Survey
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
Golden ratio wikipedia , lookup
Multilateration wikipedia , lookup
Euler angles wikipedia , lookup
Apollonian network wikipedia , lookup
Rational trigonometry wikipedia , lookup
Trigonometric functions wikipedia , lookup
Reuleaux triangle wikipedia , lookup
History of trigonometry wikipedia , lookup
Euclidean geometry wikipedia , lookup
Incircle and excircles of a triangle wikipedia , lookup
Concept: Givenatriangle,findtheappropriateanglemeasure,sidelength,orpointaccording totheproblemposed. Objectives: Giventhemeasuresoftwointerioranglesofatriangle,findthemeasureofthethird angleinthetriangle. Givenanytriangle,statewhetheritisanisoscelestriangleornotandgivetwo differentmeasurementstoproveso(fourtotal). Givenarighttrianglewiththelengthsoftwosidesknown,findthelengthofthe thirdside. Givenanytriangle,findthecentroidofit. Standards: PS.5:Useappropriatetoolsstrategically. Mathematicallyproficientstudentsconsidertheavailabletoolswhensolvinga mathematicalproblem.Thesetoolsmightincludepencilandpaper,models,aruler,a protractor,acalculator,aspreadsheet,acomputeralgebrasystem,astatistical package,ordynamicgeometrysoftware.Mathematicallyproficientstudentsare sufficientlyfamiliarwithtoolsappropriatefortheirgradeorcoursetomakesound decisionsaboutwheneachofthesetoolsmightbehelpful,recognizingboththeinsight tobegainedandtheirlimitations.Mathematicallyproficientstudentsidentifyrelevant externalmathematicalresources,suchasdigitalcontent,andusethemtoposeorsolve problems.Theyusetechnologicaltoolstoexploreanddeepentheirunderstandingof conceptsandtosupportthedevelopmentoflearningmathematics.Theyuse technologytocontributetoconceptdevelopment,simulation,representation, reasoning,communicationandproblemsolving. Withthislesson,studentswillbeusingGeometer’sSketchpad,andusingthe toolswithinthegeometrysoftwaretoexploreandlearn.Theywillhaveto manipulatecertainaspectswithintrianglesinordertolearnaboutthemand makeassumptionsthattheywilllaterseeprovedtobetrue. PS.7:Lookforandmakeuseofstructure. Mathematicallyproficientstudentslookcloselytodiscernapatternorstructure.They stepbackforanoverviewandshiftperspective.Theyrecognizeandusepropertiesof operationsandequality.Theyorganizeandclassifygeometricshapesbasedontheir attributes.Theyseeexpressions,equations,andgeometricfiguresassingleobjectsor asbeingcomposedofseveralobjects. Studentswillfirsthavetoexploreontheirownthegiventrianglesforthis assignment,drawingconclusionsfromwhattheyareseeingandwhatthey previouslyhavelearned.Thiswillmakethemfindcertainpatternsor structuresoftrianglesthatareactuallytheorems. G.T.1:Proveandapplytheoremsabouttriangles,includingthefollowing:measuresof interioranglesofatrianglesumto180°;baseanglesofisoscelestrianglesare congruent;thesegmentjoiningmidpointsoftwosidesofatriangleisparalleltothe thirdsideandhalfthelength;themediansofatrianglemeetatapoint;alineparallel