#2LessonOverview:Thislessonisadaptedfromthemodellessonbasedonplacingnumbersofthesamevalueonaray.Itisdesignedtoleadkidstotheconclusionthatthesamenumbercanbeexpressedindifferentways,butitwillalwaysshareonesinglepointonaray.Thelessonplansbeginsbyhavingstudentsplacenumbersonaray,thenshowsthemthatthosenumbersallendeduponthesamepointoftheray.Studentsarethenencouragedtocounttothatpointusingdifferentmethods,toelucidatethattherearemanydifferentwaystogettothesamepointontheray.Studentsshouldrecognizethatthepointonaraymodelisbothproofthatthenumbersareequivalent(implicit)andthatnumbersexpressedmultiplewayscanstillrepresentthesamevalue(explicit).MockUps,withdescription:

1

Thenumbers2.3,2.30,23/10,and23/10areallequal.Eventhoughtheymaylookdifferent,theyreallreallythesame!

2

Whatifwetriedtoplotthosenumbersonaray?Howmanypointsdoweneedtorepresentallofthenumbersonthesamenumberray?Studentcanselecttheoptionsatthebottom(one,two,three,four)

Ifastudentanswerscorrectly,jumpto#6(flagforenrichment).

Ifastudentanswersincorrectlycontinueto#3,donottellstudenttheywerewrong.

3

Letstryplottingthepointstoseewhathappens.Firstwellmarkeachpointonadifferentnumberray.Havestudentsdragdotsontotheraytherayscanbemarkedinwholes(1,2,3)andtengthsbetweenthat.Thepointsshouldsnaptothehashmarks.

4

Nowwewillputourpointonthesamenumberray.Watchwhathappenswhenwecombinethoseraystogether.Animationshouldshowfournumberraysalignontopofeachother,thenonebyonecollapsetogether,clearingshowingthateachpointisatthesamespotontheray.Newtextfadesintobottom:Noticethatweendedupwithonlyonepointontheray.(ImadeareallybadgiftohopefullyillustratewhatImeanifitdoesntload,youcanfindithere.)

5

Great!Letstryourquestionagain:howmanypointsdoweneedtorepresentallofthenumbersonthesamenumberray?

Regardlessofanswer,moveto#6.

6

Letsgothrougheachnumberandworkitout.First,wellcountto2.3.(A)Animationhighlights2.3tostandoutagainstotheroptions.Tocountto2.3,wewouldneedtocounttwowholeunits(B)Animationshowsbluepointbouncetwounits,thenstop.Nowwellneedtocounttwotengths.Thatmeansweneedtodivideournextunitintotengths.(C)Animationaddshashmarksintengths.Then,wecancount.One.Two.Three.Animationmovesbluepointthreetenths.Remembertherearetwowaystowrite2and3tengths.Asadecimal,2.3,andasafraction,23/10.(D)Animationaddshighlightingforboth,highlightingalongwiththewords.

7

Wecouldalsohavejustcountedbytengthsfromthebeginning.Letscutallofourunitsintotengths.(A)Animationshowshashmarksallalongnumberray.Nowwellcountjustthetengths.(B)Greenballmovesonetengthatatimetoeventuallycovertheblueballwhilethevoicecounts.There,23tengths.Animationhighlights23/10.

8[A]

[B]

[A]Toget2.30,wecanmakeoursegmentsevensmaller.Weknowwellneedtwowholeunits(A)Animationshouldshowpurpleballmovetwounits.Now,letsdivideournextunitintotengths.(B)Animationaddshashmarkstodivideunitbetween23intotengths.Thatwasfun,soletsdoitagain!Thistime,weregoingtodivideoursmallersegmentsintoevenSMALLERones!Letsdividethesmallersegmentsintotenevensegments.Animationmustbeveryspecificheretoscaffoldunderstanding![B]Zoominondistancebetween2and3,andshowhasmarkssplittingitintotengths,then

[C]

[C]slideanimationtoshowsplittingintohundredthsbetween2and2.1,thenslideanimationupto2.12.2,then2.2to2.3andshowitspliteachtime.

9

Zoombackouttoshowthatthedistancebetween2and2.3hasbeensplitinto30segments.Woah.Thatsalotoflinestocount!Maybeyoucanhelphowmanylinesaretherebetween2andthegreenpoint?Displaytext,allowanswerasafreeinput.

Ifastudentanswerscorrectly,animationshouldmovepurpleballquicklytothegreenballat2.30.Greatjob!Thatmeansthepointisalsoat2.30!Animationshouldhighlight2.30.Continueto#10.

