South Georgia College
Quantway™ I
Module 2
Student
+++++
This Module is part of QUANTWAY™, A Pathway Through College-Level Quantitative Reasoning, which is a product of a Carnegie Networked Improvement Community that seeks to advance student success. The original version of this work, version 1.0, was created by The Charles A. Dana Center at The University of Texas at Austin under sponsorship of the Carnegie Foundation for the Advancement of Teaching. This version and all subsequent versions, result from the continuous improvement efforts of the Carnegie Networked Improvement Community. The network brings together community college faculty and staff, designers, researchers and developers. It is a research and development community that seeks to harvest the wisdom of its diverse participants through systematic and disciplined inquiry to improve developmental mathematics instruction. For more information on the QuantwayTM Networked Improvement Community, please visit .
+++++
Quantway™ is a trademark of the Carnegie Foundation for the Advancement of Teaching. It may be retained on any identical copies of this Work to indicate its origin. If you make any changes in the Work, as permitted under the license [CC BY NC], you must remove the service mark, while retaining the acknowledgment of origin and authorship. Any use of Carnegie’s trademarks or service marks other than on identical copies of this Work requires the prior written consent of the Carnegie Foundation.
This work is licensed under a Creative Commons Attribution-NonCommercial 3.0 Unported License. (CC BY-NC)
[pic]
Table of Contents
Module 2
|Lesson |Title |Theme |Page |
|2.1 |How Crowded Are We? |Citizenship | |
| | Student Handout | |5 |
| | Out-of-Class Experience | |9 |
|2.2 |1185.3 Is a Crowd |Citizenship | |
| | Student Handout | |15 |
| | Out-of-Class Experience | |20 |
| | Student Data Sheet | |26 |
|2.3 |Measuring Population Change |Citizenship | |
| | Student Handout | |27 |
| | Out-of-Class Experience | |31 |
|2.4 |Picturing Data with Graphics |Citizenship | |
| | Student Handout | |37 |
| | Out-of-Class Experience | |43 |
|2.5 |Risk Reduction |Medical Literacy | |
| | Student Handout | |52 |
| | Out-of-Class Experience | |60 |
|2.6 |What Is Average? |Personal Finance | |
| | Student Handout | |67 |
| | Out-of-Class Experience | |72 |
|2.7 |Making Good Decisions with Good Statistics |Personal Finance, Citizenship| |
| | Student Handout |Citizenship |79 |
| | Out-of-Class Experience | |84 |
|2.8 |Has the Minimum Wage Kept Up? |Citizenship | |
| | Student Handout | |93 |
| | Out-of-Class Experience | |98 |
|2.9 |How Does Your City Stack Up? |Personal Finance, | |
| | Student Handout |Citizenship |105 |
| | Sorting Data in Excel | |110 |
| | Support Spreadsheet | |113 |
|Culminating |Student Overview | |114 |
|Activity |Part 1 | |117 |
| |Part 2 | |119 |
| |Student Rubric | |124 |
| |Review | |125 |
| | | | |
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 1.2, 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:
[pic]
You call this equation a proportion because the equation shows that two ratios are equal. You can also state the relationship in words:
One person per 4 square feet is proportional (equal) to 0.25 person per (1) square foot.
How would the population density change if 1 billion people each stood on a 2-foot by 2-foot square?
The following questions will help you understand how to calculate population density for different areas. You will start by doing an activity with your class.
(1) Calculate the population density based on the small rectangle. Be sure to include units.
(2) Calculate the population density based on the large rectangle. Be sure to include units.
(3) Imagine that a billion people stand on adjacent 2-foot by 2-foot squares. Calculate the population density per square mile. Be ready to explain your reasoning after working with your group members. (1 mile = 5,280 feet)
Problem Situation 2: Making Comparisons with Population Density
How crowded is China, compared to the United States?
(4) In 2010, in the United States, approximately 309,975,000 people occupied 3,717,000 square miles of land. In China, approximately 1,339,190,000 people lived on 3,705,000 square miles of land. Use this information to answer the following questions.
(a) A student carefully calculates the population densities of China and the United States. He decides that China is less dense than the United States. Using your estimation skills, decide if you think this student’s calculation is correct.
(b) At a lecture, you hear someone claim that, in terms of population, China is more than four times as dense as the United States. Using your estimation skills, decide if you think this statement is correct.
(c) Calculate more precisely the densities (per square mile) of the Chinese and U.S. populations. Based on your calculation, how many times more dense is the more crowded population? Be ready to share your calculations during the class discussion.
Making Connections
Record the important mathematical ideas from the discussion.
Further Applications
(1) The out-of-class experience contains information about the population of Alaska. Explain how the statements “Anchorage has more than 40% of the Alaskan population” and “Ketchikan has the most dense population” might both be correct.
Student Notes
Making Connections to the Lesson
(1) Which of the following was one of the main mathematical ideas of the lesson? The questions refer to the following quantities:
|50 people/mi2 |20 ft/5 sec |34% |
(i) All three of the figures are ratios.
(ii) The first two figures are ratios.
(iii) The second figure could be written as 4 ft/sec.
(iv) The first figure is a population density.
In Module 2, you will be asked to write statements about connections among the mathematical ideas in lessons. For this first assignment in Module 2, you are given statements to choose from. In future assignments, you may want to refer back to this as an example of how to write about connections.
(2) Refer back to Lesson 1.6. Choose the statement that explains how the mathematical ideas in
Question 8 connect to the work in Lesson 2.1.
(i) In Lesson 1.6, you used the water footprints of the United States. and of China. In Lesson 2.1, you used the population density of the United States and China. Both lessons compared the United States and China.
(ii) In Lesson 1.6, you had to calculate the water footprint per person in the United States. This is like splitting up all the water equally among the people. In Lesson 2.1, you calculated the population density of the United States. This is like splitting up all the people equally over an area of land. These are both ratios.
(iii) In Lesson 1.6, you used the population of a country to find the water footprint per person. In Lesson 2.1, you used the population of a country to find a population density. Both lessons used population for calculations.
Developing Skills and Understanding
(3) Use the picture below to answer the questions.
[pic]
(a) What is the density of the stars in Rectangle B?
(b) What is the density of the stars in Rectangle A (Note: The stars in Rectangle B are also in Rectangle A). Round to the nearest hundredth of a star per square foot.
(c) Suppose three new stars were added in the gray part of Rectangle A. Which of the following statements would be correct?
(i) The density of Rectangle A would increase. The density of Rectangle B would stay the same.
(ii) The density of Rectangle A would decrease. The density of Rectangle B would stay the same.
(iii) The density of Rectangle A would increase. The density of Rectangle B would increase.
(iv) The density of Rectangle A would decrease. The density of Rectangle B would decrease.
(v) The density of Rectangle A would stay the same. The density of Rectangle B would increase.
(4) Wikipedia states the following about Anchorage, Alaska: “The city constitutes more than 40 percent of the state’s total population.”[1]
(a) Calculate the population density for Anchorage, based on a 2010 population of 291,826 people living on 1,961.1 square miles. Round to the nearest person per square mile.
(b) Wikipedia also says that the small Alaskan town of Ketchikan has the densest population in Alaska. Ketchikan had a population of 7,368 in 2010 and an area of 4.1 square miles. Calculate the population density of Ketchikan. Round to the nearest person per square mile.
(5) The following information comes from the lesson:
| |Population |Land Area (sq. miles) |
|United States |309,975,000 |3,717,000 |
|China |1,339,190,000 |3,705,000 |
In India, about 1,184,639,000 people live on 1,269,000 square miles of land.[2] Which of the following statements is false?
(i) The population density of India is approximately three times that of China.
(ii) The population of India is approximately 11 times the population density of the United States.
(iii) The population densities of these three countries ranked from smallest to largest are United States, India, and China.
(iv) The population density of the United States is approximately 83.4 people per square mile. That is smaller than the population densities for China and India.
(6) Which of the following population densities are equivalent to a density of 20 people/mi2? There may be more than one correct answer.
(i) 1 person/0.2 mi2
(ii) 1 person/0.05 mi2
(iii) 1 person/0.5 mi2
(iv) 200 people/2 mi2
(v) 100 people/5 mi2
(7) One way to measure a country’s economy is per capita gross domestic product, or per capita GDP. This is the value of all the products and services produced in a country over the course of a year divided by its population.
(a) According to the CIA’s The World Factbook, in 2010, the United States had a per capita GDP of $47,400. If the population was about 309 million, which of the following is a reasonable estimate for the GDP of the United States?
(i) $60 trillion
(ii) $60 billion
(iii) $15 billion
(iv) $15 trillion
(v) $2.3 trillion
(b) Also according to The World Factbook, in 2010, China had a per capita GDP of $7,400 and a population of around 1,337,000,000. Which of the following is a reasonable estimate for the GDP of China?
(i) $60 trillion
(ii) $60 billion
(iii) $15 billion
(iv) $2.3 trillion
(v) None of the above
Making Connections Across the Course
(8) The population of Nebraska is 1,826,341, and its population density is 23.8 people per mi2. The population of New Hampshire is 1,316,470, and its population density is 146.8 people per mi2. Which of the following statements is a valid interpretation of this information?
(i) Nebraska is approximately six times more densely populated than New Hampshire.
(ii) The population of Nebraska is approximately 72% of the population of New Hampshire.
(iii) New Hampshire is approximately six times more densely populated than Nebraska.
(iv) The population of New Hampshire is approximately 139% of the population of Nebraska.
(9) Cholera, a serious intestinal disease, broke out in London in the mid-19th century. People at that time believed cholera was caused by bad air. A physician named John Snow discovered that 61 victims either used the water pump on Broad Street or lived nearby. His research became the basis for the theory that germs cause disease. In the case of cholera, the germs are transmitted by polluted water.
A vaccine for cholera was developed in the late 1800s. A vaccine is a drug that helps a person become immune to a disease. Scientists studied 818 people to determine the effects of a cholera vaccine. The study lasted from 1894 to 1896.
| |Infected |Not Infected |
|Vaccinated |3 |276 |
|Not vaccinated |66 |473 |
Use the information in the table to complete the following four statements. Round to the nearest tenth of a percent.
(a) ______% of those vaccinated were infected
(b) ______% of those not vaccinated were infected
(c) ______% of those infected were vaccinated
(d) ______% of those infected were not vaccinated
(e) Which of the following statements are correct based on the information in the table? There may be more than one correct answer.
(i) A person who was not vaccinated was 12 times more likely to get cholera than someone who got the vaccine.
(ii) A person who was not vaccinated was four times more likely to get cholera than someone who got the vaccine.
(iii) If a person was infected, it was four times more likely that (s)he was not vaccinated rather than vaccinated.
(iv) If a person was infected, it was more than 20 times more likely that (s)he was not vaccinated rather than vaccinated.
Preparing for the Next Lesson (2.2)
(10) The following problems have to do with multiplying and dividing by powers of 10. Look for patterns and ways to find the answers mentally without a calculator or writing the problem down. Check your answers with a calculator if you wish.
(a) 0.32 x 10
(b) 3.2 x 10
(c) 32 x 10
(d) 32 x 100
(e) 51,000 x 10,000
(f) 900 x 104
(g) 1.3 x 107
(h) 0.32 ÷ 10
(i) 3.2 ÷ 10
(j) 3,200,000 ÷ 10
(k) 5,500,000 ÷ 1,000
(l) 83,000,000 ÷ 10,000,000
(m) 67 ÷ 104
(11) You multiply 58,000,000,000 x 10,000 and your display reads: 5.8 E14. Which of the following represent(s) the same number as the number displayed on your calculator? There may be more than one correct answer.
(i) 5.8 x 14
(ii) 5.8 x 1014
(iii) 58 x 1014
(iv) 580,000,000,000,000
(v) 5,800,000,000,000,000
The following information will be used in Lesson 2.2. The states of the United States vary greatly in both size and population. Some states, especially on the east coast, are small and mostly urban, meaning that most people live in cities. Other states in the west and plains are larger and more rural, with people living in small towns or in the countryside. Population density is one way to measure how crowded a state is.
If an area is densely populated, it will need more services such as schools and hospitals. Some states, such as Washington, use the population density of counties to classify them as urban or rural in state law. This can affect whether residents qualify for certain kinds of assistance programs.[3]
(12) Use the information given below to calculate the population density for each of the states listed. Round to the nearest tenth. A State Population Density Table will be posted with this lesson. Record each state in the table with its population density. Bring this to your next class.
|State |Land Area |Population |Answers |
| |Square Miles | | |
|Alaska |571,951 |710,231 | |
|Idaho |82,747 |1,567,582 | |
|Kentucky |39,728 |4,339,367 | |
|Louisiana |43,562 |4,533,372 | |
|Nebraska |76,872 |1,826,341 | |
|New Hampshire |8,968 |1,316,470 | |
|New Mexico |121,356 |2,059,179 | |
|South Dakota |75,885 |814,180 | |
|Washington |66,544 |6,724,540 | |
|Wisconsin |54,310 |5,686,986 | |
(13) You will be expected to do the following things for the next class. Rate how confident you are on a scale of 1–5 (1 = not confident and 5 = very confident).
Before beginning Lesson 2.2, you should understand the concepts and demonstrate the skills listed below:
|Skill or Concept: I can … |Rating from 1 to 5 |
|Multiply by 10 by moving a decimal one place to the right and divide by 10 by moving | |
|a decimal one place to the left. | |
|Interpret an answer given in scientific notation on their calculator. | |
|Calculate and interpret population density. | |
Specific Objectives
Students will understand
the concept of population density as a ratio.
what is meant by proportional or change based on a constant ratio.
Students will be able to
estimate between which two powers of 10 a quotient of large numbers lies.
calculate a unit rate.
solve a proportion by first finding a unit rate and then multiplying appropriately.
Problem Situation: Estimating Population Densities
You will compare the populations of different states and explore how population density affects a states’ representation in the U.S. Congress. You calculated population densities of some states in your out-of-class experience. Now, you will develop strategies for estimating population densities.
