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The World of Math Through English de Carlota Petit, Teresa Socias i Oriol Pallarés està subjecta a una llicència de Reconeixement-NoComercial-CompartirIgual 4.0 Internacional de Creative Commons !!!!!!Carlota'Petit'Olivella,'Oriol'Pallarés'Monge'i'Teresa'Socias'Garriga'''' The World of Math through English The Bases of Geometry

The World of Math Through English de Carlota Petit, Teresa Socias i Oriol Pallarés està subjecta a una llicència de Reconeixement-NoComercial-CompartirIgual 4.0 Internacional de Creative Commons !10 ideas and hints to help you succeed 1. You!must!observe,!identify!and!describe!geometrical!elements!in!nature!2. You!will!discuss!the!kinds!and!functions!of!logos!and!why!they!are!so!important!3. You!must!desi gn!a!logo!fo r!a!bilingual!project!u sin g!a!ruler!and!a!p air!of!compasses!or!any!digital!tool!such!as!Geogebra!4. You!will!lear n!specific!geo metry!vocabulary,!defi nitions,!conventions!and!you!must!classify!angles,!triangles!and!other!polygons!according!to!their!properties!5. You!must!draw! basic!geometr ic!elements!( points,!angle s,!parallel!lines,!perpendicular!bisectors,!angle!bisectors)!with!the!appropriate!instruments!6. You!must!deduce!and!apply!geometric!theorems!and!properties!7. You!will!use!your!body!to!represent!geometric!properties!8. You!must!use!English!as!much!as!possible!in!your!team!discussions!and!to!present!your!poster!and!logo!to!the!class!9. You!must!work!in!teams!providing!ideas,!discussing!properties!and!hypothesis,!inferring!formulas!to!learn!together!10. You!will!represent!basic!geometric!concepts,!properties!and!formulas!and!three!examples!of!geometry!in!nature!in!a!poster!to!show!to!the!class!and!comment!and!assess!your!classmates'!logos!

The World of Math Through English de Carlota Petit, Teresa Socias i Oriol Pallarés està subjecta a una llicència de Reconeixement-NoComercial-CompartirIgual 4.0 Internacional de Creative Commons !The world of Math through English ©!Carlota'Petit'Olivella,''Oriol'Pallarés'Monge'i'Teresa'Socias'Garriga'!Carlota!Petit!Olivella,!professora!de!matemàtiques!Oriol!Pallarés!Monge,!professor!d'anglès!Teresa!Socias!Garriga,!professora!d'anglès!!Correcció!lingüística:!Daniel!L.!Casacuberta!Fotografia:!Teresa!Socias!Garriga!Disseny:!Alejandro!Domínguez!Gonzàlez!i!Marc!Julià!Gil!!!Aquest!material!està!protegit!per!la!legislació!de!protecció!de!la!propietat!intelVlectual!i!el!seu!ús!indegut!pot!ser!objecte!de!sancions,!fins!i!tot!penals,!d'acord!amb!l'article!270!del!vigent!Codi!Penal.!!Les!fotografies!utilitzades!són!propietat!de!Teresa!Socias!Garriga!o!són!de!domini!públic.!Altrament,!se!n'indica!l'autor.!!,Els!autors:!,Carlota,Petit,Olivella,(cpetit@xtec.cat),,professora!de!matemàtiques!Va!iniciar!la!seva!carrera!professional!com!a!professora!de!matemàtiques!a!diferents!instituts!de!secundària!fins!que!el!curs!2006^2007!comença!juntament!amb!Teresa!Socias!i!Oriol!Pallarés!a!l'INS!Salvador!Espriu!de!Barcelona!un!projecte!AICLE.!Ha!creat!amb!ells!tots!els!materials!per!al!currículum!de!1r,!2n!i!3r!d'ESO!de!matemàtiques!amb!aquest!enfocament,!sempre!com!a!fruit!de!la!investigació,!reflexió!i!pràctica.!Des!del!curs!2007^2008!imparteix!aquests!cursos!en!anglès!amb!enfocament!AICLE!en!el!mateix!institut.!ColVlabora!amb!la!Universitat!Autònoma!de!Barcelona!impartint!un!taller!AICLE!en!el!màster!d'Educació!i!és!formadora!AICLE.!!Oriol,Pallarés,Monge,(opallare@xtec.cat),!professor!d'anglès,Va!iniciar!la!seva!carrera!professional!com!a!mestre!i!professor!de!música!i!d'anglès!en!l'ensenyament!primari!i!secundari!a!Catalunya.!Des!d'aleshores,!ha!treballat!en!escoles!bilingües,!ha!participat!en!diferents!programes!AICLE!(Aprenentatge!Integrat!de!Contingut!i!Llengua)!i!ha!exercit!com!a!professor!de!castellà!a!l'estat!de!Califòrnia!també!a!l'ensenyament!secundari.!En!l' actualitat,!és!professor!associat!del!De partament! de!Didàctica!de!la!Llengua!de !la!Universitat!Autònoma!de!Barcelona,!on!!també!és!membre!del!grup!de!recerca!col·laborativa!CLILCSI!sobre!AICLE,!professor!del!Departament!d'Ensenyament!de!la!Generalitat!de!Catalunya!a!les!escoles!oficials!d'idiomes!i!formador!del!professorat.!,Teresa,Socias,Garriga!(tsocias@xtec.cat),!professora!d'anglès!A!partir!d'un!curs!de!postgrau!de!l'ICE!de!la!UPC,!va!entrar!en!un!"Seminary!on!Content!Teaching"!de!l'ICE!de!la!UAB!(1991^94)!i!es!va!interessar!per!l'ensenyament!cooperatiu!i!el!treball!per!projectes!que!formadors!com!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!professors!de!matemàtiques,!art,!educació!física!i!ciències!socials!dins!el!projecte!plurilingüe!de!l'INS!Salvador!Espriu!(Barcelona)!iniciat!el!curs!2007^08.

