1
SECONDARY MATH I // MODULE 8
CONNECTING ALGEBRA & GEOMETRY – 8.1
A Develop Understanding Task
TheperformancesofthePodunkHighSchooldrillteamare
verypopularduringhalf-timeattheschool’sfootballand
basketballgames.WhenthePodunkHighSchooldrillteamchoreographsthedancemovesthat
theywilldoonthefootballfield,theylayouttheirpositionsonagridliketheonebelow:
Inoneoftheirdances,theyplantomakepatternsholdinglong,wideribbonsthatwillspanfrom
onedancerinthemiddletosixotherdancers.Onthegrid,theirpatternlookslikethis:
Thequestionthedancershaveishowlongtomaketheribbons.Gabriela(G)isstandinginthe
centerandsomedancersthinkthattheribbonfromGabriela(G)toCourtney(C)willbeshorter
thantheonefromGabriela(G)toBrittney(B).
1. Howlongdoeseachribbonneedtobe?
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8.1 Go the Distance
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SECONDARY MATH I // MODULE 8
CONNECTING ALGEBRA & GEOMETRY – 8.1
2. Explainhowyoufoundthelengthofeachribbon.
Whentheyhavefinishedwiththeribbonsinthisposition,theyareconsideringusingthemtoform
anewpatternlikethis:
3. WilltheribbonstheyusedinthepreviouspatternbelongenoughtogobetweenBritney
(B)andCourtney(C)inthenewpattern?Explainyouranswer.
Gabrielanoticesthatthecalculationssheismakingforthelengthoftheribbonsremindsherof
mathclass.Shesaystothegroup,“Hey,Iwonderifthereisaprocessthatwecoulduselikewhat
wehavebeendoingtofindthedistancebetweenanytwopointsonthegrid.”Shedecidestothink
aboutitlikethis:
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SECONDARY MATH I // MODULE 8
CONNECTING ALGEBRA & GEOMETRY – 8.1
“I’mgoingtostartwithtwopointsanddrawthelinebetweenthemthatrepresentsthedistance
thatI’mlookingfor.Sincethesetwopointscouldbeanywhere,InamedthemA(x1,y1)andB(x2,y2).
Hmmmmm....whenIfiguredthelengthoftheribbons,whatdidIdonext?”
B
(x2,y2)
A
(x1,y1)
4. Thinkbackontheprocessyouusedtofindthelengthoftheribbonandwritedownyour
stepshere,intermsof(x1,y1)and(x2,y2).
5. Usetheprocessyoucameupwithin#4tofindthedistancebetweentwopointslocatedfar
enoughawayfromeachotherthatusingyourformulafrom#4ismoreefficientthan
graphingandcounting.Forexamplefindthedistancebetween(-11,25)and(23,-16)
6. Useyourprocesstofindtheperimeterofthehexagonpatternshownin#3.
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SECONDARY MATH I // MODULE 8
CONNECTING ALGEBRA & GEOMETRY – 8.1
8.1 Go the Distance – Teacher Notes
A Develop Understanding Task
NotetoTeachers:Calculatorsfacilitatetheworkforthistask.
Purpose:Thepurposeofthistaskistodevelopthedistanceformula,baseduponstudents’
understandingofthePythagoreantheorem.Inthetask,studentsareaskedtocalculatedistances
betweenpointsusingtriangles,andthentoformalizetheprocesstothedistanceformula.Atthe
endofthetask,studentswillusethedistanceformulatofindtheperimeterofahexagon.
CoreStandardsFocus:
G.GPE.4Usecoordinatestoprovesimplegeometrictheoremsalgebraically.
G.GPE.7Usecoordinatestocomputeperimetersofpolygonsandareasoftrianglesandrectangles,
e.g.,usingthedistanceformula.
StandardsforMathematicalPracticeofFocusintheModule:
SMP1–Makesenseofproblemsandpersevereinsolvingthem.
SMP7–Lookforandmakeuseofstructure.
TheTeachingCycle:
Launch(WholeClass):
Beginthetaskbyensuringthatstudentunderstandtheproblemsituation.Projectthedrawingin
#1andaskstudentswhichribbonlookslonger,GB orGC. Askhowtheycantesttheirclaims.
SomestudentsmaysuggestusingthePythagoreanTheoremtofindthelengthofGB.Askwhatthey
wouldneedtousethePythagoreanTheorem.Atthispoint,setstudentstoworkonthetask.
Explore(SmallGroup):
Duringtheexplorationperiod,watchforstudentsthatarestuckonthefirstpartoftheproblem.