(1) Check your answers from the previous out-of-class experience with your group. Now discuss strategies you can use to estimate the population density of the states without using a calculator. Use your strategies to divide the states into the categories shown in the table below.
|Density > 1,000 people/mi2 |100–1,000 people/mi2 |10–100 people/mi2 |Density < 10 people/mi2 |
| |Washington |South Dakota |Alaska |
| |Louisiana |New Mexico | |
| |Wisconsin |Idaho | |
| |Kentucky |Nebraska | |
| |New Hampshire | | |
| | | | |
| | | | |
| | | | |
| | | | |
| | | | |
| | | | |
| | | | |
| | | | |
| | | | |
| | | | |
| | | | |
| | | | |
| | | | |
| | | | |
| | | | |
| | | | |
| | | | |
| | | | |
(2) Find which state has the greatest population density. What is that population density? Round to the nearest tenth.
(3) Find which state has the least population density. What is that population density? Round to the nearest tenth.
(4) If your campus had the same population density as the state with the greatest population density, how many people would be on campus? What if your campus had the same population density as the state with the least population density?
(5) Most of the world outside the United States uses the metric system of measurement, so it is often useful to be able to make comparisons between the American system and the metric system. Bangladesh has a population density of 1,127 people/square kilometer. (Note: 1 kilometer =
0.62 mile[4])
(a) If you converted the density of Bangladesh to square miles, would the measure be larger or smaller than 1,127? Explain your reasoning.
(b) Which of the following statements is the most accurate description of the relationship between a square kilometer and a square mile?
(i) A square kilometer is about one-sixth of a square mile.
(ii) A square kilometer is about two-thirds of a square mile.
(iii) A square kilometer is about one-third of a square mile.
(iv) A square kilometer is about six-tenths of a square mile.
(c) How many people would be on your campus if the population density were the same as Bangladesh?
Making Connections
Record the important mathematical ideas from the discussion.
Further Applications
(1) The OCE assignments for Lessons 1.8 and 1.9 had examples of people collecting data to answer questions. In Lesson 1.8, the example was about a teacher collecting data on student grades, and in Lesson 1.9, 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 1.8:
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
Student Notes
Making Connections to the Lesson
(1) Which of the following was one of the main mathematical ideas of the lesson?
(i) Population density measures how crowded a state is.
Example: 2.3 people per square mile is less dense than 8.7 people per square mile.
(ii) You can find a new value that is changing by a constant rate by adding a ratio.
Example: A car’s gas mileage is 20 miles/gallon. How far can it drive on 5 gallons of gasoline?
20 mi/gal + 5 = 25 miles.
(iii) A percent is a ratio compared to 1.
Example: 12% increase in population means that the number of people increases by 12 for every 1 person in the original population.
(iv) You can find a new value that is changing by a constant rate by multiplying by a ratio.
Example: A person’s wage is $10.35/hour. How much does the person earn in 40 hours? $10.35/hour x 40 hours = $414.
(2) Explain a connection between a concept in this lesson and at least one of the following lessons: 1.3, 1.6, 1.7. You can use one of the sentence stems given below if you wish. You can also refer back to Module 1 lessons as examples.
Question number________ in Lesson 2.2 connects to Question number______ in Lesson _____ because ________________.
The idea of ___________ in Lesson 2.2 connects to Lesson _________. An example of the connections is ________.
Developing Skills and Understanding
(3) In the lesson, you used ratios in the form of population densities. A population density is a ratio because it is a comparison of two measures: Number of people per number of square miles. Which of the following are ratios? There may be more than one correct answer.
(i) 252 miles
(ii) 67 hours
(iii) 10 miles/hour
(iv) 5 lb/$3
(v) $98
(4) Calculate the gas mileage of a car that drives 283 miles on 12.3 gallons of gas. Round to the nearest tenth of a mile/gallon.
(5) A car drives 630 miles on 35 gallons of gas. How far can it drive on 12 gallons?
(6) A jar holds 128 fluid ounces of juice. The label says the jar has 16 servings. How many fluid ounces are needed for 80 servings?
(7) According to the oil company BP, in 2010, the United States used 19,148,000 barrels of oil a day, and worldwide, people used around 87,382,000 barrels of oil per day.[5] This includes oil used for (among other things) fuel and manufacturing.
(a) If there were 309 million people in the United States in 2010, what was the daily consumption rate per person in the United States? Round to the nearest hundredth of a barrel.
(b) If there were 6.89 billion people in the world in 2010, which of the following statements would be correct?
(i) The U.S. rate of oil consumption per person was about five times the world rate.
(ii) About half the people in the world lived in the United States.
(iii) If the world used oil at the same rate as the United States, it would have used about 93,450,000 barrels of oils per day.
(iv) About one-fifth of the oil used in the entire world in one day is used in the United States.
(c) There are 42 gallons in a barrel of oil. Which of the following statements is true?
(i) The American rate of oil consumption is 5 more gallons of oil per day than the world rate.
(ii) The American rate of oil consumption is 7,452,380 gallons of oil per day.
(iii) If the world used oil at the same rate as the United States, it would use about 426,957,000 gallons of oil per day.
(iv) The average American is responsible for about 2.5 gallons of oil use per day.
(8) Approximately 6.9 billion people now inhabit the earth. The surface area of the earth is
510,065,600 km2.
(a) What is the surface area of the earth in mi2? Round to the nearest million square miles. Hint: If
1 km = 0.6214 mi, then 1 km2 = how many mi2?
(i) 197,000,000 mi2
(ii) 317,000,000 mi2
(iii) 122,000,000 mi2
(iv) 82,000,000 mi2
(b) The surface area above includes both land and water. Approximately 139 million square miles of the earth’s surface area is water. Using your answer from Part (a), determine what percentage of the surface area is land. Round to the nearest tenth of a percent.
(c) Approximately 1/3 of the land is uninhabitable, meaning people cannot live on it. How much land on the earth is inhabitable (can be lived on)? Round to the nearest million square miles.[6]
(i) 106,000,000 mi2
(ii) 54,000,000 mi2
(iii) 39,000,000 mi2
(iv) 27,000,000 mi2
(d) Estimate the population density of the earth in people per square mile of habitable land. Round to the nearest person per square mile.
Making Connections Across the Course
(9) People often confuse the words million, billion, and trillion when speaking. An estimate can help you decide if the speaker uses the correct word. Consider this situation: A speaker says, “The U.S. federal debt is $14 billion dollars. That’s over $45,000 for every person in the country.”
Select the correct statement from the choices below. Note: When you say the numbers are consistent, you mean that they make sense in relationship to each other.
(i) The two numbers in the statement are consistent with each other.
(ii) The two numbers in the statement are not consistent. If the debt is $45,000 per person, the total debt must be $14 million.
(iii) The two numbers in the statement are not consistent. If the debt is $45,000 per person, the total debt must be $14 trillion.
(10) Terrence is very careful about tracking his gas mileage. Every time he fills his gas tank, he records how much gas he buys and the number of miles he has driven. He puts this information into a spreadsheet so he can easily calculate his gas mileage.
(a) Select the formula that would calculate Terrence’s gas mileage.
(i) = (A2 + A3 + A4 + A5) / (B2 + B3 + B4 + B5)
(ii) = A2 + A3 + A4 + A5 / B2 + B3 + B4 + B5
(iii) = (B2 + B3 + B4 + B5) / (A2 + A3 + A4 + A5)
(iv) = B2 + B3 + B4 + B5 / A2 + A3 + A4 + A5
(b) Terrence is planning a long road trip of about 1,000 miles. The average price of gas is $3.85/gallon. Based on the data in the spreadsheet, estimate how much he should budget
for gas. Round to the nearest dollar.
(11) Consider the reflect phase reading at the end of the last module.
(a) Give an example of two internal factors that contributed to how well you did on your last exam. Remember that these factors can have a positive or a negative contribution.
(b) In the plan phase preparing for your next exam, what would you do differently? The plan phase reading is in OCE 1.4.
In many of the future lessons, you will need to work with negative numbers. Negative numbers arise naturally when one computes the difference between two quantities. If the difference is negative, you not only know the difference between the two quantities, you automatically know which value was the larger of the two!
(12) The following table was created to compare two cell phone plans, Plan A and Plan B. The monthly bill for each plan was computed based on the number of minutes used that month. While the actual monthly bills are not given, their differences are given in the last column.
|Minutes |Plan A |Plan B |Plan B – Plan A |
|50 |* |* |–7.5 |
|100 |* |* |–5 |
|150 |* |* |–2.5 |
|200 |* |* |0 |
|250 |* |* |2.5 |
|300 |* |* |5 |
|350 |* |* |7.5 |
|400 |* |* |10 |
(a) The first entry in the last column is –7.5. Which of the following statements explains what this tells you about Plan A and Plan B?
(i) Plan A costs $7.50 more than Plan B for someone who uses 50 minutes.
(ii) Plan B costs $7.50 more than Plan A for someone who uses 50 minutes.
(iii) The Plan A customer used the phone for 7½ minutes less than the Plan B customer.
(iv) The Plan B customer used the phone for 7½ minutes less than the Plan A customer.
(b) Further down the last column is the entry 7.5. What does this tell you about Plan A and Plan B? Write your answer as a complete sentence using the Writing Principle from Module 1.
(c) When are the two plans equal? Write your answer as a complete sentence using the Writing Principle from Module 1.
Preparing for the Next Lesson (2.3)
(13) Calculate the following values.
(a) What percent of 20 is 2?
(b) 28 is what percent of 80?
(c) What percent is 18/24?
(14) A school principal of 810 students needs to determine what percent of students passed and did not pass a statewide examination. Round to the nearest percent.
(a) If 550 students passed the exam, what percent passed the test?
(b) What percent did not pass the test?
(15) A laptop computer that you want to purchase was originally priced at $1,225. You will receive a 20% student discount, and the sales tax rate is 8%. How much money will you pay for the laptop?
(i) $245
(ii) $1,058.40
(iii) $980
(iv) $1,323
(16) You will be expected to do the following things for the next class. Rate how confident you are on a scale of 1–5 (1 = not confident and 5 = very confident).
Before beginning Lesson 2.3 you should understand the concepts and demonstrate the skills listed below:
|Skill or Concept: I can … |Rating from 1 to 5 |
|Find a percent of a number. | |
(17) Self-Regulated Learning: Reflect
How much time and effort per week is this course taking? Is it what you expected? Is there anything you need to adjust in your weekly schedule to make sure you are successful?
|State |Land Area (Square Miles) |2010 Population |Population Density (People/mi2) |
|Alabama |50,744 |4,779,736 | |
|Alaska |571,951 |710,231 | |
|Arizona |113,635 |6,392,017 | |
|Arkansas |52,068 |2,915,918 | |
|California |155,959 |37,253,956 | |
|Colorado |103,718 |5,029,196 | |
|Connecticut |4,845 |3,574,097 | |
|Delaware |1,954 |900,877 | |
|District of Columbia |61 |601,723 | |
|Florida |53,927 |18,801,310 | |
|Georgia |57,906 |9,687,653 | |
|Hawaii |6,423 |1,360,301 | |
|Idaho |82,747 |1,567,582 | |
|Illinois |55,584 |12,830,632 | |
|Indiana |35,867 |6,483,802 | |
|Iowa |55,869 |3,046,355 | |
|Kansas |81,815 |2,853,118 | |
|Kentucky |39,728 |4,339,367 | |
|Louisiana |43,562 |4,533,372 | |
|Maine |30,862 |1,328,361 | |
|Maryland |9,774 |5,773,552 | |
|Massachusetts |7,840 |6,547,629 | |
|Michigan |56,804 |9,883,640 | |
|Minnesota |79,610 |5,303,925 | |
|Mississippi |46,907 |2,967,297 | |
|Missouri |68,886 |5,988,927 | |
|Montana |145,552 |989,415 | |
|Nebraska |76,872 |1,826,341 | |
|Nevada |109,826 |2,700,551 | |
|New Hampshire |8,968 |1,316,470 | |
|New Jersey |7,417 |8,791,894 | |
|New Mexico |121,356 |2,059,179 | |
|New York |47,214 |19,378,102 | |
|North Carolina |48,711 |9,535,483 | |
|North Dakota |68,976 |672,591 | |
|Ohio |40,948 |11,536,504 | |
|Oklahoma |68,667 |3,751,351 | |
|Oregon |95,997 |3,831,074 | |
|Pennsylvania |44,817 |12,702,379 | |
|Rhode Island |1,045 |1,052,567 | |
|South Carolina |30,109 |4,625,364 | |
|South Dakota |75,885 |814,180 | |
|Tennessee |41,217 |6,346,105 | |
|Texas |261,797 |25,145,561 | |
|Utah |82,144 |2,763,885 | |
|Vermont |9,250 |625,741 | |
|Virginia |39,594 |8,001,024 | |
|Washington |66,544 |6,724,540 | |
|West Virginia |24,078 |1,852,994 | |
|Wisconsin |54,310 |5,686,986 | |
|Wyoming |97,100 |563,626 | |
|50 states + DC |3,537,438 |308,745,538 | |
Specific Objectives
Students will understand that
a relative change is different from an absolute change.
a relative measure is always a comparison of two numbers.
Students will be able to
calculate a relative change.
explain the difference between relative change and absolute change.
Problem Situation: How the Census Affects the House of Representatives
Every 10 years, the United States conducts a census. The census tells how many people live in each state. You can also find how much population has changed over time from the census data. The original purpose of the census was to decide on the number of representatives each state would have in the House of Representatives. Census data continue to be used for this purpose, but now have many other uses. For example, governments may use the data to plan for public services such as fire stations and schools. You will be given a list of states in a census region and their populations in 2000 and 2010. You will be asked to calculate the population growth in people as a percentage for each state in the region and for the region as a whole. You will examine how this affects the number of representatives each state has in the House of Representatives. You will start by looking at changes in representation based on the 2010 census.
