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?