The integers are the positive whole numbers 0 and negative numbers For any two integers on a number line the number to the right is greater
THE INTEGERS 7 8 YEARS The Improving Mathematics Education in Schools (TIMES) Project NUMBER AND ALGEBRA Module 15 A guide for teachers - Years 7–8
Addition and Subtraction of Integers Integers are the negative numbers zero and positive numbers Addition of integers An integer can be represented or
Integers are numbers that are positive negative and zero but not fractions/decimals All mathematical operations can be applied to integers Signs of the two
30 jan 2019 · integers and addition-subtraction operations in light of the domain in primary school mathematics teaching and it is
INTEGERS 45 MATHEMATICS Example 12: Write the digits 0 1 2 3 4 5 6 7 8 and 9 in this order and insert '+ 'or '–' between them to get the result
MATHEMATICS 118 6 2 2 Ordering of integers Raman and Imran live in a village where there is a step well There are in all 25 steps down to the bottom of
When you subtract a positive integer, move to the left along the number line Page 7 {7} The Improving Mathematics Education in Schools (TIMES) Project
05 Illustrate the relationship between adding integers and subtracting integers N06 06 Solve a given problem involving the addition and subtraction of integers
MATHEMATICS 118 6 2 2 Ordering of integers Raman and Imran live in a village where there is a step well There are in all 25 steps down to the bottom of
The numbers that include natural numbers and zero Not a fraction or decimal {0, 2, 3, 4, 5 6, 7, 8, 9, 10, 11 } Integer A counting number, zero, or the negative
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Mathematics7
Unit1:Integers
N06
YearlyPlanUnit1:GCON06
Mathematics7,ImplementationDraft,June20151
SCON06:Studentswillbeexpectedtodemonstrateanunderstandingofadditionandsubtractionof integers,concretely,pictorially,andsymbolically. [C,CN,PS,R,V] [C]Communication[PS]ProblemSolving[CN]Connections[ME]MentalMathematicsandEstimation [T]Technology[V]Visualization[R]Reasoning
PerformanceIndicators
Usethefollowingsetofindicatorstodeterminewhetherstudentshaveachievedthecorresponding specificcurriculumoutcome. N06.01Explain,usingconcretematerialssuchasintegertilesanddiagrams,thatthesumofopposite integersiszero. N06.02Illustrate,usinganumberline,theresultsofaddingorsubtractingnegativeand positive integers. N06.03Addtwogivenintegers,usingconcretematerialsand/orpictorialrepresentations,andrecord theprocesssymbolically. N06.04Subtracttwogivenintegers,usingconcretematerialsand/orpictorialrepresentations,and recordtheprocesssymbolically. N06.05Illustratetherelationshipbetweenaddingintegersandsubtractingintegers.
N06.06Solveagivenprobleminvolvingthe
additionandsubtractionofintegers.
ScopeandSequence
Mathematics6
N07Studentswillbeexpectedto
demonstrateanunderstandingof integerscontextually,concretely, pictorially,andsymbolically.
Mathematics7
N06Studentswillbeexpectedto
demonstrateanunderstandingof additionandsubtractionof integers,concretely,pictorially, andsymbolically.Mathematics8
N07Studentswillbeexpectedto
demonstratean understandingof multiplicationanddivisionof integers,concretely,pictorially, andsymbolically.
