can you really diffferentiate instruction, develop problem ... · can you really diffferentiate...

Canyoureallydiffferentiateinstruction,developproblemsolversandthinkers,addresstheTEKSandstillhavefuninmath?

Marian SmallAugust, 2016

WhoDoesn’tBelong

• 3/5 3/8 7/3 5/9

Agenda

• UsingOpen-endedQuestions• UsingParallelTasks• FocusingonProblemSolving

• OpenQuestions

PlaceValue

• Anumberhasa3anda9astwoofitsdigits.• Butthe3isworthALOTmorethanthe9.• Whatcouldthenumberbe?• Howbigcoulditbe?• Howsmall?

• 30009OR0.003009OR….

PlaceValue

• Anumberhadthree5sinitasdigits.• One5is1000timesasmuchasanother.• One5is10timesasmuchasanother.• Whatcouldthenumberbe?

• 5005 forjustclueone• 5055 OR50055forbothclues

WhoDoesn’tBelong?

• 412• 107• 1005• 635

ComparingDecimals

• A:0.[][]• B:2.[]3• C:[].4[]

• Fillintheblankssothattheorderfromleasttogreatestiseachofthese:

• A,B,C•A,C,B• C,A,B•WhynotC,B,A?

ComparingDecimals

• A:0.[][]B:2.[]3C:[].4[]

• A,B,C:0.12, 2.43,3.46• A,C,B:0.12,1.45,2.33• C,A,B:0.43,0.88,2.53• C,B,AimpossiblesinceA<B

DecomposingFractions

• Drawarectangulargrid.• Colourafractionofit.• Useasum(acombination)offractionstodescribewhatyouhavecoloured.

• Trytoshowitasasumalotofways.

DecomposingFractions

I see 1/3 + 4/15 OR5/15 + 2/15 + 2/15 OR2/5 + 3/15

Adding/SubtractingFractions

• Thedifferenceoftwofractionsis3/4.• Whatcouldthefractionsbe?

• 1– 1/4• 2/3– 1/12


• Thesumoftwofractionsis2/3morethanthedifference.• Whatcouldthefractionsbe?

• Anythingand1/3since


•

+ 1/3– 1/3

2/3


• Whobelongswithus?

• 3/5+2/3 • 4/15+7/15

• E.g.1/3and8/15 (denominatorof15)

DividingWholeNumbers

• Youdividea4-digitnumberbya1-digitnumberandthequotienthas4digits.

• Whatcouldthedivisionhavebeen?

• E.g.4812÷ 2OR9418÷ 3OR9000÷ 9OR…

IdentifyTriangleTypes

• Oneoftheanglesinatriangle is45°.• Whatkindoftrianglecanitbe?• WhatkindcanitNOTbe?

• Itcouldbeisosceles,orscalene• NOTequilateral• Couldberightorobtuseoracute

MeasurementConversions

• 6.2____=6200 ____

• Whatmeasurementunitscangointheblanks?

• E.g.,kilometersandmeters• Metersandmillimeters

Stem-and-LeafPlots

• Choose16numbers.• Theleasthastobelessthan20andthegreatesthastobemorethan70.

• Createastem-and-leafplottoshowyourdata.

• E.g.8,9,10,11,12,17,24,35• 38,39,40,50,52,53,78,78

Stem-and-LeafPlots

• 089• 10127• 24• 3589• 40• 5023• 6• 788

Fermiproblem

• Howmanyapplesdoyouthinkyoucannormallybuyfor$100?

• 10for$4so250for$100

DividingDecimals

• IdividedtwodecimalsandthequotientwasjustSLIGHTLYlessthan1.• WhatdecimalsmightIhavebeendividing?

• 4.2÷ 4.3OR10.00÷ 11.1

PrimeNumbers

• YoucreateaLOTofrectangleswithsquaretiles.• Youdeterminetheirareas.• Listsomeareasyouwillprobablygetmoreoftenandsomeyouwillprobablygetlessoften.

• E.g.often:24,48,36less:1719

VolumeofaPrism

• YoucreateaSHORTrectangularprismwithavolumeof150 in3.• Whatcoulditslength,width,andheightbe?

• E.g.2,15,5

CoordinatePlane

• Togetfromonepointonacoordinateplanetoanother,youmove4stepstotherightand3stepsdown.

• Whatcouldthecoordinatesforthetwopointsbe?

• (x,y)and(x+4,y– 3)

RepresentingIntegers

• Youusesomepositiveintegerchips.• Youput5timesasmanynegativeintegerchipswiththem.• Youdeterminethevalueofyourinteger.• Whatcoulditbe?• Whatcoulditnotbe?

