The World of Math through English. Measuring the World. !! Carlota'Petit'Olivella,'Oriol'Pallarés'Monge'i'Teresa'Socias'Garriga' ' ' '

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1 The World of Math through English Measuring the World Carlota'Petit'Olivella,'Oriol'Pallarés'Monge'i'Teresa'Socias'Garriga' ' ' ' ' ' ' ' The World of Math Through English de Carlota Petit, Teresa Socias i Oriol Pallarés està subjecta a una llicència de Reconeixement-

2 10 ideas and hints to help you succeed 1. Youwillobservepictures/realitytoidentifygeometricelements,shapesand volumestryingtodeducetheirrealdimensionsanddecidewhichunitisbetterin eachsituation 2. YouwilllearntomeasurelengthsanddistanceswithGoogle'Earth' 3. Youwilldescribe2Dfiguresnumberofsides,lengthofsides,angles)orally 4. Youwillanalyzethefunctionofdifferentkindsofshapesandfiguresinyour everydayenvironmentandconstructcompoundfigurestoshowothergeometric options 5. Youwilllearndifferentunitsofmeasurementlength,area,volume)oftheSI InternationalSystemUnits)andtheirconversiontablesandpracticeconversions indifferentcases 6. Youcanhaveextrapracticewithformulasofpolygonsinwww.math.com 7. Youwillexperimentwithpolygonstodeduceformulasforperimetersandareas 8. YoumustuseEnglishasmuchasyoucanatalltimestoimproveyourfluency 9. Youwillworkinteamstoanalyzegeometricalelementsinreallifeandshowyour reflectionsinaglogsterandclasspinterest 10. Youwillwriteareporttocommentonyouranalysisofgeometryinyour surroundingandprepareasurveyusingwww.monkeysurvey.com)tovoteand assessyourclassamtes regeometrizedproductions,, The World of Math Through English de Carlota Petit, Teresa Socias i Oriol Pallarés està subjecta a una llicència de Reconeixement-

3 The world of Math through English Carlota'Petit'Olivella,''Oriol'Pallarés'Monge'i'Teresa'Socias'Garriga' CarlotaPetitOlivella,professoradematemàtiques OriolPallarésMonge,professord anglès TeresaSociasGarriga,professorad anglès Correcciólingüística:DanielL.Casacuberta Fotografia:TeresaSociasGarriga Disseny:AlejandroDomínguezGonzàleziMarcJuliàGil AquestmaterialestàprotegitperlalegislaciódeprotecciódelapropietatintelZlectualielseuúsindegutpotserobjectede sancions,finsitotpenals,d'acordambl'article270delvigentcodipenal. LesfotografiesutilitzadessónpropietatdeTeresaSociasGarrigaosóndedominipúblic.Altrament,sen indical autor.,, Elsautors:, Vainiciarlasevacarreraprofessionalcomaprofessoradematemàtiquesadiferentsinstitutsdesecundària finsqueelcurs2006a2007començajuntamentambteresasociasioriolpallarésal'inssalvadorespriude Barcelona un projecte AICLE. Ha creat amb ells tots els materials per al currículum de 1r, 2n i 3r d'eso de matemàtiquesambaquestenfocament,semprecomafruitdelainvestigació,reflexióipràctica.desdelcurs 2007a2008imparteixaquestscursosenanglèsambenfocamentAICLEenelmateixinstitut.ColZlaboraambla UniversitatAutònomadeBarcelonaimpartintuntallerAICLEenelmàsterd'EducacióiésformadoraAICLE. Oriol,Pallarés,Monge,opallare@xtec.cat),professord anglès, Vainiciarlasevacarreraprofessionalcomamestreiprofessordemúsicaid anglèsenl ensenyamentprimarii secundariacatalunya.desd aleshores,hatreballatenescolesbilingües,haparticipatendiferentsprogrames AICLE Aprenentatge Integrat de Contingut i Llengua) i ha exercit com a professor de castellà a l estat de Califòrniatambéal ensenyamentsecundari. En l actualitat, és professor associat del Departament de Didàctica de la Llengua de la Universitat AutònomadeBarcelona,ontambé és membre del grup de recerca col laborativa CLILCSIsobreAICLE, professor del Departament d Ensenyament de la Generalitat de Catalunya a les escoles oficials d idiomes i formadordelprofessorat., Teresa,Socias,Garrigatsocias@xtec.cat),professorad anglès Apartird'uncursdepostgraudel'ICEdelaUPC,vaentrarenun"SeminaryonContentTeaching"del'ICEdela UAB1991a94)iesvainteressarperl'ensenyamentcooperatiuieltreballperprojectesqueformadorscom Margarita Ravera, Lew Barnett i Núria Vidal van donar a conèixer. Des d'aleshores ha coordinat diferents projectes d'innovació educativa ORATOR, PELE), creat materials AICLE ciències, tecnologia, matemàtiques, educació física) i n'ha implementat amb ensenyament en tàndem. Actualment coordina un equip AICLE de professorsdematemàtiques,art,educaciófísicaiciènciessocialsdinselprojecteplurilingüedel'inssalvador EspriuBarcelona)iniciatelcurs2007a08. The World of Math Through English de Carlota Petit, Teresa Socias i Oriol Pallarés està subjecta a una llicència de Reconeixement-

