quantway instructor’s notes lesson 1.1.1: …...productive persistence focus level of productive...

Quantway Instructor’s Notes

Lesson 1.1.1: Introduction to Quantitative Reasoning Theme: Citizenship

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

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Main Math Topic Main Quantitative Reasoning Context

Productive Persistence Focus

Level of Productive Struggle

Defining quantitative reasoning

Statistical data in media Group norms Level 1

Prerequisite Assumptions Before beginning this lesson, students should be able to

• understand the concept of doubling. • do calculations involving doubling or multiplication with numbers under 1,000. • understand the structure of place value.

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.

Explicit Connections [Connections to future lessons] • The foundation of quantitative reasoning is the mindset to ask questions about information. • Estimation and understanding and using large numbers are two valuable skills that students will see

used repeatedly in lessons.

Notes to Self One thing I want to do during this lesson …

One thing I want to pay attention to in my students’ thinking …

One connection or idea I want to remember …




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Suggested Timeline

Duration Activity Suggested Structure

10 minutes Set norms for group work and classroom discussion

Class discussion

5 minutes Question 1 Students work/discuss in small groups with full-class discussion as indicated in lesson

10 minutes Questions 2 and 3

8 minutes Question 4

4 minutes Question 5 Class discussion

8 minutes Making Connections Class discussion

5 minutes Discussing the out-of-class experience (OCE) Class discussion

Special Notes This lesson has two overarching goals. The first is to introduce students to the concept of quantitative reasoning and to set the stage for this course. The problem situation utilizes math calculations and even has extensions to exponential growth, but the calculations are not the objective. Instead, the problem situation is designed to demonstrate to students that mathematics can be used to examine a quantitative situation. In teaching this lesson, you need to allow students to do enough of the mathematics to make sure they understand the situation and start to recognize that mathematical tools (such as exponents) can be useful, but not allow them to get bogged down or frustrated with calculations. The productive struggle of this lesson is around thinking about the situation and about which skills would be necessary to answer the question. This lesson also motivates the skills of rounding and naming large numbers, but again, you do not need to spend a lot of class time on these skills as the out-of-class experiences (OCEs) provide support for them.

The second goal is to establish a culture of discourse. Careful attention to how you facilitate the group work and class discussion will set the stage for the nature of the discourse in future classes. Be prepared with strategies that encourage students to speak up in class. For example:

• Listen to groups for observations, comments, or questions that students can share with the class.

• Allow wait time for questions. Do not answer your own questions. • Do not allow individuals to dominate. You can avoid this by calling on individual students or by

asking for a response from a certain part of the room (this can be less stressful than calling on a student). For example: “Now I’d like someone in the back row to answer.” Make it clear that blurting out an answer is not acceptable.

• Encourage students to respond to each other. • Do not stand in the front of the room—if you are in the front, you are the focus. • Honor any serious contribution. Thank students in class and after class for their comments.




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While this is the first lesson, it is assumed that the beginning-of-term “business” (i.e., talking about the syllabus, policies, etc.) will be handled in another class period or that this lesson will be split over two days to allow for the business items to be spread over several days.

Regarding calculator use: Because this is the first lesson and students may not have calculators yet, this lesson is written so that students may or may not use calculators. The notes below indicate some of the choices you might make based on the amount of calculator use.

[Student Handout]

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.

Point out to students that the objectives for the lesson are on their student handouts. Explain to them that these are the things they should understand and be able to do after the lesson (which includes the OCE). Use the last objective to start the discussion on group norms. Explain the term norms.

Discuss the role of group work and class discussions. Points to consider: • Why talking about concepts in both small and large groups is important • Norms for group work and discussion • Importance of respect • Value of “wrong” answers • How groups will be assigned

Put students into groups (since the group portion of the lesson is short, you may just want to do pairs). Emphasize that students should introduce themselves.




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[Student Handout]

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:

(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?

Discuss for one to two minutes in groups and then as a class. Students may not be accustomed to thinking in this way, so you may need to give an example to get them started. Here are some ideas:

• Gun control laws • To convince people that America is a violent place • To convince people that America is becoming more violent due to …

o violent video games o drug use o breakdown of the family o more immigrants

• To convince people to spend more money on police

Depending on your student population, you may choose to read this as a class. Check for understanding on the reading. Here are some prompts you might use:

• Ask for an example of doubling. • What examples can you think of in which someone presents information to get you to vote a

certain way? To donate money? To understand a health risk? (Choose one or two.) • What is a reasonable answer to the question, “How many students are in this class?” What

would not be reasonable?

[Student Handout]

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.

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

has doubled.




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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.

(2) In 1995,1 an article 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.

Give students a few minutes to discuss Questions 2 and 3. This early in the course, they may struggle to come up with ideas, so monitor very carefully and do not allow the discussions to disintegrate or go off track. Move on to class discussion and prompt the students towards a strategy. In order to move on to Question 3, they need to start working with some numbers by choosing a starting value and doubling over time. Here are some suggestions about how to facilitate a discussion by building on student ideas.

Facilitation Prompts

Students may suggest that they have to know the starting number for 1950 … • Suppose you do know a starting number. What would you do with it? • What if you made up a starting number, just to get an idea of the change? What kind of number

would you want to use? (A number that is easy to work with, especially for doubling.) • If you are testing to see if the statement is reasonable, would you use a very large number or a

small number? (Both could have merits—might need to try both; a small number would be easier to work with so it is a good starting place.)

• If using a small number, would your calculations lead to overestimating or underestimating?

Students may think they have to know more information in general or have no ideas … • What do you know? (Years, doubling.) • If you wanted to start exploring with numbers, what would you need to know? (A starting

number. Go to questions above about making up a starting number.)

After you get some ideas out, have students start working with numbers. Depending on how much scaffolding they seem to need, you might suggest they start with increments of one year as shown in the table below or have them move directly on to Question 3.

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




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The example answers below are based on starting with the smallest possible number for 1950 (1 child). However, you can also allow students to select a starting number. In this case, you want everyone to use the same starting number so that you can discuss the rounding.

Year Number of Children

1950 1

1951 2

1952 4

1953 8

1954 16

1955 32

[Student Handout]

(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 1

1960 1,024 1 thousand

1970 1,048,576 1 million

1980 1,073,741,824 1 billion

1990 1,099,000,000,000 (rounded to nearest billion) 1 trillion

1995 35,184,000,000,000 35 trillion

Answers (in italics) do not appear on the student handout.

You will have to decide how long to allow students to work on this section in groups. Remember, the goal of the lesson is not the calculations themselves or the concepts about exponents. You want them to do enough work that they understand the calculations, but after that, the goal is to discuss the ideas of quantitative literacy. If groups have calculators, some may move to using exponents, but if that does not happen, you can move into class discussion and make the following points. As much as possible, get students to bring out each point.

• Calculations are based on repeated multiplication. For example, 1953—1 x 2 x 2 x 2. • A shorthand way to write that is 23. (Note that the 3 is the number of years after 1950.) • The number for 1995 is 245. (Can do other interim steps as needed)




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• As you fill in the numbers, ask students to round and name the numbers. You will probably need to model this for them, and it is okay if you have to prompt them with the names. You want to avoid having the discussion bog down because they are trying to read the long list of digits.

Tell students that it is okay if they did not get every single step in this work. Today’s lesson is more about the ideas behind the course. They will focus on math more in the future.

You may want to use a Think/Pair/Share strategy for some or all of the next questions. This strategy allows students to think about a question on their own for a short time, then discuss with a partner, then share with the class.

[Student Handout]

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

Elicit ideas from students. At some point, you can give them the information that the current U.S. population is around 300 million. (Explain to students that this is a number they will use often in the course, so they should record it in their notes.) This makes it easy to see that the statement cannot be true. As a matter of interest, you may want to share with students that this error occurred when an author reworded the following statement: “The number of American children killed each year by guns has doubled since 1950.” (This information is also in the OCE).

Note: The Making Connections section is the most important part of this lesson. Be sure to leave time for it. Explain to students that this section will occur in every lesson. There will always be classroom discussion about the important mathematical ideas. Students should take notes. You can support this by recording a few bullet points on the board. The first question of the OCE will always relate to this section of the lesson. Make sure the students realize this!




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[Student Handout]

Making Connections

Record the important mathematical ideas from the discussion.

Making Connections: Main Ideas to Highlight The foundation of quantitative reasoning is the mindset to ask questions about information.

Estimation and understanding and using large numbers are two valuable skills that students will see used repeatedly in lessons.


What skills or information did you need to determine if the statement was reasonable?

You will elicit many ideas from the students. Here are some points that are particularly important in this course. Use these points to bring out the two main ideas listed above.

• Used estimation and rounding. • Needed some understanding of large numbers. (Note that large numbers occur in a lot of

situations: population, national budget, environmental debates, even housing prices!) • Needed a “benchmark” to measure the numbers by, some basic fact that is easy to

remember and easy to verify for accuracy. • Most importantly, needed the mindset to ask the question. • What was not needed: a bunch of research, complex calculations, a lot of time.

Explain that this is a preview of what students can expect from this course—working with quantitative situations that help them understand their world.

Further Applications Because this is the first lesson and the OCE contains extra reading, there are no Further Application questions included in this lesson.




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Preparing Students for the OCE The first OCE contains information to help students understand how the assignments are structured. Stress to them that it is important that they read it carefully. As time allows, you may want to go over some of the information in class.

Students can read the following information outside of class if you do not have time in class.

[Student Handout]

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 four 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.

• Physical World: You will learn how to understand scientific information about how the world works.

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.


Lesson 1.1.2: Seventeen Billion and Counting Theme: Citizenship


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Large numbers Budget Deficit Follow up on group norms from Lesson 1.1.1

Level 1

Prerequisite Assumptions

Before beginning this lesson, students should be able to

• understand place value to the trillions. • read a table of numbers. • Add, subtract, multiply, and divide numbers.

Specific Objectives

Students will understand that

• 1 billion = 1,000 x 1,000 x 1,000. • the representations 1 billion, 1,000,000,000, and 109 have the same meaning.

Students will be able to

• calculate quantities in the billions. • convert units from inches to feet and feet to miles. • Compare large numbers

Explicit Connections

• Large numbers occur in many quantitative contexts (this point is made in Lesson 1.1.1). It is helpful to find ways to make sense of the size of large numbers and to understand how they compare to each other (relative magnitude).

Notes to Self

One thing I want to do during this lesson …






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Suggested Timeline


5 minutes Read Scenario 1 Individual

10 minutes Work on Questions in Problem Statement 1 Small groups

5 minutes Discuss calculation methods and relative value Class discussion

5 minutes Read Problem Situation 2 Individual

8 minutes Questions 2 and 3 Small groups

5 minutes Discuss Question 2 Class discussion

7 minutes Question 4 Individual or groups with discussion


Special Notes

Questions 2-4 introduce students to the basics of writing statements using quantitative information. There are two resources you can use to support this introduction to writing. The instructor support materials document, Writing About Quantitative Information, gives you an overview of how to support students in writing. There is also a student version of this document. The authors suggest that you give this to students during this lesson or as a part of the OCE assignment. They should keep it as a reference.

[Student Handout]

Specific Objectives


• 1 billion = 1,000 x 1,000 x 1,000. • the representations, one billion, 1,000,000,000, and 109 have the same meaning.


• calculate quantities in the billions. • convert units from inches to feet and feet to miles. • Compare large numbers

Problem Situation 1: How Big Is a Billion?

A large economic and political concern is the federal deficit, the amount of money spent by the federal government in excess of revenue collected. The federal budget deficit for 2013 was approximately $901 billion. The total accumulated federal debt is about $17.5 trillion.




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It is difficult to understand just how big a billion or a trillion is. Here is a way to help you think about it.

1 million = 1,000 x 1,000 = 1,000,000 = 106

1 billion = 1,000 x 1,000 x 1,000 = 1,000,000,000 = 109

1 trillion = 1,000 x 1,000 x 1,000 x 1,000 = 1,000,000,000,000 = 1012

The following questions will give you another way to think about how big these numbers are.

(1) Imagine a stack of 1,000 one-dollar bills, which is about 4.3 inches tall. Complete the following steps, and in each write your calculations clearly so that someone else can understand your work.

(a) Imagine combining 1,000 stacks of 1,000 one-dollar bills. How much money is in the stack? How tall would that stack be? How tall is the stack measured in feet? (12 in = 1 ft).

(b) Imagine combining 1,000 stacks like the one in Part (a). How much money is in the stack? How tall is the stack measured in feet? How tall is the stack measured in miles? (5280 feet = 1 mile)

(c) Imagine combining 1,000 stacks like the one in Part (b). How much money is in the stack? How tall is the stack measured in miles?

(d) Which of the stacks of money (in a, b, or c) is closest to the federal budget deficit in 2013?

Answers:

a) 1000 * 1000 =$ 1 million. 1000 * 4.3 inches = 4300 inches 4300 inches / 12 = 358.333 feet (it would be fine if the students round to 358 ft)

b) 1000 * 1 million = $1 billion. 1000* 358.333 feet = 358,333 feet 358,333 feet / 5280 = 67.866 miles

c) 1000 * 1 billion = $1 trillion. 1000 * 67.866 = 67,866 miles

d) The federal deficit is closest to the value from part c. $1 trillion, since $901 billion is close to $1000 billion = $1 trillion.




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• Students stuck at the beginning: How tall would 2 stacks of bills be? 10? 100? • Students stuck on conversion to feet or miles:.

o Use estimation—how many feet would 120 inches be? 1200 inches? 2400 inches? 4800 inches? o Some students might find a visual helpful—Here is a pile of 1000 stacks. Here is ruler 1 foot long.

How many feet would it take to equal the height of the stacks? How could you calculate that? o Suppose you had a stack 24 inches tall. How many feet is that? (more than 24? Less than 24?)

Use this to prompt for division.

Have groups share their calculations. You may need to help groups figure out how to write their calculations so that they are clear to others. Give as much time as possible to find an answer, but make sure to leave time for the discussion of relative magnitude. If some groups are completely stuck, you can intervene early on by asking a group to present on Part (a).

It is important for students to note the relative magnitude of the answers. Note how careless many people are with the words million, billion, and trillion. These words are often used carelessly, as if they are interchangeable, but they mean very different things.

To illustrate the relative sizes of numbers, here are a couple resources you could use:

An interactive chart showing various money items of varying sizes: http://xkcd.com/980/huge/

A video (somewhat political perhaps) showing stacks of $100 bills, up to the federal debt:

https://www.youtube.com/watch?v=jKpVlDSIz9o

Picture form of the video above: http://demonocracy.info/infographics/usa/us_debt/us_debt.html

[Student Handout]

Problem Situation 2: Comparing the Sizes of Numbers

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 2, which gives you an example 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.




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(2) A headline in 2014 read “Scott vetoes $69 million in $77-billion state budget”1. Is the $69 million a small or large portion of the total state budget? Which of the following statements best describes the relationship?

(a) The portion vetoed is a very small part of the entire state budget (b) $77 billion about 1000 times larger than $69 million (c) The $69 million vetoed is a very small part of the entire state budget of $77 billion (d) The total $77-billion state budget is about 1000 times bigger than the part vetoed.

It is very important to take time to discuss this question. This modeling of good writing technique is essential to supporting students who struggle with writing. Ask students to evaluate each statement using the writing principle. Example a does not give complete contextual and quantitative information. Example b gives quantitative information, but not contextual. Example c provides context and quantitative values, but does not make a specific comparison. Also note that students are expected to write statements in complete sentences.

[Student Handout]

(3) The federal budget in 2012 included $471 billion for Medicare and $47 billion for International Affairs. Write a statement that compares the two quantities.

(4) The total federal debt is about $17.5 trillion. Think back to problem (1), and write a statement comparing the federal debt to one of these measurements:

Distance around the equator: 8 thousand miles Distance from the earth to the moon: 239 thousand miles Distance from the earth to mars: 140 million miles Distance from the earth to the sun: 93 million miles

3) Discuss examples, such as “The budget of $471 billion for Medicare is about 10 times larger than the budget for International Affairs.”

4) Some possible comparisons, based on $17.5 trillion = about 1.19 million miles: The stack of bills would wrap around the earth 149 times The stack would reach to the moon 5 times The distance to the mars is 100 times further than the height of this stack.

1 http://www.tallahassee.com/story/news/2014/06/03/scott-vetoes-million-billion-state-budget/9901117/




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[Student Handout]

Making Connections


Making Connections: Main Ideas to Highlight

Large numbers occur in many quantitative contexts (this point is made in Lesson 1.1.1). It is helpful to find ways to make sense of the size of large numbers and to understand how they compare to each other (relative magnitude).


• What contexts or situations do you know about that use large numbers? (i.e., measurements in space, federal or state deficits/budgets, population, economic information such as the value of imports or the amount spent on a certain type of product such as cell phones, amount of trash produced in a year, amount of water used, amount of oil spilled in the Gulf of Mexico, etc.).

• What do you usually do when you hear or read a large number? Do you pay attention to a billion versus a trillion? Is that different from when you read a small number (under 1,000)?

• Is it easier to understand how fast the human population is growing by examining population amounts (in millions and billions) or by examining the changes in doubling times? Explain.

The focus of this lesson is on identifying different ways to make sense of large numbers. However, the first step in making sense of large numbers is to pay attention to the large number (did the reporter just say a “billion dollar deficit” or a “trillion dollar deficit”?). Comparisons of magnitude help remind you that a billion and a trillion are really different—a trillion is 1,000 times a billion. So it is important to develop an understanding of that relative comparison. It is like understanding the difference between 1 and 1,000.

[Student Handout]

Further Applications

(1) Explain the relationship among these terms: million, billion and trillion. Imagine you are explaining the difference to a friend. You may use words, symbols, and pictures. Your explanation should follow the Writing Principle.


Lesson 1.1.3: Percentages in Many Forms Theme: Personal Finance


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Estimating and calculating percentages

Varied Multiple strategies are valued

Level 1

Prerequisite Assumptions Before beginning this lesson, students should

• understand the meaning of percent. (reviewed in OCE) • be able to convert among fractions, decimals, and percentages. (reviewed in OCE) • have previous experience calculating what percent one number is of another. • have previous experience calculating the percentage of a number. • be able to round numbers to a given place value.


• 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.

Explicit Connections • A percentage is a ratio out of 100.







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Suggested Timeline


12 minutes Work on and discuss strategies for Question 1 Group work with class discussion



10 minutes Instruction on calculating percent Class discussion/direct instruction

10 minutes Work on and discuss strategies for Questions 4–6 Group work with class discussion


Special Notes Students are often reluctant to use estimation because their previous experience in math classes is that exact answers are always better. You can help counteract this by using the following strategies:

• Students often have surprising estimation skills that have not been recognized in prior math classes. It may seem counterintuitive that this lesson focuses on estimation before calculation. This is an opportunity to draw on students’ own ideas before doing more cut-and-dried calculation that sometimes signals to students that their thinking is not valuable.

• Spend time on estimation throughout the course. Keep coming back to it. If you have a longer class period, it is very helpful to do one or two quick estimation problems at the beginning of class. In this first lesson, it may seem that too much time is devoted to relatively simple problems, but that is to allow time to hear all strategies.

• If students are reluctant to suggest strategies, you can broaden the discussion in a few ways. Think of different strategies before the lesson. As students work in small groups, prompt for strategies that have not been used yet. Encourage students to come up with multiple strategies. If one strong student in a group has all the strategies, have different group members present the “group work.” This helps encourage more people to contribute. You can also suggest strategies in class discussion by saying something like, “I had a student use this strategy …” If you present a strategy as your own, students tend to think it is the “right” one.

• Focusing on the explanation of the strategy rather than on the answer also encourages students to think about strategies rather than just doing the calculation.

• Take time to discuss multiple estimation strategies. Students gain a sense of confidence when their own strategies are acknowledged and honored. The discussion also helps students with lower skills understand that there are different ways to think about problems and encourages them to develop their own strategies.

• Do not pass judgment on strategies unless they are absolutely wrong. If a student suggests a strategy that is not very accurate, they will often find better strategies over time. Especially avoid saying that some strategies are “easier” or “more efficient.” Estimation strategies depend greatly on the individual’s understanding of the concept and on his or her way of thinking.

Note: Introductory material on percentages and benchmarks are included in the previous OCE (Lesson 1.1.2).




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[Student Handout]

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.




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(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) 25% off

(b) 70% off

Answers will vary. Examples are given: (a) $66; (b) $27.

Possible Strategies

Part (a) • 25% is 1/4, so round $87.99 to $88. 1/4 of 88 is 22. Subtract 22 from 88 to get 66. • Round 87.99 to 90. 10% of 90 is 9. 2.5 x 9 is about 22. Subtract 22 from 90 to get 68. • Sale price is 75%, or 3/4. Round 87.99 to 88. 1/4 of 88 is 22. 3 x 22 is 66.

Part (b) • Round 87.99 to 90. 10% of 90 is 9. 7 times 9 is 63. Subtract 63 from 90 to get 27. • Round 70 to 75%—This makes the sale price 25%. Round 87.99 to 88. 1/4 of 88 is 22. Since the

original rounding was a big jump, increase the price a little to get $25.

Watch for students calculating the discount instead of the sale price.

The price and the percentages are chosen to point out that rounding for estimation depends on the strategy and situation, unlike rounding to the nearest, in which there is only one correct answer. For example, if using a strategy of finding 10%, rounding to $90 makes sense because 10% of 90 is a whole number. However, if estimating 25% by dividing by 4, it may be easier to round to $88.

Another point to make is that this work could be done by subtracting the discount or estimating the percentage left. For example, Part (b) could be done by subtracting 70% or even 75%. Another strategy is to estimate 30%.




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[Student Handout]

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.

Precinct Number of Violent Incidents Number of Violent Incidents Involving a Weapon

1 25 5

2 122 18

Answer: Precinct 1 has a higher rate of violent incidents. Explanations will vary. • Precinct 1 has a rate of 5/25, which is 20%. Precinct 2 has a rate of 18/122, which is about 20/120

(reduces to 1/6), which is about 17%. Students might be more familiar with 1/3 = 33%, so they might think of this as 1/6 being half of 1/3 and 16–17% being half of 33%.

• Precinct 2 has a rate of 18/122, which is about 20/120. Since 20/100 = 20%, this rate must be lower than the 20% for Precinct 1, so you do not need to estimate any further.

• Precinct 1 has a rate of 5/25 = 1/5. Precinct 2 has a rate of 18/122, which is about 20/120 = 1/6. 1/5 is greater than 1/6.

If students are unsure how to get a percentage, encourage them to think in fractions first and then use the benchmarks to change to percentages—or they may just use the fractions, as in the third strategy above.

When talking about percentages, you could mention that in this context they represent a relationship between two values: the comparison value (which appears in the numerator) and the reference value (which appears in the denominator). This relationship is often referred to as a percentage rate as well.

[Student Handout]

(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.

Answers will vary. Example: $2.30

Possible Strategy

1% of $462 is $4.62, rounded to $4.60. 0.5% is half of 1%. Half of $4.60 is $2.30.




6

[Student Handout]

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.

You will need to gauge how much direct instruction your students need on calculating percentages. It is suggested that you ask students to discuss Part 4a first, and then allow them to share their methods of calculation. This will give you a sense of how much you need to explain. Then do the same thing with the first part of Question 5.

[Student Handout]

(4) If the coat’s regular price is $87.99, what is the exact sale price based on each of the following discounts?

(a) 25% off

(b) 70% off

Answers: (a) $65.99; (b) $26.40

[Student Handout]

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

Answers: Precinct 1: 20%; Precinct 2: 15%

[Student Handout]

(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.

Answer: $2.31




7

Making Connections


Making Connections: Main Ideas to Highlight Percentages are a ratio out of 100.

Facilitation Prompts • What is the meaning of 35%?

Answer: 35 out of 100. • In Part 4a, what does the discount of 35% mean in the context?

Answer: Reduce price by 35 cents out of every 100 cents or $0.35 out of $1.00. • These are examples of ratios, meaning that one number is compared to another number by

division. A percent is a ratio out of 100. You use 100 to standardize over different sizes of groups. In the law enforcement example, there were not 100 incidents, but the 20% means that, if there were 100 incidents, 20 would involve a weapon. This can be illustrated as follows:

Imagine that the ratio of 5 incidents with weapons per 25 total incidents happens repeatedly until you reach 100 incidents. This is, of course, an artificial situation. You know that real data does not exactly fit these patterns or generalizations. The limitations of models will be discussed later in the course. You can also relate this to equivalent fractions:

525

× 44

= 20

100

Total

Number of incidents with weapons

5 5 5 5 20

Number of incidents 25 25 25 25 100

Introduce the term ratio. One way to think of a ratio is as a relationship of two quantities that remains constant as in the example above. Tell students that they will build on the concept of ratio throughout the course.

You may also find these two visual explanations of ratio useful: • Stair steps—Vertical increase of 6 inches for every horizontal change of 9 inches. This lays a

foundation for slope. • Enlarging a picture—Picture starts with measurements 2 inches by 3 inches. To increase

proportionally, both dimensions must be multiplied by the same factor.




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[Student Handout]


(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

Estimates will vary. The explanations are the most important part. Example estimates are (a) 60; (b) 44; (c) 33%; (d) 1.5%


Lesson 1.1.4 : The Flexible Quantitative Thinker Theme: Personal Finance


1




Mathematical operations, order of operations

Household finances Encouraging multiple strategies

Level 1


Before beginning this lesson, students should

• have been introduced to the concepts of common fraction/percent equivalencies (from Lesson 1.1.3).

• understand rounding to a given place value (from Lesson 1.1.1). • be able to perform calculations using a calculator.

There are many calculation skills used in this lesson, but do not assume that all students have all these skills. The lesson is structured so that students will help each other with different approaches. It is assumed that at least some students will have the following skills and that all students will have some level of familiarity with most of them.

• Understand the relationship between multiplication/division (dividing by 3 is the same as multiplying by 1/3).

• Have experience converting between fractions and decimals. • Have experience with multiplying fractions.

Specific Objectives


• flexibility with calculations is an important quantitative skill. • different methods of calculation are often possible and helpful.


• 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 (not technically defined in lesson). o order of operations. o the Distributive Property.




2


• The ability to perform calculations in multiple ways is a valuable skill. • Mathematical rules determine if calculations are equivalent. This curriculum does not emphasize

memorizing rules (like the Commutative Property), but encourages students to use reasoning to think about calculations.

Notes to Self




Suggested Timeline


10 minutes Students work on Questions 1–2 Groups of three or four (This is not a good lesson for partners or individual work since it focuses on different ways to perform calculations—groups tend to spark more thinking.)

10 minutes Class discussion of Question 2 Full-class discussion

10 minutes Students work on Questions 3–5 Groups

10 minutes Class discussion of Questions 4 and 5 Full-class discussion

10 minutes Making Connections Full-class discussion/lecture




3

[Student Handout]

Specific Objectives


• flexibility with calculations is an important quantitative skill. • different methods of calculation are often possible and helpful.


• 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

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




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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.)

You may not have a lot of time to discuss this problem. However, in keeping with encouraging different estimation strategies, you might make note of some of the strategies you see among the groups and write them on the board—then start the discussion of Question 2 with a quick acknowledgement of different approaches.

[Student Handout]

(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.

Possible answers: 1/3 x 62,857; 62,857/3; .33 x 62,857

Try to elicit the following responses. The most likely is division by 3. If no one suggests the other two, you may want to use these prompts:

• The Problem Situation uses the number 1/3. Is there a way to use that number in your mathematical statement?

• What if this Problem Situation had used a percentage instead of a fraction? (Reference benchmarks from the previous lesson.)

• For each anticipated response, important discussion points are shown below:

Operations Questions to Ask Important Ideas

62,857 ÷ 3

62,857 x

62,857 x 0.33

Can numbers be reversed? Are all the statements exactly equivalent? (Distinction between 1/3 and 0.33). Suggestions for motivating the questions:

• If you see students who did the calculations in different ways (such as a different order), ask them to share their work and ask the class if they are the same.

• If there are no examples in class, you can say: “a student in another class did …” or “what if I did …”

Some operations (x, +) can be reversed and others cannot (–, ÷). Make the connection between multiplication and division. Note that the result of multiplication is smaller than the original number. Make the connection between decimal, fraction, and percent.




5

(Note: This is an opportunity to discuss rounding in context. Since the units are dollars, it might make sense to round to the hundredth [nearest penny]. On the other hand, given the size of the number, it also makes sense to round to the ones [nearest dollar]. It does not make sense to give a result beyond the hundredth place.)

Have students work on Questions 3–5 and then discuss.

[Student Handout]

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

Answer: 2/3

[Student Handout]

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

Discussion of Questions 4: Try to elicit the following responses, although you may have to suggest Method 4. The most common response will probably be Method 5, but you should discuss it last. The order below shows a progression from the ideas of the previous question to introducing order of operations.

Responses to Elicit and Discuss

Method 1: 62,857 x 0.67 Connection to percent; order can be reversed.

Method 2: 62,857 x These three methods should be discussed together. While mathematically equivalent, they each indicate a different way of thinking of the calculation. Method 2 thinks of 2/3 as a part of a whole. Method 3 unitizes and then doubles. Method 4 doubles and then breaks into thirds. Note that Methods 3 and 4 allow for changing the order of division and multiplication.

Method 3: 62,857 ÷ 3 x 2

Method 4: 62,857 x 2 ÷ 3

Method 5: Subtract the amount found in Question 2 from 62,857

Students probably see this as two steps:

• Step 1: 62,857 ÷ 3 = 20,952 • Step 2: 62,857 − 20,952.33 = 41,905 (opportunity to review

rounding up) Ask if there would be a way to write this as one calculation:

62,857 − 62,857 ÷ 3

This introduces the idea of order of operations. Ask students how they would know that division should be done first in the statement 62,857 − 62,857 ÷ 3.




6

Review order of operations. Note that Methods 3 and 4 illustrate that multiplication and division should be done left to right. Make sure students take notes.

[Student Handout]

(5) The Montero family, an average American family, has the following average monthly expenses. Calculate how much they spend on housing (this includes rent and utilities) in one year. Think of as many different ways to calculate the answer as you can.

Rent $1,250

Electricity $85

Gas $120

Water and sewer $72

Answer: $18,324

Discussion of Question 5: Discuss different methods to this problem. Students will probably use Method 1, and you may have to prompt for or suggest Method 2.

Method 1: Add monthly expenses together and multiply the result by 12

As with Method 5 from the previous question, students probably view this as two steps:

• Step 1: 1,250 + 85 + 120 + 72 = 1,527 • Step 2: 1,527 x 12 = 18,324

Ask if there is a way to write as one calculation: 12(1,250 + 85 + 120 + 72)

Note that the parentheses determine the order of operations.

Method 2: Multiply each item by 12 and then add together

Introduce the Distributive Property.

[Student Handout]

(6) Look at your answer in Question 5. Does it seem reasonable, considering the consumer expenditure survey? 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.

Answer: Since the average family makes $62,857, it seems reasonable that the Montero family could be paying $18, 324.




7

[Student Handout]

Making Connections



The ability to perform calculations in multiple ways is a valuable skill.

Mathematical rules determine if calculations are equivalent. This curriculum does not emphasize memorizing rules (like the Commutative Property), but encourages students to use reasoning to think about calculations.


• Why might it be useful to do the calculations from Question [… reference a problem from class] in different ways?

If students do not have ideas about the previous question, you can suggest a metaphor such as o The best football or basketball team can change between man-to-man and zone

defenses. o A good parent uses different disciplinary strategies in different situations. o A carpenter selects the right cutting tool for the specific task.

This lesson demonstrates some of the tools for making calculations.

• When you are thinking about ways to perform a calculation, how do you decide if a method is “legal”? Some basic mathematical ideas to consider:

o Relationships among operations Division is the same as multiplication by a reciprocal (see Question/Method? 2) Changing the order of operations (see Question/Method? 2) Equivalent forms of fractions, decimals and percentages (see Question/Method? 2) Order of operations (see Question/Method? 4) Distributive Property (see Question/Method? 5)




8

[Student Handout]


(1) The graph below 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

Transportation 3%

Books and Supplies

4% Travel and Vacation

5% Entertainment

6%

Apparel and Services

7% Tuition and Fees 19%

Room and Board 26%

Other 30%

Student Budget


Lesson 1.1.5: The Credit Crunch Theme: Personal Finance


1




Review of prior lessons Credit cards Check-in with students Level 2

Prerequisite Assumptions Students have completed Lessons 1.1.1–1.1.4 and have been introduced to the concepts in those lessons (listed below), but they have not necessarily fully mastered the concepts and skills.

