http://bridgingmath.com

2016PDRelease

June18,2018

6G: Coordinate Geometry

6thGradeCoordinateGeometrySRIInternational©2016OnthecoverUntitled,1973.VerenaLowensberg.ThesematerialsareprovidedthroughagrantfromtheNationalScienceFoundationoftheUnitedStates(DRL-1417895).Anyopinions,findings,andconclusionsorrecommendationsexpressedinthispublicationarethoseoftheauthor(s)anddonotnecessarilyreflecttheviewsoftheNationalScienceFoundation.WritingandEditing:JenniferKnudsen,Hee-JoonKim,TeresaLara-MeloyRevisionsandEditing:EileenBehr,HarrietteStevens

TableofContents

Preface................................................................................................................................................1Learninggoals....................................................................................................................................1JustificationStages..............................................................................................................................1TipsforArgumentation.......................................................................................................................1

Lesson1:ExactlyWhere?.....................................................................................................................3Background ....................................................................................................................................3WarmUp 15min.............................................................................................................................3MainActivity 30min.........................................................................................................................4Lesson1:Homework............................................................................................................................5

Lesson2:PatternsinCoordinates........................................................................................................7Background ....................................................................................................................................7WarmUp 10min..............................................................................................................................7MainActivity 30min.........................................................................................................................8Lesson2:Homework..........................................................................................................................11

Lesson3:Lengthsofsegments...........................................................................................................13Background 13WarmUp 5min..............................................................................................................................13MainActivity 30min.......................................................................................................................14Lesson3:Homework..........................................................................................................................18

Glossary.............................................................................................................................................19

Preface 1

Preface

LearninggoalsLesson Essential Question Learning Objectives CCSS

1 Exactly Where?

Whatisacoordinatesystemandhowisitusedtorepresentmathematicalfigures?

Studentswill● understandtheutilityofthe

coordinategridforlocatingpoints● knowtheconventionsofgraphing

inthecoordinateplane

5.G.1

2 Patterns in Coordinates

Howcanrectanglesberepresentedinacoordinatesystem?

Studentswill● createpolygonsgivencoordinates● generateconjecturesaboutthe

relationshipsbetweencoordinatesinarectangleswithsidesparalleltotheaxes,andjustifythem

6.G.3.

3 Lengths of a segment

Howdoyoufindthelengthofalinesegmentinacoordinatesystem?

Studentswill● generateconjecturesabouthowto

calculatethelengthsofsidesofrectangleswithsidesparalleltotheaxes,usingthecoordinatesofthevertices

6.G.3.6.NS.6.c6.NS.8

JustificationStages 1. Conjecturesarejustified/defendedwithoutevidenceorbysimplycitingvisualcues(becauseit

lookslikeit),orrelyingontheteacherastheauthority.2. Conjecturesarejustified/defendedbyusingoneortwocases.Thismayincludeextremecases—

wheretheexamplesaretheonlyevidence.3. Conjecturesarejustified/defendedbybuildingonacasesupportedbygeneralizedreasoning.

Thismaystartwithanexample,butthenstudentsdevelopabroaderjustificationusingthepatternsthatemergeintheexample.

4. Conjecturesarejustified/defendedwithgeneralizedmathematicalreasoningthatconnectstheconjectureandprovidesarebuttalofcounter-arguments.Counter-examplesmaybeused.

TipsforArgumentation Generating Cases

● Tohelpstudentsgeneratecases,tellthemtocreatemorethanoneexample.● Tohelpstudentsseegeneralpatterns,ensurethattheylookatarangeoftypicalcases.● Tosupportstudentsexaminingawiderangeofcases,askstudentstolookatspecialcases(e.g,

squaresarespecialrectangles.)

2 Preface

Conjecturing ● Toencouragestudentstogenerateconjectures,includingboldconjecturesforwhichthetruthis

unknown,asko Whatareyouprettysureistrue?o Whatdoyouthinkistrue?o Whatdoyouthinkmightbetrue?o Whatdoyouthinkisfalse?

● Toencouragestudentstomakeageneralizedconjecture,ask,o Whatdoyourcaseshaveincommon?o Canyoumakeaconjecturethatappliestomorethanoneexample?

● Toencouragemoremathematicalconjectures,focusstudents'attentionontherelationshipsbetweenrepresentationssuchasnumbers,graphs,equations.

Justifying ● Tostartthejustificationprocess,askstudents,

o Isyourconjecturealwaystrue?o Whatmakesyouthinkso?

