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LESSON 10Overview Add and Subtract Positive and Negative NumbersMATH FOCUSObjectivesPrior KnowledgeFocus StandardsContent Objectives 7.NS.A.1 Apply and extend previousunderstandings of addition and subtractionto add and subtract rational numbers;represent addition and subtraction on ahorizontal or vertical number line diagram.d. Apply properties of operations asstrategies to add and subtractrational numbers. See Unit 2 Pacing Guide for developing andapplied standards.STANDARDS FOR MATHEMATICALPRACTICE (SMP)SMP 1, 2, 3, 4, 5, and 6 are integrated into theTry-Discuss-Connect routine.*This lesson provides additional support for:Language Objectives 2 Reason abstractly and quantitatively.7 Look for and make use of structure.* See page 1q to learn how every lesson includesthese SMP.Solve addition and subtraction problemsinvolving negative numbers, includingrational numbers.Interpret addition expressions withrational numbers as subtracting theopposite and subtraction expressions withrational numbers as adding the opposite.Rewrite and reorder problems involvingboth addition and subtraction to makethem easier to solve. Explain solution strategies to subtractionproblems with negative numbers usinglesson vocabulary and academiclanguage.Explain how to rewrite and reorder tosolve problems involving addition orsubtraction of negative numbers duringclass discussions.Respond to clarifying questions aboutpositive and negative numbers byaccurately using the lesson vocabulary inspeech and writing. Subtract positive rational numbers forwhich the difference is positive or zero.Add rational numbers in any form.Understand that subtracting an integer isthe same as adding its opposite,p 2 q 5 p 1 (2q).Understand the distance between twointegers on the number line as theabsolute value of their difference.Model adding and subtracting integersusing integer chips and horizontal andvertical number lines.VocabularyMath VocabularyThere is no new vocabulary. Review thefollowing key terms.absolute value a number’s distance from0 on the number line. Absolute value isnever negative.opposite numbers numbers that are thesame distance from 0 but in oppositedirections. Opposite numbers have thesame numeral, but opposite signs. Theopposite of a number is also called theadditive inverse of that number.Academic Vocabularyrepresent to use as a sign, symbol, orexample for something.Learning ProgressionEarlier in Grade 7, students representedthe addition of positive and negativeintegers, fractions, and decimals on anumber line, and they applied theproperties of operations as strategies tofind sums of rational numbers.In the previous lesson, they usedinteger chips and number lines tounderstand the subtraction of a negativeinteger as the addition of its opposite.181aIn this lesson, students extend theirknowledge of subtraction to includenegative fractions and decimals. Theysolve real-world problems that involveboth addition and subtraction withrational numbers. They apply a varietyof strategies, including reorderingaddends, to make calculatingmore efficient.LESSON 10 Add and Subtract Positive and Negative NumbersLater in Grade 7, students will solvereal-world problems involving all fouroperations. They will also add andsubtract rational numbers to simplifylinear expressions and solve multi-steplinear equations.In Grade 8, students will apply theirabilities to calculate with positiveand negative numbers to solvelinear equations and systems oflinear equations. Curriculum Associates, LLCCopying is not permitted.

