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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 States by four states in 2011is shown.CA1100TX950ND350AL325Which state produced the least oil?1CA 100189TX 50 10036ND 50 100312AL 25 1006.NS.7cUnderstand the absolute valueof a rational number as itsdistance from 0 on the numberline; interpret absolute valueas magnitude for a positive ornegative quantity in a real-worldsituation.Key Vocabularyabsolute value (valor absoluto)A number’s distance from 0 onthe number line.Visit my.hrw.comto see all CACommon CoreStandardsexplained.my.hrw.com46Unit 1What It Means to YouYou can use absolute value to describe a number’s distance from 0on a number line and compare quantities in real-world situations.EXAMPLE 6.NS.7CUse the number line todetermine the absolutevalues of -4.5 F and -7.5 Fand to compare thetemperatures.-10-8-6-4 -4.5 4.5The absolute value of –4.5 is 4.5. -7.5 7.5The absolute value of –7.5 is 7.5.-24.57.5-7.5 is farther to the left of 0 than -4.5, so -7.5 -4.5 and-7.5 F is colder than -4.5 F.0 Houghton Mifflin Harcourt Publishing Company Image Credits: Karl Naundorf/FotoliaCalifornia (CA) produced the least crude oil in 2011.
LESSON3.1?Classifying RationalNumbers6.NS.6Understand a rational numberas a point on the number line.Extend number line diagramsand coordinate axes familiar fromprevious grades to represent pointson the line and in the plane withnegative number coordinates.ESSENTIAL QUESTIONHow can you classify rational numbers?Prep for 6.NS.6EXPLORE ACTIVITYRepresenting Division as a FractionAlicia and her friends Brittany, Kenji, and Ellis are taking a potteryclass. The four friends have to share 3 blocks of clay. How much claywill each of them receive if they divide the 3 blocks evenly?A The top faces of the 3 blocksof clay can be representedby squares. Use the modelto show the part of eachblock that each friend will receive. Explain your method. Houghton Mifflin Harcourt Publishing Company Image Credits: Digital Vision/AlamyB Each piece of one square is equal to what fraction of a block of clay?C Explain how to arrange the piecesto model the amount of clay eachperson gets. Sketch the model.AAAAliciaBrittanyKenjiEllisD What fraction of a square does each person’s pieces cover? Explain.E How much clay will each person receive?F Multiple Representations How does this situation represent division?Lesson 3.147
EXPLORE ACTIVITY (cont’d)Reflect1.Communicate Mathematical Ideas 3 4 can be written 34. How arethe dividend and divisor of a division expression related to the partsof a fraction?2.Analyze Relationships How could you represent the division as afraction if 5 people shared 2 blocks? if 6 people shared 5 blocks?Rational NumbersA rational number is any number that can be written as ba , where a and b areintegers and b 0.Math On the Spotmy.hrw.comEXAMPLE 16.NS.6Math TalkMathematical PracticesWhat division isrepresented by the34fraction1?A 3 25Convert the mixed number to a fractiongreater than 1.173 25 5B0.6The decimal is six tenths. Write as afraction.60.6 10C34Write the whole number as a fractionwith a denominator of 1.3434 1D -7Write the integer as a fraction with adenominator of 1.YOUR TURNWrite each rational number as ba.PersonalMath TrainerOnline Practiceand Helpmy.hrw.com48Unit 13. -154. 0.315. 4 596. 62-7-7 1 Houghton Mifflin Harcourt Publishing CompanyWrite each rational number as ba.