● Toencourageusingvisualrepresentationstosupportlanguage,suggest,“Showhowyouknowitistrue.”

● Toadvancejustification,encouragestudentstothinkaboutextremecasesthatwillchallengetheirconvictionaboutaconjecture.

● Tochallengeajustification,encouragestudentstothinkofacasethatmightnotwork.● Toadvancefromanexample-basedjustificationtoageneralizedone,askstudents“howdoyou

knowyourconjectureworksinANYcase,beyondthoseyouhaveusedsofar?”

Concluding ● Toaidstudentsindrawingaconclusion,askthemtosummarizetheirargumentsandwrite

downwhattheyknowistrueorfalseasaresult.● Toprovideclosure,stamporlabelaconjectureas“true”,“false”or“notenoughinformationto

decide.”● Toleadtothenextargument,askfornewconjecturesthatfollowfromwhattheyalreadyhave

justified.

Lesson1:ExactlyWhere? 3

Lesson 1: Exactly Where?

Background Essential Question Whatisthecoordinatesystemandhowisitusedtorepresentmathematicalfigures?

Learning Objectives Studentswill

● understandtheutilityofthecoordinategridforlocatingpoints● knowtheconventionsofgraphinginthecoordinateplane

Common Core Alignment 5.G.1.Graphpointsonthecoordinateplanetosolvereal-worldandmathematicalproblem;Useapairofperpendicularnumberlines,calledaxes,todefineacoordinatesystem,withtheintersectionofthelines(theorigin)arrangedtocoincidewiththe0oneachlineandagivenpointintheplanelocatedbyusinganorderedpairofnumbers,calleditscoordinates.Understandthatthefirstnumberindicateshowfartotravelfromtheorigininthedirectionofoneaxis,andthesecondnumberindicateshowfartotravelinthedirectionofthesecondaxis,withtheconventionthatthenamesofthetwoaxesandthecoordinatescorrespond(e.g.,x-axisandx-coordinate,y-axisandy-coordinate.

WarmUp 15min

1. Introducemathematicaltalkandnormsforargumentation.2. Introduceimprovasawaytofocusonthosenorms.3. PlayZipZapZop.Purposes:Makemistakes,payattention,sayyes,keeptheballintheair,maketheensemblelookgood,bepresent.Normsdeveloped:Payattention,listen,makeeyecontact,makesureyouareheard,participate.Instructions:

● Standinacircle.● Inturn,eachpersonmakesaverbaloffertoanother

personbythrowinghim/heraninvisibleball,sayingzip,zap,orzop(oneeach,inthatorder).

● Thecatcherthenbecomesthethrowerandsendsthenextwordtosomeoneelseinthecircle.

● Playcontinuesinanyorder.Questionsfordiscussionaboutnorms:

● Whatwasitliketoplaythatgame?● WhatdidyoulearnabouttheClassroomRightsand

Requirements?

4 Lesson1:ExactlyWhere?

MainActivity 30minActivity 1.1. Without a grid 1.Sitwithapartnersothatyoucannotseeeach

other’swork.A. Onapieceofblankpaper,drawaline

segmentofanylengthinanyposition.B. Onepersondescribeshowtoduplicate

his/herdrawing.C. Theotherfollowstheinstructions.

• Donotusenumbers.• Donotuseunitssuchasinches,

centimeters.• Donotdrawaxes.

pairworkMakesurestudentsdonotusestandardunitsormeasurementtools.Encouragethemtobeascreativeaspossibleincommunicatingabouttheirdrawingstotheirpartners.

2.Whenyouaredone,compareyourdrawings:A. Areyourlinesegmentscongruent?B. Aretheyplacedinthesamelocationon

eachpage?C. Howdoyouknow?

wholeclassAskquestions:

• Whyarethedrawingsthesameornot?• Whatwordsdidyouusetohelpcommunicate

yourdrawing?

Collectontheboardthewordsstudentsusedtodescribeposition,length,andslantoftheirlinesegment.

Activity 1.2. With a grid 1.Switchtheroleswithyourpartner.

A. Onapieceofgridpaper,drawalinesegmentofanylength,inanyposition.

B. Onepersondescribeshowtoduplicatehis/herdrawing.

C. Theotherfollowstheinstructions.● Usethegridtohelpyou.Thismeansyou

canusenumberstocountstheboxes.

pairworkMakesurestudentsdonotusestandardunitsormeasurementtools.Encouragethemtobeascreativeaspossibleincommunicatingabouttheirdrawingstotheirpartners.