Before beginning this lesson, students should be able to • name and understand large numbers written in different forms. • use benchmarks to estimate percentages, including percents less than 1%. • use order of operations and the distributive property to write expressions in different forms.


• 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.

Explicit Connections • Quantitative reasoning starts with making sense of numbers. (Detailed connections to previous

lessons are listed in the Making Connections section. This is the overarching idea.)







2

Suggested Timeline (Note: Times may be reallocated depending on needs of class. You may choose to shorten or delete discussion on some topics to give more time for others.)


8 minutes Reflection on course or review of previous lessons— choose discussion topic based on assessment of needs of students

Class discussion, possibly combined with small groups

5 minutes Question 1 Individual or small-group work followed by class discussion

8 minutes Reading and discussing credit card information Class discussion

14 minutes Questions 2–4 Individual or small-group work followed by class discussion

5 minutes Question 5 with discussion of writing explanation Small groups or partners followed by class discussion


Special Notes The purpose of this lesson is to deepen the understanding of previously covered material and demonstrate to students that the mathematical skills and concepts used in previous lessons can be applied to other quantitative contexts. This lesson also provides you with an opportunity to individualize a lesson based on the needs of your students. The questions cover skills and concepts from the first four lessons.

To prepare for this lesson, you should review the work on assignments and your classroom observations to identify areas for which further discussion is needed. Based on this analysis, you should plan to focus more intensive discussion on the problems in the lesson that will address gaps in your students’ understanding and skills. The suggested timeline allows for discussion on all problems, but you may choose to limit discussion on one question in order to focus time on another. This is also an opportunity to review specific problems from the out-of-class experiences.

Your choice of structure may also depend on your assessment of the students. If you feel students are ready to work independently, you might have them do problems individually and then discuss with a partner. If students are struggling with a concept, you will probably keep them in groups.

In addition, this lesson introduces students to reading authentic text and writing about a mathematical context. Class time is devoted to supporting reading and writing.

This is an opportunity for you to address some of the noncontent issues or structures of the course. Here are some suggestions of what you might focus on:

• Check-in about how students feel about the class. • Discuss pedagogy (little or no lecture); productive struggle. • Motivate quantitative reasoning. • Revisit class norms: group work, discussion, shared responsibility for learning.




3

• Review structure and purpose of the out-of-class experience.

One strategy you might use for framing your discussion is to have each student write a response to two to three questions regarding your topic. For example, What has been most interesting? What are you worried about? Rate your confidence that you will be successful from 1 to 10. You can use this in a couple of ways: You can collect the reflections before class, which has the advantage of allowing you to use the information to prepare for this discussion. You can have a discussion and then ask students to write a response to the prompt. (An advantage here is that some students will find it easier to articulate their thoughts if they hear others do so.) You might also consider whether the written responses should be anonymous. Another strategy for the actual discussion is to allow students to talk in small groups first and then report out. This can help depersonalize feedback. Yet another option is to review the SRL responses in the OCE.

Note: This excerpt of a credit card disclosure form is in the OCE. Students should bring this to class for use in discussion.

Interest Rates and Interest Charges

Annual Percentage Rate (APR) for Purchases

0.00% introductory APR for 6 months from the date of account opening.

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 Transfers

0.00% introductory APR for 24 months after the first transaction posts to your account under this offer.


APR for Cash Advances

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

Penalty APRs and When It Applies

Between up to 16.99% and up to 26.99% based on your creditworthiness and other 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 Paying Interest on Purchases

Your due date is at least 25 days after the close of each billing period (at least 23 days for billing periods that begin in February). We will not charge you any interest on purchases if you pay your entire balance by the due date each month.

Minimum Interest Charge

If you are charged interest, the charge will be no less than $0.50.




4

[Student Handout]

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.

Note: Question 1 connects to Lessons 1.1.1 and 1.1.2.

[Student Handout]

(1) The statements below came from two websites that report predictions about credit card debt by the end of 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.

Answer: The projected amounts are the same. 1.177 trillion is the same as 1,177 billion.

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




5

Check for understanding on the reading from the OCE (disclosure form): • When do you have to pay interest on purchases?

Answer: When you do not pay your balance at the end of the month. Make sure students understand the word balance.

• What is a credit score and how does it affect your credit?

Answer: A credit score is a measure of how likely you are to pay your bills. A high credit score means the company considers you to be reliable and will give you a lower rate.

Discuss the disclosure. The purpose here is for students to understand that eventually they will need to be able to pick relevant information out of a full document. However, since this an early lesson, you will break up this reading to give them the information that they need for specific questions. Explain that you do not expect them to know all the terms used—they will look up some of them in the OCE. There are a few things you should highlight before moving on:

• How does the format and use of bolding and type size help you pick out important information? • The ability to decide which information is important (and which is less important) is a valuable

skill. Look at the sentence in the first row: “This APR will vary with the market based on the Prime Rate.” What is important in that sentence? The word vary tells you that the rate can change. That is important to you, as the person paying the bill. It is less important to know what the Prime Rate is.

Questions 2–4 connect to Lesson 1.1.3. Questions 2 and 3 promote estimating percents with benchmarks. Question 3 uses a percentage less than 1%, which is often challenging for students. Question 4 uses calculation for percentages.

[Student Handout]

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


0.00% introductory APR for 6 months from the date of account opening.


(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.

Answers will vary. A little more than $500 (10% of 5,000). Strategies will vary.




6

Giving time to discuss different strategies encourages students to use and develop their own estimation strategies. While facilitating small groups, encourage groups to think of multiple strategies.

[Student Handout]

(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.

Answer: $1,199.50

[Student Handout]

(c) What are some things that might affect your credit score?

You might choose to address this as a large group discussion if you do not have enough time to do small group discussions first. Students might not have many ideas, so you can prompt them with the question: What would you want to know about a person before you decided to loan them money?

Credit scores are affected by the following: • Late payments on bills, credit cards, and loans • How late the overdue payments are • Amounts owed on other loans and credit cards • Length of time one has had credit—it helps your credit score if you have had an account in good

standing for a long period of time; people who have never had any credit actually have lower credit scores

• Receiving additional credit cards (these can be store-specific, like Lowes, that are pushed on customers in the guise of promotional savings)

[Student Handout]

(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 in January?

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

(c) Explain your answer to Part (b).

Answers: (a) 0.92%; (b) $5–10; (c) Answers will vary. Example: 1% of $982 is 9.82 so 0.92% will be less than $10.




7

Discuss estimation strategies. This may need extra discussion since percentages less than 1 are often challenging for students.

[Student Handout]

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.



APR for Cash Advances

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

(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) Lois pays the lowest interest rate for purchases. If she purchased a $400 TV using a cash advance, she would pay about two-and-a-half times as much interest as she would if she used the card as a regular purchase.

Answers:

(a) Reasonable. Emphasize the meaning of the percent in the context and as a ratio.

(b) Reasonable. The two-and-a-half comes from comparing 10% to 25%. Discuss other ways to compare. Would it be reasonable to say “three times as much” or better to say “a little less than three times”? What about “more than two times as much”—This is true but not very accurate.

Point out that these two statements give good quantitative information because they contain specific information about the comparison even though one is a rounded estimate. This is more informative than a statement like, “You pay a lot more for cash advances than for purchases.” This was discussed in OCE 1.1.1.

Question 5 relates to Lesson 1.1.4. If you think your students need extra explanation about spreadsheets, you might take time to demonstrate with this example. You could demonstrate entering the formula. Even though this is in the OCE, some students may need to see an example. You can also review the information from the previous OCE by asking students to identify the cell names and discussing that the cell names are used as variables in formulas.

Tell students that they will need to use a spreadsheet program in the OCE. Tell them what program is available on lab computers, mention Excel if your school has something different.




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[Student Handout]

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.

)5432(122399.0 BBBB +++

(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.

Answer: iii

In this course, you will use the instruction to “explain in terms of the context” to indicate that students need to make meaning of the numbers and operations. You will want to take some time to discuss this and explain why iii is the best choice. Note how the statement not only describes the order of operations, but also gives meaning to the calculations. It does not describe every calculation.

Explain that this is a model that students can use for explaining calculations in context. Students will be asked to do this in the OCE, so you want to make sure they know to use this as a reference.

A good follow-up question: Do the parentheses matter in this expression? Why?




9

Optional extended work for review on order of operations, distributive property, or understanding variables:

• Note that this expression is similar in structure to the expression in Question 6 in Lesson 1.1.4: 12(1,250 + 85 + 120 + 72). Here, you are using variables instead of numbers. This allows you to use this expression even if the price of the items changes.

• Have students perform the calculation and discuss order of operations and/or how the factor could be distributed.

[Student Handout]

Making Connections


Making Connections: Main Ideas to Highlight Quantitative reasoning starts with making sense of numbers.

The purpose of the first lessons in this module is to build the skills or tools needed to work with more in-depth problems. These lessons have covered a lot of skills, so this is an opportunity to help your students see that these skills are all connected—they all help make sense of quantitative information. Here are some examples you may use to demonstrate this point.

Understanding the size of numbers, especially in relationship to each other.

Lesson 1.1.2: Using the image of the line of people to visualize the relative size of large numbers.

Understanding what numbers (or variables representing numbers) mean in a situation.

Lesson 1.1.5: Explaining the meaning of the expression in Question 5.

Using estimation to make sense of numbers. Lesson 1.1.1: Estimating the result of the statistic to decide if it was reasonable.

Understanding the relationships of numbers. Lesson 1.1.3: Using benchmarks to make estimation easier.

Understanding ways you can work with numbers. Lesson 1.1.4: Flexibility in performing calculations.

Facilitation Prompts • Refer to a specific problem in this lesson and ask students how it connects back to previous

lessons. • Give students the points made above and ask them to find examples. You could give each group

one and then have them share. Model one example for them. • Refer to a specific lesson and ask how it connects to this lesson.




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[Student Handout]


(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?


Lesson 1.2.1: Whose Footprint Is Bigger? Theme: Citizenship





Large numbers, ratio Water footprint Not applicable Level 1



• calculate a quotient (one number divided by another). • use a calculator to divide numbers. • use scientific notation. • convert among fractions, percents, and decimals.

Specific Objectives


• 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.


• 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).


• Numbers can be compared in multiple ways. • Strategies for estimating and calculating with large numbers.

Notes to Self







Suggested Timeline


12 minutes Problem Situation 1 Work on Questions 1 and 2 Discuss Question 2

Guided discussion

10 minutes Work on Questions 3 and 4 followed by a brief discussion Small groups

15 minutes Work on Questions 5–9 Small groups

5 minutes Discuss Questions 5–9 Class discussion


Special Notes

Information about “water footprint” is included in the previous OCE (Lesson 1.1.5).

[Student Handout]

Specific Objectives


• 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.


• 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).




[Student Handout]

Problem Situation I: 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.

Students will have reviewed scientific notation in OCE, but you may want to review the definition. It is the first introduction to a definition using a variable and the term integer. You should briefly explain the use and meaning of a and n as integers.

[Student Handout]

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 2014, the population of the United States was 317,000,000. Earth’s population was about 7.2 billion. Write these numbers in scientific notation.

Answer: The population of the United States in 2014 is 3.17 x 108. The population of Earth in 2014 is 7.2 x 109.

The use of scientific notation in this lesson is to give students an alternative way to write calculations with large numbers, especially when using a calculator. Make sure students see the option in the following problems, but do not try to force it as a method. They should decide if it is useful. You will probably need to point out the importance of parentheses in using a calculator to divide numbers in scientific notation.




The purpose of Question 2 is to get students to think about using ratios in different forms. This is probably best done as a guided discussion. Give students a minute to talk with a partner and then discuss ideas.

[Student Handout]

(2) What are some 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.

Answer: The ratio is (3.17 x 108)/(7.2 x 109) = 0.044028. The United States makes up 4.4% of the world’s population. One way to write it as a fraction is: The United States makes up 44/1,000 of the world’s population.


• Use a simpler question—How would you compare the number of women in this class to the total number of students?

• Prompt for both fractions and percents—think about forms of numbers used previously. • Prompt for the idea of estimating how many times the U.S. population goes into Earth’s

population—about how many U.S. populations would it take to make up Earth population?

Note: Help students make the connection that 44/1,000 means that 44 out of every 1,000 people in the world reside in the United States. Also, leave room for the students to make different fractions (such as 4.4/100, which would mean that out of every 100 people in the world, 4 and 0.4—if you can have 0.4 of person!—would be from the United States.)

For estimation: Students need to use all millions or billions. Using millions, they can reason that the population of the world is 7,000 million and the population of the United States is about 300 million. Therefore, they need to reason how many times 300 goes into 7,000. Help students see this is the same as estimating how many times 3 goes into 70. Since 3 goes into 70 about 20 times (rounding down from 317 to 300 million; the low estimate of 20 times is reasonable). Then you see it would take about 20 United States to equate to the entire world population. In other words, the world population is 20 times the U.S. population.

Students may wonder how the figures of 4.4% and 20 times can both describe the same relationship. Note that the United States is 1/20 of Earth: 1/20 = 0.05 (5%).

Introduce the term ratio—a measure of the relative size of two numbers or a comparison of two numbers by division: the comparison value (numerator) divided by the reference value (denominator). It is important to establish the use of these terms as they are fundamental to many of the concepts of the course. Ratio has been defined previously, but this lesson establishes formal use of the term.

Have students complete Questions 3 and 4.




[Student Handout]

(3) In 2014, the population of China was 1.39 billion. Compare China’s 2014 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.

Answers:

Question 3: The ratio is (1.39 x 109)/(7.2 x 109) = 0.193056. China makes up 19.31% of the world’s population. One way to write it as a fraction is that China makes up 19/100 of the world’s population.

Question 4: The ratio is (1.39 x 109)/(3.17 x 108) = 4.38. So the population of China is 4.38 times the population of the United States.

[Student Handout]

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

1Retrieved from www.waterfootprint.org




Country Population (in thousands)

Total Water Footprint* (in 109 cubic meters per year)

China 1,257,521 883.39

India 1,007,369 987.38

United States 280,343 696.01

Note: Students likely will not be familiar with cubic meters. If questions regarding the units come up, you can explain that a cubic meter is the volume of one cube with each side equal to 1 meter, or about 3.28 feet, on each side. To visualize, see if you have an object about 3-1/4 feet on each side. Some teacher desks are approximately that long. Explain that a cubic meter would be the volume of a box that is 3.28 (about the length of the desk) on each side. Or get a meter stick and draw a square meter on the board to help students visualize the size.)

[Student Handout]

(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.

Answers:

Question 5—From highest to lowest:

• India (987.38; 109 cubic meters per year) • China (883.39; 109 cubic meters per year) • United States (696.01; 109 cubic meters per year)




Question 6: Let’s first compute the water footprint per person for each country:

• China

(883.39 109 cubic meters per year)/(1,257,521,000 people) = 702.5 m3/yr per person.

• India (987.38 109 cubic meters per year)/(1,007,369,000 people) = 980.2 m3/yr per person.

• United States

(696.01 109 cubic meters per year)/(280,343,000 people) = 2,482.7 m3/yr per person.

Consumption per person, from highest to lowest:

• USA (2,482.7 cubic meters per year per person) • India (980.2 cubic meters per year per person) • China (702.5 cubic meters per year per person)

Note: The United States, while third in total between these countries for total consumption, became first when looking at the figures per person. Students may need guidance finding the values per capita. You may want to review that this is a fraction or proportion.

Question 7: The ratio of the population of China to that of the United States is 4.48565. This means that China has about 4.5 times as many people as the United States.

Question 8: The water footprint ratio of United States/China (per capita) is 3.5341.

Question 9: An “average” person in the United States uses about 3.5 times as much water as the “average” person in China. Another wording: One “average” person in the United States uses as much water as 3.5 “average” Chinese people.

Although China’s population is about 4.5 times the population of the United States, each resident of the United States uses a considerable amount more water each year (3.5 times the amount used by each resident of China).

You may want to discuss with students whether an estimation or calculation is needed for this fact. Is an estimation enough to get the point across? Also, you may want to have students consider this fact and reflect upon the impact each country has on the world’s water supply. Have students consider what it would mean if each person in China had a water footprint similar to that of a person in the United States.

Extension: This discussion could be extended in many ways.

• Students could calculate their own water footprint (see waterfootprint.org for information as well as a downloadable chart, or see www.waterfootprint.org/?page=cal/WaterFootprintCalculator for an online calculator).




Return to the table of data for the United States, India, and China.

Country Population (in thousands)

Total Water Footprint* (in 109 cubic meters per year)

China 1,257,521 883.39

India 1,007,369 987.38

United States 280,343 696.01

[Student Handout]

Making Connections



Numbers can be compared in multiple ways.

The importance of having multiple strategies in working with numbers has recurred in work with estimation strategies and methods of calculation. In this lesson, you saw multiple ways to compare numbers.

• Percentages: the United States makes up 4.5% of the world population. • Fractions: the United States is 1/20 of the world population. • Ratio: 45 out of every 1,000 people in the world live in the United States. • Multiples: The world population is 20 times the U.S. population.


• What are ways that you compared numbers in this lesson? • Did some comparisons make more sense to you than others? • What are examples of ratios in this lesson?

Answer: The ratio of China/U.S. population = 1.3/0.3, 702,585 cubic meters per year per person; the United States makes up 4.5% of the world’s population. The point of this question is to help students recognize that they are using ratios in different forms. Note different ways that units are used.

• What is similar about all these ratios? What is different?

Answer: All are ways to compare measures by division; forms are different; units are used differently. This prepares for Lesson 2.1.1. You do not have to achieve full understanding here.




Strategies for estimating and calculating with large numbers.

• Importance of paying attention to magnitude (for students, you might refer to this as place value); comparing 300 million to 7 billion.

• Different forms of numbers: word/number combinations (300 million); scientific notation; standard notation

• Estimation strategies

One goal of this discussion is to identify and highlight the strategies students used. Estimation with large numbers resurfaces in Lesson 2.1.1. You may wish to keep a record of strategies used in this lesson.


• You use many different ways of writing large numbers (words, standard notation, scientific notation). What are advantages and disadvantages of each? What do you like to work with? Why?

• Did you use different forms when estimating versus when you used a calculator? • What makes comparisons with large numbers challenging? • Which strategies for comparisons and calculations did you use today? Which were the most

helpful?

[Student Handout]


(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.

Answer: The average person in the United States uses 2,483 cubic meters of water each year.

(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?

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

Answers to Parts (b)–(d) will vary.

Quantway Instructor’s Notes November 18, 2011 (Version 1.0)

Lesson 1.2.2: A Taxing Set of Problems Theme: Personal Finance


1




Multistep calculations Taxes Supporting increased challenge

Level 2

Prerequisite Assumptions Before beginning this lesson, students should know how to

• follow the order of operations. (from Lesson 1.1.4) • find a percent of a number. • estimate 1% of a number. (from Lesson 1.1.3)


• 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.

Explicit Connections • Quantitative reasoning requires skills in reading, interpreting, and using information. • Quantitative reasoning requires understanding skills with regard to calculations, including

understanding what they mean, order of operations, and flexibility in calculation methods.







2

Suggested Timeline


5 minutes Lesson Introduction Class discussion

25 minutes Questions 1–5 Small groups



Special Notes This is the first lesson that is designed to allow students to work through a whole series of questions without intervening discussion. This is transitioning students into accepting higher levels of productive struggle.

Students were given the definitions for revenue, net profit, and net loss in the previous OCE.

[Student Handout]

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.




3

Students may or may not have had experience working for an employer and receiving a detailed paycheck. You can introduce this by asking students why their take-home pay is not the same as their earned pay. Students who have worked for someone else off campus are probably aware of Social Security and Medicare deductions. Explain that the self-employment tax is just a way for self-employed people to pay their Social Security and Medicare taxes. Form the students into small groups. Define the terms net profit and revenue. Students may be curious about the term FICA.

Generally speaking, self-employment taxes include Social Security tax—which is a tax on the first $100,000 or so of income—and Medicare tax, which is a tax on all income. To reflect that people employed by others have a portion of their taxes paid by the employer, the net profit is multiplied by 92.35 percent first. Also, those with profits below $400 do not have to pay self-employment taxes.

To support reading the form, remind students of reading the credit card disclosure in Lesson 1.1.5. Suggest they scan the full document before starting to use it. Remind them of the skill of deciding which information is important and which is less important. For struggling readers, you can point out that, in Line 1a, the phrase “Net farm profit” immediately informs the reader that this does not apply to Sundos because she does not own a farm. There is no need to read the rest of the line. In Line 2, “Net profit” tells you necessary information, but you know you do not have Schedule C or the other forms, so you can ignore that information. You also know that Sundos is not a minister, so you can ignore the last sentence.

A common student error will be to use revenue and not net profit on Line 2.




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[Student Handout]

(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.

Answer: 0.153(0.9235)($11,385 – $3,862) = $1,062.97. 1 Most students will follow the flowchart on the tax form rather than write a single expression.

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




5

[Student Handout]

(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?

Answer: Raven owes no self-employment taxes because 0.9235($1,050 – $630) < $400.

[Student Handout]

(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.

Answer: 0.153(0.9235)($11,385 – $3,862) is one possibility.

Students may be unclear about what is meant by a “single expression.” You may want to have an example prepared showing how separate steps can be combined into one expression.

Facilitation Prompts • What did you do first when doing the calculation? Write the calculation. • What did you do next? How can you write that calculation with what you wrote for the first step?




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[Student Handout]

(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?

Answer: The subtraction gives the net profit, which you also might call Sundos’ income. Multiplying by 0.9235 lowers the amount of money that will be taxed, so that 92.35% of the income is taxed instead of 100%. (This is done because people who are not self-employed are not taxed on the employer contribution.) Multiplying by 0.153 means that the remaining income is taxed at 15.3% or 15.3 cents on the dollar. To summarize: Sundos pays 15.3% tax on 92.35% of her income.

This is a challenging question, especially Parts (b) and (c). Remind students of their work in Lesson 1.1.4 in which they explained a calculation in context. They might want to look at Question 5 from that lesson as a model.

Facilitation Prompts • When you multiply 0.9235 times the income, how does the result compare to the income? Students

are likely to start by saying it is smaller. Ask if they can be more specific (it is 92.35% of the income). o Indicate the (0.9235)($11,385 − $3,862) in the calculation (this is 92.35% of the income). What

did you do to that amount? (Took 15.3% of the amount.) • If students are really stuck, you may want to show the connection between the calculation and the

meaning on the board:

($11,385 – $3,862) Net profit or income

(0.9235)($11,385 – $3,862) 92.35% of the income Amount being taxed

0.153(0.9235)($11,385 – $3,862) 15.3% of 92.35% of the income 15.3% is the tax rate

Question 3 connects back to the central theme of Lesson 1.1.4—Order of Operations. That is why the question is phrased “that someone else could.” It re-emphasizes that there is an agreement in mathematics about the order in which calculations are done. Try to ensure that more than one correct expression is displayed. Writing expressions in this way will help prepare students to write variable expressions in Modules 3 and 4.

Question 5 assesses if students understood the process well enough to see how a 2% change in the tax rate would affect the total tax. It starts to get them thinking about variables, as 0.153 is replaced by 0.133 in their expressions and nothing else changes. It also allows you to see which students understand 1% as 1/100 well enough to use it as a benchmark in estimations.




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[Student Handout]

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) Estimate 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 a calculator, and try to do the calculation in your head. Write down your predictions of how much they will save.

(6) Calculate exactly 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, and no other tax changes are enacted.

Answer: Raven still owes nothing, so she saves nothing. Sundos saves about 2% of her profit, or $160. One way to estimate this is to estimate the difference between earnings and income and divide by 100 to find 1%. Then double that to find 2%. You could improve the calculation by moving it down a bit to account for the 0.9235 factor used on Line 4 of Short Schedule SE. The important fact for students to carry away is that finding 1% is the same as dividing by 100. Dividing by 100 is easy because you can move the decimal point mentally. This connects back to the benchmarks in Lesson 1.1.3.

Calculation: Raven saves nothing, as she owes nothing. Sundos owes $924.03 instead of $1,062.97, for a savings of $138.95.

Discuss the following questions with students: • How close were your predictions? • Are you satisfied with them? • How could you improve a prediction like this next time?

If you think you will have extra time in class, you might choose to include the questions about Martin Binford from the OCE. The questions are given here for your reference:

[From OCE]

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

Answer: 0.029(0.9235)($143,380 – $3563) + $13,243.20 = $16,987.71

(4) Which of the following expressions could be used to compute the self-employment tax owed by Martin Binford?

(i) 0.029 + 0.0235(143,380 – 3,563) + 13,243.20 (ii) 0.029 x 0.9235 x 143,380 – 3,563 + $13,243.20 (iii) 0.029(0.9235)(143,380 – $3,563) + 13,243.20 (iv) 0.029(0.9235)(143,380 – $3,563 + 13,243.20)

Answer: iii




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(5) Another way to calculate Martin’s tax is shown below. Based on this expression, select the statement that describes how Martin’s income is taxed.

0.153(106,800) + 0.029(129,121.43 – 106,800)

(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.

Answer: iv

The final question about how his income is taxed is interesting for discussion. This demonstrates the true structure of the tax system, which is that different levels of income are taxed differently. Unfortunately, the instructions in the form obscure this meaning. You will probably have to explain to students how this relates to the calculations in the form. First, the calculations take 2.9% of the total income. Then you add $13,243.20, which is 12.4% of $106,800. So you get 15.3% of the first $106,800.

[Student Handout]

Making Connections


Making Connections: Main Ideas to Highlight Quantitative reasoning requires skills in reading, interpreting, and using information.

Facilitation Prompts • Was this difficult? If so what made it difficult? (The answer is likely to be reading the instructions

on the form.) Emphasize that intelligent, well-educated people often have trouble reading forms like this. Students should not feel bad if they found it difficult. However, it is possible to develop skills to make understanding such forms easier. o What other things have you done in class in which you had to read, interpret, and use

quantitative information? Lesson 1.1.1: Reading wording carefully, thinking about the meaning of the stats Lesson 1.1.4: Making sense of information from survey Lesson 1.1.5: Interpreting information from the credit card disclosure form, interpreting

meaning of calculations Lesson 1.2.1: Interpreting information about countries to compare water use




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Quantitative reasoning requires understanding skills with regard to calculations, including understanding what they mean, order of operations, and flexibility in calculation methods.

Facilitation Prompts • Why do you think Question 4 was asked? (to understand the calculations/numbers) • Why might it be important for a business owner to understand the meaning of the

calculations/numbers? (to understand how taxes affect earnings) • What work have you done in class that is similar to this?

o Lesson 1.1.4: Order of operations, flexibility with calculations o Lesson 1.1.5: Making sense of calculations

[Student Handout]


(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.)

Answers:

(a) Answers will vary: 0.25(63,500 – 34,500) + 4,750 = $12,000

(b) The 4,750 is the tax on the first 34,500 of income: • Everyone is taxed 10% on the first $8,500 of income: $850 • In the second tax bracket, you are taxed 15% for amounts between $8,500 and $34,500. If your

income is $34,500, this amount is 0.15(34,500 + 8,500) = 3,900 • 3,900 + 850 = 4,750


Lesson 1.2.3: Interpreting Statements About Percentages Theme: Medical Literacy





Percentages Prevalence of smoking Not applicable Level 2


• have a basic understanding of the word percent and the notation used to describe percents (%). • be able to use a calculator to divide two numbers and interpret the resulting decimal

representation as a percent. • be able to calculate and estimate percentages.


• 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 the decimal representation of a number and a percent. • read and use information presented in a two-way table.

Explicit Connections • Every percent is really a statement about two quantities, specifically a comparison value and a

reference value.







Suggested Timeline


5 minutes Read the Problem Situation and begin working on Question 1

Individual

8 minutes Question 1 Small groups

5–7 minutes Groups present their reasoning behind Question 1 Class discussion

3–5 minutes Question 2 Small groups or class discussion

12 minutes Question 3 and 4 Small groups

10 minutes Groups report out on their solutions to Questions 3 and 4

Class discussion


Special Notes This lesson has two overarching goals. The first is to understand that two numbers are used to calculate any percentage (the numerator and the denominator). The second is to understand that careful reading of words or tables is necessary to correctly identify these two numbers.

While no algebraic manipulation is necessary in this lesson, the practice of using variables to name unknown quantities is introduced. Additionally, the use of the inequality symbols is introduced briefly in this lesson. This extra layer of symbolism may cause some students some angst, but try to reassure those students that the variable and symbols are just used to more efficiently communicate an idea. It becomes cumbersome to keep repeating the phrase “the percentage of women who smoke” over and over, so you will just refer to this as Q1 (Quantity 1).

Productive struggle in this lesson occurs mostly with Questions 1 and 2, as students are asked to grapple with some percentages expressed in terms of unknown amounts.

[Student Handout]

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

Before students start, note the use of variables here. Q is used as a common variable to indicate that you are talking about two similar quantities. The numbers 1 and 2 indicate that the quantities represent two different quantities, but their values might still be the same. For example, if C1 represents students in one class and C2 represents students in a second class, it is possible that C1 = C2, even though they represent different things. This is also an opportunity for a quick review of inequality symbols.

The purpose of Question 1 is for students to realize that the phrases used to describe Q1 and Q2 sound very similar, but the phrases are referring to two completely different percentages, or ratios. It is fine for students to come to the conclusion that there is not enough information to determine how these two quantities are related. Encourage these students to be prepared to explain their reasoning to the class and, if there is still time remaining before the class discussion, have them move on to Questions 2 and 3.

[Student Handout]

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.

Answer: These two quantities refer to two different ratios and are typically not equal to each other. In fact, Q1 could be greater than Q2, or Q2 could be greater than Q1 (as later examples will show).

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




Facilitation Prompts • The reference value under consideration is not explicit in this problem. If this is causing difficulties,

suggest that students work with a fixed set of people (i.e., adults in the United States, students enrolled at their college, students in the classroom).

• If students are having trouble with the language, encourage them to interpret these percentages numerically (as in Questions 2–3) or visually. Pie charts make for a nice visual interpretation. To help visualize Q1, the whole pie would represent all women in the group under consideration and Q1 could be the “slice” of the pie that represents those women who smoke (the remainder of the pie represents women who do not smoke). To help visualize Q2, the whole pie would represent all smokers in the group under consideration and Q2 would be the “slice” of the pie representing those smokers who are women. Note: In both pies, the slice represents women smokers, but a percentage compares an absolute value (number of women smokers) to the value of the whole (which is different for each pie).

[Student Handout]

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

Answer: To compute Q1, you need the total number of women in the group under consideration and the number of these women who smoke. To compute Q2, you need the total number of smokers in the group under consideration and the number of these smokers who are women.

[Student Handout]

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?

Answer: There are (12,736 + 5,362) = 18098 women in the study. 5,362/18,098 = 0.296, so approximately 30 percent of women smoke.

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

Answer: There are (4,572 + 5,362) = 9,934 smokers, of which 5,362 are women. Since, 5,362/9,934 = 0.54, approximately 54% of smokers are women.




Note on language: The use of the word suppose may be confusing to students. Explain that you use this word to indicate that you are setting up an example or an imagined situation, which is different from when you give actual data. The use of the word here indicates that this is not a real study.

Notice that the numerators for both Parts (a) and (b) are the same, the number of women smokers, but the denominators have changed. Try soliciting this observation from students. You might ask, “What is the same about these two calculations? What is different?” Follow up with a discussion around how the use of language helped determine the correct numerator and denominator. For instance, both phrases deal with women smokers (the single entry of 5,362 in the table). However, one phrase uses the language “percentage of women,” which indicates you want to relate the number of women smokers to the entire group of women. The second phrase uses “percentage of smokers,” which indicates you want to relate the number of women smokers to the total number of smokers.

Reinforce the use of the terminology comparison value for the numerator and reference value for the denominator.

Further, note that in this example, Q1 < Q2.

[Student Handout]

(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?

Answer: There are (536 + 1,053) = 1,589 women in the study. 536/1,053 = 0.337, so approximately 34 percent of women smoke.

[Student Handout]

(b) What percentage of smokers at Northwest College are women?

Answer: There are (1,256 + 536) = 1,792 smokers, of which 536 are women. Since, 536/1,792 = 0.299, approximately 30% of smokers are women. In this example Q1 > Q2.

[Student Handout]

(c) Consider the following statement based on the information in the table without doing any calculations. A newspaper stated that 40% of the male students at Northwest College smoked. Is that claim reasonable? Explain why or why not.

Answer: No. The number of men who smoke is greater than the number of men who do not smoke. The smokers must be more than 50% of the total population of men.




[Student Handout]

(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?

Answer: Since the number of smokers needs to be an integer, you might expect to have about three or four women out of 100 who smoke.

