Integers What, Me Negative? 10.2 Number Sets Ordering and Absolute Value

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1 Integers At 282 feet below sea level, Badwater Basin in Death Valley, California, is the lowest spot in the United States. Since terrain elevation is measured in comparison with sea level, Death Valley's elevation is expressed as a negative number: _282 feet What, Me Negative? Introduction to Negative Integers Number Sets Number Systems Ordering and Absolute Value Ordering the Rational Numbers Elevators, Making Money Redux, and Water Level Solving Problems with Rational Numbers

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3 What, Me Negative? Introduction to Negative Integers Learning Goals In this lesson, you will: Plot integers on a number line. Solve problems where quantities increase and decrease. Calculate the differences between quantities. Represent quantities using positive and negative numbers. Key Terms negative numbers infinity negative sign positive sign integers ellipses The classic science-fiction novel Twenty Thousand Leagues Under the Sea, written by Jules Verne, tells the story of the underwater adventures of Captain Nemo and his crew aboard a submarine called the Nautilus. The title of this novel, however, does not refer to the depth that the men travel, but the distance. A league, as used in the novel, was equal to exactly 4000 meters, or about 2.5 miles. A depth of 220,000 leagues would be equal to about 250,000 miles, but the diameter of the entire Earth is only about 8,000 miles. The greatest depth mentioned in the novel was only about 24 leagues. Negative numbers are often used to refer to depths. How else are negative numbers used in the real world? 10.1 Introduction to Negative Integers 681

4 Problem 1 Going Up! An office building in downtown Los Angeles has 32 floors above ground. It also has 6 floors below ground for parking. Henry parks on the third floor below ground and works on the 25th floor. Janis works on the top floor and parks on the lowest floor. Albert works on the 15th floor but parks on whatever floor he can. 1. Let s consider a day of traveling up and down the building for Henry. a. Henry decides to go to lunch with Albert, but he needs to go to his car first because he forgot his wallet. How many floors does Henry go down to get to his car? Explain your reasoning. You might want to draw a picture for yourself to help answer these questions. b. After Henry gets his wallet, he goes up to Albert s office. How many floors does he go up to get to Albert s office? Explain your reasoning. c. If Henry did not go to his car, how many floors would he need to go up or down to get to Albert s office? Explain your reasoning. d. Albert and Henry go to lunch, and then Henry returns to his office. How many floors in total did Henry travel by the time he got back to his office? Explain your reasoning. 682 Chapter 10 Integers

5 e. List the total number of floors that Henry has gone up or down for the entire day, including his trip to the parking garage at the end of the day. 2. Let s consider a day of traveling up and down the building for Janis. a. After Janis parks her car, how many floors does she go up to get to her office? Explain your reasoning. b. Janis leaves her office to eat lunch at the café on the 10th floor. She then returns to her office. How many floors in total does she go up and down to eat lunch? Explain your reasoning. c. List the total number of floors that Janis has gone up or down for the entire day, including her trip to the parking garage at the end of the day Introduction to Negative Integers 683

6 3. On Monday, Albert eats his lunch at his desk and parks on the fifth level below ground. On Tuesday, he takes the bus to work and decides to go out for lunch. On which day does Albert go up and down the most floors? Explain your reasoning. When I think about the total number of floors traveled in a day, should I start counting from the moment Albert enters the parking garage or when he gets out of his car? 4. On which level below ground should Albert park if he only wants to go up 19 floors to get to his office? 5. On which level below ground should Albert park if he only wants to travel up and down a total of 32 floors to get to his office and back to his car? 6. In a normal day, if both Janis and Henry ate lunch at their desks, who would travel more floors? How many more floors would this person travel? Explain your reasoning. 684 Chapter 10 Integers

7 Problem 2 Easy Come, Easy Go! Helen and Grace started a company called Top Notch. They calculated the company s profit and loss each week. The table shown represents the first 10 weeks of operation. Losses are represented by amounts within parentheses. For example, ($25) denotes a loss of $25. Amounts that are not in parentheses are profits. Week Total Profit or Loss Profit or Loss $159 ($201) $231 ($456) ($156) ($12) $281 $175 $192 $ In which week did Top Notch show: a. the largest profit? b. the largest loss? 2. Between which two weeks did Top Notch have the largest gain in money? What was this gain? 3. Between which two weeks did Top Notch have the largest loss? What was the loss? 4. What was the difference between the company s best week and its worst week? 5. Calculate the total profit or loss for Top Notch, and record the answer in the table Introduction to Negative Integers 685

