Quantitative Literacy

Quantitative Literacy Sherry-Anne McLean

This book was edited by Sherry-Anne McLean, Lake Washington Institute of Technology

The Quantitative Literacy Toolkit, Working With Data, and Algebraic Reasoning chapters are largely based on:

Quantway Version 1.0. The original version of this work was developed by the Charles A. Dana Center at the University of Texas Austin under sponsorship of the Carnegie Foundation for the Advancement of Teaching. This work is used (or adapted) under the Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported (CC BY-NC- SA 3.0) license: creative commons.org/licenses/by-nc-sa/3.0. For more information about Carnegie’s work on Quantway I, www.carnegiefoundation.org/quantway see ; for information on the Dana Center’s workThe on New Mathways Project, see www.utdanacenter.org/mathways.

The Skills Quiz review chapters (A through E) contain portions taken and derived from:

Worksheets created by David Lippman and released under a Creative Commons Attribution license. The worksheets were created to supplement the textbook Arithmetic for College Students. This work by Monterey Institute for Technology and Education (MITE) 2012 and remixed by David Lippman is licensed under a Creative Commons Attribution 3.0 Unported License

This work is licensed under the Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License.

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Contents

Module 1: The Quantitative Literacy Toolkit…………… 1 Module 2 Working with Data…………………………... 113 Module 3: Algebraic Reasoning………………………… 215 Strengthening Skills A: Order of Operations…………….……………... 302 Strengthening Skills B: Fractions…………………………………..…… 312 Strengthening Skills C: Signed Numbers…………………………..…… 328 Strengthening Skills D: Percentages and Proportions………………..…. 338 Strengthening Skills E: Solving Equations…………………………..….. 346

1 Quantway Student Handout November 18, 2011 (Version 1.0) Lesson 1.1.1: Introduction to Quantitative Reasoning Theme: Citizenship

Specific Objectives Students will understand that • quantitative reasoning is the ability to understand and use quantitative information. It is a powerful tool in making sense of the world. • relatively simple math can help make sense of complex situations. Students will be able to • identify quantitative information. • round numbers (based on homework). • name large numbers (based on homework). • work in groups and participate in discussion using the group norms for the class.

Problem Situation: Does This Information Make Sense? In this lesson, you will learn how to evaluate information you see often in society. You will start with the following situation. You are traveling down the highway and see a billboard with this message:

Every year since 1950, the number of American children gunned down has doubled.

(1) You do not see the name of the organization that put up the billboard. What groups might have wanted to publish this statement? What are some social issues or political ideas that this statement might support?

The information in this statement is called quantitative. Quantitative information uses concepts about quantity or number. This can be specific numbers or a pattern based on numerical relationships such as doubling. You hear and see statements using quantitative information every day. People use these statements as evidence to convince you to do things like • vote a certain way • donate or give money to a cause • understand a health risk You often do not know whether these statements are true. You may not be able to locate the information, but you can start by asking if the statement is reasonable. This means to ask if the statements make sense. You will be asked if information is “reasonable” throughout this course. This lesson will help you understand what is meant by this question.

The Carnegie Foundation for the Advancement of Teaching Quantway frontmatter available at www.quantway.org/kernel and The Charles A. Dana Center at the University of Texas at Austin or www.utdanacenter.org/mathways/index.php 1 2 Quantway Student Handout November 18, 2011 (Version 1.0) Lesson 1.1.1: Introduction to Quantitative Reasoning Theme: Citizenship

(2) In 1995,1 a group published the statement in the Problem Situation. Do you think this was a reasonable statement to make in 1995? Discuss with your group. (3) You only have the information in the statement. Using only that information, how can you decide if the statement is reasonable? Talk with your group about different ways in which you might answer this question. (4) In Question 3, you thought about ways to decide if the statement was reasonable. One approach is to start with a number for the first year. Put this number into the table below. Complete the other values in the second column of the table. Do not complete the third column right now.

Year Number of Children Rounded (using words) 1950 1960 1970 1980 1990 1995

(5) Does the number you predicted for the number of children shot in 1995 seem reasonable? What kind of information might help you decide?

Making Connections Record the important mathematical ideas from the discussion.

About This Course This course is called a quantitative reasoning course. This means that you will learn to use and understand quantitative information. It will be different from many other math classes you have taken. You will learn and use mathematical skills connected to situations like the one you discussed in this lesson. You will talk, read, and write about quantitative information. The lessons will focus on three themes: • Citizenship: You will learn how to understand information about your society, government, and world that is important in many decisions you make. • Personal Finance: You will study how to understand and use financial information and how to use it to make decisions in your life. • Medical Literacy: You will learn how to understand information about health issues and medical treatments.

1Best, J. (2001). Damned lies and statistics. University of California Press: Berkeley and Los Angeles.

The Carnegie Foundation for the Advancement of Teaching Quantway frontmatter available at www.quantway.org/kernel and The Charles A. Dana Center at the University of Texas at Austin or www.utdanacenter.org/mathways/index.php 2 3 Quantway Student Handout November 18, 2011 (Version 1.0) Lesson 1.1.1: Introduction to Quantitative Reasoning Theme: Citizenship

This lesson is part of the Citizenship theme. You learned about ways to decide if information is reasonable. This can help you form an opinion about an issue. Today, the goal was to introduce you to the idea of quantitative reasoning. This will help you understand what to expect from the class. Do not worry if you did not understand all of the math concepts. You will have more time to work with these ideas throughout the course. You will learn the following things: • You will understand and interpret quantitative information. • You will evaluate quantitative information. Today you did this when you answered if the statement was reasonable. • You will use quantitative information to make decisions.

The Carnegie Foundation for the Advancement of Teaching Quantway frontmatter available at www.quantway.org/kernel and The Charles A. Dana Center at the University of Texas at Austin or www.utdanacenter.org/mathways/index.php 3 4 Quantway Out-of-Class Experience November 18, 2011 (Version 1.0) Lesson 1.1.1: Introduction to Quantitative Reasoning Theme: Citizenship

Introduction Since this is your first assignment, the authors will be explaining how your daily assignments will be structured. An assignment is referred to as an Out-of-Class Experience (OCE). Each OCE has the same four sections: • Making Connections to the Lesson • Developing Skills and Understanding • Making Connections Across the Course • Preparing for the Next Lesson and/or Assessment

Making Connections to the Lesson The purpose of this section is to help make sure you understand the most important ideas of the lesson. Sometimes it is hard to know what to focus on when you are in class. The authors have designed this curriculum to help you identify and remember important ideas through the following steps: • Every lesson ends with a discussion. During this discussion, the class identifies the important mathematical ideas of the lesson. • The Student Handout always ends with a section called Making Connections. In this section, you write down the important mathematical ideas. • This section of your OCE always starts with a question that asks you to identify a main mathematical idea of the lesson. You are given four statements to choose from. • In future OCEs, you will describe how mathematical ideas connect across lessons. A main mathematical idea means that the idea is an important concept that helps explain how to do many different types of problems and helps connect different problems together. It may take you a while to be able to identify the main mathematical ideas of lessons. Your instructor will help you at first by making sure these ideas are discussed at the end of the lesson.

(1) Which of the following statements correctly illustrates one of the main mathematical ideas of the lesson? (i) Asking good questions about quantitative information is important in quantitative reasoning. (ii) Doubling means to multiply by 2. (iii) Gun violence is a problem in the United States. (iv) You should not use estimation.

Since this is your first time with this type of question, the authors are going to explain the answer to Question 1. The answer is (i). because asking questions about quantitative information is important in many different problem situations. The other answers may or may not be true, but they are not main mathematical ideas for this lesson. Specifically, • (ii) is true, but it only applies to one type of procedure: doubling. • (iii) is an opinion. You cannot say if it is true or false, and it is not a mathematical idea. • (iv) is not true. As you saw in the lesson, estimation is a valuable skill.

The Carnegie Foundation for the Advancement of Teaching Quantway frontmatter available at www.quantway.org/kernel and The Charles A. Dana Center at the University of Texas at Austin or www.utdanacenter.org/mathways/index.php 1 5 Quantway Out-of-Class Experience November 18, 2011 (Version 1.0) Lesson 1.1.1: Introduction to Quantitative Reasoning Theme: Citizenship

Developing Skills and Understanding The purpose of this section of the OCE is for you to practice with the skills and concepts from the lesson. You will see questions directly related to the lesson. You will also see questions that apply the skills and concepts to different situations. The section will sometimes have reading material that helps explain the topics from the lesson. Later in the course, you can look back at this information as you review what you have learned. Questions 2 and 3 highlight important quantitative reasoning skills that you will learn in this course.

Quantitative Reasoning Skill: Reading and interpreting quantitative information The lesson from class focused on a statement about children “gunned down” in America. How was such an inaccurate statement published? It was based on another statement published earlier.1 Both statements are shown below. Read them carefully and decide what each means mathematically.

Original Statement: The number of American children killed each year by guns has doubled since 1950.

Reworded Statement (circa 1995): Every year since 1950, the number of American children gunned down has doubled.

(2) Based on the original statement and the reworded statement, which of the following comments is valid? (i) Both the original and reworded statements are interpreted to mean that the number of children gunned down has doubled every year between 1950 and 1995. (ii) The interpretation of the reworded statement implies that the number of children gunned down has doubled once between 1950 and 1995. (iii) Assume that the original statement is true. If approximately 100 children were killed by guns in 1950, the number of children killed by guns in 1995 was about 200. (iv) The phrase “children killed” has the same meaning as “children gunned down.”

This highlights the importance of reading and writing carefully about quantitative information. The original and reworded statements look very similar, but mean entirely different things: • The original statement says that the number has doubled once from 1950 to the published date (1995). • The reworded statement says that the number has doubled every year between 1950 and the published date (1995).

1Best, J. (2001). Damned lies and statistics. University of California Press: Berkeley and Los Angeles.

The Carnegie Foundation for the Advancement of Teaching Quantway frontmatter available at www.quantway.org/kernel and The Charles A. Dana Center at the University of Texas at Austin or www.utdanacenter.org/mathways/index.php 2 6 Quantway Out-of-Class Experience November 18, 2011 (Version 1.0) Lesson 1.1.1: Introduction to Quantitative Reasoning Theme: Citizenship

Quantitative Reasoning Skill: Identifying information that can be verified (checked to see if it is true) (3) Which of the following statements contain quantitative information? There may be more than one correct answer. (i) Many Americans have diabetes. (ii) ABC News reported that the number of Americans that have diabetes could triple in the next 40 years.2 (iii) About a fourth of Americans with diabetes are over 65 years old according to the American Diabetes Association.3 (iv) Diabetes is a terrible disease. One characteristic of quantitative information is that it contains numerical information. Another is that it has information that can be checked or evaluated. The statement “Many Americans have diabetes” sounds quantitative. Many implies a number, but it is also a judgment. How much is many? There is no way to verify this statement because you could have different opinions about the meaning of “many.” “Diabetes is a terrible disease” is also a judgment. You can offer quantitative information to support the statement, but you cannot verify that this is true or false. Being able to evaluate a claim based on quantitative information is an important quantitative reasoning skill.

Quantitative Reasoning Skill: Naming and estimating large numbers Large numbers often occur in real-life situations, but it is hard to make sense of them. It is difficult to imagine the distinction between a million and a billion. You will do more work with understanding the size of these numbers in Lesson 1.1.2, but first you will work on recognizing the numbers and names. If you need some review on place value, you can view the following videos: • www.khanacademy.org/video/place-value-1?playlist=Developmental%20Math • www.khanacademy.org/video/place-value-2?playlist=Developmental%20Math • www.khanacademy.org/video/place-value-3?playlist=Developmental%20Math

2Retrieved from http://abcnews.go.com/WN/diabetes-rise-america-slow-growth-world-news-question/story?id=11945648. 3Retrieved from www.diabetes.org/diabetes-basics/diabetes-statistics.

The Carnegie Foundation for the Advancement of Teaching Quantway frontmatter available at www.quantway.org/kernel and The Charles A. Dana Center at the University of Texas at Austin or www.utdanacenter.org/mathways/index.php 3 7 Quantway Out-of-Class Experience November 18, 2011 (Version 1.0) Lesson 1.1.1: Introduction to Quantitative Reasoning Theme: Citizenship

(4) The following place-value chart is partially labeled.

Place-Value Chart

Tens Ones Hundreds Thousands Ten billions Ten millions Ten thousands Ten Hundred trillions Hundred Hundred millions Hundred

102 101 100 Power of 10

(a) Fill in the missing name labels on the Place-Value Chart.

(b) Fill in the power of 10 that corresponds to each position on the Place-Value Chart. (Entries for 100 in the ones place, 101 in the tens place, and 102 in the hundreds place have already been entered.)

(5) Which of the following represents the number: “Six billion, nine hundred ten million, one hundred fifty-two thousand, eight hundred twenty-four”? (i) 6,910,152,824 (ii) 600,910,152,824 (iii) 6,000,910,152,824 (iv) 6,910,001,052,824

In Question 4 of the lesson, you practiced estimating and naming large numbers. Large numbers are also estimated in another way that combines numbers and words. Look at the examples below. 35,432,000 rounded to 35.4 million Think of 35.4 million as a multiplication problem of 35.4 times 1 million: 35.4 x 1,000,000 = 35,400,000 This gives the same result as estimating 35,432,000 in millions. Here is another example: 1,452,900,812 rounded to 1.5 billion

The Carnegie Foundation for the Advancement of Teaching Quantway frontmatter available at www.quantway.org/kernel and The Charles A. Dana Center at the University of Texas at Austin or www.utdanacenter.org/mathways/index.php 4 8 Quantway Out-of-Class Experience November 18, 2011 (Version 1.0) Lesson 1.1.1: Introduction to Quantitative Reasoning Theme: Citizenship

(6) Select the number/word combination that best estimates each number. (a) 87,300,000 (i) 8.7 million (ii) 87.3 billion (iii) 87.3 million

(b) 2,670,000,000,000 (i) 2.7 trillion (ii) 2.7 billion (iii) 2700 million

(7) Following are some data about diabetes in the United States from the American Diabetes Association.4 Complete the table either by writing the words as a number or as a combination of words and numbers.

Number Word/Number Combination Number of children and adults with 25.8 million diabetes in 2010 Number of children under age 20 215,000 with diabetes in 2010 Cost due to diagnosed diabetes cases in 2007—includes medical $174,000,000,000 costs, disability payments, loss of work, and premature death

4Retrieved from www.diabetes.org/diabetes-basics/diabetes-statistics.

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Quantitative Reasoning Skill: Rounding numbers Another important skill you used in this lesson is rounding. You often round numbers when you are trying to make sense out of them or make comparisons and do not need exact numbers. In this lesson, you found that the statement predicted that trillions of children were gunned down in 1995. This was enough to know that the statement was not reasonable because that was more than the entire population of the United States (and in fact, the world). You did not need to have exact numbers. If you need review on rounding, you can view the following videos: • www.khanacademy.org/video/rounding-whole-numbers-1?playlist=Developmental%20Math • www.khanacademy.org/video/rounding-whole-numbers-2?playlist=Developmental%20Math • www.khanacademy.org/video/rounding-whole-numbers-3?playlist=Developmental%20Math

(8) The following website has two population clocks that update every minute to show the estimated populations of the United States and the world (www.census.gov/main/www/popclock.html). At 7:29 p.m. (central standard time) on April 5, 2011, the clocks showed the following values.

Rounded Number Estimated Population Name of Rounded (round to the place Count from Website Number value indicated) 311,000,000 U.S. population 311,105,182 (round to nearest 311 million million) 7,000,000,000 World population 6,910,152,824 (round to nearest 7 billion billion)

(a) Go to the population clock website. Record the current population estimates and the time at which you recorded them. Complete the table as indicated.

Time recorded: ______

Rounded Number Estimated Population Name of Rounded (round to the place Count from Website Number value indicated)

U.S. population (round to nearest million)

World population (round to nearest billion)

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(b) Wait at least 10 minutes and go back to the population clock (either close and reopen the website or refresh the website). Record the new values. Time recorded: ______

Rounded Number Estimated Population Name of Rounded (round to the place Count from Website Number value indicated)

U.S. population (round to nearest million)

World population (round to nearest billion)

(d) Did the rounded numbers change? (e) If you were making a calculation based on population, would you use the population count or the rounded number? Be prepared to justify your answer.

Making Connections Across the Course This section of the OCEs will help you make connections between concepts across the course. In Making Connections, you will be using concepts, skills, and situations from previous assignments and previewing topics you will use in later assignments. There are five lessons in the first unit of the course: 1.1.1–1.1.5. These lessons will help you develop some very important skills you will use throughout the course. These include the following: • Reading quantitative information. • Writing statements using quantitative information. • Understanding large numbers: o place value. o reading and writing large numbers in both words and digits. o the size of numbers. o comparing the relative size of numbers. • Estimation. • Understanding, estimating, and calculating percentages. • Fundamentals of calculations: o order of operations. o different ways to write and perform calculations.

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By the end of this module, you should also understand some important points about this course: • What quantitative reasoning is. • Your responsibility for: o creating and contributing to the classroom learning environment. o being prepared for class. o completing your work. o planning and monitoring your own learning and course progress. • How to be an effective member of a work group. • Strategies for working on difficult problems.

The following questions will help you prepare for this course: (9) What are your goals for this class?

(10) What academic and nonacademic strengths do you bring to the class? Examples: time to work in the tutoring center or to meet with classmates, good support at home so you can focus on your studies, confidence in yourself based on your past experiences either in school or in other aspects of your life. (11) Do you have any questions or concerns you want to ask your instructor?

Preparing for the Next Lesson (1.1.2) Your instructor expects you to be prepared for the next class. This section tells you what you need to know and be able to do to be prepared. You will be asked to rate how confident you are that you can do certain things. Be honest when you rate yourself. You will not be graded on the rating. If you do not feel confident, get help on the topic before class. Talk to your instructor about ways you can get help on campus. Reread the information from Lesson 1.1.1 that describes this course: This course is called a quantitative reasoning course. This means that you will learn to use and understand quantitative information. It will probably be different from any other math class you have ever taken. You will learn and use mathematical skills, but they will be connected to situations like the one you discussed in this lesson. You will talk, read, and write about quantitative information. The lessons will focus on three themes: • Issues of citizenship: understanding your society, government, and world (the situation from today’s lesson is an example) • Personal finance: understanding financial information and how to use it to make decisions • Medical literacy: understanding the meaning of information about risk of disease and effectiveness of treatment

The Carnegie Foundation for the Advancement of Teaching Quantway frontmatter available at www.quantway.org/kernel and The Charles A. Dana Center at the University of Texas at Austin or www.utdanacenter.org/mathways/index.php 8 12 Quantway Out-of-Class Experience November 18, 2011 (Version 1.0) Lesson 1.1.1: Introduction to Quantitative Reasoning Theme: Citizenship

The purpose of today’s lesson was to introduce you to the idea of quantitative reasoning and give you a picture of what the class will be like. Do not worry if you did not understand all of the math concepts. You will have more time to work with these ideas throughout the course. Some other skills you will learn are • how to understand and make sense of quantitative information. • how to evaluate quantitative information (like you did in this lesson when you were asked if the statement was reasonable). • how to use quantitative information to make decisions.

(12) Give one example for each theme that would be of particular interest to you (possibly an experience or a question that you have encountered). (a) Issue of citizenship

(b) Issue of personal finance

Self-Regulating Your Learning—An Introduction One goal of this course is to increase your ability to learn efficiently and effectively. This means learning faster and learning smarter—what scientists call being a “self-regulated learner.” The following section explains what this means. Self-regulating your learning means you plan your work, monitor your work and progress, and then reflect on your planning and strategies and what you could do to be more effective. These are the three phases of Self-Regulated Learning (SRL). They are introduced below, and will be followed up on later on in the course. Plan: Before doing a problem or assignment, self-regulated learners plan. They think about what they already know or do not know, decide what strategies to use to finish the problem, and plan how much time it will take. Research has shown that math experts often spend much more time planning how they will do a problem than they do actually completing it. Novices, the people who are just starting out, often do the opposite. Work: Self-regulated learners use effective strategies as they work to solve problems. They actively monitor what study strategies are working and make changes when they are not working. When they do not know which strategy would be better, they ask for help. Self-regulated learners also keep themselves focused while they are working and pay attention to their feelings to avoid getting frustrated. Reflect: Usually after an assignment or problem is done, self-regulated learners take time to reflect about what worked well and what did not. Based on that reflection, they think about how to change their approach in their future. The reflect phase helps self-regulated learners understand more about how they learn so they can become more efficient and more effective the next time. Reflecting is important for doing a better job next time you plan for a new problem or assignment.

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You can think of these three phases as a cycle. You incorporate what you learned during the reflect stage in your next plan phase, making you a more effective learner as you repeat this process many times. The most effective students get in the habit of working this way:

For most people, self-regulating takes time, practice, and hard work, but it is always possible. People can improve even if, in the beginning, they did not self-regulate their learning very well. The more you practice something and the more you train your brain to think in certain ways, the easier it becomes. Since thinking this way takes practice, you will have opportunities to practice some of these skills as you progress through this course. As you read through the lessons and homework assignments, you will encounter activities that are designed to help you to incorporate the Plan, Work, and Reflect phases in specific ways. Take the time to thoughtfully complete these exercises. The payoff will be worth it!

Self-Regulated Learning—Plan Part of effectively planning for what could be new material for you is figuring out how much you already know. In Lesson 1.1.2, you will need to be able to do the following things: • Double values in contextual situations. • Identify place value to the trillions. • Read a table of numbers. • Add and subtract numbers.

(13) Effectively plan and use your time wisely, it helps to think about what you know and do not know. For each of the following, rate how confident you are that you can do successfully do that skill. Use the following descriptions to rate yourself: 5—I am extremely confident I can do this task. 4—I am somewhat confident I can do this task. 3—I am not sure how confident I am. 2—I am not very confident I can do this task. 1—I am definitely not confident I can do this task.

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Before beginning Lesson 1.1.2, you should understand the concepts and demonstrate the skills listed below:

Skill or Concept: I can… Rating from 1 to 5 Double values in contextual situations. Identify place value to the trillions. Read a table of numbers. Add and subtract numbers.

The Carnegie Foundation for the Advancement of Teaching Quantway frontmatter available at www.quantway.org/kernel and The Charles A. Dana Center at the University of Texas at Austin or www.utdanacenter.org/mathways/index.php 11 15 Quantway Student Handout November 18, 2011 (Version 1.0) Lesson 1.1.2: Seven Billion and Counting Theme: Citizenship

Specific Objectives Students will understand that • 1 billion = 1,000 x 1,000 x 1,000. • the representations, one billion, 1,000,000,000, and 109 have the same meaning. • population growth can be measured in terms of doubling time. • doubling times can be used to compare population growth during different periods. Students will be able to • calculate quantities in the billions. • convert units from feet to miles. • use data to estimate a doubling time. • compare and contrast population growth via population doubling times.

Problem Situation 1: How Big Is a Billion? Scientists have worried about human population growth for nearly 200 years. The population of Earth has grown over time and is still growing. You do not know how many people Earth can support. In this lesson, you will get a sense of how many people there are and how that number has changed over time. The world population is estimated to be about 7 billion people. That is seven times as many people as there were 200 years ago. It is difficult to understand just how big a billion is. Here is a way to help you think about it. 1 billion = 1,000 x 1,000 x 1,000 = 1,000,000,000 = 109 The following questions will also help you think about how big 1 billion is. (1) Imagine a line of 1,000 people standing shoulder to shoulder. How long is the line? Complete the following steps to answer this question. For each step, write your calculations clearly so that someone else can understand your work. (a) Estimate the shoulder width of an “average” person. Use that estimate in the following calculations. Calculate how far a line of 1,000 people, standing shoulder to shoulder, would measure in miles (5,280 feet = 1 mile). Record your answer in the table below. (b) Imagine 1,000 lines of 1,000 people. How many people would be in line? How long is the line in measured in miles? Record your answers in the table below. (c) Imagine 1,000 lines like the one in Part (b). How many people would be in line? How long is the line in measured in miles? Record your answers in the table below.

Number of People Length of Line (miles) (a) 1,000 (b) (c)

The Carnegie Foundation for the Advancement of Teaching Quantway frontmatter available at www.quantway.org/kernel and The Charles A. Dana Center at the University of Texas at Austin or www.utdanacenter.org/mathways/index.php 1 16 Quantway Student Handout November 18, 2011 (Version 1.0) Lesson 1.1.2: Seven Billion and Counting Theme: Citizenship

Problem Situation 2: Measuring Population Growth In the next section, you will look at doubling times to determine how the human population of the earth has changed over time. The doubling time of a population is the amount of time it takes a population to double in size. Calculating doubling time helps you understand how fast a population is growing. Comparing doubling times helps you understand how growth is changing over time.

Example Table 1 gives historical estimates of the human population. The population in 8,000 BCE was estimated to be 5 million people. Two-thousand years later, in 6,000 BCE, the population had doubled to 10 million people. Therefore, the population doubling time for 8,000 BCE is about 2,000 years.

Table 1 Population Estimates Throughout History1

World Population World Population Year (Lower bound, in Year (Lower bound, in millions) millions) 10,000 BCE 1 1850 1,262 9,000 BCE 3 1900 1,650 8,000 BCE 5 1950 2,519 7,000 BCE 7 1955 2,756 6,000 BCE 10 1960 2,982 5,000 BCE 15 1965 3,335 4,000 BCE 20 1970 3,692 3,000 BCE 25 1975 4,068 2,000 BCE 35 1980 4,435 1,000 BCE 50 1985 4,831 500 BCE 100 1990 5,263 AD 1 200 1995 5,674 1000 310 2000 6,070 1750 791 2005 6,454 1800 978 2008 6,707

1 Retrieved from U.S. Census Bureau, www.census.gov/ipc/www/worldhis.html.

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(2) Use Table 1 to estimate the doubling times of Earth’s human population. Start with the year given below and estimate how long it took for the population in that year to double. The first entry is done for you. Be prepared to explain how you got your answers.

(3) Discuss the results from Question 2 with your group. What do you notice about the doubling times? What does this tell you about how the human population has changed over time?

One of the skills you will learn in this course is how to write quantitative information. A writing principle that you will use throughout the course is given below followed by Question 4, which gives you examples of how to use this principle. Writing Principle: Use specific and complete information. The reader should understand what you are trying to say even if they have not read the question or writing prompt. This includes • information about context, and • quantitative information. (4) Which of the following statements best describes the change in doubling times before 1800 AD? (a) The doubling times decreased. (b) Before 1800 AD, estimated population doubling times decreased from 2,000 to 1,000. (c) The doubling times decreased from 2,000 to 1,000.

(5) Write a statement that describes the change in doubling times after 1800 AD.

Making Connections Record the important mathematical ideas from the discussion.

Further Applications (1) Imagine that you are explaining the relationship of million, billion and trillion to someone else. You may use words, symbols, and pictures. Your explanation should follow the Writing Principle.

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Making Connections to the Lesson (1) Which of the following was one of the main mathematical ideas of the lesson? (i) The human population is quickly growing. (ii) It is important to take time to make sense of large numbers because they occur in many important situations. (iii) There are 5,280 feet in 1 mile. (iv) There is not much difference between a million and a billion. The purpose of the next question is to help you review previous lessons and understand how the mathematical ideas connect across lessons.

(2) Communication is an important skill in quantitative literacy. In Lesson 1.1.1, you saw how changes in wording can change the meaning of a statement. In Lesson 1.1.2, you learned about a writing principle to be used when writing about quantitative information. Two statements are given below. Give at least two reasons why Statement 2 is better than Statement 1.

Statement 1: The population doubled in about 40 years.

Statement 2: The world population doubled from 1960 to 2000 from about 3 billion people to 6 billion.

Developing Skills and Understanding (3) Which of the following statements is true? (i) A trillion is 100 billion. (ii) A trillion is 10 billion. (iii) A trillion is 1,000 billion. (iv) A trillion is 1010.

(4) Refer back to the table in Question 2 of Lesson 1.1.2. One of your classmates estimates the doubling time to be 500 years in 1000 AD. Does that answer seem reasonable? Meaning, does that number fit in with the numbers you see in your table? Write one or two sentences supporting your statement. Use the Writing Principle from the lesson.

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(5) Some types of investments—such as Certificates of Deposit—earn interest based on a percentage rate. People often estimate the doubling time of investments to predict how much money the investment will be worth in the future. An investment that earns 4% interest will double in value about every 18 years. Use this information to complete the missing values in the table below for $2,500 invested at 4% interest.

Year Value of Investment 2000 $2,500 $5,000 $7,500 2054

(6) Which of the following is the best estimate for the amount of time it would take the investment in Question 5 to reach a hundred thousand dollars? (i) Less than 85 years (ii) Between 85 and 95 years (iii) Between 95 and 105 years (iv) More than 105 years

Making Connections Across the Course The OCE for Lesson 1.1.1 explained the purpose of the different sections of the assignments. Refer back to that information to answer the following questions. (7) The first section of every assignment is called “Making Connections to the Lesson.” The purpose of this section is to (i) help you identify and remember the important mathematical ideas of the lesson. (ii) help you make a personal connection to the material in the lesson. (iii) help you review all the work you did in the lesson.

(8) Which of the following are ways to use the “Developing Skills and Understanding” section to support your learning? There may be more than one correct answer. (i) To earn points to improve your grade. (ii) This section is not important unless you did not understand the work in class. (iii) To assess how well you understand the new material from class. (iv) To review information from previous lessons.

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(9) Why do you rate yourself in the “Preparing for the Next Lesson” section? There may be more than one correct answer. (i) So you can show the instructor how much you know. (ii) To honestly assess if you are ready for the next class. (iii) To get the best rating in class. (iv) So you know what is expected in the next class. It is not enough to complete the rating in the “Preparing for the Next Lesson” section of the assignment. First, you should use the rating to get ready for your next lesson. If your rating is a 3 or below, you should get help with the material before class. Remember, your instructor is going to assume that you are confident with the material and will not take class time to answer questions about it. If you need help, you should see your instructor or a tutor before class. You might also consider setting up a study group with classmates so you can help each other. Second, you should use this rating to help you get better at self-assessment. Just like any other skill, being good at self-assessment takes practice. If you rate yourself as confident but then find that you are not prepared for class, you are not doing a good job of self-assessment. In this case, it is a good idea to talk to your instructor or a tutor about how you can do a better job of assessing yourself and preparing for class. (10) Self-Regulated Learning: Reflect Self-regulating your learning includes looking back and reflecting on what you understand. At the end of OCE 1.1.1, you rated your confidence in applying the mathematical skills listed below. After applying those skills in this assignment, has your confidence that you can successfully apply those skills changed? Use the following descriptions to rate yourself: 5—I am extremely confident I can do this task. 4—I am somewhat confident I can do this task. 3—I am not sure how confident I am. 2—I am not very confident I can do this task. 1—I am definitely not confident I can do this task.

Skill or Concept: I can … Rating from 1 to 5 Double values in contextual situations. Identify place value to the trillions. Read a table of numbers. Add and subtract numbers.

Did your ratings change? If so, why?

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Preparing for the Next Lesson (1.1.3) Make sure you bring this work to class in case you need to refer back to it. Read the following introduction to Lesson 1.1.3. In this course, you will talk about different types of estimation. • Educated guess: One type of estimation might be called an “educated guess” about something that has not been measured exactly. In Lesson 1.1.2, you used estimations of the world population. This quantity cannot be measured exactly—it would be impossible to count how many people live on the earth at any given time. Scientists can use good data and mathematical techniques to estimate the population, but it will always be an estimate. • Convenient estimation: Sometimes estimations are used when it is inconvenient or not worthwhile to make an exact count. Imagine that you need to know how much paint to buy to paint the baseboard trim in your house. (The baseboard trim is the piece of wood that follows along the bottom of the walls.) You need to know the length of the baseboard. You could measure the length of each wall to the nearest 1/8 inch and carefully subtract the width of halls and doors. It would be much quicker and just as effective to measure to the nearest foot or half foot. If you were cutting a piece of baseboard to go along the floor, however, you would want an exact measurement. • Estimated calculation: This usually involves rounding numbers to make calculations simpler. Lesson 1.1.3 focuses on estimating and calculating percentages. You will find in this course that percentages are used in many contexts. One of the most important skills you will develop is understanding and being comfortable working with percentages in a variety of situations. The following questions will help you prepare for Lesson 1.1.3 by reviewing some concepts about percentages. (11) Each large square below represents 100%. Use the squares to shade the indicated percentages and/or answer the questions. (a) Shade 35% of the square below. What percentage is not shaded?

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(b) What percentage of the square below is shaded? What percentage is not shaded?

(d) Shade 0.5% of the square. What percentage is not shaded?

You may want to view the following videos to review the meaning of percent: • www.khanacademy.org/video/describing-the-meaning-of-percent?playlist=Developmental Math • www.khanacademy.org/video/describing-the-meaning-of-percent-2?playlist=Developmental %20Math

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(12) Complete the following table of equivalent percentages, fractions, and decimals. Equivalent means the expressions are equal to each other. These values are benchmarks commonly used in estimation. This means that knowing these equivalent values can help you with estimation. For example, if estimating 33% of a number, it can be helpful to know that 33% is approximately one third. You may want to view the following videos to review this concept. The first row in the table is done for you. Recall that the first numeral to the right of the decimal point is referred to as the “tenth” place while the second numeral to the right of the decimal point is the “hundredth” place. • www.khanacademy.org/video/representing-a-number-as-a-decimal--percent--and- fraction?playlist=Developmental%20Math • www.khanacademy.org/video/representing-a-number-as-a-decimal--percent--and- fraction-2?playlist=Developmental%20Math

Simplified Fraction Percent Decimal

1% 0.01

0.2

25%

round to the nearest one percent round to nearest hundredth

0.5

round to the nearest one percent round to nearest hundredth

0.75

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(13) Since money and percent are both based on 100, it is easy to think in terms of money and convert to fractions and decimals. For example, a dime is 10 cents, or $0.10, and 10 dimes is 1 dollar, so 1 dime is 1/10 of a dollar. Therefore, the expression “1/10 is 0.10” is the benchmark. Use money ideas to write similar benchmarks: (a) penny (b) nickel (c) quarter (d) half dollar (e) dollar

(14) The connection between money and percent is similar. In the same way that 1 cent is 1/100 of a dollar, then 1% is 1/100 of the unit 1. Think “% can be replaced by 1/100.” Similarly, 100 cents is 1 dollar and 100% is the same as the number 1 (100% = 1). This is helpful in converting between decimals and percents. For example, Percent to decimal: 35% = = 0.35 Decimal to percent: 0.72 = 0.72(1) = 0.72(100%) = 72% For each of the following, convert between percent and decimal forms. (a) Convert 45% to a decimal. (b) Convert 0.125 to a percent. (c) Convert 0.5% to a decimal.

(15) You should be able to do the following things for the next class. Rate how confident you are on a scale of 1–5 (1 = not confident and 5 = very confident). Before beginning Lesson 1.1.3, you should understand the concepts and demonstrate the skills listed below:

Skill or Concept: I can... Rating from 1 to 5 Understand the meaning of percent. Convert between fractions, decimals, and percentages. Round numbers to a given place value.

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Specific Objectives Students will understand that • estimation is a valuable skill. • standard benchmarks can be used in estimation. • there are many strategies for estimating. • percentages are an important quantitative concept. Students will be able to • use a few standard benchmarks to estimate percentages (i.e., 1%, 10%, 25%, 33%, 50%, 66%, 75%). • estimate the percent one number is of another. • estimate the percent of a number, including situations involving percentages less than one. • calculate the percent one number is of another. • calculate the percent of a number, including situations involving percentages less than one.

Problem Situation: Estimations with Percentages In your out-of-class experience, you read about the importance of estimation. Strong estimation skills allow you to make quick calculations when it is inconvenient or unnecessary to calculate exact results. You can also use estimation to check the results of a calculation. If the answer is not close to your estimate, you know that you need to check your work. In this course, you will make estimations and explain the strategies you used to generate estimations. There is not one best strategy. It is important that you develop strategies that make sense to you. A strategy is wrong only if it is mathematically incorrect (like saying that 25% is 1/2). In the following section, you will practice your use of estimation strategies to answer the questions and calculate percentages. Use estimation to answer the following questions. Try to make your estimation calculations mentally. Write down your work if you need to, but do not use a calculator. First, complete the problem yourself. When you complete the problem, discuss your estimation strategy with your group. Your group should discuss at least two different strategies for each problem. (1) You are shopping for a coat and find one that is on sale. The coat’s regular price is $87.99. What is your estimate of the sale price based on each of the following discounts? (a) 20% off (b) 25% off (c) 35% off (d) 70% off

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Estimations help you make calculations quickly in daily situations. This next problem shows how estimates of percentages can be used to make comparisons among groups of different sizes. (2) A law enforcement officer reviews the following data from two precincts. She makes a quick estimate to answer the following question: “If a violent incident occurs, in which precinct is it more likely to involve a weapon?” Make an estimate to answer this question and explain your strategy.

Number of Violent Incidents Precinct Number of Violent Incidents Involving a Weapon 1 25 5 2 122 18

(3) You have a credit card that awards you a “cash back bonus.” This means that every time you use your credit card to make a purchase, you earn back a percentage of the money you spend. Your card gives you a bonus of 0.5%. Estimate your award on $462 in purchases.

From Estimation to Exact Calculation Being able to calculate with percentages is also very important. In the situation in Question 1, an estimate of the sale price will help you decide whether to buy the coat. However, the storeowner needs to make an exact calculation to know how much to charge. In Question 2, an estimate helps the officer get a sense of the situation, but if she is writing a report, she will want exact figures. Calculate the exact answers for the situations in Questions 1–3. You may use a calculator. Show your work. (4) If the coat’s regular price is $87.99, what is the exact sale price based on each of the following discounts? (a) 35% off (b) 25% off (c) 70% off

(5) For each precinct, what is the exact percentage of incidents that involve a weapon? Round your calculation to the nearest 1%.

(6) Calculate the exact amount of your “cash back bonus” if your credit card awards a 0.5% bonus and you charge $462 on your credit card.

Making Connections Record the important mathematical ideas from the discussion.

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Further Applications (1) Estimate an answer to each of the following. Explain your estimation strategy. (a) 62% of 87 (b) 22% of 203 (c) 37 is what percent of 125 (d) 2 is what percent of 310

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Making Connections to the Lesson (1) Which of the following was one of the main mathematical ideas of the lesson? (i) Percentages are used to calculate sale prices. (ii) To calculate 35% of a number, multiply the number by 0.35. (iii) Percentages are a ratio of a number out of 100. For example, 16% means 16 out of 100. (iv) You should always calculate percentages exactly.

(2) Refer back to Lessons 1.1.1 and 1.1.2. Which statement below is a good description of how the important mathematical ideas of Lessons 1.1.1 and 1.1.2 connect to this lesson (1.1.3)? (i) Many people worry that the world population is growing too rapidly. The rate of growth has been increasing throughout history. (ii) Estimation is used in quantitative reasoning for many things, including estimating measurements, understanding large numbers, and making quick mental calculations. (iii) Large numbers are hard to understand. (iv) Calculating percentages is an important skill in quantitative reasoning because percentages are used in many situations.

Developing Skills and Understanding Reference Information on Percentages, Fractions, and Estimation There are many ways to do calculations with percents. The following videos and websites show some examples of methods. • Calculate the percentage rate: www.khanacademy.org/video/solving-percent-problems-2?playlist=Developmental Math • Finding the percent of a number: www.worsleyschool.net/science/files/percentofa/number.html • Both problems types: www.purplemath.com/modules/percntof.htm

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Language of Percentages and Fractions There are several important vocabulary words you should know and use. • A ratio is a comparison of two numbers by division. You will see many different types of ratios in this course. In this lesson, you worked with a special type of ratio called a percentage. A percentage is a ratio because it is a number compared to 100. • Percentages are a relationship between two values: the comparison value and the reference value. The relationship is described as a percentage rate, which is shown with a percentage symbol (%). This indicates that the rate is out of 100. Example: 10 is 20% of 50. 10 is the comparison value. 50 is the reference value. 20% is the percentage rate; it can be written as a decimal by using the relationship to 100:

• Fractions have two parts:

• Every fraction can be written in equivalent forms (e.g., ). It is often useful to write the fraction in the form with the smallest numbers. This is called simplified or reduced. In the example, is in simplest form.

The Language of Estimation Certain words or phrases are often used to indicate that a number is an estimate rather than an exact figure. Read the following statement: “Almost 30% of the patients had less pain.” The word almost indicates that the percentage was a little less than 30. Some words and phrases that are commonly used to signal estimates are shown below.

almost about approximately more than less than close to just over just under nearly

(3) At Gillway Community College, 43 out of 381 students earned honors. At Montessa Valley Community College, 17 out of 108 students earned honors. (a) Estimate the rate at which Gillway CC students earned honors. (b) Estimate the rate at which Montessa Valley CC students earned honors. (c) Which school had a higher rate of students earning honors? (d) Write a statement about the estimated percentage of students who earned honors at Gillway CC. (Use a word or phrase from the list above.) You may want to refer to the Writing Principle from Lesson 1.1.2 and the handout on writing about quantitative information.

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(4) Select all of the options that are either exactly equal to the given ratio or a good estimate of the ratio. There may be more than one correct answer. (a) 60% (i) 1 out of 6 (ii) 1 out of 60 (iii) 6 out of 10

2 (iv) close to 3 (v) 6 out of 100

(b) 8 out of 1,000 (i) less than 1% (ii) about 8%

1 (iii) about 8 (iv) more than 8% (v) 0.8%

(c)

(i) less than 1% (ii) almost 10% (iii) 8 out of 10 (iv) 2 out of 25 (v) 80%

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For the situations in Questions 5–8, decide if it would be more appropriate to make an estimate or to do an exact calculation. Give your answer, and specify if the number represents an estimate or a calculation.

(5) Your bill at a restaurant is $23.17. You want to leave about 20% for a tip. (a) How much should you leave? (b) Is the answer an estimate or calculation?

(6) You are completing a tax form. The tax is 15.3% of $47,000. (a) How much do you have to pay? (b) Is the answer an estimate or calculation?

(7) During an election for city council, you hear a candidate say that 68% of children in the city live in poverty. You know that your children’s school has about 1,200 students. (a) Based on the candidate’s statement, about how many children in the school live in poverty? (b) Is the answer an estimate or calculation?

(8) You are a teacher and are grading a test. A student got 42 out 58 points. (a) What is the student’s grade as a percentage? (b) Is the answer an estimate or calculation?

(9) Some checking accounts pay a small amount of interest on the money in the account. In this case, interest is money that is paid to the account holder by the financial institution issuing the checking account. The interest is a percentage of the amount of money in the account. The percentage is called the annual interest rate. Compare the following two offers. • Bank of Avalon pays 0.8% with no annual fee. • Cypress Savings pays 1.5%, but charges a $10 annual fee. Which would be the better offer if you have $1,000 in an account for 1 year?

Making Connections Across the Course (10) There are about 300 million people in the United States. A 2007 report1 claimed that the richest 1% of Americans controlled 42% of the nation’s wealth. About how many people is this? (i) 1,000,000 (ii) 3,000,000 (iii) 126,000,000

1Retrieved from http://sociology.ucsc.edu/whorulesamerica/power/wealth.html.

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(11) The same report claims that the poorest 80% of Americans controlled only 7% of the nation’s wealth. About how many people is this? (i) 7,000,000 (ii) 21,000,000 (iii) 240,000,000

(12) The nation’s wealth in 2007 was about $72 trillion dollars. About how much money did the richest 1% of Americans control? (Recall that they controlled 42% of the nation’s wealth.) (i) $720,000,000 (ii) $5,000,000,000 (iii) $30,000,000,000,000 (iv) $56,000,000,000 Note: In later lessons, you will be asked to compute things such as the average wealth per person among the richest 1% of Americans.

Working with Large Numbers Large numbers such as those used in Question 12 can be hard to read when written out as a number. In Lesson 1.1.2, you used exponents to write powers of 10. For example, 100,000,000,000 = 1011. You can use this idea to write other large numbers in the form of a number multiplied by a power of 10. For example, the number 124,000 can be written as 1.24 x 105. You can check that this is true by multiplying this expression out: 1.24 x 105 → 1.24 x 100,000 → 124,000 There are other ways that 124,000 could be written as a power of 10. For example, 12.4 x 104 is also equal to 124,000.

(13) Which of these are equal to 135,230,000,000? There may be more than one correct answer. (i) 1.3523 x 1011 (ii) 1.3523 x 107 (iii) 13.523 x 1010 (iv) One hundred thirty-five billion, two hundred thirty million (v) One hundred thirty-five million, two hundred thirty thousand

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(14) Write 68,000,000 in equivalent forms as instructed below. (a) as a number times 105 (b) as a number times 107 (c) in words

(15) Based on your self-regulated learning reading from OCE 1.1.1, when self-regulating your learning, what are the three phases you should go through?

(16) On which phase do experienced math students or mathematicians usually spend the most time?

(17) How does reflecting on how solving a problem went help you become a more efficient learner?

Preparing for the Next Lesson (1.1.4) (18) Match the fractions to their equivalent percent form (rounded to the nearest percent).

(a) (i) 10%

(b) (ii) 20%

(d) (iv) 33%

(e) (v) 50%

(f) (vi) 67%

(g) (vii) 75%

(19) In the OCE for Lesson 1.1.1, you used a place-value chart for place values greater than 1. Place value also extends to the right of the decimal to represent numbers less than 1. (a) Place the decimal point in the correct position in the place chart below. Complete the missing names in the chart.

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ousands Hundred thousands Hundred Thousands Ten Th Hundreds Tens Ones Tenths thousandths Hundred

(b) What is the name in words for the number 0.035? (c) What is the name in words for the percent 0.02%?

(20) Round 54,927.2382 to the specified place value. (a) hundredth (b) hundreds (c) tens (d) thousands (e) tenths (f) thousandths

(21) Convert each fraction to a decimal. Round to the nearest thousandth.

(a)

(b)

(c)

(d)

(e)

(f)

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Lesson 1.1.4 focuses on skills needed to be a “flexible quantitative thinker.” One of these skills is recognizing when calculations can be done in different ways. In the following question, you will be asked if two number expressions are equivalent. An example of a number expression is 3 + 4. Equivalent means that the two expressions mean the same thing. So 3 + 4 is equivalent to 4 + 3 because it does not matter in what order you add the numbers. (22) Complete the table by marking whether the first and second expressions are equivalent.

Equivalent? First Expression Second Expression Yes No 5 x 7 7 x 5 8 – 4 4 – 8 10 ÷ 2 2 ÷ 10

20 ÷ 2

Multiplying Fractions You can think of multiplying fractions in terms of area. Look at the square below. The product can be represented by the dark gray area found by dividing a square into thirds horizontally (shade 2/3) and fifths vertically (shade 4/5) as shown. The region that is shaded twice is darker than the rest. Notice that the square is now divided into 15 regions (15 = 3 x 5), and the number of those regions that are dark gray is 8 (8 = 2 x 4). So 8 out of 15 pieces of the square are dark gray, or 8/15. This prompts the rule for multiplying fractions: multiply the numerators (2 x 4) and multiply the denominators (3 x 5), and

simplify if possible. Therefore,

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(23) Multiply the following fractions. Write answer in simplified form.

(a)

(b)

(24) The following information will be used in Lesson 1.1.4. The 2009 Consumer Expenditure Survey studied how Americans spend their income. (An expenditure is something you spend money on.) The survey is summarized on the website www.creditloan.com/infographics/how-the-average- consumer-spends-their-paycheck. Use the diagram on the website to answer the following questions. (a) What are the average annual expenditures per household? (b) What percentage of a household’s expenditures is used to pay for housing? (c) Which fraction would you use to approximate the percentage of a household’s income that is used to pay for housing?

(i)

(ii)

(iii)

(iv)

(v)

Are you prepared for Lesson 1.1.4? (25) Did you read and understand the information to be used in class (Question 24)?

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(26) You should be able to do the following things for the next class. Rate how confident you are on a scale of 1–5 (1 = not confident and 5 = very confident). Before beginning Lesson 1.1.4, you should understand the concepts and demonstrate the skills listed below:

Skill or Concept: I can … Rating from 1 to 5 Recognize common fraction benchmarks and equivalent percent form. Round a whole number to a given place value. Perform calculations using a calculator. Understand the relationship of multiplication and division (dividing by 3 is the same as multiplying by 1/3). Convert between a fraction and its decimal form. Multiply fractions.

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Specific Objectives Students will understand that • flexibility with calculations is an important quantitative skill. • different methods of calculation are often possible and helpful. Students will be able to • write a calculation in at least two different ways based on o equivalent forms of fractions/decimals. o relation of multiplication and division. o the Commutative Property. [knowing when the order of numbers can be reversed, such as 3 + 4 = 4 + 3, but 3 – 4 ≠ 4 – 3] o order of operations o the Distributive Property. [5(3 + 4) = 5 x 3 + 5 x 4]

Problem Situation: Performing Calculations in Multiple Ways The ability to solve problems in multiple ways is an important quantitative reasoning skill. Today’s lesson asks you to brainstorm different ways to find the answer to a question. This flexibility is important because different strategies are often useful in different situations. You saw in Lesson 1.1.3 that estimation strategies often depend on the specific numbers. This can also be true in calculations. Sometimes changing the order of operations or grouping operations in other ways can be helpful. It is important to know when you can make changes such as these and still make the correct calculations. You will use information from the 2009 Consumer Expenditure Survey for today’s lesson. This survey provides detailed information about how American consumers spend money. It contains information about individuals and what they purchase. The survey also has information about a typical family’s income and what that family uses its money to buy. The survey refers to each family as an “average household.” The 2009 Consumer Expenditure Survey studied how Americans spend their income. (An expenditure is something you spend money on.) The survey found that the average household had an income of $62,857. The survey also found that the average household spent about one-third (1/3) of its income on housing. This expenditure was either rent, if the family rented a home, or mortgage payments, if the family owned its home.1 You will use the information summarized above to answer the following questions. (1) Estimate how much the average household spent on housing. Try to do the estimate mentally (without writing it down or using a calculator) if you can. Explain your strategy for your estimation. (Note: It is okay if people in your group use different strategies and for your estimates to be different.) (2) How would you write a mathematical expression to find how much the average household spends on housing? (16.4 x 32 is an example of a mathematical expression.) Try to find as many different statements as possible.

1 Retrieved from www.creditloan.com/infographics/how-the-average-consumer-spends-their-paycheck.

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(3) If one-third of expenditures went to housing, what fraction went toward other expenses?

(4) How could you calculate the amount spent on expenses other than housing? Think of as many different ways as you can.

(5) Which of the methods from Question 4 makes the most sense to you? Explain why. (Note: Your answer does not have to agree with your group.)

(6) The Montero family has the following average monthly expenses. Calculate how much they spend on housing (this includes rent and utilities) in one year.

Rent $1,250 Electricity $85 Gas $120 Water and sewer $72

(7) Look at your answer in Question 6. Does it seem reasonable? Reasonable often means that your answer is not too big or too small to make sense. Write a short sentence about why your answer is reasonable. If your answer is not reasonable, check your calculations.

Making Connections Record the important mathematical ideas from the discussion.

Further Applications (1) The graph on the following page represents the budget of an average college student according to Westwood College.2 Write three questions about these data that require calculations or estimation. You may refer to Questions 1–4 in the lesson for examples. Include the answers to your questions. (Note: You may need to make up amounts to represent a student budget, as that information is not given.)

2Retrieved from www.westwood.edu/resources/student-budget

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Making Connections to the Lesson (1) Which of the following was one of the main mathematical ideas of the lesson? (i) You can change a calculation in any way that you think will make it easier to do. (ii) Calculations can often be performed in different ways based on mathematical rules.

2 (iii) Multiplying by is the same as multiplying by 2 and then dividing by 3. 3 (iv) The average household spends about 33% of its income on housing.

(2) Lessons 1.1.3 and 1.1.4 both emphasized that there are different ways to approach problems. This is true for both estimation and calculations. The best strategy depends on the situation, the numbers used, and the way you think. Select one question from each lesson that is an example of this idea.

Question Lesson Show at least two ways to do the problem. Number 1.1.3

1.1.4

Developing Skills and Understanding In Lesson 1.1.4, you used several important mathematical rules and relationships to perform calculations in different ways. Those rules are summarized for you here so you can refer back to them. The authors are also introducing the formal names for the rules. You do not have to memorize these names for this course, but you may use them in other math classes. If you want more help with any of the rules, use the formal names to find resources on the Internet. Mathematical rules are defined in terms of variables. The variables are symbols, usually letters, that represent numbers. You use variables to show that the rule can apply to multiple numbers. This is called generalizing because it shows that a rule can be used in general and not just in specific cases. The rule using both variables and numbers will be shown. While mathematical rules are very important, in this course, the authors emphasize reasoning over memorizing rules. As you review the rules, try to make sense of the rules so that they will become a part of your thinking.

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Commutative Property The order of addition and multiplication can be changed.

General Rule Example a + b = b + a 8 + 3 = 3 + 8 a x b = b x a 5 x 6 = 6 x 5

It is important to remember that the Commutative Property does not apply to subtraction and division.

Order of Operations The order of operations defines the order in which operations are performed.

