Activity #3 - Google Sites · p164 #15 OYO p179 #1 p180 ... Foldable Properties, ... "I use the Law...
Transcript of Activity #3 - Google Sites · p164 #15 OYO p179 #1 p180 ... Foldable Properties, ... "I use the Law...
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Activity#3Howmanythingsaregoingoninthissimpleconfiguration?
WhenyourTEAMhas10ormorethings,getALLofyourstuffstampedoff!
EO2Level2AnswersID:AID:B
F
H
CompleteMRforLevel3'sonly 2availableAcademicSupportPeriods
Turnitin(staplecherryformontopofyourcorrections/workandputthetestintheback)
Ifyouare100%correctontheMasteryReform,youareeligiblefortheretake.
EveryonetakingtheretakemustcompleteboththeLevel3andLevel4sections Dobetter?Lowergradeisdropped
Doworse?Level3scoresareaveragedandthelowerLevel4scoreisdropped
MasteryReform
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Level2Opp#2
EO1OneVariableInequalities 12%
EO2TwoVariableInequalities 13%
EO3Similarity&Congruence 25%
EO4Polynomials 25%
Q3Midterm
EO1Level3Mastery
ReformDueMon,Jan30th
EO1Opp#1Thu,Jan19th
EO1Opp#2Tue,Jan31st@B211
EO2Level3Mastery
ReformDueTue,Feb7th
EO2Opp#2Thu,Feb9th
EO3Level3Mastery
ReformDueFri,Feb24th
EO3Opp#2Tue,Feb28th
Q3FinalFri,March10th
EO2Opp#1Fri,Jan27th
EO3Opp#1Tue,Feb14th
EO4Opp#1Wed,Mar8th
ML
Math3Quarter3Overview
MLML
130 131 21 22 23
SignedProgressReportduelastFriday!
p163TATSInv311Inv311p164#15
OYOp179#1p180#3
PropertiesofSimilarPolygons(basedonInv1#5)
p167STM,CYU,EQ
OYOp40#1,p43#9
AngleVocabularyParallelLinesCut
byaTransversalFoldable
Properties,Postulates,&Theorems
Inv121p30#13,EQ
Similarity&ProportionPractice
Inv312p168#15,STM,CYU,EQ
Inv313p175#3,4,5***,6
OYOp41#3,5p45#13*,
p46#14,15**
OYOp180181#4Writeproofs!
*p45#13AngleAdditionPostulate
**p46#15ExteriorAngleTheoremforaTriangle
***p176#5MidpointConnectorTheoremforTriangles
EO1Opp#2Tue,Jan31st
SolvinganSASTriangle
SimilarityTheorems
TipsforWritingSimilarityProofs
EO2MasteryReformdueTuesday
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26 27 28 29 210
Inv322p202#4STM,CYU,EQ(organizer)
TriangleCenters(handout)
TipsforCongruenceWritingProofs
TriangleCenters
PropertiesofQuadrilaterals
ProofsInvolvingSimilarTriangles
TriangleCongruenceFlipBook
p195TATSInv321p196#24,67,STM,CYU,EQ
EO3Quiz#2Fri,Feb10thEO3Quiz#1
Mon,Feb6th
TeamHomeworkDiscussion OYOp27#31,p125126#30
OYOp179#1,p180#3
Similarity&ProportionPractice
OYOp40#1,p43#9
OYOp41#3,5,p45#13*,p46#14,15**
Fullcredit(2stamps)beforeteamdiscussion.Partialcreditafter.
AnswersinslidesjustbeforeToolkitentries.
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p168
EssentialQuestionsWhatcombinationsofsideoranglemeasuresaresufficientto
determinethattwotrianglesaresimilar?
LanguageObjectiveMathematicianswillbeableto......useAA,SSS,andSAStoprove2trianglessimilar.