4 ε Geometry surrounds us Platosaid, Geometryexistedbeforethecreation. TheworldhasdevelopedalotsincePlato s times.letthesepicturesinspireyouandawakenyourgeometriceye.

5 ε

6 ε

7 ε TEAMWORK Step 1: Observingrealitytoidentifygeometry 1. Lookbackatthepicturesandmark%the%following%dimensional%measuresonthem: Page1:markallthelengthsyoucanidentify Page2:dothesamewiththeareas Page3:nowmarkthevolumes 2. Make%a%list%of%all%the%measuresyouidentifiedinthedifferentpictures.Example: Picture1:theheightofatree/thelengthofaleaf... Picture2: Herearesomewordsyoumayneed: length height width distance area volume shape Size 3. Nowtrytoapproximate%the%measurementsyou vemarked.expressyouranswersusinga reasoningsimilarto: Asatreecanmeasure3%or%4%times%our%height,thetreeinthepicturecouldmeasure7m Asabottlecanbe2:3%times%the%size%of%a,thevolumeofthebottleinthepicturecan be Thefollowingtablewillhelpyou: m% dm% cm% hm% km% dam% m 2 % dm 2 % cm 2 % hm 2 % km 2 % dam 2 % m 3 % dm 3 % cm 3 % hm 3 % km 3 % dam 3 % % Notice%thatyousaymetersm),squaremetersm 2 )andcubicmetersm 3 ).

8 ε Organize%your%ideasinthefollowingtableuseadictionaryifnecessary): Image% 1% 2% 3% % What%can%you%see?% What%measurements% can%we%take?% Which%unit%would% you%use?% % % Step 2: MeasuringlengthsonGoogleEarth 1. Watch%the%videoMeasuring*Distances*with*Google*Earth*inthelinkbelow.Youwilllearnhow tousegoogleearthtomeasurelengthsusingthe Addpath and Showruler tools Havealookattheunits%of%lengthbelow.Findobjectsorplaceswhicharethislongorwhich arethesedistancesapartfromeachother.mark%the%lengthsonthemapandtake%a%snapshot ofit.write%a%descriptionfortheeachofthesnapshots. a) 1m d) 1km b) 1dam e) 10km c) 1hm f) 100km g) 1,000km The World of Math Through English de Carlota Petit, Teresa Socias i Oriol Pallarés està subjecta a una llicència de Reconeixement

9 ε Takealookatthesetwoexamples: ThismapshowsthedistancebetweenPlaça*de*les*Glòriesandthesea. Theyare2kmapart. Thismapshowsmyneighborhood. Thelengthofasidewalkinmyneighborhoodisalittlelessthanahm.

10 ε Step 3. Pairdescriptionofa2]Dfigure Student%A:Bringa2]Dfiguretoclass.Withoutshowingittoyourclassmate,describe%itto himorherfocusingonthepatterns%of%the%figure thenumberofsides,iftheseareequal ornot,theangles,etc.makesureyoudon tmentionanyimportantkeywords,otherwise yourpartnerwillfindtheguessingtooeasy. Student%B:Listencarefullytoyourpartner sdescriptionofafigureandtrytodraw%itas accuratelyaspossible.ifyoudon tunderstandhisorherindications,askforclarification orformorespecificdetails,forexample:is*this*side*over*the*square*you*mentioned* before?*are*these*two*sides*the$same*length? Finallycomparetheoriginalfigureandthedrawnone.Aretheyreasonablysimilar?Howarethey different? Intheoriginalfigure. Inthedrawnfigure. Thetreeissmaller Thetreeisbigger