LESSON 10OverviewPacing GuideItems marked withSESSION 1 are available on the Teacher Toolbox.MATERIALSDIFFERENTIATIONExplore Adding and Subtracting with Integers (35–50 min)Start (5 min) Try It (5–10 min )Discuss It (10–15 min)Connect It (10–15 min)Close: Exit Ticket (5 min)Math Toolkit grid paper, integerchips, number linesPresentation SlidesPREPARE Interactive TutorialRETEACH or REINFORCE Hands-On ActivityMaterials For each pair: 2 counters, ActivitySheet Number LinesAdditional Practice (pages 185– 186)SESSION 2 Develop Subtracting Positive and Negative Fractions and Decimals (45–60 min)Start (5 min) Try It (10–15 min )Discuss It (10–15 min)Connect It (15–20 min)Close: Exit Ticket (5 min)Math Toolkit grid paper, numberlines, place-value chartsRETEACH or REINFORCE Visual ModelREINFORCE Fluency & Skills PracticeEXTEND Deepen UnderstandingPresentation SlidesAdditional Practice (pages 191– 192)SESSION 3 Develop Adding and Subtracting Positive and Negative Fractions and Decimals (45–60 min)Start (5 min) Try It (10–15 min )Discuss It (10–15 min)Connect It (15–20 min)Close: Exit Ticket (5 min)Math Toolkit grid paper, numberlines, place-value chartsPresentation SlidesREINFORCE Fluency & Skills PracticeAdditional Practice (pages 197– 198)SESSION 4 RETEACH or REINFORCE Hands-On ActivityMaterials For each pair: 2 colored pencils(1 each of two different colors), 2 copies ofActivity Sheet Fraction BarsEXTEND Deepen UnderstandingRefine Adding and Subtracting Positive and Negative Numbers (45–60 min)Start (5 min) Monitor & Guide (15–20 min )Group & Differentiate (20–30 min)Close: Exit Ticket (5 min)Math Toolkit Have items fromprevious sessions available forstudents.Presentation SlidesRETEACH Hands-On ActivityMaterials For each pair: 1 counter, sticky notes,Activity Sheet Number LinesREINFORCE Problems 4–9EXTEND ChallengePERSONALIZELesson 10 Quiz orDigital Comprehension CheckRETEACH Tools for InstructionREINFORCE Math Center ActivityEXTEND Enrichment Activity Curriculum Associates, LLCCopying is not permitted.LESSON 10 Add and Subtract Positive and Negative Numbers181b

LESSON 10Overview Add and Subtract Positive and Negative NumbersConnect to Culture Use these activities to connect with and leverage the diverse backgroundsand experiences of all students. Engage students in sharing what theyknow about contexts before you add the information given here.SESSION 1Try ItAsk students to discuss their experiences either diving or jumping into thewater. When gymnasts began performing their routines in the water, the sport ofdiving was born. Diving was first included in the Olympic Games over 100 years ago.The sport has remained popular and competitive ever since. Divers are judged ontheir movements through the air and their entry into the water.10 m abovewater’s surfaceSESSION 2Try ItPeople consider many things when deciding where to live. One of thosethings might be the average temperature. In Kuwait City, Kuwait, the average hightemperature in July is 99 F, but the average low temperature in January is 45 F. InDudinka, Russia, the average high temperature in July is 65 F, but the average lowtemperature in January is 219 F. Ask students to think about their own preferencesand choose ideal average high and average low temperatures for their future home.Have them mark their preferred high and low temperatures on a number line withsticky notes and discuss how they compare across the class.SESSION 3Try ItAsk students familiar with Lantern Festivals to talk about their experiences.Communities in China have been celebrating the Lantern Festival for over 2,000years, and the festival has religious, social, and cultural meanings. Lanterns are alsoused for decoration and celebration in cultures around the world. A lantern with acandle inside it will rise into the air much like a hot-air balloon. The heated air risesbecause it is less dense than the colder air around it. Ask students if they know ofother festivals or holidays that have been celebrated for thousands of years.SESSION 4Apply It Problem 4 Submarines are vehicles that travel far under water. Thecrew members on a submarine are able to control the submarine’s vertical positionbecause they can alter its density. Releasing compressed air into the submarine’sballast tanks makes it less dense and causes it to rise in the water. Venting the airand flooding the ballast tanks with water causes the submarine to sink. Ask studentsto describe their experiences with changes in elevation, such as taking an elevator,climbing on steep trails, or flying in airplanes.181cLESSON 10 Add and Subtract Positive and Negative Numbers Curriculum Associates, LLCCopying is not permitted.

