansing, MI Yvonne Slanger-Grant, Academic Outreach Specialist, Michigan State University, East Lansing, MI Chris Waston, Academic Outreach Specialist, Michigan State University, East Lansing, MI SAMPLE
viii Acknowledgments ASSESSMENT TEAM Mary Bouck, Mathematics Education Consultant, Michigan State University, East Lansing, MI Valerie Mills, Mathematics Education Consultant, Ypsilanti, MI WEBSITE AND TECHNOLOGY CONSULTANTS Tyler Knowles, Technology Lead, San Antonio, TX Amie Lucas, Information Technologist, Lansing, MI Emma Craig, Graphic Art/Editing, Detroit, MI CURRICULUM DEVELOPMENT CONSULTANTS Melanie Del Grosso, Mathematics Teacher Consultant, Phoenix, AZ Teri Keusch, Mathematics Teacher Consultant, Lansing, MI Jennifer Kruger, Teacher Guide Consultant, Rochester, NY PROGRAM IN MATHEMATICS EDUCATION GRADUATE STUDENTS (2019–PRESENT) Kate Appenzelle Knowles David Bowers Eli Claffey Ashley Fabry Chuck Fessler Nic Gilbertson Funda Gonulates Ahmad Kohar Merve Kursav Kevin Lawrence Rileigh Luczak Jen Nimtz Michael Quail Molade Osibodu Amy Ray Sasha Rudow Visala Rani Satyam Amit Sharma Brady Tyburski Samantha Wald SAMPLE
Acknowledgments ix CONNECTED MATHEMATICS ® 4 UNDERGRADUATE ASSISTANTS Tyler Boyd Jacob Disbro Shayna Evans Autumn Eyre Cora Haddad Emma Herrera Maya Herrera Sarah Ingemunson Shannon McHugh Maggie Ozias Matthew Phillips Josh Pullen PUBLISHING TEAM Director of Mathematics Publications for Lab-Aids: Denise A. Botelho Project Coordination, Production, Cover and Interior Designs, and Composition: Six Red Marbles Illustrations, Graphics, and Art: Six Red Marbles CONNECTED MATHEMATICS ® 4 REVIEWERS Illinois Jenesis Byrne, Jennifer Leimberer, and Farah Mahimwalla, UIC/MCMI, Chicago Carolyn Droll, Community Consolidated School District 21, Wheeling Robert Reynolds, Peterson Elementary School, Chicago Public Schools, Chicago Maine Sally Bennett, Chris Driscoll, Joyce Hebert, Sara Jones, and Shawn Towle, Falmouth Middle School Michigan Michelle Bortnick, Hillel Day School, Dearborn Gerri Devine, Oakland County Schools, Oakland County Anne Marie Nicoll-Turner, Ann Arbor Public Schools, Ann Arbor Meredith Pelchat, Clague Middle School, Ann Arbor Brian Powell, Ionia Public Schools, Ionia Mary Beth Schmitt, Traverse City West Junior High, Traverse City Dr. Jamie Wernet, Lansing Christian School, Lansing SAMPLE
x Acknowledgments New York Michaela Marino, East Lower School, Rochester Chi-Man Ng, I.S. 289: Hudson River Middle School, New York City Jennifer Perillo, Brighton Central School District, Rochester Ohio Jim Mamer, Springfield FIELD TESTERS Alabama Rashad Bell, McIntosh High School, McIntosh Arizona Melanie Del Grosso, St. John Bosco School, Phoenix California Estasia Barrientosi and Esther Centers, Santa Cruz Waldorf School, Santa Cruz Anthony Bayro and Traci Jackson, Poway Unified Schools, San Diego Illinois Bryan Becker, Melissa Denton, Erika Inka, Meagan Stass MacDonald, and Josephine Mazzola, Barrington Community Unit 220 School District, Barrington Nancy Kay Berkas, Patrick Black, Laura Bubel, Alexander Laube, Kristy Lutton, Breanna McCann, and Amy Rendino, The Cove School, Northbrook Michael Bryant, Carolyn Droll, Maureen Gannon, Kristen Hale, Debbie Rein, Summer Riordan, and Christopher Schieffer, Community Consolidated School District 21, Wheeling Jenesis Byrne, Jennifer Mundt Leimberer, Farah Mahimwalla, Kathleen Pitvorec, and Margie Pligge, University of Illinois at Chicago, Chicago Mary McKenna Corrigan, Brian Lacey, Fannie Lawson-Rondo, Logan Hammerberg, Héctor Orlando Martinez M., Michael R. Martini, Carla Sever, Gail Smith, Melissa Talaber, and Candice M. Usauskas, Archdiocese of Chicago, Chicago Christine Czarnecki, Samia Haan, and Kristy Regan, Alsip District 126, Alsip Catherine Ditto, Jorge Prieto Math and Science Academy, Chicago Public Schools Aaron Mesh, Chicago Waldorf School, City of Chicago School District 299 Morgan Miller and Margaret Nugent, Oak Lawn-Hometown Middle School, Oak Lawn SAMPLE
Acknowledgments xi Robert Reynolds, Mary Gage Peterson Elementary School, Chicago Public Schools Courtney Southward, Percy Julian Middle School, Oak Park Indiana Andrea Leahy, St. Thomas More School, Munster Maine Sally Bennett, Christopher Driscoll, Joyce Hebert, Sara Jones, Craig Shain, and Shawn Towle, Falmouth Middle School, Falmouth Michael H. Hagerty, Jay Harrington, Sr., and Kellie McMahon, Frank H. Harrison Middle School, Yarmouth Massachusetts Andrea Hurley, Hanover Middle School, Hanover Michigan Anna Assaf, Sue Chipman, Jennifer Johnson, and Brian Powell, Saranac Community Schools, Saranac Michelle Bortnick and Abbe Luther, Hillel Day School, Farmington Hills Tom Brighton, Jordan Brown, Amber Proctor, Stacey Schrauben, Kim Schumacher, and Lisa Wandell, Ionia Middle School, Ionia Jenny Douglas and Lisa Roderique, Forest Hills Public Schools, Grand Rapids Sheri Gunns, Marian Murembya, and Heidi Nussdorfer, Okemos Public Schools, Okemos Jesica Eby, Katherine Oberdorf, and Jill Wilson, Monroe Public Schools, Monroe Courtney Henige and Melissa Jacobs, New Lothrop Area Public Schools, New Lothrop Amy Hurley and Jennifer LaCross, Walled Lake Consolidated School District, Walled Lake Mary Lovejoy and Sara Melnik, Holt Junior High, Holt Anne Marie Nicoll-Turner and Meredith Pelchat, Ann Arbor Public Schools, Ann Arbor Mary Beth Schmitt, Traverse City West Middle School, Traverse City Rachel Stelman, Oakview Middle School, Lake Orion Dr. Jamie Wernet, Lansing Christian School, Lansing New Jersey Danielle Dorn, Upper Saddle River School District, Upper Saddle River SAMPLE