The absolute change in a state’s population tells by how many people the population has changed. The relative change is the change as it compares to the earlier population. Often relative change is given as a percentage.
Use the following data for Questions 1–6.
South Atlantic States
| |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 | | | | |
(1) For your group of states, calculate the absolute change in the population of each state.
(2) For your group of states, calculate the relative change in the population of each state. Express your answer as a percentage.
(3) List in order the three states that changed most in absolute population.
(4) List in order the three states that had the largest relative increase in population.
(5) Explain why the lists in Question 3 and Questions 4 are not the same.
(6) For the region you are given, calculate the absolute change in total population from 2000 to 2010. Calculate the relative change in total population between 2000 and 2010.
While most states that lost representatives did so because their population became smaller relative to other states, Michigan’s population actually fell between 2000 and 2010.
(7) Michigan’s population changed to 9,833,640 from 9,938,444. What was the absolute decrease in Michigan’s population? What was the relative change in Michigan’s population? Round your answer to the nearest hundredth of a percent.
Making Connections
Record the important mathematical ideas from the discussion.
Further Applications
(1) In 2011, the U.S. Congress had a major debate over cutting the federal budget mid-year. The goal was to reduce the national debt, which was $14 trillion.
(a) One group wanted to reduce the budget by $100 billion. How large is this change relative to the national debt?
(b) Another group wanted to reduce the budget by $40 billion. How large is this change relative to the national debt?
(c) If a politician wanted to argue for the larger cut, would he or she use the absolute or the relative change to justify his or her position? Why?
(d) If a politician wanted to argue for the smaller cut, would he or she use the absolute or the relative change to justify his or her positions? Why?
Student Notes
Making Connections to the Lesson
(1) Which of the following was one of the main mathematical ideas of the lesson?
(i) Absolute change is measured as a quantity (for example, an increase of $3). Relative change is measured as a percentage compared to the reference value (for example, an increase of 3%).
(ii) To find relative change, subtract the original number from the new number and divide by the original.
(iii) Consider this situation: Quantity 1 increases by 15%. Quantity 2 increases by 20%. Quantity 2 must have increased by a larger amount than Quantity 1.
(iv) The population of a state determines how many representatives that state has in the House of Representatives.
(2) Explain a connection between a concept in this lesson and at least one of the following lessons: 1.8 and 1.9. You can use one of the sentence stems given below if you wish. You can also refer back to Module 1 lessons as examples.
Question number ________ in Lesson 2.3 connects to Question number ______ in Lesson _____ because ________________.
The idea of ___________ in Lesson 2.3 connects to Lesson _________. An example of the connections is ________.
Developing Skills and Understanding
(3) The following headlines all refer to change. Identify the change as absolute or relative.
(a) “Enrollments at Northeastern University are expected to increase by 1,500!”
(i) Absolute change
(ii) Relative change
(b) “Another 14% tuition increase is expected.”
(i) Absolute change
(ii) Relative change
(c) “A new proposal has sales tax rates dropping from 3% to 1%, a drop of only two percent.”
(i) Absolute change
(ii) Relative change
(d) “A new proposal has sales tax rates dropping from 3% to 1%, a 67 percent drop!”
(i) Absolute change
(ii) Relative change
Questions 4 and 5 refer to data taken from the U.S. Census.[7] The dollar values take into account the changes in the economy over the years (i.e., inflation). Inflation is a complicated issue, but for Questions 4 and 5, you do not need to worry about it.
(4) A typical high-income household in 1980 earned $125,556. A similar household in 2009 earned $180,001. What was the relative increase in income for these households from 1980 to 2009? Round to the nearest one percent.
(5) A typical middle-income household in 1980 earned $34,757. A similar household in 2009 earned $38,550. What was the relative increase in income for these households from 1980 to 2009? Round to the nearest one percent.
(6) Due to temporary tax cuts in 2010, a person with typical deductions earning $50,000 per year would have saved 2% of their income plus $850 in federal taxes.
(a) How much money did a typical person save?
(b) What percent did this person save on her income? Round to the nearest tenth of a percent.
(7) Due to the same law, a person earning $500,000 per year with typical deductions would save 2% of the first $106,800 they earned plus $14,250 in federal taxes. Fill in the blanks to complete the statement below.
A person earning $500,000 a year saved $___________ or ___________% of their income. Round to the nearest dollar and to the nearest tenth of a percent.
Making Connections Across the Course
If you need to review how to read a pie or circle graph such as the ones below, you may want to view the following video: .
You may also want to refer to the handout, Understanding Visual Displays of Information.
(8) In Lesson 1.4, you used results from the 2009 Consumer Expenditure Survey on how Americans spend their income. A summary of this information is given in Table 1.[8]
Table 1: Percentages of Average
Annual Housing Expenditures
|Housing |34.43% |
|Food |12.99% |
|Transportation |15.61% |
|Everything Else |36.97% |
Which pie graph best represents the data given in Table 1?
|(i) |(iii) |
|[pic] |[pic] |
|(ii) |(iv) |
|[pic] |[pic] |
(9) Many egg producers keep chickens in small cages that do not allow the chickens to move. Some people believe that this is unhealthy, so they buy eggs from chickens that are not caged 24-hours a day. These are sometimes called “free-range” chickens. The United States Department of Agriculture (USDA) allows chickens to be called free-range as long as the chickens spend some of their time outside. The European Union (EU), however, has several additional restrictions. One of these is that the farmers must provide enough outside area so that if all the chickens were outside, the density of chickens would be no more than 0.25 chickens/sq meters.[9]
(a) How many square meters does the EU require for one chicken?
(b) A farmer in the United States wants to meet the EU guidelines. She measures her area in yards. How many square yards does she need for 1,100 chickens? (1 m = 1.0936 yd)
(i) 301 square yards
(ii) 4,400 square yards
(iii) 4,812 square yards
(iv) 5,262 square yards
(v) None of the above
Preparing for the Next Lesson (2.4)
If you need to review how to read line graphs like the one shown below, you may want to review this video:
You will use the following work in your next class. Be sure to take the work to class.
(10) Use the graph to answer the following questions. Note that the vertical axis starts at $48,000 instead of $0.
[pic]
(a) Select the best phrase to complete this sentence: The numbers on the horizontal axis, the line across the bottom of the graph, represent
(i) income from $1,999 to $2009.
(ii) income from $1,999,000 to $2,009,000.
(iii) years from 1999 to 2009.
(iv) years and income.
(b) Select the best phrase to complete this sentence: The numbers on the vertical axis, the line along the left of the graph, represent
(i) money from $48,000 to $53,000.
(ii) income for a household or family from $48,000 to $53,000.
(iii) income for one person from $48,000 to $53,000.
(iv) income in 2009 from $48,000 to $53,000.
(c) A good estimate of the average household income in 1999 is
(i) $52,400
(ii) $52,600
(iii) $52,100
(iv) $52,000
(d) A good estimate of the average household income in 2009 is
(i) $49,000
(ii) $50,000
(iii) $49,250
(iv) $49,750
(e) Use the estimates from Parts (c) and (d) to calculate the relative change in the average household income from 1999 to 2009. Round to the nearest one percent. Indicate if the change is an increase or decrease.
(11) Jeff’s Housing: Two pairs of statements are given below. How can both pairs of statements be true? When did Jeff spend more on housing? Be prepared to discuss your answers in class.
|In 1990, Jeff spent $600 per month on housing. |In 1990, Jeff spent 20% of his income on housing. |
|In 2010, Jeff spent $1,200 per month on housing. |In 2010, Jeff spent 10% of his income on housing. |
(12) Decide if the following statement is true or false based on the two graphs below. Be prepared to discuss your answer in class.
[pic][pic]
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.
(13) You will be expected to do the following things for the next class. Rate how confident you are on a scale of 1–5 (1 = not confident and 5 = very confident).
Before beginning Lesson 2.4, you should understand the concepts and demonstrate the skills listed below:
|Skill or Concept: I can … |Rating from 1 to 5 |
|Read a line graph. | |
|Read a bar graph. | |
|Read a pie graph. | |
|Calculate relative change. | |
(14) Self-Regulated Learning: Reflect
(a) How confident are you that you correctly answered the problems you were assigned in OCE 2.3? Rate your confidence on a scale from 1–5 (1 = not confident; 5 = very confident).
(b) Which problems from this lesson do you feel you understood well? Which ones might you find it beneficial to talk to your teacher or someone else about?
(c) When planning to do this homework assignment, did you accurately predict how long the assignment was going to take?
(d) Name two strategies you used when solving this assignment.
Specific Objectives
Students will understand that
the scale on graphs can change perception of the information they represent.
to fully understand a pie graph, the reference value must be known.
Students will be able to
calculate relative change from a line graph.
estimate the absolute size of the portions of a pie graph given its reference value.
use data displayed on two graphs to estimate a third quantity.
Graphs are a helpful way to summarize data. Often there are many ways to portray information graphically. Sometimes one form is easier to read than another. Sometimes the way a graph is
made can affect the impression it gives. Today, you will look at three examples of such graphs.
Problem Situation 1: Reading Line Graphs
(1) Compare Graph 1 from your OCE (2.3) and Graph 2 below. What do you notice?
[pic]
Graph 2
(2) Based on these two graphs, would it be fair to say that the median household income was significantly lower in 2009 than it was in 1999?
Problem Situation 2: Reading Bar Graphs
Your class will discuss how the “Jeff’s Housing” question from your OCE assignment can be used to understand the gross domestic product (GDP) of a country.
[pic]
Graph 3
[pic]
Graph 4
(3) Think about the statement, “The 2010 national debt is way out of hand and has never been higher.” Use Graphs 3 and 4 to evaluate the statement. Is it true? Based on what information?
Problem Situation 3: Reading Pie (Circle) Graphs
The following questions refer to the graphs on the Hispanic population.
(4) The U.S. population in 2010 was around 310,000,000. In 2050, the U.S. population is expected to be around 439,000,000. Estimate the number of Hispanic and non-Hispanic Americans at each time.
(5) Does your work in Question 4 confirm or contradict your prediction from your out-of-class experience? Explain.
Making Connections
Record the important mathematical ideas from the discussion.
Further Applications
(1) Question 5 in the OCE (2.4) 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 2.4 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?
Student Notes
Making Connections to the Lesson
(1) Which of the following was one of the main mathematical ideas of the lesson?
(i) It is important to consider the gross national product when considering the size of the national debt. They relate to each other just as a person’s personal debt relates to his/her income.
(ii) In Graph A, you can find the quantity represented by Part A1 by multiplying the percentage for the section times the total quantity represented by the circle.
(iii) In the graphs below, you know that Part A2 represents a larger quantity than Part B2 because the piece of the graph is larger.
(iv) In the graphs below, you cannot compare the quantities represented by the parts because you do not know the reference values.
[pic][pic]
(2) Explain a connection between a concept in this lesson and Lesson 2.3. You may use one of the sentence stems given below if you wish. You may also refer back to Module 1 lessons as examples.
Question number ________ in Lesson 2.3 connects to Question number ______ in Lesson 2.4 because ________________.
The idea of ___________ in Lesson 2.3 connects to Lesson 2.4. An example of the connections is ________.
Developing Skills and Understanding
(3) Use the graph[10] on the next page to answer the following questions.
[pic]
(a) Estimate the percentage increase in new housing prices from 1999 to 2007. (Choose the best answer.)
(i) The prices increased from around $160,000 to around $245,000, about a 53% increase.
(ii) The prices increased from around $160,000 to around $220,000, about a 38% increase.
(iii) The prices increased from around $160,000 to around $245,000, about a 65% increase.
(iv) The prices increased from around $160,000 to around $220,000, about a 35% increase.
(b) Estimate the percentage increase in new housing prices from 2004 to 2007. (Fill in the blanks.)
The prices increased from ___________ to ___________. This is about a ________% increase.
(c) Estimate the percentage decrease from 2007 to 2009. (Fill in the blanks.)
The prices decreased from __________ to __________. This is about a ________% decrease.
(4) South Central Bank has a policy that limits the amount of debt customers may have in order to receive a loan. The following pie chart shows the highest percentage of debt that the bank will allow.
[pic]
(a) What is the reference value in this situation?
(i) Other expenses
(ii) Debt
(iii) Total of other expenses and debt
(iv) None of the above
(b) Three graphs are given below. Each graph represents a loan customer. The customers’ debt is broken into three categories: Car, Credit Card, and Mortgage (the loan on a house). Which customer(s) meet the bank policy on the limit of the amount of debt? There may be more than one correct answer.
(i)
[pic]
(ii)
[pic]
(iii)
[pic]
(5) The following two pie graphs show how two families spend their money. The Alvarez family has a take-home pay of $3,650 per month and the Martinez family has a take-home pay of $7,300 per month.
[pic]
[pic]
(a) Select the statement that best compares how much the families spend on gasoline.
(i) The Alvarez Family spent more on gasoline than the Martinez Family.
(ii) The two families both spent $200 on gasoline.
(iii) Both families spend around $200 on gasoline. This is 6% of the Alvarez budget, but only 3% of the Martinez budget because the Martinez family starts with about twice as much money as the Alvarez family.
(iv) The Alvarez family spent more on gasoline than the Martinez family because 3% of $7,300 is more than 6% of $3,650.
(b) Estimate the actual spending on food and housing for each family. (Fill in the tables.)
|Alvarez Family | |
|Food | |
|Housing | |
|Martinez Family | |
|Food | |
|Housing | |
(c) Both families spend about the same percentage of their income on housing. The family with the ________ income can afford a house that has twice the payment and maintenance costs.
(i) Higher
(ii) Lower
Making Connections Across the Course
(6) From 2000 to 2006, a total of 19,076 teens ages 15–19 were killed in car crashes in the United States. The number of teen males who were killed was 12,479 and the number of teen females who were killed was 6,597.[11]
A reporter used this information to write, “About 34%, or one out of every three girls will be killed in a car crash.” Which of the following statements is the best critique of this statement?