toonesideofatriangledividestheothertwoproportionally,andconversely;the PythagoreanTheorem,usingtrianglesimilarity;andtheisoscelestriangletheorem anditsconverse. Materials: Forthislesson,studentswillneedGeometer’sSketchpad,whichcanbedownloaded for$10onanycomputerforayearlicense. Therearesixfilesattachedforthislesson,fourareaworddocumentsandtheother twoareGSPdocuments.Downloadandcompletethesefiles. Therearetwoadditionalfilesthatarethestudentworkcompletedfilesspecifically forthislessonplan—thestudentswillnotgetthese.Oneisaworddocument,the otherisaGSPfile.Thesetwofilesare“Worksheets—StudentWork.docx”and “Worksheets—StudentWork.gsp.” Engagement: Studentswillbegivenanactivityprojectedontheboard.Westartthisactivitynow, andcomebacktoitlaterintheElaborationportionofthelesson.Studentswillwork ingroupsof2or3students(dependingonhowmanyareintheclass).Iwillallow thestudentstoplayaroundwiththisactivityforatleasthalfoftheperiod (dependingonhowlongclasseslast),andtheneaseintothelesson.Theactivitywill read: Constructatriangle(aslargeasyouwant,thelargerthebetter)onapieceof cardboardprovided.Cutthisout.Seeifyoucanmeasurethetriangleonaflatendederaseronapencil. Exploration: EachstudentwillreceiveaGSPfilethathassixworksheetswithinit;thisfileistitled “Worksheets.gsp.”Someofthesheetswillhavefiguresalreadyproduced,where somewillbecompletelyemptyinwhichtheywillhavetoproducethefigure.There isacorrespondingworksheetthatgoeswiththeGSPfileinwhichthestudentsare toldwhattodooneachsheet,andithasquestionsthatgoalongwithit.This documentistitled“Worksheets.docx.” Whilestudentsareworkingontheirtasks,Iwillbewalkingaroundaskingquestions whilethestudentswork.Thesequestionscouldbe: 1. Beforeyoumovethisvertex,whatdothinkwillhappentoyour measurements? 2. Whathappensifyoutrytomovebothoftheseverticesatthesametime? 3. Trytomovethisvertex(ofarighttriangle).Nowmovethisone.Whydoyou thinktheydodifferentthingstothetriangle? 4. Whydoyouthinkthisisimportant?Doyouthinkit’llleadintosomething elselater? Withthistool,itwillbeeasytodifferentiateintheclassroom.Withtheuseofthe tool,weakerstudentscanconstructthefigureseasilyandusethemeasurementtool tomakeconjectures.Iftheycannoteasilyseeconjecturestomake,Icanfurtherhelp themwithhintingatthemusethecalculationtoolsotheycanstillmakethese connectionsbythemselves,justwithalittlemoreguidance.Forthestronger students,theycanmaketheconjecturesandthenusethetechnologytotesttheir conjecture.Evenwiththis,theycanmoveonandcreatetheirownshapesandtest thesepropertiesoftriangleswithevenmoreshapesandseewhathappens. Explanation: Studentscanusethistimetoworkwithapartnerandcomparewhattheygotfor eachofthequestionsontheworksheet.Aftertheyhavecometoasingleanswerfor eachofthem,wewillcometogetherasaclasstoshareourideas.Duringthistime,I willbepointingoutkeyideasthatstudentsreached,ordrawingoutthoseideasI