! The World of Math Through English de Carlota Petit, Teresa Socias i Oriol Pallarés està subjecta a una llicència de Reconeixement-NoComercial-CompartirIgual 4.0 Internacional de Creative Commons !!δ!GEOMETRY IS EVERYWHERE! Let's&listen&to&an&extract&from&the&book&The$Man$Who$ Counted,&by&Malba&Tahan&(1990).&In&this&passage&the&man&who&counted,&Beremiz&Samir,&talks&about&geometry.&&&Did$you$know$that...$$The$Man$Who$ Counted&(original&Portuguese&title:&O$Homem$que$Calcu lava)&is &a&book&on&recreational&mathematics&and&curiou s&word&problems &by&Brazilian&writer&Júlio&César&de&Mello&e&Souza,&published&under&the&pen&name&Malba%Tahan.&Since&its&first&publication&in&1938,&the&book&has&been&immensely&popular&in&Brazil&and&abroad,¬&only&among&mathematics&teachers&but&among&the&general&public&as&well.&(from:&Wikipedia.&Last&accessed:&January&5,&2014).&$Cover$of$the$Catalan$version$of$the$book.&&& TEAMWORK!&&1. Listen&to&the&passage&from&the&book.&As&you&listen,&write&down&as&many&words&as&you&understand.&Then,&compare&with&yo ur&teammates&to&see&ho w&many&words&did&you& all&understand.&&&2. Each&team&is&given&an&envelope&with&different&pictures.&Distribute&the&pictures&among&all&the&members&of&the&team.&Listen&to&the&extract&again&and&pay&special&attention&to&the&images.&Put&them&in&the&order&they&appear&in&the&text.&&3. Look&at&the&following&images.&Can&you&see&any&points,&lines,&planes&and&figures&in&them?&&&hive&made&by&bees&&&&&&&&&&&&&&&&&&&&sand&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&leaf&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&butterfly&&4. Your&teacher&is&going&to&project&more&pictures.&Observe&them&and&identify&any&geometric&figures&and&elements&you&see&in&them.&Discuss&your&findings&with&your&teammates.&&The&flower&has...&In&this&picture,&I&can&see...&There&is&&a...&/&There&are...&

! The World of Math Through English de Carlota Petit, Teresa Socias i Oriol Pallarés està subjecta a una llicència de Reconeixement-NoComercial-CompartirIgual 4.0 Internacional de Creative Commons !!δ!!I&can&also&see...&5. Listen&to&the&fragment&from&the&book&again&to&check&whether&the&order&of&the&pictures&is&correct.&Do&you&agree&on& the&order ?&If¬,&disc uss&about&the&pic tures&that&are &more&difficult&to&place.&Hint:%You&can&send&a&student&from&your&team&to&"spy"&on&the&other&teams'&sequences&of&pictures&to&check&whether&the&order&is&the&same.&Then,&they&go&back&to&their&teams,&explain&the&different&options&and&decide&which&order&seems&better.&&6. Your&teacher&will&project&the&correct&sequence.&Make&sure&your&sequence&is&in&the&correct&order.&&&7. Now,&orally&and&in&teams,&answer&the&f ollowing &question&using&as&many&ideas&as&y ou&remember&from&Malba&Tahan's&story:&Where&can&you&find&geometry?&&Geometry$is$everywhere.$In$the...&&8. Finally,&write&your&own&text.&You&can&either&try&to&write&Malba&Tahan's&text&using&the&words&and&sentences&you&remember&and&adding&your&personal&touch&or&you&make&up&your&own&text&about&geometry&in&nature.&Give&it&an&original&title.&&&&&9. How&good&are&you&at&storytelling?&Take&your&text&home&and&read&it&to&your&family.&You&can&give&them&a&set&of&pictures&and&ask&them&to&put&them&in&order,&just&as&you&did&in&class;&this&will&help&them&understand&your&text.&Videotape&yourselves&being&a&storyteller.&You&can&later&share&your&videos&with&you&classmates&and&English&teacher.&& !

! The World of Math Through English de Carlota Petit, Teresa Socias i Oriol Pallarés està subjecta a una llicència de Reconeixement-NoComercial-CompartirIgual 4.0 Internacional de Creative Commons !!δ!Let's read in little more detail &10. Your&teacher&is& going&to&give&the&or iginal&pas sage.&Read&the&tex t&individually.&Then,&in&teams,&discuss&and&answer&the&following&questions.&&&a. What&geometric&shape&does&a&turban&have,&more&or&less?&&b. What&elements&in&nature&reveal&amazing&symmetries?&&c. What&geometric&figures&can&you&"see"&in&the&following&items?&Draw&them.&&• the&solar&disc&• a&butterfly&• a&diamond&• a&starfish&• a&grain&of&sand&&d. The&text&also&mentions&"a&raven&gently&flying&in&the&sky,&the&blood&running&through&a&camel's&veins&and&the&stone&thrown&at&the&annoying&jackal."&What&kind&of&lines&do&these&movements&describe?&Try&to&draw&them.&&e. Do&you&think&these&animals&-&the&raven,&the&camel&and&the&jackal&-&are&common&or&unusual&in&the&Persian&man's&everyday&life?&&f. According&to&the&text,&why&are&bees&expert&architects?&&g. "Geometry&is,&as&the&philosop her&said,&eve rywhe re.&But&we &need&eyes&to&see&it,&intelligence&to&understand&it,&and&soul&to&admire&it."&According&to&the&text,&does&the&author&think&that&having&eyes&is&essential&to&"identify&and&analyze"&geometric&figures&everywhere?&What&do&you&think&about&it?&&h. Do&you&think&that&the&Persian&man&believes&that&mathematicians&admire&geometry&in&nature?&&i. Who,&according&to&the&text,&admires&geometry&in&nature?&Why?&&j. Why&do&you&think&the&Persian&man&says&that&"God&was&the&great&geometer"?&