11 ε PROJECT: Regeometrizing the World Introduction Haveyouevernoticedthatyourneighborhoodisfullofpolygons,areasandgeometricrelations? In this project you will learn to observe and analyze your neighborhood. You will first focus on somebuildings,objectsorplacestodescribetheminageometricway.then,youwillbeaskedto bring in your creativity and propose alternative geometric shapes for the previously studied objects.inotherwords,youwillbecomeareal%geometer Do%you%remember%that% %?% TheLatinprefixestoexpressdifferentordersofmeasurementare TheInternationalSystemofUnitsofmeasurementare Thebasicgeometricshapesare % Did%you%know%that% %?% As philosophers said, everything can be described geometrically. So from today until the end of thisprojectyouwillhavetobegoodobserversofyoursurroundings. Let s%start%then%% Toconvinceyouonceandforall,watchthevideoGeometryAllAroundUsbyAbbeyBarr,which willgiveyouveryinterestingexamplesofwhatwearetalkingabout.

12 ε TEAMWORK % % Step 1:Observingyoursurrounding 1. Eachgrouphastochooseandtakeapictureof: a) Abuilding. b) Adetail,suchasamailbox,astreetdrain,atree,atrafficsign,adrawingonthefloor,etc. c) Abigplace,suchasastadium,astreet,amarket,apublicsquare,etc. 2. Justifyyourchoices: Herearesomeideas: Thisbuildingisvery%simple/sophisticated Idon tthinkitisappropriate Thisdetailistoo%complicatedandhaslotsofirregularities Thisisa%good%picture%of because Makesurethatthepicturesaretakenfrom%a%good%perspectiveandthattheyareof%enough% qualitytobeanalyzed.onceyouhavechosenthem,writeadescriptionof: a) Whenandwhereyoutookthepictures. Thisisapictureofa...Itwastakenin Wetookthepicturelast.

13 ε b) Whatisinthepictures.Ifnecessary,lookforinformationaboutanyitemsinthepictures whichmaybeunknowntoyou. Thepictureshowsa Youcanseea Thereis/thereare Inthebackground/intheforeground/ontheright/ontheleft Step 2:Detectingthegeometricshapes 1. Studythepicturescarefully.Then,markallthegeometric%shapes%and%relationsyoucanseein them.beaspreciseaspossible. Lookattheexamplebelow.It sawellxknownhouseincanterbury,england.canyouseewhy ithasbecomesofamous?fillinthemissingnamesfortheshapesmarkedwithanarrow. Use%geometric%toolstodrawallthefiguresyoufindinyourpictures.Thennamethem.

14 ε Step 3:Deducingdimensions Finally,calculateapproximatelythedimensionsofeachofthethreefiguresyouhavestudied.You willhavetoestimatethedatayouwillneedtousetheformulas. Inthe%exampleabovewecouldsay: Thedoorofthehousemay$bearound2mhigh,andtheheightofthehouseis5$times$the$ height$of$the$door.therefore,theheightofthehousemust$be$approximately5:2=10m. Thewidthofthedoormay$be80cm, Step 4:Nowit stimeforyourcreativity. Itwouldbeveryinterestingtothinkabout: 1. Whythebuilding,detailandbigplaceyouhaveworkedonhavetheseshapesandnot others.isitaquestionofphysicsorfashion? 2. Whodecidestheshapesofobjects architects,engineers,mathematicians,physicists, designers? 3. Ifothershapesarepossible,trytoredesign, regeometrize,yourfigurestoshowother geometricoptions. Step 5:Shareyourcreationswiththeworld 1. Presentyourthreeregeometrizedobjectstoyourclassmates. 2. Yourpresentationmustincludetheoriginalpicturesaswellasthenewgeometricoptions. 3. Don tforgettojustifyyourdesignsandthedecisionsyoutook. 4. Useatoollikewww.glogster.comtopresentyourideas. 5. Then,makeaclasspinterestwww.pinterest.com)witheveryone screationsandvote 6. Rememberyoucanmakeyourglogstersandpinterestpublictosharethemwiththeworld.

15 ε REINFORCEMENT ACTIVITIES IntheGeometryProjectwetalkedaboutsomeconceptsthatwewillnowpractice,revieworlearn moreaboutwithafewexercisesandreflections. 1. Units of Measurement Sincethe1960stheInternational*Systemof*UnitsorSI,fromtheFrenchSystèmeInternational d Unités)hassetthestandardmetricsystemthateverybodyknowsandcanusetoday.Itoperates withthreemagnitudes: * Lengthunit:meterm Areaunit:squaremeterm 2 Volumeunit:cubicmeterm 3 Butdependingonthedimensionoftheobjectorthedistanceyouhavetomeasure,ametercould betoo*muchortoo*little.therefore,eachunitofmeasurementhasitsorder*of*units.theseare expressedwithlatinprefixes: mili%centi%deci%...%deca%hecto%kilo m/m 2 /m 3 How Does the Conversion of Units of Length Work? Steps: 1. Labelthefollowingdistancesontherulerbelow:

16 ε 2. Completetheconclusions: Conclusion:* 1m= dm= cm= mm 1m= dam= hm= km How Does the Conversion of Units of Area Work? Steps: 1. Drawasquarewithasideof1dm.Thisisadm 2.Dividethesidesofthesquareinto 10equalparts.Jointhepointsanddraw10verticallinesand10horizontallines. Howmanylittlesquaresarethere?Thesearecm Drawacm 2.Dividethesidesofthesquareinto10equalparts.Howmanylittle squaresarethere?thesearemm 2. Conclusion:* 1m 2 = dm 2 = cm 2 = mm 2 1m 2 = dam 2 = hm 2 = km 2 How Does the Conversion of Units of Volume Work? Steps: 1. Drawandcutoutapatternofacubicdm,thatis,a dm Beforepastingthestructuretogether,drawthe divisionsincmoneachfaceofthecubeasinthe picture. 3. Thenanswerthesequestions: Howmanycm 3 doesyourdm 3 haveatthebase? Howmanylayersdoesthedm 3 have? Howmanycm 3 doesthedm 3 thenhave?

17 ε LET S PRACTICE Now,practiceconvertingthefollowingmeasurestocm/dm 2 /dam 3 whereappropriate: km= m 2 = 3. 78,675km 3 = ,876mm 2 = mm= dm 3 = 7. 25hm 3 = km²= 9. 34,564.34dm= 10. 3dkm²= 2. Measuring Objects in the Classroom TEAMWORK Inyourteams,measureoneofthefollowingobjectsintheclassroom: 1. Theclassroomdoor 2. TheblackboardorIWB 3. Thebaseofachair 4. Theteacher stable 5. Astudent stable 6. Thecupboard 7. Awall 8. Awindow Youwillneedatape*measureandacalculator.Herearethesteps.Rememberthateachstudent willhaveoneofthefollowing4roles:

18 ε Roles: Student*1:Determinewhichpolygonyourobjectisandlookforthe correspondingformulaforitsarea. Student*2:Measuretheobjectanddictateitsmeasurestotherecorder. Student*3:Applytheformulawiththemeasuresoftheobjectanddictate theresultdisplayedonthecalculatortotherecorder Student*4:Drawtheobjecttogetherwithitsmeasuresonapieceofpaper. Whenthespeakertellsyou,writetheareaaswell. Ifyouneedalittlehelpandextrapracticewiththeformulasofpolygons,youcanvisit

19 ε EXTENSION ACTIVITIES 1. Area calculation practice Calculatetheareasofthefollowingcompoundfigures. b=6cm h=2.5cm The triangle is equilateral r=3cm s=5cm The missing corners are equilateraltriangles: s=1cm Hint: you can add the areas ofthetwotrianglestogether s=3cm Hint:thestarismadeup of 5 triangles and an inner pentagon 2. Experiment Isthereanyrelationshipbetweendiameterandcircumference? TEAMWORK Materials 1. 4 different cylindrical objects tins, cans, rolls, boxes, wastepaper bins ) of different diametersbutgreaterthan6cm). 2. Apieceofstringortapeofabout25cm. 3. Aruler. 4. Acalculator.

20 ε Steps 1. Drawtheapproximatecenterofeachofyourcylindersandmeasuretheirdiameters. 2. Measurethelengthofthecircumferenceofeachcylinderwithapieceofstring. 3. Writeallthemeasurementsintothegridbelow. 4. CalculatetheratioC/dineachcase. 5. Fillinthisgridwithallthedatafromtheothermembersofyourteam. Nameoftheobject Diameterd) Circumference perimeterc) 6. Comparetheratios.Aretheyallaroundacertainnumber? Yes,. No,. RatioC/d Noticethat Ifyouransweris no youshouldreviseyourmeasurements. What the preceding experiment shows is a quite exact truth: in any circle with circumference C and radius r the ratio is equal to the constant. So we can deduce the formula for the perimeterofthecircle: "$% = 2" Ifwehavetousethenumberinaformula,wewillusetheapproximation3.14. Conclusion Theratioofthelengthtothediameterofanycircumferenceis aroundthenumber =.Thisnumberiscalled pi.