[Student Handout]

(b) Out of 1,000 adult women in China, about how many are smokers?

Answer: Probably around 37.

[Student Handout]

(c) Out of 100 adult women in the United States, about how many are smokers?

Answer: About 19.

[Student Handout]

(d) Out of 1,000 adult women in the United States, about how many are smokers?

Answer: About 190.

[Student Handout]

(e) Are there more women smokers in China or the United States?

Answer: You cannot tell without knowing the number of women in each country. While the percentage in the United States is much higher, China is a much larger country.

[Student Handout]

(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?

Answer: 590/1,000 = 0.59 = 59%. About 60% of men in China smoke.




[Student Handout]

Making Connections


Making Connections: Main Ideas to Highlight Every percent is really a statement about two quantities, specifically a comparison value and a reference value

Verbal descriptions of percents can be very difficult to understand without taking time to think carefully about the two quantities being compared. In general, when working with percentages, it is a good habit to ask what quantities are being compared.

Facilitation Prompts • Suppose you are reading the school newspaper and come across the statement, “62% of our

entering students are women.” How was this single number (62%) computed? How do you know what number to use for the numerator and what number to use for the denominator?

Answer: 62% was computed by dividing the number of entering students who are women by the total number of entering students. The phrase “of our entering students” indicates that the number of entering students constitute the reference value.

• Is this different than saying “62% of women on campus are entering students?” How do the numerator and denominator compare with the last example?

Answers: The reference value (denominator) is total number of women on campus and the comparison value is the number of entering students who are women.




[Student Handout]


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?”

Answers: Part (a) is Table 2. You can tell because the columns add to 100%, so the reference value is the grade. This means that 44.4% of the students who got an A were in the morning class. Part (b) is Table 1. The rows add up to 100%, so the reference value is the students in each class. 12.5% of the students in the morning class received an A.


Lesson 1.2.4: Percents and Probabilities - OPTIONAL Theme: Medical Literacy





Calculating probabilities as a percent

Medical testing OCE 1.2.3: Strategies to increase confidence

Level 2


• have a basic understanding of the meaning of percent and the notation used to describe percents (%).

• be able to use a calculator to divide two numbers and interpret the resulting decimal representation as a percent.

• calculate percentages.


• 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.

Explicit Connections • Correctly identifying the comparison value and the reference value is very important in calculating

percents. • Percentages have multiple uses with different terminology, but the meaning is the same.







Suggested Timeline


5–8 minutes Introductory discussion (if desired) Class discussion

5 minutes Read Problem Situation 2 and struggle with reading the table

Individual

5 minutes Clarify table as necessary Direct instruction

10–15 minutes Work on Questions 1–7 Small groups

5 minutes Go around the room to solicit answers for Questions 1–5; go over any discrepancies

Class discussion

10 minutes Groups report out on Questions 6 and 7; make sure groups explain their reasoning

Class discussion/group presentations

5–8 minutes Making Connections Class discussion

Special Notes This lesson builds on and extends the concepts in Lesson 1.2.3. As with Lesson 1.2.3, this lesson emphasizes the importance of taking care when working with percentages, including taking the time to carefully identify the appropriate numerator and denominator. The main goal of this lesson is to continue to increase students’ proficiency surrounding the calculation and interpretation of percents within authentic contexts. This lesson also introduces students to the use of percentages to quantify statements about chance—probabilities. The authentic context is that of drug testing for athletes. This is an issue that arises frequently in the media (consider baseball’s Barry Bonds or some of the recent Tour de France controversies) and often becomes a critical issue for colleges and high schools as they struggle to balance fair athletic practices and individual civil liberties. The terminology and mathematics in this lesson is not restricted to drug testing for athletes. Any time a medical test is administered (HIV/AIDS, pregnancy, cancer screening) a patient must be aware of the likelihood for false readings. Patients must have the quantitative literacy skills to ask their doctor the right questions and fully understand the responses.

While Lesson 1.2.3 introduced students to data presented in a two-way table, students may continue to struggle with interpreting what the numerals in each “cell” represent. Struggling students should be encouraged to read the row and column headings to help interpret each value. Productive struggle in this lesson will be around calculating and interpreting the many different possible percentages. There are four important percentages or probabilities that students will encounter:

1. The probability that this test correctly identifies an athlete who is using a performance-enhancing drug (PED).

2. The probability that this test correctly identifies an athlete who is not using a PED. 3. The probability that this test incorrectly gives a negative result. 4. The probability that this test incorrectly gives a positive result.

The first two relate to the accuracy of the test, while the last two quantify inaccuracies in the test. All four of these probabilities are different and independent of one another. That is, knowing one of the




above does not allow you to easily compute any other. The out-of-class experience is crucial for a more complete understanding of the subtleties that surround the notion of false positives/negatives. In the OCE, the same drug test is administered to a different group of athletes. Notice that the accuracy (the first two probabilities) in the OCE is the same as the in-class assignment. Nonetheless, the probabilities for false-positive and false-negative readings have changed because the probabilities are affected by a difference in the group of people being tested.

Instructors may wish to begin this lesson with a brief classroom discussion regarding the implications of a test that is not 100% accurate. For example, a high school principal would not want to falsely accuse a student of drug use (false positive). On the other hand, an athletic organization would not want to allow athletes who are using PEDs to continue to participate in the sport (false negative). If the test being administered is a test for the presence of cancer or HIV/AIDS, then the implications of a false positive or false negative become even graver.

See the notes at the end of the lesson regarding reviewing for the Module 1 assessment.

[Student Handout]

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 PEDs

Athletes Not Using PEDs Total




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.

Check to make sure that students understand that probability is a measurement of “chance.”

[Student Handout]

(1) The table is missing five entries. Fill in the missing entries.

Answer: Shown in table in italics

[Student Handout]

(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?

Answers: (a) 10; (b) 9; (c) 9/10 = 0.9 = 90% chance of receiving positive test result if one is using PEDs.

[Student Handout]

(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?

Answers: (a) 490; (b) 485; (c) 485/490 = 0.9898 = 99% chance of receiving a negative test result if one is not using PEDs.




[Student Handout]

(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.

Be very careful here! Out of all the negative test results (486), only one of them was incorrect! So the chance of receiving a false negative would be 1/486 = 0.002 = 0.2% (or 2/10 %)—a very small chance.

[Student Handout]

(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?

Answer: The test produced only 14 positive results. However, five of these were incorrect. So, the chance of this test producing a false positive is 5/14 = 0.357 = 35.7%. So, over a third of the time, you should expect a student to be falsely accused of using PEDs! Note that this figure is not obvious until you think carefully about the numerator and denominator that go into the calculation. The principal may not wish to accuse a student of using PEDs with such a high probability of being wrong.

[Student Handout]

(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.

Answer: You might report the figure from Question 2. The test correctly identified those students who do not use PEDs 99% of the time. Thus, focusing just on this value, the test seems quite accurate.

Alternately, you might report the figure from Question 4, which indicates that only 0.2% of the time, an athlete who is using PEDs is not identified with this test.




[Student Handout]

(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.

Answer: The false positive rate from Question 5 is probably the best figure to report if you are showing how inaccurate the test really is. More than one-third of students who receive a positive test result are not using PEDs.

[Student Handout]

Making Connections


Making Connections: Main Ideas to Highlight Correctly identifying the comparison value and the reference value is very important in calculating percents.

As with Lesson 1.2.3, the key here is to be very careful with numerators and denominators when computing percentages.

Facilitation Prompts • What sort of errors did you make in calculating the percentages? (You might want to refer to a

specific problem if you observed students using the wrong comparison or reference value.) • What was most important (or most helpful) in deciding which numbers to use in calculating

percentages?

Percentages have multiple uses with different terminology, but the meaning is the same.

In previous lessons (1.1.3, 1.1.5, 1.2.3), percents were used as a way to standardize ratios with different reference values. For example, “20% of women smoke” means a rate of 20 out of 100, no matter if the group includes 50 women or 10,000 women. It is also important to point out that often percentages are used to describe probabilities. That is, probabilities are just ratios too! If students are familiar with illustrating a certain ratio (or percent) as a shaded region in a pie chart, one can then think about throwing darts at the pie chart. If you’re not a very good dart player (i.e., you throw a dart randomly at the pie chart), the likelihood of the dart hitting the shaded region is the original percentage. However, the foundational meaning of the percentage as a comparison to 100 is still the same.

Facilitation Prompts • Give three examples of how percent was used in this class before this lesson. • How was percent used in today’s lesson? • How can a percent, or ratio, be represented graphically? • How can this same graphic be used to help visualize a probability?




Notes on the Context The language of “false positive” and “false negative” may be new to many students, yet it is heard often in the media and on certain product packaging (pregnancy tests, for example). It may be surprising that a seemingly very accurate test can have a high false-positive rate. While it is beyond the scope of this lesson, if students are interested in this result, you might point out that that the false-positive rate depends not only on how accurate the test is, but also on the prevalence of the condition being tested for. For example, in the above table, only 2% of the athletes in the population were using PEDs. If you used the same test on a population with a higher number of athletes using PEDs, the false-positive rate would actually decrease. See, for example, the table in the out-of-class experience. When reporting the accuracy of such tests in the medical literature, the terms sensitivity and specificity are used. Again, it is beyond the scope of this lesson to introduce this additional terminology, but some instructors or students may encounter this language during extensive research on the topic.

Note about Further Applications: This question is designed to highlight the relationship between prevalence rates and false positive rates described in the instructor notes above. It is also an excellent writing prompt because it requires students to include background information. However, you may feel it is too complex for students this early in the course. Here are some ideas of how you might use it:

• Assign it later in the course. • Assign it in stages. Have students do Parts (a) and (b). Then either check their answers or have

them check with other students. Assign Part (c). • Make peer review of Part (c) a part of the assignment. Supply a rubric for students to use to

evaluate themselves. • Discuss in class if there is time.

Notes on Reviews for Module Tests A concept-and-skill list is given at the end of each module. The list is in a separate document so that you can give it to students at any time. This list should be given to students to self-assess their readiness for the test. The end-of-module reviews can be handled in various ways; a few implementation possibilities are listed below.

Students should have some choice of problems based on their self-assessments. This allows students to match their self-assessment to the course material. However, at the beginning of the course, many students may need guidance in making good review choices. One option is starting with a more prescriptive review process and increasing student responsibility throughout the course.

Here are some ways to organize your class review: • Selecting review problems No. 1: Select a number of problems that all students are required to

do. In addition, require students do a minimum number of additional problems based on their self-assessment. For example, students might be required to select a minimum number of problems from previous OCE that they redo, based on their lowest rated skill and concept areas. Writing explanations for their thinking and showing all work should be required, especially important for multiple-choice questions.

• Selecting review problems No. 2: Give students a problem bank with a selection of problems from each module. Require that each student do a minimum number of problems that they




select based on their self-assessment. For example, students might be required to complete 15 problems selected from either the problem bank or previous work.

• In-class review: Require students to turn in their self-assessments before the in-class review day. Select problems to assign for review based on their self-assessments. Let students work on the problems in class and then finish on their own.

• Comprehensive review assignment: Have students complete the following end-of-module review assignment: o Front page: All important vocabulary words with definitions/explanations in your own

words. o Front page: All formulas or strategies with an explanation about what each is used for. Since

this course does not focus on standardized formulas, this might be a record of student-created strategies.

o Front page: Ten sample problems you think are important and that seem likely exam questions. Include directions.

o Back page: Solutions to the 10 sample problems from the front page. o Back page: A list of questions or concerns you have about the module material (questions

you want to ask a classmate, your instructor, or the learning center). • Preparing a Practice Exam (1–2 days)

o Day 1—Have students bring 10 questions to class that both represent the skills and concepts they feel represent what they should know at the end of the module and that they rated their understanding the lowest on. Students should have complete solutions for their 10 problems on a separate sheet of paper.

Put students in groups of four. Have students choose the best 10–12 that represent what they feel they should know and what will be on the exam. You can bring scissors and tape and have students piece their questions together to make an exam.

o Day 2 – Have students take each other’s practice exams in a similar exam setting. If you have time, students can grade each other’s exams and provide feedback and explanation. This exercise is especially good for the first exam in a term.

Important: Students must be able to check their work to accurately assess their readiness. You can post answers outside your office, at your college learning center, or online. Or you can have peer-checking and feedback incorporated into your review assignment.




Incorporating the End-of-Module Review Students will have regular OCE assignments to complete plus their review for exams. Consider how you want to schedule this work. You might consider separating the Prepare for the Next Lesson section from the OCE and assigning that after the test. Two possibilities for Module 1 are shown below.

With a Day for Test Review Without a Day for Test Review

Day 1 In class: Teach Lesson 1.2.4. Out of class:

• Assign OCE 1.2.4, but do not assign the Prepare for Lesson 2.1.1 section.

• Assign the Module 1 self-assessment. Day 2 In class: Review for Module 1 exam. Students do self-selected problems, problems selected by the instructor, or the creating a practice exam activity. Out of class: Students continue working on review problems.

Day 3 In class: Module 1 exam Out of class: Assign the Prepare for Lesson 2.1.1 section.


• Assign OCE 1.2.3. • Assign first half of the self-assessment skill

or concepts along with associated review problems.


• Assign OCE 1.2.4, but do not assign the Prepare for Lesson 2.1.1 section.

• Assign second half of the self-assessment skill or concepts along with associated review problems.

Day 3 In class: Module 1 exam Out of class: Assign the Prepare for Lesson 2.1.1 section.




[Student Handout]


(1) Refer to the problem situation used in this lesson and to Question 5 in the OCE for this lesson. You will call the population used in the lesson P1 and the population used in OCE Question 5 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.

Answers will vary. Example: Two groups of athletes were tested for the use PEDS. In both cases, 90% of those using PEDs tested positive. In the population in which 10% of the athletes used PEDs, 35.7% of the positive tests were false. In the population in which 20% of the athletes used PEDs, only 4.3% of the positive tests were false. This seems to indicate that a lower prevalence rate leads to higher rates of false positives.


Lesson 2.1.1: How Crowded Are We? Theme: Citizenship





Ratios and proportional reasoning

Population density OCE 1.2.4: Reflect on learning

Level 2


• understand the concept of area in square feet and square miles. • comprehend numbers up to the billions place. • be able to use a calculator to divide numbers. • be able to interpret fractions as division. • be able to interpret a decimal number.


• population density is a ratio of the number of people per unit of 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.

Explicit Connections • Relationship of ratios and percents—A percent is a special type of ratio in which a number is

compared to 100.







Suggested Timeline


5 minutes Read scenario, definition, and Example 1 Individual

10 minutes Work on Questions 1 and 2 Whole class and small groups

5 minutes Discuss Questions 1 and 2 Class discussion

5 minutes Work on Question 3 Small groups





Special Notes Scaffolding for Productive Struggle

You may find it helpful to draw a grid of 2-foot by 2-foot squares to introduce this lesson. Better yet, a floor tiled with 1-foot squares lends itself to demonstrate the example. Encourage students to draw a grid on paper, if needed, as a strategy for solving these problems.

[Student Handout]

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 4 people each stood on his or her own 2-foot by 2-foot square?

How would the population density change if 1 billion people each stood on his or her own 2-foot by 2-foot square?

Answer: The density is the same. It is still 1 person per 4 square feet or 0.25 person per 1 square feet.

Explain to students that you are going to do an activity to help them understand what you mean by population density. You will need a yardstick (or a tile floor) to measure the number of feet. Measure to the nearest foot, or make decisions about multiplying mixed numbers or using decimal approximations. Consider taking measurements in two different scenarios (crowded versus uncrowded) to contrast “more dense” with “less dense.” Ask students to stand close together in a rectangular area. They do not have to stand uncomfortably close, but it needs to feel different than how they are usually spaced in the classroom. Measure the length and width of the rectangle. Then, have students spread out evenly around the room or into a much larger rectangle. If you can use the full room, you could have the measurements beforehand. If you use a rectangle, measure the sides of a new rectangular footprint. Have students calculate footprint areas, and then have them calculate densities via proportions.

[Student Handout]

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.

(1) Calculate the population density in the classroom based on a small rectangle that the instructor designates. Be sure to include units.

(2) Calculate the population density based on a large rectangle that the instructor designates. Be sure to include units.

Concentrate on mathematically correct proportions and correct use of units (people per square foot). Check for understanding with the question, “How could you compare the densities of several populations, knowing the number of people in each population and where they live?” Here you are




looking for understanding of the process of proportional reasoning in the calculation of population density. A student’s comment, “Divide the number of people by the number of square units in which they live,” would indicate understanding of the process.

Two Important Points to Discuss • The density based on the large rectangle (2) had a lower density compared to the first density (1)

because you had the same number of people in a larger area. The point is that the densities are different even though the “crowdedness” of the actual people did not change. This demonstrates that a limitation of population density is that it measures an “average” as if the population were spaced out evenly. Throughout this course, the authors emphasize the need to understand the limitations of the measures and tools used.

• A ratio such as population density is useful because it helps you compare groups and areas of different sizes. You could compare the density of your classroom to the density of a group in the gymnasium because you are taking the ratio down to a unit rate (number of people per 1 square foot). This standardizes the measurement. This is a common use of ratios.

[Student Handout]

(3) Consider one billion people, each standing on his or her own 2-foot-by-2-foot square, where the squares are adjacent. Calculate the population density per square mile. Be ready to explain your reasoning after working with your group members. (1 mile = 5,280 feet)

Answer: One solution is 5,280 feet x 1 person/2 feet = 2,640 people. Therefore, there are 2,6402 people/square mile = 6,969,600 people/square mile.

Scaffolding for Productive Struggle

Students may need explicit help on Question 3 in the form of an intermediate problem. Ask students to consider the question: How many people stand side by side in 1 mile?

Prepare to intervene in the conceptualization of 1 square mile = 5,280 feet x 5,280 feet, but let students struggle with the concept first, mentally constructing the notion of a square mile from 1-foot squares. A visual representation can be helpful. Do not undermine the problem-solving process.

Some students may employ an alternate solution, based on calculating the area needed to situate one billion people. First, 109 people stand on an area measuring 4 by 109 square feet, and 4 x 109 ft2 ≈ 143.5 mi2. Next, note that 109 people/143.5 square miles ≈ 6,968,641 people/square mile.

Pose the following question for argument’s sake, “Is it important that one billion people were mentioned in the problem?”

Answer: The fact you have 1 billion people in the population at hand is not meaningful, but the fact that 6,969,600 of them occupy a square mile when spread out on 2-feet by 2-feet squares is meaningful. In other words, the supposed uniform spacing between people makes all the difference, even if this fact is not likely in the real world! Nevertheless, you suppose equal spacing when you compare populations by way of density.




[Student Handout]

(4) A more realistic example would be for us to consider a billion people standing on his or her own 100-foot-by-100-foot square. Calculate the population density per square mile.

Answer: One solution is 5,280 feet x 1 person/100 feet = 52.8 people. Therefore, there are 52.82 people/square mile = about 2,788 people/square mile. (That this is approximately the population density of Puyallup and Lakewood, less than half the density of Seattle, and 1/10th the density of New York City)

[Student Handout]

Problem Situation 2: Making Comparisons with Population Density

How crowded is China, compared to the United States?

(5) 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.

Answer: China is denser. Estimates of population densities will vary. In the United States, about 300 million people live on about 4 million square miles, leading, roughly, to a density of 75 people per square mile. In China, approximately 1,300 million people live on about 4 million square miles, implying an approximate density of 300 people per square mile. Some students may note that the answer is apparent because China has far more people in approximately the same space. This is a valid justification—in fact, it is highly desirable. Even though students did not do any actual “estimating,” this shows that they are really thinking about the situation. If no one brings this up, you should point it out.

[Student Handout]

(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.

Answer: China’s population is approximately four times as dense as that of the United States.

In both Parts (a) and (b), the notion of proportion is vital here. Encourage estimates that are easy, reasonable, and intuitive, even if they are not the most accurate.




[Student Handout]

(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.

Answer: The population density in the United States is approximately 83.4 people per square mile. The population density of China is approximately 361.5 people per square mile. Therefore, China is about 4.3 times as dense in humans as the United States. Facilitation Prompts • How closely did your estimates match your more precise calculations? • How many times more dense is the Chinese population than the American population? • When making density comparisons, why is it important to use a standard area measurement?

Answer: The number of people per square mile, for example, is used to compare densities of Chinese and American populations. Neither population really spreads out evenly in its respective country, but you suppose so for comparison. A standard area measurement, such as the square mile, fixes area so that the number of people assumed living within it determines density.




[Student Handout]

Making Connections


Making Connections: Main Ideas to Highlight Relationship of ratios and percents: A percent is a special type of ratio in which a number is compared to 100.

Module 1 focused on percentages with an introduction to ratios in Lesson 1.2.1. In this lesson, you are developing more understanding of ratios and specifically working with unit rates—a ratio with a denominator of 1. The relationship between percentages and ratio has been discussed before. This is a good point at which to assess student understanding of this connection. Both percentages and ratios are a comparison of two numbers by division. Units tend to be more important with ratios.

Facilitation Prompts • In this lesson, you used ratios such as 83 people per square mile. What did these ratios mean? • How is the ratio 1 person per 4 square feet alike and different from 0.25 person per 1 square

foot.

Answer: They both represent the same measurement of density. The first gives you a good way to understand the density. It is easier to think in terms of one whole person. The second is a unit rate, which tends to be easier for calculations. o Note that not all ratios are unit rates. The population density of the United States could

also have been described as 309,975,000 people per 3,717,000 square miles. What are advantages and disadvantages of the different forms?

Answer: The unit rate is easier to conceptualize and use for comparisons, but the other ratio gives you more specific data. The unit rate could refer to 166 people in 2 square miles as easily as the very large population of the United States.

• In Lesson 1.2.4, you had statements such as “there is a 90% chance of a positive test.” What does that mean?

Answer: On average, 90 tests out of 100 tests will be positive. Note that the units are the same. • How are ratios similar or different to percents?

Answer: Both compare two numbers so the measurement is relative to the reference value; they compare to different standards. Percent is to 100; unit rate is to 1; other ratios can have different reference values; units in a ratio can be different.

• Emphasize the importance of units in all ratios. Example: 50 miles per hour is very different from 50 miles per minute.




[Student Handout]


(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.


Lesson 2.1.2: 1185.3 Is a Crowd Theme: Citizenship


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Estimation, ratios, and proportional reasoning

Population density Not applicable Level 2

Prerequisite Assumptions Before beginning this lesson, students should know

• that you can 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.

• how to interpret an answer given in scientific notation on their calculator. • how to calculate and interpret population density.

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.

Explicit Connections • Estimation strategies with large numbers. • Rates and proportional reasoning can be used to predict values.







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Suggested Timeline


3 minutes Lesson introduction Class discussion

5 minutes Question 1 Groups

8 minutes Discuss strategies on Question 1 Class discussion

8 minutes Questions 1–4 Groups

5 minutes Discuss answers to Questions 1–4 Class discussion

8 minutes Question 5 Groups



Special Notes There is a separate handout with the data for students. They are given information on population density and the states in the previous PNL (2.1.1). Students are also asked to calculate the population density for Alaska, Idaho, Kentucky, Louisiana, Nebraska, New Hampshire, New Mexico, Washington, and Wisconsin. Students should have this information recorded in a table like the one shown below.

State Land Area (Square Miles) 2010 Population Population Density

(People/Mile2) Alabama 50,744 4,779,736 94.2 Alaska 571,951 710,231 1.2 Arizona 113,635 6,392,017 56.3 Arkansas 52,068 2,915,918 56.0 California 155,959 37,253,956 238.9 Colorado 103,718 5,029,196 48.5 Connecticut 4,845 3,574,097 737.7 Delaware 1,954 900,877 461.1 District of Columbia 61 601,723 9,800.0 Florida 53,927 18,801,310 348.6 Georgia 57,906 9,687,653 167.3 Hawaii 6,423 1,360,301 211.8 Idaho 82,747 1,567,582 18.9 Illinois 55,584 12,830,632 230.8 Indiana 35,867 6,483,802 180.8 Iowa 55,869 3,046,355 54.5 Kansas 81,815 2,853,118 34.9 Kentucky 39,728 4,339,367 109.2 Louisiana 43,562 4,533,372 104.1 Maine 30,862 1,328,361 43.0




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State Land Area (Square Miles) 2010 Population Population Density

(People/Mile2) Maryland 9,774 5,773,552 590.7 Massachusetts 7,840 6,547,629 835.2 Michigan 56,804 9,883,640 174.0 Minnesota 79,610 5,303,925 66.6 Mississippi 46,907 2,967,297 63.3 Missouri 68,886 5,988,927 86.9 Montana 145,552 989,415 6.8 Nebraska 76,872 1,826,341 23.8 Nevada 109,826 2,700,551 24.6 New Hampshire 8,968 1,316,470 146.8 New Jersey 7,417 8,791,894 1,185.3 New Mexico 121,356 2,059,179 17.0 New York 47,214 19,378,102 410.4 North Carolina 48,711 9,535,483 195.8 North Dakota 68,976 672,591 9.8 Ohio 40,948 11,536,504 281.7 Oklahoma 68,667 3,751,351 54.6 Oregon 95,997 3,831,074 39.9 Pennsylvania 44,817 12,702,379 283.4 Rhode Island 1,045 1,052,567 1,007.3 South Carolina 30,109 4,625,364 153.6 South Dakota 75,885 814,180 10.7 Tennessee 41,217 6,346,105 154.0 Texas 261,797 25,145,561 96.0 Utah 82,144 2,763,885 33.6 Vermont 9,250 625,741 67.7 Virginia 39,594 8,001,024 202.1 Washington 66,544 6,724,540 101.1 West Virginia 24,078 1,852,994 77.0 Wisconsin 54,310 5,686,986 104.7 Wyoming 97,100 563,626 5.8 50 states + DC 3,537,438 308,745,538 87.3




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[Student Handout]

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.

Hand out the student data sheet. Students will have already calculated the densities for some states in the previous OCE. Start by working with students to find the population density of the state that you are currently in.

Give students 1 minute to look over the list to try to find the state with the highest and lowest population density. Ask for predictions. This can connect with Lesson 2.1.3 where students will learn that the percent increase depends both on the change and the base from which you are changing.

Have students work on the first question. Emphasize that they will now be using estimation instead of calculation. You may have to enforce this! Point out that for Question 1, they only have to place the states in the different categories. Both to save time and practice estimation, make sure students are dividing the states into groups before calculation. That is, make sure that the students do Question 1 before continuing on to the other questions. If you prefer the students to work with fewer states, you can divide the states into groups of 10 or so and give each group a subset, then combine the groups’ work in a class discussion. If you do this, make sure to show the calculation for one of the densest states and one of the least dense states when you go over Question 1, as not all groups will have one of these states.

[Student Handout]

Problem Situation: Estimating Population Densities

In this lesson, you will compare the populations of different states. 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. Divide those states into the categories shown in the table below.

(2) Now discuss strategies you can use to estimate the population density of the states without using a calculator. Use your strategies to divide these states into the categories shown in the table below: Georgia, Kansas, Montana, Nevada, New Jersey, Oregon, Rhode Island, South Carolina, Tennessee, Wyoming.




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(Note: States that are shaded are done as part of the OCE.)

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

people/mi2 Rhode Island Washington South Dakota Alaska New Jersey Louisiana New Mexico Wyoming Wisconsin Idaho Montana Kentucky Nebraska New Hampshire Nevada South Carolina Kansas Tennessee Oregon Georgia

After a few minutes, stop to discuss which estimation strategies students are using. Note that the District of Columbia is not a state.

One strategy students may use is to compare both the number of extra digits in the numerator and whether the first few digits of each number are larger. For instance, South Carolina has a density greater than 100 people per square mile because it has two more digits in the numerator: 4,625,364, than the denominator: 30,109 (and 46 > 30). Oregon, on the other hand, falls in the 10–100 category because 3,821,074 has two more digits than 95,997 (but 38 < 97).

Another strategy students may use is to write the numbers in scientific notation. For instance, using Oregon as an example, 3.82 x 106 is being divided by 9.6 x 104. The division of 3.85 by 9.6 gives a quotient that is smaller than 1. 106 divided by 104 gives 102. So the complete quotient is slightly less than 102 or 100 because multiplying by a number between 0 and 1 reduces the size of the number. In the South Carolina case, 102 is being multiplied by a number larger than 1, so its population falls between 100 and 1,000.

Another strategy could involve crossing out all but the first few digits of the two numbers. This is equivalent to rounding or truncating the numbers and dividing.

Other strategies are possible. Make sure each group provides its strategy and that it is effective. If the strategy is not effective, ask the group to choose one of the other groups’ strategies. If students are willing, they can even exchange students among groups with effective and ineffective strategies so that all groups are using an effective method.

If all the groups are having trouble getting started, you can show how to use front-end estimation to round the numbers before dividing. For instance, Oregon could become 3,800,000/96,000. Then give the groups another couple of minutes to come up with their strategy. You can also remind them of strategies they used in Lesson 1.2.1.

You will want to somehow take note of the strategies students are using for reference in the Making Connections section later in the lesson.




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As groups finish, have students put one of the columns of states on a list on the board. Allow other groups to adjust if they feel they should. Usually students resolve disagreements about how to classify a state successfully themselves. Be prepared to intervene.

When going over the classification of the states in the four categories of Question 1, the states of South Carolina and Oregon are illustrative. Show how those work as a class if you did not do so in the earlier discussion.

[Student Handout]

(3) Find which state from your table has the greatest population density. Calculate that population density. Round to the nearest tenth.

Answer: New Jersey has a population density of 1,185.3 people/square mile.

Students should see that they need only calculate the population densities in the first column of the chart in Question 1 to find this answer.

[Student Handout]

(4) Find which state from your table has the least population density. What is that population density? Round to the nearest tenth.

Answer: Alaska has a population density of 1.2 people/square mile.

Students should see that they need only calculate the population densities in the fourth column of the chart in Question 1 to find this answer.

[Student Handout]

(5) Tacoma, WA has a population of about 200,000 and covers approximately 62 square miles. New York City has a population of about 8.4 million, and covers approximately 469 square miles. If Tacoma had the same population density as New York City, what would the population of Tacoma be?

Answer:

The density of NYC is 8.4million/469sq.mi = 17,910 people/square mile. Multiply the area of Tacoma by that density: 17,910 people/square mile * 62 sq mi = 1,110,448 people. Write the calculation on the board with units and point out that you are finding a number of people. This is laying the foundation for dimensional analysis in Module 3. You may want to talk about accuracy; since the 8.4million and 62 only have 2 significant figures, it would make sense to report our result with the same accuracy: 1.1 million.




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Point out that, among scientists, it is a convention to use the smaller number of significant digits in a number when reporting a product. (Informally, significant digits are units for which you are sure the measurement is correct.) Otherwise a reader might think you know the answer better than you do.

When going over this, you can point out the units and how they cancel in an almost fraction-like way. This can be a connection to dimensional analysis in Module 3. Other proportion-solving techniques, such as cross-multiplication, may be discussed if students use them.

Question 5 is an extension question that develops both number sense and proportional reasoning. Part (a) may seem simplistic, but students often have trouble with this type of thinking. Encouraging different explanations of how students view the problem can be very valuable. Part (c) brings the problem back to the lesson to reinforce the ideas in Question 4.

[Student Handout]

(6) 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 mile)

(a) If you converted the density of Bangladesh to square miles, would the measure be larger or smaller than 1,127? Explain your reasoning.

Answer: The measure would be larger. A square mile is larger, so it contains more people than a square kilometer with the same ratio.

[Student Handout]

(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.

Answer in italics above. Discussion: How can you find out exactly? Draw a picture and calculate the area. Note that the ratio for linear length is different from the ratio for area. This is a good example of why you need to check your “common sense.”




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[Student Handout]

(c) How many people would be in Tacoma if the population density were the same as Bangladesh?

Answer: Depending on time, you can have the class convert from square kilometer to square mile or just give them the conversion, which is:

1127 ppl/sq km * (1km/0.62mi)2 = 2,931 people/square mile.

2,931 people/square mile * 62 sq mi = 181,774 people.

[Student Handout]

Making Connections


Making Connections: Main Ideas to Highlight Estimation strategies with large numbers.

The goal is to help students think about estimation strategies in terms of usefulness in different situations—in this case, with large numbers. Refer back to the list of strategies recorded earlier in the lesson and strategies used in Lesson 1.2.1.

Facilitation Prompts • Did you change your strategies as you worked? In what way? • What strategies were most useful? Why? • Would you have used different strategies if you had been working with smaller numbers?

Rates and proportional reasoning can be used to predict values.