8 Problem 3 Temperature Connection Most of the numbers you have worked with in school have been numbers that are greater than or equal to zero. However, you have heard about numbers that are less than zero. Discuss the meaning of each of these statements: The weather forecaster predicts the temperature will be below zero. A submarine travels at 3000 feet below sea level. Badwater Basin in Death Valley, California, is 282 feet below sea level. 1. Mark each temperature on the thermometer shown. a. The highest temperature on record in the United States is 134 F. It occurred in 1913 in Death Valley, California. b. The lowest temperature on record is 280 F. It occurred at Prospect Creek Camp, Alaska. c. The lowest temperature recorded in the contiguous 48 states is 270 F. It occurred in Montana. d. The highest winter average temperature in the United States is 78 F, which occurs in Honolulu, Hawaii ºF 686 Chapter 10 Integers

9 2. Use the thermometer to answer each question. a. One day in 1918 in North Dakota, the temperature went from 233 F to 50 F. How many degrees did the temperature rise? b. Within 24 hours in Montana in 1916, the temperature went from 44 F to 256 F. How many degrees did the temperature fall? c. Within fifteen minutes one day in South Dakota, the temperature went from 55 F to 8 F. How many degrees did the temperature fall? ºF 3. Order these temperatures from least to greatest. 25 F 233 F 0 F 125 F 250 F 25 F 67 F 10.1 Introduction to Negative Integers 687

10 Problem 4 To Negative Infinity A thermometer is like a vertical number line labeled with numbers. You have been using the positive number line since early elementary school A number line can be created by reflecting the positive numbers across zero. Numbers to the left of zero on the number line are called negative numbers and are labeled with a negative sign. The symbol means infinity which means a quantity without bound or end. Negative infinity Positive infinity Describe the change in the values of the integers as you move to the right on the number line. 2. Describe the change in the values of the integers as you move to the left on the number line. Each positive number has a corresponding opposite number that is the same distance away from zero. Similarly, each negative number has a corresponding opposite number that is the same distance away from zero. Two different numbers, such as 5 and 25, are each the same distance from zero and are opposites of one another. The opposite of 5 is 25, and the opposite of 25 is 5. The opposite of the opposite of a number is the number itself. The opposite of 0 is 0. Calculators have a special negative sign key (_). 688 Chapter 10 Integers

11 Attaching a negative sign to a number means reflecting that number across zero on the number line. A negative number is always written with a negative sign. You can write a positive number with a positive sign or without any sign. For example, positive 5 can be written as 15 or 5. The integers are the set of whole numbers with their opposites. The integers can be represented by the set {...,25, 24, 23, 22, 21, 0, 1, 2, 3, 4,...}. Notice the three periods before and after the number set. These three periods are called an ellipsis and they are also used to represent infinity in a number set. 3. Sort the numbers shown into the appropriate set. Explain how you determined your sorting. 22.6, 126, 226, 213, 5, 1228, 13.25, 285, 20, Not Integers Negative Integers Positive Integers 10.1 Introduction to Negative Integers 689

12 4. Name the opposite of each integer given. Then, graph each integer and its opposite on the number line provided. a b c d e. Describe the distances the numbers in each pair are from zero on the number line. Talk the Talk Now that you have learned about positive and negative numbers, explain how the use of these numbers could have been helpful in analyzing each problem. 1. Problem 1 Going Up! 2. Problem 2 Easy Come, Easy Go! Be prepared to share your solutions and methods. 690 Chapter 10 Integers

13 Number Sets Number Systems Learning Goal In this lesson, you will: Classify sets of numbers. Key Terms fractional numbers rational numbers Density Property Just like scientists study and classify different kinds of plants, animals, rocks, chemicals, and so on, mathematicians study and classify different kinds of numbers. For example, the number 142,857 has six digits and when you multiply that number by 2, 3, 4, 5, or 6, you get a number with the same digits but in a different order. 142, , , ,571 These are just two examples. A number like this is called a cyclic number. A whole number that is equal to the sum of its factors except for itself is known as a perfect number. The number 6 is a perfect number, because A triangular number can be represented by dots in the shape of a triangle. Three, six, and ten are triangular numbers: Can you identify other triangular numbers or perfect numbers? 10.2 Number Systems 691

