Grade 5 Mathematics. Support Document for Teachers

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1 Grade 5 Mathematics Support Documet for Teachers

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3 G RADE 5 MATHEMATICS Support Documet for Teachers 2009 Maitoba Educatio

4 Maitoba Educatio Cataloguig i Publicatio Data Grade 5 mathematics : support documet for teachers Icludes bibliographical refereces ISBN-13: Mathematics Study ad teachig (Elemetary). 2. Mathematics Study ad teachig (Elemetary) Maitoba. I. Maitoba. Maitoba Educatio II. Title: Grade five mathematics : support documet for teachers. Copyright 2009, the Govermet of Maitoba, represeted by the Miister of Educatio. Maitoba Educatio School Programs Divisio Wiipeg, Maitoba, Caada Every effort has bee made to ackowledge origial sources ad to comply with copyright law. If cases are idetified where this has ot bee doe, please otify Maitoba Educatio. Errors or omissios will be corrected i a future editio. Sicere thaks to the authors ad publishers who allowed their origial material to be used. All images foud i this documet are copyright protected ad should ot be extracted, accessed, or reproduced for ay purpose other tha for their iteded educatioal use i this documet. Ay websites refereced i this documet are subject to chage. Educators are advised to preview ad evaluate websites ad olie resources before recommedig them for studet use. Copies of this resource ca be purchased from the Maitoba Text Book Bureau (stock umber [prit] ad stock umber [electroic]). Order olie at < This resource is also available o the Maitoba Educatio website at < Websites are subject to chage without otice.

5 C ONTENTS Itroductio 1 Number 1 Patters ad Relatios 1 Patters 1 Variables ad Equatios 1 Shape ad Space 1 Measuremet 1 3-D Objects ad 2-D Shapes 1 Trasformatios 1 Statistics ad Probability 1 Data Aalysis 1 Chace ad Ucertaity 1 Bibliography 1 Cotets iii

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7 A CKNOWLEDGEMENTS Maitoba Educatio wishes to ackowledge the cotributio of ad to thak the members of the Grade 5 to Grade 8 Mathematics Support Documet Developmet Team. Their dedicatio ad hard work have made this documet possible. Writer Betty Johs Uiversity of Maitoba Grade 5 to Grade 8 Mathematics Support Documet Developmet Team Maitoba Educatio School Programs Divisio Staff Heidi Holst Lida Girlig Darlee Willetts Holly Forsyth Chris Harbeck Steve Hut Ja Jebse Diaa Kiceko Kelly Kuzyk Judy Maryiuk Greg Sawatzky Heather Aderso Cosultat (util Jue 2007) Carole Bilyk Project Maager Lee-Ila Bothe Coordiator Ly Harriso Desktop Publisher Heather Kight Project Leader Grat Moore Publicatios Editor Lord Selkirk School Divisio Louis Riel School Divisio Evergree School Divisio Fort La Bosse School Divisio Wiipeg School Divisio St. Gerard School Kelsey School Divisio Evergree School Divisio Moutai View School Divisio Lord Selkirk School Divisio Haover School Divisio Developmet Uit Istructio, Curriculum ad Assessmet Brach Developmet Uit Istructio, Curriculum ad Assessmet Brach Documet Productio Services Uit Educatioal Resources Brach Documet Productio Services Uit Educatioal Resources Brach Developmet Uit Istructio, Curriculum ad Assessmet Brach Documet Productio Services Uit Educatioal Resources Brach Ackowledgemets v

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9 INTRODUCTION Backgroud Kidergarte to Grade 8 Mathematics: Maitoba Curriculum Framework of Outcomes resulted from ogoig collaboratio with the Wester ad Norther Caadia Protocol (WNCP) ad its resultig mathematics documet, The Commo Curriculum Framework for K 9 Mathematics. I its work, WNCP emphasizes commo educatioal goals the ability to collaborate ad achieve commo goals high stadards i educatio plaig a array of educatioal activities removig obstacles to accessibility for idividual learers optimum use of limited educatioal resources The growig effects of techology ad the eed for techology-related skills has become more apparet i the last half cetury. Mathematics ad problem-solvig skills are becomig more valued as we move from a idustrial to a iformatioal society. As a result of this tred, mathematics literacy has become icreasigly importat. Makig coectios betwee mathematical study ad daily life, busiess, idustry, govermet, ad evirometal thikig is imperative. The Kidergarte to Grade 12 Mathematics curriculum is desiged to support ad promote the uderstadig that mathematics is a way of learig about our world part of our daily lives both quatitative ad geometric i ature Purpose of the Documet Grade 5 Mathematics: Support Documet for Teachers provides various istructioal activities, assessmet strategies, ad learig resources meat to promote the meaigful egagemet of math learers. It is meat to be used as a aid to teachers as they work with studets i achievig the prescribed outcomes ad achievemet idicators as outlied i the Kidergarte to Grade 8 Mathematics: Maitoba Curriculum Framework of Outcomes. Itroductio 1

10 Beliefs about Studets ad Mathematics Learig The Kidergarte to Grade 8 Mathematics curriculum is desiged with the uderstadig that studets have uique iterests, abilities, ad eeds. As a result, it is imperative to make coectios to all studets prior kowledge, experieces, ad backgrouds. Studets are curious, active learers with idividual iterests, abilities, ad eeds. They come to classrooms with uique kowledge, life experieces, ad backgrouds. A key compoet i successfully developig umeracy is makig coectios to these backgrouds ad experieces. Studets lear by attachig meaig to what they do, ad eed to costruct their ow meaig of mathematics. This meaig is best developed whe learers ecouter mathematical experieces that proceed from the simple to the complex ad from the cocrete to the abstract. The use of maipulatives ad a variety of pedagogical approaches ca address the diversity of learig styles ad developmetal stages of studets, ad ehace the formatio of soud, trasferable mathematical cocepts. At all levels, studets beefit from workig with a variety of materials, tools, ad cotexts whe costructig meaig about ew mathematical ideas. Meaigful studet discussios ca provide essetial liks amog cocrete, pictorial, ad symbolic represetatios of mathematics. The learig eviromet should value ad respect all studets experieces ad ways of thikig, so that learers are comfortable takig itellectual risks, askig questios, ad posig cojectures. Studets eed to explore problem-solvig situatios i order to develop persoal strategies ad become mathematically literate. Learers must realize that it is acceptable to solve problems i differet ways ad that solutios may vary. First Natios, Métis, ad Iuit Perspectives First Natios, Métis, ad Iuit studets i Maitoba come from diverse geographic areas with varied cultural ad liguistic backgrouds. Studets atted schools i a variety of settigs icludig urba, rural, ad isolated commuities. Teachers eed to recogize ad uderstad the diversity of cultures withi schools ad the diverse experieces of studets. First Natios, Métis, ad Iuit studets ofte have a whole-world view of the eviromet; as a result, may of these studets live ad lear best i a holistic way. This meas that studets look for coectios i learig, ad lear mathematics best whe it is cotextualized ad ot taught as discrete cotet. May First Natios, Métis, ad Iuit studets come from cultural eviromets where learig takes place through active participatio. Traditioally, little emphasis was placed upo the writte word. Oral commuicatio alog with practical applicatios ad experieces are importat to studet learig ad uderstadig. 2 Grade 5 Mathematics: Support Documet for Teachers

11 A variety of teachig ad assessmet strategies are required to build upo the diverse kowledge, cultures, commuicatio styles, skills, attitudes, experieces, ad learig styles of studets. The strategies used must go beyod the icidetal iclusio of topics ad objects uique to a culture or regio, ad strive to achieve higher levels of multicultural educatio (Baks ad Baks, 1993). Affective Domai A positive attitude is a importat aspect of the affective domai that has a profoud effect o learig. Eviromets that create a sese of belogig, ecourage risk takig, ad provide opportuities for success help studets develop ad maitai positive attitudes ad self-cofidece. Studets with positive attitudes toward learig mathematics are likely to be motivated ad prepared to lear, participate willigly i classroom activities, persist i challegig situatios, ad egage i reflective practices. Teachers, studets, ad parets eed to recogize the relatioship betwee the affective ad cogitive domais, ad attempt to urture those aspects of the affective domai that cotribute to positive attitudes. To experiece success, studets must be taught to set achievable goals ad assess themselves as they work toward these goals. Strivig toward success ad becomig autoomous ad resposible learers are ogoig, reflective processes that ivolve revisitig the settig ad assessig of persoal goals. Middle Years Educatio Middle Years is defied as the educatio provided for youg adolescets i Grades 5, 6, 7, ad 8. Middle Years learers are i a period of rapid physical, emotioal, social, moral, ad cogitive developmet. Socializatio is very importat to Middle Years studets ad collaborative learig, positive role models, approval of sigificat adults i their lives, ad a sese of commuity ad belogig greatly ehace adolescets egagemet i learig ad commitmet to school. It is importat to provide studets with a egagig ad social eviromet withi which to explore math ad costruct meaig. Adolescece is a time of rapid brai developmet whe cocrete thikig progresses to abstract thikig. Although higher-order thikig ad problem-solvig abilities develop durig Middle Years, cocrete, exploratory, ad experietial learig is most egagig to adolescets. Itroductio 3

12 Middle Years studets seek to establish their idepedece ad are most egaged whe their learig experieces provide them with a voice ad choice. Persoal goal settig, co-costructio of criteria, ad participatio i assessmet, evaluatio, ad reportig, help adolescets take owership of their learig. Clear, descriptive, ad timely feedback ca provide importat iformatio to the math studet. Askig ope-eded questios, acceptig multiple solutios, ad havig studets develop persoal strategies will help to develop their mathematical idepedece. Adolescets who see the coectios betwee themselves ad their learig, ad betwee the learig iside the classroom ad life outside the classroom, are more motivated ad egaged i their learig. Adolescets thrive o challeges i learig but their sesitivity at this age makes them proe to discouragemet if the challeges seem uattaiable. Differetiated istructio allows teachers to tailor learig challeges to adolescets idividual eeds, stregths, ad iterests. It is importat to focus your istructio o where studets are ad to see every cotributio to math class as valuable. The eergy, ethusiasm, ad ufoldig potetial of youg adolescets provide both challeges ad rewards to educators. Those educators who have a sese of humour ad who see the woderful potetial ad possibilities of each youg adolescet will fid teachig i Middle Years excitig ad fulfillig. Goals for Studets The mai goals of mathematics educatio are to prepare studets to use mathematics cofidetly to solve problems commuicate ad reaso mathematically appreciate ad value mathematics make coectios betwee mathematics ad its applicatios commit themselves to lifelog learig become mathematically literate adults, usig mathematics to cotribute to society Studets who have met these goals will gai uderstadig ad appreciatio of the cotributios of mathematics as a sciece, philosophy, ad art exhibit a positive attitude toward mathematics egage ad persevere i mathematical tasks ad projects cotribute to mathematical discussios take risks i performig mathematical tasks exhibit curiosity 4 Grade 5 Mathematics: Support Documet for Teachers

13 CONCEPTUAL FRAMEWORK FOR KINDERGARTEN TO GRADE 9 MATHEMATICS The chart below provides a overview of how mathematical processes ad the ature of mathematics ifluece learig outcomes. STRAND GRADE K NATURE OF MATHEMATICS CHANGE, CONSTANCY, NUMBER SENSE, PATTERNS, RELATIONSHIPS, SPATIAL SENSE, UNCERTAINTY Number Patters ad Relatios Patters Variables ad Equatios Shape ad Space Measuremet 3-D Objects ad 2-D Shapes Trasformatios Statistics ad Probability Data Aalysis Chace ad Ucertaity GENERAL OUTCOMES, SPECIFIC OUTCOMES, AND ACHIEVEMENT INDICATORS MATHEMATICAL PROCESSES: COMMUNICATION, CONNECTIONS, MENTAL MATHEMATICS AND ESTIMATION, PROBLEM SOLVING, REASONING, TECHNOLOGY, VISUALIZATION Mathematical Processes There are critical compoets that studets must ecouter i a mathematics program i order to achieve the goals of mathematics educatio ad ecourage lifelog learig i mathematics. Studets are expected to commuicate i order to lear ad express their uderstadig coect mathematical ideas to other cocepts i mathematics, to everyday experieces, ad to other disciplies demostrate fluecy with metal mathematics ad estimatio develop ad apply ew mathematical kowledge through problem solvig develop mathematical reasoig select ad use techologies as tools for learig ad solvig problems develop visualizatio skills to assist i processig iformatio, makig coectios, ad solvig problems The commo curriculum framework icorporates these seve iterrelated mathematical processes that are iteded to permeate teachig ad learig. Itroductio 5

14 Commuicatio Studets commuicate daily (orally, through diagrams ad pictures, ad by writig) about their mathematics learig. This eables them to reflect, to validate, ad to clarify their thikig. Jourals ad learig logs ca be used as a record of studet iterpretatios of mathematical meaigs ad ideas. Coectios Mathematics should be viewed as a itegrated whole, rather tha as the study of separate strads or uits. Coectios must also be made betwee ad amog the differet represetatioal modes cocrete, pictorial, ad symbolic (the symbolic mode cosists of oral ad writte word symbols as well as mathematical symbols). The process of makig coectios, i tur, facilitates learig. Cocepts ad skills should also be coected to everyday situatios ad other curricular areas. Metal Mathematics ad Estimatio The skill of estimatio requires a soud kowledge of metal mathematics. Both are ecessary to may everyday experieces ad studets should be provided with frequet opportuities to practise these skills. Problem Solvig Studets are exposed to a wide variety of problems i all areas of mathematics. They explore a variety of methods for solvig ad verifyig problems. I additio, they are challeged to fid multiple solutios for problems ad to create their ow problems. Reasoig Mathematics reasoig ivolves iformal thikig, cojecturig, ad validatig these help childre uderstad that mathematics makes sese. Childre are ecouraged to justify, i a variety of ways, their solutios, thikig processes, ad hypotheses. I fact, good reasoig is as importat as fidig correct aswers. Techology The use of calculators is recommeded to ehace problem solvig, to ecourage discovery of umber patters, ad to reiforce coceptual developmet ad umerical relatioships. They do ot, however, replace the developmet of umber cocepts ad skills. Carefully chose computer software ca provide iterestig problem-solvig situatios ad applicatios. Visualizatio These are the metal images eeded to develop cocepts ad uderstad procedures. Images ad explaatios help studets clarify their uderstadig of mathematical ideas. These processes are outlied i detail i Kidergarte to Grade 8 Mathematics: Maitoba Curriculum Framework of Outcomes (2008). Strads The learig outcomes i the Maitoba curriculum framework are orgaized ito four strads across Kidergarte to Grade 9. Some strads are further subdivided ito substrads. There is oe geeral outcome per substrad across Kidergarte to Grade 9. 6 Grade 5 Mathematics: Support Documet for Teachers

15 The strads ad substrads, icludig the geeral outcome for each, follow. Number Develop umber sese. Patters ad Relatios Patters Use patters to describe the world ad solve problems. Variables ad Equatios Represet algebraic expressios i multiple ways. Shape ad Space Measuremet Use direct ad idirect measure to solve problems. 3-D Objects ad 2-D Shapes Describe the characteristics of 3-D objects ad 2-D shapes, ad aalyze the relatioships amog them. Trasformatios Describe ad aalyze positio ad motio of objects ad shapes. Statistics ad Probability Data Aalysis Collect, display, ad aalyze data to solve problems. Chace ad Ucertaity Use experimetal or theoretical probabilities to represet ad solve problems ivolvig ucertaity. Outcomes ad Achievemet Idicators The Maitoba curriculum framework is stated i terms of geeral outcomes, specific outcomes, ad achievemet idicators. Geeral outcomes are overarchig statemets about what studets are expected to lear i each strad/substrad. The geeral outcome for each strad/substrad is the same throughout the grades. Specific outcomes are statemets that idetify the specific skills, uderstadig, ad kowledge studets are required to attai by the ed of a give grade. Achievemet idicators are oe example of a represetative list of the depth, breadth, ad expectatios for the outcome. Achievemet idicators are pedagogy- ad cotextfree. Itroductio 7

16 I this documet, the word icludig idicates that ay esuig items must be addressed to fully meet the learig outcome. The phrase such as idicates that the esuig items are provided for illustrative purposes or clarificatio, ad are ot requiremets that must be addressed to fully meet the learig outcome. Summary The coceptual framework for Kidergarte to Grade 9 mathematics describes the ature of mathematics, mathematical processes, ad the mathematical cocepts to be addressed i Kidergarte to Grade 9 mathematics. The compoets are ot meat to stad aloe. Activities that take place i the mathematics classroom should stem from a problem-solvig approach, be based o mathematical processes, ad lead studets to a uderstadig of the ature of mathematics through specific kowledge, skills, ad attitudes amog ad betwee strads. The Grade 5 Mathematics: Support Documet for Teachers is meat to support teachers to create meaigful learig activities that focus o formative assessmet ad studet egagemet. 8 Grade 5 Mathematics: Support Documet for Teachers

17 ASSESSMENT Authetic assessmet ad feedback are a drivig force for the Suggestios for Assessmet sectio of this documet. The purpose of the suggested assessmet activities ad strategies is to parallel those foud i Rethikig Classroom Assessmet with Purpose i Mid (2006). These iclude: assessig for, as, ad of learig ehacig studet learig assessig studets effectively, efficietly, ad fairly providig educators with a startig poit for reflectio, deliberatio, discussio, ad learig Assessmet for learig is desiged to give teachers iformatio to modify ad differetiate teachig ad learig activities. It ackowledges that idividual studets lear i idiosycratic ways, but it also recogizes that there are predictable patters ad pathways that may studets follow. It requires careful desig o the part of teachers so that they use the resultig iformatio to determie ot oly what studets kow, but also to gai isights ito how, whe, ad whether studets apply what they kow. Teachers ca also use this iformatio to streamlie ad target istructio ad resources, ad to provide feedback to studets to help them advace their learig. Assessmet as learig is a process of developig ad supportig metacogitio for studets. It focuses o the role of the studet as the critical coector betwee assessmet ad learig. Whe studets are active, egaged, ad critical assessors, they make sese of iformatio, relate it to prior kowledge, ad use it for ew learig. This is the regulatory process i metacogitio. It occurs whe studets moitor their ow learig ad use the feedback from this moitorig to make adjustmets, adaptatios, ad eve major chages i what they uderstad. It requires that teachers help studets develop, practise, ad become comfortable with reflectio, ad with a critical aalysis of their ow learig. Assessmet of learig is summative i ature ad is used to cofirm what studets kow ad ca do, to demostrate whether they have achieved the curriculum outcomes, ad, occasioally, to show how they are placed i relatio to others. Teachers cocetrate o esurig that they have used assessmet to provide accurate ad soud statemets of studets proficiecy, so that the recipiets of the iformatio ca use the iformatio to make reasoable ad defesible decisios. Itroductio 9

18 Overview of Plaig Assessmet Assessmet for Learig Assessmet as Learig Assessmet of Learig Why Assess? to eable teachers to determie ext steps i advacig studet learig to guide ad provide opportuities for each studet to moitor ad critically reflect o his or her learig ad idetify ext steps to certify or iform parets or others of studet s proficiecy i relatio to curriculum learig outcomes Assess What each studet's progress ad learig eeds i relatio to the curricular outcomes each studet's thikig about his or her learig, what strategies he or she uses to support or challege that learig, ad the mechaisms he or she uses to adjust ad advace his or her learig the extet to which studets ca apply the key cocepts, kowledge, skills, ad attitudes related to the curriculum outcomes What Methods? a rage of methods i differet modes that make studets skills ad uderstadig visible a rage of methods i differet modes that elicit studets learig ad metacogitive processes a rage of methods i differet modes that assess both product ad process Esurig Quality accuracy ad cosistecy of observatios ad iterpretatios of studet learig clear, detailed learig expectatios accurate, detailed otes for descriptive feedback to each studet accuracy ad cosistecy of studet's self-reflectio, selfmoitorig, ad self-adjustmet egagemet of the studet i cosiderig ad challegig his or her thikig studets record their ow learig accuracy, cosistecy, ad fairess of judgemets based o high-quality iformatio clear, detailed learig expectatios fair ad accurate summative reportig Usig the Iformatio provide each studet with accurate descriptive feedback to further his or her learig differetiate istructio by cotiually checkig where each studet is i relatio to the curricular outcomes provide parets or guardias with descriptive feedback about studet learig ad ideas for support provide each studet with accurate, descriptive feedback that will help him or her develop idepedet learig habits have each studet focus o the task ad his or her learig (ot o gettig the right aswer) provide each studet with ideas for adjustig, rethikig, ad articulatig his or her learig provide the coditios for the teacher ad studet to discuss alteratives studets report about their learig idicate each studet's level of learig provide the foudatio for discussios o placemet or promotio report fair, accurate, ad detailed iformatio that ca be used to decide the ext steps i a studet's learig Source: Maitoba Educatio, Citizeship ad Youth. Rethikig Classroom Assessmet with Purpose i Mid: Assessmet for Learig, Assessmet as Learig, Assessmet of Learig. Wiipeg, MB: Maitoba Educatio, Citizeship ad Youth, 2006, Grade 5 Mathematics: Support Documet for Teachers

19 INSTRUCTIONAL FOCUS The Maitoba curriculum framework is arraged ito four strads. These strads are ot iteded to be discrete uits of istructio. The itegratio of outcomes across strads makes mathematical experieces meaigful. Studets should make the coectio betwee cocepts both withi ad across strads. Cosider the followig whe plaig for istructio: Itegratio of the mathematical processes withi each strad is expected. By decreasig emphasis o rote calculatio, drill, ad practice, ad the size of umbers used i paper-ad-pecil calculatios, more time is available for cocept developmet. Problem solvig, reasoig, ad coectios are vital to icreasig mathematical fluecy, ad must be itegrated throughout the program. There is to be a balace amog metal mathematics ad estimatio, paper-ad-pecil exercises, ad the use of techology, icludig calculators ad computers. Cocepts should be itroduced usig maipulatives ad gradually developed from the cocrete to the pictorial to the symbolic. DOCUMENT FORMAT This documet cosists of two parts, the itroductio ad the support piece. The support piece cosists of eight sectios: 1. Edurig Uderstadig 2. Specific Learig Outcome(s) ad Achievemet Idicators 3. Prior Kowledge 4. Backgroud Iformatio 5. Mathematical Laguage 6. Learig Experieces 7. Puttig the Pieces Together Edurig Uderstadig summarizes the core idea of a particular outcome(s). Each statemet provides a coceptual foudatio for the outcome. It ca be used as a pivotal startig poit i itegratig other mathematical outcomes or other subject cocepts. The itegratio of cocepts, skills, ad strads remais of utmost importace. Itroductio 11

20 Specific Learig Outcome(s) ad Achievemet Idicators cotaied i this documet are take directly from Kidergarte to Grade 8 Mathematics: Maitoba Curriculum Framework of Outcomes. Achievemet idicators are examples of a represetative list of the depth, breadth, ad expectatios for the outcome. The idicators may be used to determie whether studets uderstad the particular outcome. These achievemet idicators will be met through the activities that follow. Prior Kowledge has bee listed to give teachers a referece to what studets may have experieced previously. Backgroud Iformatio has bee provided to give teachers kowledge about specific cocepts ad skills related to the particular outcome. Mathematical Laguage is foud after outcomes, icludig a list of terms studets will ecouter i the particular outcome(s). These words ca be placed o math word walls or used i a classroom math dictioary. Kidergarte to Grade 8 Mathematics Glossary: Support Documet for Teachers provides teachers with a uderstadig of key terms foud i Kidergarte to Grade 8 mathematics. The glossary is available o the Maitoba Educatio website at < math/supports.html>. Learig Experieces cotai suggested teachig strategies ad assessmet ideas of the specific outcomes ad achievemet idicators. I geeral, activities ad teachig strategies related to specific outcomes are developed idividually, except i cases where it seems more logical to develop two or more outcomes together. Suggestios for assessmet cotai iformatio that ca be used to assess studets prior kowledge ad observatios used to assess studets progress of the uderstadig of a particular outcome or learig experiece. It is expected that you will use your kowledge of math ad your studets to determie which learig experieces to use ad i what order to use them. Puttig the Pieces Together tasks are foud at the ed of some outcomes ad cosist of a variety of assessmet strategies. They may assess oe or more outcomes across oe or more strads ad may cotai cross-curricular coectios. 12 Grade 5 Mathematics: Support Documet for Teachers

21 Compoets Edurig Uderstadig summarizes the core idea of a particular outcome(s). Each statemet provides a coceptual foudatio for the outcome. It ca be used as a pivotal startig poit i itegratig other mathematical outcomes or other subject cocepts. The itegratio of cocepts, skills, ad strads remais of utmost importace. Prior Kowledge has bee listed to give teachers a referece to what studets may have experieced previously. Backgroud Iformatio has bee provided to give teachers kowledge about specific cocepts ad skills related to the particular outcome. Learig Experieces cotai suggested teachig strategies ad assessmet ideas of the specific outcomes ad achievemet idicators. I geeral, activities ad teachig strategies related to specific outcomes are developed idividually, except i cases where it seems more logical to develop two or more outcomes together. Notes idicate suggestios that will help i the teachig of the outcomes. SPECIFIC LEARNING OUTCOME(S): 5.N.2 Apply estimatio strategies, PRIOR KNOWLEDGE Studets should be able to do the follow (4.N.3) Add whole umbers wi LEARNING EXPERIENCES Note: Notice that studets ar square facts to 81. Other facts studets are expected to us th Edurig Uderstadigs: Computatioal estimatios pro Geeral Outcome: Develop umber ses BACKGROUND INFORMATION: Computatioal estimatio is the process computatioal problems. Studets Assessig Prior Kowledge Materials: BLM 5.N.11 Orgaizatio: Idividual a) Tell studets that i the ex umbers greater tha 10 0 out what they already k b) Ask studets to comp Observatio Check Use stude Materials: Situatio Cards (BLM 5.N.2.1) ad Orgaizatio: Small groups dure: ACHIEVEMENT INDICATORS: Provide a cotext for w MATHEMATICAL LANGUAGE: Approximate C tibl b Write a umeral usig proper spac Express a give umeral i expad Write the umeral represeted i e Mathematical Laguage is foud after outcomes that iclude a list of terms studets will ecouter i the particular outcome(s). These words ca be placed o math word walls or used i a classroom math dictioary. Kidergarte to Grade 8 Mathematics Glossary is available to support teachers with uderstadig of key terms foud i the documet. Assessig Prior Kowledge ad Observatio Checklist are suggestios for assessmet prior to lessos ad assessmet checklists to observe durig lessos to direct istructio. Achievemet Idicators are examples of a represetative list of the depth, breadth, ad expectatios for the outcome. They have bee take directly from the Kidergarte to Grade 8 Mathematics: Maitoba Curriculum Framework of Outcomes. The idicators may be used to determie whether studets uderstad the particular outcome. These achievemet idicators will be met through the activities that follow. Blacklie Masters (BLMs) are foud throughout the documet. The BLMs are i Microsoft Word format so they ca be altered to meet teachers eeds. PUTTING THE PIECES TOGETHER Purpose: The itet of this ivestigatio is to have measuremet cocepts to a real-world desiged to reiforce studets ab apply estimatio st Puttig the Pieces Together tasks are foud at the ed of some outcomes ad cosist of a variety of assessmet strategies. They may assess oe or more outcomes across oe or more strads ad may cotai crosscurricular coectios. Itroductio 13

22 N OTES 14 Grade 5 Mathematics: Support Documet for Teachers

23 G RADE 5 MATHEMATICS Number

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25 Grade 5: Number Edurig Uderstadigs: The positio of a digit i a umber determies its value. Each place value positio is 10 times greater tha the place value positio to its right. Geeral Outcome: Develop umber sese. SPECIFIC LEARNING OUTCOME(S): ACHIEVEMENT INDICATORS: 5.N.1 Represet ad describe whole umbers to [C, CN, T, V] Write a umeral usig proper spacig without commas (e.g., ad ot 934,567). Describe the patter of adjacet place positios movig from right to left. Describe the meaig of each digit i a umeral. Provide examples of large umbers used i prit or electroic media. Express a give umeral i expaded otatio (e.g., = [4 x ] + [5 x 1000] + [3 x 100] + [2 x 10] + [1 x 1] or ). Write the umeral represeted i expaded otatio. PRIOR KNOWLEDGE Studets should be able to do the followig: (4.N.1) Represet ad describe whole umbers to pictorially ad symbolically (4.N.2) Compare ad order whole umbers to (4.N.3) Demostrate a uderstadig of additio of umbers with sums to (4.N.3) Demostrate a uderstadig of subtractio of 3- ad 4-digit umbers Number 3

26 BACKGROUND INFORMATION For studets to work effectively with large umbers, they eed to have a good uderstadig of the structure of our umeratio system. The Hidu-Arabic, or base-10, umeratio system that we use today origiated i Idia aroud 500 CE, ad was carried to other parts of the world by Arab people. The system gradually replaced the use of Roma umerals ad the abacus i trade ad commerce i Europe ad, by the 16th cetury, was predomiat. The features of the system that led to its acceptace ad the computatioal procedures we use today iclude the followig: 1. It cosists of 10 digits (symbols), 0, 1, 2, 3, 4, 5, 6, 7, 8, ad 9, that are used i combiatio to represet all possible umbers. 2. It has a base umber. I this system, 10 oes are replaced by oe group of 10, 10 tes are replaced by oe hudred, 10 hudreds are replaced by oe thousad, ad so o. The umber of objects grouped together is called the base of the system. Thus, the Hidu-Arabic system is a base-10 system. 3. It has place value. Each place i a umeral has its ow value. For ay place i the system, the ext positio to the left is 10 times greater ad the positio to the right is oe-teth as large. 4. It has a symbol for zero. The symbol has two fuctios. It is a placeholder i umerals like 5027, where it idicates there are o hudreds (or 50 hudreds). It is also the umber that idicates the size of the set that has o objects i it. 5. It is additive ad multiplicative. The value of a umeral is foud by multiplyig each place value by its correspodig digit ad the addig all the resultig products. Expressig a umeral as the sum of its digits times their respective place values is called expaded otatio. For example, the expaded otatio for 8273 is (8 x 1000) + (2 x 100) + (7 x 10) + (3 x 1) or Cosequetly, the focus of the learig experieces that follow is o helpig studets coceptualize the magitude of large umbers ad uderstadig the characteristics of our umeratio system that allow us to read, write, ad iterpret the umerals for these umbers. MATHEMATICAL LANGUAGE Base Digit Expaded otatio Hudred thousad Oe millio Place value Te thousad 4 Grade 5 Mathematics: Support Documet for Teachers

27 LEARNING EXPERIENCES Assessig Prior Kowledge Materials: BLM 5.N.1.1: Place Value Orgaizatio: Idividual a) Tell studets that i the ext few lessos they will be learig about umbers greater tha , but before they begi you eed to fid out what they already kow about large umbers. b) Ask studets to complete the activity foud o BLM 5.N.1.1. Observatio Checklist Use studets resposes to the questios to determie whether they ca do the followig: compare ad order whole umbers i the thousads write umbers i words idetify the place value positio of the digits i a umeral idetify the value of each digit i a umeral Provide examples of large umbers used i prit or electroic media. Materials: A Millio Dots by Adrew Clemets (ISBN-13: ), calculators, stopwatch or timer with a secod had. Orgaizatio: Whole class/small groups a) Ask studets, How may dots do you thik you ca draw i oe miute? If we couted all the dots everyoe i the class makes i oe miute, how may dots do you thik we would have altogether? Do you thik we would have a millio dots? b) Explai that a millio is a big umber ad they are goig to fid out what a millio dots looks like. c) Read A Millio Dots. d) After readig the book, ask studets whether they wat to chage their estimates of the umber of dots that they ca draw i oe miute Have studets draw dots for oe miute. Whe they fiish, have them suggest ways to cout the dots. Ecourage them to thik about makig groups of tes to facilitate the coutig process. Number 5

28 e) Have studets use the total umber of dots that they make i oe miute to determie how log it would take oe perso to make a millio dots the class to make a millio dots f) Have each group decide what else they could do to show how big a millio is. Help them devise ad carry out a pla for showig the magitude of the umber. For example, studets could determie the legth of 1 millio looies laid ed to ed the umber of pages a telephoe book would eed to have to list 1 millio people the umber of boxes of toothpicks they would eed to make a millio g) Have each group share their plas ad what they foud out about 1 millio with the other members of the class. Observatio Checklist Observe studets resposes to determie whether they ca do the followig: make reasoable estimates solve computatioal problems with ad without usig techology develop ad carry out a pla for solvig a problem idicate that they have a sese of the magitude of 1 millio Describe the patter of adjacet place positios movig from right to left. Materials: Blak Hudred Square (BLM 5 8.6), scissors, tape or stapler. Orgaizatio: Small groups a) Write the umbers 600 ad 60 o the board or o a overhead. Ask studets, Would you rather have 600 peies or 60 peies? Have studets explai their reasoig. b) Explai that the place of a digit withi a umber is importat because it tells us the value of the digit, ad today they will be learig more about place value. 6 Grade 5 Mathematics: Support Documet for Teachers