General Rule Example

1. Operations within grouping symbols, 15 + [12 – (3 + 2) ] – 2 × 32 ÷ 6 innermost first. Grouping symbols include 15 + [12 – (5) ] – 2 × 32 ÷ 6 • Parentheses ( ) 15 + [7] – 2 × 32 ÷ 6 • Brackets [ ] • Fraction Bar

2. Exponents 15 + [7] – 2 × 9 ÷ 6

3. Multiplication and division, left to right 15 + [7] – 18 ÷ 6 15 + [7] − 3

4. Addition and subtraction, left to right 22 – 3 19

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Distributive Property The Distributive Property is easiest to understand by looking at examples.

General Rule Example a (b + c) = a × b + a × c 4 (3 + 1) = 4 × 3 + 4 × 1 Note about subtraction: Subtraction is related to To demonstrate that these two calculations are addition. The Distributive Property is shown using equivalent, each side is done separately. addition, but it also works with subtraction as Left side: Using order of operations, the operation shown below: inside the parentheses is done first. 8 (5 – 1) = 8 × 5 – 8 × 1 4 (3 + 1) 4 (4) Notation: The operation of multiplication is shown 16 in many ways. You have already seen the use of the Right side: Using the Distributive Property, the multiplication symbol (x). Another way to indicate multiplication is distributed over the addition. multiplication is a number or variable in front of parenthesis with no other symbol. For example: 4 (3 + 1) 6(2) = 6 x 2 4 × 3 + 4 × 1 a(b) = a x b Order of operations tells you to multiply first. You will learn other symbols for multiplication later 12 + 4 in the course. 16

Division Division is the same as multiplication by the reciprocal. You get the reciprocal of a number when you write the number as a fraction and reverse the numerator (the top number) and the denominator (bottom number).

General Rule Example

a ÷ b = a × 1 15 ÷ 5 = 15 × 1 b 5

a ÷ b = a × c 10 ÷ 3 = 10 × 5 c b 5 3

(3) In Lesson 1.1.4, you saw that there was a relationship between multiplication and division. Refer back to this work to complete the following statement.

62,857 ÷ 3 is the same as

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(4) Using the concept from the previous question, fill in the blanks to create equivalent statements.

Multiplication Division

85 × 1

5 85 ÷

1.23 × 1.23 ÷ 7

1.23 ÷ 2 1.23 × 3

(5) Which expressions are equivalent to 16 x ? There may be more than one correct answer. (i) 16 x 3 ÷ 4 (ii) 16 ÷ 0.75 (iii) 3 x 16 ÷ 4 (iv) 3 ÷ 4 x 16 (v) 16 x 0.75 (vi) 16 ÷ 4 x 3 (vii) 0.75 x 16 (viii) 16 x 4 ÷ 3

(6) According to the Consumer Expenditure Survey, the average American household spent $6,372 on food in 2009. About two-fifths of that was spent on eating out at restaurants. Calculate two-fifths of $6,372 to estimate the amount that was spent on eating out.

Introduction to Spreadsheets A spreadsheet is a computer program used to organize and analyze data. In the example below, Lisa has created a spreadsheet for her monthly budget. Data is entered into cells, like the boxes in a table. The cells are named by the letter of the column along the top and numbered rows down the side. Note the cell that contains the word income is labeled as A2, not 2A.

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Use this spreadsheet to answer Questions 7–9. (7) What is in Cell B4?

(8) What does the number in Cell B4 represent in Lisa’s budget? (i) The money she plans to spend on rent each month. (ii) The money she plans to spend on utilities each month. (iii) The money she plans to spend on food each month. (iv) The money she plans to spend on insurance each month. (v) The money she plans to spend on gas for her car each month.

Formulas can be used to perform calculations in spreadsheets. The formulas use the cell name as a variable that represents the value in that cell. For example, in the spreadsheet above, the formula =B3+B4 would result in the calculation 750 + 230, and $980 would be displayed. Spreadsheets are a valuable tool because once a formula is written, its result changes when the values change. So if Lisa’s rent increases, she can change the number in Cell B3. The formulas calculate the new results automatically.

(9) Lisa put the following formula in her spreadsheet: = B2 – B3 – B4 – B5 – B6 – B7. (a) Calculate the result of this formula. (b) What does this value represent for Lisa? (i) The amount of money she expects to lose each month. (ii) The amount of money she expects to have left after paying bills each month. (iii) The percentage of her income that she will be able to save each month. (iv) The value has no meaning for Lisa.

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(c) Which of the following expressions would give the same result as Lisa’s formula? There may be more than one correct answer. (i) = B2 – B3 + B4 + B5 + B6 + B7 (ii) = B2 – (B3 + B4 + B5 + B6 + B7) (iii) = (B3 + B4 + B5 + B6 + B7) – B2

Making Connections Across the Course (10) Which of these expressions show ways to calculate 25% of 2,310? There may be more than one correct answer. (i) 2,310 ÷ 4 (ii) 2,310 x 4 (iii) 2,310 ÷ 25

(iv) 2,310 x 25 (v) 2,310 x 0.25 (vi) 2,310 ÷ 0.25

(vii) x 2,310

(viii) ÷ 2,310

(ix) 0.25 x 2,310 (x) 0.25 ÷ 2,310

(11) Which expression is the same as 20% of a billion? There may be more than one correct answer. (i) 0.2 x 1,000,000,000 (ii) 0.2 x 1,000,000 (iii) 109 ÷ 5 (iv) 109 ÷ 20 (v) 106 ÷ 5 (vi) One-fifth of 1,000 million (vii) 20,000,000 (viii) 20 ÷ 100 x 1,000,000,000

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Scientific Notation In OCE 1.1.3, you saw that a large number can be written as a number times a power of 10 in many different ways. For example, the number 124,000 can be written as 1.24 x 105 or 12.4 x 104. These different forms are all equivalent. Scientific notation is a very specific way to write a large number as a power of 10. The purpose of scientific notation is to make it easier for people to use and communicate with large numbers. It would be confusing if two people working together on one project wrote the same number in two different ways. To avoid this, people decided that numbers in scientific notation would always be written in the same way: a number between 1 and 10 times a power of 10. From the previous example: • 1.24 x 105 is in scientific notation because 1.24 is a number between 1 and 10. • 12.4 x 104 is not in scientific notation because 12.4 is larger than 10. Write each of the following numbers in scientific notation. (12) 16,900,000 (13) 4,275,000,000

Self-Regulating Your Learning: The Plan Phase At the start of this module, the authors briefly described what it means to be a “self-regulated learner.” As you already learned, being a self-regulated learner involves going through three phases when you are working on a problem or an assignment. The phases are 1. Plan 2. Work 3. Reflect In this lesson, you will look at what you should be doing during the Plan phase. As you might imagine, the planning phase involves thinking about all the things you need to do to successfully complete a problem or assignment before you begin working on it. As was said previously, researchers who study how people learn found that experts often spend a lot more time planning how they are going to finish a task than they spend actually doing the task. The planning phase involves several important aspects. The following are some that will be explored in this course: • How much confidence you have that you can successfully complete the problem. • The amount of time and effort you think it will take to understand and work on the problem. • The strategies you might use to solve the problem. • The goals you have as you try to work on the problem. The authors will now describe each aspect in a little more detail. You will also continue to revisit them throughout the rest of the course.

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Confidence: People who study how you learn have found that your beliefs regarding your ability to do a given task, like work a particular math problem, often predicts how well you actually do. Here is one way to think about it: If you really believe you can succeed at a problem, you are more likely to keep trying and keep working on that problem even if you get stuck. Because you invest more effort, you are more likely to be successful. On the other hand, if you look at a problem and immediately think “I cannot do this,” then when you do get stuck or confused, you might be more likely to give up and not be successful. Researchers call your beliefs about your abilities your self- efficacy. In this course, you will be asked to rate your self-efficacy on certain problems. If you rate yourself low, then you might want to allow more time to do that problem, plan to go get help, or try being more patient than you might normally be. Thinking about your confidence can help you plan your time and effort when you work on a problem or task. Time and Effort: Obviously, some problems or assignments take more time than others. Some assignments require more effort than others. It can be frustrating to jump into an assignment thinking you can finish it easily or quickly only to discover it is harder or takes way more time than you thought it would. You can avoid some or all of that frustration if you have a realistic idea of how hard the assignment will be. Also, having a good idea of how much time and effort will be needed helps you manage your time. For example, you might need to allocate time to discuss the assignment with your instructor, classmates, or tutors. For these reasons, approximating the time and effort needed before starting work on an assignment is a good planning tool. Strategies: When you start working on a problem or assignment, you often have to try several different strategies before you find an approach that will help you complete it successfully. Sometimes, it is the first strategy you think of, but often it is not. If you think about possible strategies before you begin working, you immediately have another one to try if your first one does not work. Self-regulated learners think about many different possible strategies, and then begin trying to solve a problem. Goals: Education researchers have shown that students who have “learning goals,” are more likely to succeed than students who have what are called “performance goals.” If you have learning goals, you are trying to understand what you are learning and trying to make connections between ideas and concepts. If you have performance goals, you care most about finishing an assignment to get points or have it done; you are not focused on understanding the material. Self- regulated learners try to have learning goals more than performance goals. This helps them stay focused and motivated to learn when the problems are challenging. Good planning means making an effort to change your thinking so you have learning goals as often as possible. In future lessons and assignments, you will have opportunities to practice the planning ideas presented here. Before then, start incorporating the planning phase whenever you start an assignment. If you do, you will be better prepared and more likely to succeed.

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Preparing for the Next Lesson (1.1.5) (14) Which expressions are ways to write the number 5,200,000? There may be more than one correct answer. (i) 52 million (ii) 5.2 billion (iii) 5.2 million (iv) Five million, two hundred thousand (v) Fifty-two million (vi) 5,200

(15) Estimate the following percentages without using a calculator. (a) 10.1% of 7,800 (b) 0.99% of 83,583 (c) 20% of 5,008,340 (d) 0.52% of 472,028

(16) Which expressions are equivalent to ? There may be more than one correct answer.

(i)

(ii)

(iii)

(iv)

(v)

The following information will be used in Lesson 1.1.5. You will be given some information about how much you have to pay to borrow money on a credit card. The company charges interest on the amount that you do not pay off each month. This is called your balance. Interest is based on a percentage of the amount you have borrowed. The annual interest rate is the APR (annual percentage rate.)

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Creditworthiness is how likely it is that you will pay your bills on time. It is measured by a credit score that can range from 300 to 850. Someone with a high credit score has good credit and will get a lower interest rate than someone with a low credit score. Credit cards are very complicated. Because it is important for people to understand how much credit cards charge, the U.S. government has a law called the Credit Card Accountability, Responsibility, and Disclosure Act of 2009, which requires companies to publish information about rates and fees in a standard format. This is called a disclosure or the pricing and terms. The disclosure begins with a summary like the one shown below. Scan this form. (This means to read it quickly to get a general sense of the information without trying to understand every detail.) This information will be discussed in more detail in Lesson 1.1.5.

Interest Rates and Interest Charges

Annual Percentage 0.00% introductory APR for 6 months from the date of account opening. Rate (APR) for Purchases After that, your APR will be 10.99% to 23.99% based on your creditworthiness. This APR will vary with the market based on the Prime Rate.

APR for Balance 0.00% introductory APR for 24 months after the first transaction posts to your account Transfers under this offer. After that, your APR will be 10.99% to 23.99% based on your creditworthiness. This APR will vary with the market based on the Prime Rate.

APR for Cash 28.99%. This APR will vary with the market based on the Prime Rate. Advances

Penalty APRs and Between up to 16.99% and up to 26.99% based on your creditworthiness and other When It Applies factors. This APR will vary with the market based on the Prime Rate. This APR may be applied to new purchases and balance transfers on your account if you make a late payment. How long will the penalty APR apply?: If your APRs for new purchases and balance transfers are increased for a late payment, the Penalty APR will apply indefinitely. How to Avoid Your due date is at least 25 days after the close of each billing period (at least 23 days for Paying Interest on billing periods that begin in February). We will not charge you any interest on purchases Purchases if you pay your entire balance by the due date each month. Minimum Interest If you are charged interest, the charge will be no less than $0.50. Charge

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Are you prepared for Lesson 1.1.5? (17) SRL: Plan You should be able to do the following things for the next class. Rate how confident you are on a scale of 1–5 (1 = not confident and 5 = very confident). Before beginning Lesson 1.1.5, you should understand the concepts and demonstrate the skills listed below:

Skill or Concept: I can … Rating from 1 to 5 Name and understand large numbers written in different forms. Use benchmarks to estimate percentages including percentages less than 1%. Use order of operations and the Distributive Property to write expressions in different forms.

(18) SRL: Plan The next lesson (1.1.5) will be a review of concepts in the course so far. After reading the information above about the Plan phase, think about what you might do to be well prepared for your next class session. Try to incorporate the ideas of confidence (self-efficacy), time and effort, strategies, and your goals. Write out your planning ideas. Your instructor may ask you to discuss this in class.

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Specific Objectives Students will understand that • quantitative reasoning and math skills can be applied in various contexts. • creditworthiness affects credit card interest rates and the amount paid by the consumer. • reading quantitative information requires filtering out unimportant information (introductory level). • course expectations regarding writing about mathematics in context. Students will be able to • recognize common mathematical concepts used in different contexts. • apply skills and concepts from previous lessons in new contexts. • identify a complete response to a prompt asking for connections between mathematical concepts and a context. • write a formula in a spreadsheet.

Problem Situation: Understanding Credit Cards When you use a credit card, you can pay off the amount you charge each month. If you do not pay the full amount, you are borrowing money from the credit card company. This is called credit card debt. Many people in the United States are concerned about the amount of credit card debt both for individuals and for society in general. In this lesson, you will use skills and ideas from previous lessons to think about some issues related to credit cards. You may want to refer back to the previous lessons.

(1) The statements below came from two websites that report predictions about credit card debt in 2010: • “In 2010, the U.S. census bureau is reporting that U.S. citizens have over $886 billion in credit card debt and that figure is expected to rise to $1.177 trillion this year.”1 • The debt in 2010 is “expected to grow to a projected 1,177 billion dollars.”2 Do these two websites project the same amount of debt? Or did one of the websites make an error? Justify your answer with an explanation.

1Retrieved from www.hoffmanbrinker.com/credit-card-debt-statistics.html 2Retrieved from www.money-zine.com/Financial-Planning/Debt-Consolidation/Credit-Card-Debt-Statistics

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You will use the following information from the disclosure for Questions 2 and 3.

(2) Creditworthiness is measured by a “credit score,” with a high credit score indicating good credit. In the following questions, you will explore how your credit score can affect how much you have to pay in order to borrow money. Juanita and Brian both have a credit card with the terms in the disclosure form given above. They have both had their credit cards for more than 6 months. (a) Juanita has good credit and gets the lowest interest rate possible for this card. She is not able to pay off her balance each month, so she pays interest. Estimate how much interest Juanita would pay in a year if she maintained an average balance of $5,000 each month on her card. Explain your estimation strategy. (b) Brian has a very low credit score and has to pay the highest interest rate. He is not able to pay off his balance each month, so he pays interest. Calculate how much interest he would pay in a year if he maintained an average balance of $5,000 each month. Show your calculation. (c) What are some things that might affect your credit score?

(3) The APR is an annual rate, or a rate for a full year. The APR is divided by 12 to calculate the interest for a month. This is called the periodic rate. (a) What is the periodic rate for Juanita’s card? Round to two decimal places. (b) Juanita has a balance of $982 on her January statement. Which of the following is the best estimate of how much interest she will pay?

Less than a dollar $5–$10 $10–$20 More than $20

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You will use the following information from the disclosure for Question 4. A cash advance is when you use your credit card to get cash instead of using it to make a purchase.

Annual Percentage After that, your APR will be 10.99% to 23.99% based on your Rate (APR) for creditworthiness. This APR will vary with the market based on the Prime Rate. Purchases

APR for Cash 28.99%. This APR will vary with the market based on the Prime Rate. Advances

(4) Discuss each of the following statements. Decide if it is a reasonable statement. (a) Jeff pays the highest interest rate for purchases. For a cash advance, he would pay $0.05 more for each dollar he charges to his card. (b) The interest for cash advances is about two-and-a-half times as much as for the lowest rate for purchases.

Brian used a spreadsheet to record his credit card charges for a month.

Brian used the following expression to calculate his interest for these charges for one month.

(5) Which of the following statements best explains what the expression means in terms of the context? (i) Brian added his individual charges. Then he divided 0.2399 by 12. Then he multiplied the two numbers. (ii) Brian found the interest charge for the month by dividing 0.2399 by 12 and multiplying it by the sum of Column B. (iii) Brian added the individual charges to get the total amount charged to the credit card. He found the periodic rate by dividing the APR by 12 months and multiplied the rate by the total charges. This gave the interest charge for the month.

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Making Connections Record the important mathematical ideas from the discussion.

Further Applications (1) Refer to Question 6 in the Lesson 1.1.5 OCE. Write an explanation of at least one estimation strategy that could have been used for each correct statement. (2) Refer to the expression given in Question 3 of the Lesson 1.1.5 OCE. Why do you do the addition in the numerator before dividing by 12?

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Making Connections to the Lesson (1) Which of the following was one of the main mathematical ideas of the lesson? (i) The number $1.177 trillion can also be written as $1,177 billion. (ii) Credit cards are expensive to use if you do not pay off your balance each month. You pay more interest for cash advances than for the balance on purchases. Credit card debt is a problem in the United States. (iii) A percentage is always a number greater than 1.

(iv) Understanding numbers includes knowing how numbers compare in size, knowing what numbers represent in situations, and using estimation to answer questions about numbers.

(2) Four lessons are listed below. In each lesson, you were asked to make sense of numbers in different ways. Find a specific example from the lessons. Use the first two as examples.

Lesson We made sense of numbers when we … 1.1.1 Asked if the statistic on the sign was reasonable. 1.1.2 Used the idea of lines of people to compare the size of a million to a billion. 1.1.4 1.1.5

Developing Skills and Understanding (3) Refer back to Question 5 in the lesson. (a) A student used a different expression to calculate Brian’s monthly interest. Choose the sentence that best explains what the expression means in terms of the context and the order in which the calculations were done. Spreadsheets use an asterisk (*) to indicate multiplication: 3 * 4 means 3 times 4.

0.2399 ∗ B2 + 0.2399 ∗ B3 + 0.2399 ∗ B4 + 0.2399 ∗ B5

(i) Find the annual interest for each individual charge and then add to find the total annual interest. Divide by 12 months to find the interest for 1 month. (ii) Distribute 0.2399 to the sum of the charges and then divide by 12. (iii) Divide the annual interest rate by 12 to find the monthly interest rate and then multiply by each of the charges to find the monthly interest for each charge. Add the monthly interest for each charge to find the total monthly interest. (iv) Multiply each entry in the B column by 0.2399. Add the results and divide by 12 to find the final answer.

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(b) Open a spreadsheet program. Enter the information shown in Question 5 from the lesson. In which cell(s) should the formula for calculating the monthly interest be entered? (i) C2 through C5 (ii) B7 (iii) B6 (iv) C7 (v) A6 (c) Enter the formula given in Question 5 into the correct cell. To do this, click on the cell. First type =. (A formula in a spreadsheet always starts with an = sign.) Type the formula. Notice as you type that your formula appears in the cell and also in the formula bar above the spreadsheet cells. Press enter. Record the result (what appears in the cell) when you are done.

(4) Refer back to Question 1 in the lesson. (a) Write the projected debt in standard form (written as a number like 374,000). (b) What is the projected debt in scientific notation? (i) 1,177 x 109 (ii) 11.77 x 1012 (iii) 1.177 x 1012 (iv) 1.177 x 1011 (v) 11.77 x 1011

(5) The Federal Reserve has useful consumer information about credit cards. Go to the website www.federalreserve.gov/creditcard. Select the option, “Learn more about your offer.” This is an interactive site in which h you can get information by clicking on parts of the offer form. Use the information to answer the following questions. (a) Which of the following can trigger a penalty annual percentage rate (APR)? There may be more than one correct answer. (i) You are late in paying your bill. (ii) You pay your bill too early. (iii) You do not use the credit card for six consecutive months. (iv) You go over your credit limit.

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(b) Which of these statements are true? There may be more than one correct answer. (i) A man has a car loan and a credit card with Great American Bank. He misses a payment on his car loan. Great American can charge the penalty APR on his credit card. (ii) All credit cards charge an annual fee. (iii) You do not pay interest on a cash advance until 25 days after the advance is made. (iv) If you pay your bill late, in addition to paying a higher penalty rate, you will also pay a penalty fee. (c) How can you avoid paying interest on purchases? (i) Always make the minimum payment on time. (ii) Avoid late fees. (iii) Pay the entire balance by the due date. (iv) Pay the minimum interest charge. (d) Use the introductory APR shown in the disclosure on the website. How much more in interest would you pay in one year for a balance of $5,000 if you have a very high credit score compared to having a very low credit score?

(6) A college student is talking to her family about a February 1, 2010, news story she read at msnbc.com.1 It states: Florida college students could face yearly 15 percent tuition increases for years, and University of Illinois students will pay at least 9 percent more. The University of Washington will charge 14 percent more at its flagship campus. And in California, tuition increases of more than 30 percent have sparked protests reminiscent of the 1960s. The student attends the University of California and paid about $7,800 in tuition in 2009. Which of the following statements is a good quantitative description of how her tuition will change based on the news story? There may be more than one correct answer. (i) My tuition is going to increase by almost a third! (ii) My tuition will go up by more than $2,000. (iii) My tuition is going up a lot! (iv) My tuition will be around $9,000. (v) My tuition will be around $8,500.

1Retrieved from www.msnbc.msn.com/id/35185920/ns/us_news-life/t/coast-to-coast-double-digit-college-tuition-hikes

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Making Connections Across the Course Big budget movies are tracked by investors and consumers. The following table gives data on the six movies with the largest budgets that had been released as of June 20, 2010.2 The data includes an estimate of the U.S. gross earnings and worldwide gross earnings of movies. Gross earnings is the amount of money that a movie takes in.

Release U.S. Gross Gross Earnings Movie Distributor Budget Date Earnings Outside U.S.

Pirates of the 5/25/2007 Caribbean: At Buena Vista $300,000,000 $309,420,425 $651,576,067 World’s End 11/24/2010 Tangled Buena Vista $260,000,000 $200,821,936 $385,760,000 5/4/2007 Spider-Man 3 Sony $258,000,000 $336,530,303 $554,345,000 Pirates of the 5/20/2011 Caribbean: On Buena Vista $250,000,000 220,746,502 $731,900,000 Stranger Tides Harry Potter and 7/15/2009 the Half-Blood Warner Bros. $250,000,000 $301,959,197 $632,000,000 Prince 20th Century 12/18/2009 Avatar $237,000,000 $760,507,625 $2,023,411,357 Fox

(7) Write the name in words for the gross earnings outside the United States for Avatar.

(8) Which of the following calculations shows a correct method to estimate the net earnings for Pirates of the Caribbean: At World’s End? Net earnings is the total amount the movie makes after expenses (the budget) are taken out. There may be more than one correct answer. (i) ($310,000,000 + $650,000,000) – $300,000,000 = $660,000,000 (ii) $650,000,000 – ($310,000,000 + $300,000,000) = $660,000,000 (iii) The budget and the U.S. gross earnings are about the same and cancel each other out. The net earnings would be about the same as the gross earnings outside the United States, or about $650 million dollars. (iv) The gross earnings are about $600 million plus $300 million, or $900 million. The expenses are about $300 million. So, the net earnings are about $600 million dollars.

2Retrieved from www.the-numbers.com/movies/records/budgets.php.

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(9) Refer to the data for Harry Potter and the Half-Blood Prince. (a) Estimate the net earnings. (b) Write two statements to explain a way to estimate the net earnings. One should be a numeric expression (as in 8i) and the other should be in words (as in 8iv). (10) The return on investment is the percentage that the net earnings are of the budget. Which of the following statements best estimates the return on investment for Pirates of the Caribbean: At World’s End? There may be more than one correct answer. (i) The net earnings for Pirates of the Caribbean: At World’s End were more than triple the investment. (ii) The net earnings for Pirates of the Caribbean: At World’s End were more than double the investment. (iii) The return on investment for Pirates of the Caribbean: At World’s End was more than 300%. (iv) The return on investment for Pirates of the Caribbean: At World’s End was more than 200%.

(11) In OCE 1.1.4, you read about self-regulating your learning during the plan phase. Explain briefly why it is important to evaluate your confidence before planning on working a problem.

(12) From your previous reading about the plan phase, what is the difference between performance goals and learning goals? Explain why students with learning goals are often more successful.

Preparing for the Next Lesson (1.2.1) (13) Which of the following can be used to represent 5 out of 20? There may be more than one correct answer. (i)

(ii) 0.25 (iii) 25% (iv) 0.25% (v) 25

(14) What is another way to represent 3/5? There may be more than one correct answer. (i) 1.66 (ii) 166% (iii) 60% (iv) 0.6 (v) 6/10 (vi) 3 out of 5

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(15) Which of the following is the standard form of 1.23 x 1011? (i) 1,230,000,000,000 (ii) 123,000,000,000 (iii) 12,300,000,000 (iv) 1,230,000,000

Scientific Notation and Calculators Scientific notation is useful because it is easy to make mistakes when working with numbers that contain a lot of zeros. You can use scientific notation with a calculator that has an exponent feature. Instructions for using exponents with two different types of calculators are given as follows.

Scientific Calculator: These calculators have a key that looks like one of the following:

xy yx or

The keystrokes for entering 108 are

1 0 xy 8

Graphing Calculators and Computers: Graphing calculators and computers have a key that looks like the picture shown below. This is called a caret symbol. It is also used for exponents in computer programs, including spreadsheets. ^

The keystrokes for entering 108 are 1 0 ^ 8

Calculators automatically display results to calculations in scientific notation when the numbers have too many digits to be displayed on the screen. The way that these are displayed varies slightly with different calculators. One common display is 2.34 E9, which represents 2.34 x 109.

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(16) As of 2011, the world population was estimated to be about 6.93 x 109. (Recall the discussion of scientific notation from the previous assignment.) About 4.5% of the world’s population lives in the United States. Approximately how many people live in the United States? (i) 1.54 x 108 (ii) 3.1 x 109 (iii) 3.1 x 108 (iv) 1.54 x 109 (v) 3.1 x 107 (vii) 1.54 x 107 (17) Water usage varies greatly in different countries, from as little as 20 liters a day per person in some third world countries, to 600 liters a day in the U.S. How much water would be needed for one day if every person in the world used 50 liters of water a day? (i) Write your answer in scientific notation. (ii) Write your answer in standard notation (as a number). (iii) Write your answer in words.

Background Information for the Upcoming Lesson The following information will be used in Lesson 1.2.1. You will again examine the situation of the earth’s population. Recall in Lesson 1.1.2 that you looked at how the population has grown and is currently growing. As stated in Lesson 1.1.2, “Numerous scientists have conjectured about how long we can sustain ourselves, as we cruise the solar system in our self-contained environment.” One of the most important natural resources that humans need for survival is water. An influential United Nations report published in 2003 predicted severe water shortages will affect 4 billion people by 2050. This report also said that 40 percent of the world’s population did not have access to adequate sanitation facilities in 20033. You need clean water not just for drinking, but for necessary tasks such as sanitation, growing food, and producing goods. You will use a measure of water consumption, called a “water footprint” that includes all of the ways that people use fresh water. According to Waterwiki.net, “The water footprint of an individual, business or nation is defined as the total volume of freshwater that is used to produce the goods and services consumed by the individual, business, or nation.”4 Goods are physical products such as food, clothes, books, or cars. Services are types of work done by other people. Examples of services are having your hair cut, having a mechanic fix your car, or having someone provide day care for your children. Fresh water is often used to make goods and to provide you with services.

3Retrieved from Rajan, A. Forget carbon: you should be checking your water footprint. Monday, 21 April 2008. Link [http://www.independent.co.uk/environment/green-living/forget-carbon-you-should-be-checking-your-water-footprint- 812653.html] 4Retrieved from http://waterwiki.net/index.php/Water_footprint

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To prepare for your class, make sure you understand this information and understand the term water footprint. For more information, you can do an Internet search for “definition of water footprint.” Or, review the following two resources: • “Forget carbon: you should be checking your water footprint” by Amol Rajan, April 21, 2008. www.independent.co.uk/environment/green-living/forget-carbon-you-should-be-checking- your-water-footprint-812653.html • http://waterwiki.net/index.php/Water_footprint

Are you prepared for Lesson 1.2.1? (18) Did you read and understand the information to be used in class?

(19) You should be able to do the following things for the next class. Rate how confident you are on a scale of 1–5 (1 = not confident and 5 = very confident). Before beginning Lesson 1.2.1, you should understand the concepts and demonstrate the skills listed below:

Skill or Concept: I can … Rating from 1 to 5 Calculate a quotient (one number divided by another). Use calculator to divide numbers. Use scientific notation. Convert between fractions, percents, and decimals.

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Specific Objectives Students will understand that

• the magnitude of large numbers is seen in place value and in scientific notation. • proportions are one way to compare numbers of varying magnitudes. • different comparisons may be needed to accurately compare two or more quantities.

Students will be able to

• express numbers in scientific notation. • estimate ratios of large numbers. • calculate ratios of large numbers. • use multiple computations to compare quantities. • compare and rank numbers including those of different magnitudes (millions, billions).

Problem Situation 1: Comparing Populations In your out-of-class experience, you read about a “water footprint.” In this lesson, you are going to compare the populations of China, the United States, and India. You will go on to look at the water footprint for each nation as a whole and per person (“per capita”) to make some comparisons and to consider what this information might mean for the planet’s sustainability—that is, Earth’s ability to continue to support human life. While there is no one definition of sustainability, most agree that “sustainability is improving the quality of human life while living within the carrying capacity of supporting eco-systems.” Carrying capacity refers to how many living plants, animals, and people Earth can support into the future.

You will begin by thinking of various ways you can compare different countries’ populations. Scientific notation might be a useful tool because it is a way to write large numbers. A number in scientific notation is written in the form: a x 10n where 1 ≤ a < 10; and n is an integer.

Examples

• 28,930,000 can be written in scientific notation as 2.893 x 107. • In 2011, the population of the world was approximately 6.9 billion people. You can write this as 6,900,000,000 or you can use scientific notation to write it as 6.9 x 109 people.

(1) In 2011, the population of the United States was 311,000,000. Earth’s population was about 7 billion. Write these numbers in scientific notation.

1 65 Quantway Student Handout November 18, 2011 (Version 1.0) Lesson 1.2.1: Whose Footprint Is Bigger? Theme: Citizenship

(2) What are some other ways you could compare the population of the United States to the population of Earth? Think about forms of comparisons using both estimation and calculation.

(3) In 2011, the population of China was 1.341 billion. Compare China’s 2011 population to the world population with a ratio. Write your answer as a percent and as a fraction. Consider how many decimals to give in your final answer.

(4) Compare China’s population with the population of the United States using a ratio with the U.S. population as the reference value. Write a sentence that interprets this ratio in the given context.

Problem Situation 2: Comparing Water Footprints The population of the United States is smaller than many other major countries in the world. However, the people who live in the United States consume (or use up) a larger percentage of some natural resources, such as water. This means that the United States has a large “water footprint.”

According to the website www.waterfootprint.org, “People use lots of water for drinking, cooking, and washing, but even more for producing things such as food, paper, cotton clothes, etc. The water footprint is an indicator of water use that looks at both direct and indirect water use of a consumer or producer. The water footprint of an individual, community, or business is defined as the total volume of freshwater that is used to produce the goods and services consumed by the individual or community or produced by the business.”

The table below gives the population and water footprints of China, India, and the United States from 1997–2001.1

Population Total Water Footprint* Country (in thousands) (in 109 cubic meters per year) China 1,257,521 883.39 India 1,007,369 987.38 United States 280,343 696.01

1Retrieved from www.waterfootprint.org

2 66 Quantway Student Handout November 18, 2011 (Version 1.0) Lesson 1.2.1: Whose Footprint Is Bigger? Theme: Citizenship

(5) Notice that the countries are listed in the table above from highest to lowest population. Using the data on Total Water Footprint, rank the countries (from highest to lowest) according to their total water footprint.

(6) Rank the countries in order of water footprint per person (“per capita”) from highest to lowest. Be careful with the units on your numbers and final answer.

(7) How many times larger is the population of China compared with the population of the United States? Write your answer in a sentence. (You may want to refer back to Question 4.)

(8) Calculate how many times more water the average person in the United States uses compared to the average person in China.

(9) Write a sentence to explain the meaning of your answer to Question 8. (Remember the Writing Principle: Use specific and complete information.) Someone who reads what you wrote should understand what you are trying to say, even if they have not read the question or writing prompt.

Making Connections Record the important mathematical ideas from the discussion.

Further Applications (1) According to the data in this lesson, the per-person water footprint for the United States for 1997–2001 was 2,482.7 cubic meters per year per person.

(a) Write a sentence explaining what this number means.

(b) Find the current population of the United States. One good site is www.census.gov/main/ www/popclock.html. Use this information and the given water footprint to estimate the current total water footprint of the United States.

(c) Look at the water footprint you calculated in Part (b). Does your answer seem reasonable given what you know about the size of water footprints?

3 67 Quantway Student Handout November 18, 2011 (Version 1.0) Lesson 1.2.1: Whose Footprint Is Bigger? Theme: Citizenship

(d) Now compare this number to the U.S. water footprint given in this lesson. How many times larger is it now?

4 68 Quantway Out-of-Class Experience November 18, 2011 (Version 1.0) Lesson 1.2.1: Whose Footprint Is Bigger? Theme: Citizenship

Making Connections to the Lesson (1) Which of the following was one of the main mathematical ideas of the lesson? (i) Ratios are a way to compare measurements in different situations. (ii) A water footprint measures the amount of water used by a person or group. This includes water used for cooking, drinking, cleaning, and to produce all the goods and services used by the person. (iii) The number 311,000,000 can be written in scientific notation as 3.11 x 108. (iv) A nation’s water footprint can be calculated by dividing the nation’s population by the amount of water used in that nation.

(2) In Lesson 1.2.1, you used scientific notation for large numbers. Understanding large numbers has been an important concept in previous lessons. Find specific examples from your previous OCEs in which you used the skills listed below. The lesson number is listed. You must give the number of the question. In question number ______in the OCE for Lesson 1.1.1, you identified the names of large numbers. In question number ______in the OCE for Lesson 1.1.2, you compared the sizes of large numbers.

Developing Skills and Understanding (3) The website for the nonprofit organization Charity: Water1 discusses the need for clean water around the world. (a) The website states that worldwide, “90% of the 30,000 deaths that occur every week from unsafe water and unhygienic living conditions are of children under five years old.” The following statements are all correct interpretations of this statistic. Which gives the most complete information? (i) 27,000 children die every week from unsafe water and unhygienic living conditions. (ii) 27,000 out of the 30,000 deaths that occur every week from unsafe water and unhygienic living conditions are of children under five years old. (iii) 90% of deaths from unsafe water and unhygienic living conditions are of children under five years old. (iv) 90 out of 100 deaths that occur every week from unhealthy living conditions are of children under five years old. (b) The website also states: “Almost a billion people on the planet don’t have access to clean drinking water.” If there are 6.9 x 109 people in the world, then approximately what percent of them live without clean drinking water? Round to the nearest tenth of a percent.

1www.charitywater.org

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(4) According to data on the website for the Centers for Disease Control and Prevention (CDC)2, 28% of adults were obese in 2010. (a) According to the U.S. Census Bureau, about 63% of Americans are adults (18 and over). Use the U.S. population estimate of 311 million people to calculate the number of American adults in 2010. Round to the nearest thousand adults. (b) According to the CDC, about how many of these adults were obese? Round to the nearest hundred thousand adults. (c) What is this number in scientific notation? (i) 5.49 x 107 adults (ii) 5.49 x 106 adults (iii) 0.549 x 107 adults (iv) 54.9 x 106 adults (v) 0.549 x 108 adults

Over the past several years, there has been a dramatic increase in obesity rates in the United States. Use the following website to answer the following questions about adult obesity in the United States: http://apps.nccd.cdc.gov/brfss/list.asp?cat=OB&yr=1995&qkey=4409&state=All (d) In 1995, approximately ___ out of 100 adults in the United States were obese. Round to the nearest adult. (e) If there were 165 million American adults in 1995, about how many of them were obese? Round to the nearest million adults. (f) About how many more American adults were obese in 2010 than in 1995? (i) 2.3 x 107 adults (ii) 2.9 x 107 adults (iii) 4.1 x 107 adults (iv) 2.3 x 108 adults (v) 2.9 x 108 adults

2http://apps.nccd.cdc.gov/brfss/list.asp?cat=OB&yr=2010&qkey=4409&state=All

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Making Connections Across the Course (5) Tannika has a health insurance plan that will reimburse her for 60% of her family’s health expenses after she pays a $2,000 deductible. A deductible is the amount a person pays (to a hospital, for example) before an insurance company will begin to pay for a percentage of the remaining expenses. Tannika has to pay the deductible and the percentage not covered by the insurance company. These are called “out-of-pocket expenses” because they are paid by the person who owns the policy. Tannika records the total of her health care expenses in the spreadsheet below.

(a) Which of the following formulas could Tannika use in Cell E1 to calculate the amount paid by her insurance? (i) =0.6(B2 + B3 + B4 + B5) – 2000 (ii) =0.6 * B2 + B3 + B4 + B5 – 2000 (iii) =2000 − 0.6(B2 + B3 + B4 + B5) (iv) =0.6(B2 + B3 + B4 + B5 – 2000)

(b) Which of the following formulas could Tannika use in Cell E2 to calculate her out-of-pocket expenses? There may be more than one correct answer. (i) =0.4(B2 + B3 + B4 + B5 – 2000) + 2000 (ii) =0.4 * B2 + B3 + B4 + B5 + 2000 (iii) =2000 + 0.4(B2 + B3 + B4 + B5) (iv) =(B2 + B3 + B4 + B5) – E1

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Preparing for the Next Lesson (1.2.2) (6) According to the order of operations, what is one correct way to solve this problem? 8 + 6 x (3 + 6) ÷ 2 – 4 (i) 8 + 6 x (3 + 6) ÷ 2 – 4 → 8 + 6 x (3 + 6) ÷ 2 → 8 + 6 x (3 + 3) → 8 + (18 + 3) → 8 + 21 → 29 (ii) 8 + 6 x (3 + 6) ÷ 2 – 4 → 8 + 6 x 9 ÷ 2 – 4 → 8 + 54 ÷ 2 – 4 → 8 + 27 – 4 → 35 – 4 → 31 (iii) 8 + 6 x (3 + 6) ÷ 2 – 4 → 8 + (18 + 6) ÷ 2 – 4 → 8 + (18 + 3) – 4 → 8 + 21 – 4 → 29 – 4 → 25 (iv) 8 + 6 x (3 + 6) ÷ 2 – 4 → 8 + 6 x (3 + 3) – 4 → 8 + (18 + 3) – 4 → 8 + 21 – 4 → 29 – 4 → 25 (v) 8 + 6 x (3 + 6) ÷ 2 – 4 → 14 x (9) ÷ 2 – 4 → 14 x (9) ÷ 2 → (126) ÷ 2 → (63) → 63

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(7) Miguel has a coupon for 20% off any purchase in a furniture store. He decides to purchase a desk for $80. Excluding tax, how much does Miguel save on his purchase? (i) $16 (ii) $40 (iii) $2 (iv) $4

(8) Sylvia is charged 8% tax for her $2 cheeseburger. How much does Sylvia owe the cashier? (i) $2.08 (ii) $2.80 (iii) $2.16 (iv) $1.84

The following terms will be used in the next class. Make sure you understand what they mean. Revenue: This is the amount of money that a business receives when it sells a product or service. Net profit: The net profit is the actual amount of money a business makes after expenses. The expression for this is: Net profit = Revenue − Expenses For example, a restaurant might charge a customer $10 for a meal, but it cost the restaurant $4 for the food, $1 for the waiter’s paycheck, and $1 for the building. You need to add up all the restaurant’s expenses ($4 + $1 + $1 = $6). Then you subtract it from the total amount they make ($10) to figure out the net profit. The expression would be 10 – (4 + 1 + 1) = 4. The restaurant’s net profit is $4. Net loss: A net loss is similar to net profit, but a business has a net loss if the net profit is a negative number. For example, if the restaurant’s expenses were higher than the revenue, they would have a net loss. They could pay the waiter more ($5) and the building could cost more ($3). The expression would be 10 – (4 + 5 + 3) = −2. The restaurant’s net loss is $2.

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(9) You will be expected to do the following things for the next class. Rate how confident you are on a scale of 1–5 (1 = not confident and 5 = very confident).

Before beginning Lesson 1.2.2, you should understand the concepts and demonstrate the skills listed below:

Skill or Concept: I can … Rating from 1 to 5 Follow the order of operations. Find a percent of a number. Estimate 1% of a number.

(10) If your confidence ratings are below 3 for any of these skills/concepts, what are three things you might do to increase your confidence in these areas?

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Specific Objectives Students will understand that • order of operations is needed to communicate mathematical expressions to others. Students will be able to • perform multistep calculations using information from a real-world source. • rewrite multistep calculations as a single expression. • explain the meaning of a calculation within a context.

Problem Situation: FICA Taxes The United States government requires that businesses pay into two national insurance programs— Social Security and Medicare—which help senior citizens pay for their personal expenses and health care. Businesses take money out of their employees’ paychecks in order to pay the government. If you work for a business, your employer deducts Social Security and Medicare taxes from your paycheck. Also, the business pays a portion of the taxes for you. These taxes are called Federal Insurance Contributions Act (FICA) taxes. People who own their own businesses are self-employed. They have to pay their own taxes. This is called the self-employment tax. In this lesson, you will use a tax worksheet called the Short Schedule SE. This is an Internal Revenue Service (IRS) tax form. The IRS is the part of the government that collects taxes. It has many different types of forms for individuals and businesses to figure out how much they owe in taxes. With the Short Schedule SE, you will calculate how much two self-employed individuals owe in self-employment tax.

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(1) Sundos Allianthi sells crafts such as jewelry and baskets for extra money. She does not have a farm or get any of the benefits on Line 1b. In 2010, she sold $11,385 in crafts and her expenses totaled $3,862. Expenses are the things she needed to buy for her business. Fill out Section A—Short Schedule SE below for Sundos. How much self-employment tax does Sundos owe? Assume that Line 29 of her 1040 form has a 0 amount. This is asked for on Line 3 of the Short Schedule SE.

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(2) Raven Craig started a tutoring business at the end of 2010. She has no income to report on Line 1a or Line 1b of Schedule SE. She earned $1,050 and her expenses totaled $630. How much self- employment tax does Raven Craig owe?

(3) In Question 1, you learned about Sundos Allianthi. You used the Short Schedule SE form to figure out how much self-employment tax she owes. Now, write your answer in a single expression that someone else could use and understand.

(4) Look back at the expression you wrote for Question 3. Imagine you have to explain the expression and how you calculated the tax to Sundos. Answer these questions about the expression: (a) What does the operation $11,385 − $3,862 mean in the context? In other words, what does the result of this operation represent for Sundos? (b) What does the operation of multiplying by 0.9235 mean in this context? (c) What does the operation of multiplying by 0.153 mean in this context?

In 2010, the U.S. Congress passed the Tax Relief, Unemployment Insurance Reauthorization, and Job Creation Act of 2010. The act reduced the self-employment tax rate from 15.3% to 13.3%. This changes the amount in the first bullet under Line 5 of the Short Schedule SE. (5) Predict how much Raven Craig and Sundos Allianthi will save in taxes in 2011 if their incomes and expenses are the same as they were in 2010. Do not use pencil and paper or a calculator. Write down your predictions of how much they will save.

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Making Connections Record the important mathematical ideas from the discussion.

Further Applications (1) In Question 7c of the Lesson 1.2.2 assignment, you were asked to calculate the income tax for a person earning $63,500. (a) Write a single expression for this calculation. (b) The $4,750 in the third line of the table is based on information from the previous two lines. Explain how the $4,750 is calculated. (Hint: Start by thinking about where the $850 in Line 2 came from.)

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Making Connections to the Lesson (1) Which of the following was one of the main mathematical ideas of the lesson? (i) In order of operations, you do operations in this order: 1) Within parentheses; 2) Exponents; 3) Multiplication; 4) Division; 5) Addition; 6) Subtraction. (ii) Taxes are very complicated, and tax forms are hard to complete. (iii) Part of quantitative reasoning is being able to read, interpret, and use quantitative information to perform a task. (iv) It does not matter how you write your calculations as long as you get the correct answer.

(2) Refer back to Question 5 in Lesson 1.1.5 and Question 4 in this lesson (1.2.2). What important quantitative reasoning skill was used in both of these questions? Choose the best answer from the following. (i) Both questions related to money. (ii) Both questions related to making sense of numbers and calculations. (iii) Both questions were about personal finance.

Developing Skills and Understanding (3) Martin Binford is an author. He has no income he would report on line 1a or line 1b of his Schedule SE1. He earned $143,380 in 2010 from his books. He had $3,563 in expenses. How much self- employment tax does he owe?

1Retrieved from http://www.irs.gov/pub/irs-pdf/f1040sse.pdf

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(4) Which of the following expressions can be used to compute how much self-employment tax Martin Binford owes? (i) 0.029 + 0.9235(143,380 – 3,563) + 13,243.20 (ii) 0.029 x 0.9235 x 143,380 – 3563 + $13,243.20 (iii) 0.029(0.9235)(143,380 – $3563) + 13,243.20 (iv) 0.029(0.9235)(143,380 – $3563 + 13,243.20)

(5) The expression below shows another way to calculate Martin’s tax. 0.153(106,800) + 0.029(129,121.43 – 106,800) Based on this expression, select the statement that describes how Martin’s income is taxed. (i) Martin pays 15.59% tax on his income. (ii) Martin pays 44.3% tax on his income. (iii) Martin pays 15.3% in tax on the first $106,800 of his income. He pays 29% on his income over $106,800. (iv) Martin pays 15.3% in tax on the first $106,800 of his income. He pays 2.9% on his income over $106,800.

(6) Miguel is moving and wants to estimate what his electricity bill will be in his new apartment. He looks at old bills and sees that he uses around 700 kWh of electricity each month. The utility company charges $6 each month plus 6.726 cents for the first 500 kWh and 8.136 cents for any kilowatt-hours above 500. (a) How much will Miguel pay for 700 kWh of electricity? People often make a common error in situations like the one Question 6a. The purpose of the next two questions is to help you recognize this error and correct your work in part (a) if necessary. (b) If someone bought three items for $1.50, 37 cents, and 5 cents, how much did they spend? (c) Which of the following is most likely the common error in part (b)? (i) Making an addition error such as 37 + 5 = 45 cents (ii) Forgetting to change the cents to dollars: 1.50 + 37 + 5 = $43.50 (iii) Leaving off the decimal: 1.50 + 37 + 5 = $4,350

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(7) Workers in the U.S. pay several types of taxes on income. The lesson discussed the FICA taxes. You also have to pay federal income tax. Your federal income tax rate is based on the amount of money you make. Income is broken into levels called tax brackets. The table below shows the tax brackets for 2011.2

Taxable Income Tax $0–$8,500 10% of taxable income $8,500–$34,500 $850 plus 15% of excess over $8,500 $34,500–$83,600 $4,750 plus 25% of excess over $34,500 $83,600–$174,400 $17,025 plus 28% of excess over $83,600 $174,400–$379,150 $42,449 plus 33% of excess over $174,400 $379,150 plus $110,016.50 plus 35% of excess over $379,150

(a) What tax rate does everyone pay on the first $8,500 of income? (b) Calculate the income tax for a person earning $25,000. (c) Calculate the income tax for a person earning $63,500. (d) Refer to your answer for part (c). The total tax in part (c) is what percentage of the person’s income? Round to the nearest one percent. Making Connections Across the Course (8) In Lesson 1.2.1, it was determined that the water footprint for a typical American is 2,483 m3/year. (a) A family of three would like to reduce their water footprint so that it is 75% of the typical American’s water footprint. Which calculation shows how they can estimate their target water footprint for one day? There may be more than one correct answer. (i) (3 × 2,500 × 3 ) ÷ 365 4 (ii) 2,500 × 3 × 0.75 (iii) 3 × 2,500 × 0.75 ÷ 365 (iv) 2,500 × 3 × 4 ÷ 365 3 (v) 2,500 × 3 ÷ 0.75 (vi) 2,500 ÷ (365 × 0.75) × 3 (vii) 3 × (2,500 × 0.75) ÷ 365

2http://www.bargaineering.com/articles/federal-income-irs-tax-brackets.html

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(b) One thing a person can do to reduce his or her water footprint is to use less water every day. If each American were to reduce his or her daily water use by 10 m3 (2,642 gallons), how would you calculate the new annual water footprint for a typical American? There may be more than one correct answer. (i) 2,483 − 10 × 365 (ii) (2,483 – 10) × 365 (iii) 2,483 – (10 × 365) (iv) 2,483 × 365 – 10 (v) 2,483 – 365 × 10 (vi) (2,483 ÷ 365 – 10) × 365 (vii) 2,483 ÷ 365 – 10 (c) Which of the following would cause the greatest decrease in the American water footprint? (i) Each American decreases his or her daily water footprint by 300 m3. (ii) Each American decreases his or her daily water footprint to 95% of what it is now. (iii) Each American decreases his or her annual water footprint by 120,000 m3. (iv) Each American decreases his or her annual water footprint to 94% of what it is now.

Preparing for the Next Lesson (1.2.3) (9) Which of the following represents 0.02%? There may be more than one correct answer. (i) 2 out of 100 (ii) 0.2 out of 100 (iii) 0.02 out of 100 (iv) 2 (v) 0.02 (vi) 0.0002 (vii) 2 out of 1,000 (viii) 2 out of 10,000

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(10) Which of the following is equivalent to 4%? There may be more than one correct answer. (i) 0.04 (ii) 1/25 (iii) 4/100 (iv) 40/100 (v) 4 out of 100 (vi) 2 out of 5

(11) Which of the following is correct? There may be more than one correct answer. (i) A percent is one part in every 100. (ii) A percent can be converted into a decimal number by dividing that percent number by 100. (iii) A percent can be converted into a decimal number by moving the decimal point two places to the left and removing the % sign. (iv) 50% means 50 per 100 or 50/100 = 0.5.

(12) Which of the following is the percent estimate of 1/3, rounded to the nearest hundredth of a percent? (i) 3.3% (ii) 0.33% (iii) 33.33% (iv) 33.3% (13) 1,352 is what percent of 40,929? Round to the nearest tenth of a percent.

(14) You will be expected to do the following things for the next class. Rate how confident you are on a scale of 1–5 (1 = not confident and 5 = very confident). Before beginning Lesson 1.2.3, you should understand the concepts and demonstrate the skills listed below:

Skill or Concept: I can … Rating from 1 to 5 Have a basic understanding of the word percent and the notation used to describe percentages (%). Use a calculator to divide two numbers and interpret the resulting decimal representation as a percent. Calculate and estimate percentages.

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Self-Regulating Your Learning—The Work Phase In an earlier lesson, you read in detail about what it means to effectively plan for your learning. That involved accounting for time and effort, your confidence (self-efficacy), study strategies, and learning goals. In this lesson, we will discuss the second phase of self-regulated learning (SRL), the work phase. As the name implies, the work phase of SRL is where you are actually working on the problem or assignment. However, it is more than just getting the assignment done. In this phase, you monitor or pay attention to a variety of things. For example: • What you are or are not understanding (and when). • Which strategies you are using; which ones are working, and which ones are not. • What emotions and feelings you are experiencing, both positive and negative. • When you should seek help from others. Let’s explore each of these in a little more detail. Understanding: Self-regulated learners monitor what they understand and what they do not. This is done by frequently asking yourself: “Do I understand this?” or “Could I explain this to someone?” The goal is to monitor your understanding so that you may adapt your strategies, especially if you get stuck. Being honest about your understanding is important because it can help you progress successfully on a problem, or make you aware of your learning strengths and weaknesses. Sometimes, people talk about this as “thinking about your thinking.” Researchers call it metacognition. Strategies: In learning about the SRL planning phase, you discovered that it can be useful to think about multiple strategies before you start working on a problem. In the work phase of SRL, having multiple strategies in mind (both those you have used before and those you have planned to try) can help when you get stuck. You can stop, think about how the problem is progressing, and try another strategy that you think might work. Self-regulated learners often make mental notes about which strategies work in which situations, and which ones are easiest to use. Evaluating strategies allows you to become better at solving a variety of problems. Emotions: Self-regulated learners know how to monitor their emotions—especially negatives ones such as frustration or anger—so that these emotions do not cause them to give up on a problem. When they start feeling frustrated, self-regulated learners often do things such as trying new strategies, seeking help, or engaging in positive self-talk. This is saying things to yourself such as: “I know I can do this if I choose the right strategies and put in the effort, even if it is challenging.” The opposite is called negative self-talk, which involves saying things such as: “I am never going to get this! What is the point?” Monitoring and controlling your emotions, especially the negative ones, can be challenging and may require a lot of practice, but the benefits are worth it. Seeking Help: With practice and experience, self-regulated learners know when it is beneficial to stop working and find someone else with whom they can discuss the problem. There is nothing wrong with getting help when you are learning something new. Some people think that asking for help means you do not have ability, but the truth is that knowing when to seek help is part of being an effective learner. Seeking help can save you time because you avoid the added frustration of making a lot of effort without making any progress. If you are spending a lot of time on a problem, you have tried several strategies without success, or you are not able to control negative emotions, stop and write down your

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questions. Bring the written questions with you and discuss the problem with someone else as soon as you can. Help could come from your instructor during his or her office hours, a campus learning center, or other classmates. During the work phase, you are required to juggle two things at once: (1) Working on the problem or assignment and (2) monitoring your progress (e.g., thinking about your thinking). This process takes practice, however, it is important to master if you want to become a self-regulated learner. Thinking about how you are working makes the work easier and gives you information for the SRL reflect phase. You will explore the reflect phase in an upcoming lesson. Until then, practice these work strategies while you are working on problems and class assignments.