Activities#15,STM,CYU,EQ
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GivenSAS
information
UsetheLawOfCosinestofindthe3rdside
UseLawOfSinestofindthe2ndangle
UsetheTriangleSumPropertytofindthe3rdangle
TOOLKIT:SolvingaSAStriangleReference:Investigation2SufficientConditionsforSimilarityofTrianglesp169#1
Example: UsetheLawOfCosinestofindthe3rdside
UseLawOfSinestofindthe2ndangle
UsetheTriangleSumPropertytofindthe3rdangle
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2m 4m
Couldyousaythetrianglesweresimilar?
3m 6m83o
83o
SideAngleSidesimilaritytheorem!
Ifyousawthis:
17m
51m
Couldyousaythetrianglesweresimilar?
10m 15m 30m 45m
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LawofCosines
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17m
51m
Couldyousaythetrianglesweresimilar?
10m 15m 30m 45m
SideSideSidesimilaritytheorem!
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#2SSSSimilarityTheorem!
TOOLKIT:SimilarityTheorems#1SASSimilarityTheorem!
#3AASimilarityTheorem!
NOTE:WhyareASAandSAAnotincludedassimilaritytheorems???
ANSWER:Whenconsidering"sufficientconditions"wearelookingfortheminimumcriteriathatconsistentlyprovessimilarity.SincewehaveprovedAAasasimilaritytheorem,that'stheminimumcriteria.BothASAandSAAaddanadditionalcriteriaofaS(side).CanyouseethatbothASAandSAAhaveAAinthem?
SSS~Thm
SAS~Thm
AA~Thm
Whatissufficientinformation?Correspondencepatternsthatworkforsimilarity:
SAS SSS AA
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57o38o
57o
95o
28o
85o
Similar? NO!
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35o
Similar? YES!
35o
12
15
16
20
Similar? YES!
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Similar? NO!
416
18
1040
48
12
1263o
83o
63o34o
83o34o
Similar? YES!
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Math1
TOOLKITS
TOOLKIT:PythagoreanTheorem&ItsConverse
ThePythagoreanTheoremcanbeusedtofindanymissingsideofaRIGHTtrianglesolongastwosidesofthetriangleareknown.
Examples:
Findthehypotenuse:
Findtheleg:
TheConverseofthePythagoreanTheorem:Foraanytrianglewithsidesa,b,c,ifa2+b2=c2,thenit'sarighttriangle
1.Writea2+b2=c22.Fillinthe3partsthataregiventoyou.
Makesuretheygointherightspot!cisalwaysthelongestside.3.Solveeachsideoftheequation.
Ifequal,thenit'sarighttriangle.IfNOTequal,thenit'sNOTarighttriangle
4.Writeyouranswerinacompletesentence
?
1.Writea2+b2=c2
2.Fillinthe3partsthataregiventoyou.
Makesuretheygointherightspot!cisalwaysthelongestside.
3.Solveeachsideoftheequation.Ifequal,thenit'sarighttriangle.IfNOTequal,thenit'sNOTarighttriangle
4.Writeyouranswerinacompletesentence
?
5in
12in13in
Example#1 1.Writea2+b2=c2
2.Fillinthe3partsthataregiventoyou.
Makesuretheygointherightspot!cisalwaysthelongestside.
3.Solveeachsideoftheequation.Ifequal,thenit'sarighttriangle.IfNOTequal,thenit'sNOTarighttriangle
4.Writeyouranswerinacompletesentence
?
3cm5cm
7cm
Example#2
Askingthequestion"Isittrue?"
ThePythagoreanTheorem:Forarighttriangle,thesumofthetwoleglengthssquaredisequaltothelengthofthehypotenusesquared.
a
b
c
a2+b2=c2
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TOOLKIT:TriangleInequalityTheorem
Thesumofanytwosidelengthsofatriangleisgreaterthanthethirdsidelength.
ac
b
Examples:
Activity#2:Canthesesidelengthsformatriangle?Quadrilateral?Explain.