Facilitation Prompts • Recall the following example from the Making Connections section of Lesson 1.1.3 and discuss it

in relationship to Question 4:

Total

Number of incidents with weapons

5 5 5 5 20

Number of incidents

25 25 25 25 100




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Note that in this example, the ratio of 25 to 5 could be expressed as 5 to 1 (or 1 to 5). There are a total of five incidents for every one incident with a weapon. Population density uses the same reasoning. The population increases by 1,185.3 people for every 1 square mile. You can use the rate to find new values (as in Question 4) by considering how the change in the first value (the number of square miles) would change the second value (the number of people) assuming the rate remains constant.

[Student Handout]


(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


Lesson 2.1.3: Measuring Population Change Theme: Citizenship


1




Absolute and relative change

Appointment of House of Representatives

OCE 2.1.2: Reflect on time and effort

Level 2



• know how to find a percent of a number.

Specific Objectives


• a relative change is different from an absolute change. • a relative measure is always a comparison of two numbers.


• calculate a relative change. • explain the difference between relative change and absolute change.


• Change can be discussed as absolute change or relative change. • A relative measure is always a comparison of two numbers.

Notes to Self







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Suggested Timeline


10 minutes Lesson introduction Class discussion

5 minutes Students should calculate absolute and relative change for at least one of their states (as in Questions 1 and 2)

Small groups

5–10 minutes Each group will share a strategy for computing absolute and relative change

Class discussion

10 minutes Answer Questions 1–7 Small groups

5–10 minutes Discuss answers to Questions 3–7; refer back to map from lesson introduction

Class discussion


Special Notes

The next lesson (2.1.4) uses graphs. There are support materials for both instructors and students on “Understanding Visual Displays of Information.” You may want to assign the student version as part of the OCE for this lesson. Students are expected to interpret graphs in this OCE in preparation for Lesson 2.1.4.

You will need to project the apportionment map given below unless you can make color copies for students to use. The map can also be found at the following link:

www.2010.census.gov/news/pdf/apport2010_map1.pdf

The key for the data for all the states is given here:

[Data and answers for all states]

Region State 2010 Population 2000 Population Raw Change Percent Change

ENC Illinois 12,830,632 12,419,293 411,339 3.31%

ENC Indiana 6,483,802 6,080,485 403,317 6.63%

ENC Michigan 9,883,640 9,938,444 -54,804 –0.55%

ENC Ohio 11,536,504 11,353,140 183,364 1.62%

ENC Wisconsin 5,686,986 5,363,675 323,311 6.03%

East North Central Region 46,421,564 45,155,037 1,266,527 2.80%




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ESC Alabama 4,779,736 4,447,100 332,636 7.48%

ESC Kentucky 4,339,367 4,041,769 297,598 7.36%

ESC Mississippi 2,967,297 2,844,658 122,639 4.31%

ESC Missouri 5,988,927 5,595,211 393,716 7.04%

ESC Tennessee 6,346,105 5,689,283 656,822 11.54%

East South Central Region 24,421,432 22,618,021 1,803,411 7.97%

M Arizona 6,392,017 5,130,632 1,261,385 24.59%

M Colorado 5,029,196 4,301,261 727,935 16.92%

M Idaho 1,567,582 1,293,953 273,629 21.15%

M Montana 989,415 902,195 87,220 9.67%

M Nevada 2,700,551 1,998,257 702,294 35.15%

M New Mexico 2,059,179 1,819,046 240,133 13.20%

M Utah 2,763,885 2,233,169 530,716 23.77%

M Wyoming 563,626 493,782 69,844 14.14%

Mountain Region 22,065,451 18,172,295 3,893,156 21.42%

MA Connecticut 3,574,097 3,405,565 168,532 4.95%

MA New Jersey 8,791,894 8,414,350 377,544 4.49%

MA New York 19,378,102 18,976,457 401,645 2.12%

MA Pennsylvania 12,702,379 12,281,054 421,325 3.43%

Middle Atlantic Region 44,446,472 43,077,426 1,369,046 3.18%

NE Maine 1,328,361 1,274,923 53,438 4.19%

NE Massachusetts 6,547,629 6,349,097 198,532 3.13%

NE New Hampshire 1,316,470 1,235,786 80,684 6.53%

NE Rhode Island 1,052,567 1,048,319 4,248 0.41%

NE Vermont 625,741 608,827 16,914 2.78%

New England Region 10,870,768 10,516,952 353,816 3.36%




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P Alaska 710,231 626,932 83,299 13.29%

P California 37,253,956 33,871,648 3,382,308 9.99%

P Hawaii 1,360,301 1,211,537 148,764 12.28%

P Oregon 3,831,074 3,421,399 409,675 11.97%

P Washington 6,724,540 5,894,121 830,419 14.09%

Pacific Region 49,880,102 45,025,637 4,854,465 10.78%

SA Delaware 900,877 783,600 117,277 14.97%

SA Florida 18,801,310 15,982,378 2,818,932 17.64%

SA Georgia 9,687,653 8,186,453 1,501,200 18.34%

SA Maryland 5,773,552 5,296,486 477,066 9.01%

SA North Carolina 9,535,483 8,049,313 1,486,170 18.46%

SA South Carolina 4,625,364 4,012,012 613,352 15.29%

SA Virginia 8,001,024 7,078,515 922,509 13.03%

SA Washington, D.C. 601,723 572,059 29,664 5.19%

SA West Virginia 1,852,994 1,808,344 44,650 2.47%

South Atlantic Region 59,779,980 51,769,160 8,010,820 15.47%

WNC Iowa 3,046,355 2,926,324 120,031 4.10%

WNC Kansas 2,853,118 2,688,418 164,700 6.13%

WNC Minnesota 5,303,925 4,919,479 384,446 7.81%

WNC Nebraska 1,826,341 1,711,263 115,078 6.72%

WNC North Dakota 672,591 642,200 30,391 4.73%

WNC South Dakota 814,180 754,844 59,336 7.86%

West North Central Region 14,516,510 13,642,528 873,982 6.41%




5


WSC Arkansas 2,915,918 2,673,400 242,518 9.07%

WSC Louisiana 4,533,372 4,468,976 64,396 1.44%

WSC Oklahoma 3,751,351 3,450,654 300,697 8.71%

WSC Texas 25,145,561 20,851,820 4,293,741 20.59%

West South Central Region 36,346,202 31,444,850 4,901,352 15.59%




6

Retrieved from

• U.S. Census Bureau (http://2010.census.gov/2010census/data/apportionment-pop-text.php) as the primary source.

• Wikipedia (http://en.wikipedia.org/wiki/List_of_U.S._states_and_territories_by_population) as a secondary source.




7

[Student Handout]

Specific Objectives


• a relative change is different from an absolute change. • a relative measure is always a comparison of two numbers.


• 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.

If possible, give each student a copy of the census apportionment map or project the map on a screen to show the apportionment of representatives after the 2010 census. The map also uses color to show which states are gaining representatives and which states are losing representatives.

Discussion Questions

• Which states have the largest and smallest populations?

Answer: States with a large number of representatives have large populations.

• Which states have small populations?

Answer: States with small populations have one representative.




8

• Which states are growing fastest and growing slowest? • What does it mean to be growing fastest in terms of representation?

Students may mention absolute change in people, or they may mention a growth rate. Simply acknowledge that they should have the answer by the end of the lesson. (Absolute change has the biggest impact on an individual states’ representatives.)

One misconception students may have is that states that lose representatives actually have lost population. By the end of the lesson and the out-of class experience, students should realize that these states are simply growing at a slower rate (except for Michigan).

One way to approach this lesson is to give half the groups in the class the states in the South Atlantic Region. Give the other half of the students the Mountain Region. This divides the states equally between groups. It will also allow you to make the point that the South Atlantic States are gaining more people in an absolute sense, but they are growing at a slower relative rate because more people live there. In case you want to have a broader project, or use data from your local area, the data for the entire country is included.

[Student Handout]

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 in the tables on the following page for Questions 1–6. Your instructor will tell you which table to work with.




9

South Atlantic States

2010 Population 2000 Population Absolute Change

Percentage 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

Mountain States

2010 Population 2000 Population Absolute Change

Percentage 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




10

(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.

Students should work through Question 1 quickly. Monitor work on Question 2. Give time to try methods and prompt. When a few groups have discovered methods, you can stop and discuss.


• Remind students that they have found relative measures before. You might refer back to the problem in OCE: 550 out of 810 students passed an exam. What percent passed? o Note the comparison value and the reference value. o What is the comparison value and reference value in this question?

• Make sure that students are using the 2000 populations and not the 2010 populations as the base. A good prompt is to remind them that they are finding the change relative to the 2000 population.

If some groups struggle you might spend the first few minutes of the next group section working with them or see if two groups want to shuffle members to put someone who has a correct method in each group.

Answers for all 50 states and the District of Columbia are given above.

[Student Handout]

(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.

Answers:

Question 3: In the Mountain Region: Arizona, Colorado, Utah; In the South Atlantic Region: Florida, Georgia, North Carolina.

Question 4: In the Mountain Region: Nevada (35.15%), Arizona (24.59%), Utah (23.77%); In the South Atlantic Region: North Carolina (18.46%), Georgia (18.34%), Florida (17.64%).




11

Question 5: States that have a smaller population in the beginning change by a higher percentage given the same absolute increase.

Question 6: Answers are given on the same page that shows the changes in state population. Students will have to total the 2000 and 2010 populations as an intermediate step in finding this answer. Students cannot just average the percentage changes for the states because the bases of those percentages are different.

[Student Handout]

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.

Answer: Michigan’s population decreased by 54,804 people or 0.55%. Students will sometimes want to report this as −0.55%. If the word decrease is used, then the negative is at best redundant and at worst implies an increase. If the words percentage change or relative change are used, then the minus sign tells the reader that it is a decrease.

So, “Michigan’s population decreased by 0.55%” and “Michigan’s population changed by −0.55%” are both correct, unambiguous statements.

When it is time to go over this, call on a group to share their lists of three states. Ask other groups if they agree. Make corrections until there is agreement. When discussing Question 5, it may be helpful to compare the calculation for percent increase for Nevada with that for Virginia. You can perform the calculation on the board or display student work using a document camera and data projector.

Virginia gained more people, but Nevada increased by a much larger percentage because its population was lower in 2000.

Compare the population changes in the two regions as a whole and the percentage changes. Students should notice that the Mountain Region is growing faster in one sense—the percentage rate is higher. In terms of actual people, however, the South Atlantic Region is growing faster. The South Atlantic Region gained four representatives and the Mountain Region gained three representatives. This shows the importance of absolute change in determining representatives.

To use states instead of regions, Florida and Nevada are illustrative of the fact that absolute change matters more in the representation context. Nevada grew at a higher relative rate, but Florida added more people. Hence, Florida got two new representatives while Nevada got one representative.




12

Discuss how to report the change in population in Michigan. Students may differ on whether to use a negative in front of Michigan’s population change.

[Student Handout]

Making Connections



Change can be discussed as absolute change or relative change.

One way to show a relative change is to use percentage increases or decreases. If the number to which the change is being compared is large, then the relative change can be small. For instance, $10,000 is 20% of a typical family budget in 2011, but $10,000 would represent only a fraction of a percent of the U.S. budget. Adding 1 million people to California would only change the relative population of California by a couple percentage points. Adding 1 million people to Wyoming would change the relative population by more than 100%.


• If you were going to buy something, and someone offered you $50 off the price, would you think you were getting a good deal? What if it were an $80 MP3 player? What if it were a $14,000 car? How does this relate to the ideas of absolute and relative change?

A relative measure is always a comparison of two numbers.

Using percentages to show relative change is the third way you have used percents in this class. Before this lesson, percentages were used to show probabilities (as in drug tests given to athletes) and to compare the relative sizes of two numbers (as in the percentage of smokers in a group).


• What are examples of using percentages from this class? • How are those examples alike and different?

Answers:

Different purposes: Relative size, probability, relative measure of change (in this lesson)

Alike: Always a comparison of two numbers




13

[Student Handout]


(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?

Answers:

(a) 100 billion is about 0.7% of the national debt.

(b) 40 billion is about 0.3% of the national debt.

(c) A politician wanting larger cuts would argue with absolute change because a multibillion-dollar cut is large compared to the amounts of money most U.S. households and businesses are used to seeing.

(d) To argue for the smaller cut, a politician might say that the difference between 0.7% and 0.3% of the national debt is such a small amount that it is not worth making deeper short-term cuts.


Lesson 2.1.4: Picturing Data with Graphs Theme: Citizenship


1




Graphical displays Household income, U.S. debt, population comparisons

Not applicable Level 1


• know how to read a line graph. • know how to read a bar graph. • know how to read a pie graph. • be able to calculate relative change.


• 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.

Explicit Connections • It is important to ask questions about and make sense of data. • In all relative measures, be aware of the reference value.







2

Suggested Timeline


5 minutes Check for understanding on OCE 2.1.3 Class discussion

8 minutes Work on Question 1 (discuss Question 2 as follow up)

Group or individual work followed by brief reporting out and discussion of Question 2

8 minutes Discuss OCE (“Jeff” question) and introduce GDP Class discussion and direct instruction

10 minutes Work on Question 4 Group work followed by brief discussion

8 minutes Discuss OCE (pie graphs) and work on Questions 5–7

Group work



Special Notes You may wish to refer to the instructor support materials on “Understanding Visual Displays” to think about how to help students read graphs.

[Student Handout]

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.

Check for understanding on the OCE (2.1.3) work with Graph 1. You may want to explain that “2009 dollars” means that the income has been adjusted to account for the change in the value a dollar (inflation). Students will learn more about this concept later. Then have students work on the following problem.




3

[Student Handout]

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 1 from OCE:

Graph 2:

Hopefully, students notice that the two graphs appear to show the same data. If they do not notice this, ask them to compare the data in the two graphs. Push them to focus on the data rather than the display.

Facilitation Prompts • Why do the graphs appear so different if they represent the same data?

Answer: They use different vertical scales, which makes the change in the first graph appear more dramatic.

• What is it about the way things are presented that makes the graphs appear so different?

Discuss the difference in scales. Since both graphs represent the same data, the relative change is the same. Graph 2 shows the entire vertical scale, making 6% appear to be a small portion of the total. With the truncated vertical axis, the change looks greater.

$48,000$48,500$49,000$49,500$50,000$50,500$51,000$51,500$52,000$52,500$53,000

1999 2000 2001 2002 2003 2004 2005 2006 2007 2008 2009

Aver

age

Hous

ehol

d In

com

e

Year

Average Household Income (2009 dollars)

$0

$10,000

$20,000

$30,000

$40,000

$50,000

$60,000

1999 2000 2001 2002 2003 2004 2005 2006 2007 2008 2009

Aver

age

Hous

ehol

d In

com

e

Year

Average Household Income (2009 dollars)




4

[Student Handout]

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

Most students will probably answer, “No,” based on the second graph. Point out that truncated scales are often used in the media, but they can be deceptive and are often not good representations of data.

Make sure students recognize that the same data is pictured in each graph and that the change in vertical scale can be used to emphasize or minimize the relative change.

The next section deals with Gross Domestic Product. Use the following question from the OCE to help students understand the concept.

[From OCE]

Two pairs of statements are given below. How can both pairs of statements be true? When did Jeff spend more on housing?

In 1990, Jeff spent $600 per month on housing. In 2010, Jeff spent $1200 per month on housing.

In 1990, Jeff spent 20% of his income on housing. In 2010, Jeff spent 10% of his income on housing.

Ask students about their answers from the OCE. If they have trouble getting started, ask a couple of students how much they earn now and how much they hope to earn 20 years from now. Generally, students will predict a higher future income. Both pairs of statements could be true if Jeff’s income has increased over time.

(In this example, Jeff’s income has grown from $30,000 to $120,000. The idea of growth is the important point here, not the specific values. Using specific values can be helpful for demonstration purposes.)

This refers back to the idea of an absolute measure versus a relative measure. One comparison is not right or wrong. They both give true information about the situation. The most complete way to answer the second question is to say that Jeff spent more money on housing in 2010, but he spent a higher percentage of his income on housing in 1990.

Give background on the next context by comparing it to the “Jeff situation.” GDP, or Gross Domestic Product, can be thought of as the country’s income. It is the value of all goods and services the country produces. The national debt is how much the country owes. Just as Jeff’s spending on housing can be calculated as a percent of income, a country’s national debt can be calculated as a percent of its GDP.




5

[Student Handout]

Problem Situation 2: Reading Bar Graphs

Your class will discuss how the “Jeff’s Housing” question from your OCE assignment to understand the relationship between the national debt and GDP.

Graph 3

Graph 4

0

2

4

6

8

10

12

14

16

1950 1960 1970 1980 1990 2000 2010

Trill

ions

of D

olla

rs

US National Debt in Trillions of Dollars

0

10

20

30

40

50

60

70

80

90

1950 1960 1970 1980 1990 2000 2010

Perc

ent o

f GDP

US National Debt as a Percentage of GDP




6

(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?

To answer this question, students should make parallels to arguments made by the class when they were comparing the amount of money an individual spent on housing over time. If a group is having trouble, point out that Graph 3 can be thought of as “total spending on housing over time” and the second graph can be thought of as “the percent spent on housing over time.”

As a total amount, the debt has never been higher, but as a percentage of income, debt was not at the highest level ever in 2010. It is a matter of opinion whether it is out of hand. The country has had higher levels of debt relative to GDP. Some students might emphasize the upward trend in the graph.

This is a good time to discuss the meaning of the question, “Is it true?” Part of the above statement is true, but that does not mean it gives complete information about the situation. Students should begin to realize that a careful consideration of multiple types of data is important.

Students have the following two graphs and questions in their OCE. It is important to discuss this question. It was given in OCE to give students time to think about it, but do not expect students to have a solid understanding of these concepts before class. Ask them to share their responses in their groups and then complete Questions 4 and 5.

[From OCE]

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.

Answer: You cannot tell because you do not know the reference value.

Students will have differing opinions. Capture some reasons why the students think this statement is true or false. Students may make a connection to the earlier lesson that featured 2010 census data. Make sure students understand that the size of the section in the graph is proportional to the percentage represented. More area represents more of the reference value in that category.

Note on time: If you run short on time, do Question 4 as a demonstration to illustrate the point in the question above and skip Question 5.




7

[Student Handout]

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.

Possible methods:

In 2010, around 16%—or 1 in 6—Americans was Hispanic. 1/6 of 300,000,000 is 50 million. The rest (260,000,000) were non-Hispanic.

In 2050, 30%—or around 1/3—of Americans will be Hispanic. 1/3 of 439,000,000 is around 140,000,000. The rest (around 300,000,000) will be non-Hispanic.

[Student Handout]

(5) In the OCE, you were asked to determine if this statement was 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

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

Answer: The work supports the notion that the claim (that the pair of graphs predicts that the number of non-Hispanics in the United States is expected to decline between 2010 and 2050) is false. The Hispanic population will grow much faster (around 180%) than the non-Hispanic population (around 15%), but both populations will grow.




8

[Student Handout]

Making Connections


Making Connections: Main Ideas to Highlight It is important to ask questions about and make sense of data.

In previous work, you have used this skill in relationship to numerical data. Here you extend this to graphical presentation of data. Point out to students that this is a “quantitative mindset” or “habit of mind.”

• Lesson 1.1.1: Questioning data, using estimates • Lesson 1.1.2: Making sense of large numbers with estimates and comparisons • Lessons 1.2.3 and 1.2.4: Making sense of statistical statements by using examples • Lessons 1.1.4 and 2.1.3: Making sense of calculations

Facilitation Prompts • What should you watch for when you read a graph? (Scales; truncated scales; is data in

“absolute” or “relative” terms; titles and labels; units) • What can you do to make sense of graphical information? How is estimation useful in this? • Refer back to the questions in the lesson: Did it help make sense of the data when you answered

questions about it? You can make it a habit to ask your own questions. • How is this similar to previous work with data in numerical form?

In all relative measures, be aware of the reference value.

This lesson also reinforced the idea of understanding relative and absolute measures. The reference value is important in relative measures. This connects back to Lessons 2.1.3, 1.2.3, and 1.2.4 in which students had to carefully identify the reference value. It also connects to the previous point about carefully reading graphs. The titles of the pie charts in this lesson give the year in which the populations were being compared. Since the years were different, the population bases would be different.

Facilitation Prompts • How was the “Jeff” question similar to the question with the pie graphs?

Answer: In these situations, you could not compare relative values directly because the reference values changed.

• How does this connect to the comparison in Lesson 1.2.3 (the percentage of women who smoke versus the percentage of smokers who are women)?

Answer: Determining the difference in the two quantities meant understanding the reference value in each situation




9

Note on future lesson: Assign Further Applications (2) if you want to use class data on average number of credit cards in Lesson 2.2.1.

[Student Handout]


(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?


Lesson 2.2.1: What Is Average? Theme: Personal Finance





Measures of central tendency

Credit cards OCE 2.1.5: Reflect on learning

Level 3



• perform basic operations using quantities such as integers, fractions, or decimals with the aid of technology.

• identify mean, median, and mode of a small data set (based on the reading provided in the previous PNL).

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.


• calculate the mean, median, and mode for numerical data. • create a data set that meets certain criteria for measures of central tendency.


• Measures of central tendency are tools—that have advantages and limitations—for summarizing data.

• It is important to ask questions about data.

Notes to Self







Suggested Timeline


15 minutes Questions 1 and 2 Introduction—Credit Cards

Class discussion Discuss the questions as you read through the first paragraph of the Problem Situation with students

5 minutes Check for understanding on measures of central tendency from OCE

Teacher directed

15 minutes Question 3 Individual work with class discussion

10 minutes Question 4

Pair and share Class discussion

5 minutes Making Connections

Special Notes

Students were given information about measures of central tendency in the previous PNL (2.1.4). The Further Applications from Lesson 2.1.4 includes a survey about credit cards that you may want to use for this lesson. If you are using this information, be prepared to show data and the mean to students.

This lesson includes an Excel spreadsheet for demonstration. The file name is QW 2.2.1 support spreadsheet.

[Student Handout]

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.


• calculate the mean, median, and mode for numerical data. • create a data set that meets certain criteria for measures of central tendency.




[Student Handout]

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

Helping Students Read the Information

It is important for students to learn how to read, digest, and interpret information with lots of numbers. You may want to go through the second paragraph line by line talking about the numbers in the information.

Point out the last sentence in the third paragraph: “In 2004, three-fourths of all American families have at least one credit card, but only 58% carried a balance.” What is ambiguous about this statement? The reference value is unclear. It could be “all American families” or it could be American families with “at least one credit card.” In this case, it is the latter. It is common to see statements like this in which the reference value is not clear. It is important to recognize this and understand that the information is meaningless without the reference value.

Encourage students to give meaning to the numbers. For example, quantify the average debt; that is, answers questions like: How many months salary is this? For a monthly salary of $5,000, this is almost three months salary. This discussion will lead into formal use of measures of center. Before going on, check for understanding on how to identify the mean, median and mode (from the previous OCE).

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




[Student Handout]

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?

Answers:

Average number of credit cards per person: 600,000,000 ÷ 300,000,000 = 2 Average number of credit cards per household: 600,000,000 ÷ 100,000,000 = 6


• According to www.CreditCards.com, the average credit card holder has 3.5 cards. Ask students, “Why is 3.5 (the average number of credit cards held by a card holder) different from 2 (the average number of credit cards per person)?”

Answer: The two numbers represent different averages. The first is the average number of cards for only people with a credit card. The second represents the average number of cards per person (all people, with or without credit cards). The two fractions will have the same numerator, but different denominators.

• If you have collected student data on credit cards, show it to students now. “Why is the average number of cards owned by students in this class not 3.5?” (most likely)

Answer: Students are not as likely to have accumulated as many cards as older adults. Students may also have credit restrictions put upon them by parents. A very small sample that is not representative of the entire population (the class) does not usually have the same mean as the population as a whole.

[Student Handout]

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

Answer: This is probably a reference to the mean, so if all the credit card debt were combined and then spread evenly among the households with a credit card, each household would owe $14,743.


• What is meant by “average credit card debt”? • Does everyone have this much debt? • Consider the data from above that 42% of credit cardholders carry no debt. This means some of the

other 58% cardholders owe much more than $14,743.




[Student Handout]

(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.

Class Discussion

• How can such different groups have the same mean? Discuss each set separately.

Answers: A and C have a variety of values that “balance” each other out. Group A has more extremes—one very high and one low.

Each value in Group B is the mean.

Group D has one very high value balanced by four low values.

• When you look at the data in each group, does a mean that is close to $4,000 seem reasonable? Explain your reasoning.

Answer: Students may say that $4,000 is near “the middle” of the numbers in Group A. Alternatively, they may round the numbers in Group A to the nearest thousand and add: 11 + 1 + 6 + 3 = 21 (thousand). Dividing by five (21 ÷ 5) is just a little more than 4 (thousand). Again for Group C, $4,000 is near the middle of the five numbers. The sum of all five numbers in Group D is easily found. Divide $20,000 by 5 and you get $4,000.

• What do you think the real data for college seniors looks like? Do most college seniors have around $4,100 in debt? Are there some that have much higher debt and others with none at all?

Answer: Answers will vary.

• The information in the introduction said that one-fifth of college seniors owe more than $7,000 in credit card debt. In which group(s) was this the case?

Answer: Groups A and D




[Student Handout]

(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

Answers may vary. Inquire as to how students arrived at the other five numbers. One student may have just put in four numbers and then “figured out” what the fifth number had to be in order for the average to be $4,100. Ask the student to explain what he or she means by “figured out.” The sum needed is $20,500.

Another strategy is to slightly modify the other data sets by lowering one number and raising another by the same amount.

Another strategy is to start with a number representing the debt of one student, then subtract that from the sum required, divide by 4, and use those four numbers. (20,500 – 7,500 = 13,000; 13,000 ÷ 4 = 3,250; 7,500; 3,250; 3,250; 3,250; 3,250)

[Student Handout]

(c) Find the median of each set of data including the one you created.

Group A Group B Group C Group D Your Data

$4,200 $4,100 $4,000 $0 Answers vary

(Answers do not appear in student handout.)




[Student Handout]

(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.

Answer: A data set of six elements that have a median of $939 may be created by making the middle two elements $939 or by making the average of the middle two elements $939 (e.g., $978 and $900). The value of the other elements do not matter, as long as two of them are greater than the middle two and two of them or less than the middle two.

Class Discussion

• What does knowing both the mean and the median tell you about the data that you did not know from knowing only one of those values (either the median or the mean?)

Answer: The closer the mean is to the median, the more evenly spread the data are. When the mean is larger (or smaller) than the median, there are “outliers” that causes the mean to misrepresent the rest of the data. What the mean and median do not tell you is how spread out the data are. As you compare Group A with Group C, you see that the mean and median are close together. Group A’s mean is $4,100, and the median is $4,200. Group C’s mean is $4,100, and the median is $4,000. However, the numbers in Group A have a greater spread than the numbers in Group C.




[Student Handout]

Making Connections



Measures of central tendency are tools—that have advantages and limitations—for summarizing data.

Graphs are another type of summary previously used. All summaries can help you understand data, but they can also obscure meaning. As with graphs, you have to understand the type of summary (such as what a mean is) to really make sense of the data.


• Measures of central tendency are tools for summarizing data. What other tools have you used to summarize data?

Answer: Graphs, tables

o What things are important to keep in mind with graphs?

Answer: Reading carefully; thinking about the type of graph; considering what information you can and cannot get from the graph.

o How do these ideas relate to measures of central tendency?

Answer: Understand the meaning of the statistic; consider what information you can and cannot get from the statistic.

It is important to ask questions about data.


• You have worked with data in similar situations in previous lessons. Cite examples that you think students will respond to: Lesson 1.1.1 (statistic on shooting deaths), Lesson 1.2.1 (taxes), Lesson 1.2.3 (reducing medical risk), Lesson 1.2.4 (medical tests). What are some common themes among different ways of thinking about quantitative information?

You are likely to get a wide range of answers. Highlight any that are important quantitative skills. Make sure you point out that, in all these examples, they made sense of the data by asking questions that prompted them to think about the meaning of the data.




[Student Handout]


(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?


Lesson 2.2.2: Making Good Decisions with Good Statistics Theme: Personal Finance, Citizenship





Measures of central tendency

Varied Not applicable Level 3


• 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.


• 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.

Explicit Connections • It is important to ask questions about quantitative information. • Additive change maintains absolute differences. Multiplicative change does not.







Suggested Timeline


8 minutes Work on Questions 1 and 2 Groups or partners

10 minutes Discuss Questions 1 and 2 Class discussion

5 minutes Work on Question 3 Groups or partners


5 minutes Work on Question 4 Groups or partners



[Student Handout]

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

Available!

We have immediate need for five

enthusiastic self-starters who love the outdoors and who love people. Our salespeople make an average of $1,000 per week. Come join

the winning team. Call 555-0100 now!

Are you above average?

Our company is hiring one person this month—will you be that person? We pay the top percentage commission and supply

you leads. Half of our sales force makes over $3,000

per month. Join the Above Average Team! Call 555-0127 now!

We are!

NEED A NEW CHALLENGE?

Join a super sales force and make as much as you want. Five of our

nine salespeople closed FOUR homes

last month. Their average commission was $1,500 on each

sale. Do the math—this is the job for you.

Making dreams real— 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) Consider the set of monthly salaries below. Which company could these salaries have come from?

$1500, $2000, $2000, $2500, $2500, $2500, $6000, $8000, $9000

(c) For the other two advertisements, 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.

Have groups of students work on Questions 1–7. Ask students to make sure each person in the group understands the logic behind their conclusions and calculations.

Make sure students are clear on the instructions for Part (c). If they seem to be struggling with the meaning of the question (versus the mathematics), ask them to start with a number of employees and then prompt them to make a list of salaries that fit the description. You can also remind them of the similar work they did in Lesson 2.2.1.

Answers:

(a) The average is used in the first, the median in the second, and the mode and average in the third.

(b) This data comes from the first job. The average of the salaries is $4,000 ($1000/wk*4).

(c) Are you above average?—Scenarios may vary. One scenario might be the following: • There are eight salespeople. They make, respectively, $2,000, $2,200, $2,400, $2,900,

$3,100, $3,300, $3,500, and $3,800 per month. Four of the eight make more than $3,000 per month. (The way the advertisement is stated does not require the median to be $3,000, as it is here.)

Need a new challenge?—Scenarios may vary. One scenario might be the following: • The nine salespeople sold 0, 1, 2, 4, 4, 4, 4, 4, and 6 homes. If the commission on each sale is

$1,500, then the salespeople made $0, $1,500, $3,000, $6,000, $6,000, $6,000, $6,000, $6,000, and $9,000 last month. However, even keeping the average commission $1,500, the commissions could be more complicated. Another scenario is shown below:




[Student Handout]

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

Answer: Based on the three scenarios, it is most likely that a person would make the most money in the job described in the third advertisement.

[Student Handout]

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.

This question serves several purposes: • Practice reading graphs. • Reinforce writing about quantitative information. • Notice the trend about the mean and median to build to the next question.

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




Start by focusing on the first two goals in the previous list. Accept any statement that is accurate, but prompt students if the statement does not contain complete information. Statements should be specific: “The average sales price of a new home in the United States in 1997 was approximately $180,000.” Depending on your facilities and resources, you might think about ways to have students share statements in a time-efficient manner. For example:

• Each group writes and displays statements on chart paper. • Each group writes at least one statement on the board or on a transparency.

Prompt for statements about the relationship of the mean and median as needed to highlight that the mean and median are close together in the 1960s and farther apart in the 2000s, and that the values seem to go up and down together.

Background information: In the early 1960s, it appears there were not as many very expensive homes to make the average price much different from the median price. This conjecture is proven true in an article (footnoted here).2 The range between the 10% and 90% level of home prices in 1960 was (roughly) $27,000 to $98,000. By 1981, the range had risen to $56,000 to $207,000. By 1989, the range was $34,000 to $203,000. The values move together because the economic factors that cause the median price of homes to increase will also cause the average price of homes to increase.

[Student Handout] Table 1 gives a sample data set of home prices that corresponds with the graph for the year 1977, since the mean (average) and median of the data set corresponds to the points on the graph for 1977.

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

$40,000 $45,000 $50,000 $56,000 $67,000 $75,000 $112,000

2Retrieved from www.knowledgeplex.org/kp/text_document_summary/scholarly_article/relfiles/jhr_0401_gyourko.pdf




(4) 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 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 have the same mean and median shown in the graph for

2005?

