Rich Mathematics Task Exemplars

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<ul><li><p>Educational Designer November 2009</p><p>Burkhardt, H. On Strategic Design 1 Task Exemplars</p><p>Assessment Task Exemplars</p><p>Here we set out to show something of the range of tasks designed and developed by theShell Centre for Mathematical Education for various projects in the US and the UK overthe last 25 years.</p><p>Animals.......................................................................................................................2Skeleton tower ...........................................................................................................3Cross the Chasm .........................................................................................................4Sports Bag ..................................................................................................................6Consecutive sums .......................................................................................................7Traffic Jam .................................................................................................................8Ponzi Pyramid ............................................................................................................9Perspectives .............................................................................................................10Polluted river ............................................................................................................11Reducing Road Accidents ........................................................................................12Smoking ...................................................................................................................14Security camera ........................................................................................................15Machines ..................................................................................................................16Fly Fast ....................................................................................................................18Oxygen ....................................................................................................................20Table tennis tournament ...........................................................................................22</p><p>These tasks were chosen to illustrate a diversity of styles, methodology and topics,including:</p><p> Computer-delivered and traditional tasks Medium length (10 minutes) to extended (several lessons) Group work or individual work A variety of contexts, including pure mathematics, science, design, planning and</p><p>finance Tasks designed to assess the regular mathematics curriculum, others with a</p><p>focus on cross-curricular problem solving and scientific reasoning</p></li><li><p>Educational Designer November 2009</p><p>Burkhardt, H. On Strategic Design 2 Task Exemplars</p><p>AnimalsThis problem gives you the chance to: find the median, mode and range of a set of tabulated data interpret graphs</p><p>Hughworksinanaturereserve.Hehascaughtanumberofanimalsofonespeciesinanareaofwoodland.Herecordstheirweightsingrams,thenheputstheirweightsingroupedfrequencytables.</p><p>Males FemalesWeightingrams Numberof</p><p>animalsWeightingrams Numberof</p><p>animals2029 1 2029 33039 4 3039 84049 8 4049 55059 10 5059 26069 7 6069 47079 5 7079 78089 2 8089 5</p><p>Hughusesthesetablestoplotthetwographsshownbelow.</p><p>20-29 30-39 40-49 50-59 60-69 70-79 80-89Weight in grams</p><p>Numberofanimals</p><p>10 9 8 7 6 5 4 3 2 1 0</p><p>20-29 30-39 40-49 50-59 60-69 70-79 80-89Weight in grams</p><p>Numberofanimals</p><p>10 9 8 7 6 5 4 3 2 1 0</p><p>Males Females</p><p>1. Findthemedianweightsofthemaleandfemaleanimals.Inwhichgroupsdotheylie?</p><p>2. Usingthegraphsandtablesforevidence,writethreedifferentstatementscomparingthesimilaritiesanddifferencesbetweentheweightsofthemaleandfemaleanimals.