wantedthemtofindbuttheydidn’tquietgetthere. Specifically,Iwantthemtofindthefollowing:thesumofthemeasuresofthe interioranglesofatriangleis180°,anisoscelestrianglehastwocongruentsides andtheanglesoppositethosesidesarecongruent,arighttrianglehasoneright angle,thePythagoreanTheorem,andthemediansofatriangleintersectatapoint insidethetrianglecalledthecentroid.Tohelpguidethem,Icanaskthemquestions like: 1. Whenyoumovedavertexofatriangle,whathappenedtothemeasurements thatyouhadfound? 2. Whentwomeasurementsarethesameinatriangle,whatelsedidyou notice? 3. Ifoneangleofatriangleisright,howaretheshortersidesrelatedtothe longerside?Wasthiseasytoseebeforethehintofusingthesquaresofeach ofthesides? 4. Nomatterwhereyoumovedthevertices,whatremainedthesamewithall threeofthemedians? Oncewehavediscussedthesethings,Iwillformallydefinethefollowingterms: isoscelestriangle,righttriangle,mediansofatriangle,andcentroid.Asaclass,we willthengooverthefollowingproofs:TriangleAngleSumTheorem,Isosceles TriangleTheorem,andthePythagoreanTheorem.Thestudentswillnothavetodo theseontheirownrightnow,butIwillaskthemtosharewhattheythinkgoesinto theproof.Imayaskthemtoreproducesomeofthemonatestorfinal.Theproofs areattachedinaworddocumenttitled“ExplanationProofs.docx.” Elaboration: StudentswillbegivenanotherGSPfileandcorrespondingworksheetwith quadrilaterals,thesefilesare“ElaborationWorksheet.docx”and“Elaboration.gsp.” Theywillbeaskedtousewhattheyknowaboutinterioranglesinatriangletomake conjecturesabouttheanglesintheseotherpolygons. Afterwehavegoneovermorepolygons,Iwillallowthemtimetogetouttheir cardboardtrianglesagain.Thistime,Iwillaskthemtomarkapointonthetriangle wheretheythinktheycanbalanceit.Iwillrefertothisasthecenterofgravity. Withoutlettinggroupstesttheirpointfirst,Iwillhaveeachgroupcometothefront oftheclassandexplainwhattheydid,thentheywilltestitout.Forthosegroups thatsuccessfullyfoundthecenterofgravity,wewillgobacktotheirreasoningfor choosingthatpoint.Thiswillleadustomaketheconnectionthatacentroidofa triangleisitscenterofgravity. Evaluation: Therewillbea13-questionquizattheendoftheunittoseeifstudentshavefully masteredthematerial.Thequizistheworddocument“EvaluationQuiz.docx.” References: Gray,Dawson(2008).UsingTheGeometer'sSketchpadintheMathClassroomto ImproveEngagement,TransformtheLearningEnvironment,andEnhance Understanding.http://discoverarchive.vanderbilt.edu/handle/1803/571. Serra,Michael(1989).DiscoveringGeometry:AnInductiveApproach. Serra,Michael(2013).DiscoveringGeometry:AnInvestigationApproach,Fourth Edition(Teacher’sEdition). Name:___________________________________________ OpentheGSPfilethatIsentyou. Activity1:InteriorAnglesofaTriangle Createatriangleonpage1,“InteriorAngles,”ofthesketch. Inthefile,findthemeasuresofeachoftheinteriorangles. Movetheverticesaround.Whathappenstothemeasuresoftheinteriorangles? Whatisalwaystrueaboutthemeasuresofthethreeinterioranglesnomatterthe measureoftheindividualangles?