! The World of Math Through English de Carlota Petit, Teresa Socias i Oriol Pallarés està subjecta a una llicència de Reconeixement-NoComercial-CompartirIgual 4.0 Internacional de Creative Commons !!δ!Let's find out more &1. Look&for&a&short&biography&of&the&author&of&the&book&The$Man$Who$Counted$on&the&Internet&and&answer&this&question:&Why&do&you&think&he&is&interested&in&math?&2. Why&did&Malba&Tahan&choose&an&Arabian&man&to&explain&important&parts&of&mathematics?&3. Read&the&fol lowing&d efinition&from&www.mathwords.com&(last&accessed &on&March&21,&2008)&and&compare&this&mathematical&definition&with&the&explanation&of&the&Persian&about&geometry&in&the&text.&&&&&4. Look&at&this&picture&of&a&megalithic&monument,&the&Taula$de$Talatí$de$Dalt&(Minorca,&5th¢ury&B.C.).&It's&a&temp le&structure&shaped&lik e&an&enor mou s&T&made&of&gigantic &rectangular&stone&slabs.&This&taula&is&unique&because&of&the&slab&supporting&the&upright&stone&diagonally.&Three&different&planes&can&be&clearly&appreciated.&&&a. What&would&the&Persian&say&about&its&shapes?&&b. What&does&this&monument&tell&you&about&the&people&who&built&it&in&prehistoric×?&&&Photo:!Paul!Stephenson!Geometry:&(Greek&γεωμετρία;&geo&=&earth,&metria&=&measure)&is&a&part&of&mathematics&that&studies&geometric&figures&in&two&dimensions&(plane&geometry)&and&three&dimensions&(solid&geometry).&It&includes&the&study&of&points,& lines,&tr iangles,&quadrilaterals,&other&polygons,&circles,&spheres,&prisms,&pyramids,&cones,&cylinders,&and&polyhedra. www.mathwords.com, 21/3/08 !Notice%that...&This&definition&corresponds&to&classical&geometry:&Euclidean&geometry.&Among&the&various&types&of&geometry&are&analytic&geometry,&Euclidean&geometry&and&noniEuclidean&geometry,&projective&geometry,&algebraic&geometry...&

! The World of Math Through English de Carlota Petit, Teresa Socias i Oriol Pallarés està subjecta a una llicència de Reconeixement-NoComercial-CompartirIgual 4.0 Internacional de Creative Commons !!δ!Passage script: &"Don't$wonder,$my$friend,"$continued$the$intelligent$Persian,$"about$the$fact$that$I$want$to$see$turbans$with$geometric$shapes.$Geometry$is$everywhere.$Try$to$observe$the$regular$and$perfect$shapes$that$many$objects$present.$Flowers,$leaves,$and$a$lot$of$animals$reveal$amazing$symmetries$that$fill$our$soul$with$light."$$"Geometry,$I$insist,$is$everywhere.$In$the$solar$disc,$in$the$butterfly,$in$the$diamond,$in$the$starfish,$even$in$a$grain$of$sand.$There$is,$in$short,$an$infinite$variety$of$geometric$shapes$spread$in$Nature.$A$raven$gently$flying$in$the$sky$traces$wonderful$shapes$with$the$black$stain$of$its$body;$the$blood$running$within$a$camel's$veins$does$not$avoid$the$accurate$geometric$principles;$the$stone$thrown$to$the$annoying$jackal$traces$a$perfect$curve!$The$bee$builds$the$cells$of$its$hive$in$hexagonal$prisms$and$adopts$this$geometric$shape$to$take$the$most$advantage$of$its$building$material."$$"Geometry$is,$as$the$philosopher$said,$everywhere.$But$we$need$eyes$to$see$it,$intelligence$to$understand$it,$and$soul$to$admire$it."$$$"The$rustic$Bedou in$sees$the$geo metric$shapes,$but$he$doesn 't$understand$them; $the$Sunn ite$understands$them,$but$doesn't$admire$them;$the$artist,$finally,$foresees$the$perfection$in$shapes,$understands$Beauty$and$admires$Orde r$and$Harmony!$God$ was$the$great$ geometer.$He$geometrized$the$Earth$and$the$Sky."$$&&Did%you%know%that...&&Malba$Tahan$(Rio$de$Janeiro,$1895d1974)$was$a$pedagogue$and$a$mathematician$who$was$very$good$at$the$art$of$telling$stories.$$&His$book$t ells$Beremiz$S amir's$adventures$ facing$situations$where $his$mathematical$abili ties$are$needed.$His$wit$and$method$always$help$him$to$solve$them.$This$is$the$author's$excuse$to$present$a$collection$of$classical$problems$of$mathematical$entertainment$and$other$curiosities$(magic$squares,$perfect$numbers,$friendly $numbers,$the$chess$legend... )$and$to$show$his$admirat ion$for$classical$mathematicians,$especially$Arabian.&!

! The World of Math Through English de Carlota Petit, Teresa Socias i Oriol Pallarés està subjecta a una llicència de Reconeixement-NoComercial-CompartirIgual 4.0 Internacional de Creative Commons !!δ!PROJECT: DESIGNING A LOGO FOR YOUR SCHOOL !Introduction Do#you#like#logos?!Do#you#ever#pay#attention#to#the#logos#on#the#objects#you#use#every#day?#In#this#project#you#are#going#to#reflect#about#the#meaning!and#the!process!of!creation!of!a!logo,#and#you#will#design#one:#a. for#your#school's#plurilingual#project#b. using#geometric#constructions##Did you remember that... • ...a#straight#line#is...#• ...two#parallel#lines#are...#• ...a#right#angle#is...# Did you know that... • The#construction#of#geometric#figures#using#no!more!than!a!ruler!and!a!pair!of!compasses#is#a#fundamental#part#of#geometry.#We#know#that#some#constructions#can#be#drawn#this#way#-#and#also#that#others#cannot.#Designers#deal#with#these#constructions#every#day,#although#today#most!of!them!work!with!computers.!!• We#can#define#a#logo#as#a#symbol#that#represents#an#organization#or#a#company#and#that#is#used#in#advertisements#or#on#products#to#distinguish#this#organization#or#company#from#similar#ones,#and#to#help#people#remember#and#choose#it#instead#of#another.#A#logo#can#be:###1. A#graphic!symbol!or!icon#which#projects!an!idea#that#everybody#associates#with#the#organization#or#company#that#it#stands#for.#It#can#be:##a. A#representational!mark,#for#example#Apple,#The'Rolling'Stones,#Firefox,#Shell,#Forum'Barcelona'2004#b. An#abstract#mark,#like!Ferrocarrils'de'la'Generalitat,#Mitsubishi'Motors,#Mercedes'Benz,#Audi,#Consorci'del'Besòs,#London'Underground#

