CorrectionKey A Rational Numbers MODULE 3

Rational Numbers?MODULEESSENTIAL QUESTIONHow can you use rationalnumbers to solve realworld problems?3LESSON 3.1Classifying RationalNumbers6.NS.6LESSON 3.2Identifying Oppositesand Absolute Valueof Rational Numbers6.NS.6, 6.NS.6a,6.NS.6c, 6.NS.7,6.NS.7cLESSON 3.3Comparing andOrdering RationalNumbers6.NS.7, 6.NS.7a, Houghton Mifflin Harcourt Publishing Company Image Credits: Tetra Images / Alamy6.NS.7bReal-World Videomy.hrw.commy.hrw.comIn sports like baseball, coaches, analysts, and fanskeep track of players' statistics such as battingaverages, earned run averages, and runs batted in.These values are reported using rational numbers.my.hrw.comMath On the SpotAnimated MathPersonal Math TrainerGo digital with yourwrite-in studentedition, accessible onany device.Scan with your smartphone to jump directlyto the online edition,video tutor, and more.Interactively explorekey concepts to seehow math works.Get immediatefeedback and help asyou work throughpractice sets.43

Are YOU Ready?PersonalMath TrainerComplete these exercises to review skills you will needfor this module.Write an Improper Fractionas a Mixed NumberEXAMPLE11 33 33 33 233 1 1 1 23 3 23 3 23my.hrw.comOnline Practiceand HelpWrite as a sum using names for one plusa proper fraction.Write each name for one as one.Add the ones.Write the mixed number.Write each improper fraction as a mixed number.1. 72122.5154.4113.7Write a Mixed Number as an Improper FractionEXAMPLE3 34 1 1 1 34Write the whole number as a sum of ones. 44 44 44 34Use the denominator of the fraction towrite equivalent fractions for the ones.15 4Add the numerators.Write each mixed number as an improper fraction.6. 4 355. 2 127. 3 498. 2 57EXAMPLE Houghton Mifflin Harcourt Publishing CompanyCompare and Order DecimalsOrder from least to greatest: 7.32, 5.14, 5.16.Use place value to7.32 is greatest.compare numbers,5.14 5.16starting with ones, thenThe order is 5.14, 5.16, 7.32. tenths, then hundredths.Compare the decimals.9. 8.868.6510. 0.7320.7512. Order 0.98, 0.27, and 0.34 from greatest to least.44Unit 111. 0.220.022

Reading Start-UpVisualize VocabularyUse the words to complete the web. You may put more thanone word in each box.-15, -45, -6025, 71, 102IntegersVocabularyReview Wordsabsolute value (valorabsoluto)decimal (decimal)dividend (dividendo)divisor (divisor)fraction (fracción)integers (enteros) negative numbers(números negativos) opposites (opuestos) positive numbers(números positivos) whole number (númeroentero)Preview Words-20 and 209rational number (númeroracional)Venn diagram (diagramade Venn)Understand Vocabulary Houghton Mifflin Harcourt Publishing CompanyFill in each blank with the correct term from the preview words.1. Aratio of two integers.is any number that can be written as a2. Abetween groups.is used to show the relationshipsActive ReadingTri-Fold Before beginning the module, createa tri-fold to help you learn the concepts andvocabulary in this module. Fold the paper intothree sections. Label the columns “What I Know,”“What I Need to Know,” and “What I Learned.”Complete the first two columns before you read.After studying the module, complete the thirdcolumn.Module 345

GETTING READY FORRational NumbersUnderstanding the standards and the vocabulary terms in the standardswill help you know exactly what you are expected to learn in this module.6.NS.7bWrite, interpret, and explainstatements of order for rationalnumbers in real-world contexts.Key Vocabularyrational number(número racional)Any number that can beexpressed as a ratio of twointegers.What It Means to YouYou can order rational numbers to understand relationshipsbetween values in the real world.EXAMPLE 6.NS.7BThe fraction of crude oil produced inthe United Sta mber thatis not an integer but is a whole number? Explain.52Unit 1 Houghton Mifflin Harcourt Publishing Company19. Analyze Relationships Explain how the Venn diagrams in this lessonshow that all integers and all whole numbers are rational numbers.