1) 5,7,13
2) 6,12,9
3) 25,10,15
4) 3,5,7,13
5) 12,6,9,25
6) 24,2,14,8
TOOLKIT:QuadrilateralInequalityTheoremThesumofanythreesidelengthsofaquadrilateralisgreaterthanthefourthsidelength.
ac
b
dExamples:
45
8
7
2 35
12Canformaquadrilateral
Cannotformaquadrilateral
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TOOLKIT:TriangleCongruencyTheorems
S.A.S.S.S.S.A.A.S.A.S.A.H.L
S.S.A A.A.A
So,howmanycombinationsofanglesandsidesarepossible?Six!
ButtwoofthemcanNEVERbeusedtoprovecongruence.
Becarefultorecognizethattheorderoftheselettersrepresentstheshortestconsecutivepatharoundthetriangle.Doyouunderstand???
CONGRUENTFIGURESAllcorrespondingsides
andallcorrespondinganglesarecongruent(equal)!
So,itmustbearighttriangletohaveanhypotenuse!
Triangleshave3anglesand3sidelengths.Toprovethattrianglesarecongruent,youmustshowthatthereare3congruentparts.
AB DE XYAC DF XZBC EF YZ
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TOOLKIT:CorrespondingPartsofCongruentTrianglesareCongruent
(CPCTC)
CPCTC:CorrespondingPartsofCongruentTriangles
areCongruent
Whendevelopingaprooffortriangleparts,youmustfirstprovethatthetrianglesarecongruent,andthenyoucanuseCPCTCasthereasonthetriangle'sotherpartsarecongruent.
Besurethatyouunderstandeachpartofthisstatement.
TriangleSumProperty
Thesumofallanglesofatriangleisequalto180o
A
BC
Ifthevalueof2quantitiesareknowntobeequal,thenthevalueofonequantitycanbereplacedbythevalueoftheother.
Subtractingthesamenumberfromeachsideofanequationgivesusanequivalentequation.
Whentwolinesintersect,oppositeanglesthatsharethesamevertexorcornerpointarecongruent.
ReflexiveProperty
Congruenttoitself(appliestobothsegmentslengthsandangles)
sharedline
D
A
C
B
VerticalAnglesTheorem
SubstitutionPropertyofEquality
A
B
C
DE
sharedangle
SubtractionPropertyofEquality
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TOOLKIT:LawofSines
"IusetheLawofSineswhenIknowSAAorSSA."
SetUpTips1.Beginwiththefractionbars&equalsign2.Putyourunknowninthefirstnumerator3.Puttheoppositeterminthedenominator4.Puttheknownangle&oppositesidepairontheright
or
Side
Angle Angle
Side,Angle,Angle
Side
SideAngle
Side,Side,AngleInbothcasesIknow1angle&oppositesidepair!
Math2
TOOLKITS
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a
b
c
A
B
C
TOOLKIT:PythagoreanTheorem
Whensolvingforthehypotenuse:Whensolvingforoneofthelegs:
Ifthesquareofonesideofatriangleisequaltothesumofthesquaresoftheothertwosides,thenthetriangleisarighttriangle.
PythagoreanTheoremConverse
3,4,5 5,12,13 8,15,17
TherearecertainsetsofnumbersthathaveaveryspecialpropertyinconnectiontothePythagoreanTheorem.NotonlydothesenumberssatisfythePythagoreanTheorem,butanymultiplesofthesenumbersalsosatisfythePythagoreanTheorem.