! The World of Math Through English de Carlota Petit, Teresa Socias i Oriol Pallarés està subjecta a una llicència de Reconeixement-NoComercial-CompartirIgual 4.0 Internacional de Creative Commons !!δ!2. Or#a#special#logotype#with:###a. #The#name#of#the#company#or#institution.#For#example,#everybody#can#recognize#the#special#logotypes#of#Facebook,'CocaFCola,'Canon#or#Harry'Potter'merchandising.###b. The#first!letter#of#the#word#or#an#acronym,#for#example#S#for#Superman,#e#for#Internet'Explorer,#the#V#and#W#of#Volkswagen,'007#for#James'Bond'merchandising#or#CNN#for#Cable'News'Network##3. The#combination#of#both:#Bristish'Petroleum,#Expo'Zaragoza'2008,#I'love'New'York...#### TEAMWORK!###Step 1: Analyzing Logos #Companies#use#their#logo#as#a#trademark#or#brand!for#people#to#identify#all#their#products#at#first#sight.#All#these#designs#are#registered#and#protected#by#law.##1. Go#to#Google#images#and#look#for#the#logos#of#five#of#the#organizations#mentioned#here.#Print#one#of#each#type.##2. What#kind#of#logos#are#these?#Representational,#abstract#marks#or#logotypes?#Comment#on#the#idea#or#meaning#of#each#one.##There#are#many#logos#around#us#-#in#the#supermarket,#in#books,#on#TV...#Almost#each#product#or#concept#has#its#own#logo#to#identify#it.#Let's#take#the#Ferrocarrils'de'la'Generalitat#logo#as#an#example:###

! The World of Math Through English de Carlota Petit, Teresa Socias i Oriol Pallarés està subjecta a una llicència de Reconeixement-NoComercial-CompartirIgual 4.0 Internacional de Creative Commons !!δ!a. Describe#the#logo#geometrically.#b. Draw#it#using#a#ruler#and#a#pair#of#compasses.#c. Write#the#steps#you#have#followed#to#draw#it.#Be#as#precise#as#in#the#examples#seen#in#section#8#of#Extension#or#section#7#of#Reinforcement.###Step 2: Designing A Logo For Your School Your#school#is#taking#part#in#a#plurilingual#project#where#students#use#English#and/or#other#languages#in#some#subjects,#and#it#needs#a#logo#to#identify#this#project#and#the#school.#You!are!going!to!be!the!designers!###Follow#these#instructions:####Once#the#teacher#has#all#your#designs,#the#class#or#the#school#can#vote#for#the#most#appropriate#logo#for#the#school's!plurilingual!project,#which#involves#teaching#different#subjects#in#different#languages.#And#-#who#knows?#-#this#might#be#the#beginning#of#a#successful#career#as#a#graphic!designer.##Notice that...##Nowadays,#most#professionals#who#use#geometry#in#their#jobs#don't#use#physical#geometry#tools#but#digital#ones.#So,#of#course,#you#can#design#your#logo#with#a#geometry#software#like#Geogebra,'Paint#or#any#other#you#come#across.##!Instructions:!1.Use#a#DIN#A4#sheet#to#draw#your#design.#Do#not#write#anything#else#on#it.#2.All#the#lines#must#be#drawn#using#a#ruler#and#a#pair#of#compasses.#3.As#you#design#the#logo,#think#of#some#kind#of#connection#with#your#school's#plurilingual#project.##4.Write#all#the#steps#you#follow#to#draw#your#logo.#5.You#must#color#it.#Think#of#the#most#suitable#colors.#6.Explain#the#idea#that#inspired#your#logo,#why#you#have#chosen#your#geometric#construction#and#what#it#means.!By!the!way!!This!is!the!LOGO!of!the!Geogebra!program!!

! The World of Math Through English de Carlota Petit, Teresa Socias i Oriol Pallarés està subjecta a una llicència de Reconeixement-NoComercial-CompartirIgual 4.0 Internacional de Creative Commons !!δ!#The#Geogebra#software##Here#are#some#examples#of#logos#designed#by#students#using#digital#tools:##########Marc#Castelló,#2008###############Albert#València,#2009##And#by#hand:####################Irene#Segarra,#2009##################Oriol#Carreras,#2009#

! The World of Math Through English de Carlota Petit, Teresa Socias i Oriol Pallarés està subjecta a una llicència de Reconeixement-NoComercial-CompartirIgual 4.0 Internacional de Creative Commons !!δ!REINFORCEMENT ACTIVITIES Let's&now&practice,&review&and&learn&more&about&geometry. 1. Warming Up: Basic Vocabulary &Match&each&word&with&its&corresponding&definition.&&vertices(&A&__________&is&a&straight&path&that&extends&without&end&in&opposite&directions.&vertex(&When&two&__________&lines&cross&at&a&point,&they&are&said&to&__________.&equidistant(&The&point&at&which&they&cross&is&called&the&__________.&straight(&A&__________&is&a&line&that&has&one&endpoint&(or&origin&point,&in&Catalan).¶llel(&A&line&with&endpoints&A&and&B#is&called&the&line&__________&AB.&polygon(&__________&lines&are&the&same&distance&apart&all&along&their&length.&sides(&When&two&lines&are&equidistant&and&never&cross,&they&are&called&__________&lines.&intersect(&When&two&line&segments&meet&at&a&point,&the&measure&of&rotation&of&one&of&the&line&segments&to&the&other&is&called&the&....................&.&ray(&An&angle&is&formed&by&two&rays&with&a&common&endpoint&that&is&called&__________&.&line(&An&angle&that&measures&90°&is&called&a&__________&angle.&plane(&When&two&lines&intersect&to&form&right&angles&they&are&__________&lines.&