xii Acknowledgments New York Karri Ankrom, Scott Dobbs, and Danielle Levy, Village Community School, New York Catherine Klein, West Hempstead Secondary School, West Hempstead Shannon Johnson, Christopher Longwell, Michaela Marino, Sarah Meade, Beth Merritt Jennifer Rees, Liana Spencer, and Tom Street, Rochester City School District, Rochester Mike McNall, Holy Family School, Diocese of Syracuse John Ottomanelli, P.S./I.S. 30, New York Department of Education, New York City Dawn Schafer, I.S. 276 Battery Park City School, Manhattan Jennifer Perillo, Brighton Central School District, Rochester North Carolina Karen Abraham, Darla Jones, and Jessica Wallace, C.C. Griffin STEM Middle School, Cabarrus County Schools, Concord Teresa Fulk, Nancy Lynn Green, and Kim Mann, Canterbury School, Greensboro Ohio Rachael Brookshire, Jamie Chaney, Lynnette Flannery, Megan Hammond, Rebecca Hoffman, Kimberly Kraft, Gaby Tagliamonte, and Kristin Whitt, Hamilton City School District, Hamilton Colin Dietrich, Alexandria Ferguson, Heather S. Gust, and Jennifer Linn, Sylvania City Schools, Sylvania Blake Garberich, Clark-Shawnee Local Schools, Springfield Karin Lauterbach, Leslie Liebig, Braden Short, and Ilona Webel, Oakwood City Schools, Oakwood Pennsylvania Nykeesha Brown, Chester Community Charter School, Aston Danielle Crossey, Mt. Lebanon School District, Pittsburgh Vermont Gerald Bailey, Bellows Free Academy, Franklin West Supervisory Union, Fairfax Heather Estey, Bristol Elementary School, Mt. Abraham Unified School District, Bristol Lorrene Palermo, Fletcher Elementary School, Franklin West Supervisory Union, Fletcher SAMPLE
Acknowledgments xiii Virginia Jessica Khawaja, Sabot School, Richmond Tim Malloy and Katrien Vance, North Branch School, Afton Wisconsin Brenda Carlborg, Leah Enwright, and Bianca Gloria, Kenosha Unified School District, Kenosha INTERNATIONAL FIELD TESTERS Brazil Jodie Greve, Pan American Christian Academy, São Paulo Colombia Sandra Moreno Cárdenas, Colegio Los Nogales The Netherlands Jill Broderick, Nadine Galante, Lea anne Windham, The American School of The Hague in Wassenaar United Kingdom Jill Broderick, Laura Brown, Stephanie McBride-Bergantine, and Katherine Muir, American School in London, England Vietnam Lia Garcia Halpin and Jennifer Zimbrick, Concordia International School Hanoi SAMPLE
SAMPLE
xv Mathematical Goals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xvii Looking Ahead . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xviii Investigation 1. Exploring Data Patterns in Building Bridges . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 Problem 1.1 Bridge Thickness and Strength Experiment: Linear or Nonlinear? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Problem 1.2 Bridge Length and Strength Experiment: Linear or Nonlinear? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 Problem 1.3 Custom Construction Parts: More Patterns . . . . . . . . . . . . . . 9 Mathematical Reflection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 Applications—Connections—Extensions (ACE) . . . . . . . . . . . . . . . . . . . . . . 14 Investigation 2. Linear Models and Equations . . . . . . . . . . . . . . . . . .22 Problem 2.1 Treetop Fun: Equations for Linear Functions . . . . . . . . . . . .24 Problem 2.2 Boat Rental: Finding Solutions Using Tables, Graphs, and Symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 Problem 2.3 Up and Down the Staircase Again: Exploring Slope . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 Problem 2.4 Exploring Patterns with Lines . . . . . . . . . . . . . . . . . . . . . . . . . 36 Mathematical Reflection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 Applications—Connections—Extensions (ACE) . . . . . . . . . . . . . . . . . . . . . . 39 Investigation 3. Inverse Variation: Linear or Nonlinear? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .50 Problem 3.1 Rectangles with Fixed Area . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 Problem 3.2 Distance, Speed, and Time . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 CONTENTS SAMPLE
Problem 3.3 Planning a Fundraising Event: Finding Individual Costs. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 Mathematical Reflection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 Applications—Connections—Extensions (ACE) . . . . . . . . . . . . . . . . . . . . . .60 Investigation 4. Variability in Numerical and Categorical Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .66 Problem 4.1 Lines of Best Fit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67 Problem 4.2 Wood or Steel? Relationships in Categorical Data . . . . . 73 Problem 4.3 School Team-Building Activity: Setting Up a Two-Way Table . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77 Problem 4.4 Linear, Bivariate, and Categorical Data . . . . . . . . . . . . . . . 79 Mathematical Reflection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .83 Applications—Connections—Extensions (ACE) . . . . . . . . . . . . . . . . . . . . . .84 English/Spanish Glossary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .99 Index. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .107 xvi Contents SAMPLE
xvii Linear and Nonlinear Functions. Understand linear functions. • Represent data patterns using mathematical models such as graphs, tables, word descriptions, equations, and algebraic expressions • Investigate the nature of linear functions in contexts • Use mathematical models to answer questions about linear relationships • Recognize and write equations for linear functions represented by verbal, numerical, or graphical information • Write, analyze, and solve linear equations using tables, graphs, or symbolic methods • Investigate the nature of inverse variation, a nonlinear relationship • Compare linear relationships to nonlinear situations such as inverse variation relationships Bivariate Data. Understand bivariate data. • Recognize and model patterns in bivariate data • Use data to make predictions • Fit a line to data that show a linear trend and measure closeness of fit • Analyze scatterplots of bivariate data to determine the strength of the linear association between the two variables • Distinguish between categorical and numerical variables • Use two-way tables and analysis of cell frequencies and relative frequencies to decide whether two variables are related MATHEMATICAL GOALS SAMPLE