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Specific Objectives Students will understand that • percents involve a numerator (comparison value) and a denominator (reference value). Students will be able to • correctly identify the quantities involved in a verbal statement about percents. • convert between ratios and percents. • convert between decimal the representation of a number and a percent. • read and use information presented in a two-way table.

Problem Situation: The Language of Percentages The World Health Organization (www.who.org) is the part of the United Nations that oversees health issues in the world. The WHO leads numerous studies on tobacco use around the world. In its study on Gender and Tobacco, the organization learned that tobacco use among women is increasing. For example, recent research shows that just as many young girls smoke as young boys. The report is filled with information about percentages of women who smoke, percentages of men who smoke, and the percentage of smokers who start smoking by age 10. The language used to describe this information can be difficult to understand. Pay close attention to the language used to describe a percent at the beginning of this lesson. This will help you to understand new findings in the relationship between tobacco use and gender.1 Consider the following two quantities: • Quantity 1 (Q1): The percentage of women who smoke. • Quantity 2 (Q2): The percentage of smokers who are women. (1) Are these two quantities equal (Q1 = Q2)? Could Q1 be greater than Q2 (Q1 > Q2)? Could Q1 be less than Q2 (Q1 < Q2)? Be prepared to explain your reasoning.

(2) What information would you need to compute these percentages?

1Retrieved from www.who.int/tobacco/research/gender/about/en/index.html

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Questions 3 and 4 present two situations with data. You can use these situations to test your ideas from Questions 1 and 2. (3) Suppose a study on smoking was conducted at Midland University. The following table indicates the results of the study.

Men Women Smokers 4,572 5,362 Nonsmokers 10,284 12,736

(a) What percentage of women smoke at Midland University? (b) What percentage of smokers at Midland University are women?

(4) Suppose a study was conducted at Northwest College. The following table indicates the results of the study:

Men Women Smokers 1,256 536 Nonsmokers 1,028 1,053

(a) What percentage of women smoke at Northwest College? (b) What percentage of smokers at Northwest College are women? (c) A newspaper stated that 40% of the male students at Northwest College smoked. Is that claim reasonable? Explain why or why not.

(5) In 2006, the World Health Organization conducted a study about smoking in the United States and China. The organization reports that 3.7% of the adult women in China smoke tobacco products. In the United States, 19% of adult women smoke. (a) Out of 100 adult women in China, about how many are smokers? (b) Out of 1,000 adult women in China, about how many are smokers? (c) Out of 100 adult women in the United States, about how many are smokers? (d) Out of 1,000 adult women in the United States, about how many are smokers? (e) Are there more women smokers in China or the United States? (f) Suppose you read that 590 out of 1,000 men in China smoke. Based on these data, what percentage of men in China smoke?

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Making Connections Record the important mathematical ideas from the discussion.

Further Applications The following question is included in the out-of-class experience for this lesson. Write an explanation for your answers to Parts (a) and (b). (1) A teacher has collected data on the grades his students received in his two classes. The following tables show two different ways to represent the same data.

Table 1

Grades A B C D F Morning Class 12.5% 25.0% 37.5% 6.3% 18.8% Afternoon Class 14.3% 20.0% 37.1% 8.6% 20.0%

Table 2

Grades A B C D F Morning Class 44.4% 53.3% 48.0% 40.0% 46.2% Afternoon Class 55.6% 46.7% 52.0% 60.0% 53.8%

(a) Which table could be used to answer the following question: “What percentage of the students who received an A are in the morning class?” (b) Which table could be used to answer the following question: “What percentage of the students in the morning class received an A?”

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Making Connections to the Lesson (1) Which of the following was one of the main mathematical ideas of the lesson? (i) A percentage is calculated by dividing one number by another number. (ii) Smoking is a major health problem in the United States and China. (iii) A percentage is a comparison of two numbers. To understand the meaning of a percentage, it is important to know what two quantities are being compared. (iv) The percentage of students who are parents is the same as the percentage of parents who are students. (2) You have worked with percentages in Lessons 1.1.3, 1.1.5, 1.2.1, and 1.2.2. Select one or two examples that helped you understand percentages from one of these lessons. Write a short explanation of how they helped further your understanding.

Developing Skills and Understanding (3) Data from the National Postsecondary Student Aid Study (NPSAS)1 provides a statistical snapshot of the proportion of community college students who majored in different fields of study in 2003–04. A total of 25,000 community college students were included in the study. Table 1 displays the total number of community college students who majored in each of the following fields of study in 2003–04. Table 1

Number of Students who Percentage of Students who Field of Study Majored in Field Majored in Field Humanities 3,700

Social/Behavioral Sciences 1,250

Mathematics and Science 900

Computer/Information Science 1,525

Engineering 1,025

Education 2,025

Business/Management 4,600

Health 5,975

Vocational/Technical 1,225

Other Technical/Professional 2,775

1Retrieved from http://nces.ed.gov/pubsearch/pubsinfo.asp?pubid=2006184

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(a) Complete the table by filling in the percent of community college students who majored in each field of study. Round to the nearest one percent. (b) What was the most popular major in 2003–04? (i) Humanities (ii) Business/Management (iii) Health (iv) Education (c) Fill in the blanks to complete the following statements. About ______out of every 100 community college students in 2003–04 majored in the most popular field. About ______out of every 1,000 community college students in 2003–04 majored in the most popular field.

(4) Select the answers that correctly complete the statement from the list below: A New York Times story2 reported that 10% of male high school dropouts are in jail or detention centers. According to this statistic, about ______in every ______male high school dropouts is (are) in jail or juvenile detention. There may be more than one correct answer. (i) 1 in every 10 (ii) 10 in every 100 (iii) 1 in every 10% (iv) 1 in every 100 (v) 0.1 in every 100

2Retrieved from http://www.nytimes.com/2009/10/09/education/09dropout.html

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(5) A teacher has collected data on the grades his students received in his two classes. The tables below show two different ways to represent the same data as percentages.

Table 2

Grades A B C D F

Morning Class 12.5% 25.0% 37.5% 6.3% 18.8%

Afternoon Class 14.3% 20.0% 37.1% 8.6% 20.0%

Table 3

Grades A B C D F

Morning Class 44.4% 53.3% 48.0% 40.0% 46.2%

Afternoon Class 55.6% 46.7% 52.0% 60.0% 53.8%

(a) Which table could be used to find out what percentage of the students who received an A are in the morning class? (b) Which table could be used to find out what percentage of the students in the morning class received an A? (c) What are the reference values in Table 2? (i) The number of students in a certain class. (ii) The number of students who got a certain grade. (iii) The number of students in a certain class who got a certain grade. (iv) The total number of students in both classes. (d) What are the reference values in Table 3? (i) The number of students in a certain class. (ii) The number of students who got a certain grade. (iii) The number of students in a certain class who got a certain grade. (iv) The total number of students in both classes.

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Making Connections Across the Course (6) What are the four things you should monitor when you are self-regulating your learning during the work phase? Try to think of all four aspects when you solve Question 7.

(7) Congratulations! You won a lottery prize and have a taxable income of $1,025,400. Use the table below to answer the following questions.

Taxable Income Tax $0−$8,500 10% of taxable income $8,500−$34,500 $850 plus 15% of excess over $8,500 $34,500−$83,600 $4,750 plus 25% of excess over $34,500 $83,600−$174,400 $17,025 plus 28% of excess over $83,600 $174,400−$379,150 $42,449 plus 33% of excess over $174,400 $379,150 plus $110,016.50 plus 35% of excess over $379,150

(a) How much tax will you pay on your winnings? (b) Which of the following expressions can be used to calculate the tax? (i) $1,025,400 – 0.35($379,150 – $110,016.50) (ii) 0.35 + $379,150($1,025,400 – $110,016.50) (iii) ($1,025,400 + $110,016.50) – ($37,650 × 0.35) (iv) $110,016.50 + 0.35($1,025,400 – $379,150) (v) $1,025,400 – 0.35($1,025,400 – $379,150)

(8) The work phase of regulating your learning includes checking your understanding. One of the ways to do this is by asking yourself: “Can I explain this to someone?” Check your understanding by explaining your answer to Question 7b.

Preparing for the Next Lesson (1.2.4) According to the World Health Organization (WHO), “Every person is at risk of foodborne illnesses.”3 A foodborne illness is an illness that a person gets from eating food that has spoiled or been contaminated in some way. (9) In industrialized countries, such as the United States, up to 30% of the population suffers from foodborne diseases each year. This means that 30 out of ______people living in industrialized countries will likely suffer from foodborne diseases each year.

3 Retrieved from http://www.who.int/mediacentre/factsheets/fs237/en/

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(10) In 1994, an outbreak of illness due to ice cream contaminated with the bacteria salmonella occurred in the U.S. The outbreak affected an estimated 224,000 people. (a) If the total population in the United States at that time was 260,000,000, which is the best estimate for the percentage of people who were affected? (i) About 0.1% (ii) About 1% (iii) About 5% (iv) About 10% (v) About 25% (b) Complete the following statement: Approximately 86 out of every ______people in the United States were affected by the 1994 salmonella outbreak.

(11) A report about foodborne illnesses indicated, “About 1 egg out of every 20,000 contains salmonella inside the shell.” This means that (i) 0.005% of eggs contain salmonella. (ii) about 1% of eggs contain salmonella. (iii) more than 1% of eggs contain salmonella. (iv) approximately 50 out of every 100,000 eggs contain salmonella.

(12) In 2011, Germany had an outbreak of illness caused by the bacteria called e. coli. As of June 15 of that year, 3,235 people in Germany had become sick and 36 had died.4 What percentage of those who got sick also died? Round to the nearest tenth of percent.

(13) You will be expected to do the following things for the next class. Rate how confident you are on a scale of 1–5 (1 = not confident and 5 = very confident). Before beginning Lesson 1.2.4, you should understand the concepts and demonstrate the skills listed below:

Skill or Concept: I can … Rating from 1 to 5 Have a basic understanding of the word percent and the notation used to describe percents (%). Use a calculator to divide two numbers and interpret the resulting decimal representation as a percent. Calculate percentages.

4Retrieved from http://health.usnews.com/health-news/managing-your-healthcare/articles/2011/06/15/health-highlights- june-15-2011

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Specific Objectives Students will understand that • a percent has different uses, including being used to express the likelihood (or probability) of a certain event. • the importance of selecting the correct comparison value and reference value in calculating percentages. Students will be able to • extract relevant information from a table. • select the appropriate values to calculate probabilities.

Problem Situation: Using Percentages to Describe the Accuracy of Medical Tests Some athletes use performance-enhancing drugs (PEDs) to improve how they do in sports. Schools, sports leagues, and other sports organizations usually do not allow the use of PEDs. These groups can administer or give athletes a blood or urine test to determine if the athletes are using drugs. In this situation, 500 athletes have undergone a test to determine if they use PEDs. A positive (+) test result indicates or shows that the athlete is using a PED. A negative (–) test result indicates the athlete is not using these drugs. However, this test is not 100% accurate. This means that some errors may have occurred in the test results. The table below shows how often the test correctly determined if athletes used PEDs.

Athletes Using Athletes Not Total PEDs Using PEDs Positive test result 9 5 14 Negative test result 1 485 486 Total 10 490 500

Use the figures or numbers in the table to answer the questions below. You will use the figures in the table to decide on the probability that this test gives correct and incorrect results. Probability means the chance that something happens. Report probabilities in percents (%). Be careful what figures you use for the numerator and denominator in your calculations.

(1) The table is missing one row total and one column total. Fill in the missing totals.

(2) Correctly identify the presence of PEDs using the steps below. (a) How many athletes are using PEDs? (b) How many of the athletes using PEDs received a positive test result? (c) If an athlete is using PEDs, what is the chance this test gives a positive result?

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(3) Correctly identify the absence of PEDs using the steps below. (a) How many athletes are not using PEDs? (b) How many of the athletes not using PEDs received a negative test result? (c) If an athlete is not using PEDs, what is the chance that this test gives a negative result?

(4) False Negatives: Did you see how one athlete using PEDs received a negative test result? This means the test incorrectly identified this single athlete. This is called a false-negative test result. Think about this situation: An athlete gets a negative result on a test. What is the chance the result is a false negative? Hint: Think about the ratio of incorrect negative results compared to all negative results.

(5) False Positives: The test also produced false positives. This means the test gave some athletes not using PEDs positive results. Think about a situation in which a school principal finds that an athlete gets a positive result on the test. Answer these questions: (a) What is the chance the result is a false positive? (b) How should the principal think about this percentage? What should the principal do with this information?

(6) You can use different percentages to show how accurate the test was. A test is accurate when it produces very few mistakes or errors. Pick one figure or percentage that you think best describes how accurate the test was. Explain what this figure says about the test and why you picked this figure.

(7) Now, think about how to use a figure or percentage to show how inaccurate the test was. A test is inaccurate if it produces many errors. Pick one figure to show how inaccurate the test was. Explain what this figure says about the test and why you picked this figure.

Making Connections Record the important mathematical ideas from the discussion.

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Further Applications (1) Refer to the problem situation used in this lesson and to Question 6 in the OCE for this lesson. You will call the population used in the lesson P1 and the population used in OCE Question 6 P2. (a) A prevalence rate is the percentage of people in a population who have a certain disease or behave in a certain way. Find the prevalence rate of using PEDs for P1 and P2. Another way to say this is, “What percent of the population used PEDs?” Put your answers in the following table.

P1 P2 Prevalence rate 10% 20% True positive rate 90% 90% False Positive rate 35.7% 4.3%

(b) Complete the table with the true positives (the percentage that were correctly identified as using PEDs) and false positive rates for each population. You already have that information in your lesson and OCE work. (c) Based on the information in the table, what appears to affect the rate of false positives? Write your answer using the Writing Principle.

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Making Connections to the Lesson (1) Which of the following was one of the main mathematical ideas of the lesson? (i) A probability is a percentage in which the chance of an event is measured as a ratio out of 100. (ii) Medical tests are far less accurate than most people think. (iii) To calculate a percentage, divide the comparison value by the reference value. (iv) Probabilities are not related to percentages.

(2) In the problem situation in Lesson 1.2.4, there was a 90% chance that an athlete who used Performance Enhancing Drugs (PEDs) would have a positive test result. You could explain this by saying 90 out every 100 athletes who use PEDs will have a positive test result. Refer to previous lessons. Find two lessons that use percentages. Choose one question using a percentage from each lesson. In the table below, list the lesson and the question number then write an interpretation of the percentage similar to the example above. Suggested lessons: 1.1.3, 1.1.5, 1.2.2, and 1.2.3.

Lesson Question Number Interpretation of the percentage

Developing Skills and Understanding (3) About 16% of drivers are uninsured. There were approximately 196 million drivers in the United States in 2003.1 How many of these drivers were likely uninsured? (i) 31,360 (ii) 31,360,000 (iii) 313,600,000 (iv) 3,136,000,000

1Retrieved from http://wiki.answers.com/Q/Number_of_drivers_in_US

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(4) Determine whether each of the following is an example of a false negative or an example of a false positive. (a) A woman has breast cancer. Her test indicates that she does not have breast cancer. This is an example of … (i) A false negative (ii) A false positive (iii) A true positive (b) A woman does not have breast cancer. Her test indicates that she does have breast cancer. This is an example of … (i) A false negative (ii) A false positive (iii) An accurate result

(5) A test is administered to 500 athletes to determine if they are using performance-enhancing drugs (PEDs). A positive test result indicates that the athlete is using performance-enhancing drugs; a negative test result indicates that the athlete is not using these drugs. However, this test is not 100% accurate, so some errors occur. The following table shows the test results for a group of athletes.

Athletes using PEDs Athletes not using PEDs TOTAL Positive test result 90 4 94 Negative test result 10 396 406 TOTAL 100 400 500

Use the information in the table to answer the following questions. (a) How many athletes are using PEDs? (b) How many of these received a positive test result? (c) If an athlete is using PEDs, which of the following describes the chance that this test will return a positive result? There may be more than one correct answer. (i) 90 out of 100 chance of receiving a positive test result if one is using PEDs. (ii) 90% chance of receiving a positive test result if one is using PEDs. (iii) 9% chance of receiving a positive test result if one is using PEDs. (d) What is the chance that a positive test result is a false positive? Round to the nearest tenth of the percent.

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(e) If an athlete is not using PEDs, what is the chance that this test will return a negative result? (i) 99% chance of receiving a negative test result if one is not using PEDs. (ii) 4 out of 400 chance of receiving a negative test result if one is not using PEDs. (iii) 39.6% chance of receiving a negative test result if one is not using PEDs. (f) What is the chance that a negative test result is a false negative? Round to the nearest tenth of a percent.

(6) A hospital tracks the number of cases that come into its Emergency Room during each eight-hour shift. The cases are listed in categories based on the severity of the illness or injury. The categories from least severe to most severe are: stable, serious, and critical. The following table gives the data for a week. (a) Complete the missing blanks in the table.

Stable Serious Critical Total 8:00 am–3:59 pm 250 120 45 415 4:00 pm–11:59 pm 270 230 105 12:00 am–7:59 am 175 95 460 Total 710 245 1480

A nursing supervisor ranks the shifts based on two different criteria. (b) Which shift received the highest percentage of the total critical cases? Rank the shifts from highest to lowest. Round to the nearest one percent.

Shift Percentage of Total Critical Cases

Shift Percentage of Total Cases in Shift

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(d) The hospital schedules nursing staff based on the number and severity of expected cases. One goal of scheduling is that more experienced nurses should work on critical cases. The nursing supervisor considers the overall number of experienced nurses and the ratio of experienced nurses to the less experienced nurses. Based on these data, which of the following conclusions could be drawn? (i) The highest number and the highest ratio of experienced nurses should be scheduled during the 12:00 am–7:59 am shift. (ii) The highest number and the highest ratio of experienced nurses should be scheduled during the 4:00 pm–11:59 pm shift. (iii) The highest number of experienced nurses should be scheduled during the 4:00 pm–11:59 pm shift. The highest ratio of experienced nurses should be scheduled during the 12:00 am– 7:59 am shift.

Making Connections Across the Course (7) We can save time and work by using spreadsheets to perform calculations. To do this, you use formulas as you saw in your OCE for Lesson 1.1.4. The spreadsheet below could be used to calculate the self-employment taxes from Lesson 1.2.2.

(a) You are going to set up a spreadsheet to calculate Sundos Allianthi’s self-employment tax. Use the information given in Lesson 1.2.2 to fill in the blanks for in cells B2–B5.

B2: B3: B4: B5:

(b) Create an actual spreadsheet like the one shown above. Enter the information from Part (a). (c) Write a formula for cell B7 that will calculate the self-employment tax. Remember to start the formula with an “=” sign. In a spreadsheet, you use an asterisk ( * ) to indicate multiplication. So B3 times B4 would be written as B3*B4. Record your formula below. (Note: You can check if your formula is correct by comparing the result of the calculation with your work in the lesson.) (d) Suppose Sundos made a mistake in calculating her expenses. The expenses should be $4,371. Enter this new value into the spreadsheet. What is the new amount for the self-employment tax?

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(e) Why did the spreadsheet specify that the entries for B4 and B5 be written as decimals? Select the best explanation. (i) Percentages are easier to work with when written in decimal form because the numbers are smaller. (ii) It is a rule that you always move the decimal two places to the left with a percentage. (iii) The spreadsheet uses the number in the cell for the formula. The percentages had to be written as decimals so they could be used in the calculation. (iv) It did not matter that the spreadsheet asked for decimals. The percentages could have been entered without the change. Nothing would have changed in the spreadsheet because percentages are equivalent to the decimal.

Self-Regulating Your Learning: The Reflect Phase The last phase in regulating your learning is the reflect phase. In this phase, after you finish a problem or assignment, you intentionally reflect on how your learning and problem solving went. You gather information about yourself, about studying, and about learning in general. This information is then used to improve future learning situations. Here are some things that self-regulated learners reflect upon: • Confidence (self-efficacy) • Strategy selection • Time and effort • Emotions • Causes of success and setbacks Confidence (Self-Efficacy): Regulating your learning means that you continuously pay attention to how much you believe that you can succeed at what you are trying accomplish. This is important after you complete a problem because it allows you to plan for what you will need as you move forward. If you rate your confidence low, then you would benefit from spending more time practicing and studying, and you may decide to get additional help. If you do not stop to think about how things went, it is easy to just move on to the next concept, thinking that you understand something you do not. Also, it is important to give yourself credit for your successes. You will feel better about spending the time and effort and you will be more motivated to continue working hard. Strategy Selection: When regulating your learning, look back at the strategies you used when working on a problem or assignment. Make note of what seemed to work well for certain problems and what strategies seemed easier. This information will guide your plan phase in preparing for a new assignment. You can practice this by asking yourself: “What worked well and what did not?” Time and Effort: Pay attention to the time and effort it took to complete an assignment. Reflect on whether you planned accurately. When doing this, try to identify the types of things you may need more time for in the future, or what might be more of a challenge. This information will be used during the plan phase. Emotions: For successful learning, it is important to manage your emotions, especially your frustration and anxiety. One way to do this is to ask yourself what caused you to become frustrated or anxious and

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think about what helped you overcome those feelings. This gives you tools to deal with those feelings when they come up again. Causes of Success and Failure: One of the most common challenges for students is correctly identifying the reasons for both their successes and their setbacks. Students often blame external factors for shortcomings. When faced with challenges, students often blame the teacher (“She does not explain well”); or maybe the book (“It is hard to read and it is confusing”); or the test itself (“It was full of trick questions.”) The problem with blaming external factors is that it gives you little control over your own learning outcomes. On the other hand, by considering internal factors—ones you can control—then you are in charge of your learning outcomes. For example, when facing a setback, ask yourself: • “Did I spend enough time on this assignment?” • “Did I use the right strategies?” • “Did I seek help when I needed it?” • “Did I put in the work and effort that was really required?” These types of self-reflection questions help you understand yourself better and assist you in becoming a more effective learner.

Preparing for the Next Lesson (2.1.1) (8) The Medical Center in Houston, Texas, is bound by US-59 to the north, I-610 to the west (west loop), I-610 to the south (south loop), and Hwy 288 to the east. The region is roughly a 3-mile by 4-mile rectangle. (a) What is the area of the Medical Center? (b) What are the correct units for the area? There may be more than one correct answer. (i) miles (ii) square miles (iii) mi2

(9) Which of the following represent(s) the number 8.4 billion? There may be more than one correct answer. (i) 840,000,000,000 (ii) 8,400,000,000 (iii) 8.4 x 109 (iv) 8.4 x 1010 (v) 8,400 million

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(10) Certain words and phrases indicate that division is required to compute an answer. For example, per, ratio, and divide into indicate division. There are also different symbols for division—including ÷ and the fraction bar (/). Which of the following does not indicate division? (i) To compare gasoline usage, compute miles per gallon. (ii) Convert 1/4 to a decimal. (iii) To compute a bill, total all charges. (iv) To compute how fast water flows past a meter, compute gallons per minute. (v) To compute how to share tips among five waiters, compute dollars per person.

(11) You drive 310 miles on 15 gallons of gas. Select the statement(s) that correctly summarize(s) the situation. There may be more than one correct answer. (i) Your gas mileage is about 15 miles per gallon. (ii) Your gas mileage is about 20 miles per gallon. (iii) You can drive a little more than 15 miles on a gallon of gas. (iv) You can drive a little more than 20 miles on a gallon of gas.

(12) You will be expected to do the following things for the next class. Rate how confident you are on a scale of 1–5 (1 = not confident and 5 = very confident). Before beginning Lesson 2.1.1, you should understand the concepts and demonstrate the skills listed below:

Skill or Concept: I can … Rating from 1 to 5 Understand the concept of area. Comprehend numbers up to the billions place. Use a calculator to divide numbers. Interpret fractions as division. Interpret a decimal number.

(1) In 1915, the population of the United States was approximately 100,000,000 people. Which of the following is not the same number as 100,000,000 people? (1.1.1, 1.1.2) (a) 0.1 billion people (b) 100 million people (c) 108 people (d) 1,000,000 thousand people (e) One hundred million people

(2) Determine which of the following three statements is true. (1.1.2) Statement I: A billion is 1,000 million. Statement II: A trillion is 1,000 million. Statement III: A trillion is 1,000 billion. (a) Only Statement I is true. (b) Statements I and II are true, and Statement III is false. (c) Statements I and III are true, and Statement II is false. (d) Statements II and III are true, and Statement I is false. (e) Statements I, II, and III are all true.

(3) In 1995, the U.S. federal government debt totaled 5 trillion dollars. In 2008, the total debt reached 10 trillion dollars.1 Which statement about the doubling time of the U.S. federal debt is true based on this information? (1.1.2) (a) The doubling time is getting bigger. (b) The doubling time of the U.S. federal debt in 1995 was 13 years. (c) The doubling time of the U.S. federal debt is always 13 years. (d) If the federal deficit continues to increase with the same doubling time as in 1995, the federal deficit will be 20 billion dollars in 2013. (e) The doubling time is getting smaller.

1Retrieved from www.treasurydirect.gov/govt/reports/pd/histdebt/histdebt.htm

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(4) Data from 2010 show an average of 1 in 110 children have an autism spectrum disorder. Which of the following numbers is the best estimate for the percent of children who have an autism spectrum disorder? (1.1.3) (a) 0.009% (b) 0.09% (c) 0.9% (d) 1.1% (e) 11%

(5) The tutoring center director is studying the preferences of students at her college to determine when to offer tutoring sessions for extra help. Every student in Course A and B completed the survey and each student preferred either the afternoon or evening help session. (1.2.3 and 1.2.4)

Number of Students Who Prefer Number of Students Who Prefer Course Sessions in the Afternoon Sessions in the Evening Course A 15 35 Course B 30 20

Consider the following three statements. Statement I: 30% of students in Course B prefer sessions in the afternoon. Statement II: 50% of all students in the survey are in Course B. Statement III: More than 50% of all students surveyed prefer sessions in the afternoon. Which of the statements is true based on the information in the table? (a) Only Statement I (b) Only Statement II (c) Only Statement III (d) None of the three statements are true. (e) All three statements are true.

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(6) The Discover credit card is advertising a special 5% cash-back bonus on all purchases in grocery stores and restaurants. Discover Card users also receive the regular 1% cash-back bonus on all other purchases. If you typically charge a total of $2,000 a month, 1/4 of which is at grocery stores and restaurants, what is the amount of your monthly cash-back bonus? (1.1.3 and 1.1.4) (a) $30 (b) $40 (c) $100 (d) $120 (e) $400

(7) Americans earning between $34,500 and $83,600 fall into the 25% tax bracket, which means they pay $4,750 plus 25% of income over $34,500 in taxes. If you earn $60,000, which calculation is the correct one to estimate how much money you will pay in taxes? (1.2.2) (a) 4,750 + 0.25(60,000 – 34,500) (b) 0.25 x 60,000 (c) 4,750 + 0.25 x 60,000 (d) 4,750 + 0.25 x 60,000 – 34,500 (e) 4,750 + 0.25 x 34,500

(8) Tuition at Penn State’s main campus was $7,500 in Fall 2010. Officials have announced a 6% increase for Fall 2011. Which statement gives the most exact description of the new tuition? (1.1.3) (a) Tuition will increase. (b) Tuition will go up by roughly $400. (c) Tuition is high. (d) Tuition will cost $7,950 in Fall 2011. (e) Tuition in Fall 2011 will be less than $8,000.

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(9) A new car is estimated to lose 15 to 20% of its original value during the first year after buying it.2 If you bought a new car for $24,000, what is a reasonable estimate for what the car would be worth after one year? (1.1.3) (a) $8,000 (b) $12,000 (c) $15,000 (d) $17,000 (e) $20,000

(10) Two-thirds of undergraduate students who graduated with a bachelor's degree incurred some debt in 2007–2008. The average student loan debt among these graduating seniors was $23,186.3 How much interest will a student with this balance need to pay in the first year if the average interest rate is 8.5%? (1.1.3) (a) $197.08 (b) $1,970.81 (c) $3,477.90 (d) $19,708.10 (e) $25,156.81

(11) It is estimated that the world population reached 7 billion in 2011. What is this number written in scientific notation? (1.2.1) (a) 7.0 x 103 (b) 7.0 x 106 (c) 7.0 x 109 (d) 7.0 x 1012 (e) None of the above

2Retrieved from www.buyingadvice.com/featured-car-articles/ownership-survey 3Retrieved from www.finaid.org/loans

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(12) Winning the U.S. Open Tennis championship can be quite profitable. One of the largest paychecks was given to Roger Federer in 2007 when he won $2.4 million.4 Even the tennis player who loses the first round of the tournament makes $3,000. Given these amounts, you can calculate that Federer made ______times as much as the loser of the first round. (1.2.1) (a) 1/8 (b) 8 (c) 80 (d) 800 (e) 8,000

(13) A popular Internet video calling service has one plan that charges $0.02 per minute plus a connection fee of $0.04 for each call. Pria makes four calls each lasting 18 minutes. Myan makes three calls each lasting 22 minutes. Assuming they are both using this service plan, who spent the most on their calls? (1.2.2) (a) Pria (b) Myan (c) The both spend the same amount.

(14) A state government is implementing a new annual fee on each vehicle registered in the state to fund road safety construction projects. The fee schedule is given in the table below. Calculate the Road Safety Fee for someone driving a 2006 Toyota Corolla that weighs 3,585 pounds. (1.2.2)

Weight of Car Road Safety Fee 2,000 pounds and less $16 plus $0.01 per pound 2,001 to 5,000 pounds $23 plus $0.01 per pound 5,001 to 10,000 pounds $28 plus $0.02 per pound 10,001 to 16,000 pounds $37 plus $0.02 per pound More than 16,000 pounds $39 plus $0.03 per pound

(a) Less than $20 (b) Between $20 and $40 (c) Between $40 and $60 (d) Between $60 and $80 (e) More than $80

4Retrieved from www.usopen.org/en_US/news/articles/2011-08-04/201108041312495773418.html

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(15) In 2007, 32.2% of college students were minorities.5 If you took a random group of 500 college students in 2007, how many would you expect to be minorities? (1.2.3) (a) Less than 60. (b) Between 60 and 90. (c) Between 90 and 120. (d) Between 120 and 150. (e) More than 150.

(16) Results from a national study of science and engineering graduates are given below.6 What percentage of those who attended community college were married? Round to the nearest tenth of a percent. (1.2.3)

Attended Did Not Attend Marital Status Community College Community College Married 126,700 120,700 Separated, Divorced, or Widowed 49,100 9,400 Never Married 242,400 379,100

(a) 13.7% (b) 30.3% (c) 33.4% (d) 51.2% (e) 52.2%

(17) In 1998, 63% of women aged 16–24 worked outside of the home. Which of the following statements would be accurate interpretations of these data? (1.2.3) (a) In 1998, in a group of 100 workers aged 16–24, approximately 63 of them would be women. (b) In 1998, in a group of 100 women aged 16–24, approximately 63 of them would work outside of the home. (c) Both of these interpretations are correct. (d) Neither of these interpretations is correct.

5Retrieved from http://nces.ed.gov/fastfacts/display.asp?id=98 6Retrieved from nces.ed.gov/fastfacts/display.asp?id=98

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(18) Some pregnant women are screened to see if their babies are at risk for Down syndrome. The table below presents data on 1,000 tested women. If a test is positive for Down syndrome, what is the chance the baby actually has this syndrome? (1.2.4)

Baby does not Baby has Down have Down Total syndrome. syndrome. Test Positive for Down 15 35 50 syndrome Test Negative for Down 5 945 950 syndrome Total 20 980 1,000

(a) There is a 30% chance that the baby has Down syndrome if the test is positive. (b) There is a 20% chance that the baby has Down syndrome if the test is positive. (c) There is a 15 out of 1,000 chance that the baby has Down syndrome if the test is positive. (d) There is a 15 out of 20 chance that the baby has Down syndrome if the test is positive. (e) None of the above.

Use the table below to answer Questions 19 and 20.

Population Total Water Footprint Country (in thousands) (in 109 cubic meters per year) Mexico 112,322 222.17

(19) Calculate Mexico’s water footprint per person in cubic meters per year. (1.2.1) (a) 1.978 m3/year per capita (b) 197.78 m3/year per capita (c) 1,978 m3/year per capita (d) 0.002 m3/year per capita (e) 505.57 m3/year per capita

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(20) If the world population is 6.9 million people, what percentage of the world’s population lives in Mexico? (1.1.3) (a) 0.61% (b) 6.1% (c) 0.016 % (d) 0.16 (e) 1.6%

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During any class, it is important to frequently and accurately assess what you do and do not know. This is especially true before a quiz or test or when ending a module or chapter. Math is different from many subjects. In math, you often have to show you can complete a problem, not just remember facts or choose the right answer. Every math student has had the experience of looking at work they have previously completed or examples done in the book and thinking, “I know how to do that,” only to get home or into a test and not be able to do a similar problem. To check your understanding accurately, you must do problems that represent the concepts and skills you need to know. If you take time to accurately assess what you know, you can cut down on your study time. You can dedicate your study time to learning only the concepts and skills you need to understand better.

Assessing Your Understanding The table on the following page lists the Module 1 concepts and skills you should understand. This exercise helps you assess what you understand. After completing it, you will be able to prioritize your review time more effectively. 1. Assess your understanding. • Go through the topics list and locate each concept or skill in the Module 1 in-class or OCE materials. • If you have not used the skill in a while, do two or more problems to check your understanding. • If you have recently used the skill and feel confident that you did it correctly, rate your understanding a 4 or 5. • If you remember the topic but could use more practice, rate your understanding a 3. • If you cannot remember that skill or concept, rate your understanding a 1 or 2. Now that you have done an initial rating of your understanding, it is time to begin reviewing. Complete the remaining steps. The goal is to have a confidence rating of 4 or 5 on all the topics in the table when you have finished your review of Module 1. 2. Start at the beginning of module and reread the material in the lessons, the OCE, and your notes on the skills and concepts you rated 3 or below. 3. Select a few problems to do. Do not look at the answer or your previous work to help you. 4. Once you have finished the problems, check your answers. If you are not sure if you have done the problems correctly, check with your instructor, other classmates, and your previous work or work with a tutor in the learning center. 5. Rate your confidence on this skill again. If you understand the concept better, rate yourself higher. Begin a list of topics that you want to review more thoroughly. 6. If you have time, do one or two problems on skills or concepts you rated 4 or above. 7. For topics that you need to review more thoroughly, make a plan for getting additional assistance by studying with classmates, visiting your instructor during office hours, working with a tutor in the learning center, or looking up additional information on the Internet.

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Module 1 Concept or Skill Rating Working with and Understanding Large Numbers Place value and naming large numbers (1.1.1) Scientific notation (1.2.1) Calculations with large numbers (1.2.1) Relative magnitude and comparison of numbers (1.2.1) Estimation and Calculation Rounding (1.1.1) Fractions and decimals (1.1.3) Relationship of multiplication and division (1.1.4) Order of operations (1.1.4) Properties that allow flexibility in calculations: Distributive Property, Commutative Property (1.1.4) Perform multistep calculations (1.2.2) Percentages and Ratios Estimations with fraction and percent benchmarks (1.1.2, 1.1.3) Calculate percentages (1.1.3) Write and understand ratios (1.2.1) Calculate percentages from a two-way table (1.2.3, 1.2.4) Use percentages as probabilities and ratios (1.2.4)

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Specific Objectives Students will understand that • population density is a ratio of the number of people per unit area. • population density may be described proportionately to compare populations. Students will be able to • calculate population densities. • calculate population density proportions from density ratios. • compare and contrast populations via their densities.

Problem Situation 1: Using Ratios to Measure Population Density In Lesson 1.1.2, you learned that Earth’s human population has grown from about 1 billion people to nearly 7 billion in the last two centuries. However, populations in different regions do not always grow uniformly. For example, populations tend to increase in areas where people already live close enough to one another to find mates. On the other hand, crowded populations decrease when deadly diseases such as smallpox or Ebola virus, sweep through them. In this lesson, you will compare geographic regions by their population densities. Definition: The population density of a geographic region is a ratio of the number of people living in that region to the area of the region. Population density ratios are “reduced” by division in order to compare them with a standard area measurement.

Example Imagine 100 people standing on a parking lot that measures 20 feet by 20 feet. The people are spaced so that each person stands on a 2-foot by 2-foot square. The population density could be thought of as 100 people per 400 square feet or as 1 person per 4 square feet, or it could be expressed as fractions:

You call this equation a proportion because the equation shows that two ratios are equal. You can also state the relationship in words: One person per 4 square feet is proportional (equal) to 0.25 person per (1) square foot. How would the population density change if 1 billion people each stood on a 2-foot by 2-foot square? The following questions will help you understand how to calculate population density for different areas. You will start by doing an activity with your class. Your teacher will create a small taped off area for the "small rectangle". You can measure your classroom for the "large rectangle". (1) Calculate the population density based on the small rectangle. Be sure to include units. (2) Calculate the population density based on the large rectangle. Be sure to include units. (3) Imagine that a billion people stand on adjacent 2-foot by 2-foot squares. Calculate the population density per square mile. Be ready to explain your reasoning after working with your group members. (1 mile = 5,280 feet)

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Problem Situation 2: Making Comparisons with Population Density How crowded is China, compared to the United States? (4) In 2010, in the United States, approximately 309,975,000 people occupied 3,717,000 square miles of land. In China, approximately 1,339,190,000 people lived on 3,705,000 square miles of land. Use this information to answer the following questions. (a) A student carefully calculates the population densities of China and the United States. He decides that China is less dense than the United States. Using your estimation skills, decide if you think this student’s calculation is correct. (b) At a lecture, you hear someone claim that, in terms of population, China is more than four times as dense as the United States. Using your estimation skills, decide if you think this statement is correct. (c) Calculate more precisely the densities (per square mile) of the Chinese and U.S. populations. Based on your calculation, how many times more dense is the more crowded population? Be ready to share your calculations during the class discussion.

Making Connections Record the important mathematical ideas from the discussion.

Further Applications (1) The out-of-class experience contains information about the population of Alaska. Explain how the statements “Anchorage has more than 40% of the Alaskan population” and “Ketchikan has the most dense population” might both be correct.

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Making Connections to the Lesson (1) Which of the following was one of the main mathematical ideas of the lesson? The questions refer to the following quantities: 50 people/mi2 20 ft/5 sec 34% (i) All three of the figures are ratios. (ii) The first two figures are ratios. (iii) The second figure could be written as 4 ft/sec. (iv) The first figure is a population density. In Module 2, you will be asked to write statements about connections among the mathematical ideas in lessons. For this first assignment in Module 2, you are given statements to choose from. In future assignments, you may want to refer back to this as an example of how to write about connections.

(2) Refer back to Lesson 1.2.1. Choose the statement that explains how the mathematical ideas in Question 8 connect to the work in Lesson 2.1.1. (i) In Lesson 1.2.1, you used the water footprints of the United States. and of China. In Lesson 2.1.1, you used the population density of the United States and China. Both lessons compared the United States and China. (ii) In Lesson 1.2.1, you had to calculate the water footprint per person in the United States. This is like splitting up all the water equally among the people. In Lesson 2.1.1, you calculated the population density of the United States. This is like splitting up all the people equally over an area of land. These are both ratios. (iii) In Lesson 1.2.1, you used the population of a country to find the water footprint per person. In Lesson 2.1.1, you used the population of a country to find a population density. Both lessons used population for calculations.

Developing Skills and Understanding (3) Use the picture below to answer the questions.

Area of Rectangle A = 12 ft2 Rectangle A Area of Rectangle B = 4 ft2

Rectangle B

(a) What is the density of the stars in Rectangle B?

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(b) What is the density of the stars in Rectangle A (Note: The stars in Rectangle B are also in Rectangle A). Round to the nearest hundredth of a star per square foot. (c) Suppose three new stars were added in the gray part of Rectangle A. Which of the following statements would be correct? (i) The density of Rectangle A would increase. The density of Rectangle B would stay the same. (ii) The density of Rectangle A would decrease. The density of Rectangle B would stay the same. (iii) The density of Rectangle A would increase. The density of Rectangle B would increase. (iv) The density of Rectangle A would decrease. The density of Rectangle B would decrease. (v) The density of Rectangle A would stay the same. The density of Rectangle B would increase.

(4) Wikipedia states the following about Anchorage, Alaska: “The city constitutes more than 40 percent of the state’s total population.”1 (a) Calculate the population density for Anchorage, based on a 2010 population of 291,826 people living on 1,961.1 square miles. Round to the nearest person per square mile.

(b) Wikipedia also says that the small Alaskan town of Ketchikan has the densest population in Alaska. Ketchikan had a population of 7,368 in 2010 and an area of 4.1 square miles. Calculate the population density of Ketchikan. Round to the nearest person per square mile.

(5) The following information comes from the lesson:

Population Land Area (sq. miles) United States 309,975,000 3,717,000 China 1, 339, 190,000 3,705,000

In India, about 1,184,639,000 people live on 1,269,000 square miles of land.2 Which of the following statements is false? (i) The population density of India is approximately three times that of China. (ii) The population of India is approximately 11 times the population density of the United States. (iii) The population densities of these three countries ranked from smallest to largest are United States, India, and China. (iv) The population density of the United States is approximately 83.4 people per square mile. That is smaller than the population densities for China and India.

1 Retrieved from http://en.wikipedia.org/wiki/Anchorage,_Alaska 2 Retrieved from www.worldatlas.com/aatlas/populations/ctypopls.htm

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(6) Which of the following population densities are equivalent to a density of 20 people/mi2? There may be more than one correct answer. (i) 1 person/0.2 mi2 (ii) 1 person/0.05 mi2 (iii) 1 person/0.5 mi2 (iv) 200 people/2 mi2 (v) 100 people/5 mi2

(7) One way to measure a country’s economy is per capita gross domestic product, or per capita GDP. This is the value of all the products and services produced in a country over the course of a year divided by its population. (a) According to the CIA’s The World Factbook, in 2010, the United States had a per capita GDP of $47,400. If the population was about 309 million, which of the following is a reasonable estimate for the GDP of the United States? (i) $60 trillion (ii) $60 billion (iii) $15 billion (iv) $15 trillion (v) $2.3 trillion (b) Also according to The World Factbook, in 2010, China had a per capita GDP of $7,400 and a population of around 1,337,000,000. Which of the following is a reasonable estimate for the GDP of China? (i) $60 trillion (ii) $60 billion (iii) $15 billion (iv) $2.3 trillion (v) None of the above

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Making Connections Across the Course (8) The population of Nebraska is 1,826,341, and its population density is 23.8 people per mi2. The population of New Hampshire is 1,316,470, and its population density is 146.8 people per mi2. Which of the following statements is a valid interpretation of this information? (i) Nebraska is approximately six times more densely populated than New Hampshire. (ii) The population of Nebraska is approximately 72% of the population of New Hampshire. (iii) New Hampshire is approximately six times more densely populated than Nebraska. (iv) The population of New Hampshire is approximately 139% of the population of Nebraska. (9) Cholera, a serious intestinal disease, broke out in London in the mid-19th century. People at that time believed cholera was caused by bad air. A physician named John Snow discovered that 61 victims either used the water pump on Broad Street or lived nearby. His research became the basis for the theory that germs cause disease. In the case of cholera, the germs are transmitted by polluted water. A vaccine for cholera was developed in the late 1800s. A vaccine is a drug that helps a person become immune to a disease. Scientists studied 818 people to determine the effects of a cholera vaccine. The study lasted from 1894 to 1896.

Infected Not Infected Vaccinated 3 276 Not vaccinated 66 473

Use the information in the table to complete the following four statements. Round to the nearest tenth of a percent. (a) ______% of those vaccinated were infected (b) ______% of those not vaccinated were infected (c) ______% of those infected were vaccinated (d) ______% of those infected were not vaccinated (e) Which of the following statements are correct based on the information in the table? There may be more than one correct answer. (i) A person who was not vaccinated was 12 times more likely to get cholera than someone who got the vaccine. (ii) A person who was not vaccinated was four times more likely to get cholera than someone who got the vaccine. (iii) If a person was infected, it was four times more likely that (s)he was not vaccinated rather than vaccinated. (iv) If a person was infected, it was more than 20 times more likely that (s)he was not vaccinated rather than vaccinated.

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Preparing for the Next Lesson (2.1.2) (10) The following problems have to do with multiplying and dividing by powers of 10. Look for patterns and ways to find the answers mentally without a calculator or writing the problem down. Check your answers with a calculator if you wish. (a) 0.32 x 10 (b) 3.2 x 10 (c) 32 x 10 (d) 32 x 100 (e) 51,000 x 10,000 (f) 900 x 104 (g) 1.3 x 107 (h) 0.32 ÷ 10 (i) 3.2 ÷ 10 (j) 3,200,000 ÷ 10 (k) 5,500,000 ÷ 1,000 (l) 83,000,000 ÷ 10,000,000 (m) 67 ÷ 104

(11) You multiply 58,000,000,000 x 10,000 and your display reads: 5.8 E14. Which of the following represent(s) the same number as the number displayed on your calculator? There may be more than one correct answer. (i) 5.8 x 14 (ii) 5.8 x 1014 (iii) 58 x 1014 (iv) 580,000,000,000,000 (v) 5,800,000,000,000,000

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The following information will be used in Lesson 2.1.2. The states of the United States vary greatly in both size and population. Some states, especially on the east coast, are small and mostly urban, meaning that most people live in cities. Other states in the west and plains are larger and more rural, with people living in small towns or in the countryside. Population density is one way to measure how crowded a state is. If an area is densely populated, it will need more services such as schools and hospitals. Some states, such as Washington, use the population density of counties to classify them as urban or rural in state law. This can affect whether residents qualify for certain kinds of assistance programs.3 (12) Use the information given below to calculate the population density for each of the states listed. Round to the nearest tenth. A State Population Density Table will be posted with this lesson. Record each state in the table with its population density. Bring this to your next class.

Land Area State Population Answers Square Miles Alaska 571,951 710,231 Idaho 82,747 1,567,582 Kentucky 39,728 4,339,367 Louisiana 43,562 4,533,372 Nebraska 76,872 1,826,341 New Hampshire 8,968 1,316,470 New Mexico 121,356 2,059,179 South Dakota 75,885 814,180 Washington 66,544 6,724,540 Wisconsin 54,310 5,686,986

(13) You will be expected to do the following things for the next class. Rate how confident you are on a scale of 1–5 (1 = not confident and 5 = very confident). Before beginning Lesson 2.1.2, you should understand the concepts and demonstrate the skills listed below:

Skill or Concept: I can … Rating from 1 to 5 Multiply by 10 by moving a decimal one place to the right and divide by 10 by moving a decimal one place to the left. Interpret an answer given in scientific notation on their calculator. Calculate and interpret population density.

3Retrieved from http://www.ofm.wa.gov/pop/popden/rural.asp

The Carnegie Foundation for the Advancement of Teaching Quantway frontmatter available at www.statway.org/kernel and The Charles A. Dana Center at the University of Texas at Austin or www.utdanacenter.org/mathways/index.php 6 Lesson 2.1.2: Student Data (Retrieved from http://en.wikipedia.org/wiki/U.S._state#List_of_states, May 8, 2011) 121 State Land Area (Square Miles) 2010 Population Population Density (People/mi2) Alabama 50,744 4,779,736 Alaska 571,951 710,231 Arizona 113,635 6,392,017 Arkansas 52,068 2,915,918 California 155,959 37,253,956 Colorado 103,718 5,029,196 Connecticut 4,845 3,574,097 Delaware 1,954 900,877 District of Columbia 61 601,723 Florida 53,927 18,801,310 Georgia 57,906 9,687,653 Hawaii 6,423 1,360,301 Idaho 82,747 1,567,582 Illinois 55,584 12,830,632 Indiana 35,867 6,483,802 Iowa 55,869 3,046,355 Kansas 81,815 2,853,118 Kentucky 39,728 4,339,367 Louisiana 43,562 4,533,372 Maine 30,862 1,328,361 Maryland 9,774 5,773,552 Massachusetts 7,840 6,547,629 Michigan 56,804 9,883,640 Minnesota 79,610 5,303,925 Mississippi 46,907 2,967,297 Missouri 68,886 5,988,927 Montana 145,552 989,415 Nebraska 76,872 1,826,341 Nevada 109,826 2,700,551 New Hampshire 8,968 1,316,470 New Jersey 7,417 8,791,894 New Mexico 121,356 2,059,179 New York 47,214 19,378,102 North Carolina 48,711 9,535,483 North Dakota 68,976 672,591 Ohio 40,948 11,536,504 Oklahoma 68,667 3,751,351 Oregon 95,997 3,831,074 Pennsylvania 44,817 12,702,379 Rhode Island 1,045 1,052,567 South Carolina 30,109 4,625,364 South Dakota 75,885 814,180 Tennessee 41,217 6,346,105 Texas 261,797 25,145,561 Utah 82,144 2,763,885 Vermont 9,250 625,741 Virginia 39,594 8,001,024 Washington 66,544 6,724,540 West Virginia 24,078 1,852,994 Wisconsin 54,310 5,686,986 Wyoming 97,100 563,626 50 states + DC 3,537,438 308,745,538

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Specific Objectives Students will understand • the concept of population density as a ratio. • what is meant by proportional or change based on a constant ratio. Students will be able to • estimate between which two powers of 10 a quotient of large numbers lies. • calculate a unit rate. • solve a proportion by first finding a unit rate and then multiplying appropriately.

Problem Situation: Estimating Population Densities You will compare the populations of different states and explore how population density affects a states’ representation in the U.S. Congress. You calculated population densities of some states in your out-of-class experience. Now, you will develop strategies for estimating population densities. (1) Check your answers from the out-of-class experience with your group. Now discuss strategies you can use to estimate the population density of the states without using a calculator. Use your strategies to divide the states into the categories shown in the table below.

Density > 1,000 Density < 10 100–1,000 people/mi2 10–100 people/mi2 people/mi2 people/mi2

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(2) Find which state has the greatest population density. What is that population density? Round to the nearest tenth.

(3) Find which state has the least population density. What is that population density? Round to the nearest tenth.

(4) If your campus had the same population density as the state with the greatest population density, how many people would be on campus? What if your campus had the same population density as the state with the least population density?

(5) Most of the world outside the United States uses the metric system of measurement, so it is often useful to be able to make comparisons between the American system and the metric system. Bangladesh has a population density of 1,127 people/square kilometer. (Note: 1 kilometer = 0.62 mile1) (a) If you converted the density of Bangladesh to square miles, would the measure be larger or smaller than 1,127? Explain your reasoning. (b) Which of the following statements is the most accurate description of the relationship between a square kilometer and a square mile? (i) A square kilometer is about one-sixth of a square mile. (ii) A square kilometer is about two-thirds of a square mile. (iii) A square kilometer is about one-third of a square mile. (iv) A square kilometer is about six-tenths of a square mile. (c) How many people would be on your campus if the population density were the same as Bangladesh?

Making Connections Record the important mathematical ideas from the discussion.

1Retrieved from http://en.wikipedia.org/wiki/List_of_sovereign_states_and_dependent_territories_by_population_density

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Further Applications (1) The OCE assignments for Lessons 1.2.3 and 1.2.4 had examples of people collecting data to answer questions. In Lesson 1.2.3, the example was about a teacher collecting data on student grades, and in Lesson 1.2.4, the example was about a hospital collecting data about the types of cases that came into the emergency room. Give an example of a situation in which you might collect data to answer a question. You can think of your own situation based on your interests or you may use the examples given below. For the situation you choose: (a) Identify two questions you would ask. (b) Identify the data you would collect to answer the question.

For example, this is how you would present the situation from Lesson 1.2.3: Situation: Teacher comparing performance of two classes Questions: Do your afternoon students do better on exams than your morning students? What percentage of students in each class gets passing grades on exams? Data: Letter grades on the exams for students in your two classes. Situations you might use: • someone buying a new car • a parent concerned about a child’s eating habits • a contractor who has to make estimates on jobs • a commuter considering driving versus taking the bus to work

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Making Connections to the Lesson (1) Which of the following was one of the main mathematical ideas of the lesson? (i) Population density measures how crowded a state is. Example: 2.3 people per square mile is less dense than 8.7 people per square mile. (ii) You can find a new value that is changing by a constant rate by adding a ratio. Example: A car’s gas mileage is 20 miles/gallon. How far can it drive on 5 gallons of gasoline? 20 mi/gal + 5 = 25 miles. (iii) A percent is a ratio compared to 1. Example: 12% increase in population means that the number of people increases by 12 for every 1 person in the original population. (iv) You can find a new value that is changing by a constant rate by multiplying by a ratio. Example: A person’s wage is $10.35/hour. How much does the person earn in 40 hours? $10.35/hour x 40 hours = $414.