(i) The statement is correct because 6,597 is 34% of 19,076, which is very close to one-third. The ratio of 1 out of 3 is a good estimate of the percentage.
(ii) The statement is incorrect because 6,597 is 34% of 19,076, which is very close to three-tenths. The ratio of 1 out of 3 is not a good estimate of the percentage.
(iii) The statement is incorrect because the reference value is the number of teens killed, not all teen girls. It would be correct to say that 34% of teens killed in car crashes are female.
(7) Two different graphs showing how much two families spend on professional baseball tickets each year are given below. Choose the true statement based on the graphs.
(i) The Wagners must have a higher income than the Cobbs. They spend less on baseball tickets, but it is a higher portion of their total income.
(ii) The Cobbs must have a higher income than the Wagners. They spend more on baseball tickets, but it is a smaller portion of their total income.
(iii) The Wagners must have a higher income than the Cobbs. They spend more on baseball tickets, but it is a smaller portion of their total income.
(iv) The Cobbs must have a lower income than the Wagners. They spend more on baseball tickets, but it is a smaller portion of their total income.
[pic]
[pic]
Preparing for the Next Lesson (2.5)
(8) A digital camera is regularly priced at $350 and it is marked 30% off the regular price. What is the new price for the digital camera?
(i) $320 is the new price for the digital camera.
(ii) $338.40 is the new price for the digital camera.
(iii) $105 is the new price for the digital camera.
(iv) $245 is the new price for the digital camera.
(9) The American Diabetes Association reports the following information:
“In general, if you are a man with type-1 diabetes, the odds of your child getting diabetes are 1 in 17. If you are a woman with type-1 diabetes and your child was born before you were 25, your child's risk is 1 in 25; if your child was born after you turned 25, your child's risk is 1 in 100.”
Rank the risk of having a child with diabetes from 1 (highest risk) to 3 (lowest risk).
___ Mother with type-1 diabetes who gave birth after turning 25
___ Father with type-1 diabetes
___ Mother with type-1 diabetes who gave birth before turning 25
(10) Self-Regulated Learning: Work
An aspect of the work phase is always checking your understanding. One way to check your understanding is to explain how you solved a problem. Explain how you solved Question 9.
(11) Self-Regulated Learning: Reflect
After checking your understanding, on a scale from 1 to 5, how confident are you that you answered Question 9 correctly? Be sure to incorporate this confidence rating in the table at the end of the assignment.
(12) Mark each statement true or false.
(a) A man with type-1 diabetes has a child. The probability that the child will also have diabetes is about 6%.
(i) True
(ii) False
(b) A 20 year-old woman with type-1 diabetes has a child. The probability that the child will also have diabetes is about 25%.
(i) True
(ii) False
(c) A 30-year-old woman with type-1 diabetes has a child. The probability that the child will also have diabetes is about 1%.
(i) True
(ii) False
(13) You will be expected to do the following things for the next class. Rate how confident you are on a scale of 1–5 (1 = not confident and 5 = very confident).
Before beginning Lesson 2.5, you should understand the concepts and demonstrate the skills listed below:
|Skill or Concept: I can … |Rating from 1 to 5 |
|Read and interpret bar graphs. | |
|Have a basic understanding of the word percent and the notation used to describe | |
|percents (%). | |
|Use a calculator to divide two numbers and interpret the resulting decimal | |
|representation as a percent. | |
|Understand that a percent may be used to express the likelihood (or probability) of a| |
|certain event. | |
Specific Objectives
Students will understand that
the change in a quantity can be expressed as an absolute change and a relative change.
there is often ambiguity in the English language when talking about the change of a quantity that is represented by a percent (e.g., many rates).
Students will be able to
create graphs that show both absolute and relative changes in a rate (percent).
compute absolute and relative changes.
Problem Situation: The Effect of Reducing Risk
A new drug advertises that it “reduces the risk of heart attack by 50%.” In order to better understand the benefits of this drug, you will examine heart attack risk for two different groups.
Group 1 consists of individuals in Africa who are
40 years old,
do not have Diabetes Mellitus,
smoke tobacco,
have high cholesterol, and
have high blood pressure.
Group 2 consists of individuals in Africa who are
40 years old,
do not have Diabetes Mellitus,
do not smoke,
have low cholesterol, and
have high blood pressure.
The World Health Organization reports that individuals in Group 1 have a greater than 40% chance (or risk) of suffering a heart attack within 10 years. The same report indicates that individuals in Group 2 have a less than 10% risk of suffering a heart attack within 10 years.
(1) What are the differences between individuals in Group 1 and Group 2?
(2) Understanding Risk in Group 1
(a) If 500 people from Group 1 are observed for 10 years, how many individuals would you expect to suffer a heart attack within this time period?
(b) Which of the following graphs best show the risk of heart attack for Group 1 individuals?
|[pic] |[pic] |
|Graph 1 |Graph 2 |
|[pic] |[pic] |
|Graph 3 |Graph 4 |
(c) If 500 people from Group 1 are treated with the new drug and are observed for 10 years, how many individuals would you expect to suffer a heart attack within this time period?
(d) Create a graph that shows the risk of heart attack for Group 1 individuals who are taking the new drug.
(e) By placing the correct graph from Part (b) next to the graph you created in Part (d), you can show the effect of the new drug on this group of people. What, specifically, is being reduced by 50% in these graphs?
(f) There are two ways to report the drop in heart attacks due to the new drug. The absolute change refers to the actual number of fewer individuals suffering heart attacks as a result of using the new drug, while the relative change (or percent change) refers to this change as a percentage. Report both the absolute and relative changes in the number of individuals who suffered heart attacks.
(3) Understanding Risk in Group 2
(a) If 500 people from Group 2 are observed for 10 years, how many individuals would you expect to suffer a heart attack within this time period?
(b) Which of the following graphs best show the risk of heart attack for Group 2 individuals?
|[pic] |[pic] |
|Graph 5 |Graph 6 |
|[pic] |[pic] |
|Graph 7 |Graph 8 |
(c) If 500 people from Group 2 are treated with the new drug and are observed for 10 years, how many individuals would you expect to suffer a heart attack within this time period?
(d) Create a graph that shows the risk of heart attack for Group 2 individuals who are taking the new drug.
(e) By placing the correct graph from Part (b) next to the graph from Part (d), you can show the effect of the new drug on this group of people. What, specifically, is being reduced by 50% in these graphs?
(f) There are two ways to report the drop in heart attacks due to the new drug. The absolute change refers to the actual number of fewer individuals suffering heart attacks as a result of using the new drug, while the relative change (or percent change) refers to this change as a percentage. Report both the absolute and relative change in the number of individuals who suffered heart attacks.
(4) Suppose a third group of people have only a 0.5% chance of suffering a heart attack. Report the absolute and relative change in the number of individuals (out of 500) who are expected to suffer a heart attack if they take the new drug.
(5) Based on the previous calculations, critique the statement, “A drug which reduces the risk of heart attack by 50% will most likely save many lives.”
Making Connections
Record the important mathematical ideas from the discussion.
Further Applications
(1) Write a complete response, including mathematical examples, to Question 5 from the lesson. Critique the statement, “A drug that reduces the risk of heart attack by 50% will most likely save many lives.”
Student Notes
Making Connections to the Lesson
(1) Which of the following was one of the main mathematical ideas of the lesson?
(i) To calculate a percentage, move the decimal in the percentage rate two places to the left and multiply times the reference value.
(ii) Smoking increases your chance of a heart attack.
(iii) Consider this situation: 20% of a population gets a certain illness. A vaccine can reduce the infection rate by 10%. Out of 100 people, the reference value for the 10% reduction would be the full population (or 100 people).
(iv) Consider this situation: 20% of a population gets a certain illness. A vaccine can reduce the infection rate by 10%. Out of 100 people, the reference value for the 10% reduction would be the 20 people who are projected to get the illness.
(2) Explain a connection between a concept in this lesson and a concept in Lesson 2.3 or Lesson 2.4. You can use one of the sentence stems given below if you wish.
Question number________ in Lesson 2.5 connects to Question number ______ in Lesson _____ because ________________.
The idea of ___________ in Lesson 2.5 connects to Lesson _________. An example of the connections is ________.
Developing Skills and Understanding
(3) If you did not have time to do Question 4 during class, do this problem now.
(4) You either did Question 2 or Question 3 in the lesson. Refer back to your work for the following questions.
(a) In class, you used a group with 500 people. Repeat the work with a group of 100 people.
Group 1 has a 40% risk of heart attack and Group 2 has a 10% risk.
| |Group 1 |Group 2 |
|Number in group |100 |100 |
|Number who are expected to have a heart attack without the drug | | |
|Number who are expected to have a heart attack with the drug | | |
|What percentage of the group is expected to have a heart attack with the drug? | | |
(b) Select the true statement:
(i) The size of the group does not change anything about the results for the group.
(ii) The size of the group changes the number of people who will have a heart attack with the drug, but it does not change the percentage.
(iii) The size of the group changes the number and the percentage of people who will have a heart attack with the drug.
(5) The University of Washington is a large public university, while the University of Puget Sound is a small private university.
(a) If both of these universities are expecting 12% more freshmen next year, which university will see the greatest absolute change in the number of freshmen? (Choose the best answer.)
(i) Since the University of Washington is larger, they have more freshmen to begin with. Thus, a 12% increase will not translate to a larger absolute change.
(ii) Since the University of Washington is larger, they have more freshmen to begin with. Thus, a 12% increase will translate to a larger absolute change.
(iii) Both will undergo the exact same absolute change.
(iv) Since the University of Puget Sound is smaller, they have fewer freshmen to begin with. Thus, a 12% increase will translate to a larger absolute change
(b) Which university will see the greatest relative change in the number of freshmen?
(i) University of Washington
(ii) University of Puget Sound
(iii) Both universities will undergo the same relative change.
(6) Suppose that the populations of the United States and China both undergo the same absolute change in one year. Which undergoes the larger relative change?
(i) The United States would undergo the smaller relative change, since the population of China is so much larger than that of the United States.
(ii) China would undergo the larger relative change since the population of the United States is much smaller than that of China.
(iii) The United States would undergo the larger relative change since the population of the United States is so much smaller than that of China.
(iv) The two countries would undergo the exact same relative change.
(7) Suppose that the populations of the United States and China both increase by 12 million people in one year. What would be the relative changes that each country underwent? (Fill in the blanks.)
(a) If you assume that the U.S. population is about 300 million, an increase of 12 million would result in a relative change of _______%.
(b) If you assume that China has about 1 billion people, an increase of 12 million would result in a relative change of _______%.
(8) Employees at a certain company currently have to pay 3% of their health care costs, while the company pays the rest. Next year, however, employees will have to pay 6% of their health care costs. Express this change as an absolute change (in percentage points) and a relative change.
(a) Absolute change: _______ percentage points.
(b) Relative change: _______%.
Making Connections Across the Course
(9) The graph[12] below is based on information from the 2000 U.S. Census. Use the graph to answer the questions below.
[pic]
(a) Which is the best estimate of the total population for age group 0–9?
(i) 400
(ii) 40,000
(iii) 40,000,000
(iv) 400,000,000
(b) Which is the best estimate for the total population of age group 70–79?
(i) one hundred and sixty
(ii) one hundred and sixty thousand
(iii) one hundred and sixty million
(iv) sixteen million
(c) Which statement is the most accurate?
(i) There are fewer than 10,000,000 people age 70 and older.
(ii) There are fewer than 20,000,000 people age 70 and older.
(iii) There are between 20,000,000 to 30,000,000 people age 70 and older.
(iv) There are between 30,000,000 to 40,000,000 people age 70 and older.
(d) If one wanted to compare the number of people in their forties (age 40–49) to the number of people in their fifties (age 50–59), one could consider the absolute difference in the populations or the relative difference. Which of the following statements are true? There may be more than one correct answer.
(i) The absolute change in population from forties to fifties is about the same as the absolute change from fifties to sixties.
(ii) The absolute change in population from forties to fifties is greater than the absolute change from fifties to sixties.
(iii) The relative change in population from forties to fifties is about the same as the relative change from fifties to sixties.
(iv) The relative change in population from forties to fifties is less than the relative change from fifties to sixties.
(e) An advertiser is considering two advertising campaigns for a product. Campaign A is most effective for ages 20–39, while campaign B is most effective for ages 30–49. Which campaign should the advertiser choose to reach the most people?
(i) Campaign A
(ii) Campaign B
Preparing for the Next Lesson (2.6)
(10) Find the sum of each set of numbers, as well as the size n of each set of numbers (i.e., the number of numbers in the set). Use these exercises to practice techniques for adding numbers quickly. Try to do the problems without a calculator. Think about ways in which grouping the numbers might make them easier to add.
(a) 25, 35, 19, 31
(i) Sum:
(ii) n:
(b) 101, 73, 49, 27, 24, 36,
(i) Sum:
(ii) n:
(c) 25, 25, 25, 30, 30, 30, 32, 32, 32
(i) Sum:
(ii) n:
The following information will be used in Lesson 2.6.
People often talk about “averages,” and you probably have an idea of what is meant by that. Now, you will look at more formal mathematical ways of defining averages. In mathematics, you call an average a measure of center because an average is a way of measuring or quantifying the center of a set of data. There are different measures of center because there are different ways to define the center.
[pic]
Think about a long line of people waiting to buy tickets for a concert. (Figure A shows a line about
100-feet long and each dot represents a person in the line.) In some sections of the line people are grouped together very closely, while in other sections of the line people are spread out. How would you describe where the center of the line is?
Would you define the center of the line by finding the point at which half the people in the line are on one side and half are on the other (see Figure B)? Is the center based on the length of the line even though there would be more people on one side of the center than on the other (see Figure C)? Would you place the center among the largest groups of people (see Figure D)? The answer would depend on what you needed the center for. When working with data, you need different measures for different purposes.