2.Whenyouaredone,compareyourdrawings:A. Areyourlinescongruentandinthesame

place?B. Howdoyouknow?

wholeclassHavestudentscomparethetaskswithandwithoutthegrid.• Howdidyouusethegridtoexplaintoyour

partnerhowtocreatethelinesegment?• Howwasthistaskdifferentorsimilartothetask

withoutthegrid?• Wasiteasyordifficulttocommunicatewithyour

partnercomparedtothetaskwithoutgrid?why?

Lesson1:ExactlyWhere? 5

3.Iftimeallows,switchrolesandrepeat,thistimedrawingaxesonthegrid.

Ifneeded,reviewvocabularyforcoordinateplane;axes,origin,coordinates,quadrants.

Lesson1:Homework1. Youarethecommanderinacontroltower.Youhavetoindicatewhererocksaresothatsail

boatsdon’trunintothem.Helpthemtotravelsafelybygivingthemthecoordinatesoftherocks.

PointA:(,)

PointB:(,)

PointC:(,)

PointD:(,)

PointE:(,)

PointF:(,)

6 Lesson1:ExactlyWhere?

Lesson2:PatternsinCoordinates 7

Lesson 2: Patterns in Coordinates

Background Essential Question Howcanrectanglesberepresentedusingacoordinatesystem?

Learning Objectives Studentswill

● generateconjecturesabouttherelationshipsbetweencoordinatesinrectangleswithsidesparalleltotheaxesandjustifythem.

Common Core Alignment 6.G.3.Drawpolygonsinthecoordinateplanegivencoordinatesforthevertices;usecoordinatestofindthelengthofasidejoiningpointswiththesamefirstcoordinateorthesamesecondcoordinate.Applythesetechniquesinthecontextofsolvingreal-worldandmathematicalproblems.

WarmUp 10minGiftGiving

Seeppt.

Purpose:Makingoffers,acceptingoffersregardlessoftheirclarity,makingyourpartnerandtheensemblelookgood,sayingyes.Normsdeveloped:Listeningandbuildingoffofeachother’sideas.

Instructions:• Partnersstandfacingeachotherwithahugeclosetof

unlimitedgiftsbehindthem.• Onepartnerofferstheotheragiftfromthecloset,byhanding

him/herawrappedbox.Thisgiftcanbeofanydimension,andtheofferergivesanideaofthesize,weight,and/orshape.

• Thereceiverthenopensthegiftandnamesitbythankingthegiver(e.g.,“Thankyouforthisgrapefruit”)ashe/shepicksupandhandlesthegift.

• Thegiverthenrespondswithhowhe/sheselectedthegiftandwhyhe/sheknewthereceiverwouldenjoyit.Therolesarethenswitched.

Questionsfordiscussionaboutnorms:• Whatwasitliketoplaythatgame?• WhatdidyoulearnabouttheClassroomRightsand

Requirements?• Whatdidyoulearnaboutmakingconjectures?(i.e.,

Conjecturesarelikegifts.)

8 Lesson2:PatternsinCoordinates

MainActivity 30minActivity 2.1.

Gen

erat

ing

case

s

1.DragverticesA,B,andCoftheredrectangletomakerectanglesofdifferentsizesandatdifferentlocations.a.Whatseemstohappentothecoordinatesasyoudrageachvertex?b.Makethefollowingrectangles.Recordthecoordinatesoftheirverticesinyournotebooks.

• Along,skinnyrectangle• Asquare• Arectanglewithvertex(0,0)• Arectanglethatisintwo

quadrantsofthecoordinategrid

groupwork

• Havestudentsdrageachpointoneatatimeandmakeobservationsabouthowdraggingeachpointchangestherectangle.

• MovingpointDdoesnothingtotherectangle.• MakesurestudentsdragpointsbeyondQuadrant1.• Makesurestudentsdragpointsoneatatimeand

observehowthecoordinatesarechanging.• Encouragestudentstostartwritingthepatterns

theyobserve.• Provideatablesothatstudentscanorganize

coordinatesofeachrectangle• Askstudentstoobservewhatchangesandwhat

doesn’tchangeincoordinatesastheydragvertices.

2.Describepatternsthatyouseeinthecoordinatesofthevertices.

• Havestudentsfocusonthecoordinatesofthevertices.