xviii LOOKING AHEAD How is the thickness of a steel beam or bridge related to its strength? How is the length of a beam or bridge related to its strength? Design an experiment to answer these two questions. The equation c = 0.40t + 10.50 gives the charge c in dollars for renting a canoe for t minutes. How much time can you buy to rent a canoe if you have $12.00? The scatterplot shows the number of roller coaster riders and their ages on a given day. The pink dots represent wood-frame roller coasters. The green dots represent steel-frame coasters. Describe the association between the variables in the data. Write a claim about the data. Explain why your claim is true. How might your claim be useful in making predictions about the situation? Age of Riders 10 20 30 40 50 60 70 80 90 0 10 20 30 40 50 60 70 80 Number of Riders Riders and Their Ages x y SAMPLE
Looking Ahead xix People in many professions make important predictions every day. Sometimes millions of dollars and often people’s lives depend on the accuracy of these predictions. › A company may develop a wonderful new product, but without accurate predictions about the number and location of potential customers and the price they would be willing to pay, the company could lose a great deal of money. › When designing a bridge, a civil engineer must make accurate predictions about how much weight the bridge can hold and about how it will stand up to strong winds and earthquakes. › Stockbrokers use data to forecast growth of investments over time. › Epidemiologists use data to predict the spread of viruses. › Weather forecasters use data to predict the path of a hurricane. People construct mathematical models by gathering data about a situation and then finding a graph or equation that fits the data. They use the mathematical models to make predictions. In prior Connected Mathematics® units, we explored relationships between two variables. We identified relationships from tables and graphs; sometimes we wrote equations to represent the data. The representations (tables, graphs, and equations) we created are also called mathematical models. SAMPLE
xx Thinking with Mathematical Models In this unit, we will solve problems, conduct experiments, analyze data, and write equations that summarize, or model, the data patterns. Then we will use the models to make predictions. The skills we develop in this unit will help us ask questions like the ones in the Looking Ahead. When encountering a new problem, ask yourself the following questions: What are the key variables in the situation? What is the pattern that describes the relationship between the variables? If there is a pattern, is it strong enough to allow me to make predictions? What kind of model (table, graph, or equation) will express the relationship? How can I use the model to make predictions or answer questions about the relationship? xx Looking Ahead SAMPLE
1 People in many professions use data and mathematical reasoning to solve problems and make decisions. For example, engineers analyze data from lab tests to determine the amount of weight a beam can support related to its thickness, length, and design. In this investigation, we will conduct experiments and collect data to make predictions about bridge strength. As we analyze the data, we will be interested to know if the relationship is linear or nonlinear. In the Moving Straight Ahead unit, we explored linear relationships and found that they can be represented by a straight line. We also know that as the independent variable changes by a constant amount, the dependent variable changes by a constant amount for linear relationships. These linear relationships are also called linear functions. The Business Club at Roseville Middle School decides to investigate various businesses in the area. They are interested in how a business is developed, marketed, and distributed. One of the local businesses builds supports for bridges. During a visit to the company, the Business Club was told that to build supports for bridges, the company needs to consider various features: › How long is the bridge? › How high is the bridge? › How much traffic is on the bridge in a day? › What is the weight of the vehicles? › What are the weather conditions in the area? Exploring Data Patterns in Building Bridges INVESTIGATION 1 SAMPLE
2 Thinking with Mathematical Models PROBLEM 1.1 Bridge Thickness and Strength Experiment: Linear or Nonlinear? Many bridges are built with frames of steel beams. Steel is very strong, but any beam will bend or break if you put too much weight on it. To answer the question What variables determine the strength of a bridge?, engineers often use scale models to test their designs. INITIAL CHALLENGE The Business Club decides to do an experiment to discover mathematical patterns needed to build strong supports for bridges. Make a Prediction • How do you think the strength of a beam is related to its thickness? Conduct the Experiment Equipment › several 11 inch-by-4 __1 4 inch strips of paper › a small paper cup (approximately 3 ounces) › about 50 pennies › two books of the same thickness SAMPLE
Investigation 1 Exploring Data Patterns in Building Bridges 3 Analyze the Data • What patterns do you observe in the data? Do they support your original prediction? Explain. • Does the relationship between the number of layers and the breaking weight appear to be linear or nonlinear? How do the graph and the table show this relationship? WHAT IF . . . ? Situation A. Carlos’s Prediction Carlos wanted to know if the predictions the Business Club made in the Initial Challenge would work if you split the layers of paper. What do you think? 1. Suppose you could split layers of paper in half. What breaking weight would you predict for a bridge 2.5 layers thick? Explain. Directions › Start with one of the paper strips. Make a “bridge” by folding up 1 inch on each long side. › Suspend the bridge between two books. Place a paper cup in the center of the bridge. › Put pennies in the cup, one at a time, until the bridge collapses. Record the number of pennies added to the cup. This number is the breaking weight of the bridge. › Repeat the experiment with two, three, four, and five strips of new paper. Record your data. › Represent the data in a table and graph. 1 in. 2––– in. 1 4 1 in. 1.1 SAMPLE