PythagoreanTriples:
TOOLKIT:StandardPositionAngle
InitialSide
TerminalSide
66o
Anangleformedbyrotating(counterclockwiseispositive,clockwiseisnegative)arayfromitsinitialposition(vertexattheoriginandlayingonthexaxis)totheterminalposition.
y
x66o
P(x,y)Foranystandardpositionanglewhoseterminalsidepassesthroughthepoint(x,y),weknowthe3sidelengthsare...
x
y
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y
P(x,y)
Hypotenuse
Opposite
Adjacentx
TOOLKIT:TrigFunctions:Sine,Cosine,andTangentyrsin==
Opposite
Hypotenuse
xrcos==
AdjacentHypotenuse
yxtan==
Opposite
Adjacent
SohCahToaMemorize! NotetoStudents:This
Toolkitissufficientbyitself,butyoucanalsoaddthenext3slidesifyouwant.
TOOLKIT:InverseTrigFunctions
Anytimewe'resolvingfortheANGLEweusetheINVERSEtrigonometryfunction.
Taketheinversetrigfunctionofeachsideoftheequation
Anyvaluetimesitsinverseequals1
Useyourcalculator(inDEGREEmode)tofindtheanswer
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TOOLKIT:"Special"RightTrianglesTherearetwo"special"righttriangleswithwhichyouneedtobefamiliarthe454590triangleandthe306090triangle.The"special"natureofthesetrianglesistheirabilitytoyieldexactanswersinsteadofdecimalapproximationswhendealingwithtrigonometricfunctions.
Thischartshowsthevalueswithrationalizeddenominatorsaswellasthe123and321tricktohelpthosestudentswholikememorizingtables.
Whenyouthenrationalizethedenominator,you'llget
TOOLKIT:AnglesElevation&Depression
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TOOLKIT:LawofSines ,Part1
or
"IusetheLawofSineswhenIknowSAAorSSA."
InbothcasesIknow1angle&oppositesidepair!
TOOLKIT:LawofSines ,Part2
"IusetheLawofSineswhenIknowSAAorSSA."
SetUpTips1.Beginwiththefractionbars&equalsign2.Putyourunknowninthefirstnumerator3.Puttheoppositeterminthedenominator4.Puttheknownangle&oppositesidepairontheright
or
Side
Angle Angle
Side,Angle,Angle
Side
SideAngle
Side,Side,AngleInbothcasesIknow1angle&oppositesidepair!
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GivenSASinformation
UseLOCtofindthe3rdside
UseLOStofindthe2ndangle
UsetheTriangleSumPropertytofindthe3rdangle
TOOLKIT:LawofCosines(LoC)
SolvingaSASTriangle:
SolvingaSSSTriangle:
GivenSSSinformation
UseLoCtofindoneangle
UseLoStofinda2ndangle
UsetheTriangleSumPropertytofindthe3rdangle
Solvingforthe3rdsidewhengivenSAS
Solvingforthe1stanglewhengivenSSS
Equal,EquivalentForms
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#2SSSSimilarityTheorem!
TOOLKIT:SimilarityTheorems#1SASSimilarityTheorem!
#3AASimilarityTheorem!
NOTE:WhyareASAandSAAnotincludedassimilaritytheorems???
ANSWER:Whenconsidering"sufficientconditions"wearelookingfortheminimumcriteriathatconsistentlyprovessimilarity.SincewehaveprovedAAasasimilaritytheorem,that'stheminimumcriteria.BothASAandSAAaddanadditionalcriteriaofaS(side).CanyouseethatbothASAandSAAhaveAAinthem?
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TOOLKIT:MidpointConnectorTheoremforTriangles.
Ifalinesegmentjoinsthemidpointsoftwosidesofatriangle,thenitishalfthelengthofthethirdsideofthetriangleand
paralleltothethirdside.
