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45 SECONDARY MATH 1 // MODULE 1 CCBYJDHancock 1.10 Geometric Meanies A Practice Understanding Task Eachofthetablesbelowrepresentsageometric sequence.Findthemissingtermsinthesequence, showingyourmethod. Table1 x y 1 3 2 3 12 Isthemissingtermthatyouidentifiedtheonlyanswer?Whyorwhynot? Table2 x y 1 7 2 3 4 875 Arethemissingtermsthatyouidentifiedtheonlyanswers?Whyorwhynot? Table3 x y 1 6 2 3 4 5 96 Arethemissingtermsthatyouidentifiedtheonlyanswers?Whyorwhynot? Mathematics Vision Project Licensed under the Creative Commons Attribution CC BY 4.0 mathematicsvisionproject.org https://flic.kr/p/eodTxP SEQUENCES – 1.10 46 SECONDARY MATH 1 // MODULE 1 SEQUENCES – 1.10 Table4 x y 1 4 2 3 4 5 6 972 Arethemissingtermsthatyouidentifiedtheonlyanswers?Whyorwhynot? A. Describeyourmethodforfindingthegeometricmeans. B. Howcanyoutelliftherewillbemorethanonesolutionforthegeometricmeans? Mathematics Vision Project Licensed under the Creative Commons Attribution CC BY 4.0 mathematicsvisionproject.org SECONDARY MATH 1 // MODULE 1 SEQUENCES – 1.10 1.10 Geometric Meanies – Teacher Notes A Practice Understanding Task Purpose: Thepurposeofthistaskistosolidifystudentunderstandingofgeometricsequencestofindmissing termsinthesequence.Studentswilldrawupontheirpreviousworkinusingtablesandwriting explicitformulasforgeometricsequences. CoreStandards: A.REI.3Solvelinearequationsandinequalitiesinonevariableincludingequationswithcoefficients representedbyletters. ClusterswithInstructionalNotes:Solveequationsandinequalitiesinonevariable. Extendearlierworkwithsolvinglinearequationstosolvinglinearinequalitiesinonevariableand tosolvingliteralequationsthatarelinearinthevariablebeingsolvedfor.Includesimple exponentialequationsthatrelyonlyonapplicationofthelawsofexponents, suchas5x=125or2x=1/16. StandardsforMathematicalPracticeofFocusintheTask: SMP7–Lookforandmakeuseofstructure. TheTeachingCycle: Launch(WholeClass):Explaintostudentsthattoday’spuzzlesinvolvefindingmissingterms betweentwonumbersinageometricsequence.Thesenumbersarecalled“geometricmeans”.Ask themtorecalltheworkthattheyhavedonepreviouslywitharithmeticmeans.Ask,“What informationdoyouthinkwillbeusefulforfindinggeometricmeans?Somemayrememberthatthe firsttermandthecommondifferencewereimportantforarithmeticmeans;similarlythefirstterm andthecommonratiomaybeimportantforgeometricmeans. Mathematics Vision Project Licensed under the Creative Commons Attribution CC BY 4.0 mathematicsvisionproject.org SECONDARY MATH 1 // MODULE 1 SEQUENCES – 1.10 Ask,“Howdoyoupredictthemethodsforfindingarithmeticandgeometricmeanstobesimilar?” Ideally,studentswouldthinktousethecommonratiotomultiplytogetthenextterminthesame waythattheyaddedthecommondifferencetogetthenextterm.Theymayalsothinkabout writingtheequationusingthenumberof“jumps”thatittakestogetfromthefirsttermtothenext termthattheyknow. Ask,“Howdoyouthinkthemethodswillbedifferent?”Ideally,studentswillrecognizethatthey havetomultiplyratherthansimplyadd.Theymayalsothinkthattheequationshaveexponentsin thembecausetheformulasthattheyhavewrittenforgeometricsequenceshaveexponentsinthem. Explore(SmallGroup):Theproblemsinthistaskgetlargerandrequirestudentstosolve equationsofincreasinglyhigherorder.Somestudentswillstartwithaguessandcheckstrategy, whichmayworkformanyofthese.Evenifitisworking,youmaychoosetoaskthemtoworkona strategythatismoreconsistentandwillworknomatterwhatthenumbersare.Studentsarelikely