! The World of Math Through English de Carlota Petit, Teresa Socias i Oriol Pallarés està subjecta a una llicència de Reconeixement-NoComercial-CompartirIgual 4.0 Internacional de Creative Commons !!δ!right(&A&__________&is&a&2HD&shape&made&from&line&segments&on&the&same&flat&surface.&arc(&The&line&segments&of&a&polygon&are&called&the&__________.&The&endpoints&of&these&sides&are&called&__________.&intersection(&An&__________&is&a&part&of&a&circle.&&(2. Conventions For Labeling Lines, Angles And Shapes &Look&at&the&pictures&below&and&complete&the&following&sentences.&(1. Points&are&always&named&with&__________&2. Lines&are&always&named&with&__________&3. Line&segments&are&named&with&__________&4. Angles&are&named&__________&or&__________&or&__________&5. A&right&angle&is&expressed&with&a&__________&6. A&vertex&is&named&with&a&__________&&&&&&&&&&&&&(The&conventions&for&labeling(triangles&are&as&follows:&&• The&vertices&are&labeled&using&capital(letters,&going(round(in(order&(clockwise&or&anticlockwise).&&&• The&corresponding&lower7case&letter&is&given&to&the(side(opposite(each(vertex.&&((

! The World of Math Through English de Carlota Petit, Teresa Socias i Oriol Pallarés està subjecta a una llicència de Reconeixement-NoComercial-CompartirIgual 4.0 Internacional de Creative Commons !!δ!!! LET'S PRACTICE Following&the&conventions&we&have&just&seen,&label&the&following&triangles.!(((((((((((((((((((((((((((3. Intersecting Parallel Lines &Observe&this&special&construction&and&do&the&exercises&below:&((((((1. Name&the&lines:&the&orange&one&is&l,&the&blue&one&is&m&and&the&green&one&is&n.&2. Draw&the&eight&angles&formed&by&the&lines&and&name&them&!,&!,&!,&!,&!,&!,&!&and&!.(3. What&is&the&name&of&the&two¶llel&lines?(4. And&name&of&the&third&line?(5. Which&angles&are&equal?(6. Which&angles&are&supplementary?((Look&at&the&figure&below.&Name&all&its&angles.&Without&using&the(protractor,&say&which&angles&are&equal&and&why.&Hint:(Can&you&see&any¶llel&lines?(&&

! The World of Math Through English de Carlota Petit, Teresa Socias i Oriol Pallarés està subjecta a una llicència de Reconeixement-NoComercial-CompartirIgual 4.0 Internacional de Creative Commons !!δ!4. Drawing Basic Geometric Figures In&geometry&it&is&very&important&to&understand&and&follow&the&steps&indicated,&and&also&to&be&very&careful&and&precise.&For&example,&a&measurement&with&a&pair(of(compasses&is&more&precise&than&one&with&a&protractor(or&a&ruler.&First&of&all,&let's&present&the&instruments&we&need.&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&Photo:&Luigi&Chiesa&&&&&&&&&&&&&&&&&&&&&&&&Foto:&Luigi&Chiesa&&&&&&&&&&Look&up&their&names&in&a&dictionary.&Now&we&are&ready&to&draw&basic(geometric(constructions.&Notice(that&a&set&square&is&called&a&triangle&in&American&English.&&!Now,(try(to(draw(the(following(geometric(constructions((Construction #1.!Drawing!a"point:!We!can!draw!it!as!the!intersection!of!two!very!small!line!segments.!"

! The World of Math Through English de Carlota Petit, Teresa Socias i Oriol Pallarés està subjecta a una llicència de Reconeixement-NoComercial-CompartirIgual 4.0 Internacional de Creative Commons !!δ!Construction #2.#Drawing#an#angle#with#a#protractor:#• Draw&a&ray&with&O&as&an&endpoint.&• Place&the&protractor&in&such&a&way&that&its¢er&corresponds&to&the&endpoint.&• Draw&a&little&mark&on&the°ree&of&the&angle&you&want&to&draw.&• Join&point&O&and&the&little&mark&using&your&ruler.&• Mark&the&angle&with&an&arc.&&&&&&&&&&&&&&&Construction #3.#Drawing#parallel!lines#with#the#ruler#and#the#triangle.&• Place&the&triangle&on&the&line&drawn.&• Place&the&ruler&along&one&of&the&edges&that&form&the&right&angle&of&the&triangle.&• Move&the&triangle&by&sliding&it&along&the&ruler.&Stop&at&the&point&where&you&want&to&draw&a¶llel&line.!!&&&&&&&&&&&&&&&&&&&Parallel#and#perpendicular#lines#can#be#found#everywhere.&

! The World of Math Through English de Carlota Petit, Teresa Socias i Oriol Pallarés està subjecta a una llicència de Reconeixement-NoComercial-CompartirIgual 4.0 Internacional de Creative Commons !!δ!Construction #4.#Drawing#the#perpendicular!bisector#of#a#line#segment#AB:#• Draw&an&arc&with¢er&at&A&and&an&opening&greater&than&half&the&segment.&• Do&the&same&with&B&as&a¢er&-&the&opening&must&be&the&same&as&in&the&first&step.&• Two&points,&C&and&D,&appear.&The&line&going&through&C&and&D&is&the&perpendicular&bisector&of&AB.&Notice(that...&the&Catalan&name&for&the&perpendicular(bisector&of&a&line&segment&is&mediatriu.&&Once&you&have&drawn&this&construction,&complete&the&following&definition:&&&Construction #5.&Drawing#an#angle!bisector:#• Draw&an&arc&with¢er&at&vertex&A&of&the&angle.&• You&will&get&two&new&points,&B&and&C.&• With¢er&at&B&and&a&suitable&opening,&draw&an&arc&within&the&angle&and&do&the&same&for¢er&C&(with&the&same&opening).&• Both&arcs&must&cross&each&other&at&one&point,&D.&• The&ray&AD&is&the&angle&bisector.&&After&you&have&drawn&this&construction,&complete&the&following&definition:&&& 5. Classifying Geometric Objects In&math&-&just&like&in&science&in&general&-&it&is&very&important&to&make&appropriate&classifications.&These&provide&an&easier,&more&precise&way&for&mathematicians&to&communicate&with&each&other.&Let's&look&at&the&criteria&for&classifying&angles,&triangles,&and&quadrilaterals.&( Definition:(&A&perpendicular&bisector&of&a&segment&is&a&ray&______________________________________&.&Definition:(&An&angle&bisector&is&a&ray&_______________________________________________________&.&