• Couldbe:–4,– 8,– 12

Ratio

• Youdrawtwopicturesthatshowtheratio2:3,butyouusedifferentnumberofitemsineachpicture.

• Whatcouldthepictureslooklike?

Percent

• Fillintheblankstomakethistruelotsofways.• 8is___%of______.

• E.g.10%of80• 50%of16• 8%of100

AreaFormulas

• Theareaofaparallelogramis1/4oftheareaofatriangle.• Whatcouldthebasesandheightsofeachbe?

• b,hAND2b,4h• ORb,hAND8b,h

Solvingequations

• Yousolvedanequationthatyouthoughtwassupereasytosolve.• Whatmightthatequationhavebeen?

• E.g.x=12• OR2x=20• OR2x=x+10

Solvingequations

• Yousolvedanequationbydividingandthensubtracting.• Whatmightthatequationhavebeen?

• E.g.3x+90=333

MedianandMean

• Themedianofasetofdataisonehalfofthemean.• Whatcouldthedatasetbe?

• E.g.4,5,5,8,28

MedianandMean

• Youknowthatthemeanofasetofdataisverysmall.• Whatelsedoyouknowaboutthedata?

• E.g.nobigvaluesORnotmanybigvaluesbutalotofnumbers

Rates

• Kyledrove17milesin16minutes.• LiannedrovejustaLITTLEslower.Whatmightherspeedhavebeeninmph?

• E.g.1mileperminute,so60mph

Rates

• Abouthowfastdomostpeoplegrowfromages0to20?

• E.g.20inchesto70 inchesso2.5inchesperyear

Circlemeasures

• Onemeasurementofacircleis30inches.• Whatcouldtheothermeasuresbe?

• E.g.60inOR90π inOR…

Probability

• Fillintheblanks.• Itisabouttwiceaslikelythatyou_____thanyou______.

• E.g.rollatotalof7thanatotalof4

Probability

• Theprobabilityof_____happeningand_____happeningis3/8.• Whatmighttheeventsbe?

• E.g.rollanevenonadieandmultiplythevaluesontwodiceandgetaneven

Areasofcompositeshapes

• Ashapecomposedofparallelogramsandtriangleshasatotalareaof40squareinches.

• Whatcouldtheshapelooklike?Includethedimensions

Areasofcompositeshapes

Algebra

• Whatstorymighteachequationrepresent?• 3x+8=59?

• 5x+2y=60

• E.g.Ibought3shirtsandsomethingfor$8andspent$59.Howmuchpershirt?

Algebra

• Whatstorymighteachequationrepresent?• 3x+8=59?

• 5x+2y=60

• E.g.Ihad60toothpicksandmadesomepentagonsandangles.Howmanyofeach?

Squareroots

• Drawapicturetoshowwhy√18isabout4.25.

Dilations

• Youdilateashapeandthevertexthatusedtobeat(4,6)movesto(6,9).

• Whatcouldtheshapeanddilationlooklike?

Linearity

• Thinkofasituationthatrelatestwovariables.• Isittruethatifonevariabledoubles,sodoestheother?• Whydidthathappen?

• E.g.Arms=2xPeople

Pythagorean theorem

• Onesideofarighttriangleis10cm.• Whatcouldtheothersidelengthsbe?

• E.g.6– 8-10• OR10- 10- 14.14

Pythagorean theorem

• Twosidesofarighttrianglearecloseinlength.• Whatcouldallthreesidelengthsbe?

• E.g.3-4-5• 100- 101- 14.2

Transformations

• Youperformareflectiononacoordinategridandthepoint(4,5)movesdownandtotheright.

• Whatcouldthereflectionlinehavebeen?

• E.g.y=x(movesto(5,4)) or• Y=0.5x

Transformations

• Youperformareflectiononacoordinategrid.Onepointontheshapemoves8unitsandanothermoves20units.

• Drawapossiblesituation.

Strategies

• Givetheanswer.Studentscreateaquestion.• Askresimilaritiesanddifferences.• Whodoesn’tbelong?• Whobelongswithus?• Use”soft”words.• Askwhatelseyouknow.

•ParallelTasks

FocusonProblemSolving

Thoughtful,notcomplicated

• Comparethetwo:

Complicated

• Ibought3shirtsthateachcost$12.45.• Ibought4sweatersthateachcost$39.95.• Ibought5pairsofpantsthatcost$9.95, $12.95, $22.95, $17.95and$18.95.

• Howmuchofthe$500 IhadbudgetedforclothesdoIhaveleft?