MN||AC2MN=AC
EO3TOOLKITSSimilarity
Math3
CalculatingScaleFactors
PropertiesofSimilarPolygons(seeInv1#5slides)
ParallelLinesCutbyaTransversalFoldable
ParallelLinesCutbyaTransversalProperties
Properties,Postulates,&Theorems
AngleDefinitions
SolvingaSASTriangle
SimilarityTheorems
TipsforWritingSimilarityProofs
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TOOLKIT:CalculatingScaleFactors
sidelengthofthelargerpolygonsidelengthofthesmallerpolygon=Smallertolargerpolygon
sidelengthofthesmallerpolygonsidelengthofthelargerpolygon=Largertosmallerpolygon
ScaleFactor>1
ScaleFactor
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TOOLKIT:ParallelLinesCutbyaTransversal
FOLDABLE
LinearPairPostulate
VerticalAnglesTheorem
AngleAdditionPostulateSeep45#13
MidpointConnectorTheoremforTrianglesSeep176#5
SubstitutionPropertyofEquality
Addition,Subtraction,MultiplicationorDivisionPropertyofEquality*
Ifthevalueof2quantitiesareknowtobeequal,thenthevalueofonequantitycanbereplacedbythevalueoftheother.
Adding,subtracting,multiplying,ordividing*thesamenumberfromeachsideofanequationgivesusanequivalentequation.
*Pickjustoneoperation!
Twoadjacentangleswhoseunsharedsidesformastraightanglearesupplementary(equalto180o)
180o
Whentwolinesintersect,oppositeanglesthatsharethesamevertexorcornerpointarecongruent.
CorrespondingAnglesPostulate
AlternateInteriorAnglesTheorem
AlternateExteriorAnglesTheorem
SamesideInteriorAnglesTheorem
SamesideExteriorAnglesTheorem
CAsareCongruent
AIAsareCongruent
AEAsareCongruent
SSIAsareSupplementary
SSEAsareSupplementary
Anexteriorangleofatriangleisequaltothesumofthetworemoteinteriorangles
ExteriorAngleTheoremforaTriangleSeep46#15
Reflexiveproperty Congruenttoitself
TOOLKIT:Properties,Postulates,&Theorems
NameDescriptionDiagram
Ifthetwomidpointsofatriangleareconnected,thenthemidlineisparalleltoandhalfthelengthofthethirdside.
IfPisapointintheinteriorofthen
B
AP
C
or
p31
p32
p31
ConverseoftheCorrespondingAnglesPostulate
ConverseoftheAlternateInteriorAnglesTheorem
ConverseoftheAlternateExteriorAnglesTheorem
ConverseoftheSamesideInteriorAnglesTheorem
ConverseoftheSamesideExteriorAnglesTheorem
Iftwolinesareintersectedbyatransversalandcorrespondinganglesarecongruent,thenthelinesareparallel.
Iftwolinesareintersectedbyatransversalandalternateinterioranglesarecongruent,thenthelinesareparallel.
Iftwolinesareintersectedbyatransversalandalternateexterioranglesarecongruent,thenthelinesareparallel.
Iftwolinesareintersectedbyatransversalandsamesideinterioranglesarecongruent,thenthelinesareparallel.
Iftwolinesareintersectedbyatransversalandsamesideexterioranglesarecongruent,thenthelinesareparallel.
NOTE:WewilladdtothisToolkitasnecessary.
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TOOLKIT:AngleVocabularyPoint
Line
Plane
Angle
Right Angle
Obtuse Angle
Acute Angle
Straight Angle
Adjacent Angles
Linear Pair
Vertical Angles
Complementary Angles
Supplementary Angles
Perpendicular LinesParallel Lines
Transversal
Postulate (or axiom)Theorem
NOTE:WewilladdtothisToolkitasnecessary.
|| lines in a plane do not intersect
Define the following terms. Diagrams would be helpful.
Anglescanbenamedwithnumbers.
13
42
WU
V
YX
Notethatisnotspecific.Itcouldbereferringtoanyof4angles.