(2) Explain a connection between a concept in this lesson and at least one of the following lessons: 1.1.3, 1.2.1, 1.2.2. You can use one of the sentence stems given below if you wish. You can also refer back to Module 1 lessons as examples. Question number______in Lesson 2.1.2 connects to Question number______in Lesson _____ because ______. The idea of ______in Lesson 2.1.2 connects to Lesson ______. An example of the connections is ______.

Developing Skills and Understanding (3) In the lesson, you used ratios in the form of population densities. A population density is a ratio because it is a comparison of two measures: Number of people per number of square miles. Which of the following are ratios? There may be more than one correct answer. (i) 252 miles (ii) 67 hours (iii) 10 miles/hour (iv) 5 lb/$3 (v) $98

(4) Calculate the gas mileage of a car that drives 283 miles on 12.3 gallons of gas. Round to the nearest tenth of a mile/gallon.

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(5) A car drives 630 miles on 35 gallons of gas. How far can it drive on 12 gallons?

(6) A jar holds 128 fluid ounces of juice. The label says the jar has 16 servings. How many fluid ounces are needed for 80 servings?

(7) According to the oil company BP, in 2010, the United States used 19,148,000 barrels of oil a day, and worldwide, people used around 87,382,000 barrels of oil per day.1 This includes oil used for (among other things) fuel and manufacturing. (a) If there were 309 million people in the United States in 2010, what was the daily consumption rate per person in the United States? Round to the nearest hundredth of a barrel. (b) If there were 6.89 billion people in the world in 2010, which of the following statements would be correct? (i) The U.S. rate of oil consumption per person was about five times the world rate. (ii) About half the people in the world lived in the United States. (iii) If the world used oil at the same rate as the United States, it would have used about 93,450,000 barrels of oils per day. (iv) About one-fifth of the oil used in the entire world in one day is used in the United States. (c) There are 42 gallons in a barrel of oil. Which of the following statements is true? (i) The American rate of oil consumption is 5 more gallons of oil per day than the world rate. (ii) The American rate of oil consumption is 7,452,380 gallons of oil per day. (iii) If the world used oil at the same rate as the United States, it would use about 426,957,000 gallons of oil per day. (iv) The average American is responsible for about 2.5 gallons of oil use per day.

(8) Approximately 6.9 billion people now inhabit the earth. The surface area of the earth is 510,065,600 km2. (a) What is the surface area of the earth in mi2? Round to the nearest million square miles. Hint: If 1 km = 0.6214 mi, then 1 km2 = how many mi2? (i) 197,000,000 mi2 (ii) 317,000,000 mi2 (iii) 122,000,000 mi2 (iv) 82,000,000 mi2 (b) The surface area above includes both land and water. Approximately 139 million square miles of the earth’s surface area is water. Using your answer from Part (a), determine what percentage of the surface area is land. Round to the nearest tenth of a percent.

1http://www.bp.com/liveassets/bp_internet/globalbp/globalbp_uk_english/reports_and_publications/statistical_energy_revie w_2011/STAGING/local_assets/pdf/oil_section_2011.pdf

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(c) Approximately 1/3 of the land is uninhabitable, meaning people cannot live on it. How much land on the earth is inhabitable (can be lived on)? Round to the nearest million square miles.2 (i) 106,000,000 mi2 (ii) 54,000,000 mi2 (iii) 39,000,000 mi2 (iv) 27,000,000 mi2 (d) Estimate the population density of the earth in people per square mile of habitable land. Round to the nearest person per square mile.

Making Connections Across the Course (9) People often confuse the words million, billion, and trillion when speaking. An estimate can help you decide if the speaker uses the correct word. Consider this situation: A speaker says, “The U.S. federal debt is $14 billion dollars. That’s over $45,000 for every person in the country.” Select the correct statement from the choices below. Note: When you say the numbers are consistent, you mean that they make sense in relationship to each other. (i) The two numbers in the statement are consistent with each other. (ii) The two numbers in the statement are not consistent. If the debt is $45,000 per person, the total debt must be $14 million. (iii) The two numbers in the statement are not consistent. If the debt is $45,000 per person, the total debt must be $14 trillion.

2Annenburg Learner, http://www.learner.org/courses/envsci/unit/text.php?unit=7&secNum=2

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(10) Terrence is very careful about tracking his gas mileage. Every time he fills his gas tank, he records how much gas he buys and the number of miles he has driven. He puts this information into a spreadsheet so he can easily calculate his gas mileage.

(a) Select the formula that would calculate Terrence’s gas mileage. (i) = (A2 + A3 + A4 + A5) / (B2 + B3 + B4 + B5) (ii) = A2 + A3 + A4 + A5 / B2 + B3 + B4 + B5 (iii) = (B2 + B3 + B4 + B5) / (A2 + A3 + A4 + A5) (iv) = B2 + B3 + B4 + B5 / A2 + A3 + A4 + A5 (b) Terrence is planning a long road trip of about 1,000 miles. The average price of gas is $3.85/gallon. Based on the data in the spreadsheet, estimate how much he should budget for gas. Round to the nearest dollar.

(11) Consider the reflect phase reading at the end of the last module. (a) Give an example of two internal factors that contributed to how well you did on your last exam. Remember that these factors can have a positive or a negative contribution. (b) In the plan phase preparing for your next exam, what would you do differently? The plan phase reading is in OCE 1.1.4.

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In many of the future lessons, you will need to work with negative numbers. Negative numbers arise naturally when one computes the difference between two quantities. If the difference is negative, you not only know the difference between the two quantities, you automatically know which value was the larger of the two! (12) The following table was created to compare two cell phone plans, Plan A and Plan B. The monthly bill for each plan was computed based on the number of minutes used that month. While the actual monthly bills are not given, their differences are given in the last column.

Minutes Plan A Plan B Plan B – Plan A 50 * * –7.5 100 * * –5 150 * * –2.5 200 * * 0 250 * * 2.5 300 * * 5 350 * * 7.5 400 * * 10

(a) The first entry in the last column is –7.5. Which of the following statements explains what this tells you about Plan A and Plan B? (i) Plan A costs $7.50 more than Plan B for someone who uses 50 minutes. (ii) Plan B costs $7.50 more than Plan A for someone who uses 50 minutes. (iii) The Plan A customer used the phone for 7½ minutes less than the Plan B customer. (iv) The Plan B customer used the phone for 7½ minutes less than the Plan A customer. (b) Further down the last column is the entry 7.5. What does this tell you about Plan A and Plan B? Write your answer as a complete sentence using the Writing Principle from Module 1. (c) When are the two plans equal? Write your answer as a complete sentence using the Writing Principle from Module 1.

Preparing for the Next Lesson (2.1.3) (13) Calculate the following values. (a) What percent of 20 is 2? (b) 28 is what percent of 80? (c) What percent is 18/24?

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(14) A school principal of 810 students needs to determine what percent of students passed and did not pass a statewide examination. Round to the nearest percent. (a) If 550 students passed the exam, what percent passed the test? (b) What percent did not pass the test?

(15) A laptop computer that you want to purchase was originally priced at $1,225. You will receive a 20% student discount, and the sales tax rate is 8%. How much money will you pay for the laptop? (i) $245 (ii) $1,058.40 (iii) $980 (iv) $1,323

(16) You will be expected to do the following things for the next class. Rate how confident you are on a scale of 1–5 (1 = not confident and 5 = very confident). Before beginning Lesson 2.1.3 you should understand the concepts and demonstrate the skills listed below:

Skill or Concept: I can … Rating from 1 to 5 Find a percent of a number.

(17) Self-Regulated Learning: Reflect How much time and effort per week is this course taking? Is it what you expected? Is there anything you need to adjust in your weekly schedule to make sure you are successful?

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Specific Objectives Students will understand that • a relative change is different from an absolute change. • a relative measure is always a comparison of two numbers. Students will be able to • calculate a relative change. • explain the difference between relative change and absolute change.

Problem Situation: How the Census Affects the House of Representatives Every 10 years, the United States conducts a census. The census tells how many people live in each state. You can also find how much population has changed over time from the census data. The original purpose of the census was to decide on the number of representatives each state would have in the House of Representatives. Census data continue to be used for this purpose, but now have many other uses. For example, governments may use the data to plan for public services such as fire stations and schools. You will be given a list of states in a census region and their populations in 2000 and 2010. You will be asked to calculate the population growth in people as a percentage for each state in the region and for the region as a whole. You will examine how this affects the number of representatives each state has in the House of Representatives. You will start by looking at changes in representation based on the 2010 census. The absolute change in a state’s population tells by how many people the population has changed. The relative change is the change as it compares to the earlier population. Often relative change is given as a percentage. Use the following data for Questions 1–6. South Atlantic States

Absolute Percentage 2010 Population 2000 Population Change Change Delaware 900,877 783,600 Florida 18,801,310 15,982,378 Georgia 9,687,653 8,186,453 Maryland 5,773,552 5,296,486 North Carolina 9,535,483 8,049,313 South Carolina 4,625,364 4,012,012 Virginia 8,001,024 7,078,515 Washington, D.C. 601,723 572,059 West Virginia 1,852,994 1,808,344 South Atlantic Region

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Mountain States

Absolute Percentage 2010 Population 2000 Population Change Change Arizona 6,392,017 5,130,632 Colorado 5,029,196 4,301,261 Idaho 1,567,582 1,293,953 Montana 989,415 902,195 Nevada 2,700,551 1,998,257 New Mexico 2,059,179 1,819,046 Utah 2,763,885 2,233,169 Wyoming 563,626 493,782 Mountain Region

(1) For your group of states, calculate the absolute change in the population of each state.

(2) For your group of states, calculate the relative change in the population of each state. Express your answer as a percentage.

(3) List in order the three states that changed most in absolute population.

(4) List in order the three states that had the largest relative increase in population.

(5) Explain why the lists in Question 3 and Questions 4 are not the same.

(6) For the region you are given, calculate the absolute change in total population from 2000 to 2010. Calculate the relative change in total population between 2000 and 2010.

While most states that lost representatives did so because their population became smaller relative to other states, Michigan’s population actually fell between 2000 and 2010. (7) Michigan’s population changed to 9,833,640 from 9,938,444. What was the absolute decrease in Michigan’s population? What was the relative change in Michigan’s population? Round your answer to the nearest hundredth of a percent.

Making Connections Record the important mathematical ideas from the discussion.

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Further Applications (1) In 2011, the U.S. Congress had a major debate over cutting the federal budget mid-year. The goal was to reduce the national debt, which was $14 trillion. (a) One group wanted to reduce the budget by $100 billion. How large is this change relative to the national debt? (b) Another group wanted to reduce the budget by $40 billion. How large is this change relative to the national debt? (c) If a politician wanted to argue for the larger cut, would he or she use the absolute or the relative change to justify his or her position? Why? (d) If a politician wanted to argue for the smaller cut, would he or she use the absolute or the relative change to justify his or her positions? Why?

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Making Connections to the Lesson (1) Which of the following was one of the main mathematical ideas of the lesson? (i) Absolute change is measured as a quantity (for example, an increase of $3). Relative change is measured as a percentage compared to the reference value (for example, an increase of 3%). (ii) To find relative change, subtract the original number from the new number and divide by the original. (iii) Consider this situation: Quantity 1 increases by 15%. Quantity 2 increases by 20%. Quantity 2 must have increased by a larger amount than Quantity 1. (iv) The population of a state determines how many representatives that state has in the House of Representatives.

(2) Explain a connection between a concept in this lesson and at least one of the following lessons: 1.2.3 and 1.2.4. You can use one of the sentence stems given below if you wish. You can also refer back to Module 1 lessons as examples. Question number ______in Lesson 2.1.3 connects to Question number ______in Lesson _____ because ______. The idea of ______in Lesson 2.1.3 connects to Lesson ______. An example of the connections is ______.

Developing Skills and Understanding (3) The following headlines all refer to change. Identify the change as absolute or relative. (a) “Enrollments at Northeastern University are expected to increase by 1,500!” (i) Absolute change (ii) Relative change (b) “Another 14% tuition increase is expected.” (i) Absolute change (ii) Relative change (c) “A new proposal has sales tax rates dropping from 3% to 1%, a drop of only two percent.” (i) Absolute change (ii) Relative change (d) “A new proposal has sales tax rates dropping from 3% to 1%, a 67 percent drop!” (i) Absolute change (ii) Relative change

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Questions 4 and 5 refer to data taken from the U.S. Census.1 The dollar values take into account the changes in the economy over the years (i.e., inflation). Inflation is a complicated issue, but for Questions 4 and 5, you do not need to worry about it. (4) A typical high-income household in 1980 earned $125,556. A similar household in 2009 earned $180,001. What was the relative increase in income for these households from 1980 to 2009? Round to the nearest one percent.

(5) A typical middle-income household in 1980 earned $34,757. A similar household in 2009 earned $38,550. What was the relative increase in income for these households from 1980 to 2009? Round to the nearest one percent.

(6) Due to temporary tax cuts in 2010, a person with typical deductions earning $50,000 per year would have saved 2% of their income plus $850 in federal taxes. (a) How much money did a typical person save? (b) What percent did this person save on her income? Round to the nearest tenth of a percent.

(7) Due to the same law, a person earning $500,000 per year with typical deductions would save 2% of the first $106,800 they earned plus $14,250 in federal taxes. Fill in the blanks to complete the statement below. A person earning $500,000 a year saved $______or ______% of their income. Round to the nearest dollar and to the nearest tenth of a percent.

Making Connections Across the Course If you need to review how to read a pie or circle graph such as the ones below, you may want to view the following video: http://www.youtube.com/watch?v=YFyqueQRCG8&feature=related. You may also want to refer to the handout, Understanding Visual Displays of Information. (8) In Lesson 1.1.4, you used results from the 2009 Consumer Expenditure Survey on how Americans spend their income. A summary of this information is given in Table 1.2

Table 1: Percentages of Average Annual Housing Expenditures

Housing 34.43% Food 12.99% Transportation 15.61% Everything Else 36.97%

1 www.census.gov/hhes/www/income/data/historical/household/index.html 2 www.creditloan.com/infographics/how-the-average-consumer-spends-their-paycheck

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Which pie graph best represents the data given in Table 1?

(i) (iii)

(ii) (iv)

(9) Many egg producers keep chickens in small cages that do not allow the chickens to move. Some people believe that this is unhealthy, so they buy eggs from chickens that are not caged 24-hours a day. These are sometimes called “free-range” chickens. The United States Department of Agriculture (USDA) allows chickens to be called free-range as long as the chickens spend some of their time outside. The European Union (EU), however, has several additional restrictions. One of these is that the farmers must provide enough outside area so that if all the chickens were outside, the density of chickens would be no more than 0.25 chickens/sq meters.3 (a) How many square meters does the EU require for one chicken? (b) A farmer in the United States wants to meet the EU guidelines. She measures her area in yards. How many square yards does she need for 1,100 chickens? (1 m = 1.0936 yd) (i) 301 square yards (ii) 4,400 square yards (iii) 4,812 square yards (iv) 5,262 square yards (v) None of the above

3Wikipedia: Free Range Eggs: http://en.wikipedia.org/wiki/Free_range_eggs

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Preparing for the Next Lesson (2.1.4) If you need to review how to read line graphs like the one shown below, you may want to review this video: http://www.youtube.com/watch?v=Opd1GDJjx4s You will use the following work in your next class. Be sure to take the work to class. (10) Use the graph to answer the following questions. Note that the vertical axis starts at $48,000 instead of $0.

(a) Select the best phrase to complete this sentence: The numbers on the horizontal axis, the line across the bottom of the graph, represent (i) income from $1,999 to $2009. (ii) income from $1,999,000 to $2,009,000. (iii) years from 1999 to 2009. (iv) years and income. (b) Select the best phrase to complete this sentence: The numbers on the vertical axis, the line along the left of the graph, represent (i) money from $48,000 to $53,000. (ii) income for a household or family from $48,000 to $53,000. (iii) income for one person from $48,000 to $53,000. (iv) income in 2009 from $48,000 to $53,000.

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(c) A good estimate of the average household income in 1999 is (i) $52,400 (ii) $52,600 (iii) $52,100 (iv) $52,000 (d) A good estimate of the average household income in 2009 is (i) $49,000 (ii) $50,000 (iii) $49,250 (iv) $49,750 (e) Use the estimates from Parts (c) and (d) to calculate the relative change in the average household income from 1999 to 2009. Round to the nearest one percent. Indicate if the change is an increase or decrease.

(11) Jeff’s Housing: Two pairs of statements are given below. How can both pairs of statements be true? When did Jeff spend more on housing? Be prepared to discuss your answers in class.

In 1990, Jeff spent $600 per month on housing. In 1990, Jeff spent 20% of his income on housing. In 2010, Jeff spent $1,200 per month on housing. In 2010, Jeff spent 10% of his income on housing.

(12) Decide if the following statement is true or false based on the two graphs below. Be prepared to discuss your answer in class.

True or False: This pair of graphs predicts that the number of non-Hispanics in the United States is expected to decline between 2010 and 2050.

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(13) You will be expected to do the following things for the next class. Rate how confident you are on a scale of 1–5 (1 = not confident and 5 = very confident). Before beginning Lesson 2.1.4, you should understand the concepts and demonstrate the skills listed below:

Skill or Concept: I can … Rating from 1 to 5 Read a line graph. Read a bar graph. Read a pie graph. Calculate relative change.

(14) Self-Regulated Learning: Reflect (a) How confident are you that you correctly answered the problems you were assigned in OCE 2.1.3? Rate your confidence on a scale from 1–5 (1 = not confident; 5 = very confident). (b) Which problems from this lesson do you feel you understood well? Which ones might you find it beneficial to talk to your teacher or someone else about? (c) When planning to do this homework assignment, did you accurately predict how long the assignment was going to take? (d) Name two strategies you used when solving this assignment.

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Specific Objectives Students will understand that • the scale on graphs can change perception of the information they represent. • to fully understand a pie graph, the reference value must be known. Students will be able to • calculate relative change from a line graph. • estimate the absolute size of the portions of a pie graph given its reference value. • use data displayed on two graphs to estimate a third quantity.

Graphs are a helpful way to summarize data. Often there are many ways to portray information graphically. Sometimes one form is easier to read than another. Sometimes the way a graph is made can affect the impression it gives. Today, you will look at three examples of such graphs.

Problem Situation 1: Reading Line Graphs (1) Compare Graph 1 from your OCE (2.1.3) and Graph 2 below. What do you notice?

Graph 2

(2) Based on these two graphs, would it be fair to say that the median household income was significantly lower in 2009 than it was in 1999?

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Problem Situation 2: Reading Bar Graphs Your class will discuss how the “Jeff’s Housing” question from your OCE assignment can be used to understand the gross domestic product (GDP) of a country.

US National Debt Trillions of Dollars

Trillions of Dollars 6

0 1950 1960 1970 1980 1990 2000 2010

Graph 3

US National Debt as a Percent of GDP

40 Percent of GDP of Percent

0 1950 1960 1970 1980 1990 2000 2010

Graph 4

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(3) Think about the statement, “The 2010 national debt is way out of hand and has never been higher.” Use Graphs 3 and 4 to evaluate the statement. Is it true? Based on what information?

Problem Situation 3: Reading Pie (Circle) Graphs The following questions refer to the graphs on the Hispanic population in the OCE. (4) The U.S. population in 2010 was around 310,000,000. In 2050, the U.S. population is expected to be around 439,000,000. Estimate the number of Hispanic and non-Hispanic Americans at each time.

(5) Does your work in Question 4 confirm or contradict your prediction from your out-of-class experience? Explain.

Making Connections Record the important mathematical ideas from the discussion.

Further Applications (1) Question 5 in the OCE (2.1.4) gives information about how the Alvarez and Martinez families spend their money. Write a comparison about how much the two families spend on housing. Use the statements in OCE 2.1.4 Question 5a as examples.

(2) Write the answers to the following questions on a piece of paper and bring it to the next class period. If you do not own a credit card, answer Part (a) only. Keep your responses anonymous by writing only the answers to the questions. (Do not write your name.) (a) How many credit cards do you possess? (b) Do you normally pay the entire balance on the credit card statement? (c) What is the approximate balance (total) on your card(s) right now?

The Carnegie Foundation for the Advancement of Teaching Quantway frontmatter available at www.quantway.org/kernel and The Charles A. Dana Center at the University of Texas at Austin or www.utdanacenter.org/mathways/index.php 3 143 Quantway Out-of-Class Experience November 18, 2011 (Version 1.0) Lesson 2.1.4: Picturing Data with Graphics Theme: Citizenship

Making Connections to the Lesson (1) Which of the following was one of the main mathematical ideas of the lesson? (i) It is important to consider the gross national product when considering the size of the national debt. They relate to each other just as a person’s personal debt relates to his/her income. (ii) In Graph A, you can find the quantity represented by Part A1 by multiplying the percentage for the section times the total quantity represented by the circle. (iii) In the graphs below, you know that Part A2 represents a larger quantity than Part B2 because the piece of the graph is larger. (iv) In the graphs below, you cannot compare the quantities represented by the parts because you do not know the reference values.

(2) Explain a connection between a concept in this lesson and Lesson 2.1.3. You may use one of the sentence stems given below if you wish. You may also refer back to Module 1 lessons as examples. Question number ______in Lesson 2.1.3 connects to Question number ______in Lesson 2.1.4 because ______. The idea of ______in Lesson 2.1.3 connects to Lesson 2.1.4. An example of the connections is ______.

Developing Skills and Understanding (3) Use the graph1 on the next page to answer the following questions.

1Retrieved from www.census.gov/const/uspriceann.pdf

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(a) Estimate the percentage increase in new housing prices from 1999 to 2007. (Choose the best answer.) (i) The prices increased from around $160,000 to around $245,000, about a 53% increase. (ii) The prices increased from around $160,000 to around $220,000, about a 38% increase. (iii) The prices increased from around $160,000 to around $245,000, about a 65% increase. (iv) The prices increased from around $160,000 to around $220,000, about a 35% increase.

(b) Estimate the percentage increase in new housing prices from 2004 to 2007. (Fill in the blanks.) The prices increased from ______to ______. This is about an 11% increase.

(c) Estimate the percentage decrease from 2007 to 2009. (Fill in the blanks.) The prices decreased from ______to ______. This is about a ______% decrease.

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(4) South Central Bank has a policy that limits the amount of debt customers may have in order to receive a loan. The following pie chart shows the highest percentage of debt that the bank will allow.

(a) What is the reference value in this situation? (i) Other expenses (ii) Debt (iii) Total of other expenses and debt (iv) None of the above (b) Three graphs are given below. Each graph represents a loan customer. The customers’ debt is broken into three categories: Car, Credit Card, and Mortgage (the loan on a house). Which customer(s) meet the bank policy on the limit of the amount of debt? There may be more than one correct answer. (i)

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(ii)

(iii)

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(5) The following two pie graphs show how two families spend their money. The Alvarez family has a take-home pay of $3,650 per month and the Martinez family has a take-home pay of $7,300 per month.

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(a) Select the statement that best compares how much the families spend on gasoline. (i) The Alvarez Family spent more on gasoline than the Martinez Family. (ii) The two families both spent $200 on gasoline. (iii) Both families spend around $200 on gasoline. This is 6% of the Alvarez budget, but only 3% of the Martinez budget because the Martinez family starts with about twice as much money as the Alvarez family. (iv) The Alvarez family spent more on gasoline than the Martinez family because 3% of $7,300 is more than 6% of $3,650. (b) Estimate the actual spending on food and housing for each family. (Fill in the tables.)

Alvarez Family Food Housing

Martinez Family Food Housing

(c) Both families spend about the same percentage of their income on housing. The family with the ______income can afford a house that has twice the payment and maintenance costs. (i) Higher (ii) Lower

Making Connections Across the Course (6) From 2000 to 2006, a total of 19,076 teens ages 15–19 were killed in car crashes in the United States. The number of teen males who were killed was 12,479 and the number of teen females who were killed was 6,597.2 A reporter used this information to write, “About 34%, or one out of every three girls will be killed in a car crash.” Which of the following statements is the best critique of this statement? (i) The statement is correct because 6,597 is 34% of 19,076, which is very close to one-third. The ratio of 1 out of 3 is a good estimate of the percentage. (ii) The statement is incorrect because 6,597 is 34% of 19,076, which is very close to three-tenths. The ratio of 1 out of 3 is not a good estimate of the percentage. (iii) The statement is incorrect because the reference value is the number of teens killed, not all teen girls. It would be correct to say that 34% of teens killed in car crashes are female.

2Retrieved from CDC website: http://www.cdc.gov/MotorVehicleSafety/Teen_Drivers/data.html

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(7) Two different graphs showing how much two families spend on professional baseball tickets each year are given below. Choose the true statement based on the graphs. (i) The Wagners must have a higher income than the Cobbs. They spend less on baseball tickets, but it is a higher portion of their total income. (ii) The Cobbs must have a higher income than the Wagners. They spend more on baseball tickets, but it is a smaller portion of their total income. (iii) The Wagners must have a higher income than the Cobbs. They spend more on baseball tickets, but it is a smaller portion of their total income. (iv) The Cobbs must have a lower income than the Wagners. They spend more on baseball tickets, but it is a smaller portion of their total income.

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Preparing for the Next Lesson (2.1.5) (8) A digital camera is regularly priced at $350 and it is marked 30% off the regular price. What is the new price for the digital camera? (i) $320 is the new price for the digital camera. (ii) $338.40 is the new price for the digital camera. (iii) $105 is the new price for the digital camera. (iv) $245 is the new price for the digital camera.

(9) The American Diabetes Association reports the following information: “In general, if you are a man with type-1 diabetes, the odds of your child getting diabetes are 1 in 17. If you are a woman with type-1 diabetes and your child was born before you were 25, your child's risk is 1 in 25; if your child was born after you turned 25, your child's risk is 1 in 100.” Rank the risk of having a child with diabetes from 1 (highest risk) to 3 (lowest risk). ___ Mother with type-1 diabetes who gave birth after turning 25 ___ Father with type-1 diabetes ___ Mother with type-1 diabetes who gave birth before turning 25

(10) Self-Regulated Learning: Work An aspect of the work phase is always checking your understanding. One way to check your understanding is to explain how you solved a problem. Explain how you solved Question 9.

(11) Self-Regulated Learning: Reflect After checking your understanding, on a scale from 1 to 5, how confident are you that you answered Question 9 correctly? Be sure to incorporate this confidence rating in the table at the end of the assignment.

(12) Mark each statement true or false. (a) A man with type-1 diabetes has a child. The probability that the child will also have diabetes is about 6%. (i) True (ii) False (b) A 20 year-old woman with type-1 diabetes has a child. The probability that the child will also have diabetes is about 25%. (i) True (ii) False

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(c) A 30-year-old woman with type-1 diabetes has a child. The probability that the child will also have diabetes is about 1%. (i) True (ii) False

(13) You will be expected to do the following things for the next class. Rate how confident you are on a scale of 1–5 (1 = not confident and 5 = very confident). Before beginning Lesson 2.1.5, you should understand the concepts and demonstrate the skills listed below:

Skill or Concept: I can … Rating from 1 to 5 Read and interpret bar graphs. Have a basic understanding of the word percent and the notation used to describe percents (%). Use a calculator to divide two numbers and interpret the resulting decimal representation as a percent. Understand that a percent may be used to express the likelihood (or probability) of a certain event.

9 152 Quantway Student Handout November 18, 2011 (Version 1.0) Lesson 2.1.5: Risk Reduction Theme: Medical Literacy

Specific Objectives Students will understand that • the change in a quantity can be expressed as an absolute change and a relative change. • there is often ambiguity in the English language when talking about the change of a quantity that is represented by a percent (e.g., many rates). Students will be able to • create graphs that show both absolute and relative changes in a rate (percent). • compute absolute and relative changes.

Problem Situation: The Effect of Reducing Risk A new drug advertises that it “reduces the risk of heart attack by 50%.” In order to better understand the benefits of this drug, you will examine heart attack risk for two different groups. Group 1 consists of individuals in Africa who are • 40 years old, • do not have Diabetes Mellitus, • smoke tobacco, • have high cholesterol, and • have high blood pressure. Group 2 consists of individuals in Africa who are • 40 years old, • do not have Diabetes Mellitus, • do not smoke, • have low cholesterol, and • have high blood pressure. The World Health Organization reports that individuals in Group 1 have a greater than 40% chance (or risk) of suffering a heart attack within 10 years. The same report indicates that individuals in Group 2 have a less than 10% risk of suffering a heart attack within 10 years.

(1) What are the differences between individuals in Group 1 and Group 2?

The Carnegie Foundation for the Advancement of Teaching Quantway frontmatter available at www.quantway.org/kernel and The Charles A. Dana Center at the University of Texas at Austin or www.utdanacenter.org/mathways/index.php 1 153 Quantway Student Handout November 18, 2011 (Version 1.0) Lesson 2.1.5: Risk Reduction Theme: Medical Literacy

(2) Understanding Risk in Group 1 (a) If 500 people from Group 1 are observed for 10 years, how many individuals would you expect to suffer a heart attack within this time period? (b) Which of the following graphs best show the risk of heart attack for Group 1 individuals?

Group 1 600

500

400

300

200 Group 1 100

0 Number of Group 1 individuals Number of Group 1 individuals who suffer heart attack who do not suffer a heart attack

Graph 1 Graph 2

Group 1 Group 1 500 400 450 350 400 300 350 300 250 250 200 200 150 150 Group 1 Group 1 100 100 50 50 0 0 Number of Group 1 individuals Number of Group 1 individuals Number of Group 1 individuals Number of Group 1 individuals who suffer heart attack who do not suffer a heart who suffer heart attack who do not suffer a heart attack attack

Graph 3 Graph 4

(c) If 500 people from Group 1 are treated with the new drug and are observed for 10 years, how many individuals would you expect to suffer a heart attack within this time period? (d) Create a graph that shows the risk of heart attack for Group 1 individuals who are taking the new drug.

The Carnegie Foundation for the Advancement of Teaching Quantway frontmatter available at www.quantway.org/kernel and The Charles A. Dana Center at the University of Texas at Austin or www.utdanacenter.org/mathways/index.php 2 154 Quantway Student Handout November 18, 2011 (Version 1.0) Lesson 2.1.5: Risk Reduction Theme: Medical Literacy

(e) By placing the correct graph from Part (b) next to the graph you created in Part (d), you can show the effect of the new drug on this group of people. What, specifically, is being reduced by 50% in these graphs?

Group 1 taking new drug 450 400 350 300 250 200 150 Group 1 taking new drug 100 50 0 Number of Group 1 Number of Group 1 individuals who suffer individuals who do not heart attack suffer a heart attack

(f) There are two ways to report the drop in heart attacks due to the new drug. The absolute change refers to the actual number of fewer individuals suffering heart attacks as a result of using the new drug, while the relative change (or percent change) refers to this change as a percentage. Report both the absolute and relative changes in the number of individuals who suffered heart attacks.

The Carnegie Foundation for the Advancement of Teaching Quantway frontmatter available at www.quantway.org/kernel and The Charles A. Dana Center at the University of Texas at Austin or www.utdanacenter.org/mathways/index.php 3 155 Quantway Student Handout November 18, 2011 (Version 1.0) Lesson 2.1.5: Risk Reduction Theme: Medical Literacy

(3) Understanding Risk in Group 2 (a) If 500 people from Group 2 are observed for 10 years, how many individuals would you expect to suffer a heart attack within this time period? (b) Which of the following graphs best show the risk of heart attack for Group 2 individuals?

Group 2 Group 2 500 450 450 400 400 350 350 300 300 250 250 200 200 150 Group 2 150 Group 2 100 100 50 50 0 0 Number of Group 2 individuals Number of Group 2 individuals Number of Group 2 individuals Number of Group 2 individuals who suffer heart attack who do not suffer a heart who suffer heart attack who do not suffer a heart attack attack

Graph 5 Graph 6

Group 2 Group 2 350 350 300 300 250 250 200 200 150 150 100 Group 2 100 Group 2 50 50 0 0 Number of Group 2 individuals Number of Group 2 individuals Number of Group 2 individuals Number of Group 2 individuals who suffer heart attack who do not suffer a heart who suffer heart attack who do not suffer a heart attack attack

Graph 7 Graph 8

(c) If 500 people from Group 2 are treated with the new drug and are observed for 10 years, how many individuals would you expect to suffer a heart attack within this time period? (d) Create a graph that shows the risk of heart attack for Group 2 individuals who are taking the new drug.

The Carnegie Foundation for the Advancement of Teaching Quantway frontmatter available at www.quantway.org/kernel and The Charles A. Dana Center at the University of Texas at Austin or www.utdanacenter.org/mathways/index.php 4 156 Quantway Student Handout November 18, 2011 (Version 1.0) Lesson 2.1.5: Risk Reduction Theme: Medical Literacy

(e) By placing the correct graph from Part (b) next to the graph from Part (d), you can show the effect of the new drug on this group of people. What, specifically, is being reduced by 50% in these graphs?

Group 2 500 450 400 350 300 250 200 150 Group 2 100 50 0 Number of Group 2 individuals Number of Group 2 individuals who suffer heart attack who do not suffer a heart attack

Group 2 taking new drug 500 450 400 350 300 250 200 150 Group 2 taking new drug 100 50 0 Number of Group 2 Number of Group 2 individuals who suffer individuals who do not heart attack suffer a heart attack

(f) There are two ways to report the drop in heart attacks due to the new drug. The absolute change refers to the actual number of fewer individuals suffering heart attacks as a result of using the new drug, while the relative change (or percent change) refers to this change as a percentage. Report both the absolute and relative change in the number of individuals who suffered heart attacks.

(4) Suppose a third group of people have only a 0.5% chance of suffering a heart attack. Report the absolute and relative change in the number of individuals (out of 500) who are expected to suffer a heart attack if they take the new drug.

(5) Based on the previous calculations, critique the statement, “A drug which reduces the risk of heart attack by 50% will most likely save many lives.”

The Carnegie Foundation for the Advancement of Teaching Quantway frontmatter available at www.quantway.org/kernel and The Charles A. Dana Center at the University of Texas at Austin or www.utdanacenter.org/mathways/index.php 5 157 Quantway Student Handout November 18, 2011 (Version 1.0) Lesson 2.1.5: Risk Reduction Theme: Medical Literacy

Making Connections Record the important mathematical ideas from the discussion.

Further Applications (1) Write a complete response, including mathematical examples, to Question 5 from the lesson. Critique the statement, “A drug that reduces the risk of heart attack by 50% will most likely save many lives.”

The Carnegie Foundation for the Advancement of Teaching Quantway frontmatter available at www.quantway.org/kernel and The Charles A. Dana Center at the University of Texas at Austin or www.utdanacenter.org/mathways/index.php 6 158 Quantway Out-of-Class Experience November 18, 2011 (Version 1.0) Lesson 2.1.5: Risk Reduction Theme: Medical Literacy

Making Connections to the Lesson (1) Which of the following was one of the main mathematical ideas of the lesson? (i) To calculate a percentage, move the decimal in the percentage rate two places to the left and multiply times the reference value. (ii) Smoking increases your chance of a heart attack. (iii) Consider this situation: 20% of a population gets a certain illness. A vaccine can reduce the infection rate by 10%. Out of 100 people, the reference value for the 10% reduction would be the full population (or 100 people). (iv) Consider this situation: 20% of a population gets a certain illness. A vaccine can reduce the infection rate by 10%. Out of 100 people, the reference value for the 10% reduction would be the 20 people who are projected to get the illness.

(2) Explain a connection between a concept in this lesson and a concept in Lesson 2.1.3 or Lesson 2.1.4. You can use one of the sentence stems given below if you wish. Question number______in Lesson 2.1.5 connects to Question number ______in Lesson _____ because ______. The idea of ______in Lesson 2.1.5 connects to Lesson ______. An example of the connections is ______.

Developing Skills and Understanding (3) If you did not have time to do Question 4 during class, do this problem now.

(4) You either did Question 2 or Question 3 in the lesson. Refer back to your work for the following questions. (a) In class, you used a group with 500 people. Repeat the work with a group of 100 people. Group 1 has a 40% risk of heart attack and Group 2 has a 10% risk.

Group 1 Group 2 Number in group 100 100 Number who are expected to have a heart attack without the drug Number who are expected to have a heart attack with the drug What percentage of the group is expected to have a heart attack with the drug?

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(b) Select the true statement: (i) The size of the group does not change anything about the results for the group. (ii) The size of the group changes the number of people who will have a heart attack with the drug, but it does not change the percentage. (iii) The size of the group changes the number and the percentage of people who will have a heart attack with the drug. (5) The University of Washington is a large public university, while the University of Puget Sound is a small private university. (a) If both of these universities are expecting 12% more freshmen next year, which university will see the greatest absolute change in the number of freshmen? (Choose the best answer.) (i) Since the University of Washington is larger, they have more freshmen to begin with. Thus, a 12% increase will not translate to a larger absolute change. (ii) Since the University of Washington is larger, they have more freshmen to begin with. Thus, a 12% increase will translate to a larger absolute change. (iii) Both will undergo the exact same absolute change. (iv) Since the University of Puget Sound is smaller, they have fewer freshmen to begin with. Thus, a 12% increase will translate to a larger absolute change (b) Which university will see the greatest relative change in the number of freshmen? (i) University of Washington (ii) University of Puget Sound (iii) Both universities will undergo the same relative change. (6) Suppose that the populations of the United States and China both undergo the same absolute change in one year. Which undergoes the larger relative change? (i) The United States would undergo the smaller relative change, since the population of China is so much larger than that of the United States. (ii) China would undergo the larger relative change since the population of the United States is much smaller than that of China. (iii) The United States would undergo the larger relative change since the population of the United States is so much smaller than that of China. (iv) The two countries would undergo the exact same relative change. (7) Suppose that the populations of the United States and China both increase by 12 million people in one year. What would be the relative changes that each country underwent? (Fill in the blanks.) (a) If you assume that the U.S. population is about 300 million, an increase of 12 million would result in a relative change of ______%. (b) If you assume that China has about 1 billion people, an increase of 12 million would result in a relative change of ______%.

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(8) Employees at a certain company currently have to pay 3% of their health care costs, while the company pays the rest. Next year, however, employees will have to pay 6% of their health care costs. Express this change as an absolute change (in percentage points) and a relative change. (a) Absolute change: ______percentage points. (b) Relative change: ______%.

Making Connections Across the Course (9) The graph1 below is based on information from the 2000 U.S. Census. Use the graph to answer the questions below.

1Retrieved from http://www.census.gov/popest/national/asrh/NC-EST2009-sa.html

3 161 Quantway Out-of-Class Experience November 18, 2011 (Version 1.0) Lesson 2.1.5: Risk Reduction Theme: Medical Literacy

(a) Which is the best estimate of the total population for age group 0–9? (i) 400 (ii) 40,000 (iii) 40,000,000 (iv) 400,000,000 (b) Which is the best estimate for the total population of age group 70–79? (i) one hundred and sixty (ii) one hundred and sixty thousand (iii) one hundred and sixty million (iv) sixteen million (c) Which statement is the most accurate? (i) There are fewer than 10,000,000 people age 70 and older. (ii) There are fewer than 20,000,000 people age 70 and older. (iii) There are between 20,000,000 to 30,000,000 people age 70 and older. (iv) There are between 30,000,000 to 40,000,000 people age 70 and older. (d) If one wanted to compare the number of people in their forties (age 40–49) to the number of people in their fifties (age 50–59), one could consider the absolute difference in the populations or the relative difference. Which of the following statements are true? There may be more than one correct answer. (i) The absolute change in population from forties to fifties is about the same as the absolute change from fifties to sixties. (ii) The absolute change in population from forties to fifties is greater than the absolute change from fifties to sixties. (iii) The relative change in population from forties to fifties is about the same as the relative change from fifties to sixties. (iv) The relative change in population from forties to fifties is less than the relative change from fifties to sixties. (e) An advertiser is considering two advertising campaigns for a product. Campaign A is most effective for ages 20–39, while campaign B is most effective for ages 30–49. Which campaign should the advertiser choose to reach the most people? (i) Campaign A (ii) Campaign B

4 162 Quantway Out-of-Class Experience November 18, 2011 (Version 1.0) Lesson 2.1.5: Risk Reduction Theme: Medical Literacy

Preparing for the Next Lesson (2.2.1) (10) Find the sum of each set of numbers, as well as the size n of each set of numbers (i.e., the number of numbers in the set). Use these exercises to practice techniques for adding numbers quickly. Try to do the problems without a calculator. Think about ways in which grouping the numbers might make them easier to add. (a) 25, 35, 19, 31 (i) Sum: (ii) n: (b) 101, 73, 49, 27, 24, 36, (i) Sum: (ii) n: (c) 25, 25, 25, 30, 30, 30, 32, 32, 32 (i) Sum: (ii) n:

The following information will be used in Lesson 2.2.1. People often talk about “averages,” and you probably have an idea of what is meant by that. Now, you will look at more formal mathematical ways of defining averages. In mathematics, you call an average, a measure of center because an average is a way of measuring or quantifying the center of a set of data. There are different measures of center because there are different ways to define the center.

Think about a long line of people waiting to buy tickets for a concert. (Figure A shows a line about 100-feet long and each dot represents a person in the line.) In some sections of the line people are grouped together very closely, while in other sections of the line people are spread out. How would you describe where the center of the line is?

5 163 Quantway Out-of-Class Experience November 18, 2011 (Version 1.0) Lesson 2.1.5: Risk Reduction Theme: Medical Literacy

Would you define the center of the line by finding the point at which half the people in the line are on one side and half are on the other (see Figure B)? Is the center based on the length of the line even though there would be more people on one side of the center than on the other (see Figure C)? Would you place the center among the largest groups of people (see Figure D)? The answer would depend on what you needed the center for. When working with data, you need different measures for different purposes.

Mean (Arithmetic Average) Find the average of numeric values by finding the sum of the values and dividing the sum by the number of values. The mean is what most people call the “average.” Example Find the mean of 18, 23, 45, 18, 36 Find the sum of the numbers: 18 + 23 + 45 + 18 +36 = 140 Divide the sum by 5 because there are 5 numbers: 140 ÷ 5 = 28 The mean is 28. Median Find the median of numeric values by arranging the data in order of size. If there is an odd number of values, the median is the middle number. If there is an even number of values, the median is the mean of the two middle numbers. Example (data set with odd number of values) Find the median of 18, 23, 45, 18, 36. Write the numbers in order: 18, 18, 23, 36, 45 There is an odd number of values, so the median is the number in the middle. The median is 23. Example (data set with even number of values) Find the median of 18, 23, 45, 18, 12, 50. Write the numbers in order: 12, 18, 18, 23, 45, 50 There is an even number of values, so there is no one middle number. Find the median by finding the mean of the two middle numbers: 18 + 23 = 41 41 ÷ 2 = 20.5 The median is 20.5.

Mode Find the mode by finding the number(s) that occur(s) most frequently. There may be more than one mode. Example Find the mode of 18, 23, 45, 18, 36. The number 18 occurs twice, more than any other number, so the mode is 18.

6 164 Quantway Out-of-Class Experience November 18, 2011 (Version 1.0) Lesson 2.1.5: Risk Reduction Theme: Medical Literacy

(11) Write the answers to the following questions on a piece of paper and bring it to the next class period. If you do not own a credit card, answer Part (a) only. Keep your responses anonymous by writing only the answers to the questions. (Do not write your name.) (a) How many credit cards do you possess? (b) Do you normally pay the entire balance on the credit card statement? (c) What is the approximate balance (total) on your card(s) right now?

(12) You will be expected to do the following things for the next class. Rate how confident you are on a scale of 1–5 (1 = not confident and 5 = very confident). Before beginning Lesson 2.2.1, you should understand the concepts and demonstrate the skills listed below:

Skill or Concept: I can … Rating from 1 to 5 Perform basic operations using quantities as integers, fractions, or decimals with the aid of technology. Identify the mean, median, and mode of a small data set.

(13) In the question above, if you or another student had any ratings below 3, what are some things that you or the student could do to increase your/their confidence while planning for Lesson 2.2.1? Please describe at least two ways.

7 165 Quantway Student Handout November 18, 2011 (Version 1.0) Lesson 2.2.1: What Is Average? Theme: Personal Finance

Specific Objectives Students will understand • that numerical data can be summarized using measures of central tendency. • how each statistic—mean, median, and mode—provide a different snapshot of the data. • that conclusions derived from statistical summaries are subject to error. • that a spreadsheet can be used to organize data. Students will be able to • calculate the mean, median, and mode for numerical data. • create a data set that meets certain criteria for measures of central tendency.

Problem Situation: Summarizing Data About Credit Cards A revolving line of credit is an agreement between a consumer and lender that allows the consumer to obtain credit for an undetermined amount of time. The debt is repaid periodically and can be borrowed again once it is repaid. The use of a credit card is an example of a revolving line of credit. According to www.CreditCards.com, U.S. consumers own more than 600 million credit cards. About 98% of the total U.S. revolving debt is made up of credit card debt. Average credit card debt per household with a credit card is $14,743. Worldwide, there are more than $2.5 trillion in transactions annually. It is estimated that there are 10,000 card payment transactions made every second. According to the U.S. government (2009), 15% of college freshmen had a zero balance on their credit card. The median debt carried by freshmen was $939. Seniors graduated with an average credit card debt of more than $4,100, and one-fifth of seniors owed more than $7,000 on their credit cards. In 2004, three-fourths of all American families had at least one credit card, but only 58% carried a balance. If a credit card user carries a balance (e.g., does not pay the monthly debt in full) the credit card company assesses a finance charge (interest) for the use of their money. This can be avoided by paying the balance in full.1

In the first part of this lesson, you will use the information about credit cards given above to learn about some ways to summarize quantitative information. (1) The population of the United States is slightly more than 300 million people. There are about 100 million households in the United States. What is the average number of credit cards per person? What is the average number of credit cards per household?

(2) Consider the statement, “Average credit card debt per household with a credit card is $14,743.” What does this mean?

1Retrieved from www.creditcards.com/credit-card-news/credit-card-industry-facts-personal-debt-statistics-1276.php

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(3) The introduction states that “college seniors graduated with an average credit card debt of more than $4,100.” Imagine you ask four groups of five college graduates what their credit card debt is. The amount of debt for each senior in each group is shown in the table.

(a) Find the mean debt of each group of college graduates. Make sure the value you found is reasonable given the values in the table. (b) Complete the data set called “Your Data” so that it represents the debt of five college seniors with a mean debt of $4,100.

Your Data

Group A Group B Group C Group D Your Data

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(4) The introductory information gives data about the median debt carried by freshmen. Create a data set of six freshmen so that the data set has the same median reported for all college freshmen.

Debt of College Freshmen

Note on language: The word mean is used in mathematics. There are actually several different kinds of means. The one that you have discussed in this course is the arithmetic mean. You will also see people use the word average when referring to the mean. You should be familiar with both terms.

Making Connections Record the important mathematical ideas from the discussion.

Further Applications (1) This course draws on examples from three themes: Citizenship, Personal Finance, and Medical Literacy. Choose at least two different lessons with the Personal Finance theme. Answer the following questions. (a) What new information did you learn about personal finance? (b) How will you use this information or why is it important to know this information?

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Making Connections to the Lesson (1) Which of the following was one of the main mathematical ideas of the lesson? (i) Any of the three measures of central tendency (mean, median, and mode) are good representations of data. It does not matter which one you use. (ii) U.S. college students carry far too much credit card debt. (iii) The mean is calculated by adding all the numbers and dividing by the number of data points. (iv) The mean, median, and mode all give important information about a data set, but they do not give a complete picture of the data set.

(2) In Lesson 2.1.4, you learned about reading graphs. Describe a connection between interpreting a graph and interpreting measures of central tendency. You can use one of the sentence stems given below if you wish. Question number______in Lesson 2.1.4 connects to Question number ______in Lesson 2.2.1 because…

The idea of ______in Lesson 2.1.4 connects to Lesson 2.2.1. An example of the connections is…

Developing Skills and Understanding (3) Use the following data set to answer the questions.

13 15 20 20 20 20 20 20 20 23 27 31

(a) What is the mean? (b) What is the mode? (c) What is the median? (d) What fraction of the numbers in the data set are less than the median? (e) What fraction of the numbers in the data set are greater than the median? (f) Which of the following statements are correct? (i) The median is the middle of a data set. Half of the data points are always less than the median, and half are always greater than the median. (ii) The median is the middle of a data set. Half of the data points are either less than or equal to the median. (iii) The median is the middle of a data set. At least half of the data points are always equal to the median. (iv) The median is not the middle of a data set. You cannot predict the distribution of the numbers in relationship to the median.

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(4) Consider the statement “Worldwide, there are more than $2.5 trillion in credit card transactions annually.” (a) What is the daily average dollar amount of transactions? Round to the nearest hundred million dollars. (b) How many dollars in credit card transactions are made each day? (i) $6,849,315 (ii) $6,849,315,068 (iii) $1,460,000,000 (iv) $1,460,000 (v) It is impossible to know.

(5) Students at Dover Community College (DCC) have a mean credit card debt of $3,600 with a median of $1,500. Students at Ralton Community College (RCC) have a mean credit card debt of $3,000 with a median of $2,800. Which statements about the two groups are true based on this information? There may be more than one correct answer. (i) Less than half of DCC students have debt in excess of $3,600. (ii) No more than half of RCC students have debt less than $2,800. (iii) About three-fourths of DCC students have debt less than $3,600. (iv) The largest debt of the RCC students is less than the largest debt of the DCC students. (v) The total debt of RCC students is less than the total debt of DCC students. (vi) The total debt of RCC students is larger than the total debt of DCC students.

(6) Decide whether the following statements must be true or not, based on the information provided. If the statement must be true, write True; otherwise, write False. Be prepared to explain your reasoning. (a) The median of 25 numbers is 13. Twelve of the numbers must be greater than 13. (b) The average of 11 numbers is 130. None of the 11 numbers are more than 260. (c) The average of 25 numbers is 100, and the median of those 25 numbers is also 100. The mode of the 25 numbers must be 100. (d) The mean of 45 numbers is 70. If you pick any group of 10 numbers from the 45, the mean will be 70. (e) The average of 42 numbers is 20. The sum of all 42 numbers is 840. (f) The average of 49 numbers is 100. If a 50th number is added and the average remains at 100, the 50th number must have been 100.

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(7) Rio Blanca City Hall publishes the following statistics on household incomes of the town’s citizens. The mode is given as a range. Mean: $257,000 Median: $65,000 Mode: $20,000–$30,000 Which measure would be the most useful for each of the following situations? (a) State officials want to estimate the total amount of state income tax paid by the citizens of Rio Blanca. (b) The school district wants to know the income level of the largest number of students. (c) A businesswoman is thinking about opening an expensive restaurant in the town. She wants to know how many people in town could afford to eat at her restaurant.

Making Connections Across the Course (8) Use Figures 1 and 2 for the following questions. (a) Shade 40% of each figure.

Figure 1 Figure 2

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(b) Which of the following statements are correct? There may be more than one correct answer. (i) The shaded area of Figure 1 is larger than the shaded area of Figure 2 because Figure 1 is larger than Figure 2. (ii) The shaded area of Figure 1 is the same as the shaded area of Figure 2 because they are both 40% of the square. (iii) The shaded area of Figure 1 is the same proportion of the figure as the shaded area of Figure 2 because they are both 40% of the square. (c) These figures illustrate what important concept? (i) Percentages cannot be used for comparisons unless the reference values are equal. (ii) Percentages compare measures relative to the size of the reference values, but do not give information about absolute measures. (iii) Percentages are a ratio out of 100, so they can always be compared directly. In other words, 60% of one value is equal to 60% of another value.

(9) Animal population densities are important to those who manage wildlife. This is particularly true when animals pose risks to humans. Managing the population of grizzly bears in Yellowstone National Park is one example. Grizzly bears roam Yellowstone Park, which is 3,472 square miles. Additionally, the bears roam a surrounding area that is 40% as large as Yellowstone. Each male grizzly bear needs a territory to roam that is about 300 square miles, while each female bear needs a roaming territory of about 100 square miles. Although their territories overlap considerably, each bear needs at least 10 square miles to himself/herself to call home.1 For the following questions, assume that bears roam freely within and outside Yellowstone without being captured or shot. (a) Calculate the total available land in and around Yellowstone for grizzly bears to roam. Round to the nearest square mile. (b) Estimate the maximum number of bears that can be supported on the available land. (c) If there are about 1.5 adult female bears for every adult male bear, then approximately how many male bears can live in and around Yellowstone Park? How many female bears?

The bar graph below shows the number of bear sightings (all bears, not only grizzlies) in Yellowstone National Park from 1998 to 2002.2 Because the number of sightings is affected by the number of visitors to the park, these data are shown as well.3

1Retrieved from www.nrmsc.usgs.gov/files/norock/products/IGBST/2009report. 2Retrieved from www.nps.gov/yell/naturescience/bearsighttable.htm. 3Retrieved from www.nature.nps.gov/stats/park.cfm?parkid=421.

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Note: Because the number of visitors each year is much larger than the number of bear sightings, the data are plotted on different scales. The scale on the left (ranging from 0 to 3,500,000) is for the number of visitors, shown in gray. The scale on the right (ranging from 0 to 980) is for the number of bear sightings, shown in black.

(d) In what year was the percentage of visitors who saw bears the highest? (e) In what two years was the percentage of visitors who saw bears the lowest? (f) Estimate the average number of bear sightings over these five years. (i) 600 (ii) 700 (iii) 800 (iv) 900

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Preparing for the Next Lesson (2.2.2) (10) Answer the questions for the graph shown below.

