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LESSON3.2?Identifying Oppositesand Absolute Value ofRational NumbersESSENTIAL QUESTIONCOMMONCORE6.NS.6cFind and position integersand other rational numberson a horizontal or verticalnumber line diagram Also6.NS.6, 6.NS.6a, 6.NS.7,6.NS.7cHow do you identify opposites and absolute value of rationalnumbers?COMMONCOREEXPLORE ACTIVITY6.NS.6, 6.NS.6cPositive and Negative Rational NumbersRecall that positive numbers are greater than 0. They are located tothe right of 0 on a number line. Negative 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. Houghton Mifflin Harcourt Publishing Company Image Credits: Anna Blume/AlamyA 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.Time4 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 the same way you found theopposites of integers. Two rational numbers are opposites if they are the samedistance 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 1COMMONCORE6.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 0.3210-1-2-31-4is between4-4 and -5. It iscloser to -4.-4-5The opposite of -4 14 is 4 14.The opposite of the change in stock value for Wednesday is 4 14.YOUR TURNPersonalMath TrainerOnline Assessmentand Interventionmy.hrw.com54Unit 12.What are the opposites of 7, -3.5, 2.25, and 9 13? Houghton Mifflin Harcourt Publishing Company Image Credits: Image Source/Getty ImagesThe table shows the change in value of astock over two days. Graph the change instock value for Wednesday and itsopposite on a number line.

Absolute Values of Rational NumbersYou can also find the absolute value of a rational number the same wayyou found the absolute value of an integer. The absolute value of a rationalnumber is the number’s distance from 0 on the number line.Math On the Spotmy.hrw.comEXAMPL 2EXAMPLECOMMONCORE6.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)-3.2-5.4-0.83.8-243STEP 12Graph the 2009 average January low temperature.1The 2009 average January low is -5.4 C.Graph a point 5.4 units below 0.STEP 20-1Find the absolute value of -5.4.-2-5.4 is 5.4 units from 0.-3 -5.4 -5-4 5.4-6Reflect3.Communicate Mathematical Ideas What is the absolute value ofthe average January low temperature in 2011? How do you know?Math Talk Houghton Mifflin Harcourt Publishing CompanyMathematical PracticesHow 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.-4.5; -4.5 4; 4 055.7. 1 12 ; 1 12 -3 14 ; -3 14 PersonalMath TrainerOnline Assessmentand Interventionmy.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)12. 5.23213. -41114. 015. -6 3516. -2.1217. 8.2?ESSENTIAL QUESTION CHECK-IN18. How do you identify the opposite and the absolute value of a rationalnumber?56Unit 1 Houghton Mifflin Harcourt Publishing CompanyFind the absolute value of each number. (Example 2)

NameClassDate3.2 Independent PracticeCOMMONCOREPersonalMath Trainer6.NS.6, 6.NS.6a, 6.NS.6c, 6.NS.7, 6.NS.7cmy.hrw.comOnlineAssessment andIntervention19. 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. Houghton Mifflin Harcourt Publishing Company20. Multistep Trina and Jessie went on a vacation to Hawaii. Trina wentscuba diving and reached an elevation of -85.6 meters, which is belowsea level. Jessie went hang-gliding and reached an altitude of 87.9meters, which is above sea level.a. 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. The object ofthe game is to make the least possible number by arranging the givendigits inside absolute value bars on a card. In the first round, each playerwill use the digits 3, 5, and 7 to fill in the card.753a. One student arranges the numbers on the card as shown. What wasthis 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.a. Write the elevation of the Java Trench.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 Company25. 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.

Activity3.2INSTRUCTIONSSTEP 1Get a game card from your teacher. Cards contain either a number or areal-world situation. Search the room to find the person with the realworld situation or number that matches your card.The interest ratedropped 1.5%.-1.5 Houghton Mifflin Harcourt Publishing CompanySTEP 2STEP 3Once you and your partner have found each other, fill in a row of yourtable using the information from your cards.NumberAbsoluteValue(Magnitude)Distance From 0on theNumber LineReal-WorldSituation–1.51.51.5The interest ratedropped 1.5%.Plot the number on the appropriate number line below your table. Makesure your number line graph agrees with your partner’s graph.-3- 2.5-2- 1.5-1- 0.500.511.522.53STEP 4Turn your cards back in to your teacher, who will shuffle them and thengive each of you a new card so that you may repeat Steps 1–3.STEP 5Once you have filled in all the rows in your table, answer the Reflectquestions.Activity 3.258A

-3-3- 2.5-212-2-2- 1.5-112-1-1Distance From 0on theNumber Line- 0.5-12Real-World Situation00.5101211.521221Reflect1. What relationship do you see between the absolute value (magnitude)of a number and the distance of that number from 0 on a number line?2. What relationship do you see between a number and its absolute value(magnitude)?58BUnit 12.531232 Houghton Mifflin Harcourt Publishing CompanyValueAbsolute Value(Magnitude)