LESSON 10OverviewConnect to Family and Community After the Explore session, have students use the Family Letter to let theirfamilies know what they are learning and to encourage family involvement.LESSON 10 ADD AND SUBTRACT POSITIVE AND NEGATIVE NUMBERSActivity Thinking About Positive andNegative Numbers Around YouLESSON10This week your student is learning about adding and subtracting positive andnegative decimals and fractions.Your student has already learned to add and subtract integers. The strategies foradding and subtracting positive and negative decimals and fractions are similar tothose for adding and subtracting 612 2 7 5 2566····20.2 1 0.6 5 0.46··Your student will be solving problems like the one below.A manatee is swimming at 25.6 feet relative to sea level. It swims down3.8 feet. What is the manatee’s new elevation? ONE WAY to find the manatee’s new elevation is to use a number line.23.82102928272625 ANOTHER WAY is to rewrite a subtraction problem as an addition problem. Do this activity together to investigate positive andnegative numbers in the real world.The hottest temperature recorded in the United States was inCalifornia in 1913. It was 134.1 F!Add and Subtract Positive and Negative NumbersDear Family,In 1971, a settlement in Alaska reached 279.8 F. That isthe coldest temperature recorded in the United States.The difference between the hottest and coldest temperaturesis 134.1 2 (279.8), or 213.9 F!Where else do you see positive and negativefractions and decimals around you?25.6 2 3.8 5 25.6 1 (23.8)5 [25 1 (23)] 1 [20.6 1 (20.8)]5 28 1 (21.4)5 29.4Both ways show that the manatee’s new elevation is 29.4 feet.Use the next page to start a conversationabout positive and negative numbers. Curriculum Associates, LLC Copying is not permitted.LESSON 10 Add and Subtract Positive and Negative Numbers181182LESSON 10 Add and Subtract Positive and Negative Numbers Curriculum Associates, LLC Copying is not permitted.Connect to Language For English language learners, use the Differentiation chart to scaffold thelanguage in each session. Use the Academic Vocabulary routine for academicterms before Session 1.DIFFERENTIATION ENGLISH LANGUAGE LEARNERSMATH TERMSA non-integer isnot a wholenumber or aninteger. Fractionsand decimals arenon-integers.Distance is themeasurementbetween twopoints. Curriculum Associates, LLCUse with Session 1Connect ItLevels 1–3: Listening/WritingLevels 2–4: Listening/WritingLevels 3–5: Listening/WritingPrepare students to write responsesto Connect It problem 2. Create aCo-Constructed Word Bank afterreading the problem aloud. Beginwith distance between two elevationsand use a sketch to clarify its meaning.Add integers and non-integers andgive students examples to label. Pointout that non- means “not.” Guidestudents to circle the non-integersin problem 2a, label them on thenumber line, and mark the distancebetween the points.Guide students to identify keyterms for problems 2b and 2c. Helpstudents write explanations using theword bank.Prepare students to write responsesto Connect It problem 2. Readthe problem aloud and use aCo‑Constructed Word Bank tohelp clarify words and phrases, suchas positive, negative, integers, nonintegers, and opposite numbers.Have students use the number lineto discuss the meaning of distancebetween. Ask for examples of integersand non-integers and guide studentsto explain that non- means not.Encourage partners to work togetherto write their explanations. Havestudents use the word bank tohelp them write using precisemathematical and academic language.Prepare students to write responsesto Connect It problem 2. Readthe problem aloud and begin aCo‑Constructed Word Bank withdistance between two elevationsrepresented by integers.Have partners discuss each part of theproblem and add to the word bankbefore writing their explanations.Have them decide how the twomodels in problem 2b are the sameand different.Have students write their explanationsusing complete sentences. Remindthem to pay attention to theprepositions used with the termsdistance and opposite.Copying is not permitted.LESSON 10 Add and Subtract Positive and Negative Numbers181–182