</p></li><li><p>Educational Designer November 2009</p><p>Burkhardt, H. On Strategic Design 3 Task Exemplars</p><p>SKELETON TOWER</p><p>1. Howmanycubesareneededtobuildthistower?2. Howmanycubesareneededtobuildatowerlikethis,but12cubeshigh?</p><p>Explainhowyouworkoutyouranswer.3. Howwouldyoucalculatethenumberofcubesneededforatowerncubes</p><p>high?</p></li><li><p>Educational Designer November 2009</p><p>Burkhardt, H. On Strategic Design 4 Task Exemplars</p><p>CROSS THE CHASM(Computerbased,grouptaskwithindividualreporting)</p><p>1. Describehowthewidth,thespanandthethicknessoftheplankbridgeaffectthemaximumweightitwillsupport.Usewords,tables,graphsandformulae.</p><p>2. Useyourresultstopredicthowmuchweightthefollowingbridgeswillsupport:</p><p>a) thickness=4cm width=50cm span=2m</p><p>b) thickness=12cm width=20cm span=1m</p><p>c) thickness=2cm width=30cm span=6m</p><p>d) thickness=12cm width=20cm span=10m</p><p>Thebridgestrengthcalculatorwon'ttellyou!</p></li><li><p>Educational Designer November 2009</p><p>Burkhardt, H. On Strategic Design 5 Task Exemplars</p><p>Samplestudentreport</p></li><li><p>Educational Designer November 2009</p><p>Burkhardt, H. On Strategic Design 6 Task Exemplars</p><p>SPORTS BAG</p><p>Youhavebeenaskedtodesignasportsbag.</p><p> Thelengthofthebagwillbe60cm. Thebagwillhavecircularendsofdiameter25cm. Themainbodyofthebagwillbemadefrom3piecesofmaterial;apiecefor</p><p>thecurvedbody,andthetwocircularendpieces. Eachpiecewillneedtohaveanextra2cmallarounditforaseam,sothat</p><p>thepiecesmaybestitchedtogether.</p><p>1. Makeasketchofthepiecesyouwillneedtocutoutforthebodyofthebag.Yoursketchdoesnothavetobetoscale.Onyoursketch,showallthemeasurementsyouwillneed.</p><p>2. Youaregoingtomakeoneofthesebagsfromarollofcloth1metrewide.Whatistheshortestlengththatyouneedtocutfromtherollforthebag?Describe,usingwordsandsketches,howyouarriveatyouranswer.</p></li><li><p>Educational Designer November 2009</p><p>Burkhardt, H. On Strategic Design 7 Task Exemplars</p><p>CONSECUTIVE SUMS</p><p>Thenumber15canbewrittenasthesumofconsecutivewholenumbersinexactlythreedifferentways:</p><p>15=7+815=1+2+3+4+515=4+5+6</p><p>Thenumber9canbewrittenasthesumofconsecutivewholenumbersintwoways:</p><p>9=2+3+49=4+5</p><p>Thenumber16cannotbewrittenasaconsecutivesum.</p><p>Nowlookatothernumbersandfindoutallyoucanaboutwritingthemassumsofconsecutivewholenumbers.</p><p>Writeanaccountofyourinvestigation.Ifyoufindanypatternsinyourresults,thentrytoexplainthemfully.</p><p>Whichnumberscannotbewrittenasthesumofconsecutivewholenumbers?</p></li><li><p>Educational Designer November 2009</p><p>Burkhardt, H. On Strategic Design 8 Task Exemplars</p><p>TRAFFIC JAM</p><p>1. Last Sunday an accident caused a traffic jam 12 miles long on a two lanemotorway. How many cars do you think were in the traffic jam?Explain your thinking and show all your calculations.Write down any assumptions you make.</p><p>2. When the accident was cleared, the cars drove away from the front, one carevery two seconds. Estimate how long it took before the last car moved.</p></li><li><p>Educational Designer November 2009</p><p>Burkhardt, H. On Strategic Design 9 Task Exemplars</p><p>PONZI PYRAMID SCHEMES</p><p>Maxhasreceivedthisemail.Itdescribesaschemeformakingmoney.</p><p>From: ACrookDate: Thursday15thJanuary2009To: BCarefulSubject:Getrichquick!</p><p>Dearfriend,</p><p>Doyouwanttogetrichquick?Justfollowtheinstructionscarefullybelowandyoumayneverneedtoworkagain:</p><p>1.Atthebottomofthisemailthereare8namesandaddresses.Send$5tothenameatthetopofthislist.2.Deletethatnameandaddyourownnameandaddressatthebottomofthelist.3.Sendthisemailto5newfriends.</p><p>1. Ifthatprocessgoesasplanned,howmuchmoneywouldbesenttoMax?</p><p>2. Whatcouldpossiblygowrong?Explainyouranswerclearly.</p><p>3. WhydotheymakePonzischemeslikethisillegal?</p></li><li><p>Educational Designer November 2009</p><p>Burkhardt, H. On Strategic Design 10 Task Exemplars</p><p>PERSPECTIVES</p><p>Simon is standing on the deck of a ship.To the south he sees four mountain peaks;Mounts Axe, Bob, Col and Don.The diagrams below show what Simon sees.