task 3: part 2 open problem solving activitiesszalonta.hu/mm/resources/task3/task3-problems.pdf ·...
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TASK3:Part2
OpenProblemSolvingActivitiesHereisaselectionofproblemsthatarelessstructuredthanusualandcould,forexample,beusedtoencouragemathematicalthinkingaswellasdevelopingmathematicaltechniquesorconceptsthatmaybenewtothelearners.PleasenotethatwehaveorderedtheproblemsroughlyintermsofsuitabilityforKeyStagesBUTmanyoftheproblemscouldbeusedatavarietyofages.TheverylastquestionisbasedonFermiestimationtechniques,whichmaybeanewtopicformanyofyou(see*below).Alsonotethat:
• Itisimportanttouseaproblemthatyouhavenottackledbeforeand/orarenotfamiliarwith;
• Tackletheproblem,notingdownyourworkingandalsothatofcolleaguesthatyouarecollaboratingwith;
• Nowplanyourresearchlessonthatincorporatesoneoftheseproblems(orsimilar)withyourexpectationsofwhatyourlearnerswillproduceandthemisconceptionsormisunderstandingsthatmightarise;
• Whenyougivethelesson,payparticularattentiontowhatactuallyhappensandhowwellyouranticipatedsolutionsorstrategiesaremet.
Again,thisisbestachievedworkingwithcolleaguesinlessonstudymode,gettingtheirfeedbackfromobservationsmadeonthelessonanddiscussionafterwardsinthereview.
Insummary,donotdeliveran“OfSTED”lessonbuttakerisks,innovateandtryoutnewideasandstrategiesbutatallstages,reviewandevaluateprogressmadeorissuesthatarise.
Theproblemsareprovidedintwoformats,pdf(fromwhichyoucancutandpasteanyproblemthatyouwanttouse)andWORD(fromwhichyouwillbeabletoeditthewordsandquestionsposed,etc).
*Seeforexample:http://en.wikipedia.org/wiki/Fermi_problemhttp://lesswrong.com/lw/h5e/fermi_estimates/
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Problem1–MakingaDifferenceThedifferencebetweentwowholenumbersis5.
Whatmightthenumbersbe?
Trytofindasmanypossibleanswersasyoucan.
Problem2–OddOneOut
Amongthesenumbers,chooseanumberthatisdifferentfromtheothers.
Canyouexplainwhyitisdifferent?
1,2,4,6,8,12Trytofindasmanypossibleanswersasyoucan.
Problem3–EqualTeams
84childreninYear5arearrangedintoteamswiththesamenumberineachteam.
Howmanyteamsarethereandhowmanychildrenwouldbeineachteam?
Trytofindasmanypossibleanswersasyoucan.
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Problem4–HailstoneNumbersChooseapositiveinteger.Ifitiseven,halveit;ifitisodd,multiplyby3andadd1.
Repeatthisprocess;forexample,
9 28 14 7 22 11 34 ...Whathappens?Doesitmatterwhatnumberyoustartwith?
Problem5–Oneup,OnedownLookatthesemultiplicationsums.
Whathappensifyoumakethefirstnumber1moreandthesecondnumber1less?
Doesthisalwayswork?
Whathappensifthefirsttwonumbersarenotthesame?
Exploreanddevelopyourownideas.
Problem6–SquaresandRectangles
Weknowthisinformationaboutacertainsquareandcertainrectangle:
• Theirareasareequal.• Theperimeterofthesquareis4fifthsoftheperimeteroftherectangle.• Thelongsideoftherectangleis4timesthelengthofitsshortside.• Theperimeters,areasandsidesofthebothshapesarewholenumbersless
than100.
Whatcouldbethelengthsofthesidesofthesquareandtherectangle?
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Problem7–Integers
Theintegersxandysatisfytheequation
Howmanypossiblepairsofvaluesforxandycanyoufindthatmakethisequationtrue?
Problem8–Cuboids
Howmanydifferentcuboidscouldbebuiltfrom40smallunitcubes?
Whichonehasminimumvolume?
Whataboutcuboidswith24unitcubes?
Canyougeneralise?
Problem9–ReadingAge
‘Readingage’isthelevelofreadingabilitythatapersonhasincomparisontoanaveragechildofaparticularage.
Sothatpupils’readingagescanbeassessed,itisimportanttohaveanestimateofthereadingagesofbookswrittenforschool‐agereaders.Therearemanywaysofdoingthis.
Designaformulaorproceduretoestimatethereadingageofatextusingasamplepassage.
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Problem10–IceSkatingInaniceskatingcompetition,JennaandKimwerethetoptwocompetitors.
Thefivejudgesgavethemthefollowingscores.
CanyougivegoodreasonswhyJennawasdeclaredthewinnerorshouldit havebeenKim?Problem11–ASquareProblem
Squaresusingmatchsticksareshownabove.
Howmanydifferentwayscanyoufindofcountingthematchsticksneededfor5squares?Whatabout10squares?
Canyougeneraliseyourresult?
Judge 1 Judge 2 Judge 3 Judge 4 Judge 5
Jenna 8 6 10 9 7
Kim 9 9 7 8 7
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Problem12–HotShot
Ninecompetitorstookplaceinashootingcompetition.
Rankthecompetitors,explainingyourreasoning.
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Problem13–DartBoardMaths
Whatdifferentscoresbetween50and60canyouscorewithONEdart?