• Checkstudentunderstandingoftheterm“vertex.”

Con

ject

urin

g

3.Makeconjecturesabouttherelationshipsbetweenthecoordinatesoftheverticesofallrectangleswithsidesparalleltothexandy-axis.

wholeclassElicitstudentconjectures:• Whatdoyoumeanbythisparticularword?Isthere

anotherwaytosayit,mathematically?• Whatdowealreadyknowforsure?• Doyouthinkthatthepatternholdsforanycaseyou

couldgenerate?Ifso,howcanwedescribethat,mathematically?Ifnot,trytodescribejustthecasesforwhichitdoeshold.

• Canyouusemoremathematicallypreciselanguageforstatingthepattern?

Buildonstudentconjectures:

• Whatelseistrueaboutthecoordinatesofthevertices?

• Whatrepeatsandwhatchangesasyougofromvertextovertex?

Lesson2:PatternsinCoordinates 9

Just

ifyin

g 4.Justifyoneofyourconjecturesusingwhatyouknowaboutrectanglesandthecoordinategrid,andwhatyoucanseewhenyoumakechangeswiththeredrectangle.

wholeclassElicitstudentunderstanding• Whatisyourdefinitionofarectangle?• Canasquarebearectangle?Canarectanglebea

square?• Howdidyouuseyourdefinitiontoexplainthe

patternsyouobservedfromthecoordinates?• Whatcanyoutellaboutthecoordinatesofpoints

thatareonalineparalleltox-axis?y-axis?

Con

clus

ions

5.Conclude:Whatdoweknowaboutthecoordinatesofverticesofrectangleswithsidesparalleltotheaxes?

wholeclass• Markconjecturestrueorfalseasyoujustifythem.• Summarizetheargumentandmakesurethereis

consensusaboutthetruthorfalsityoftheconjectures.

• Makesurethatstudentsknowwhichconjecturesaretrueandwhicharefalse(whetherbyconsensusorothermeans).

• Encouragethestudentstowritedowntheirconclusions.

Arguments

Conjecture Justification Conclusion

Thex-coordinatesarealwaysthesameonthesameverticalside.Similarly,they-coordinatesarealwaysthesameonthesamehorizontalside.

Ifanytwopointsareonthesameverticalsides,theirx-coordinatesarethesamebecausethesideisparalleltothey-axisandthepointonthesamesideisthesamedistanceawayfromthey-axis.

True

Youneed4differentnumbersatmosttofillinthecoordinatesoftheverticesforanyrectanglewithsidesparalleltotheaxes.

Coordinatesofthetwoverticesofthesamesidehavecommonx-ory-coordinates.Youcanhaveonly2differentx-coordinatesand2differenty-coordinates.Therefore,youneedatmost4differentnumberstousetomakecoordinatespairsfortheverticesofarectangle.

True

Youneedonly2numberstomakecoordinatesoftheverticestomakearectangle.

Sometimesyoucanuseonly2numberstocreatearectangleasshownintheexample,butitisnotnecessarilytrueforanyrectangle.Example.(1,1),(2,1),(1,2),(2,2)-square.

False

10 Lesson2:PatternsinCoordinates

Iftwoverticesaretheendpointsofthediagonal,theydonothaveacommonx-ory-coordinatesbetweenthevertices.

Proofbycontradiction:Iftheyhavecommonx-ory-coordinates,theywouldbeonthesameverticalorhorizontalline.Therefore,theycan’tbeonadiagonal.Note:Proofsbycontradictionarenoteasyforstudentstoconstructandunderstand.Makecleartheproofstartsbycontradictingtheconclusion,i.e.“Whatiftheverticesdidhavethecommoncoordinates?”

True

Iftwopointsonalinehavethesamex-value,thenthelineisparalleltothey-axis.Similarly,iftwopointsonalinehavethesamey-value,thenthelineisparalleltothex-axis.

Iftwopointshavethesamex-coordinatebutdifferenty-coordinates,theywillbelinedupvertically.Therefore,thepointswillbeonthelinethatisparalleltothey-axis.

True

Giventwolines,ifonelineisparalleltothey-axisandtheotherlineisparalleltothex-axis,thelinesarealsoperpendiculartoeachother.

Thisistruebecausetheaxesareperpendiculartoeachother.