4 Thinking with Mathematical Models 2. Predict the breaking weight for a bridge 6 layers thick. Explain your reasoning. 3. Test your prediction of strength for the 6-layer bridge. Explain why results from such a test might not exactly match predictions. Situation B. Annette Analyzes Graph Models Annette notices that the points do not lie exactly on a line. She thought a line might approximate the data pattern. The Business Club could use the line to make predictions. However, they could not agree which line was the best fit. 1. Which line best fits the data gathered by the Business Club? Explain why. 2. Using each line, what is the breaking weight for a bridge 6 layers thick? Is the answer the same for all four lines? Explain. Luke’s Line Number of Layers 10 20 30 40 50 60 70 0 1 2 3 4 5 6 7 y x Breaking Weight (pennies) Maria’s Line Number of Layers 10 20 30 40 50 60 70 0 1 2 3 4 5 6 7 y x Breaking Weight (pennies) Billie’s Line Number of Layers 10 20 30 40 50 60 70 0 1 2 3 4 5 6 7 y x Breaking Weight (pennies) Chris’s Line Number of Layers 10 20 30 40 50 60 70 0 1 2 3 4 5 6 7 y x Breaking Weight (pennies) NOW WHAT DO YOU KNOW? Describe the relationship between the bridge strength and the bridge thickness revealed by your experiment. 1.1 SAMPLE
Investigation 1 Exploring Data Patterns in Building Bridges 5 1.2 Bridge Length and Strength PROBLEM Experiment: Linear or Nonlinear? In the last problem, we gathered data by testing the strength of some paper bridges. We represented the relationship in a coordinate graph model. We found that as the number of layers increased, so did the strength. That is, bridges with more layers are stronger than bridges with fewer layers. We represented the data in a graph to show the relationship between the two variables. We used the data to make predictions. In the following experiment, we continue to investigate data about bridges. INITIAL CHALLENGE There are other variables involved in building bridges, such as the length of a bridge. The Business Club decides to investigate how the length and strength of a bridge are related. Make a Prediction • How do you think the length and strength of a bridge are related? Conduct the Experiment Equipment › 4__1 4inch-wide paper strips with lengths 4 inches, 6 inches, 8 inches, 9 inches, and 11 inches › a small paper cup (approximately 3 ounces) › about 50 pennies › two books of the same thickness SAMPLE
6 Thinking with Mathematical Models WHAT IF . . . ? Situation A. Sunny’s and Matt’s Claims Sunny and Matt remember doing different experiments in previous years. They said, “We were using models in other units.” They find examples of using models in both grades 6 and 7. 1. Can their models be used to make predictions? Explain. 2. Does the graph or the table represent a linear relationship? Explain why. 1.2 Analyze the Data • What patterns do you observe in the data? Do they support your original prediction? Explain. • Does the relationship between the bridge length and the breaking weight appear to be linear or nonlinear? How do the graph and the table show this relationship? • Use your data to predict the breaking weights for bridges of lengths 3 inches, 5 inches, 10 inches, and 12 inches. Explain how you made your predictions. Directions › Fold the paper strips to make bridges as shown below. 11 in. 9 in. 8 in. 6 in. 4 in. › Start with the 4-inch bridge. Suspend the bridge between the two books as you did in Problem 1.1. The bridge should overlap each book by 1 inch. › Put pennies into the cup, one at a time, until the bridge collapses. Record the number of pennies you added to the cup. As in the first experiment, this number is the breaking weight of the bridge. › Repeat the experiment to find breaking weights for the other bridges. › Represent the data in a table and graph. SAMPLE
Sunny’s Jumping Jacks Example In sixth grade, we did a jumping jacks experiment. I had a consistent pace for the first 20 seconds. I slowed down, increased, slowed down. In the last 30 seconds, I wanted to get more done, so my pace increased a lot. Here is my graph. Time (seconds) 10 20 30 40 50 60 70 80 90 100 110 120 0 10 20 30 40 50 60 70 80 90 y x Total Number of Jumping Jacks Sunny’s Jumping Jack Data 100 110 120 Matt’s Dripping Water Example In seventh grade, we did an experiment to look at how much water is lost if you have a leaky faucet. We kept track of how much water dripped out of a cup. Here is my data. Matt’s Dripping Water Data Time (seconds) Water (millileters) 5 4 10 10 15 13 20 18 25 21 30 24 35 27 40 33 45 36 50 40 55 45 60 48 NOW WHAT DO YOU KNOW? How does the relationship between the number of layers in a bridge and its breaking weight compare to the relationship between bridge length and breaking weight? How does a graph or table model the relationship between the variables? 1.2 Investigation 1 Exploring Data Patterns in Building Bridges 7 SAMPLE
8 Thinking with Mathematical Models Did You Know? According to the U.S. National Science Foundation funded researchers at New York University, one way scientists learn more about aerodynamics and flight stability is from the properties that make a paper airplane fly. Inspiration for the study came from the scientists’ curiosity about the design of a paper airplane that makes for smooth gliding. They conducted a series of experiments using paper planes to make their conclusions. Their research could influence the development of airborne vehicles like drones. 1.2 SAMPLE
Investigation 1 Exploring Data Patterns in Building Bridges 9 1.3 Custom Construction Parts: PROBLEM More Patterns A common structure used in building is called a truss. A truss is a formation of materials to create a rigid structure. We might see a truss holding up a bridge, the roof of a building, or some other structure. Spartan Stadium Jumbotron Video Board Installation, Michigan State University INITIAL CHALLENGE One month, the Business Club invited the manager of Custom Steel Products (CSP) to talk to the club. The company supplies materials to builders. The manager presented a challenge to the club members. The following is a picture of a 7-foot truss. It is made by fastening together steel rods that are 1 foot long. 1-foot steel rod 7-foot truss made from 27 rods This truss has an overall bottom length of 7 feet (ft.). The manager at CSP needs to know the number of rods for any bottom length of truss a customer might order. • How many rods are in a 12-foot truss? • If 51 rods are used, how long is the truss? • Describe a pattern that the manager can use to predict the number of rods for any length of truss. • Does the pattern represent a linear relationship? Explain. SAMPLE