Mean (Arithmetic Average)
Find the average of numeric values by finding the sum of the values and dividing the sum by the number of values. The mean is what most people call the “average.”
Example
Find the mean of 18, 23, 45, 18, 36
Find the sum of the numbers: 18 + 23 + 45 + 18 +36 = 140
Divide the sum by 5 because there are 5 numbers: 140 ÷ 5 = 28
The mean is 28.
Median
Find the median of numeric values by arranging the data in order of size. If there is an odd number of values, the median is the middle number. If there is an even number of values, the median is the mean of the two middle numbers.
Example (data set with odd number of values)
Find the median of 18, 23, 45, 18, 36.
Write the numbers in order: 18, 18, 23, 36, 45
There is an odd number of values, so the median is the number in the middle.
The median is 23.
Example (data set with even number of values)
Find the median of 18, 23, 45, 18, 12, 50.
Write the numbers in order: 12, 18, 18, 23, 45, 50
There is an even number of values, so there is no one middle number. Find the median by finding the mean of the two middle numbers:
18 + 23 = 41
41 ÷ 2 = 20.5
The median is 20.5.
Mode
Find the mode by finding the number(s) that occur(s) most frequently. There may be more than one mode.
Example
Find the mode of 18, 23, 45, 18, 36.
The number 18 occurs twice, more than any other number, so the mode is 18.
(11) Write the answers to the following questions on a piece of paper and bring it to the next class period. If you do not own a credit card, answer Part (a) only. Keep your responses anonymous by writing only the answers to the questions. (Do not write your name.)
(a) How many credit cards do you possess?
(b) Do you normally pay the entire balance on the credit card statement?
(c) What is the approximate balance (total) on your card(s) right now?
(12) You will be expected to do the following things for the next class. Rate how confident you are on a scale of 1–5 (1 = not confident and 5 = very confident).
Before beginning Lesson 2.6, you should understand the concepts and demonstrate the skills listed below:
|Skill or Concept: I can … |Rating from 1 to 5 |
|Perform basic operations using quantities as integers, fractions, or decimals with the aid of | |
|technology. | |
|Identify the mean, median, and mode of a small data set. | |
(13) In the question above, if you or another student had any ratings below 3, what are some things that you or the student could do to increase your/their confidence while planning for Lesson 2.6? Please describe at least two ways.
Specific Objectives
Students will understand
that numerical data can be summarized using measures of central tendency.
how each statistic—mean, median, and mode—provide a different snapshot of the data.
that conclusions derived from statistical summaries are subject to error.
that a spreadsheet can be used to organize data.
Students will be able to
calculate the mean, median, and mode for numerical data.
create a data set that meets certain criteria for measures of central tendency.
Problem Situation: Summarizing Data About Credit Cards
A revolving line of credit is an agreement between a consumer and lender that allows the consumer to obtain credit for an undetermined amount of time. The debt is repaid periodically and can be borrowed again once it is repaid. The use of a credit card is an example of a revolving line of credit.
According to , 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.[13]
In the first part of this lesson, you will use the information about credit cards given above to learn about some ways to summarize quantitative information.
(1) The population of the United States is slightly more than 300 million people. There are about 100 million households in the United States. What is the average number of credit cards per person? What is the average number of credit cards per household?
(2) Consider the statement, “Average credit card debt per household with a credit card is $14,743.” What does this mean?
(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.
[pic]
(a) Find the mean debt of each group of college graduates. Make sure the value you found is reasonable given the values in the table.
(b) Complete the data set called “Your Data” so that it represents the debt of five college seniors with a mean debt of $4,100.
|Your Data |
| |
| |
| |
| |
| |
(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) The introductory information gives data about the median debt carried by freshmen. Create a data set of six freshmen so that the data set has the same median reported for all college freshmen.
|Debt of College Freshmen |
| |
| |
| |
| |
| |
| |
Note on language: The word mean is used in mathematics. There are actually several different kinds of means. The one that you have discussed in this course is the arithmetic mean. You will also see people use the word average when referring to the mean. You should be familiar with both terms.
Making Connections
Record the important mathematical ideas from the discussion.
Further Applications
(1) This course draws on examples from three themes: Citizenship, Personal Finance, and Medical Literacy. Choose at least two different lessons with the Personal Finance theme. Answer the following questions.
(a) What new information did you learn about personal finance?
(b) How will you use this information or why is it important to know this information?
Student Notes
Making Connections to the Lesson
(1) Which of the following was one of the main mathematical ideas of the lesson?
(i) Any of the three measures of central tendency (mean, median, and mode) are good representations of data. It does not matter which one you use.
(ii) U.S. college students carry far too much credit card debt.
(iii) The mean is calculated by adding all the numbers and dividing by the number of data points.
(iv) The mean, median, and mode all give important information about a data set, but they do not give a complete picture of the data set.
(2) In Lesson 2.4, you learned about reading graphs. Describe a connection between interpreting a graph and interpreting measures of central tendency. You can use one of the sentence stems given below if you wish.
Question number______ in Lesson 2.4 connects to Question number ______ in Lesson 2.6 because…
The idea of _________________ in Lesson 2.4 connects to Lesson 2.6. An example of the connections is…
Developing Skills and Understanding
(3) Use the following data set to answer the questions.
|13 |15 |
|Perform basic operations using quantities as such integers, fractions, or decimals | |
|with the aid of technology. | |
|Find the mean, median, and mode of a set of numeric data. | |
|Read a line graph. | |
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! |Are you above average? |NEED A NEW CHALLENGE? |
| | |Join a super sales force and make as |
|We have immediate need for five |Our company is hiring one person this |much as you want. Five of our nine |
|enthusiastic self-starters who love the |month—will you be that person? We pay the top |salespeople closed FOUR homes last |
|outdoors and who love people. Our |percentage commission and supply you leads. |month. Their average commission was |
|salespeople make an average of $1,000 per|Half of our sales force makes over $3,000 per |$1,500 on each sale. Do the math—this|
|week. Come join the winning team. |month. Join the |is the job for you. |
|Call 555-0100 now! |Above Average Team! |Making dreams real— |
| |Call 555-0127 now! |call 555-0199 |
| | | |
| |We are! | |
(1) Examine the three advertisements shown in the problem situation.
(a) Identify any measures of central tendency and how they are used in each advertisement.
(b) For each advertisement, create a scenario that fits the information provided. Scenario means to create a set of data that fits the description. You did this in the previous lesson when you made a list of credit card debts for the five college students.
(2) In which job would you expect to earn the most money?
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.[17] Examine the graph. Write at least three statements about the data. Recall the Writing Principle: Use specific and complete information.
[pic]
(4) Table 1 gives a sample data set of home prices that matches the data shown in the graph for the year 1977. Five possible data sets for the year 2005 are given in Table 2. Use your knowledge of mean and median to answer the following questions without calculating the mean of the data sets. There may be more than one correct answer to any of the questions.
Table 1: Sales Prices of New Homes
Sold in United States in 1977
|1977 Sales Price |
|$40,000 |
|$45,000 |
|$50,000 |
|$56,000 |
|$67,000 |
|$75,000 |
|$112,000 |
Table 2: Possible Data Sets for 2005
|Set A |Set B |Set C |Set D |Set E |
|$240,000 |$84,000 |$120,000 |$74,000 |$74,000 |
|$245,000 |$105,000 |$135,000 |$95,000 |$90,000 |
|$250,000 |$125,000 |$150,000 |$105,000 |$120,000 |
|$256,000 |$240,000 |$168,000 |$240,000 |$240,000 |
|$267,000 |$245,000 |$201,000 |$242,000 |$250,000 |
|$275,000 |$469,000 |$225,000 |$250,000 |$635,000 |
|$312,000 |$810,000 |$336,000 |$251,000 |$669,000 |
(a) Which of the data sets could represent the data in the graph?
(b) Which of the data sets would likely have a mean that is less than the median?
(c) Which of the data sets would likely have a mean and median that are close together?
Making Connections
Record the important mathematical ideas from the discussion.
Further Applications
(1) This course draws on examples from three themes: Citizenship, Personal Finance, and Medical Literacy. Choose at least two different lessons with the Citizenship theme. Answer the following questions.
(a) What new information did you learn about citizenship?
(b) How will you use this information or why is it important to know this information?
Student Notes
Making Connections to the Lesson
(1) Which of the following was one of the main mathematical ideas of the lesson?
(i) Home prices in 2007 were more than 10 times what they were in the 1960s.
(ii) The mean and median of a data set are always very close together, but the mode might be very different.
(iii) When using averages or measures of central tendency, it is always important to ask questions about what the different types of measurements do and do not tell you about the data set.
(iv) The median is the middle number when a data set is listed in order. If there is an even number of points in the data set, the median is found by finding the mean of the middle two numbers.
(2) Refer back to the OCE for Lesson 2.4. Question 3 contains a graph about housing prices.
(a) Explain what the data for 2007 tells you about the cost of a house.
(b) Compare this graph to the graph of house prices used in Lesson 2.7. What is similar about the information? What is different?
Developing Skills and Understanding
(3) The first advertisement discussed in class states that the salespeople make an average of $1,000 per week. Suppose there are nine salespeople. What would the ninth person need to earn for the mean to be $1,000 if the other eight salespeople earned $550, $600, $600, $800, $950, $950, $1,000, and $1,100?
(4) The second advertisement states that half the salespeople make more than $3,000 per month. Suppose there are eight salespeople. What would the eighth person need to earn for the median to be $3,000 if the other seven salespeople earned $2,400, $2,500, $2,800, $2,800, $3,400, $3,400, and $3,800?
(5) Which statistic (mean, median, or mode) is most appropriate in each of the following situations?
(a) Tables in the dining hall are numbered 1 through 12 for students who eat there. The principal calls out a number for the table that will go through the buffet line first. The other tables follow in order of the table numbers. One student is sure the principal calls certain tables more often. She keeps track of which numbers are called over a 21-day period.
(i) Mean
(ii) Median
(iii) Mode
(b) The offensive line of a football team is larger than in previous years. The program will list a statistic to show this fact.
(i) Mean
(ii) Median
(iii) Mode
(c) A reporter is doing a story on the falling prices of homes in a large neighborhood. The reporter wants to demonstrate how the prices have fallen for all homes, not just the most expensive houses.
(i) Mean
(ii) Median
(iii) Mode
(6) Lines at the Department of Motor Vehicles are so long! A supervisor decided to do a study on the number of people standing in line. At the beginning of each hour for an entire week, the supervisor counted the number of people in line and recorded the number. At the end of the week, the supervisor made the frequency table below. Note that the first column shows the number of people in line at the beginning of the hour. The second column shows the number of times that length of line occurred in the 40 observations.
|Number of People in |Frequency |
|Line | |
|1 |1 |
|2 |0 |
|3 |2 |
|4 |2 |
|5 |2 |
|6 |4 |
|7 |4 |
|8 |4 |
|9 |8 |
|10 |4 |
|11 |9 |
(a) How many times did the supervisor observe six people in line?
(b) Which calculation could be used to determine how many people the supervisor observed standing in line all together?
(i) 1 x 1 + 2 x 0 + 3 x 2 + 4 x 2 + 5 x 2 + 6 x 4 + 7 x 4 + 8 x 4 + 9 x 8 + 10 x 4 + 11 x 9
(ii) 1 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10 + 11
(iii) 3 + 2 + 2 + 2 + 4 + 4 + 4 + 8 + 4 + 9
(iv) None of the above
(c) How many observations did the supervisor make?
(d) Determine the mode of the data in the frequency table.
(e) Determine the median of the data in the frequency table.
(f) Determine the mean of the data in the frequency table.
(g) The supervisor made observations again the following week. The table below shows the observations.
|Number of People in |Frequency |
|Line | |
|1 |3 |
|2 |4 |
|3 |5 |
|4 |4 |
|5 |5 |
|6 |1 |
|7 |2 |
|8 |1 |
|9 |2 |
|10 |6 |
|11 |8 |
The mean for the second week was 6.366, the median was 5, and the mode was 11. The supervisor wanted to make the argument that additional personnel were needed. Which of these arguments is correct?
(i) Most of the time, there are 11 people in line!
(ii) Most of the time, there are 5 or more people in line!
(iii) Most of the time, there are more than 6 people in line!
(7) Three descriptions of measures of central tendency are given below. They are labeled A, B, and C. Descriptions of data sets are listed below that. Match each data set with a description of measures of central tendency by writing the letter in the blank. Choices may be used more than once.
A The mean and median are close together.
B The mean is much higher than the median.
C The median is much higher than the mean.
(a) _____ The data have a large range with some very high numbers and many small numbers.
(b) _____ The data set has a large range with the numbers evenly spaced.
(c) _____ The data set has a small range with most of the numbers grouped in the middle.
(d) _____ The data set has a large range with a few very low numbers.
(8) If you lived in Canada in 2008, you might have seen the following headline:
“Canada Below G7 Average for Productivity!”
Here is some information to help you understand this headline.
Productivity is a way to measure the economy of a nation. One way to measure productivity is by Gross Domestic Product (GDP) per worker. You may recall from Lesson 2.4 that GDP is the value of all the goods and services produced in a country.
The G7 is a coalition of the major industrial democracies in the world: United States, United Kingdom, France, Germany, Italy, Canada, and Japan.
(a) Which of the following is most likely what the author of the headline wanted the reader
to think?
(i) Canada’s economy is weak and is falling behind other countries in the G7.
(ii) Canada’s economy is strong and is leading other countries in the G7.
(iii) Canada’s economy is very similar to other countries in the G7.
(iv) Canada’s economy should not be compared to other countries.
(b) Which of the following can you conclude from the headline?
(i) Canada is less productive than half of the G7 nations.
(ii) There is at least one G7 nation that is more productive than Canada.
(iii) There is at least one G7 nation that is less productive than Canada.
(iv) None of the above.