(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? Answers:

(a) B, E

(b) D

(c) A

The mean and median of the data sets are given below:

1977 Data Set A Set B Set C Set D Set E

Mean $63,571 $263,571 $296,857 $190,714 $179,571 $296,857

Median $56,000 $256,000 $240,000 $168,000 $240,000 $240,000

The goal of these questions is to get students to think about the relationship of the mean and median to the data. Make sure they do not just calculate the mean. You can give them the results as a part of the discussion. Some groups might identify Set C as a possible representation of the graph. This is a good conjecture because this set does have a mean that is higher than the median. It takes some judgment to decide if the data have enough extremes to separate the median and mean so much. You should not expect students to have that level of judgment. The set is included for the purposes of discussion. Points to make in the discussion:




• The median and mean moved apart in the 2005 data because a few houses rose in value by much more than other houses. These very expensive homes pull the mean up. You see the opposite effect in Set D in which the top three values are very close to the median, but the mean is pulled down by a few numbers on the low end.

• Sets B and E have the same mean and median, but have some different characteristics. Set E has more extreme values on both ends. This demonstrates that summaries of data, such as measures of central tendency, are valuable, but only give part of the picture.

• Ask if students see the relationship between the original data and Set A. (Each value increased by $200,000.) Notice that the mean and median also increased by $200,000, so the absolute difference between the mean and median was preserved. This points out that an additive change shifts the data set together; it can be thought of as sliding all the data points up the scale together.

• Ask if students see the relationship between the original data and Set C. (Each value is multiplied by three.) Notice that the mean and median are also multiplied by three, but now the absolute difference has increased. This points out that a multiplicative change does not increase every data set by the same amount. The increase is relative to the size of the original number.

[Student Handout]

Making Connections


Making Connections: Main Ideas to Highlight It is important to ask questions about quantitative information.

This is a repeat of Lesson 2.2.1. Build off the discussion in the previous day, discussing specific questions about measures of central tendency.

Facilitation Prompts • If you see a measure of central tendency in an article, what questions would you ask about it?

o How big is the population? (This might actually be a sample, but that concept is not covered in this course.)

o Would you expect this population to have a large range? o Would you expect a lot of the data to be grouped together? In what way? o Some of these questions are an opportunity to draw rough sample distributions. You do not

need to make a formal presentation about this type of graph, but it can help students visualize data and it previews a type of graphical display that they might see in later courses.

Additive change maintains absolute differences. Multiplicative change does not.

This concept previews work with additive and multiplicative change in Module 4. Discuss it as you have time. It is not an expectation that students will master the concept.




[Student Handout]


(1) This course draws on examples from four themes: Citizenship, Personal Finance, Medical Literacy, and Physical World. 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?


Lesson 3.1.1: The Cost of Driving Theme: Personal Finance





Dimensional analysis Cost of driving Not applicable Level 2


• be able to multiply two fractions. • be able to divide two fractions. • understand that a fraction can be simplified by “canceling” or dividing common factors in the

numerator and denominator. • understand that multiplying by 1 does not change a value. • be familiar with basic units of measure of length (feet, miles) and time (seconds, hours,

minutes).


• 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.

Explicit Connections • Canceling units is based on the same mathematical concept as canceling common factors in the

numerator and denominator of a fraction. This is actually a form of division and is based on the fact that anything (except 0) divided by itself is equal to 1 and that multiplying a number by 1 does not change the value of a number. Note: You might consider if you want to use the term canceling as it tends to obscure this understanding—it is important to get students to explain the process mathematically. Students will often use the term canceling, but it is more correct to refer to the operation as division.

• All ratios have a common meaning but can represent different types of relationships. • Units are important in problem solving.







Suggested Timeline


8 minutes Discussion of Questions 1–2 Small groups, class discussion

15 minutes Work on and discuss Question 3 Small groups, class discussion

6 minutes Work on and discuss Question 4 Small groups, class discussion

11 minutes Work on and discuss Question 5 Class discussion


Special Notes In this lesson, rates are purposely written in different forms so that students will become familiar with the different forms. You may need to clarify with them that these forms are equivalent (e.g., miles per gallon, miles/gallon, mi/gal).

[Student Handout]

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. In this lesson 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 amount of information students will be able to contribute depends on the student population and how familiar they are with the elements of car ownership. Ideally, students should generate the list without comment from the instructor. Students will add or delete items as they work through the problems. However, if students have little or no experience with driving, the instructor may have to give them enough information to get started. Students may also suggest items that are not necessary in the problem. Allow these to be included.

Cost of Jenna’s car: • gas mileage • cost of maintenance of the car • car registration and fees • insurance • price of gas • miles driven

There may be some disagreement about whether to include fixed costs such as registration and insurance. In one sense, this does not affect the comparison of costs because Jenna has to pay those costs even if she drives a rental. These have been included in this lesson because it gives a more comprehensive view of the true cost of owning a car.

Students may not be sure if the price of gas and the miles driven will matter. The miles driven do not affect the per-mile cost of Jenna’s car, but they do for the rental.

Parking is a major expense in densely populated areas such as New York City and San Francisco, where people often pay to park even when at home. In other areas, parking is free at one’s home. Since Jenna is only traveling for work and it is assumed that she will be reimbursed for parking while she is traveling, you are not including those costs. You may want to add parking costs if appropriate for your location.

Cost of rental: • price of rental—a fixed price plus the cost of gas • gas mileage • price of gas • total miles driven




[Student Handout]

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 22 miles/gallon. If Jenna rents, she can request a small, fuel-efficient car such as the Hyundai Elantra, which gets 40 miles/gallon. What is the cost of the gas for each vehicle?

Answers: 4Runner: $25.02; Elantra: $13.76

Many students will be able to answer these two questions without using the units; however, you should suggest they write the numbers with units (it is likely that you will need to encourage them to think about units in the discussion). If they are stuck, prompt them to think about the discussion of units in the previous OCE. Demonstrate setting up the problem using dimensional analysis as:

$3.67 1 gallon 150 miles $25.02gallon 22 miles 1

• • =

Important Points • 150 miles is a measurement of length, not a rate or ratio, so it does not have units in the

denominator. • The units of gallons and miles cancel out. • If you only look at units, you can see that the problem is set up correctly. • Knowing that you want to end with dollars helps set up the problem. The unit of dollars must be in

the numerator, meaning that gallons will be in the denominator in the first ratio. That, in turn, means that gallons must be in the numerator in some other ratio to make the gallons cancel. Continue this line of thinking to set up the problem.

• This answer is given as a cost for a specific trip—or dollars per 150 miles. Ask students if there would be another way to describe the cost. (Some may have already come up with it.) Note that a unit rate could also be used to compare the costs. Jenna’s car costs $0.17 per mile for gas while the rental costs $0.09 per mile.

Students are often resistant to using dimensional analysis because it seems unnecessarily complex—particularly when they can reason through simpler problems such as the ones above without it. Have students read the next section and discuss it with them. As always with this course, it is important to honor different strategies, but it is also important that students be challenged to attempt this method.




[Student Handout]

(4) 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)

Answers will vary due to rounding, approximately $147.90.

(5) 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 gallon2 • 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 miles3

Answers will vary due to rounding. Approximately $115–117.

1Retrieved from www.fueleconomy.gov 2 Retrieved from www.fueleconomy.gov 3 Retrieved from www.edmunds.com/toyota/4runner/2006/tco.html?style=100614746




[Student Handout] (6) Suppose Jenna will be taking a series of trips of varying lengths. What factors would affect

the cost of the driving her own car vs renting?

This question is intended to get them thinking about what is constant and what may vary, to start identifying the variables in the scenario. In this case, the miles driven is the biggest factor, and the number of days of car rental, along with the cost of gas, which is likely to change from trip to trip.

At this point, walk the students through generalizing their work in questions 4 and 5, introducing variables for the cost of gas and total miles driven. For simplicity, assume that the number of days the car is rented for (and consequently the price of the rental car) doesn't change. There are a couple approaches that could be taken:

• You could ask the students to describe in words how they would determine the cost of using Jenna's car and the cost of using a rental car. Show them how to translate the verbal description of the process into an equation.

• Write out the details of the calculation as one expression, possibly writing out several specific cases, for different distance trips. Show how we can replace the values with a variable. For example:

100 miles: Rental: 100miles$114.12 $3.67/gal

40miles/gal+ ⋅

J = Cost of using Jenna’s car R = Cost of using a rental car g = Cost of gas (dollars/gallon) m = total miles

$114.1240mR g= + ⋅ 0.146

22mJ m g= + ⋅

Point out we could also compare the two options by looking at cost per mile, rather than total cost. These could be obtained by dividing the equations above by m, or by building them from scratch.

J = Cost of Jenna’s car in dollars per mile R = Cost of rental car in dollars per mile

= + 0.14622g

J

and = +114.12

40g

Rm

These two equations, particularly the first one, is referenced in several later lessons, so try to make sure it’s mentioned.




[Student Handout]

Making Connections


Making Connections: Main Ideas to Highlight Canceling units is based on the same mathematical concept as canceling common factors in the numerator and denominator of a fraction. This is actually a form of division and is based on the fact that anything (except 0) divided by itself is equal to 1 and that multiplying a number by 1 does not change the value of a number. Note: You might consider if you want to use the term canceling as it tends to obscure this understanding—it is important to get students to explain the process mathematically. Students will often use the term canceling, but it is more correct to refer to the operation as division.

Facilitation Prompts • What does it mean to “cancel” units? Why is it allowed? • How does this relate to simplifying or multiplying fractions?

All ratios have a common meaning but can represent different types of relationships.

Facilitation Prompts • Review some of the types of ratios used in this course. For example, in Lesson 1.2.1, the water

footprint of the United States was given as 2,482,709 cubic meters per year, per person. Lesson 2.1.2 stated that New Jersey has a population density of 1,185.3 people per square mile. Today’s lesson used ratios such as 60 minutes per hour. How are these ratios alike and different? o The water footprint and the population density are based on a measurement that could

change. The conversion factor of minutes to hour is based on the definition of minutes and hour. It is not going to change.

o Both types of ratios express a relationship between two values. Another way to think of the relationship is to say, “For every ________, there are ___________.” This also indicates a relationship in which the two values change together in a predictable way.

o Dimensional analysis uses both types of ratios.

Units are important in problem solving.

Facilitation Prompts • Why was it important to keep track of units in this lesson? • What other examples of using units have you seen in this class?

o Lesson 1.2.1: Water usage was measured in cubic meters per year and in cubic meters per person per year.

o Lesson 2.1.2: Population density could be measured in people per square mile or people per square kilometer.




[Student Handout]


(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.


Lesson 3.2.1: The Fixer Upper Theme: Personal Finance





Geometric reasoning Home improvements Not applicable Level 2


• know basic formulas for area and volume (area of a rectangle and circle; volume of a rectangular box; finding the volume of certain solids by taking the product of the base and the height as a right circular cylinder).

• use units in calculations that contain rates and measures.


• 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.

Explicit Connections • Generalizing relationships and rules is one of the powerful tools of mathematics. Variables,

expressions, equations, and formulas are all tools you can use to generalize.







Suggested Timeline


5 minutes Discuss units of length, area, and volume Class discussion

5 minutes Question 1 Discuss estimation skills—exact may not be possible

Class discussion guided by teacher questions

10 minutes Question 2 Group work




Special Notes Some students do not have a firm grasp of the units used to measure area or volume. The previous OCE helps address this, but you will also want to make sure students have a good understanding of the concepts. Use ceiling or floor tiles to point out how a one-dimensional measure cannot be used to measure a two-dimensional area. The language of algebra should help students understand how to use units of area and volume. They know that (x)(x) = x2. Point out that (inches)(inches) = inches2 because of the same rules of algebra. Also point out that (x)(y) ≠ (xy)2. When multiplying two quantities, the units matter. Two quantities can only be added when they have the same units, that is, when they are like terms. You can add (2x) + (5x) and get 7x but you cannot add (2x) + (7y) and get 9xy. You can add (3 inches) + (5 inches) and get 8 inches, but cannot add (2 feet) + (7 inches) and get 9 feet-inches. Many students lack a deep understanding of area and volume. One common misconception is that since there are 3 feet in a yard, then 3 square feet must equal 1 square yard.

Students may have a wide range of working knowledge with geometry. In addition to the problems in the previous OCE that help prepare students for this lesson, you may be able to identify students who need additional preparation. Assigning them videos from YouTube may help. Some URLs to videos are listed below. No doubt you, or they, can find more. One way to accomplish this task is to assign each student the task of identifying one video on geometric figures (areas or volumes) that they like. Bring the URL in or post it where others can view the video.

• Area: Square Units and Rectangles www.Youtube.com/watch?v=epeFZ6v7u_U

• Area: Parallelograms and Triangles www.Youtube.com/watch?v=vQC10PPmuoA&feature=related

• Area of Composite Shapes www.Youtube.com/watch?v=by9-_95Zn08&feature=related

• Geometric Solids: Lesson Hook www.Youtube.com/watch?v=RGlU_SqgjEg

• GMAT Prep: Math, Geometry, Rectangular Solids by Knewton www.Youtube.com/watch?v=SJycga8V02M&feature=related




[Student Handout]

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?

The purpose of this question is to have students think about and make sense of the scale drawings before delving into the lesson. You should also use the figures to discuss the idea of estimation. Some of the drawings are not used in the lesson, but are used in the OCE.

One approach is to divide the class into groups, assigning each group a different question (see sample questions below), then have the groups switch questions to critique each other’s results. You may need to limit the number of questions based on time. Another possible scenario is to give each group the same question and have them check each other’s answers.

Sample Questions • Estimate the dimensions of the lot. (Answer: 235 feet x 115 feet) • Estimate the number of square feet in the house. (Answer: 2,200 square feet) • Estimate the length of the driveway. (Answer: 95 feet) • What percentage of the lot is covered by the house? (Answer: 8.14%) • What is the area of the lot? (Answer: 27,025 square feet) • Which do you believe is a closer to the actual measure, the area of the lot or the area of the house?

(Answer: Since the borders of the house seem to align with the grid, the area of the house may be closer—however, all of these dimensions are approximate. Discuss how to approximate the dimensions of the lot and how the product of the length and width can compound error.)




[Student Handout]

(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

advertisement

Answers will vary due to rounding: The area of the region to be reseeded is about 15,500 square feet (130 x 115 + 55 x 10). The job should require 64 pounds of grass seed (16,000 square feet x 4 pounds/1,000 square feet = 64 pounds) at a cost of $80. The fertilizer costs $32.50 (15,500 square feet x 50 pounds/12,000 square feet = 65 pounds). The labor costs $180, making the total cost $292.50.

Students will approximate the dimensions of the backyard from the scale drawing (exact dimensions are not provided). It is important only that the dimensions they use are in the ballpark and that their calculations support their conclusions.

Note: It is assumed that Gerry will sell the grass seed and fertilizer in 1-pound increments.

With this assumption, Gerry’s estimate is more than twice what it should cost. Ask students what they would do in this situation. Ask, “How many people do you believe would check behind the contractor’s estimate?”




[Student Handout]

(3) The Mazurskys want to build a 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. The cost for the materials is shown in the advertisement below. The total cost will include 9.4% sales tax. Calculate the cost of the materials required to fence in the backyard.

DO IT YOURSELF SPECIAL — Chain-Link Fence chain-link fence—$21 per linear yard 48-inch wide gate—$75 each Inline posts—$12.50 each Corner/gate posts—$20 each

HIGH Home Improvement—Your Fencing Specialist

Students need to make decisions here regarding the inline posts.

Consider the two horizontal sections first: • The 140-foot section requires 19 posts, 17 of which are inline. (140/8 = 17.5) • The 130-foot section requires 18 posts, 16 of which are inline. (130/8 = 16.25)

The back section of the yard is 115 feet wide and requires no corner posts since they are included in the other sections; only 14 inline posts. (115/8 = 14.375)

The two sections that join the house both include 4-foot gates; one is a total of 25 feet wide and the other is 40 feet wide. The 25-foot section (25 feet - 4 feet for the gate = 21 ft) requires two corner/gate posts, and two inline posts. The 40-foot section (40 feet - 4 feet for the gate = 36 ft)requires four inline posts, along with the two corner posts.

Answers

All together, the project requires 53 inline posts and 8 corner posts, totaling $822.50.

The two gates cost $150.

The amount of fencing required is 432 feet (140 + 130 + 115 + 21 + 36 = 442) at a cost of $3,094.

The total cost of materials is $4,126.50.

The tax is $387.89, making the total cost $4,514.39.




[Student Handout]

Making Connections


Making Connections: Main Ideas to Highlight Generalizing relationships and rules is one of the powerful tools of mathematics. Variables, expressions, equations, and formulas are all tools you can use to generalize.

Lessons 3.2.1 and 3.2.2 have this same connection—a focus on how algebraic symbolism is used to generalize. In this lesson, you will probably need to direct instruction more than usual to make this point. In Lesson 3.2.2, students will be better prepared to discuss on their own. Important points to bring out:

• Call attention to the use of variables in the geometric formulas. Because students are familiar with many of these formulas, they may not have really thought about the fact that they are essentially evaluating expressions. Connect this to more abstract expressions such as x – 9.

• Note the difference between a variable and a constant such as π. • Note the importance of knowing what each variable represents. Since most of the algebraic

work in this course is done in context, this is very important. • Discuss the role of order of operations in something like the formula for the area of a circle.

Note that even formulas with many operations and variables are based on the order of operations.

[Student Handout]


(1) 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?

As students tackle this problem, they may feel overwhelmed with all the information. Your guidance is critical—do not tell them what to do first—rather, ask them what they need to know in order to proceed.

Facilitation Prompts • What is a semicircle? • What is a slab of concrete? • Can you sketch the design? • Are you talking about length, area, or volume? • If the top and bottom of a three-dimensional figure are the same, is there an easy way to find its

volume?




The shape of the patio is a semicircle with a radius of 3 feet. The volume of concrete required is V = (1/2)(πr2h) = (1/2)π (3 feet)2(2/12 feet) = 2.36 cubic feet. However, students do not necessarily need to use the formula for volume. You can encourage them to think about the area of the top surface and reason out that a depth of 1 foot would mean multiplying by one; a depth of 2 feet would mean multiplying by two, etc.

Answers: Eight 40-pound bags of mix are required. (2.36 cubic feet) x (1 bag/0.30 cubic feet) = 7.8 bags. The cost of the eight bags is $52, making the total cost, with tax, $55.90.

[Student Handout]

(2) 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.

(3) Dimensional 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.

Answers will vary. For example: The U.S. Census Bureau states that the area of Ohio is about 41,000 square miles.

22

2 2

27,878,400 feet 1 acre41,000 miles = 26,240,000 acres

1 mile 43,560 feet

The Wallow Fire was about 2% of the area of Ohio.

1Retrieved from http://en.wikipedia.org/wiki/List_of_wildfires#North_America




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 and reseeded. 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




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


Lesson 3.2.2: Breaking Down Variables Theme: Citizenship





Using formulas Effects of speed on braking distance



• be able to evaluate expressions containing parentheses and exponents of two. • understand dimensional analysis and how to use units—including squared units—in calculations. • understand subscripts.


• 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.

Explicit Connections • Generalizing relationships and rules is one of the powerful tools of mathematics. Variables,

expressions, equations, and formulas are tools you can use to generalize. • Order of operations defines how to perform calculations in formulas.







Suggested Timeline

The timing of this lesson depends a lot on the amount of discussion about the context and the formula. You may have to de-emphasize some problems if your discussions run long. Question 4 could be handled more quickly or skipped. There are notes in the lesson about options for Question 5.



10 minutes Question 2 and discussion of formula Small group and/or class discussion

10 minutes Work on and discuss Question 3 Small group, class discussion




[Student Handout]

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?

This lesson is about variables. Students are accustomed to using x and y as variables in mathematics classes. This is an opportunity to broaden their understanding of the variables to include not just the actual letters or symbols used to represent numbers, but the factors or elements of a situation that change.




As a general rule in this course, you use symbols for variables that have meaning in the context. Letters such as x and y do not mean much to students. These abstractions are often a barrier to using variables to represent physical relationships and to understanding formulas that contain many letters (none of which are x or y). In this course, you should attempt to use meaningful variables that arise naturally from the problem situation.

Variables that might affect the stopping distance of a vehicle include the following: • The type of road surface and the condition of the tires. These conditions determine the

coefficient of friction between the tires and the roadway. Promote more discussion about friction by asking questions such as o “Do you stop more quickly when the tires lock and skid or when they do not?” (Answer: The

skid is a loss of friction and the deceleration is slower. The skid also causes a loss of control.) o “What conditions cause better or worse friction?” (Answer: Conditions change from one

situation to another. Within this experiment, you will consider these variables to remain fixed. Otherwise, the problem becomes too complex. Ask, “What does the statement ‘variable to remain fixed’ mean?”)

• The grade of the road. A car traveling on a positive grade (uphill) stops more quickly than a car traveling on a level road, or a car traveling downhill (a negative grade). o “What is meant by the grade of a road?” [Answer: The grade is a percent derived from

dividing the change in vertical position by the change in the horizontal distance. A stretch of road that is 1,000 feet long in horizontal distance and is 20 feet higher on one end has a grade of (20 feet)/(1,000 feet) = 0.02 = 2% (if the travel is uphill), or –2% (if the travel is downhill). See the problem in the previous OCE. Again, grade is a variable from one situation to another. Within this experiment, the grade remains fixed.]

• The speed of the vehicle. Speed is an obvious variable; the faster you are going, the more time it takes to stop. The speed at which the vehicle was traveling when the brakes were applied does not change during the period of deceleration. Again, initial speed is a variable that will be fixed for any one experiment.

• Driver reaction time. Reaction time is definitely a variable, but it is difficult to assess and will not be taken into consideration. Instead, consider the time it takes to stop once the brakes have been engaged (braking distance). Students should understand that during the time it takes to see the need to decelerate and engage the brakes, the car will travel (speed)(time) feet.

Some students might suggest that the weight of the car has an effect on the stopping distance. The weights of most cars are close enough to one another that the weight does not matter as long as the coefficient of friction is relatively high. Remember the inclined plane problems you did in physics? The weight of the block did not matter. The more the block weighed, the more frictional force it had to overcome. At the same time, the more the block weighed, the greater the component of the weight down the plane. When the suggestion is made that the weight of the car matters, use this opportunity to talk about the acceleration due to gravity (32.2 feet per second per second). Acceleration due to gravity enters into the calculation when considering the grade of the road as well as the frictional force used to stop the vehicle. It should be obvious that a tractor-trailer filled with logs takes longer to safely




stop than a car. Limit the discussion here to cars that are about the same size. If more questions arise about weight, consider referencing the article (link below) from Motor Trend magazine:1

www.motortrend.com/roadtests/suvs/112_0101_chevrolet_tahoe/torque_test.html

This article compares three brands of sport utility vehicle: a Chevrolet Tahoe, a Ford Expedition, and a Toyota Sequoia. The tests conducted for the article found that it took 142 feet for the Tahoe to stop with only a driver on board. With three full seats of passengers (around one-half ton of extra weight), only four additional feet were required to stop the vehicle from an initial speed of 60 miles per hour.

[Student Handout]

(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. For example, what do you think will happen to the braking distance if you double the speed? Would the answer be different for very low speeds or very high speeds?

Present the question and allow time for discussion.

Discuss the formula for braking distance. Talk to students about the importance of knowing the meaning of each variable in a formula. Students have seen subscripts in the previous OCE (Lesson 3.2.1), but you should make sure they understand the use of the symbol in the formula. Also note that it is important to know what units are being used in the context. In this context, you are measuring distance in feet so the velocity must be in feet per second. Another context might use meters.

You will want to have a discussion about g, acceleration due to gravity, which is 32.2 feet per second per second. These units refer to a change in speed (fps) over time (fps/sec). Decide how detailed an explanation to give based on your students. In this context, students should understand that g accounts for the force of gravity acting on the car to slow it down. Students may wonder why you use a “variable” for the acceleration due to gravity if it is really a “constant.” This goes back to units. The formula is written so it can be used for all units: g = 32.2 ft/sec2. If you were using different units, g would have a different value.

Note the change from speed to velocity. Velocity measures direction in addition to a rate of speed. Speed is always positive, but a negative velocity means moving in an opposite direction. Because all the velocities in this context are positive, the two terms are interchangeable for this situation. However, since the formula uses velocity, you will switch to that term.

1SUV Showdown—Chevrolet Tahoe vs. Ford Expedition vs. Toyota Sequoia. Retrieved September 22, 2011, from www.motortrend.com. Source Interlink Media.




[Student Handout]

The formula for the braking distance of a car is 2

0 = 2 ( + )

Vd

g f G 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 provide 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.

Note: This coefficient of friction is what could be expected with good tires on a good road. Ask students what a grade of 0.05 means (see previous OCE).

( ) ( )( )( )= = = =

+ +

2 2 22 20 0 0

022

sec 0.018268 ftft2 ft 54.74 2 32.2 0.8 0.05 secsec

V V Vd V

g f G

Either expression, the last or next to last, may be considered a simplified form of the formula.

Assuming that your students predicted that the braking distance would increase as speed increases, this is an opportunity to link that intuitive, contextual reasoning to mathematical reasoning. Ask students how they could tell that d increases as V0 increases based on the formula. As V0 gets larger, it is divided or multiplied by a constant (depending on the form used) so the result gets larger.

Depending on how well students have done on previous work with order of operations, you may want to use this opportunity to talk about order of operations in the formula.




[Student Handout] (4) Calculate the absolute and relative change in braking distance if: (a) the speed doubles from 10 mph to 20 mph (b) the speed doubles from 40 mph to 80 mph

× × =10 miles 1 hour 5,280 feet

14.67 feet / secondhour 3,600 sec 1 mile

= =

22sec ft 0.018268 14.67 3.93 feet

ft secd

Velocity (mph) Velocity (fps) Braking distance (ft)

10 14.67 3.93

20 29.33 15.72

40 58.67 62.88

80 117.33 251.50

(a) From 10mph to 20mph:

Absolute change: 11.8 ft. Relative change: 400%

(b) From 40mph to 90mph:

Absolute change: 188.6 ft. Relative change: 400%

Note that students need to convert miles per hour into feet per second in order to use the formula.

[Student Handout] (5) What will be the relative change in braking distance if the speed were to triple?

Some students might recognize the effect of the squaring operation in the formula and be able to deduce that the braking distance will increase by a factor of 9. For other students, it may help to suggest they pick specific values and perform the calculations.

If it hasn't already come up in the discussion, help students see how the results of problems 4 and 5 follow from the squaring operation in the formula.

Emphasize the use of units throughout the problem and note the connection to dimensional analysis. Mention the number of decimal places required in the simplified formula. What if the formula had been simplified rounding to three decimal places? The result would have been 217.8 feet. The reason you need more decimal places is because you are likely to be using rather large numbers in the product.




[Student Handout]

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.

Making Connections


Making Connections: Main Ideas to Highlight Generalizing relationships and rules is one of the powerful tools of mathematics. Variables, expressions, equations, and formulas are all tools you can use to generalize.

Facilitation Prompts • Compare the formula for the area of a rectangle to the formula for braking distance. (Put both

on the board.) How could you express these relationships without variables and mathematical symbols? o Students will probably note that it is easy to say, “area equals length times width,” but

braking distance is quite complex to express verbally. Point out how valuable the use of symbols is in expressing this relationship.

• What other uses have you seen for using variables and symbols? o In Module 1 (1.1.4 OCE), you used symbols to show that rules such as the distributive

property were true for all numbers. o In Module 1 (Lesson 1.2.3), you used Q1 and Q2 as shorthand to distinguish between two

different quantities. o You have used variables and symbols to write formulas for spreadsheets. o Why is the use of variables sometimes challenging? When does using variables make things

easier?




Order of operations defines how to perform calculations in formulas.

Facilitation Prompts • How did you know which operation to do first in using this formula? • Can order of operations be used with any formula?

The OCE contains information for students about the terminology of equations, expressions, formulas and models. If you have time, you may want to discuss some of this in class.

[Student Handout]


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.

=2

ft16

sec

dt

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.

Trial Distance R (inches)

Distance L (inches)

1

2

3

Average

Average of both hands


Lesson 3.2.3: Comparing Apples to Apples Theme: Citizenship





Dimensional analysis, geometric formulas, misleading graphs

Apple import Not applicable Level 3


• be able to use dimensional analysis. • understand and use variables. • be able to evaluate expressions and formulas.


• 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.

Explicit Connections • Many quantitative skills are used to analyze a complex situation.




NOTE: This lesson will require students to have access to rulers with millimeter markings




Suggested Timeline


5 minutes Introduce the scenario Class discussion

15 minutes Work on and discuss Question 1 Small groups followed by class discussion

10 minutes Work on and discuss Question 2 Small groups followed by class discussion

15 minutes Work on and present Question 3 Ask groups to pair up and share their blog entry with another group


Special Notes This is a formative assessment lesson that reviews skills with dimensional analysis, geometric concepts, and use of formulas and variables in Lessons 3.1.1, 3.1.2, 3.2.1, and 3.2.2. It also returns to the topic of misleading graphs, first presented in Lesson 2.1.4. In this lesson, students analyze a misleading use of pictographs.

If you need time in this lesson for extra review of previous material or discussing OCE items, you can consider modifying Question 3. This question provides an excellent opportunity for students to use quantitative information to present an argument, but if you cannot devote the full time needed for the question, you could have students discuss briefly in class and then write their responses as a part of the OCE for this lesson.

You will also need to have rulers for this lesson. If you cannot provide students with a way to make the measurements required in Question 1, the authors suggest you give the height of the objects and allow them to figure out the rest of the information.

[Student Handout]

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.

Note: Recent reports have cited the presence of arsenic, E. coli, and other contaminants in imported apple juice. You may want to mention a news article to emphasize the importance of this issue, but be careful not to take sides on whether imported juice is good or bad. It should also be noted that while the percentage of consumed apple juice from other countries is quite high, most fresh apples still come from U.S. farms.

This exercise asks students to apply geometric formulas and to simplify equations.

[Student Handout]

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) 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?

Answer: Students will probably say “a lot.” Ask them to quantify this hunch by asking, “How many times more?” They will probably cite the size of the apple graphic as their comparison factor. Although this is the intended comparison, other estimation and good graph-reading skills should be appreciated and recognized if students use them. Point out that many media consumers often do not have such skills.

[Student Handout]

(2) 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. Use a ruler to measure the height of the apples in the pictograph, in millimeters. 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 sphere is given by V =

43πr3 where r is the radius).

Height of

Apple Approximate

Radius Approximate Area of Apple Graphic

Approximate Volume of Apple Graphic

1998 little apple

2008 big apple

Ratio: (2nd value/1st value)

Answers

height (mm) radius (mm) area (mm2) volume (mm3)

1998 little apple 18 9 254 3,052

2008 big apple 40 20 1,256 33,493

Ratios 2.22 2.22 4.94 10.97




[Student Handout]

(3) Which of these ratios accurately represents the actual change in apple juice imports over this 10-year period?

Answer: The ratio of the actual imports is 2.33. The first two ratios in the table (2.22) accurately represent the data and imply that the imports more than doubled in those 10 years. However, the area gives the impression that imports are almost five times larger in 2008 as 1998 (which is a huge distortion in the data). Using the volume might lead you to conclude that imports in 2008 are nearly 11 times the amount in 2008.

[Student Handout]

(4) Make a graph that would portray the data more accurately.

Answer: Students might only think to make a bar graph, but you can point out that pictographs can be used without being misleading. In this case, you could use stacked apples of the same size to represent a larger quantity (see the answer in the Further Application section).

The three-dimensional effect of the apple is more obvious in color, so you may want to project the graph for students to see. Note that the measurements given in the solutions are based on the size of the graph in the original document. If you make any changes, you should check to make sure the measurements are still correct.

Students may ask how to measure the apples. The important thing is that they are consistent in how they measure the two figures. It could be interesting to compare how close the ratios are with different measurement methods. Students may also measure both the height and the radius. This is fine for their work, but you should discuss the fact that if the apple is really a circle, the radius is half the height. This is probably more accurate since it would be difficult to pinpoint the center of the apple.

Make sure students use the correct units for the different measures.




[Student Handout]

Making Connections


Making Connections: Main Ideas to Highlight Many quantitative skills are used to analyze a complex situation.

Facilitation Prompts • Discuss the many quantitative skills from this course that students used in these two situations:

o reading graphs o using units o using formulas and variables o dimensional analysis o ratios o organizing information o communicating with quantitative information

[Student Handout]


(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.

One Possible Solution: This graph was made using a marketing year that spanned the calendar years: 1997/98 instead of 1997. The authors decided to simplify the problem by changing the marketing years to a single year.