14 Problem 1 Sets of Numbers During your years in school you have worked with different sets of numbers, including the set of counting numbers and the set of whole numbers. 1. What numbers are included in the set of counting numbers, c? Use brackets, { }, to represent sets. 2. What numbers are included in the set of whole numbers, W? You have also worked with the set of fractional numbers, which is the set of all numbers that can be written as a, where a and b are whole numbers and b fi 0. b 3. Using this definition of fractional numbers, rewrite each number as a fractional number. Then, identify the values of a and b. a. 2 b. 3 7 c. 3 1 d e In the definition of fractional numbers, b fi 0. Why? 5. Are the counting numbers fractional numbers? Explain your reasoning. You might want to go back and reread the definition of fractional numbers. 692 Chapter 10 Integers

15 6. Are the whole numbers fractional numbers? Explain your reasoning. 7. You have recently been introduced to the set of integers. Are all integers fractional numbers? Explain your reasoning. As with the integers, if fractional numbers are graphed on a number line and then are reflected across zero, a new set of numbers called the rational numbers occurs. Negative infinity Positive infinity The definition of the set of rational numbers is the set of all numbers that can be written as a, where a and b are integers and b fi 0. b 8. Rewrite each number as a rational number using the definition of rational numbers. Then, identify the values of a and b. a. 2 b c d e Number Systems 693

16 Problem 2 Classifying Numbers There are many ways you can classify numbers. Many of the classifications are subsets of other classifications. The diagram shows the different sets of numbers you have encountered in your mathematical experiences. Rational Integers Whole Natural Natural numbers are a subset of whole numbers. Whole numbers are a subset of integers. Integers are a subset of rational numbers. Natural Numbers Whole Numbers Integers Rational Numbers Examples 1, 2, 3, 0, 1, 2, 3,, 23, 22, 21, 0, 1, 2, 3, 26.5, 22, 2 2 3, 0, 2 3, 2, 6.5 Description Counting numbers Natural numbers and 0 Whole numbers with their opposites Integers and fractions, including repeating and terminating decimals, with their opposites 694 Chapter 10 Integers

17 1. Write all the sets to which each number belongs. a. 3 b c. 0 d e f. 54 g. 25 h i j. 21,364, Explain how rational numbers are related to integers. 3. Can a number be an integer and a whole number? Give examples, if possible Number Systems 695

18 Problem 3 Density The Density Property states that between any two rational numbers there is another rational number. The property is not true for natural numbers, whole numbers, or integers. For example, there is no integer between 25 and 26. There is no whole number or natural number between 12 and Plot each number on the number line a b c. 1 5 d e f. 2 3 g h i. 1 j Plot and label a rational number between each pair of rational numbers given. a. 4 1 and b. 5.5 and c and d and Chapter 10 Integers

19 Talk the Talk Determine if each statement is true or false. Explain your reasoning. 1. All integers are rational numbers. 2. All whole numbers are integers. 3. All rational numbers are integers. 4. All integers are whole numbers. Be prepared to share your solutions and methods Number Systems 697

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21 Ordering and Absolute Value Ordering the Rational Numbers Learning Goals In this lesson, you will: Order rational numbers. Define the absolute value of a number. Calculate the absolute value of a number. Key Term absolute value Chances are you have heard of the equator from your social studies class. The equator is the imaginary line that horizontally circles the earth. But, did you know that the equator can be used as a basis to give a specific location to any place on earth? This is because the equator is a line of latitude. Lines of latitude are many imaginary lines that horizontally circle the earth north and south of the equator. The equator is known as being at 0 latitude because it splits the earth in half. Lines of latitude north of the equator have a positive measure of degrees and lines of latitude south of the equator have a negative measure of degrees. For example, the North Pole is at 190 latitude and the South Pole is at 290 latitude. This means that the North Pole and the South Pole are equal distance away from the equator. How do the lines of latitude relate to the use of number lines in mathematics? 10.3 Ordering the Rational Numbers 699

22 Problem 1 Comparing and Ordering Rational Numbers 1. Use the number line and the number set to answer each question. a. Plot each value on the number line b. Which of the numbers has the least value? How do you know? c. Which of the numbers has the greatest value? How do you know? 2. Plot each rational number on the number line. Then, insert a.,,, or 5 symbol to make each number sentence true. a b c d Chapter 10 Integers

23 e f g h Explain how you knew which rational number was greater in Question Explain how you knew which rational number was the lesser in Question What is the least nonnegative rational number? 6. Give an example of two rational numbers that are the same distance from zero Ordering the Rational Numbers 701