A graph of the GDP per worker of the G7 nations is shown below.[18]
[pic]
(c) Find the mean of the GDP per worker for the G7 nations. Round to the nearest hundred dollars.
(d) Is the headline correct?
(e) Which of the seven G7 nations have “above average” productivity?
(f) Which of the following are correct conclusions based on the data in the graph? There may be more than one correct answer.
(i) Canada is in the top half of the G7 in productivity.
(ii) Canada’s productivity is relatively close to all the G7 nations except for the United States and Japan.
(iii) Canada is far behind the G7 nations in productivity.
(iv) None of the above.
Making Connections Across the Course
Themes explored in Lesson 2.8 include buying power of money, the role of the Urban Consumer Price Index, and the concept of index numbers.
(9) Buying power of money: Gasoline costs have varied significantly in recent months. The American Petroleum Institute posted an update on gasoline prices for June 15, 2011.[19]
U.S. PUMP PRICE UPDATE—JUNE 15, 2011
The average U.S. retail price for all grades of gasoline fell this week for the fifth week in a row by 6.6 cents from the prior week to $3.767 per gallon, according to the Energy Information Administration (EIA). This was at the highest level since August 2008 with the exception of the recent highs in the prior two months. Compared with the December 29, 2008 low of $1.670, the all-grade average was higher by $2.097 per gallon, or 125.6 percent. The average has been above $3.50 per gallon since the beginning of March 2011. Nominal prices have been above the year-ago average for 66 weeks—and were up by 101.1 cents or 36.7 percent, from the year-ago average of $2.756 per gallon.
(a) How much was the average retail price for one gallon of gasoline a week before this article was published?
(i) $3.701
(ii) $3.518
(iii) $3.833
(iv) None of the above
(b) What is the relative change in the retail price for one gallon of gasoline from
December 29, 2008, to June 15, 2011?
(i) 225.6%
(ii) 125.6%
(iii) 25.6%
(iv) None of the above
(c) If the retail price for one gallon of gasoline was $2.756 a year before, then what is the absolute change in the retail price on June 15, 2011?
(i) $2.097
(ii) $1.086
(iii) $1.670
(iv) None of the above
(10) A key idea of the previous question is that the buying power of a dollar is not constant. For example, the price of gasoline varies greatly, so the amount of gas you can buy with $1 varies over time. The Urban Consumer Price Index (CPI-U) is a tool designed to compare the price of goods and services in terms of base-year dollars. You will be using the CPI-U in your next lesson. The following questions will help you learn about it.
Refer to the website cpi to answer the following questions.
(a) Complete the following description of the Urban Consumer Price Index.
The Consumer Price Indexes (CPI) program produces monthly data on …
Go to the CPI Overview.
(b) Read the Data Available section. What percentage of the population is represented by the
CPI-U?
(c) Read the Coverage section. Are user fees for services like water and sewer included in the CPI?
(d) Are income taxes included in the CPI?
(e) Read the Uses section. The website states that the CPI affects the income of almost 80 million people because their incomes are tied to changes in the CPI. Which option is the best estimate for the percentage of these people who receive Social Security benefits?
(i) A little less than 48%
(ii) More than 50%
(iii) Around 80%
(f) Which option is the best estimate for the fraction of these people who receive food stamps?
(i) About a third
(ii) About three-fourths
(iii) About half
(g) State one use for the CPI-U, based on information from the website.
Preparing for the Next Lesson (2.8)
The following information will be used in the next lesson.
The United States has a federal minimum wage. This means that there is a law that requires employers to pay employees a certain amount. Many states also have their own minimum wage laws that require a higher wage than the federal law. The minimum wage started in 1938 as a part of a law that protected the rights of workers in many ways. The minimum wage in 1938 was set at $0.25 per hour.[20]
Since 1938, Congress has increased the minimum wage many times to account for inflation. Inflation is when prices increase over time. Maybe you have heard stories from older people about how a cup of coffee used to cost a dime or a gallon of gasoline was less than a dollar. Prices for these items have increased due to inflation.
When deciding if the minimum wage should be increased, people often talk about the buying power of the wage. Buying power refers to how much you can actually buy with a dollar. Because of inflation, a dollar bought more in 1938 than it did in 2009. In Lesson 2.8, you will answer the question: Did the minimum wage in 1938 have more buying power than the minimum wage in 2009? You will start by thinking about the specific example of buying a movie ticket.
(11) The minimum wage from 1997 through 2006 was $5.15 per hour. In 2009, the minimum wage increased to $7.25 per hour.
(a) What was the absolute change in the minimum wage from 1997 to 2009?
(b) What was the relative change in the minimum wage from 1997 to 2009? Round to the nearest tenth of one percent.
(12) The average price of a movie ticket in 1997 was $4.59.[21] Which statement is correct? Be prepared to explain your answer in class.
(i) In 1997, a person earning minimum wage had to work less than an hour to earn enough for a movie ticket.
(ii) In 1997, a person earning minimum wage had to work more than an hour to earn enough for a movie ticket.
(13) The average price of a movie ticket in 2009 was $7.50. Which statement is correct? Be prepared to explain your answer in class.
(i) In 2009, a person earning minimum wage had to work less than an hour to earn enough for a movie ticket.
(ii) In 2009, a person earning minimum wage had to work more than an hour to earn enough for a movie ticket.
(14) Did you read and understand the information to be used in class?
_____Yes _____No
(15) Did you complete the work that you need to take to class?
_____Yes _____No
(16) You should be able to do the following things for the next class. Rate how confident you are on a scale of 1–5 (1 = not confident and 5 = very confident).
Before beginning Lesson 2.8, you should understand the concepts and demonstrate the skills listed below:
|Skill or Concept: I can … |Rating from 1 to 5 |
|Calculate and interpret relative change. | |
|Understand the minimum wage. | |
Specific Objectives
Students will understand that
ratios provide a way of comparing the relative increase or decrease of two variables.
index numbers are a way of comparing the relative size of a variable over time.
Students will be able to
use proportional reasoning to find the size of a variable that remains a constant proportion of another variable.
use index numbers to find the value of a variable relative to time (or another variable).
Problem Situation: The Buying Power of the Minimum Wage
You found from your previous work that the minimum wage did not increase enough from 1997 to 2009 to keep up with the cost of a movie ticket. A movie ticket is only a single product that someone might buy. To compare the buying power of the minimum wage from 1938 to 2009, you need more information about more products. You will use a tool called an index number.
Index numbers are a way to compare the relative change or difference in a data set, such as the prices
of products. The change can be measured over time or over different geographic regions. There are different types of index numbers. One type is a measure of average relative change in data. One data point is called the base and is assigned the value of 100 (meaning it represents 100%). The other figures are adjusted in proportion to the base. The plural form of the word index is indices.
The following two questions will help you understand index numbers. You will begin by looking at one product: the Big Mac.
(1) You are going to use 1992 as the base year for our Big Mac Price Index. Prices of the Big Mac vary with time. In 1992, a Big Mac cost about $2.19.
(a) In 1968, a Big Mac cost about $0.49. Find the percentage that the 1968 price compared to the 1992 price.
(b) In 2010, a Big Mac cost about $3.25. Find the percentage that the 2010 price compared to the 1992 price.
Big Mac Price Index
|Year |Index Number |
|1968 | |
|1992 | |
|2010 | |
Now you have seen how an index is created and what it means. You are now going to look at a real index that the U.S. government uses to track changes in the prices of many items over time. It is called the Consumer Price Index (CPI). You will use a version called the Urban Consumer Price Index (CPI-U).
The Bureau of Labor Statistics publishes the CPI-U each month. It is often used to compare the buying power of a dollar in one year as compared to another (it compares prices over time). The CPI is a measure of the weighted average of a “basket of consumer goods and services.” The basket includes transportation, food, medical care, housing, apparel, recreation, and education. The more this basket of goods and services costs, the less of it you can purchase with the dollar. The CPI is one of the most frequently used statistics for identifying periods of deflation or inflation. If the CPI is more than 100, then prices have increased compared to the base year. This means there has been inflation since the base year. If CPI is less than 100, then prices have decreased compared to the base year. This is called deflation. The CPI-U represents the cost of a basket of goods and services in base-year dollars. The table shows the indices for selected years from 1913 to 2010.
|Year |Index | |Year |Index |
|1913 |9.9 | |1981 |90.9 |
|1914 |10.0 | |1982 |96.5 |
|1915 |10.1 | |1983 |99.6 |
|1921 |17.9 | |1984 |103.9 |
|1922 |16.8 | |1985 |107.6 |
|1923 |17.1 | |1996 |156.9 |
|1924 |17.1 | |1997 |160.5 |
|1925 |17.5 | |1998 |163.0 |
|1936 |13.9 | |1999 |166.6 |
|1937 |14.4 | |2000 |172.2 |
|1938 |14.1 | |2001 |177.1 |
|1939 |13.9 | |2002 |179.9 |
|1940 |14.0 | |2003 |184.0 |
|1946 |19.5 | |2004 |188.9 |
|1947 |22.3 | |2005 |195.3 |
|1948 |24.1 | |2006 |201.6 |
|1949 |23.8 | |2007 |207.3 |
|1950 |24.1 | |2008 |215.3 |
| | | |2009 |214.5 |
| | | |2010 |218.1 |
(2) What year is used as the base for the CPI-U? What does the base mean?
(3) How does the 2010 dollar compare in value to the dollar in 1983?
(4) How does the 1949 dollar compare in value to the dollar in 1983?
(5) Who had more buying power? Provide quantitative information to support your answer.
The person making minimum wage ($0.25 per hour) in 1938.
The person making minimum wage ($7.25 per hour) in 2010.
Making Connections
Record the important mathematical ideas from the discussion.
Further Applications
(1) Find the price of gasoline in 1981 and in 2010. Give the sources for your information. Use the CPI-U table in the lesson to evaluate the statement, “Gasoline was more expensive in 2010 than in 1981.” Provide mathematical information to justify your explanation.
Student Notes
Making Connections to the Lesson
(1) Which of the following was one of the main mathematical ideas of the lesson?
(i) The Consumer Price Index (CPI) is an example of an index number that measures the cost of a basket of goods and services over time. It can be used to show inflation and deflation of prices.
(ii) Inflation is calculated by finding the relative change in the CPI.
(iii) The CPI is a ratio that compares the cost of goods to a base year in the form of a percentage. This is an example of how ratios are used for comparisons.
(iv) The CPI always increases over time.
(2) Explain a connection between a concept in this lesson and Lesson 1.6 or 2.2. You can use one of the sentence stems given below if you wish.
Question number________ in Lesson 2.8 connects to Question number______ in Lesson _____ because …
The idea of ___________ in Lesson 2.8 connects to Lesson _________. An example of the connections is …
Developing Skills and Understanding
(3) The price of a first-class stamp has increased rapidly since the 1970s. The graph below shows the price (in cents) of a first-class stamp since 1917.
[pic]
Source: United States Postal Service
(a) Estimate the absolute increase in the price of a first-class stamp from 1992 to 2007.
(i) 10%
(ii) $0.10
(iii) 25%
(iv) $0.14
(v) $0.25
(vi) 33%
(b) Estimate the relative increase in the price of a first-class stamp from 1992 to 2007.
(i) 10%
(ii) $0.10
(iii) 25%
(iv) $0.14
(v) $0.25
(vi) 33%
(c) Suppose the absolute increase in the price of a first-class stamp from 2007 to 2022 is the same as it was from 1992 to 2007. What would the cost of a first-class stamp be in 2022?
(i) $0.44
(ii) $0.50
(iii) $0.53
(iv) $0.59
(v) $0.65
(d) Suppose the relative increase in the price of a first-class stamp from 2007 to 2022 is the same as it was from 1992 to 2007. What would the cost of a first-class stamp be in 2022?
(i) $0.44
(ii) $0.50
(iii) $0.53
(iv) $0.59
(v) $0.65
Refer to the table of CPI-U given in the lesson for Questions 4 and 5. This table gives the cost of a representative basket of goods and services in base-year dollars (1983). The table shows the indices for selected years from 1913 to 2010. The Consumer Price Index (CPI-U) provides an indication of how the price of consumer goods changes over time.
(4) In 2006, a 12-ounce coffee at Starbucks cost $1.46. In 2010, that same cup of coffee was $1.94.
(a) What was the relative increase in the price of the cup of coffee from 2006 to 2010? Round to the nearest one percent.
(b) What was the relative increase in the CPI-U from 2006 to 2010? Round to the nearest one percent.
(c) Which of the following statements is true?
(i) From 2006 to 2010, the price of coffee at Starbucks increased at a lower rate than the CPI-U.
(ii) From 2006 to 2010, the price of coffee at Starbucks increased at the same rate as the CPI-U.
(iii) From 2006 to 2010, the price of coffee at Starbucks increased at a higher rate than the
CPI-U.
(5) The rent on a building in 2008 was $1,350 per month. The landlord changes the rent relative to changes in the CPI-U. Use the CPI-U to calculate the rent in 2010. Round to the nearest dollar.
(6) The OCE for Lesson 2.2.2 had a question using productivity data for the G7 nations. The graph below shows another way of comparing productivity data using an index. Use the graph to answer the following questions. Note that these data are not the same as in the earlier problem.
[pic]
(a) How does Japan’s productivity compare to that of the United Kingdom?
(i) Japan has a higher rate of productivity.
(ii) Japan has a lower rate of productivity.
(iii) Japan has about the same rate of productivity.
(b) Which two countries have about the same rate of productivity?
(c) Which country is used as the base?
(d) What does the vertical scale represent?
(i) Productivity in U.S. dollars
(ii) Productivity in Euros
(iii) Index numbers (percentages relative to a base)
(e) Which of the following statements are true? There may be more than one correct answer.
(i) The U.S. productivity is about $132.00 per worker.
(ii) The U.S. productivity is about $32.00 per worker more than the U.K. productivity.
(iii) The U.S. productivity is 32% higher than the U.K. productivity.