(Youmayneedtousethecalculatetoollocated underthenumberstabforthisone.) Activity2:IsoscelesTriangles Onpage2,“IsoscelesTriangles,”ofthesketchyouare giventhetriangleinfigure1.Thisiscalledanisosceles triangle. InGSP,providetwodifferenttypesofmeasurements thatareuniquetoisoscelestriangles.Whatarethose characteristics? Figure1 Movetheverticesaround.Whathappenstoyour measurements? Continueontopage3oftheGSPfile.Providethetwodifferentmeasurementsfor eachofthetrianglesshown.Movetheverticesaroundandstatewhethereachoneis alwaysanisoscelestriangle.GiveyouranswersintheGSPfile. Activity3:RightTriangle Page4,“RightTriangle,”ofthesketchcontainsthetriangle inFigure2.Thistriangleiscalledarighttriangle. IntheGSPfile,providemeasurementsthatmakethis triangleuniquefromothertriangles.Whatisthat characteristic? Now,moveyourverticesaround.Whathappenstoyour Figure2 measurements? Onpage5,“PythagoreanTheorem,”anotherrighttriangleisgiven,thistimewith certainmeasurements.Whatdoyounoticeaboutthesemeasurements?(Again,you mayneedtousethe“calculate”toolunderthenumberstabacoupletimesforthis, inparticularly,withsquaresand/orsums.) Activity4:MediansofaTriangle Onpage6ofthefile,“Medians,”constructatriangle. Findthemidpointoneachofthesidesofthetriangle. Connecteachofthemidpointstothevertexoppositethatsidewithalinesegment. Thesearethemediansofthetriangle. Whatdoyounoticeaboutthemedians? Changethelengthsofthesidesofthetriangle.Whatstaysthesamewiththe medians? PossibleStudentWork Activity1:InteriorAnglesofaTriangle OpentheGSPfilethatIsentyouandcreateatriangleonpage1,“InteriorAngles.” Inthefile,findthemeasuresofeachoftheinteriorangles. Movetheverticesaround.Whathappenstothemeasuresoftheinteriorangles? Astheverticesmove,themeasuresoftheinteriorangleschange,theyeither increaseordecreasebutnotall3ofthemdothesamething. Whatisalwaystrueaboutthemeasuresofthethreeinterioranglesnomatterthe measureoftheindividualangles?(Youmayneedtousethecalculatetoollocated underthenumberstabforthisone.) Nomatterthemeasuresoftheindividualinteriorangles,thesumofthethree anglesalwaysaddsupto180°. Activity2:IsoscelesTriangles Onpage2,“IsoscelesTriangles,”ofthesketchyouare giventhetriangleinfigure1.Thisiscalledanisosceles triangle. InGSP,providetwodifferenttypesofmeasurements thatareuniquetoisoscelestriangles.Whatarethose characteristics? Twosidesofthetrianglearecongruent,andtwo anglesofthetrianglearecongruent. Figure1 Movetheverticesaround.Whathappenstoyour measurements? Asthetrianglechangesinsize,thelengthsofthesidesandthemeasuresofthe angleschange,butthetwosidesandthetwoanglesstillremaincongruent. Continueontopage3oftheGSPfile.Providethetwodifferentmeasurementsfor eachofthetrianglesshown.Movetheverticesaroundandstatewhethereachoneis alwaysanisoscelestriangle.GiveyouranswersintheGSPfile. Activity3:RightTriangle Page4,“RightTriangle,”ofthesketchcontainsthetriangle