Classifying Rational NumbersA Venn diagram is a visual representation used to show the relationshipsbetween groups. The Venn diagram below shows how rational numbers,integers, and whole numbers are related.Math On the Spotmy.hrw.comRational numbersinclude integers andwhole numbers.Rational NumbersIntegersIntegers includewhole numbers.Whole NumbersEXAMPL 2EXAMPLE6.NS.6Place each number in the Venn diagram. Then classify each number byindicating in which set or sets each number belongs.Rational Numbers0.35 Houghton Mifflin Harcourt Publishing Company-3IntegersMy Notes3475Whole NumbersA 75The number 75 belongs in the sets of whole numbers, integers,and rational numbers.B -3The number -3 belongs in the sets of integers and rational numbers.C 343The numberbelongs in the set of rational numbers.4D 0.35The number 0.35 belongs in the set of rational numbers.Reflect7.Analyze Relationships Describe how the Venn diagram models therelationship between rational numbers, integers, and whole numbers.Lesson 3.149
YOUR TURNPersonalMath TrainerOnline Practiceand Helpmy.hrw.comPlace each number in the Venndiagram. Then classify each numberby indicating in which set or sets itbelongs.Rational NumbersIntegers8. 14.1Whole Numbers9. 7 1510. -811. 101Guided Practice1. Sarah and four friends are decorating picture frames with ribbon.They have 4 rolls of ribbon to share evenly. (Explore Activity 1)a. How does this situation represent division?b. How much ribbon does each person receive?Write each rational number in the form ba, where a and b are integers. (Example 1)2. 0.73. -294. 8 13Place each number in the Venn diagram. Then classify each numberby indicating in which set or sets each number belongs. (Example 2)106. 511?ESSENTIAL QUESTION CHECK-IN7. How is a rational number that is not an integer differentfrom a rational number that is an integer?50Unit 1Rational NumbersIntegersWhole Numbers Houghton Mifflin Harcourt Publishing Company5. -15
NameClassDate3.1 Independent PracticePersonalMath Trainer6.NS.6my.hrw.comList two numbers that fit each description. Then writethe numbers in the appropriate location on theVenn diagram.Online Practiceand HelpRational NumbersIntegers8. Integers that are not whole numbersWhole Numbers9. Rational numbers that are not integers10. Multistep A nature club is having its weekly hike. The table showshow many pieces of fruit and bottles of water each member of theclub brought to share.MemberBaxterHendrickMaryKendraPieces of Fruit3245Bottles of Water5237a. If the hikers want to share the fruit evenly, how many pieces shouldeach person receive?b. Which hikers received more fruit than they brought on the hike? Houghton Mifflin Harcourt Publishing Companyc. The hikers want to share their water evenly so that each member hasthe same amount. How much water does each hiker receive?11. Sherman has 3 cats and 2 dogs. He wants to buy a toy for each of hispets. Sherman has 22 to spend on pet toys. How much can he spendon each pet? Write your answer as a fraction and as an amount in dollarsand cents.12. A group of 5 friends are sharing 2 pounds of trail mix. Write a divisionproblem and a fraction to represent this situation.13. Vocabulary Adiagram can represent set relationships visually.Lesson 3.151
Financial Literacy For 14–16, use the table. The table shows Jason’sutility bills for one month. Write a fraction to represent the divisionin each situation. Then classify each result by indicating the set orsets to which it belongs.14. Jason and his 3 roommates share the cost of the electric bill evenly.March BillsWater 35Gas 14Electric 10815. Jason plans to pay the water bill with 2 equal payments.16. Jason owes 15 for last month’s gas bill also. The total amount of the twogas bills is split evenly among the 4 roommates.17. Lynn has a watering can that holds 16 cups of water, and she fills it halffull. Then she waters her 15 plants so that each plant gets the sameamount of water. How many cups of water will each plant get?Work AreaFOCUS ON HIGHER ORDER THINKING18. Critique Reasoning DaMarcus says the number6 belongs only to theset of rational numbers. Explain his error.2420. Critical Thinking Is it possible for a number to be a rational number 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.
Math On the Spot my.hrw.com Math Trainer Online Practice and Help Personal my.hrw.com Rational Numbers A rational number is any number that can be written as a_ b, where a and b are integers and b 0. Write each rational number as a_ b. 3 2_ 5 Convert the mixed number to a fraction g