(iv) The U.S. productivity is 132% higher than the U.K. productivity.
(v) The U.S. productivity is about 50% higher than Japan’s productivity.
(7) Data from the Census of Population and Housing surveys of 1960, 1974, and 1989, the National Association of Realtors, and the U.S. Census Bureau are shown below.
| |1960 |1974 |1989 |2009 |
|Median Family Income |$5,500 |$13,020 |$34,400 |$60,600 |
|Median Price of House |$60,052 |$86,751 |$84,622 |$185,000 |
In which of these four years was a house most affordable?
(i) 1960
(ii) 1974
(iii) 1989
(iv) 2009
Making Connections Across the Course
There is now an expectation that you will be able to conduct an Internet search to find information. You will begin seeing questions in your assignments that require you to identify the information you need and find the information from a reliable source.
(8) According to the CIA’s The World Factbook, around 320,000 people in Botswana had HIV/AIDS in 2009. About 5,800 people in Botswana died of AIDS that year. In the United States, around 1.2 million Americans were infected with HIV/AIDS in 2009, and about 17,000 of those people died.
(a) Use estimation to identify which of the following statements are true. There may be more than one correct answer.
(i) There are more HIV/AIDS cases in Botswana than in the United States.
(ii) There are more HIV/AIDS cases in the United States than in Botswana.
(iii) The HIV/AIDS infection rate is higher in Botswana than in the United States.
(iv) The HIV/AIDS infection rate is lower in Botswana than in the United States.
(v) The death rate among people infected with HIV/AIDS is much larger in Botswana than in the United States.
(vi) In 2009, the United States had about four times as many HIV/AIDS cases as Botswana and about 150 times the population.
(vii) In 2009, the United States had about 40 times as many HIV/AIDS cases as Botswana and about 15 times the population.
(viii) The death rate among people infected with HIV/AIDS is much smaller in Botswana than in the United States.
(ix) The death rate among people infected with HIV/AIDS is about the same in Botswana as it is in the United States.
(b) Suppose that a new experimental drug being tested in Botswana showed that it can decrease the death rate among people infected with HIV/AIDS by 18%. If this drug had been used in 2009 by all Botswanans infected with HIV/AIDS, how many people would you have expected to die from HIV/AIDS? Round to the nearest hundred people.
(c) Suppose that the drug from Part (b) has the same influence on Americans infected with HIV/AIDS. If the drug had been used in 2009 by all Americans infected with HIV/AIDS, how many people would you have expected to die from HIV/AIDS? Round to the nearest thousand people.
(9) The type of comparison shown below is called relative magnitude because it is a comparison of the magnitude, or size, of one quantity relative to another quantity.
In 2009, the United States had about four times as many HIV/AIDS cases as Botswana and about 150 times the population.
Write a statement comparing the relative magnitude of the populations of the United States and Mexico in 2011.
Preparing for the Next Lesson (2.9)
In Lesson 2.9, you will explore an extensive data set related to factors that form the basis for the 2010 list of “Best Cities for the Next Decade,” published by Kiplinger.[22] The data are based on 367 cities. A sample of the data set is given below.
Notes on the table: COL is cost of living, and % Workforce in Creative Class refers to the percentage of people who are employed as scientists, engineers, educators, writers, artists, and entertainers.
|Metro Area |Population |
|Perform basic operations using quantities such as integers, fractions, or decimals | |
|with the aid of technology. | |
|Calculate a ratio and write the result as a percent. | |
|Calculate and apply the measures of central tendency. | |
Specific Objectives
Students will understand that
multiple sources and types of data are important for making judgments and decisions.
Students will be able to
draw appropriate conclusions from data.
apply previous learning to new contexts.
Problem Situation: Data on the Best Cities
In this lesson, you will compare cities based on data from the 2010 list of Best Cities for the Next Decade. This list was published by a national financial news organization called Kiplinger.[23] The list covers 367 cities. Your instructor will project the full list. Then you will work with a shorter list of 20 cities given on the next page. You will be viewing six of the categories from the list. One of these categories needs some explanation. “Percentage of Workforce in Creative Class” refers to the percentage of working people who are, for example, scientists, engineers, educators, writers, artists, and entertainers.
|Metro Area |Population |Cost of Living|Percentage of |Median Household|Income Growth from|Employment Growth |
| | |Index |Workforce in |Income |2005 to 2007 | |
| | | |Creative Class | | | |
|Albany-Schenectady-Troy, NY |851,925 |100.00 |20.7 |54,755 |4.25% |1.62% |
|Austin-Round Rock, TX |1,590,744 |94.93 |25.1 |54,827 |3.46% |5.54% |
|Boston-Cambridge-Quincy, MA-NH |4,649,838 |129.53 |35.8 |66,870 |4.01% |1.95% |
|Boulder, CO |289,005 |124.74 |33.1 |63,064 |5.82% |4.04% |
|Canton-Massillon, OH |407,530 |100.00 |22.4 |44,530 |3.19% |-1.60% |
|Fresno, CA |895,357 |119.17 |28.2 |44,979 |3.99% |2.63% |
|Great Falls, MT |81,888 |100.00 |26.8 |41,802 |2.98% |1.19% |
|Ithaca, NY |100,535 |103.31 |28.4 |46,225 |-1.63% |39.84% |
|Jackson, MS |533,870 |95.82 |24.6 |42,921 |4.44% |4.01% |
|Jonesboro, AR |115,787 |85.47 |22 |36,527 |4.29% |0.65% |
|Kokomo, IN |99,631 |100.00 |25.9 |47,040 |4.77% |0.69% |
|McAllen-Edinburg-Mission, TX |706,039 |86.90 |28 |28,328 |2.58% |3.73% |
|New Bedford, MA |173,441 |129.53 |38.2 |66,870 |3.43% |0.64% |
|Oklahoma City, OK |1,189,529 |89.92 |27.4 |43,652 |3.65% |1.69% |
|Seattle-Tacoma-Bellevue, WA |3,299,005 |114.64 |33.9 |61,740 |4.06% |3.79% |
|Springfield, IL |206,445 |86.20 |30.6 |49,116 |4.82% |-2.07% |
|Waterbury, CT |199,412 |129.53 |27.1 |66,870 |2.01% |0.11% |
|Wausau, WI |129,849 |96.18 |21.9 |52,241 |3.13% |0.66% |
|Yuma, AZ |189,682 |103.64 |28.9 |38,502 |3.82% |2.17% |
(1) McAllen, Texas, is the largest city in Hidalgo County, one of the fastest growing counties in the United States. The US News lists McAllen as one of the top places to retire.[24] Kiplinger listed Boulder, Colorado, as one of its top 10 cities in 2010.[25]
(a) Use the information on your worksheet to decide in which city you are most likely to “live well” financially. Use estimation skills to justify your decision.
(b) Calculate how much a person in McAllen would need to make in order to have the same “buying power” as someone earning the median income in Boulder.
(2) Use the Kiplinger information to answer the following questions comparing Waterbury, Connecticut, and Springfield, Illinois.
(a) In which city are there more families with an income above $50,000?
(b) In which city are there more families with an income between $50,000 and $67,000?
(3) Review the information for Ithaca, New York. What do you think was happening in the job market from 2005 to 2007?
Making Connections
Record the important mathematical ideas from the discussion.
Further Applications
(1) This course draws on examples from three themes: Citizenship, Personal Finance, and Medical Literacy. Choose at least two different lessons with the Medical Literacy theme. Answer the following questions.
(a) What new information did you learn about medical literacy?
(b) How will you use this information or why is it important to know this information?
Student Notes
Two methods of sorting are shown here.
Example 1 uses the Sort option in the toolbar
This is an example.
(1) Click on the upper-left cell and drag diagonally downward to the lower-right cell (indicated by the red arrow) to highlight all of the cells. Make sure to include the cells that contain the column names at the top.
(2) Click on Data from the menu at the top of the screen, and select Sort from the pop-up menu, as shown below. Note that this example uses the toolbar at the very top of the window. Some versions of Excel also have Tabs that include a Data option.
[pic]
(3) In the window that pops up, click on the [pic] button in the Sort by section at the top and select the category that you want to sort. In the figure below, the category called “Price” is selected. Also, make sure that the circular button labeled “Ascending” is selected, which will sort the data from the lowest to highest Price.
[pic]
(4) Click OK. The pop-up window will close and the data will now be sorted according to Price.
Example 2 uses the Sort Tab
This example is from Lesson 2.9.
• Click in the spreadsheet.
• Select the Data tab.
• Click on the small black triangle by the sort icon on the right side of the toolbar:
[pic]
• Select Custom Sort
• Make sure the box by “My list has headers” is checked.
• Click on the arrows under the column to select the first category. Note that in some cases, high values are desired and in others, low values are desired. So you might sort Cost of Living from smallest to largest, but sort Income from largest to smallest.
• Click the “+” button at the bottom of the dialog box to add another level of sorting.
• Click on the arrows under the column to select the second category.
• Repeat the last two steps to add the third category.
• Click “OK.”
[pic]
This assignment will give you an opportunity to demonstrate your learning on many skills you have learned in the first two modules of the course, including:
Mathematical concepts
Interpreting quantitative information
Writing about quantitative information
In addition, you will be using new skills introduced in the Culminating Activity Parts 1 and 2: finding information through an Internet search and making graphs.
The overall task is to research a question and create a poster to answer the question. In addition to the poster, you will also give a two- to three-minute presentation about your work. Choose one of the following three questions for your project.
Option 1
Your business has decided to advertise on three social networking sites: Facebook, LinkedIn, and Twitter. You have been asked to present data to your boss on how to split a budget of $250,000 among the three sites.
Option 2
You are a hospital administrator. Your hospital board is concerned about future staffing problems. Present information on the availability of nurses in the United States in the next
few years. Do they have reason to be concerned about the availability of nurses to staff their hospital? Present information that details whether your hospital board should be concerned
or not.
Option 3
You are a representative of a student organization at your college. You have been asked to make a presentation to your college’s board of trustees regarding recent changes in tuition at public colleges and universities. You decide to focus on the question, “Have recent tuition increases at public colleges and universities been larger than in previous years?”
This work will take place over three class days. The purpose of each class is explained below. You should fill in the dates for each class based on information from your instructor.
Culminating Activity Part 1: Internet Searches
Purpose: Learn how to find relevant and reliable sources on the Internet.
Preparation: None
Assignment: Find two to three relevant and reliable resources to answer your question for the Culminating Activity, Part 2.
Date: _____________________________
Culminating Activity Part 2: Evaluate Resources and Making Graphs
Purpose: Review your resources; learn how to make graphs.
Preparation: Bring information about the resources you found.
Assignment: Prepare poster and presentation for Culminating Activity, Part 3.
Date: _____________________________
Culminating Activity Part 3: Poster Session
Purpose: Present your poster to classmates. Discuss what makes communication about quantitative information effective.
Preparation: Poster presentation.
Date: _____________________________
Your poster does not have to be a certain size, but you do want to make sure it is large enough that people can read it easily. You may tape paper together to make a larger space. You do not have to buy special materials for this assignment. Your poster and presentation must meet the following criteria.
Poster
Organization and Neatness
• Title is informative.
• Main points stand out.
• The order of the material is easy to follow and makes sense.
• Text and graphics are easy to read.
• Appearance is neat.
Note: Text should be typed, but hand-drawn graphs are acceptable; sophisticated use of technology will not be rated more highly than carefully prepared work done by hand.
Meets All the Requirements of the Assignment
• Poster contains a graph.
• Poster contains an answer to the question (at least one paragraph).
• Contains references to at least two sources.
Completely Answers the Question
• Answer to the question is clear.
• Appropriate and accurate quantitative information supports the answer.
• Reasoning is explained, makes sense, and is adequate to support the answer.
• Answer shows that the student thought about multiple aspects of the question.
• Sources are relevant and reliable.
Graphics
• The type of graph is appropriate for the data and the purpose.
• Graph supports the answer and makes sense as a part of the answer.
• Graph contains all necessary elements (title, axis labels, scales or legend, etc.).
Grammar
• Uses correct grammar, punctuation, and spelling.
Presentation
• Speaker is easy to hear.
• Speaker gives a complete summary.
• Information is clear and well organized.
In addition to making a presentation, you will be expected to evaluate at least two posters and presentations from other students. Your instructor will explain how to use a rubric for the evaluation. You will be asked to give specific feedback on the work. Specific feedback should tell the presenter exactly what you think is good or what needs to be improved. Some examples are given below.
|NOT good feedback |Good feedback |
|I liked the graph. |The graph helped me understand because it showed the trend over time very |
| |clearly. |
|The explanation was not good. |The explanation was confusing because it started with the reasons for the |
| |answer before stating the answer. |
Specific Objectives
Students will understand that
search engines return more useful results when given more specific queries.
some Internet sites are more reliable than others.
Students will be able to
conduct a search to answer a quantitative question.
determine whether a source found online is relevant and reliable (introductory level).
Problem Situation: Searching for Information on the Internet
The Internet has made tremendous amounts of data available. There is so much data out there that sometimes it is hard to know what to trust. When you search for data to answer a question, you want to make sure the data is relevant to the topic. In other words, the data should help you answer the question you are asking. You also want the data to be reliable, meaning that you can trust that the data is accurate. This is the first part of three lessons about using the Internet to find data. In the second and third lessons, you will research and prepare a brief presentation about a specific question.
In this lesson, you will talk about how to conduct an Internet search and how to evaluate the results. You will consider the specific question, “Has air quality improved in Los Angeles?”
Air pollution was a growing problem in the United States throughout the 19th and 20th centuries. The U.S. and state governments have made attempts to limit air pollution, especially in the last third of the 20th century. As a class, you will search for information that will answer the question.