totrythesamestrategiesastheyusedforarithmeticsequences.Thesestrategieswillbesuccessful iftheythinktomultiplybythecommonratio,ratherthanaddthecommondifference.Iftheyare writingequations,theymayhavesimilarthinkingtothis: Table2 x 1 2 3 4 y 7 875 “Ineedtostartat7andmultiplybyanumber,r,togettothesecondterm,thenmultiplybyragain togettothethirdterm,andbyragaintoendupat875.So,Iwillwritetheequation: 7 ∙ ! ∙ ! ∙ ! = 875 or 7! ! = 875 Somestudentsmayhavedifficultysolvingtheequationthattheywrite.Itmaybehelpfultoget themtodivideby7andthenthinkaboutthenumberthatcanberaisedtothethirdpowertoget 125.Calculatorswillalsobehelpfulifstudentsunderstandhowtotakerootsofnumbers.Allofthe numbersusedinthistaskarestraightforward,makingthemaccessibleformoststudentstothink Mathematics Vision Project Licensed under the Creative Commons Attribution CC BY 4.0 mathematicsvisionproject.org SECONDARY MATH 1 // MODULE 1 SEQUENCES – 1.10 aboutwithoutacalculator.Watchforstudentsthatcanexplainthestrategyshownabove,along withotherproductivestrategiessothatyoucanhighlighttheirworkinthediscussion. Alongwithcompletingthetable,studentsareaskedifthesolutiontheyfoundistheonlysolution. Thisisareminderforthemtothinkofboththepositiveandnegativesolutioniftheexponentis even.Asyouaremonitoringstudentwork,watchtoseeiftheyareansweringthisprompt appropriately.Ifnot,youmaywanttoaskaquestionlike,“Iseethatyouhave! ! = 4, ! = 2.Is thereanothernumberthatyoucanmultiplybyitselftoget4?” Discuss(WholeClass):StartthediscussionwithTable1.Selectastudentthatguessedatthe commonratioandaskhowhe/shefigureditoutandthenhowthecommonratiowasusedtofind themissingterm.Sincemostguessersonlygetthepositiveratio(2),youwillprobablyhave studentsthatwanttoaddthenegativesolution(-2)also.Great!Askhowtheyfiguredthesolution outandtoverifyfortheclassthatacommonratioof-2alsoproducesatermthatworksinthe sequence.Somestudentsmayhavesimplyreasonedthat-2wouldworkasacommonratio.Be suretoselectastudentthathaswrittenanequationlike:3! ! = 12.Usethisasanopportunityfor studentstoseethatthealgebraicsolutiontothisequationis! = ±2. Next,askastudenttowriteandexplainanequationforTable2,asshownabove.Astheysolvethe equation,7! ! = 875,! ! = 125,askstudentsifthereismorethanonenumberthatcanbecubedto obtain125?Theyshoulddecidethat5isasolution,but-5isnot.Haveastudentdemonstrate usingthecommonratiothatwasfoundtocompletethetable. Nowthatstudentshaveworkedacoupleofproblemsasaclass,theyhavehadachancetocheck theirwork,askthemtoreconsiderthegeneralizationsattheendofthetask.Givestudentsafew minutestomodifytheirresponsestoquestionsAandB.Then,askafewstudentstosharetheir procedureforfindinggeometricmeansandknowingthenumberofsolutionswiththeentireclass. Studentsmaynoticethatoneofthestrategiestheyareusingisanalogoustotheprocesstheywere usingtofindthecommondifferenceinanarithmeticsequence.Withthearithmeticsequence,the processledtotheslopeformula.Withageometricsequence,theprocessbecomes: Mathematics Vision Project Licensed under the Creative Commons Attribution CC BY 4.0 mathematicsvisionproject.org SECONDARY MATH 1 // MODULE 1 SEQUENCES – 1.10 • Dividethelasttermbythefirstterm • Takethen-1rootoftheresulttogetthecommonratio(Studentswillprobablynotusethis terminology) Ifthisoccursinclass,itmaybeproductivetocomparehowto“undo”anarithmeticsequenceanda geometricsequence.Thedifferencesresultfromthedifferentnatureofthetwosequencetypes; arithmeticsequencesareadditiveandgeometricsequencesaremultiplicative. AlignedReady,Set,GoHomework:Sequences1.10 Mathematics Vision Project Licensed under the Creative Commons Attribution CC BY 4.0 mathematicsvisionproject.org 47 SECONDARY MATH I // MODULE 1 1.10 SEQUENCES – 1.10 READY, SET, GO! $$$$$$Name$ $$$$$$Period$$$$$$$$$$$$$$$$$$$$$$$Date$ READY ! Topic:!Arithmetic!and!geometric!sequences! ! For$each$set$of$sequences,$find$the$first$five$terms.