(a) What was the average or mean price of a new home in 2005? (b) What was the median price of a new home in 1984? (c) Which of the following intervals had the largest increase in the median price? (i) 1965–1968 (ii) 1981–1984 (iii) 1989–1992 (iv) 1997–2000 (d) In what two-year period was the largest drop in the average price?

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(11) You should be able to do the following things for the next class. Rate how confident you are on a scale of 1–5 (1 = not confident and 5 = very confident). Before beginning Lesson 2.2.2, you should understand the concepts and demonstrate the skills listed below:

Skill or Concept: I can … Rating from 1 to 5 Perform basic operations using quantities as integers, fractions, or decimals with the aid of technology. Find the mean, median, and mode of a set of numeric data. Read a line graph.

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Specific Objectives Students will understand that • each statistic—the mean, median, and mode—is a different summary of numerical data. • conclusions derived from statistical summaries are subject to error. • they can use the measures of central tendency to make decisions. Students will be able to • make good decisions using information about data. • interpret the mean, median, or mode in terms of the context of the problem. • match data sets with appropriate statistics.

Problem Situation 1: Making Sense of Measures of Central Tendency

Employment Opportunities Sales Positions Are you above average? NEED A NEW Available! CHALLENGE? Our company is hiring one Join a super sales force We have immediate person this month—will and make as much as need for five you be that person? We you want. Five of our pay the top percentage nine salespeople enthusiastic self-starters commission and supply closed FOUR homes who love the outdoors you leads. Half of our sales last month. Their and who love people. force makes over $3,000 average commission Our salespeople make per month. Join the was $1,500 on each an average of $1,000 Above Average Team! sale. Do the math—this per week. Come join Call 555-0127 now! is the job for you. the winning team. Making dreams real— Call 555-0100 now! We are! call 555-0199

(1) Examine the three advertisements shown in the problem situation. (a) Identify any measures of central tendency and how they are used in each advertisement. (b) For each advertisement, create a scenario that fits the information provided. Scenario means to create a set of data that fits the description. You did this in the previous lesson when you made a list of credit card debts for the five college students.

(2) In which job would you expect to earn the most money?

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Problem Situation 2: Understanding Trends in Data (3) The median and average sales price of new homes sold in the United States from 1963–2008 is shown in the following graphic.1 Examine the graph. Write at least three statements about the data. Recall the Writing Principle: Use specific and complete information.

1Data retrieved from the U.S. Census Bureau.

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(4) Table 1 gives a sample data set of home prices that matches the data shown in the graph for the year 1977. Five possible data sets for the year 2005 are given in Table 2. Use your knowledge of mean and median to answer the following questions without calculating the mean of the data sets. There may be more than one correct answer to any of the questions.

Table 1: Sales Prices of New Homes Sold in United States in 1977

1977 Sales Price $40,000 $45,000 $50,000 $56,000 $67,000 $75,000 $112,000

Table 2: Possible Data Sets for 2005

Set A Set B Set C Set D Set E $240,000 $84,000 $120,000 $74,000 $74,000 $245,000 $105,000 $135,000 $95,000 $90,000 $250,000 $125,000 $150,000 $105,000 $120,000 $256,000 $240,000 $168,000 $240,000 $240,000 $267,000 $245,000 $201,000 $242,000 $250,000 $275,000 $469,000 $225,000 $250,000 $635,000 $312,000 $810,000 $336,000 $251,000 $669,000

(a) Which of the data sets could represent the data in the graph? (b) Which of the data sets would likely have a mean that is less than the median? (c) Which of the data sets would likely have a mean and median that are close together?

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Making Connections Record the important mathematical ideas from the discussion.

Further Applications (1) This course draws on examples from three themes: Citizenship, Personal Finance, and Medical Literacy. Choose at least two different lessons with the Citizenship theme. Answer the following questions. (a) What new information did you learn about citizenship? (b) How will you use this information or why is it important to know this information?

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Making Connections to the Lesson (1) Which of the following was one of the main mathematical ideas of the lesson? (i) Home prices in 2007 were more than 10 times what they were in the 1960s. (ii) The mean and median of a data set are always very close together, but the mode might be very different. (iii) When using averages or measures of central tendency, it is always important to ask questions about what the different types of measurements do and do not tell you about the data set. (iv) The median is the middle number when a data set is listed in order. If there is an even number of points in the data set, the median is found by finding the mean of the middle two numbers.

(2) Refer back to the OCE for Lesson 2.1.4. Question 3 contains a graph about housing prices. (a) Explain what the data for 2007 tells you about the cost of a house. (b) Compare this graph to the graph of house prices used in Lesson 2.2.2. What is similar about the information? What is different?

Developing Skills and Understanding (3) The first advertisement discussed in class states that the salespeople make an average of $1,000 per week. Suppose there are nine salespeople. What would the ninth person need to earn for the mean to be $1,000 if the other eight salespeople earned $550, $600, $600, $800, $950, $950, $1,000, and $1,100?

(4) The second advertisement states that half the salespeople make more than $3,000 per month. Suppose there are eight salespeople. What would the eighth person need to earn for the median to be $3,000 if the other seven salespeople earned $2,400, $2,500, $2,800, $2,800, $3,400, $3,400, and $3,800?

(5) Which statistic (mean, median, or mode) is most appropriate in each of the following situations? (a) Tables in the dining hall are numbered 1 through 12 for students who eat there. The principal calls out a number for the table that will go through the buffet line first. The other tables follow in order of the table numbers. One student is sure the principal calls certain tables more often. She keeps track of which numbers are called over a 21-day period. (i) Mean (ii) Median (iii) Mode

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(b) The offensive line of a football team is larger than in previous years. The program will list a statistic to show this fact. (i) Mean (ii) Median (iii) Mode

(c) A reporter is doing a story on the falling prices of homes in a large neighborhood. The reporter wants to demonstrate how the prices have fallen for all homes, not just the most expensive houses. (i) Mean (ii) Median (iii) Mode

(6) Lines at the Department of Motor Vehicles are so long! A supervisor decided to do a study on the number of people standing in line. At the beginning of each hour for an entire week, the supervisor counted the number of people in line and recorded the number. At the end of the week, the supervisor made the frequency table below. Note that the first column shows the number of people in line at the beginning of the hour. The second column shows the number of times that length of line occurred in the 40 observations.

Number of Frequency People in Line 1 1 2 0 3 2 4 2 5 2 6 4 7 4 8 4 9 8 10 4 11 9

(a) How many times did the supervisor observe six people in line?

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(b) Which calculation could be used to determine how many people the supervisor observed standing in line all together? (i) 1 x 1 + 2 x 0 + 3 x 2 + 4 x 2 + 5 x 2 + 6 x 4 + 7 x 4 + 8 x 4 + 9 x 8 + 10 x 4 + 11 x 9 (ii) 1 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10 + 11 (iii) 3 + 2 + 2 + 2 + 4 + 4 + 4 + 8 + 4 + 9 (iv) None of the above

(d) Determine the mode of the data in the frequency table.

(e) Determine the median of the data in the frequency table.

(f) Determine the mean of the data in the frequency table.

(g) The supervisor made observations again the following week. The table below shows the observations.

Number of Frequency People in Line 1 3 2 4 3 5 4 4 5 5 6 1 7 2 8 1 9 2 10 6 11 8

The mean for the second week was 6.366, the median was 5, and the mode was 11. The supervisor wanted to make the argument that additional personnel were needed. Which of these arguments is correct? (i) Most of the time, there are 11 people in line! (ii) Most of the time, there are 5 or more people in line! (iii) Most of the time, there are more than 6 people in line!

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(7) Three descriptions of measures of central tendency are given below. They are labeled A, B, and C. Descriptions of data sets are listed below that. Match each data set with a description of measures of central tendency by writing the letter in the blank. Choices may be used more than once. A The mean and median are close together. B The mean is much higher than the median. C The median is much higher than the mean.

(a) _____ The data have a large range with some very high numbers and many small numbers.

(b) _____ The data set has a large range with the numbers evenly spaced.

(d) _____ The data set has a large range with a few very low numbers.

(8) If you lived in Canada in 2008, you might have seen the following headline: “Canada Below G7 Average for Productivity!” Here is some information to help you understand this headline. Productivity is a way to measure the economy of a nation. One way to measure productivity is by Gross Domestic Product (GDP) per worker. You may recall from Lesson 2.1.4 that GDP is the value of all the goods and services produced in a country. The G7 is a coalition of the major industrial democracies in the world: United States, United Kingdom, France, Germany, Italy, Canada, and Japan.

(a) Which of the following is most likely what the author of the headline wanted the reader to think? (i) Canada’s economy is weak and is falling behind other countries in the G7. (ii) Canada’s economy is strong and is leading other countries in the G7. (iii) Canada’s economy is very similar to other countries in the G7. (iv) Canada’s economy should not be compared to other countries.

(b) Which of the following can you conclude from the headline? (i) Canada is less productive than half of the G7 nations. (ii) There is at least one G7 nation that is more productive than Canada. (iii) There is at least one G7 nation that is less productive than Canada. (iv) None of the above.

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A graph of the GDP per worker of the G7 nations is shown below.1

(d) Is the headline correct?

(e) Which of the seven G7 nations have “above average” productivity?

(f) Which of the following are correct conclusions based on the data in the graph? There may be more than one correct answer. (i) Canada is in the top half of the G7 in productivity. (ii) Canada’s productivity is relatively close to all the G7 nations except for the United States and Japan. (iii) Canada is far behind the G7 nations in productivity. (iv) None of the above.

1Retrieved from www.bls.gov/fls/flsgdp.pdf.

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Making Connections Across the Course Themes explored in Lesson 2.2.3 include buying power of money, the role of the Urban Consumer Price Index, and the concept of index numbers. (9) Buying power of money: Gasoline costs have varied significantly in recent months. The American Petroleum Institute posted an update on gasoline prices for June 15, 2011.2

U.S. PUMP PRICE UPDATE—JUNE 15, 2011 The average U.S. retail price for all grades of gasoline fell this week for the fifth week in a row by 6.6 cents from the prior week to $3.767 per gallon, according to the Energy Information Administration (EIA). This was at the highest level since August 2008 with the exception of the recent highs in the prior two months. Compared with the December 29, 2008 low of $1.670, the all-grade average was higher by $2.097 per gallon, or 125.6 percent. The average has been above $3.50 per gallon since the beginning of March 2011. Nominal prices have been above the year-ago average for 66 weeks—and were up by 101.1 cents or 36.7 percent, from the year-ago average of $2.756 per gallon.

(a) How much was the average retail price for one gallon of gasoline a week before this article was published? (i) $3.701 (ii) $3.518 (iii) $3.833 (iv) None of the above (b) What is the relative change in the retail price for one gallon of gasoline from December 29, 2008, to June 15, 2011? (i) 225.6% (ii) 125.6% (iii) 25.6% (iv) None of the above (c) If the retail price for one gallon of gasoline was $2.756 a year before, then what is the absolute change in the retail price on June 15, 2011? (i) $2.097 (ii) $1.086 (iii) $1.670 (iv) None of the above

2Retrieved from www.api.org/aboutoilgas/gasoline/upload/PumpPriceUpdate.pdf

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(10) A key idea of the previous question is that the buying power of a dollar is not constant. For example, the price of gasoline varies greatly, so the amount of gas you can buy with $1 varies over time. The Urban Consumer Price Index (CPI-U) is a tool designed to compare the price of goods and services in terms of base-year dollars. You will be using the CPI-U in your next lesson. The following questions will help you learn about it. Refer to the website www.bls.gov/cpi to answer the following questions. (a) Complete the following description of the Urban Consumer Price Index. The Consumer Price Indexes (CPI) program produces monthly data on …

Go to the CPI Overview. (b) Read the Data Available section. What percentage of the population is represented by the CPI-U?

(d) Are income taxes included in the CPI?

(e) Read the Uses section. The website states that the CPI affects the income of almost 80 million people because their incomes are tied to changes in the CPI. Which option is the best estimate for the percentage of these people who receive Social Security benefits? (i) A little less than 48% (ii) More than 50% (iii) Around 80%

(f) Which option is the best estimate for the fraction of these people who receive food stamps? (i) About a third (ii) About three-fourths (iii) About half

(g) State one use for the CPI-U, based on information from the website.

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Preparing for the Next Lesson (2.2.3) The following information will be used in the next lesson. The United States has a federal minimum wage. This means that there is a law that requires employers to pay employees a certain amount. Many states also have their own minimum wage laws that require a higher wage than the federal law. The minimum wage started in 1938 as a part of a law that protected the rights of workers in many ways. The minimum wage in 1938 was set at $0.25 per hour.3 Since 1938, Congress has increased the minimum wage many times to account for inflation. Inflation is when prices increase over time. Maybe you have heard stories from older people about how a cup of coffee used to cost a dime or a gallon of gasoline was less than a dollar. Prices for these items have increased due to inflation. When deciding if the minimum wage should be increased, people often talk about the buying power of the wage. Buying power refers to how much you can actually buy with a dollar. Because of inflation, a dollar bought more in 1938 than it did in 2009. In Lesson 2.2.3, you will answer the question: Did the minimum wage in 1938 have more buying power than the minimum wage in 2009? You will start by thinking about the specific example of buying a movie ticket.

(11) The minimum wage from 1997 through 2006 was $5.15 per hour. In 2009, the minimum wage increased to $7.25 per hour. (a) What was the absolute change in the minimum wage from 1997 to 2009? (b) What was the relative change in the minimum wage from 1997 to 2009? Round to the nearest tenth of one percent.

(12) The average price of a movie ticket in 1997 was $4.59.4 Which statement is correct? Be prepared to explain your answer in class. (i) In 1997, a person earning minimum wage had to work less than an hour to earn enough for a movie ticket. (ii) In 1997, a person earning minimum wage had to work more than an hour to earn enough for a movie ticket.

(13) The average price of a movie ticket in 2009 was $7.50. Which statement is correct? Be prepared to explain your answer in class. (i) In 2009, a person earning minimum wage had to work less than an hour to earn enough for a movie ticket. (ii) In 2009, a person earning minimum wage had to work more than an hour to earn enough for a movie ticket.

3Retrieved from www.dol.gov/oasam/programs/history/flsa1938.htm. 4Retrieved from www.natoonline.org.

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(14) Did you read and understand the information to be used in class? _____Yes _____No

(15) Did you complete the work that you need to take to class? _____Yes _____No

(16) You should be able to do the following things for the next class. Rate how confident you are on a scale of 1–5 (1 = not confident and 5 = very confident). Before beginning Lesson 2.2.3, you should understand the concepts and demonstrate the skills listed below:

Skill or Concept: I can … Rating from 1 to 5 Calculate and interpret relative change. Understand the minimum wage.

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Specific Objectives Students will understand that • ratios provide a way of comparing the relative increase or decrease of two variables. • index numbers are a way of comparing the relative size of a variable over time. Students will be able to • use proportional reasoning to find the size of a variable that remains a constant proportion of another variable. • use index numbers to find the value of a variable relative to time (or another variable).

Problem Situation: The Buying Power of the Minimum Wage You found from your previous work that the minimum wage did not increase enough from 1997 to 2009 to keep up with the cost of a movie ticket. A movie ticket is only a single product that someone might buy. To compare the buying power of the minimum wage from 1938 to 2009, you need more information about more products. You will use a tool called an index number. Index numbers are a way to compare the relative change or difference in a data set, such as the prices of products. The change can be measured over time or over different geographic regions. There are different types of index numbers. One type is a measure of average relative change in data. One data point is called the base and is assigned the value of 100 (meaning it represents 100%). The other figures are adjusted in proportion to the base. The plural form of the word index is indices. The following two questions will help you understand index numbers. You will begin by looking at one product: the Big Mac. (1) You are going to use 1992 as the base year for our Big Mac Price Index. Prices of the Big Mac vary with time. In 1992, a Big Mac cost about $2.19. (a) In 1968, a Big Mac cost about $0.49. Find the percentage that the 1968 price compared to the 1992 price. (b) In 2010, a Big Mac cost about $3.25. Find the percentage that the 2010 price compared to the 1992 price. Big Mac Price Index

Year Index Number 1968 1992 2010

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Now you have seen how an index is created and what it means. You are now going to look at a real index that the U.S. government uses to track changes in the prices of many items over time. It is called the Consumer Price Index (CPI). You will use a version called the Urban Consumer Price Index (CPI-U). The Bureau of Labor Statistics publishes the CPI-U each month. It is often used to compare the buying power of a dollar in one year as compared to another (it compares prices over time). The CPI is a measure of the weighted average of a “basket of consumer goods and services.” The basket includes transportation, food, medical care, housing, apparel, recreation, and education. The more this basket of goods and services costs, the less of it you can purchase with the dollar. The CPI is one of the most frequently used statistics for identifying periods of deflation or inflation. If the CPI is more than 100, then prices have increased compared to the base year. This means there has been inflation since the base year. If CPI is less than 100, then prices have decreased compared to the base year. This is called deflation. The CPI-U represents the cost of a basket of goods and services in base-year dollars. The table shows the indices for selected years from 1913 to 2010.

Year Index Year Index 1913 9.9 1981 90.9 1914 10.0 1982 96.5 1915 10.1 1983 99.6 1921 17.9 1984 103.9 1922 16.8 1985 107.6 1923 17.1 1996 156.9 1924 17.1 1997 160.5 1925 17.5 1998 163.0 1936 13.9 1999 166.6 1937 14.4 2000 172.2 1938 14.1 2001 177.1 1939 13.9 2002 179.9 1940 14.0 2003 184.0 1946 19.5 2004 188.9 1947 22.3 2005 195.3 1948 24.1 2006 201.6 1949 23.8 2007 207.3 1950 24.1 2008 215.3 2009 214.5 2010 218.1

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(2) What year is used as the base for the CPI-U? What does the base mean?

(3) How does the 2010 dollar compare in value to the dollar in 1983?

(4) How does the 1949 dollar compare in value to the dollar in 1983?

(5) Who had more buying power? Provide quantitative information to support your answer. • The person making minimum wage ($0.25 per hour) in 1938. • The person making minimum wage ($7.25 per hour) in 2010.

Making Connections Record the important mathematical ideas from the discussion.

Further Applications (1) Find the price of gasoline in 1981 and in 2010. Give the sources for your information. Use the CPI-U table in the lesson to evaluate the statement, “Gasoline was more expensive in 2010 than in 1981.” Provide mathematical information to justify your explanation.

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Making Connections to the Lesson (1) Which of the following was one of the main mathematical ideas of the lesson? (i) The Consumer Price Index (CPI) is an example of an index number that measures the cost of a basket of goods and services over time. It can be used to show inflation and deflation of prices. (ii) Inflation is calculated by finding the relative change in the CPI. (iii) The CPI is a ratio that compares the cost of goods to a base year in the form of a percentage. This is an example of how ratios are used for comparisons. (iv) The CPI always increases over time.

(2) Explain a connection between a concept in this lesson and Lesson 1.2.1 or 2.1.2. You can use one of the sentence stems given below if you wish. Question number______in Lesson 2.2.3 connects to Question number______in Lesson _____ because …

The idea of ______in Lesson 2.2.3 connects to Lesson ______. An example of the connections is …

Developing Skills and Understanding (3) The price of a first-class stamp has increased rapidly since the 1970s. The graph below shows the price (in cents) of a first-class stamp since 1917.

Source: United States Postal Service

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(a) Estimate the absolute increase in the price of a first-class stamp from 1992 to 2007. (i) 10% (ii) $0.10 (iii) 25% (iv) $0.14 (v) $0.25 (vi) 33%

(b) Estimate the relative increase in the price of a first-class stamp from 1992 to 2007. (i) 10% (ii) $0.10 (iii) 25% (iv) $0.14 (v) $0.25 (vi) 33%

(c) Suppose the absolute increase in the price of a first-class stamp from 2007 to 2022 is the same as it was from 1992 to 2007. What would the cost of a first-class stamp be in 2022? (i) $0.44 (ii) $0.50 (iii) $0.53 (iv) $0.59 (v) $0.65

(d) Suppose the relative increase in the price of a first-class stamp from 2007 to 2022 is the same as it was from 1992 to 2007. What would the cost of a first-class stamp be in 2022? (i) $0.44 (ii) $0.50 (iii) $0.53 (iv) $0.59 (v) $0.65

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Refer to the table of CPI-U given in the lesson for Questions 4 and 5. This table gives the cost of a representative basket of goods and services in base-year dollars (1983). The table shows the indices for selected years from 1913 to 2010. The Consumer Price Index (CPI-U) provides an indication of how the price of consumer goods changes over time. (4) In 2006, a 12-ounce coffee at Starbucks cost $1.46. In 2010, that same cup of coffee was $1.94. (a) What was the relative increase in the price of the cup of coffee from 2006 to 2010? Round to the nearest one percent. (b) What was the relative increase in the CPI-U from 2006 to 2010? Round to the nearest one percent. (c) Which of the following statements is true? (i) From 2006 to 2010, the price of coffee at Starbucks increased at a lower rate than the CPI-U. (ii) From 2006 to 2010, the price of coffee at Starbucks increased at the same rate as the CPI-U. (iii) From 2006 to 2010, the price of coffee at Starbucks increased at a higher rate than the CPI-U.

(5) The rent on a building in 2008 was $1,350 per month. The landlord changes the rent relative to changes in the CPI-U. Use the CPI-U to calculate the rent in 2010. Round to the nearest dollar.

(6) The OCE for Lesson 2.2.2 had a question using productivity data for the G7 nations. The graph below shows another way of comparing productivity data using an index. Use the graph to answer the following questions. Note that these data are not the same as in the earlier problem.

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(a) How does Japan’s productivity compare to that of the United Kingdom? (i) Japan has a higher rate of productivity. (ii) Japan has a lower rate of productivity. (iii) Japan has about the same rate of productivity.

(b) Which two countries have about the same rate of productivity?

(d) What does the vertical scale represent? (i) Productivity in U.S. dollars (ii) Productivity in Euros (iii) Index numbers (percentages relative to a base)

(e) Which of the following statements are true? There may be more than one correct answer. (i) The U.S. productivity is about $132.00 per worker. (ii) The U.S. productivity is about $32.00 per worker more than the U.K. productivity. (iii) The U.S. productivity is 32% higher than the U.K. productivity. (iv) The U.S. productivity is 132% higher than the U.K. productivity. (v) The U.S. productivity is about 50% higher than Japan’s productivity.

(7) Data from the Census of Population and Housing surveys of 1960, 1974, and 1989, the National Association of Realtors, and the U.S. Census Bureau are shown below.

1960 1974 1989 2009 Median Family $5,500 $13,020 $34,400 $60,600 Income Median Price of $60,052 $86,751 $84,622 $185,000 House

In which of these four years was a house most affordable? (i) 1960 (ii) 1974 (iii) 1989 (iv) 2009

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Making Connections Across the Course There is now an expectation that you will be able to conduct an Internet search to find information. You will begin seeing questions in your assignments that require you to identify the information you need and find the information from a reliable source. (8) According to the CIA’s The World Factbook, around 320,000 people in Botswana had HIV/AIDS in 2009. About 5,800 people in Botswana died of AIDS that year. In the United States, around 1.2 million Americans were infected with HIV/AIDS in 2009, and about 17,000 of those people died. (a) Use estimation to identify which of the following statements are true. There may be more than one correct answer. (i) There are more HIV/AIDS cases in Botswana than in the United States. (ii) There are more HIV/AIDS cases in the United States than in Botswana. (iii) The HIV/AIDS infection rate is higher in Botswana than in the United States. (iv) The HIV/AIDS infection rate is lower in Botswana than in the United States. (v) The death rate among people infected with HIV/AIDS is much larger in Botswana than in the United States. (vi) In 2009, the United States had about four times as many HIV/AIDS cases as Botswana and about 150 times the population. (vii) In 2009, the United States had about 40 times as many HIV/AIDS cases as Botswana and about 15 times the population. (viii) The death rate among people infected with HIV/AIDS is much smaller in Botswana than in the United States. (ix) The death rate among people infected with HIV/AIDS is about the same in Botswana as it is in the United States. (b) Suppose that a new experimental drug being tested in Botswana showed that it can decrease the death rate among people infected with HIV/AIDS by 18%. If this drug had been used in 2009 by all Botswanans infected with HIV/AIDS, how many people would you have expected to die from HIV/AIDS? Round to the nearest hundred people. (c) Suppose that the drug from Part (b) has the same influence on Americans infected with HIV/AIDS. If the drug had been used in 2009 by all Americans infected with HIV/AIDS, how many people would you have expected to die from HIV/AIDS? Round to the nearest thousand people.

(9) The type of comparison shown below is called relative magnitude because it is a comparison of the magnitude, or size, of one quantity relative to another quantity. In 2009, the United States had about four times as many HIV/AIDS cases as Botswana and about 150 times the population. Write a statement comparing the relative magnitude of the populations of the United States and Mexico in 2011.

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Preparing for the Next Lesson (2.2.4) In Lesson 2.2.4, you will explore an extensive data set related to factors that form the basis for the 2010 list of “Best Cities for the Next Decade,” published by Kiplinger.1 The data are based on 367 cities. A sample of the data set is given below. Notes on the table: COL is cost of living, and % Workforce in Creative Class refers to the percentage of people who are employed as scientists, engineers, educators, writers, artists, and entertainers.

% Workforce Median COL Income Employment Metro Area Population in Creative Household Index Growth Growth Class Income Abilene, TX 159,137 86.83 23.6% $39,371 5.13% 0.36% Akron, OH 698,964 96.49 32% $47,336 3.85% 2.33% Albany, GA 164,504 88.96 25.4% $39,166 3.82% –0.02% Albany-Schenectady- 851,925 100.00 20.7% $54,755 4.25% 1.62% Troy, NY Albuquerque, NM 833,506 97.97 26.8% $45,634 3.91% 4.42% Alexandria, LA 153,689 100.00 33.4% $36,753 3.18% 3.73% Allentown-Bethlehem- 801,712 100.00 29.2% $54,420 2.99% 1.19% Easton, PA-NJ Altoona, PA 125,348 100.00 27.2% $40,196 3.57% –0.66% Amarillo, TX 241,849 89.04 37.4% $41,944 4.04% 4.17% Ames, IA 85,188 97.69 29.9% $45,991 3.64% 2.19% Anchorage, AK 361,201 125.44 26.7% $65,534 2.97% 2.97%

(10) Based on the sample given, identify each of the following values: (a) Which city has the highest COL, and which has the lowest COL? (b) What is the median household income in Ames, Iowa? What does the median tell you about the population? (c) Which city has had the largest decrease in employment? (d) The U.S. Census Bureau estimated that in 2008, there were approximately 42,500 Native Americans living in Albuquerque, New Mexico. What percentage of Albuquerque’s population is Native American? Round to the nearest tenth of a percent.

1Retrieved from www.kiplinger.com/tools/bestcities_sort/index.php?si=1

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Find the mean and median populations of the cities in the list. You may use any appropriate technology, such as a spreadsheet or a calculator. (e) What is the mean population of the selected cities? (f) What is the median population of the selected cities?

(11) You will be expected to do the following things for the next class. Rate how confident you are on a scale of 1–5 (1 = not confident and 5 = very confident). Before beginning Lesson 2.2.4, you should understand the concepts and demonstrate the skills listed below:

Skill or Concept: I can … Rating from 1 to 5 Perform basic operations using quantities as integers, fractions, or decimals with the aid of technology. Calculate a ratio and write the result as a percent. Calculate and apply the measures of central tendency.

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Two methods of sorting are shown here.

Example 1 uses the Sort option in the toolbar This is an example. (1) Click on the upper-left cell and drag diagonally downward to the lower-right cell (indicated by the red arrow) to highlight all of the cells. Make sure to include the cells that contain the column names at the top.

(2) Click on Data from the menu at the top of the screen, and select Sort from the pop-up menu, as shown below. Note that this example uses the toolbar at the very top of the window. Some versions of Excel also have Tabs that include a Data option.

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(3) In the window that pops up, click on the button in the Sort by section at the top and select the category that you want to sort. In the figure below, the category called “Price” is selected. Also, make sure that the circular button labeled “Ascending” is selected, which will sort the data from the lowest to highest Price.

(4) Click OK. The pop-up window will close and the data will now be sorted according to Price.

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Example 2 uses the Sort Tab This example is from Lesson 2.2.4. • Click in the spreadsheet. • Select the Data tab. • Click on the small black triangle by the sort icon on the right side of the toolbar:

• Select Custom Sort • Make sure the box by “My list has headers” is checked. • Click on the arrows under the column to select the first category. Note that in some cases, high values are desired and in others, low values are desired. So you might sort Cost of Living from smallest to largest, but sort Income from largest to smallest. • Click the “+” button at the bottom of the dialog box to add another level of sorting. • Click on the arrows under the column to select the second category. • Repeat the last two steps to add the third category. • Click “OK.”

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Specific Objectives Students will understand that • multiple sources and types of data are important for making judgments and decisions. Students will be able to • draw appropriate conclusions from data. • apply previous learning to new contexts.

Problem Situation: Data on the Best Cities In this lesson, you will compare cities based on data from the 2010 list of Best Cities for the Next Decade. This list was published by a national financial news organization called Kiplinger.1 The list covers 367 cities. Your instructor will project the full list. Then you will work with a shorter list of 20 cities given on the next page. You will be viewing six of the categories from the list. One of these categories needs some explanation. “Percentage of Workforce in Creative Class” refers to the percentage of people who are, for example, scientists, engineers, educators, writers, artists, and entertainers.

1Retrieved from www.kiplinger.com/tools/bestcities_sort/index.php?si=1

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Percentage Income Cost of Median of Workforce Growth Employment Metro Area Population Living Household in Creative from 2005 Growth Index Income Class to 2007

Abilene, TX 159,137 86.83 23.6 39,371 5.13% 0.36%

Albany-Schenectady-Troy, NY 851,925 100.00 20.7 54,755 4.25% 1.62%

Austin-Round Rock, TX 1,590,744 94.93 25.1 54,827 3.46% 5.54%

Boston-Cambridge-Quincy, MA-NH 4,649,838 129.53 35.8 66,870 4.01% 1.95%

Boulder, CO 289,005 124.74 33.1 63,064 5.82% 4.04%

Canton-Massillon, OH 407,530 100.00 22.4 44,530 3.19% -1.60%

Fresno, CA 895,357 119.17 28.2 44,979 3.99% 2.63%

Great Falls, MT 81,888 100.00 26.8 41,802 2.98% 1.19%

Ithaca, NY 100,535 103.31 28.4 46,225 -1.63% 39.84%

Jackson, MS 533,870 95.82 24.6 42,921 4.44% 4.01%

Jonesboro, AR 115,787 85.47 22 36,527 4.29% 0.65%

Kokomo, IN 99,631 100.00 25.9 47,040 4.77% 0.69%

McAllen-Edinburg-Mission, TX 706,039 86.90 28 28,328 2.58% 3.73%

New Bedford, MA 173,441 129.53 38.2 66,870 3.43% 0.64%

Oklahoma City, OK 1,189,529 89.92 27.4 43,652 3.65% 1.69%

Seattle-Tacoma-Bellevue, WA 3,299,005 114.64 33.9 61,740 4.06% 3.79%

Springfield, IL 206,445 86.20 30.6 49,116 4.82% -2.07%

Waterbury, CT 199,412 129.53 27.1 66,870 2.01% 0.11%

Wausau, WI 129,849 96.18 21.9 52,241 3.13% 0.66%

Yuma, AZ 189,682 103.64 28.9 38,502 3.82% 2.17%

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(1) McAllen, Texas, is the largest city in Hidalgo County, one of the fastest growing counties in the United States. The US News lists McAllen as one of the top places to retire.2 Kiplinger listed Boulder, Colorado, as one of its top 10 cities in 2010.3 (a) Use the information on your worksheet to decide in which city you are most likely to “live well” financially. Use estimation skills to justify your decision. (b) Calculate how much a person in McAllen would need to make in order to have the same “buying power” as someone earning the median income in Boulder.

(2) Use the Kiplinger information to answer the following questions comparing Waterbury, Connecticut, and Springfield, Illinois. (a) In which city are there more families with an income above $50,000? (b) In which city are there more families with an income between $50,000 and $67,000?

(3) Review the information for Ithaca, New York. What do you think was happening in the job market from 2005 to 2007?

Making Connections Record the important mathematical ideas from the discussion.

Further Applications (1) This course draws on examples from three themes: Citizenship, Personal Finance, and Medical Literacy. Choose at least two different lessons with the Medical Literacy theme. Answer the following questions. (a) What new information did you learn about medical literacy? (b) How will you use this information or why is it important to know this information?

2Retrieved from http://money.usnews.com/money/blogs/the-best-life/2010/11/05/how-to-find-your-best-place-to-retire 3Retrieved from www.kiplinger.com/magazine/archives/10-best-cities-2010-for-the-next-decade.html

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(1) Which of the following are equivalent to a population density of 2,000 people/100 mi2? There may be more than one correct answer. (2.1.1) (a) 200 people/10 mi2 (b) 2 people/mi2 (c) 2 people/10 mi2 (d) 20 people/mi2 (e) 0.2 people/mi2

(2) According to 2010 census data, the two largest cities in Washington State are Seattle and Spokane. The data indicate that 612,000 people lived in Seattle on 83.78 square miles of land and 206,900 people lived in Spokane on 60.22 square miles of land.1 Which of the following is true? (2.1.1) (a) The population density of Seattle is approximately three times that for Spokane. (b) The population density of Seattle is approximately 1.5 times that of Spokane. (c) The population density of Seattle is approximately 7,304.8 people per square mile. (d) The population density of Spokane is approximately 4,436.0 people per square mile. (e) The population density of Seattle is approximately the same as that for Spokane.

(3) In 2010, the 313,000,000 people in the United States consumed 12,040 thousand metric tons of beef per year. The 6,920,000,000 people in the world consumed 56,544 thousand metric tons of beef per year.2 If the entire population of the world ate beef at the same rate as the United States, then what would the world consumption rate be? (2.1.2) (a) Between 50,000 and 100,000 thousand metric tons of beef per year (b) Between 100,000 and 150,000 thousand metric tons of beef per year (c) Between 150,000 and 200,000 thousand metric tons of beef per year (d) Between 200,000 and 250,000 thousand metric tons of beef per year (e) Between 250,000 and 300,000 thousand metric tons of beef per year

1Retrieved from www.ofm.wa.gov/pop/popden/default.asp 2Retrieved from www.census.gov/compendia/statab/cats/international_statistics.html

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(4) A school lunch program orders 3,250 pounds of chicken to serve 13,000 students. Which of the following correctly expresses this situation as a unit rate? (2.1.2) (a) ¼ pound (b) ¼ pound per student (c) 4 pounds (d) 4 pounds per student (e) 3,250 pounds per 13,000 students

(5) The average price of a house in Tucson, Arizona, was $172,500 in 2009 and decreased to $156,600 in 2010. In Seattle, Washington, the average price of a house in 2009 was $306,200 and $295,700 in 2010.3 Which of the following statements is true? (2.1.3) (a) Tucson had the larger absolute change and relative change in the average price of a house from 2009 to 2010. (b) Seattle had the larger absolute change and relative change in the average price of a house from 2009 to 2010. (c) Tucson had the larger absolute change and Seattle had the larger relative change in the average price of a house from 2009 to 2010. (d) Seattle had the larger absolute change and Tucson had the larger relative change in the average price of a house from 2009 to 2010. (e) Seattle had the larger absolute change and both cities had the same relative change in the average price of a house from 2009 to 2010.

(6) Tuition at universities and colleges in Minnesota increased by 4.8% for the 2011–2012 academic year. Which type of college will have the greatest absolute increase in tuition? (2.1.3) (a) The community colleges whose tuition is between $2,480 and $4,708. (b) The public four-year universities whose tuition is between $3,258 and $6,639. (c) The private universities whose tuition is between $27,450 and $52,110. (d) All will have the same absolute increase.

3Retrieved from www.realtor.org/research/research/metroprice

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(7) In 1995, 11% of children of age 12–19 were obese. Five years later, in 2000, the percentage had increased to 22%. Express this change as an absolute change (in percentage points) and a relative change. (2.1.3) (a) The absolute change was 11 percentage points, and the relative change was 50%. (b) The absolute change was 11 percentage points, and the relative change was 100%. (c) The absolute change was 50 percentage points, and the relative change was 100%. (d) The absolute change was 50 percentage points, and the relative change was 50%. (e) The absolute change was 100 percentage points, and the relative change was 11%.

(8) In 2008, the Nelson study reported that teens send an average of 1,742 text messages per month. In 2010, they repeated the study and found that teens send an average of 3,339 texts per month.4 The relative change in the number of text messages from 2008 to 2010 was ______. (2.1.3) (a) 1,597 (b) 15.97% (c) 91.67 (d) 91.67% (e) Not enough information was provided to determine relative change.

4Retrieved from http://news.cnet.com/8301-1035_3-10048257-94.html and http://mashable.com/2010/10/14/nielsen-texting- stats

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Use the graph below for Questions ___and ____. It shows the Consumer Price Index for all urban consumers (CPI-U) from 1915 to 2010. (2.1.4)

(9) Which is the best estimate for the CPI-U in 2000? (a) 150 (b) 160 (c) 170 (d) 190 (e) 200

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(11) Above is a chart of the world’s population distribution by continent using 2010 data.5 There were approximately 6,853,000,000 people in the world in 2010. Consider the following statements: Statement I: In 2010, Asia’s population was approximately four times that of Africa. Statement II: Asia’s population was approximately 4.1 billion in 2010. Statement III: Oceania having 0% in the graph means it has no population. Based on this information, determine which of the following options is valid. (2.1.4) (a) Only Statement I is true. (b) Only Statement II is true. (c) Both Statements I and II are true, and Statement III is not true. (d) Both Statements II and III are true, and Statement I is not true. (e) All three statements are true.

5Retrieved from www.census.gov/compendia/statab/cats/international_statistics.html

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(12) A 2009 review found that mammograms reduced the risk of death due to breast cancer in women aged 50–75 by 15%.6 If you wanted to figure out how many lives could be saved by mammograms, you would need to know: (2.1.5) (a) The number of women in the population. (b) The number of women in the population between the ages of 50 and 75. (c) The number of women in the population between the ages of 50 and 75 who are likely to get breast cancer. (d) The number of women in the population between the ages of 50 and 75 who are likely to get any kind of cancer.

(13) The Centers for Disease Control and Prevention estimates that 0.2% of the U.S. population died of heart disease in 2006.7 Suppose a new treatment plan reduced the risk of dying of heart disease by 12%. How many people out of the population of 311 million Americans could be saved in a year? (2.1.5) (a) 485,160 (b) 48,516,000 (c) 74,640 (d) 746,400 (e) 37,320,000

(14) Five friends independently studied for a math quiz. After the quiz was over, they reported the following study times: 25, 30, 50, 75, and 100 minutes. Which of the following statements is true? (2.2.1) Statement I: The mean study time was 61 minutes. Statement II: The median study time was 50 minutes. Statement III: There is no distinct mode for this data set. (a) Both Statements I and II are true. (b) Both Statements I and III are true. (c) Both Statements II and III are true. (d) All three statements are true. (e) None of the three statements is true.

6Retrieved from en.wikipedia.org/wiki/Breast_cancer_screening 7Retrieved from www.cdc.gov/dhdsp/data_statistics/fact_sheets/fs_heart_disease.htm

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(15) Which of the following statements are true? (2.2.1 and 2.2.2) Statement I: The mode of 25 numbers is 376. There must be more numbers with the value of 376 out of the 25 than any other. Statement II: The median of 25 numbers is 376. If the list of numbers is sorted from smallest to largest, the 13th number in the list must be 376. Statement III: The mean of 25 numbers is 376. Half of the numbers must be more than 376. (a) Only Statement I is true. (b) Only Statement II is true. (c) Only Statement III is true. (d) Two of the statements are true. (e) All three statements are true.

(16) Television ownership is often compared across countries to identify cultural differences. Consider the data in Table 1 for the year 2003.8 Which country has the lowest average number of televisions per person? (2.2.1 and 2.2.2) Table 1 Number of Country Population Televisions China 400,000,000 1,307,000,000 France 34,800,000 60,180,000 Malta 280,000 400,000 (a) China (b) France (c) Malta

(17) A baseball supply store has many reasonably priced baseball gloves. In fact, over half of the gloves in stock cost less than $50. However, there are also a few high-end gloves in stock. Some of those cost over $300. The owner is preparing a flyer for the local little league club and wants to advertise her store as a place to obtain reasonably priced items. Which number should she use in the advertisement for the gloves? (2.2.1 and 2.2.2) (a) Mean baseball glove price (b) Median baseball glove price (c) Mode baseball glove price

8Retrieved from www.nationmaster.com/graph/med_tel-media-televisions

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(18) Figure 1 is a graph with the number of hours that 14 college students slept on a given night. Use this graph to calculate the mean, rounded to one decimal place, of the nightly sleep times for these students. (2.2.1 and 2.2.2) (a) The mode is 8 hours. (b) The mean is 6.5 hours. (c) The mean is 6.8 hours. (d) The median is 7 hours. (e) The mean cannot be determined from the graph.

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(19) In 1936, the cost of a Hershey’s chocolate bar was 5 cents. In 1980, the cost of a Hershey’s chocolate bar was 25 cents.9 By comparison, the CPI-U in 1936 was 13.9, and the CPI-U in 1980 was 82.4. From 1936 to 1981, how did the price of the Hershey bar change compare to the change in CPI-U? (2.2.3)

Cost of a Hershey Bar CPI–U (Consumer Price Index Year (in Cents) for Urban Consumers) 1936 5 13.9 1980 25 82.4

(a) The relative increase in the price of the Hershey’s bar was lower than the relative increase in the CPI-U. (b) The relative increase in the price of the Hershey’s bar was the same as the relative increase in the CPI-U. (c) The relative increase in the price of the Hershey’s bar was higher than the relative increase in the CPI-U.

(20) The CPI-U for the years 2000-2002 are given in the table below. Select the statement that correctly describes the rate of inflation (increase in prices) during this period. (2.2.3)

Year CPI-U 2000 172.2 2001 177.1 2002 179.9

(a) The rate of inflation and prices were the same in 2001 and 2002. (b) Both the rate of inflation and prices were higher in 2001 than in 2002. (c) The rate of inflation was higher in 2001 than in 2002, but prices were lower. (d) Both the rate of inflation and prices were higher in 2002 than in 2001. (e) The rate of inflation was higher in 2002 than in 2001, but prices were lower.

9Retrieved from www.foodtimeline.org/foodfaq5.html#candybar

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As with Module 1, you should assess your understanding of Module 2 to prepare for the Module 2 test. Your instructor may give you specific assignments for your review in addition to this self-assessment.

Assessing Your Understanding The table on the following page lists the Module 2 concepts and skills you should understand. This exercise helps you assess what you understand. After completing it, you will be able to prioritize your review time more effectively. 1. Assess your understanding. • Go through the topics list and locate each concept or skill in the Module 2 in-class or OCE materials. • If you have not used the skill in a while, do two or more problems to check your understanding. • If you have recently used the skill and feel confident that you did it correctly, rate your understanding a 4 or 5. • If you remember the topic but could use more practice, rate your understanding a 3. • If you cannot remember that skill or concept, rate your understanding a 1 or 2. Now that you have done an initial rating of your understanding, it is time to begin reviewing. Complete the remaining steps. The goal is to have a confidence rating of 4 or 5 on all the topics in the table when you have finished your review of Module 2. 2. Start at the beginning of module and reread the material in the lessons, the OCE, and your notes on the skills and concepts you rated 3 or below. 3. Select a few problems to do. Do not look at the answer or your previous work to help you. 4. Once you have finished the problems, check your answers. If you are not sure if you have done the problems correctly, check with your instructor, other classmates, and your previous work or work with a tutor in the learning center. 5. Rate your confidence on this skill again. If you understand the concept better, rate yourself higher. Begin a list of topics that you want to review more thoroughly. 6. If you have time, do one or two problems on skills or concepts you rated 4 or above. 7. For topics that you need to review more thoroughly, make a plan for getting additional assistance by studying with classmates, visiting your instructor during office hours, working with a tutor in the learning center, or looking up additional information on the Internet.

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Module 2 Concept or Skill Rating Using Ratios Understand meaning of equivalent ratios in context (2.1.1) Use units with ratios (2.1.1) Calculate a unit rate (2.1.2) Use ratios and proportionality to calculate new values (2.1.2) Interpret and use index numbers to calculate new values (2.2.3, 2.2.4) Applications of Percentages Calculate and interpret absolute change between two quantities (2.1.3, 2.1.5) Calculate and interpret relative change between two quantities (2.1.3, 2.1.5) Calculate and interpret absolute change between two percentages (2.1.3) Calculate and interpret relative change between two percentages (2.1.3) Make and interpret comparisons of absolute measurements versus relative measurements (2.1.4, 2.1.5) Graphical Displays Read and interpret pie graphs, bar graphs, and line graphs (2.1.4) Recognize distortion of graphs due to different scales (2.1.4) Calculate absolute and relative change from a graph (2.1.4) Measures of Central Tendency Calculate mean, median, and mode of a data set (2.2.1, 2.2.2) Interpret the meaning of and differences between the mean, median, and mode (2.2.1, 2.2.2)

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Note: Because Module 2 ends with the Culminating Activity and Module Assessment, there is not the usual OCE assignment covering a full lesson. This is the material to prepare you for the next lesson, 3.1.1. Preparing for Lesson 3.1.1 Equivalent Fractions Two fractions are equivalent if they have the same value or represent the same part of an object. For example, the figure shows that 1/2, 2/4, and 4/8 all represent the same part of a whole. They are equivalent fractions. Recall that the denominator of a fraction represents the number of parts into which the whole has been divided. The numerator represents a count of the number of parts. So, means that the whole is divided into 8 equal parts, and 4 of these parts are counted. The same part of an object: 1/2 = 2/4 = 4/8

(1) Write two more fractions that are equivalent to 1/2.

Simplest Form of a Fraction The fraction 50/100 is equivalent to 1/2. Note that you can write 50 1 * 50 1 = = * 1 100 2 * 50 2 The above calculation shows that both 50 and 100 can be written as a number times 50. You say that 50 and 100 have a “common factor of 50.”

50 50 Another way to think of this is that the number 1 (written as ) is embedded in the fraction . 50 100

“1” is a special number in mathematics because if you multiply any number by 1, you get a result that is equivalent to the original.

50 50 1 By dividing to get 1, you simplify to . 50 100 2

In this case, the word, simplify, means that the fraction has been written in an equivalent form with smaller numbers. The simplest form means that the fraction is written using the smallest possible numbers. In general, answers should always be given in simplest form unless the question specifically calls for a different form.

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Caution! It is common to say that you “cancel” the 50 from the numerator (top number) and denominator (bottom number) and write the fraction in “reduced form.” This language is misleading for two reasons. Numbers can be “canceled” by dividing to “1” or adding to “0”—it is better to think about the operation (add, subtract, multiply, divide) so you understand why it works. Second, the value of the simpler fraction is the same as the original fraction, but the word “reduced” implies that the “reduced fraction” represents a smaller quantity. It can be confusing, so the terminology “simplest form” makes more sense. (2) Write each fraction in its simplest form.

24 (i) 36 9 (ii) 12 20 (iii) 30 35 (iv) 28 (3) Compare the two fractions to determine if they are equivalent.

3 6 (i) , 8 16 3 6 (ii) , 4 9 10 5 (iii) , 8 4 10 5 (iv) , 24 8

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Multiplying and Dividing Fractions The fact that common factors in the denominator and numerator of a number can be divided to make 1 can be used to make multiplying fractions easier. Consider the following multiplication problem.

2 7 14 7 * 2 7 * → → → 3 8 24 12 * 2 12 If you see that there is a common factor of 2 in the numerator and denominator before multiplying, you can divide the common factors first. This makes the multiplication easier because you have smaller numbers to work with and the simplification is complete.

2 7 2 7 2 7 1 * 7 7 * → * → * → → 3 8 3 4 * 2 3 4 * 2 3 * 4 12 This is an important concept when working with ratios with units. You will learn more about this below.

Many people struggle with dividing fractions because it is difficult to visualize. A full explanation of the mathematics behind dividing fractions is beyond what the authors can do in these materials. Instead, the authors are providing you with a context that might help you remember how to divide fractions. Suppose you have $48 to spend on going to the movies during a month. How many tickets can you buy in a month? A movie ticket costs $8. One way to think about this is that you want to know how many groups of $8 there are in $48 or 48 ÷ 8. In the same way, suppose you had $10 to spend on downloading songs for a 1/2 dollar. (For the sake of the mathematics, you are going to express “a half of a dollar” as a fraction instead of as a decimal.) This means you want to know how many 1/2 dollars there are in $10. Your common sense probably tells you that the answer is 20 because every 1 dollar has two halves. So you multiplied 10 x 2. Look at this written as a calculation: 1 2 10 ÷ is the same as 10 * 2 1 1 2 and are called reciprocals. 2 1 So you say that division is the same as multiplying by the reciprocal. Here are more examples:

1 2 4 ÷ → 4 * → 8 2 1

2 3 36 12 ÷ →12 * → →18 3 2 2

4 4 5 20 ÷ 2 → * → →2 5 5 2 10

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Perform the calculations indicated in each problem.

2 3 (4) 10 * = (5) 8 ÷ = 5 4

5 7 5 8 (6) ÷ = (7) * = 8 8 8 7

(8) Jarrod is helping with his daughter’s school assembly. Every student will receive a decorative ribbon to wear. The ribbons have to be 2/3 of a foot long. A local store donates 30 feet of ribbon. How many decorative ribbons can Jarrod make?

(9) Lorinda has started to think about saving for retirement. She reads a recommendation that says she should save at least 3/10 of her income because she is over 40 years old. Lorinda makes $42,000 a year. How much should she save in one year according to this recommendation?

For more information about working with fractions, you might want to refer to the following Internet videos: • Equivalent fractions: http://www.khanacademy.org/video/equivalent- fractions?playlist=Arithmetic • Multiplying fractions: http://www.khanacademy.org/video/multiplying- fractions?playlist=Developmental%20Math • Dividing fractions: http://www.khanacademy.org/video/dividing-mixed- numbers?playlist=Developmental%20Math

Ratios and Unit Rates Unit rates are ratios with a denominator of 1, although they are not always written as fractions. For 60 miles example, 60 mph is the same as . 1 hour The language “miles per hour” implies that the operation is miles divided by 1 hour.

(10) Write each expression as a unit rate in fractional form. (i) 23 mpg (miles per gallon) (ii) 12 ft/sec (feet per second) (iii) 5 gal/min (gallons per minute) (iv) $7.15/hr (dollars per hour)

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(11) Convert each rate into a unit rate.

(i)

(ii)

(iii)

Conversion Factors A fraction that is a ratio of quantities can be equivalent to one even when the numerator and denominator are not the same number. However, it is necessary that the numerator and denominator represent equivalent quantities. For example, the following fractions are all forms of one:

These types of ratios are sometimes called conversion factors because they can be used to covert between units. (12) Complete the following fractions to make a conversion factor. Look up the conversion factor in a reference book or on the Internet if you do not know it.

(i)

(ii)

(iii)

A rate, like 35 miles per hour, can be expressed as a fraction: .

You can use the conversion factor of 60 minutes per 1 hour to convert the rate to the units of miles per 1 minute.

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The example below shows how to set up a multiplication problem with the rate and the conversion factor to convert miles per hour to miles per minute.

35 miles 1 hour * 1 hour 60 minutes Notice that the conversion factor is written so that the units of “hours” are in the numerator. This is because you want the “hours” to divide out in the same way that common factors divided out in the multiplication problems above. This leaves the units of miles/minute as shown below.

35 miles 1 hour 35 miles 0.58 mile → * → → 1 hour 60 minutes 60 minutes 1 minute You will use this concept in Lesson 3.1.1. This assignment included a large number of vocabulary words that are important. Make sure you fully understand each term listed below: • equivalent fraction • simplify; simplest form • common factor • reciprocal • unit rate • conversion factor (13) You are expected to be able to do the following things for the next class. Rate how confident you are on a scale of 1–5 (1 = confident and 5 = very confident). Before beginning Lesson 3.1.1, you should understand the concepts and demonstrate the skills listed below:

Skill or Concept: I can … Rating from 1 to 5 Multiply two fractions. Divide two fractions. Understand that a fraction can be simplified by dividing common factors in the numerator and denominator (simplify fractions). Understand that multiplying by 1 doesn’t change a value. Be familiar with basic units of measure of length (feet, miles) and time (seconds, hours, minutes).

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Specific Objectives Students will understand that • the units found in a solution may be used as a guide to the operations required in the problem— that is, factors are positioned so that the appropriate units cancel. • units provide meaning to the numbers they get in calculations. Students will be able to • write a rate as a fraction. • use a unit factor to simplify a rate. • use dimensional analysis to help determine the factors in a series of operations to obtain an equivalent measure.

Problem Situation: Cost of Driving Jenna’s job requires her to travel. She owns a 2006 Toyota 4Runner, but she also has the option to rent a car for her travel. In either case, her employer will reimburse her for the mileage using the rate set by the Internal Revenue Service. In 2011, that rate was 55.5 cents/mile. Over the next two lessons you will explore the question of whether it would be better for Jenna to drive her own car or to rent a car. (1) What do you need to know to calculate the cost of Jenna driving her own car?