Lesson 3.2.4: Balancing Blood Alcohol Theme: Medical Literacy





Solving linear equations Blood alcohol content Not applicable Level 3


• understand the use of variables in mathematical equations. • be able to substitute a value for a variable in a mathematical equation and simplify the

equation. • understand that an equation is a statement of equality.


• 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 evaluation a given equation.

Explicit Connections • Solving equations is based on the principles of undoing operations (and steps) and on balancing

each operation.







Suggested Timeline


5 minutes Problem Situation and Question 1 Class discussion

20 minutes Work on and discuss Questions 2–4 Small groups, class discussion

10 minutes Work on and discuss Question 5 Small groups, class discussion, demonstration

10 minutes Practice by varying Question 5 and exploring the scenario in Question 6

Small groups


[Student Handout]

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

1Retrieved from http://en.wikipedia.org/wiki/Blood_alcohol_content. 2Retrieved from www.icadts2007.org/print/108widmarksequation.pdf




− +

2.84 = 0.015

N

B tW g

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.

Carefully discuss each variable, determining why it might be important to the BAC calculation. The variables B, N, W, g, and t change depending on the person. Ensure that students understand that the units of B are a percentage. So, 0.08 is not 8%, but eight-hundredths of a percent.

Notes on the Widmark Equation: • This equation has been simplified from the one found in the reference. • The numbers 0.015 and 2.84 are constants based on the average person. The value 0.015 is the

average rate of elimination of alcohol, and the value 2.84 is a conversion factor between weight (in pounds) of an individual, density of alcohol in a standard drink, and density of water in an average person.

• Many high-end beers have twice as much alcohol as the “standard beer,” which assumes 4% alcohol.

[Student Handout]

(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?

This problem should be a review of earlier modules. The simplified equation is

= − +0.015 0.104B t

Be sure, though, that everyone has the same simplified equation before moving on. In addition, point out that, in this case, you have held some variables (N, W, and g) fixed as a way of further investigating the model. Now, there are only two unknowns in the equation.




[Student Handout]

(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?

Answers: When t = 1, BAC = 0.089; when t = 3, BAC = 0.059; and when t = 5, BAC = 0.029.

Notice that 0.089 is above the legal limit for driving and 0.059 is still high (0.05 is the legal limit in Canada). Point out that they “solved” for B when they substituted in t. That is, B was the only unknown left in the equation. Students should notice that BAC decreases over time. Students might try substituting in more hours. If they try any time greater than seven hours (t > 7), the BAC is negative. Talk about how to interpret this value. After a certain point, the BAC is 0 because the alcohol has been metabolized by the liver, even if the equation gives a negative number for BAC.

[Student Handout]

(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.

Answer

Students should outline something like this:

• First, replace t with the given hours in the equation (i.e., plug in the value for t). • Second, multiply the hours by −0.015. • Third, add 0.104. • The result is the BAC.

This may seem tedious, but it helps students when they try to figure out how to solve for t when B is known. Ensure that students are being explicit about the order of operations and remind them how to determine this from the equation. It is important that they see the relationship between the equation and the steps. Also, reinforce that the steps are based on the order of operations.

[Student Handout]

(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?

Answer: t = 1.6 hours for BAC = 0.08 and t = 6.93 hours for BAC = 0.0.

Let students struggle with this question a bit. Some will plug in hours until they come close to 0.08 and 0.0, but they will not get those BAC values exactly. Direct them back to the equation, asking them what




is unknown in this case. Ask them to look at their algorithm above and discuss how finding the hours when given the BAC is really working backwards through their algorithm and undoing each step. To get students started, you may need to ask what undoes addition and what undoes multiplication. Introduce the term inverse operation in talking about undoing. You can ask them to write out the undoing steps such as:

• First, start with the known value of BAC for B in equation. • Second, subtract 0.104 (the other way, they had added 0.124). • Third, divide by −0.015 (because division undoes multiplication). • The result is t.

Although some students may have solved similar equations algebraically in the past, you should explicitly and carefully model how this undoing works, emphasizing the importance of keeping both sides of the equation balanced using the B = 0.08 case. Then, give them a chance to work through the case of B = 0 on their own before moving to the next question. A demonstration on the board that links their algorithm with the algebraic operations might look like this:

Undoing Algorithm Looks Like with the Equation Notes

Step 1: Replace given BAC for B

− 0.08 = 0.015 + 0.104 t Substituting B into the equation.

Step 2: Subtract 0.104 − − −0.08 0.104 = 0.015 + 0.104 0.104t and simplified:

− = − 0.024 0.015 t

To maintain equality, any operation you do on one side must also be done to the other side of the equation.

Step 3: Divide by −0.015 − −=

− −0.024 0.0150.015 0.015

t

or 1.6 = t

Again, dividing both sides of the equation by −0.015 to maintain a balanced and equal equation.

The result is t t = 1.6 hours By convention, you rewrite so that the variable is on the left of the equation.

Undoing (inverse operations) and balancing equations are two key ideas that students can use to solve for an unknown variable nested in an equation. It is also important for students to realize that undoing is directly related to the order of operations—in essence, performing the order of operations backwards. One strategy in deciding what to do first in solving an equation is to first put a value in for the variable and simplify as modeled in this lesson. The steps in simplifying (the order of operations) are reversed for solving.

Make sure to remind students that 1.6 hours is not 1 hour and 60 minutes. If there is time, you could ask them to convert 1.6 hours into 1 hour and 36 minutes. Ask students to check the reasonableness of their answers with the numbers they got in the “going forward” model above.

Ample time should be spent on Question 5. For students ready to move on, direct them to Question 6. For students who need more practice, stick with the above model and provide them varying values for B.




[Student Handout]

(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

= − +2.84

0.03 60.5

NB

(a) Compare her BAC for one glass of wine versus three glasses of wine at the time she will leave.

Answer: When N = 1, B = 0.017; when N = 3, B = 0.111.

Point out how, in this case, you have held t fixed and let N be the variable. Students should notice that if they fix N and let time increase, BAC goes down (as seen above). If they fix t and let N increase, BAC goes up (as seen in this case). Talk about why this makes sense. Finally, if needed, remind students that “adding a negative 0.03” is equivalent to “subtracting positive 0.03.” This might help them in the undoing process in Part (b).

[Student Handout]

(b) In this scenario, determine how many drinks she can have so that her BAC remains less than 0.08.

When B = 0.08, N = 2.34 drinks. Ask students to estimate what 2.34 drinks means.

Solving for N in this equation is a bit trickier than before because the unknown is in the second half of the equation and has been multiplied and divided by different numbers. If students get stuck, remind them about order of operations and encourage them to write out the steps going from N to B and then back from B to N. Some students may recognize that they can divide the coefficient of N to get rid of the fraction. This is fine, but it is important that you discuss how to solve the equation by undoing the division. They will use this when solving proportions in Lesson 3.2.5.




[Student Handout]

Making Connections


Making Connections: Main Ideas to Highlight Solving equations is based on the principles of undoing operations (and steps) and balancing each operation.

In Lesson 1.1.4, students were introduced to the idea of creating equivalent expressions based on mathematical rules. This lesson builds on that idea. Each step of solving an equation creates a new equivalent form of the original equation. Students must understand that the rules that govern solving equations apply to all equations. This will be emphasized over the next two lessons.

Facilitation Prompts • In an equation with two operations (use an example from the lesson), how do you decide which

operation to do first? It can be useful to demonstrate that the result is different if the work is done in a different order.

• How do you decide which operation to use? What if negatives are involved? • Why do you have to perform the same operation on both sides of the equation? • How can you check if a solution to an equation is correct?

[Student Handout]


(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.

Answers: (a) Substitute −3 in for x. Multiply −4 times −3. Subtract 2. y = 10

(b) Substitute −3 in for y. Add 2 to both sides of the equation. Divide both sides by −4. x = 1/4


Lesson 3.2.5: A Return to Proportional Reasoning Theme: Citizenship





Proportions Graphic design Not applicable Level 2


• 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.


• 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 with algebraic methods.

Explicit Connections • Proportionality preserves the ratio between values. • Consistency of mathematical rules—foundational rules for solving equations can be used on any

equation.







Suggested Timeline


7 minutes Demo and discussion Class discussion

10 minutes Work on and discuss Question 1 Small group work, class discussion



8 minutes Work on an discuss Question 4 Small group work, class discussion

5 minutes Work on Questions 5–7; discuss Question 7 Small group work, class discussion


Special Notes In this lesson, you will manipulate a graphic to demonstrate the concept of proportionality. The cloud graphic shown below is provided in a separate document, where you will find two copies so that you can make a visual comparison between the original and the changed version. You may wish to find a different picture that will be more interesting for students. If you are not familiar with how to manipulate graphics, see the instructions below (it is assumed that you have a version of Word that supports these manipulations). If you do not have the proper software in the classroom to do the demonstration on the computer, print out different versions of the graphic with the dimensions and make overheads.

1. Select the graphic.

2. A Format tab will appear on the tool bar between Home and Layout. Select Format.

3. The Format tool bar appears as shown below. The dimensions of the graphic appear in the far right corner.




4. To change the size of the graphic, drag the boxes and circles outlining the box. To change only one dimension at a time, drag one of the boxes in the middle of the line (make sure that the small box by the dimensions—the Lock Aspect Ratio—on the toolbar is not checked as shown above). To change both dimensions, drag the circles at the corners.

5. If you have a newer version of Word, you can enlarge or shrink the figure and maintain the

proportions of the dimensions by checking the Lock Aspect Ratio box by the dimensions on the toolbar as shown below. Now when you drag the circles at the corners of graphic, the ratio between the dimensions is fixed.

For older versions of Word, right-click on the graphic and select Format Object. Select Size from the

choices on the left side of the dialog box. Select Lock Aspect Ratio from the selections under Rotate and scale:




[Student Handout]

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.

As you go through the demonstration, ask students to predict what will happen to the picture and to the dimensions. Discuss how the graphic is distorted when only one dimension is changed. Most students will be very familiar with this type of graphic manipulation. The goal is to get them to connect what they know intuitively to the mathematical concept of proportionality.

a. Have three blank data tables on the board, one for each of the following: Changing the width; Changing the length and width without maintaining proportionality; and Changing both dimensions together to maintain proportionality.

b. Record the starting dimensions of the graphic. Start each step with the original dimensions. (You can either change the graphic back to the original dimensions or close and reopen the document without saving.)

c. Change the width of the graphic (make sure the Lock Aspect Ratio box on the tool bar is unchecked as explained in Steps 3 and 4 above). Record at least three dimensions other than the original.

d. Change the length and width of the graphic without maintaining proportionality. Record at least three dimensions other than the original.

e. Change the width and length of the graphic while maintaining proportionality (see Step 5). Record at least three dimensions other than the original. Explain why these versions of the graphic are proportional to each other. You can see this visually because the image is not distorted. A more technical definition of proportionality is given below.




[Student Handout]

(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.

Give students time to work in groups and then bring the class together for a discussion. The goal is to have students recognize that the ratio between length and width is constant when proportionality is maintained. There may be small differences in the ratios of the proportional dimensions due to measurement error. This is a good opportunity to talk about measurement error. Even though there are differences, they will be much smaller than the differences in the other ratios.

In this discussion, it is important that you record examples of findings in the form of ratios. For example, for the table values show below, you can write ratios.

Width Length Width Length

3 6 3 6

4.13 8.26 8 8.5

=

3 4.13

6 8.26 ≠

3 8

6 8.5

Students may not think to check ratios because they may have more intuitive explanations. The following prompts may help guide their thinking.

Facilitation Prompts • Is there a way to measure the relationship between the two numbers? • What if the width is doubled? This can be related to W/L = 2W/2L.

Students may try to find the differences. If they do not suggest this, you should bring it up in the discussion, as it is a common misconception. You can demonstrate that adding or subtracting the same amount to both dimensions creates a distorted image. This is easily done by typing in new values for the height and width. For example, start with the original dimension of 1.82 inches x 3.63 inches. Subtracting 1 from both dimensions for a result of 0.82 inches x 2.63 inches creates an obvious distortion. Adding 1 also results in an obvious distortion.

Guide the discussion to write a definition of proportionality: a relationship based on equal ratios.

For the questions below, you might replace the given dimensions with the dimensions of the graphic you used or something related to your college such as the college logo or mascot.




[Student Handout]

(2) You are a graphic artist hired to make a billboard for a college. The original logo is 12

4 inches

(width) by 33

8 inches (length). You need to enlarge it to a length of 6 feet. How wide will the

enlarged version be?

Answer: 4 feet

Students are likely to use similar strategies to those used in Module 2 to find proportional values. A common mistake here may be to take the ratio of length/width (1.5) and multiply it times the length. Questioning whether the width or length should be longer alerts students to the error.

The goal of this question is to introduce setting up proportions as equal ratios. How you prompt this depends on the methods your students propose. Some students may have already been using proportions, so it may come up naturally. You can also prompt students to find multiple strategies for solving the problem—refer them back to the discussion about equal ratios in Question 1 and to the idea of using a variable to represent an unknown. If the method does not arise in at least one group, you can bring it out in the class discussion to show:

=2.25

3.375 6

x (where x is the width in feet)

Students may write the proportions in different ways. This is explicitly addressed in Question 3.

Note the relationships in the proportion. For example, in the above example: • Numerators are both widths; denominators are both lengths. • The numerator and denominator of each individual fraction represent measurements of the

same figure.

Ask students to solve the equation using what they learned in the previous lesson. It is not recommended that you show them cross-multiplying. Students need to understand that the same rules apply to solving all equations, including those that look unfamiliar. If they later discover the shortcut of cross multiplication, that is fine. If you have students who use cross-multiplication, try to get them to demonstrate other methods first.

Be sure to relate the method of setting up and solving the proportion to any other strategies used by students. For example, the steps of finding the ratio (0.6) and multiplying it times the length are the same as the steps to solving the equation. At some point, this needs to be made very explicit. You may choose to return to this in the Making Connections section.

Your discussion depends somewhat on the different methods that students have used with proportions through the course. It is likely that they have simplified a ratio and multiplied it times a known value to find a new value (Lesson 2.1.2). Explain that the methods they have used previously to solve this type of problem are still good methods. The purpose here is to show them another strategy. Ask them what they do or do not like about the different methods. Note that setting up a proportion can help you determine the relationship between the different values. As noted above, the error of using the ratio of 1.5 is avoided by keeping the relationships correct.




Discuss why different units can be used in the proportion (inches for the small copy and feet for the large). If the numbers are going to be added or subtracted, the units need to be the same because in that case the numbers represent actual measurements. In this case, the ratio of 2.25 to 3.375 represents a relationship between measurements with like units. Therefore, this ratio is constant for all comparisons with like units: 2.25 and 3.375 are both measurements in inches and the ratio is 0.6 inch of width for 1 inch of length. The 6 and the 4 need to have the same units so that the ratio of 0.6 unit to 1 unit is maintained. Question 7 below has a relationship in which the ratio has different units. This is discussed below.

[Student Handout]

(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?

=2.25 4

3.375 6

Answer: The proportion can be written in several ways as long as the relationships between the numbers are maintained:

=3.375 6

2.25 4

=6 4

3.375 2.25

=2.25 3.375

4 6

The following is not correct:

=2.25 6

3.375 4

Discuss that there are both contextual and mathematical ways to understand this. The proportions maintain the relationship of like measurements. For example, this equation

=6 4

3.375 2.25

could be thought of as ratio of the large length to the small length; it is the same as the ratio of the long width to the short width. Mathematically, you can show the two ratios are equal by taking the decimal approximations, reducing the fractions, or finding cross products.

[Student Handout]

(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?

=3.375 6

2.25 x

Answers will vary. Students are often intimidated by seeing the variable in the denominator. They may recognize that they can flip the proportion over to create the same proportion used in the original




problem, but the goal is for students to recognize that the rules they have used to solve other equations still apply here. Give students a chance to think this through before showing them how to solve it. Prompt them by asking what operation they used to take the 6 out of the denominator in the original proportion. Point out that since x represents a number, it can be used as a number. So it is possible to multiply both sides of the equation by x.

The following problems are used to check for understanding of setting up and solving proportions.

[Student Handout]

Solve each equation. Round to the nearest tenth.

(5) =12.7 0.2

3x

(6) =8,500

4,200 5

x

(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?

Answers: (5) 190.5; (6) 10.1; (7) 32 ounces

Return to the issue of units raised in Question 2. Because the ratio here is given in a comparison of gallons to ounces, the result also needs to be in gallons to ounces. You can convert the gallons to ounces and find a ratio with the same units that can be applied to all situations.

[Student Handout]

Making Connections


Making Connections: Main Ideas to Highlight Proportionality preserves the ratio between values.

Facilitation Prompts • How can you tell if a relationship is proportional? You may want to refer back to Question 1. • How does this relate to Lesson 2.1.2 in which you found a population of an area based on the

population density?

Answer: The population density was a ratio that was preserved for different areas of land.




Consistency of mathematical rules—foundational rules for solving equations can be used on any equation.

Facilitation Prompts • What are the basic rules to think about when solving any equation? • If you have time, it can be very empowering to students to realize that they have the tools to

solve some very “scary-looking” equations. For example:

o =−

5 3

2 8 3x

x

o x2 + 15.3 = 39

Some students may know about taking a square root, but others will not. This example points out to students that even when equations require tools that they may not yet have, the basic rules do still apply. Recognizing that they need another tool—knowing what you cannot do and why—is also an important skill! Students’ basic knowledge tells them that they need to “undo” or “reverse” adding 15.3 by subtracting it from both sides of the equation—so they at least have a starting point. Next, students should recognize that they need to “undo” the power of 2. Even if they are stopped here, they understand the idea of what they are looking for.

This is also a good problem to push students to use their number sense. Consider the equation x2 = 23.7. What numbers squared give a result close to 23.7? Consider whether you want to discuss both the positive and negative result.

[Student Handout]


(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

1Retrieved from www.widescreen.org/widescreen.shtml




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

Answers:( a) Answers will vary. Students may show that the decimal approximations are not equal or that the cross products of the ratios are not equal. (b) Answers will vary. If a picture with dimensions of 2.35:1 is resized, the width must increase by 2.35 inches for every inch that the height increases. The width of a picture with a ratio of 4:3 increases by 1.33 inches for ever inch that the height increases.

Lesson 3.2.5.2: Inaccuracy of Measurement Instructor Theme: Physical World

Pierce College Math Department


• Measurement accuracy is affected by the scale used • Errors in measurement can get magnified when measured are used in formulas

Students will be able to • Describe the accuracy of a measurement for a given scale • Calculate a margin of error for the output of a formula

The focus of this lesson is to build off the proportional geometry from Lesson 3.2.5 using similar triangles, and to discuss how inaccuracy of measurement can propagate in formulas. For problem situation 1, you can either use the photos included, or give students rulers and yardsticks and have them perform the experiment themselves.

[Student Handout] In Lesson 3.2.5, you used proportions to scale artwork, and used proportions to solve problems. If our proportion involves measured quantities, or we are using measured quantities in other formulas, the errors present in the measurement can get magnified. It is important to be aware of how errors can magnify, both so we can better judge the accuracy of decisions made based on calculations, and so we can determine the necessary accuracy for our initial measure to avoid excessive error in the end.

Problem Scenario 1 Sometimes you need to estimate a distance across a river or how far you are away from a landmark, and it is not practical to measure it directly. Luckily, you can estimate the distance if you have a ruler, and an object of known height in the distance. To illustrate this approach, consider the two people below. To estimate how far apart they are, David holds a ruler out at arm’s length.

David lines up the top of the ruler with Sharon’s head, and measures the distance the ruler shows to Sharon’s feet. Next, David gets help to measure how far the ruler is from his eyes, 25”. The last important piece of information is Sharon’s height – 5’3”.



1. Estimate the distance between Sharon and David using the given information.

Students may have no idea how to start on this. Encourage them to draw a diagram of the situation, and label the known distances. Students may come up with their answers in different ways. If no one sets up a proportion as in 3.2.5, help them do so. Answer:

After drawing the diagram like this, show them the de-contexualized version with just two triangles. Discuss how the similar triangles are related to keeping the aspect ratio on rectangles in the previous lesson; both are about keeping the dimensions proportional.

53 63825 x

= , so x is about 434.5 inches, or 36.2 feet.

2. Determine a margin of error for your measurement of Sharon’s apparent height using the ruler.

How much would that affect the estimate of distance? If different students in the class estimate the measurement from the picture differently, this might help motivate why we’d want to consider the margin of error. Answer may vary here, but a very generous margin of error would be 3.5 < height < 3.75. Students may argue for it to be narrower, which is fine. For the second part of the question, students could calculate the distance estimate using the endpoints of that interval: A measurement of 3.5 leads to a distance of 37.5 feet A measurement of 3.75 leads to a distance of 35 feet.

5’3” = 63”

25”

About 3 5/8” ?



3. Write an inequality showing the range of estimates for the distance. 35 ft < distance < 37.5 ft Help them notice, if they haven’t already, that a very small change in the measurement (1/8”) magnifies to a change of over a foot in the determined distance. Ask them to explain why that’s happening. The next problem situation continues the idea from above, and brings in formula evaluation as well.

[Student Handout] Problem Scenario 2 The Tacoma Dome is an arena in Tacoma, Washington, and is the world’s largest arena with a wooden dome in terms of total volume and seating capacity. The dome has a diameter of 530 feet, the height to the tallest point of the dome is 152 feet, and the side wall around the dome is 36 feet tall. 1. In 2012 the dome was cleaned by workers rappelling

from the top of the dome. How many person-hours do you think it took to clean it? Make a guesstimate.

Students will likely have a wide range of guesses. If they’re hesitant to guess, ask them “what’s a number you know is too low? Too high?”

2. The dome has the shape of a spherical cap; a portion of a sphere.

The surface area of a spherical cap is given by the formula

( )2 2A a hπ= + . If a worker can clean about 4 square feet a

minute, how many worker hours will it take to clean the whole dome?

Answer: From the given info a = 265 (the radius of the dome), and h = 116 feet; the 152 height minus the 36 feet side wall.

( )2 2265 116A π= + = 262,892 sq ft.

2

2

1 1262,892 4 ft 60 min hrft

min⋅ ⋅ = 1095 hours

3. Suppose the estimate for cleaning rate could be off by ½ square foot per minute. How far off

might the estimate for cleaning time be? 3.5 sq ft / min = 1252 hours 4.5 sq ft / min = 974 hours



Making Connections Remind the students about Lesson 3.2.2, with the formula for braking distance for a car (below). Ask them which values they are likely to know with good precision, and which are likely to be known with less precision.

The formula for the braking distance of a car is 2

0 = 2 ( + )

Vd

g f G 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 provide a coefficient of friction of about 0.8 to 0.85.

Remind the students of Lesson 3.2.3 when they measured the height of the apple graphics, and calculated the corresponding area and volume. Ask student whether if the measurement for height was off a bit, would calculating area and volume magnify that error, or hide the error (it would magnify it).


Lesson 3.2.6: Solving More Equations Theme: Personal Finance, Medical Literacy, Citizenship





Solving equations Varied Not applicable Level 3


• understand order of operations when simplifying an expression. • be able to substitute a value for a variable in a mathematical model and simplify the model. • be able to square a number. • be able to solve a two-step linear equation such as 2 = x/3 + 5. • understand the meaning of the word term in the context of an algebraic expression.


• 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.

Explicit Connections • The process of solving equations is always the same, deciding how to “undo” what has been done to

the variable you want to solve for and keeping the equation balanced by undoing both sides of it. The process is based on the order of operations.







Suggested Timeline


5 minutes Discuss the importance of solving equations Small groups, class discussion

25 minutes Work on Questions 1–3 Have pairs of students work on the scenarios; if groups finish early, they can start OCE Questions 1 and 2 to further practice their equation-solving skills

15 minutes Discuss Questions 1–3 Ask for volunteers to present their solutions and strategies for each problem; ask other class members to offer additional strategies


Special Notes See the notes at the end of the lesson about the OCE and preparing for the Module 3 assessment.

The goal of this lesson is for students to practice the equation-solving skills introduced in Lesson 3.2.4 and Lesson 3.2.5. Each question provides slightly more complex equations than students have seen before. Students should work in pairs on the following questions as you circulate and answer questions. As you circulate, remind students to check that their answers make sense with the context of the equation and that they are remembering to include units in their final answers.

[Student Handout]

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:

( )= + +1

0.082 0.22 252

C m m

(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?

Answers:

(a) 1/2 is because Paula is paying half of the car cost; 0.082m is the total maintenance costs for m miles, 0.22m is the total gas cost for m miles, and 25 is the weekly parking fee.

(b) For one week, m = 42 miles (7 x 2 x 3); C = $18.84 per week.

(c) m = 62.92 when C = $22. With 14 miles per trip, this equates to 4.5 trips per week. So, if she went to school five or more days a week, the bus would be cheaper.

In this problem, students need to simplify the equation before solving for m. If they have trouble, prompt by asking if they can write the equation in a simpler form.

In the discussion of this problem, be sure that the different ways of working with the 1/2 are noted. It can be distributed, or both sides of the equation can be multiplied by 2. Also note that when simplifying, you do not have to perform the same operation on both sides of the equation. When adding like terms, you are rewriting the equation in an equivalent form, so the equation is still balanced. This is often confusing to students. They see it as, “the rules are different.” You can use examples such as the ones below to demonstrate this point.

Start with an obvious equation such as 3 = 2 + 1. If you combine the two terms to get 3 = 3, it is still a true equation because all you have done is to rewrite 2 + 1 in an equivalent form. However, if you subtract the 1 from the right side 3 = 2 + 1 – 1, the equation is out of balance because 2 + 1 ≠ 2 + 1 – 1. So in this case, you must subtract 1 from both sides: 3 – 1 = 2 + 1 – 1.




[Student Handout]

(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.

Answers

(a) N = 5.45 or between 5 and 6 drinks

(b) +=

0.015

0.022B t

N

(c) N = 8.75 or approximately 9 drinks

Although students have solved this equation before when B and t were given, this may be more challenging when the fixed variables are not being replaced with numbers. Remind students that the process remains the same. You might suggest that students think about and apply the steps they used in Part (a) to Part (c).

1Retrieved from www.cdc.gov




[Student Handout]

Making Connections


Making Connections: Main Ideas to Highlight The process of solving equations is always the same, deciding how to “undo” what has been done to the variable you want to solve for and keeping the equation balanced by undoing both sides of it. The process is based on the order of operations.

Facilitation Prompts • What was the same about the process of solving each equation in this lesson? • What was different? Note that simplifying the equation before starting by distributing and/or

combining terms may make the process easier, but the rules of solving equations are still the same.

OCE and Preparing Students for the Module 3 Assessment As with the previous modules, students should complete a review and self-assessment for Module 3.You will need to decide when to assign this based on your scheduling of the assessment and whether you will have a day for review in class. You may choose to replace all or part of OCE 3.2.6 with a review assignment.

[Student Handout]


(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:

π= 34

3V r

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?

Answer: Given the dimensions of the slab, V = 48 in3. So, π

= =336

2.2545r or about 2.25 inches.

Students might also try to plot the volume equation to estimate r.

Module 3 Wrapup: What’s this Algebra Stuff Good For? Instructor Theme: Personal Finance

Pierce College Math Department. Licensed CC-BY.

For each piece of this problem, try to solve it both numerically and algebraically. For each, which is easier? Walk students through part 1. Help them understand what it means to solve numerically vs solving algebraically. Encourage students to try to solve each problem using both techniques, but let them know that if one approach becomes too difficult, it’s OK to only use one. 1. Angela runs a small business selling gift baskets for $30 each. Given only this info, how many baskets will she need to sell to earn $30,000 profit for the year? Numerically 30,000 divided by 30 = 1000 baskets

Algebraically n = number of baskets sold 30n = 30,000 n = 1000

2. In addition to the information above, each basket costs Angela $10 to assemble. Considering this, how many baskets will she need to sell to earn $30,000 profit for the year? Numerically $20 profit per basket 30,000 divided by 20 = 1500 baskets

Algebraically 20n = 30,000 or 30n – 10n = 30,000 This is a good opportunity to point out that these are equivalent, and just different ways of writing the same thing.

3. In addition to the information above, Angela also has to pay $300 a month rent for the storage and work space she rents. Considering this, how many baskets will she need to sell to earn $30,000 profit for the year? Numerically Needs to make 30,000 + 12*300 = 33,600 now 33,600 divided by 20 = 1680 baskets

Algebraically Some might write 20n = 33600, and that’s OK. But have them consider the equation 30n – 10n – 3600 = 30,000 Talk about how this allows you to write the equation considering each piece of info, rather than trying to pre-calculate pieces.

Module 3 Wrapup: What’s this Algebra Stuff Good For? Instructor Theme: Personal Finance

Pierce College Math Department. Licensed CC-BY.

4. In addition to the information above, Angela also has to pay 5% business taxes on her revenue (the money she brings in, before considering costs). Considering this, how many baskets will she need to sell to earn $30,000 profit after taxes for the year? Numerically This is getting to the point where numerically is REALLY hard to do. If they get stuck, encourage them to just try doing it algebraically

Algebraically 30n – 0.05(30n) – 10n – 3600 = 30,000 18.5n = 33600 n = 1816 baskets

5. In addition to the information above, Angela also has to pay about 20% income tax on her income (the amount she brings in, after costs). Considering this, how many baskets will she need to sell to earn $30,000 profit after taxes for the year? Numerically Algebraically

Two ways to do this: the “take away 20% of profit” way: 30n – 0.05(30n) – 10n – 3600 – 0.20(30n – 0.05(30n) – 10n – 3600 ) = 30,000 Or, the “keep 80% of the profit” way: 0.80(30n – 0.05(30n) – 10n – 3600) = 30,000 n = 2222 baskets

Pierce College Math Department. Licensed CC-BY-SA-NC.

Lesson 4.1.0: Seeing Patterns Instructor The idea behind this assignment is to emphasize different ways of seeing the patterns. Really encourage students to identify different ways they are coming up with the patterns. 1) Using the pattern shown

Stage 1 Stage 2 Stage 3

a) Complete the table

Stage What I See Here Number of dots 1 5 2 9 3 13 4 17 10 41 n 1 dot in the middle + 4 arms,

each with n dots Start with 5 dots, add 4 more each stage

1+4n 5 + 4(n-1)

Discuss why the different representations are equivalent, going to back to simplifying expressions. b) How many dots will be in stage 20? 81 c) What stage will have 137 dots? 34


2) Using the pattern shown

Stage 1 Stage 2 Stage 3 a) Complete the table

Stage What I See Here Number of boxes 1 8 2 13 3 18 4 23 10 53 n (n+1) stacks of 3, and n stacks

of 2 A stack of 3 on the right, plus n C-shaped groups of 5 Start with 8, add 5 each stage

(n+1)*3 + n*2 3 + 5n 8 + 5(n-1)

b) How many boxes will be in stage 20? 103 c) What stage will have 1268 boxes? 253



Stage 1 Stage 2 Stage 3 Stage 4

a) Complete the table Stage What I See Here Number of boxes 1 2 2 4 3 6 4 8 10 n N stacks of 2

Start with 2, add 2 each stage

2n 2 + 2(n-1)

b) How many boxes will be in stage 20? 40 4) Using the pattern shown


a) Complete the table Stage What I See Here Number of boxes 1 1 2 4 3 9 4 16 n n by n square

n2

b) How many boxes will be in stage 20? 400 c) In which stage will there be 625 boxes? 25




a) Complete the table Stage What I See Here Number of boxes 1 2 2 6 3 12 4 20 n n wide by n+1 tall

an n by n square, with another row of n on top

n(n+1) n2 + n

b) How many boxes will be in stage 20? 420 4) (Challenge) Using the pattern shown:

Stage 1 Stage 2 Stage 3 Stage 4

a) Complete the table Stage What I See Here Number of boxes 1 1 2 3 3 6 4 9 n They will probably say “add n

each time”, but that’s not a closed form. If you rearrange the blocks into a right triangle, you can see it is ½ of and n by n+1 rectangle

0.5n(n+1)

b) How many boxes will be in stage 20? 210


Lesson 4.1.1: Lining Up Theme: Personal Finance





Linear equations Estimating costs Not applicable Level 2


• understand the basic meaning and use of variables. • be able to solve for an unknown variable in a one-variable equation. • understand the basic idea of a rate of change as a ratio or proportion. • be able to graph points on a coordinate plane.


• linear models are appropriate when the situation has a constant increase/decrease. • slope is the rate of change. • the rate of change (slope) has units in context. • different representations of a linear model can be used interchangeably.