$Then$compare$the$growth$of$the$arithmetic$ sequence$and$the$geometric$sequence.$Which$grows$faster?$$ $When?$ $ 1.!! Arithmetic!sequence:!! 1 = 2, common!difference, ! = 3! !!!!!!!! Geometric!sequence:!! 1 = 2, common!ratio, ! = 3! Arithmetic! Geometric! ! 1 =! ! 1 =! ! 2 =! ! 2 =! ! 3 =! ! 3 =! ! 4 =! ! 4 =! ! 5 =! ! 5 =! ! a)! Which!value!do!you!think!will!be!more,!! 100 !or!!(100)?!! b)!!!Why?! 2.!! !!!!!!!!! ! Arithmetic!sequence:!! 1 = 2, common!difference, ! = 10! ! Geometric!sequence:!! 1 = 128, common!ratio, ! = ! ! Arithmetic! Geometric! ! ! ! ! ! ! ! ! ! ! 1 2 3 4 5 =! =! =! =! =! 1 2 3 4 5 =! =! =! =! =! ! a)! Which!value!do!you!think!will!be!more,!! 100 !or!!(100)?!! b)!!!Why?! ! 3.!! Arithmetic!sequence:!! 1 = 20, ! = 10! Geometric!sequence:!! 1 = 2, ! = 2! Arithmetic! ! 1 =! ! 2 =! ! 3 =! ! 4 =! ! 5 =! Geometric! ! 1 =! ! 2 =! ! 3 =! ! 4 =! ! 5 =! ! a)! Which!value!do!you!think!will!be!more,!! 100 !or!!(100)?!! b)!!!Why?! ! ! Mathematics Vision Project Licensed under the Creative Commons Attribution CC BY 4.0 mathematicsvisionproject.org 48 SECONDARY MATH I // MODULE 1 1.10 SEQUENCES – 1.10 4.!! Arithmetic!sequence:!! 1 = 50, common!difference, ! = −10! Geometric!sequence:!! 1 = 1, common!ratio, ! = 2! Arithmetic! Geometric! ! 1 =! ! 1 =! ! 2 =! ! 2 =! ! 3 =! ! 3 =! ! 4 =! ! 4 =! ! 5 =! ! 5 =! ! a)! Which!value!do!you!think!will!be!more,!! 100 !or!!(100)?!! b)!!!Why?! ! ! 5.!! Arithmetic!sequence:!! 1 = 64, common!difference, ! = −2! ! Geometric!sequence:!! 1 = 64, common!ratio, ! = ! ! Arithmetic! ! 1 =! ! 2 =! ! 3 =! ! 4 =! ! 5 =! Geometric! ! 1 =! ! 2 =! ! 3 =! ! 4 =! ! 5 =! ! a)! Which!value!do!you!think!will!be!more,!! 100 !or!!(100)?!! b)!!!Why?! ! ! 6.!! ! ! ! Considering!arithmetic!and!geometric!sequences,!would!there!ever!be!a!time!that!a! geometric!sequence!does!not!out!grow!an!arithmetic!sequence!in!the!long!run!as!the! number!of!terms!of!the!sequences!becomes!really!large?!!!!!!!!!!!!!!!!!!!!!!!!!Explain.! SET Topic:!Finding!missing!terms!in!a!geometric!sequence! ! Each$of$the$tables$below$represents$a$geometric$sequence.$$Find$the$missing$terms$in$the$ sequence.$Show$your$method.$ $ 7.!!Table!1! x" 1! 2! 3! y" 3! ! 12! ! ! ! ! ! ! Mathematics Vision Project Licensed under the Creative Commons Attribution CC BY 4.0 mathematicsvisionproject.org ! 49 SECONDARY MATH I // MODULE 1 1.10 SEQUENCES – 1.10 8.!!Table!2! x) 1! 2! 3! 4! y$ 2! ! ! 54! ! ! ! ! ! ! ! ! 9.!!Table!3! x) 1! 2! 3! 4! ! y$ 5! ! 20! ! ! ! ! ! ! ! 10.!!Table!4! x) y$ 1! 4! 2! ! 3! ! 4! ! 5! 324! GO ! ! Topic:!Writing!the!explicit!equations!of!a!geometric!sequence! $ Given$the$following$information,$determine$the$explicit$equation$for$each$geometric$ sequence.$ $ 11.!!! 1 = 8, !"##"$!!"#$%!! = 2! ! ! ! ! 12.!!! 1 = 4, ! ! = 3!(! − 1)! ! ! ! ! ! 13.!!! ! = 4! ! − 1 ; !! 1 = ! ! ! ! ! ! ! 14.!!Which!geometric!sequence!above!has!the!greatest!value!at!!! 100 !?! ! ! ! Mathematics Vision Project Licensed under the Creative Commons Attribution CC BY 4.0 mathematicsvisionproject.org !