10 Thinking with Mathematical Models Maria Uses a Table I worked out some cases for truss lengths. I recorded them in a table. I noticed that as the length increased by 1 foot, the number of rods increased by 4. So the number of rods is increasing at a rate of change of 4 rods for every 1 additional foot of length. I made predictions from the table. Maria’s Truss Data Truss Length (ft.) 2 3 4 5 6 7 8 Number of Rods 7 11 27 1.3 Walter Uses Triangles In the picture, I see triangles with connecting rods. Each triangle except the first one has a connecting rod on top. For the length of 7 feet, I see the first triangle and 6 triangles with a connecting rod. So the number of rods needed is 3 + 6 • 4. I can keep using the pattern to make predictions. 10 Thinking with Mathematical Models WHAT IF . . . ? Situation A. Student Strategies Students in the Business Club claim that the relationship between the truss length and number of rods is a linear relationship. They use different strategies to find a model they could use to make predictions about the length and number of rods needed to make a truss. 1. Can each strategy be used to make predictions? Explain. 2. How are the strategies the same? Different? Sangha Uses a Graph I made a graph of the data for some of the trusses. The data points appear to lie on a straight line. I don’t think it goes through the origin. I can sketch a line that represents the data. I can use the line to make predictions. Truss Length (ft.) Number of Rods 8 16 24 32 20 CSP Trusses 4 6 8 y x Jenna Uses a Picture I see triangles and connecting rods. There is 1 triangle (or 3 rods) for each foot of truss. For the connecting rods, there is one less rod than the number of feet. So for 7 feet, the number of rods needed is 7 × 3 + 6 = 27. I can use this pattern to make predictions. SAMPLE
NOW WHAT DO YOU KNOW? How does having a table or a graph for a linear pattern help you make predictions? Investigation 1 Exploring Data Patterns in Building Bridges 11 1.3 Gia Looks for a Starting Point I used Maria’s table. I counted backward by 4 to find that the number of rods for a truss with a length of 1 foot. It is 3 rods. The number of rods for a truss with a length 0 is −1 rod. This does not make sense. But if I start with −1 and add keep adding 4, I can make predictions. Gia’s Truss Data Truss Length (ft.) 0 1 2 3 4 5 6 7 8 Number of Rods −1 3 7 11 27 −1−1−1 −4 −4 −4 Situation B. Another Structure: Staircase Frames Custom Steel Products also makes staircase frames like those shown here. 1 step made from 4 rods 2 steps made from 10 rods 3 steps made from 18 rods 1. Describe the pattern of change in the number of rods as the number of steps increases. Explain how you found the pattern. 2. How would the pattern you described be shown in a table? Graph? 3. Does this pattern represent a linear relationship? Explain. 4. How many steel rods are in a staircase frame with 12 steps? Explain how you could find this number without drawing the staircase frame. SAMPLE
12 Thinking with Mathematical Models 1.3 Did You Know? When designing a bridge, engineers need to consider the load, or the amount of weight, the bridge must support. The dead load is the weight of the bridge and fixed objects on the bridge. The live load is the weight of moving objects on the bridge. On many city bridges in Europe— such as the famous Ponte Vecchio in Florence, Italy—dead load is very high because tollbooths, apartments, and shops are built right onto the bridge surface. Local ordinances can limit the amount of automobile and rail traffic on a bridge to help control live load. SAMPLE
Investigation 1 Exploring Data Patterns in Building Bridges 13 In this unit, we are exploring linear and nonlinear relationships or functions and how they are used to model real-life phenomena and solve problems. At the end of this investigation, ask yourself: What are the characteristics of linear functions? How might you use what you know about linear functions to model a situation? Mathematical Reflection MR SAMPLE
14 Thinking with Mathematical Models INVESTIGATION 1 APPLICATIONS 1. A group of students conducted the bridge thickness experiment with construction paper. The following table contains their results. APPLICATIONS—CONNECTIONS—EXTENSIONS (ACE) a. Make a graph of the (number of layers, breaking weight) data. Describe the relationship between breaking weight and number of layers. b. Suppose it is possible to use half-layers of construction paper. What breaking weight would you predict for a bridge 3.5 layers thick? Explain. c. Predict the breaking weight for a construction-paper bridge of 8 layers. Explain how you made your prediction. 2. The graph shows the amount of money Jake earned while babysitting for several hours. a. Put scales on the axes that make sense. Explain why you chose your scales. b. What would the equation of the graph be, based on the scale you chose in ACE 2a? c. If the line on this graph were steeper, what would it tell about the money Jake is making? Write an equation for such a line. 3. A group of students conducted the bridge length experiment with construction paper. The table below contains their results. Hours Jake’s Babysitting Money Jake Made x y Bridge Length Experiment Length of Bridge (in.) 4 6 8 9 11 Breaking Weight (pennies) 25 15 12 10 7 Bridge Thickness Experiment Number of Layers 1 2 3 4 5 6 Breaking Weight (pennies) 12 20 29 42 52 61 SAMPLE