True

Lesson2:PatternsinCoordinates 11

Lesson2:HomeworkForeachsetofcoordinatepairs,

● Plotpointsusingthecoordinatepairsprovided.● Connectthepointsinthesameorderlisted.● Whatshapeisit?Explainhowyouknow.

A. (3,5),(3,7),(5,7),(5,5)

rectangle (square)

B. (−2, 2), (−2,6), (−8,6), (−8,2)

rectangle

C. (−5,-3), (−4, −3), (−4, −7), (−5, −7)

rectangle

D. (0,2), (2,0), (0, −2), (−2,0)

rectangle (square)

Teacher note: Provide grid paper with axes. Students may explain that they know a shape is a rectangle by visual inspection only. Press them to think about definitions of rectangles and how they can use grids to figure out angles. d would be a challenge to explain how there is a 90° angle at each vertex.

Answer:

12 Lesson2:PatternsinCoordinates

Lesson3:Lengthsofsegments 13

Lesson 3: Lengths of segments

Background Essential Question Howdoyoufindthelengthofalinesegmentinacoordinatesystem?

Learning Objectives Studentswill

● generateconjecturesaboutastrategytodeterminelengthofasegmentusingcoordinatepairs.

Common Core Alignment 6.G.3.Drawpolygonsinthecoordinateplanegivencoordinatesforthevertices;usecoordinatestofindthelengthofasidejoiningpointswiththesamefirstcoordinateorthesamesecondcoordinate.Applythesetechniquesinthecontextofsolvingreal-worldandmathematicalproblems.6.NS.6.c.Findandpositionintegersandotherrationalnumbersonahorizontalorverticalnumberlinediagram;findandpositionpairsofintegersandotherrationalnumbersonacoordinateplane.6.NS.8.Solvereal-worldandmathematicalproblemsbygraphingpointsinallfourquadrantsofthecoordinateplane.Includeuseofcoordinatesandabsolutevaluetofinddistancesbetweenpointswiththesamefirstcoordinateorthesamesecondcoordinate.

WarmUp 5minPatternGame

PlayPatternGamePurposes:Makemistakes,payattention,sayyes,keeptheballintheair,maketheensemblelookgood,bepresent.Normsdeveloped:Payattention,makeboldconjectures,lookforpatterns,buildoffeachother’sideas,participate.Instructions:

• Playersstandaroundtheperimeteroftheroom,sothateveryonecanseeeveryoneelse.Inthefirstround,tellstudentsthey'llmakethepatternAABB.Thefirststudent(A)candoorsayanything(e.g.,reachhandsuphigh).Thesecondstudentwouldhavetorepeatthesamemotion.Thethirdstudentwouldmakeanewpattern(B,e.g.,mooslikeacow).Thefourthstudentwouldfollow.Thefollowingstudentsfollowthepattern.

• Forthesecondround,suggestanABCpattern.• Remembertouse"circusbow"to.

Questionsfordiscussionaboutnorms:

• Whatwasitliketoplaythatgame?• WhatdidyoulearnabouttheClassroomRightsandRequirements?

14 Lesson3:Lengthsofsegments

MainActivity 30minActivity 3.1

Gen

erat

ing

case

s

1.Youaregoingtoplayagame.Use1,8,and5tomakethecoordinatesoftwoendpointsofalinesegment.Thelinesegmentmustbeparalleltoeitherthex-ory-axis.Thepersonwhomakesthelongestsegmentbetweentheirtwopointswins.DragpointsAandBtomakesegmentswiththecoordinatepairsyoucameupwith.Checktheirlengthsandseewhichcoordinatepairsproducethelongestsegment.

groupsof4Askquestions:• Howmanynumbersdoyouneedtocreate

coordinatepairsinordertomakealinesegmentthatisparalleltotheaxes?Why?(Thisshouldbeareviewfromthepreviouslesson)

• Whatpatternsdoyousee?

Ifneeded,havestudentsrecordtheircoordinatesandtheirlengthsintheirnotebook.

2.InRound2,eachgroupchoosesanythreepositivewholenumbers.Thistime,playerscanchangethesignofoneofthenumbers.Playagaintomakethelongestlinesegment.Dragpointsineachsegmenttomakesegmentswiththecoordinatepairsyoucameupwith.Checktheirlengthsandseewhichcoordinatepairsproducethelongestsegment.

groupsof4Ifneeded,havestudentsrecordtheircoordinatesandlengthsintheirnotebooks.

Con

ject

urin

g 3.Makeconjectures:A. Makeaconjectureabouthowyou

canfindthelengthofalinesegmentgiven2endpointswiththeircoordinates.