(2) What do you need to know to calculate the cost of Jenna renting a car?

The next section introduces skills that will help you with the problem situation. You will start by working with more focused questions and specific information. (3) Gas mileage is rated for either city driving or highway driving. Most of Jenna’s travel will take place on the highway. For one trip, she drives 150 miles and the price of gas is $3.67/gallon. Her 4Runner gets 18 miles/gallon. If Jenna rents, she can request a small, fuel-efficient car such as the Hyundai Elantra, which gets 40 miles/gallon. (a) Use your estimation skills to compare the cost of gas for the two vehicles. Which one costs more? How much more? Explain your answer. (b) What is the cost of the gas for each vehicle?

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Dimensional analysis is a method of setting up problems that involves converting between different units of measurement. It is also called unit analysis or unit conversion. Many professionals—including pharmacists, dieticians, lab technicians, and nurses—use unit analysis. It is also useful for everyday conversions in cooking, finances, and currency exchanges. Many people can do simple conversions without dimensional analysis; however, they will likely make mistakes on more complex problems. The advantage of using dimensional analysis is that it is a way to check your calculations. While it is always important that you develop your own methods to solve problems, this is a time when you are encouraged to learn and use a specific method. Once you have learned dimensional analysis, you can decide when to use it and when to use other methods. To help build this skill, you will now leave the problem situation to practice dimensional analysis. You will come back to the situation of Jenna and the cost of driving in the OCE and the next lesson. (4) Many states have banned texting while driving because it is dangerous, but many people do not think that texting for a few seconds is that harmful. Suppose you are driving 60 miles/hour. You take your eyes off the road for 4 seconds. How many feet will you travel in that time? (a) A student set up the calculation below to convert miles/hour into feet/second. Only the units are shown. Use the units to decide if the problem is set up correctly. If not, correct it.

(b) How many feet will you travel in 4 seconds if you are traveling at 60 miles/hour?

Now, you will return to the problem situation. If you do not complete this work in class, finish it as a part of the OCE. (5) Using the information below, calculate Jenna’s total cost of driving a rental car for a round trip. • Price of gas: $3.50/gallon • Length of trip (one way): 193 miles • Gas mileage of rental car: 40 miles/gallon1 • Price of the rental car: $98.98 plus 15.3% tax (gas is not included in the rental price and the car must be returned to the rental agency with a full tank)

1Retrieved from www.fueleconomy.gov

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Making Connections Record the important mathematical ideas from the discussion.

Further Applications (1) Show your work for Question 7 from the OCE (Lesson 3.1.1). Write an explanation for how you set up the problem.

(2) Do an Internet search for dimensional analysis or unit analysis. Find at least one site that provides examples of how to make conversions using this technique. (a) Record the site name and URL address. (b) Copy one example as shown on the site.

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Making Connections to the Lesson (1) Which of the following was one of the main mathematical ideas of the lesson? (i) Units can be used to set up conversion problems by using the fact that common factors in the numerator and denominator of a fraction divide out to 1. (ii) There are many hidden costs in driving such as insurance, registration, and maintenance. (iii) Units can be used to set up conversion problems by using the fact that common factors in the numerator and denominator of a fraction can be subtracted to equal 0. (iv) It is possible to convert miles per hour to feet per second. (2) Find examples of three ratios from three different lessons in the course, not including Lesson 3.1.1. For each ratio, do the following: (a) Record the lesson in which the ratio was used. (b) Write a statement explaining what the ratio means in context. (c) Identify if the ratio is a unit rate.

Developing Skills and Understanding (3) Use Figure 1 for the following questions.

Figure 1

(a) What fraction of Figure 1 is shaded? (b) Shade the same fraction of the area in Figures 2 and 3.

Figure 2

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Figure 3

(4) As of September, 2011, Florence Griffith-Joyner held the women’s world record for the 100-meter dash. She set the record with a time of 10.49 seconds in 1988.1 Which of the following calculations are correctly set up to convert this speed into miles per hour?

10.49 seconds 1 meter 5,280 feet 1 minute 1 hour (i) x x x x 100 meters 3.28 feet 1 mile 60 seconds 60 minutes

100 meters 3.28 feet 1 mile 60 seconds 60 minutes (ii) x x x x 10.49 seconds 1 meter 5,280 feet 1 minute 1 hour

10.49 seconds 1 meter 5,280 feet 60 seconds 60 minutes (iii) x x x x 100 meters 3.28 feet 1 mile 1 minute 1 hour

(5) Find the answer to the conversion in Question 4. Round to the nearest tenth of a mile per hour.

(6) A 2010 Toyota Prius hybrid vehicle gets 48 mpg for highway driving. The tank holds 11.9 gallons of fuel.2 Typically the low fuel warning light comes on when approximately two gallons of fuel remain in the tank. Which of the following calculations can be used to find the distance that can be traveled after the fuel light comes on and before the car runs out of gasoline?

1 48 miles (i) x = 24 miles 2 gallons 1 gallon

2 gallons 48 miles (ii) x = 96 miles 1 1 gallon

11.9 gallons 48 miles (iii) x = 571.2 miles 1 gallon

2 gallons 1 gallon 1 (iv) x = miles 1 48 miles 24

1Retrieved from http://en.wikipedia.org/wiki/100_metres#Women 2Retrieved from Kelley Blue Book: http://www.kbb.com/toyota/prius/

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Making Connections Across the Course (7) In Lesson 1.2.1, you learned about a water footprint. Part of a person’s water footprint is the water used for cleaning. In this question, you will calculate the cost of water for laundry and bathing. You will use the City of New York 2011 rate of $7.64/100 cubic feet of water. Calculate the cost of each of the following based on this rate. Use the conversion factor of 7.48 gallons per cubic foot.3 (a) A standard washing machine uses approximately 50 gallons of water per load.4 A household washes one load of laundry per week for 52 weeks. Find the total cost per year. (b) According to one study, the average American shower lasts for 8.2 minutes and uses 17.2 gallons. A person showers once a day for a year. Find the total cost per year.5

Preparing for the Next Lesson (3.1.2) If you did not finish Question 5 from the lesson during class, complete it. You will need this information and the answer for the following question for Lesson 3.1.2. (8) Using the information below, calculate the total cost of Jenna driving her own car for a round trip. Note that the cost of insurance, vehicle registration, and taxes varies greatly with location and individual. • Price of gas: $3.50 per gallon • Length of trip (one way): 193 miles • Gas mileage of Jenna’s car: 22 miles per gallon6 • Insurance, registration, taxes: Jenna spends $2,000 a year on these expenses and last year she drove about 21,600 miles Maintenance costs for Jenna’s car: • General maintenance (oil and fluid changes): $40 every 3,000 miles • Tires: Tires for Jenna’s car cost $920; they are supposed to be replaced every 50,000 miles • Repairs: The website Edmonds.com estimates that repairs on a three-year-old 2009 4Runner will be approximately $328 per year; this is based on driving 15,000 miles7

3 Retrieved from www.nyc.gov/html/dep/html/residents/wateruse.shtml 4 Retrieved from www.nyc.gov/html/dep/html/residents/wateruse.shtml 5 Retrieved from www.allianceforwaterefficiency.org/Residential_Shower_Introduction.aspx 6 Retrieved from www.fueleconomy.gov 7 Retrieved from www.edmunds.com/toyota/4runner/2006/tco.html?style=100614746

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Lesson 3.1.2 builds on the use of dimensional analysis from Lesson 3.1.2, so there are no additional skills to practice at this time. (9) You are expected to be able to do the following things for the next class. Rate how confident you are on a scale of 1–5 (1 = not confident and 5 = very confident). Before beginning Lesson 3.1.2, you should understand the concepts and demonstrate the skills listed below:

Skill or Concept: I can … Rating from 1 to 5 Use dimensional analysis in a contextual problem.

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Specific Objectives Students will understand that • units can be used in dimensional analysis to set up calculations. • precision should be based on several factors, including the size of the numbers used and the precision of the original values. Rounding can produce large differences in results. Students will be able to • solve a complex problem with multiple pieces of information and steps. • use dimensional analysis. • investigate how changing certain values affects the result of a calculation.

Problem Situation: Comparing Costs Jenna’s job requires her to travel. She owns a 2009 Toyota 4Runner, but she also has the option to rent a car for her travel. In either case, her employer will reimburse her for the mileage using the rate set by the Internal Revenue Service. In 2011, that rate was 55.5 cents per mile. (1) Discuss the cost for Jenna to drive her own car (from the OCE [Lesson 3.1.1]). (a) Identify different strategies used. Make sure everyone in the group understands the different strategies and agrees on the answer. (b) Jenna’s employer will reimburse her at a rate of 55.5 cents per mile. Calculate how much profit Jenna makes after she pays her expenses in each situation.

(2) Since her trips vary in length, it is useful for Jenna to compare the cost per mile of renting a car to the cost per mile of driving her own car. Find the cost per mile for each option.

(3) Will it cost Jenna less to use her own car for every trip? What factors would affect the relative cost of the two options? Explain your answer.

(4) After discussing and exploring Question 3 as a class, write to Jenna, explaining how she can decide if it is better to drive her own car or to get a rental. Your explanation should include information about what factors affect the cost of driving and why.

Making Connections Record the important mathematical ideas from the discussion.

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Further Applications (1) If you drive a car or plan to get a car, complete Part (a). If you do not have a car, complete Part (b). Show your work. (a) Estimate the cost per mile of driving your car based on what you actually pay for insurance and gas mileage. You may use the cost of maintenance from the lesson or research the costs on your own. You should have references for any information you give, which can include information from your own insurance and maintenance records. (b) If you do not have a car, find the cost per year of some activity or item that you pay for at least twice a week on average. For example, buying a cup of coffee or energy drink, downloading music, going to a movie, paying a babysitter, riding a bus, etc.

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Making Connections to the Lesson (1) Which of the following was one of the main mathematical ideas of the lesson? (i) Renting a car is more expensive than driving your own car. (ii) A small difference in rounding does not make much difference in results. (iii) A small difference in rounding can significantly change results when working with large numbers. (iv) You can calculate the cost of gas per mile by dividing the price of gas by the gas mileage of the car. (2) In this lesson, you were asked to figure out what factors would affect the cost of two different options. This type of problem solving is used in many situations and will be used in future lessons. Write a brief description of how you approached this problem or how you might have approached it more effectively.

Developing Skills and Understanding (3) In planning a Thanksgiving vacation, you want to rent a car for a week and travel the Pacific Highway from San Francisco to San Diego. You want to return to San Francisco via Las Vegas, Death Valley, and Yosemite National Park. This trip covers approximately 1,500 miles. You plan to return the car with a full tank of gasoline. You are considering two options advertised by Hertz: • Toyota Camry, 33 mpg highway, costs $465.59 plus taxes and fees, totaling $553.28. • Toyota Prius Hybrid, 48 mpg highway, costs $634.49 plus taxes and fees, totaling $804.61. In both cases, you must purchase gasoline, which costs approximately $3.80 per gallon in California. (a) Find the total cost of renting and driving the Camry for the trip. (b) Find the total cost of renting and driving the Prius for the trip.

(4) The advertised mpg for new cars is based on a speed of 55 mph. As speed increases above 55 mph, the efficiency reduces dramatically.1 • 3% less efficient at 60 mph • 17% less efficient at 70 mph (a) Compare the efficiency of a Toyota Camry (33 mpg highway) versus a Toyota Prius (48 mpg highway) at speeds of 55 mph, 60 mph, and 70 mph. Round to the nearest tenth.

55 mph 60 mph 70 mph

Camry 33 mpg

Prius 48 mpg

1Retrieved from www.mpgforspeed.com

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(b) If gasoline costs $3.67 per gallon, how much money would you save by going 60 mph versus 70 mph on the trip of 1,500 miles in the Camry? In the Prius? (c) How much longer, to the nearest minute, would it take to travel 300 miles at 60 mph versus 70 mph?

Making Connections Across the Course (5) The National Center for Children in Poverty (NCCP) posted the following information in 2011: Nearly 15 million children in the United States—21% of all children—live in families with incomes below the federal poverty level ($22,050 a year for a family of four). Research shows that, on average, families need an income of about twice that level to cover basic expenses. Using this standard, 42% of children live in low-income families.2 (a) The graph below can be used to illustrate the statement from the NCCP. Identify what each section and the full circle represent in terms of the context and the percentage. Section 1 is the darker section. The full circle represents ______and is ______%. Section 1 represents ______and is ______%. Section 2 represents ______and is ______%.

2Retrieved from www.nccp.org/topics/childpoverty.html

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(b) Can the statement from the NCCP be restated to say that 21% of people below the federal poverty level in the United States are children? Select the best answer. (i) Yes, because the reference value in both statements is the children in the United States. (ii) Yes, because both statements are about children living in poverty so the percentages are the same. The reference values do not matter. (iii) No, because the percentage of children is really much higher because families really need much more money than the federal poverty level. (iv) No, because the reference value in the first statement is the total number of children. The reference value in the second statement is the total number of people below the poverty level. (c) Based on the information in the paragraph, which of the following is the best estimate for the number of children in the United States? (i) 30 million (ii) 75 million (iii) 100 million (iv) There is not enough information to make an estimate.

Preparing for the Next Lesson (3.2.1) The next lesson explores calculations needed for repairs and improvements to a house and lot. The problems will require that you understand concepts of length, area, and volume and appropriate units of measure, based on customary U.S. units. Length Length is one-dimensional. In the house remodeling context, an example would be the total length of baseboard needed to trim the walls of a room. Examples of units of measure for length are inches, feet, yards, or miles. A number line can be used to model lengths. The thicker segment on each number line is 3 units long. A Number Line If the scale is in inches, each line segment is 3 inches long. If the scale is in feet, each line segment is 3 feet long. Area Area is two-dimensional and is measured in square units. The total number of one-foot square tiles needed to cover the floor of a room is an illustration of area measured in square feet. A rectangle is one shape that can be used to model area. Recall the formula for the area of a rectangle: A = L x W. The area of a rectangle is the product of the length and the width, which is a shortcut for counting the number of square units needed to cover the rectangle. A Coordinate Axis

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Each of the two shaded areas on the coordinate axis has an area of 12 square units. If the horizontal and vertical scales are in inches, each area is 12 square inches. If the scales are in feet, each area is 12 square feet. Notice that the regions measured do not have to be squares, yet the area is measured in square units. Notice how the units in the calculation determine the units in the result: A = (2 inches) x (6 inches) (2 x 6) (inches x inches) 12 square inches or inches2 If the units are feet, the area of the rectangle on the top is A = (3 feet) x (4 feet) = 12 square feet or feet2. Note 1: It is common to abbreviate the units of measure using exponents. Since the area might be 12 feet x feet, write A = 12 ft2. Notice the connection to algebra here! Multiplying (3 feet) by (4 feet) is similar to multiplying (3x) by (4x). You multiply the numbers in front of the variables (coefficients), and then multiply the variables: (3 • 4) (x • x) = 12x2. Note 2: It is common to confuse length and area formulas. To calculate the length of the line surrounding the rectangle, which is called the perimeter, simply add the total number of units as if traveling around the area. For example, if the units are in feet, then the length of the line around the bottom rectangle is 2 feet + 6 feet + 2 feet + 6 feet = 16 feet. The arithmetic operation is addition, and the unit of measure is feet. By comparison, the arithmetic operation to compute area is multiplication and the unit of measure is square feet. Again, this connects to algebra. To add algebraic terms, you must have like terms, meaning terms with the same variables: 2x + 6x +2x + 6x = 16x. You cannot add 2x + 3y just as you cannot add 2 feet + 3 inches. The area of the circle is given by the formula, A = πr2. • π is a constant that is approximately 3.14159 (you probably learned 3.14, but that can lead to rounding errors). • r is the radius of the circle, which varies. • A is the area of the circle, which varies.

(6) Write each of the following products (the result to a multiplication problem) using exponents to express the results in a simpler form. (a) (3a)(5a) (b) (5p)(2p) (c) (3 inches)(5 inches) (d) (5 feet)(2 feet)

(7) How many square inches are in 1 square foot?

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(8) Find the unknown quantity in each situation. (a) A rectangle with an area of 72 square inches has a length of 6 inches. What must the height of this rectangle be? (b) A rectangle has a length of 9 inches and a height of 1.5 feet. What is the area of the rectangle in square inches? What is the area of the rectangle in square feet? (c) A rectangle with an area of 1 square foot has a height of 6 inches. What is the length of the rectangle?

(9) What is the area of a circle with a diameter of 5 feet?

(10) If the radius of a circle is doubled, will the area also double? Hint: Compare the areas of two circles: one with a radius of 10 inches and the other with a radius of 20 inches. Volume Volume is three-dimensional and is measured in cubic units. The formula to calculate the volume of a box is V = L  W  H. The shaded volume is 3 units wide, 4 units tall, and 5 units long, a volume of 60 cubic units. If the scales are measured in inches, then the volume is V = (5 inches)  (3inches)  (4 inches) = 5 • 3 • 4 cubic inches, or 60 in3. If the scales are in feet, the volume is 60 ft3 or 60 cubic feet. Note: The power of 3 is often called the cube of a number just as the power of 2 is called the square of a number. So 53 can be called 5 cubed. (11) A cubic foot is a cube that is one foot on each side. How many cubic inches are in one cubic foot? (12) You are expected to be able to do the following things for the next class. Rate how confident you are on a scale of 1–5 (1 = not confident and 5 = very confident). Before beginning Lesson 3.2.1, you should understand the concepts and demonstrate the skills listed below:

Skill or Concept: I can … Rating from 1 to 5 Use basic formulas for area and volume (area of a rectangle and circle; volume of a box). Use appropriate units in calculations for length, area, and volume.

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Specific Objectives Students will understand that • they can find formulas through the Internet and reference books. • a variable can be used to represent an unknown. • using a formula requires knowing what each variable represents. • they must know the appropriate units for length, area, and volume. Students will be able to • use formulas from geometry and perform calculations that involve rates and measures to support financial decisions. • evaluate an expression.

Problem Situation: Home Improvements Bob and Carol Mazursky have purchased a home and they want to make some improvements to it. In the following few problems, you will calculate the costs of those improvements. You will use scale drawings of the house and lot to assist you. (1) Review the drawings of the house and lot (Figure 1). What does the scale mean for each drawing?

(2) The Mazurskys are expecting their first child in several months and want to get the backyard fertilized and reseeded before little Ted or Alice comes along. They found an ad for Gerry’s Green Team lawn service (see below). Gerry came to their house and said that the job would take about half a day and would cost about $600. Is Gerry’s estimate consistent with his advertisement?

Gerry’s Green Team Itemized Costs: Grass seed 4 pounds per 1,000 sq. ft. @$1.25 per pound Fertilizer 50 pounds per 12,000 sq. ft. @ $0.50 per pound Labor 4 hours @ $45 per hour

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(3) The Mazurskys want to build a 48-inch-tall chain link fence around the backyard. The fence would have two gates on either side of the house. They decide to do the work themselves. They need an inline post at least every 8 feet along the fence, a corner post at each corner, and a corner post on each side of the gates. They have a coupon they can use for the materials (shown below). The total cost will include 7.5% sales tax. Calculate the cost of the materials required to fence in the backyard.

DO IT YOURSELF SPECIAL — Chain-Link Fence

48-inch chain-link fence—$21 per yard 10% OFF 48-inch gate—$75 Inline posts—$12.50 each Corner/gate posts—$20 each SPECIAL HIGH Home Improvement—Your Fencing Specialist

(4) There is a brick grill in the backyard. Bob and Carol are going to make a concrete patio in the shape of a semicircle next to the grill. The concrete slab needs to be at least 2 inches thick. They will use 40-pound bags of premixed concrete. Each 40-pound bag makes 0.30 cubic feet of concrete and costs $6.50. How much will the materials cost, including the 7.5% tax?

Making Connections Record the important mathematical ideas from the discussion.

Further Applications (1) In 2011, the Wallow Fire burned 538,049 acres in Arizona and New Mexico.1 At the time, it was the largest wildfire in Arizona history. How does this compare with the area of the state in which you live? State your answer as a comparison such as, “The Wallow Fire was twice as large as ___” or “The Wallow Fire was one-tenth the size of ___.”List references for any information that you find to solve the problem.

(2) Dimension analysis is one way of checking whether your calculations are correct. Show your conversion factors, dimensional analysis, and calculations for the problem above. Make sure that all units cancel, leaving only the one that should be included in your answer.

1Retrieved from http://en.wikipedia.org/wiki/List_of_wildfires#North_America

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Drawings House and Lot: This scale drawing shows the rectangular lot (dark border), the house (dark shade), and the driveway (lighter shade). Figure 1

Fenced-in Backyard: The light shaded area to the rear of the house represents the backyard that is to be fenced in. The fence is to enclose the entire area, except for the area adjacent to the house. Each corner requires a “corner post” and each gate requires two corner posts. The gates are adjacent to the house. Regular posts need to be set along each side and should be no more than eight feet apart. Figure 2

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Outdoor Grill: Bob is going to add a semicircular patio adjacent to the outdoor grill in the backyard. The shaded area is to be concrete, 2-inches deep. Figure 3

Concrete Patio: Bob will add a concrete patio on the side of the house adjacent to the driveway. Figure 4

New Sod: Bob and Carol will add new sod next to the house and driveway. Figure 5

(8) 4,200 ft2

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Making Connections to the Lesson (1) Which of the following was one of the main mathematical ideas of the lesson? (i) Formulas are useful because they allow us to generalize a rule to many different situations. (ii) Formulas use variables. (iii) Geometry can be useful in home improvement projects. (iv) When using a formula, you do not really need to know what the variables mean.

(2) Give an example of a formula that was not used in this lesson that you have used in this course or elsewhere. Define all the variables in the formula.

Developing Skills and Understanding (3) The formulas for finding the area of two-dimensional geometric figures that occur in everyday use are published in reference books or available online. Use the Internet or some other reliable source to find a formula for the area of each figure. Define each variable in the formula, and label the figure with the variables to indicate the correct meaning of the variable. You may have to add to the figure to indicate all variables.

For example: Rectangle L

Area of a rectangle = L × W W Variables: L = length; W = width

(a) Parallelogram Area of a parallelogram = Variables:

(b) Triangle Area of triangle = Variables:

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(4) The formulas for finding the volume of three-dimensional geometric figures that occur in everyday use are published in reference books or available online. Use the Internet or some other reliable source to find a formula for the volume of each figure. Define each variable in the formula, and label the figure with the variables to indicate the correct meaning of the variable. Note: The volume of an object that is the same on the top and the bottom is typically found by determining the area of the base figure (two-dimensional) and “stretching” that base to the desired height.

For example: Box Volume: V = L × W × H Variables: V = volume; L = length; W = width; H = height The base figure is a rectangle with area = L × W, which is multiplied by the height (H) to get the volume of the figure.

(a) Cylinder Volume of the cylinder: Variables:

(b) Pyramid with a square base Volume of the pyramid: Variables:

(5) Refer to the figure of the box in Question 4. Which of the following would be appropriate units of measurement for the different parts of the figure. (i) Bottom edge (L), the area of the top, and the volume are all measured in inches. (ii) Bottom edge (L) is measured in square inches; the area of the top is measured in inches and the volume is measured in cubic inches. (iii) Bottom edge (L) is measured in inches; the area of the top is measured in square inches, and the volume is measured in cubic inches. (iv) Bottom edge (L), the area of the top and the volume are all measured in square inches.

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Making Connections Across the Course (6) Bob and Carol want to hire Able Refinishing to sand and refinish the dining room floor to match the floor in the living room. Able charges $2.89 per square foot to sand and refinish a hardwood floor. The dining room is rectangular and measures 17 feet 8 inches by 11 feet 8 inches. Find the area of the dining room floor, rounded up to the next square foot, and the cost of the work.

(7) After doing some work in the house, Bob and Carol want to put a concrete patio on the side of the house to keep people from tracking mud inside. They decide to hire someone to do the work. The dimensions of the rectangular patio are 23 feet 9 inches by 10 feet 1 inch. The patio will need to be at least 4 inches deep. Rachel’s Ready-Mix bid on the job based on the information provided. • Calculate the volume of concrete needed, in cubic yards, adding 5% to allow for spillage and an uneven base, and round up to the nearest 1/4 cubic yard. • The delivered cost of the concrete is “$150 per yard (in increments of 1/4-yard) plus a $50 surcharge for orders less than four yards.” Find the total cost of the job. Select the best answer from the options below: Figure 4: Concrete patio (i) Order 3.25 yd3; total cost is $537.50 (ii) Order 1.25 yd3; total cost is $237.50 (iii) Order 3.75 yd3; total cost is $612.50 (iv) Order 11.25 yd3; total cost is $1,687.50

Preparing for the Next Lesson (3.2.2) Use the formulas you found earlier in the lesson to answer Questions 8 and 9. (8) A courtyard is shaped like a trapezoid as shown below. Find the area of the courtyard. 50 ft

60 ft

90 ft

(9) Use the formula for the volume of a cylinder to find the volume of a water cistern with a radius of 10 feet and a height of 40 feet.

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The following information will be used in Lesson 3.2.2.

A subscript is a symbol that is written in small type below a variable in regular type. For example: P0 is read as P sub-zero. Subscripts are used to distinguish between variables that represent similar quantities. For example, if you were working with a problem in which there were different prices over time, you might want to use P to represent all of those prices, but you also want to be able to tell the difference between them. So you could use P0 for the initial price, then P1 (P sub-one) for the second price. The subscript is only a label. It is not an operation. The grade of a road is important information for drivers of large trucks in mountainous terrain. If a trucker begins to travel too fast going downhill, then it is possible for brakes to fail. Of course, as the driver of a car, you might be frustrated with a truck that is traveling up a hill with a steep grade, especially if you cannot pass. Runners and bicyclists who compete in hilly terrain, also consider the grade of the hill in predicting their stamina and speed. The grade of the road quantifies the rate of increase (or decrease) in height over some horizontal span. The grade is written as a fraction with the numerator being the change in height (vertically) and the denominator being the change in distance (horizontally). Typically, the grade is reported as a positive value (even though mathematically “uphill” is positive and “downhill” is negative). In practice, if one car is traveling uphill on a road, then oncoming cars are traveling downhill, so it would be confusing to report the sign as + or –. Notice that the units will divide-out, leaving a number that is dimensionless. That number is then written as a percent. Example: Determine the grade of a road that decreases 72 feet in height over a horizontal distance of 600 feet.

Answer:

(10) You are expected to be able to do the following things for the next class. Rate how confident you are on a scale of 1–5 (1 = not confident and 5 = very confident). Before beginning Lesson 3.2.2, you should understand the concepts and demonstrate the skills listed below:

Skill or Concept: I can… Rating from 1 to 5 Evaluate expressions containing parentheses and exponents of two. Understand dimensional analysis and use of units in calculations including squared units. Understand the use of subscripts.

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Specific Objectives Students will understand that • a variable is a symbol that is used in algebra to represent a quantity that can change. • many variables can be present in a scenario or experiment, but some can be held fixed in order to analyze the effect that the change in one variable has on another. Students will be able to • evaluate an expression. • informally describe the change in one variable as another variable changes.

Problem Situation: Calculating the Braking Distance of a Car Experts agree that driving defensively saves lives. Knowing how far it takes your vehicle to come to a complete stop is one aspect of safe driving. For example, when you are going only 45 miles per hour (mph), you are traveling about 66 feet every second. This means that to be a safe driver, you need to drive “in front of you” (i.e., you need to know what is going on ahead of you so that you can react accordingly). In this lesson, you will learn more about what it takes to drive defensively by examining the braking distance of a vehicle. Braking distance is the distance a car travels in the time between when the brake is applied and when it comes to a full stop.

(1) What are some variables that might affect the braking distance of a car?

(2) For this lesson, you will examine how speed affects braking distance. In the OCE (Lesson 3.2.2), you will consider the effects of other variables. Discuss with your group how you think the speed will affect the braking distance. Think of some specific questions you might ask. For example, what happens to the braking distance if you double the speed? Would the answer be different for very low speeds or very high speeds?

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The formula for the braking distance of a car is where

V0 = initial velocity of the car (feet per second). That is, the velocity of the car when the brakes were applied. The subscript, zero, is used customarily to represent time equaling zero. So, V0 is the velocity when t = 0. d = braking distance (feet) G = roadway grade (percent written in decimal form). Note: There are no units for this variable, as explained in the previous OCE. f = coefficient of friction between the tires and the roadway (0 < f < 1). Note: Good tires on good pavement provides a coefficient of friction of about 0.8 to 0.85. Constant: g = acceleration due to gravity (32.2 ft/sec2) Since g is a constant, this formula has four variables. To understand the relationships between the variables, you will hold two of them fixed. That leaves you with two variables—one that will affect the other. Since you want to see how speed affects braking distance, you will hold the other two variables, f and G, fixed.

(3) Let f = 0.8 and G = 0.05. Write a simplified form of the formula using these values for the two variables.

(4) How can you verify your predictions about the relationship between velocity and braking distance?

(5) Now you will explore the question(s) developed by the class. (a) Record the question(s). (b) Create a strategy for exploring the question with your group. Record your strategy. (c) Use your strategy to answer the question. Write a complete statement about your results.

You have now used several different formulas in this course. In Lesson 3.2.1, you used common geometric formulas for area and volume. You had probably seen those formulas before. In this lesson, you used a formula that was more complex and probably less familiar to you. Almost every field has specialized formulas, but they all depend on three basic skills: • Understanding and knowing how to use variables, including the use of subscripts. • Understanding and knowing how to use the order of operations. • Understanding and knowing how to use units, including dimensional analysis. With these three skills, you will be able to use formulas in any field.

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Making Connections Record the important mathematical ideas from the discussion.

Further Applications Find your reaction time! Ask a friend to help with this experiment. Have him or her hold a ruler (or yard stick) vertically while you position your thumb and first finger about 1 inch apart and on either side of the bottom of the ruler. Ask your friend to drop the ruler without warning while you attempt to catch it with your thumb and finger as quickly as possible. Take note of where you catch the ruler (the distance from the bottom of the ruler). Repeat the experiment three times and record your results. Find the average distance of the three trials. Then repeat the experiment again, using your other hand. Find the average distance for both hands. Use the following formula where d is the average distance (in feet) for both hands.

Distance R Distance L Trial (inches) (inches)

Average

Average of both hands

Note: The reaction time to catch a ruler with your fingers is going to be about a third of the time needed to apply your brakes.

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Making Connections to the Lesson (1) Which of the following was one of the main mathematical ideas of the lesson? (i) When using variables, it is important to know what they represent and what units should be used with them. (ii) When using variables, it is only important to know what numbers to substitute in for them. (iii) A subscript is a label on a variable. (iv) Braking distance is affected by many factors. (2) How did you use order of operations in this lesson?

Developing Skills and Understanding Some Notes About Mathematical Terminology Formulas are a type of an algebraic equation. You have probably seen algebraic equations such as “y = x + 3” in previous math classes. Each side of the equation is called an algebraic expression. So “x + 3” is an expression and “y” is an expression. An equation is a statement that two expressions are equal. The purpose of such an equation is to define a sequence of calculations using a shortcut language. In this example, the equation “y = x + 3” means: (1) Start with x. (2) Add three to x. (3) The result is y. The word formula is usually used to express important and nonchanging relationships, especially in contexts such as science, business, medicine, sports, or statistics. For example the area of a rectangle, A = L  W, is a formula because the relationship between area and the length and width of a rectangle is always the same. It is also an equation, but that wording is less common. Suppose you had a situation in which you make $12 per hour. This relationship could be written algebraically as P = 12h where P is your pay in dollars, and h is the number of hours you work. This would be called an equation instead of a formula because if you got a raise, the relationship would change. You also might call the equation a model because it models a situation mathematically. In other math classes, you might see problems like the one shown below. Each line represents a simplification of the line above. Evaluate the expression 3x2+ 2y if x = 3 and y = 5. 3(3)2 +2(5) 3(9) + 10 27 + 10 37

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These types of problems are not included in this course because the focus is on using mathematics in a meaningful context. However, you should recognize that this type of expression uses the same skills that you used when working with formulas. Units are not involved, so the first step is to recognize that the values can replace the variables and the order of operations is applied to simplify the form of the problem.

(3) In the lesson, you investigated the relationship between velocity and braking distance. You will now investigate the relationship between the coefficient of friction and the braking distance.

Recall that the formula for the braking distance of a car is

(a) Define each variable including its units if applicable. State if there are no units.

(i) V0 (ii) d (iii) G (iv) f (v) g (b) Which of the variables listed in Part (a) represents a constant? (c) To investigate the relationship between the coefficient of friction and the braking distance, you need to hold the other variables fixed. Let G = 0.02. Which of the following is a correct interpretation of the value G = 0.02? (i) The grade of a road is 0.02%, which is a vertical increase of 0.02 feet for every 1 foot of horizontal increase. (ii) The grade of a road is 0.02%, which is a vertical increase of 0.02 feet for every 100 feet of horizontal increase. (iii) The grade of a road is 2%, which is a vertical increase of 2 feet for every 1 foot of horizontal increase. (iv) The grade of a road is 2%, which is a vertical increase of 2 feet for every 100 feet of horizontal increase.

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(d) Let V0 = 72 mph and use the value of G in Part (c). Which of the following expressions represents the simplified form of the formula using these values?

11,151.36 (i) d = ft (64.4 f + 0.02) 11,151.36 (ii) d = + 8,657.89 ft 64.4 f 11,151.36 (iii) d = ft (64.4 f + 1.288)

(4) Use the formula you found in Question 3d. (a) Complete the table of values for f and d (in feet). Use the values of f given in the table. Perform one of the calculations f d (feet) on paper showing the units. You may then use technology to 0.30 complete the table. 0.50 (b) The four values of f correspond to the coefficient of friction for 0.70 four road conditions: an icy road, a very good road with great 0.90 tires, an asphalt road with worn tires, and a wet road with fair tires. Match the coefficients of friction to the appropriate conditions by looking at the braking distance required. (i) Icy road, f = (ii) Very good road with great tires, f = (iii) Medium quality road with fair tires, f = (iv) Wet road with fair tires, f = (c) The coefficient of friction (f) is increasing at a constant rate, since each value is 0.2 more than the previous value. How is d changing as f increases at a constant rate? (i) The stopping distance is decreasing. (ii) The stopping distance is constant. (iii) The stopping distance is increasing.

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(5) Lenders such as banks, credit unions, and mortgage companies make loans. The person receiving the loan usually pays the loan off in small payments over a long period of time. The lender earns money by charging interest, which is based on a percentage of the amount that is borrowed. There are different types of interest. Car loans are usually calculated using the formula for simple interest. The total amount repaid is based on the value of the original loan, called the principal, and the interest. The formula for the total dollars needed to repay the loan, with interest, is found using the formula

where • A is the amount (total principal plus interest) required to repay the loan • P is the amount borrowed, the principal • r is the annual interest rate, quoted as a percent, but used as a decimal • t is the time, in years (has to be a full year, so six months would be 1/2 year) Suppose you get a loan of $5,000 at an annual interest rate of 4.25%. (a) Use the given information to write the formula for the total amount to be repaid in t years. (b) Make a table of values that shows the payoff amount (A) for 4 months, 6 months, 1 year, 3 years, and 6 years. t (years) A ($) 0 4 months

6 months 1 year 3 years 6 years

Making Connections Across the Course (6) The tuition at a daycare center is based on family income. A reduced tuition has a subsidy. There are three levels of tuition: • Full subsidy—the family does not pay any tuition • Partial subsidy—the family pays part of the tuition • No subsidy—the family pays the full tuition The data for the daycare center for each age level is given below. Answer the questions below. Round to the nearest whole percent.

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Full Subsidy Partial Subsidy No Subsidy Total 3 year-olds 17 13 8 4 year-olds 22 14 15 5 year-olds 15 16 11 Total

(a) Complete the last column and last row. (b) What percentage of 3-year-olds received a full or partial subsidy? (c) What percentage of those who receive no subsidy are 5 years old? (d) What percentage of the students are 3 years old? (e) The daycare center’s funding for one term comes from federal funding for the subsidy and the tuition paid by families based on the formula below. Find the funding for the center. Funding = 1,530F + 1,750P + 1,875N where F = number of children receiving a full subsidy P = number of children receiving a partial subsidy N = number of children receiving no subsidy

(7) In Lesson 1.2.2, you used a formula that was written as steps in a form to calculate self-employment taxes for different people. Formulas are often written in this way. One example is the Expected Family Contribution (EFC) Formula, which is used to determine if a college student is eligible for financial aid. The EFC has many different sections that each use different calculations. One section of the 2010–11 form is shown below.

Student’s Contribution from Assets Cash, savings, and checking Net worth of investments If negative, enter zero + Net worth of business and/or investment farm If negative, enter zero + Net worth (sum of lines 45 through 47) Assessment rate x 0.20 Students Contribution from Assets =

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(a) Calculate the Student’s Contribution from Assets given the following information. Cash: $500 Investments: loss of $2,000 Savings: $3,240 Business: $0 Checking: $732 (b) Write a formula that summarizes the calculation in this form using the following variables: C = Cash including savings and checking

NI = Net worth of investment

NB = Net worth of business or farm S = Student’s contribution from assets

Preparing for the Next Lesson (3.2.3) (8) It is often necessary to change money into different currencies when traveling or doing business in different countries. Exchange rates, which change constantly, are used to make these conversions. Consider the following situation. Sonia is traveling to Mexico. Answer the following questions about her trip. (a) Sonia starts with $100 in cash and changes it into pesos at a rate of 1 peso = $0.075. Which of the following would be a correct way to use dimensional analysis to make this conversion?

(i)

(ii)

(iii)

(iv)

(b) Sonia spends 984 pesos while in Mexico. She converts her remaining pesos back into dollars. How many dollars does she have? Round to the nearest dollar. (c) While traveling in Mexico, Sonia wanted to be able to estimate how much things cost in dollars. Explain a strategy that could be used to estimate the conversion.

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(9) You are expected to be able to do the following things for the next class. Rate how confident you are on a scale of 1–5 (1 = not confident and 5 = very confident). Before beginning Lesson 3.2.3, you should understand the concepts and demonstrate the skills listed below:

Skill or Concept: I can … Rating from 1 to 5 Use dimensional analysis to make conversions. Understand the use of variables in formulas. Evaluate expressions and formulas.

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Specific Objectives Students will understand that • pictographs can be misleading because areas and heights of figures do not increase proportionally. Students will be able to • solve dimensional analysis scenarios involving multiple conversion factors. • analyze misrepresentations in graphs related to area and volume. • evaluate formulas and use the results to make a decision.

Problem Situation: Analyzing Data on Apple Juice Imports In the United States, in recent years, there has been an increased consumption of apple juice from foreign countries. Apple juice has been shown to have many health benefits1 and because imported apple juice is potentially cheaper, and therefore available to a larger percentage of the population, importing it can be seen as a positive thing. However, importing food from other countries also causes some concerns, including an increased reliance on food from other countries, a loss of control over the quality of imported food, and a reduction in business for U.S. farmers. You will examine some of these issues below. (1) The graph below is similar to ones commonly seen in media reports. It is called a pictograph because it uses pictures (instead of bars) to represent quantitative changes. Using the data above, this pictograph was created to show the changes in apple juice imports over the 10-year time period from 1998 to 2008.2

1Retrieved from http://en.wikipedia.org/wiki/Apple_juice#Health_benefits 2Retrieved from http://usda.mannlib.cornell.edu/MannUsda/viewDocumentInfo.do?documentID=1825

1 254 Quantway Student Handout November 18, 2011 (Version 1.0) Lesson 3.2.3: Comparing Apples to Apples Theme: Citizenship

(a) Based on the graph, would you say that apple juice imports grew a little, some, or a lot over this time period? What are you looking at when you make this comparison? (b) People who study how to make visual displays of data (like graphs) are called data scientists. Data scientists caution about the use of pictographs because if they are not carefully constructed, they can be misleading. In the graph above, for example, it is unclear if you should compare the height of the apples, the area, or the volume. (In this case, the volume of a three- dimensional apple represented by the graphic.) Fill in the table below to see the comparison of these different comparisons. Assume that the area of an apple is approximately the area of a circle and the volume is approximately the volume of a sphere. The area of a circle is given by the equation A = πr2 and the volume of a 4 sphere is given by V = πr 3 (where r is the radius). 3

Height of Approximate Approximate Area Approximate Volume Apple Radius of Apple Graphic of Apple Graphic 1998 little apple 2008 big apple Ratio: (2nd value/1st value)

2 255 Quantway Student Handout November 18, 2011 (Version 1.0) Lesson 3.2.3: Comparing Apples to Apples Theme: Citizenship

(d) How could you make a graph that would portray the data more accurately?

(2) An American company has been criticized for using imported apple juice. The company’s public relations department is asked to prepare a press release defending the use of imports. As a member of this department, you are asked to calculate how much it costs to pick enough apples to make 1 gallon of apple juice in China and in the United States. You find the following information: • In a 2007 article, The New York Times reported, “China’s advantage is its cheap labor. A picker makes about 28 cents an hour, or $2 a day, according to the U.S. Apple Association. In 2005, workers in Pennsylvania made about $9 to $10 per hour, and those in Washington State about $14 per hour, the association said.”3 • It takes 36 apples to create 1 gallon of juice.4 • One bushel of apples contains about 126 medium apples.5 • An experienced apple picker can harvest about 2-1/2 bushels of apples per hour.6 (3) Select one of the two prompts below. Write a paragraph presenting your argument and supporting it with specific quantitative information. • Use the information in Question 1 to make the argument that the United States is importing too much apple juice. Include your reasons explaining why this is a problem. • Use the information in Question 2 to make the argument that it is good that the United States is importing apple juice. Include your reasons explaining how this benefits the country.

Making Connections Record the important mathematical ideas from the discussion.

Further Applications (1) Total U.S. apple juice consumption for the marketing years 1997 and 2007 were 422.4 million gallons and 686.4 million gallons, respectfully. Use this information and the data from the in-class exercises to create a new graph of the import data, by changing the vertical axis to “percentage of total apple juice consumption.” Because pictographs can be eye-catching and make the data memorable, you can use a pictograph in your graph, but choose one that preserves the integrity of the data.

3Retrieved from www.nytimes.com/2007/06/25/business/worldbusiness/25iht-apples.1.6312540.html 4Retrieved from www.pickyourown.org/applepicking.htm 5Retrieved from www.applejournal.com/ref.htm 6Retrieved from www.uky.edu/Ag/CDBREC/introsheets/apples.pdf

3 256 Quantway Out-of-Class Experience November 18, 2011 (Version 1.0) Lesson 3.2.3: Comparing Apples to Apples Theme: Citizenship

Making Connections to the Lesson (1) Which of the following was one of the main mathematical ideas of the lesson? (i) Pictographs are a type of graph in which pictures are used to represent quantities. (ii) Units are important in dimensional analysis, but do not matter in using formulas except when giving the answer to a problem. (iii) Units are important in many mathematical skills including graphs, dimensional analysis, and formulas. (iv) Imports of apple juice have increased due to the high cost of labor in the United States.

(2) How was the work you did with graphs in this lesson similar to what you learned about graphs in Lesson 2.1.4?

Developing Skills and Understanding (3) Over the last decade, the use of bottled water has increased dramatically in the United States and around the world. In 1999, the annual U.S. consumption of bottled water was 16.2 gallons per capita. In 2007, this had increased to 29 gallons.1 The graphic below is designed to illustrate this increase.

US Bottled Water Consumption

1999 16.2 gallons per capita 2007 29.0 gallons per capita

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The table below gives the dimensions of the figures. Complete the table as instructed. (a) Approximate the area of each figure. To do this, think of the rectangle that contains the figure with the given dimensions. Calculate the area of the rectangle. (b) The figures give the illusion of three dimensions. Using the given dimensions, approximate the volume of the cylinder represented by each figure. (If you do not know the formula for a cylinder, you should look it up.) (c) Calculate the ratios of the large bottle to the small bottle. For the dimensions, find the ratio of the heights. (d) Write a statement about whether the graphic is an accurate depiction of the data. Explain your answer.

Dimensions Actual data Area of rectangle Volume of cylinder (height x width) Small bottle 16.2 gallons/ capita 1.5 x 0.59 inches Large bottle 29.0 gallons/ capita 2.55 x 0.99 inches Ratio Ratio of heights: Large/Small

(4) In addition to paying attention to distortions of data, data scientists also rely on artistic ideas to design visually appealing graphs. One theory they rely on is the golden ratio for rectangles. The ratio is given by the formula l g = w where l is the length of the longest side of the rectangle and w is the length of the shortest side of the rectangle. When g is close to 1.6, the rectangle is thought to be visually appealing.2 Identify if each item listed below has dimensions that match the golden ratio. You will have to look up the dimensions of some of the items. (a) Small water bottle graphic from Question 3. (b) Large water bottle graphic from Question 3. (c) Credit card (d) Football field (e) Sheet of notebook paper

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Making Connections Across the Course (5) The total U.S. apple juice consumption for the marketing years 1997 and 2007 were 422.4 million gallons and 686.4 million gallons, respectfully. (a) Calculate the average consumption of apple juice in fluid ounces per person per week for 1997 and 2007. Use the following U.S. population figures: 266.5 million people in 1997 and 302 million people in 2007. (b) Using your calculations, describe how apple juice consumption changed over this 10-year period.

When comparing quantities that change over time, you usually compute the difference in values according to: New value – Old value

If the average price of a gallon of gasoline in 2009 was $2.92 and the average price in 2011 was $3.59, one might compare these two prices by computing $3.59 – $2.92 = $0.67. Since this number is positive, you can say that the average price of gasoline has increased 67 cents in two years, which is an annual change of 33.5 cents (67 cents divided by 2 years). If the average price for a loaf of bread was $2.90 in 2007 and $2.50 in 2009, then the change in this basic item of food would be –$0.40. The negative sign tells us that the price of bread has decreased over this two-year period.

(6) Write a sentence describing the annual change in the average price of a loaf of bread from 2007 to 2009.

(7) The total change in the price of a basic iPod from 2001 to 2007 was –$150.3 (a) The price in 2007 was $249. What was the price in 2001? (b) Write a sentence describing the total change in the price of an iPod in this period.

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(8) The following pie charts represent how a typical household budget might be broken into categories.

(a) For Chart A, which of the following statements best describes the comparison of the categories for Food and Other? (i) Food is a smaller percentage than Other. (ii) Food is a greater percentage than Other. (iii) Food is about the same percentage as Other.

(b) For Chart B, which of the following statements best describes the comparison of the categories for Food and Other? (i) Food is a smaller percentage than Other. (ii) Food is a greater percentage than Other. (iii) Food is about the same percentage as Other.

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(c) The two charts are actually created from the same data (shown below). Do they both accurately represent these data? Why or why not?

Housing 40% Food 20% Medical 5% Paying debt 10% Other 25%

Preparing for Module 4 As in all OCE, the next section will help you prepare for the next lesson, 3.2.4. In addition to that, this OCE is used to start preparing you for Module 4. In the next module, you will make graphs on a coordinate plane like the one shown below. Over the next few OCE assignments, you will learn about using a coordinate plane and practice how to graph points. This material is spread over several OCEs to give you time to be fully prepared for Module 4.

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You will begin with some vocabulary. A coordinate plane has two axes that measure distance in two dimensions. The horizontal axis goes from left to right. In previous classes, you may have called this the x-axis. The vertical axis goes up and down. This is sometimes called the y-axis. The axes are two number lines that create a grid on the coordinate plane. Note: Axis is singular and axes is plural. The point at which the two axes intersect or cross is called the origin. This point represents 0 for both axes. To the left of this point, the horizontal axis is negative; to the right it is positive. Below the origin, the vertical axis is negative; above the origin it is positive. You can see this in the numbers along each axis above. These numbers are called the scale. Each location or point on the coordinate plane is defined by an ordered pair. You can think of this as the address of a point. Ordered pairs are written in a set of parentheses ( ). An ordered pair must contain two numbers. The first number is the distance and direction going left or right from the origin and the second number is the distance and direction going up or down. The ordered pair for the origin is (0, 0). Follow these steps to find the point represented by the ordered pair (2, 3): Step 1 First, think about the “address” of the point. If this were a street address, the ordered pair tells you to walk 2 blocks horizontally in the positive direction (right) and then walk 3 units vertically in the positive direction (up). Step 2 Start at the origin. Go 2 units to the right because this is the positive side of the horizontal axis. Step 3 Go 3 units up.

Point A on the graph above is the point (2, 3). A few other examples from the graph are given below: Point B: (–3, 1) Point E: (0, 1) Point F: (4, 0) (9) Write the ordered pairs for the following points on the graph. Point C: Point D: Point G: Point H: Point I: Point J:

Preparing for the Next Lesson (3.2.4) An equation is a statement of equality, meaning that it tells you that two expressions are equal to each other. An equation can be as simple as 3 = 3 or it can have complicated expressions with multiple terms on one or both sides of the equal sign. One of the most important things to remember is that if the value of one side of an equation is changed, then it is no longer an equation because the two sides are no longer equal. If you need to change the value of one side and you want to keep the equation true, you must change the value of the other side in the same way.

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(10) Each of the following examples starts with the equation 3 = 3. Then a new operation is performed to one or both expressions. The new operations are shown in bold. If the new operations maintain the statement of equality, put an equal sign (=) in the blank. If the new operations do not maintain the statement of equality, put a not-equal sign (≠) in the blank. (a) 3 = 3 (b) 3 = 3 3 + 2 ____ 3 + 2 2 + 3 ____ 3 + 2

(e) 3 = 3 (f) 3 = 3 3 ÷ 2 ____ 3 × 2 6 – 3 ____ 3 – 6

It is important to note that you are talking about changes to the value of an expression. Remember that there are multiple ways to write expressions without changing their value. For example, if you change to , you have not changed the value because you have changed the fraction into an equivalent form.

(11) Each of the following examples start with the equation 2x + 3x + 1 = 11. Then an operation, shown in bold, is performed on one or both expressions. If the new expressions maintain the statement of equality, put an equal sign (=) in the blank. If the new operations do not maintain the statement of equality, put a not-equal sign (≠) in the blank.

(a) 2x + 3x + 1 = 11 (b) 2x + 3x + 1 = 11 5x + 1 ____ 11 x + x + 3x + 1 ____ 11

(e) 2x + 3x + 1 = 11 2x + 3x + 1 – 1 ____ 11 – 1

The equation in Question 11 contains a variable that represents an unknown value. In an equation like this, there may be one or more values that can be substituted in for the variable to make a true equation. This is called a solution to an equation.

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(12) Determine if each of the following is a solution to the equation 2x + 3x + 1 = 11. (i) x = –2 (ii) x = 2 (iii) x = 0

(13) You are expected to be able to do the following things for the next class. Rate how confident you are on a scale of 1–5 (1 = not confident and 5 = very confident). Before beginning Lesson 3.2.4, you should understand the concepts and demonstrate the skills listed below:

Skill or Concept: I can … Rating from 1 to 5 Understand the use of variables in mathematical equations. Substitute a value for a variable in a mathematical equation and simplify the equation. Understand that an equation is a statement of equality.

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Specific Objectives Students will understand that • addition/subtraction and multiplication/division are inverse operations. • solving for a variable includes isolating it by “undoing” the actions to it. Students will be able to • solve for a variable in a linear equation. • explicitly write out order of operations to evaluate a given equation.

Problem Situation: Calculating Blood Alcohol Content Blood alcohol content (BAC) is a measurement of how much alcohol is in someone’s blood. It is usually measured as a percentage. So, a BAC of 0.3% is three-tenths of 1%. That is, there are 3 grams of alcohol for every 1,000 grams of blood. A BAC of 0.05 impairs reasoning and the ability to concentrate. A BAC of 0.30% can lead to a blackout, shortness of breath, and loss of bladder control. In most states, the legal limit for driving is a BAC of 0.08%.1 BAC is usually determined by a breathalyzer, urinalysis, or blood test. However, Swedish physician, E.M.P. Widmark developed the following equation for estimating an individual’s BAC. This formula is widely used by forensic scientists:2

where B = percentage of BAC N = number of “standard drinks” (A standard drink is one 12-ounce beer, one 5-ounce glass of wine, or one 1.5-ounce shot of liquor.) N should be at least 1. W = weight in pounds g = gender constant, 0.68 for men and 0.55 for women t = number of hours since the first drink

(1) Looking at the equation, discuss why the items on the right of the equation make sense in calculating BAC.

(2) Consider the case of a male student who has three beers and weighs 120 pounds. Simplify the equation as much as possible for this case. What variables are still unknown in the equation?

1Retrieved from http://en.wikipedia.org/wiki/Blood_alcohol_content. 2Retrieved from www.icadts2007.org/print/108widmarksequation.pdf

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(3) Using your simplified equation, find the estimated BAC for this student one, three, and five hours after his first drink. What patterns do you notice in the data?

(4) Discuss with your group how you arrived at the BAC values mathematically. For example, did you multiply, add, subtract, etc., and what did you do first? Outline the steps that you took to get from the time to the BAC.

(5) How long will it take for this student’s BAC to be 0.08, the legal limit? How long will it take for the alcohol to be completely metabolized resulting in a BAC of 0.0?