! The World of Math Through English de Carlota Petit, Teresa Socias i Oriol Pallarés està subjecta a una llicència de Reconeixement-NoComercial-CompartirIgual 4.0 Internacional de Creative Commons !!δ!Classifying Angles &Angles&are&classified&according&to&their&measurements:&(• An´(angle&measures&less(than(90°.&• A&right(angle&measures&90°.&• An&obtuse(angle&measures&more(than(90°&and&less(than(180°.&• A&straight(angle&measures&exactly(180°.&&!&&Classifying Triangles &a) According&to&their&sides&they&are:&equilateral:&..........................&scalene:&..........................&isosceles:&..........................&b) According&to&their&angles&they&are:&right:&..........................&obtuse:&..........................´:&..........................& ACTING OUT MATH: &Four&students&in&the&class&will&model&the&four&kinds&of&angles,&and&another&student&will&model&angle&0°.&The&rest&of&the&class&has&to&guess&which&is&which.&!

! The World of Math Through English de Carlota Petit, Teresa Socias i Oriol Pallarés està subjecta a una llicència de Reconeixement-NoComercial-CompartirIgual 4.0 Internacional de Creative Commons !!δ! THINK AND DISCUSS.!Reason&whether&the&following&sentences&are&possible&or¬:!!&a) A&triangle&can&have&more&than&one&right&angle.&b) A&right&triangle&can&be&scalene.&c) An&equilateral&triangle&can&be´.&d) An&equilateral&triangle&can&be&obtuse.&&6. Analyzing Shapes a) Describe&the&figure&in&the&p icture&-&a&de tail&of&a&church&in&Aragó&Street,&Barcelona.&&b) Now&draw&it&usin g&the& ruler&and&the&pair&of &compasses.&The&circumferences&on&your&drawing&have&to&be&of&3cm&of&radius.&&&7. Analyzing Shapes Made With Your Body Take&a&picture&of&your&body&in&a&specific&posture.&Then,&draw&it&and&analyze&all&the&angles&you&can&see&in&it.&&

! The World of Math Through English de Carlota Petit, Teresa Socias i Oriol Pallarés està subjecta a una llicència de Reconeixement-NoComercial-CompartirIgual 4.0 Internacional de Creative Commons !!δ!EXTENSION ACTIVITIES 1. Eratosthenes. The Diameter Of The Earth Do#you#remember#what#method#Eratosthenes#devised#to#determine#prime&numbers?##Now#we#are#go ing#to#stu dy#another# of#his#relevant#inv entions .#Eratosthenes#(c .#194#B.C.#in#Alexandria,#Egypt)#was#the#first#pe rson#to#calculate#the#Earth's#diame ter#quite#ap proximately#without#satellites#or#telescopes!#He#based#his#proof#upon#two#hypotheses:##• The#Earth#is#a#perfect#sphere.##• The#sunrays#which#reach#the#Earth's#surface#are#all#parallel#(that#is,#he#supposed#that#the#sun#was#far#enough#from#the#Earth#for#that#statement#to#be#true).# Eratosthenes#saw#that#the#inclination#of#the#sun#at#the#summer#solstice#(21st#June)#was#null#at#noon#(12:00#at#midday,#solar#time)#in#Syene#(now#Aswan),#in#the#south#of#Egypt,#and#that#the#inclination#of#the#sun#in#Alexand ria#at#the#sam e#moment#w as#7.2º#degr ees.#Ther efore,#the#curvature#of#the#Earth#between#those#two#cities#described#an#arc.#As#the#distance#from#Alexandria#to#Syene#was#794#km,#he#was#able#to#easily#calculate#the#diameter#of#the#Earth#-#assuming#that#it#was#a#sphere.###Eratosthenes#applied#a#rule#of#three#to#these#data#and#came#up#with#the#value#of#4,000#km#in#diameter.###

! The World of Math Through English de Carlota Petit, Teresa Socias i Oriol Pallarés està subjecta a una llicència de Reconeixement-NoComercial-CompartirIgual 4.0 Internacional de Creative Commons !!δ!!1. Draw#a#picture# wi th#all#the#data#mentioned#and #mark#the#paral lel#lines #which#ar e#intersected#by#another#line.##2. After#that,#write#th e#rule#of #three#that#Eratosthe nes#follo wed#to#calculate#th e#Earth's#diameter.##Source:#http://outreach.as.utexas.edu/marykay/assignments/eratos1.html#(last&accessed&on&January&5,&2014)& 2. Think About... If#we#draw#the#angle&bisector#of#an#angle#of#89º,#how#much#will#the#two#new#angles#measure?# 3. Drawing##Follow#the#instructions#above,#answer#the#final#question#and#reason#it!##• Draw#a#point#B#wherever#you#want#on#a#piece#of#paper.##• Draw#a#circumference#with#a#radius#of#4cm#and#center#at#B.##• Draw#3#different#lines#so#that:#i. one#of#them,#l,#does#not#intersect#the#circumference#at#all.##ii. the#second#line,#l',#intersects#the#circumference#just#in#one#point#(that#is,#the#line#is#tangent#to#the#circumference).#iii. the#third,#l",#intersects#the#circumference#in#two#points.##• Can#you#draw#a#line#intersecting#the#circumference#in#more#than#two#points?##Hint:#The#point#of#tangency#is#the#point#of#intersection#between#a#circle#and#its#tangency#line#or#tangent#segment.####!!!