24 7. Which is greater, a negative or a positive rational number? Explain your reasoning. 8. Which is greater, zero or any positive rational number? Explain your reasoning. 9. Which is greater, zero or any negative rational number? Explain your reasoning. 10. How do you decide which of two numbers is greater if both numbers are positive? 11. How do you decide which of two numbers is greater if both numbers are negative? 12. Order the rational numbers 27, 13.75, , , and 34 8 from least 9 to greatest. For any two rational numbers a and b, one and only one of the following is true: a, b a. b a 5 b A similar rule was stated in Chapter 9, but now we can say "for any two ra t i o n a l numbers." Our knowledge of numbers is expanding! 702 Chapter 10 Integers

25 Problem 2 The Distance from Zero The absolute value of a number is its distance from zero on a number line. Since distance can never be negative, absolute value is always positive or zero. The symbol for absolute value is. You can read the expression shown as the absolute value of a number n. n Consider the expression 25 and its distance from zero on the number line. Imagine you are standing at 0 holding one end of a rope and your friend is standing at 25 holding the other end of the rope. If your friend held onto the rope and walked in a semi-circle to the positive side of the number line, the end of the rope would put him at 15. The distance from 0 to 25 and from 0 to 5 is 5 in both cases. Your friend 5 You Determine each absolute value. a b c. 0 5 d Explain whether two different rational numbers can have the same absolute value. 3. What two numbers have an absolute value of 11? 4. What can you say about the absolute value of any positive number? 10.3 Ordering the Rational Numbers 703

26 5. What can you say about the absolute value of any negative number? 6. What can you say about the absolute value of zero? 7. Name all of the integers that are 10 units away from zero on the number line. 8. Name all the rational numbers that are 10 units away from 25 on the number line. 9. Insert a.,,, or 5 symbol to make each number statement true. a b c d e f Determine each sum or difference. a b c d e f Chapter 10 Integers

27 Talk the Talk Absolute values are used in real-world applications when you are interested in only the number and not in the sign of the number. When you look at temperature change, you could say the temperature fell by, decreased by, or increased by an absolute value. 1. Complete the table. Situation Absolute Value Statement Rational Number The temperature went from 45 F to 10 F. The temperature fell by 35 F. 235 F The value of his stock went from $500 to $250 last year. He lost $250 last year. Her savings account increased from $1250 to $2350 last year The water level dropped from 10 feet to 2 feet. His savings account decreased from $1500 to $100. Be prepared to share your solutions and methods Ordering the Rational Numbers 705

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29 Elevators, Making Money Redux, and Water Level Solving Problems with Rational Numbers Learning Goals In this lesson, you will: Solve problems using rational numbers. Use positive and negative numbers to represent quantities in real-world situations. Do you remember when you first learned to do something, like riding a twowheel bike, swimming, or surfing? When you began, it seemed that you would never be able to do it. Perhaps, even when you finally were able to ride the bike for a few feet without falling, or swim a few strokes, or get up on the surfboard for a few seconds, it still seemed that you could never be good at it. However, after practicing, it became easier and easier until it became almost automatic. Learning mathematics is very similar: after practicing for a while it almost becomes automatic. Think of some mathematics that you have learned that was hard at first, but now is almost automatic Solving Problems with Rational Numbers 707

30 Problem 1 Elevators Again A building has 10 floors of offices above ground and 4 floors below ground as shown. Ground floor 1. Draw a vertical number line next to the picture. Label the ground floor 0, and each level with the appropriate integer. 2. Melanie has an office on the 9th floor. a. How many floors must she go up from the ground floor? What is the absolute value of this number? b. If she parks on the 3rd floor below ground, what integer represents this floor? What is the absolute value of this number? c. How many floors must she go up from her car to reach the ground floor? d. How many floors must she go up from her car to her office? Write a numeric expression that would represent this situation using absolute values. 708 Chapter 10 Integers

31 3. Write an expression using absolute value for each situation, and then calculate the answer. a. Patrick parks his car on the 2nd floor below ground and works on the 9th floor. How many floors must he go up to get to his office? b. A woman working on the 8th floor goes down to her car on the 4th floor below ground, and then back up to the ground floor to go to lunch. How many floors does the woman travel? c. After lunch on the ground floor, Cassie goes down to her car on the 2nd floor below ground and then back up to ground floor. How many floors does she travel? What number is she on the number line when she is back on the ground floor? d. Jamal is working on the 10th floor and goes down to the ground floor for lunch. He then goes back up to the 5th floor for a meeting. How many floors does he travel? e. If Damon goes from his office on the 10th floor to a meeting on the 5th floor, how many floors does he travel and in which direction? 10.4 Solving Problems with Rational Numbers 709