(6) A female student, weighing 110 pounds, plans on going home in two hours. Using the formula above, the simplified equation for this case is

(a) Compare her BAC for one glass of wine versus three glasses of wine at the time she will leave. (b) In this scenario, determine how many drinks she can have so that her BAC remains less than 0.08.

Making Connections Record the important mathematical ideas from the discussion.

Further Applications (1) Solve the following equation for the values given in Parts (a) and (b). In each case, write the steps you used as you did in Question 4 from the lesson. y = −4x – 2 (a) Solve for y if x = −3. Write your steps. (b) Solve for x if y = −3. Write your steps.

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Making Connections to the Lesson (1) Which of the following was one of the main mathematical ideas of the lesson? (i) Blood Alcohol Content (BAC) is affected by many different variables. (ii) Multiplication undoes division. (iii) The way you solve an equation that contains addition has nothing to do with the way to solve a different equation that contains subtraction. (iv) An equation is a statement saying that two expressions are equal so an operation that changes the value of one side must also be done to the other side of the equation.

(2) How is the idea of keeping an equation balanced similar to finding equivalent fractions?

Developing Skills and Understanding (3) Find the solution to each of the following: (a) 3x + 5 = 14 (b) 6x – 5 = 10 (c) 2x – 1 = –7 (d)

(4) Recall that Blood Alcohol Content (BAC) is a measurement of how much alcohol is in someone’s blood as a percentage. However, police and the public typically omit the language for % when quoting the BAC and simply say, “BAC is 0.04.” Write an interpretation of what each of the following BAC values means in terms of how much alcohol is in the bloodstream in the form of the amount of alcohol per 1,000 grams of blood. You may want to refer back to the example in the lesson. (a) BAC = 0.1 (b) BAC = 0.02

(5) Use information from the website http://en.wikipedia.org/wiki/Blood_alcohol_content to list effects on an individual having a BAC as given. Give at least three effects for each. (a) BAC = 0.1 (b) BAC = 0.5 (c) BAC = 0.05

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Use the Widmark Equation, to solve Questions 6–8. Recall that g = 0.68 for men and g = 0.55 for women.

(6) A male student had five glasses of wine at a party. He weighs 160 pounds. How long it will take before his BAC is 0.08? (i) 3.33 hours (ii) 1.31 hours (iii) –3.33 hours

(7) Look up the BAC limit for the state in which you live. (a) How long should you wait after consuming two margaritas to ensure that your BAC is less than the legal limit for your state? (b) If you drink alcohol over a period of 5 hours, how many drinks would you be able to consume and still ensure that your BAC is less than the legal limit for your state?

(8) The percentage of Americans who are retired has been increasing over the last decade. This is causing some concern because health care, social security, and other costs will be the responsibility of a smaller group of people. That is, as the percentage of retired people increases, the percentage of working-age people decreases. The following model predicts the percentage of retired people based on demographic data:1

where R is the percentage (as a decimal) of Americans who are retired in the year t. Use this model to complete the table below.

Year % of Retired People 10%

15%

20%

1Retrieved from www.census.gov/population/www/projections/2008projections.html

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Making Connections Across the Course (9) Crown molding is a decorative trim installed over the joint between the walls of a room and the ceiling. It is similar to a baseboard used on the bottom joint between the walls and the floor, but there are no gaps, since doors do not extend to the full ceiling height. (If you are not sure that you understand the idea, do an Internet search to find an example picture of crown molding). Andy intends to install crown molding around the four sides of the dining room. The dining room ceiling is a rectangle with dimensions 14 feet 9 inches by 13 feet. The crown molding is sold in eight-foot lengths that cost $24 for each 8-foot piece. He decides to purchase enough to allow for 10% waste due to possible loss in the corners. (a) What is the perimeter of the dining room? Perimeter is distance around the room. (b) How many 8-foot boards are needed? (c) If sales tax is 8-1/4%, then what is the total cost?

(10) For the following questions, you will need the formula for the perimeter of a rectangle. You can write your own or look one up. (a) Formula: Variables: (b) Andy’s house is on a large lot. He got 100 yards of chain-link fence on sale. He wants to use all of the material to fence in an area in his backyard. He can only make the fenced area 60 feet wide and he wants it to be as long as possible. What is the longest length possible for the sides?

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(11) Now, you will return to graphing on a coordinate plane in preparation for Module 4. Label the following items on the coordinate plane given below. For the points, place a dot at the location of the point and label it with the ordered pair. (a) Horizontal axis (b) Vertical axis (c) (–2, 4) (d) (2, –4) (e) (–4, 2) (f) (0, 3) (g) (3, 0)

(h) ( , 1)

(i) (3.2, 3.7)

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Preparing for the Next Lesson (3.2.5) (12) Ben has $75 in his savings account. He plans to deposit $35 per week to build his account balance. (a) Complete the following equation to represent the amount of money (A) Ben will have in his account after any number of weeks. A = (b) What does your variable represent in this problem? (c) Which of the following values could be the value of the variable in this context? (i) 4.2 (ii) 3 (iii) 18 (iv) –5 (d) Ben wants to use his savings to buy a computer for $740. Use your algebraic expression to determine the number of weeks it will take him to save enough money to buy the computer. The dimensions of a figure can be written as a ratio. The rectangle below has a length of 10 inches and a width of 3 inches.

width = 3 in.

length = 10 in. 10 You can say that the ratio of the length to width is 10:3 or . 3 3 It is also correct to say that the ratio of the width to length is 3:10 or . 10 The important thing is to be consistent once you have set up your ratio.

20 You have learned previously that a fraction can be written in many equivalent forms (i.e., = ). 6 However, a rectangle with a length of 20 inches and a width of 6 inches is obviously not the same as the first rectangle. width = 3 in. width = 6 in.

length = 10 in.

length = 20 in.

While these figures are not equivalent, they are proportional to each other because their dimensions have the same ratio.

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(13) Use the figure below to answer the following questions. 5 ft

9 ft

(a) Write the ratio of the dimensions of the cylinder shown above in the form of diameter to height. (b) Give the dimensions of a cylinder that would be proportional to the one shown. Diameter: Height:

(14) Which of the following fractions has a ratio of 4:3? There may be more than one correct answer.

24 (i) 18

(ii)

(iii)

(iv)

(v)

(15) You are expected to be able to do the following things for the next class. Rate how confident you are on a scale of 1–5 (1 = not confident and 5 = very confident). Before beginning Lesson 3.2.5, you should understand the concepts and demonstrate the skills listed below:

Skill or Concept: I can … Rating from 1 to 5 Interpret the meaning of ratios including when written as fractions. Understand the use of a variable to represent an unknown. Solve a two-step equation such as 2x + 9 = 13.

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Specific Objectives Students will understand that • proportional relationships are based on a constant ratio. • rules for solving equations can be applied in unfamiliar situations. Students will be able to • set up a proportion based on a contextual situation. • solve a proportion using algebraic methods.

Problem Situation: Proportions in Artwork Many professionals such as graphic artists, architects, and engineers work with objects that are enlarged or shrunk. It is usually important that the objects have the same appearance despite the change in size. For example, a business logo on a billboard needs to look the same as a logo on a coffee mug. In this lesson, you will explore the mathematics behind these changes in size. Your instructor will start the lesson with a demonstration.

(1) Suppose you were given the three tables showing these dimensions without seeing the graphics. How could you tell which changes were proportional and which were not? Remember that, in a proportional relationship, the image is not distorted.

(2) You are a graphic artist hired to make a billboard for a college. The original logo is inches (width)

by inches (length). You need to enlarge it to a length of 6 feet. How wide will the enlarged version be?

(3) In Question 2, you could have used the following proportion to represent the relationship between the original and enlarged objects. Could this proportion be written in other ways?

(4) Suppose you had set up the following proportion to solve the original problem in Question 2. What steps would you use to solve the equation?

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Solve each equation. Round to the nearest tenth.

(5)

(6)

(7) Many small engines for saws, motorcycles, and utility tractors require a mixture of oil and gas. If an engine requires 20 ounces of oil for 5 gallons of gas, how much oil would be needed for 8 gallons of gas?

Making Connections Record the important mathematical ideas from the discussion.

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Further Applications (1) You have probably watched movies on TV in a “letterbox” format. This means that there is a dark band above and below the image. This format is used to compensate for the difference between the dimensions of a movie screen compared to a TV screen. The following information is excerpted from Widescreen.org.1 Since 1955, most movies were (and are) filmed in a process where the width of the visual frame is between 1.85 to 2.4 times greater than the height. This means that for every inch of visual height, the frame as projected on the screen is between 1.85 to 2.4 times as wide. This results in a panoramic view that when used properly can add a greater breadth and perception of the environment and mood of a movie. This formula is called an “aspect ratio.” A movie that is 1.85 times wider than it is high has an aspect ratio of 1.85:1. Similarly, a movie that is 2.35 times wider than it is high has an aspect ratio of 2.35:1. Modern televisions come in two aspect ratios—1.33:1 (or 4:3), which has been the standard since television became popular—or 1.77:1 (more commonly known as 16:9), which is quickly becoming the new standard. However, neither of these aspect ratios is as wide as the vast majority of modern movies, most of which are either 1.85:1 or 2.35:1. “When you watch a movie on your television screen, you’re not necessarily seeing it the way it was originally intended. As a director, when I set up a shot and say that there are two people in the frame, with the wide screen, I can hold both with one person on each end of the frame. When that shot is condensed to fit on your TV tube, you can't hold both [actors] … and the intent of the scene is sometimes changed as a result.” —Leonard Nimoy, Commentary for the Director's Edition of Star Trek IV: The Voyage Home

(a) Demonstrate mathematically that an aspect ratio of 2.35:1 for a movie is not proportional to the ratio of 4:3 for a TV. Provide written explanation as needed. (b) Explain why a picture with dimensions of 2.35:1 cannot be resized to have dimensions of 4:3 without changing the picture.

For more information about how this affects what a movie looks like on a television screen, see the YouTube video titled “Turner Classic Movies: Letterbox” at www.youtube.com/watch?v=5m1-pP1-5K8

1Retrieved from www.widescreen.org/widescreen.shtml

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Making Connections to the Lesson (1) Which of the following was one of the main mathematical ideas of the lesson? (i) Graphic artists have to be very aware of proportionality and know how to solve proportions. (ii) The rules for solving equations are the same for all types of equations. (iii) The rules for solving equations depend on the type of equation. (iv) To solve for a variable in the denominator of a fraction, multiply both sides of the equation by the variable.

(2) In Module 1, you learned that a statement such as 30% of voters support Candidate A can be interpreted as 30 out of 100. (a) How many voters out of 1,000 support Candidate A? (b) How many voters out of 1,500 support Candidate A? (c) Is this a proportional relationship? Explain your answer.

Developing Skills and Understanding (3) The tables below give dimensions of different rectangles. Circle the correct choice of Proportional or Not Proportional to correctly describe the relationship between the rectangles. (a) Circle one: (b) Circle one:

Proportional Not Proportional Proportional Not Proportional

Width Length Width Length 18 120.6 7.4 16.2 23.4 156.8 17 45 33 221.1 23.4 64.2 52.2 349.7 36.2 102.6

(4) A marine biologist would like to feed some dolphins a mix of fish that consists of 9 parts cod to 4 parts mackerel. List three combinations that would be an acceptable mixture of these fish.

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(5) Identify the proportions that have the same solution as the one below:

(i)

(ii)

(iii)

(iv)

(v)

(6) Solve the following proportions:

(a)

(b)

(7) Erica would like to bake an 8-pound roast for a family gathering. The cookbook tells her to bake a 5-pound roast for 135 minutes. Create and solve a proportion that would allow Erica to cook her 8-pound roast.

(8) Cefaclor is a medication used for infections. It is often given in liquid form. A pharmacist is mixing a dosage for a child. The instructions indicate that 125 mg of the medication should be mixed with 5 ml of fluid. If the child only requires a dosage of 100 mg of Cefaclor, how much fluid should the pharmacist use?

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Making Connections Across the Course (9) A company is making pennants or flags for a sports team. The team wants small versions for fans and large versions that will fly over the stadium. The dimensions of the small version are shown below.

8 in.

20 in.

(a) The large version needs to be 12.5 feet across the base (the short side of the triangle). How long should it be? (b) How much material will be needed to make the large version of the flag? Round to the nearest tenth.

(10) A staircase is made up of individual steps that should be consistent in height and width. The height of each step is called the rise, and the width of the step is called the run. (a) The staircase below is made up of four steps with a rise of 6.5" and a run of 8.25". Find the height (H) and depth (D) of the entire staircase.

(b) Builders have to follow guidelines on the rise and run of stairs when building a staircase to meet a code. One acceptable ratio is a rise of 7-3/4 inches for a run of 9-3/4 inches. If a builder is using this ratio to build a staircase that is 15 feet high, how deep will the staircase need to be (d in the drawing below)? Note that the drawing does not show the correct number of steps. Round to the nearest tenth.

15’

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Now, you will once again return to graphing on a coordinate plane. You may have noticed that the two axes split the coordinate plane into four sections. These are called quadrants and are numbered using Roman numerals as shown below. For practical reasons, only a small part of the coordinate plane can be shown, but understand that the axes can go on into infinity in all four directions. The scale of the grid tells you which numbers are included in the portion of the plane that is shown. You can change the scale to make graphs with very large or very small numbers. The scale on a single axis must be consistent. In other words, if the distance between the gridlines represents 5 units on one part of the horizontal axis, then that same distance must always represent 5 units on that axis. However, the vertical and horizontal axes can have different scales as in the example below. As you have seen with other types of graphs, it is important to pay close attention to the scale.

Quadrant II Quadrant I

Quadrant III Quadrant IV

(11) Place the following points on the graph above. Label each with its ordered pair. (a) (–90, 7) (b) (0, 19) (c) (63, –16)

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(12) Indicate if each statement is true or false. (a) The point (–7, –5) is in Quadrant II. (b) The point (0, 5) lies on the vertical axis. (c) All the points in Quadrant IV have a positive horizontal coordinate and a negative vertical coordinate. (d) The points (20, 12) and (20, 200) lie on the same horizontal line.

Many applications use only positive numbers. In these cases, only Quadrant I of the graph is usually shown because that is the only quadrant that is used. An example of this is given below. (13) You learned about the golden ratio in OCE 3.2.3. A rectangle whose dimensions match the golden ratio is called a golden rectangle. The graph below shows the width and lengths of golden rectangles.

(a) Based on the graph, is a rectangle with a width of 17 inches and a length of 30 inches a golden rectangle? (b) Use the graph to complete the table of values below. Estimate to the nearest whole number.

Width Length 5 8 20 31

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Preparing for the Next Lesson (3.2.6) (14) Which expressions are equivalent to the expression below? There may be more than one correct answer. 2(5x + 4) – 3x + 1 (i) 10x + 8 – 3x + 1 (ii) 18x – 3x + 1 (iii) 15x + 1 (iv) 7x + 9 It is important to pay careful attention to notation in working with negatives. For example, –52 is not the same as (–5)2. You can verify this by evaluating each expression on a calculator. The reason has to do with order of operations. The negative in each expression can be thought of as multiplication by a –1. Look at the expressions rewritten with a –1 and think about what operation you would perform first.

2 -1 * 52 (–1 * 5)

In the first expression, the exponent is done first: -1 * 52 → -1 * 25 → –25. In the second expression, the multiplication is done first: (-1 * 5)2 → (–5)2 → 25.

(15) Simplify each of the following: (a) 52 (b) –32 (c) (–4)2 (d) –42 (e) –(–6)2

(16) Solve each equation:

(a)

(b)

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In algebra, a term is an individual part of an algebraic expression. Terms are separated + or – signs, and can consist of numbers, variables (letters), or the product of numbers and one or more variables. In instances where a number and variables are being multiplied, the number is called a coefficient. For example, the expression below has three terms as shown in boxes. Notice that the last term is (–5). In the original expression, this was written as minus 5, but this can be rewritten as adding negative 5 as shown below. When breaking an expression into terms, you ask, what is being added?

The coefficients of each term are –a3 –1 2b 2 –5 Does not have a coefficient because there is no variable. This is called a constant term because it never changes.

(17) State the number of terms in each expression: (a) 3x + 4 (b) 5x – 4x2 + 2 (c) 5

(18) What is the coefficient of the x2 term in Question 17b?

(19) Recall the formula from Lesson 3.1.2 used to find Jenna’s cost to drive her own car for work. In this formula, J = Cost of driving Jenna’s car in $/mile and g = Cost of gas in $/gallon.

(a) Find the cost of driving Jenna’s car (J) when the price of gas is $3.56/gallon. (b) Find the cost of gas (g) when the cost of driving Jenna’s car is $0.31/mile.

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(20) You are expected to be able to do the following things for the next class. Rate how confident you are on a scale of 1–5 (1 = not confident and 5 = very confident). Before beginning Lesson 3.2.6, you should understand the concepts and demonstrate the skills listed below:

Skill or Concept: I can … Rating from 1 to 5 Understand order of operations when simplifying an expression. Substitute a value for a variable in a mathematical model and simplify the model. Square a number. Solve a two-step linear equation such as 2 = x/3 + 5. Understand the meaning of the word term as in a term in an algebraic expression.

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Specific Objectives Students will understand that • solving all equations follows the basic rules of undoing and keeping the equation balanced. Students will be able to • solve linear equations that require simplification before solving. • solve for a variable in a linear equation in terms of another variable. • solve for a variable in a single-term quadratic equation.

Problem Situation: Solving Equations of Different Forms Solving equations such as the Widmark equation for blood alcohol content (BAC) and proportional equations for resizing graphics is an important skill. Mathematical models are often constructed to represent real-life situations. Being able to use these equations fully includes being able to solve for various unknown variables in the equation. Below, are three scenarios for you to practice and enhance your equation-solving skills. With each answer, check that the answer is reasonable given the context and that you have included the correct units with your solution.

(1) Paula has two options for going to school. She can carpool with a friend or take the bus. Her friend estimates that driving will cost 22 cents per mile for gas and 8.2 cents per mile for maintenance of the car. Additionally, there is a $25 parking fee per week at the college. If Paula carpools, she would pay half of these costs. The cost of the carpool can be modeled by the following equation where C is cost of carpooling per week and m is the total miles driven to school each week:

(a) Explain what each term in the equation represents. (b) Find the total weekly carpooling cost if the commute to school is 7 miles each way and Paula goes to school three times a week. (c) A weekly bus pass costs $22.00 dollars. How many total miles must Paula commute to school each week for the carpool cost to be equal to the bus pass? How many trips to school each week must Paula make for the bus pass to be less expensive than carpooling?

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(2) Recall Widmark’s equation for BAC. In the case of the average male who weighs 190 pounds,1 you can simplify Widmark’s formula to get B = −0.015t + 0.022N Forensic scientists often use this equation at the time of an accident to determine how many drinks someone had. In these cases, time (t) and BAC (B) are known from the police report. The crime lab uses this equation to estimate the number of drinks (N). (a) Find the number of drinks if the BAC is 0.09 and the time is 2 hours. (b) Since they use the formula to solve for N over and over, it is easier if the formula is rewritten so that it is solved for N. In other words, so that N is isolated on one side of the equation and all other terms are on the other side. Solve for N in terms of t and B. (c) Use the new formula to find the number of drinks if the BAC is 0.17 and the time is 1.5 hours.

(3) You volunteer for a nonprofit organization interested in women’s issues. The logo for your nonprofit organization is three identical squares arranged as follows:

(a) The organization wants to make banners of different sizes. Find an equation that can be used to find the total area of the logo based on the length of the side of one of the squares. (b) The organization is sponsoring a walk-a-thon to raise funds for breast cancer research. You want to recreate this logo in the middle of the racetrack with bras that have been collected at multiple drop-off sites around the city. You estimate that approximately 1,500 square feet of bras have been donated. How long should you make each side of the square?

Making Connections Record the important mathematical ideas from the discussion.

1Retrieved from www.cdc.gov

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Further Applications (1) An artist is creating a sculpture using a sphere made of clay to represent Earth. The volume of a sphere is given by the equation:

where r is the radius of the sphere. The artist has a rectangular slab of clay that is 4 inches wide, 6 inches long, and 2 inches high. What is the radius of the largest sphere the artist can create with this clay?

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Making Connections to the Lesson (1) Which of the following was one of the main mathematical ideas of the lesson? (i) The order of operations is used in determining the order of steps in solving an equation. (ii) The order of operations is not related to solving equations. (iii) Weight, gender, and time are all important factors in Blood Alcohol Content (BAC). (iv) You can undo addition by subtracting.

(2) Lesson 1.1.4 was called, “The Flexible Quantitative Thinker.” Review Lesson 1.1.4 and briefly describe how you used ideas from that lesson in Lesson 3.2.6.

Developing Skills and Understanding (3) Solve the following equations for the unknown variable in each: (a)

(b)

(c)

18 + n (d) 3 = 44

(4) Solve for the specified variable in each equation. (a) A group of French students plan to visit the United States for two weeks. They are trying to pack appropriate clothing, but are not familiar with Fahrenheit. One student remembers this formula:

where F is the temperature in Fahrenheit and C is the temperature in Celsius. Solve the equation for C. (b) Recall using the simple interest formula, A = P + Prt, from the OCE in Lesson 3.2.2. In the formula: A = the full amount paid for the loan P = the principle or the amount borrowed r = the interest rate as a decimal t = time in years A car dealership wants to use the formula to find the rate needed for certain values of the other variables. Solve the formula for r.

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(5) Akiko earns $935 per month at her full time job. She also works part-time on weekends and evenings for $10.70/hour. (a) Write an equation for Akiko’s monthly income, M. Define your variables. (b) Akiko would like to set up a spreadsheet that will calculate how many hours she has to work to earn different amounts. Her spreadsheet is shown below. Write a formula that Akiko can use in cell B2 to calculate the hours. Test your formula on a calculator or spreadsheet to make sure it is correct.

(6) Recall the simplified formula for the braking distance of a car:

where V0 is the initial velocity of the car (miles per hour) and d is the braking distance (ft). In this model, the roadway grade is kept constant at 5% and the coefficient of friction at 0.8. In a school zone, you want the maximum breaking distance to be 10 feet since this seems like a reasonable distance to see a child who might be in the way of a driver. (a) What should you set as the speed limit (i.e., initial velocity of the car) so that the breaking distance is 10 feet or less? (b) There are two solutions to the equation, . What is the second solution and why is it not an answer to the question in Part (a)?

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Making Connections Across the Course (7) Refer to Lesson 1.2.1 in which you compared the water footprint of different countries. The following information was given.

Population Total Water Footprint1 Country (in thousands) (in 109 cubic meters per year) China 1,257,521 883.39 India 1,007,369 987.38

(a) What does 109 cubic meters mean? (i) one trillion cubic meters (ii) one billion cubic meters (iii) one million cubic meters (iv) one hundred thousand cubic meters (b) The following equation is based on information from the table. What does x represent?

(i) x represents the water footprint of China if it used water at the same rate as India. (ii) x represents the population of China if it used water at the same rate as India. (iii) x represents the water footprint of India if it used water at the same rate as China. (iv) x represents the population of India if it used water at the same rate as China. (c) Calculate China’s water footprint if it used water at the same rate as India.

1Retrieved from www.waterfootprint.org/

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Preparing for the Next Lesson (4.1.1) Recall a problem you examined in Lesson 3.1.2 when you tried to see if it cost more for Jenna to drive her own car or a rental car. As a part of that lesson, you looked at the relationship between the cost of gas and the cost for Jenna to drive her car in $/mile. You have also learned in previous lessons how algebra can be used to express relationships between variables. Now you are going to expand this idea to talk about four representations of a mathematical relationship. Model or Equation In Lesson 3.1.2, your class wrote a mathematical equation for the relationship. An equation is useful because it can be used to calculate the cost values. As you saw with the formula for braking distance in Lesson 3.2.2, equations are also useful for communicating complex relationships. In writing equations, it is always important to define what the variables represent, including units. For example, in Lesson 3.1.2, the variables were defined as shown below. Note that each definition includes what the variable represents, such as cost of Jenna’s car, and the units in which this quantity is measured, such as $/mile. J = Cost of Jenna’s car in $/mile g = Price of gas ($/gal)

These variables were used in the mathematical equation, .

Table Another way that you could have represented this relationship between the price of gas and the cost of driving the car is in a table that shows values of g and D as ordered pairs. An ordered pair is two values that are matched together in a given relationship. You used this representation in Lesson 3.2.2 when you explored how one variable affected another. Tables are helpful for recognizing patterns and general relationships or for giving information about specific values. A table should always have labels for each column. The labels should include include units when appropriate.

Price of Gas ($/gal) Cost of Driving Jenna’s Car ($/mile) 3.00 0.28 3.50 0.31 4.00 0.33 4.50 0.35

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Graph A graph provides a visual representation of the situation. It helps you see how the variables are related to each other and make predictions about future values or values in between those in your table. The horizontal and vertical axis of the graph should be labeled, including units.

Verbal Description A verbal description explains the relationship in words. As you discussed in Lesson 3.2.2, some relationships are very difficult to put into words, but in other cases, a verbal description can help you make sense of what the relationship means in the context. The verbal description for Jenna’s car is too complex to discuss here. You will see examples of verbal descriptions in the next lesson. Summary Throughout this course, you have learned that having the skill to move between different forms and tools is important in problem solving. Alternating among the four representations of mathematical relationships is another example of this. In some cases, you may struggle writing an equation, but find starting with a table helpful. You might want a graph for a visual representation, but also need to express a relationship in words. It is important that you can translate one form into another and also that you can choose which form is most useful in a specific situation. You will practice these skills with the following questions using the above situation of finding the cost of driving Jenna’s car.

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(8) Complete the three missing entries in the table:

Price of Gas ($/gal) Cost of Driving Jenna’s Car ($/mile) 2.00 3.00 0.28 3.20 3.50 0.31 4.00 0.33 4.50 0.35 0. 40

(9) Plot the points that you added to the table in Question 8 to the graph shown above. Extend the line to include the points.

(10) Use the graph to estimate the cost of driving if gas is $2.50/gallon.

(11) You are expected to be able to do the following things for the next class. Rate how confident you are on a scale of 1–5 (1 = not confident and 5 = very confident). Before beginning Lesson 4.1.1, you should understand the concepts and demonstrate the skills listed below:

Skill or Concept: I can … Rating from 1 to 5 Understand the basic meaning and use of variables. Solve for an unknown variable in a one-variable equation. Graph points on a coordinate plane.

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(1) According to the U.S. Bureau of Labor Statistics, in May of 2008 police officers had an average yearly wage of $52,810. Which of the following could be used to calculate the average hourly wage for police officers? Assume that police officers generally work 40 hours a week,1 and there are 52 weeks in the year. (3.1.1)

(a)

(b)

(c)

(d)

(2) Assume regular unleaded fuel costs $3.75 per gallon. A Volkswagen Jetta that uses unleaded fuel gets about 30 miles per gallon. How much money will the driver of the Jetta spend on gas in one year if he drives 15,000 miles? (3.1.1) (a) $1,750 (b) $1,810 (c) $1,875 (d) $1,905

(3) What are the units for the result of the following calculation? (3.1.1)

50 miles 1 hour 1 minute 5,280 feet * * * 1 hour 60 minutes 60 seconds 1 mile (a) feet per mile (b) miles per hour (c) hours per mile (d) feet per second (e) feet per hour

1 Retrieved from http://www.bls.gov/k12/law01.htm

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(4) According to the U.S. Department of Agriculture, 13.9 pounds of ice cream were consumed per person over an entire year.2 Ben loves ice cream, but limits himself to one 2-ounce serving each week. Does he eat more or less than the U.S. annual average cited above? Assume 1 pound of ice cream is equivalent to 16 ounces and there are 52 weeks in a year. (3.1.2) (a) Ben consumes less than the annual average. (b) Ben consume about the same as the annual average. (c) Ben consumes more than the annual average.

(5) Kitchen tiles come in 18" square tiles. If you want to tile a kitchen floor that is rectangular in shape with dimensions 10' x 16', how many tiles are required? (You can cut tiles, but you do not want to glue leftover pieces of tiles together to cover any part of the floor.) (3.2.1) (a) 9 tiles (b) 72 tiles (c) 77 tiles (d) 81 tiles (e) 107 tiles

Use the following information for Questions ____ and ____. (3.2.1) The Padilla family has a plastic wading pool that their son loves to use every day in the summer. He always wants his parents to fill it up to the very top with water. However, his parents are worried that it takes too much water to fill up the pool every day. The round pool is 5 feet across. When full, the water is 1.5 feet high. (6) Which of the following units could be used to measure the amount of water the pool holds? There may be more than one correct answer. (a) feet (b) square feet (c) cubic feet (d) ft2 (e) ft3

2Retrieved from www.ers.usda.gov/data/foodconsumption

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(7) How much water does it take to fill the pool in terms of the units you chose above? The formula for the volume of a cylinder is V = πr2h where r is the radius of the base and h is the height. (a) About 18 (b) About 24 (c) About 29 (d) About 35 (e) About 118

(8) You are landscaping your backyard and want to fertilize the lawn. Home Depot sells SuperGreen lawn fertilizer in 15-lb bags, and each bag costs $16.78 (tax included). Each 15-lb bag is estimated to cover about 3,500 square feet. How much will it cost to fertilize your lawn that covers 1/4 of an acre? Assume you cannot purchase partial bags. Note: 1 acre is 43,560 square feet. (3.2.1) (a) $16.78 (b) $33.56 (c) $50.34 (d) $67.12 (e) $83.90

(9) Basal Metabolic Rate (BMR) is the amount of calories expended by a person at rest for one day. One estimate of a person’s BMR is given by Mifflin’s equation: BMR = 4.536w + 15.875h – 5a + g where w is weight in pounds, h is height in inches, and a is age in years. The variable g is a gender adjustment: g is 5 for males and –161 for females.3 Gina weighs 130 pounds, is 62 inches tall, and is 30 years old. Estimate her BMR using this formula. (3.2.2) (a) Less than 1,000 calories (b) Between 1,000 and 1,500 calories (c) Between 1,500 and 2,000 calories (d) More than 2,000 calories (e) Not enough information is provided to determine BMR.

3 Retrieved from http://en.wikipedia.org/wiki/Basal_metabolic_rate

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Use the following information for Questions ____, _____ and ____. (3.2.2)

The formula for the braking distance of a car is , where

d is the braking distance (in feet),

V0 is the initial velocity of the car (in feet per second), G is the roadway grade (written in decimal form), f is the coefficient of friction between the tires and the roadway (0 < f < 1), and g is the acceleration constant due to gravity (32.2 ft/sec2).

(10) Which of the following are the correct entries in the table below if G = 0.1 and f = 0.8?

Initial Velocity (ft/sec) Braking Distance (ft) 45 60 90 120

(a) 35; 62; 140; 248 (b) 73; 97; 146; 194 (c) 3,261; 5,797; 13,042; 23,185 (d) 280; 497; 1,119; 1,989 (e) None of the above

(11) If the initial velocity doubles and f and G remain fixed, the braking distance will (a) decrease. (b) remain the same. (c) double (be multiplied by 2). (d) triple (be multiplied by 3). (e) quadruple (be multiplied by 4).

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(12) Melissa explains how to use the braking distance formula to a friend. She says, “To find the braking distance for a speed of 50 miles per hour, you have to put 50 in for V0.” Which of the following statements is correct? (a) Melissa’s method will always work. (b) Melissa’s method will work sometimes, depending on the values of f and G.

(c) Melissa’s method will not work because V0 is a velocity and 50 miles per hour is a speed. (d) Melissa’s method will not work because the units are not correct.

(13) A desktop printer advertises that it takes 2 minutes to warm up and can then print 5 pages per minute. The following equation can be used to calculate the number of pages (p) that can be printed in 15 minutes. p 15 = 2 + 5 What is the first step in solving for p? (3.2.4) (a) Subtract 2 from both sides of the equation. (b) Divide both sides of the equation by 2. p (c) Multiply 15 and by 5. 5 (d) Subtract 5 from both sides of the equation. (e) None of the above

(14) An extreme sports center offers party packages. They advertise a flat rate of $300 for the first 10 guests and charge $15 per person for each additional guest. A formula for the cost of the party is determined to be: C = 300 + 15p where p is the number of guests more than 10. (So, if 12 people come to the party, p = 2.) Suppose you want to go all out and have a huge birthday celebration this year. However, you have budgeted $500 for your party. How many guests can you invite if you decide to have your party at this sports center? (3.2.4) (a) 13 guests total (b) 14 guests total (c) 23 guests total (d) 24 guests total (e) 7,800 guests total

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(15) An artist needs to paint a mural on a wall. The mural needs to be 15 feet high. The mural is to be drawn to scale from a small picture on paper that is 10 inches high and 8 inches wide. Which of the following proportions can be used to calculate the width, w, of the mural? (3.2.5)

10 w (a) = 8 15 10 w (b) = 15 8 15 8 (c) = w 10 15 w (d) = 8 10 (e) None of the above

(16) A table below shows the dimensions of different images. Select the answer that best explains if the dimensions are proportional or not. The width and length for each image are measured in the same units. (3.2.5)

Width Length Image 1 5 17.5 Image 2 10 35 Image 3 15 52.5

(a) Cannot tell because the units are not given. (b) Yes, because the length goes up by 5 units each time. (c) Yes, because the ratio of length to width is always the same. (d) No, because the width does not increase by the same percentage each time. (e) No, because the difference between the length and width is different.

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(17) Jose wants to make a big batch of cookies to take to a party. The chocolate chip cookie recipe calls for 2-1/4 cups of flour and makes four dozen cookies. If he wants to make 10 dozen cookies, how many cups of flour does he need to use? (3.2.5) (a) 4-1/2 cups (b) 5-5/8 cups (c) 6-1/8 cups (d) 6-3/4 cups (e) 10 cups

(18) Many popular diets are low in fat. The percentage of fat (P) in a food can be calculated using the following equation:

where f is the grams of fat and c is the total calories in a food. Solve this formula for f. (3.2.6) (a)

(b)

c (c) f = 9P

(d)

(e)

The Carnegie Foundation for the Advancement of Teaching Quantway frontmatter available at www.quantway.org/kernel and The Charles A. Dana Center at the University of Texas at Austin or www.utdanacenter.org/mathways/index.php 7 299 Quantway Assessment November 18, 2011 (Version 1.0) Problems for Module 3

(19) A free-falling object is one that is dropped from a certain height. Physicists often use the following equation for free-falling objects: 1 d = g t2 2 where d is the distance the object has traveled after t seconds. The constant g is 9.8 meters per second2 and is a measure for the acceleration of gravity on Earth. If an object is dropped from the top of the Empire State Building, how long will it take before it reaches the ground, traveling the entire distance of the building’s height? According to Wikipedia, the Empire State Building is 381 meters high.4 (3.2.6) (a) Less than 5 seconds (b) Between 5 and 10 seconds (c) Between 10 and 20 seconds (d) More than 20 seconds (e) Not enough information is given to determine time

(20) Solve the equation for n: 67 = 3(n–1) + 2n. (3.2.6) (a) n = 13.6 (b) n = 14.0 (c) n = 14.4 (d) n = 32.0 (e) n = 35.0

4 Retrieved from http://en.wikipedia.org/wiki/Empire_State_Building

The Carnegie Foundation for the Advancement of Teaching Quantway frontmatter available at www.quantway.org/kernel and The Charles A. Dana Center at the University of Texas at Austin or www.utdanacenter.org/mathways/index.php 8 300 Quantway Student Handout November 18, 2011 (Version 1.0) Module 3 Review

As with Modules 1 and 2, you should assess your understanding of Module 3 to prepare for the Module 3 test. Your instructor may give you specific assignments for your review in addition to this self-assessment.

Assessing Your Understanding The table on the following page lists the Module 3 concepts and skills you should understand. This exercise helps you assess what you understand. After completing it, you will be able to prioritize your review time more effectively. 1. Assess your understanding. • Go through the topics list and locate each concept or skill in the Module 3 in-class or OCE materials. • If you have not used the skill in a while, do two or more problems to check your understanding. • If you have recently used the skill and feel confident that you did it correctly, rate your understanding a 4 or 5. • If you remember the topic but could use more practice, rate your understanding a 3. • If you cannot remember that skill or concept, rate your understanding a 1 or 2. Now that you have done an initial rating of your understanding, it is time to begin reviewing. Complete the remaining steps. The goal is to have a confidence rating of 4 or 5 on all the topics in the table when you have finished your review of Module 3. 2. Start at the beginning of module and reread the material in the lessons, the OCE, and your notes on the skills and concepts you rated 3 or below. 3. Select a few problems to do. Do not look at the answer or your previous work to help you. 4. Once you have finished the problems, check your answers. If you are not sure if you have done the problems correctly, check with your instructor, other classmates, and your previous work or work with a tutor in the learning center. 5. Rate your confidence on this skill again. If you understand the concept better, rate yourself higher. Begin a list of topics that you want to review more thoroughly. 6. If you have time, do one or two problems on skills or concepts you rated 4 or above. 7. For topics that you need to review more thoroughly, make a plan for getting additional assistance by studying with classmates, visiting your instructor during office hours, working with a tutor in the learning center, or looking up additional information on the Internet.

The Carnegie Foundation for the Advancement of Teaching Quantway frontmatter available at www.quantway.org/kernel and The Charles A. Dana Center at the University of Texas at Austin or www.utdanacenter.org/mathways/index.php 1 301 Quantway Student Handout November 18, 2011 (Version 1.0) Module 3 Review

Module 3 Concept or Skill Rating Making Conversions Understand use of units in making conversions (3.1.1, 3.1.2) Use dimensional analysis to make a conversion involving multiple conversion factors (3.1.1, 3.1.2) Geometric Reasoning Understand concepts of and units for linear measurement, area, and volume (3.2.1) Identify and use the appropriate geometric formula to apply in a given situation (3.2.1) Using Formulas and Algebraic Expressions Understand the use of variables in formulas and algebraic expressions, including the appropriate way to define a variable (3.2.2) Understand the role of a constant in a formula (3.2.2) Use a formula to solve for a value (3.2.2, 3.2.4, 3.2.6) Using Graphical Displays Read and interpret a pictograph (3.2.3) Understand the limitations and potential for distortion in pictographs (3.2.3) Creating and Solving Equations Solve a linear equation in one variable (3.2.4, 3.2.6) Interpret the solution to an equation (3.2.4, 3.2.5, 3.2.6) Solve an equation or formula for a variable (3.2.6) Write and solve proportions (3.2.5) Solve complex equations with multiple variable terms and variables in the denominator (3.2.6) Solve or estimate the solution to equations with a variable raised to the power of 2 (3.2.6)

The Carnegie Foundation for the Advancement of Teaching Quantway frontmatter available at www.quantway.org/kernel and The Charles A. Dana Center at the University of Texas at Austin or www.utdanacenter.org/mathways/index.php 2 302 A.1 Place Value, Rounding, Comparing Whole Numbers

Place Value Example: The number 13,652,103 would look like Millions Thousands Ones Hundreds Tens Ones Hundreds Tens Ones Hundreds Tens Ones 1 3 6 5 2 1 0 3

We’d read this in groups of three digits, so this number would be written thirteen million six hundred fifty two thousand one hundred and three

Example: What is the place value of 4 in 6,342,105? The 4 is in the ten-thousands place

Example: Write the value of two million, five hundred thousand, thirty six 2,500,036

Rounding When we round to a place value, we are looking for the closest number that has zeros in the digits to the right.

Example: Round 173 to the nearest ten. Since we are rounding to the nearest ten, we want a 0 in the ones place. The two possible values are 170 or 180. 173 is closer to 170, so we round to 170.

Example: Round 97,870 to the nearest thousand. The nearest values are 97,000 and 98,000. The closer value is 98,000.

Example: Round 5,950 to the nearest hundred. The nearest values are 5,900 or 6,000. 5,950 is exactly halfway between, so by convention we round up, to 6,000.

Comparing To compare to values, we look at which has the largest value in the highest place value.

Example: Which is larger: 126 or 132? Both numbers are the same in the hundreds place, so we look in the tens place. 132 has 3 tens, while 126 only has 2 tens, so 132 is larger. We write 126 < 132, or 132 > 126.

Example: Which is larger: 54 or 236? Here, 54 includes no hundreds, while 236 contains two hundreds, so 236 is larger. 54 < 236, or 236 > 54

These worksheets were created by David Lippman, and are released under a Creative Commons Attribution license. 303 Worksheet – A.1 Place value, rounding, comparing Name: ______

1) Write out in words: 13,904

2) Write out in words: 30,000,000 3) Write out in words: 13,000,000,000

4) Write the number: sixty million, three hundred twelve thousand, two hundred twenty five

5) Round to the nearest ten: 83,974 6) Round to the nearest hundred: 6,873

Round 8,499 to the nearest 7) ten 8) hundred 9) thousand

Determine which number is larger. Write < or > between the numbers to show this. 10) 13 21 11) 91 87

12) 136 512 13) 6,302,542 6,294,752

14) six thousand five hundred twenty three six thousand ninety five

304 A.2 Introducing Order of Operations

Order of Operations (Standard Operations) When we combine multiple operations, we need to agree on an MD: Multiplication and Division order to follow, so that if two people calculate ⋅+ 432 they will AS: Addition and Subtraction get the same answer. If we consider just our standard operations (+, - ,×, ÷), it is important to compute multiplication and division work first, and then complete addition and subtraction operations.

Example: Simplify 2 + 3 4 We start with any multiplication or division: =⋅ 1243 : 2 + ∙ Now we can add or subtract: 14 𝟏𝟏𝟏𝟏 Example: Simplify 12 ÷ 6 2 We start with any multiplication or division: 12 ÷ 6 = 2: 2 − Now we can add or subtract: 0 𝟐𝟐 − IMPORTANT!! Notice that multiplication and division have the SAME precedence, as do addition and subtraction. When you have multiple operations of the same level, you work left to right.

Example: Simplify 3 + 5 2 5 We start with any multiplication or division – we only have: 5 2 = 10: 3 + 5 ∙ − Now we add or subtract from left to right, so do 3 + 10 = 13 first: 5 ∙ 𝟏𝟏𝟏𝟏 − 8 𝟏𝟏𝟏𝟏 − Example: Simplify 30 ÷ 5 6 21 + 4 Start with any multiplication or division- the first we see is: 30 ÷ 5 = 6 : 6 21 + 4 ∙ − There’s still a multiplication, so that needs to come next: 6 6 = 36 : 21 + 4 𝟔𝟔 ∙ − Now we add or subtract, again left to right. So do 36 21 = 15 first: + 4 ∙ 𝟑𝟑𝟑𝟑 − Then finish up with the last addition: 19 − 𝟏𝟏𝟏𝟏 Order of Operations (adding in parentheses) What if you really want to do operations in a different order? We can use parentheses to over-ride normal the rules above. Anything in parentheses gets priority and must be completed first! If there are multiple things to do inside the parentheses, be sure to follow order of operations inside.

Example: Simplify 6 + 40 ÷ (2 + 3) Always start by doing what’s in parenthesis first. So 2 + 3 = 5: 6 + 40 ÷ Then do multiply or divide before adding or subtracting, so 40 ÷ 5 = 8: 6 + 𝟓𝟓 14 𝟖𝟖 Example: Simplify 2(5 + 2 3) Looking inside the parentheses, do the multiplication first: 2 3 = 6 : 2(5 + ) ∙ Finishing what’s in the parentheses, now do 5 + 6 = 11: 2( ) ∙ 𝟔𝟔 Which leaves us with multiplying the 2 outside by the parentheses result: 𝟏𝟏𝟏𝟏 𝟐𝟐𝟐𝟐 305 Worksheet – A.2 Introducing Order of Operations Name: ______

Evaluate. Only perform one step at a time. Show all your work vertically.

1) 5 + 2 6 2) 20 10 ÷ 2 3) 5 7 3

∙ − ∙ −

4) 4 2 + 10 3 5) 5 2 9 6) 24 ÷ 2 3

− − ∙ ∙ ∙

7) 3 + (9 5) 8) 5 (32 ÷ 4) 9) 3 + 4 2 1

− ∙ ∙ −

10) 5(1 + 6 2) 11) 2 + (15 ÷ 3) 12) 50 (20 + 25)

∙ −

13) 4 + (3 2 5) 14) 3(5 2) + 1 15) 2(10 4 1) 5

∙ − − − ∙ −

306 A.3 Using Exponents and Roots in Order of Operations

Exponents and Roots If we have repeated multiplication, like ⋅⋅⋅ 5555 we can write this more simply using exponents: 54 Example: Write ⋅⋅⋅⋅ 33333 using exponents Since we are multiplying 3 times itself 5 times, the base is 3, and the exponent is 5: 35

Example: Evaluate 63 3 =⋅=⋅⋅= 2166366666

Undoing squaring a number is finding the square root, which uses the symbol . It’s like asking “what number times itself will give me this value?” So = 636 since 2 = 366 Example: Find 81 = 981 because 2 = 819

Order of Operations When we combine multiple operations, we need to agree on an P: Parentheses order to follow, so that if two people calculate ⋅+ 432 they will E: Exponents and roots get the same answer. To remember the order, some people use MD: Multiplication and Division the mnemonic PEMDAS: AS: Addition and Subtraction

IMPORTANT!! Notice that multiplication and division have the SAME precedence, as do addition and subtraction. When you have multiple operations of the same level, you work left to right.

Example: Simplify ÷+−⋅+ 2106432 We start with the multiplication and division: =⋅ 1243 and =÷ 5210 : +−+ 56122 Now we add and subtract from left to right: 14 – 6 + 5 8 + 5 13

Example: Simplify +−+ 496)25(4 2 We start with the inside of the parentheses: 5+2=7: +− 496)7(4 2 Next we evaluate the exponents and root: 2 == 164,39 : +⋅− 1636)7(4 Next we do the multiplications: +− 161828 Lastly add and subtract from left to right: +1610

307 Worksheet – A.3 Using Exponents and roots in order of operations Name: ______

Evaluate. 1) 43 2) 49 3) ÷− 31218 4) +− 5310

100 5) ⋅⋅ 532 6) ⋅÷ 5210 7) ⋅32 2 8) 2

9) ()⋅−− 32826 10) ()2 ++ 4313 11) − 4254 2

12) For a rectangle, the formula Perimeter = 2L+2W is often used, where L is the length and W width. Use this formula to find the perimeter of a rectangle 10 feet long and 4 feet wide.

Write out the mathematical expression that would calculate the answer to each question: 13) A family of four goes out to a buffet, and pays $10 each for food, and $2 each for drinks. How much do they pay altogether?

14) Don bought a car for $1200, spent $300 on repairs, and sold it for $2300. How much profit did he make?

308 A.4 More Order of Operations Practice

Order of Operations To remember the order, some people use the mnemonic P: Parentheses PEMDAS: E: Exponents and roots MD: Multiplication and Division IMPORTANT!! Notice that multiplication and division have AS: Addition and Subtraction the SAME precedence, as do addition and subtraction. When you have multiple operations of the same level, you work left to right.

Example: Simplify (3 + 2) + 5 2 We start with what’s in parenthesis, so 3 + 2 = 5: + 5 Now we can do exponents. So 5 = 25: 252 + 5 𝟓𝟓 Then finish by doing the remaining addition:2 30

Example: Simplify 3 + (20 ÷ 4) 3 2 We start with the inside of the parentheses: 20 ÷ 4 = 5: 3 + Next we evaluate the exponents left to right: 3 = 27: 3 + 52 𝟓𝟓 Still another exponent to go: 53 = 25: 27 + 2 𝟐𝟐𝟐𝟐 Lastly do the remaining addition:2 52 𝟐𝟐𝟐𝟐

Nested Parentheses Sometimes you will see parentheses inside of other parentheses. We call these “nested” parentheses. Always start with the inside-most parentheses first.

Example: Simplify ((2 + 1) + 1) 2 2 We start with the inside-most of the parentheses: 2 + 1 = 3: ( + 1) Now we want to start working on the next parentheses. Do exponents 2 2 𝟑𝟑 before addition: 3 = 9: ( + 1) Finish the parentheses before adding: 9 + 1 2= 10: 2 𝟗𝟗 Now that we are done with the parentheses, we can do the exponent: 1002 𝟏𝟏𝟏𝟏

Example: Simplify 35 + (4(1 + 4) ) 2 We start with the inside-most of the parentheses: 1 + 4 = 5: 35 + (4 ) Now we want to start working on the next parentheses. Do exponents 2 ∙ 𝟓𝟓 before multiplication: 5 = 25: 35 + (4 ) Finish the parentheses first: 4 25 2= 100: 35 + ∙ 𝟐𝟐𝟐𝟐 Now that we are done with the parentheses, we can do the addition: 135 ∙ 𝟏𝟏𝟏𝟏𝟏𝟏 309 Worksheet – A.4 More Order of Operations Practice Name: ______

Evaluate. Only perform one step at a time. Show all your work vertically.

1) 4 × 10 2) 32,000 ÷ 10 3) 1,500,000 ÷ 10 3 2 3

4) (38 6) ÷ 16 + 5 5) (12 2 + 3 ) + 5 6) (8 4) + (8 ÷ 2) 2 2 2 − ∙ −

7) 2 + (18 ÷ 2 4) 8) 5(13 2) + 6 9) 4 + (16 (4 + 5))

− − −

10) 4 + (2 + 24 ÷ 3) 11) 3 + (9(10 7) ) 12) 15 + ((10 2) + 3 ) 2 2 2 2 − −

13) 8((2 + 1) + (20 ÷ 10) ) 14) 5 + [3(2 + 1)] 15) ((6 + 4) + 2) 11 + 3 2 2 2 −

310 Practice Test for Skills Quiz A – Order of Operations

1.) Simplify: 15 + (5-2)2 ·2 – 1

2.) Simplify: 22 + [3+ 5(8-6)]

3.) Simplify: 18 – 2 · 5 + 5 · 22

4.) Simplify: 18 ÷ 2 · 4 + 7 · 22

Write the following numbers in strictly numeric or standard form

5.) 7 billion

6.) 227 million

Simplify. Write the answer without exponent.

7.) 23

8.) 32∙42

9.) 7 x 102

10.) 55,000 ÷102 311 Answer Key for Sample Order of Operations Skills Test

1.) 15 + (5-2)2 ·2 – 1 15 + (3)2 ·2 – 1 15 + 9 ·2 – 1 15 + 18 – 1 32 2.) 22 + [3+ 5(8-6)] 22 + [3+ 5(2)] 22 + [3+ 10] 22 + 13 35 3.) 18 – 2 · 5 + 5 · 22 18 – 2 · 5 + 5 · 4 18 – 10 + 20 28 4.) 18 ÷ 2 · 4 + 7 · 22 18 ÷ 2 · 4 + 7 · 4 9 · 4 + 7 · 4 36 + 28 64 Write the following numbers in strictly numeric or standard form 5.) 7,000,000,000 6.) 227,000,000 Simplify. Write the answer without exponent. 7.) 2x2x2 =8 8.) 3x3 x 4x4 = 9x16 = 144 9.) 700 10.) 550

312 B.1 Fractions and Mixed Numbers

Fractions are a way of representing parts of a whole. For example, if pizza is cut into 8 pieces, and Sami takes 3 3 pieces, he’s taken of the pizza, which we read as “three eighths.” 8

The number on the bottom is called the denominator, and indicates how many pieces the whole has been divided into. The number on top is the numerator, and shows how many pieces of the whole we have.

Example: What fraction of the large box is shaded? 6 The box is divided into 10 pieces, of which 6 are shaded, so is shaded. 10

If we have more than one whole, we often write mixed numbers. Example: In the picture shown, we have two full circles, and a part 1 of a third circle. We commonly write this as 2 , indicating that we 4 have two wholes, and 1 additional quarter.

This mixed number could also be written as an improper fraction, which is what we call a fraction where the numerator is equal to or bigger than the denominator. In our circle picture above, we could write the shaded 9 part as , indicating that if we divide all the circles into quarters, there are 9 shaded quarters altogether. A 4 proper fraction is a fraction where the numerator is smaller than the denominator.

Converting from mixed number to improper fraction - Multiply the whole number by the denominator of the fraction to determine how many pieces we have in the whole. - Add this to the numerator of the fraction - Use this sum as the numerator of the improper fraction. The denominator is the same. 2 Example: Convert 5 to an improper fraction. 7 If we had 5 wholes, each divided into 7 pieces, that’d be =⋅ 3575 pieces. 37 Adding that to the additional 2 pieces gives 35+2 = 37 total pieces. The fraction would be 7

Converting from improper fraction to mixed number - Divide: numerator ÷ denominator - The quotient is the whole part of the mixed number - The remainder is the numerator of the mixed number. The denominator is the same. 47 Example: Write as a mixed number. Dividing, 47÷6 = 7 remainder 5. So there are 7 wholes, and 5 6 5 remaining pieces, giving the mixed number 7 6 313 Worksheet – B.1 Intro to Fractions Name: ______

1) Out of 15 people, four own cats. Write the fraction of the people who own cats.

For each picture, write the fraction of the whole that is shaded 2) 3) 4)

For each picture, write the shaded portion as a mixed number and as an improper fraction 5) 6) 7)

Convert each mixed number to an improper fraction 3 7 1 5 8) 4 9) 1 10) 15 11) 23 4 16 2 8

Convert each improper fraction to a mixed number or whole number 35 15 23 164 12) 13) 14) 15) 2 6 10 4

Measure the length of each bar in inches using a ruler. Record your answer as a mixed number.