! The World of Math Through English de Carlota Petit, Teresa Socias i Oriol Pallarés està subjecta a una llicència de Reconeixement-NoComercial-CompartirIgual 4.0 Internacional de Creative Commons !!δ!!4. The Triangle Inequality Theorem THINK AND DISCUSS!##Objective:#What#is#the#relation#between#the&sum&of&two&sides#of#a#triangle#and#the&third&one?###Step 1.#Each#member#of#the#team#draws#a&different&kind&of&triangle#(see#previous#page)#and#completes#his/her#chart#with#the#information#corresponding#to#his/her#triangle.##After#that,#dictate#your#results#to#the#rest#of#the#team#and#complete#the#other#charts.##Student&1's&triangle&Sum&Third&side&Compare&the&sum&with&the&third&side&a+b&&&&&c####b+c&&&&&a####a+c&&&&&b####Student&2's&triangle&Sum&Third&side&Compare&the&sum&with&the&third&side&a+b&&&&&c####b+c&&&&&a####a+c&&&&&b####Student&3's&triangle&Sum&Third&side&Compare&the&sum&with&the&third&side&a+b&&&&&c####b+c&&&&&a####a+c&&&&&b####Student&4's&triangle&Sum&Third&side&Compare&the&sum&with&the&third&side&a+b&&&&&c####b+c&&&&&a####a+c&&&&&b#####Step 2.#Analyze#the#results#in#the#chart.#Look#at#the#fourth#column.#Can#you#see#any#similarity#between#the#different#triangles?###

! The World of Math Through English de Carlota Petit, Teresa Socias i Oriol Pallarés està subjecta a una llicència de Reconeixement-NoComercial-CompartirIgual 4.0 Internacional de Creative Commons !!δ!##Step 3.#State#the#conclusion:###! LET'S PRACTICE ##Using#this#result,#deduce#whether#the#following#triangles#are#possible:##a) a&=&5cm,&b&=&6cm,&c&=&13cm&b) a&=&20cm,&b&=&15cm,&c&=&18cm&##5. Guessing A Shape #Listen#to#and#follow#your#teacher's#instructions.# a) Draw#a#right#triangle#ABC#with#a#blue#pen#so#that#angle#B#is#the#right#angle.##b) After#that,#with#a#red#pen,#draw#a#line#parallel#to#segment#AB#going#through#vertex#C.#c) Then,#also#with#a#red#pen,#draw#another#line#parallel#to#BC#going#through#vertex#A.##d) Finally,#still#with#a#red#pen,#draw#a#line#parallel#to#CA#going#through#vertex#B.## What#shape#do#you#get#in#red? Language Tip.#These#are#possible#questions#you#can#ask#in#your#discussion.#Take#notes.##•Look&at&the&fourth&column.&Can&you&see&any&similarity&between&the&different&triangles?#•Is&it&possible&to&draw&a&triangle&of&sides&3,&4,&and&9&cm?&Why&(not)?#•What&does&a&triangle&of&4,&4,&and&8&cm&look&like?&What&are&its&angles?#•Let's#assume#that#vertex#A#is#your#home,#B#is#your#school,#and#C#is#a#park.#What#is#the#shortest#way#to#go#from#your#home#to#school?##!Conclusion&#The#triangle#inequality#says#that#for#any#triangle#the#sum#of#two#sides#of#this#triangle#is#always#..................#than#the#third#side.#Or,#in#other#words,#the#measure#of#a#given#side#must#be#..................#than#the#sum#of#the#other#two#sides.#!

! The World of Math Through English de Carlota Petit, Teresa Socias i Oriol Pallarés està subjecta a una llicència de Reconeixement-NoComercial-CompartirIgual 4.0 Internacional de Creative Commons !!δ!&6. Essential Geometric Constructions. In#this#section#we#will#practice#some#wellaknown#constructions#that#can#be#made#with#just#a#pair#of#compasses#and#a#ruler.#Look#for#the#construction#steps#on#the#Internet:#&• http://www.acanoumea.nc/maths/amc/polyhedr/polyg_draw_.htm#• http://en.wikipedia.org/wiki/Pentagon#Construction#• ...#&a) Drawing&a&circle#passing#through#three#fixed#points,#A,#B#and#C.#b) Construction#of#a®ular&pentagon.#On#http://en.wikipedia.org/wiki/Pentagon#Construction#you#have#an#ani mation#of#the# construction#of#a#regular#pentagon.#c) Construction#of#a®ular&hexagon.#On#this#page#http://www.acanoumea.nc/maths/amc/polyhedr/polyg_draw_.htm#there#are#different#constructions#of#regular#polygons.###7. Deducing A Formula In#this#section#we#are#going#to#prove#a#property#that#you#already#learned#in#primary#school.#We#are#going#to#relate#the#sum#of#the#angles#of#a#polygon#to#the#number#of#sides#of#that#polygon.#To#deduce#the#general#result,#we#will#follow#five#steps.##Step 1.#Three(sided#polygons:#Triangles#• Do#you#remember#how#much#the#three#angles#of#a#triangle#add#up#to?##• Measure#the#angles#of#these#triangles#and#calculate#their#sums.###

! The World of Math Through English de Carlota Petit, Teresa Socias i Oriol Pallarés està subjecta a una llicència de Reconeixement-NoComercial-CompartirIgual 4.0 Internacional de Creative Commons !!δ!It#se ems#that#the#sum#of#the #three#angles#of #a#triangle#equals________.#But,# as#young#mathematicians,#you#must#make#sure#that#this#statement#is#always#true.#Checking#a#property#with#only#three#triangles#is#not#enough.#Let's&prove&it&with&a&picture.#### TEAMWORK!##1. Each#member#of#the#team#draws#a#different#type#of#triangle.#2. Label#the#angles#and#the#sides#as#in#the#drawing#below.#3. Draw#a#line#parallel#to#side#c#of#the#triangle#so#that#it#passes#through#the#opposite#vertex.#4. You#will#obtain#3#new#adjacent#angles#that#you#can#name#A,#B#and#C.#Can#you#reason#why#they#can#be#named#A,#B#and#C?#5. Draw#conclusions#about#the#three#angles#of#your#triangle.##After#the#construction,#we#have#obtained#____________________###Now#that#you#have#proved#this,#we#can#state#the#following#property:#&##Step 2.#Four(sided#polygons:#Quadrilaterals#How#can#we#find#the#sum#of#the#angles#of#any#quadrilateral?#As#mathematicians#often#do,#let's#change#the#situation#so#that#we#can#use#already#known#results.##Conclusion&#The#sum#of#the#three#angles#of#any#triangle#always#equals#_________.#!