Investigation 1 Exploring Data Patterns in Building Bridges 15 a. Make a graph of the (length, breaking weight) data. Describe the relationship between breaking weight and length. b. What breaking weight would you predict for a bridge with a length of 5 inches? Explain. c. Predict the breaking weight for a construction-paper 12-inch bridge. Explain how you made your prediction. 4. The table shows the maximum weight a crane arm can lift at various distances from its cab. Construction Crane Data Distance from Cab to Weight (ft.) 12 24 36 48 60 Weight (lb.) 7,500 3,750 2,500 1,875 1,500 a. Describe the relationship between distance and weight for the crane. b. Make a graph of the (distance, weight) data. Explain how the graph’s shape shows the relationship you described in ACE 4a. c. Estimate the weight the crane can lift at distances of 18 feet, 30 feet, and 72 feet from the cab. d. How, if at all, are the data for the crane similar to the data from the bridge experiments in Problems 1.1 and 1.2? 5. A truss or staircase frame from Custom Steel Products costs $2.25 for each rod, plus $50 for shipping and handling. a. Refer to your data from the Initial Challenge of Problem 1.3. Copy and complete the table below to show the costs of trusses of different lengths. Cost of CSP Trusses Truss Length (ft.) 1 2 3 4 5 6 7 8 Number of Rods Cost of Truss ($) b. Make a graph of the (truss length, cost) data. c. Describe the relationship between truss length and cost. ACE SAMPLE
16 Thinking with Mathematical Models 6. A truss or staircase frame from Custom Steel Products costs $2.25 for each rod, plus $50.00 for shipping and handling. a. Refer to your data from What If . . . ? Situation B of Problem 1.3. Copy and complete the table below to show the costs of staircase frames with different numbers of steps. Cost of CSP Staircase Frames Number of Steps 1 2 3 4 5 6 7 8 Number of Rods Cost of Frame ($) b. Make a graph of the (number of steps, cost) data. c. Describe the relationship between number of steps and cost. 7. ACE 7a–f refer to relationships between variables you have studied in this investigation. Tell whether each is linear or nonlinear. a. Cost depends on truss length (ACE 5). b. Cost depends on the number of rods in a staircase frame (ACE 6). c. Bridge strength depends on bridge thickness (Problem 1.1). d. Bridge strength depends on bridge length (Problem 1.2). e. Number of rods depends on truss length (Problem 1.3). f. Number of rods depends on the number of steps in a staircase frame (Problem 1.3). g. Compare the patterns of change for all the nonlinear relationships in ACE 7a–f. 8. a. Assume that this pattern continues beyond Figure 3. Write an equation that represents the number of squares S in figure n. Figure 1 Figure 2 Figure 3 b. Explain how you know your equation will work for any figure number. c. Write two different equations that represent the perimeter P for any given figure number n. ACE SAMPLE
Investigation 1 Exploring Data Patterns in Building Bridges 17 CONNECTIONS For ACE 9 and 10, tell whether the statement is true or false. Explain your reasoning. 9. 6(12 − 5) > 50 10. 10 − 5 · 4 > 0 11. The following rectangle has a perimeter of 70 feet. 20 ft. 15 ft. a. Make drawings of at least three other rectangles with a perimeter of 70 feet. b. What is the width of a rectangle with a perimeter of 70 feet if its length is 1 foot? 2 feet? x feet? c. What is the width of a rectangle with a perimeter of 70 feet if its length is _1 2 foot? _ 3 2 feet? d. Give the dimensions of rectangles with a perimeter of 70 feet and length-to-width ratios of 3 to 4, 4 to 5, and 1 to 1. e. Suppose the length of a rectangle increases but the perimeter remains 70 feet. How does the width change? 12. The following rectangle has an area of 300 square feet. 20 ft. 15 ft. a. Make drawings of at least three other rectangles with an area of 300 square feet. b. What is the width of a rectangle with an area of 300 square feet if its length is 1 foot? 2 feet? 3 feet? c. What is the width of a rectangle with an area of 300 square feet and a length of l feet? d. How does the width of a rectangle change if the length increases but the area remains 300 square feet? e. Make a graph of (width, length) pairs for rectangles with an area of 300 square feet. Explain how your graph illustrates your answer for ACE 12e. ACE SAMPLE
18 Thinking with Mathematical Models In ACE 13 and 14, each pouch holds the same number of coins. The coins all have the same value. Find the number of coins in each pouch. Explain your method. 13. = 14. = 15. Refer to ACE 13 and 14. a. For each ACE, write an equation to represent the situation. Let x represent the number of coins in a pouch. b. Solve each equation. Explain the steps in your solutions. c. Compare your strategies with those you used in ACE 13 and 14. 16. In each pair of equations, solve the first, and graph the second. a. 0 = 3x + 6 y = 3x + 6 b. 0 = x − 2 y = x − 2 c. 0 = 3x + 10 y = 3x + 10 d. In each pair, how is the solution related to the graph? For ACE 17–20, tell which graph matches the equation. –4 –2 0 2 4 2 4 –2 –4 Graph A y x –4 –2 0 2 4 2 4 –2 –4 Graph B y x ACE SAMPLE
Investigation 1 Exploring Data Patterns in Building Bridges 19 –4 –2 0 2 4 2 4 –2 –4 Graph C y x –4 –2 0 2 4 2 4 –2 –4 Graph D y x 17. y = 3x + 1 18. y = −2x + 2 19. y = x − 3 20. y = 1 + _1 2 x EXTENSIONS 21. The budget for the Grant Center for Outdoor Education assumes a linear relationship between the number of student visitors and the daily operating cost of the center. Some sample values (number of students, operating cost) are given in the table. ACE Sample Values for Number of Students and Operating Cost Number of Students 0 10 20 40 Daily Operating Costs ($) 450 600 750 1,050 a. For what number of student visitors will the daily operating cost be $690? Show how you found your answer. b. What will be the operating cost on a day with 12 student visitors? Show how you found your answer. c. Use the data in the table to write an equation showing how the operating cost, C, depends on the number of students, x. Explain or show how you found your equation. SAMPLE
20 Thinking with Mathematical Models 22. Study the patterns in this table. Note that the numbers in the x column may not be consecutive after x = 6. a. Use the patterns in the first several rows to find the missing values. x p q y z 1 1 1 2 1 2 4 8 4 _1 2 3 9 27 8 _1 3 4 16 64 16 __1 4 5 25 125 32 _1 5 6 1,024 2,048 1,728 n b. Are any of the patterns linear? Explain. 23. The table gives data for a group of middle school students. Some soccer players at Lab-Aids University decided to keep track of their individual practice during Spring Break. They kept track of their total time practicing (time), the number of times they tried to kick a long-distance goal (shots on goal), and how many goals they made (goals). Player Time (hours) Shots on Goal Goals Thomas Petes 11 126 23 Michelle Hughes 14 117 21 Shoshana White 13 112 17 Faith Locke 12 127 21 Tonya Stewart 12 172 32 Richard Mood 11 135 22 Tony Tung 8 130 20 Simone Robinson 10 134 21 Bobby King 9 156 29 Camila Flores 14 164 28 ACE SAMPLE