3.3Getting ReadyCOMMONCOREInterpret statements ofinequality as statementsabout the relative position oftwo numbers on a numberline diagram. Also 6.NS.6,Compare and Order?6.NS.7a6.NS.6cESSENTIAL QUESTIONWhat strategies can you use to compare andorder fractions and decimals?EXPLORE ACTIVITY 1COMMONCORE6.NS.6, 6.NS.6c, 6.NS.7aEquivalent FractionsYou can order fractions by rewriting the fractions with a common denominator, thenplotting them on a number line.Four students are running a race and trying to reach a goal time. The table shows thestudents’ times in relation to the goal time. Order the time differences from least to greatest.StudentABCDTime Difference1016- 143238A Write the fractions as equivalent fractions with a common denominator of 8.10 3 12 1 --41623 3888B Plot the fractions on the number line. Houghton Mifflin Harcourt Publishing Company-2802848168128148168C Use the relative positions of the points on the number line to write thefractions in order from least to greatest.Sinceleft to right,is the fraction furthest to the left, it has the least value. Readingis the next fraction, then , then. D When writing the fractions in order, be sure to use the original forms of thefractions given at the beginning of the problem. The time differences fromleast to greatest are,,, and.Reflect1.Justify Reasoning Is 8 the only common denominator that could have been used?Getting Ready 3.358C

EXPLORE ACTIVITY 2COMMONCORE6.NS.6, 6.NS.6c, 6.NS.7aOrdering Rational NumbersYou can order decimals by plotting them on a number line.0.6Order -0.2, 0.15, 0.4, -0.35, and 0.55 from greatest to least.0.50.4A Plot the decimals on the number line.0.3B Use the relative positions of the points on the number line to write the decimalsin order from greatest to least.Since0.10is closest to the top of the number line, it has the-0.1greatest value.-0.2Reading from the top down,then0.2is the next decimal,, then, then-0.3-0.4.The decimals in order from greatest to least are,, and.,,PracticeOrder the numbers from least to greatest using the number line.-5502. -2.5, 1.25, 4, and -3.25-4-3-2-1012345 ,3 from least to greatest using a number line.1 , 11 ,3. Order4 2 8 1658DUnit 1 Houghton Mifflin Harcourt Publishing Company5 , 4, and -319 , 11. -42 10

LESSON3.3?Comparing andOrdering RationalNumbersCOMMONCORE6.NS.7aInterpret statements ofinequality as statementsabout the relative position oftwo numbers on a numberline diagram. Also 6.NS.7,6.NS.7bESSENTIAL QUESTIONHow do you compare and order rational numbers?EXPLORE ACTIVITYCOMMONCOREPrep for 6.NS.7aEquivalent Fractions and DecimalsFractions and decimals that represent the same value are equivalent. Thenumber line shows equivalent fractions and decimals from 0 to 1.A Complete the number line by writingthe missing decimals or fractions.0B Use the number line to find a fractionthat is equivalent to 0.25. Explain.0.2 0.3 0.4110310250.6 0.7120.9354519103414 Houghton Mifflin Harcourt Publishing Company7C Explain how to use a number line to find a decimal equivalent to 1.10D Use the number line to complete each statement.0.2 3 100.75 1.25 Reflect1.Communicate Mathematical Ideas How does a number line representequivalent fractions and decimals?2.Name a decimal between 0.4 and 0.5.Lesson 3.359

Ordering Fractions and DecimalsYou can order fractions and decimals by rewriting the fractions as equivalentdecimals or by rewriting the decimals as equivalent fractions.Math On the Spotmy.hrw.comEXAMPLE 1COMMONCORE6.NS.7, 6.NS.7aA Order 0.2, 34, 0.8, 12, 14, and 0.4 from least to greatest.STEP 1Write the fractions as equivalent decimals.1 0.254AnimatedMathSTEP 2Use the number line to write the decimals in order.0my.hrw.com3 0.7541 0.520.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.910.2 0.25 0.4 0.5 0.75 0.8The numbers from least to greatest are 0.2, 41 , 0.4, 12 , 34 , 0.8.1B Order, 2 , and 0.35 from least to greatest.12 3Write the decimal as an equivalentfraction.35 70.35 10060 is a multiple ofthe denominatorsof all three fractions.20STEP 2Find equivalent fractions with 60 as the common denominator. 3 5 2054012721 12603602060 20 3 5STEP 3Order fractions with common denominators by comparing thenumerators.5 21 40540, 21,.The fractions in order from least to greatest are60 60 601The numbers in order from least to greatest are, 0.35, and 23.12YOUR TURNPersonalMath TrainerOnline Assessmentand Interventionmy.hrw.com60Unit 1Order the fractions and decimals from least to greatest.373. 0.85, 5, 0.15,10 Houghton Mifflin Harcourt Publishing CompanySTEP 1