Using1,2or3darts,whatisthelowestscorethatitisimpossibletoobtain?
Problem14–FaultLines
Considerbricksmadeinthescaleratio2:1,forexample, .
Youcanfitthemtogethertoformdifferentrectangularshapes,forexample:
Butboththeseshapeshave‘fault’linesandcouldbeunstableindifficultconditions.Whatisthesmallestrectangle,excludingthatmadefromasinglebrick,thatcanbeconstructedfrom2by1bricks,whichhasnofaultline?Using2by1bricks,canyoudesignan8by8shapewithnofaultline?
Fault line
Fault line
Fault line
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Problem15–Braille
Brailleisamethodofrepresentingletters,etc.byraiseddotswhichblindandvisuallyimpairedpeoplecanreadbytouch.
Itwasinventedin1824bytheFrenchman,LouisBraille,wholostthesightinoneeyewhileplayingwithaknifebelongingtohisfather,andsoonlosthissightcompletely.
AnotherFrenchman,CharlesBarbier,hadearlierdevelopedasystemknownas‘nightwriting’usingraiseddots,forsoldierspassingmessagesinthedark.Thissystemusedasmanyas12dotstorepresentasinglesymbol.Eachletterwasmadeupofapatternofraiseddots,‘read’bypassingthefingerslightlyoverthemanuscript.
LouisBrailleadaptedandtransformedBarbier’ssystem,usingabaseofsixpositions(3verticalin2rows)fortheraiseddotsanddevelopingthesystemusedtoday,knownasBraille.
HowmanydifferentpatternscanbemadeusingtheBraillesystemandhowmanypatternsdoyouneedtocodeletters,capitalletters,digits,punctuation,etc.?CantheBrailledesignmeettheserequirements?
Problem16–PatrioticDesignAgroupofexpertsindesignwereaskedtochoosethetop10iconicdesignsthatrepresenttheUK.Thelistinalphabeticalorderisgivenopposite.
Doyouagreewiththislist?Whatdesignsaremissing?Putthelistinwhatyouthinkistherankorderofimportance,with1beingyourhighestorderdownto10.Ifyounowweregiventherankorderchosenbytheexperts,howcouldyoucompareyouranswerwiththischosenorder?Howwouldyouchoosetheclosestanswergivenbystudentsinyourclass?
Braille’s system
Concorde
London taxi
Mini car
Red phone box
Red pillar box
Rolls Royce car
Routemaster bus
Spitfire plane
Tube map
Union Jack
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Problem17–SpecialDice
AWizardhasdesignedagamefor2players,using3specialdice.
ThefacesoftheREDdicearemarked1,4,4,4,4and4.
ThefacesoftheBLUEdicearemarked2,2,2,5,5and5.
ThefacesoftheGREYdicearemarked3,3,3,3,3and6.
Anyplayerchallengingthewizardhasthefirstchoiceofcolour.WhentheplayerhaschosenacolourtheWizardchooseshis.Theykeeptheircolourthroughoutthegame.
ThegameconsistsofthechallengerandtheWizardthrowingtheir
dicesimultaneously,notingwhohasthehigherscoreeachtime.
Throwsarerepeateduntiloneoftheplayerswins10games.Theyarethewinner!TheWizardclaimstobetheWorldChampionplayerofthegameashehasneverbeenbeaten.
WhydoyouthinktheWizardcanbeconfidentofremainingWorldChampion?
Problem18–Marbles
Threestudents,Alisha,BenandCatherine,eachthrewfivemarbles,whichcametorestasshown.Inthisgame,thewinneristhestudentwiththesmallestscatteringofmarbles.ThedegreeofscatteringseemstodecreaseintheorderA,B,C.
Deviseasmanywaysasyoucantoexpressnumericallythedegreeofscattering.
4 4 1
3 6 3
5 2 2
A
B
C
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Problem19–VotingSystems
Therearemanyvotingsystemsinusearoundtheworld;inUKelectionsforParliamentweusewhatiscalledthe“firstpastthepost”model.Itisverystraight‐forwardtounderstandandadministerbutithasonemajordrawback.
Ifyouvoteinaconstituencythathasapredeterminedwayofvoting,thatis,thepartycurrentlyinpowerhasalargemajoritythatisunlikelytochange,youmayfeelthatyourvotewouldbewastedandnotbothertovoteatall.
Somepeoplethinkthatproportionalrepresentationisanimprovedmethodbutitalsohasitsproblems.
Forexample,consideracityelectionwhenthereare5seatstoallocatewithvotescastasshowninthetable.
Howcanyoufairlyallocatethe5seats?
Itiseasytoshowhow3ofthe5seatsshouldbeallocatedbutwhichpartydeservestheremaining2?
Problem20–Birthdays Firsttrythisexperiment.Findoutthebirthdaysof30differentpeople(e.g.class,friends,relatives).
Doanyofthemhavebirthdaysonthe
samedayoftheyear?
Perhapssurprisingly,theprobabilityofthishappeningisabout0.7.
Yourtaskistoseehowlikelyitisthattwomembersofagroupofanysize
havethesamebirthdayand,forexample,howmanypeopleareneededinthegrouptobe95%certainthattherewillbeatleasttwowiththesamebirthday.
Party Votes
A 17920
B 11490
C 11170
D 4420
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Problem21–FermiEstimation
Howmanyfootballscouldyoufitintoyoursportshall?
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