inFigure2.Thistriangleiscalledarighttriangle. IntheGSPfile,providemeasurementsthatmakethis triangleuniquefromothertriangles.Whatisthat characteristic? Oneoftheinterioranglesisarightangle,thatis,it measures90°. Now,moveyourverticesaround.Whathappenstoyour Figure2 measurements? Themeasurementsofthetwointerioranglesthatarenotthe rightanglechange.However,thesetwoanglesstilladdupto90°. Onpage5,“PythagoreanTheorem,”anotherrighttriangleisgiven,thistimewith certainmeasurements.Whatdoyounoticeaboutthesemeasurements?(Again,you mayneedtousethe“calculate”toolunderthenumberstabacoupletimesforthis, inparticularly,withsquaresand/orsums.) Thesumofthesquaresofthetwoofthesidesisequaltothesquareofthethirdside (thehypotenuse). Activity4:MediansofaTriangle Onpage6ofthefile,“Medians,”constructatriangle. Findthemidpointoneachofthesidesofthetriangle. Connecteachofthemidpointstothevertexoppositethatsidewithalinesegment. Thesearethemediansofthetriangle. Whatdoyounoticeaboutthemedians? Theyallintersectatonepointinsideofthetriangle. Changethelengthsofthesideofthetriangle.Whatstaysthesamewiththe medians? Theycontinuetointersectatapointinsideofthetriangle. TriangleAngleSumTheorem Theorem:Thesumofthemeasuresoftheinterior anglesofatriangleis180°. Given:∆𝐴𝐵𝐶 Prove:𝑚∠𝐴𝐵𝐶 + 𝑚∠𝐵𝐶𝐴 + 𝑚∠𝐶𝐴𝐵 = 180° Construct𝐶𝐷throughpoint𝐶andparallelto Constructionofalinethrougha point 𝐴𝐵 𝑚∠𝐵𝐶𝐴 + 𝑚∠𝐵𝐶𝐷 = 𝑚∠𝐴𝐶𝐷 Angleadditionpostulate 𝑚∠𝐴𝐶𝐷 + 𝑚∠𝐴𝐶𝐸 = 180° Linearpairpostulate 𝑚∠𝐵𝐶𝐴 + 𝑚∠𝐵𝐶𝐷 + 𝑚∠𝐴𝐶𝐸 = 180° Substitution 𝑚∠𝐶𝐴𝐵 ≅ 𝑚∠𝐴𝐶𝐸 Alternateinteriorangles 𝑚∠𝐴𝐵𝐶 ≅ 𝑚∠𝐵𝐶𝐷 theorem 𝑚∠𝐶𝐴𝐵 = 𝑚∠𝐴𝐶𝐸 Definitionofcongruence 𝑚∠𝐴𝐵𝐶 = 𝑚∠𝐵𝐶𝐷 𝑚∠𝐴𝐵𝐶 + 𝑚∠𝐵𝐶𝐴 + 𝑚∠𝐶𝐴𝐵 = 180° Substitution IsoscelesTriangleTheorem Theorem:Iftwosidesofatrianglearecongruent, thentheanglesoppositethosesidesarecongruent. Given:𝐴𝐶 ≅ 𝐵𝐶 Prove:𝑚∠𝐶𝐴𝐵 ≅ 𝑚∠𝐴𝐵𝐶 Proof1: Given 𝐴𝐶 ≅ 𝐵𝐶 Everyanglehasanangle Construct𝐶𝐷suchthatitbisects∠𝐵𝐶𝐴 bisector 𝑚∠𝐴𝐶𝐷 ≅ 𝑚∠𝐵𝐶𝐷 Definitionofanglebisector Reflexiveproperty 𝐶𝐷 ≅ 𝐶𝐷 ∆𝐴𝐶𝐷 ≅ ∆𝐵𝐶𝐷 SAS 𝑚∠𝐶𝐴𝐵 ≅ 𝑚∠𝐴𝐵𝐶 CPCTC Proof2: Given 𝐴𝐶 ≅ 𝐵𝐶 Constructpoint𝐷suchthat𝐷isthemidpoint Everylinesegmenthasa midpoint of𝐴𝐵 Definitionofmidpoint 𝐴𝐷 ≅ 𝐵𝐷 ReflexiveProperty 𝐶𝐷 ≅ 𝐶𝐷 ∆𝐴𝐶𝐷 ≅ ∆𝐵𝐶𝐷 SSS 𝑚∠𝐶𝐴𝐵 ≅ 𝑚∠𝐴𝐵𝐶 CPCTC PythagoreanTheorem Theorem:Thesquareofthehypotenuseofaright triangleisequaltothesumofthesquaresofthetwo legs. Given:∆𝐴𝐵𝐶isright Prove:𝑎! + 𝑏 ! = 𝑐 ! TherearesomanyproofsforthePythagoreanTheorem.Wewilllookatonlyone. However,itwillnotbeinthesameformattingoftheotherones,butitisstillaproof! Usingthefiguretotherightletsfirststartby findingtheareaofthewhole,largersquare: 𝐴= 𝑎+𝑏 𝑎+𝑏 Whichexpandsinto: 𝐴 = 𝑎! + 2𝑎𝑏 + 𝑏 ! Now,letsfindtheareaofthefourrighttriangles andthesmallersquare: ! 𝐴∆! = 4 ! 𝑎𝑏 and𝐴∎ = 𝑐𝑐 Thesesimplifyto: 𝐴∆! = 2𝑎𝑏and𝐴∎ = 𝑐 ! Thus,ifwearelookingattheareaofthewhole,largersquarebyaddingthefour righttrianglesandsmallersquaretogether,weget: 𝐴 = 2𝑎𝑏 + 𝑐 ! Sincewefoundtheareaofthewhole,largersquaretwodifferentways,thoseareas mustbeequaltoeachother.Thus, 𝑎! + 2𝑎𝑏 + 𝑏 ! = 2𝑎𝑏 + 𝑐 ! Whichsimplifiesto: 𝑎! + 𝑏! = 𝑐 ! Name:___________________________________________ OpenthesecondGSPfile,“Elaboration.” Activity1:EquilateralTriangle Onpageoneofthesketch,withoutusingthemeasure tool,findthemeasuresofeachoftheangles. 