! The World of Math Through English de Carlota Petit, Teresa Socias i Oriol Pallarés està subjecta a una llicència de Reconeixement-NoComercial-CompartirIgual 4.0 Internacional de Creative Commons !!δ!####This#method#of#dividing#a#polygon#into#triangles#is#called#triangulation.#Now#we#can#mark#the#6#angles#in#the#two#triangles#we#have#obtained#and#say#how#much#they#will#add#up#to#-#by#applying#the#property#of#triangles#that#we#have#just#proved.#That#is:#####Step 3.#Five(sided#polygons:&Pentagons##Now#we#can#use#the#method#of#triangulation#of#a#polygon#again.#Write#your#own#explanation.#• First,!I!divide!the!pentagon!into!________!triangles.!!• Second,!I!count!________!!• Then,!I!apply!the!property!of!triangles!________!!• After!that,!________#!!##Step 4.#Six(sided#polygons:#Hexagons#The#same#triangulation&method#again...#Conclusion&#The#sum#of#the#four#angles#of#any#quadrilateral#always#equals!_________.#!Conclusion&#The!sum!of!the!five!angles!of!any!pentagon!always!equals!_________.#!

! The World of Math Through English de Carlota Petit, Teresa Socias i Oriol Pallarés està subjecta a una llicència de Reconeixement-NoComercial-CompartirIgual 4.0 Internacional de Creative Commons !!δ!Write#your#own#explanation.#• First!...!!• !!• !!• #!###Step 5.#Polygons#of#n&sides:#The#general#case#This#property#of#the#sum#of#the#angles#of#a#polygon#seems#to#work#well!#So#it#may#be#easy#to#find#a#formula&that&calculates&the&sum&of&all&the&angles&of&a&polygon&according&to&the&number&of&sides.#Let's#summarize#the#results#we#have#so#far.#Complete#this#table.##Number#of#sides&of#the#polygon&Number#of#triangles&in#the#triangulation&Sum#of#the#angles&3&&180º&4&&360º&=&2&Q&180º&5&&____&=&__&Q&180º&6&&&7&&&...&...&...&n&&___&=&___&Q&180º&!! LET'S PRACTICE You#have#deduced#a#formula#that#is#used#to#find#the#sum#of#all#the#angles#of#any#polygon#according#to#the#number#of#sides#of#the#polygon.#Now#use#the#formula#to#calculate:#Conclusion&#The!sum!of!the!six!angles!of!any!hexagon!always!equals!_________.#!

! The World of Math Through English de Carlota Petit, Teresa Socias i Oriol Pallarés està subjecta a una llicència de Reconeixement-NoComercial-CompartirIgual 4.0 Internacional de Creative Commons !!δ!#a) the#angles#of#a#heptagon#if#it#is#regular#-#that#is,#if#all#its#angles#are#equal.#b) the#angles#of#a#regular#decagon.#c) the#angles#of#a#regular#15asided#polygon.#&!Final&Conclusion&#The#sum#of#the#angles#of#any#polygon#always#equals:#(n#-&2)&Q&180.#Where#n#is#the#number#of#sides#of#the#polygon.#

! The World of Math Through English de Carlota Petit, Teresa Socias i Oriol Pallarés està subjecta a una llicència de Reconeixement-NoComercial-CompartirIgual 4.0 Internacional de Creative Commons !!δ!Assessment of/for/as learning !1. A potser !To!make!sure!that!you!have!understood!what!we!have!done!these!days,!you!have!to!design!a!poster!in!teams!for!all!the!new!concepts!to!be!displayed!and!connected.!!!As!the!main!objective!of!this!unit!was!to!design!a!logo,!this!can!be!the!center!of!the!poster!with!all!the!other!concepts!we!have!learned!around!it:!!!1) Basic!geometric!concepts!!• parallel!and!perpendicular!lines!• angles!• classification!of!polygons!• how!to!label!figures!• ...!!2) History!or!general!concepts!of!geometry.!!3) What!a!logo!is!and!why!it!is!useful.!!!4) The!purpose!and!the!design!process!of!your!logo.!!5) Geometry!in!nature:!a!few!examples!and!your!opinions.!!!6) ...!!!

! The World of Math Through English de Carlota Petit, Teresa Socias i Oriol Pallarés està subjecta a una llicència de Reconeixement-NoComercial-CompartirIgual 4.0 Internacional de Creative Commons !!δ!Assessment of/for/as learning !!End$of$unit*Learning*Progress*Checklist*!In!this!unit,!we!have:!!!! Observed,!identified!and!described!geometrical!elements!in!nature!! Discussed!the!kinds!and!functions!of!logos!and!why!they!are!so!important!! Designed!a!logo!for!a!bilingual!project!using!a!ruler!and!a!pair!of!compasses!or!any!digital!tool!such!as!Geogebra!! Learned!specific!geometry!vocabulary,!definitions,!conventions!and!classified!angles,!triangles!and!other!polygons!according!to!their!properties!! Drawn!basic!geometric!elements!(points,!angles,!parallel!lines,!perpendicular!bisectors,!angle!bisectors)!with!the!appropriate!instruments!! Deduced!and!applied!geometric!theorems!and!properties!! Used!your!body!to!represent!geometric!properties!! Used!English!as!much!as!possible!in!your!team!discussions!and!to!present!your!poster!and!logo!to!the!class!! Worked!in!teams!providing!ideas,!discussing!properties!and!hypothesis,!inferring!formulas!to!learn!together!! Represented!basic!geometric!concepts,!properties!and!formulas!and!examples!of!geometry!in!nature!in!a!poster!to!show!to!the!class!and!commented!and!assessed!your!or!our!classmates'!logos.!!!!!

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