Background
Integersarethesetofnumbersconsistingofthenaturalnumbers(1,2,3,...),theiropposites(-1,- 2, -3,...),andzero.Theyarealsoreferredtoasthewholenumbersandtheiropposites.Integers indicatebothaquantity(magnitude)andadirectionfromzero.Positiveintegersare greater thanzeroandarelocatedtotherightofzeroonthenumberline.Theyarerepresentedbya positivesymbol(+)beforetheinteger,suchas(+5).Negativeintegersarelessthanzeroandare locatedtotheleftofzeroonthenumberline.Theyarerepresentedby anegativesymbol(-) beforetheinteger,suchas(-3).Therearetwocommonnotationsforintegers.Thesymbolsare writtenwiththe+or-signprecedingtheinteger,asin-5or+3orenclosedwithinparentheses, asin(+5),(-3).Theparenthesesarecommonlyusedinstudent materialstoavoidanyconfusion betweentheintegersignandthenotationsforadditionandsubtraction.Intheexpression(+5)- (-3)theparenthesesindicatethenumbersinsideareintegersanddistinguishtheinteger symbolsfromthesubtractionsymbol. Understandingandworkingwithintegersisimportantindailylife.Integers areregularly encounteredincontextssuchasfinances(networth,balancesheets,andprofitorloss),
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Mathematics7,ImplementationDraft,June20152
investments,temperatures,elevations,timerelevanttoevents,andsports.Providestudents withcontextualizedproblemsofthisnature.Proficiencywithintegersisnecessarywhen evaluatingalgebraicexpressionsandsolvingequations.Itallowsstudentstographrelations usingallfourquadrants.Workwithintegerswillbeappliedtofuturestudyofrational expressions,andextendedtoirrationalandrealnumbers.Itcontinuestobuildnumbersense, preparingstudentsforawiderangeofproblemͲsolvingactivities.Operationswithintegersbuild onoperationswithwholenumbers. Thebalanceofpositiveandnegativevaluesisknownasthezeroprinciple,anditisthefoundation formanycomputationsinvolvingintegers.Emphasismustbeplacedonthezeroprincipleandits applicationinadditionandsubtractionsituations.Forexample,(-1)+(+1)=0,(-3)+(+3)=0,(-
17)+(+17)=0.Thesumofanypairofoppositeintegersiszero.Basedon
thisprinciple,zeroand otherintegerscanbeexpressedinmultipleways. Atileorcounterrepresenting(+1)andonerepresenting(-1)formazeropair.Whencombined, thesetilesmodelthenumberzero.Agivenintegercanbemodelledinmanydifferentways.For example,onewayto represent-3usingintegertilesisshownbelow. Usingintegertilesorcountersandthezeroprincipletomodelthevariouswaysanumbercanbe representedwillhelpstudentsconcludethataddingazeropairdoesnotchangethevalueofthe integerbeingmodelled.Workwith zeropairswillbeimportantasstudentsconcretelyaddand subtractintegers.Studentswillprogresstoaddingandsubtractingintegerssymbolically.They willgeneralizeandapplyrulesfortheseintegeroperations. Concretemodelscommonlyusedforrepresentingintegersarecolouredcounters,algeͲtiles,and numberlines.Digitalmodelsalsoexistand simulationsallowstudentstoworkwith representationsofpositiveandnegativecounters.Studentsshouldbeexposedtomultiple models.Paralleldevelopment,usingbothmodelsatthesametime,maybethemostconceptual approach.Integersinvolvetwoconcepts,"quantity"and"opposite."Quantityismodelledbythe numberofcountersor lengthofthearrowsinavectormodel.Oppositeisrepresentedas differentcoloursordifferentdirections.Anexampleofaddingintegersusingeachmodelis provided. (+4)+(-1)=+3 Johnwasa+4inhisfirstgameofahockeytournament.Hewasa-1 inthesecondgameofthe tournament.Whatishisplus/minusattheendofthesecondgame? Billwas+4attheendofthefirst9holesofgolf.Inthenext9holes,hisscorewas-1.Whatishis scoreattheendof18holes ofgolf?
YearlyPlanUnit1:GCON06
Mathematics7,ImplementationDraft,June20153
Afteraheavyrain,theSackvilleRiverwas4metresaboveitsusuallevel.Afteroneday,theriver dropped1m.Whatwastheleveloftheriveronthesecondday? Startat0.Move4unitstotherighttorepresent+4.Fromthere,move1 unittothelefttorepresent -1. ItwasadrysummerinShubenacadie.Theriverwas2mbelowitsnormallevel.DuringAugust,there wasnorain,andthewaterlevelwentdownanothermetre.
Howfaristheriverbelowthenormallevelnow?
(-2)+ (-1)=(-3) Thedistanceanintegerisfromzerorepresentsthemagnitudeoftheinteger,andthedirectionfrom zerorepresentswhethertheintegerispositiveornegative.Whileteachersmaymodelcorrect useofabsolutevalue(thedistancefromanintegertozeroonanumberline),it isnotan expectationforstudentstoknowtheterm.Onaverticalnumberline,thedistanceabovezero representspositiveintegers.Distancesbelowzerorepresentnegativeintegers.Numbers increaseinvalueabovezeroonaverticalnumberline,anddecreaseinvaluebelowzeroona verticalnumberline.On ahorizontalnumberline,numbersincreaseinvaluetotherightofzero, anddecreaseinvaluetotheleftofzero.Valuesalwaysincreasefromlefttoright,anddecrease fromrighttoleft. Theuseofmodelshelpsstudentsdevelopaconceptualunderstandingofthefollowingprinciplesfor addingintegers.