LESSON3.2?Identifying Oppositesand Absolute Value ofRational NumbersESSENTIAL QUESTION6.NS.6cFind and position integersand other rational numberson a horizontal or verticalnumber line diagram; findand position pairs of integersand other rational numberson a coordinate plane. Also6.NS.6, 6.NS.6a, 6.NS.7,6.NS.7cHow do you identify opposites and absolute value of rationalnumbers?6.NS.6, 6.NS.6cEXPLORE ACTIVITYPositive and Negative Rational NumbersAll rational numbers can be represented as points on a number line.Positive rational numbers are greater than 0. They are located to theright of 0 on a number line. Negative rational numbers are less than 0.They are located to the left of 0 on a number line.Water levels with respect to sea level, which has elevation 0, may bemeasured at beach tidal basins. Water levels below sea level arerepresented by negative numbers.A The table shows the water level at a tidal basin at different timesduring a day. Graph the level for each time on the number line. Houghton Mifflin Harcourt Publishing Company Image Credits: Anna Blume/AlamyTime4 A.M.A8 A.M.BNoonC4 P.M.D8 P.M.E3.52.5-0.5-2.50.5Level (ft)-5 -4 -3 -2 -10 1 2 3 4 5B How did you know where to graph -0.5?C At what time or times is the level closest to sea level? How do you know?D Which point is located halfway between -3 and -2?E Which point is the same distance from 0 as D?Reflect1.Communicate Mathematical Ideas How would you graph -2.25?Would it be left or right of point D?Lesson 3.253

Rational Numbers and Oppositeson a Number LineMath On the Spotmy.hrw.comYou can find the opposites of rational numbers that are not integers the sameway you found the opposites of integers. Two rational numbers are opposites ifthey are the same distance from 0 but on different sides of 0.2 34 and - 2 34 are opposites.-5 -4 -3 -2 -10 1 2 3 4 5EXAMPLE 16.NS.6a, 6.NS.6cUntil June 24, 1997, the New York Stock Exchange priced the value of ashare of stock in eighths, such as 27 18 or at 41 34. The change in value of ashare of stock from day to day was also represented in eighths as a positiveor negative number.STEP 1DayChange invalue ( )Graph the change in stock value forWednesday on the number line.1 58-4 14541.The change in value for Wednesday is 441units below 0.Graph a point 44STEP 2Tuesday WednesdayGraph the opposite of -4 14 .The opposite of -4 14 is the samedistance from 0 but on the otherside of 0.The opposite of -4 14 is 4 14.3210-1-2-31-4is between4-4 and -5. It iscloser to -4.-4-5The opposite of the change in stock value for Wednesday is 4 14.YOUR TURNPersonalMath TrainerOnline Practiceand Helpmy.hrw.com54Unit 12.What are the opposites of 7, -3.5, 2.25, and 9 31? Houghton Mifflin Harcourt Publishing Company Image Credits: Image Source/Getty ImagesThe table shows the change in value of astock over two days. Graph the changein stock value for Wednesday and itsopposite on a number line.