Anglescanbenamedwithletters.
two rays with a common starting point
identifies a position, has no dimension, labeled with a single capital letter
determined by two points, infinite length, no thickness or width, labeled with two capital letters with a line above or one single lower case letter
a two-dimensional serface determined by 3 points, infinite length and width but no thickness, labeled with 3 capital letters or one italicized capital letter
adjacent angles whose measures add up to 180 degrees (supplementary)
two congruent angles formed by intersecting lines
2 angles whose measures add up to 180 degrees
line that intersects parallel lines
lines in a plane that intersect at 90 degree angles
2 angles whose measures add up to 90 degrees
13
42
equals 90 degrees
greater than 90 degrees
less than 90 degrees
equals 180 degrees
two angles with a common vertex and shared side (ray)
GivenSAS
information
UsetheLawOfCosinestofindthe3rdside
UseLawOfSinestofindthe2ndangle
UsetheTriangleSumPropertytofindthe3rdangle
TOOLKIT:SolvingaSAStriangleReference:Investigation2SufficientConditionsforSimilarityofTrianglesp169#1
Example: UsetheLawOfCosinestofindthe3rdside
UseLawOfSinestofindthe2ndangle
UsetheTriangleSumPropertytofindthe3rdangle
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#2SSSSimilarityTheorem!
TOOLKIT:SimilarityTheorems#1SASSimilarityTheorem!
#3AASimilarityTheorem!
NOTE:WhyareASAandSAAnotincludedassimilaritytheorems???
ANSWER:Whenconsidering"sufficientconditions"wearelookingfortheminimumcriteriathatconsistentlyprovessimilarity.SincewehaveprovedAAasasimilaritytheorem,that'stheminimumcriteria.BothASAandSAAaddanadditionalcriteriaofaS(side).CanyouseethatbothASAandSAAhaveAAinthem?
SSS~Thm
SAS~Thm
AA~Thm
TOOLKIT:TipsforWritingSimilarityProofs
GIVEN:Couldbeinlist,asentence,oradiagram.PROVE:Typicallyfouroptions Trianglesaresimilar~ Correspondinganglesarecongruent Sidesarerelatedbythesamescalefactor Linesareparallel
STATEMENTREASON
Yourfinalstatementmustbeexactlythesameaswhatyou'reaskedtoproveintheprompt.
Ifprovingthattrianglesaresimilar,thenyourlaststatementisasimilaritystatement
IfprovingthatProvethatcorrespondinganglesarecongruent,thenyourlaststatementisananglecongurencestatement.
Ifprovingthatsidesarerelatedbythesamescalefactor,thenyourlaststatementisascalefactorstatementwherekisthescalefactor.
Ifprovingthatlinesareparallel,thenyourlaststatementisaparallellinesstatement.
Thebodyoftheproofmustleadtothesimilaritytheoremthatyou'reusing:
AA~Thm:Writestatements&reasonsthatshowthatthetwopairsofcorrespondinganglesarecongruent.
SSS~Thm:Writestatements&reasonsthatshowthatallcorrespondingsidesarerelatedbythesamescalefactor.
SAS~Thm:Writestatementsandreasonsthatshowthatthetwopairsofcorrespondingsidesarerelatedbythesamescalefactoreandthattheincludedanglesarecongruent.
1.Proofstypicallybeginbystatingthegiveninformation
1.Given
AA~ThmorSSS~ThmorSAS~Thm
CASTC*(CorrespondingAnglesofSimilarTrianglesareCongruent)
CSSTSSSF*(CorrespondingSidesofSimilarTrianglesSharetheSameScaleFactor)
If......correspondinganglesarecongruent......alternateinterioranglesarecongruent......alternateexterioranglesarecongruent......samesideinterioranglesaresupplementary......samesideexterioranglesaresupplementary...
...thenthetwolinescutbythetransversalareparallel.
Pickthereasonthatfitstheproof
{
*CASTCandCSSTSSSFaretheacronymsforthephrasesinparenthesis.Ifyoucanrecalltheacronymcorrectly,thenuseit.Ifnot,youneedtowriteoutasimilarlywordedreason.
NOTE:Youmusthaveshownthatthetrianglesaresimilarfirst,thenyoucanusethisreason!