𝑚∠𝐶𝐴𝐵 = 𝑚∠𝐴𝐵𝐶 = 𝑚∠𝐵𝐶𝐴 = Howdidyoucomeupwiththesemeasurements? Now,usingthemeasuretool,checkyouranswers.Whereyoucorrect? Thisnewtypeoftriangleiscalledanequilateraltriangle.Whatelsedoyounotice aboutequilateraltriangles?Providethreemoremeasurementsbelow. MovevertexAofthetriangle.Whathappenstoallofyourmeasurements? Activity2:Parallelogram OnpagetwooftheGSPfile,thereisaparallelogram. Createtwotriangleswithintheparallelogramby connectingtwoofthevertices.Thissegmentisknown asa“diagonal.” Didyouformspecialtypesoftriangles? Usewhatyouknowabouttrianglestomakeaconjectureaboutthesumofthe interioranglesoftheparallelogram. Usethemeasuretooltoseeifyouwerecorrect.Wereyou? Activity3:Rectangle OnpagethreeoftheGSPfile,thereisarectangle. Createtwotriangleswithintherectanglebyconnectingtwoof thevertices.Thissegmentisknownasa“diagonal,”justlikein thepreviousactivity. Didyouformspecialtypesoftriangles? Usewhatyouknowabouttrianglestomakeaconjectureaboutthesumofthe interioranglesoftherectangle. Usethemeasuretooltoseeifyouwerecorrect.Wereyou? Activity4:Square OnpagefouroftheGSPfile,thereisasquare. Createtwotriangleswithinthesquarebyconnectingtwoof thevertices.Thissegmentisknownasa“diagonal,”justlike intheprevioustwoactivities. Didyouformspecialtypesoftriangles? Usewhatyouknowabouttrianglestomakeaconjectureaboutthesumofthe interioranglesofthesquare. Usethemeasuretooltoseeifyouwerecorrect.Wereyou? Activity5:Trapezoid OnpagefiveoftheGSPfile,thereisatrapezoid. Createtwotriangleswithinthetrapezoidby connectingtwoofthevertices.Thissegmentis knownasa“diagonal,”justlikeintheprevious threeactivities. Didyouformspecialtypesoftriangles? Usewhatyouknowabouttrianglestomakeaconjectureaboutthesumofthe interioranglesofthetrapezoid. Usethemeasuretooltoseeifyouwerecorrect.Wereyou? Activity6 Usingyourconjecturesinactivities2-5,makeaconjectureaboutthesumofthe interioranglesinanyquadrilateral. Name:___________________________________________ Quiz—Triangles Forproblems1-8,findthemissingmeasurements. 1. x=___________ 5. x=___________ 2. x=___________ 6. c=___________ 3. x=___________ 7. c=___________ y=___________ 4. x=___________ 8. a=___________ y=___________ b=___________ 9. Explainthereasoningyouusedforeachofthefiguresinproblems1-8. 10. Thinkbacktowhenwecutatriangleoutofcardboard.Wereyouableto balanceitontheeraserofyourpencil?Wheredoyouneedtoputyoureraser sothatyoucanbalanceit? 11. D,E,andFarethemidpointsofthecorrespondingsidesof△ABC.Findthe centroidof△ABC. 12. Gisthecentroidof△ABC.Findthemidpointsofeachofthesidesof△ABC. 13. Findthemeasurementsofunknownangles. a=___________ c=___________ b=___________ d=___________