Absolute Values of Rational NumbersYou can find the absolute value of a rational number that is not an integer thesame way you found the absolute value of an integer. The absolute value of arational number is the number’s distance from 0 on the number line.Math On the Spotmy.hrw.comEXAMPL 2EXAMPLE6.NS.7, 6.NS.7cThe table shows the average low temperatures in January in one locationduring a five-year span. Find the absolute value of the average Januarylow temperature in 2009.My Notes6Year200820092010201120125Temperature ( C)-3.2-5.4-0.83.8-24STEP 13Graph the 2009 average January low temperature.The 2009 average January low is -5.4 C.Graph a point 5.4 units below 0.STEP 210-1Find the absolute value of -5.4.-2-5.4 is 5.4 units from 0.-3 -5.4 -5-4 5.4-6Reflect3.2Communicate Mathematical Ideas What is the absolute value ofthe average January low temperature in 2011? How do you know?Math TalkMathematical Practices Houghton Mifflin Harcourt Publishing CompanyHow do you knowwhere to graph -5.4?YOUR TURNGraph each number on the number line. Then use your number line to findeach absolute value.-54.6.05-4.5; -4.5 5.4; 4 7. 1 12; 1 12 -3 14; -3 14 PersonalMath TrainerOnline Practiceand Helpmy.hrw.comLesson 3.255

Guided PracticeGraph each number and its opposite on a number line. (Explore Activity and Example 1)1. -2.8-52. 4.303. -3 45-5-5505054. 1 1305-5Find the opposite of each number. (Example 1)5. 3.7856. -7127. 08. 4.29. 12.110. 2.611. Vocabulary Explain why 2.15 and -2.15 are opposites. (Example 1)Find the absolute value of each number. (Example 2)12. 5.23213. -41114. 015. -6 3516. -2.1217. 8.2ESSENTIAL QUESTION CHECK-IN18. Use absolute value to write a definition of the opposite of a nonzerorational number. Give an example.56Unit 1 Houghton Mifflin Harcourt Publishing Company?

NameClassDate3.2 Independent PracticePersonalMath Trainer6.NS.6, 6.NS.6a, 6.NS.6c, 6.NS.7, 6.NS.7cmy.hrw.comOnline Practiceand Help19. Financial Literacy A store’s balance sheet represents the amountscustomers owe as negative numbers and credits to customers as rBalance ( )-85.2320.44-116.3313.50-9.85a. Write the opposite of each customer’s balance.b. Mr. Yuan wants to use his credit to pay off the full amount thatanother customer owes. Which customer’s balance does Mr. Yuanhave enough money to pay off?c. Which customer’s balance would be farthest from 0 on a numberline? Explain.20. Multistep Trina went scuba diving and reached an elevation of-85.6 meters, which is below sea level. Jessie went hang-glidingand reached an altitude of 87.9 meters, which is above sea level. Houghton Mifflin Harcourt Publishing Companya. Who is closer to the surface of the ocean? Explain.b. Trina wants to hang-glide at the same number of meters above sealevel as she scuba-dived below sea level. Will she fly higher thanJessie did? Explain.21. Critical Thinking Carlos finds the absolute value of -5.3, and then finds theopposite of his answer. Jason finds the opposite of -5.3, and then finds theabsolute value of his answer. Whose final value is greater? Explain.Lesson 3.257

22. Explain the Error Two students are playing a math game. Theobject of the game is to make the least possible number byarranging given digits inside absolute value bars on a card. Inthe first round, each player will use the digits 3, 5, and 7.753a. One student arranges the numbers on the card as shown.What was this student’s mistake?b. What is the least possible number the card can show?Work AreaFOCUS ON HIGHER ORDER THINKING23. Analyze Relationships If you plot the point -8.85 on a number line,would you place it to the left or right of -8.8? Explain.24. Make a Conjecture If the absolute value of a negative number is 2.78,what is the distance on the number line between the number and itsabsolute value? Explain your answer.25. Multiple Representations The deepest point in the Indian Ocean is theJava Trench, which is 25,344 feet below sea level. Elevations below sealevel are represented by negative numbers.b. A mile is 5,280 feet. Between which two integers is the elevationin miles?c. Graph the elevation of the Java Trench in miles.-50526. Draw Conclusions A number and its absolute value are equal. If yousubtract 2 from the number, the new number and its absolute valueare not equal. What do you know about the number? What is a possiblenumber that satisfies these conditions?58Unit 1 Houghton Mifflin Harcourt Publishing Companya. Write the elevation of the Java Trench.