29 c) Poit to each digit i the umeral 600 ad ask studets, What is the place value positio of this digit? Write studets resposes o the board or overhead, ad show studets that the oes ca be represeted with a square, the tes with a strip of 10 squares, ad hudreds with a grid of 100 squares. d) Ask studets, What place value positio comes ext? How ca we use the hudred squares to show 1000? Let studets explore differet ways to arrage the hudred squares to make Each studet should the make a 1000-strip by tapig or staplig te of the hudred squares together. e) Ask studets questios about the relatioship betwee the differet place value positios. For example: How may hudreds i oe thousad? How may times larger is oe thousad tha oe hudred? How may tes are i oe hudred? How may times larger is oe hudred tha te? How may tes are i oe thousad? How may times larger is oe thousad tha te? f) Ask studets, What place value positio comes ext? How much larger tha the thousads positio should the ew place value positio be? Why do you thik this? How ca we use the 1000-strips to show the ext place value positio? Have groups of 10 studets staple or tape their 1000 strips together. Whe studets fiish makig their square strips, ask them questios about the relatioship betwee the differet place value positios similar to the oes i part (e). g) Have studets i each group work together to aswer these questios: What place value positio comes ext? What is the relatioship of this positio to the other place value positios? What would a model of this place value positio look like? Have each group share its aswers with the other members of the class. Ecourage studets to explai their reasoig. h) Repeat part (g) to itroduce studets to the millios positio. Observatio Checklist Observe studets resposes to determie whether they ca do the followig: recogize that each place is 10 times greater tha the place to its right describe the various relatioships amog the place value positios (e.g., is 100 times greater tha 100 ad 1000 times greater tha 10) describe place value positios to the millios Number 7

30 Write the umeral usig proper spacig without commas. Materials: Paper ad pecils, overhead copy of place value chart whole umbers (BLM 5 8.7) Orgaizatio: Whole class/pairs a) Write a umber o the board or overhead (e.g., ), ad ask studets, How do you read the umber? How do place value patters help us read umbers? b) Show studets a place value chart. Explai that whe we read ad write large umbers, we group the digits ito threes. Each group of three forms a family. Each family has a differet last ame ad is separated from the other families by a space. The family o the far right is the oes. The family to its immediate left is the thousads. The ext family o the left is the millios. I each family, there is a place for oes, tes, ad hudreds. c) Tell studets that, for the remaiig time, they will be focusig o umbers i the thousads family. Record a umber i the place value chart (e.g., ), ad explai how to read the umber ad what each digit i the umber meas. Do three or four more examples. d) Have studets work with their parter. Studets eed to sit so oe perso i a pair ca see the board ad the other oe caot. Write a umber o the board (e.g., ). Studets facig the board read the umber to their parter. Their parter writes the umber dow. Studets the compare the umber they wrote dow with the umber o the board. Repeat the activity several times, givig each studet a opportuity to be both the reader ad the writer. Observatio Checklist Observe studets resposes to determie whether they ca do the followig: read a umber correctly write a umeral correctly with proper spacig 8 Grade 5 Mathematics: Support Documet for Teachers

31 Describe the meaig of each digit i a umeral. Materials: Number cards with the umbers 0 through 9 (BLM 5 8.5) with oe umber per card, ad large strips of paper showig the place value headigs (BLM 5.N.1.2), oe for each group Orgaizatio: Small groups (group size depeds o the size of the umbers) a) Put the place value colum headigs o the walls so they are just above the studets heads. b) Say a umber (e.g., ). Studets i each group must fid the appropriate umber cards, the arrage themselves ito a lie udereath the colum headigs showig the umber you said. Ecourage studets to tell what each digit i the umber meas. Have studets repeat the activity several more times. c) Expad the place value colum headigs to iclude hudred thousads ad have studets form 6-digit umbers. Observatio Checklist Observe studets resposes to determie whether they ca do the followig: idetify the place value positio of each digit i a umber describe the meaig of each digit i a umber Number 9

32 Describe the meaig of each digit i a umeral. Materials: Noe Orgaizatio: Whole class a) Tell studets they will be doig some umber calistheics. Explai that they will be actig out a umber you write o the board or overhead projector. They must act out the umber by doig i sequece: as may hops o their left foot as specified by the value of the digit i the hudred-thousads as may jumpig jacks as the value of the digit i the te-thousads positio as may clap-your-hads as i the value i the thousads positio as may touch-your-toes as the value i the hudreds positio as may hops o their right foot as the value of the digits i the tes place as may figer saps as the value of the digits i the oes positio For example, for the umber , studets would do 2 hops o their left foot 4 jumpig jacks 3 clap-your-hads 1 touch-your-toes 6 hops o their right foot 7 figer saps b) Have studets act out ; ; ad Whe studets are familiar with the movemets for each place value positio, have them act out 3-digit, 4-digit, 5-digit, ad 6-digit umbers. c) Vary the activity by havig studets choose the umbers that they act out. Observatio Checklist Observe studets resposes to determie whether they ca do the followig: idetify the correct place value positio of each digit i the umber idetify the value of each digit i a umber 10 Grade 5 Mathematics: Support Documet for Teachers

33 Write the umeral usig proper spacig without commas. Materials: Paper ad pecils Orgaizatio: Whole class a) Write four or five umbers o the board that use differet arragemets of the same digits. For example: b) Read oe of the umbers (e.g., te thousad five hudred thirty). Ask the studets to tell you which oe you chose. Have studets explai how they kew which umber you read. Ecourage studets to describe what the zeros i each umeral mea. c) Cotiue readig the umbers ad havig studets idetifyig them. Whe they fiish idetifyig all the umbers, have them order the umbers from smallest to largest ad the write the umbers i expaded otatio. d) Repeat the activity usig differet sets of umbers. e) Vary the activity by usig six-digit umbers istead of five-digit umbers. Observatio Checklist Observe studets resposes to determie whether they ca do the followig: describe the meaig of each digit i a umeral describe the patter of place value positios movig from right to left idetify the place value positio of each digit i a umeral express a umeral i expaded otatio Number 11

34 Write a umeral usig proper spacig without commas. Express a give umeral i expaded otatio. Materials: Dice Orgaizatio: Small groups a) Distribute six dice to each group. Tell studets that they will be tossig the dice five times. The first time they roll the dice they should create a six-digit umber with the umbers that they roll. They should record the umber ad the write it i expaded otatio. Next, they should remove oe die ad roll the remaiig five dice to create a five-digit umber. Agai, they should write the umber i both stadard otatio ad expaded otatio. Studets should cotiue removig a die, creatig a umber with the umbers that are rolled, ad recordig the umbers i stadard otatio ad expaded otatio util they have oe die left. b) Have each group share its results with the other members of the class. c) Vary the activity by havig studets write the umber i stadard otatio ad i words. Observatio Checklist Observe studets resposes to determie whether they ca do the followig: write a umeral usig proper spacig without commas express a give umeral i expaded otatio write a umeral i expaded otatio 12 Grade 5 Mathematics: Support Documet for Teachers

35 Write a umeral usig proper spacig without commas. Express a give umeral i expaded otatio. Write the umeral represeted i expaded otatio. Materials: Calculators, overhead of the excerpt from Which Do You Prefer Chuky or Smooth? (BLM 5.N.1.3) Orgaizatio: Idividual a) Have studets read the excerpt show below. Ask them to rewrite the umber words usig umerals ad rewrite the umerals usig umber words. I additio, have studets write each of the umbers i expaded otatio. I her book called Which Do You Prefer Chuky or Smooth?, Heather Brazier tells us the followig: O a average day i Caada we cosume eighty thousad, eight hudred forty-ie kilograms of peaut better. Of the total, kg are chuky (46) b) Have studets figure out how much smooth peaut better must be eate by Caadias o a average day. Have them write their aswer as a umeral ad i words. c) Throughout the year, have studets brig i examples of large umbers that they fid i ewspapers or magazies. Keep a class chart that shows the umber i umerals, expaded form, ad words. Observatio Checklist Observe studets resposes to determie whether they ca do the followig: write umbers i word form write the umeral for a umber writte i words write a umeral i expaded otatio provide examples of large umbers used i prit or electroic media Number 13

36 Describe the meaig of each digit i a umeral. Materials: Calculators Orgaizatio: Whole class Procedures: a) Ask studets to show o their calculators. Tell them that their goal is to chage the 2 to 0 (zero it) by subtractig oe umber. Whe studets fiish, ask them the followig: What umber do you have o your calculator ow? What umber did you subtract to wipe out the 2? Why did you subtract that umber? b) Cotiue askig studets to show five-digit ad six-digit umbers o their calculators. After you ame a umber for them to show o their calculators, ask them to zero a digit i oe of the place value positios. Ecourage studets to describe what they did ad why they did it to zero a digit. c) Ask studets to add a umber to wipe out a digit (e.g., addig 4 ca wipe out the 6 i 506). d) Vary the activity by tellig studets that they ca use either additio or subtractio to wipe out a digit. Observatio Checklist Observe studets resposes to determie whether they ca do the followig: idetify the place value positio of each digit i a umeral idetify the value of each digit i a umeral use techology to compute sums ad differeces 14 Grade 5 Mathematics: Support Documet for Teachers

37 Materials: Number cards (BLM 5 8.5) Orgaizatio: Groups of three a) Tell studets that they will be playig a place value game. Explai how to play the game. 1. Shuffle the cards ad lay them face dow i the playig area. 2. Players take turs drawig five cards from the deck. 3. Players arrage the cards i their hads so that they have the largest possible umber. 4. Oe player says, Let s see the umbers ad everyoe lays their cards face up i frot of them. A card caot be moved after it has bee placed face-up o the playig surface. 5. Players take turs readig the umber that they created. The player who has the largest umber ad reads the umber correctly wis a poit. 6. The wier is the perso with the most poits after five rouds of the game. b) Demostrate how to play the game ad aswer ay questios that studets may have. Have studets play the game. c) Vary the game by havig studets draw six cards istead of five create the smallest possible umber with their cards Observatio Checklist Observe studets resposes to determie whether they ca do the followig: read 5-digit ad 6-digit umbers correctly use place value cocepts to determie which of two or more umbers is the largest use place value cocepts to determie which of two or more umbers is the smallest Number 15

38 Write a umeral usig proper spacig without commas. Describe the meaig of each digit i a umeral. Materials: Calculators, paper ad pecils Orgaizatio: Pairs a) Tell studets that they will be playig a game called Give ad Take. Explai how to play the game. 1. Players write dow a six-digit umber cotaiig o zeros ad o idetical digits. Players keep their umbers hidde from each other throughout the game. 2. Players take turs beig the giver ad the taker. Each player tries to icrease his or her umber by takig digits from the other player. 3. A tur begis whe the asker says: Give me your x s, where x ca be ay digit from 1 through 9. (e.g., Give me your 7s. ). 4. If that digit is i the giver s umber, the giver aouces its place value (e.g., You get 700. If the digit is ot i the giver s umber, the giver aouces this by sayig, You get zero. ). Note that the value of a digit that is asked for depeds o its positio i the giver s umber. If 7 is asked for ad the umber is , the the giver says, You get 700. If the giver s umber is , the the giver says you get As soo as the giver respods with the umber, the asker adds that amout to his or her umber (e.g., + 700) ad the giver subtracts that amout from his or her umber (e.g., 700). 6. Players umbers chage with each ew additio or subtractio. Players always use the most recet form of their umbers whe addig, subtractig, or aoucig the place value of a digit. Players keep track of their chagig umber by addig ad subtractig from their origial umber ad its successors. For example: = = If the same digit appears two or more times i a giver s umber durig play, the giver ca say either of its values (e.g., for 845, 218, the giver ca say 8 ad ot metio the ). 8. The game eds after each player has had five turs as asker. Players check each other s additio ad subtractios. The player with the largest umber is the wier. 16 Grade 5 Mathematics: Support Documet for Teachers

39 Observatio Checklist Observe studets resposes to determie whether they ca do the followig: idetify the correct place value positio of each digit idetify the value of each digit i a umber use calculators correctly to determie sums ad differeces write a umeral with the proper spacig with o commas Number 17

40 N OTES 18 Grade 5 Mathematics: Support Documet for Teachers

41 Grade 5: Number Edurig Uderstadigs: Computatioal estimatios produce approximate aswers. Geeral Outcome: Develop umber sese. SPECIFIC LEARNING OUTCOME(S): ACHIEVEMENT INDICATORS: 5.N.2 Apply estimatio strategies, icludig frot-ed roudig compesatio compatible umbers i problem-solvig situatios. [C, CN, ME, PS, R, V] Provide a cotext for whe estimatio is used to make predictios check reasoableess of a aswer determie approximate aswers Describe cotexts i which overestimatig is importat. Determie the approximate solutio to a problem ot requirig a exact aswer. Estimate a sum or product usig compatible umbers. Estimate the solutio to a problem usig compesatio, ad explai the reaso for compesatio. Select ad use a estimatio strategy to solve a problem. Apply frot-ed roudig to estimate sums (e.g., is more tha = 800) differeces (e.g., is close to = 700) products (e.g., the product of 23 x 24 is greater tha 20 x 20 or 400 ad less tha 25 x 25 or 625) quotiets (e.g., the quotiet of is greater tha or 200) Number 19

42 PRIOR KNOWLEDGE Studets should be able to do the followig: (4.N.3) Add whole umbers with sums less tha (4.N.3) Subtract whole umbers with differeces less tha (4.N.3) Use differet strategies to estimate sums ad differeces (4.N.6) Multiply a 1-digit whole umber times a 2-digit or 3-digit whole umber (4.N.6) Use a persoal strategy to estimate a product (4.N.7) Divide a 2-digit whole umber divided by a 1-digit whole umber divisor (4.N.7) Use a persoal strategy to estimate a quotiet RELATED KNOWLEDGE Studets should be able to do the followig: (5.N.5) Demostrate a uderstadig of multiplicatio of a 2-digit whole umber by a 2-digit whole umber (5.N.6) Demostrate a uderstadig of divisio of a 3-digit whole umber by a 1-digit whole umber BACKGROUND INFORMATION: Computatioal estimatio is the process of determiig approximate aswers to computatioal problems. Studets who are skillful estimators have a good grasp of basic facts, place value, ad the operatios of additio, subtractio, multiplicatio, ad divisio. They are also adept at metal mathematics ad flexible i their use of estimatio strategies, such as the oes described below. Frot-Ed Estimatio: Frot-ed roudig ivolves idetifyig the most sigificat (left-most) digits i a questio, performig the appropriate operatio, ad determiig the place value of the digits. For example: is more tha 1700 sice = 17 (ad aex the zeros) (or sice = 1700) is approximately 300 sice 5 2 = 3 (ad aex the zeros) (or sice = 300) 4 x 728 is more tha 2800 sice 4 x 7 = 28 (ad aex the zeros) (or sice 4 x 700 = 2800) is more tha 300 sice 9 3 = 3 (ad aex the zeros) (or sice = 300) 20 Grade 5 Mathematics: Support Documet for Teachers

43 Note: It is importat for teachers to emphasize estimatio skills. Discourage studets from calculatig first, the estimatig (e.g., I kow is 7.2, so I will estimate it is close to 7. ). Although frot-ed roudig ca be used with ay operatio, it is most powerful whe addig ad multiplyig. With these two operatios, the computatio is always uderestimated. Compatible Numbers: This strategy ivolves searchig for pairs of umbers that are easy to compute. Whe usig this strategy, studets look at all the umbers i a problem, ad chage or roud the umbers so they ca be paired usefully with aother umber. It is particularly effective for divisio. For example, i the questio , roudig the divided to 2300 (the closest hudred) or 2000 (the closest 1000) does ot facilitate the estimatio process. However, roudig it to 2400 (a compatible umber because it is divisible by 6) makes estimatig the quotiet easier. This strategy is also useful for additio. For example, whe addig several umbers, studets look for umbers that ca be paired or grouped together to make multiples of 10. about about 100 about Therefore, the sum of is about 200 or about 100 Compesatio: Compesatio ivolves refiig, or adjustig, a origial estimate that was obtaied with aother strategy. For example, the frot-ed estimatio of 220 for the sum ca be adjusted to 240, sice ad (the umbers i the oes positio) are both close to 10. Similarly, the frot-ed estimatio of 2400 for 43 x 62 ca be adjusted to 2600 sice (3 x 60) + (2 x 40) would be greater tha 200. Also, for 44 x 54, for example, you ca roud oe umber up ad oe umber dow ad compute 50 x 50 for a estimate of 2500, rather tha frot-ed roudig for a estimate of I may istaces, differet strategies ca be applied to the same problem. The choice of strategies depeds o the studets, the umbers, ad the operatios ivolved. Teachers eed to help studets become aware of the various strategies ad help them develop cofidece i their ability to estimate. To do this, they eed to egage studets i discussios about the strategies they used to estimate the solutio to a computatioal problem (Sharig strategies ca lead to the developmet ad use of ew strategies.) accept a rage of estimates i order to help studets uderstad that there is o oe right estimate ecourage studets to idetify real-world situatios that ivolve estimatios Number 21

44 icorporate estimatio throughout their istructioal programs (Like problem solvig, estimatio should ot be taught i isolated uits.) MATHEMATICAL LANGUAGE: Aex Approximate Compatible umbers Compesatio Estimate Estimatio Frot-ed roudig LEARNING EXPERIENCES: Assessig Prior Kowledge Materials: Noe Orgaizatio: Idividual a) Tell studets you eed to kow what they kow about computatioal estimatio so you ca help them become better estimators. To fid out what they kow, give them some problems to estimate. Tell studets that you will show them several problems, oe at a time, ad they will have a appropriate amout of time (decide this based o idividual studets approximately 30 secods) to estimate the solutio to each problem. They must record their estimate before their time is up. b) Give studets the followig problems: x x c) Have studets share their estimates ad the strategies they used to determie them. 22 Grade 5 Mathematics: Support Documet for Teachers

45 Observatio Checklist Check studets resposes to the problems to determie whether they ca estimate the solutios to additio, subtractio, multiplicatio, ad divisio problems. Use the class discussio to fid out what strategies studets use to make their estimates. Provide a cotext for whe estimatio is used. Materials: Markers ad ewspapers/magazies Orgaizatio: Whole class a) Tell studets that they are goig to ivestigate the use of estimated ad exact umbers. b) Give studets copies of differet ewspapers. Ask them to circle the umbers used i the headlies ad articles. Next, have studets review the cotext for the use of each circled umber to determie whether the umbers i the headlies or articles refer to exact or estimated (approximate) values. For example, have studets decide whether these statemets take from a ewspaper refer to exact or estimated values: Oe millio people evacuated from New Orleas The codo resold for $ Last year, tos of scrap metal were recycled c) Egage studets i a discussio about the umbers they foud i the ewspapers. Ecourage them to explai why they thik a give umber is exact or estimated. Have studets discuss why estimated umbers are ofte used i ewspaper articles (e.g., estimated umbers are easier to iterpret ad use). Observatio Checklist Observe studets resposes to determie whether they ca do the followig: idetify real-world examples of estimated umbers distiguish betwee exact ad estimate (approximate) umbers give a reasoable explaatio why a umber is either exact or estimated Number 23

46 Materials: Situatio Cards (BLM 5.N.2.1) ad idex cards Orgaizatio: Small groups a) Give each group a card with oe of the situatios o it. b) Ask studets to decide whether the situatio o their card requires a estimated aswer or a exact aswer, ad to list the reasos for their respose. c) Have each group read its situatio to the other members of the class, ad explai why they thik the situatio requires a estimated or exact aswer. d) Have each group create a situatio card. Each group should record, o a separate piece of paper, the reasos why they thik the situatio they created requires a estimated or exact aswer. Have the groups exchage cards ad decide whether the ew situatio they were give requires a estimated or exact aswer ad why they thik so. Each group should compare its respose with the respose of the group who created the situatio. Observatio Checklist Observe studets resposes to determie whether they ca do the followig: provide a cotext for whe estimatio is used to approximate a aswer provide a cotext for whe estimatio is used to predict a aswer distiguish betwee situatios that require a exact aswer ad those that require a estimated aswer give reasoable explaatios of why a situatio requires either a exact or estimated aswer 24 Grade 5 Mathematics: Support Documet for Teachers

47 Provide a cotext for whe estimatio is used. Determie the approximate solutio to a problem ot requirig a exact aswer. Materials: Noe Orgaizatio: Pairs/Whole class/small groups a) Preset studets with the followig problem: The 28 studets i Mr. Nelso s fifth-grade class are plaig a Hallowee party. The studets decided to make a fruit puch for everyoe to drik at the party. They kow that a ca of juice makes eight cups of puch. How may cas of juice should they buy? b) Give studets time to solve the problem, ad the have them share their solutios with their parter. c) Have studets discuss their solutios ad share their reasoig with the other members of the class. Help them recogize that there are times we eed a estimate because there is ot eough iformatio to determie a exact aswer (e.g., we do ot kow how thirsty studets will be). d) Ask each group to idetify other situatios that require a estimate because there is ot eough iformatio to compute a exact aswer. Observatio Checklist Observe studets resposes to determie whether they ca do the followig: idetify situatios that require a estimate because there is ot eough iformatio to compute a exact aswer provide a cotext whe estimatig is used to make predictios provide reasoable estimates give reasoable explaatios for their estimates Number 25

48 Describe cotexts i which overestimatig is importat. Materials: Copies of estimatio situatios (BLM 5.N.2.2) Orgaizatio: Pairs/Large group a) Ask studets to read each of the situatios, ad decide whether a overestimate or uderestimate is eeded. b) Have studets discuss their aswers with their parters. The have studets share their aswers ad the reasos for them with the other members of the class. c) Have studets describe other estimatio situatios ad decide whether a uderestimate or a overestimate is eeded. Observatio Checklist Observe studets resposes to determie whether they ca do the followig: idetify ad describe cotexts i which overestimatig is importat idetify ad describe cotexts i which uderestimatig is importat 26 Grade 5 Mathematics: Support Documet for Teachers

49 Determie the approximate solutio to a problem ot requirig a exact aswer. Materials: Number tiles or umber cards (BLM 5 8.5), calculators Orgaizatio: Idividual/Pairs/Whole class a) Ask studets to complete the followig activity: Explai that they must use the umber tiles 4 9 to create 3-digit by 1-digit multiplicatio problems. The products of the problems must be as close to the target as possible. They get three tries for each target umber. They should record each problem they create ad its solutio. Tell them they ca use their calculators to fid the solutio to the problems they create. Target x = 5000 x = 8000 x = 7000 x = 4000 x = 4000 x = 7000 b) Whe studets fiish, have them compare their estimates with a parter ad explai the strategies they used to create the problems. c) Have studets share the strategies they used to create the problems with the other members of the class. Observatio Checklist Observe studets resposes to determie whether they ca do the followig: select ad use a estimatio strategy to estimate the product of two umbers make reasoable adjustmets to their estimates of the product of two umbers explai the strategy they used to estimate the product of two umbers Number 27

50 Materials: Dice, calculators, paper ad pecil Orgaizatio: Pairs a) Tell studets that they will be playig a estimatio game with their parter. Explai how the game is played. 1. Oe player tosses three dice ad creates a 3-digit umber with the umbers that are rolled. This umber becomes the divided. 2. The other player tosses oe die, ad the umber that is rolled becomes the divisor. Both players should record the problem. 3. Players record the problem, the quickly ad siletly write a estimate of the quotiet of the two umbers. Players should ot take more tha 10 or 15 secods to write their estimates. 4. Players use a calculator to fid the quotiet of the two umbers. Each player s score is the differece betwee his or her estimate ad the exact aswer. 5. The perso with the lowest score after five rouds of the game is the wier. b) Demostrate how to play the game ad aswer ay questios studets may have. Have studets play the game. c) Have studets share some of the problems they created ad the strategies that they used to estimate the quotiet. Observatio Checklist Observe studets resposes to determie whether they ca do the followig: select ad use a estimatio strategy for divisio use techology to determie the solutio to a divisio problem describe the strategy they used to determie a estimate 28 Grade 5 Mathematics: Support Documet for Teachers

51 Apply frot-ed roudig to estimate sums, differeces, products, ad quotiets. Estimate the solutio to a problem usig compesatio, ad explai the reaso for the compesatio. Materials: Paper ad pecil Orgaizatio: Whole class a) Preset studets with the followig problem: Mrs. Marti s class was estimatig the sums of additio problems. Matty said that the sum of is about 150. How did Matty get her estimate? b) Have studets share their ideas about how the sum was estimated. The ask, Is the sum of over or uder 150? How do you kow? How could you get a closer estimate of the sum? c) Explai that whe we use the frot-ed roudig strategy to estimate the sum of two or more umbers, we ca always adjust our estimate by lookig at the other digits i the problem. For example, if we look at the digits i the oes positio i the problem , is close to 10. So the sum of the oes is greater tha 10. Therefore, we ca adjust our estimate to 160. d) Do two or three more examples, ad the ask studets to use the frot-ed roudig strategy to estimate the sum of the followig problems, ad the adjust their estimates to get a closer approximatio of the solutio d) Have studets share their estimates. Ecourage them to describe how they adjusted their estimates to get a closer approximatio. e) Have studets use frot-ed roudig to estimate the sums of problems with 3- ad 4-digit umbers, ad adjust their estimates to get a closer approximatio. f) Use a similar approach to help studets lear how to adjust problems ivolvig subtractio, multiplicatio, ad divisio. Observatio Checklist Observe studets resposes to determie whether they ca do the followig: use frot-ed roudig to estimate the sum (differece, product, quotiet) of two or more umbers estimate the solutio to a problem usig compesatio, ad explai the reasos for compesatig Number 29

52 N OTES 30 Grade 5 Mathematics: Support Documet for Teachers

53 Grade 5: Number Edurig Uderstadigs: Proficiecy with the basic facts facilitates estimatio ad computatio with larger ad smaller umbers. Multiplicatio ad divisio are iverse operatios. Geeral Outcome: Develop umber sese. SPECIFIC LEARNING OUTCOME(S): ACHIEVEMENT INDICATORS: 5.N.3 Determie multiplicatio facts (to 81) ad related divisio facts. [C, CN, ME, R, V] Describe the metal mathematics strategy used to determie a basic fact, such as skip-cout up by oe or two groups from a kow fact (e.g., If 5 x 7 = 35, the 6 x 7 is equal to ad 7 x 7 is equal to ) skip-cout dow by oe or two groups from a kow fact (e.g., 8 x 8 = 64, the 7 x 8 = 64 8 ad 6 x 8 is equal to ) doublig (e.g., for 8 x 3 thik 4 x 3 = 12 ad 8 x 3 = ) patters whe multiplyig by 9 (e.g., for 9 x 6, thik 10 x 6 = 60, ad 60 6 = 54; for 7 x 9, thik 7 x 10 = 70, ad 70 7 = 63) repeated doublig (e.g., if 2 x 6 is equal to 12, the 4 x 6 is equal to 24, ad 8 x 6 is equal to 48) repeated halvig (e.g., for 60 4, thik 60 2 = 30 ad 30 2 = 15) Recall the multiples of 0, 1, 2, 3, ad 5 to 81 ad related divisio facts. Recall the multiplicatio facts that are squares: 1 x 1, 2 x 2, up to 9 x 9. Number 31

54 PRIOR KNOWLEDGE Studets should be able to do the followig: (4.N.4) Describe the properties of 0 ad 1 for multiplicatio ad the property of 1 for divisio (4.N. 5) Describe ad apply metal mathematics strategies, such as skip-coutig from a kow fact usig doublig or halvig usig doublig ad addig oe more group usig patters i the ie facts usig repeated doublig to develop recall of the basic facts (3.N.11) Use arrays to represet multiplicatio facts BACKGROUND INFORMATION Calculatios people do o a daily basis ivolve kowig basic math facts. For this reaso, basic facts cotiue to be a itegral part of the mathematics curriculum. I the Early Years, studets are expected to recall some facts ad use strategies to determie others i order to help them lear more sophisticated mathematics. If studets eterig the Middle Years do ot have the strategies to determie or recall the basic facts, the teachers eed to teach the strategies ad help studets work towards these skills. Thikig strategies provide studets with differet approaches for arrivig at a aswer. Studets also stregthe their umber sese ad lear to adapt these strategies whe workig with larger umbers. Oce studets have had time to practice these strategies i game or activity settigs, the differet methods ca be implemeted to help studets develop ad maitai the ability to recall ad be able to determie the facts that are appropriate for the grade level. 32 Grade 5 Mathematics: Support Documet for Teachers

55 MATHEMATICAL LANGUAGE: Divide Divided Divisio Divisor Factor Multiplicatio Multiply Product Quotiet LEARNING EXPERIENCES: Assessig Prior Kowledge Materials: Math jourals, Metal Math Strategies checklist (BLM 5 8.8) Orgaizatio: Idividual/Whole class a) Ask studets to solve each of the followig problems i two differet ways: Rosa is plaig to arrage 48 books o six shelves. If she puts a equal umber of books o each shelf, how may books will she put o each shelf? Mark has a six-page photo album. How may pictures does Mark have if each page holds eight pictures? b) Have studets share their solutios with the other members of the class. Ecourage studets to explai their reasoig by askig questios, such as: Which strategy did you use to solve the problem? What is aother strategy you could use to solve the problem? Will the strategy work for other problems ivolvig divisio (multiplicatio)? Show me. Which strategy do you prefer to use? Why? Number 33

56 Observatio Checklist Use studets resposes ad BLM to determie which strategies studets kow. Also, examie their resposes to determie whether they ca do the followig: idetify problem situatios that call for the operatio of multiplicatio idetify problem situatios that call for the operatio of divisio describe ad apply a thikig strategy to determie the product or quotiet of two whole umbers describe ad apply more tha oe thikig strategy to determie the product or quotiet of two whole umbers Describe the metal math strategy used to describe a basic fact. Materials: Activity sheets that show the same array repeated about eight times. For example: 1. xxxxxxx xxxxxxx xxxxxxx xxxxxxx xxxxxxx xxxxxxx xxxxxxx xxxxxxx 2. xxxxxxx xxxxxxx xxxxxxx xxxxxxx xxxxxxx xxxxxxx xxxxxxx xxxxxxx 3. xxxxxxx xxxxxxx xxxxxxx xxxxxxx xxxxxxx xxxxxxx xxxxxxx xxxxxxx 4. xxxxxxx xxxxxxx xxxxxxx xxxxxxx xxxxxxx xxxxxxx xxxxxxx xxxxxxx 5. xxxxxxx xxxxxxx xxxxxxx xxxxxxx xxxxxxx xxxxxxx xxxxxxx xxxxxxx 6. xxxxxxx xxxxxxx xxxxxxx xxxxxxx xxxxxxx xxxxxxx xxxxxxx xxxxxxx 7. xxxxxxx xxxxxxx xxxxxxx xxxxxxx xxxxxxx xxxxxxx xxxxxxx xxxxxxx 8. xxxxxxx xxxxxxx xxxxxxx xxxxxxx xxxxxxx xxxxxxx xxxxxxx xxxxxxx 34 Grade 5 Mathematics: Support Documet for Teachers

57 Orgaizatio: Idividual/Whole class a) Show studets a 8 x 7 array. Ask them to split the array to show a strategy for fidig the product of 8 x 7 ad have them describe the correspodig umber seteces. For example: xxxxxxx xxxxxxx xxxxxxx xxxxxx xxxxxxx xxxxxxx xxxxxxx xxxxxxx 7 x 7 = 49 1 x 7 = 7 Therefore, 8 x 7 = = 56 Ask studets to split the remaiig arrays i differet ways to show the various strategies that ca be used to fid the product of 8 x 7. Have them record the correspodig umber seteces for each strategy that they fid. b) Have studets share the strategies they foud with the other members of the class. Ask, What strategy is the easiest? Why do you thik this? c) Repeat the activity for other multiplicatio facts ad have studets compare the strategies that they fid for each fact. Observatio Checklist Observe studets resposes to determie whether they ca do the followig: describe the metal math strategy used to determie a multiplicatio fact such as skip coutig by 1 or 2 groups from a kow fact idetify ad apply differet metal math strategies to determie a multiplicatio fact such as doublig, repeated doublig (e.g., 8 x 7 = [2 x 7] + [2 x 7] + [2 x 7] + [2 x 7] = 56), or doublig plus oe or two groups (e.g., 8 x 7 = [3 x 7] + [3 x 7] + [2 x 7] = 56) Number 35

58 Materials: Paper ad pecils Orgaizatio: Whole class/pairs a) Have studets study the followig array ad the ask them how kowig the facts of 5 ca help them with other facts. x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x 5 x 6 = 30 2 x 6 = 12 Therefore, 7 x 6 = 42. b) Have studets describe how usig a thik 5 facts strategy ca help them determie these fact problems. If ecessary, have studets draw the correspodig arrays. 9 x 6 = o 8 x 3 = o 6 x 4 = o 7 x 7 = o c) Have studets make a list of other facts that they could determie easily usig a thik 5 facts strategy. Have them share their lists with their parter ad describe how they would use the strategy to solve each fact that they listed. Observatio Checklist Observe studets resposes to determie whether they ca do the followig: describe how the facts of 5 strategy ca be used to determie a basic multiplicatio fact idetify multiplicatio facts that ca be determied usig a facts of 5 strategy 36 Grade 5 Mathematics: Support Documet for Teachers

59 Materials: Paper ad pecil Orgaizatio: Pairs a) Give studets a fact problem such as 6 x 8. Have the first studet i each pair do oe part of the problem (e.g., 4 eights is 32). The secod studet must fiish the problem i this case, 2 eights is 16. The first studet the adds the two parts together to determie the product. Have the studets record the strategy that they used. Ecourage studets to split the problem ito parts that are easy to fid. b) Repeat the activity several times, but have studets switch roles. c) Have studets share the strategies they used with the other members of the class ad discuss which strategies are the easiest ad most efficiet to use. Observatio Checklist Observe studets resposes to determie whether they ca do the followig: idetify ad apply differet metal math strategies to determie a multiplicatio fact describe the metal math strategies used to determie a multiplicatio fact Materials: Paper ad pecil Orgaizatio: Whole class a) Ask studets to write four umber seteces usig oly the umbers 3, 5, ad 15. Whe they fiish, ask them to show the facts (3 x 5 = 15, 5 x 3 = 15, 15 3 = 5, ad 15 5 = 3) o umber lies. For example: 3 x 5 = = Number 37