Investigation 1 Exploring Data Patterns in Building Bridges 21 a. Make graphs of (time, shots on goal) data, the (time, goals) data, and the (shots on goal, goals) data. b. Look at the graphs you made in ACE 23a. Which seem to show linear relationships? Explain. c. Estimate the average shots on goal to goal ratio. How many footlengths tall is the typical student in the table? d. Which student has the greatest shots on goal to goal ratio? Which student has the least shots on goal to goal ratio? 24. A staircase is a type of prism. This is easier to see if the staircase is viewed from a different perspective. In the prism shown here, each of the small squares on the top has an area of 1 square unit. Top Right Bottom a. Sketch the base of the prism. What is the area of the base? b. Rashid tries to draw a flat pattern that will fold up to form the staircase prism. The following is the start of his drawing. Finish Rashid’s drawing, and give the surface area of the entire staircase. Hint: You may want to draw your pattern on grid paper and then cut it out and fold it to check. Left Rear Suppose the prism has six stairs instead of three. Assume each stair is the same width as those in the prism above. Is the surface area of this six-stair prism twice that of the three-stair prism? Explain. ACE SAMPLE
22 In Investigation 1, we used tables and graphs to study relationships between variables. In Problem 1.1, the relationship between the strength of the bridge and the thickness of the bridge appeared to be linear. In Problem 1.3, the relationship between the number of rods needed to build a triangular truss and the length of the truss is a linear relationship. These relationships are also linear functions. A function assigns exactly one value of the dependent variable to each value of the independent variable. We use a rule or expression to define the relationship between the independent variable x and the dependent variable y. In Problem 1.3, the number of rods needed to build a triangular truss depends on the length of the truss. This is a linear function. For each length of a truss, there is exactly one number for the total number of rods needed. In the Moving Straight Ahead unit, we learned that in a linear relationship, as the independent variable changes by a constant amount, the dependent variable changes by a constant amount. For example, the relationship between the number of rods and the length of a truss is a linear function. As the length increased by 1, the number of rods increased by 4, a constant amount. Graphs of linear functions are straight lines. Linear Models and Equations INVESTIGATION 2 SAMPLE
› In the Moving Straight Ahead unit, we learned that linear relationships or linear functions can be represented by equations of the form y = mx + b. In these equations, the variable y depends on the variable x. › A coefficient is the number that multiplies a variable in an equation. In y = mx + b, m is the coefficient of x, so mx means m times x. The coefficient is the rate of change between the two variables in the relationship. It also tells the steepness and direction of the line. In some situations, when b = 0, the rate of change is also the constant of proportionality or unit rate. › The point at which a graph crosses the y-axis is called the y-intercept. › The coordinates of the y-intercept for the graph are (0, b). To save time, we sometimes refer to the number b as the y-intercept. We will continue to explore linear functions and ways they can be used to model data. If we decide that a linear function is a good model for a situation, then we can find an equation that fits our data. In any problem that calls for a linear model, the goal is to find the values of m and b for an equation y = mx + b whose graph fits the data pattern well. 0 y x y = mx + b (0, b) y-intercept Graph of a Linear Relationship Investigation 2 Linear Models and Equations 23 SAMPLE
24 Thinking with Mathematical Models Did You Know? A zip line is a cable that starts at a higher point than it ends. Using the natural decline of the cable, a person or cargo can travel down the cable on a pulley system that minimizes friction to help the rider accelerate. For example, for a 200 footlong zip line that is 7 feet high in the middle (when loaded with a rider), should sag 5 percent, or 10 feet. This means that the end attachments are 17 feet high each. At this point, the longer the zip line, the higher in the air you’ll need to be to start. You can travel internationally on zip lines. The Límite Zero zip line crosses the Guadiana River from Spain to Portugal. PROBLEM 2.1 Treetop Fun: Equations for Linear Functions Treetop Fun (TTF, for short) runs adventure sites with zip lines, swings, rope ladders, bridges, and trapezes. The company uses mathematical models to predict and manage how much income and profit it might make at its many locations. The total money collected from customers is called the income or revenue. But the company also has costs or expenses to set up and run the activities. These costs or expenses are paid from the income or revenue. What is left after the costs are paid is called the profit. SAMPLE