LESSON 10 SESSION 1Explore Adding and Subtracting with IntegersPurpose LESSON 10 SESSION 1Explore the addition and subtraction of integersrepresenting real-world situations.Understand that strategies for subtracting integers canbe applied to subtracting rational numbers.STARTExplore Adding and Subtractingwith IntegersPreviously, you learned how to add positive and negativenumbers. In this lesson, you will learn about subtracting positiveand negative fractions and decimals.CONNECT TO PRIOR KNOWLEDGEStart Use what you know to try to solve the problem below.Same and Different14 – 9A pool’s diving platform is 10 m above the water’s surface. Thebottom of the pool is at 25 m, relative to the surface of thewater. What is the distance between the diving platform andthe bottom of the pool?–14 9A BC D–3 (–2)3 2TRYIT Curriculum Associates, LLC Copying is permitted.Possible SolutionsPossible work:All expressions have a value of 5 or 25.SAMPLE AMath Toolkit grid paper, integer chips, number linesSAMPLE BDistances:A is the only subtraction expression.Platform to surface of water: 10 mSurface of water to bottom of pool: 25 mB is the only expression with a negative integerand a positive integer.15 m25 mDISCUSS ITSMP 2, 3, 6Support Partner DiscussionAfter students work on Try It, have them respond toDiscuss It with a partner. Listen for understanding of: the distance is represented by the sum u10u 1 u5uor the difference 10 2 (25). the surface of the water is represented by 0.Share: In my work . . .represents . . .Learning TargetsSMP 1, SMP 2, SMP 3, SMP 4, SMP 5, SMP 6Apply and extend previous understandings of addition and subtraction to add and subtract rationalnumbers; represent addition and subtraction on a horizontal or vertical number line diagram. Apply properties of operations as strategies to add and subtract rational numbers.SMP 1, 2, 3, 4, 5, 6See Connect to Culture to support studentengagement. Before students work on Try It, use SayIt Another Way to help them make sense of theproblem. Ask students for a show of thumbs up ordown to agree with a rephrasing of the problem andto offer a revision if they disagree.Ask: How does yourwork represent thesurface of the water?It is 15 m from the diving platform to thebottom of the pool.WHY? Support students’ facility at classifyingand calculating the addition of positive andnegative integers.Make Sense of the Problem5 15The distance is 15 m.DISCUSS ITD is the only addition expression with two negative integers.TRY IT10 2 (25) 5 10 1 510 mC is the only addition expression with two positiveintegers.18310 m abovewater’s surface Curriculum Associates, LLC Copying is not permitted.183LESSON 10 Add and Subtract Positive and Negative Numbers183Common Misconception Listen for students who state the distance as 5 meters,perhaps by finding the sum of 10 and 25. These students may be confusing themeaning of addition and subtraction in modeling real-life situations. As studentsshare their strategies, ask them to draw a diagram to illustrate the problem. Ask themto clarify how their model shows the distance between the diving board and thebottom of the pool.Select and Sequence Student StrategiesSelect 2–3 samples that represent the range of student thinking in your classroom.Here is one possible order for class discussion: LESSON 10 Add and Subtract Positive and Negative Numbersusing integer chips to model the distance(misconception) finding 10 1 (25) as the distanceusing a number line to model the distanceusing an equation to calculate the distance Curriculum Associates, LLCCopying is not permitted.

LESSON 10 SESSION 1ExploreFacilitate Whole Class DiscussionLESSON 10 SESSION 1Call on students to share selected strategies. Askstudents to make sure that they describe thepositions of the diving board and pool bottom usingthe positive and negative numbers included in theproblem statement. To confirm understanding, callon another student to reword the description usingmathematical language as necessary.1Guide students to Compare and Connect therepresentations. Prompt students to refer to theirmodels or diagrams to help explain why theirstrategies make sense for the problem.2CONNECT ITLook Back What is the distance between the diving platform and the bottomof the pool? How do you know?15 m; Possible explanation: The diving platform is 10 m above the surface,the bottom of the pool is 5 m below the surface, and 10 1 5 5 15.Look Ahead In the Try It, you found the distance between two elevationsrepresented by integers. You can also find the distance between non-integers.a. Explain how you can use the number line to find the distance between4.5 and 23.75.ASK How do the models show whether you shouldfind the sum or difference of 10 and 25?Possible explanation: The number line shows that there are 4.5 unitsbetween 4.5 and 0. There are 3.75 units between 0 and 23.75. In total,4.5 1 3.75 5 8.25, so the distance between 4.5 and 23.75 is 8.25 units.LISTEN FOR The models show that to find thedistance between 10 and 25, you find thedifference between 10 and 25.CONNECT IT1b. You can use both the expression u4.5 2 (23.75)u and the expressionu23.75 2 4.5u to find the distance between 4.5 and 23.75. Why?Possible explanation: Distance is always positive and the distancebetween two points is the same no matter which point you start from.Look Back Look for understanding thatc. You can subtract to find the difference between 4.5 and 23.75. Explain why23.75 2 4.5 is the opposite of 4.5 2 (23.75).distance along a number line is represented asthe difference between two numbers and thatsubtracting a negative number has the sameeffect as adding its opposite.121222324Possible answer: You find both of them by subtracting. When you finddistance, you use absolute value, so the order in which you subtract doesnot matter. When you find a difference, you do not use absolute value andthe order does matter.Hands-On Activity Use a number line to modelsubtraction.Copying is not permitted.2Reflect How is finding the distance between two numbers on the number linelike finding the difference between two numbers? How is it different?DIFFERENTIATION RETEACH or REINFORCE Curriculum Associates, LLC3Possible answer: You travel the same distance with both expressions, butin opposite directions.3Materials For each pair: 2 counters, Activity SheetNumber Lines Invite students to model a subtraction problem withpositive integers and a positive result, such as 7 2 4.Have them demonstrate the subtraction by placinga counter at 7 and moving the second counter4 units to the left or down on the number line to 3. Ask: How does this show finding 7 2 4? [The numberyou end at, 3, is the difference.] Have students model 10 2 5. Ask: Suppose, instead of 5, you want to subtract 25.Which direction should you move on the numberline? [In the positive direction: to the right or up.]If needed, remind students that addition andsubtraction are inverse operations. Prompt themto think about the movement that would undoadding 25. Repeat with other examples of subtracting anegative integer from a positive integer, such as8 2 (23) or 5 2 (22).40SMP 2, 4, 5If students are unsure about subtracting negativenumbers, then use this activity to help them visualizethe process.51841842LESSON 10 Add and Subtract Positive and Negative Numbers Curriculum Associates, LLC Copying is not permitted.Look Ahead Point out how the subtraction problem 4.5 2 (23.75) involvessubtracting a negative number from a positive number, which was also done inTry It. Students should recognize that the strategy they used in Try It could beadapted to solve this problem.CLOSE3EXIT TICKETReflect Look for understanding of the difference between two numbersas modeled by the distance between the numbers on a number line.Common Misconception If students think that adding a positive number anda negative number shows the distance between them, then have them plottwo opposite numbers, such as 6 and 26, on a number line. Then ask them tocompare the sum of the two numbers, which is 0, with the distance betweenthem on the number line, 12.LESSON 10 Add and Subtract Positive and Negative Numbers184