NOTE:Youmusthaveshownthatthetrianglesaresimilarfirst,thenyoucanusethisreason!
ConverseoftheCorrespondingAnglesPostulate
ConverseoftheAlternateInteriorAnglesTheorem
ConverseoftheAlternateExteriorAnglesTheorem
ConverseoftheSamesideInteriorAnglesTheorem
ConverseoftheSamesideExteriorAnglesTheorem
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Attachments
M3211Organizer.docx
M3311ParallelLines&TransversalFoldable.pdf
SelectorTools.exe
EO2 Inequalities in 1 Variable Name __________________
Unit 2 Lesson 1 With calculator Date: _______________ Period ____
Warm-Up
Match the following number line graphs with the correct symbolic inequality solutions.
1) A)
2) B)
3) C)
4) D)
5) E)
6) F)
Reminders:
7) When you have an open dot (non-filled in circle) you are using the _____ or _____ sign.
8) When you have a closed dot (filled in circle) you are using the _____ or _____ sign.
9) When reading an inequality, like x > 2, it is explaining the answer to some rule of possible x-values that can be plugged in and solve to be true.
a. Therefore, some x-values that do NOT work in the example x > 2 are: __________________________
b. And some x-values that do work in the example x > 2 are: _______________________
c. Can x = 2 here? Why or why not?
Interval Notation: Mathematicians (like yourself) can used interval notation as a kind of shorthand to also describe the solutions like the ones above. Interval notation uses parentheses () and brackets [], like symbolic inequalities and closed and open circles on a number line. Examine the following rules and examples below.
Rules:
When using greater than (>) and less than ( 2
Number Line:
Interval Notation: (2, )
When thinking about Interval Notation, think of it as the range in which all the solutions falls. The parentheses before 2 is telling us that the solutions start as any number larger than 2. The infinity symbol,, is telling us that the solution goes on forever for any number greater than 2. ALL infinity symbols, - or , must have a parentheses next to it since we can never actually reach it.
Example 2: The 3 ways to represent solutions of all x-values less than or equal to -1 or all x-values greater than 2.
Symbolically:
Number Line:
Interval Notation: (, 1] (2, )
The symbol, , means the union of the two intervals. So, the numbers that are in one interval (negative infinity to -1 included) or the numbers in the other interval (any number greater than 2) are the solutions.
Practice writing interval notation from a number line graph: Write the following solutions in interval notation.
1) 2)
3) 4)
Practice writing interval notation from symbolic notation: Write the following solutions in interval notation.
6) 7)
8) 9)
10)
Practice writing solutions symbolically AND on a number line graph when given interval notation:
11) Solution: (, 5]Symbolically:
Number Line:
12) Solution: [, 1)Symbolically:
Number Line:
13) Solution: (, 2) [5, ) Symbolically:
Number Line:
Name: ___________________________ Date: _________________ Period: ____ Page 109
Unit 2 Lesson 1 Investigation 1 Getting the Picture Activities #1-2, STM, CYU, EQ
Essential Questions
Essential Question(s)
Language Objective(s): Mathematicians will be able to
On page 107 of your book:Unit 2: Inequalities and Linear Programing
For some people, athletes and astronauts in particular, selection of a good diet is a carefully planned scientific process. Each person wants maximum performance for minimum cost. The search for an optimum solution is usually constrained by available resources and outcome requirements.
Lesson 1: Inequalities in One Variable
Use numeric and graphic estimation methods and algebraic reasoning to solve problems that involve linear and quadratic inequalities in one variable.
Page 108: Lesson 1 Inequalities in One Variable.
In previous courses, you learned how to solve a variety of problems by representing and reasoning about them with algebraic equations and inequalities. For example, suppose that the plans for a fundraising raffle show that profit P will depend on ticket price x according to the function. A graph of profit as a function of ticket price is shown here.
