Download 4.4 Greater Than?

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SECONDARY MATH I // MODULE 4
EQUATIONS AND INEQUALITIES – 4.4
A Develop Understanding Task
Foreachsituationyouaregivenamathematicalstatement
andtwoexpressionsbeneathit.
1. Decidewhichofthetwoexpressionsisgreater,iftheexpressionsareequal,orifthe
relationshipcannotbedeterminedfromthestatement.
2. Writeanequationorinequalitythatshowsyouranswer.
3. Explainwhyyouransweriscorrect.
Watchout—thisgetstricky!
Example:
Statement:! = 8
Whichisgreater?! + 5 or 3! + 2
Answer:3x+2>x+5becauseifx=8, 3! + 2 = 26, ! + 5 = 13 and 26 > 13.
Tryityourself:
1. Statement:! < !
Whichisgreater?! − ! or ! − !
2. Statement:2! − 3 > 7
Whichisgreater?5 or !
3. Statement:10 − 2! < 6
Whichisgreater?! or 2
4. Statement:4! ≤ 0
Whichisgreater?1 or !
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4.4 Greater Than?
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SECONDARY MATH I // MODULE 4
EQUATIONS AND INEQUALITIES – 4.4
5. Statement:nisaninteger
Whichisgreater?! or − !
6. Statementx>y
Whichisgreater?x+aory+a
7. Statement:x>y
Whichisgreater?x–aory–a
8. Statement:5 > 4
Whichisgreater?5! or 4!
9. Statement:5 > 4
Whichisgreater?
5
4
or x
x
€ €
10. Statement:0 < x< 10 and 0 < y< 12
Whichisgreater?! or !
n+2
11. Statement: 3 ≥ 27 Whichisgreater?! or 1
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SECONDARY MATH I // MODULE 4
EQUATIONS AND INEQUALITIES – 4.4
4.4 Greater Than? – Teacher Notes
A Develop Understanding Task
Purpose:Thepurposeofthistaskistochallengestudentstoreasonaboutinequalityrelationships
andtodevelopanunderstandingofthepropertiesofinequalities.Eachproblemrequiresreasoning
aboutnumbers,includingnegativenumbersandfractions,andthinkingmathematicallyaboutthe
variouspossibilitiesinthegivenproblemsituation.
CoreStandardsFocus:
A.REI.1Explaineachstepinsolvingasimpleequationasfollowingfromtheequalityofnumbers
assertedatthepreviousstep,startingfromtheassumptionthattheoriginalequationhasasolution.
a.Constructaviableargumenttojustifyasolutionmethod.
b.Solveequationsandinequalitiesinonevariable.
A.REI.3Solvelinearequationsandinequalitiesinonevariable,includingequationswith
coefficientsrepresentedbyletters.
MathematicsINote:Extendearlierworkwithsolvinglinearequationstosolvinglinearinequalities
inonevariableandtosolvingliteralequationsthatarelinearinthevariablebeingsolvedfor.
StandardsforMathematicalPracticeofFocusintheTask:
SMP1–Makesenseofproblemsandpersevereinsolvingthem
SMP2–Reasonabstractlyandquantitatively
SMP8–Lookforandexpressregularityinrepeatedreasoning
TheTeachingCycle:
Launch(WholeClass):
Explaintostudentsthatthistaskisabiglogicpuzzle.Alloftheproblemsrequirethinkingaboutall
thedifferentpossibilitiestodecidewhichexpressionisgreater.Therearesomethatcannotbe
determinedfromtheinformationgiven.Whenthathappens,studentsshouldwritedownwhat
informationtheywouldneedinordertohaveadefiniteanswerforthequestion.(Youmaychoose
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SECONDARY MATH I // MODULE 4
EQUATIONS AND INEQUALITIES – 4.4
nottotellstudentsthisinadvancesothattheyhaveanopportunitytowrestlewiththeideasandto
justifytheirposition.)Startbyaskingstudentstoreadtheexamplegiven.Toconfirmthe
instructions,askhowtheyseethethreerequiredpartsoftheexplanationintheanswer.Next,refer
studentstoproblem#1.Givethemafewminutestoanswerandwritetheirownexplanation.Ask
theclassfortheiranswersandexplanationsandmodelhowtowriteananswerwithacomplete
explanation.Youmayalsowanttomodelthinkingaboutpossiblevaluesforxandy,like:“Ifxisa
negativenumber,thenymustalsobeanegativenumberbecauseitislessthanx.”
Explore(SmallGrouporPairs):Monitorstudentsastheywork.Encouragethemtothinkabout
thevariouspossibilitiesforxandyineachcase.Besurethattheirwrittenexplanationsadequately
communicatetheirlogic.Watchforproblemsthatgeneratedisagreementordifficultyfortheclass
discussion.Alsolookforstudents’explanationsthatdemonstratesoundmathematicallogicor
goodcommunicationtobehighlightedinthediscussion.Ifyounoticeacommonmisconception
occurringduringtheexploration,plantoraiseitasanissueinthediscussion.
