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Unformatted text preview: MATH 114 QUANTITATIVE REASONING FOR PROFESSIONALS An Inquiry-Based Approach Ferris State University Semester 2 2nd Edition: 2019-20 1 Authors, Contributors, and Consultants Victor Piercey Lead Author and Project Manager Mathematics Michael Dekker Author and Technical Typesetter Mathematics Jerome Trouba Author and Online Homework Designer Mathematics Erin Militzer Author Mathematics Anil Venkatesh Author Mathematics Rhonda Bishop Lead Consultant and Application Contributor Nursing Mischelle Stone Lead Consultant and Application Contributor Social Work Mary Beaudry Consultant and Scenario Contributor Nursing Lianne Birggs Consultant and Scenario Contributor Hospitality Management Lauren Cavner-Williams Consultant and Assignment Contributor Mathematics Becky Johnson-Hines Consultant and Scenario Contributor Nursing Elizabeth Post Consultant and Scenario Contributor Social Work Christopher Smith Consultant and Scenario Contributor Social Work James Shimko Consultant and Scenario Contributor Accounting Acknowledgement: © Supported by National Science Foundation grant DUE 1625321 2019 Victor Piercey 2 Contents To the Student 5 Introduction 6 Resources 9 Resource 1: Problem Solving Strategies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 Resource 2: Reading Strategies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 Resource 3: Question Starters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 Resource 4: Setting Up Your Computer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 Resource 5: Reading a Spreadsheet . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 Resource 6: Making Graphs and Charts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 Resource 7: Programming Formulas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 Resource 8: Formatting and Reordering Columns . . . . . . . . . . . . . . . . . . . . . . . . . . 21 . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 Resource 9: Autosum and Predened Formulas . . . . . . . . . . . . . . . . . . . . 25 Resource 11: Paste Special and Paste Link . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 Resource 12: Other Spreadsheet Features . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 Resource 13: Signicant Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 Resource 14: Downloading the Solver and Analysis Toolpaks . . . . . . . . . . . . . . . . . . . . 35 Resource 15: Linear Programming in Excel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 Resource 16: Running Regression in Excel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 Resource 17: Laws of Exponents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 Resource 10: Absolute, Relative, and Mixed Cell References Quantitative Reasoning for Professionals Contents 68 Preview of MATH 114 45 5 6 7 P.1 Quantiables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 P.2 Reasoning Graphically . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 P.3 The Rule of Four . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 P.4 All in the Family . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64 Linear Functions 68 5.1 Introduction to Linear Functions and Slope . . . . . . . . . . . . . . . . . . . . . . . . . . . 70 5.2 Practical Interpretations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75 5.3 Linearity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80 5.4 Graphs of Linear Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86 5.5 Equations of Linear Functions 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91 5.6 Equations of Linear Functions 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95 Exponential Functions 101 6.1 Introduction to Exponential Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104 6.2 Laws of Exponents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110 6.3 Exponential Growth and Decay . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116 6.4 Exponentiality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123 6.5 Half-Life . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131 6.6 Annuities and Lump Sums 6.7 Summation Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143 6.8 Annuities with Geometric Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148 6.9 The Credit Card Problem 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152 Logarithms 158 7.1 How Many Zeros Are There? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160 7.2 Meet the Logarithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163 7.3 A Second Date with Logarithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169 7.4 Logarithm Identities 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174 7.5 Doubling Time . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178 Quantitative Reasoning for Professionals Contents 8 9 68 7.6 Logarithms and Half-Life . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184 7.7 The Credit Card Problem 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 190 7.8 The Birth Control Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194 Linear Analysis 199 8.1 Linear Systems 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201 8.2 Linear Systems 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 206 8.3 A First Look at Linear Programming . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213 8.4 An Overview of Linear Programming . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 216 8.5 Feasibility Regions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 222 8.6 Linear Programming . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 230 Models from Data 239 9.1 Linear Regression . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 241 9.2 Power Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 248 9.3 Logarithm Identities 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253 9.4 The Log Transform 258 9.5 Semi-Log Plots and Log-Log Plots 9.6 Nonlinear Regression . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 264 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 271 5 To the Student For many of you, your recent mathematical experience has seemed an endless set of disconnected problems that you solved by applying a sequence of steps. A sequence of steps for each problem was told to you by your teacher and you memorized these steps for your tests. During tests, you struggled with varying degrees of success to match the steps you memorized to each problem. Once the exam was over, you deleted those steps from your memory. If this was your experience, and you believe that algebra is pointless, you are right. This course is dierent. The focus will be on constructing meaning and understanding your own learning. The content for this course has been trimmed by mathematics and partner discipline faculty to only what is essential. Much time will be spent analyzing questions of why  not just how. Most of the content is directly connected with authentic professional contexts. Throughout the course, you will be constructing your own understanding of math  guring everything out as we go, together. Expect to read and write in this course. Reading is critical to learning. As with any other subject, math can be read. Your writing will unlock the secrets and assumptions in your thinking. This is a critical component of the learning process as well as our most powerful method of communicating our thoughts. While there are few mathematical prerequisites for success in this course, a certain depth of character are absolutely essential. By depth of character, we mean attitude toward learning. We will get stuck, we will ounder. But if we greet every doubt and every obstacle with a smile and a fresh resolve, together we will make the most of our learning opportunities. This course will require a sense of accountability. You are responsible for your own education. The slate is wiped clean. The opportunity to solidify your future is placed here before you. Commit to it and seize it. Believe that it is possible for one learning experience to forever change the course of your academic and professional life. Believe that it is this course  this experience. This course is your secret weapon that will separate you from other professionals. This course is an open road and a tank full of gas. This course was designed for you. (partially adapted from A Note to the Student by Dylan Retsek, California State Polytechnic College) 6 Introduction A dierential equations textbook once declared this course is about predicting the future! How exciting! The course that you are about to embark upon has the same mission. We want to predict the future. Isn't that impossible? No, not if you have reliable mathematical models. In this course we will examine and construct mathematical models and use them to make predictions. We begin by setting the stage in a Preview chapter. Afterwards, we explore two types of mathematical models  linear and exponential functions. Linear and exponential functions are the most common types of mathematical models and cover a wide range of applications. We will briey look at some other types of models at a couple of points during the course. In the second half of the course, we will use linear and exponential functions to solve specic professional problems, from how long you should stay in a shelter after a nuclear bomb falls to how long before the drinking water in Flint will be safe to drink. This text contains two primary types of materials: resources and explorations. The resources are just what they say they are  resources. They are items that you can return to throughout the course for help. The explorations are the primary lessons for our class sessions and are organized into the chapters described above. Think of each exploration as a section in a textbook. You should read each exploration before beginning work. It helps to understand the organization of each. Most explorations follow a similar pattern: 1. A narrative introduction, usually about 1 page in length. 2. A list of objectives for the exploration. You should be able to understand what these mean by the time you nish the exploration. 3. A key question. The key question is something you should be able to answer by the end of the exploration. These tell you what you should be thinking about as you are working. 4. Introductory questions. These are like initial breadcrumbs as you make your way into the forest. They begin by having you apply your prior knowledge in a way that leads to new ideas. 5. Critical questions. These questions take your observations from the introductory questions and require you to dive deeper to discover the new concept(s). 6. Questions that redirect the new concepts. These questions require you to apply what you learned in new scenarios. 7. Summary questions. These questions are designed to get you to look back over the exploration and summarize what you have learned. They are often tied to the key question. 8. Supplements. Some explorations have supplements that include important reference information you will need during the lesson. To aid you in seeing this rough outline as it is applied to each exploration, some questions are preceded by question ags. The ags in the explorations are: Quantitative Reasoning for Professionals Introduction 1. ¤ 7 KEY QUESTION: This is the key question  it tells you what you should be thinking about as you work through the exploration. 2. o CRITICAL QUESTION: These questions tie back to the course objectives and direct you to uncover the new concept(s) in the exploration. They are tied to the objectives listed at the beginning of each exploration. 3. + SUMMARY QUESTION: This is where you will write a summary of what the exploration was about in order to close out the lesson. There are other question ags that serve specic purposes for some explorations: 4.  THE ROLE OF UNITS: Units are a critical component in quantitative reasoning, and while we specically address units and conversions in Exploration 2.5, they come up in important ways elsewhere in the course (especially early on). 5. ù REVIEW QUESTION: A review question leads back to a prior exploration to help you make connections across the course, or to bring back prior knowledge necessary for the exploration. 6. Ò WRITING TO UNDERSTAND: These are additional opportunities to use writing to explain your thinking. 