Students will be able to • label units on variables used in a linear model. • make a linear model when given data or information in context. • make a graphical representation of a linear model. • make a table of values based on a linear relationship. • identify and interpret the vertical and horizontal intercepts in context.

Explicit Connections • The four representations of a mathematical relationship help you understand, organize, and use

mathematical information in different ways. • A linear equation is a special mathematical relationship in which there is a constant rate of change

between two variables.

Notes to Self One thing I want to do during this lesson…

One thing I want to pay attention to in my students’ thinking…




One connection or idea I want to remember…

Suggested Timeline

Note: Some aspects of the timeline vary depending on how you handle Questions 2 and 3.


5 minutes Discuss Problem Situation 1 (Question 1) Small groups with class discussion or Think/Pair/Share

30 minutes Questions 2 and 3 The Instructor’s Notes offer different options for structuring this portion of the lesson; how the time is used depends on the structure you choose

10 minutes Questions 4 and 5 Small groups with class discussion


Special Notes Students need grid or graph paper for making graphs. Ideally, it is best to make graph paper available in the room for students to use. This creates the option of using graphs without giving the message that a graph is the only or best tool to use. However, it may not be possible to provide graph paper. In that case, you can insert a coordinate grid into the student materials or have students print some copies of a grid. A template is provided at the end of the Instructor’s Notes.

Note that there are multiple instructional options for Question 2.

Introduction for Teachers The goal of this lesson is for students to understand that situations with a constant rate of change can be modeled by linear equations. The rate of change can be an increase or decrease, but it needs to be assumed to be constant to write a model of this form. Lessons 4.1.2 and 4.1.3 extend these ideas to more complex situations where the rate of change varies and where the data do not increase or decrease by exactly a constant amount. This lesson also introduces students to using the four representations for a model and has them practice using various representations. The productive struggle of this lesson is around thinking about the situation and how to model it with any of the four representations to arrive at an answer for the problem.




[Student Handout]

Specific Objectives

Students will understand that • linear models are appropriate when the situation has a constant increase/decrease. • slope is the rate of change. • the rate of change (slope) has units in context. • different representations of a linear model can be used interchangeably.

Students will be able to • label units on variables used in a linear model. • make a linear model when given data or information in context. • make a graphical representation of a linear model. • make a table of values based on a linear relationship. • identify and interpret the vertical and horizontal intercepts in context.

In this lesson, you will learn about how linear models (linear equations in context) can be useful in examining some situations encountered in real life. A model is a mathematical description of an authentic situation. You can also say that the mathematical description “models” the situation. You will practice using the four representations you read about in the OCE to express the linear models for two situations.

Problem Situation 1: Cell Phones

Cell phone service can be purchased in two different ways. One is an unlimited service where the customer signs a one- or two-year contract and pays a monthly fee for unlimited minutes. The other is a prepaid service where the customer purchases the phone and a specified number of minutes and can stop paying at any time.

You want to have your own phone and need to decide which option costs less. • Per-Minute Pricing: There is a monthly fee of $15.99 plus $0.13 per minute. • Unlimited Plan: The plan costs $39.99 per month. The phone is free and unlimited minutes

of talk time are included, but a two-year contract is required.

(1) Which plan do you think is less expensive? What factors affect this decision?

Answers will vary.

[Student Handout]

(2) Determine which plan is less expensive. Be prepared to justify your answer.

The question comparing the plans is the motivator for learning about linear models and not a learning goal in itself. There is not an expectation that students will solve a system using formal algebra—intuitive approaches to answering the question are fine. These might include guess and check,




estimation from a graph, or interpolating from a table. Some may simply reason the answer out by realizing they can subtract 15.99 from 39.99 and divide by 0.13. Examples with equations, tables, and graphs are shown on the following pages.

The goal of the lesson is to build an understanding of linear models. You will use the models in the situation as a basis for this discussion. Students are given a list of terms they need to know in Question 3.

Following are some options for the level of productive struggle to use depending on your students.

Level 2 Productive Struggle

a. Provide students with structure by first having them model the prepaid plan. You can motivate this by saying that one way to approach solving this problem is to first build a thorough understanding of the situation by focusing on one part.

b. Discuss the characteristics of a linear model using the prepaid plan as an example.

c. Use the unlimited plan either in groups or as a class discussion depending on time. As a part of the discussion, talk about the idea of a constant function and show the graph of a horizontal line.

Level 3 Productive Struggle

a. Let students answer the question using whatever methods they want.

b. Prompt students in groups or class discussion to develop the remaining three representations (they start with verbal). Eventually, every student should write an equation, graph, and table.

c. Discuss the general characteristics of a linear model and the special characteristics of a constant function.

Facilitation Prompts • How many minutes need to be used each month to make the per-minute plan more/less

expensive? • Encourage students to organize their thinking and number crunching into a table/chart. • Remind students about the four representations of a model and/or remind them how to write

equations like those developed in the previous lesson. • You may help students get started on writing the equations for each cell phone plan by defining

their variables. • You may suggest writing some points as ordered pairs in order to use a table or graph.

Answer: The two plans are equal at approximately 184 minutes.

Note for the students that the descriptions of these options in the problem situation are examples of verbal representations of the mathematical relationships.




Using Equations

Let P = cost of per-minute plan, m = number of minutes used, and U = cost of unlimited plan. If you include the cost of the phone, the cost of the prepaid plan is given by P = 15.99 + 0.13m.

Note: If students do not mention it, point out to them that $15.99 is the fixed cost (what they pay even if they do not use the phone) and the $0.13 per minute is the cost to talk (what is called the variable cost in economics). In addition, mention that this $0.13 per minute is a ratio of cost per minute and also the slope.

The total cost of contract plan per month is $39.99; so, U = 39.99 dollars. (You may want to point out that this is a constant monthly cost rather than one that increases/decreases.)

Answer with Table

Students may choose not make a table for the contract plan. If they do not, prompt for that in the discussion. A solution method for the question is to make a table for the prepaid plan using intervals until a value close to 39.99 is reached and then using guess and check to narrow down to the solution.

Minutes Talked Cost of Per Minute Plan

Cost of Unlimited Plan

30 $19.89 $39.99

60 $23.79 $39.99

90 $27.69 $39.99

120 $31.59 $39.99

150 $35.49 $39.99

180 $39.39 $39.99

210 $43.29 $39.99

You may need to guide students to plot points for the prepaid plan by charting different values of m (probably using increments of 10 or 100 minutes rather than 1 minute).

Answer with Graph

To get students started on the graph, you can remind them of drawing and labeling axes, plotting points (using points from the table the class just did), and then drawing a line through the points. It may be helpful to remind them that all the points should be on a line (not a jagged line or a triangle or other shape). Students may ask about labeling the vertical axes since they will be graphing U and P; the units are in dollars for both variables, so discuss if it makes sense to graph both on the same axes.

A graph of the costs C and P (for one month) for talking m minutes is given on the following page. Students are unlikely to graph the unlimited plan as a horizontal line, but they might think to find the point at which the prepaid plan costs $39–40.

To model the unlimited plan, students hopefully realize from the table that it is a constant cost of $39.99. There is no increase or decrease. Students should note that this gives a horizontal line rather




than an increasing (or decreasing) line. Guide students to finding the point of intersection as where the costs are equal.

By finding the point of intersection, they are finding where the costs are equal. This happens when the number of minutes used in the month is about 184 minutes. So, another solution some students may give is that if you talk more than 185 minutes per month, the unlimited plan is cheaper.

Explain to students that this graph only shows positive values for minutes and cost because the negative values do not have meaning in the context. Mathematically, they could use the equations and find negative values. Most graphs in this course are limited to the first quadrant.

[Student Handout]

(3) The two mathematical relationships in the cell phone plans are linear. They have certain important characteristics. The following terms are important vocabulary in talking about linear models and other types of mathematical relationships. You will discuss these in class. Make sure you take good notes about what each means. • constant rate of change • slope • vertical intercept • horizontal intercept

Constant rate of change: This is the characteristic that defines a linear relationship. Demonstrate the constant change with the table and the graph of the per-minute plan. If your table is set up in increments of 10, note that the cost increases by $1.30 for every 10 minutes. Then discuss the rate of change for U = 39.99. The rate of change is 0, which is constant.




Slope: This is a ratio that describes the rate of change. The slope of P can be written in different ways: $1.30/10 minutes or $0.13/minute. Note the units. Connect the slope to the graph as rise/run. Connect the concept of slope as the rate of change to its position in the equation. The slope is multiplied by the variable because you use it as a rate to find a new value. Students probably used this strategy when working with population densities in Lesson 2.1.2. Units demonstrate the relationship:

= +

=

$0.13$ min $15.99

1 min

$0.13$

1 min

m

minm +

= +

$15.99

$ $0.13 $15.99m

Vertical intercept: Explain the general concept of an intercept first and then explain vertical intercept. Note that this often has important meanings in context. In the case of P, it is a fixed cost or monthly fee. It is sometimes called the starting value. Connect the concept of the intercept to the position in the formula: The plan costs $15.99 before any minutes are used.

Horizontal intercept: P does not have a horizontal intercept in context, but you can ask students if it would mathematically (relating back to the discussion of the graph in the first quadrant). You can also ask if U would have a horizontal intercept to draw the contrast between the two models.

[Student Handout]

Problem Situation 2: Daily Latte

A local coffee shop offers a Coffee Card that you can preload with any amount of money and use like a debit card each day to purchase coffee. At the beginning of the month (when you get your paycheck), you load it with $50. Each day, your short soy latte costs $2.63.

(4) Estimate if the Coffee Card will last until the end of the month if you purchase a latte every weekday.

(5) If you purchase a latte every weekday, calculate if your $50 Coffee Card will last until the end of the month. If not, how many days will you have to go without coffee before you can reload it with money after your next paycheck (beginning of next month)? Or, if so, how much money is left at the end of the month?

The primary purpose of these questions is to demonstrate a negative slope and a horizontal intercept and reinforce the concepts about linear models. If time allows, have students work in groups, but if it does not you can go through this as a class discussion.

Answers to Question 4 will vary. Students hopefully estimate how many weekdays there are each month (at least 4 weeks = at least 20 weekdays). Then, 20 • $2.63 = $52.60; you are over the amount of the card. Thus, it will not last until the end of the month.

Notes: Encourage students to think about Question 5 using one of the three representations (equation, table, or graph) to be more exact than just estimating (Question 4). Regardless of what representation students use, ensure that they understand the idea that the slope of the line is the rate of change. Here the slope is –2.63 because the price of the coffee = rate of decrease of coffee card.




For Question 5, you may want a more exact estimate of the number of weekdays in a month, although you can also use an estimate. There are approximately 365 days a year. If five out of seven days are weekdays, then 365 • 5/7 = 260.71. If you divide by 12 months, you get 21.7 days per month. Thus, you can say any given month has about 22 weekdays (could be as few as 20 in February or as many as 23 some other months).

Using C = money on the Coffee Card and n = number of lattes, • solve using an equation:

C = 50 – 2.63n

So if you put in n = 22, then C = 50 – 2.63(22) = –7.86, which leaves about three days without coffee.

• solve using a table:

Number (n) of Lattes Money on Card (C)

0 50.00

1 47.37

2 44.74

3 42.11

… …

19 0.03

After going to 19 days buying coffee, you find the balance is only $0.03. Therefore, you cannot buy more coffee until you get paid again. That means (on average) you have three more days without coffee (could be from one to four days).

• solve using a graph:

Ask students to predict how this graph compares with the graph in the first situation. Encourage them to think about the graph in terms of slope and intercept. Students can also use the points in the table to graph the line, but they should also connect to the visual representation of rise/run.




Note: Talk to students about the importance of the intercepts. Where the graph crosses the n-axis is when they are out of money! So, it looks like they are out of money at Day 19, leaving three days without coffee.

[Student Handout]

Making Connections


Making Connections: Main Ideas to Highlight The four representations of a mathematical relationship help you understand, organize, and use mathematical information in different ways.

You may not have time for an in-depth discussion, but point out to students that the four representations give them different tools for working with mathematical relationships. If students are stuck on one representation, it is often useful to try another.

A linear equation is a special mathematical relationship in which there is a constant rate of change between two variables.

[Student Handout]


(1) Explain how you can tell from a graph if a linear model has a negative slope, positive slope, or slope of 0. Include sketches of each type of graph.

Answers will vary. A line with a positive slope goes up from left to right. A line with a negative slope goes down from left to right. A line with a slope of 0 is flat or horizontal.




Grid Template


Lesson 4.1.2: Comparing Change Theme: Personal Finance and Citizenship





Linear models and slope Comparing prices; beverage consumption



• understand a rate of change as a ratio or proportion. • be able to solve for an unknown variable in a linear relationship. • be able to create a linear model in authentic context with units.


• that linear models are appropriate when the situation has a constant rate of increase/decrease or can be approximated by a constant rate.

• that the rate of change (slope) has units in context. • the difference between a positive slope and a negative slope. • that the linear models for authentic situations have limitations in using them to make

predictions.

Students will be able to • make a linear model when given data or information in context. • calculate a slope given data or information in context. • estimate the value that makes two linear models equivalent.

Explicit Connections • A slope is a rate with units. It can be thought of as the rate of change between two data points. In

cases of nonlinear data, this is an average rate of change.

Notes to Self One thing I want to do during this lesson…

One thing I want to pay attention to in my students’ thinking…

One connection or idea I want to remember…




Suggested Timeline


20 minutes Give groups time to answer Problem Situation 1: Milk and Soft Drink Consumption

Small groups and discussion

15 minutes Problem Situation 2: Moving Truck Rental Small groups and discussion

10 minutes Groups share answers; class discussion (with instructor guidance) at places where answers vary

Class discussion


Special Notes This lesson builds from Lesson 4.1.1 to help deepen students’ understanding of linear equations. This includes calculating a slope, which is the major goal of the lesson. In the first situation, students work with two linear equations where one is increasing and one is decreasing. They find where the two lines intersect (where the values are equal). Point out that the slope is the average rate of change. You may need to remind students to label units on their slopes and models. In the second situation, students use three different but similar linear models to determine which size moving truck they should rent from U-Haul to make their moving costs the least expensive.

The productive struggle of this lesson is around thinking about the authentic situation and how to model it with an equation to answer the question.

This lesson looks at two situations of creating and analyzing linear models in context. The first situation (consumption of milk and soft drinks) is the more difficult problem, but it is also more important because it introduces calculating slope and negative slope. In this problem, the consumption increases/decreases by a constant amount, but students need to find the rate of change (it is not given explicitly). Students need to figure out the average rate of change and then make the linear model for each beverage. After making the models, they use the models (in any representation) to find when the consumption rates were the same.

The second situation compares three moving truck options that have different size capacities as well as different costs. After estimating the volume of each truck to estimate the number of trips needed for each size truck, students need to consider three different models. Students should be able to use the equations for the moving truck costs to consider which is the better choice; however, students may choose to approach the problem using a different representation.

It is important to cover the concept of slope thoroughly before moving on to Problem Situation 2. If you do not have time to complete Problem Situation 2 in class, students can probably do it on their own.




[Student Handout]

Specific Objectives

Students will understand • that linear models are appropriate when the situation has a constant rate of

increase/decrease or can be approximated by a constant rate. • that the rate of change (slope) has units in context. • the difference between a positive slope and a negative slope. • that the linear models for authentic situations have limitations in using them to make

predictions.

Students will be able to • make a linear model when given data or information in context. • calculate a slope given data or information in context. • estimate the value that makes two linear models equivalent.

Problem Situation 1: Milk and Soft Drink Consumption

Over the last 60 years, the U.S. per-person consumption of milk and soft drinks has changed drastically. For example, in 1950, the number of gallons of milk consumed per person was 36.4 gallons; in 2000 that number had decreased to 22.6 gallons. Meanwhile, the number of gallons of soft drinks consumed per person in 1950 was 10.8 gallons. By 2000, this number had increased to 49.3 gallons per person.1

Is there a time when the consumption (per person) of milk equaled or will equal the consumption (per person) of soft drinks? If so, find when they were equal or when they will be equal. For this problem, assume that the change in consumption is linear.

Facilitation Prompts • Which of the four representations might be useful? • What information do you need to create the model in that representation?

A table can help students make sense of the data without the complication of text. It can also prompt them to make a graph. These are the easiest starting points for most students, but eventually they need an equation. This is the first time they have encountered a situation where the slope has to be calculated as an average rate of change (rather than be given it as a rate). Once students recognize that they need to calculate the slope, you can prompt them on strategies:

• A slope is a rate of change with units. What are the units for this slope? Another way to think of this is, “What is changing?”

• How much did the milk consumption change? Over how much time? (Remind students that they have worked with rates before.)

1Retrieved from www.ers.usda.gov/data/foodconsumption/spreadsheets/beverage.xls




At some point, stop the class to discuss calculating slope. While this course does not emphasize the use of conventional formulas, it is important that students think about how to generalize what they are doing. Discuss ways they could generalize their strategy. Some may mention the slope formula, while others might think of it as a visual of the vertical change divided by the horizontal change. Make sure this concept is well established and well understood before allowing students to continue with the question. You may wish to start by discussing the soda situation since it has a positive slope.

You can also connect this back to the following staircase problem from the Lesson 3.2.5 OCE:

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.

Another important point to discuss either here or in the problem wrap-up is that although this line goes through these two points, it may not perfectly model the situation for every year between (or after). Students were told to assume they could use a linear model, but it is extremely unlikely that the change is exactly linear. Therefore, this is called an average rate of change. You should also make sure the units on the slope are discussed since the units have a rate within a rate. Then you can proceed to have students write equations for milk consumption and soft drink consumption.

To find the rate of change of milk: Rate of change = (22.6 – 36.4)/50 years = –0.276 gallons (per person) per year. So, the linear model that estimates milk consumption can be given by M = –0.276t + 36.4 gallons t years after 1950. You may need to help students understand that they can use 36.4 as the base amount in 1950 since that is Year 0. In addition, emphasize that the negative slope means that the amount consumed has been decreasing.

To find the rate of change of soft drinks: Rate of change = (49.3 – 10.8)/50 years = 0.77 gallons (per person) per year. So, a linear model that estimates soft drink consumption can be given by: S = 0.77t + 10.8 gallons t years after 1950. Note that the slope here is positive, meaning the consumption has been increasing.

As with Lesson 4.1.1, the purpose is not to solve the system with formal algebraic methods. Encourage students to use a variety of strategies. To find out if/when the consumption is equal, they can solve using the equations, a graph, or a table.

• Using equations: Set the equations equal and solve: –0.276x + 36.4 = 0.77x + 10.8. You get x = 24.474 years since 1950. This means that in 1974, the consumption of milk and soft drinks was equivalent (at 29.6 gallons per person per year).

• Using a table: Make a table showing the consumption of each beverage.

H

D




Calendar Year Years Since 1950 Milk Soft Drinks

1950 0 36.4 10.8

1960 10 33.64 18.5

1970 20 30.88 26.2

1980 30 28.12 33.9

1990 40 25.36 41.6

2000 50 22.6 49.3

The table shows that the consumption levels switched at some point between 1970 and 1980. You could estimate it being around 1974 or 1975, but you would have to make a yearly table to predict exactly.

• Solving graphically: Graph both equations and find where they intersect. Be sure that students include labels and units on their axes.

Point out that it is difficult to find the exact solution graphically when graphed by hand; however, if an estimated answer is enough, then a graph can be a useful tool.

This is a good time to bring up the limitations of the models. Although the vertical intercepts have meaning (the amounts consumed in 1950 if 1950 is time = 0), the horizontal intercepts do not have meaning in reality. When the linear model for milk equals 0 (crosses the horizontal axis), this means that the milk consumption per person is none. However, in reality, the consumption will not likely go to 0 any time soon. So, there are limits to the time for which the model is reliable and accurate. Point out that there are limitations to using mathematical models (linear or nonlinear), but also emphasize how useful the models are—as long as they are used for a period of time.

Discuss the use of number of years versus calendar years. You rescale the years to start at 0 so the vertical intercept is the starting value. In the course materials, you maintain mathematical consistency




between models and use the same values in graphs and tables. However, in real-world contexts, graphs and tables usually use the calendar year. This is an opportunity to make students aware of this. They will see problems later in which they have to rescale the years.

Note about context: The data are from raw data from the United States Department of Agriculture (USDA) Economic Research Service. The Table of Contents page states, “Data for the carbonated soft drinks (1947–2003) are from the Census of Manufactures, replacing data previously provided by the Beverage Marketing Corporation of New York. At their request, ERS has removed the Beverage Marketing Corporation’s data series on carbonated soft drinks, bottled water, fruit drinks, and vegetables juices from the website and thus no longer provides this data in the Food Availability (Per Capita) Data System.” No other information on the data collection methods was given. A good discussion to have with students is regarding the accuracy of the data depending on who was collecting the data (e.g., the dairy industry or soft drink industry).

Another topic for discussion is the implications of these data. Milk and soft drinks have very different nutritional values. The increase in soft drink consumption has been an issue for many in the health and medical fields. You may ask students if they know how Americans’ health issues such as weight and health (e.g., diabetes) changed from 1950 to 2000.

[Student Handout]

Problem Situation 2: Four friends decided to do a joint weight loss challenge. They weighed in at the start, and again after 3 or 4 weeks. Their results are shown in the graph to the right.

2) Calculate each person's rate of change.

3) How does the rate of change relate to the graph of the line?

4) Find a linear model for Jamie's weight. If she continues losing weight at the same rate:

a) How much will she weigh after 5 weeks?

b) How long will it take until she reaches her target weight of 173 pounds?

Answers:

2. Carlie: 0 lb/wk. Jamie: -1.25 lb/wk. Marcel: -3.33 lb/wk. Jonas: 1.25 lb/wk

3. Positive rate of change: increasing, negative rate of change: decreasing. The further from the zero the rate of change is, the steeper the line.

4. W = 185 - 1.25t. a) 178.75 lb. b) 9.6 weeks

2 3 4 5 1

165

170

175

180

185

190

195

200

Weeks

Wei

ght,

poun

ds

Marcel

Jonas

Jamie

Carlie




[Student Handout]

Making Connections


Making Connections: Main Ideas to Highlight A slope is a rate with units. It can be thought of as the rate of change between two data points. In cases of nonlinear data, this is an average rate of change.

Facilitation Prompts • How are the slopes in this lesson similar to rates used previously such as conversion factors in

Lesson 3.1.1: 60 seconds/1 minute, and water footprint in Lesson 1.2.1: 883 x 109 m3/year? Note the use of units. You could use those other rates to graph points or make a table, which would create a linear model.

• How do you know which units go in the numerator? Encourage students to make sense of this intuitively by thinking about how the units make sense in the ratio.

• How does this relate to a graph? Make sure students realize that the units in the numerator are represented on the vertical axis and the denominator is on the horizontal axis. This sets up the concept of rise over run or vertical change divided by horizontal change.

• How does the slope relate to the equation? Students should note that the slope is multiplied by the variable in the equation.

• How does the slope relate to the table? The slope defines the change between the points listed in the table. The slope can be found from the table and can also be used to create a table.

[Student Handout]


(1) In Lesson 3.2.2, you probably created a table of values like the one below for the relationship between the velocity of a car in miles per hour (mph) and the braking distance in feet.

(a) Calculate the slope between each of the two points in the table. The first answer is shown as an example.

Velocity (mph) Braking Distance (ft) Slope Between Two Points

5 0.98

15 8.84 The slope between (5, 0.98) and (15, 8.84) is 0.786.

20 15.72 Find the slope between (15, 8.84) and (20, 15.72).






(b) Is this a linear relationship? Explain your answer.

(c) Which of the following is the best explanation for the meaning of the first slope in the table?

(i) At speeds between 5 and 15 miles per hour, the braking distance increases exactly 0.786 feet for every increase of 1 mile per hour in speed.

(ii) On average, at speeds between 5 and 15 miles per hour, the braking distance increases 0.786 feet for every increase of 1 mile per hour in speed.

(iii) At speeds between 5 and 15 miles per hour, the braking distance decreases exactly 0.786 feet for every increase of 1 mile per hour in speed.

(iv) On average, at speeds between 5 and 15 miles per hour, the braking distance decreases 0.786 feet for every increase of 1 mile per hour in speed.

(d) Use the trend of the data in the table to make predictions about the braking distance for speeds between 50 and 70 mph. Which of the following is a correct statement?

(i) The braking distance between 50 and 70 mph will increase by about 3.144 feet per mile.

(ii) The braking distance between 50 and 70 mph will increase by exactly 3.144 feet per mile.

(iii) The braking distance between 50 and 70 mph will increase by more than 3.144 feet per mile.

(iv) The braking distance between 50 and 70 mph will increase by less than 3.144 feet per mile.

(e) Explain your answer to Part (d).

Answers

(a)

Velocity (mph) Braking Distance (ft) Slope Between Two Points

5 0.98

15 8.84 The slope between (5, 0.98) and (15, 8.84) is 0.786.

20 15.72 1.376

30 35.37 1.965

50 98.24 3.144

(b) It is not linear because the slope is not the same between each set of points. (c) ii (d) iii (e) Answers will vary. The change in the rate of change gets larger as the velocity increases.




[Student Handout]

Problem Situation 2: Moving Truck Rental

Suppose you and your roommates need to move, so you look into renting a moving truck for the day. U-Haul (www.uhaul.com) charges the following amounts for three different trucks. The dimensions for the trucks are given in length x width x height.

• $19.95 plus $0.79 per mile for a cargo van with inside dimensions: 9'2" x 5'7-1/2" x 4'5" • $29.95 plus $0.99 per mile for a 14' truck with inside dimensions: 14'6" x 7'8" x 7'2" • $39.95 plus $0.99 per mile for a 20' truck with inside dimensions: 19'6" x 7'8" x 7'2"

You estimate that you and your roommates have three rooms full of furniture and boxes, equivalent to about 1,000 cubic feet. Your new apartment is about 10 miles (one way) from your old apartment.

As a group, decide which vehicle size is a better deal to rent. Provide work to support your answer.

Estimations for Number of Trips

Cargo van: 222 cubic feet; requires five trips

14' truck: 761 cubic feet; requires two trips

20' truck: 1,023 cubic feet; requires one trip

Cost Calculations Based on These Estimations Using Equations

Need five trips in the van: So, cost = 19.95 + 0.79(5 trips • 2 • 10 miles) = $98.95

Need two trips in 14' truck: So, cost = 29.95 + 0.99(2 trips • 2 • 10 miles) = $69.55

Need one trip in 20' truck: So, cost = 39.95 + 0.99(1 trip • 2 • 10 miles) = $59.75

Prompt students to consider the difference in cost and the number of trips (and per-mile costs are different). Encourage them to start by estimating the volume that each vehicle can hold. This is an opportunity to discuss estimation. Calculating an exact volume is not necessary and adds work because of having to convert inches to feet. However, rounding to the nearest foot is not very accurate. Since it is easy to round to the nearest half foot, this is a better choice. It adds accuracy without adding a lot of work.

Ask if these situations are all linear. Discuss the slope of each equation, including using units. Answer: 0.79 dollars per mile; 0.99 dollars per mile. Make sure students understand that these are rates for the increase in price per mile driven. Also, note that the vertical intercept is the fixed cost of the rental.

The 20' truck is the cheapest overall since it only requires one trip (even though the daily cost and the mileage cost is the most expensive of the three).

Students may say that they can solve this problem easily without an equation. Acknowledge that this is true for many situations, but the real power of algebra comes when they need to generalize or project.




If they need to make calculations for many different values, an equation is valuable. If they want a quick visual reference, a graph is useful. If they want a quick reference for specific commonly used values, a table is appropriate.

Students may not make graphs or tables to solve this question. Depending on time, you may want to ask students to graph the equations, but you can also project the following graph to discuss the slopes and intercepts of the graphs. Place points on the graphs at the number of miles needed for each truck, and compare the costs at those points. Ask if the medium truck ever costs more than the large truck. Slopes of parallel lines are not part of the content of this course, but this is a good opportunity to point out the relationship of the lines and slopes.

If students try to use different representations of a graph or table, discuss what you would have to do to solve (include various number of trips). However, point out that these representations still can be used and the resulting conclusion is the same regardless of which representation is chosen. (Also, you can mention the problem is given with the verbal representation.)


Lesson 4.1.3: That Is Close Enough Theme: Citizenship





Linear models Social Security and life expectancy




• understand the meaning and use of variables. • be able to solve for an unknown variable in a linear relationship. • be able to calculate a rate of change (slope) from data. • be able to make a linear model given linear data in context.

Specific Objectives


• linear equations can approximate nearly linear data.


• find the equation of a line that estimates nearly linear data by calculating the rate of change and estimating the vertical intercept of the line.

• use approximate linear models to interpolate and extrapolate.


• A model for data is a type of summary or approximation that is useful but has limitations. • An algebraic model is a powerful tool.

Notes to Self







Suggested Timeline


15 minutes Introduction and Question 1 Class discussion

5 minutes Question 2 Individual work

15 minutes Questions 3–5 Small-group work

10 minutes Discuss Questions 3–5 and close Class discussion


Special Notes

This lesson uses different data sets (graphs) that are split among groups. The data sets are included at the end of the lesson in both the instructor materials and student handouts. You need to decide how to handle these materials depending on your delivery method for the materials. Students can print out all the data sets or you can make copies for groups.

The following material is included in the previous OCE.

Your next lesson focuses on the Social Security program. Social Security provides income for people who are elderly or disabled. Currently, people pay 6.2% of their wages up to $90,000 into Social Security. When they reach “full retirement age,” they can claim a full retirement benefit. Those that retire before their full retirement age receive a lower benefit.

(1) Use Social Security Online’s frequently asked questions webpage (http://ssa-custhelp.ssa.gov/app/answers/detail/a_id/14) to answer Parts a, b, and d.

(a) Who can retire at age 65 with full retirement benefits?

(b) At what age can someone born in 1950 retire with full benefits?

(c) In what year were you born?

(d) What is your full retirement age?

Answer: (a) Someone born in 1937 or earlier; (b) 66; (c) and (d) Answers will vary.

(2) Think of a question you have about Social Security. Find the answer from a reliable source. Record your work.

(a) Your question:

(b) The answer:

(c) Your source:

Answers will vary.




There is concern about the Social Security program because it is projected to eventually run out of money. Based on the current program, Social Security is projected to be able to pay all benefits through 2036. After that, it will only take in enough money to pay three-fourths of benefits.1 There is broad agreement that the program should be reformed now to avoid this future budget problem, but there is not consensus about how that should be done. The following are possible solutions that have been proposed:

• Increase the tax rate for paying into Social Security. • Increase the limit on wages so that people pay Social Security on wages above $90,000. • Decrease future benefits. • Increase the retirement age. • Change the program into a system of private retirement accounts in which individuals

invest their own money.

Projections about Social Security are based on many variables. One of these is life expectancy, which is a prediction about how long people live on average. It is important to understand that this is a mean. When students studied mean in Module 2, they learned that a data set can have values that can be far above or below the mean. If a large group of people has a life expectancy of 63 years, some people will die very young, even as infants, and some will live to be over 100.

This might lead you to ask if using a mean to measure life expectancy is very accurate. In Lesson 2.2.2, you saw data sets with home prices in which the mean was not a good representation of the data because there were a few very high home prices that made the mean much higher than the rest of the data. Prices of luxury homes can be 10 or 20 times higher than the price of average homes. This makes the mean a poor representation of the data. The range of life expectancy is different because it has more defined limits. As of 2011, the person known to have lived the longest was Jeanne Calment, who died at the age of 122 in 1997.2 This is an extremely high and rare age, but it is not that much higher relative to what you might consider an average in the range of 50–60 years. This indicates that the mean is a fairly accurate way to summarize the data. This is useful when making projections about a large population, which is why life expectancy statistics are used for projecting costs for Social Security. However, you should always remember that these types of statistics are not good predictors for individuals.

[Student Handout]

Specific Objectives


• linear equations can approximate nearly linear data.


1Retrieved from www.socialsecurity.gov/OACT/TRSUM/index.html 2Retrieved from http://en.wikipedia.org/wiki/Oldest_people




• find the equation of a line that estimates nearly linear data by calculating the rate of change and estimating the vertical intercept of the line.

• use approximate linear models to interpolate and extrapolate.

Problem Situation: Life Expectancy and Social Security

As you read in the OCE work, one proposal about reforming Social Security is to increase the retirement age. This raises concerns about fairness. No matter what the retirement age is, some people will pay into Social Security but die before retirement and never receive a benefit. This happens to more people when the retirement age is increased. In this lesson, you will examine the effects of raising the retirement age to 75. Specifically, you will answer the question of whether this change would have a greater impact on some groups than others.