Ifastudentanswersincorrectly,offertoletthemtryagain:Notquite.Tryagain!Ifstudentanswersincorrectlyasecondtime,animationshouldslowlymovepurpleballfrommarktomark,countingalonguntilitreachesthegreenpointandthenumber30.Theboxinthebottomright,wherestudentsanswersgo,shouldcountalongwiththemovingball,toshowthefinalansweras30.Greatjob!Thatmeansthepointisalsoat2.30!Animationshouldhighlight2.30.(Flagstudentforinterventionaftersecondwronganswer.)Continueto#10.

10

Sinceallfournumbersarethereallythesame,theyalsoshareonesinglepointontheray.Animationshouldzoombackouttoshownumberraywithpurpledot,thenpullapartallfourrayswitheachdifferentcolordotasastack.2.3,2.30,23/10,and23/10arenotonlythesamenumber...Animationshouldhighlightray/numberpairsasthevoicesaysthenumber....theyrealsothesamepoint!Animationshouldfinishbycollapsingthestackbacktogethertoemphasizethatthelocationsarethesame.

11 [Thisslideshoulddisplaytheclosingquestion/answersinastandardmultiplechoiceformat.]

Basedonwhatwelearnedtoday,whatcanweconclude?Optionsshouldbemultiplechoicesentences:*(A)Ifanumberiswrittendifferently,itwillcorrespondtoadifferentpointonthenumberray.(B)Thesamenumbercanbewritteninmanydifferentways,butitwillcorrespondtoonlyonepointonthenumberray.*(C)Anumbercanonlybewritteninoneway,eveniftherearemanydifferentcorrespondingpointsonthenumberray.*(D)Asinglepointonthenumberraymeansthereisonlyonewaytowriteanumber.

Ifcorrect(B),congratulatestudentonajobwelldonewithcelebratoryanimation!

Ifincorrect,explainswhyitisnotthatanswer,thengiveanothertry:

(A)Thinkbacktoourexample.2.3and23/10arewrittendifferently,buttheybothcorrespondedthe

SAMEpointonthenumberray! (C)Thinkbacktoourexample.2.3,23/10,2.30,

and23/10areallthesamenumber,andtheyrewritteninmorethanoneway!

(D)Ourexampleshowedusthatonepointcanoftencorrespondtomorethanonewaytowriteanumber.

Ifincorrecttwice,proceedto#12(flagforintervention).12 Notquite.Remember:eventhoughanumbercanbewrittenin

differentways,itsstillthesamenumber!Thatmeansitwillcorrespondtoonlyonepointonthenumberray.Studentshouldbeaskedifhedliketoretrythelessonorproceedwithcaution!

Advantages:

Instantgeneraldifferentiationforstudentswhoarenotgraspingconcept Allowsstudentstomoveattheirownpacing,withoutbeingpushedalongbyinclusioninthegroupdynamic Allowseasierscaffoldingofcontentthroughanimatedmodeling Eliminatesfearoffailurefromstudentswhomightnotbewillingtoparticipateduringwholeclassinstruction

Disadvantages/Limitations:

Lossofspecificdifferentiationforstudentsparticularneedsnormallygainedbyhavinganactiveteachermonitoringcommonmistakes,astudentspersonalstruggles,andotherextentuatingcircumstances

Studentscannotaskspecificquestionsduringthelesson note:thisissolvedbythestudentusingRMduringclass,andteachersbeingavailabletohelpstudentswhohave

specificquestionsaboutthematerialasitispresented Inabilitytouseopenendedquestionsforstudentstorecordandkeeptheirresponses

note:thiscouldbeslightlyremediedbyaddinganopenendedanswersystemtoRMwhichsearchesforkeywordsintheanswer,e.g.givingcreditwhenitdetectsthewordssame,different,onepoint,ray,butthisisnotwidelyusedandcouldbefinnicky

Ifstudentsarestrugglingaftermultipleattemptswiththelesson,itwillnothelpthemsincethelessonisrelativelystatic(i.e.studentswillbereading/watchingthesamethingoverandoveragainiftheydidntgetitthefirstfewtimes,itsunlikelyitwillsuddenlyclickwithoutadifferentversionoftheexplanation)

note:thiscouldberemediedbyoffering(1)adynamicmethodofchangingtheexamplenumbersandnumberraylocationsand(2)offerringdifferentversionsofthesamelessonforwhenstudentshavenotsucceededafterassessmentandreteaching

Difficultyintegratingaverifiablenotetakingsystemforstudents