Ordering Rational NumbersYou can use a number line to order positive and negative rational numbers.EXAMPL 2EXAMPLECOMMONCORE6.NS.7a, 6.NS.7bMath On the Spotmy.hrw.comFive friends completed a triathlon that included a 3-mile run, a 12-mile bikeride, and a 12 -mile swim. To compare their running times they created a tablethat shows the difference between each person’s time and the averagetime, with negative numbers representing times less than the average.RunnerTime above or belowaverage (minutes)John12SueAnnaMikeTom1.4 1 14 2.01.95Order the numbers from greatest to least.STEP 1Write the fractions as equivalent decimals.1 0.52STEP 2 1 14 1.25Use the number line to write the decimals in order.- 2.0 - 1.5 - 1.0 - 0.50.00.51.01.52.0Math TalkAverage TimeMathematical Practices Houghton Mifflin Harcourt Publishing Company Image Credits: ImageStateRoyalty Free/Alamy1.95 1.4 0.5 -1.25 -2.0The numbers in order from greatest to least are 1.95, 1.4, 12, -1 14, -2.0.Who was the fastestrunner? Explain.Reflect4.Communicate Mathematical Ideas Describe a different way to orderthe numbers.YOUR TURN5. To compare their bike times, the friends created a table that shows thedifference between each person’s time and the average bike time. Orderthe bike times from least to greatest.BikerTime above or belowaverage (minutes)John 1.8SueAnnaMikeTom11 259110-1.25PersonalMath TrainerOnline Assessmentand Interventionmy.hrw.comLesson 3.361

Guided PracticeFind the equivalent fraction or decimal for each number.(Explore Activity 1)1. 0.6 1 2.43. 0.9 4. 0.1 3 5.106. 1.4 4 7.58. 0.4 6 9.8Use the number line to order the fractions and decimals from least togreatest. (Example 1)10. 0.75, 12 , 0.4, and 1500.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.911. The table shows the lengths of fish caught by threefriends at the lake last weekend. Write the lengths inorder from greatest to least. (Example 1)1Lengths of Fish (cm)EmmaAnneEmily12.712 3512 3412. 2.3, 2 45 , 2.635, 0.75,13. 0.5,164812,414. 0.5, 15 , 0.35,25 587, 34 ,15. 34 , 10105, 0.65, 2416. 38 ,1617. 2.3, 2 45 , 2.67 0.7218. 0.6, 58 , 1219. 1.45, 1 12 , 1 13 , 1.220. 0.3, 0.5, 0.55, 0.35?ESSENTIAL QUESTION CHECK-IN21. Explain how to compare 0.7 and 58 .62Unit 1 Houghton Mifflin Harcourt Publishing CompanyList the fractions and decimals in order from least to greatest.(Example 1, Example 2)

NameClassDate3.3 Independent PracticeCOMMONCOREPersonalMath Trainer6.NS.7, 6.NS.7a, 6.NS.7bmy.hrw.com22. Rosa and Albert receive the same amount of allowanceeach week. The table shows what part of their allowancethey each spent on video games and pizza.a. Who spent more of their allowance on video games?Write an inequality to compare the portion spent onvideo games.OnlineAssessment tb. Who spent more of their allowance on pizza? Write an inequality tocompare the portion spent on pizza.c. Draw Conclusions Who spent the greater part of their totalallowance? How do you know?23. A group of friends is collecting aluminum for a recycling drive. Each personwho donates at least 4.25 pounds of aluminum receives a free movie coupon.The weight of each person’s donation is shown in the table. Houghton Mifflin Harcourt Publishing CompanyWeight(lb)BrendaClaireJimMicahPeter4.35.56 161544 38a. Order the weights of the donations from greatest to least.b. Which of the friends will receive a free movie coupon? Which will not?c. What If? Would the person with the smallest donation win a moviecoupon if he or she had collected 21 pound more of aluminum? Explain.Lesson 3.363

24. Last week, several gas stations in a neighborhood all charged the sameprice for a gallon of gas. The table below shows how much gas priceshave changed from last week to this week.Gas StationChange from lastweek (in cents)Gas andGoSamsonGas 6.65.8Star GasCornerStoreTip TopShop 6 34275 5 58a. Order the numbers in the table from least to greatest.b. Which gas station has the cheapest gas this week?c. Critical Thinking Which gas station changed their price the leastthis week?FOCUS ON HIGHER ORDER THINKINGWork Area25. Analyze Relationships Explain how you would order from least to greatestthree numbers that include a positive number, a negative number, and zero.27. Communicate Mathematical Ideas If you know the order from least togreatest of 5 negative rational numbers, how can you use that information toorder the absolute values of those numbers from least to greatest? Explain.64Unit 1 Houghton Mifflin Harcourt Publishing Company26. Critique Reasoning Luke is making pancakes. The recipe calls for 0.5 quart18of milk and 2.5 cups of flour. He has 38 quart of milk and8 cups of flour.Luke makes the recipe with the milk and flour that he has. Explain his error.

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Personal Math Trainer Interactively explore key concepts to see how math works. Animated Math Go digital with your write-in student edition, accessible on any device. Scan with your smart phone to jump directly to the online edition, video tutor, and more.