To explore this question, you will use life expectancy data from the Center for Disease Control. Real data rarely fall on a straight line, but sometimes data show a definite trend. If the trend is close to linear, the data can be approximated by a linear model. This means that a linear model gives good estimates of what the data will be if the trend continues. A model can also be used to estimate values between data points. In this lesson, you will learn to create linear models from data.

Discuss the information from the OCE and the Problem Situation. Make sure students understand the information.


• Ask students to share the questions and answers they researched in the OCE. • Ask students to take a position on whether the Social Security retirement age should be raised

or whether other methods should be used to reform the system. Ask them to justify their reasons.

• Do you think the life expectancy is different for certain populations in the United States?




[Student Handout]

(1) The following data show the life expectancy of African American males in the United States at birth. Working with your teacher, find a linear model to approximate these data, letting L be the life expectancy at birth and y the year of birth measured in years after 1900.

The graphs in this lesson use the rescaled number of years on the horizontal axis rather than the calendar years. This choice was made to simplify the students’ work in this initial lesson. Point this out and tell students that published graphs are more likely to show the calendar years. They will see data in the OCE given in terms of calendar years.

Begin by having students use a straightedge to indicate where the best line to model the data lies. A clear straightedge works best so students can see points both above and below their lines. If you bring a box of dry spaghetti, each student can use a strand to represent a line.

You could then call a few students up to argue why their lines are the best. You will probably want to have a projection of the graph so that students can reference it in the discussion. Generally speaking, a line that works well should follow the general trend of the data. Since this line rises from left to right, a line with positive slope should be used. The slope should be around 1/3. You can have students adjust the slope of the projected line until the class has consensus on the best line. Record the slope.

One challenge when working with life expectancy is that the units of the slope are year/year for the slope. If that is causing confusion, suggest students write something like “year of life/calendar year.” For instance, African American men are gaining ____ years of life expectancy/calendar year.

The sample line should have about as many points above the line as below the line. Students usually readily agree a line has a better fit to the data if it has about as many points above as below instead of having lots above or below. You can slide the straightedge up and down without changing its slope until there is consensus in the class that you have nearly the best line. Have students estimate the vertical intercept. They may not notice the instruction “years after 1900.” You may wish to point this out so that it is (much) easier to find the vertical intercept.

Using (80, 64) and (30, 48) to get the rate and 40 as the vertical intercept, a possible model is L = 0.32y + 40. Keep in mind the answer may vary based on points and intercept selected.

0

10

20

30

40

50

60

70

80

0 20 40 60 80 100 120

Life

Exp

ecta

ncy

at B

irth

Years since 1900

African American Male Life Expectancy at Birth




[Student Handout]

(2) When will African American males have a life expectancy of 75 years at birth? When will these men begin collecting Social Security if the retirement age is raised to 75?

Seventy-five is a life expectancy. So, using the model 75 = 0.32y + 40 and solving for y gives 109 years. It may already have happened in 2009. (Some students may use the slope to go forward from 2006’s 70 years. Here they would need 5 years/0.32 = 16 years, so 16 years from 2006. However, the model might not go through any given data point.)

Therefore, using the model in Question 1, babies born in 2009 will have an average lifespan of 75 years. These children would start drawing Social Security in 2084. (Students should add 75 to whatever they have as an answer to the first part of Question 2.) Make sure students understand the meaning of the numbers. You may want to wait for further discussion about the context until they have worked with the other data.

Assign the other three data sets (Graphs 1–3) to different groups. Make sure that groups with the same data set work independently so that you have the opportunity to discuss different decisions about the best fit.

If you are short on time, skip Question 5.

[Student Handout]

(3) Your instructor will assign you data about the life expectancy of a population of Americans. Working with your group, find a linear model to approximate these data, letting L be the life expectancy at birth and y the year of birth measured in years after 1900. Be prepared to show the line you feel best represents the data graphically and how you found the equation of this line.

(4) When does your model predict your population will first have a life expectancy of 75 years at birth? When will these people begin collecting Social Security if the retirement age is raised to 75? Be prepared to share your results with the class.

(5) When does your model predict your population will first have a life expectancy of 90 years at birth?

Answers will vary. Example solutions:




African American Females

L = 0.38y + 40 is a reasonable line using the points (40, 54) and (90, 73).

• It predicts 1992 as when life expectancy was 75. So, 2067 is when these people will collect social security on average.

• It predicts 2031 as when life expectancy is 90. So, 2116 is when these people will collect social security on average.

White Female

L = 0.25y + 57 is a reasonable line using the points ( 30, 64) and (90, 79).


• It predicts 2032 as when life expectancy is 90. So, 2107 is when these people will collect social security on average.

White Male

L = 0.23y + 52 is a reasonable line using the points (30, 60) and (100, 76).


• It predicts 2165 as when life expectancy is 90. So, 2240 is when these people will collect social security on average. This answer does not seem realistic compared with white female and African American male answers unless there is a reason white males would continue to add life expectancy at a lower rate (a reason why the slope of their line is less.)

Have each group share the results by showing the line of best fit and their calculations one at a time. Or if there is not enough time for the groups to present all their work, have each group present one piece. Find examples of different lines to discuss.

Calculate the answer for Question 5 for the African American males (or have an early finishing group do it). Record the answers for Questions 4 and 5 for each population in a table. Make some minor corrections if students commit algebra errors. Question 5 is really included just as a way to force all students to extrapolate. You can tie it to the question at hand by noting that this is the first year when the typical (average) person will have collected Social Security for 15 years.

Revisit the initial question, “Is it fair to raise the Social Security retirement age?” Based on the new data, many opinions will not change. Some students might now argue that society should be color-blind, and the demographic data should not influence the decision. Another possible argument is that part of the difference in average ages is a higher infant mortality rate in African American communities. These babies pay no Social Security taxes. Others may argue that present and past discrimination causes the change to hit certain communities harder than others. This is obviously a loaded topic, so remind students that you are just looking at the data and it is up to them and the leaders they elect to make the final decision.




You also need to discuss the limitations of the model. A lot of things could change in the next 70 years. Life expectancy could increase dramatically if treatments for heart disease improve. Some people predict that life expectancies will begin to decrease because of the increased rate of obesity. Is it reasonable to think that life expectancies will continue to increase indefinitely? Social Security may change substantially. The United States itself may change in dramatic ways. There are dangers in extrapolating too far with any model. The limitations of models should always be considered. On the other hand, this is not just an issue that is projected into the future. The current data show that certain groups are already affected differently by the retirement age.

The data for these questions come from the Centers for Disease Control and Prevention (www.cdc.gov/nchs/data/nvsr/nvsr58/nvsr58_21.pdf).

Summarizing the Lesson

A line that estimates data should

• follow the general trend of the data. • not be too far from any one point. • have about as many points above and below the line.

A line that estimates data might not actually pass through any data points.

The description above is informal. Some students may wish to know that there are formal ways to do this. Students who take statistics classes or algebra classes will learn more about best-fit lines. Once you have a model, it can be used to interpolate (i.e., find data between existing input values). The outputs will be close to existing data points, but may not agree completely.

You can also extrapolate beyond the existing data points. Care should be given when extrapolating to whether the model will change in some way if the input values change substantially. For instance, life expectancy could change substantially if treatments for heart disease or cancer improve.




[Student Handout]

Making Connections



A model for data is a type of summary or approximation that is useful but has limitations.


• What methods of summarizing data have you used in this course? (graphical representations, measures of center)

• How is a model such as an equation similar to these? (is not exact, gives an approximation or an average value, is more accurate with data with less variability)

• In what other situations would models of life expectancy be useful? (planning for the future needs of a society such as care for the elderly, housing needs, pricing for insurance)

An algebraic model is a powerful tool.


• In what other ways have you used algebra? (formulas, generalizing rules, spreadsheets) • What is useful about algebraic tools? (simplify describing complex relationships, a way to

generalize, technology, predicting values)

[Student Handout]


(1) Gather data on the U.S. population from 1900 to 2010 from a reliable source(s). You should have data for the endpoints (1900 and 2010) and at least four other years spread between the endpoints.

(a) Record your source(s).

(b) Make a table with your data.

(c) Make a graph of your data.

(d) Create a linear model for your data.

(e) Use your model to project the population in 2050.

(f) Do you think your model is an accurate tool for prediction? Your answer can be based on whether a linear model is appropriate for the data or factors that you think might change historical trends. Give specific reasons for you answer.

Answers will vary.




Graph 1

Graph 2

Graph 3

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Lesson 4.1.4: Spring Scale Instructor Theme: Physical World Note: This is a replacement for the original 4.1.4


Specific Objectives

Students will understand that • linear models can created based on collected or given data

Students will be able to • write the equation for a linear model given two pairs of values

The primary focus of this lesson is to talk about finding a linear model through any two arbitrary points. This is a good time to emphasize the 𝑦 = 𝑚𝑥 + 𝑏 general equation of a line. The approach used for finding the intercept is the substitution approach: Once the slope is found, substitute a known ordered pair for x and y and solve for the intercept. This is a more intuitive approach than using point-slope form of a line. Problem Situation 1 asks the students to find the intercept given a slope and a point.

[Student Handout]

In Lesson 4.1.3, you created linear models based on data by approximating the vertical intercept. In this lesson, you will learn how to find the equation of a linear model given the rate of change and an input/output pair, then tie that to the ideas from Lesson 4.1.3.

Problem Situation 1 A custom t-shirt printing company charges a set amount per shirt and a setup charge. The general equation for the cost, C, of printing n shirts can be written as 𝐶 = 𝑚 ⋅ 𝑛 + 𝑏

1. Identify the meaning of the parameters m and b in the equation.

2. If the per-shirt charge is $6, what information in the equation does this provide? Update the general equation to use this information.

3. Suppose you paid $500 for an order of 70 custom-printed t-shirts. What information does this provide? Can you use this to solve for the unknown parameter?

4. How much would an order of 120 custom-printed t-shirts cost?

Answers 1. m is the slope, with units cost per shirt. b is the vertical intercept, the initial cost, in dollars 2. The $6 is the slope. C = 6n + b 3. This gives us a pair of values for n and C. 500 = 6*70 + b -> b = 80. C = 6n+80 4. C = 6*120 + 80 = $800



Depending on time, you may want to only do problem situation 2 or 3. If you have the equipment, you can do the experiment in class. Otherwise, you can use this video: http://youtu.be/Dfrl-EcU0kQ. After about 5 seconds of the video, you can pause and ask some questions:

• If we were to graph height above ground vs weight, what kind of shape do you think we’ll get? • Is the slope going to be positive or negative?

If you don't have audio, or want to write it out for students, the data is this:

50 g weight: 25.8 cm 100g weight: 21 cm 150g weight: 16.2 cm Mystery block: 24.5cm

[Student Handout]

Problem Situation 2 When weights are hung from a spring, the amount the spring stretches is linearly related to the amount of weight applied. This principle in physics is called Hooke’s Law.

1. Plot the data collected from the spring scale. What kind of relationship does this appear to be?

2. Find a model for the relationship.

3. Interpret the slope of the model. What is it telling us? What are the units?

4. What would the height of the scale be with a mass of 30 grams?

5. Use the model to predict the mass of the mystery block. Consider the accuracy of the height measurement to determine a margin of error for the prediction.



Answers

1. . The graph appears linear

2. Using the first two points (21-25.8)/50 = -0.096. H = -0.096w + b. Substituting in (100, 21) gives 21 = -0.096(100)+b 21 = -9.6 + b b = 30.6 H = -0.096w + 30.6, where H is height about ground, and w is the weight in grams.

3. The height decreases by 0.096 cm for each additional gram of weight

4. H = -0.096*30 + 30.6 = 27.6 cm

5. 24.5 = -0.096w + 30.6 -> 64 grams

Problem situation 3 uses almost-linear data from a graph, in which trying to extend the line to estimate the intercept will be hard, since the x-axis starts at 2. This problem situation also ties into writing inequalities.

[Student Handout]

Problem Situation 3 The chart to the right is a growth chart for boys, aged 2 to 15 years1. The different curves show different percentiles for growth; the top curve shows the 95th percentile, where 95% of boys are that height or shorter. The middle curve shows the 50th percentile (the median height), and the bottom curve shows the 5th percentile.

1. Identify two pairs of data from the median height curve, and use those to determine an approximate linear model for the curve.

1 Based on data from http://www.cdc.gov/growthcharts/clinical_charts.htm#Set1

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2. Use your model to predict the height of a 10 year old boy. How well does your model’s prediction agree with the data in the chart?

3. Use your model to predict the height of a 16 year old boy. Are you more or less confident in this prediction than the prior one?

4. Use your model to predict the height of a 30 year old boy. How accurate is this prediction likely to be?

5. Determine a reasonable domain for this model. That is, determine an interval of ages for which this model is likely to be reasonably accurate.

6. Ninety percent of children fall between the 5th percentile and the 95th percentile. We’ll call this the “typical” interval. Write an inequality for the “typical” interval of heights for a 9 year old boy.

7. If a lost child is found who is 145 cm tall, determine an interval of likely ages for the child. Answers 1. Using (3, 95) and (13, 156), slope = 6.1. H = 6.1t + b. Substituting (3,95), 95 = 6.1*3 + b, gives b = 76.7. H = 6.1t + 76.7

2. Model predicts H = 6.1*10 + 76.7 = 137.7 cm Graph value appears to be about 138, so the model agrees well with the actual value.

3. Model predicts H = 6.1*16 + 76.7 = 174.3 cm Less confident, since it's outside of domain of the graph

4. Model predicts H = 6.1*30 + 76.7 = 259.7 cm (that's 8.5 feet) This is very unlikely to be accurate; people don't continue growing at the same rate into adulthood

5. A reasonable domain would be 2 < t < 15. Possibly up to 16-18. 6. 124 < H < 144 7. 9 < t < 13

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Quantway Instructor’s Notes Lesson 4.1.5: Compounding Interest Makes Cents Theme: Personal Finance

The Carnegie Foundation for the Advancement of Teaching and The Charles A. Dana Center at the University of Texas at Austin




Exponential models Compound interest Not applicable Level 2



• understand the connection between percents and decimals. • be able to convert percents to decimals and vice versa. • be able to calculate a percent of a value. • be able to add, subtract, multiply, and divide decimals. • understand how certificates of deposit (CDs) and compound interest work, including the

vocabulary of principal, term, and compounding period.

Specific Objectives


• compounding is repeated multiplication by a compounding factor. • compounding is best expressed in terms of exponential growth, using exponential notation. • exponential growth models the compounding of interest on an initial investment.


• calculate the earnings on a principal investment with annual compound interest. • write a formula for annual compound interest. • compare and contrast linear and exponential models.


• Generalizing rules is an important algebraic skill.

Notes to Self






Suggested Timeline


10 minutes Check for understanding and work on Question 1 Class discussion, small groups


8 minutes Work on Questions 2 and 3 Small groups

15 minutes Discuss Questions 2 and 3 Class discussion and lecture



Special Notes

Students received information on certificates of deposit and compound interest in the previous OCE. Check for understanding of this material before students start working on the Problem Situation.

[Student Handout]

Specific Objectives


• compounding is repeated multiplication by a compounding factor. • compounding is best expressed in terms of exponential growth, using exponential

notation. • exponential growth models the compounding of interest on an initial investment.


• calculate the earnings on a principal investment with annual compound interest. • write a formula for annual compound interest. • compare and contrast linear and exponential models.

Problem Situation: The Five-year CD

Suppose you invest $1,000 principal into a certificate of deposit (CD) with a five-year term that pays a 2% annual percentage rate (APR) interest. The compounding period is one year.

(1) How much money will you have in your account after one year? Find a way to compute the answer using only one multiplication.

Make sure students understand why 1000+0.02*1000 can also be calculated as 1000*1.02.



(2) How much money will you have in your account at the end of the five-year term? (Be ready to explain your calculations.)

As students work in groups, prompt them to think of ways to simplify and generalize their work.


• Can you use the concepts from part 1 to simplify the calculations? (multiply by 1.02 instead of a two-step calculation)

• Are there any steps you are repeating over and over? Is there a way to simplify that?

When you move to group discussion, organize the work so that students see the repeated multiplication and connect this to exponents. You may need to provide instruction about how to do calculations with powers other than 2 on their calculators.

Answer:

Term Calculation Amount Accrued

1 year $1,000 + $1,000 x 0.02 = $1,000(1 + 0.02) = $1,000*1.02 $1,020.00

2 years ($1,000*1.02)(1.02) = $1,000(1.02)2 $1,040.40

3 years ($1,000*1.02*1.02)(1.02) = $1,000(1.02)3 $1,061.21

4 years ($1,000*1.02*1.02*1.02)(1.02) = $1,000(1.02)4 $1,082.43

5 years ($1,000*1.02*1.02*1.02*1.02)(1.02)= $1,000(1.02)5 $1,104.08

[Student Handout]

(3) Using patterns discovered in answering Question 1, develop a formula for the total amount accrued in a CD with annual compounding after n years, if the principal = $1,000 and the APR = 2%. Then use your formula to fill out the following table.

Answer: A = 1,000(1.02)n


10 years A = 1,000(1.02)10 $1,218.99

20 years A = 1,000(1.02)20 $1,485.95

50 years A = 1,000(1.02)50 $2,691.59

100 years A = 1,000(1.02)100 $7,244.65



[Student Handout]

(4) Plot your results from Question 3. Label the vertical axis appropriately.

Answer:

It may be interesting to also show them the graph for the first 5 years, to show that it's not always obvious from a graph over a short timeframe whether the growth is linear or exponential.

[Student Handout]

(5) Is your formula from Question 3 linear? Explain your reasoning.

In this discussion, build on the student ideas to introduce the term exponential model and cover the following concepts. Make sure students take careful notes as this is both discussion and direct instruction. Project a graph (sample given below) of the function and reference it during the discussion. You may want to show a linear example for contrast. An example is included at the end of this lesson.

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Compare and Contrast Linear to Exponential

• Constant rate of change versus increasing (in this case) rate of change. o Constant additive change versus constant multiplicative change.

• Forms of the equation and how that relates to the rate of change. • Both have vertical intercepts that can be identified from the equation and represent the

starting value in context.

Behavior of Exponential Growth

• Note that the growth is very small at first, but then it changes dramatically. • Doubling time is not a specific learning goal, but it is worth a brief explanation since

students may have heard references to it. • You may decide to discuss asymptotes if it seems appropriate for your students; it is not

an expectation of the course.

Note: You will discuss the answer to Question 5 as a part of the Making Connections section. If you do not have time, use the previous formula from Question 3 for Making Connections and ask students to do Question 5 for homework.



[Student Handout]

(6) Write a general formula that could be used to find the accrued amount (A) for a CD with annual compounding. Let P = the principal, r = the APR as a decimal, and n = number of years.

Answer: A = P(1 + r)n

[Student Handout]

Problem Situation 2: The Value of a CD

In the last problem, the CD was compounded annually. In this problem, we will extend that for compounding periods may be of various durations.

Suppose you invest $1,000 principal in a two-year CD, advertised with an annual percentage rate (APR) of 2.4%, where compounding occurs monthly.

(1) If the APR is 2.4% per year, how much is it per month? Use your answer to complete the following table. Record how you found the results in the middle column.

• Principal = $1,000 • APR = 2.4% • Term = two years • Compounding period = one month

Period Calculation Amount Accrued

1 month

2 months

3 months

6 months

12 months

24 months

Use your judgment about whether students need to discuss the calculations for each month before generalizing in Question 2. If students work through the calculations easily, allow them to go on.



Answers:

Period Calculation Amount Accrued

1 month A = $1,000 + $1,000 x (0.024/12) = $1,000 + $1,000 x 0.002 = $1,000(1 + 0.002) = $1,000(1.002)

$1,002.00

2 months A = $1,000(1 + 0.002)2 = $1,000(1.002)2 $1,004.00

3 months A = $1,000(1 + 0.002)3 = $1,000(1.002)3 $1,006.01

6 months A = $1,000(1 + 0.002)6 = $1,000(1.002)6 $1,012.06

12 months A = $1,000(1 + 0.002)12 = $1,000(1.002)12 $1,024.27

24 months A = $1,000(1 + 0.002)24 = $1,000(1.002)24 $1,049.12

[Student Handout]

(2) Write a general formula that can be used to calculate the value of any CD. Define your variables.

Answer: The standard formula is = +

1 nt

rA P

nwhere A = accumulated amount, P = principal,

r = rate, n = compounding period, and t = time in years.

Students may struggle with how to write the power. It is not expected that students will get the notation completely correct on their own, but they should recognize the general form and have some idea of what factors are involved.


• Think about the model from the previous problem situation. • How could you set up a calculation for 24 months using only the numbers in the starting

information (listed above the table). • Instead of thinking of the combination of nt, students might think in terms of another variable

that represents the number of compounding periods.

Once you have discussed students’ ideas, give them the standard formula. Note that while it looks more complicated than the exponential models in problem situation 1, it is really the same formula. The expression in parentheses includes the conversion of the annual rate to the rate for the compounding period. Connect this to the verbal formula, or introduce the formal version of y = Cax.

New value = Starting value x (1 + rate)Number of repeated multiplications



If you are concerned about time, you may skip Question 3 or handle it as a quick class discussion. Students will work with this concept more in the OCE.

[Student Handout]

(3) How does the compounding period affect the accumulated amount? Work in your group to investigate this question.

Answer: The more compounding periods, the higher the accumulated amount.

Based on previous experience, students should realize that they need to hold the other variables fixed to explore this question. If they do not, ask them to think back to previous investigations such as calculating the braking distance or blood alcohol content.

You may not need to discuss this result as it is fairly intuitive.

[Student Handout]

Making Connections



Generalizing rules is an important algebraic skill.


• How did you find the general rule (or equation) for these calculations? • Why is having a general rule or equation useful? • What are examples of other general rules you have used in this course? (geometric formulas,

braking distance and the blood alcohol content formula, definitions and rules like place value and order of operations)



[Student Handout]


(1) Find the current interest rates for three CDs with different terms. Give the following information:

(a) Your source

(b) The name of the institution that offers the CD

(c) The interest rate

(d) The term

(e) Any other information given (e.g., minimum deposit, special restrictions)

(2) Compare the return on a CD with annual compounding that pays 2% interest to one that pays 4% interest. Are the returns doubled? Does the comparison vary over time? Explain how you came to your conclusions.

Answers: (1) Answers will vary. (2) The returns for 4% are double the returns for 2% for the first year, but after that the returns for 4% increase more quickly than those for 2%.



Linear Example to Support Discussion

You have $1,000 in savings and add $20 each year.

Equation: A = 1,000 + 20n where A = amount ($) and n = number of years

Table of values with additive change demonstrated:


1 year 1,000 + 20 $1,020

2 years (1,000 + 20) + 20 = $1,000 + 2*20 $1,040

3 years (1,000 + 20 + 20) + 20 = $1,000 + 3*20 $1,060

4 years (1,000 + 20 + 20 + 20) +20 = $1,000 + 4*20 $1,080

5 years (1,000 + 20 + 20 + 20 +20) + 20 = $1,000 + 5*20

Graph

The linear-only graph is useful for demonstrating the additive growth. You can see that the amount increases by $100 every 5 years ($20 per year). The graph with both the linear and exponential can be used to compare the two models.


Lesson 4.1.6: Beyond Compounding Theme: Personal Finance





Exponential functions Depreciation & Growth Not applicable Level 2


• write a basic exponential model. • calculate values raised to powers using a calculator.


• the differences and similarities between exponential growth and decay.

Students will be able to • use the compound interest formula for different compounding periods. • write an exponential decay model.

Explicit Connections • Part of mathematics is categorizing relationships based on shared characteristics.

Note to Self One thing I want to do during this lesson …






Suggested Timeline


3 minutes Discuss compounding periods Class discussion

17 minutes Work on Questions 1 and 2 and discuss Small groups, class discussion



10 minutes Discuss Question 4 and decay models Class discussion


Special Notes Before students start work, discuss different compounding periods. Ask students which of the following should have a greater yield: annual, quarterly, or monthly compounding.

[Student Handout]

Specific Objectives

Students will understand • the differences and similarities between exponential growth and decay.

Students will be able to • use the compound interest formula for different compounding periods. • write an exponential decay model.

In this lesson, you will connect the exponential mathematics of compounding to related applications, such as automobile depreciation and spread of disease.

Problem Situation 1: Understanding Depreciation

Depreciation is a process of losing value, opposite to that of accruing interest. For example, new automobiles lose 15 to 20% of their value each year for the first few years you own them.

(1) Based on this fact, develop a depreciated-value (D) formula for a $26,000 automobile, based on 15% depreciation per year. Use the table below to guide your calculations.

Age of the Automobile Calculation Value

New $26,000

1 year old

2 years old

5 years old

t years old




Answers:

Age of the Automobile Calculation Value

New $26,000

1 year old D = 26,000(1 – 0.15) = 26,000(0.85) $22,100

2 years old D = 26,000(1 – 0.15)2 = 26,000(0.85)2 $18,78.50

5 years old D = 26,000(1 – 0.15)5 = 26,000(0.85)5 $11,536.34

t years old D = 26,000(1 – 0.15)t = 26,000(0.85)t 26,000(1 – 0.15)t or 26,000(0.85)t

Remind students of Lesson 4.1.5 if they are struggling with simplifying the factor to 0.85.

Project a graph of this function. Note the similarities and differences to the growth model. Introduce the terms growth and decay.

• Both have a vertical intercept that represents the starting value and are in the equation. • The growth model has a factor >1. Decay has a factor <1. You can either use the term factor or

call it the base of the exponent. • Help students see that the compound interest formula (exponential growth) and the decay

model have the same basic form. You can use the verbal formula given above (adjusting for subtracting the rate from 1 for decay) or the more formal formula of y = Cax. You might want to write the general form and examples from the problems on the board. Show how each has a starting value (C) that is multiplied by a factor raised to a variable power. Especially note that the compound interest formula looks different, but the factor can be simplified down to one number.




• In growth, rate of change starts slow and increases. In decay situations, the rate of change initially represents a steep negative decline but becomes less steep (but still negative) as time goes on.

• Both are based on multiplicative (or relative) change.

[Student Handout] Problem Situation 2: A Spreading Disease How quickly a disease spreads is very important to containing an outbreak.

Suppose an outbreak of a flu strain has been detected. On February 1, there were 120 reported cases. A week later, there were 160 reported cases.

(2) Find the absolute and relative change between the two reported values.

Answers: Absolute change: 160 - 120 = 40 Relative change: (160 - 120)/120 = 0.25 = 33.3%

[Student Handout]

(3) Let C represent the number of cases after t weeks. Find a linear model based on the data given. Use that model to predict the number of cases after 10 weeks.

Answers: C = 40t + 120 Prediction for 10 weeks: C = 40*10 + 120 = 520 cases

[Student Handout]

(4) The relative change you found in question 2 is the percent increase, or percent growth rate. Use it to find an exponential model based on the data given. Use that model to predict the number of cases after 10 weeks.

Answers: C = 120(1.333)t Prediction for 10 weeks: C = 120(1.333)10 = 2126 cases

[Student Handout]

(5) Which model is more likely to be accurate? What are the limitations of the models?

Lead a discussion with students. Discuss why exponential might be more appropriate by sharing the idea of 1 person infecting 2, then those 2 each infecting 2 others, etc. For limitations, discuss things like reaching saturation (everyone's gotten infected) or people getting immunized.




It may be helpful to show a graph of the two models:

[Student Handout]

Making Connections


Making Connections: Main Ideas to Highlight Part of mathematics is categorizing relationships based on shared characteristics.

There probably is not time for a discussion of this idea. Point out to students that they have now seen two general types of models: linear and exponential. You might tell them that you call these families because they share certain characteristics. Within each family, there are subgroups like growth and decay. Thinking about models in terms of their families helps you generalize your understanding of individual models. You can look at an equation and know it has certain characteristics without having to do any calculations or graphing.

[Student Handout]


(1) Your assignment is to find examples of exponential growth and decay that go beyond the contexts you saw in this lesson. You may find your own references or use one of the sources below.

(a) Identify your source.

(b) Explain the context.

(c) Explain how the situation is related to exponential growth or decay.




Population Growth • www.learner.org/interactives/dailymath/population.html • www.ecofuture.org/pop/facts/exponential70.html • www.worldpopulationbalance.org/exponential-growth-tutorial/bacteria-exponential-

growth.html

Carbon 14 Dating • www.physlink.com/education/askexperts/ae403.cfm

Radioactive Decay • www.colorado.edu/physics/2000/isotopes/radioactive_decay3.html

Compound Interest with Debt and the Rule of 72 • www.streetdirectory.com/travel_guide/11087/debts_loans/the_wonders_of_compound_

interest.html • www.allbusiness.com/banking-finance/personal-finance-personal-debt/7067882-1.html

Lesson 4.1.7: Finding Time Instructor Theme: Physical World Note: This is not the original 4.1.7.


Specific Objectives

Students will understand • The relationship between a logarithm and an exponent

Students will be able to • Use logarithms to solve an exponential equation • Read and place values on a logarithmic scale

This lesson focuses on using logarithms to solve problems. The PNL will have introduced the basic idea of solving a 𝑏𝑥 = 𝑐 type equation using logs, but you may need to spend some time ensuring the students understood the process. The goal here is not for students to have a deep understanding of logs, just to understand that they un-do an exponential, and how to use them to solve the equation. Problem situation 2 introduces the idea of log scaling, which is useful in Astronomy and Chemistry.

[Student Handout] In Lesson 4.1.5 and Lesson 4.1.6, you created increasing exponential models for compound interest and an exponential decay model for depreciation. In this lesson, you will create exponential models for population growth, and use logarithms to help predict when the population will reach a value. Additionally you will explore how logarithms can simplify representing very large and very small numbers.

Problem Situation 1: When ancient artifacts are found, scientists determine their age using a technique called radioactive dating. Carbon 14 is a radioactive isotope that is in living organism. All living organisms have approximately the same ratio of carbon 14 to carbon 12. After dying, the carbon 14 decays to carbon 12, changing the ratio of carbon 14 to carbon 12. The fraction of the original carbon 14 remaining in an organism after t thousand years is modeled by 𝐴 = 0.8861𝑡.

1. Complete the table for the amount of carbon 14 remaining after the given numbers of years.

Years 1000 2000 3000 4000 5000 6000 7000 C-14 remaining

2. Textbooks will common report that the “half-life” of carbon 14 is 5730 years. Based on your calculations, what do you think “half-life” means?

3. Just using the idea of half-life, complete the table

Years 0 5730 11460 17190 22920 C-14 remaining 100%



4. An artifact is found, and scientists determine that it contains 14% of its original carbon 14.

a. Estimate the age of the artifact, using the results from #3.

b. Determine the approximate age of the artifact using algebra.

Answers: 1.

Years 1000 2000 3000 4000 5000 6000 7000 C-14 remaining 1 0.886 0.785 0.696 0.617 0.547 0.485

2. After 5730 years, half the original amount is left 3.

Years 0 5730 11460 17190 22920 C-14 remaining 100% 50% 25% 12.5% 6.25%

4. a) about 15000 years, +- 2000 b) 0.14 = 0.8861𝑡, so log(0.14) = 𝑡 log(0.8861), so t is about 16.259 thousand years, or 16,259 years

[Student Handout]

Problem Situation 2: When numbers are very small or very large, or a set of values varies greatly in size, it can be hard to represent those values. Consider the distance of the planets of our solar system from the sun:

Planet Distance (millions of km) Mercury 58 Venus 108 Earth 150 Mars 228 Jupiter 779 Saturn 1430 Uranus 2880 Neptune 4500

Placed on a linear scale – one with equally spaced values – these values get bunched up.



However, representing each value as a power of 10, and using a scale spaced by powers of 10, the values are more reasonably spaced.

1. The scale above shows the dwarf planet Pluto. How many kilometers from the sun is Pluto?

2. Ida is a large asteroid in asteroid belt, approximately 429 million kilometers from the sun. Add Ida to the scale above.

Answers:

1. The value for Pluto looks like it's about 103.72 = about 5248 million km

2. 429 = 10n, then n = log(429) = 2.63. So it would be placed at 102.63

Mercury Venus Neptune

101.5

102 10

2.5 10

3

103.5

Earth Mars Jupiter Saturn Uranus

104

Pluto Ida

Mercury Venus Neptune

101.5

102 10

2.5 10

3

103.5

Earth Mars Jupiter Saturn Uranus

104

Pluto

0 500 1000 1500 2000 2500 3000 3500 4000 4500

Mercury Venus Earth Mars

Jupiter Saturn Uranus Neptune

distance

quantway instructor’s notes lesson 1.1.1: …...productive persistence focus level of productive...

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