60 c) Ask studets what they otice about the facts ad how you showed them o the umber lies. (Studets should otice the iverse relatioship betwee multiplicatio ad divisio, although they may use the terms opposite or backwards. ) d) Have studets write four math seteces for each of the followig triplets of umbers: e) Ask studets how a divisio fact ca be determied by thikig a multiplicatio fact. The have them use the relatioship betwee multiplicatio ad divisio to describe the thikig strategy for solvig the followig divisio problems. For example, for 36 9 = o, thik some umber x 9 = 36 sice 4 x 9 = 36, 36 9 must equal = o 16 4 = o = o 27 3 = o = o 40 8 = o = o 49 7 = o = o 72 9 = o Observatio Checklist Observe studets resposes to determie whether they ca do the followig: idetify the multiplicatio ad divisio facts that ca be associated with a give triplet of umbers idetify the iverse of a multiplicatio fact idetify the iverse of a divisio fact describe how a thik multiplicatio strategy ca be used to recall a divisio fact recogize the iverse relatioship betwee multiplicatio ad divisio 38 Grade 5 Mathematics: Support Documet for Teachers

61 Describe the metal mathematics strategy used to determie a basic fact. Recall the multiples for 0, 1, 2, 3, ad 5 to 81 ad related divisio facts. Recall the multiplicatio facts that are squares. Note: Notice that studets are required to recall oly multiples of 0, 1, 2, 3, ad 5, ad square facts to 81. Studets should be give more time to compute other facts (e.g., 7 x 8), as they are expected to use strategies to determie the aswer (e.g., kowig it is 7 x 7 plus aother group of 7 ). Materials: A deck of cards with the face cards removed. Orgaizatio: Small groups a) Tell studets that they are goig to play a variatio of the game I spy with the members of their group. Explai how the game is played. 1. Lay the cards o the playig surface face up i five rows of Players take turs challegig each other to fid two cards that have a specific product. The two cards must be ext to each other horizotally, vertically, or diagoally. For example, if a 5 ad a 3 are ext to each other, a player could say, I spy two cards whose product is The other players look for the cards. The player who fids the right combiatio takes the two cards. If the combiatio caot be foud the the player who posed the I spy questio takes the two cards. 4. If a player makes a error or there is o such combiatio of cards, obody collects ay cards ad the ext player takes his or her tur. 5. As cards are removed, the remaiig cards should be rearraged to fill i the spaces. 6. The game is over whe all the cards have bee picked up. The wier is the player with the most cards. b) Demostrate how the game is played ad aswer ay questios studets might have. Have studets play the game. Observatio Checklist Observe studets resposes to determie which facts that they ca recall easily facts that they have difficulty recallig Number 39

62 Materials: Tic-tac-toe grids (BLM 5.N.3.1) Orgaizatio: Pairs Cautio: I some commuities, playig cards are see as a form of gamblig ad discouraged. Please be aware of local sesitivities before itroducig this activity. a) Tell studets that they will be playig Multiplicatio Tic-tac-toe. Explai how the game is played. 1. Players decide who will be X ad who will be O. 2. Each player lists the umbers from 1 to 9 i a colum beside the multiplicatio grid. 3. The first player crosses out ay umber i his or her colum of umbers. 4. Begiig with the secod player, the game proceeds i this maer. Durig a tur, a player crosses off ay umber i his or her colum of 9 umbers that has ot bee crossed off. The player the multiplies that umber by the last umber crossed off by his or her oppoet. If the product is o the tic-tac-toe board ad ot yet crossed off the player places a x or O over the product. For example: player 1 crosses off the umber 9 i his or her colum of umbers player 2 crosses off the umber 7 i his or her colum of umbers (Sice 7 x 9 = 63, player 2 places a X [or O] over the 63 o the grid.) player 1 crosses off 5 i his or her colum of umbers (Sice 5 x 7 = 35, player 1 places a O [or X] over the 35 o the grid.) 5. The game eds whe ay of the followig occur: a player gets three marks i a row (as i tic-tac-toe) all of the umbers o the grid are marked off with either a X or a O all ie umbers i a player s colum of umbers are marked off b) Demostrate the game ad aswer ay questios that studets might have. Have the studets play the game. c) Have studets play the game usig these grids d) Have studets make their ow multiplicatio tic-tac-toe grids ad use them to play the game with their parter. e) Have studets discuss the strategies that they used to wi the game. Observatio Checklist Observe studets resposes to determie which facts that they ca recall easily facts that they have difficulty recallig strategies they are usig to wi the game 40 Grade 5 Mathematics: Support Documet for Teachers

63 Materials: Copies of the divisio puzzle (BLM 5.N.3.2) Orgaizatio: Idividual/Pairs a) Tell studets that their task is to fid the te divisio facts that are hidde horizotally ad vertically i the puzzle. Explai that two adjacet squares ca be used to form a 2-digit umber ad show them that the umbers 8, 1, 9, ad 9 i the first row form the fact 81 9 = 9 b) Have studets fid the remaiig facts. They should circle each fact that they fid ad the compare their aswers with their parter. c) Vary the activity by creatig multiplicatio puzzles or by creatig combied multiplicatio ad divisio puzzles. d) Have studets create their ow fact puzzles ad exchage them with the other members of the class. Observatio Checklist Check studets resposes for the followig facts: 81 9 = = = = = = = = = = 5 Number 41

64 Materials: A set of 27 cards for each group, two cards for each of the umbers 1 through 9 (BLM 5 8.5), ad ie cards with the word everyoe o it ad a umber from 1 through 9 uder it (BLM 5.N.3.3); oe-miute timers, paper ad pecils Orgaizatio: Small groups a) Tell studets that they will be playig a game ivolvig the basic facts for divisio. Explai how the game is played. 1. Shuffle the cards ad place them face dow o the playig area. 2. The player whose birthday comes first starts the game. 3. The first player turs over the top card. The player has two miutes to write as may divisio facts as he or she ca that have the umber o the card as a quotiet. The secod player acts as the timer ad says Stop! whe two miutes are up. 4. The first player receives oe poit for each correct divisio fact that has the umber o the card as the quotiet. 5. The secod player turs over the ext card ad writes as may divisio facts as he or she ca that have the umber o the card as a quotiet. The first player acts as the timer, ad says Stop! whe two miutes are up. The secod player receives oe poit for each correct fact. 6. If a player turs over a everyoe card, all players write dow as may divisio facts as they ca that have the umber o the card as a quotiet. Oe player voluteers to be the timekeeper, ad says Stop! whe two miutes are up. Each player receives oe poit for each fact that has the umber o the card as a quotiet. 7. The first player to get 50 poits is the wier. b) Demostrate how the game is played ad aswer ay questios studets might have. Have studets play the game. Observatio Checklist Observe studets resposes to determie which facts they recall easily have difficulty recallig 42 Grade 5 Mathematics: Support Documet for Teachers

65 Materials: Number cards (BLM 5 8.5), observatio form (BLM 5 8.1) Orgaizatio: Groups of 3 a) Tell studets that they will be playig a game ivolvig the basic facts for multiplicatio ad divisio. Explai how the game is played. 1. Shuffle the cards ad place them face dow i a pile i the cetre of the playig area. 2. Two studets sit facig each other while the third studet sits so they ca see the other two. The two players who are facig each other are the guessers. The third studet is the caller. 3. Each guesser chooses oe card from the deck without lookig at the card ad holds it up to his or her forehead. 4. The caller states the product of the umbers o the cards. 5. The guesser who is the first to figure out what umber is o his or her card wis both cards. 6. The player who has the most cards after 10 rouds is the wier. 7. The wier becomes the caller for the ext game. b) Demostrate how to play the game ad aswer ay questios studets might have. Have studets play the game. Observatio Checklist Observe studets resposes to determie which facts they recall easily have difficulty recallig use a strategy for have difficulty usig a strategy for Use the observatio form (BLM 5 8.1) to assess how well studets work together. Number 43

66 N OTES 44 Grade 5 Mathematics: Support Documet for Teachers

67 Grade 5: Number Edurig Uderstadigs: There are may strategies that ca be used to compute the aswers to computatioal problems. Strategies for computig the aswers to computatioal problems ivolve takig apart ad combiig umbers i a variety of ways. Geeral Outcome: Develop umber sese. SPECIFIC LEARNING OUTCOME(S): ACHIEVEMENT INDICATORS: 5.N.4 Apply metal mathematics strategies for multiplicatio, such as aexig or addig zeros halvig ad doublig usig distributive property [C, ME, R] Determie the products whe oe factor is a multiple of 10, 100, or 1000 by aexig zero or addig zeros (e.g., for 3 x 200, thik 3 x 2 ad the add two zeros). Apply halvig ad doublig whe determiig a product (e.g., 32 x 5 is the same as 16 x 10). Apply the distributive property to determie a product ivolvig multiplyig factors that are close to multiples of 10 [e.g., 98 x 7 = (100 x 7) (2 x 7)]. PRIOR KNOWLEDGE Studets should be able to do the followig: (4.N.6) Model a multiplicatio problem usig the distributive property (4.N.6) Multiply a 1-digit umber times a 2-digit whole umber or 3-digit whole umber (4.N.6) Use arrays to represet multiplicatio problems (4.N.3) Add ad subtract whole umbers less tha (4.N.6) Coect cocrete represetatios of multiplicatio problems with symbolic represetatios Number 45

68 RELATED KNOWLEDGE Studets should be able to do the followig: (5.N.3) Determie multiplicatio facts to 81 ad the related divisio facts BACKGROUND INFORMATION The term metal math is most commoly used to describe computatio that is doe without paper ad pecil or ay calculatio device such as a computer or calculator. A focus o metal math ca help studets become more adept at reasoig with umbers ad eable them to gai ew isights ito operatios ad umber relatioships. It ca also help them become adept at estimatig, a skill that has become more importat because of its practicality ad the widespread use of computers ad calculators. Metal math usually ivolves the use of ostadard algorithms such as repeated doublig or doublig ad halvig. Perhaps the most commoly used metal math strategy is the droppig ad reattachig of commo zeros. For example, to fid the product of 3 x 70, thik 3 x 7 = 21 ad the tack o the zero that was dropped to get 210. The termiology add zero should be avoided, sice it is misleadig is 21 ot 210. Teachers ca help studets become adept at metal computatio by makig metal math a itegral part of their istructioal programs. I particular, they eed to help studets develop metal math strategies that make sese to them provide frequet practice sessios that are about 10 miutes i duratio help studets develop cofidece by gradually icreasig the complexity of the metal computatios ecourage studets to use metal math wheever possible ecourage studets to develop their ow metal math strategies make sure that studets kow the differece betwee metal math ad estimatio MATHEMATICAL LANGUAGE Doublig Factor Halvig Product Multiple Multiplicatio 46 Grade 5 Mathematics: Support Documet for Teachers

69 LEARNING EXPERIENCES Assessig Prior Kowledge Note: This activity is also used to assess studets readiess for outcome 5.N.5. Materials: Paper ad pecils Orgaizatio: Idividual/Whole class a) Ask studets to solve the followig problem i two differet ways. There are eight rows of chairs i the school auditorium. If each row has 45 chairs, how may chairs are there altogether? b) Have studets share their solutios ad strategies with the other members of the class. Record the strategies studets use o the board or overhead, ad ecourage discussio by askig questios, such as: Is there aother strategy you could use to solve the problem? What is it? Which strategy is easier to use? Why do you thik it is easier? Will the strategy work for other problems ivolvig multiplicatio? Show me. Which strategy do you prefer to use? Why? Observatio Checklist Use studets resposes to determie which strategies they kow. Also, examie their resposes to determie whether they ca do the followig: idetify problem situatios that require the operatio of multiplicatio determie the correct product of a 1-digit whole umber times a 2-digit whole umber use more tha oe strategy to solve a multiplicatio problem ivolvig a 1-digit whole umber times a 2-digit whole umber. determie the product of a 1-digit umber times a 2-digit umber usig the distributive property. Number 47

70 Determie the products whe oe factor is a multiple of 10, 100, or 1000 by aexig zero or addig zeros. Materials: Base-10 blocks, strigs of 100 beads, or other place-value materials, cetimetre grid paper (BLM 5 8.9), copies of multiplicatio problems (BLM 5.N.4.1) Orgaizatio: Pairs/Whole class a) Have studets explore multiplyig by powers of te by askig them to solve the problems. Let studets kow that they ca use materials or draw diagrams to help them solve the problems. b) Have studets explai the strategies that they used to solve the problems. Ecourage studets to discuss the similarities amog the problems ad ay patters that they see by askig them questios such as: How are the problems alike? How are the solutios alike? Why do you thik this is so? Observatio Checklist Observe studets resposes to determie whether they ca do the followig: recogize that the problems describe multiplicatio situatios use models (materials or diagrams) appropriately to solve the problems represet the problems symbolically (e.g., 4 x 20 = 80) recogize that the problems ivolve calculatig the products of multiples of 10 recogize that the solutios ivolve a power or multiple of Grade 5 Mathematics: Support Documet for Teachers

71 Materials: Base-10 blocks ad calculators Orgaizatio: Whole class a) Show studets these arrays. 4 x 3 oes 4 x 3 tes 4 x 3 = 12 4 x 3 tes = 12 tes = 120 Have studets compare the two arrays ad fid the products. Ecourage discussio by askig: How are the two arrays alike? (Both have 4 rows of three.) How are the two arrays differet? (The first has oes i each row, the secod has tes.) How are the products differet? (I the first we have oes; i the secod we have tes.) b) Repeat part (a) several times. For example, have studets compare these arrays ad record the correspodig multiplicatio seteces. 8 x 2 oes ad 8 x 2 tes 8 x 2 = 16 ad 8 x 2 tes = 16 tes = x 8 oes ad 3 x 8 tes 3 x 8 = 24 ad 3 x 8 tes = 24 tes = x 5 oes ad 7 x 5 tes 7 x 5 = 35 ad 7 x 5 tes = 35 tes = x 9 oes ad 2 x 9 tes 2 x 9 = 18 ad 2 x 9 tes = 18 tes = x 6 oes ad 4 x 6 tes 4 x 6 = 24 ad 4 x 6 tes = 24 tes = x 3 oes ad 9 x 3 tes 3 x 9 = 27 ad 9 x 3 tes = 27 tes = 270 Help studets geeralize their fidigs by askig them questios such as: What patters do you see? What coclusios ca you draw? What do you kow about the product of a umber times a multiple of 10? What rule ca you use whe multiplyig a umber of oes by a multiple of 10? Number 49

72 c) Provide studets with a variety of metal math exercises. For example, ask studets to solve these problems metally: 8 x 60 3 x x 4 60 x 5 9 x x 5 10 x 7 d) Exted studets kowledge of multiplyig by a multiple of te by havig them use their calculators to solve problems like the followig: 15 x x x x x x x x x x x x x x x x x x 90 Have studets record ay patters that they see ad determie a rule for multiplyig a umber by a multiple of 10. e) Use a similar procedure for multiplyig by oes times hudreds ad oes times thousads. Observatio Checklist Observe studets resposes to determie whether they ca do the followig: recogize that oes times tes is tes recogize that whe multiplyig by te, the product has a zero i the oes positio fid the product of te times a umber recogize the rule that to fid the product of a umber times a multiple of te, multiply the umber times the digit i the tes positio, the tack o a zero to show tes Use similar criteria for multiplyig by 100 ad Grade 5 Mathematics: Support Documet for Teachers

73 Materials: Base-10 blocks, base-10 grid paper (BLM ), or graph paper Orgaizatio: Whole class a) Ask studets to use the base-10 blocks or graph paper to make a 2-tes by 3-tes array. Whe studets fiish, ask the followig questios: How may rows are there? (2 tes) How may colums are there? (3 tes) How may hudreds are there? (6 hudreds) What multiplicatio setece does the array illustrate? (2 tes x 3 tes = 6 hudreds; 20 x 30 = 600) b) Repeat part (a) several times. For example, have studets make these arrays ad record the correspodig multiplicatio seteces. 1 te x 6 tes 1 tes x 6 tes = 6 hudreds 10 x 60 = tes x 3 tes 5 tes x 3 tes = 15 hudreds 50 x 30 = tes x 4 tes 8 tes x 4 tes = 32 hudreds 80 x 40 = tes x 1 te 3 tes x 1 te = 3 hudreds 10 x 30 = tes x 5 tes 9 tes x 5 tes = 45 hudreds 90 x 50 = tes x 6 tes 6 tes x 6 tes = 36 hudreds 60 x 60 = 3600 Number 51

74 Help studets geeralize their fidigs by askig them questios such as: What patters do you see? What coclusios ca you make? What do you kow about the product of a multiple of te times a multiple of te? What ca you use to fid the product of a multiple of te times a multiple of te? c) Provide studets with a variety of metal math activities. For example, ask studets to solve these problems metally: 20 x x x x x x x x 60 Observatio Checklist Observe studets resposes to determie whether they ca do the followig: recogize that tes x tes is hudreds recogize that whe multiplyig a multiple of te times a multiple of te, there are zeros i the oes ad the tes positio of the product fid the product of a multiple of te times a multiple of te recogize the rule that to fid the product of a multiple of te times a multiple of te, fid the product of the digits i the tes positio, the tack o two zeros to show hudreds 52 Grade 5 Mathematics: Support Documet for Teachers

75 Materials: Calculators ad copies of the game sheet cut ito strips (BLM 5.N.4.2) Orgaizatio: Pairs a) Tell studets that they will be playig a game with their parter. Explai how to play the game. 1. Players take turs selectig tasks from the pile. 2. The other player uses his or her calculator to carry out the task. If the player ca perform the task, he or she scores a poit. For example, suppose a player is give the task to chage 2 ito 120 usig multiplicatio, i oe iput. The player eters 2 ito his or her calculator, ad the iputs x 60 = to get The player with the most poits after 10 rouds of the game is the wier. b) Demostrate how to play the game ad aswer ay questios studets might have. Have studets play the game. Observatio Checklist Observe studets resposes to determie whether they ca do the followig: recogize multiples of 10 recogize multiples of 100 determie the missig factor i multiplicatio problems ivolvig multiples of 10 ad 100 Number 53

76 Apply halvig ad doublig whe determiig a product. Materials: Couters Orgaizatio: Whole class a) Have studets make a rectagular array with eight rows of three couters. Ask them how may couters they have altogether. What strategy did you use to fid the total? What is aother way you could fid the total? Focus o partitioig strategies, such as double four threes or five threes ad three threes. Record the appropriate umber setece(s) o the board. Record the umber setece 8 x 3 = 24 o the board or overhead. x x x x x x x x x x x x x x x x x x x x x x x x b) Ask studets to work out how they ca chage their array to show 4 x 6. If ecessary, show studets how the 8 x 3 array ca be partitioed to illustrate 4 x 6. Ask, How may couters do you have altogether? Record the umber setece 4 x 6 = 24 o the board ad ote that 8 x 3 ad 4 x 6 have the same product. Record 8 x 3 = 4 x 6. Ask studets why they thik this is so. Help studets make the coectio betwee their actios o the materials ad the geeralizatio that oe set of factors ca be chaged ito the other by doublig ad halvig. x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x 54 Grade 5 Mathematics: Support Documet for Teachers

77 c) Have studets use their couters to test if the followig statemets are true: 4 x 5 = 2 x 10 2 x 8 = 4 x 4 4 x 3 = 2 x 6 4 x 10 = 8 x 5 d) Ask studets to match each of the followig multiplicatio problems with a equivalet problem. Have studets use their couters to check their aswers x 16 6 x x 3 3 x x 4 6 x x 4 14 x x x x 2 18 x 2 Observatio Checklist Observe studets resposes to determie whether they ca do the followig: double a umber fid half of a umber recogize that the product is the same if oe factor is doubled ad the other factor is halved determie a equivalet multiplicatio problem by doublig ad halvig its factors Materials: Paper ad pecil Orgaizatio: Whole class/small groups a) Tell studets that they will be doig a metal math activity. Explai that you will be givig them some multiplicatio problems. You will oly say a problem oce ad they will oly have eough time to record the aswer. Number 55

78 b) Give studets the followig problems: 1. What is 6 doubled? 2. What is half of 14? 3. What is 10 doubled? 4. What is 8 doubled? 5. What is half of 24? 6. What is half of 18? 7. What is 3 doubled plus half of 10? 8. What is half of 4 plus 11 doubled? 9. What is 12 doubled plus 5 doubled? 10. What is half of 30 plus half of 20? c) Have studets share their aswers with the other members of the class. d) Have each group discuss the followig questio: What metal math strategies ca you use to fid the product of 15 x 18? Have the groups share their strategies. Ecourage studets to discuss the advatages ad disadvatages of usig each of the suggested strategies. e) If o oe suggests doublig ad halvig, show studets the strategy. Explai that oe way to determie the product of 15 x 18 is to double 15 ad multiply it by half of 18. Sice 15 doubled is 30 ad half of 18 is 9, the product of 15 times 18 is the same as 30 times 9, which is equal to 270. Do two or three more examples. f) Ask studets to use the doublig ad halvig strategy to determie the followig products: x x x x x 25 Observatio Checklist Observe studets resposes to determie whether they ca do the followig: double a umber fid half a umber determie the product of two umbers by applyig a doublig ad halvig strategy 56 Grade 5 Mathematics: Support Documet for Teachers

79 Apply the distributive property to determie a product ivolvig multiplyig factors that are close to multiples of 10. Materials: Base-10 blocks or couters Orgaizatio: Whole class a) Preset studets with the followig problem: Elle keeps the stamps she collects i a book. There are 38 stamps o each page of her book. If there are six pages i her book, how may stamps does she have altogether? b) Ask studets to idetify a strategy that they would use to solve the problem. As studets explai their strategy, model it with the materials. If o oe suggests usig the distributive property of multiplicatio over subtractio, explai the strategy while modellig it with materials. There are 6 pages with 38 stamps o each page, so we eed to make a 6 x 38 array. If we add 2 to each 38, we have six 40s or 240. Now take the 12 that we added away, which leaves 228. c) Do two or three more examples ad the ask studets to solve the followig problems usig the strategy. Let studets use materials to help them solve the problems x 29 = 2. 7 x 18 = 3. 4 x 49 = 4. 2 x 58 = 5. 8 x 28 = Number 57

80 d) Have studets share their aswers ad explai how they used the strategy to solve each of the problems. e) Work with studets to represet their solutios cocretely (i.e., with base-10 blocks), pictorially (as above), ad symbolically (i.e., 6 x 38 = 6 x 40 6 x 2 or 6 groups of 38 is the same as 6 groups of 40 subtract 6 groups of 2). Observatio Checklist Observe studets resposes to determie whether they ca do the followig: apply the distributive property to determie a product whe oe of the factors is close to a multiple of te calculate the correct product of a 1-digit umber times a multiple of te calculate the correct differece betwee two whole umbers explai how to use the strategy to fid the product whe oe of the factors is close to a multiple of te Materials: Copies of the Products activity (BLM 5.N.4.3) ad base-10 materials Orgaizatio: Idividual/Parter a) Have studets complete the activity. b) Have studets check their aswers with their parter. If discrepacies arise, have studets use materials to determie the correct aswer. c) Have studets discuss the strategies that they used to fid the products i part B. Ecourage studets to describe the strategies that they used ad explai why they choose them. Ecourage studets to use cocrete, pictorial, ad symbolic represetatios. Observatio Checklist Observe studets resposes to determie whether they ca do the followig: apply the distributive property to determie a product whe oe of the factors is close to a multiple of 10 calculate the correct product of a 1-digit umber times a multiple of 10 calculate the correct differece betwee two whole umbers recogize whe to use the distributive strategy ad the halvig ad doublig strategy 58 Grade 5 Mathematics: Support Documet for Teachers

81 Grade 5: Number Edurig Uderstadigs: There are a variety of strategies that ca be used to compute the aswers to computatioal problems. Strategies for computig the aswers to computatioal problems ivolve takig apart ad combiig umbers i a variety of ways. Geeral Outcome: Develop umber sese. SPECIFIC LEARNING OUTCOME(S): ACHIEVEMENT INDICATORS: 5.N.5 Demostrate a uderstadig of multiplicatio (2-digit x 2-digit umerals) to solve problems. [C, CN, PS, V] Illustrate partial products i expaded otatio for both factors [e.g., for 36 x 42, determie the partial products for (30 + 6) x (40 + 2)]. Represet both 2-digit factors i expaded otatio to illustrate the distributive property [e.g., to determie the partial product of 36 x 42, (30 + 6) x (40 + 2) = 30 x x x x 2 = = 1512]. Model the steps for multiplyig 2-digit factors usig a array ad base-10 blocks, ad record the process symbolically. Describe a solutio procedure for determiig the product of two 2-digit factors usig a pictorial represetatio, such as a area model. Solve a multiplicatio problem i cotext usig persoal strategies, ad record the process. Number 59

82 PRIOR KNOWLEDGE Studets should be able to do the followig: (4.N.6) Use persoal strategies to solve multiplicatio problems ivolvig a 1-digit umber times a 2-digit or 3-digit umber (4.N.6) Use arrays to represet multiplicatio problems (4.N.6) Coect cocrete represetatios of multiplicatio problems with symbolic represetatios (4.N.6) Model a multiplicatio product usig the distributive property (4.N.6) Estimate the product of a 1-digit umber ad a 2-digit or 3-digit umber (4.N.3) Add ad subtract whole umbers less tha RELATED KNOWLEDGE Studets should be able to do the followig: (5.N.3) Determie multiplicatio facts to 81 (5.N.3) Determie the product of two umbers whe oe of the factors is a multiple of 10, 100, or 1000 (5.N.4) Determie the product of a multiple of 10 times a multiple of 10 BACKGROUND INFORMATION A algorithm is a system of fiite procedures for solvig a particular class of problems. The best kow algorithms are the traditioal paper-ad-pecil procedures for addig, subtractig, multiplyig, ad dividig. The focus o these algorithms is beig replaced by a emphasis o metal mathematics, estimatio, the use of techology, the developmet of iveted procedures, ad the use of alterative algorithms, such as area model multiplicatio, ad addig up to solve subtractio problems. By ecouragig studets to develop their ow computatio strategies ad allowig them to use alterative algorithms, the emphasis i mathematics istructio is shifted to reasoig, problem solvig, ad coceptual uderstadig. Providig studets with opportuities to ivet their ow strategies ad use alterative algorithms ehaces their umber ad operatio sese. Studets become more flexible i their thikig, more aware of the differet ways to solve a problem, ad more adept at selectig the most appropriate procedure for solvig a problem. Discussio of these algorithms ca also help studets develop better reasoig ad commuicatio skills. 60 Grade 5 Mathematics: Support Documet for Teachers

83 Although teachig the traditioal algorithms for computatio is ot icorrect, please be ecouraged to follow the cocrete, pictorial, symbolic sequece of teachig. The importat idea is to allow studets to costruct meaig, ot memorize procedures without uderstadig. Studets miscoceptios (or their fuzzy uderstadigs) ca be reiforced by a poorly uderstood algorithm. Teachers ca facilitate studets uderstadig ad use of a variety of computatioal strategies by providig a supportig ad acceptig eviromet allowig time for exploratio ad experimetatio embeddig computatioal tasks i real-life situatios allowig studets to discuss, aalyze, ad compare their solutio strategies uderstadig that a child eeds to be efficiet at computatio ad that this looks differet for each studet MATHEMATICAL LANGUAGE Array Expaded otatio Factor Multiplicatio Multiply Partial product Product Number 61

84 LEARNING EXPERIENCES Assessig Prior Kowledge Note: This activity is also used to assess studets readiess for outcome 5.N.4. Materials: Paper ad pecils Orgaizatio: Idividual/Whole class a) Ask studets to solve the followig problem i two differet ways: There are eight rows of chairs i the schools auditorium. If each row has 45 chairs, how may chairs are there altogether? b) Have studets share their solutios ad strategies with the other members of the class. Record their strategies o the board or overhead, ad ecourage discussio by askig questios, such as the followig. Is there aother strategy you could use to solve the problem? What is it? Which strategy is easier to use? Why do you thik it is easier? Will the strategy work for other problems ivolvig multiplicatio? Show me. Which strategy do you prefer to use? Why? Observatio Checklist Use studets resposes to determie which strategies they kow. Also, examie their resposes to determie whether they ca do the followig: idetify problem situatios that require the operatio of multiplicatio determie the correct product of a 1-digit umber times a 2-digit umber use more tha oe strategy to solve a multiplicatio problem ivolvig a 1-digit umber times a 2-digit umber determie the product of a 1-digit umber times a 2-digit umber usig the distributive property 62 Grade 5 Mathematics: Support Documet for Teachers

85 Model the steps for multiplyig 2-digit factors usig a array ad base-10 blocks, ad record the process symbolically. Describe the solutio procedure for determiig the product of 2-digit factors usig a pictorial represetatio, such as a area model. Materials: Dot paper Orgaizatio: Whole class Note: Before begiig this task, you may eed to determie the studets uderstadig of area as the amout of space take up by a 2-D shape. a) Preset studets with the followig situatio ad ask them what they eed to do to solve the problem: There are 23 rows of cars i the shoppig mall s parkig lot. Each row has 27 cars i it. How may cars are parked i the lot? b) Have studets model the scearios usig base-10 blocks. Each uit cube represets oe parkig stall. c) Have studets draw a border aroud a array (23 x 27) that represets the cars i the parkig lot. Ask them to partitio the array i a way that will make it easier to fid the total umber of dots. d) Have studets share how they partitioed the array. Itroduce the followig ways of partitioig the array if studets do ot suggest them, ad discuss how they are related. Studets should ot yet be required to record the process symbolically. Sice 20 x 27 = 20 x (20 + 7) = (20 x 20) + (20 x 7) ad 3 x 27 = 3 x (20 + 7) = (3 x 20) + (3 x 7) the (20 + 3)27 = (20 x 20) + (20 x 7 ) + (20 x 3) + (3 x 7) Number 63

86 Emphasize that the whole umbers ca be broke apart ito more coveiet pieces i order to make the computatio easier. Make sure that studets uderstad that this will ot affect their aswer. e) Do aother example of usig place value to fid the product of two umbers that is, show studets how the product of 35 x 45 ca be foud by partitioig a array ito four parts that ca be associated with the expaded forms of the factors ad fidig the sum of the parts. Oce studets become comfortable, they ca draw the represetatio without eedig the appropriate umber of dots. Use the sum of the idividual (partial) products to fid the total value of the origial product: 35 x 45 = (30 + 5) x (40 + 5) = (30 x 40) + (30 x 5) + (5 x 40) + (5 x 5) = = 1575 f) Ask studets to solve each of the followig problems by partitioig a array ad fidig the sum of the parts, as illustrated i part (e). Have them represet these seteces cocretely ad pictorially i as may ways as possible. 42 x x x x Grade 5 Mathematics: Support Documet for Teachers

87 Observatio Checklist Examie studets resposes to determie whether they ca do the followig: illustrate partial products i expaded otatio for both factors represet both 2-digit factors i expaded otatio to illustrate the distributive property describe a solutio procedure for determiig the product of two 2-digit umbers usig pictorial represetatios calculate correct sums ad products model a 2 x 2 digit multiplicatio cocretely ad pictorially Examie studets resposes to determie whether ay errors are due to carelessess ot kowig a basic multiplicatio or additio fact a procedural error (e.g., reamig ad regroupig icorrectly whe fidig the sum of two umbers or ot fidig all the partial products i a multiplicatio problem) Illustrate partial products i expaded otatio for both factors. Represet both 2-digit factors i expaded otatio to illustrate the distributive property. Model the steps for multiplyig 2-digit factors usig a array ad base-10 blocks, ad record the process symbolically. Describe a solutio procedure for determiig the product of two 2-digit factors usig a pictorial represetatio, such as a area model. Materials: Base-10 blocks Orgaizatio: Whole class/pairs a) Ask studets to solve the followig problem: The studets erolled i the commuity cetre s swimmig program are lied up i rows to get their picture take. There are 12 rows with 13 studets i each row. How may studets are erolled i the swimmig program? Number 65

88 b) Have studets share their solutios to the problem. Ecourage them to discuss the strategies they used by askig the followig questios: How did you fid your aswer? Will your strategy work for other problems? Is there aother strategy you could use? Which strategy is more efficiet? c) If studets do ot suggest usig the distributive property, illustrate the strategy by makig a array with the base-10 blocks. The array should cosist of 12 rows of 13 blocks (or 13 rows of 12 blocks). 12 x 13 = 12 (10 + 3) Sice 10 logs equal 1 flat ad 10 uits equal 1 log, the array ca be simplified by exchagig these blocks. d) Model several more 2 x 2 digit multiplicatio scearios usig base-10 blocks. 66 Grade 5 Mathematics: Support Documet for Teachers

89 e) Emphasize that the array has bee partitioed ito four parts. Each part represets a partial product. These four parts ca be show i a diagram called a area model: The product of the origial questio is simply the sum of each for the four sectios. f) The partial products are the result of expressig each factor i expaded otatio, ad multiplyig each added i the first factor by each added i the secod factor. The product is the sum of the partial products ad ca be expressed symbolically: 12 x 13 = (10 + 2) x (10 + 3) = (10 x 10) + (10 x 3) + (2 x 10) + (2 x 3) = 156. g) Do oe or two more examples, the have studets use the base-10 blocks to solve these problems. Ecourage studets to record the procedure they used for each problem, cocretely, pictorially, ad symbolically. 15 x x x x x 31 Number 67