</p><p>MountAxe</p><p>MountBob</p><p>MountCol</p><p>MountDon</p><p>A guidebook lists the heights of these Mountains as:</p><p>Mount Don: 3,000m Mount Bob: 2,000mMount Axe: 2,000m Mount Col: 1,000m</p><p>1. Which mountain is furthest away from Simon? Explain how you know.</p><p>2. Which mountain is nearest to Simon? Explain how you know.To the north Simon can see two mountains, Mount Bart and Mount Homer.Mount Bart is 20 km away and 1,200m high,Mount Homer is 30 km away and 1, 500m high.</p><p>3. Which mountain looks taller to Simon? Explain how you know.</p></li><li><p>Educational Designer November 2009</p><p>Burkhardt, H. On Strategic Design 11 Task Exemplars</p><p>POLLUTED RIVER</p><p>Afactoryissituatedonthebankofariver.Itpourstoxicwasteintotheriver.Thegraphsbelowshowhowthewasteaffectsplantandanimallifeintheriver.Theanimalsaretinycreatureswhichlivebyeatingtheplants.</p><p>Distance downstream (metres) </p><p>Amount of toxic waste in the water (units) </p><p>50 100 150 200 250 300 0 </p><p>Factory </p><p>Pipe pours toxic waste into the river here </p><p>River flow </p><p>Distance downstream (metres) </p><p>Number of animals in the water </p><p>50 100 150 200 250 300 0 </p><p>Distance downstream (metres) </p><p>Number of plants in the water </p><p>50 100 150 200 250 300 0 </p><p>A lot </p><p>None </p><p>A lot </p><p>None </p><p>1 </p><p>2 </p><p>3 </p><p>4 </p><p>0 </p><p>Animals </p><p>Plants </p><p>Pollution </p><p>Explainfullywhatishappeningtothenumbersofanimalsandplantsintheriverasyougodownstream.Usethegraphstosuggestreasonsforthesechanges.</p></li><li><p>Educational Designer November 2009</p><p>Burkhardt, H. On Strategic Design 12 Task Exemplars</p><p>REDUCING ROAD ACCIDENTS(Computerbased,grouptask)</p><p>Imagineyouliveinthissmalltown.Inthistown,thenumberofpeopleinjuredorkilledontheroadshasincreasedsteadilyoverthepastfouryears.Eachaccidentlocationisshownbyareddotonthemap.</p><p>Yourjobistoinvestigatewhatcouldbedoneaboutthisworryingsituation.Youhaveabudgetof100,000tospendonreducingroaddeathsandinjuries.Yourjobistoplanthemosteffectivewayofspendingthemoney.</p><p>Somepossiblewaysofspendingthemoneyareshownonthenexttwopages.</p><p>Usethecomputersoftwaretoexplorethedataanddiscoverpossiblecausesoftheaccidents.Writeandpresentaconvincinggroupreport,showinggraphsandtablestosupportyourargument.</p></li><li><p>Educational Designer November 2009</p><p>Burkhardt, H. On Strategic Design 13 Task Exemplars</p><p>Possiblemeasuresforimprovingroadsafety</p></li><li><p>Educational Designer November 2009</p><p>Burkhardt, H. On Strategic Design 14 Task Exemplars</p><p>SMOKING</p><p>The chart below resulted from a study of the smoking habits of men.It shows data for about 1,000 men in each of four categories: non-smokers, and those who smoke 1 to 9, 10 to 39, or more than 40cigarettes a day.It shows how many men would be expected to survive to each age.</p><p>For example, of 1,000 men aged 25 who do smoke more than 40cigarettes per day, about 856 will survive to the age of 50.</p><p>Age Number of survivors</p><p> number of cigarettes smoked per day</p><p>Zero 1 to 9 10 to 39 More than 40 </p><p>25 1,000 1,000 1,000 1,000</p><p>30 994 991 991 988</p><p>35 987 981 981 973</p><p>40 978 966 965 951</p><p>45 964 942 939 910</p><p>50 944 906 869 856</p><p>55 909 859 831 777</p><p>60 855 778 744 671</p><p>65 777 673 622 540</p><p>70 667 524 468 400</p><p>Source: E.C. Hammond, Journal of the National Cancer Institute, 43 (951-962) 1969.</p><p>Use the data in the table to write comments on the following four opinions.You should try to reply to each statement as fully and informatively as possible.</p><p>a) I am 25 years old. I only smoke 5 cigarettes per day.Smoking isn't going to affect me much at all.</p><p>b) I am also 25. I am a heavy smoker (about 50 per day). I reckon that I mightreduce my lifespan by two or three years, but its not that much really.</p><p>c) I am 45 and smoke about 20 per day. I guess I stand about a 70% chance ofreaching the age of 70. That is little different to a non-smoker.</p><p>d) This table alone proves that smoking is a cause of early death.