32 Problem 2 Making Money Again Remember that Helen and Grace started a company called Top Notch. They calculated the company s profit and loss each week. The table shown represents the first 10 weeks of operation. Losses are represented by amounts within parentheses. Week Total Profit or Loss Profit or Loss $ ($201.35) $ ($456.45) ($156) ($12.05) $ $175 $ $ Rewrite the loss amounts in the table as negative numbers. Calculate the total profit or loss for Top Notch, and record the answer in the table. 2. Between which two weeks did Top Notch have the largest gain in money? What was this gain? 3. Write an absolute value equation to represent this gain. 4. Between which two weeks did Top Notch have the largest loss? What was the amount of this loss? 5. Write an absolute value equation to represent this loss. 6. What was the difference between the company s best week and their worst week? 7. Write an absolute value equation to represent this difference. 710 Chapter 10 Integers

33 Problem 3 Water Level As part of a long-term science experiment, two rulers were connected at zero and used to measure the water level in a pond. The connected rulers were placed in the pond so that the water level aligned at zero. The water level was measured each week for 10 weeks. The water level above zero was recorded as a positive number, and the water level below zero was recorded as a negative number. Week Water level Between which two weeks did the water level change the most? Write an absolute value equation for this problem situation and calculate the change. 2. Between which two weeks did the water level change the least? Write an absolute value equation for this problem situation and calculate the change. 3. How much did the water level change between Weeks 4 and 5? Write an equation for this situation and calculate the change Solving Problems with Rational Numbers 711

34 4. How much did the water level change between Weeks 8 and 9? Write an absolute value equation for this problem situation and calculate the change. 5. How much did the water level change between Weeks 9 and 10? Write an absolute value equation for this problem situation and calculate the change. Be prepared to share your solutions and methods. 712 Chapter 10 Integers

35 Chapter 10 Summary Key Terms negative numbers (10.1) infinity (10.1) negative sign (10.1) positive sign (10.1) ellipses (10.1) integers (10.1) fractional numbers (10.2) rational numbers (10.2) Density Property (10.2) absolute value (10.3) Solving Problems Where Quantities Increase and Decrease Negative and positive numbers can be used in real-world problems. Depths, heights, temperatures, gains and losses, all involve working with positive and negative numbers. Example The morning temperature was 26 F and reached a high of 12 F. The temperature rose 18 degrees during the day. Plotting Integers on a Number Line The integers are the set of whole numbers with their opposites. Did you know every time you learn something new, you grow more synapses, and expand your brain? Your brain is growing every day! The integers can be represented by the set {, 25, 24, 23, 22, 21, 0, 1, 2, 3, 4, 5, }. Example The opposite of 8 is shown and graphed on the number line. Opposite of 8: Chapter 10 Summary 713

36 Classifying Numbers There are many ways to classify numbers. Many of these classifications are subsets of other classifications. Number Sets Natural Numbers Whole Numbers Integers Rational Numbers Examples 1, 2, 3, 0, 1, 2, 3,, 23, 22, 21, 0, 1, 2, 3, 26.5, 22, 2 2 3, 0, 2, 2, Descriptions Counting Numbers Natural numbers and 0 Whole Numbers with their opposites Integers and fractions, including repeating and terminating decimals, with their opposites Example Write all the sets to which each number belongs. a. 5 rational numbers, integers, whole numbers, natural numbers b rational numbers c. 1 4 rational numbers Understanding the Density Property The Density Property states that between any two rational numbers there is another rational number. The property is not true for natural numbers, whole numbers, or integers. Example A rational number between 2.3 and 2.4 has been plotted and labeled on the number line shown Chapter 10 Integers

37 Comparing and Ordering Integers A number line can be used to compare and order integers. A number is greater than another number if it is to the right of that number on a number line. Likewise, a number is less than another number if it is to the left of that number on a number line. Example The integers in order from least to greatest are 213, 29, 21, 2, and 11. Each is graphed on the number line shown Determining Absolute Value The absolute value of a number is its distance from zero on a number line. Absolute value is always either positive or zero. The symbol for absolute value is. Example Solving Problems Using Rational Numbers Absolute value can be used to write mathematical expressions to represent a change between real-world integers, such as money, distance, temperature, etc. Example Karen s checking account balance is $ After a trip to the bookstore, her new balance is 2$3.85. The absolute value equation shown represents the amount of money Karen spent at the bookstore Karen spent $31.35 at the bookstore. Chapter 10 Summary 715

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