As you conduct Internet research and discuss the question “Has air quality improved in Los Angeles?” you will need to understand the following terms that have to do with types of sources:
Reliable
Relevant
Primary source
Secondary source
OCE Assignment
This lesson has a different type of out-of-class experience from other lessons. You will receive a student overview of the Module 1 and 2 Culminating Activity that summarizes the full assignment. Specifically, for this assignment, you need to do the following:
(1) Find two to three Internet resources that could be used to answer the question you selected from the list provided on the Student Overview.
(2) For each resource:
Record the URL.
Record the search terms you used.
Summarize the information on the site that you might use to answer the question.
Explain why you think the site is reliable and relevant.
You will need to bring this information to class for the Culminating Activity Part 2. Your instructor will give you the due date.
Further Applications
(1) The Internet is an important source of information, but there are other sources. Give at least three sources for information other than the Internet.
Specific Objectives
Students will understand that
line graphs and bar graphs effectively show change over time.
pie charts effectively compare portions of a whole.
Students will be able to
choose whether a pie chart or a bar graph better displays data in support of the answer to a quantitative question.
choose an appropriate resource with quantitative data to answer a question.
In this lesson, you will report back on the resources you found on your first attempt to research the poster project introduced in the Culminating Activity Part 1.
You will then further consider which graph to use to best present the data you have found, or hope to find for your poster project.
Problem Situation 1: What Resources Did You Find?
The purpose of this activity is to
get feedback from your classmates about the reliability and relevance of the resources you have found.
share ideas about finding additional resources and suggest solutions to any problems you or your classmates have had.
Group members should briefly report the following about their resources:
Search terms used
Information on sites
Why the site is relevant and reliable
As you have only 15 minutes to complete this task, it might be best to have each student report on what they found and where they found it for one minute per group member before you start asking questions. As other students report on their sites, think about questions you might want to ask them about the relevance and reliability of their website. You should also record ideas about your resources for later use.
Record ideas and questions about your resources below.
Questions about the relevance of my resources
Questions about the reliability of my resources
Search terms I want to try
Resources I want to try
Problem Situation 2: Creating Good Graphs
The purpose of the next activity is to help you think about how to make graphs of data.
The following data represent the populations of the three largest counties in the Detroit Metropolitan Area over the last century. [26] Your teacher will assign your group one of the five questions below to answer.
Your group must create a rough graph that clearly answers your question using the data in the table. A rough graph does not need exact measurements, but it should be qualitatively correct. For example, in a pie chart, the piece of data with the largest value should get the largest slice. If one piece of data is more than half the total, it should take up more than half the circle. In a bar graph, if one value is twice another, the bar should look approximately twice as high. Your graph should also include correct titles, labels, and scales.
Part of answering these questions is for your group to choose the most appropriate type of graph to display the data.
|Wayne |Oakland |Macomb |
|Year |Population |Year |Population |Year |Population |
|1900 |348793 |1900 |44792 |1900 |33244 |
|1910 |531591 |1910 |49576 |1910 |32606 |
|1920 |1177645 |1920 |90050 |1920 |38103 |
|1930 |1888946 |1930 |211251 |1930 |77146 |
|1940 |2015623 |1940 |254068 |1940 |107638 |
|1950 |2435235 |1950 |396001 |1950 |184961 |
|1960 |2666297 |1960 |690259 |1960 |405804 |
|1970 |2666751 |1970 |907871 |1970 |625309 |
|1980 |2337891 |1980 |1011793 |1980 |694600 |
|1990 |2111687 |1990 |1083592 |1990 |717400 |
|2000 |2061162 |2000 |1194156 |2000 |788149 |
|2010 |1820584 |2010 |1202362 |2010 |840978 |
(1) When did Wayne County reach its highest population?
(2) What proportion did each of the three counties (Wayne, Oakland, and Macomb) make up of the Detroit tri-county area in 1960?
(3) What were the relative sizes of Wayne, Oakland, and Macomb Counties in 1960?
(4) What were the relative sizes of Wayne, Oakland, and Macomb Counties in 2010?
(5) When was Wayne County growing in population? Shrinking in population?
OCE Assignment
Your assignment is to prepare for the poster session in the Culminating Activity Part 3 as explained in the Student Overview. Your instructor will give you the date for the poster session.
The following information is provided to help you use technology to make a graph for your poster. You are not required to use this. Talk to your instructor if you have any questions.
A Brief Overview Of Creating A Graph In Googledocs
1. Go to . Under the More tab at the top of the screen, select Documents. If you have a gmail account, use your information to log in. (If you do not have a gmail account, you will need to create one to use Googledocs.)
2. Select the Create New tab. Select spreadsheet.
3. Type the data you wish to use to create a graph into cells in adjacent columns. Use the mouse to select cells, or type enter when done typing in a cell to go to the one underneath it.
[pic]
4. Highlight the cells you wish to use to make your graph. One way to do this is to select a cell in one corner, press and hold down the left mouse button, and drag to a corner so that all the data you wish to use is highlighted in light blue.
5. Press the button that looks like a bar graph.
[pic]
6. Experiment with the settings until you see a chart you like.
[pic]
7. You can customize further with options under the Customize tab if desired.
[pic]
8. Press Insert.
9. Save your work. (Save is under the file menu.)
10. Print your chart. If you do not wish to print the entire spreadsheet you can click on the graph and one of the options under Chart 1 will be to save as an image. You can save that file and later insert it in a word processing program. Or, select the chart by clicking on it and choose selection on the spreadsheet print menu.
[pic]
Reviewer Name: Presenter Name:
Place an “X” in the box for the rating of each category.
|Categories |Exemplary |Good |Needs Improvement |Unsatisfactory |
|APPEARANCE |
|Organization and neatness |Meets all criteria |Meets all criteria, |Missing one element or major |Missing multiple elements or |
|Title is informative. |fully. |but has minor problems|problems with one or two of the|major problems with more than |
|Main points stand out. | |with some. |criteria. |two of the criteria. |
|The order of the material is easy to follow and makes sense. | | | | |
|Text and graphics are easy to read. | | | | |
|Appearance is neat. | | | | |
|Note: Text should be typed, but hand-drawn graphs are acceptable; sophisticated use of technology | | | | |
|should not be rated more highly than carefully prepared work done by hand. | | | | |
|CONTENT |
|Completely answers the question |Meets all criteria |Meets all criteria, |Missing one element or major |Missing multiple elements or |
|Answer to the question is clear. |fully. |but has minor problems|problems with one or two of the|major problems with more than |
|Appropriate and accurate quantitative information supports the answer. | |with some. |criteria. |two of the criteria. |
|Reasoning is explained, makes sense, and is adequate to support the answer. | | | | |
|Answer shows that the student thought about multiple aspects of the question. | | | | |
|Sources are relevant and reliable. | | | | |
|Graphics |Meets all criteria |Meets all criteria, |Missing one element or major |Missing multiple elements or |
|The type of graph is appropriate for the data and the purpose. |fully. |but has minor problems|problems with one or two of the|major problems with more than |
|Graph supports the answer and makes sense as a part of the answer. | |with some. |criteria. |two of the criteria. |
|Graph contains all necessary elements (title, axis labels, scales or | | | | |
|legend, etc.). | | | | |
|Presentation |Meets all criteria |Meets all criteria, |Missing one element or major |Missing multiple elements or |
|The speaker is easy to hear. |fully. |but has minor problems|problems with one or two of the|major problems with more than |
|The speaker gives a complete summary, but does not just read from the poster. | |with some. |criteria. |two of the criteria. |
|Information is clear and well organized. | | | | |
Reviewer: On the back of this form, write at least two comments that give specific feedback to the presenter.
As with Module 1, you should assess your understanding of Module 2 to prepare for the Module 2 test. Your instructor may give you specific assignments for your review in addition to this self-assessment.
Assessing Your Understanding
The table on the following page lists the Module 2 concepts and skills you should understand. This exercise helps you assess what you understand. After completing it, you will be able to prioritize your review time more effectively.
1. Assess your understanding.
Go through the topics list and locate each concept or skill in the Module 2 in-class or OCE materials.
If you have not used the skill in a while, do two or more problems to check your understanding.
If you have recently used the skill and feel confident that you did it correctly, rate your understanding a 4 or 5.
If you remember the topic but could use more practice, rate your understanding a 3.
If you cannot remember that skill or concept, rate your understanding a 1 or 2.
Now that you have done an initial rating of your understanding, it is time to begin reviewing. Complete the remaining steps. The goal is to have a confidence rating of 4 or 5 on all the topics in the table when you have finished your review of Module 2.
2. Start at the beginning of module and reread the material in the lessons, the OCE, and your notes on the skills and concepts you rated 3 or below.
3. Select a few problems to do. Do not look at the answer or your previous work to help you.
4. Once you have finished the problems, check your answers. If you are not sure if you have done the problems correctly, check with your instructor, other classmates, and your previous work or work with a tutor in the learning center.
5. Rate your confidence on this skill again. If you understand the concept better, rate yourself higher. Begin a list of topics that you want to review more thoroughly.
6. If you have time, do one or two problems on skills or concepts you rated 4 or above.
7. For topics that you need to review more thoroughly, make a plan for getting additional assistance by studying with classmates, visiting your instructor during office hours, working with a tutor in the learning center, or looking up additional information on the Internet.
|Module 2 Concept or Skill |Rating |
|Using Ratios |
|Understand meaning of equivalent ratios in context (2.1) | |
|Use units with ratios (2.1) | |
|Calculate a unit rate (2.2) | |
|Use ratios and proportionality to calculate new values (2.2) | |
|Interpret and use index numbers to calculate new values (2.8, 2.9) | |
|Applications of Percentages |
|Calculate and interpret absolute change between two quantities (2.3, 2.5) | |
|Calculate and interpret relative change between two quantities (2.3, 2.5) | |
|Calculate and interpret absolute change between two percentages (2.3) | |
|Calculate and interpret relative change between two percentages (2.3) | |
|Make and interpret comparisons of absolute measurements versus relative measurements (2.4, 2.5) | |
|Graphical Displays |
|Read and interpret pie graphs, bar graphs, and line graphs (2.4) | |
|Recognize distortion of graphs due to different scales (2.4) | |
|Calculate absolute and relative change from a graph (2.4) | |
|Measures of Central Tendency |
|Calculate mean, median, and mode of a data set (2.6, 2.7) | |
|Interpret the meaning of and differences between the mean, median, and mode | |
|(2.6, 2.7) | |
+++++
This Module is part of QUANTWAY™, A Pathway Through College-Level Quantitative Reasoning, which is a product of a Carnegie Networked Improvement Community that seeks to advance student success. The original version of this work, version 1.0, was created by The Charles A. Dana Center at The University of Texas at Austin under sponsorship of the Carnegie Foundation for the Advancement of Teaching. This version and all subsequent versions, result from the continuous improvement efforts of the Carnegie Networked Improvement Community. The network brings together community college faculty and staff, designers, researchers and developers. It is a research and development community that seeks to harvest the wisdom of its diverse participants through systematic and disciplined inquiry to improve developmental mathematics instruction. For more information on the QuantwayTM Networked Improvement Community, please visit .
+++++
Quantway™ is a trademark of the Carnegie Foundation for the Advancement of Teaching. It may be retained on any identical copies of this Work to indicate its origin. If you make any changes in the Work, as permitted under the license [CC BY NC], you must remove the service mark, while retaining the acknowledgment of origin and authorship. Any use of Carnegie’s trademarks or service marks other than on identical copies of this Work requires the prior written consent of the Carnegie Foundation.
This work is licensed under a Creative Commons Attribution-NonCommercial 3.0 Unported License. (CC BY-NC)
[pic]
-----------------------
[1] Retrieved from
[2] Retrieved from aatlas/populations/ctypopls.htm
[3]Retrieved from
[4]Retrieved from
[5]
[6]Annenburg Learner,
[7] hhes/www/income/data/historical/household/index.html
[8] graphics/how-the-average-consumer-spends-their-paycheck
[9]Wikipedia: Free Range Eggs:
[10]Retrieved from const/uspriceann.pdf
[11]Retrieved from CDC website:
[12]Retrieved from
[13]Retrieved from credit-card-news/credit-card-industry-facts-personal-debt-statistics-1276.php
[14]Retrieved from nrmsc.files/norock/products/IGBST/2009report.
[15]Retrieved from yell/naturescience/bearsighttable.htm.
[16]Retrieved from nature.stats/park.cfm?parkid=421.
[17]Data retrieved from the U.S. Census Bureau.
[18]Retrieved from fls/flsgdp.pdf.
[19]Retrieved from aboutoilgas/gasoline/upload/PumpPriceUpdate.pdf
[20]Retrieved from oasam/programs/history/flsa1938.htm.
[21]Retrieved from .
[22]Retrieved from tools/bestcities_sort/index.php?si=1
[23]Retrieved from tools/bestcities_sort/index.php?si=1
[24]Retrieved from
[25]Retrieved from magazine/archives/10-best-cities-2010-for-the-next-decade.html
[26]Retrieved from population/cencounts/mi190090.txt ; ; and county Wikipedia pages.
-----------------------
LESSON 2.1
OCE 2.1
Rectangle A
Rectangle B
Area of Rectangle B = 4 ft2
Area of Rectangle A = 12 ft2
LESSON 2.2
OCE 2.2
LESSON 2.3 2.3
OCE 2.3
LESSON 2.4
OCE 2.4
LESSON 2.5
OCE 2.5
[pic]
LESSON 2.6 2.6
[pic]
OCE 2.6
LESSON 2.7
OCE 2.7
LESSON 2.8
OCE 2.8
LESSON 2.9
REVIEW
................
................
In order to avoid copyright disputes, this page is only a partial summary.
To fulfill the demand for quickly locating and searching documents.
It is intelligent file search solution for home and business.
Related searches
- south college tuition
- university of south florida college of medicine
- south college enroll
- south college catalog
- home walb com south georgia news weather sports
- south georgia deer hunts
- georgia college transcript request
- west georgia technical college transcript
- savannah georgia college of art
- georgia college rankings
- georgia college and state university
- free college for georgia residents