YoumayaskthemtodrawthetrianglethatwillhelpthemtousethePythagoreanTheoremand
howtheymightfindthelengthofthelegsofthetrianglesotheycanfindthehypotenuse.Asyou
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SECONDARY MATH I // MODULE 8
CONNECTING ALGEBRA & GEOMETRY – 8.1
monitorstudentthinkingon#3,watchforstudentswhoarenoticinghowtofindthelengthofthe
legsofthetrianglewhenithasbeenmovedawayfromtheorigin.Lookforstudentsthathave
writtenagoodstep-by-stepprocedurefor#4.Itwillprobablybedifficultforthemtousethe
symbolsappropriately,sowatchforwordsthatappropriatedescribetheprocedure.
Discuss(WholeClass):
StartthediscussionbyhavingagroupshowhowtheyfoundthelengthofBCinproblem#3.Move
nextto#4andhaveagroupthathaswrittenastepbystepprocedure.Trywalkingthroughthe
group’sprocedurewiththenumbersfromproblem#3andseeifitgivestheappropriateanswer.
Ifnecessary,workwiththeclasstomodifytheproceduresothatthelistofstepsiscorrect.Once
thestepsareoutlinedinwords,gothroughthestepsusingpointsA(x1,y1)andB(x2,y2)and
formalizetheprocedureswiththesymbols.Anexample:
Stepsinwords
Stepsinsymbols
Findthelengthofthehorizontallegofthe
triangle
Findthelengthoftheverticallegofthetriangle
UsethePythagoreanTheoremtowritean
equation
Solveforc
Takethesquarerootofbothsidesofthe
equation
Simplify
x2-x1
(!!
y2-y1
!
− !! ) + ( !! − !! )! = ! ! (!! − !! )! + ( !! − !! )! = ! ! (!! − !! )! + ( !! − !! )! =
!!
(!! − !! )! + ( !! − !! )! = !(cbeingthe
desireddistance)
Usingalgebraicnotationtomodelacorrectprocessthatisgivenverballywillresultinderivingthe
distanceformula.Aftergoingthroughthisprocess,applytheformulausingthepointsin#5.
AlignedReady,Set,Go:ConnectingAlgebraandGeometry8.1
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SECONDARY MATH I // MODULE 8
8.1
CONNECTING ALGEBRA & GEOMETRY - 8.1
READY, SET, GO!
Name
PeriodDate
READY
Topic:Findingthedistancebetweentwopoints
Usethenumberlinetofindthedistancebetweenthegivenpoints.(ThenotationABmeansthe
distancebetweenthepointsAandB.)
1.AE
2.CF
3.GB
4.CA
5.BF
6.EG
A B
C
–4
–2
D
E
0
2
F
G
4
7.Describeawaytofindthedistancebetweentwopointsonanumberlinewithoutcountingthe
spaces.
A
8.
a.FindAB.
b.FindBC.
B
C
c.FindAC.
9.WhyisiteasiertofindthedistancebetweenpointAandpointBandpointBandpointCthanitis
tofindthedistancebetweenpointAandpointC?
10.ExplainhowtofindthedistancebetweenpointAandpointC.
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SECONDARY MATH I // MODULE 8
8.1
CONNECTING ALGEBRA & GEOMETRY - 8.1
SET
Topic:Slopetrianglesandthedistanceformula
TriangleABCisaslopetriangleforthelinesegmentABwhereBCis
theriseandACistherun.NoticethatthelengthofsegmentBChasa
correspondinglengthonthey-axisandthelengthofAChasa
correspondinglengthonthex-axis.Theslopeformulaiswrittenas
! !!
! = ! ! wheremistheslope.
!! !!!
10
8
y2
B
6
y1 A
C
4
11.a.Whatdoesthevalue !! − !! tellyou?
2
b.Whatdoesthevalue !! − !! tellyou?
x2
x1
5
Inthepreviousunityoufoundthelengthofaslantedlinesegmentbydrawingtheslopetriangleand
thenusingthePythagoreantheoremonthetwosidesofthetriangle.Inthisexercise,trytodevelopa
moreefficientmethodofcalculatingthelengthofalinesegmentbyusingthemeaningof !! − !! and
!! − !! combinedwiththePythagoreantheorem.
12.FindAB. 13.FindAB.
14.FindAB. 15.FindAB.
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SECONDARY MATH I // MODULE 8
CONNECTING ALGEBRA & GEOMETRY - 8.1
8.1
GO
Topic:Rectangularcoordinates
Usethegiveninformationtofillinthemissingcoordinates.Thenfindthelengthoftheindicatedline
segment.
H( ,
)
K(
,
)
B ( , 6)
16.a)FindHB.
G( , )
C(
, )
A (0 , 0)
b)FindBD.
F (-10,
)
D (10 ,
)
E(
, -4)
17.a)FindDB
b)FindCF
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