Thoughtful

• Ibought7itemsthatcostunder$20andbought5itemsthatcostmorethan$20.Ispentalmost$300.

• Whatarepossiblepricesforthe12 items?Explainyourthinking.

Whatotherconditions?

• Misinterpretationisallowed.• Mistakesarenormal.• Tasksareengaging.

• Youareonthenumber5onanumberline.• YoumoveSOMEstepsforward.• ThenyoumoveSOMEstepsback.• Yourepeatbothmovestwomoretimeseach.• Youlandat11.• Howmanystepseachway?

• Thesamenumberofballsareineachyellowbox.• Thesamenumberareineachredbox.• Howmanymightbeineachyellowbox?Howmanyineachred?

• Ihadmorethan20counters.• Isplitthemupinto2smallpilesand2largepiles.• Thelargepileshadtwiceasmanycountersasthesmallones.• Howmanymighthavebeenineachpile?

• Youbuildarectanglewithsquaretiles.• Youcutitinhalf(basedonarea).• Couldthenewperimeterbehalfoftheoldone?• Couldthenewperimeterbe3/4oftheoldone?2/3?

Area = 16 squaresPerimeter = 16 side lengths

• Usepatternblocks.Buildadesignwhere:• Thereis3timesasmuchredareaasyellowareaand• Twiceasmuchgreenareaasbluearea.• Tellwhatfractionoftheareaiseachcolour.

• Youcanaddasmanycopiesofthenumbers6,9and20asyouwish.• Whatareallthenumbersyoucanget?• Whatareallthenumbersyoucannotget?

Cannotget

• 1,2,3,4,5

Then

• 6=6• No7,8• 9=9• No10,11• 12=6+6• No13,14

Then

• 15=9+6• No16,17,• 18=9+9• No19• 20=20• 21=9+6+6• No22,23

Then

• 24=6+6+6+6• No25• 26=20+6• 27=9+9+9• No28• 29=20+9• 30=6+9+6+9

Then

• No31• 32=20+6+6• 33=9+9+9+6• No34• 35=20+9+6• 36=6+6+6+6+6+6• No37

Then

• 38=20+6+6+6• 39=9+9+6+9+6• 40=20+20• 41=20+9+6+6• 42=6+6+6+6+6+6+6• No43

Then

• 44=20+6+6+6+6• 45=9+9+9+9+9• 46=20+20+6• 47=20+6+6+6+9• 48=6+6+6+6+6+6+6+6• 49=20+20+9

Then

• WhyamIdone?

• Google“43McNuggets”

• 43McNuggets

• https://www.youtube.com/watch?v=vNTSugyS038

• Youmultiplytwo2-digitnumbersusingbasetenblocks.• Ittakesyou36blockstoshowthemultiplication.• Whatmightyouhavebeenmultiplying?

• Makeapatternblockdesignthatis1/2greenand1/5blueinarea.

• Youmultiplytwofractions:a/bandc/d.• Theresultisalotmorethana/b,butabitlessthanc/d.• Whatcouldthefractionsbe?

• Maybe9/10x95/2

• Youstartwiththeintegera.• Whatintegerbcouldyouusesothata+bis:• 12morethana– b• 12lessthana– b• 3timesasmuchasa– b

• Youbuyajacketat40%off.• Youbuyshoesat20%off.• Youpaythesameforbothitems.• Whatdoyouknowabouttherelationshipbetweenthetwooriginalprices?

WhatitmightlooklikeinI

• If60%ofA=80%ofB,• thenB=3/4ofA.

• 0.6x=0.8ymeans6x=8yory=3/4 xOR

0 10 20 30 40 50 60 70 80 90 100

0 10 20 30 40 50 60 70 80 90 100

• Analgebraicexpressionisworth20morewhenx=5thanwhenx=3.• Howmuchmorewillitbeworthwhenx=20thanwhenx=12?

Maybe10x+3

• Thequotientoftwofractionsis4timesgreatertheproduct.Whatmightthefractionsbe?

• 5÷ ½=10but5x½=2.5• 4÷1/2=8but4x1/2 =2

• Thequotientoftwofractionsis2¼timesgreaterthantheproduct.Whatmightthefractionsbe?

• Youhavetwoalgebraicexpressions• Whenx=6, thevalueofthefirstexpressionis1/4thevalueofthesecondone.

• Butwhenx=2, thevalueofthefirstexpressionis1/6thevalueofthesecondone.

• Whatcouldtheexpressionsbe?

Tosumitup

• Goodproblemsdon’tneedtobelong-winded.• Goodproblemsoftenleadtogeneralizations.

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