90 Observatio Checklist Check studets resposes to determie whether they ca do the followig: illustrate partial products i expaded otatio for both factors represet both 2-digit factors i expaded otatio to illustrate the distributive property model the procedure for fidig the product of two 2-digit factors usig a array ad base-10 blocks record the procedure symbolically solve a multiplicatio problem i cotext usig persoal strategies, ad record the procedure cocretely, pictorially, ad symbolically Examie studets resposes to determie if a mistake is due to carelessess ot kowig a basic multiplicatio or additio fact a procedural error (e.g., reamig ad regroupig icorrectly whe fidig the sum of two umbers or ot fidig all the partial products i a multiplicatio problem) Illustrate partial products i expaded otatio for both factors. Represet both 2-digit factors i expaded otatio to illustrate the distributive property. Materials: Number cards (BLM 5 8.5), paper ad pecils. Orgaizatio: Groups of 2 4/Whole class a) Tell studets that they will be playig a game ivolvig multiplicatio of 2-digit umbers. Explai how the game is played. 1. Shuffle the cards ad place them face dow i a pile i the cetre of the playig area. 2. Each player draws four cards ad arrages them to form two 2-digit whole umbers that will give him or her the largest possible product. 3. Players place their umbers face up i frot of them. Each player writes his or her umbers as the sum of tes ad oes, the multiplies each member of the first umber times each member of the secod umber. For example, if 1, 3, 5, ad 7 are draw, a player ca form the umbers 71 ad 53, the fid the product of the two umbers by multiplyig (70 + 1)(50 + 3) that is, (70 +1)(50 + 3) = (70 x 50) + (70 x 3) + (1 x 50) + (1 x 3) = = Grade 5 Mathematics: Support Documet for Teachers

91 4. Players check each other s calculatio. The player with the largest product gets oe poit. 5. The wier is the player with the most poits after five rouds. b) Demostrate how the game is played ad aswer ay questios studets might have. Have studets play the game. c) Have studets discuss the strategies that they used to wi the game. Begi the discussio by askig questios, such as What four cards did you draw? How did you decide which umbers to form with the cards you drew? What other umbers could you have made? How did you decide which umbers would give you the largest product? Observatio Checklist Observe studets resposes to determie whether they ca do the followig: illustrate partial products i expaded otatio for both factors preset both 2-digit factors i expaded otatio to illustrate the distributive property determie the correct product of two 2-digit whole umbers give reasoable estimates of the product of two 2-digit umbers recogize that the larger the factors, the greater the product Multiply 2-digit umbers. Materials: Copies of the basic facts multiplicatio table (BLM ) Orgaizatio: Whole class a) Show studets a copy of the multiplicatio table ad ask them to describe ay patters they see. b) Show studets the square labelled A ad have them fid the product of the opposite pairs of vertices (2 x 6 ad 3 x 4). Ask them what they otice about the products. Number 69

92 c) Now show studets the square labelled B ad have them fid the product of the opposite pairs of vertices (18 x 54 ad 36 x 27). Agai, ask studets what they otice about the products. Ask studets if they thik this would be true for other squares. d) Have studets work with their parters to fid the product of the opposite pairs of vertices of 10 differet squares o the multiplicatio table. Make sure studets vary the size of the squares. e) Have studets discuss their fidigs with the other members of the class. f) Exted the activity by askig studets to determie whether the patter they foud for squares also holds for rectagles. Observatio Checklist Observe studets resposes to determie whether they ca do the followig: recall the basic facts for multiplicatio determie the correct product of a 1-digit umber times a 2-digit umber determie the correct product of a 2-digit umber times a 2-digit umber use a strategy for calculatig the product of two umbers that is mathematically correct ad efficiet ad ca be geeralized (applied to other multiplicatio problems) recogize that the products of the opposite pairs of vertices of squares (ad other rectagles) o the multiplicatio table are equal 70 Grade 5 Mathematics: Support Documet for Teachers

93 Materials: Number cards (BLM 5 8.5), dice, paper ad pecils Orgaizatio: Pairs a) Tell studets that they will be playig the game target. Explai how the game is played. 1. Shuffle the cards ad place them face dow i a pile i the cetre of the playig area (the cards will eed to be reshuffled after each roud of play). 2. Players take turs rollig a die ad drawig four cards. 3. O a tur, a player rolls the die ad uses the chart show below to determie the target rage of the product. Number Rolled Target Rage or less more tha The player the draws four cards ad uses them to form two umbers whose product the player thiks falls withi the target rage. The player does ot have to use all four cards. A umber caot begi with a zero. 5. The player multiplies the two umbers. The other player checks his or her parter s calculatios. If the product is i the target rage, he or she gets oe poit. If the product is outside the rage, o poits are awarded. 6. The first player to get five poits is the wier. b) Demostrate how the game is played ad aswer ay questios studets might have. Have studets play the game. Observatio Checklist Observe studets resposes to determie whether they ca do the followig: give reasoable estimates of the product of a 2-digit umber times a 1-digit or 2-digit umber determie the product of 2-digit umbers usig the distributive property use cocrete or pictorial represetatios to determie the product of two umbers use a efficiet strategy for fidig the product of two umbers Number 71

94 N OTES 72 Grade 5 Mathematics: Support Documet for Teachers

95 Grade 5: Number Edurig Uderstadigs: There are may strategies that ca be used to determie the aswers to computatioal problems. Strategies for computig the aswers to computatioal problems ivolve takig apart ad combiig umbers i a variety of ways. Geeral Outcome: Develop umber sese. SPECIFIC LEARNING OUTCOME(S): ACHIEVEMENT INDICATORS: 5.N.6 Demostrate a uderstadig of divisio (3-digit umerals by 1-digit umerals) with ad without cocrete materials, ad iterpret remaiders to solve problems. [C, CN, PS] Model the divisio process as equal sharig usig base-10 blocks, ad record it symbolically. Explai that the iterpretatio of a remaider depeds o the cotext: Igore the remaider (e.g., makig teams of 4 from 22 people) Roud up the quotiet (e.g., the umber of five passeger cars required to trasport 113 people) Express remaiders as fractios (e.g., five apples shared by two people) Express remaiders as decimals (e.g., measuremet or moey) Solve a divisio problem i cotext usig persoal strategies, ad record the process. Number 73

96 PRIOR KNOWLEDGE Studets should be able to do the followig: (4.N.7) Solve divisio problems ivolvig 2-digit whole umber divideds by 1-digit whole umber divisors (4.N. 7) Use persoal strategies to solve divisio problems ivolvig 1-digit whole umber divisors ad 2-digit whole umber divideds (4.N.7) Estimate the quotiets of divisio problems ivolvig 1-digit whole umber divisors ad 2-digit whole umber divideds (4.N.7) Relate divisio to multiplicatio (4.N.3) Add ad subtract whole umbers to RELATED KNOWLEDGE Studets should be able to do the followig: (5.N.3) Determie basic multiplicatio ad divisio facts to 81 (5.N.3) Use strategies such as frot-ed roudig, compesatio, ad compatible umbers to estimate the aswers to computatioal problems (5.N.4) Divide multiples of 10 ad 100 by whole umbers less tha 10 MATHEMATICAL LANGUAGE Divisio Divisible Divisor Divided Quotiet Remaider 74 Grade 5 Mathematics: Support Documet for Teachers

97 LEARNING EXPERIENCES Assessig Prior Kowledge Materials: Math jourals Orgaizatio: Idividual/Whole class a) Preset studets with the followig situatio: Mauel s calculator is broke. He eeds to fid the quotiet of 57 6, but he has forgotte how to divide. Help him out by explaiig how he ca fid the quotiet. b) Have studets share their aswers. Record their strategies o the board or overhead ad ecourage discussio of them by askig questios, such as the followig: What strategy could Mauel use to fid the quotiet of 57 6? Will your strategy work for other divisio problems? Show me. What is aother strategy he could use? How are the strategies alike? How do they differ? Which strategy do you prefer? Why? Observatio Checklist Observe studets resposes to determie whether they ca do the followig: correctly use the terms divided, divisor, quotiet, remaider, ad divide recogize that divisio ivolves partitioig ito equal parts recogize that divisio ivolves formig equal-sized groups describe a strategy that is mathematically correct use a efficiet strategy that ca be used for all divisio problems recogize that the divisio is ueve (i.e., there is a remaider of 3) Number 75

98 Model the divisio process as equal sharig usig base-10 blocks, ad record it symbolically. Materials: Base-10 materials such as base-10 blocks or digi-blocks Orgaizatio: Whole class a) Ask studets to solve the followig problem: Marcy works i the library. She eeds to put the same umber of books o four shelves. There are 128 books. How may books should she put o each shelf? b) Have the studets share their solutios with the other members of the class. Ecourage discussio by askig questios, such as the followig: What strategy did you use to determie the solutio to the problem? Is there aother strategy you could use to solve the problem? How are the strategies alike? Which strategy is more efficiet? Why do you thik this? How do you kow your solutio is correct? c) Model the divisio process as equal sharig if studets do ot suggest this strategy. For example, show studets 128 with the blocks. Explai oe strategy by tellig studets that 128 eeds to be partitioed ito four equal parts. Sice there are ot eough flats (100s) to put oe i each part, the flat ca be exchaged for 10 logs so 128 becomes 12 tes ad 8 oes. 76 Grade 5 Mathematics: Support Documet for Teachers

99 Next, partitio the tes ito four equivalet parts. The partitio the oes ito four equivalet parts. Now, put the pieces together to make four parts with 32 i each part. Record the process as = 12 tes ad 8 oes 4 = 3 tes ad 2 oes, or 32, or as d) Do two or three more examples, ad the have studets use the strategy to solve the followig problems. Have studets record their aswers symbolically = o 3 tes, 2 oes ) tes, oes or ) = o = o = o = o Observatio Checklist Observe studets resposes to determie whether they ca do the followig: model the divisio process as equal sharig usig base-10 blocks record the divisio process symbolically Number 77

100 Model the divisio process as equal sharig usig base-10 blocks, ad record it symbolically. Explai that the iterpretatio of a remaider depeds o the cotext. Solve a divisio problem i cotext usig persoal strategies, ad record the process. Materials: Base-10 materials such as base-10 blocks or digi-blocks. Orgaizatio: Whole class/small groups a) Have studets solve the followig problem: Max ad Joshua are resposible for settig up tables for the school s baquet. Eight people ca sit at each table. How may tables do Max ad Joshua eed to set up if there are 150 people goig to the baquet? b) Have studets share their aswers ad discuss how the remaider should be iterpreted. Ecourage studets to explai why the quotiet eeds to be icreased by 1. c) Explai that remaiders are a commo occurrece i divisio, ad how they are iterpreted depeds o the problem statemet. d) Have each group of studets solve the followig problems ad decide how the remaider should be iterpreted. Some studets will be wearig sashes for the sprig school dace festival. Four metres of ribbo are eeded for each sash. Mr. Sachez has 135 metres of ribbo. How may sashes ca he make? Sally has 26 cookies to share evely with three frieds. How may cookies does each get? Marshall saved $ He spet half of the moey he saved to buy three video games. How much did he sped o the video games? Sevety-four people are goig o a campig trip. Six people ca go i each car. How may cars are eeded? e) Have the groups share their aswers ad discuss how they iterpreted the remaider. Ecourage studets to explai their reasos for how they hadled the remaider. Summarize the discussio by stressig that there are four ways the remaider ca be hadled. Depedig o the problem statemet, the remaider ca be expressed as a fractio or a decimal; it ca be dropped or it ca require the quotiet to be icreased by 1. f) Assig each group oe of the ways that a remaider ca be iterpreted, ad ask them to write a problem that ivolves the iterpretatio that you gave them. Have the groups give their problem to the other members of the class to solve ad decide how to iterpret the remaider. 78 Grade 5 Mathematics: Support Documet for Teachers

101 Observatio Checklist Observe studets resposes to determie whether they ca do the followig: model the divisio process as equal sharig usig base-10 materials solve a divisio problem usig persoal strategies, ad record the process iterpret the remaider with respect to the cotext write a divisio problem that ivolves droppig the remaider, icreasig the quotiet by 1, expressig the remaider as a fractio, or expressig the remaider as a decimal Model the divisio process as equal sharig usig base-10 blocks, ad record it symbolically. Explai that the iterpretatio of a remaider depeds o the cotext. Materials: Dice, base-10 blocks or digi-blocks Orgaizatio: Pairs a) Tell studets that they will be doig a activity ivolvig divisio. Explai that they will be rollig a die four times ad usig the four umbers that they roll to create as may divisio problems as they ca that have o remaider. Each problem that they create must ivolve a 3-digit whole umber divided ad a 1-digit whole umber divisor. Let studets kow that they ca use the materials to help them determie whether there is a remaider, ad that they should record both the umbers that they rolled ad the problems that they created that have o remaiders. If they are coviced that there are o problems that ca be created that have a remaider of zero, they should record the umbers that they rolled, ad explai i writig why they thik this is so. b) Do a example of the activity with the studets. For istace, suppose 2, 5, 6, ad 4 are rolled. Oe problem that has o remaider that ca be made with the umbers is 564 2; aother is c) Have studets repeat the activity, but this time have them create problems with the largest possible remaider. Number 79

102 Observatio Checklist Aalyze studets resposes to determie whether they ca do the followig: devise a strategy determiig all the possible problems that ca be cosidered solve divisio problems i cotext usig persoal strategies, ad record the process use a strategy that is efficiet, mathematically correct, ad ca be geeralized (work for all problems) recogize divisio problems that have o remaiders recogize divisio problems that have remaiders idetify the largest possible remaider for a give divisor Solve a divisio problem i cotext usig persoal strategies, ad record the process. Materials: Divisio problem cards (BLM 5.N.6.1), timer or clock with secod had, calculator for each group. Orgaizatio: Groups of 3/Whole class a) Tell studets that they will be playig a game that ivolves estimatig quotiets. Explai how the game is played. 1. Shuffle the cards ad place them face dow i the middle of the playig area. 2. Oe studet i each group acts as the timer. The other two studets play agaist each other. 3. The timer turs over a card ad says, Go! The other two studets ow have 10 secods to write dow their estimate of the quotiet. 4. Whe the 10 secods are up, the timer says, Stop! ad the players must put their pecils dow. The timer the uses the calculator to fid the quotiet. 5. The player whose estimate is closest to the actual aswer wis the card. If there is a tie, o player receives a card. 6. The game is over whe there are o cards left. The player with the most cards wis the game. b) Demostrate how the game is played ad aswer ay questios that studets might have. Have studets play the game. 80 Grade 5 Mathematics: Support Documet for Teachers

103 c) Have studets discuss the strategies they used to estimate the quotiets. Begi the discussio by askig questios, such as the followig: How did you estimate the quotiet of 748 2? How ca you get a closer estimate? Is there aother strategy you could use to estimate the quotiet? d) Have the wier of the game become the timer ad play the game agai. e) Have studets make up their ow divisio problem cards ad use them to play the game. Observatio Checklist Observe studets resposes to determie whether they ca do the followig: give reasoable estimates of the quotiets use techology to solve divisio problems use strategies such as frot-ed roudig ad compesatio to determie reasoable estimates of the quotiet determie basic facts for divisio Model the divisio process as equal sharig usig base-10 blocks, ad record the process symbolically. Explai that the iterpretatio of a remaider depeds o the cotext. Solve a divisio problem i cotext usig persoal strategies. Materials: Base-10 blocks or digi blocks for studets who eed them; paper ad pecils Orgaizatio: Whole class/small groups a) Ask studets to solve the followig problem: 24 3 = o. Explai that there are other problems related to 24 3 that also have a quotiet of 8, but these problems have a remaider. For example: 25 3 is 8 with a remaider of 1. Ask studets to fid aother problem related to 24 3 that has a quotiet of 8 ad a remaider (26 3). Have studets fid all the divisio problems with a remaider related to 36 4 = o (37 4 = o, 38 4 = o, ad 39 4 = o). Number 81

104 b) Assig each group oe of the divisio problems show below, ad ask them to fid all the related divisio problems as well as the remaider for each problem = o (151 5 = o, = o, = o, = o) = o = o = o = o c) Have each group share its fidigs with the other members of the class ad record their aswers o the board. Ecourage discussio by askig questios, such as the followig: What patters do you see? If you divide a 634 by 12, how may related divisio problems could you write? Why? What do you kow about the remaiders whe you divide by 12? Whe you divide a umber, what is the smallest remaider you ca have? What is the largest remaider you ca have? Observatio Checklist Observe studets resposes to determie whether they ca do the followig: model the divisio process as equal sharig usig base-10 blocks solve divisio problems i cotext usig a persoal strategy ad record the process symbolically recogize that the larger the divisor, the more related divisio problems recogize that the smallest possible remaider is 0 ad the largest possible remaider is always 1 less tha the divisor recogize that the umber of possible remaiders is the same as the divisor (e.g., whe dividig by 3, there are 3 possible remaiders: 0, 1, ad 2) 82 Grade 5 Mathematics: Support Documet for Teachers

105 Explai that the iterpretatio of a remaider depeds o the cotext. Solve a divisio problem i cotext usig persoal strategies ad record the process. Materials: Two dice ad a playig board for each group (BLM 5.N.6.2). Oe die should have the umbers 1, 3, 5, 7, 7, ad 9 writte o it ad the other die should have the umbers 1, 2, 4, 6, 6, ad 8 writte o it. Orgaizatio: Groups with 3 or 4 studets a) Tell studets that they will be playig a game ivolvig remaiders i divisio. Explai how the game is played. 1. Players take turs, rotatig clockwise. 2. Durig a tur, a player crosses off ay uused umber o the playig board. He or she the chooses oe of the dice ad rolls it. Next, the player divides the umber rolled ito the umber that was crossed off ad fids the remaider. 3. The remaider for the divisio problem is the player s score for that roud. For example, a player crosses off 113 ad chooses to roll the die with the umbers 1, 3, 5, 7, 7, ad 9 o it. If the umber rolled is a 5, the player divides 5 ito 113 ad gets a quotiet of 22 with a remaider of 3. The player s score for that roud is If a player otices a mistake that aother player makes, he or she gets that player s score for the tur. 5. The game eds whe all the umbers have bee crossed off. The player with the largest cumulative score is the wier. b) Demostrate how the game is played ad aswer ay questios studets might have. Have studets play the game. c) Have studets play the game agai, but this time the player with the smallest cumulative score is the wier. d) Have studets create their ow divisio game board ad use it to play the game with their parter. Number 83

106 Observatio Checklist Observe studets resposes to determie whether they ca do the followig: solve divisio problems i cotext usig a persoal strategy, ad record the process symbolically use a strategy that is efficiet, is mathematically correct, ad ca be geeralized develop a strategy for playig the game recogize problems that have bee solved icorrectly 84 Grade 5 Mathematics: Support Documet for Teachers

107 Grade 5: Number Edurig Uderstadigs: Equivalet fractios are fractios that represet the same value. Differet strategies ca be used to compare fractios with ulike deomiators. Geeral Outcome: Develop umber sese. SPECIFIC LEARNING OUTCOME(S): ACHIEVEMENT INDICATORS: 5.N.7 Demostrate a uderstadig of fractios by usig cocrete ad pictorial represetatios to create sets of equivalet fractios compare fractios with like ad ulike deomiators [C, CN, PS, R, V] Create a set of equivalet fractios ad explai why there are may equivalet fractios for ay fractio usig cocrete materials. Model ad explai that equivalet fractios represet the same quatity. Determie if two fractios are equivalet usig cocrete materials or pictorial represetatios. Formulate ad verify a rule for developig a set of equivalet fractios. Idetify equivalet fractios for a fractio. Compare two fractios with ulike deomiators by creatig equivalet fractios. Positio a set of fractios with like ad ulike deomiators o a umber lie (vertical or horizotal), ad explai strategies used to determie the order. Number 85

108 PRIOR KNOWLEDGE Studets should be able to do the followig: (4.N.8) Demostrate a uderstadig of fractios less tha or equal to oe usig cocrete ad pictorial represetatios (4.N.8) Name ad record fractios for the parts of a whole or a set (4.N.8) Compare ad order fractios with like umerators or like deomiators (4.N.8) Provide examples of where fractios are used (4.N.10) Relate fractios to decimals BACKGROUND INFORMATION Equivalet fractios are fractios that represet the same value. For example,,,, ad are differet ames for the same umber. Studets eed a uderstadig of 12 equivalece i order to compare, order, add, ad subtract fractios. Studets ofte fid fractios cofusig ad difficult to comprehed. Difficulties with learig fractios ca arise from istructio that emphasizes procedural kowledge rather tha coceptual kowledge. They also arise because the whole umber cocepts that studets leared do ot always apply to fractios. For example, whe the umerators are the same ad the deomiators are differet, the larger of two fractios is determied by comparig deomiators usig order ideas that are the iverse of 1 1 those for whole umbers. For example, 4 is less tha 5, but is less tha. However, 5 4 whe the deomiators are the same, the larger of two fractios is determied by comparig the umerators usig whole umber cocepts. For example, 3 is less tha so is less tha. 8 8 To help studets overcome their difficulties with learig fractios, istructio should focus o developig cocepts rather tha o abstract rules. Learig experieces that emphasize exploratio ad the maipulatio of a variety of cocrete materials ad pictorial represetatios are key to helpig studets develop meaig for fractio cocepts. Rules for maipulatig fractios should oly be itroduced after studets develop a uderstadig of the cocepts. If the rules are itroduced too soo, studets ed up memorizig them. Rules that are memorized without uderstadig are ofte forgotte or applied iappropriately. 86 Grade 5 Mathematics: Support Documet for Teachers

109 MATHEMATICAL LANGUAGE Deomiator Equivalet Fractio Greater tha Less tha Numerator LEARNING EXPERIENCES Assessig Prior Kowledge Materials: Copies of cocept descriptio sheet (BLM 5 8.2) Orgaizatio: Idividual a) Tell studets that i the ext few lessos they will be learig about fractios, but before they begi you eed to fid out what they already kow about fractios. b) Ask studets to complete the cocept developmet sheet. Let them kow that it is all right if they caot thik of aythig to put i a sectio. They will have aother opportuity to complete the sheet whe they lear more about fractios. c) Whe studets fiish, begi a discussio of fractios by askig, What is a fractio? What is a example of a fractio? As the discussio progresses, clear up ay miscoceptios studets may have ad make sure that they see a variety of examples ad o-examples. d) Have studets complete the cocept developmet sheet agai. Observatio Checklist Observe studets resposes to determie whether they ca do the followig: recogize that a whole ca be a regio or a collectio of objects recogize that the parts of a whole must be equivalet give appropriate examples ad o-examples of fractios write a symbol for a fractio kow the terms deomiator ad umerator Number 87

110 Determie if two fractios are equivalet usig cocrete materials or pictorial represetatios. Create a set of equivalet fractios ad explai why there are may equivalet fractios for ay fractio usig cocrete materials. Idetify equivalet fractios for a fractio. Materials: Two-coloured chips Orgaizatios: 3 groups of studets a) Give each studet 12 chips ad divide the studets ito three groups: A, B, ad C. 1 Give each group differet istructios. For example, tell Team A to show, Team B to show, ad Team C to show b) Whe studets fiish, have them compare their results. Studets should ote that everyoe has tured over four chips ad the istructios yield the same results Explai that,, ad are equivalet fractios because they are differet ames for the same amout = 4 6 = 12 c) Give differet istructios that yield equivalet results. For example, ask group A to show, group B to show, ad group C to show. Ask questios such as, Why is equivalet to? Is the same as eve whe you start with 18 chips? Why? Have studets verify their aswer by maipulatig the chips. d) Cotiue the activity, but chage the umber of chips ad the istructios that result i equivalet fractios. 1 e) Vary the activity by havig oe team give the istructio (e.g., show ad the other 2 two teams give istructios that they thik will yield the same results whe carried out). Have studets carry out the istructios to see if they create equivalet fractios. 88 Grade 5 Mathematics: Support Documet for Teachers

111 Observatio Checklist Observe studets resposes to determie whether they ca do the followig: determie if two fractios are equvalet usig cocrete materials create a set of equivalet fractios ad explai why they are equivalet idetify equivalet fractios for a fractio Materials: Egg cartos ad couters Orgaizatio: Pairs 1 a) Have studets use their egg cartos ad couters to show you of a doze. The ask studets to show you of a doze. Now have studets show you of a doze Whe studets show two couters i their egg cartos to illustrate of a doze, say, But you just said that is of a doze! Ask, Which is it: or? Why are they equivalet? b) Ask studets to use their egg cartos to fid as may equivalet fractios as they ca (studets should fid fractios equivalet to,,,,, ad ). Have studets record their fidigs pictorially ad symbolically = 6 6 = 12 c) Ask studets to fid two fractios that are ot equivalet ad explai why they are ot equivalet. Agai, have studets record their fidigs pictorially ad symbolically. d) Have studets share their fidigs with the other members of the class. Ecourage studets to explai why two fractios are equivalet or ot equivalet. Number 89

112 Observatio Checklist Observe studets resposes to determie whether they ca do the followig: model ad explai that equivalet fractios represet the same quatity distiguish betwee equivalet ad o-equivalet fractios idetify equivalet fractios for a fractio determie if two fractios are equivalet usig cocrete materials or pictorial represetatios Model ad explai that equivalet fractios represet the same quatity. Determie if two fractios are equivalet usig cocrete materials or pictorial represetatios. Idetify equivalet fractios for a fractio. Materials: Fractio blocks or fractio bars Orgaizatio: Small groups Note: Colours may vary based o materials used. Be sure to determie values before you begi. a) Ask studets to show you the yellow block. Tell them that this block represets oehalf of the whole pik fractio block. Have them ame each of the other blocks as a fractio of the pik block. b) Ask studets to use their blocks to fid out how may reds fit o the yellow block, ad to write a umber setece that 2 1 shows the relatioship betwee the red ad yellow blocks = 4 2 how may blues fit o the yellow block, ad to write a umber setece that shows the relatioship how may grees fit o the yellow block, ad to write a umber setece that shows the relatioship what oe-eighth would look like, ad to fid out how may eights would fit o the yellow block (Have them write a umber setece that shows this relatioship.) c) Ask studets to list all the fractios they foud that are equivalet to oe-half Aswer: = = = =, ad to describe ay patters or relatioships that they see. 90 Grade 5 Mathematics: Support Documet for Teachers

113 Observatio Checklist Observe studets resposes to determie whether they ca do the followig: ame the fractioal part of each block i relatio to the larger pik block create a set of equivalet fractios ad explai why there are may equivalet fractios usig cocrete materials model ad explai that equivalet fractios represet the same quatity idetify patters or relatioships (e.g., the deomiator is twice the umerator) Determie if two fractios are equivalet usig cocrete materials or pictorial represetatios. Idetify equivalet fractios for a fractio. Materials: Two-coloured chips, fractio blocks or fractio bars Orgaizatio: Pairs a) Tell studets that you will be givig them some fractios ad it is their job to determie if they are equivalet. Explai that they ca use chips or fractio blocks to help them decide. b) Have studets determie whether the followig fractios are equivalet: 3 9 ad ad ad ad ad 9 27 Number 91

114 c) Have studets share their aswers with the other members of the class. Ecourage discussio by askig questios, such as the followig: 3 9 How do you kow that is equivalet to? Why is ot equivalet to? What fractio is equivalet to? How do you kow? 12 Observatio Checklist Observe studets resposes to determie whether they ca do the followig: idetify equivalet fractios distiguish betwee equivalet ad o-equivalet fractios model ad explai that two equivalet fractios represet the same amout Determie if two fractios are equivalet usig cocrete materials or pictorial represetatios. Materials: A set of 40 equivalet fractio cards for each group of studets (BLM 5.N.7.1). Orgaizatio: Groups of 2 to 4 studets a) Tell studets that they will be playig a game ivolvig equivalet fractios. Explai how the game is played. 1. Deal five cards to each player. The remaiig cards are placed i a pile face dow i the middle of the playig area. 2. Players take turs askig aother player for a card. For example, Please give me 1 2 the card that shows =. If the player has the card, he or she must give it to 2 4 the asker. Whe a player receives a requested card, the player lays the two matched cards aside. 3. If the player does ot have the requested card, the asker draws a card from the pile i the middle of the playig area. If it matches a card i his or her had, the cards are set aside. If it does ot match ay of the cards i his or her had, the player keeps the card. 4. The game is over whe oe player rus out of cards. The player with the most pairs is the wier. 92 Grade 5 Mathematics: Support Documet for Teachers

115 b) Demostrate how to play the game ad aswer ay questios studets may have. Have studets play the game. Observatio Checklist Observe studets resposes to determie whether they ca do the followig: idetify pictorial represetatios of equivalet fractios match pictorial represetatios of equivalet fractios with symbolic statemets determie if two fractios are equivalet usig cocrete materials or pictorial represetatios Materials: Math jourals Orgaizatio: Idividual a) Ask studets to aswer the followig questio i their math jourals: Are these fractios equivalet? Explai how you reached your aswer b) Have studets share their aswers with the other members of the class. Ecourage them to explai their reasoig. Observatio Checklist Observe studets resposes to determie whether they ca do the followig: determie if two fractios are equivalet 3 6 explai how they kow = or demostrate usig pictorial represetatios that = (If pictorial represetatios are used, make 4 12 sure their diagrams match their explaatios.) Number 93

116 Idetify equivalet fractios for a fractio. Create a set of equivalet fractios ad explai why there are may equivalet fractios for ay fractio usig cocrete materials. Determie if two fractios are equivalet usig cocrete materials or pictorial represetatios. Formulate ad verify a rule for developig a set of equivalet fractios. Materials: Two-coloured chips ad a set of fractio cards for each group of studets (BLM 5.N.7.2) Orgaizatio: Small groups a) Ask studets to sort the cards ito piles accordig to how much of each diagram is shaded. Have studets tur over each pile ad record the fractios o the back of the cards. For example,,,,, ad. Have them describe ay patters they see, ad the ask them to ame other fractios that belog to each set ad explai how they kow they are equivalet. b) Have studets complete each of the followig patters. Tell them that they ca draw pictures or use materials to help them complete each patter ,,,,, ,,,,,, ,, , 20, 25, 30,,, c) Ask studets to share their aswers ad explai their reasoig. Observatio Checklist Observe studets resposes to determie whether they ca do the followig: idetify equivalet fractios for a fractio create a set of equivalet fractios ad explai why there are may equivalet fractios determie if two fractios are equivalet usig pictorial represetatios formulate a rule for determiig equivalet fractios 94 Grade 5 Mathematics: Support Documet for Teachers

117 Compare two fractios with ulike deomiators by creatig equivalet fractios. Materials: Egg cartos, couters, fractio bars (BLM ), Cuiseaire rods, or fractio blocks, clock face (BLM ) Orgaizatio: Idividual or pairs a) Ask studets to use their egg cartos ad couters to aswer questios, such as idetifyig which is larger amog the followig: b) Have studets share their aswers ad explai the strategies they used to determie which fractio is the larger. c) If a aalog clock is ot i the classroom, draw a picture of a clock face o the board ad the ask questios such as idetifyig which is greater amog the followig: 1 1 doze or doze doze or doze doze or doze doze or 2 12 doze 2 5 doze or doze hour or hour 6 1 hour or hour hour or hour 3 1 hour or hour 2 2 hour or hour 4 7 hour or hour 12 Number 95

118 d) Repeat the activity, but have studets use a differet material. For example, have studets use fractio bars to aswer questios such as idetifyig which is larger amog the followig: 5 2 or or or or or or or 8 2 or 6 e) Have studets share their aswers ad explai the strategies they used to determie which fractio is the larger. Observatio Checklist Observe studets to determie whether they ca do the followig: represet a give fractio with cocrete materials compare two fractios with ulike deomiators usig cocrete materials 96 Grade 5 Mathematics: Support Documet for Teachers

119 Materials: Graph paper Orgaizatio: Small groups/whole class a) Preset studets with the followig problem: Sammy had a piece of graph paper o his desk. His teacher asked, Which is 2 3 larger, or? After a few miutes, Sammy said, I thik I see a ew way to 3 4 fid out. How do you thik Sammy did it? b) Have each group explai how they thik Sammy used the graph paper to help him compare the two fractios. If oe of the groups use the graph paper to fid commo deomiators, explai that Sammy could have used the graph paper to mark off a rectagle that was four squares oe way ad three squares the other way. 1 3 Each colum is of the rectagle, so would look like this: Each row is, so would look like this: = ad =, is greater tha, so is greater tha Number 97

120 c) Do two or three more examples, the ask studets to use the method to decide which fractio is larger: 3 4 ad ad ad 6 4 ad ad 4 5 d) Have studets share their aswers. Ecourage them to describe ad show how they used the graph paper to help them fid fractios with commo deomiators. Observatio Checklist Observe studets resposes to determie whether they ca do the followig: represet fractios pictorially compare two fractios with ulike deomiators by creatig equivalet fractios describe ad show how they foud equivalet fractios Materials: Spier with the umbers 1-12 o them (BLM ), copies of the recordig sheet (BLM 5.N.7.3), fractio strips, patter blocks or other fractio maipulatives, a overhead trasparecy of the recordig sheet Orgaizatio: Pairs/Whole class a) Tell studets that they will be playig a game with their parter that ivolves comparig fractios. Explai how to play the game. 1. Players take turs spiig the spier four times. 2. After each spi, the player writes the umber i oe of the boxes or o oe of the lies beside the bottom box. Oce a umber has bee writte, it caot be chaged. After four spis, there will be a fractio (formed by the umbers i the boxes) ad two rejected umbers that ca be put i the trash ca. 98 Grade 5 Mathematics: Support Documet for Teachers