16)

17)

18) 314 B.2 Simplifying Fractions

To simplify fractions, we first will need to be able to find the factors of a number. The factors of a number are all the numbers that divide into it evenly. Example: Find the factors of 18. The factors of 18 are 1, 2, 3, 6, 9, 18, since each of those numbers divides into 18 evenly.

When we factor a number, we write it is a product of two or more factors. Example: Factor 24 There are several possibilities: ⋅122 , ⋅83 , ⋅64 , ⋅⋅⋅ 3222

The last of the factorizations above is called the prime factorization because it is written as the product of prime numbers – numbers that can’t be broken into smaller factors.

Equivalent fractions To find equivalent to fractions, we can break our fraction into more or fewer pieces. 3 6 For example, by subdividing the rectangle to the right, we see = . By doubling 8 16

the number of total pieces, we double the number of shaded pieces as well.

To find equivalent fractions, multiply or divide both the numerator and denominator by the same number. 2 Example: Write two fractions equivalent to 8 2 ⋅32 6 By multiplying the top and bottom by 3, = = 8 ⋅38 24 = 2 ÷ 22 1 By dividing the top and bottom by 2, = = 8 ÷ 28 4 3 Example: Write with a denominator of 15 5 3 ⋅33 9 To get a denominator of 15, we’d have to multiply 5 by 3. = = 5 ⋅35 15

To simplify fractions to lowest terms, we look for the biggest factor the numerator and denominator have in common, and divide both by that. 12 Example: Simplify 18 12 ÷ 612 2 12 and 18 have a common factor of 6, so we divide by 6: = = 18 ÷ 618 3 12 ⋅62 6 12 ⋅ 62 2 Alternatively, you can write = and since = 1, = = 18 ⋅63 6 18 ⋅ 63 3 24 Example: Simplify 132 24 ÷ 224 12 ÷ 612 2 If you’re not sure of the largest factor, do it in stages: = == = 132 ÷ 2132 66 ÷ 666 11 315 Worksheet – B.2 Simplifying Fractions Name: ______

Write all the factors of each number 1) 36 2) 32 3) 120

Find the biggest common factor of each pair of numbers 4) 12 and 8 5) 4 and 12 6) 10 and 25 7) 36 and 27

3 18 8) Rewrite with a denominator of 28 9) Rewrite with a denominator of 6 7 24

Simplify to lowest terms 3 10 130 18 10) 11) 12) 13) 6 12 150 24

40 28 27 70 14) 15) 16) 17) 56 49 54 126

Rewrite each pair of fractions to have the same denominator 1 1 5 3 3 7 5 3 18) and 19) and 20) and 21) and 3 4 6 8 20 32 56 42 316 B.3 Multiplying Fractions

a c ⋅ ca To multiply two fractions, you multiply the numerators, and multiply the denominators: =⋅ b d ⋅ db 2 5 Example: Multiply and simplify ⋅ 3 8 2 5 ⋅52 10 5 =⋅ = , which we can simplify to 3 8 ⋅83 24 12

⋅52 Alternatively, we could have noticed that in , the 2 and 8 have a common factor of 2, so we can divide the ⋅83 ⋅52 ⋅51 5 numerator and denominator by 2, often called “cancelling” the common factor: = = ⋅83 ⋅43 12

7 Example: Multiply and simplify ⋅ 6 8 7 6 ⋅67 It can help to write the whole number as a fraction: =⋅ . Since 6 and 8 have a factor of 2 in common, 8 1 ⋅18 ⋅ 37 21 1 we can cancel that factor, leaving = . This could also be written as the mixed number 5 . ⋅14 4 4

To multiply with mixed numbers, it is easiest to first convert the mixed numbers to improper fractions. 1 4 Example: Multiply and simplify 3 ⋅4 3 5 1 10 4 24 1 4 10 24 Converting these to improper fractions first, 3 = and 4 = , so 3 4 ⋅=⋅ 3 3 5 5 3 5 3 5 10 24 ⋅2410 ⋅242 =⋅ . Since 5 and 10 have a common factor of 5, we can cancel that factor: . 3 5 ⋅53 ⋅13 ⋅82 16 Since 3 and 24 have a common factor of 3, we can cancel that factor: == 16 ⋅11 1

Areas of Triangles 1 To find the area of a triangle, we can use the formula Area ⋅⋅= hb h 2 Example: Find the area of the triangle shown b 1 The area would be ⋅⋅ 78 2 1 8 7 56 ==⋅⋅ 28 7 2 1 1 2 8 317 Worksheet – B.3 Multiplying Fractions Name: ______

Multiply and simplify 2 3 5 1 5 9 1) ⋅ 2) ⋅ 3) ⋅ 5 4 6 3 12 10

3 2 5 2 4 4) ⋅⋅ 5) 12 ⋅ 6) 10⋅ 10 5 9 3 15

1 4 1 2 3 7) 3 ⋅ 8) 8 ⋅4 9) ⋅ 24 2 5 6 7 5

Find the area of each shape 10) 11) 12) 3 miles ft

6 mile

13) Legislature can override the governor’s veto with a 2/3 vote. If there are 49 senators, how many must be in favor to override a veto?

14) A recipe calls for 2½ cups flour, ¾ cup of sugar, and 2 eggs. How much of each ingredient do you need to make half the recipe? 318 B.4 Dividing Fractions

To find the reciprocal of a fraction, we swap the numerator and denominator 5 1 Example: Find the reciprocal of , , and 5 12 4 5 12 1 4 5 1 The reciprocal of is . The reciprocal of is = 4. The reciprocal of 5 = is . 12 5 4 1 1 5

To find the reciprocal of a mixed number, first write it as an improper fraction 1 Example: Find the reciprocal of 3 4 1 13 4 3 = , so the reciprocal is 4 4 13

To divide two fractions, you find the reciprocal of the number you’re dividing by, and multiply the first number times that reciprocal of the second number. 5 5 Example: Divide and simplify ÷ 8 6 5 5 6 ⋅65 ⋅31 3 We find the reciprocal of and change this into a multiplication problem: =⋅ = = 6 8 5 ⋅58 ⋅14 4 3 1 Example: Divide and simplify ÷ 4 8 1 3 8 ⋅83 ⋅ 23 6 We find the reciprocal of and change this into a multiplication problem: =⋅ = == 6 8 4 1 ⋅14 ⋅11 1 1 1 Example: Divide and simplify 5 ÷1 2 3 11 4 Rewriting the mixed numbers first as improper fractions, ÷ 2 3 4 11 4 ⋅411 ⋅211 22 1 We find the reciprocal of and change this into a multiplication problem: =⋅ = == 7 3 2 3 ⋅32 ⋅31 3 3

3 Example: You have 5 cups of flour, and a batch of cookies requires 1 cups of flour. How many batches can 4 you make? 3 7 4 5 4 20 6 We need to divide: ÷15 . Rewriting, 5 5 ==⋅=⋅=÷ 2 . You can make 2 batches of cookies. 4 4 7 1 7 7 7 You almost have enough for 3 batches, so you might be able to get away with 3.

Example: Making a pillow requires ¾ yard of fabric. How many pillows can you make with 12 yards of fabric? 3 3 4 12 4 4 4 16 We need to divide: 12 ÷ . Rewriting, 12 12 ==⋅=⋅=⋅=÷ 16 . 4 4 3 1 3 1 1 1 You can make 16 pillows with 12 yards of fabric. 319 Worksheet – B.4 Dividing Fractions Name: ______

Divide and simplify 3 1 7 7 9 3 2 1) ÷ 2) ÷ 3) ÷ 4) 18 ÷ 5 4 8 12 10 5 3

7 1 1 2 1 1 5) ÷14 6) 3 ÷ 7) 2 ÷ 4 8) 8 ÷ 6 8 4 6 5 3 2

Decide if each question requires multiplication or division and then answer the question 1 13) One dose of eyedrops is ounce. How many ounces are required for 40 doses? 8

1 14) One dose of eyedrops is ounce. How many doses can be administered from 4 ounces? 8

15) A building project calls for 1½ foot boards. How many can be cut from a 12 foot long board?

3 16) A cupcake recipe yielding 24 cupcakes requires 2 flour. How much flour will you need if you want to 4 make 30 cupcakes? (this may be a two-step question)

320 B.5 Add / Subtract Fractions with Like Denominator

We can only add or subtract fractions with like denominators. To do this, we add or subtract the number of a b + ba a b − ba pieces of the whole. The denominator remains the same: =+ and =− c c c c c c 1 2 Example: Add and simplify + 5 5 1 2 + 21 3 =+ = 5 5 5 5 5 3 Example: Subtract and simplify − 8 8 5 3 − 35 2 1 =− == 8 8 8 8 4

To add mixed numbers, add the whole parts and add the fractional parts. If the sum of the fractional parts is greater than 1, combine it with the whole part

7 5 Example: Add and simplify 3 + 2 9 9 7 5 + 57 12 3 1 Adding the whole parts =+ 523 . Adding the fractional parts, =+ 1 === 1 . 9 9 9 9 9 3 1 1 Now we combine these: 15 =+ 6 3 3

To subtract mixed numbers, subtract the whole parts and subtract the fractional parts. You may need to borrow a whole to subtract the fractions 4 3 Example: Subtract and simplify 8 − 3 5 5 4 3 4 3 1 4 3 1 Since is larger than , we don’t need to borrow. =− 538 , and =− , so 8 3 =− 5 5 5 5 5 5 5 5 5 1 3 Example: Subtract and simplify 5 − 3 4 4 1 3 1 1 5 Since is smaller than , we need to borrow. We can say 5 14 =+= 4 . Now we can subtract: 4 4 4 4 4 5 3 2 1 1 3 1 =− 134 and ==− , so 5 3 =− 1 4 4 4 2 4 4 2

Alternatively, you can add or subtract mixed numbers my converting to improper fractions first: 1 3 21 15 6 3 1 5 3 ===−=− 1 4 4 4 4 4 2 2

321 Worksheet – B.5 Add/Subt Fractions Like Denom Name: ______

Add or Subtract and simplify 2 1 3 5 6 4 1) + 2) + 3) + 5 5 10 10 7 7

7 5 1 5 3 4 4) + 5) 4 + 3 6) 1 + 7 8 8 8 8 5 5

1 5 2 7 2 7) 2 + 3 8) 2 + 3 9) − 6 6 9 9 9

7 5 5 1 1 3 10) − 11) 5 − 3 12) 4 − 2 8 8 6 6 4 4

1 2 2 1 13) 1 − 14) 4 − 2 15) − 36 3 3 5 4 322 B.6 Part 1 Least Common Multiple

To compare or add fractions with different denominators, we first need to give them a common denominator. To prevent numbers from getting really huge, we usually like to find the least common denominator. To do this, we look for the least common multiple: the smallest number that is a multiple of both denominators.

Method 1: Lucky guess / intuition In this approach, perhaps you look at the two numbers and you immediately know the smallest number that both denominators will divide into. 1 3 Example: Give and a common denominator. 6 10 Perhaps by looking at this, you can immediately see that 30 is the smallest multiple of both numbers; the 1 5 1 5 5 smallest number both will divide evenly into. To give a denominator of 30 we multiply by : ⋅ = . 6 5 6 5 30 3 3 3 3 9 To give a denominator of 30, we multiply by : =⋅ 10 3 10 3 30

Method 2: List the multiples In this approach, we list the multiples of a number (the number times 2, times 3, times 4, etc.) and look for the smallest value that shows up in both lists. 1 5 Example: Give and a common denominator. 12 18 Listing the multiples of each: 12: 12 24 36 48 60 72 96 18: 18 36 54 72 90 108 While they have both 36 and 72 as common multiples, 36 is the least common multiple. To give 12 a denominator of 36 we multiply top and bottom by 3; to give 18 a denominator of 36 we multiply top and bottom 1 3 3 5 2 10 by 2. =⋅ , =⋅ 12 3 36 18 2 36

Method 3: List prime factors We list the prime factors of each number, then use each prime factor the greatest number of times it shows up in either factorization to find the least common multiple. Example: Find the least common multiple of 40 and 36. Breaking each down, ⋅⋅⋅=⋅= 522210440 ⋅⋅⋅=⋅= 33229436 Our least common multiple will need three factors of 2, two factors of 3, and one factor of 5: =⋅⋅⋅⋅⋅ 360533222

Method 4: Common factors In the above approach, after noticing 40 and 36 had a factor of 4 in common, we might have noticed that 9 and 10 had no other common factors, so the least common multiple would be =⋅⋅ 3601094 . We only use common factors once in the least common multiple. 323 Worksheet – B.6p1 Least Common Multiples Name: ______

Find the least common multiple of each pair of numbers 1) 3 and 7 2) 4 and 10 3) 12 and 16

4) 20 and 30 5) 9 and 15 6) 15 and 18

Give each pair of fractions a common denominator 5 9 3 5 3 1 7) 8) 9) 7 14 4 8 8 6

1 4 11 3 5 7 10) 11) 12) 10 15 18 16 72 60

324 B.6 Part 2 Add / Subtract Fractions with Unlike Denominator

Since can only add or subtract fractions with like denominators, if we need to add or subtract fractions with unlike denominators, we first need to give them a common denominator. 1 1 Example: Add and simplify + 4 2 Since these don’t have the same denominator, we identify the least common multiple of the two denominators, 1 1 1 2 + 21 3 4, and give both fractions that denominator. Then we add and simplify. =+=+ = 4 2 4 4 4 4 5 7 Example: Subtract and simplify − 8 12 The least common multiple of 8 and 12 is 24. We give both fractions this denominator and subtract. 5 7 15 14 1 =−=− 8 12 24 24 24 3 7 Example: Add and simplify + 4 12 3 7 9 7 + 79 16 We give these a common denominator of 12 and add: =+=+ = 4 12 12 12 12 12 16 4 1 This can be reduced and written as a mixed number: == 1 12 3 3

To add and subtract mixed numbers with unlike denominators, give the fractional parts like denominators, then proceed as we did before. 2 3 Example: Add and simplify 2 + 5 3 4 8 9 Rewriting the fractional parts with a common denominator of 12: 2 + 5 12 12 8 9 17 5 Adding the whole parts + = 752 . Adding the fractional parts, ==+ 1 . 12 12 12 12 5 5 Now we combine these: 17 =+ 8 12 12 1 5 Example: Subtract and simplify 6 − 4 3 6 4 10 Rewriting the fractional parts with a common denominator of 6: 6 − 4 12 12 10 4 4 4 16 Since is smaller than , we borrow: 6 15 5 +=+= 12 12 12 12 12 16 10 6 1 4 10 1 5 – 4 = 1, and ==− , so 6 − 4 = 1 12 12 12 2 12 12 2

325 Worksheet – B.6p2 Add/Subt Fractions Unlike Denom Name: ______

Add or Subtract and simplify 2 1 1 1 3 1 1) + 2) + 3) + 5 3 2 6 8 6

9 20 1 1 2 3 4) + 5) 3 + 2 6) 8 + 6 14 21 4 2 3 4

4 7 5 2 7 1 7) − 8) − 9) 3 −1 5 10 6 9 12 4

1 5 5 1 10) 4 − 2 11) 4 − 12) − 36 3 7 6 4

326

Practice Test for Skills Quiz B – Fractions

1 3 1.) Add: + 6 12

2 2 2.) Subtract: − 3 9

4 10 3.) Multiply: 2 ⋅ 5 7

5 4.) Multiply: ⋅ 7 21

3 18 5.) Divide: ÷ 4 8

4 6.) Divide: 2 ÷14 5

2 5 7.) Subtract: − 3 18

327

Answer Key for Sample Fractions Skills Test

5 1.) 12

4 2.) 9

3.) 4

4.) 5/3 or 1 2/3

1 5.) 3

1 6.) 5

7 7.) 18

328 C.1 Signed Numbers

All the numbers we’ve looked at up until now have been positive numbers: numbers bigger than zero. If a number is less than zero, it is a negative number.

Example: The temperature is 20 degrees below 0°. If the temperature was 30 degrees above 0°, we’d just write 30°. Since the temperature is 20 degrees below 0°, we write -20°.

Example: Ben overdrew his bank account, and now owes them $50. Since his account balance is below $0, we could write the balance as -$50.

We can visualize negative numbers using a number line. Values increase as you move to the right and decrease to the left. Negative Positive

-8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8

Every number has an opposite: a number on the other side of zero, the same distance from zero. Example: Find the opposite of: a) 5 b) -3 c) ½ Since 5 is five units to the right of zero, the opposite is five units to the left: -5 Since -3 is three units to the left of zero, the opposite is three units to the right: 3 Since ½ to the right of zero, the opposite is to the left: -½

Example: Place these numbers on the number line: a) 4 b) -6 c) -3.5 d) -1¼ a) 4 is four units to the right of zero. b) -6 is six units to the left of zero c) -3.5 is halfway between -3 and -4 d) -1¼ is further left than -1; it is the opposite of 1¼ -6 -3.5 -1¼ 4

-8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8

We can compare two signed numbers by thinking about their location on the number line. A number further left on the number line is smaller than a number to its right.

Example: Write < or > to compare the numbers: a) 3 __ 5 b) -4 __ 3 c) -2 __ -5 d) -2.1 __ -2.4 a) On a number line, 3 is to the left of 5, so 3 < 5 b) On the number line, -4 is to the left of 3, so -4 < 3 c) On a number line, -2 is to the right of -5, so -2 > -5 d) On a number line, -2.1 is to the right of -2.4, so -2.1 > -2.4

329 Worksheet – C.1 Signed Numbers Name: ______

Write a signed number for each situation 1) I deposit $200 in my bank account 2) I withdraw $100 from my account

3) 20 feet above sea level 4) 40 feet below sea level

Find the opposite of each number 5) 3 6) -7 7) 3.7 8) -2.6 9) 3½ 10) -4¾

Place each number on the number line 11) -7 12) 4 13) -4.7 14) 3.2 15) 6¾ 16) -4¾

-8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8

Write < or > to compare the numbers 17) 2 __ 8 18) 207 __ 198 19) 23 __ -37 20) -15 __ 34

21) -2 __ -7 22) -8 __ -4 23) -152 __ -130 24) -1743 __ -823

25) 2.3 __ 3.1 26) -5.3 __ -5.8 27) -3.8 __ 2.3 28) -0.3 __ -0.07

2 1 2 4 1 3 1 29) __ 30) __−− 31) 3 −− 3__ 32) −− 6__6 3 3 5 5 4 4 2

2 1 5 7 3 5 5 33) __ 34) __−− 35) 1 −1__ 36) −− 1__2.1 5 3 8 10 4 7 12 330 C.2 Adding / Subtracting Signed Numbers

To add signed numbers of the same sign (both positive or both negative) • Add the absolute values of the numbers • If both numbers are negative, the sum is negative

Example: Add: –8 + (–5) Since both numbers are negative, we add their absolute values: 8 + 5 = 13 The result will be negative: –8 + (–5) = –13

To add signed numbers of opposite sign (one positive, one negative) • Find the absolute value of each number • Subtract the smaller absolute value from the larger value • If the negative number had larger absolute value, the result will be negative

Example: Add: –4 + 9 The absolute values of the two numbers are 9 and 4. We subtract the smaller from the larger: 9 – 4 = 5 Since 9 had the larger absolute value and is positive, the result will be positive. 9 + (–4) = 5

Example: Add: 5 + (–8) The absolute values of the two numbers are 5 and 8. We subtract the smaller from the larger: 8 – 5 = 3 Since 8 had the larger absolute value and is negative, the result will be negative. 5 + (–8) = –3

Notice that 5 – 3 is the same as 5 + (–3). Likewise, 5 – (–3) is the same as 5 + 3. Subtracting a number is the same as adding its opposite. To subtract signed numbers: • Rewrite subtraction as adding the opposite of the second number: a – b = a + (–b) and a – (–b) = a + b

Example: Subtract: 10 – (–3) We rewrite the subtraction as adding the opposite: 10 + 3 = 13

Example: Subtract: –5.3 – 6.1 We rewrite the subtraction as adding the opposite: –5.3 + (– 6.1) Since these have the same sign, we add their absolute values: 5.3 + 6.1 = 11.4 Since both are negative the result is negative: –5.3 – 6.1 = –11.4

1 2 Example: Subtract: − 6 3 1 4 1  4  First, we give these a common denominator: − . Next, rewrite as adding the opposite: −+  6 6 6  6  4 1 3 1 Since these are opposite signs, we subtract the absolute values: ==− 6 6 6 2 1 2 1 Since the negative number had larger absolute value, the result is negative: −=− 6 3 2 331 Worksheet – C.2 Add / Subtract Signed Numbers Name: ______

Add or Subtract: 1) –8 + 3 2) –1 + 13 3) 8 + (–6) 4) 120 + (–150)

5) –14 + (–10) 6) 7.1 + 3.6 7) –1.6 + 3.4 8) –0.4+ (–0.8)

9) 18 – 6 10) 6 – 18 11) 23 – 67 12) –10 – 8

13) –10 – (–4) 14) 26 – (–12) 15) 8.3 – 10.4 16) –3.22 – 4

4 8 1 5 1 1 5 17) − 18) − 19) 2 −− 4 20) 5 − 3 5 5 12 6 2 4 6

21) In Fargo it was -18°F, while in Tacoma it was 43°F. How much warmer was Tacoma?

22) Darrel’s account was overdrawn by $120, before he deposited $450. What is his balance now?

332 C.3 Multiplying / Dividing Signed Numbers

To multiply or divide two signed numbers • If the two numbers have different sign, the result will be negative • If the two numbers have the same sign, the result will be positive

Example: Multiply: a) ⋅− 34 b) − )6(5 c) −− )4(7 a) The factors have different signs, so the result will be negative: −=⋅− 1234 b) The factors have different signs, so the result will be negative: −=− 30)6(5 c) The factors have the same signs, so the result will be positive: =−− 28)4(7

− 36 Example: Divide: a) ÷− 1040 b) −÷ )4(8 c) − 3 a) The numbers have different signs, so the result will be negative: −=÷− 41040 b) The numbers have different signs, so the result will be negative: −=−÷ 2)4(8 − 36 c) The numbers have the same signs, so the result will be positive: = 12 − 3

The same rules apply to deciding signs when multiplying or dividing fractions and decimals −18 2 6 Example: Calculate: a) ÷− 128 b) c) ⋅− − 4 3 7 a) Since this doesn’t divide evenly, we can either use decimals or fractions. Using fractions, −8 2 128 =÷− −= . 12 3 −18 9 1 b) Since both numbers have the same sign, the result will be positive: == 4 − 4 2 2 2 6 2 2 4 c) Different signs, so the result will be negative: −=⋅−=⋅− 3 7 1 7 7

Remember the following rules for multiplying and dividing fractions (look back to the review skills for quiz #2 if you need a refresher):

*For multiplying fractions – multiply across the numerator and denominator – reduce if you can *For dividing fractions – multiply by the reciprocal of the second fraction instead *For multiplying or dividing with mixed numbers – change to improper fractions before you start

333 Worksheet – C.3 Multiply / Divide Signed Numbers Name: ______

Multiply or divide 1) ⋅− 47 2) −− )8(5 3) − )3(5 4) 3( 6)

− −

5) −÷ )4(32 6) −÷− )8(48 7) ÷− 330 8) 72 ÷ ( 9)

− −

−16 − 5 12 9) 10) 11) 12) − 4 10 −1 −30 −40

3  5  4  3  3 13) −⋅  14) −÷−  15) 10 ⋅− 16) ( 20)( 3) 8  6  5  10  4 − −

17) 18) ÷ 19) 2 ÷ ( 4) 20) 5 ÷ 3 4 8 1 5 1 1 5

5 ∙ �− 5� 12 − 6 − 2 − − 4 6

334 C.4 Order of Operations with Signed Numbers

The same order of operations we used before also applies to signed numbers – don’t forget PEMDAS!

Example: Simplify: 30 ÷ ( 5 6) Do what is in the parentheses first, so: 5 6 = 30 : 30 ÷ ( ) − ∙ Now we can do the division : − ∙ − −𝟑𝟑𝟑𝟑

−𝟏𝟏 Example: Simplify: 2 3(1 5) Do what is in the parentheses first, so: 1 5 = 1 + ( 5) = 4 : 2 3( ) − − No exponents, so do multiplication or division next: 3( 4) = 12 : 2 ( ) − − − − −𝟒𝟒 Rewrite the subtraction to add the opposite: 2 ( 12) = 2 + 12 = 14: 14 − − − −𝟏𝟏𝟏𝟏

− −

Example: Simplify: 3( 5)( 9) All multiplication, so do them in order, left to right: 3( 5) = 15 : ( 9) − − − Now do the last multiplication: − − 𝟏𝟏𝟏𝟏 −

−𝟏𝟏𝟏𝟏𝟏𝟏 A Warning about Exponents When following the order of operations, 3 and ( 3) do NOT mean the same thing. 3 means the opposite of 3 squared, which2 is equal2 to -9. ONLY the 3 is squared. − − ( 3)2 is different. Here the parentheses say you need to square ALL of -3. − 2 This means -3*-3, which gives you a POSITIVE 9. −

Example: Simplify: 5( 2) + ( 2) Although there are parentheses,2 there are no operations inside them: − − Moving on to exponents: ( 2) = 2 2 = 4 : 5( 2) + Next do the multiplication:2 5( 2) = 10: + 4 − − ∙ − − 𝟒𝟒 Finish by doing the addition: − − −𝟏𝟏𝟏𝟏

−𝟔𝟔

Example: Simplify: 3 5( 3 + 32 ÷ ( 2)) We begin with the inside of the paren2 theses, with the exponent: 3 5( + 32 ÷ 2) − − − − Still inside the parentheses, we do the division next: 3 5( 9 + ( )) − − −𝟗𝟗 − Still inside the parentheses, we add 3 5( ) − − − −𝟏𝟏𝟏𝟏 Now multiply 3 ( ) − − −𝟐𝟐𝟐𝟐 Rewrite the subtraction by adding the opposite 3 + 125 − − −𝟏𝟏𝟏𝟏𝟏𝟏 Add 122 −

335 Worksheet – C.4 Order of Operations with Signed Numbers Name: ______

Evaluate. Only perform one step at a time. Show all your work vertically.

1) 5 + 4( 2) 2) 50 ÷ ( 2) ( 5) 3) ( 2)( 3)( 4)

− − ∙ − − − −

4) 4 ( 5) 6 5) 8 + 30 ÷ 6 6) 9 5 ( 2)

− − ∙ − − − − −

7) ( 4) 2 8) 4 2 9) ( 5)( 7) + ( 2) 2 2 3 − − − − − − −

10) ( 6) 2 11) ( 5) + 1 12) ( 7)( 4) + ( 9) 2 3 2 2 − − − − − − −

13) 14) ( 8) ÷ ( 2) 15) 1 + 5( 2 14 ÷ ( 2)) 1 3 1 2 2 4 − 4 ∙ 6 − − − − − − −

336

Practice Test for Skills Quiz C – Arithmetic with Signed Numbers

1) 4 + 15

− 2) ( 9)( 2)

− −

3) ( 2) 3 −

4) 4 2 −

5) 1 2

4 − 3

6) Simplify: 6 2 ( 12) 1

− − − − − 7) Simplify: 6 9 10 + 17

− − 8) Simplify 3(−2) + (– 2)3

9) Simplify: −22 + (−50)

10) Simplify: − (−2)2 + (−9 + 1)

337

Answer Key for Sample Arithmetic of Signed Numbers Skills Test

1) 11

2) 18

3) -8

4) -16

5) 5

− 12 6) 3

7) 4

8) -14

9) -54

10) -10

338

D.1 Intro to Percents What is a percent Percent means “per hundred” or “out of 100.” The symbol % is used after a number to indicate a percent.

15 Example: 15% means 15 out of 100, or as fraction. Visually, the box 100 to the right has 15% of the squares shaded: 15 out of the 100.

Writing a percent as a decimal or fraction

To write a percent as a fraction, write the percent as a fraction of 100. 23 Example: Write 23% as a fraction. 23% = 100 50 1 Example: Write 50% as a fraction. 50% = = 100 2

To write a decimal as a percent, look to see how many hundredths you have 40 Example: Write 0.4 as a percent. 0.4 = 0.40 = = 40% 100

You may notice this is the same as moving the decimal place to the right two places Example: Write 0.057 as a percent. Moving the decimal to the right two places: 5.7%

To convert a percent to a decimal, write the decimal out of 100 and divide. Notice this is the same as moving the decimal place to the left two places. Example: Write 12.5% as a decimal and as a fraction. 5.12 As a decimal, 12.5% = = 125.0 100 125 1 To write as a fraction, we could start with the decimal: 125.0 == 1000 8

To write a fraction as a percent, there are 2 good methods you can use: 1) Create an equivalent fraction with 100 as the denominator 2) Divide the top of the fraction by the bottom to find a decimal, then write it as a percent. If you do not have a calculator, method 1 is generally easier (since you don’t have to do long division!) 2 Example: Write as a percent. 5 ? Method 1: = . To get the denominator from 5 to 100, you’d have to multiply by 20. So multiply 2 the top by 20 to get5 2*20100 = 40. So = 40% 40 ==÷ Method 2: 40.04.052 100 = 40%

Note— if you can’t come up with a nice factor to multiply to get the denominator equal to 100, just use method 2 339

Worksheet – D.1 Intro to Percents Name: ______

Rewrite as a decimal 1) 20% 2) 46% 3) 7.4% 4) 0.3% 5) 127%

Rewrite as a percent 6) 0.74 7) 0.9 8) 0.0254 9) 1.35 10) 0.05

Rewrite as a reduced fraction 11) 0.25 12) 0.3 13) 0.05 14) 0.025

Rewrite as a fraction. Reduce if possible. 15) 30% 16)13% 17) 5% 18) 43.2% 19) 0.3%

Rewrite as a percent. Round to the nearest tenth of a percent if needed. 3 1 1 7 47 20) 21) 22) 23) 24) 4 2 6 10 50

340

D.2 Solving Percent Problems Pieces of a percent problem Percent problems involve three quantities: the base amount (the whole), the percent, and the amount (a part of the whole). The amount is a percent of the base.

Example: 50% of 20 is 10. 20 is the base (the whole). 50% is the percent, and 10 is the amount (part of the whole)

In percent problems, one of these quantities will be unknown. Here are the three cases: Example: What is 25% of 80? 80 is the base (the whole) we are finding a percent of. The percent is 25%. The amount is unknown. Example: 60 is 40% of what number? The percent is 40%. The unknown is the base that we are finding a percent of. The amount (part of the whole) is 60. Example: What percent of 320 is 80? The base we are finding a percent of is 320. The percent is unknown. The amount is 80.

Solving percent problems To solve percent problems, we can use the idea of equivalent fractions. The percentage in the problem can be written as a fraction divided by 100. The other values can be written in fractional form. Then we just need to find the missing value: Amount = Percent Base 100 Example: What is 25% of 80? The base is 80 and the percent is 25%, so we can set up the following equivalent fraction problem: ? = . In this case, it’s not easy to see what to multiply 80 by to get to 100. However, this problem 25 gets80 much100 easier if you can reduce/simplify the fraction on the right. Since = , then we can solve: ? 25 1 = . To get from 4 to 80, we need to times by 20. So times the top by 20100 as well.4 1*20 = 20. So 20 1 is80 25%4 of 80.

Example: 60 is 40% of what number? The percent is 40%., the amount is 60, and the base is unknown. Using this, we can set up the following = equivalent fraction problem. ? . Again, this problem is more easily solved if we reduce to . 60 40 40 2 = So to find the base, use the relationship100 ? . To get from 2 to 60, we would have to multiply100 by 30.5 60 2 Multiplying the bottom by 30, we find 5*30 =5 150. So 60 is 40% of 150.

Example: What percent of 320 is 80? The base is 320, the amount is 80, and the percent is unknown. Using this, we can set up the following ? equivalent fraction problem. = . After reducing = , we can find the missing piece of ? 80 80 1 information: = . To get 320from 4100 to 100, we would have320 to 4multiply by 25. Multiplying the top by 1 25, we find 1*254 =100 25. So 25% of 320 is 80. Don’t forget to add in the % symbol when solving for the percentage! 341

Worksheet – D.2 Solving Percent Problems Name: ______

1) 30% of what number is 54 2) What number is 40% of 8?

3) What percent of 200 is 40? 4) 10 is 5% of what number?

5) What is 120% of 30? 6) $30 is what percent of $80?

7) Out of 300 diners, 60 ordered salads. What percent of diners ordered a salad?

8) The population of the US is around 300,000,000. How many people make up 1% of the population?

9) Bob bought a $800 TV on sale for $650. What percent savings is that? (be careful!)

10) Out of 200 people, 40 own dogs. What percent is that? Out of 550 people, how many would you expect to own dogs?

342 D.3 Proportions

Percentage problems are a special case of a bigger family of math problems. A proportion is any equation showing the equivalence of two fractions. (Note, fractions are sometimes also referred to as ratios or rates). flour cups 2 cups flour 4 cups flour Example: Is this proportion true? = cookies 03 cookies 06 cookies Yes, this proportion is true since the two fractions are equivalent since = and = . Also both rates 2 1 4 1 have the same units of “cups flour per cookies.” 30 15 60 15

gallons 2 gallons 001 miles Example: Is this proportion true? = 05 miles 4 gallons This proportion cannot be true since the rate on the left has units “gallons per mile” and the rate on the right has units “miles per gallon”. Since the units are different, these are not comparable.

Sometimes it is tricky to check if two fractions/ratios reduce to the same value. Another method is this: if we cross-multiply, both sides of a true proportion will be equal.

Example: = is a true proportion since both fractions can be reduced to the same value, . Or, 4 8 4 5 = 10 is a true proportion since ⋅=⋅ 58104 5 4 8 5 10 n 12 Example: Solve for the unknown n. = 20 80 Find the value of n to make these two fractions equivalent. To get from 80 to 20, you can divide by 4. So dividing 12 by 4, I get n = 3.

Example: A picture is taken that is 4 inches tall and 6 inches width. If you want to enlarge the photo to be 10 inches tall, how wide will it be? 4 inches tall 01 inches tall We can set up a proportion, where both ratios have the same units: = . 6 wideinches x wideinches Find the value of x that will make the proportion true. This problem is solved more easily if we can reduce

= the left proportion first. . To get from 2 to 10 in the numerator, I can multiply by 2 𝑖𝑖𝑖𝑖𝑖𝑖ℎ𝑒𝑒𝑒𝑒 𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 10 𝑖𝑖𝑖𝑖𝑖𝑖ℎ𝑒𝑒𝑒𝑒 𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 5. So if I multiply the denominator3 𝑖𝑖𝑖𝑖𝑖𝑖ℎ𝑒𝑒𝑒𝑒 𝑤𝑤𝑤𝑤𝑤𝑤𝑤𝑤 by 5,𝑥𝑥 𝑖𝑖𝑖𝑖𝑖𝑖I’llℎ 𝑒𝑒𝑒𝑒get𝑤𝑤 𝑤𝑤an𝑤𝑤𝑤𝑤 equivalent fraction: x = 3*5 = 15 inches wide

Example: A report shows 2 out of 3 students receive some financial aid. Out of 1200 students, how many would you expect to be receiving financial aid? students 2 students with financial aid n students with financial aid We can set up a proportion: = . students 3 students total 2001 students total To get from 3 to 1200 in the denominator, I can multiply by 400. So to get an equivalent fraction, I need to also multiply the top by 400. n = 2*400 = 800 students with financial aid.

NOTE: As another method, to solve for an unknown in a proportion, we can cross-multiply, then divide. 2400 312002 ⋅=⋅ n , or = 32400 n . Dividing, n == 800 students receiving financial aid. 3 343 Worksheet – D.3 Proportions Name: ______

Is each proportion true? 24 trees 48 acres 18 45 $3 50.4$ 1) = 2) = 3) = 4) = 2 acres 4 trees 6 15 21 eggs 81 eggs 3 10 4 16

Solve the proportion for the unknown. [Hint: when solving using equivalent fractions, it often helps to reduce the known fraction before trying to figure out the unknown value]

n 10 32 r 749 20 5 5) = 6) = 7) = 8) = 6 15 114 q 9 8 x

First write a proportion to represent the situation. Be sure to label units! Then solve for the requested value.

9) If 6 ounces noodles makes 4 servings, how many ounces of noodles do you need for 20 servings?

10) At 1pm, Mikayla’s shadow was 0.5 meters long. Mikayla is 1.5 meters tall. She measures a tree’s shadow to be 7 meters. How tall is the tree?

11) If you are supposed to mix 3 ounces of floor cleaner to every 2 cups of water, how much water should you mix with 8 ounces of cleaner? [hint: this proportion may be easier to solve using the “cross-multiply and divide” method]

344 Practice Test for Skills Quiz D – Percentages and Proportions

1.) 10 is 25% of what number?

2.) What is 25% of 40?

3.) 70 is what percent of 140?

4.) Write the following Fraction as a Decimal AND a Percentage

1/5

Decimal: ______Percentage: ______

5.) Write the following Decimal as a Fraction AND a Percentage

0.35

Fraction: ______Percentage: ______

6.) Write the following Percentage as a Fraction AND a Decimal

70%

Fraction: ______Decimal: ______

7.) Solve the following proportion

= 𝑥𝑥 24

8.)5 Solve30 the following proportion

4 20 = 7

Find the𝑥𝑥 length of a wall represented by 2 inches on a blueprint if 1 inch represents 8 feet.

9.) Set up the proportion to solve the problem above. Include units

10.) Solve for the actual length of the wall described in the blueprint. 345 Answer Key for Sample Percentages and Proportions Skills Test

1.) 40

2.) 10

3.) 50%

4.) Decimal: ____0.20______Percentage: ___20%______

5.) Fraction: ____35/100 or 7/20______Percentage: _____35%______

6.) Fraction: ____70/100 or 7/10______Decimal: ______0.70______

7.) x = 4

8.) x = 35

Find the length of a wall represented by 2 inches on a blueprint if 1 inch represents 8 feet.

9.) Set up the proportion to solve the problem above

1 2 = 8 𝑖𝑖𝑖𝑖 𝑖𝑖𝑖𝑖 10.)𝑓𝑓𝑓𝑓 x = 16ft𝑥𝑥

346 E.1 Combining Like Terms and the Distributive Property

Often we need to use formulas to solve problems. For example, you might remember the formula we use to find the perimeter of a rectangle: P = 2L + 2W. Formulas contain variables - letters used to represent unknown quantities. In the formula above, P, L, and W are the variables. To make formulas easier to work with, we use some rules to make the expressions simpler. You’ll learn a lot more about this next quarter, but let’s look at a few important rules for working with variables

Combining like terms is what we do when we combine similar things. Example: Simplify the formula A = 3p + 2n + 5p – n First, notice that 3p and 5p have the same kind of thing: p’s. If we have 3 of them and add 5 more, now we’ll have 8 p’s. So we can combine the “like” p terms by adding: 3p + 5p = 8p. Also, notice that there is more than one n variable in the expression above. There is a 2n and an n. The second n has no number in front of it. So we assume it is a 1n. Also, note that the n is being subtracted. It is VERY important to always keep the sign to the left of the variable when combining (note if it is a + or -). So we have 2 n’s and want to subtract 1 of those n’s. So to combine the “like” n terms: 2n – n = 2n – 1n = 1n. Note that this is a POSITIVE 1n. So: A = 3p + 2n + 5p – n can be simplified to A = 8p + 1n. The “1” in front of a variable is considered optional, so you’ll usually just see this written as: A = 8p + n.

Example: Simplify 56242 +−++− yxyx We can combine the terms with x in them: 2x + 6x = 8x. We can combine the terms with y in them: –4y + y = –3y We can combine the numbers: 2 – 5 = –3 Altogether, 56242 +−++− yxyx simplifies to

𝟖𝟖𝟖𝟖 − 𝟑𝟑𝟑𝟑 − 𝟑𝟑 The distributive property allows us to get rid of parentheses when normally the order of operations doesn’t work. For example, if you consider the expression 2(x + 3), PEMDAS says we have to do x + 3 first. This is impossible when you do not have a value for x. We can use the distributive property to get rid of the parentheses so we can move on and combine like terms. In general, )( ⋅+⋅=+ cabacba Example: xxx +=⋅+⋅=+ 62322)3(2 Example: xxx −=⋅−⋅=− 62322)3(2 Example: x x x () x x +−=−−−=⋅−−⋅−=−−=−− 23)2(32131)23(1)23( or, x x () x x +−=−−+⋅−=−+−=−− 23)2)(1(31)23(1)23(

We still need to follow the order of operations! So, when trying to combine like terms, always make sure you use the distributive property FIRST before trying to combine like terms.

Example: Simplify: 2 + 2(3 5) We need to deal with parentheses first. Since we cannot do 3x – 5, we need to distribute: 𝑥𝑥 − 2(3 5) = 2 3 2 5 = So our expression is: 2 + 6 10. Now that there are no parentheses, we can combine like terms. 𝑥𝑥 − ∙ 𝑥𝑥 − ∙ 𝟔𝟔𝟔𝟔 − 𝟏𝟏𝟏𝟏 We can combine the plain numbers 2 – 10 = - 8. The 6x is the only term with x’s. Putting those 𝑥𝑥 − expressions together, we end up with either or + as a final solution.

𝟔𝟔𝟔𝟔 − 𝟖𝟖 −𝟖𝟖 𝟔𝟔𝟔𝟔 347 Worksheet – E.1 Evaluating Formulas Name: ______

Simplify the formula by combining like terms. 1) ++−= 4326 abaaA 2) −+= 352 xxy

3) y −+−= 2062006 pp 4) R x −−+= 1010015200 x

Use the distributive property to rewrite the expressions below without parentheses.

5) − p)5(4 6) p +− )43(2 7) −− p)25(3

8) 5(4 6 ) 9) (6 + 8) 10) 4( 1) 1

𝑎𝑎 − 𝑏𝑏 2 𝑦𝑦 𝑦𝑦 −

Use distribution and combining like terms to simplify each expression 11) 3( 1) + 12) xx +− )1(32 13) xx −+ )1(45

− 𝑥𝑥 − 𝑥𝑥

14) +− 3)3(2 ww 15) 7(2 + 3) + 1 16) 3(5 7) 9

𝑥𝑥 𝑥𝑥 − −

17) 2 3 + 3( 2) 18) n −− )32(4 19) 7 6( 1) + 2

𝑥𝑥 − 𝑥𝑥 − 𝑦𝑦 − 𝑦𝑦 −

348 E.2 Solving 1-Step Equations

Much of algebra revolves around the concept of trying to solve equations. As an example, we have a formula where we know the answer we want, but don’t know one of the inputs. In these cases, we need to find the missing part of the problem. In the next few lessons, we will work on developing some of the fundamental skills needed to solve for an unknown variable.

Using the Inverse Operation To solve an equation, our goal is to get the variable (or unknown quantity) alone. To get rid of any numbers or values that are attached to our variable, we need to “undo” the operation keeping it there. At this point, we will just stick to our standard arithmetic operations. Addition and subtraction are considered inverse operations. Multiplication and division are considered inverse operations.

Remember, anything you do to one side of the equation, you have to do to the other side as well

Example: Solve: + 5 = 12 Our goal is to get the x variable by itself on one side of the equation. Right now, it is tied to a +5. To 𝑥𝑥 get rid of a + 5, we need to subtract 5. Because this is an equation we have to do this to both sides of the equation: + 5 = 12 5 5 𝑥𝑥 ------− − = 7 Notice that on the left side, the addition and subtraction undo each other, leaving just x. On the other side, after combining𝑥𝑥 like terms, we get the solution of 7. Since the x is alone, we have achieved our goal! We know that the variable x must be equal to 7 to make this equation true. In fact, if you put 7 in for x you can see that it checks out: 7 + 5 = 12. **NOTE – we are starting with simple equations. You may know the answers to these in your head by looking! However, be sure to show all inverse operation, and all of your steps. We are developing patterns and habits that will be invaluable when the formulas and equations get more complex!

Example: Solve: 10 = 6 To get the x alone, we need to get rid of a subtracted 10. We can do this by adding 10 to each side: 𝑥𝑥 − 10 = 6 + 10 + 10 𝑥𝑥 − ------= 16

𝑥𝑥 Example: Solve: 2 = 24 To get the x alone, we need to get rid of a -2 which is being multiplied. We can do this by dividing each − 𝑥𝑥 side by -2. On the left, the multiplication and division undo each other. On the right, we get our answer: 2 = 24 ÷ 2 ÷ 2 − 𝑥𝑥 ------− − = 12

𝑥𝑥 − 349 Worksheet – E.2 Solving 1-Step Equations Name: ______

Solve each equation. Be sure to show what inverse operation you did on each side of the equation as well as your final answer.

1) + 15 = 24 2) 13 + = 2 3) 12 + = 0

𝑥𝑥 𝑦𝑦 − − 𝑚𝑚

4) 3 = 14 5) 10 = 6 6) 5 = + 4

𝑥𝑥 − 𝑛𝑛 − − 𝑎𝑎

7) 3 = 27 8) 5 = 25 9) 16 = 4

𝑥𝑥 − 𝑥𝑥 − 𝑦𝑦

10) = 30 11) = 2 12) = 8 𝑥𝑥 𝑦𝑦 𝑛𝑛

5 −3 −9 −

13) 5 + = 12 14) 24 = 4 15) 10 = 120

− 𝑥𝑥 𝑥𝑥 − − − 𝑚𝑚 −

Reminders: When adding and subtracting fractions, make sure you have a common denominator. When dividing fractions, be sure to multiply by the reciprocal.

16) + = 17) + = 18) = 15 1 5 2 11 3

− 6 𝑥𝑥 6 𝑛𝑛 3 12 4 𝑦𝑦 350 E.3 Solving 2-Step Equations

In the last section, all the equations only had 1 number on the same side of the equation as the unknown variable. However, this is rarely the case! To solve equations when there is “more” going on we are going to use the same principles! We need to “undo” each operation that is with our variable. The order of “undo”-ing is very important – we need to follow the reverse of the order of operations when applying the inverse operations. So first, get rid of any added or subtracted values. Then any multiplied or divided values.

Solving a 2-Step Equation Example: Solve: 2 3 = 11

𝑥𝑥 − 2 3 = 11 we need to get the x alone! + 3 + 3 start by getting rid of the minus 3 by adding 3 𝑥𝑥 − ------2 = 14 now get rid of the times 2 by dividing by 2 ÷ 2 ÷ 2 𝑥𝑥 ------=

Again, 𝒙𝒙note 𝟕𝟕that you can check your answer by plugging x = 7 into the original equation. It should be true! 2 7 3 = 14 3 = 11

∙ − − Example: Solve: 4 2 = 8

− 𝑥𝑥 4 2 = 8 we need to get the x alone! -4 -4 start by getting rid of the positive 4 added out front − 𝑥𝑥 ------2 = 4 *NOTE* the – is part of the 2x, so it stays with the 2x ÷ -2 ÷ -2 now get rid of the times -2 by dividing by -2 − 𝑥𝑥 ------=

𝒙𝒙 −𝟐𝟐 Example: Solve: 3(2 5) = 15 *Note because the x is in parentheses, our reverse order of operations is affected. There are 2 approaches: 𝑥𝑥 −

3Method 1: reverse order of operations Method 2: use distribute property then solve

3(2 5) = 15  divide both sides by 3 first 3(2 5) = 15  distribute the 3 first ÷3 ÷3 3 2 3 5 = 15 𝑥𝑥 − 𝑥𝑥 − 2 5 = 5  add 5 to each side next 6 15 = 15  add 15 to each side ∙ 𝑥𝑥 − ∙ +5 +5 +15 +15 𝑥𝑥 − 𝑥𝑥 − 2 = 10  divide by 2 to each side 6 = 30  divide by 6 on each side ÷2 ÷2 ÷6 ÷6 𝑥𝑥 𝑥𝑥 =  THE SOLUTION =  THE SOLUTION

𝒙𝒙 𝟓𝟓 𝒙𝒙 𝟓𝟓 351 Worksheet – E.3 Solving 2-Step Equations Name: ______

Solve each equation. Be sure to show what inverse operations you did on each side of the equation (only 1 per line!) as well as your final answer.

1) 3 + 8 = 17 2) 4 + 2 = 18 3) 2 5 = 7

𝑥𝑥 − 𝑥𝑥 − 𝑦𝑦 −

4) 2 + 2 = 27 5) 37 = 8 + 3 6) 2 + = 5 𝑛𝑛

− 𝑚𝑚 − 𝑥𝑥 − 7

7) + 7 = 11 8) 2 3 = 29 9) 14 + 8 = 54 𝑛𝑛

3 − 𝑥𝑥 − − 𝑘𝑘

10) 5(2 1) = 35 11) 3( 6) = 24 12) 2( 8 5) = 6

𝑥𝑥 − − 𝑥𝑥 − − 𝑥𝑥 −

352 E.4 Solving Equations with Variables on Both Sides

So far, all the equations that we’ve solved have only had a single variable present in the equation. In this section we will learn to solve linear equations when the unknown variable shows up more than once.

Simplify each side of an equation (combining any like terms) BEFORE trying to solve. Example: Solve: 2( 5) + 3 = 35 2( 5) + 3 = 35 Notice there is more than one x in the problem 𝑥𝑥 − 𝑥𝑥 2 2 5 + 3 = 35 Use distributive property to get rid of parentheses 𝑥𝑥 − 𝑥𝑥 2 10 + 3 = 35 Simplify the distribution ∙ 𝑥𝑥 − ∙ 𝑥𝑥 5 10 = 35 Combine like terms (add the x’s together!) 𝑥𝑥 − 𝑥𝑥 *Now that there is only a single x, we can solve using the processes from the last lesson. 𝑥𝑥 − +10 +10 Undo the subtracting 10 by adding 10 to each side ------5 = 45 ÷5 ÷5 Undo the multiplying 5 by dividing each side by 5 ------𝑥𝑥 = THE SOLUTION!

𝒙𝒙 𝟗𝟗 If you end up with a variable on opposite sides of the equation, move one of the terms over to the other side by adding or subtracting. You can then combine it with the matching variable on the other side. Example: Solve: 21 + 12 = 6 3 21 + 12 = 6 3 Notice there is more than one x in the problem − 𝑥𝑥 − − 𝑥𝑥 +3x +3x Either add 3x OR add 21x to each side − 𝑥𝑥 − − 𝑥𝑥 ------18 + 12 = 6 Combine like terms (add the x’s together!) -12 -12 Undo adding 12 by subtracting 12 from each side − 𝑥𝑥 − ------18 = 18 ÷ -18 ÷ -18 Undo multiplying -18 by dividing each side by -18 ------− 𝑥𝑥 − = THE SOLUTION!

𝒙𝒙 𝟏𝟏 Example: Solve: 4 + 4 = 4(8 8) 4 + 4 = 4(8 8) Simplify each side before starting (distribute!) − 𝑘𝑘 𝑘𝑘 − 4 + 4 = 32 32 Either subtract 4k OR subtract 32k from each side − 𝑘𝑘 𝑘𝑘 − -4k -4k − 𝑘𝑘 𝑘𝑘 − ------4 = 28 32 Combine like terms (add the x’s together!) +32 +32 Undo subtracting 32 by adding 32 to each side − 𝑘𝑘 − ------28 = 28 Undo multiply 28 by dividing each side by 28 ------𝑘𝑘 = THE SOLUTION!

𝟏𝟏 𝒌𝒌 353 Worksheet – E.4 Solving Equations with Variables on Both Sides Name: ______

Solve each equation. Be sure to show what inverse operations you did on each side of the equation (only 1 per line!) as well as your final answer.

1) 2 + 6 7 = 4 2) 2 + 24 = 3 1

𝑥𝑥 − 𝑥𝑥 − 𝑦𝑦 − 𝑦𝑦 −

3) 3( 4) = 2 + 2 4) 7 + 5 = 4 + 17

− 𝑥𝑥 − 𝑥𝑥 𝑥𝑥 𝑥𝑥

5) 17 2 = 35 8 6) 5 + 4 = 3 + 16

− 𝑚𝑚 − 𝑚𝑚 𝑥𝑥 𝑥𝑥

7) 2( + 3) = 5 3 8) 5 2( + 3) = 12

𝑥𝑥 𝑥𝑥 − 𝑥𝑥 − 𝑥𝑥

9) 2 2 = + 7 10) 2(8 4) = 8(1 + )

𝑥𝑥 − −𝑥𝑥 𝑛𝑛 − 𝑛𝑛 354 Practice Test for Skills Quiz E – Solving Equations

Simplify the following expressions

1.) 2x + 7 + 3x + 2 - 14

2.) 3(7x – 3)

3.) 2 + 5 21 + 4

𝑥𝑥 − 𝑥𝑥

4.) 5 9 + 3 + 2

𝑥𝑥 − 𝑥𝑥 𝑥𝑥

Solve the following algebraic equations.

5.) 4 = 28

𝑥𝑥

6.) x/6 = 3

7.) 4 + 7 = 11 + 6

𝑥𝑥 𝑥𝑥

8.) 3x + 7 = 4

9.) 2(2x – 5) = 10

10.) 7y = 3(y – 4)

355

Answer Key for Sample Solving Equations Skills Test

1.) 5x + 5

2.) 21x – 9

3.) 6 14

𝑥𝑥 − 4.) 1 + 2

− 𝑥𝑥

Solve the following algebraic equations.

5.) = 7

𝑥𝑥 6.) x = 18

7.) = 9

𝑥𝑥 − 8.) x = -1

9.) x = 5

10.) y = – 3