Bagsofwheat
Annotation​
Aliuseshissound​ placevalueknowledgeofournumbersystemtoefficientlysolvethis​ decimalfractionproblem.
Heisabletoworkeasilybetweenwhole​ numbers,fractionsanddecimals.
Problem:​Bagsofwheat
Theteachershowsthisproblemtothe​ studentandreadsitwithhimasrequired:
Maluneeded​ 2.5k gofwheatforalargewheatbag.Shehad0.88k gand1.7k gin2​ separatebags.Ifshe
combinedthesewouldshehaveenough?
Student​response
Ali:
Teacher:
Ali:
Teacher:
Ali:
Yes,she’dhave2.58kgbecause1.7+0.88=2.58.
Tellmehowyoudidthat.
WellIcouldseethat0.8or8/10addedto1.7or1and7/10is​ 2.5.Andthere’sjustthat0.08
or8/100,whichis80grams,left​ over
Howwouldyourecordthat?
I’dprobablywrite1.7+0.88=2.58
Estimates
Annotation​
Rickworkseasilywithpositiveandnegativeintegers,hesensibly​ andreadilyestimatesthesolutionstoproblems
involvingthe​ additionandsubtractionfordecimalsandintegers,andcanexplain​ histhinking.
Problem:​Estimates​
Theteachershowsthisproblemtothestudentandreadsitwithhim​ asrequired:
YouandyourclassmateDanhavebeenmak ingquick estimatesof​ solutionstosomemoneyproblems.Whatdo
youthink ofDan’s​ estimates?
$249.42–$394.42=–$255
$78.85+$79.85+$98.60=$301.20
$71.43–$69.88=$11.60
Student​response
Rick:
Teacher:
Rick:
Danisn’tthatgreatonestimating.
Howdoyouknowthat?
Ijustknowthatitcan’tbe–255becauseifItook400​ from250it’dbe–150,notlike–250.
Youcankindofignore​ the.42bitbecausethat’sbeingsubtractedanyway.Andthatone​
wouldbelike80+80+100,which’dbemorelike260.Thelast​ one’souttoobecauseit’dbe
about$1.60not$11.60,because​ it’slike$71.50–$70.Mmm…heneedstoimprovehis​
estimationIreckon.
Pipepieces
Annotation​
Paorasolvesthis​ problembyefficientlycombiningfractionstomakewholenumbers.He​ demonstrateshisability
tomentallyapplytheadditionoperationto​ fractionsandtoworkwithdifferentdenominators,equivalent​ fractions,
andwithimproperfractions.Hissolutionandexplanation​ demonstrateastrongnumbersense.
Problem:​Pipepieces
Theteachershowsthisproblemtothe​ studentandreadsitwithhimasrequired:
Hemiwantsto​ usethesmallestnumberofleftoverpiecesofpipetomak ea3metre​ length.Thesearethe
lengths,inmetres,thathehas:1/2,7/8,​ 5/8,1/8,3/4,3/8,1/4.Whichwillheuse?
Student​response
Paora:
Teacher:
Paora:
Teacher:
Paora:
Teacher:
Paora:
He’llusethe¾and¼,the½,7/8,and5/8.
Tellmehowyoudidthat.
Well¾+¼=1and½is4/8,and4/8+5/8+7/8is16/8​ whichis2.1and2isthree.
Whydidyoudoitthatway?
It’sjusteasiertomakewholeswiththepiecesinmyhead.
Howwouldyourecordthat?
I’dprobablywriteitasafractionequationlikethis.​ (Hewrites¾+¼=1,4/8+5/8+7/8=
16/8=2,1+2=3)or​ youcouldjustshowitallaseighthsinwhichcaseitwouldtotal​ 24/8.
Sugarforjam
Annotation​
Uputauaimmediately​ respondstothenumberswithinthisproblem,seeingdifferencesand​ combiningthese
easily.Sheworksefficientlywithwholeanddecimal​ numberstocalculatea‘startunknown’problem.
Problem:​Sugarforjam
Theteachershowsthisproblemtothe​ studentandreadsitwithherasrequired:
Evawasmak ing​ jam.Shewasweighingsugarfromtwopartlyusedbags.Whenshe​ added0.67k gsugarfrom
thesecondbagthescalesshowed2.33k g.How​ muchwasinthefirstbag?
Student​response
Uputaua:
Ifirstthoughtaround1.7andit’sactually1.66kg.
Teacher:
Tellmehowyoudidthat.
Uputaua:
WellIcouldjustseethat0.67+0.33made1,another0.33​ makes1.33andsoit’sjust
another1kg.​
Teacher:
Whydidyoudoitthatway?
Uputaua:
Thenumberskindofsuggestedit.Icouldseethatthose​ two0.33swouldmake0.66,then
there’sthe1.
Teacher:
Howwouldyourecordthat?
Uputaua:
I’dwriteitthewaytheproblemis,0.67+1.66=2.33
Stillchilly
Annotation​
Foufilisolvesthis​ problembyaddingintegersandindoingsodemonstratesthathe​ understandstheirpractical
application.Withreferencetothe​ contextheisabletosuccinctlyexplaintheadditionoperationwith​ integersand
therelationshipofthesumtozero.
Problem:​​Still​chilly
Theteachershowsthisproblemtothe​ studentandreadsitwithhimasrequired:
Sincelastmonth​ thetemperatureinAnchorage,Alask a,hasincreasedby22°C.​ Itisnow5°C.Whatwasthe
temperaturelastmonth?
Student​response
Foufili:
Teacher:
Foufili:
Teacher:
Foufili:
Teacher:
Foufili:
–17°C.
Howdidyouworkthatout?
BecauseifItake22from5Iwillhave–17.
Whatdoyouknowthathelpsyou?
WellIknowifIaddapositivenumbertoanegativenumber​ itwillreducethesizeofthe
negativenumberandinthiscasethe​ additionof+22takesthetemperatureintopositive​
numbers.Thatmeanstheoriginaltemperaturehastobewarmerthan​ –22.
Howwouldyourecordthat?
IguessI’dwritethatas–17++22​ =+5becausethatexplainsquitesimplywhatis
happening​ tothetemperature.
Theleakytank
Annotation​
Arohasolvesthis​ multi-digitsubtractionproblembyusinganequaladditionsstrategy.​ Shedemonstratesand
appliesherknowledgeofcompatiblenumbersto​ 1000,herunderstandingoftheprincipleofequaladditions,and
in​ herexplanationshejustifiesherefficientstrategychoice.
Problem:​Theleakytank
Theteachershowsthisproblemtothe​ studentandreadsitwithherasrequired:
Therewere9608​ litresofwaterinalargetank .Thetank leak edandonly934litres​ remain.Howmuchwater
drainedoutofthetank ?
Student​response
Aroha:
Teacher:
Aroha:
Teacher:
Aroha:
Teacher:
Aroha:
Teacher:
Aroha:
It’s8674.
Tellmehowyoudidthat.
WhenIseethat934Ijustwanttomakeit1000becauseit’s​ easiersoIadd66to934andto
9608whichgivesmeaneasier​ subtraction:9674–1000.
Whatdoyouknowthathelpedyou?
Ijustknowcompatiblenumbersfor1000andthatwhenIadd​ numberstobothpartsofa
subtractionequationthedifferencestays​ thesame.
Tellmewhyyoudiditthatway.
Becauseit’squickanditworksforme.Icouldhave​ writtenthisasanalgorithmbutitwas
easiernotto.
Howwouldyourecordthis?
IcouldwriteeverythingIdidbuttheeasiestisjustto​ writetheequation,9608–934=8674.