121 3. After both players have created their fractios, they ca use the materials or draw pictures to model their fractios ad decide which is the larger. The players the write a setece to show the compariso e.g., > or = The player who made the larger fractio scores oe poit. If the fractios are equivalet, both players score a poit. 5. The wier is the player with the most poits after four rouds. c) Demostrate how to play the game ad aswer ay questios studets might have. Have studets play the game. d) Vary the game by havig oe player spi the spier four times ad both players use the same umbers to complete the game board. e) Have studets discuss their experiece playig the game. Ecourage them to describe ay strategy they used that worked well for them. Observatio Checklist Observe studets resposes to determie whether they ca do the followig: model fractios appropriately usig cocrete ad pictorial represetatios compare fractios with ulike deomiators by creatig equivalet fractios usig cocrete materials or pictorial represetatios usig a persoal strategy recogize equivalet fractios show symbolically which of two fractios is larger e.g., 2 1 > 3 2 recogize that if the deomiators are the same, the fractio with the larger umerator is the larger recogize that if the umerators are the same, the fractio with the smaller deomiator is the larger Number 99

122 Materials: Math jourals ad pecils Orgaizatio: Idividual/Whole class a) Write these questios o the board or overhead. 6 4 Which is larger: or? How do you kow? 8 5 b) Ask studets to thik about the questios ad the record their aswers i their math jourals. c) Whe studets fiish, have them share their aswers. Ecourage them to discuss the strategies they used by askig them questios, such as: 4 6 How do you kow that is larger tha? Is there aother strategy you could use to show that is larger tha? 5 8 Which strategy do you prefer? Why? Observatio Checklist Observe studets resposes to determie the strategy they used to fid the larger fractio. For example, some studets may fid the larger fractio by creatig equivalet fractios with like deomiators e.g., ad creatig equivalet fractios with like umerators e.g., ad comparig the fractios to a bechmark (e.g., decidig whether or 8 5 is closer to 1) Also, observe studets resposes to determie whether they ca do the followig: commuicate their ideas effectively use appropriate diagrams or pictures (whe used i their explaatio) recogize that the more pieces ito which a whole is divided, the smaller the pieces. recogize equivalet fractios use appropriate procedures to fid equivalet fractios 100 Grade 5 Mathematics: Support Documet for Teachers

123 Positio a set of fractios with like ad ulike deomiators o a umber lie (vertical or horizotal), ad explai strategies used to determie the order. Materials: Fractio cards (BLM 5.N.7.2), vertical ad horizotal umber lies marked 1 with 0,, ad 1. 2 Orgaizatio: Whole class/pairs a) Show studets the cards with fractios o them, ad explai that their task is to place the fractios i order from smallest to largest. Select oe card ad prop it o the chalkboard tray. Show the other cards oe at a time ad have a studet place them o the tray. Have studets explai their reasoig. 1 b) Draw a umber lie o the board ad mark the poits 0,, ad 1. Have studets 2 place each fractio o the umber lie ad explai their reasoig. c) Have studets work with their parter to positio each of the followig sets of fractios o the umber lie: 1 12, , 3 3, 3, 4, , 3 9 8, 7 8, 16, , 1 5 4, 2 6 3, 9, 3, d) Have studets create their ow set of fractios to order ad place o the umber lie. Have them justify, i writig, their placemet of the fractios. Observatio Checklist Observe studets resposes to determie whether they ca do the followig: use a appropriate (mathematically correct) strategy for comparig fractios with ulike deomiators positio a set of fractios with like ad ulike deomiators o a umber lie explai the strategies that they used to positio the fractios Number 101

124 N OTES 102 Grade 5 Mathematics: Support Documet for Teachers

125 Grade 5: Number Edurig Uderstadigs: Decimals are symbols for commo fractios whose deomiators are powers of te. Decimals are a extesio of the base-10 umeratio system. Fractios ad decimals ca be used iterchageably. Geeral Outcome: Develop umber sese. SPECIFIC LEARNING OUTCOME(S): ACHIEVEMENT INDICATORS: 5.N.8 Describe ad represet decimals (teths, hudredths, thousadths) cocretely, pictorially, ad symbolically. [C, CN, R, V] 5.N.9 Relate decimals to fractios (teths, hudredths, thousadths). [CN, R, V] Write the decimal for a cocrete or pictorial represetatio of part of a set, part of a regio, or part of a uit of measure. Represet a decimal usig cocrete materials or a pictorial represetatio. Represet a equivalet teth, hudredth, or thousadth for a decimal, usig a grid. Express a teth as a equivalet hudredth ad thousadth. Describe the value of each digit i a decimal. Write a decimal i fractioal form. Write a fractio with a deomiator of 10, 100, or 1000 as a decimal. Express a pictorial or cocrete represetatio as a fractio or decimal (e.g., 250 shaded squares o a thousadth grid ca be expressed as or Number 103

126 PRIOR KNOWLEDGE Studets should be able to do the followig: (4.N.8) Demostrate a uderstadig of fractios less tha or equal to oe usig cocrete ad pictorial represetatios (4.N.8) Name ad record fractios for the parts of a whole or a set (4.N.8) Compare ad order fractios with like umerators or like deomiators (4.N.8) Provide examples of where fractios are used (4.N.10) Relate fractios to decimals (to hudredths) (4.N.9) Describe ad represet decimals (teths ad hudredths) cocretely, pictorially, ad symbolically RELATED KNOWLEDGE Studets should be able to do the followig: (5.N.7) Use cocrete or pictorial represetatios to create equivalet fractios (5.SS2) Measure the legth of a object i millimetres, cetimetres, or metres (5.SS.2) State the relatioship betwee millimetres ad cetimetres, cetimetres ad metres, millimetres ad metres BACKGROUND INFORMATION Kowledge of decimals is ecessary to deal effectively with everyday situatios ivolvig moey, measuremet, probability, ad statistics. However, may studets lack the uderstadig of decimals eeded to deal with these situatios i meaigful ways. May of their miscoceptios about decimals stem from their efforts to apply whole umber cocepts to decimals. For example, some studets believe that is greater tha 0.43 because 143 is larger tha 43, while others believe that is te times larger tha 0.15 sice 150 is te times larger tha 15. Istructio that focuses o the meaig of decimals ca help studets overcome or avoid these miscoceptios. I particular, learig experieces eed to emphasize two iterpretatios of decimals: first, decimals are just aother symbol for commo fractios whose deomiators are powers of te; ad secod, decimals are a extesio of the base-10 umeratio system. Allowig studets to maipulate cocrete ad pictorial represetatios of decimals ad helpig them make coectios betwee their actios o these represetatios ad the symbols for decimals ca facilitate their uderstadig of these two iterpretatios. 104 Grade 5 Mathematics: Support Documet for Teachers

127 MATHEMATICAL LANGUAGE Decimal Decimal poit Deomiator Equivalet Fractio Hudredths Numerator Teths Thousadths LEARNING EXPERIENCES Assessig Prior Kowledge Materials: Paper ad pecil Orgaizatio: Idividual/Whole class a) Tell studets that i the ext few lessos they will be learig about decimals, but before they begi you eed to fid out what they already kow about decimals. Have studets write a letter tellig you what they kow about decimals. b) Whe studets fiish, have them share what they kow about decimals with the other members of the class. Use the discussio to clear up ay miscoceptios they might have about decimals. Observatio Checklist Observe studets resposes to determie whether they ca do the followig: relate fractios to decimals (to hudredths) describe ad represet decimals pictorially ad symbolically idetify examples ad o-examples of decimals idetify examples of where decimals are used recogize that decimals are a extesio of the place value system Number 105

128 Write the decimal for a cocrete or pictorial represetatio of a part of a set, part of a regio, or part of a uit of measure. Represet a decimal usig cocrete materials or a pictorial represetatio. Represet a equivalet teth, hudredth, or thousadth for a decimal, usig a grid. Materials: Hudred squares (BLM 5 8.6) ad tape Orgaizatio: Whole class a) Give each studet 10 hudred squares. Tell them that the squares are pieces of a whole ad that they will be usig them to lear about decimals. Ask studets to arrage the pieces i a strip to show a whole. Have them tape the pieces together, as show below. b) Name the piece oe whole strip. Ask studets, How may hudred squares make oe whole strip? If there are 10 hudred squares i a whole, what is a hudred square called (1 teth)? How may rows of te small squares are there? How do you kow? If there are 100 colums of te small squares, what is each colum of te small squares called (1 hudredth)? How may small squares are i the whole? How do you kow? If there are 1000 small squares i the whole, what is each small square called (1 thousadth)? c) Ask studets to cout parts of the strips by teths. Have them poit to each teth as they cout. Next, have them cout by hudredths. Tell studets that sice it would take too log to cout by thousadths, you wat them to cout by 10 thousadths. Ask, What part of the whole is 10 thousadths (1 hudredth)? Have studets poit to each 10 thousadth as they cout 10 thousadths, 20 thousadths,, 100 thousadths,, 1000 thousadths. d) Write the decimals 0.1 ad 0.01 o the board or overhead. Ask studets to use their strips to show what each decimal meas. The ask, How do you thik we should write the decimal for oe-thousadth? Why? Discuss that it makes sese to use the ext place to the right for thousadths. 106 Grade 5 Mathematics: Support Documet for Teachers

129 e) Write the followig decimals o the board. Have studets read each decimal ad illustrate it with their strip Note: Studets will eed this thousads strip for a later activity. Observatio Checklist Observe studets resposes to determie the strategy they used to fid the larger fractio. For example, some studets may fid the larger fractio by idetifyig teths, hudredths, ad thousadths coutig by teths, hudredths, ad 10 thousadths readig decimals correctly associatig symbols for decimals with a cocrete represetatio represetig decimals with cocrete materials Represet a decimal usig cocrete materials or a pictorial represetatio. Materials: Metre sticks (oe for each pair of studets) ad a overhead trasparecy of a metre stick Orgaizatio: Pairs a) Ask studets to fid 0.1 of their metre stick. Record the decimal o the board or overhead, ad the ask studets to explai their choice. b) Repeat the activity that is, have studets use their metre sticks to show the followig: Number 107

130 Write the decimals o the board or overhead ad have studets explai their choices. c) Vary the activity by askig studets questios such as: Is closer to 0 or to 1? Is closer to or 0.300? What decimals come betwee ad 0.780? What decimal comes immediately before 0.431? What decimal comes immediately after 0.599? Observatio Checklist Observe studets resposes to determie whether they ca do the followig: represet decimals with cocrete materials associate symbols for decimals with cocrete materials idetify the decimal that comes immediately before or after a give decimal idetify decimals that come betwee two give decimals Represet a decimal usig cocrete materials or a pictorial represetatio. Express a teth as a equivalet hudredth ad thousadth. Express a hudredth as a equivalet thousadth. Describe the value of each digit i a decimal. Materials: The thousadths strips that studets made ad used i a prior activity for these outcomes Orgaizatio: Whole class/pairs a) Have studets decide what part of their strip represets Next, ask studets to use their strip to help them ame a decimal equivalet to (0.6 ad 0.60). If studets experiece difficulty completig this task, have them cout by teths (ad the hudredths) util is reached. b) Repeat the activity, but this time ask studets to decide what part of their strip represets The have studets use their strip to fid a decimal equivalet to ( ad 0.35). Cotiue to give studets other thousadths ad ask them to fid decimal equivalets. 108 Grade 5 Mathematics: Support Documet for Teachers

131 c) Have studets use their strips to fid decimals equivalet to 0.45 (0.450, , ) ad 0.3 (0.30 ad 0.300). d) Have studets work with their parter to fid decimals that are equivalet to ( , , ) Observatio Checklist Observe studets resposes to determie whether they ca do the followig: represet a decimal usig cocrete materials describe the value of each digit i a decimal express teths as equivalet hudredths ad thousadths describe thousadths as equivalet teths ad hudredths describe hudredths as equivalet teths ad thousadths Materials: Thousadth grid (BLM ) (Studets should use oe grid for each decimal umber.), crayos or markers Orgaizatio: Whole class/small groups a) Ask studets to use the grid paper to represet these decimals Number 109

132 b) Whe studets fiish, ask questios about the relatioships amog the decimals, such as: What do you otice about two-teths, twety-hudredths, ad two-hudredthousadths? How are these decimals alike? How do they differ? c) Have studets use the grid paper to fid decimals that are equivalet to 0.10, 0.300, 0.5, 0.60, ad 0.8. Ecourage studets to discuss their fidigs by askig questios similar to the followig: What decimals are equivalet to te hudredths? How do you kow? How are the decimals alike? How do they differ? What rule do your observatios suggest? d) Have groups cosider these questios: What is the differece betwee the values of the 5 i (1) ad the values i (2)? What does this tell you? How do you kow? 1. 5, 50, , 0.50, e) Have the groups share their aswers with the other members of the class. Observatio Checklist Observe studets resposes to determie whether they ca do the followig: demostrate the meaig of a decimal by represetig it pictorially represet a equivalet teth, hudredth, or thousadth usig a grid recogize ad explai the similarities ad differeces amog equivalet decimals represet a teth as a equivalet hudredth or thousadth recogize that aexig a zero to the right of a whole umber chages the place ad value of each digit recogize that aexig a zero to the right of a decimal chages the ame but ot the place or value of each digit describe the value of each digit i a whole umber or decimal 110 Grade 5 Mathematics: Support Documet for Teachers

133 Materials: Base-10 blocks or digi-blocks ad a place value mat (BLM ) Orgaizatio: Whole class a) Show studets the large block ad tell them that, for this activity, the block will represet oe, a flat will represet oe-teth, the log will represet oe-hudredth, ad the small cube will represet oe-thousadth. To help studets get used to the ew umber ames for the pieces, ask them to use their blocks to show you: two-teths five-hudredths six-thousadths ie-teths two-hudredths three-thousadths five-teths four-hudredths ie-thousadths b) Cotiue askig studets to show you differet umbers with their base-10 blocks util they ca do it quickly ad easily. c) Represet the followig decimals with the blocks ad ask studets to write the correspodig symbols for the decimals Ecourage studets to discuss their aswers by askig them questios such as: What decimal did you write? What is the value of the 1 i 2.156? What is the place value of the 8 i 0.189? What is the value of the 2 i 3.204? I 3.420? What decimal is equivalet to 3.420? d) Vary the activity by amig a decimal (e.g., 1.254) ad havig studets represet it with their blocks. After they represet the umber ask them to record the correspodig symbol (decimal). Agai, ecourage studets to discuss their aswers by askig questios similar to the oes i part (c). Observatio Checklist Observe studets resposes to determie whether they ca do the followig: write the decimal that correspods to its cocrete represetatio represet a decimal usig cocrete or pictorial represetatios state the value of each digit i a decimal ame the place value positio of each digit i a decimal recogize equivalet decimals Number 111

134 Write a decimal i fractioal form. Write a fractio with a deomiator of 10, 100, or 1000 as a decimal. Express a pictorial or cocrete represetatio as a fractio or decimal. Materials: Thousadths grid (BLM ) ad crayos Orgaizatio: Pairs a) Give studets several copies of the thousadths grid. Have studets observe that each grid is a 20 by 50 rectagle cotaiig 1000 small squares. b) Ask studets to shade i 350 squares, ad the write uder the grid the decimal ad fractio ames for the shaded-i squares. c) Next, have studets shade i 250 squares. Ask studets to write the basic fractio 250 ad decimal ame for the shaded squares ad Ecourage studets to 1000 use their grids to help them fid equivalet umbers. Ask: 250 What decimal is equivalet to How do you kow? 1000? What fractio with a smaller umerator ad deomiator is equivalet to ? or How do you kow? What other fractio is equivalet to How do you kow? 1000? d) Repeat the activity several times. For example, have studets shade 125 squares 600 squares 184 squares 750 squares 375 squares 100 squares 500 squares 267 squares Ecourage studets to write as may decimal ad fractio ames as they ca for each shaded grid. e) Have studets share their aswers ad explai their reasoig. Observatio Checklist Observe studets resposes to determie whether they ca do the followig: express a pictorial represetatio as a fractio express a pictorial represetatio as a decimal idetify equivalet fractios for a pictorial represetatio idetify equivalet decimal for a pictorial represetatio explai their reasoig 112 Grade 5 Mathematics: Support Documet for Teachers

135 Write a decimal i fractioal form. Write a fractio with a deomiator of 10, 100, or 1000 as a decimal. Materials: Fractio ad decimal equivalet cards (BLM 5.N.8&9.1) Orgaizatio: Pairs a) Tell studets that they will be playig a game that ivolves matchig decimals with their equivalet fractios. b) Tell studets that they will eed to shuffle their cards ad spread them face up o their playig area. Whe you say go, they should begi matchig the fractio/decimal cards without sayig aythig to their parter. The first pair to match the decimal/fractio cards correctly wis. c) Demostrate how to play the game ad aswer ay questios studets might have. Have them play the game. d) Repeat the activity, but this time put a time limit o it. For example, give studets a miute to complete the activity. The pair that matches the most umber of cards correctly wis. e) Have studets make their ow equivalet fractio/decimal cards ad use them to play the game. Observatio Checklist Observe studets resposes to determie whether they ca do the followig: associate a fractio with a deomiator of 10 with a equivalet decimal associate a fractio with a deomiator of 100 with a equivalet decimal associate a fractio with a deomiator of 1000 with a equivalet decimal idetify fractios ad decimals that are ot equivalet Number 113

136 Materials: Fractio ad decimal equivalet cards (BLM 5.N.8&9.1) Orgaizatio: Small groups a) Tell studets that they will be playig cocetratio with the fractio/decimal cards. b) To play the game, have studets spread the cards face dow o the playig area. Have studets take turs turig over two cards. If the cards match, the player keeps the cards ad takes aother tur. If the cards do ot match, the player turs them back over ad the ext player takes a tur. Play cotiues util all the cards have bee matched. The player with the most cards is the wier. c) Demostrate how to play the game ad aswer ay questios studets might have. Have studets play the game. d) Have studets play the game agai with the fractios/decimal equivalet cards that they made (see previous activity). Observatio Checklist Observe studets resposes to determie whether they ca do the followig: associate a fractio with a deomiator of 10 with a equivalet decimal associate a fractio with a deomiator of 100 with a equivalet decimal associate a fractio with a deomiator of 1000 with a equivalet decimal idetify fractios ad decimals that are ot equivalet 114 Grade 5 Mathematics: Support Documet for Teachers

137 Materials: A coi ad paper ad pecils Orgaizatio: Whole class a) Tell studets that their job is to record the umbers that you will be readig to them. They ca record a umber as a decimal or as a fractio (they must choose oe represetatio, but ca chage which they choose for each ew umber). They will score a poit if they record the umber correctly. After they record the umber, you will toss a coi. If the coi lads heads up, everyoe who wrote the umber correctly as a decimal scores aother poit. If the coi lads tails up, everyoe who wrote the umber correctly as a fractio scores aother poit. Studets keep their ow score. b) Read these umbers to studets: three-thousadths six-teths oe-hudred-fiftee thousadths oe-hudred-five thousadths sixty-two thousadths ie-teths eighty-thousadths five-hudred-three thousadths forty-hudredths three-hudredths c) Select 10 more umbers to read to the class. Repeat the activity, but this time let the studet who scored the most poits read the umbers. Observatio Checklist Observe studets resposes to determie whether they ca do the followig: associate a oral represetatio of a umber with the correct decimal symbol for the umber associate a oral represetatio of a umber with the correct fractio symbol for the umber read a decimal umeral correctly read a fractio correctly Number 115

138 N OTES 116 Grade 5 Mathematics: Support Documet for Teachers

139 Grade 5: Number Edurig Uderstadigs: Decimals are a extesio of the base-10 umeratio system. Geeral Outcome: Develop umber sese. SPECIFIC LEARNING OUTCOME(S): ACHIEVEMENT INDICATORS: 5.N.10 Compare ad order decimals (teths, hudredths, thousadths) by usig bechmarks place value equivalet decimals [CN, R, V] Order a set of decimals by placig them o a umber lie (vertical or horizotal) that cotais the bechmarks 0.0, 0.5, ad 1.0. Order a set of decimals icludig oly teths usig place value. Order a set of decimals icludig oly hudredths usig place value. Order a set of decimals icludig oly thousadths usig place value. Explai what is the same ad what is differet about 0.2, 0.20, ad Order a set of decimals icludig teths, hudredths, ad thousadths usig equivalet decimals. RELATED KNOWLEDGE Studets should be able to do the followig: (5.N.8) Describe ad represet decimals to thousadths cocretely, pictorially, ad symbolically Number 117

140 MATHEMATICAL LANGUAGE Decimal Equivalet Bechmark Teths Hudredths Thousadths LEARNING EXPERIENCES Order a set of decimals icludig oly teths usig place value. Order a set of decimals icludig oly hudredths usig place value. Materials: Math jourals, base-10 blocks, place value mats (BLM ) Orgaizatio: Pairs a) Show studets a flat ad tell them that, for this activity, a flat will represet oe, a log will represet oe-teth, ad the small cube will represet oe-hudredth. To help studets get used to the ew umber ames for the pieces, ask them to use their blocks to show you the followig: ie-teths two-teths three-teths oe-hudredth six-hudredths fourtee-hudredths b) Cotiue askig studets to show you differet umbers with their base-10 blocks util they ca do it quickly ad easily. c) Have studets use their blocks to represet each pair of umbers ad decide which umber is larger. Ask studets to record their aswers. 0.5 ad ad ad ad ad ad ad ad Grade 5 Mathematics: Support Documet for Teachers

141 d) Vary the activity by askig studets to make a umber larger tha a give umber (e.g., ask studets to make a umber greater tha 2.6) make a umber smaller tha a give umber (e.g., ask studets to make a umber less tha 0.36) make a umber betwee two give umbers (e.g., ask studets to make a umber betwee 0.1 ad 0.4) look at each umber i a pair of umbers, decide which umber is greater, ad the check their aswers with the blocks (e.g., ask studets to circle the larger umber [0.51 or 0.43], ad the use their blocks to check their aswers) e) Ask studets to aswer the followig questios i their math jourals: Which umber is smaller: 2.8 or 2.3? How do you kow? Which umber is larger: 0.53 or 0.06? How do you kow? Observatio Checklist Observe studets resposes to determie whether they ca do the followig: order a set of decimals icludig oly teths usig place value reame hudredths as teths ad hudredths (e.g., Do studets recogize that fifty-three hudredths is the same as five-teths ad three-hudredths) order a set of decimals icludig oly hudredths usig place value idetify umbers greater tha (or less tha) a give umber idetify umbers that come betwee two give umbers Number 119

142 Order a set of decimals icludig oly teths usig place value. Order a set of decimals icludig oly hudredths usig place value. Order a set of decimals icludig oly thousadths usig place value. Materials: Oe die per group, recordig sheet (BLM 5.N.10.1) Orgaizatio: Small groups a) Tell studets that they will be playig a game called Less Tha. Explai how the game is played. 1. Desigate oe player to roll the die. Whe the die is rolled, players write the resultig umber i oe of their boxes before the die is rolled agai. Oce a umber is writte o a lie, it caot be chaged. 2. A roud of the game cosists of four rolls of the die. If the two umbers that are geerated from the four rolls of the die are i the correct order, the player scores oe poit. If the umbers are ot i the correct order, the player scores a zero. 3. Play four rouds. The wier is the player with the most poits. b) Demostrate how to play the game ad aswer ay questios studets may have. Have studets play the game. c) Vary the game by havig studets create ad order three decimal umbers ( < < ) create ad order decimals i the hudredths or thousadths (e.g., 0 < 0 ) Observatio Checklist Depedig o the versio of the game that is played, check studets resposes to determie whether they ca order a set of decimals icludig oly teths usig place value icludig oly hudredths usig place value icludig oly thousadths usig place value 120 Grade 5 Mathematics: Support Documet for Teachers

143 Materials: Paper ad pecils Orgaizatio: Idividual a) Have studets solve problems like the followig: The split times for Dimitri ad Euclid were ad secods respectively. Who had the fastest time? The masses of five eggs were as follows: kg, kg, kg, kg, ad kg. Place the masses i order from smallest to largest. To make a miiature toy car, you eed tires with a width betwee cm ad cm. Will a tire with a width of cm work? Explai your aswer. At a swimmig competitio, Jue scored 9.80, Nora scored 9.75, Debbie scored 9.79, ad Alexia scored What must Tia score to wi the competitio? Explai your aswer. b) Have studets share their solutios to the problems ad explai their reasoig. Observatio Checklist Check studets resposes to determie whether they ca do the followig: solve problems ivolvig the comparig ad orderig of decimals to thousadths compare ad order hudredths compare ad order thousadths Number 121

144 Materials: Paper ad pecil Orgaizatio: Idividual a) Ask studets to ame a decimal that is greater tha 5.9 ad less tha 6 greater tha 9 ad less tha 9.1 greater tha 0.63 ad less tha 0.64 greater tha 8.9 ad less tha 9.15 greater tha 7.8 ad less tha 7.62 b) Have studets use the digits 0 through 9 to complete the aswer to each statemet. A digit caot be used more tha oce. greater tha 0.52 but less tha greater tha but less tha greater tha but less tha greater tha but less tha Observatio Checklist Moitor studets resposes to determie whether they ca do the followig: idetify a umber betwee two give umbers compare ad order decimals icludig oly teths compare ad order decimals icludig oly hudredths compare ad order decimals icludig oly thousadths 122 Grade 5 Mathematics: Support Documet for Teachers

145 Order a set of decimals by placig them o a umber lie (horizotal or vertical) that cotais the bechmarks 0.0, 0.5, ad 1.0. Materials: Decimal cards (BLM 5.N.10.2), as well as grid paper, metre sticks or base-10 blocks available for studets who would like to use them Orgaizatio: Pairs or small groups/whole class a) Give each pair of studets a set of cards ad ask them to sort the cards ito three groups: those that are close to zero, those that are close to five-teths, ad those that are close to 1. b) Whe studets fiish sortig their cards, have them share their aswers with the other members of the class. Ecourage studets to explai their reasoig by askig them questios such as: How do you kow whe a decimal is close to five-teths? How do you kow whe a umber is close to zero? How do you kow whe a umber is close to oe? Are the umber of decimal places importat i determiig the size of a decimal? Why? Which decimals are hardest to order? Why? c) Draw either a horizotal or vertical umber lie o the board ad label the poits 0, 0.5, ad 1. Ask studets to estimate where oe-teth would be o the umber lie. Have a studet tape a card showig oe-teth o the umber lie. Cotiue havig studets estimate ad show where the umbers o their decimal cards would be o the umber lie. d) Ask each pair of studets to make their ow decimal cards that show hudredths ad thousadths (e.g., 0.20 ad 0.132). Have them exchage their cards with aother pair ad the sort the cards ito three groups: decimals that are close to oe, decimals that are close to five-teths, ad decimals close to zero. e) Have studets share the umbers ad their explaatios of how they grouped them with the other members of the class. Ecourage them to explai their reasoig by askig them questios similar to those i part (b). f) Select several of the umber cards that the studets created ad have studets estimate ad show where the umbers are o the umber lie. Number 123

146 Observatio Checklist Observe studets resposes to determie whether they ca do the followig: idetify decimals that are close to the bechmarks of 0, 0.5, ad 1 explai how they kow whe a decimal is close to 0, 0.5, ad 1 order a set of decimals cosistig of teths ad hudredths usig place value order a set of decimals cosistig of hudredths ad thousadths usig place value order a set of decimals by placig them o a umber lie that cotais the bechmarks 0, 0.5, ad 1 order a set of teths, hudredths, ad thousadths by usig equivalet decimals Order a set of decimals icludig teths, hudredths, ad thousadths usig equivalet decimals. Materials: Blak dice (Write the decimals 0.4, 0.3, 0.9, 0.74, 0.04, ad 0.60 o each die.) Orgaizatio: Pairs a) Tell studets that they are goig to play a place value game caller Larger. Explai how the game is played. 1. Both players roll a die at the same time. After a roll, each player records the resultig decimal umbers i a table like the oe show below. My Number My Parter s Number 2. Players circle the larger decimal umber. The player with the larger umber scores oe poit. If a tie occurs, circle both decimal umbers ad give each player a poit. 3. The wier of the game is the perso with the most poits after 20 rouds of the game. b) Demostrate how the game is played ad aswer ay questios studets may have. Have studets play the game. c) Vary the game by writig differet decimals o the dice. 124 Grade 5 Mathematics: Support Documet for Teachers

147 Observatio Checklist Observe studets resposes to the game to determie whether they ca do the followig: order sets of decimals usig place value order teths ad hudredths usig equivalet decimals Order a set of decimals icludig oly thousadths usig place value. Materials: Players averages ad battig criteria (see below), ie pieces of paper for each studet or pair of studets Orgaizatio: Idividual or pairs/large group a) Preset studets with the followig sceario: You have just bee hired to coach the school s softball team. You have to make up the battig order for the game today but there is o oe aroud who ca help you. You kow from your coachig experiece that a player s battig average, o-base average, ad sluggig average are good idicatios of where a player should bat i the lieup. These are the averages of your players: Player Battig Average O-Base Average Sluggig Average Arez Britto Brock Charles Jackso Lamar Role Satos White Number 125

148 b) Tell studets that they must make up cards that list each player s statistics, ad the arrage them i a battig order that is most closely aliged to the followig criteria. Whe they fiish, they should make a lieup card to share with the rest of the class. Player Battig Average O-Base Average Sluggig Average 1 High Highest Low Medium 2 High Medium High Medium 3 Highest High High 4 Medium Medium Highest 5 Medium Medium High 6 Medium Medium Medium 7 Low Medium Low Medium Low Medium 8 Low Medium Low Medium Low Medium 9 Lowest Lowest Lowest c) Before studets begi workig o the activity, make sure they uderstad that the higher the battig average, the more ofte the player gets a hit; the higher the obase average, the more ofte the player gets o base; ad the higher the sluggig average, the more ofte a player gets a extra base hit (doubles, triples, ad homers). d) Have studets share their lieups with the other members of the class. Ecourage them to explai why their lieups satisfy the give criteria. Observatio Checklist Observe studets resposes to determie whether they ca do the followig: order a set of decimals i the thousadths usig place value create a lieup card that fits the criteria explai the reasos for the placemet of each batter i the lieup 126 Grade 5 Mathematics: Support Documet for Teachers

149 Grade 5: Number Edurig Uderstadigs: Addig ad subtractig decimals is similar to addig ad subtractig whole umbers. Geeral Outcome: Develop umber sese. SPECIFIC LEARNING OUTCOME(S): ACHIEVEMENT INDICATORS: 5.N.11 Demostrate a uderstadig of additio ad subtractio of decimals (limited to thousadths). [C, CN, PS, R, V] Place the decimal poit i a sum or differece usig frot-ed estimatio (e.g., for , thik 6 plus 306, so the sum is greater tha 312). Correct errors of decimal poit placemets i sums ad differeces without usig paper ad pecil. Explai why keepig track of place value positios is importat whe addig ad subtractig decimals. Predict sums ad differeces of decimals usig estimatio strategies. Solve a problem that ivolves additio ad subtractio of decimals, limited to thousadths. PRIOR KNOWLEDGE Studets should be able to do the followig: (4.N.11) Use compatible umbers whe addig ad subtractig decimals (to hudredths) (4.N.11) Estimate the sums ad differeces of problems ivolvig additio ad subtractio of decimals (to hudredths) (4.N. 11) Use metal math strategies to solve problems ivolvig additio ad subtractio of decimals (to hudredths) Number 127

150 RELATED KNOWLEDGE Studets should be able to do the followig: (5.N.9) Describe ad represet decimals to thousadths cocretely, pictorially, ad symbolically (5.N.10) Compare ad order decimals to thousadths (5.SS.2) Model ad explai the relatioship betwee mm ad cm uits ad mm ad m uits MATHEMATICAL LANGUAGE Additio Differece Estimate Subtractio Sum LEARNING EXPERIENCES Assessig Prior Kowledge Materials: Paper ad pecil Orgaizatio: Idividual a) Ask studets to solve the followig problems: 1. Roberta was gettig ready for the first day of school. She bought a set of five pes for $2.57, a bider for $4.35, ad a box of three-hole paper for $5.15. What was the total cost of her school supplies? 2. Joe is savig moey to buy a ew video game. He has $19.50, ad tomorrow Mr. Mitchell is givig him $4.75 for mowig his law. If the video game cost $39.95, how much more moey does Joe eed to save? b) Have studets share their solutios to the questios. Ecourage them to explai the strategies they used to solve the problems. 128 Grade 5 Mathematics: Support Documet for Teachers

151 Observatio Checklist Observe studets resposes to determie whether they ca do the followig: idetify the operatio(s) eeded to solve a computatioal problem calculate sums ad differeces of decimals (limited to hudredths) solve oe- ad two-step problems use appropriate strategies to solve additio ad subtractio problems ivolvig decimals Solve a problem that ivolves additio ad subtractio of decimals, limited to thousadths. Materials: Base-10 blocks, two differet-coloured dies for each group ad a place value mat (BLM ) Orgaizatio: Small groups a) Show studets the large block ad explai that they will be playig a game that ivolves lettig the block represet oe. Ask, If the block represets oe, what does the flat represet? Why? What does the log represet? Why? What does the small cube represet? Why? b) To help studets become familiar with the ew ames for the base-10 block, ask them to use their blocks to show 5 teths 8 teths 6 hudredths 3 hudredths 12 hudredths 2 thousadths 7 thousadths 15 thousadths Cotiue to ask studets to represet differet decimals with the blocks util they ca do it quickly ad easily. Number 129