21 ε 3. Deduction of some areas of polygons In math the why is just as important as the what. So we will now deduce the area of a parallelogramandweinviteyoutodeducetheotherformulas. Weneedthefollowingdata: h:height b:base Steps 1. Drawaparallelogramof15cmofbaseand8cmofheight.Cutitout. 2. Then,markitsheightononeofitsvertexes. 3. Youwillobtainatriangle.Cutitoutandplaceitontheoppositesideoftheparallelogram. Whatdoyouhavenow? Thenewfigurewehaveobtainedis Conclusion Tocalculatetheareaofanyparallelogramyouhavetocalculatethe areaoftherectanglewiththesame and. Writtenasaformula,weusetheletterbtorepresentthenumberofunitsorthelength)ofthe base,andtheletterhfortheheight.sotheformulais: ""$$%$'" = h

22 ε Assessment of/for/as learning Here$are$some$assessment$guidelines$and$tools:$$ $ 1. The Written Report Written$reports$may$be$short$as$a$one6minute$paper$or$as$long$as$you$want$them$to$be.$The$can$ also$be$open$or$guided$through$a$series$of$proposed$aspects$to$cover$or$questions$to$answer.$$ $ Here$are$some$ideas$to$help$your$students$write$a$written$report$in$English$if$they$can$or$in$ Catalan)$to$show$their$learning$achievements$after$the$project:$$Regeometrizing the World$ $ You$can$write$about$any$of$the$following$aspects:$$ $ 1) How$easy$or$difficult$it$has$been$for$you$to$find$and$decide$on$the$best$objects$for$a$ geometric$analysis.$ Ithasbeendifficulttochooseanobjectinourneighborhoodbecausewefoundalotof differentoptions/wecouldn tfindanyappropriateobjects 2) The$discoveries$you$have$made$about$your$neighborhood$that$have$surprised$you$the$ most.$ Oneofthemostsurprisingfactshasbeenthat 3) The$shape$that$you$have$found$most$frequently$in$your$neighborhood$when$looking$for$ interesting$objects$to$analyze.$ 4) The$strangest$shape$you$have$found$ $maybe$an$octagon,$a$trapezium,$a$semicircle $ 5) Your$conclusions$about$the$figures$you$have$worked$on.$Write$the$specific$results$you$have$ calculated$ $areas,$distances $ 6) About$measurements$in$general:$ a) Which$do$you$think$is$the$most$frequently$used$unit$for$calculating$distances$in$ your$neighborhood$ $for$example$from$your$house$to$your$school$or$to$a$friend s$or$ a$relative s$house,$or$from$one$end$of$your$neighborhood$to$the$other?$ b) Why$do$you$think$the$SI$International$System$of$Units)$is$important?$ 7) Other$aspects$you$have$discovered$or$learnt.$ 8) New$questions$that$have$arisen$or$aspects$you$would$like$to$learn$more$about.$ 9) Areas$you$need$to$study$more$or$improve$in.$ $

23 ε 2. A survey Surveys$will$allow$your$or$your$students$to$receive$feedback$from$their$peers.$Tools$like$ $ You$can$create$a$survey$together$with$your$students$so$everyone$can$vote$their$ regeometrized $ productions.$possible$questions:$$ $ 1) Which$figure$struck$you$the$most?$ 2) Which$proposal$is$the$most$creative?$ 3) Which$ regeometrized $figure$could$easily$exist?$ $ $$ 4) $

24 ε Assessment of/for/as learning $ $ End%of%unitLearningProgressChecklist $ In$this$unit,$we$have:$$ $ Learned$different$units$of$measurement$and$seen$which$is$better$in$each$situation$ Observed$pictures/reality$to$identify$geometric$elements,$shapes$and$volumes$trying$ to$deduce$their$real$dimensions$and$decided$which$unit$was$better$in$each$situation$ Learned$to$measure$lengths$and$distances$with$Google$Earth$ Described$2D$figures$number$of$sides,$length$of$sides,$angles)$orally$ Analyzed$the$function$of$different$kinds$of$shapes$and$figures$in$our$everyday$ environment$and$constructed$compound$figures$to$show$other$geometric$options$ Learned$different$units$of$measurement$length,$area,$volume)$of$the$SI$ International$System$Units)$and$their$conversion$tables$and$practiced$conversions$in$ different$cases$ Had$extra$practice$with$formulas$of$polygons$in$ Experimented$with$polygons$to$deduce$formulas$for$perimeters$and$areas$ Used$English$to$improve$our$fluency$ Worked$in$teams$to$analyze$geometrical$elements$in$real$life$and$showed$our$ reflections$in$a$glogster$and$class$pinterest$ Written$a$report$to$comment$on$our$analysis$of$geometry$in$our$surrounding,$and$ prepared$a$survey$using$ classamtes $regeometrized$productions$