Geometry
Set6:UnderstandingthePythagoreanTheorem
Instruction
Goal:Toprovideopportunities forstudentstodevelopconceptsandskillsrelatedto understandingthatthePythagoreantheoremisastatementaboutareasofsquares on thesidesofarighttriangle
CommonCoreStandards
Geometry
UnderstandandapplythePythagoreantheorem.
8.G.6.ExplainaproofofthePythagoreantheoremanditsconverse.
8.G.7.ApplythePythagoreantheoremtodetermineunknownsidelengthsinrighttriangles inreal-worldandmathematicalproblemsintwoandthreedimensions.
8.G.8.ApplythePythagoreantheoremtofindthedistancebetweentwopointsina coordinatesystem.
StudentActivitiesOverviewandAnswerKey
Station1
Studentsdrawatriangleanddrawtheareaofeachsidesquared.Thentheytrytofittheareaof thelegssquaredintotheareaofthehypotenusesquared.Theydiscusshowthisillustratesthe Pythagoreantheorem.
Answers: Yes;theareaofthetwolegssquaredisequaltotheareaofthehypotenusesquared
Station2
StudentsusethePythagoreantheoremtoansweraquestionabouttheareaofland.Thentheyusethe Pythagoreantheoremtofinddistanceacrossasquare.ThisallowsstudentstousethePythagorean theoremintwodifferentwaysforthesameproblem.
Answers:5200squarefeet;thePythagoreantheorem;about72feet
Station3
StudentsworkthoughabasicproofofthePythagoreantheorem.Theyuseareatocomeupwith theresult.
Answers:c2;(1⁄2)ab;2ab;c2+2ab
Geometry
Set6:UnderstandingthePythagoreanTheorem
Station4
Instruction
Studentsdeterminetheareaofthesquaresoftwolegsofatriangleandcomparethattothearea ofthesquareofthehypotenuse.Thentheyexplainhowthisdemonstrationisrelated tothe Pythagoreantheorem.
Answers:25;25;theyarethesameareatotal;itshowsthatthetwolegssquaredareequaltothe hypotenusesquared
MaterialsList/Setup
Station1pairofscissors,ruler,andpiecesofpaperforeachgroupmember
Station2 none Station3 none Station4 none
Geometry
Set6:UnderstandingthePythagoreanTheorem
DiscussionGuide
Instruction
Tosupportstudentsinreflectingontheactivitiesandtogathersomeformativeinformationabout studentlearning,usethefollowingpromptstofacilitateaclassdiscussionto“debrief”thestation activities.
Prompts/Questions
1. WhydoyouusesquarestoshowthePythagoreantheoreminaphysicalway?
2. Whatisanexampleofareal-lifesituationwhenyouwouldusethePythagoreantheorem?
3. If theareaofasquarecomingoffalegofarightangleis64sqinchesandthelengthofthe hypotenuseis10 sqinches,whatistheareaofthesquarecomingofftheotherleg?
4. Explainhowyoucould cutthesquaresthatcomeoffthesidesofarighttriangleintosmaller piecestofindPythagoreantriples.
Think,Pair,Share
Havestudentsjot downtheirownresponsestoquestions,thendiscusswithapartner(whowasnot intheirstationgroup),thendiscussasawholeclass.
SuggestedAppropriateResponses
1. InthePythagoreantheorem,thelengthofthesidesaresquaredwhichislikefindingthearea ofasquare.
2. Manypossibilities—findingthedistancebetweentwoplacesifyouknowthehorizontaland verticaldistance
3. 36sqinches
4. Example:3,4,5triangle—cuteachsquareinto4squares,thenyouhavea6,8,10triangle
PossibleMisunderstandings/Mistakes
•Takingthesquare rootoftheamountoflandtheoldestbrotherowns
•Havingtroublecuttingthesquaresofthelegstofitintothesquareofthehypotenuse
•Havingtroublecompletingtheproof,i.e.,notunderstandingallthesteps
Station1
Atthisstation,youwillfindapairofscissors,aruler,andapieceofpaperforeachgroupmember. Eachpersonshouldcompletetheactivityanddiscusshisorherfindingswiththegroup.
Drawarighttriangle.Nowdrawsquaresusingeachsideofthetriangleasonesideofthesquares. Thereshouldbethreesquares.Yourfigurewilllookliketheonebelow.
Cutoutthesquaresthatareconnectedtothetwolegsofthetriangle.
Seeifyoucancutthemupsotheyfitinsidethesquarethatisconnectedtothehypotenuse. Canyou?
ThePythagoreantheoremstatesthata2+b2=c2.Howdoesthisactivitydemonstrate thattheorem?
Station2
Discussandanswerthefollowingquestionsasagroup.
Therearethreebrotherswhoownlandaroundapark.Theparkisintheshapeofarighttriangle. Theparklookslikethetrianglebelow.Theonlylandthebrothersowniswhatisdirectlytouching thepark(inwhite).
N
WE
S
Theyoungestbrothergetsthelandsouthofthepark.Hislandisaperfectsquare,andhehas
1600squarefeetofland.Themiddlebrothergetsthelandwestofthepark.Hislandisalsoaperfect square.Hehas3600squarefeetofland.Theoldest brothergetsthelandthatisalongthelongest sectionofthepark.Hislandisalsoaperfectsquare.
Howmuchlanddoestheoldestbrotherhave?
Explainyourstrategyforsolvingthisproblem.
If theoldestbrotherwalkedthediagonalacrosshisproperty,howfarwouldhewalk?
Station3
Inthisactivity,youwilluseadiagramandtheareaofthediagramtoprovethePythagoreantheorem. Lookatthediagrambelow.Itismadeupoffourtrianglesputtogether.
Forthepurposeofthisproof,lookatthetrianglebelow.
b
a c
Nowlabel thesidesofthefourtrianglesintheoriginaldiagramwitha,b,andc.Worktogether to answerthesequestions.
Whatistheareaofthesmallsquare?
Whatistheareaofoneofthetriangles?
Whatistheareaofthefourtriangles?
Whatistheareaofthelargesquare?
Station4
Atthisstation,youwillexplorethePythagoreantheoremandseehowitrelatestothearea ofsquares.Asagroup,discussandanswerthequestionsbelow.
Inthefigureabove,numbertheboxesinsquaresthatareattachedtothetwolegsoftherighttriangle
(1,2,3,etc.).
Howmanytotalsquaresarethere?
Numbertheboxesinthesquarethatisattachedtothehypotenuse.
Howmanytotalsquaresarethere?
Whatdoyounotice?
HowdoesthisrepresentthePythagoreantheorem?