152 c) Tell studets that the game they will be playig is called Race to Oe. Explai how the game is played. 1. Let oe coloured die represet hudredths ad the other coloured die represet thousadths. 2. Players take turs rollig the die ad usig their blocks to represet the umber o their place value mats. 3. Whe players get 10 cubes, they trade them for a log. Whe they have 10 logs, they trade them for a flat, ad whe they have 10 flats, they trade them for a block. 4. The first player to get a block wis the game. d) Demostrate how the game is played ad aswer ay questios studets might have. Have studets play the game. e) After studets have played the game several times, have them use a recordig sheet like the oe show below. Tur Number Rolled Total f) Repeat the activity for subtractio that is, have studets play Race to Zero. For this versio of the game, studets start with the large block ad, o each tur, remove the umber of blocks that represets the umber rolled o the die. The first player to remove all of his or her blocks is the wier. After studets have played the game several times, have them use a recordig sheet like the oe show above. Observatio Checklist Observe studets resposes to determie whether they ca do the followig: use cocrete materials to solve problems ivolvig additio ad subtractio of decimals to thousadths make groups of te to trade, whe ecessary exchage materials for smaller or larger uits whe ecessary 130 Grade 5 Mathematics: Support Documet for Teachers

153 Materials: Decimal grid paper that shows thousadths (BLM ), metre sticks, ad base-10 blocks Orgaizatio: Idividual/Large group a) Tell studets that they will be solvig some problems ivolvig decimals. Explai that they ca use a strategy of their ow choosig ad that grid paper, metre sticks, ad base-10 blocks are available if they wat to use materials to help them solve the problems. b) Preset studets with these problems: Marco bought two baaas. The mass of the first baaa was kg, ad the mass of the secod was 0.45 kg. What was the combied mass of the baaas? At the swim meet, Marsha scored poits for her high dive. Maxie scored poits for her high dive. How much higher was Marsha s score tha Maxie s? Nadia has a piece of strig that is 1.12 m log. If she uses m to tie a package, how much strig does she have ow? Dimitri ra the first leg of a race i secods, the secod leg of the race i 9.72 secods, ad the last leg of the race i secods. How log did it take him to ru the etire race? c) Whe studets fiish solvig a problem, have them share their solutios with the other members of the class. Ecourage studets to explai the strategies they use to solve the problems. d) Cotiue to give studets problems like those above. Observatio Checklist Observe studets resposes to determie whether they ca do the followig: idetify the operatio eeded to solve the problem solve problems ivolvig additio ad subtractio of decimals to thousadths use a variety of strategies to solve a problem calculate correctly Number 131

154 Place the decimal poit i a sum or differece usig frot-ed estimatio. Explai why keepig track of place value positios is importat whe addig ad subtractig decimals. Predict sums ad differeces of decimals usig estimatio strategies. Materials: Paper ad pecils Orgaizatio: Pairs/Whole class a) Tell studets that you will be givig them some problems. Explai that you do ot wat them to fid the aswers to the problems. Istead, you wat them to list everythig that they kow about them. For example, there are two thigs we kow about this problem: = The aswer is i teths ad the aswer is greater tha oe. b) Ask studets to make a list of everythig they kow about the followig problems: c) Have studets share their aswers with the other members of the class. Ecourage studets to explai their reasoig by askig them questios such as: How do you kow the aswer is about 9? How do you kow the aswer is i thousadths? Why is it importat to keep track of place value positios whe addig? Subtractig? How do you kow that the aswer is less tha 3? d) Istead of havig studets list everythig they kow about a problem, have them list what the aswer caot be. For example, the aswer to the problem caot be less tha 9. The total umber i the thousadths place caot be greater tha Grade 5 Mathematics: Support Documet for Teachers

155 Observatio Checklist Observe studets resposes to determie whether they ca do the followig: predict sums ad differeces of decimals usig estimatio strategies (e.g., frot-ed estimatio or frot-ed estimatio with compesatio) explai why it s importat to keep track of place value positios whe addig ad subtractig decimals predict the place of the decimal poit i sums ad differeces usig frot-ed estimatio Explai why keepig track of place value positios is importat whe addig ad subtractig decimals. Solve a problem that ivolves the additio ad subtractio of decimals, limited to thousadths. Materials: Number cards (BLM 5 8.5) (oe set for each studet), umber frames (BLM 5.N.11.1) Orgaizatio: Pairs a) Ask studets to arrage cards 1 6 o their umber mats to create a problem with the largest possible sum. Have them record their solutio. Have studets share their solutios with the other members of the class ad explai their reasoig. b) Repeat the activity. Have studets use cards 1 6 to create problems with the smallest possible sum largest possible differece smallest possible differece c) Have studets use the digit cards 0, 1, 4, 6, 7, ad 9 to create problems with the largest possible sum smallest possible sum largest possible differece smallest possible differece Number 133

156 Observatio Checklist Observe studets resposes to determie whether they ca do the followig: solve problems ivolvig additio ad subtractio of decimals (limited to thousadths) compare ad order decimals to thousadths idetify the place value positio of each digit i a decimal (limited to thousadths) idetify the value of each digit i a decimal (limited to thousadths) calculate sums ad differeces ivolvig decimals usig appropriate strategies Predict sums ad differeces of decimals usig estimatio strategies. Solve a problem that ivolves additio ad subtractio of decimals, limited to thousadths. Materials: Copies of the activity (BLM 5.N.11.2) Orgaizatio: Idividual a) Have studets complete the activity. Observatio Checklist Observe studets resposes to determie whether they ca do the followig: solve problems ivolvig additio ad subtractio of decimals (limited to thousadths) predict sums ad differeces ivolvig decimals usig estimatio strategies calculate sums ad differeces ivolvig decimals usig appropriate computatioal strategies 134 Grade 5 Mathematics: Support Documet for Teachers

157 Materials: Paper ad pecils, umber fa (BLM ) Orgaizatio: Pairs/Whole class a) Ask studets to make a decimal greater tha 0.8 usig their umber fas. Make a list of their suggestios to show them a variety of aswers. b) Tell studets that they will be goig o a hut for decimals. Explai that you will be givig them differet clues ad their job is to work with their parter to fid a aswer that fits the clue ad to show their respose o a umber fa. 1. A decimal betwee 3.25 ad Two decimals whose sum is Two decimals with a differece of Three decimals whose sum is c) Have studets share their aswers. Make a list of their aswers to each statemet so studets ca see a variety of solutios. Ecourage studets to share the strategies that they used to determie their aswers. d) Have each pair of studets make up clues ad exchage them with aother pair of studets. Observatio Checklist Observe studets resposes to determie whether they ca do the followig: solve problems ivolvig additio ad subtractio of decimals (limited to thousadths) order decimals icludig teths, hudredths, ad thousadths recogize the relatioship betwee operatios calculate sums ad differeces ivolvig decimals usig appropriate computatioal strategies explai the strategies they used to determie the aswers to the statemets Number 135

158 PUTTING THE PIECES TOGETHER Plaig a Party Purpose: The itet of this ivestigatio is to have studets apply their kowledge of umber ad measuremet cocepts to a real-world situatio. I particular, the ivestigatio is desiged to reiforce studets ability to apply estimatio strategies compute whole umbers demostrate a uderstadig of fractios represet decimals to thousadths compare ad order decimals add ad subtract decimals demostrate a uderstadig of uits of measure ad their relatioship to each other I additio, the ivestigatio ehaces studets ability to solve problems reaso mathematically make coectios to other subjects (sciece ad ELA) make coectios to the real world commuicate mathematically Materials/Resources: Price list ad purchase order (see below) Orgaizatio: Small groups a) Preset the followig situatio: You have bee asked to pla a party for the studets i your class. You eed to order food, beverages, ad etertaimet for the party. You must pla two hours of etertaimet. You have $25.00 plus $2.00 per perso to sped o the party. b) Tell studets that they should use the followig price list to decide which items they would order for the party. Explai that they ca specify the flavours of the drik ad food that they wat (e.g., apple juice ad chocolate cake). They ca also ame a specific movie or game (sports or video) that they would like to have at the party. Aythig else they eed for the party, such as plates ad utesils, will be provided. 136 Grade 5 Mathematics: Support Documet for Teachers

159 Price List: Beverages: No ame pop Brad ame pop No ame juice Brad ame juice Milk $0.98 for 2 L $1.98 for 2 L $1.12 for 1 L $2.04 for 1 L $5.00 for 4 L Food: Apples 4 for $1.00 Watermelo 1/2 for $3.50 No ame chips $4.99 for a 500 g bag Brad ame chips $1.25 for 100 g bag Hot dogs $2.79 per doze Hot dog bus $2.50 for 10 Microwave popcor $2.99 for 3 bags 20 cm by 30 cm cake $ cm by 60 cm cake $20.00 Cupcakes/muffis $0.50 each Large carrots $1.44 for 12 carrots Baby carrots $2.99 for 30 carrots Party sub $15.00 for 10 people Idividual subs $1.75 per perso Etertaimet: Movie retal $5.49 Game system retal $21.75 Video game retal $4.26 Karaoke machie $28.15 Commuity cetre retal $11.00 per hour Sportig equipmet retal $1.50 per perso Sports cetre retals $2.25 per perso Number 137

160 c) Explai that whe they have decided o what items they wat for the party, they should complete the followig purchase order: Purchase Order Item Reasos Amout You Will Buy? How Much Each Perso Will Get Cost Total Cost d) Have studets complete the ivestigatio ad the share their plas for the party. Ecourage studets to explai their reasoig. 138 Grade 5 Mathematics: Support Documet for Teachers

161 Observatio Checklist Observe studets resposes to determie whether they ca do the followig: give reasoable estimates of the amout of food ad drik eeded for the party calculate correctly the cost of a item calculate correctly the amout of a item eeded provide appropriate reasos for choosig a item calculate correctly the total cost of the party calculate the cost of the party to be less tha or equal to the amout of moey available pla etertaimet that is withi the two-hour limit. pla etertaimet that is fu ad iclusive for all select food ad driks that meet everyoe s eeds select food ad driks that reflect the school s policy o utritio Number 139

162 N OTES 140 Grade 5 Mathematics: Support Documet for Teachers

163 G RADE 5 MATHEMATICS Patters ad Relatios (Patters)

164

165 Grade 5: Patters ad Relatios (Patters) Edurig Uderstadigs: Number patters ad relatioships ca be represeted usig variables. Geeral Outcome: Use patters to describe the world ad solve problems. SPECIFIC LEARNING OUTCOME(S): ACHIEVEMENT INDICATORS: 5.PR.1 Determie the patter rule to make predictios about subsequet elemets. [C, CN, PS, R, V] Exted a patter with or without cocrete materials, ad explai how each elemet differs from the precedig oe. Describe, orally or i writig, a patter usig mathematical laguage, such as oe more, oe less, five more. Write a mathematical expressio to represet a patter, such as r + 1, r 1, r + 5. Describe the relatioship i a table or chart usig a mathematical expressio. Determie ad explai why a umber is or is ot the ext elemet i a patter. Predict subsequet elemets i a patter. Solve a problem by usig a patter rule to determie subsequet elemets. Represet a patter visually to verify predictios. PRIOR KNOWLEDGE Studets should be able to do the followig: (3.PR.1) Describe, exted, compare, ad create icreasig patters usig maipulatives, diagrams, ad umbers (3.PR.2) Describe, exted, compare, ad create decreasig patters usig maipulatives, diagrams, ad umbers (4.PR.1) Idetify ad describe patters foud i tables icludig a multiplicatio chart (4.N.3) Add ad subtract whole umbers less tha Patters ad Relatios (Patters) 3

166 (4.N.6) Multiply whole umbers less tha 100 by whole umbers less tha 10 (4.N.7) Divide whole umbers less tha 100 by whole umbers less tha 10 BACKGROUND INFORMATION Although there are differet types of patters, the learig experieces that follow focus o icreasig/decreasig patters. The elemets that make up these patters are called terms. Each term builds o the previous term. Cosequetly, these patters are ofte referred to as growig patters. For example, 2, 4, 6, 8, ad 1, 2, 4, 8, are two commo icreasig patters. Usig a table to model a icreasig/decreasig patter ca help studets orgaize their thikig. It ca also help them geeralize the patters symbolically. There are two types of geeralizatios (rules) that ca be made: recursive ad explicit. A recursive geeralizatio tells how to fid the value of a term give the value of the precedig term. A explicit geeralizatio expresses the relatioship betwee the value of the term ad the term umber. For example, cosider this patter: The patter ca be orgaized ito a table like this. Term Term Value The recursive geeralizatio that describes this patter is + 2, sice the value of each term is two more tha the precedig term. If the patter were cotiued, the value of the sixth term would be 11 sice = 11. However, to fid the value of the 100th term, you would eed to fid the value of each of the 99 precedig terms. It is easier to predict the value of subsequet terms with a explicit geeralizatio. Notice that i the above patter, if you double a term umber ad subtract 1, you get the value of the term. For example: The value of the third term is 2 x 3 1 = 5 ad the value of the fifth term is 2 x 5 1 = 9. Thus, the explicit geeralizatio that describes the patter is 2 1. If the patter were cotiued, the value of the 100th term would be 199, sice 2 x = 199. Whe helpig studets recogize patters, it is importat to remember that they may ot see the patter i the same way as you. Therefore, it is importat that you ask studets to explai their thikig. Havig studets describe their reasoig ca also help them realize that ofte there is more tha oe way to look at a patter. 4 Grade 5 Mathematics: Support Documet for Teachers

167 MATHEMATICAL LANGUAGE Decreasig patter Icreasig patter Patter Term Term umber Term value LEARNING EXPERIENCES Assessig Prior Kowledge Materials: K-W-L charts (BLM ) ad large chart paper sheets Orgaizatio: Pairs/Whole class a) Tell studets that i the ext few lessos they will be learig about patters, but before they begi you eed to fid out what they already kow about them. b) Pose the questio: What is a patter? Have studets thik about the questio ad the share their thoughts with their parters. Ask studets to work with their parters to complete the first two colums of their K-W-L chart. c) Post the large sheets of chart paper o the board. Make a class K-W-L chart by askig each pair of studets to share their ideas with the rest of the class ad to write their resposes o the chart paper. Ecourage studets to explai what they kow about patters by askig them questios such as What is a patter? How do you kow the example you gave is a patter? What comes ext i your patter? How do you kow? How ca you describe your patter? Is there aother way you could describe your patter? d) Ask studets to record what they leared from the discussio i the third colum of their chart. e) Have studets maitai ad revisit their chart throughout the uit o patters. Patters ad Relatios (Patters) 5

168 Observatio Checklist Observe studets resposes to determie whether they ca do the followig: defie what a patter is idetify examples ad o-examples of patters idetify differet types of patters (e.g., Do they kow what a repeatig patter is? Do they kow what a icreasig patter is?) exted a patter. describe a patter use the vocabulary associated with patters correctly (e.g., Do they use words such as core, repeatig, icreasig/decreasig, ad term correctly?) Use their resposes to clear up ay miscoceptios they might have about patters ad to ehace their ability to create, exted, aalyze, ad describe patters. Exted a patter with or without cocrete materials, ad explai how each elemet differs from the precedig oe. Describe, orally or i writig, a patter usig mathematical laguage. Write a mathematical expressio to represet a patter. Describe the relatioship i a table or chart usig a mathematical expressio. Predict subsequet elemets i a patter. Represet a patter visually to verify predictios. Materials: Coloured tiles, overhead of the problem, ad copies of the worksheet (BLM 5.PR.1.1). Orgaizatio: Whole class/pairs a) Preset problem (a) to studets. Ecourage studets to discuss the patter by askig them questios, such as the followig: What does Agela s patter look like? How may tiles will Agela eed to make the fifth term of her patter? Draw the ext two terms i Agela s patter. Was your predictio right? How may tiles do you thik Agela will eed to make the teth term? Why? 6 Grade 5 Mathematics: Support Documet for Teachers

169 b) Begi to fill i a table. Explai that a table ca help them thik about the patter. Fill i the values for the first three terms. Relate each umber you write to the patter. Term Term Value c) Tell studets that they should work with their parter to fid out how may tiles Agela eeds to make the teth term, ad record their fidigs o the followig worksheet. d) Have studets share their aswers with the other members of the class ad explai their reasoig. Draw a completed table showig the umber of tiles eeded for the first 10 terms, ad ask studets how they could fid the umber of tiles eeded for ay term i the patter. Ecourage studets to look at the relatioship betwee a term ad the umber of tiles i the term. (Studets should ote that the umber of tiles is always three more tha the term umber, so the rule for fidig the umber of tiles i ay term of the patter is + 3). Show studets how to record the rule usig a mathematical expressio. Term Term Value Observatio Checklist Observe studets resposes to determie whether they ca do the followig: exted a patter with or without cocrete materials, ad explai how each elemet differs from the precedig oe describe, orally ad i writig, a patter usig mathematical laguage, such as oe more, etc. write a mathematical expressio to represet a patter, such as + 3 describe the relatioship i a table or chart usig a mathematical expressio predict subsequet elemets i a patter solve a problem by usig a patter rule to predict subsequet elemets represet a patter visually to verify predictios Patters ad Relatios (Patters) 7

170 Materials: Coloured tiles ad patter blocks Orgaizatio: a) Ask studets to use the coloured tiles to costruct ad the draw the ext two terms i this patter. b) Ecourage studets to discuss the patters by askig them questios, such as the followig: What patters do you otice? How does the secod term differ from the first term? The third term? Are there other patters that describe how the patter grows? What are they? c) Ask studets to make a table of values for the patter, ad the predict the umber of tiles eeded to make the teth term. Term No. of Tiles d) Have studets explai the patter that they used to predict the teth term. Ask them to write a rule for fidig the value of ay term i the patter (2), ad the use their rule to predict the umber of tiles i the 25th term. e) Have studets draw or use patter blocks to create a patter that uses the same umbers as the tile patter they just discussed. Have them share their patters with the other members of the class. 8 Grade 5 Mathematics: Support Documet for Teachers

171 Observatio Checklist Observe studets resposes to determie whether they ca do the followig: exted a patter with or without the use of cocrete materials, ad explai how each elemet differs from the precedig oe describe, orally or i writig, a patter usig mathematical laguage, such as oe more or oe less write a mathematical expressio to represet a patter, such as +2 describe the relatioship i a table or chart usig a mathematical expressio predict subsequet elemets i a patter recogize that differet patters ca have the same rule create a rule that fits a give patter Materials: Paper, pecil, ad a overhead of the problem (BLM 5.PR.1.2) Orgaizatio: Whole class a) Preset the followig problem: A fece is costructed usig posts ad boards. Betwee adjacet posts is oe board, as show o the trasparecy. How may boards will you eed if you build a fece with 90 posts? b) Have studets discuss the problem ad suggest strategies for solvig it. Try their strategies. If studets do ot thik of creatig a patter, ask them to draw a picture to show what the fece would look like if it were made with three posts. Four posts? The ask, How may boards are eeded if the fece is made with three posts? Four posts? How may boards do you thik are eeded if the fece is made with five posts? Draw a picture to check your predictio. Were you right? What could you do to fid out the umber of boards eeded for a fece with 90 posts? c) Ask studets to work with their parter to solve the problem. d) Have studets share their aswers ad explai their reasoig. Write their strategies o the board ad ecourage studets to describe a rule that they could use to fid the umber of boards eeded to make ay umber of posts ( 1, where represets the umber of posts). Patters ad Relatios (Patters) 9

172 Observatio Checklist Observe studets resposes to determie whether they ca do the followig: solve a problem by usig a patter rule to determie subsequet elemets describe orally a patter usig mathematical laguage represet a patter visually to verify predictios costruct a table of values describe the relatioship i a table or chart usig mathematical expressios explai why a umber is or is ot the ext elemet i a patter Exted a patter with or without cocrete materials, ad explai how each elemet differs from the precedig oe. Describe, orally or i writig, a patter usig mathematical laguage. Write a mathematical expressio to represet a patter. Predict subsequet elemets i a patter. Materials: Patter blocks ad overhead or copies of the problem (BLM 5.PR.1.3) Orgaizatio: Whole class a) Give studets the followig activity: Miguel started to build this patter with the triagle patter blocks. 1. Costruct ad draw the ext two terms i Miguel s patter. 2. Complete the table of values: Term 1 No. of Triagles 3 3. Predict the teth term of the patter. 4. Describe usig mathematical symbols ay patter (rule) you used to determie the teth term. b) Have studets share their solutios with the other members of the class ad explai their reasoig. 10 Grade 5 Mathematics: Support Documet for Teachers

173 Observatio Checklist Observe studets resposes to determie whether they ca do the followig: exted a patter usig cocrete materials, ad explai how each elemet differs from the precedig oe describe a patter usig mathematical laguage, such as oe more write a mathematical expressio to represet a patter, such as r + 1 predict subsequet elemets i a patter Materials: Paper ad pecils Orgaizatio: Whole class a) Preset the followig problem. Mr. Olse s class was studyig additio. Corry, who likes to add, decided to create a patter ivolvig the sums of two umbers. These are the first three terms i his patter: = = = = = = = = = 3 If Corry cotiues his patter, how may differet ways will he write 75 as the sum of two whole umbers? b) Have studets discuss the problem ad suggest strategies they could use to solve the problem. Have studets try their strategies. If o oe suggests tryig to fid a rule to describe the patter, ask studets to write the ext two term terms i Corry s patter ad the create a table of values. No No. of Ways c) Ask, How may differet ways ca Corry write 10 as the sum of two whole umbers? 15? 75? What rule ca you use to describe the umber of ways ay whole umber ca be writte as the sum of two whole umbers? ( + 1) d) Exted the activity by askig: How may differet ways ca you write the umber 75 as the sum of two whole umbers if combiatios that use the same addeds but i a differet order oly cout as oe way? For example, ad would oly cout as oe way. Patters ad Relatios (Patters) 11

174 Observatio Checklist Observe studets resposes to determie whether they ca do the followig: exted a patter ad explai how each elemet differs from the precedig oe describe orally a patter usig mathematical laguage, such as oe more describe the relatioship i a table or chart usig mathematical laguage solve a problem by usig a patter rule to determie subsequet elemets predict subsequet elemets i a patter determie ad explai why a umber is or is ot the ext umber i a patter Represet a patter visually to verify predictios. Materials: Overhead of the patter (BLM 5.PR.1.4), paper ad crayos or markers Orgaizatio: Idividual/Whole class a) Ask studets to complete the followig activity: 1. Here are two rules: + 5 ad 4. Ask studets, Which rule fits this patter? How do you kow the rule fits the patter? 2. Desig a patter that fits the other rule. 3. Ask them to explai how they kow their patter fits the rule. b) Have studets share their aswers with the other members of the class. Ecourage them to explai their reasoig. c) Have studets revisit their K-W-L chart. Have them share their ideas ad add them to the class K-W-L chart. 12 Grade 5 Mathematics: Support Documet for Teachers

175 Observatio Checklist Observe studets resposes to determie whether they ca do the followig: idetify that the rule represets a give patter create a patter that fits a give rule recogize that differet patters ca have the same rule Exted a patter with or without cocrete materials, ad explai how each elemet differs from the precedig oe. Describe, orally or i writig, a patter usig mathematical laguage. Write a mathematical expressio to represet a patter. Materials: Copies of the activity sheet (BLM 5.PR.1.5) Orgaizatio: Idividual a) Ask studets to complete the activity sheet. b) Have studets share their aswers with the other members of the class. Ecourage studets to explai their reasoig. c) Have studets revisit their K-W-L chart. Have studets share their ideas ad add them to the class K-W-L chart. Observatio Checklist Observe studets resposes to determie whether they ca do the followig: exted a patter without cocrete materials explai how each elemet differs from the precedig oe describe orally a patter usig mathematical laguage, such as oe more tha write a mathematical expressio to represet a patter, such as + 2 determie ad explai why a umber is or is ot the ext elemet i a patter Patters ad Relatios (Patters) 13

176 PUTTING THE PIECES TOGETHER How Tall? Purpose: The purpose of this ivestigatio is to have studets apply their kowledge of umber cocepts, patters, ad measuremet. I particular, the ivestigatio was desiged to reiforce studets ability to apply estimatio strategies recall multiplicatio ad divisio facts demostrate a uderstadig of fractios represet decimals to thousadths relate fractios to decimals measure the legth of objects determie the rule for a patter to make predictios The specific cocepts ad skills that studets demostrate will deped o the strategy that they use to solve the problem. I additio, the ivestigatio is desiged to ehace studets ability to solve problems reaso mathematically commuicate mathematically make coectios Materials: Metre sticks ad several copies of a large had (the size of bristol board) Orgaizatio: Small groups a) Hag copies of the large had aroud the room before studets come ito the room. b) Have studets look aroud the room ad ask them what they otice. Tell studets that they just had a visitor i the room ad he left his hadprits all over the room. Ask, Who could have left these prits? How big do you thik this perso is? c) Tell studets that their job is to figure out how tall the visitor was. Have the groups discuss the problem ad devise a strategy for determiig the height of the visitor. d) Have the groups carry out their strategy for solvig the problem. e) Have the groups share their solutios with the other members of the class ad explai their reasoig. 14 Grade 5 Mathematics: Support Documet for Teachers

177 Observatio Checklist Observe studets resposes to determie whether they ca do the followig: use mathematical laguage correctly carry out mathematical procedures correctly, such as measurig ad computig use mathematics cofidetly to solve problems use a variety of strategies to esure the correctess of their solutio commuicate ad reaso mathematically cotribute to the mathematical discussio Extesio: Examie proportios i the book Jim ad the Beastalk by Raymod Briggs. The giat claims that he is goig to eat three fried boys o a piece of toast. How big would the toast have to be to fit three fifth-graders? How big would the giat be based o the size of his/her toast? Patters ad Relatios (Patters) 15

178 N OTES 16 Grade 5 Mathematics: Support Documet for Teachers

179 G RADE 5 MATHEMATICS Patters ad Relatios (Variables ad Equatios)

180

181 Grade 5: Patters ad Relatios (Variables ad Equatios) Edurig Uderstadigs: Number patters ad relatioships ca be represeted by variables. Geeral Outcome: Represet algebraic expressios i multiple ways. SPECIFIC LEARNING OUTCOME(S): ACHIEVEMENT INDICATORS: 5.PR.2 Solve problems ivolvig siglevariable (expressed as symbols or letters), oe-step equatios with whole-umber coefficiets, ad whole-umber solutios. [C, CN, PS, R] Express a problem i cotext as a equatio where the ukow is represeted by a letter variable. Solve a sigle-variable equatio with the ukow i ay of the terms (e.g., + 2 = 5, 4 + a = 7, 6 = r 2, 10 = 2c). Create a problem i cotext for a equatio. PRIOR KNOWLEDGE Studets should be able to do the followig: (4.PR.5) Express a problem as a equatio i which a symbol is used to represet a ukow umber (4.PR.6) Solve oe-step equatios ivolvig a symbol to represet a ukow umber Patters ad Relatios (Variables ad Equatios) 3

182 BACKGROUND INFORMATION The learig experieces i this sectio focus, i part, o traslatig word problems ito equatios ad the solvig them. This is ot ew to studets. I the Early Years, studets were itroduced to whole-umber operatios through routie problems. I the begiig, studets solved these problems usig cocrete ad pictorial represetatios. Later, they traslated these problems ito equatios, ofte usig a empty square to represet the ukow value. Cosequetly, the learig experieces provide studets with additioal experiece with solvig routie problems ad, at the same time, begiig the trasitio to usig letters to represet ukow quatities. They also serve as a iformal itroductio to the terms equatio, mathematical expressio, ad variable. These terms are defied as follows: A equatio is a mathematical setece statig that oe or more quatities are equal. Equatios that cotai variables, such as 3 + x = 21 ad 2y + 3 = 15, are sometimes referred to as ope seteces, while equatios that have o variables, such as = 8 ad 24 3 = 8, are referred to as closed seteces. A mathematical expressio comprises umbers, variables, ad operatio sigs, but does ot cotai a relatioal symbol such as =,, <, >,, ad. For example, 6x + 3 ad x 8 are mathematical expressios. 4 A variable is a symbol for a umber or group of umbers i a mathematical expressio or equatio. MATHEMATICAL LANGUAGE Equatio Solutio Ukow 4 Grade 5 Mathematics: Support Documet for Teachers

183 LEARNING EXPERIENCES Assessig Prior Kowledge Materials: Noe Orgaizatio: Pairs/Whole class a) Tell studets they will be solvig some problems ivolvig equatios, but before you ca give them these problems you eed to kow what they already kow about equatios. b) Pose this questio: What is a equatio? Have studets thik about the questio for a few momets, the share their aswer with their parter. c) Have studets share their aswers with the other members of the class. Ecourage discussio by askig studets questios, such as the followig: What is a example of a equatio? What does the equatio tell you? Is 6 x 7 a equatio? Why or why ot? Is 5 = 14 9 a equatio? Why or why ot? Is 4 = 16 a equatio? Whe do you use equatios? d) Write studets resposes o the board or overhead. Place their resposes uder the headigs Thigs we kow about equatios ad Thigs we eed to thik about. Use the list to help you pla subsequet lessos. Observatio Checklist Observe studets resposes to determie whether they ca do the followig: describe what a equatio is recogize examples ad o-examples of equatios provide examples of equatios that cotai variables (ukow quatities) provide examples of equatios that do ot cotai variables recogize that a expressio ca be o either side of the equal sig (e.g., 5 = 14 9 is the same as 14 9 = 5) Patters ad Relatios (Variables ad Equatios) 5

184 Express a problem i cotext as a equatio where the ukow is represeted by a letter variable. Materials: Overhead of the problems (BLM 5.PR.2.1), copies of the activity (5.PR.2.2) Orgaizatio: Pairs/idividual a) Ask studets to work with their parter to solve the followig problem: Make sure studets uderstad what the phrase best fits meas (i.e., the equatio should reflect the structure of the problem. For example, if the actio i the problem idicates items are beig combied, the equatio should express the additio of quatities i the order they are give). Also, stress that they do ot have to solve the equatio. b) Give studets Problem A. Have studets share their aswer ad explai their reasoig. c) Give studets Problem B. Have studets share their aswer ad explai their reasoig. d) Have studets complete the activity sheet. Observatio Checklist Examie studet resposes to determie whether they ca do the followig: idetify the equatio that reflects the structure of the problem express a problem i cotext as a equatio where the ukow is represeted by a letter variable 6 Grade 5 Mathematics: Support Documet for Teachers

185 Materials: Math joural Orgaizatio: Idividual/Whole class a) Preset this problem: Nacy ad Jessica were asked to write a equatio for this story. I wat to buy 35 pecils. Pecils come i packages of 7. How may packages do I eed to buy? Nacy wrote 7 x = 35 ad Jessica wrote 35 7 =. Who is right? Why? b) Ask studets to thik about the problem, ad the record their aswer i their math jourals. c) Have studets discuss their aswer with the other members of the class. Observatio Checklist Observe studets resposes to determie whether they ca do the followig: idetify a equatio that fits a give story problem recogize that both equatios fit the problem Express a problem i cotext as a equatio where the ukow is represeted by a letter variable. Solve a sigle-variable equatio with the ukow i ay of the terms. Materials: Copies of the word problems (BLM 5.PR.2.3) Orgaizatio: Whole class a) Ask studets to write a equatio to fit a problem ad the solve it. Make sure studets kow that they should use a letter to represet the ukow value. b) Have studets share their aswer ad explai their reasoig. Ecourage discussio of the problem by askig them questios, such as the followig: What equatio did you use to solve the problem? Did ayoe use a differet equatio? What is it? Do the equatios have the same solutio? How do you kow your aswer is correct? What strategy did you use to solve the problem? Did ayoe use a differet strategy? What is it? Patters ad Relatios (Variables ad Equatios) 7

186 c) Repeat the activity with similar problems. d) Ask studets to write their ow problems. Have them give their problems to the other members of the class ad ask them to solve them. Make a class booklet of the problems that they write, ad share the book with aother class. Observatio Checklist Observe studets resposes to determie whether they ca do the followig: express a problem i cotext as a equatio where the ukow is represeted by a letter variable solve a sigle variable equatio where the ukow is represeted by a letter variable create a problem i cotext for a equatio use appropriate strategies to solve a equatio justify the solutio to a equatio Express a problem i cotext as a equatio where the ukow is represeted by a letter variable. Solve a sigle-variable equatio with the ukow i ay of the terms. Create a problem i cotext for a equatio. Materials: Paper ad pecils Orgaizatio: Small groups a) Ask the groups to select oe equatio from each of the followig lists: x + 9 = y = = 2c 7 + r = 35 y 8 = 16 5x = 60 z = x = k = 32 b) Ask the groups to write oe word problem for each equatio that they chose. c) Have each group exchage its problems with aother group. Have the groups solve each other s problems. 8 Grade 5 Mathematics: Support Documet for Teachers

187 d) Have several groups share their problems ad solutios with the other members of the class. Ecourage studets to discuss their solutios by askig them questios, such as the followig: What equatio did you use to solve the problem? Why did you choose that equatio? Is there aother equatio that could be used to solve the problem? How do you kow your solutio is correct? What strategy did you use to solve the equatio? Is there aother strategy that could be used to solve the problem? What is it? Observatio Checklist Observe studets resposes to determie whether they ca do the followig: create a problem i cotext for a equatio solve a sigle variable equatio with the ukow i ay of the terms express a problem i cotext as a equatio where the ukow is represeted by a letter variable use a appropriate strategy to solve a equatio justify their solutio to a equatio Patters ad Relatios (Variables ad Equatios) 9

188 Solve a sigle-variable equatio with the ukow i ay of the terms. Materials: Copies of the activity (BLM 5.PR.2.4) Orgaizatio: Idividual/Whole class a) Have studets complete the activity. Do a example with the class before lettig studets work o their ow. b) Have studets share their aswers with the other members of the class. Ecourage studets to justify their solutios ad explai the strategy that they used to solve the equatios. Observatio Checklist Observe studets resposes to determie whether they ca do the followig: solve a sigle variable equatio with the ukow represeted by a letter variable idetify the variable i a equatio use a appropriate strategy to solve the equatio justify their solutio to a equatio 10 Grade 5 Mathematics: Support Documet for Teachers

