Download Concept: Given a triangle, find the appropriate angle measure, side

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=___________