Investigation 2 Linear Models and Equations 25 INITIAL CHALLENGE The Business Club invites the Treetop Fun manager to talk to their group. They ask the manager to talk about the amount of money needed to set up a business, how much to charge customers, and how much profit the business might make. The manager explains that she uses equations to make decisions. She has the club members investigate the following questions. Daily Income or Revenue Amount collected The standard charge per customer at TTF is $25. What equation relates the daily income I to the number of n customers? Unpaid Balance Amount owed on an item TTF bought a new rope bridge for $4,500, which it will pay for in monthly payments of $350. What equation relates the unpaid balance B after m monthly payments? Daily Profit Amount left after paying costs/expenses The operating costs of TTF are $500 per day. What equation relates the daily profit P to the number n of customers? • For each question, find the equation that models the data. • Describe how the equation can be used to make predictions. WHAT IF . . . ? Situation A. Group Admission Fees The manager of Treetop Fun offered discounts for groups. She recorded the group admission fees in a table. 1. How much money would a group expect to pay if the group size is 0? 21? 30? 2. What equation relates admission fee A to the number n in the group? 3. Use the equation to check your predictions if the group size is 0, 21, or 30. 2.1 Group Admission Fees Number in Group 1 2 3 4 5 10 15 20 Admission ($) 75 90 105 120 135 210 285 360 SAMPLE
26 Thinking with Mathematical Models 2.1 Situation B. Back to Moving Straight Ahead: Matching Representations In the Moving Straight Ahead unit, we explored a middle school walkathon. They used the following data to make predictions. Match each data set with the context and graph representation. Data Set 1 y = 2x − 1 Data Set 3 y = 3 − 3x Data Set 2 y = 2 Data Set 4 y-intercept = 0 and rate of change is 1 Context 5 The distance Alana will walk at 1 meter for second. Context 7 Gilberto receives $2 for each meter he walks in the walkathon and has to pay $1 from the total for expenses. Context 6 The amount of money Leanne gets from her walkathon sponsors is $2 regardless of how far she walks. Context 8 Emile wonders if he walks 3 miles per hour, how far he is from his friend’s house that is 3 miles away. Graph 9 0 2 31 –1 –5 –4 –3 –2 –1 1 4 5 2 3 4 5 –2 –3 –4 –5 y x Graph 10 0 2 31 –1 –5 –4 –3 –2 –1 1 4 5 2 3 4 5 –2 –3 –4 –5 y x SAMPLE
2.1 NOW WHAT DO YOU KNOW? What strategies can you use to write equations for linear functions? Investigation 2 Linear Models and Equations 27 Graph 11 0 2 31 –1 –5 –4 –3 –2 –1 1 4 5 2 3 4 5 –2 –3 –4 –5 y x Graph 13 0 2 31 –1 –5 –4 –3 –2 –1 1 4 5 2 3 4 5 –2 –3 –4 –5 y x Graph 12 0 2 31 –1 –5 –4 –3 –2 –1 1 4 5 2 3 4 5 –2 –3 –4 –5 y x Graph 14 0 2 31 –1 –5 –4 –3 –2 –1 1 4 5 2 3 4 5 –2 –3 –4 –5 y x Situation C. Luke’s New Strategy Luke’s Strategy In Problem 1.3, we had a line that represents the data that relates truss length and the number of rods needed to make the truss. I can also write an equation for the line. I can the use it to make predictions about the truss length and the number of rods. 1. Is Luke correct? What equation would he write? 2. Use the equation to find the number of rods needed for a 20-foot truss. SAMPLE
28 Thinking with Mathematical Models When we have a relationship between two quantities or variables, we can use numeric reasoning, tables, graphs, and equations to find the value of a variable. We have used these strategies in several Connected Mathematics® units in grades 6 and 7. INITIAL CHALLENGE The Business Club visits Sandy’s Boat House for more business information. Sandy’s Boat House is run by a group of college students. They rent canoes to use on the Red Cedar River. They use various models to make predictions. They challenge the club members to investigate two different options for the cost of renting canoes. They want a rental cost that will attract customers but also provide a profit. Sam’s Cost Plan Sam suggests the cost to rent a canoe is $30.00 per hour. Millie’s Cost Plan Millie suggests they use the equation CMillie = 0.40t + 12.00 to determine the cost C in dollars to rent a canoe for t minutes. • Which option would you choose? Explain. WHAT IF . . . ? Situation A. Serena’s, Rashida’s, and Omar’s Strategies To answer the question in the Initial Challenge, Which cost plan would you choose?, students use different strategies. 1. Explain how each strategy models the data. 2. Explain how each strategy would or would not work to decide which cost plan is better. PROBLEM 2.2 Boat Rental: Finding Solutions Using Tables, Graphs, and Symbols SAMPLE
Investigation 2 Linear Models and Equations 29 Serena Uses a Table Rental Time (minutes) 30 60 90 120 150 Sam’s Plan ($) 15 Millie’s Plan ($) 24 Rashida Uses a Graph 20 40 60 80 100 300 Rental Time (minutes) Rental Charge ($) 60 90 120 150 CMillie = 0.4t + 12 CSam = 0.5t Omar Uses Equations I wrote equations for the cost of each plan in hours, h. CSam = 30h, for Sam’s plan. CMillie = 24h + 12, for Millie’s plan. I set the two expressions for cost equal. I wanted to know when Sam’s = Millie’s. 30h = 24h + 12 Now, I can solve for h. To solve it, I thought of it like the pouches and coins problem in the Moving Straight Ahead unit. Situation B. River Fun Boat Rides The Business Club also visited River Fun Boats, which also rents canoes. To determine the charge C in dollars for renting a canoe for t minutes, the company uses the equation C = 30 + 0.20t. 1. What is the charge to rent a canoe for 60 minutes? 2. A customer at River Fun Boats is charged $42. How long did the customer use a canoe? 3. Suppose a group wants to spend at most $60. How long could they use a canoe? 4. How does the cost to rent a canoe from River Fun Boats compare to the cost to rent a canoe from Sandy’s Boat House? Situation C. Solving Equations: Which Strategy? Members of the club use different strategies to solve Equation 1 and Equation 2 for a given value of one of the variables. Which strategy goes with each equation? 2.2 SAMPLE
30 Thinking with Mathematical Models Equation 1 y = −x + 6 What is x if y = 4? Equation 2 y = 5x − 3 What is y if x = 4? Mateo’s Strategy x −5 −4 −3 −2 −1 0 1 2 3 4 5 y −28 −23 −18 −13 −8 −3 2 7 12 17 22 Ivan’s Strategy x −5 −4 −3 −2 −1 0 1 2 3 4 5 y 11 10 9 8 7 6 5 4 3 2 1 Max’s Strategy –4 –2 0 2 4 2 4 6 8 10 y x Bela’s Strategy –2 0 2 4 5 10 15 20 25 y x Audrey’s Strategy 4 = −x + 6 Chris’s Strategy y = 5(4) − 3 Situation D. Linear Functions 1. Do any of the cost plans in this problem represent linear functions? Explain why or why not. 2. Are any of these linear functions proportional? If so, what is the unit rate? NOW WHAT DO YOU KNOW? How do graphs, tables, and equations of functions help you solve problems? What are the advantages and disadvantages of each representation? 2.2 SAMPLE