Discuss(WholeClass):Startthediscussionwithproblems6-9.Askpreviously-selectedstudents
togivetheirexplanationsforeachoftheseproblems.Besurethattheexplanationsinclude
exampleofbothpositiveandnegativenumbers.Highlightfortheclassthatthesethreeproblems
areaskingthemtojustifythepropertiesofinequalities.Generally,studentswillhavetested
specificnumbersandmadegeneralizationsaboutallnumbersbasedupontheselectedexamples.
Asktheclassiftheycancreateanargumentastowhyeachpropertycanbegeneralizedtoallreal
numbers.Writeeachofthepropertiesofinequalities(addition,subtraction,multiplication,and
division),andaskstudentstostatethemintheirownwords.Aftergoingthrougheachofthese,
turnthediscussiontoanymisconceptionsorprovocativeproblemsthatwereselectedduringthe
explorationphase.
AlignedReady,Set,GoHomework:GettingReady4.4
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SECONDARY MATH I // MODULE 4
4.4
SOLVING EQUATIONS AND INEQUALITIES – 4.4
READY, SET, GO!
Name
PeriodDate
READY
Topic:Writeanequationfromacontext.Interpretnotationforinequalities.
Writeanequationthatdescribesthestory.Thenanswerthequestionaskedbythestory.
1.Virginia’sPaintingServicecharges$10perjoband$0.20persquarefoot.IfVirginiaearned$50
forpaintingonejob,howmanysquarefeetdidshepaintatthejob?
2.Rentingtheice-skatingrinkforapartycosts$200plus$4perperson.Ifthefinalchargefor
Dane’sbirthdaypartywas$324,howmanypeopleattendedhisbirthdayparty?
Indicateifthefollowingstatementsaretrueorfalse.Explainyourthinking.
3.Thenotation12 < ! meansthesamethingas! < 12.Itworksjustlike12 = ! !"# ! = 12.
4.Theinequality−2 ! + 10 ≥ 75saysthesamethingas−2! − 20 ≥ 75.Icanmultiplyby-2on
theleftsidewithoutreversingtheinequalitysymbol.
5.Whensolvingtheinequality10! + 22 < 2,thesecondstepshouldsay10! > −20becauseI
added-22tobothsidesandIgotanegativenumberontheright.
6.Whensolvingtheinequality−5! ≥ 45,theansweris! ≤ −9becauseIdividedbothsidesofthe
inequalitybyanegativenumber.
7.Thewordsthatdescribetheinequality! ≥ 100are“xisgreaterthanorequalto100.”
SET
Topic:Solveinequalities.Verifythatgivennumbersareelementsofthesolutionset.
Solveforx.(Showyourwork.)Indicateifthegivenvalueofxisanelementofthesolutionset.
8.2! − 9 < 3
9.4! + 25 > 13
Isthisvaluepart! = 6; !"#?
ofthesolutionset?
!"?
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Isthisvaluepart! = −5; !"#?
ofthesolutionset?
!"?
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SECONDARY MATH I // MODULE 4
4.4
SOLVING EQUATIONS AND INEQUALITIES – 4.4
10.6! − 4 ≤ −28
Isthisvaluepart! = −10; !"#?
ofthesolutionset?
11.3! − 5 ≥ −5
!"? Isthisvaluepart! = 1; !"#?
ofthesolutionset?
Solveeachinequalityandgraphthesolutiononthenumberline.
12.! + 9 ≤ 7
– 10
–5
0
!"?
5
10
5
10
13.−3! − 4 > 2
– 10
14.3! < −6
!
!
!
!"
15. > −
16.−10! > 150
17.
!
!!
≥ −5
– 25
–5
– 10
–5
0
5
10
– 10
–5
0
5
10
– 20
19.
!(!!!)
!"
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– 15
10
0
Solveeachmulti-stepinequality.
18.! − 5 > 2! + 3
0
– 10
20
≤
!!
!
–5
0
30
20.2 ! − 3 ≤ 3! − 2
40
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SECONDARY MATH I // MODULE 4
4.4
SOLVING EQUATIONS AND INEQUALITIES – 4.4
GO
Topic:Usesubstitutiontosolvelinearsystems
Solveeachsystemofequationsbyusingsubstitution.
Example:
! = 12
2! − ! = 14
Thefirstequationstatesthat! = 12.Thatinformationcanbeusedinthesecondequationtofindthe
valueofxbyreplacingywith12.Thesecondequationnowsays!" − !" = !".Solvethisnewequation
byadding12tobothsidesandthendividingby2.Theresultisx=13.
21.
!=5
−! + ! = 1
22.
!=8
5! + 2! = 0
23.
2! = 10
4! − 2! = 50
24.
3! = 12
4! − ! = 5
25.
! = 2! − 5
!=!+8
26.
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3! = 9
5! + ! = −5