B. Makeanotherconjectureabouthowyoucanalwayswinthegame.

groupsof4Ask:

A. Howdidyouusethegrid?Howcanyouexpressthisintermsofoperations?(abitbeyondgrade-levelstandards)

B. Howcanyouuseabsolutevaluetohelpyouexpressyourconjecture?

Lesson3:Lengthsofsegments 15

Just

ifyin

g 4.Justifyyourconjectures.

wholeclassorsamegroupsAskquestions:• Howdidyoufindthelengthofasegmentusing

thecoordinatepairsofthesegment?• Howdidyouselectnumberstomakethelongest

segment?• Whichnumberdidyoudecidetochangesign?

Why?Howdidthathelplengthenyourline?• Howdidyoufindthelengthofthesegment

whenyouhadbothpositiveandnegativenumbersascoordinatesoftheendpoints?

• Whatisthelengthofthesegmentwhenyouhavepositivenumbersonlyascoordinatesoftheendpoints?

Con

clud

ing 5.Writedownyourconclusion. wholeclass

• Markconjecturesastrueorfalseasyoujustifythem.

• Summarizetheargument,makingsurethereisconsensusaboutthetruthorfalsityoftheconjectures.

Arguments, Part A

Conjecture Justification Conclusion

Ifalinesegmentisparalleltothex-axisandboththex-coordinatesarepositive,yougetthelengthbysubtractingthesmallerx-coordinatefromthelargerx-coordinate.

Ifalinesegmentisparalleltothex-axis,theiry-coordinatesarethesame.Ifbothx-coordinatesarepositive,thesegmentisinQuadrant1.Thelengthisthedifferencebetweenhowfarthelargerx-coordinateandthesmallerx-coordinatearefromthey-axis.

True

Ifalinesegmentisparalleltothey-axisandboththey-coordinatesarepositive,yougetthelengthbysubtractingthesmallery-coordinatefromthelargery-coordinate.

Ifalinesegmentisparalleltothey-axis,theirx-coordinatesarethesame.Ifbothy-coordinatesarepositive,thesegmentisinQuadrant1.Thelengthisthedifferencebetweenhowfarthelargery-coordinateandthesmallery-coordinatearefromthex-axis.TeacherNote:Whenthereisanegativecoordinate,onehastotakeintoaccountabsolutevalue–inotherwords,thenegativecoordinate’sdistancefromzero.

True

16 Lesson3:Lengthsofsegments

Ifalinesegmentisparalleltothex-axisandthex-coordinateshaveoppositesigns,yougetthelengthbyaddingthepositivex-coordinateandtheabsolutevalueofthenegativex-coordinate.

Ifx-coordinateshaveoppositesigns,thentheendpointsappearontheoppositesidesofthey-axis,sothelengthofthesegmentisthesumofhowfaronepointisfromthey-axisandhowfartheotherpointisfromthey-axis.Or,5+|–3|=8

True

Ifalinesegmentisparalleltothey-axisandthey-coordinateshaveoppositesigns,yougetthelengthbyaddingthepositivey-coordinateandtheabsolutevalueofthenegativey-coordinate.

Ify-coordinateshaveoppositesigns,thentheendpointsappearattheoppositesidesofthex-axis,sothelengthofthesegmentisthesumofhowfaronepointisabovethex-axisandhowfartheotherpointisbelowthex-axis.

True

Foralinesegmentparalleltothex-axis,ifbothx-coordinatesarenegative,yougetthelengthbyaddingx-coordinatesandchangingthesign.

Ifalinesegmentisparalleltothex-axis,thentheiry-coordinatesarethesame.Ifbothx-coordinatesarenegative,thesegmentisinQuadrant2or3.Thelengthisthedifferencebetweenhowfarleftx-coordinatesarefromy-axis,orinotherwords,thedistancebetweentwocoordinates.

True

Yougetthelengthofasegmentbyaddingtwonumbers.

Counterexample.(1,3),(4,3).Ifyouadd1and4,youget5,butthelengthis3units.However,ifyouhave(–1,3)and(4,3)andcounthowmanyunitsyoumovefrom4to–1,thenyouget5units,thelengthofthesegment.Thevalidityalsodependsonwhichtwonumbersoneadds–youcan’taddx-coordinatestoy-coordinates,butonlycorrespondingx-ory-values.Teachernote:Ambiguityofthestatementallowsotherpossiblecombinationsoftwocoordinates,resultinginrejectionofthestatementforitsvalidity.