</p></li><li><p>Educational Designer November 2009</p><p>Burkhardt, H. On Strategic Design 15 Task Exemplars</p><p>SECURITY CAMERA</p><p>A shop owner wants to prevent shoplifting.He decides to install a security camera on the ceiling of his shop.The camera can turn right round through 360.The shop owner places the camera at point P, in the corner of the shop.</p><p>Plan view of the shop</p><p>Entrance</p><p>P</p><p>BA</p><p>C</p><p>E</p><p>D</p><p>FG</p><p>H J K</p><p>1. The plan shows ten people are standing in the shop.These are labelled A, B, C, D, E, F, G, H, J, K.Which people cannot be seen by the camera at P ?</p><p>2. The shopkeeper says that "15% of the shop is hidden from the camera"Show clearly that he is right.</p><p>3. (a) Show the best place for the camera, so that the it can see as much of theshop as possible.</p><p>(b) Explain how you know that this is the best place for the camera.</p></li><li><p>Educational Designer November 2009</p><p>Burkhardt, H. On Strategic Design 16 Task Exemplars</p><p>MACHINES(computerbasedindividualtask)</p><p>Screen1</p><p>Screen2</p></li><li><p>Educational Designer November 2009</p><p>Burkhardt, H. On Strategic Design 17 Task Exemplars</p><p>Screen3:</p><p>Notes:</p><p>Onthistask,studentsfreelyexploretheeffectofdraggingdifferentcogstomakedifferentmachines.Onthefirsttwoscreensthisisfulyinteractive.Onthethirdthestudentshavetogeneralisefromtheirearlierresultsasthisscreenpreventsthemfromexperimenting.Forexample,onscreen2theymayobtain:</p></li><li><p>Educational Designer November 2009</p><p>Burkhardt, H. On Strategic Design 18 Task Exemplars</p><p>FLY FAST(computerbasedtask)</p><p>1. Whichtwobirdsmaybedescribedasbothlightandfast?Youfindthenameofabirdbyclickingonthepointthatrepresentsit.</p><p>2. Herearesomestatementsaboutbirds:Heavybirdsflymoreslowlythanlighterbirds.Birdswithgreaterwingareasflyfasterthanbirdswithsmallerwingareas.Heavierbirdshavegreaterwingareasthanlighterbirds.Howvalidarethesestatements?Explainhowyouknow,givingevidencetosupporteachexplanation.</p><p>3. Ascientistwantstofindawayofpredictingthespeedofabirdfromitsweightandwingarea.Whichgraphisthebestforthis?Drawasketchofit.</p><p>4. Abirdhasaweightof65Nandawingareaof0.60m2.Whatdoyouthinkitscruisingspeedwillbeinflight?</p><p>5 Canyoufindaruleorformulathatwillenableyoutopredictthespeedofabirdfromitsweightandwingarea.?</p></li><li><p>Educational Designer November 2009</p><p>Burkhardt, H. On Strategic Design 19 Task Exemplars</p><p>Notes:</p><p>Onthistask,studentsfreelyexploretheeffectofdragginglabelstotheaxes.Forexample,theymayobtain:</p><p>or</p></li><li><p>Educational Designer November 2009</p><p>Burkhardt, H. On Strategic Design 20 Task Exemplars</p><p>OXYGEN(computerbasedtask)</p><p>1. Explorethesituation.Whatconclusionscanyoudraw?</p><p>2. JimandAnnruntheirownexperimentstodeterminehowoxygenproductiondependsontemperature.Annfoundlittleornoeffectoftemperatureonoxygenproduction.Jimfoundthatatemperaturehadalargeeffect.Couldtheybothberight?Ifnot,whatmistakemighttheyhavemade?</p><p>3. JimandAnndiscusshowtomaketheplantsproducethemostoxygen.Jimsuggeststhattheykeeptheplantsaswarmaspossible.Annsuggeststhattheykeepthelightsonfullpower.Whohasthebetteridea?Whatconditionsdoyouthinkproducethemostoxygen?</p></li><li><p>Educational Designer November 2009</p><p>Burkhardt, H. On Strategic Design 21 Task Exemplars</p><p>Notes</p><p>Onthistask,studentsfreelyexploretheeffectofdragginglabelstotheaxes.Forexample,theymayobtain:</p></li><li><p>Educational Designer November 2009</p><p>Burkhardt, H. On Strategic Design 22 Task Exemplars</p><p>PLANNING AND ORGANIZING: A TABLE TENNISTOURNAMENT</p><p>Youhavethejoboforganisingatabletennisleague.</p><p> 7playerswilltakepart Allmatchesaresingles. Everyplayerhastoplayeachoftheotherplayersonce. Therearefourtablesattheclub. Gameswilltakeuptohalfanhour. Thefirstmatchwillstartat1.00pm.</p><p>Planhowtoorganisetheleague,sothatthetournamentwilltaketheshortestpossibletime.Putalltheinformationonapostersothattheplayerscaneasilyunderstandwhattodo.</p></li></ul>