1.Thesumoftwopositiveintegersispositive.
2.Thesumoftwonegativeintegersisnegative.
3.Thesumofanegativeintegerandapositiveintegercanbenegativeorpositive.Thesumhasthe
signofthenumberthatisfurtherfromzero. As withaddition,itisimportantforstudentstohaveaconceptualunderstandingofinteger subtraction.Studentsshouldmodelsubtractionofintegersusingcolouredcountersandnumber lines.Onepossibilitywhensubtractingintegersistousea"takeaway"meaning.Thisiseasily manageablewhentheintegershavethesamesign,and thesubtrahendisclosertozerothan theminuend,orstartingamount.Forexample,(-4)-(-3)canbemodelledwithcountersas follows.
YearlyPlanUnit1:GCON06
Mathematics7,ImplementationDraft,June20154
Whentakingawayisnotimmediatelypossible,itisnecessarytoaddzeropairstoreͲrepresentthe minuendandallowthestudenttobeabletoremovethetilesthatrepresentthesubtrahend. Guidethisdevelopmentalunderstandingthroughtheuseofgoodquestions.Tosolve(-3)-(+1) askstudentshowtheycouldrepresent(-3)thatwouldallowthemtotakeaway(+1). Tosubtractintegers,youcanalsousea"thinkaddition"meaning.Todetermine(+1)-(-2),ask "Howmuchmustbeaddedto(-2)togetto(+1)?"Usinganumberline,begin at(-2)anddraw anarrowto(+1).Ithasalengthof3pointingright. Whiletherule"tosubtractaninteger,addtheopposite"allowsstudentstoreachthecorrect answer,itdoesnotpromoteconceptualunderstanding.Studentsshouldbeled tothis conclusionthroughtheuseofmodels.Agoodexamplewouldinvolveusingthenumberlineto subtractanegativefromapositive.Suchasituationshouldmakeiteasierforstudentstosee whyyoucanaddtheoppositetosubtract.Fromthepreviousexample,(+1)-(-2)tells whatto addto(-2)togetto(+1).Togofrom(-2)to(+1),move2totherighttogetto0,andthen another1totherighttogetto(+1).Thetotalamounttobeaddedis(+2)+(+1)or,since additioniscommutative, (+1)+(+2).Studentsshouldnowseethat(+1)-(-2)=(+1)+(+2). Patterningcanbeusedtodevelopthisaswell.Askstudentstostudyapatternsuchastheone givenhere. (+4)-(+2)=2(+4)+(-2)=2 (+4)-(+1)=3 (+4)+(-1)=3 (+4)-(0)=4(+4)+(0)=4
YearlyPlanUnit1:GCON06
Mathematics7,ImplementationDraft,June20155
(+4)-(-1)=5(+4)+(+1)=5 (+4)-(-2)=6(+4)+(+2)=6 Astheycompareeachcolumn,theyshouldconcludethatsubtractingresultsinthesameansweras addingtheopposite.Studentsshouldbeawarethatwhileadditionofintegersiscommutative, subtraction isnot.Infact,iftheorderoftheintegersinthesubtractionstatementchanges,the differencesareoppositeintegers. Studentsshouldeventuallymoveawayfrommodelsastheypracticeintegerarithmetic.However,it isimportantthattheynotviewtheproceduralrulesasarbitrary.Whilethecorrectansweris important,emphasisshouldbeontherationaleandnotonhowquicklytheycangetanswers.