LESSON 10 SESSION 1Prepare for Subtracting Positive and Negative NumbersSupport VocabularyDevelopmentAssign Prepare for Subtracting Positive andNegative Numbers as extra practice in class oras homework.LESSON 10 SESSION 1Name:Prepare for Subtracting Positive andNegative Numbers1If you have students complete this in class, then use theguidance below.Ask students to consider the term absolute value.Remind students that opposite numbers havethe same absolute value and ask them to provideexamples.Think about what you know about numbers and absolute value. Fill in each box.Use words, numbers, and pictures. Show as many ideas as you can.Possible answers:What Is It?What I Know About Ita number’s distance from 0 on thenumber lineAbsolute value is never negative.Have students work in pairs to complete the graphicorganizer. Invite pairs to share their completedorganizers and prompt a whole-class comparativediscussion of definitions, examples, andnon‑examples.The absolute value symbol is z z.absolutevalueHave students look at problem 2 and discuss with apartner whether 24 represents the absolute valueof 3 2 7. Encourage students to revise the questionso that the answer is yes. For example, u24u is equalto u3 2 7u.Examplesz4z 5 4Non-Examplesz24z 5 432350z0z 5 02 356Problem Notes12Students should understand that a number andits opposite have the same absolute value,which is positive for all numbers except 0.Paired vertical bars are used to show absolutevalue, such as u23u 5 3.2Is 24 the absolute value of 3 2 7? Explain.No; Possible explanation: Absolute value is always positive.Students should recognize that the absolutevalue of 3 2 7 is equal to the absolute valueof 24, which is 4.185LESSON 10 Add and Subtract Positive and Negative Numbers Curriculum Associates, LLC Copying is not permitted.185REAL-WORLD CONNECTIONWhen people travel down major highways,they can use mileposts to calculate distances.Mileposts mark the distance along a highway,starting at one end and going to the other endor to the point where the highway crosses astate border. Depending on which direction adriver travels, the numbers on the milepostsmight increase or decrease. For example, if adriver has just passed milepost 137 and wantsto eat at a restaurant at milepost 60, the driverknows that the distance to the restaurant, inmiles, is equal to the absolute value of 60 2 137,or 77 miles. Ask students to think of other realworld examples where applying the conceptof absolute value might be useful in solving aproblem with positive and negative numbers.185LESSON 10 Add and Subtract Positive and Negative Numbers Curriculum Associates, LLCCopying is not permitted.

LESSON 10 SESSION 1Additional Pr

181a LESSON 10 Add and Subtract Positive and Negative Numbers Curriculum Associates, LLC Copying is not permitted. Later in Grade 7, students will solve real-world problems involving all four operations. They will also add and subtract rational numbers to simplify linear expressions and solve multi-step linear equations.

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