Page 109: Think About This Situation: Questions important to the fundraising group can be answered by solving inequalities involving the profit function.
a) What would you learn from solutions of the following inequalities?
i.
ii.
iii.
iv.
b) How could you use the graph to estimate solutions of the inequalities in Part a?
c) In what ways could you record solutions of the inequalities in words, symbols, or diagrams?
Page 109-110: In this lesson, you will learn how to use graphical reasoning and algebraic methods to solve inequalities in one and two variables. You will also learn how to represent the solutions symbolically and graphically and how to interpret them in the contexts of the questions that they help to answer.
Investigation 1: Getting the Picture
You learned in earlier work with inequalities that solutions can be found by first solving related equations. For example, in the raffle fundraiser situation, the solutions of the equation
are approximately $1.23 and $5.44. The reasonableness of these solutions can be seen by scanning the graph of the profit function and the constant function y = 2,500. (Hint: Graph -2500+5000x-750x2 in Y1, and Graph 2500 in Y2 of your graphing calculator. Push 2nd, Push Trace, pick 5: Intersect to find the intersection points).
The solution of the inequality are at values of x between $1.23 and $5.44. Those solutions to the inequality can be represented using symbols or a number line graph.
How would you write the solutions to the inequality in interval notation?
Similarly, the solutions of the inequality are all the values of x that are either less than $1.23 or greater than $5.44. Those solutions can also be represented using symbols or a number line graph.
How would you write the solutions to the inequality in interval notation?
As you work on problems of this investigation, look for answers to these questions:
How can you solve inequalities in one variable?
How can you record the solutions in symbolic (inequality), graphic, and interval notation form?
#1) on page 110-111
The next graph shows the height of the main support cable on a suspension bridge. The function defining the curve is . Where x is horizontal distance (in feet) from the left end of the bridge and h(x) is the height (in feet) of the cable above the bridge surface.
a. Where is the bridge cable less than 40 feet above the bridge surface?
Write the equation or inequality that could be used to solve.
Mark/circle the part or parts of the graph that are solutions. (Use your graphing calculator to help you find more exact answers when writing solutions).
Express Solution Symbolically
Express Solution Graphically
Express Solution in Interval Notation
b. Where is the bridge cable at least 60 feet above the bridge surface?
Write the equation or inequality that could be used to solve.
Mark/circle the part or parts of the graph that are solutions. (Use your graphing calculator to help you find more exact answers when writing solutions).
Express Solution Symbolically
Express Solution Graphically
Express Solution in Interval Notation
c. How far is the cable above the bridge surface at a point 45 feet from the left end?
Write the equation or inequality that could be used to solve.
Mark/circle the part or parts of the graph that are solutions. (Use your graphing calculator to help you find more exact answers when writing solutions).
Express Solution Symbolically
Express Solution Graphically
Express Solution in Interval Notation
d. Where is the cable 80 feet above the bridge surface?
Write the equation or inequality that could be used to solve.
Mark/circle the part or parts of the graph that are solutions. (Use your graphing calculator to help you find more exact answers when writing solutions).
Express Solution Symbolically
Express Solution Graphically
Express Solution in Interval Notation
#2) on page 111
The graph below shows the height of a bungee jumpers head above the ground at various times during her ride on the elastic bungee cord. Suppose that h(t) gives height in feet as a function of time in seconds
For each Part a-d:
Write a question about the bungee jump that can be answered by the indicated mathematical operation.
Use the graph to estimate the answer.
Express your answer (where appropriate) with a number line graph and interval notation.
a) Evaluate h(2)
b) Solve h(t) = 10
c) Solve h(t) 10
d) Solve h(t) < 10.
Summarize the Math p111
a.
b.
c.
Check Your Understanding p112
a.
bi.
ii.
iii.
c.
Now answer the Essential Questions.
How can you solve inequalities in one variable?
How can you record the solutions in symbolic, graphic, and interval form?
SMART Notebook
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SMART Notebook
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