Assessment,Teaching,andLearning
AssessmentStrategies
ASSESSINGPRIORKNOWLEDGE
Taskssuchasthefollowingcouldbeusedtodeterminestudents'priorknowledge. Askstudents:Howmanynegativeintegersaregreaterthan-9? Tellstudentsthatanumberis14jumpsawayfromitsoppositeonanumberline.Ask,whatisthe number? Askstudentsto explainwhetheritistruethat anegativenumberfurtherfrom0islessthananegativenumberthatiscloseto0 anegativenumberisalwayslessthanapositivenumber apositivenumberisalwaysgreaterthananegativenumber
WHOLEͲCLASS/GROUP/INDIVIDUALASSESSMENTTASKS
Considerthefollowingsampletasks(thatcanbeadapted)foreitherassessmentforlearning (formative)orassessmentoflearning(summative). Useanumberlineortwocolourcounterstoexplainwhythefollowingcalculationsarecorrect. (-3)+8=5 (-5)-3=-8 (-4)-(-6) =2
9+(-2)=7
6-4=2
8-(-3)=11
Jonsaved$50duringthefall.Heowes$15tohisfriend.Heearned$20mowinglawns.WhatisJon's networth? Findintegerpairswithasumof-16where onenumberislessthan-16
YearlyPlanUnit1:GCON06
Mathematics7,ImplementationDraft,June20156
onenumberisgreaterthan+16 onenumberisgreaterthan0andlessthan+5 Canyoumodel+2withanoddnumberofcounters?Explainwhyorwhynot. Model(-2)-(-4)usinganumberline. Isthesumofa negativeintegerandapositiveintegeralwaysnegative?Explainwhyorwhynot. Can+2bemodelledwithanoddnumberofcounters?Explainyourreasoning. Answerthefollowing:Timhasadebtof$55.Heearns$30oneachoftwodaysandspends$39ona pairofpants.Howmuchmoneyordebtdoeshenowhave?Includeadiagramandanumber sentence. Isthesumofanegativenumberandapositivenumberalwaysnegative?Explainyourreasoning. Whenyouaddtwonegativeintegers,youalwaysgetanegativesum.Whenyou subtracttwo negativeintegers,doyoualwaysgetanegativedifference?Explainwiththeaidofexamples. Respondtothefollowing: Isthedifferencebetweenanegativenumberandapositivenumberalwaysnegative?Explain yourreasoning. Howdoyousubtractintegersusingtiles?Explainandgive anexample. Withoutactuallycalculatingthedifferencebetweentwointegers,howdoyouknowwhether theanswerwillbepositive,negative,orzero?Explainwiththeaidofexamples. Writeanadditionequationtodescribeeachsituation,andexplainitsmeaning. Beforeyouwenttosleeplastnight thetemperaturewas-3°C.Duringthenightthetemperature droppedby5°C.Whatwasthetemperatureinthemorning? Mrs.Brownparkedintheparkinggarage10mbelowstreetlevel.Shethengotonanelevator andwentup27mtoheroffice.Howfarabovethe streetisheroffice? Writeasubtractionequationforeachsituation. Asoccerballwaskicked5mintheforwarddirectiononthefirstplayofthegame.Theopposing teamthenkickeditback6m.Whatwasthetotalchangeindistance? Iftheaverage temperatureinYarmouthis19°CinJulyand-6°CinJanuary,whatisthe differencebetweentheaveragehighestandlowesttemperatures? AdamandAndréweredigginginthesandatthebeach.Archiedugaholethatwas22cmbelow thesurface,andAndrédugahole thatwas13cmbelowthesurface.Whatisthedifferencein thedepthsoftheholes?
PlanningforInstruction
CHOOSINGINSTRUCTIONALSTRATEGIES
Considerthefollowingstrategieswhenplanningdailylessons. Emphasizethatstudentsbeginwithconcrete,movetopictorialandfinallytosymbolic representationsforsumsanddifferencesofintegers. Givestudentsopportunitiestouseavarietyofshapeswithinamodeltorepresentonewhole.For example,whenusing patternblocks,theyellowhexagoncouldrepresentonewholeorwhen combiningpatternblockswithfractionblocks,thepinkdecagonpiececouldbeusedtorepresent onewholeandthustheyellowhexagonwouldnowrepresentoneͲhalf.
YearlyPlanUnit1:GCON06
Mathematics7,ImplementationDraft,June20157
Connectthesubtractionofintegerstotheadditionofintegers. UseproblemͲsolvingcontextsthatarerelevanttostudents. Connectproblemsapplyingadditionandsubtractionofintegerstosimilarproblemswithwhole numbers. Havestudentsexploresumsanddifferencesofintegersbyusingavarietyofmodels, suchasalgebra tiles,twoͲcolourcounters,andarrowsonanumberline.Itshouldbenotedthatitisimportantto alwaysuseakeyinpictorialrepresentationsofintegers,sostudentsknowwhichcolourrepresents positiveintegersandwhichrepresentnegativeintegers.Thereisnosetstandardfor theseand studentsshouldbeflexibleintheirthinking. Includevariousstructuresofproblemssuchas,"partͲpartͲwhole"and"comparison"fromprevious grades.Forexample,(-6)-տ =3տ +2=(-5)(-9)=(-4)+(-տ). Havestudentsjustifythestrategiesthey useinfindingsumsanddifferencesofintegersandprovide opportunitiestodiscussstrategiesusedbyothers.