7. ­ EXTEND YOUR THINKING: These questions typically ask you to modify some of the features of earlier questions and examine how things change, or to apply what you have learned in some new way. Finally, some explorations include questions that go beyond that particular exploration and connect to a bigger picture. DEVELOPING ALGEBRA INTUITION: Intuition is about understanding meaning and 8. being able to make predictions. As we work through the rst two units, which are numerical in nature, occasionally we will have questions that will help you develop intuition about algebra by connecting forward to the algebra units. 9. ¤ FOCUS ON QUANTITATIVE ETHICS: Ethics is everywhere, and it is easy to use (re- ally misuse) and communicate quantitative information in an unethical manner. These questions connect your quantitative thinking to real world consequences. 10.  CHALLENGE QUESTION: Challenge questions ask you to go above and beyond what is expected in the exploration. How they are used in your class will be up to your instructor. In addition, some parts of the explorations have visual icons in them. These are simple, visual cues to both you and your instructor about the organization of the lesson. They can roughly be interpreted as follows: / Expect to take more time than usual in attempting this question and thinking about this question. Quantitative Reasoning for Professionals Introduction ­ 8 This question is intended for reection on your learning process or on the mathematics. o This question is very important.  You will have to read something relatively substantial before proceeding to the question. ! This is a good place to pause for discussion. Your instructor will let you know if you can move on if you nish before other classmates. You will also be given several types of assignments by your instructor throughout the course. At the end of every chapter, you will work in a small team on a case study. You will also be given extension assignments (requiring you to extend the inquiry outside of the classroom), reection questions (which will require you to process and explain mathematical concepts), and written and online exercises (to practice basic skills). Together, the text and these assignments are designed to open new doors for you  the professional in the data age. Data surrounds us. We nd it everywhere. In order for a business professional to be successful, rst and foremost they must be able to process and interpret data in order to make sound decisions. In the 21st century, these skills are more critical than they ever have been before. 9 Resources 10 Resource 1: Problem Solving Strategies 1. Draw a picture or a diagram 2. Trial and error 3. Generate examples 4. Use algebra 5. Find a pattern 6. Start with a simpler problem 7. Act out the problem 8. Check for relevant or irrelevant information 9. Find smaller parts of a larger 10. Work backwards 11. Make a table, chart, or organized list 11 Resource 2: Reading Strategies 1. Actually read the problem, the entire problem, before beginning work. 2. Anticipate what you are reading, anticipate the solution. 3. Reread the problem several times. 4. Annotate as you read and reread . 5. Summarize in one sentence what you read, including detail. 6. Connect the problem or text you are reading to your experience and prior knowledge. 7. Ask questions that would help clarify the problem or text you are reading (see Resource 3 on starter questions). 8. Create images or movies (on paper or in your head) to make the reading three-dimensional. 9. Make guess about what you are reading, drawing conclusions and making predictions. 10. Separate important from unimportant information. 11. Incorporate new information into your background knowledge. 12 Resource 3: Question Starters 1. What does the word/phrase mean? 2. What should the answer look like or mean? 3. Might I gain more information by rereading the problem? 4. What data from the problem/setup/scenario do I need to use to address this question? 5. What mathematical knowledge do I need to use for this question? 6. Is there some mathematical knowledge that I need to use but do not know? 7. Do I agree with the question? Why or why not? 8. Where are these questions going? What is the big picture here? 9. Are there some assumptions I need to make? 10. Can I restate the problem in my own words? 11. How is this question similar to questions that I have already answered? How is it dierent? 12. Can I sketch a picture or make a physical model to help me with this problem? 13. Can I write down an example? 14. Can I solve a simpler problem? 15. Are there any incorrect approaches I can try and what can I learn from them? 16. What is it in this problem that I do understand? Are there any breadcrumbs leading me back to something familiar? 17. What is making this dicult for me? 13 Resource 4: Setting Your Computer Up Prior to starting Chapter 1, if you have not already done so, you will need to (a) register your device, (b) connect to FerrisWiFi (this is NOT the same as FSUStudent), and (c) download and install Microsoft Oce 365 products. Follow the links below for each step: 1. To register your device, visit: 2. To connect to FerrisWiFi, visit: Note: this is NOT the same as FSUStudent, which his available in the dorms but not neces- sarily in academic buildings. 3. To download and install Microsoft Oce 365 products, visit ce Use your Ferris email and password. 14 Resource 5: Reading A Spreadsheet If you are accustomed to word processing software, a spreadsheet may feel a little funny at rst. In this Resource we are going to learn some basics. As we proceed through this course, we will learn more about spreadsheets. Here is a screenshot from a blank spreadsheet. In Excel, this is called a workbook: Each rectangle is called a cell. A cell sits in a column and a row. The columns are labeled by capital letters and the rows are labeled by numbers. A cell is labeled by its column and row names, as in the picture above where the balloon points to Cell J13. At the top of the screen we have menus, ribbons, and function bar. Quantitative Reasoning for Professionals Resource 5: Reading A Spreadsheet 15 The menus are the selections at the very top, such as HOME,INSERT, and FORMULAS. Each menu s...
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