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6 SECONDARY MATH I // MODULE 1 1.2 Growing Dots A Develop Understanding Task https://flic.kr/p/d9Xi2T CCBYDavideDellaCasa SEQUENCES – 1.2 1. Describethepatternthatyouseeinthesequenceoffiguresabove. 2. Assumingthepatterncontinuesinthesameway,howmanydotsarethereat3minutes? 3. Howmanydotsarethereat100minutes? Mathematics Vision Project Licensed under the Creative Commons Attribution CC BY 4.0 mathematicsvisionproject.org 7 SECONDARY MATH I // MODULE 1 SEQUENCES – 1.2 4. Howmanydotsarethereattminutes?Solvetheproblemsbyyourpreferredmethod.Your solutionshouldindicatehowmanydotswillbeinthepatternat3minutes,100minutes, andtminutes.Besuretoshowhowyoursolutionrelatestothepictureandhowyou arrivedatyoursolution. Mathematics Vision Project Licensed under the Creative Commons Attribution CC BY 4.0 mathematicsvisionproject.org SECONDARY MATH I // MODULE 1 SEQUENCES – 1.2 1.2 Growing Dots– Teacher Notes A Develop Understanding Task Purpose:Thepurposeofthistaskistodeveloprepresentationsforarithmeticsequencesthat studentscandrawuponthroughoutthemodule.Thevisualrepresentationinthetaskshouldevoke listsofnumbers,tables,graphs,andequations.Variousstudentmethodsforcountingand consideringthegrowthofthedotswillberepresentedbyequivalentexpressionsthatcanbe directlyconnectedtothevisualrepresentation. CoreStandards: F-BF:Buildafunctionthatmodelsarelationshipbetweentwoquantities. 1:Writeafunctionthatdescribesarelationshipbetweentwoquantities.* a.Determineanexplicitexpression,arecursiveprocess,orstepsforcalculationfroma context. F-LE:Linear,Quadratic,andExponentialModels*(SecondaryIfocusonlinearandexponential only) Constructandcomparelinear,quadraticandexponentialmodelsandsolveproblems. 1.Distinguishbetweensituationsthatcanbemodeledwithlinearfunctionsandwith exponentialfunctions. a.Provethatlinearfunctionsgrowbyequaldifferencesoverequalintervalsand thatexponentialfunctionsgrowbyequalfactorsoverequalintervals. b.Recognizesituationsinwhichonequantitychangesataconstantrateperunit intervalrelativetoanother. Mathematics Vision Project Licensed under the Creative Commons Attribution CC BY 4.0 mathematicsvisionproject.org SECONDARY MATH I // MODULE 1 SEQUENCES – 1.2 2.Constructlinearandexponentialfunctions,includingarithmeticandgeometric sequences,givenagraph,adescriptionofarelationship,ortwoinput-outputpairs(include readingthesefromatable). Interpretexpressionforfunctionsintermsofthesituationtheymodel. 5.Interprettheparametersinalinearorexponentialfunctionintermsofacontext. ThistaskalsofollowsthestructuresuggestedintheModelingstandard: StandardsforMathematicalPracticeofFocusintheTask: SMP1:Makesenseofproblemsandpersevereinsolvingthem. SMP7:Lookforandmakeuseofstructure. TheTeachingCycle: Launch(WholeClass):Startthediscussionwiththepatternongrowingdotsdrawnontheboard orprojectedfortheentireclass.Askstudentstodescribethepatternthattheyseeinthedots (Question#1).Studentsmaydescribefourdotsbeingaddedeachtimeinvariousways,depending onhowtheyseethegrowthoccurring.Thiswillbeexploredlaterinthediscussionasstudents writeequations,sothereshouldnotbeanyemphasisplaceduponaparticularwayofseeingthe growth.Askstudentsindividuallytoconsideranddrawthefigurethattheywouldseeat3minutes (Question#2).Then,askonestudenttodrawitontheboardtogiveotherstudentsachanceto checkthattheyareseeingthepattern. Mathematics Vision Project Licensed under the Creative Commons Attribution CC BY 4.0 mathematicsvisionproject.org SECONDARY MATH I // MODULE 1 SEQUENCES – 1.2 Explore(SmallGrouporPairs):Askstudentstocompletethetask.Monitorstudentsasthey work,observingtheirstrategiesforcountingthedotsandthinkingaboutthegrowthofthefigures. Somestudentsmaythinkaboutthefiguresrecursively,describingthegrowthbysayingthatthe nextfigureisobtainedbyplacingfourdotsontothepreviousfigureasshown: Somemaythinkofthefigureasfourarmsoflengtht.withadotinthemiddle. Othersmayusea“squares”strategy,noticingthatanewsquareisaddedeachminute,asshown: Mathematics Vision Project Licensed under the Creative Commons Attribution CC BY 4.0 mathematicsvisionproject.org SECONDARY MATH I // MODULE 1 SEQUENCES – 1.2 Asstudentsworktofindthenumberofdotsat100minutes,theymaylookforpatternsinthe numbers,writingsimply1,5,9,...Ifstudentsareunabletoseeapattern,youmayencouragethem tomakeatableorgraphtoconnectthenumberofdotswiththetime: Time(Minutes) Numberof Dots 0 1 1 5 2 9 3 13 t Watchforstudentsthathaveusedagraphtoshowthenumberofdotsatagiventimeandtohelp writeanequation.Encouragestudentstoconnecttheircountingstrategytotheequationthatthey write. Forthediscussion,selectastudentforeachofthethreecountingstrategiesshown,atable,agraph, arecursiveequation,andatleastoneformofanexplicitequation. Discuss(WholeGroup):Beginthediscussionbyaskingstudentshowmanydotsthattherewillbe at100minutes.Theremaybesomedisagreement,typicallybetween100and101.Askastudent thatsaid101toexplainhowtheygottheiranswer.Ifthereisgeneralagreement,moveontothe discussionofthenumberofdotsattimet. Startbyaskingagrouptochartandexplaintheirtable.Askstudentswhatpatternstheyseeinthe table.Whentheydescribethatthenumberofdotsisgrowingby4eachtime,addadifference columntothetable,asshown. Mathematics Vision Project Licensed under the Creative Commons Attribution CC BY 4.0 mathematicsvisionproject.org SECONDARY MATH I // MODULE 1 SEQUENCES – 1.2 Difference Time(Minutes) NumberofDots 0 1 1 5 2 9 3 13 … … t >4 >4 >4 Askstudentswheretheyseethedifferenceof4occurringinthefigures.Notethatthedifference betweentermsisconstanteachtime. Continuethediscussionbyaskingagrouptoshowtheirgraph.Besurethatitisproperlylabeled, asshown. Numberofdots Time(Minutes) Askstudentshowtheyseetheconstantdifferenceof4onthegraph.Theyshouldrecognizethat they-valueincreasesby4eachtime,makingalinewithaslopeof4. Now,movethediscussiontoconsiderthenumberofdotsattimet,asrepresentedbyanequation. Startwithagroupthatconsideredthegrowthasarecursivepattern,recognizingthatthenextterm is4plusthepreviousterm.Theymayrepresenttheideaas:! + 4,withXrepresentingthe previousterm.Thismaycausesomecontroversywithstudentsthatwroteadifferentformula.Ask thegrouptoexplaintheirworkusingthefigures.Itmaybeusefultorewritetheirformulawith words,like: Mathematics Vision Project Licensed under the Creative Commons Attribution CC BY 4.0 mathematicsvisionproject.org SECONDARY MATH I // MODULE 1 SEQUENCES – 1.2 Thenumberofdotsinthecurrentfigure=thenumberofdotsinthepreviousfigure+4 Orsimply,Current=previous+4 Thismaybewritteninfunctionnotationas:! ! = ! ! − 1 + 4.(Althoughstudentshavesome exposuretofunctionnotationingrade8,theyhavenotseenitusedtowriterecursiveformulas. Youmaychoosetointroducethisnotationinlaterlessons,simplyfocusingonwritingtherecursive ideainwordsasshownabove.) Nextaskagroupthathasusedthe“fourarmsstrategy”towriteandexplaintheirequation.Their equationshouldbe:! ! = 4! + 1. Askstudentstoconnecttheirequationtothefigure.They shouldarticulatethatthereis1dotinthemiddleand4arms,eachwithtdots.The4inthe equationshows4groupsofsizet. Next,askagroupthatusedthe“squares”strategytodescribetheirequation.Theymayhave writtenthesameequationasthe“fourarms”group,butaskthemtorelateeachofthenumbersin theequationtothefiguresanyway.Inthiswayofthinkingaboutthefigures,therearetgroupsof4 dots,plus1dotinthemiddle.Althoughitisnottypicallywrittenthisway,thiscountingmethod wouldgeneratetheequation! ! = ! ∙ 4 + 1. Mathematics Vision Project Licensed under the Creative Commons Attribution CC BY 4.0 mathematicsvisionproject.org SECONDARY MATH I // MODULE 1 SEQUENCES – 1.2 Nowaskstudentstoconnecttheequationswiththetableandgraphs.Askthemtoshowwhatthe4 andthe1representinthegraph.Askhowtheysee4t+1inthetable.Itmaybeusefultoshowthis patterntohelpseethepatternbetweenthetimeandthenumberofdots: Time(Minutes) NumberofDots 0 1 1 1 5 1+4 2 9 1+4+4 3 13 1+4+4+4 … … t 1+4t Difference >4 >4 >4 Youmayalsopointoutthatwhenthetableisusedtowritearecursiveequationlike ! ! = ! ! − 1 + 4,youmaysimplylookdownthetablefromoneoutputtothenext.When writinganexplicitformulalike! ! = 4! + 1,itisnecessarytolookacrosstherowsofthetableto connecttheinputwiththeoutput. Finalizethediscussionbyexplainingthatthissetoffigures,equations,table,andgraphrepresent anarithmeticsequence.Anarithmeticsequencecanbeidentifiedbytheconstantdifference betweenconsecutiveterms.Tellstudentsthattheywillbeworkingwithothersequencesof numbersthatmaynotfitthispattern,buttables,graphsandequationswillbeusefultoolsto representanddiscussthesequences. AlignedReady,Set,GoHomework:Sequences1.2 Mathematics Vision Project Licensed under the Creative Commons Attribution CC BY 4.0 mathematicsvisionproject.org 8 SECONDARY MATH I // MODULE 1 1.2 SEQUENCES – 1.2 READY, SET, GO! Name PeriodDate READY Topic:Usingfunctionnotation Toevaluateanequationsuchas! = 5! + 1 whengivenaspecificvalueforx,replacethevariablex withthegivenvalueandworktheproblemtofindthevalueofy. Example:Findywhenx=2.Replacexwith2.! = 5 2 + 1 = 10 + 1 = 11. Therefore,y=11whenx=2.Thepoint 2, 11 isonesolutiontotheequation! = 5! + 1.Insteadof using! !"# !inanequation,mathematiciansoftenwrite! ! = 5! + 1becauseitcangivemore information.Withthisnotation,thedirectiontofind! 2 ,meanstoreplacethevalueof!with2and worktheproblemtofind! ! .Thepoint !, ! ! isinthesamelocationonthegraphas !, ! ,where !describesthelocationalongthex–axis,and! ! istheheightofthegraph. Giventhat! ! = !" − !and! ! = !" − !",evaluatethefollowingfunctionswiththeindicated values. 1.! 5 = 2.! 5 = 3.! −4 = 4.! −4 = 5.! 0 = 6.! 0 = 7.! 1 = 8.! 1 = Topic:Lookingforpatternsofchange Completeeachtablebylookingforthepattern. 9. Term Value 1st 2 2nd 4 3rd 8 4th 16 5th 32 6th 7th 8th Term Value 1st 66 2nd 50 3rd 34 4th 18 5th 6th 7th 8th Term Value 1st 160 2nd 80 3rd 40 4th 20 5th 6th 7th 8th Term Value 1st -9 2nd -2 3rd 5 4th 12 5th 6th 7th 8th 10. 11. 12. Mathematics Vision Project Licensed under the Creative Commons Attribution CC BY 4.0 mathematicsvisionproject.org 9 SECONDARY MATH I // MODULE 1 1.2 SEQUENCES – 1.2 SET Topic:Usevariablestocreateequationsthatconnectwithvisualpatterns. Inthepicturesbelow,eachsquarerepresentsonetile. Step 1 Step 2 Step 4 Step 3 Step 5 13.DrawStep4andStep5. Thestudentsinaclasswereaskedtofindthenumberoftilesinafigurebydescribinghowtheysawthe patternoftileschangingateachstep.Matcheachstudent’swayofdescribingthepatternwiththe appropriateequationbelow.Notethat“s”representsthestepnumberand“n”representsthenumberof tiles. (a)! = !" − ! + ! − ! (b)! = !" − ! (c)! = ! + ! ! − ! 14._____Danexplainedthatthemiddle“tower”isalwaysthesameasthestepnumber.Healsopointed outthatthe2armsoneachsideofthe“tower”containonelessblockthanthestepnumber. 15._____Sallycountedthenumberoftilesateachstepandmadeatable.Sheexplainedthatthenumber oftilesineachfigurewasalways3timesthestepnumberminus2. stepnumber 1 2 3 4 5 6 numberoftiles 1 4 7 10 13 16 16._____Nancyfocusedonthenumberofblocks inthebasecomparedtothenumberofblocks abovethebase.Shesaidthenumberofbase blocksweretheoddnumbersstartingat1.And thenumberoftilesabovethebasefollowedthe pattern0,1,2,3,4.Sheorganizedherworkin thetableattheright. Mathematics Vision Project Licensed under the Creative Commons Attribution CC BY 4.0 mathematicsvisionproject.org Stepnumber #inbase+#ontop 1 1+0 2 3+1 3 5+2 4 7+3 5 9+4 10 SECONDARY MATH I // MODULE 1 1.2 SEQUENCES – 1.2 GO Topic:Themeaningofanexponent Writeeachexpressionusinganexponent. 17.6×6×6×6×6 18.4×4×4 19.15×15×15×15 ! ! ! ! 20. × A)Writeeachexpressioninexpandedform.B)Thencalculatethevalueoftheexpression. 21.7! A) B) 22.3! A) B) 25.7(2)! A) B) 23.5! A) B) Mathematics Vision Project Licensed under the Creative Commons Attribution CC BY 4.0 mathematicsvisionproject.org A) B) A) B) 26.10 8! 24.10! 27.3 5 ! A) B) 28.16 A) B) ! ! !