SUGGESTEDLEARNINGTASKS
Solveadditionandsubtractionofintegersthroughtheuseofmagicsquares.Forexample, Rolltwodiceofdifferentcolours.Assignnegativetoonecolourandpositivetotheother,andfind thesumofthenumbersrolled.Rollthetwodiceagain,findthesumand addtheresulttoyour previousscore.Exchangeturnsuntilonepersonreaches+20or-20.Whywoulditbefairtoaccept +20or-20asthewinningscore? SolveandcreateproblemsusingrealͲlifesituationssuchas:timezones,temperature,heightsabove andbelowsealevel, profitsandlosses,games,sports,sharesofstocks,etc. Createcardswithoppositeintegerswrittenonthem.Distributethesecardstostudents.Locatetheir oppositeandsitinzeropairs. Usingintegertiles,demonstratethreewaystomodelanintegersuchas-7,8,-2,etc.Shareyour models withtheclass. Modelasmanyintegersasyoucanusingexactlynineintegercounters. Youearn$5andthenspend$5.Howmuchisyourprofitorloss?Drawdiagramswithinteger counterstorepresentthisproblem. Arrangestudentsinteams.Beginwitheachteamstanding.
Writeanadditionexpressiononthe
boardandhavestudentswritetheanswer.Studentswithcorrectanswersremainstanding.All otherssit.Afterthreequestions,theteamwiththemostpeoplestandingreceives10points.One memberoftheteamisselectedtoattemptafunactivity(e.g.,shootingafoam basketballintoanet) foranadditional5pointsfortheteam.Thenanewroundofplaybegins.Theteamthatreaches100 pointsfirstwins. Findthreepairsofintegerswithasumof-29. Createandsolvetheirownproblemsusingreal-lifesituationssuchas timezones,temperature, heightsaboveandbelowsealevel,profit/loss,etc. -7 -11 -9-1-2
YearlyPlanUnit1:GCON06
Mathematics7,ImplementationDraft,June20158
Findchartsinthedailynewspapershowingthelowandhightemperaturesforthedayforcities aroundtheworld.Useinformationfromthechartstomakeuptwoproblemsinvolvingsubtraction ofpositiveandnegativenumbers.Findthesolutionsandthenexchangeproblemswithother studentsandsolve. Teacherscanusemaskingtape(orothermaterials)tomakealargenumberlineonthefloor. Studentscandiscusshowanumberlinecanbeusedforsubtraction,aswellasforadditionof integers.Studentscanwalkonthenumberlinetoshowsubtractionssuchas(+7)- (-3)or(-4)- (-2). S
UGGESTEDMODELSANDMANIPULATIVES
algebratiles* horizontalnumberlines thermometers twoͲcolourcounters verticalnumberlines *alsoavailableinInteractiveMathTools(Pearsonn.d.) M
ATHEMATICALLANGUAGE
TeacherStudent
absolutevalue integer minuend negativeinteger positiveinteger sign subtrahend zeroprinciple magnitude integer negativeinteger positiveinteger sign zeroprinciple
YearlyPlanUnit1:GCON06
Mathematics7,ImplementationDraft,June20159
Resources
Print
MathMakesSense7(Garneauetal.2007)
Unit2:Integers(NSSBB#:2001640) Section2.1RepresentingIntegers Section2.2AddingIntegerswithTiles Section2.3AddingIntegersonaNumberLine Section2.4SubtractingIntegerswithTiles Section2.5SubtractingIntegersonaNumber Line UnitProblem:WhatTimeIsIt? ProGuide(CD;WordFiles)(NSSBB#:2001641) AssessmentMasters ExtraPracticeMasters UnitTests ProGuide(DVD)(NSSBB#:2001641) ProjectableStudentBookPages ModifiableLineMasters
Digital
"NationalLibraryofVirtualManipulatives:NumberandOperations(Grades3-5),"UtahState University(UtahStateUniversity2015):http://nlvm.usu.edu/en/nav/category_g_2_t_1.html (Select ColorChips - AdditionorColorChips - Subtractionfromthelistofvirtualmanipulativesprovided.)
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