189 G RADE 5 MATHEMATICS Shape ad Space (Measuremet)

190

191 Grade 5: Shape ad Space (Measuremet) Edurig Uderstadigs: There is o direct relatioship betwee perimeter ad area. Geeral Outcome: Use direct or idirect measuremet to solve problems. SPECIFIC LEARNING OUTCOME(S): ACHIEVEMENT INDICATORS: 5.SS.1 Desig ad costruct differet rectagles, give either perimeter or area or both (whole umbers), ad draw coclusios. [C, CN, PS, R, V] Costruct or draw two or more rectagles for a give perimeter i a problem-solvig cotext. Costruct or draw two or more rectagles for a give area i a problem-solvig cotext. Illustrate that for ay perimeter, the square or shape closest to a square will result i the greatest area. Illustrate that for ay perimeter, the rectagle with the smallest possible width will result i the least area. Provide a real-life cotext for whe it is importat to cosider the relatioship betwee area ad perimeter. Shape ad Space (Measuremet) 3

192 PRIOR KNOWLEDGE Studets should be able to do the followig: (3.SS. 3) Estimate, measure, ad record the legth, width, ad height of objects to the earest metre ad cetimetre (3.SS.5) Estimate, measure, ad record the perimeter of regular ad irregular shapes (3.SS.5) Costruct shapes with a give perimeter (3.SS.5) Estimate, measure, ad record the area of regular ad irregular shapes with a give perimeter (4.SS.3) Estimate, measure, ad record the area of regular ad irregular shapes (4.SS.3) Costruct shapes with a give area (3.SS.5 & 4.SS.3) Distiguish betwee perimeter ad area (4.PR.1) Idetify ad describe patters foud i tables ad charts (3.SS.6) Idetify polygos Studets should also kow the followig: (3.SS.5) Perimeter is the distace aroud a shape (4.SS.3) Area is the amout of surface withi a regio RELATED KNOWLEDGE Studets should be able to do the followig: (5.SS.6) Demostrate a uderstadig of measurig legth i millimetres, ad distiguish rectagles from other quadrilaterals (5.SS.6) Recogize that all squares are rectagles BACKGROUND INFORMATION Perimeter is the distace aroud a shape. Studets ofte cofuse this cocept with area, the amout of surface a shape covers. Ivolvig studets i actual measurig experieces ca help them distiguish betwee these two cocepts. For example, activities that have studets completely coverig shapes with square uits ca help them uderstad the meaig of area. Moreover, studets ofte have miscoceptios about the relatioship betwee perimeter ad area. Two of the most commo are the followig: 1. the loger the perimeter, the larger the area 2. perimeter ad area icrease at the same rate 4 Grade 5 Mathematics: Support Documet for Teachers

193 For example, some studets have the mistake belief that if the perimeter is doubled, the area will double. Therefore, the itet of the learig experieces i this sectio is to help studets overcome their miscoceptios by havig them explore the perimeter ad area of rectagles. The learig experieces are also desiged to help studets recogize at least five geeralizatios about the relatioship betwee these two measuremets: If oly the perimeter (area) of a rectagle is give, its area (perimeter) caot be determied. Icreasig the perimeter (area) of a rectagle does ot ecessarily icrease the area (perimeter) of the rectagle. If the legth (width) of a rectagle is fixed, the icreasig its perimeter will icrease its area. The square has the largest area amog rectagles that have the same perimeter. The square has the smallest perimeter amog rectagles that have the same area. MATHEMATICAL LANGUAGE Area Legth Perimeter Polygo Rectagle Square Width Shape ad Space (Measuremet) 5

194 LEARNING EXPERIENCES Assessig Prior Kowledge Materials: Cetimetre grid paper (BLM 5 8.9) Orgaizatio: Idividual a) Ask studets to use the cetimetre grid paper to draw the followig: A polygo with a perimeter of 10 cm A polygo with a perimeter greater tha 15 cm A polygo with a area of 12 cm 2 A polygo whose area is greater tha 15 cm 2 ad less tha 25 cm 2 Iside of each shape that they draw, have studets write the ame of the polygo ad its perimeter or area measuremet. b) Preset the studets with the followig problems: Kelly wats to make a woode frame for the picture his aut drew for him. Does Kelly eed to measure the perimeter or the area of the picture to fid out how much wood he eeds? What uit of measuremet do you thik he should use? Explai your aswers. Mr. Lie wats to cover the bulleti board i his classroom with a piece of paper. Does he eed to measure the perimeter or the area of the bulleti board to fid out how much paper he eeds? What uit of measuremet do you thik he should use? Explai your aswers. Observatio Checklist Observe the studets resposes to determie whether they ca do the followig: costruct a polygo with a give perimeter costruct a polygo with a give area distiguish betwee perimeter ad area uderstad that perimeter is the distace aroud a shape uderstad that area is the amout of surface iside a regio ame polygos accordig to the umber of sides that they have kow that perimeter is measured i liear uits ad that area is measured i square uits justify their selectio of a uit of measuremet 6 Grade 5 Mathematics: Support Documet for Teachers

195 Costruct or draw two or more rectagles for a give perimeter i a problem-solvig cotext. Illustrate that for ay perimeter, the square or shape closest to a square will result i the greatest area. Illustrate that for ay perimeter, the rectagle with the smallest possible width will result i the least area. Materials: Square tiles, cetimetre grid paper (BLM 5 8.9), ad recordig sheet (BLM 5.SS.1.1) Orgaizatio: Small group/whole class a) Preset studets with the followig problem: Mrs. Zah ad Mr. Stewart have gardes that are rectagular i shape. The perimeter of Mrs. Zah s garde is 16 metres ad the perimeter of Mr. Stewart s garde is 20 metres. Is Mr. Stewart s garde larger tha Mrs. Zah s? Make sure studets uderstad the problem by askig: What do you kow about Mrs. Zah s garde? What do you kow about Mr. Stewart s garde? What questio do you eed to aswer? b) Have studets write dow what they thik the aswer is, ad the share it with the other members of their group. c) Next, have studets draw rectagles o the cetimetre grid paper to show why they thik their aswer is correct, ad the share their drawigs with the other members of their group. d) Challege studets by askig: Are there other rectagles that have perimeters of 16 metres? Are there other rectagles that have perimeters of 20 metres? What are their areas? Have studets i each group use the square tiles or cetimetre grid paper to fid other rectagles that have perimeters of 16 metres ad 20 metres. e) Ecourage studets to orgaize their work by havig them record their fidigs i the table provided (see BLM 5.SS.1.1). f) Have studets i each group aalyze their tables ad record ay patters or relatioships that they fid. g) Ask each group to preset its fidigs to the other members of the class, as well as its coclusio regardig whose garde Mrs. Zah s or Mr. Stewart s is larger. Shape ad Space (Measuremet) 7

196 Observatio Checklist Check studets work to determie whether they ca do the followig: costruct or draw two or more rectagles for a give perimeter i a problem-solvig cotext recogize that they caot tell for sure whether Mr. Stewart s garde is larger tha Mrs. Zah s that is, if oly the perimeter of a rectagle is give, its area caot be determied recogize patters ad relatioships, such as the square has the largest area amog rectagles with the same perimeter the rectagle with the smallest width has the least area icreasig the perimeter does ot ecessarily icrease the area if the legth of a rectagle is fixed, icreasig its perimeter icreases its area Materials: Square tiles, cetimetre grid paper (BLM 5 8.9), ad recordig sheet (BLM 5.SS.1.1) Orgaizatio: Small group/whole class a) Preset studets with the followig problem: A farmer has 36 metres of fecig material. He is plaig to use all of the fecig material to make a rectagular pe for his sheep. What is the largest pe he ca make for his sheep? Make sure studets uderstad the problem by askig: What does the farmer wat to do? How much fecig material does he have? What do you eed to fid out? b) Have studets write dow what they thik the aswer to the problem is ad share their aswer with the other members of their group. c) Next, ask, How may differet rectagular pes ca the farmer make with 36 metres of fecig material? d) Have studets i each group use the square tiles or cetimetre grid paper to fid all the rectagles that have a perimeter of 36 uits. Have studets record their fidigs i the table provided (see BLM 5.SS.1.1). e) Have studets aalyze their fidigs ad record ay patters ad relatioships that they fid. f) Have each group share with the other members of the class its solutio to the problem ad ay other patters that it fids. 8 Grade 5 Mathematics: Support Documet for Teachers

197 Observatio Checklist Observe studets resposes to determie whether they ca do the followig: costruct or draw two or more rectagles with a give perimeter i a problem-solvig cotext recogize that the pe with the largest area is a square with sides ie metres i legth that is, the square has the largest area amog rectagles with the same perimeter recogize patters ad relatioships, such as the rectagle with the smallest width has the least area the closer the rectagle is to a square, the closer the area is to the maximum area Costruct or draw two or more rectagles for a give area i a problemsolvig cotext. Illustrate that for ay perimeter, the square or shape closest to a square will result i the greatest area. Illustrate that for ay perimeter, the rectagle with the smallest possible width will result i the least area. Provide a real-life cotext for whe it is importat to cosider the relatioship betwee area ad perimeter. Materials: Spaghetti ad Meatballs for All by Marily Burs (ISBN ), square tiles, ad recordig table (BLM 5.SS.1.1) Orgaizatio: Whole class/small group a) Read Spaghetti ad Meatballs for All. As you read the story, have the studets use the square tiles to model what is happeig with the tables. b) Have studets discuss the problem with the table arragemets. Begi the discussio by askig, Why does Mrs. Comfort keep sayig the table arragemets wo t work? Have studets work with their parter to fid differet ways of arragig eight tables. Have them decide which arragemet is the best. c) Pose the problem: Mrs. Comfort has 24 square tables. If she pushes the tables together to form a rectagle, what is the highest umber of people she ca sit aroud the rectagle? Shape ad Space (Measuremet) 9

198 d) Make sure the studets uderstad the problem by askig them the followig questios: How may square tables does Mrs. Comfort have? What does Mrs. Comfort do with the tables? What do you eed to fid out? What is oe way that Mrs. Comfort ca push the tables together to form a rectagle? (Make sure the studets recogize that the rectagular arragemets caot have ay spaces i the middle.) How may people ca Mrs. Comfort sit aroud the table? e) Explai that the umber of tables pushed together represets the area ad the umber of people who ca sit aroud the table represets the perimeter. The, ask, Are there other rectagles that Mrs. Comfort ca make that have a area of 24 square uits? Ca she seat the same umber of people aroud each rectagle? f) Have the studets work with their parters to determie all the rectagles that ca be made with 24 tiles. Ecourage studets to record their fidigs i the table provided (see BLM 5.SS.1.1). g) Have studets aalyze their fidigs ad record ay patters ad relatioships that they fid. h) Ask studets to share their fidigs ad their coclusio as to which rectagle Mrs. Comfort should make if she wats to seat the most people with the rest of the class. 10 Grade 5 Mathematics: Support Documet for Teachers

199 Observatio Checklist Moitor studets resposes to determie whether they ca do the followig: costruct or draw two or more rectagles for a give area i a problem-solvig cotext recogize that Mrs. Comfort ca seat the most people aroud a 1 x 24 rectagle that is, amog rectagles with the same area, the oe with the smallest width has the greatest perimeter recogize patters ad relatioships, such as the followig: The closer the rectagle is to a square, the smaller its perimeter. If two rectagles have the same area, they do ot ecessarily have the same perimeter. If oly the area of a rectagle is give, its perimeter caot be determied. Costruct or draw two or more rectagles for a give area i a problemsolvig cotext. Illustrate that for ay perimeter, the square or shape closest to a square will result i the greatest area. Materials: Math jourals, square tiles, ad square cetimetre paper (BLM 5 8.9) Orgaizatio: Idividual/Large group a) Pose the followig problem: Mr. Satos is makig a rectagular flower garde i his backyard. If the area of the garde is 36 m 2, what is the least amout of fecig that he eeds to eclose the garde? b) Make sure the studets uderstad the problem by askig them the followig questios: What is Mr. Satos makig? How big is the garde? What do you eed to fid out? c) Tell studets that they ca use the square tiles or the cetimetre grid paper to help them solve the problem. Have the studets record their solutios i their math jourals. d) Have studets share their aswers ad the strategies they used to solve the problem. Shape ad Space (Measuremet) 11

200 Observatio Checklist Check studets work to determie whether they ca do the followig: recogize that differet rectagles ca have the same area recogize that the least amout of fecig that is eeded is 24 metres PUTTING THE PIECES TOGETHER Desig a Clubhouse Purpose: The purpose of this ivestigatio is to have studets apply the cocepts of perimeter ad area to a problem-solvig situatio. I particular, it is desiged to ehace studets ability to differetiate betwee perimeter ad area costruct rectagles with a give perimeter or area maximize or miimize the area of a rectagle with a fixed perimeter maximize or miimize the perimeter of a rectagle with a fixed area I additio, the ivestigatio is desiged to ehace studets ability to commuicate mathematically coect mathematics to real-world situatios solve problems reaso mathematically Materials/Resources: Cetimetre grid paper (BLM 5 8.9), coloured cetimetre grid paper, square tiles, scissors, ad glue Orgaizatio: Whole class/small groups a) Preset studets with the followig situatio: You ad your frieds have decided to build a rectagular clubhouse. You pla to build your clubhouse i a sectio of the schoolyard with a area of 200 m 2. You have decided that your clubhouse must have the followig: The largest possible floor space Two rugs (Oe rug must have a perimeter of 24 m ad cover the largest possible area, ad the other rug must have a perimeter of 16 m that covers the least possible area.) At least two doors with a width of oe metre 12 Grade 5 Mathematics: Support Documet for Teachers

201 A play area that takes up at least ¼ of the floor space. No furiture ca be placed i the play area A rectagular seatig area with a perimeter of 12 m A rectagular table with a area of 4 m 2 You also decide that the clubhouse ca have other items as log as they are ot placed i the play area. b) Explai that each group must draw up a pla for the clubhouse that icludes the dimesios of each item i the list of specificatios. Tell studets that they ca draw their pla for the clubhouse o the white cetimetre grid paper ad use the colour cetimetre paper to idicate the furiture ad the rugs. They should let each square cetimetre o the grid paper represet oe square metre. c) Help studets develop the criteria for assessig their plas for a clubhouse. d) Have studets work o their plas for a clubhouse. e) Have each group preset its desig for a clubhouse to the other members of the class. Ecourage studets to describe the dimesios of each item i their clubhouse ad how they determied its size. Observatio Checklist Use the rubric provided ad the studet-developed criteria to assess studets attaimet of outcome 5.SS.1 durig the completio of the project. Shape ad Space (Measuremet) 13

202 3 2 1 Distiguishes betwee perimeter ad area Determies the perimeter of a rectagle by fidig the distace aroud it. Determies the area of a rectagle by fidig the umber of square uits it covers. Determies the perimeter of a rectagle or the area of a rectagle. Is ot able to determie the perimeter of a rectagle. Costructs rectagles with a give perimeter Costructs a rectagle with a give perimeter. Costructs a rectagle with a give perimeter with support. Is ot able to costruct a rectagle with a give perimeter. Recogizes relatioships Recogizes that a square has the largest area amog rectagles that have the same perimeter. Recogizes that, amog rectagles that have the same perimeter, the oe with the smallest width has the least area. Recogizes that a square has the smallest perimeter amog rectagles with the same area. Recogizes that a square has the largest area amog rectagles with the same perimeter with support. Recogizes that, amog rectagles that have the same perimeter, the oe with the smallest width has the least area with support. Recogizes that a square has the smallest perimeter amog rectagles with the same area with support. Does ot recogize that a square has the largest area amog rectagles with the same perimeter. Does ot recogize that, amog rectagles that have the same perimeter, the oe with the smallest width has the least area. Does ot recogize that a square has the smallest perimeter amog rectagles with the same area. 14 Grade 5 Mathematics: Support Documet for Teachers

203 Grade 5: Shape ad Space (Measuremet) Edurig Uderstadigs: All measuremets are comparisos. Legth, area, volume, capacity, ad mass are measurable properties of objects. The uit of measure must be of the same ature as the property beig measured. Geeral Outcome: Use direct or idirect measuremet to solve problems. SPECIFIC LEARNING OUTCOME(S): ACHIEVEMENT INDICATORS: 5.SS.2 Demostrate a uderstadig of measurig legth (mm) by selectig ad justifyig referets for the uit mm modellig ad describig the relatioship betwee mm ad cm uits, ad betwee mm ad m uits [C, CN, ME, PS, R, V] Provide a referet for oe millimetre ad explai the choice. Provide a referet for oe cetimetre ad explai the choice. Provide a referet for oe metre ad explai the choice. Show that 10 millimetres is equivalet to 1 cetimetre usig cocrete materials (e.g., ruler). Show that 1000 millimetres is equivalet to 1 metre usig cocrete materials (e.g., metre stick). Provide examples of whe millimetres are used as the uit of measure. PRIOR KNOWLEDGE Studets should be able to do the followig: (3.SS.3) Estimate, measure, ad record the legth, width, ad height of objects to the earest metre or cetimetre (3.SS.3) Describe the relatioship betwee a metre ad a cetimetre (3.SS.3) Idetify referets for a cm ad a m (4.N.8) Demostrate a uderstadig of fractios less tha oe Studets should also kow that the terms legth, width, height, ad perimeter all idicate a measure of distace. Shape ad Space (Measuremet) 15

204 RELATED KNOWLEDGE Studets should be able to do the followig: (5.N.4) Multiply ad divide whole umbers by 10s, 100s, ad 1000s (5.N.8) Read, write, iterpret, ad use decimal otatio for 10ths, 100ths, ad 1000ths (5.N.9) Relate fractios to decimals (5.PR.1) Describe orally ad i writig the rule for a patter BACKGROUND INFORMATION Measuremet is the process of comparig a uit of measure with a measurable property of a object or pheomeo. The process cosists of the followig: 1. Idetifyig the property to be measured 2. Selectig a appropriate uit of measure 3. Repeatedly matchig the uit with the property or pheomea beig measured 4. Coutig the umber of uits By the ed of the 18th cetury, uits of measure varied greatly withi ad betwee coutries. The lack of stadard uits made tradig with other cultures difficult to carry out. To remedy this situatio, the Frech Natioal Assembly i 1790 asked the Academy of Sciece to develop a commo system of measuremet. The system it developed is kow as the metric system. The uits of measure developed by the academy have evolved ito the Système Iteratioal d Uités (abbreviated SI), which was established i The SI is govered by the Geeral Coferece o Weights ad Measures, which makes chages i the system to reflect the latest advaces i sciece ad techology. Eve though there are differeces betwee the two systems, SI is still referred to as the metric system. Because of its simplicity, all but a few coutries have adopted the metric system. Its simplicity arises from its use of the followig: 1. A small umber of base uits 2. The decimal system 3. A uiform set of prefixes that apply to each area of measuremet 16 Grade 5 Mathematics: Support Documet for Teachers

205 These prefixes the most commo of which are show below idicate multiples or subdivisios of the base uits. Prefix kilo (k) hecto (h) deka (da) deci (d) ceti (c) milli (m) Meaig 1000 uits 100 uits 10 uits 0.1 uit 0.01 uit uit I the Early ad Middle Years, studets are itroduced to legth, area, volume, capacity, ad mass. Their measuremet of these properties ivolves the uits listed i the chart below, ad ca be either direct or idirect. Direct measuremets ivolve selectig a uit ad comparig it directly with the object (e.g., usig a metre stick to measure the height of a table). Idirect measuremets are made whe a uit caot be placed directly o the object (e.g., fidig the height of a flagpole or the area of a coutry). Ofte, objects ca be measured idirectly by comparig them with thigs that ca be measured (e.g., fidig the height of a tree by measurig its shadow). Quatity Uits Symbol Legth Area Volume Capacity Mass kilometre metre cetimetre millimetre square metre square cetimetre cubic metre cubic cetimetre litre millilitre kilogram gram km m cm mm m 2 cm 2 m 3 cm 3 L ml kg g Shape ad Space (Measuremet) 17

206 Learig experieces that require studets to use measurig istrumets i realistic situatios are key igrediets i helpig them uderstad the cocepts ad skills ivolved i measuremet systems. I particular, these experieces ca help studets uderstad the followig: The measure of a uit is always 1 The uit must be of the same ature as the property that is beig measured The uit must be repeatedly matched with the property beig measured without ay gaps or overlaps (This process is kow as uit iteratio.) Oe uit may be more appropriate tha aother to measure the property of a object There is a iverse relatioship betwee the umber of uits ad the size of the uit A smaller uit gives a more exact measuremet A measuremet must iclude both a umber ad a uit Whe the same uits are used, measuremets ca be easily compared Estimatig that is, makig a reasoable judgmet about the approximate amout of a quatity also plays a importat role i the developmet of studets uderstadig of measuremet systems. A focus o estimatig eables studets to create a metal frame of referece for the size of uits ad their relatioships to each other. It also helps them judge the reasoableess of their measuremets. MATHEMATICAL LANGUAGE Cetimetre Estimate Height Legth Measuremet Metre Millimetre Referet Width 18 Grade 5 Mathematics: Support Documet for Teachers

207 LEARNING EXPERIENCES Assessig Prior Kowledge Materials: Assessmet activity sheet (BLM 5.SS.2.1) ad cm rulers Orgaizatio: Idividual Have studets complete the assessmet activity sheet (BLM 5.SS.2.1). Provide a referet for oe millimetre ad explai the choice. Show that 10 millimetres is equvalet to 1 cetimetre, usig cocrete materials (e.g., ruler). Provide examples of whe millimetres are used as the uit of measure. Materials: cm rulers with mm marked o them, a cm ruler that ca be projected o the overhead or a overhead of a cm ruler, toothpicks, safety pis, idex cards, crayos, math scribblers, soda straws, ad the activity sheet (BLM 5.SS.2.2) Orgaizatio: Whole class/idividual a) Ask studets to draw a metre stick ad make sure that they iclude all the markigs. Whe studets fiish their drawigs, have them share their pictures ad explai what the markigs o their metre sticks mea. Use the discussio to determie what the studets already kow about mm so you ca clear up ay miscoceptios that they may have. b) Tell studets that they will be learig about a ew uit of liear measure called a millimetre. Place a cm ruler o the overhead ad poit out that there are 10 spaces betwee cosecutive cetimetres. Tell studets that each space represets 1 millimetre. Write the word millimetre o the board or overhead, ad show studets the symbol for the uit. c) Have studets take out their cm rulers. Ask them to fid the umber of millimetres betwee the 1 cm mark ad the 2 cm mark the 10 cm mark ad the 11 cm mark the 15 cm mark ad the 17 cm mark the 20 cm mark ad the 23 cm mark Shape ad Space (Measuremet) 19

208 d) Have studets show these poits o their rulers: 20 mm 45 mm 85 mm 120 mm e) Tell studets that millimetres are used to measure the legths of small objects. Have them idetify objects that they would measure with this uit. f) Have studets complete the activity sheet (BLM 5.SS.2.2). Remid studets of the symbol for millimetre. Observatio Checklist Observe studets resposes to determie whether they ca do the followig: make reasoable estimates read ad use their cm rulers correctly to determie the legths of objects record measuremets correctly (e.g., recorded measuremets iclude both a umber ad the uit) recogize whe it s appropriate to use mm as a uit of measure (Note: May of the objects that studets ame could also be measured i cm or m.) 20 Grade 5 Mathematics: Support Documet for Teachers

209 Provide a referet for oe millimetre ad explai the choice. Provide a referet for oe cetimetre ad explai the choice. Provide a referet for oe metre ad explai the choice. Materials: Metre sticks ad cm rulers Orgaizatio: Whole class/parters Procedure a) Discuss the importace of estimatio i measuremet. For example, talk about how good estimatig skills ca help a idividual recogize whe a error i measuremet is made ad the cosequeces of usig a icorrect measuremet. b) Have studets idetify examples of situatios whe estimatig the legths of objects would be beeficial. For example: We eed to wrap a gift. Do we have eough ribbo to wrap the package? We wat to put a ew shelvig uit i the room. Is the room high eough for the shelvig uit? We wat to store some books. Is the box we have wide eough? c) Ask studets to estimate the legth of the room to the earest metre. Record their resposes o the board. Have studets share their strategies for estimatig the legth of the room ad discuss why their estimates varied. d) Discuss the importace of persoal referets i the estimatig process. Explai that a persoal referet is a familiar object, oe that they see or use regularly whose measure is kow. They ca thik of this object whe they are estimatig the legth of a ukow object (e.g., the legth of a baseball bat is approximately 1 metre). Whe estimatig the legth of a ukow object, they ca thik of a bat ad visualize how may bats log the object is. e) Have the studets work with a parter to fid five commo objects that are approximately 1 mm i legth, width, or height 1 cm i legth, width, or height 1 m i legth, width, or height f) Have studets share their referets for each uit with the rest of the class. Observatio Checklist Moitor studets resposes to determie whether they ca do the followig: provide reasos why estimatig is a importat skill give examples of situatios i which estimatig would be beeficial idetify appropriate referets for 1 mm, 1 cm, ad 1 m Shape ad Space (Measuremet) 21

210 Materials: Decks of 20 cards (oe side of each card should have a letter o it; the other side should have a lie segmet draw o it [BLM 5.SS.2.3]), a aswer sheet listig the legth of the lie segmet o each card Orgaizatio: Pairs a) Tell studets that they will be playig a estimatig game called Metric 210 with their parter. Explai how the game is played. 1. Shuffle the cards ad lay them face dow o the playig surface. 2. Players take turs takig a card from the top of the pile util oe of them estimates that he or she has a total legth of 210 mm ad stops the game by sayig I have the lie. This player may get rid of ay oe card that pushes the total over 210 mm. The player the states a estimate for the total legth of the remaiig cards. 3. The player uses the aswer sheet to determie his or her score. The player s score is determied by addig the differece betwee the estimate ad the actual legth to the differece betwee the actual legth ad The wier is the player with the lowest total score after five rouds of the game. b) Demostrate how the game is played ad aswer ay questios studets might have. Have studets play the game. c) Vary the game by havig studets estimate the legths of the lie segmets i cm. A roud of the game is over whe a studet thiks he or she has reached a legth of 210 mm. Observatio Checklist Observe studets to determie whether they ca do the followig: uderstad the rules for playig the game give reasoable estimates of the legths of the lie segmets calculate the total legth of the lies correctly calculate their scores correctly 22 Grade 5 Mathematics: Support Documet for Teachers

211 Materials: A deck of measuremet cards (BLM 5.SS.2.4), straight edges, a cetimetre ruler, ad a piece of paper for each player Orgaizatio: Pairs a) Tell studets that they will be playig a variatio of the game Metric 210. Explai how the ew versio of the game is played. 1. Shuffle the cards ad place them face dow o the playig surface. 2. Each card i the deck represets a mm legth. 3. The first player turs over a card ad uses a straight edge to draw a lie segmet he or she estimates to be the same legth as the umber of mm o the card (e.g., if the player turs over a 30, he or she draws, without measurig, a lie segmet that he or she thiks is 30 mm i legth ad records the legth above the lie segmet). 4. The secod player turs over a card ad uses a straight edge to draw a lie segmet he or she estimates to be the same legth as the umber of mm o the card. The secod player the records the legth above the lie segmet. 5. The first player turs over a lie card, ad exteds, without measurig, his or her lie segmet the umber of mm show o the card. The first player the records the legth above the lie segmet. For example, if the first player draws a 30 ad the a 50, his or her paper would look like this: 6. Play cotiues i this fashio util oe player has a lie segmet he or she estimates is 210 mm i legth ad stops the game by sayig, I have the lie. If the player thiks the last extesio of his or her lie segmet pushes the total legth beyod 210 mm, he or she ca state a estimate greater tha 210 mm. 7. The player measures the lie segmet to determie his or her score. The player s score is determied by addig the differece betwee the actual legth ad the estimated legth to the differece betwee the actual legth ad 210 mm. 8. The wier is the player who has the lowest score after five rouds of the game. b) Demostrate how the game is played ad aswer ay questios studets might have. Have studets play the game. c) Vary the game by chagig mm to cm (BLM 5.SS.2.4). A roud of the game is over whe a studet draws a lie segmet he or she estimates is 21 cm i legth. Shape ad Space (Measuremet) 23

212 Observatio Checklist Observe studets to determie whether they ca do the followig: uderstad the rules of the game make reasoable estimates determie the legth of the lie segmets by usig ad readig their rulers correctly record measuremets correctly (e.g., recorded measuremets iclude both the umber ad the uit) calculate their scores correctly Materials: Metre sticks, cm rulers, table to record their fidigs (BLM 5.SS.2.5) Orgaizatio: Small groups Procedure a) Tell studets that they will be playig a estimatig ad measurig game with the members of their group. Explai how to play the game. 1. Players take turs amig a uit of measure (m, cm, mm) ad a object that everyoe ca see. 2. Everyoe records a estimate of the legth of the object i the stated uit. 3. The player who amed the object measures it. The player whose estimate is the closest to the actual measuremet gets oe poit. If there is a tie, all players with the best estimate get oe poit. 4. The wier of the game is the first perso to get five poits. b) Have studets record i the table provided their choice of objects, their estimates, ad the actual measuremets (5.SS.2.5). c) Demostrate how the game is played ad aswer ay questio studets might have. Have studets play the game. Observatio Checklist Observe studets to determie whether they ca do the followig: uderstad the rules of the game determie the legths of objects by usig ad readig a metre stick or cm ruler correctly record measuremets correctly (e.g., recorded measuremets iclude both a umber ad the uit of measure) make reasoable estimates select a appropriate uit of measure 24 Grade 5 Mathematics: Support Documet for Teachers

213 Show that 10 millimetres is equivalet to 1 cetimetre usig cocrete materials (e.g., ruler). Show that 1000 millimetres is equivalet to 1 metre usig cocrete materials (e.g., metre stick). Materials: cm rulers ad lie segmets (BLM 5.SS.2.6) Orgaizatio: Idividual a) Give studets a copy of BLM 5.SS.2.6, ad tell them that they should measure each lie segmet twice: The first time, they should measure the lie segmet to the earest cm; the secod time, they should measure the lie segmet to the earest mm. b) Ecourage studets to record their fidigs i the table provided i BLM 5.SS.2.6. c) Ask studets to study their tables ad record ay patters they see. d) Have studets share their fidigs with the other members of the class. Ecourage studets to state a rule that describes the relatioship betwee cm ad mm. e) Check studets uderstadig of the relatioship betwee cm ad mm by askig: How may mm are i 1 cm? 2 cm? 3 cm? 4 cm? 8 cm? 15 cm? 50 cm?? If the legth of a object is give i cm, how ca you fid how log it is i mm without measurig? If 1 cm = 10 mm, what part of a cm is 1 mm? 2 mm? 4 mm? 8 mm? 10 mm? If a object is 7 mm log, how log is it i cm? If a object is 35 mm log, how log is it i cm? If a object is 83 mm log, how log is it i cm? If the legth of a object is give i mm, how ca you fid how log it is i cm without measurig? f) Show studets how to record the relatioship betwee mm ad cm. 1 cm = 10 mm 1 mm = 0.1 cm g) Help studets uderstad the relatioship betwee m ad mm by askig: How may cm are i 1 metre? How may mm are i 1 cm? If there are 10 mm i 1 cm ad 100 cm i 1 metre, how may mm are i 1 metre? Have studets use a metre stick to show why their aswers are correct. Shape ad Space (Measuremet) 25

214 h) Check studets uderstadig of the relatioship betwee mm ad m by askig: How may mm are i 1 m? 2 m? 3 m? 5 m? 8 m? 10 m? If the legth of a object is give i m, how ca you fid how log it is i mm without measurig? If there are 1000 mm i oe m, what part of a m is 1 mm? 2 mm? 10 mm? 25 mm? 100 mm? If a object is 1000 mm log, how log is it i m? If a object is 3000 mm log, how log is it i m? If a object is 6000 mm log, how log is it i m? If the legth of a object is give i mm, how ca you fid how log it is i m without measurig? i) Show studets how to record the relatioship betwee m ad cm. 1 m = 1000 mm 1 mm = m Emphasize that milli meas thousadths so 1 mm meas 1 thousadth of a metre. Observatio Checklist Observe studets to determie whether they ca do the followig: fid the legths of lie segmets by usig ad readig their cm rulers correctly record measuremets correctly (e.g., recorded measuremets iclude both a umber ad the uit) recogize the relatioship betwee mm ad cm recogize the relatioship betwee mm ad m covert cm to mm ad vice versa covert mm to m ad vice versa 26 Grade 5 Mathematics: Support Documet for Teachers

215 Materials: I have, who has? cards (BLM 5.SS.7) Orgaizatio: Whole class a) Tell studets that they will be playig a metric versio of the game I have, who has?. Explai that each studet will get oe card (some studets may get two cards if there are fewer tha 30 studets i the class). Oe studet will start the game by readig his or her card, ad the perso who has the aswer to the questio posed by this studet reads his or her card. Play cotiues i this fashio util it gets back to the perso who started the game. b) After the studets have played the game several times, have them make their ow metric coversio I have, who has? game ad play it with the other members of the class. Variatio: Have studets work i groups of 2 or 3, givig them several of the cards. Play the game as a class as you would i (a) ad (b) above. This gives studets the opportuity to egage with more tha oe card. Observatio Checklist Moitor studets resposes to determie whether they multiply or divide by tes, hudreds, or thousads add, subtract, multiply, or divide umbers other tha powers of 10 kow the relatioship betwee m, cm, ad mm Materials: cm rulers, stir sticks, books, erasers, soda cas, ad pecil cases Orgaizatio: Pairs/Whole class a) Preset the followig problem to studets: Marti drew a lie that was 64 mm log. His fried Zack measured the lie segmet ad said that it was 6.4 cm log. Is Zack right? How do you kow? b) Make sure studets uderstad the problem by askig: How log is the lie that Marti drew? What else do you kow? What do you eed to fid out? c) Have studets work with their parter to solve the problem. Whe they fiish, have them share their solutios with the other members of the class, ad discuss why the two measuremets are the same. Note: Some studets will be able to solve the problem by reasoig while others will eed to use their cm rulers ad draw the lie. Shape ad Space (Measuremet) 27