Notalwaystrue

Yougetthelengthbysubtractingonenumberfromtheother.

Counterexample.(1,1),(1,–5).Ifyoudo1–(–5),thenyougetthelength.Ontheotherhand,ifyoudo–5–1,thenyouget–6,whichcan’tbethelengthbecauselengthcan’tbenegative.Teachernote:Ambiguityofthestatementallowsotherpossiblecombinationsoftwocoordinates,resultinginrejectionofthestatementforitsvalidity.TeacherNote:Studentsdonotneedtodocalculationswithintegers,butmakesuretoaddresswhytheywanttosubtractonenumberfromanotherandwhatthatmeansintermsofgettingadistancebetweentwopointsorlengthofasegment.

Notalwaystrue

Lesson3:Lengthsofsegments 17

Yougetthelengthbycountinggridsalongthesegment.

Countingthegridscansometimesworkifonegridrepresents1unit,butitisnottrueingeneral.

Notalwaystrue

Argument, Part B

Conjecture Justification Conclusion

Pickthetwolargestofthenumbersandchangethesignofoneofthenumbers.Thenusethetwonumbersasthex-ory-coordinatesforbothendpoints.

Thelengthofasegmentisdeterminedbyhowfaraparttwopointsare.Thex-andy-coordinatesrepresenthowfarapointisfromzeroonthex-ory-axis.Soifyouhaveonepositiveandonenegativenumber,thepointsarefarthestapart.SoIpickthetwolargestnumbers(whenbotharepositive)andchangeoneofthemtobenegative.Forexample,given2,3,5:

● Ipicked3and5andchanged5to﹣5,thenImade(2,﹣5)and(2,3)tomakeaverticalsegment.Itslengthis8units.

True

Usethesmallestnumberasthex-coordinateandthetwolargestnumbersasthey-coordinateswithoutchangingasign.

Counterexample.Given1,2,3:● Usingtherule,youcreate(1,2)and(1,3),whichhas

lengthof1.● Ontheotherhand,ifyouchange2to−2,youcan

createasegmentoflength5(e.g.(1,−2)and(1,3)).

False

Orderthenumberfromthesmallestandthelargestandpickthesmallestandthelargestwithoutchangingasign.

Counterexample.Given2,3,5:● Ifyoupick2and5withoutchangingtheirsign,you

cancreateasegmentoflength3(e.g.(23)and(5,3)● Ontheotherhand,ifyouchange5to−5,youcan

createasegmentoflength8with(2,3)and(2,−5),whichislongerthanthesegmentoflength3.

False

18 Lesson3:Lengthsofsegments

Lesson3:Homework1. Eachsetofcoordinatepairsrepresenttheverticesofasquareorarectangle.Foreachset,

determineiftheverticeswillformasquareorrectangleWITHOUTdrawing.Explainhowyoucameupwiththeanswer.Checkifyouarerightbydrawing.

A. (5,2),(5,5),(9,2),(9,5)

B. (–1,3),(1,3),(–1,5),(1,5)

C. (–4,–7),(–1,–7),(–4,–4),(–1,–4)

D. (2,2),(2,-3),(3,2),(3,-3)

Possible Answer: A, C, and D make rectangles and B makes a square. I noticed how far apart the x-coordinates were from each other as the y-coordinates were from each other. If the distances are the same, the vertices make a square, because in squares, all sides have the same length. In B, the x-coordinates (–1 and 1) are 2 units apart and the y-coordinates (3 and 5) are also 2 units apart, therefore, the coordinate pairs will form the vertices of a square with side lengths of 2 units.

Glossary 19

Glossary Axes:apairofperpendicularnumberlines.Origin:thepointwheretwoaxesintersect.Coordinates:theorderedpairsofnumbers.Thefirstnumbertellshowfartogointhedirectionofx-axisandthesecondnumbertellshowfartogointhedirectionofy-axisfromtheorigin.e.g.(2,5)representsx-coordinateas2andy-coordinateas5.Quadrants:4sectionsoftheplanedividedbythex-andy-axes.TheyarenumberedasI,II,III,IV,counterclockwisestartingfromtoprightsectionasshownbelow.

Absolutevalue:thedistancefromzeroonthenumberlineorthemagnitudeofapositiveornegativequantity.e.g.|﹣2|=2.