216 d) Check studets uderstadig of how to express measuremets to the earest teth of a cm by askig: What is 37 mm expressed i cm? What is 93 mm expressed i cm? What is 58 mm expressed i cm? What is 116 mm expressed i cm? Have studets use their cm rulers to justify their aswers. e) Have studets make ad complete the followig table: Object Estimated Legth Legth to Nearest Teth of a cm Legth to Nearest mm A stir stick Thickess of a book A eraser Distace aroud a ca of soda Width of a pecil case f) Have studets express each of the followig measuremets i mm: 25.1 cm 85.6 cm 37.9 cm 12.2 cm Observatio Checklist Moitor studets resposes to determie whether they ca do the followig: record a measuremet to the earest teth of a cm covert mm to the earest teth of a cm ad vice versa uderstad that 10 mm is the same as 1 cm read ad iterpret measuremets expressed i decimals make reasoable estimates 28 Grade 5 Mathematics: Support Documet for Teachers

217 Cautio: I some commuities, playig cards are see as a form of gamblig ad discouraged. Please be aware of local sesitivities before itroducig this activity. Materials: Deck of cards for each group Orgaizatio: Small groups of 2 to 4 studets a) Tell studets they will be playig the Metric Covert game, ad explai how it is played. 1. Shuffle the cards ad place them face dow o the playig area. 2. The umbers o the cards represet mm. Let aces = 1 mm, jacks = 11 mm, quees = 12 mm, ad kigs = 13 mm. 3. Oe player turs over a card ad places it i the cetre of the playig area so everyoe ca see it. 4. The first player to covert mm to cm correctly takes the card (e.g., if a 8 is tured over, the first player to say 0.8 [8 teths] cm wis the card). 5. If there is a tie or a error is made, the card is put back ito the deck ad the cards are reshuffled. 6. The perso who wis the cards turs over the ext card. 7. The game proceeds i this fashio util there are o cards. 8. The perso with the most cards is the wier. b) Demostrate how the game is played ad aswer ay questios studets might have. Have studets play the game. c) Vary the game by havig the studets covert from cm to mm m to cm m to mm mm to m Observatio Checklist Observe studets to determie whether they kow the relatioships betwee mm, cm, ad m calculate correctly Shape ad Space (Measuremet) 29

218 N OTES 30 Grade 5 Mathematics: Support Documet for Teachers

219 Grade 5: Shape ad Space (Measuremet) Edurig Uderstadigs: All measuremets are comparisos. Legth, area, volume, capacity, ad mass are measurable properties of objects. The uit of measure must be of the same ature as the property of the object beig measured. Geeral Outcome: Use direct or idirect measuremet to solve problems. SPECIFIC LEARNING OUTCOME(S): ACHIEVEMENT INDICATORS: 5.SS.3 Demostrate a uderstadig of volume by selectig ad justifyig referets for cm 3 or m 3 uits estimatig volume by usig referets for cm 3 ad m 3 costructig rectagular prisms for a give volume [C, CN, ME, PS, R, V] Idetify the cube as the most efficiet uit for measurig volume ad explai why. Provide a referet for a cubic cetimetre ad explai the choice. Provide a referet for a cubic metre ad explai the choice. Determie which stadard cubic uit is represeted by a give referet. Estimate the volume of a 3-D object usig persoal referets. Determie the volume of a 3-D object usig maipulatives ad explai the strategy. Costruct a rectagular prism for a give volume. Explai that may rectagular prisms are possible for a give volume by costructig more tha oe rectagular prism for the same give volume. Shape ad Space (Measuremet) 31

220 PRIOR KNOWLEDGE Studets should be able to do the followig: (K.SS.1) Use direct compariso to compare the volume of two objects (1.SS.1) Idetify attributes of objects that ca be compared (1.SS.1) Demostrate a uderstadig of measuremet as a process of comparig by fillig (4.SS.5) Describe ad costruct rectagular prisms (3. SS.3) Measure the legths of objects i m or cm BACKGROUND INFORMATION The terms volume ad capacity are ofte used iterchageably. For the purposes of the learig experieces i this sectio ad the sectio that follows, a distictio will be made. Volume is the amout of space a object occupies or, if the object is hollow, the amout of space iside the object. Volume is measured i cubic cetimetres (cm 3 ) or cubic metres (m 3 ). Capacity is the maximum amout of liquid a cotaier ca hold. Capacity is measured i litres (L) ad millilitres (ml). MATHEMATICAL LANGUAGE Cubic uit (cetimetre ad metre) Dimesio Rectagular prism Legth (width, height) Less (least) volume More (greatest) volume Same volume Volume 32 Grade 5 Mathematics: Support Documet for Teachers

221 LEARNING EXPERIENCES Idetify the cube as the most efficiet uit for measurig volume ad explai why. Determie the volume of a 3-D object usig maipulatives ad explai the strategy. Materials: A variety of small boxes, cubes, marbles, other 3-D shapes such as a triagular prism or pyramid, ad sad Orgaizatio: Whole class/small group a) Show studets the isides of two empty boxes. Ask, Which box has more space iside? How ca we tell for sure? b) Explai that volume is the amout of space iside a cotaier or the umber of uits eeded to fill the cotaier. Ask, What uit do you thik we should use to measure volume? c) Give each group a box ad three possible uits: marbles, cubes, ad triagular prisms (or ay other shape). Tell studets that their task is to determie which uit is best for measurig volume. Explai that they will be measurig the volume of their box three times. Each time they will completely fill the box with oe of the uits. Explai that whe fillig the box they should lay the uits carefully o the bottom of the box, record the umber used, ad the fill the box layer by layer. Ask studets to record the total umber of uits used as well as their observatios o the appropriateess of the uit. d) Have studets share their observatios about the differet uits. Help them recogize that the cube is the best uit to use because it is easy to stack ad there are o gaps or overlaps whe fillig the cotaiers (e.g., have studets pour sad ito a box filled with marbles to show them that there are gaps betwee the marbles). Observatio Checklist Observe studets to determie whether they ca do the followig: measure the volume of the box correctly (e.g., completely fill the box with a uit ad cout the umber of uits used) record both the umber ad the uit of measure recogize that the cube is the most efficiet uit for measurig volume ad explai why Shape ad Space (Measuremet) 33

222 Determie the volume of a 3-D object usig maipulatives ad explai the strategy. Materials: Small boxes ad cubes Orgaizatio: Pairs or small groups a) Give each group four or five boxes. Have studets label the boxes A, B, C, D. b) Have studets look at the labelled boxes, ad decide which oe they thik has the smallest volume ad which oe has the largest volume. Ask them to put the boxes i order from the smallest volume to the largest volume ad to record the order they have decided o. c) Have studets measure the volume of each box to the earest whole uit ad record their measuremets i a table like the oe show below. Have studets record the actual volume of the cotaiers ad compare it with their estimated volume. Box Estimated Volume Actual Volume A B C D d) Have each group share its fidigs with the rest of the class. Ecourage studets to discuss the strategies they used to determie the volume of the boxes, particularly whe the umber of cubes was ot a exact fit (e.g., whe there was some space aroud the layers). Observatio Checklist Moitor studets resposes to determie whether they ca do the followig: compare ad order cotaiers accordig to their volume make reasoable estimates of the volume of cotaiers determie the volume of a object usig maipulatives ad explai the strategy record measuremets that iclude both a umber ad the uit use the terms more (greatest) volume, less (least) volume, ad the same volume correctly 34 Grade 5 Mathematics: Support Documet for Teachers

223 Idetify the cube as the most efficiet uit for measurig volume ad explai why. Provide a referet for a cubic cetimetre ad explai the choice. Materials: Differet sizes of boxes, differet sizes of cubes, a white rod from a set of Cuiseaire rods, ad ceticubes Orgaizatio: Small groups a) Preset studets with the followig situatio: A box Nicky measured has a volume of 18 cubic uits. She gave the box to Cathy. Whe Cathy measured the volume of the box, she foud it had a volume of 26 cubic uits. Could both girls measuremets be correct? Why or why ot? b) Ecourage studets to devise ad carry out a pla to prove their assertios about the situatio. c) Have studets share their results ad reasoig with the other members of the class. Ecourage studets to discuss the eed for a stadard uit of measure ad the reasos why it is importat to use commo uits (e.g., to facilitate commuicatios, busiess, ad trade). d) Show studets a white rod from the set of Cuiseaire rods or a ceticube, ad explai that i the metric system a cubic cetimetre is oe of the uits used to determie the volume of a object. Show studets the word ad the symbol for the uit. e) Tell studets that they will be usig the white rods (or ceticubes) to complete the followig activity: 1. Fid a cotaier that has a volume that is greater tha 80 cm 3 less tha 40 cm 3 betwee 50 ad 60 cm 3 2. Fid as may objects as you ca that have a volume of 1 cm 3. f) Have studets share their fidigs with the rest of the class. Ecourage studets to discuss the strategies they used to fid the cotaiers ad the objects they foud that are approximately 1 cm 3. Start a class list of objects that have a volume of 1 cm 3. Ecourage studets to look outside the classroom for objects that have a volume of approximately 1 cm 3 ad add them to list. Shape ad Space (Measuremet) 35

224 Observatio Checklist Observe studets resposes to determie whether they ca do the followig: recogize the eed for a stadard uit determie the volume of a object usig maipulatives ad explai the strategy idetify objects that have a volume of approximately 1 cm 3 Estimate the volume of a 3-D object usig persoal referets. Determie the volume of a 3-D object usig maipulatives ad explai the strategy. Costruct a rectagular prism for a give volume. Explai that may rectagular prisms are possible for a give volume by costructig more tha oe rectagular prism for the same volume. Materials: Cetimetre grid paper (BLM 5 8.9), ceticubes, scissors; tape, copies of the istructios for the activity (BLM 5.SS.3.1), ad observatio form (BLM 5 8.1) Orgaizatio: Small groups Procedures: a) Tell studets that they will be usig the followig procedure to complete a ivestigatio ivolvig volume. 1. Cut a 1-cm square from each corer of a 20-cm by 20-cm sheet of grid paper (see diagram below). 36 Grade 5 Mathematics: Support Documet for Teachers

225 2. Fold up the four 1-cm sides ad tape to form a ope box that is 1 cm deep. Fid the volume of the box. 3. Predict what will happe to the volume if you cut larger ad larger squares from each corer. Record your predictios o a chart. 4. As a group, costruct as may ope boxes as possible ad record the volume of each ope box. 5. How close were your predictios to the volume of the ope boxes? 6. Explai what happeed to the volume whe larger squares were cut from the corers. 7. Write a group report about what you have leared. b) Help studets determie what should be icluded i their reports ad the criteria for evaluatig them. Ecourage studets to cosider such thigs as the accuracy of their measuremets, the strategies they used to determie the volumes of the ope boxes, ad the clarity of their explaatio of what happes to the volume as the dimesios of the ope boxes chage. Observatio Checklist Use the observatio form (BLM 5 8.1) to observe how well studets work together. Shape ad Space (Measuremet) 37

226 Estimate the volume of a 3-D object usig persoal referets. Determie the volume of a 3-D object usig maipulatives ad explai the strategy. Material: Ceticubes, small boxes such as a shoebox or a cereal box, a list of the volumes of the boxes, ad math jourals Orgaizatio: Small groups/whole class a) Give each group four or five small boxes ad oly eough ceticubes to cover the bottom of each box separately, plus eough to make oe stack the height of each box. b) Ask studets to estimate the volume of each box, ad to record their estimates i their math jourals. Explai that there are ot eough ceticubes to completely fill ay box; however, they ca use the ceticubes that they have bee give to help them make their estimates. Whe they fiish estimatig the volumes of the boxes, they should compare their estimates with the list of the volumes of the boxes that you have prepared. c) Have studets share the strategies they used to estimate the volumes of each box. Observatio Checklist Moitor studets resposes to determie whether they ca do the followig: explai the strategy they used to determie the volume of the boxes use a referet to make reasoable estimates of the volume of the boxes 38 Grade 5 Mathematics: Support Documet for Teachers

227 Costruct a rectagular prism for a give volume. Explai that may rectagular prisms are possible for a give volume by costructig more tha oe rectagular prism for the same volume. Materials: Ceticubes, or the white rods from a set of Cuiseaire rods, or multilik cubes Orgaizatio: Pairs a) Tell studets that they will be makig rectagular prisms with their ceticubes ad determiig their volumes. Show studets a rectagular prism made with 10 cubes. Ask studets what the volume of the prism is, ad how they kow. b) Have studets complete the followig activity: 1. Costruct a rectagular prism with a volume of 12 cm 3 16 cm 3 Record the dimesios of the prisms you made. 2. Build two rectagular prisms, side by side, so that oe prism has a volume of 6 cm 3 more tha aother. Record the dimesios of each prism. 3. Make two rectagular prisms with the same legth, with oe wider ad shorter tha the other, but with differet volumes. Record the dimesios of each prism. 4. Make two rectagular prisms with the same legth, with oe wider ad shorter tha the other, but with the same volume. Record the dimesios of each prism. 5. Make as may rectagular prisms as you ca that have a volume of 24 cm 3. Record the dimesios of each prism that you make. 6. Make three rectagular prisms with the followig dimesios: 6 cm x 6 cm x 6 cm 3 cm x 12 cm x 6 cm 3 cm x 9 cm x 8 cm Fid the volume of each prism. Record the dimesios of each prism ad its volume. 7. Thik about the rectagular prisms that you made. What ca you coclude about the volume of prisms? Record your observatios. Whe studets fiish each part of the activity, have them share their results with the other members of the class. Shape ad Space (Measuremet) 39

228 Observatio Checklist Observe studets resposes to determie whether they ca do the followig: costruct a rectagular prism with a give volume recogize that differet rectagular prisms are possible for a give volume recogize that rectagular prisms with differet dimesios ca have the same volume recogize that if oe dimesio of two rectagular prisms is the same, the volume of the prisms is ot ecessarily the same recogize that the volume of a rectagular prism is depedet o its dimesios Provide a referet for a cubic metre ad explai the choice. Determie which stadard cubic uit is represeted for a give referet. Materials: Metre sticks, plasticie, cardboard, scissors, ad tape Orgaizatio: Small groups/whole class a) Ask studets to thik of cotaiers they see iside ad outside of school whose volume should be measured i cm 3. Keep a list of their suggestios. Fiish the discussio by askig, Are there ay cotaiers or objects that are too large to be measured with a cm 3? b) Have each group develop a list of cotaiers or objects that would require a larger uit of measure. Whe studets fiish, have them share their list with the other members of the class. c) Explai that i the metric system the volumes of very large items or cotaiers are measured i cubic metres. Ask studets to show with their hads how large they thik a cubic metre is. d) Have each group make a model of a cubic metre. Some groups ca make their cubic metre usig 12 metre sticks (or woode dowels 1 metre i legth) joied with plasticie or maskig tape, while other groups ca draw ad cut out six 1 metre squares from heavy cardboard ad joi the squares with maskig tape. e) Have studets estimate how may studets they thik will fit ito a cubic metre. Have them try it out ad the discuss how their estimates compared with the actual umber ad the reasos why they may vary. 40 Grade 5 Mathematics: Support Documet for Teachers

229 Observatio Checklist Observe studets resposes to determie whether they ca do the followig: idetify cotaiers or objects whose volumes should be measured i cubic cetimetres idetify cotaiers or objects whose volumes should be measured with larger uits make a cubic metre Provide a referet for a cubic metre ad explai the choice. Determie which stadard cubic uit is represeted for a give referet. Estimate the volume of a 3-D object usig maipulatives ad explai the strategy. Materials: Models of cubic metres Orgaizatio: Whole class/small groups a) Have studets refer to their models of a cubic metre to estimate whether the followig objects have a volume greater tha, less tha, or about the same as a cubic metre: their desk a pop machie a filig cabiet a garbage ca a dump truck a stove Ecourage studets to explai their reasos for their estimates. b) Ask each group to make a list of items iside ad outside of the classroom whose volume could be measured i cubic metres. Have the groups share their lists ad explai the reasos for their choices. c) Have each group refer to its model of a cubic metre to estimate the volume of their classroom the school gym the pricipal s office Have the groups share their estimates ad the strategies they used to determie the volumes of the rooms. Shape ad Space (Measuremet) 41

230 d) Have studets discuss the questio: Does a cubic metre have to be a cube? Note: Studets should recogize from the previous activity that a cubic metre does ot eed to be a cube sice prisms with differet dimesios ca have the same volume. e) Have studets collect ad fill oe of the cardboard cubic metres they made with a item they would like to give to charity (e.g., studets could give a cubic metre of clothes that they have outgrow). Have studets write a letter to the commuity ad other classes i the school explaiig what they are doig ad ivitig them to help collect the item they have chose. Observatio Checklist Moitor studets resposes to determie whether they ca do the followig: idetify objects or cotaiers whose volume could be measured i cubic metres give reasoable estimates of cotaiers or items whose volume could be measured i cubic cetimetres explai the strategies they used to estimate the volumes of large cotaiers ad objects 42 Grade 5 Mathematics: Support Documet for Teachers

231 Grade 5: Shape ad Space (Measuremet) Edurig Uderstadigs: All measuremets are comparisos. Legth, area, volume, capacity, ad mass are measurable properties of objects. The uit of measure must be of the same ature as the property of the object beig measured. Geeral Outcome: Use direct or idirect measuremet to solve problems. SPECIFIC LEARNING OUTCOME(S): ACHIEVEMENT INDICATORS: 5.SS.4 Demostrate a uderstadig of capacity by describig the relatioship betwee ml ad L selectig ad justifyig referets for ml or L uits estimatig capacity by usig referets for ml or L measurig ad recordig capacity (ml or L) [C, CN, ME, PS, R, V] Demostrate that 1000 millilitres is equivalet to 1 litre by fillig a 1-litre cotaier usig a combiatio of smaller cotaiers. Provide a referet for a litre ad explai the choice. Provide a referet for a millilitre ad explai the choice. Determie which capacity uit (ml or L) is represeted by a give referet. Estimate the capacity of a cotaier usig persoal referets. Determie the capacity of a cotaier usig materials that take the shape of the iside of the cotaier (e.g., a liquid, rice, sad, beads), ad explai the strategy. Shape ad Space (Measuremet) 43

232 PRIOR KNOWLEDGE Studets should be able to do the followig: (1.SS.1) Idetify attributes of objects that ca be measured (K.SS.1) Use direct compariso to compare the capacity of two objects (1.SS.1) Demostrate a uderstadig of measuremet as a process of comparig by fillig (4.N.1) Demostrate a uderstadig of whole umbers less tha (4.N.3) Demostrate a uderstadig of additio ad subtractio of whole umbers with aswers less tha RELATED KNOWLEDGE (5.SS.2) Studets should kow what a referet is ad the role it plays i estimatig measuremets. BACKGROUND INFORMATION The terms volume ad capacity are ofte used iterchageably. For the purposes of the learig experieces i this sectio ad the previous sectio, a distictio will be made. Volume is the amout of space a object occupies or, if the object is hollow, the amout of space iside the object. Volume is measured i cubic cetimetres (cm 3 ) or cubic metres (m 3 ). Capacity is the maximum amout of liquid a cotaier ca hold. Capacity is measured i litres (L) ad millilitres (ml). MATHEMATICAL LANGUAGE Capacity More capacity Less capacity Same capacity Estimate Litre Referet Millilitre 44 Grade 5 Mathematics: Support Documet for Teachers

233 LEARNING EXPERIENCES Determie the capacity of a cotaier usig materials that take the shape of the iside of the cotaier (e.g., a liquid, rice, sad, beads), ad explai the strategy. Materials: A variety of cotaiers (some of which should be trasparet), fuels, water, sad (or ay other material that will take the shape of cotaiers), paper towels, spoges, ad markers Orgaizatio: Whole class/small groups a) Explai that we ofte hear expressios, such as the followig: The room was filled to capacity. They played to a capacity crowd. Ask, What does the word capacity mea? How ca we fid the capacity of a object? b) Explai that i math we use the term capacity to describe how much liquid a cotaier ca hold, ad to determie the capacity of a cotaier we eed a uit of measure. c) Show studets a trasparet cotaier. Show studets how to measure the capacity of the cotaier by usig aother smaller trasparet cotaier as the uit of measure. Repeat this activity two or three times to make sure studets uderstad how to measure the capacity of a cotaier. d) Give each group four or five cotaiers. Have studets select oe of their cotaiers to be the uit of measure ad label the other cotaiers A, B, C, D. e) Have studets look at the labelled cotaiers ad decide which oe they thik has the smallest capacity ad which oe has the largest capacity. Ask them to put the cotaiers i order from the smallest capacity to the largest capacity, ad to record the order they have decided o. f) Have studets give their uit a ame. Have them measure each cotaier ad record their measuremets i a table like the oe show below. Have studets record the actual order of the cotaiers, ad compare it with their estimated order. Cotaier Estimated Capacity Actual Capacity A B C D Shape ad Space (Measuremet) 45

234 g) Have each group share its fidigs with the rest of the class. Ecourage them to describe how the real order of the cotaiers compared with their estimated order. Observatio Checklist Observe studets resposes to determie whether they ca do the followig: use the terms more capacity, less capacity, ad the same capacity correctly measure correctly (studets completely fill the uit over ad over agai util the cotaier beig measured is full) record their measuremets correctly (icludes both a umber ad the uit) give reasoable estimates of capacity compare ad order cotaiers accordig to their capacities Materials: A wide variety of cotaiers, maskig tape, spoos, large plastic glasses or jars, water, markers, paper towels, spoges, ad procedure steps (BLM 5.SS.4.1) Orgaizatio: Pairs a) Have studets use the followig procedure to make their ow measurig device: 1. Put a piece of maskig tape dow the side of a glass (or jar). 2. Fill a small cotaier with spoofuls of water, keepig track of the umber of spoofuls eeded to fill it. 3. Empty the water i the small cotaier ito the glass. 4. Mark the level of the water ad the umber of spoofuls o the tape. 5. Fill the small cotaier agai. Empty the water ito the glass. Mark the level of the water ad the total umber of spoofuls. 6. Cotiue fillig ad markig the glass util the top is reached. b) Make sure the studets kow how to read ad use their measurig device. Have them use their measurig device to fid the capacity of five differet cotaiers i two differet ways. Have the studets record their fidigs i the chart provided i BLM 5.SS.4.1. c) Have studets write a paragraph describig the two differet ways they foud the capacity of their cotaiers. Have studets share ad discuss their methods with the other members of the class. d) Have studets exchage their cotaiers with aother group. Have them use their measurig device to determie the capacity of these cotaiers ad record their fidigs i a table. 46 Grade 5 Mathematics: Support Documet for Teachers

235 Observatio Checklist Moitor studets resposes to determie whether they ca do the followig: use the terms more capacity, less capacity, ad the same capacity correctly determie the capacity of a cotaier by fillig it with water usig their measurig device determie the capacity of a cotaier by fillig it, ad the pourig its cotets ito the measurig device to see how much it holds measure correctly (e.g., completely fill the cotaier) read their measurig devices correctly record their measuremets correctly (iclude both the umber ad the uit) Estimate the capacity of a cotaier usig persoal referets. Determie the capacity of a cotaier usig materials that take the shape of the iside of the cotaier (e.g., a liquid, rice, sad, beads), ad explai the strategy. Materials: Cotaiers (some cotaiers should be greater tha a litre, less tha a litre, ad equal to a litre), water, sad (ad other material that takes the shape of a cotaier), studet-made measurig devices, litre measurig devices, maskig tape, ad markers Orgaizatio: Small groups a) Give each group the same two cotaiers. Have some of the groups use their measurig devices to determie the capacity of the cotaiers. Have other groups select aother cotaier to be their uit. Have these groups ame their uit ad fid the capacity of their cotaiers. b) Have studets share their measuremets. List their measuremets o the board ad ask why they differ. Ask studets what they could do so everyoe would get the same measuremet. Help studets recogize the eed for a stadard uit of measure ad the reasos why it s importat to use stadard uits (e.g., the use of stadard uits facilitates busiess ad trade). c) Tell studets that i the metric system the litre is the stadard uit of measure for capacity. Show studets a umarked litre cotaier ad tell them that a litre is the amout of the liquid it ca hold. Also, show studets how to write the word ad the symbol for the uit. Shape ad Space (Measuremet) 47

236 d) Give each group five or six cotaiers. Have the studets label the cotaiers from A to F ad the make a list i their math joural of the cotaiers they thik are less tha a litre, the same as a litre, ad larger tha a litre. e) Have studets use the umarked litre cotaiers to measure the capacity of each cotaier. Explai that they should ot fill ay cotaier higher tha the bottom part of the eck of the cotaier. Ask studets to write the letter of each cotaier i their math joural, ad record whether its capacity is greater tha a litre, less tha a litre, or the same as a litre. Observatio Checklist Moitor studets resposes to determie whether they ca do the followig: determie the capacity of a cotaier usig materials that take the shape of the cotaier measure correctly (e.g., fill the litre-measurig cotaier ad the cotaiers they are measurig to the right levels) record measuremets properly (e.g., use the correct symbol for a litre) make reasoable estimates of capacity Materials: Litre-measurig cotaiers, a variety of cotaiers, water, sad, or ay other material that takes the shape of a cotaier, a pitcher, a water pail, ad a wastepaper basket Orgaizatio: Whole class/small groups a) Have studets provide examples of whe they would eed to estimate the capacity of a cotaier, ad discuss how they ca esure that their estimates are reasoable. b) Ask each group to fid two commo cotaiers they ca use as a referet for a litre. c) Have the groups share their referets with each other ad keep a class list of referets for a litre. d) Show studets a large pitcher, a wastepaper basket, ad a empty water pail. Ask them to thik of their referet ad the estimate the capacity of each cotaier. Have the studets check their estimates by measurig each item. e) Ask studets to estimate the capacity of a bathtub. Help them devise ad carry out a pla to check their estimates. 48 Grade 5 Mathematics: Support Documet for Teachers

237 Observatio Checklist Moitor studets resposes to determie whether they ca do the followig: provide a referet for a litre ad explai their choice make reasoable estimates of the capacities Provide a referet for a millilitre ad explai the choice. Estimate the capacity of a cotaier usig persoal referets. Determie the capacity of a cotaier usig materials that take the shape of the iside of the cotaier (e.g., a liquid, rice, sad, beads), ad explai the strategy. Materials: Beakers calibrated i ml, graduated cyliders calibrated i ml, a eyedropper, baby food jars, ti cas, small milk cartos, small soda cas, pickle jars, ketchup bottles, water, paper towels, fuels, ad spoges, capacity table (BLM 5.SS.4) Orgaizatio: Whole class/small group a) Show studets a small cotaier, such as a empty tua ca or empty baby food jar, ad ask them how they could fid the capacity of the cotaier. b) Explai that to fid the capacity of smaller cotaiers, we eed a ew uit of measure. The uit that is commoly used is the millilitre. Tell studets that the millilitre is a very small uit about the size of a drop from a eyedropper. Fill a eyedropper with water ad show studets several drops so they ca begi to coceptualize how large the uit is. c) Explai that because the uit is so small, we ofte use measurig devices that are marked off i millilitres. Show studets differet measurig devices that are calibrated i ml, ad explai how they should use them to fid the capacity of a cotaier. d) Have studets measure the capacity of each object listed below i two differet ways ad record their results i the table from 5.SS.4.1. Shape ad Space (Measuremet) 49

238 Demostrate that 1000 millilitres is equivalet to 1 litre by fillig a 1-litre cotaier usig a combiatio of smaller cotaiers. Materials: A 500 ml beaker, a 250 ml beaker, a 100 ml beaker, ad a 50 ml beaker; umarked litre cotaiers, water, fuels, paper towels, ad math jourals Orgaizatio: Small groups/whole class a) Show studets the litre cotaier ad tell them that their job is to determie the umber of ml i a litre. b) Have studets estimate the umber of 50 ml beakers of water it will take to fill the litre cotaier. Have them record their estimates i a table like the oe show below. Beaker Estimated Number of Beakers Actual Number of Beakers Total Number of ml 50 ml 100 ml 250 ml 500 ml c) Have studets check their estimates by fillig the litre cotaier with 50 ml beakers of water ad record their results i the table. d) Repeat the activity usig the 100 ml beaker, the 250 ml beaker, ad the 500 ml beaker. e) Have studets compare their results with aother group. Ask them what they ca coclude about the relatioship betwee a ml ad a litre. f) Give studets the followig problem ad have them record their solutio i their math jourals: Jessi has a cotaier that holds 1425 ml of liquid. Is Jessi s cotaier smaller tha or larger tha a litre? How do you kow? How much larger or smaller tha a litre is Jessi s cotaier? 50 Grade 5 Mathematics: Support Documet for Teachers

239 Observatio Checklist Observe studets resposes to determie whether they ca do the followig: demostrate that there are 1000 ml i a litre usig a variety of smaller cotaiers measure correctly (fill the beakers properly) record the measuremets correctly solve a problem ivolvig the relatioship betwee millilitres ad litres Materials: Cards umbered from 0 to 9 (BLM 5 8.5), paper ad pecil Orgaizatio: Small groups/whole class a) Tell studets that they will be playig a game ivolvig the relatioship betwee a litre ad a millilitre. Explai how the game is played 1. Players should make the followig grid o their papers: 2. Shuffle the cards ad place them face dow o the playig area. 3. Tur over oe card. Players decide where they wat to write that umber o their grid. Oce a umber has bee placed o the grid, it caot be chaged. 4. The ext card is tured over ad the players ow place this umber o their grids. Play cotiues util six umbers have bee tured over ad each player has placed the umbers o his or her grid. 5. The players add the millilitre quatities ad the player or players closest to 1 litre receive oe poit. 6. Reshuffle the cards, make ew grids, ad play the game agai. 7. Cotiue playig the game. The first player to reach 10 poits is the wier. b) Demostrate how the game is played ad aswer ay questios studets might have. Have studets play the game. c) Vary the game so that the player with the sum closest to 500 ml wis a poit. Shape ad Space (Measuremet) 51

240 Observatio Checklist Observe studets to determie whether they kow the relatioship betwee ml ad litres calculate correctly PUTTING THE PIECES TOGETHER Plaig a Healthy Meal Purpose: The purpose of this ivestigatio is to have studets apply their kowledge of capacity to a real-world situatio. I particular, it is desiged to reiforce studets abilities to measure the capacity of cotaiers estimate the capacity of cotaiers record the capacity of cotaiers The ivestigatio is also desiged to ehace studets abilities to commuicate mathematically solve problems reaso mathematically coect mathematics to real-world situatios ad other subject areas (PE/HE) Materials/Resources Cetimetre measurig cubes Assorted cotaiers Food groups guide (ca be foud o the Iteret) Water, sad, or other material that takes the shape of a cotaier Cyliders or beakers calibrated i ml Paper towels Markers Orgaizatio: Whole class/small groups 52 Grade 5 Mathematics: Support Documet for Teachers

241 a) Tell studets that each group will be resposible for plaig a healthy breakfast or luch. Sice the capacity of the huma stomach is approximately 1 litre, the meal they plaed should ot cotai more tha 800 ml of food. I plaig their meal, they should use the food guide to help them select foods from each food group iclude foods that are available locally idicate the quatity of each food i ml b) Have studets desig their meals. Whe they fiish plaig their meal, have them fid a cotaier with the same capacity as each item o their meu. Have studets label each cotaier by idicatig the item of food it represets ad its capacity. c) Have each group display its meu ad correspodig cotaiers. Have studets explai why their meals are utritious ad how the capacities of the differet items add up to a 800 ml meal. Observatio Checklist Observe studets resposes to determie whether they ca do the followig: pla a meal that meets the criteria specified i part (a) make reasoable estimates of the capacities of cotaiers measure the capacity of cotaiers correctly record the capacity of cotaiers correctly Shape ad Space (Measuremet) 53

242 N OTES 54 Grade 5 Mathematics: Support Documet for Teachers

243 G RADE 5 MATHEMATICS Shape ad Space (3-D Objects ad 2-D Shapes)

244

245 Grade 5: Shape ad Space (3-D Objects ad 2-D Shapes) Edurig Uderstadigs: Shapes are distiguished by their properties. Geeral Outcome: Describe the characteristics of 3-D objects ad 2-D shapes, ad aalyze the relatioship betwee them. SPECIFIC LEARNING OUTCOME(S): ACHIEVEMENT INDICATORS: 5.SS.5 Describe ad provide examples of edges ad faces of 3-D objects ad 2-D shapes that are parallel itersectig perpedicular vertical horizotal [C, CN, R, T, V] Idetify parallel, itersectig, perpedicular, vertical, ad horizotal edges ad faces o 3-D objects. Idetify parallel, itersectig, perpedicular, vertical, ad horizotal sides o 2-D shapes. Provide examples from the eviromet that show parallel, itersectig, perpedicular, ad horizotal lie segmets. Fid examples of edges, faces, ad sides that are parallel, itersectig, perpedicular, vertical, ad horizotal i prit ad electroic media, such as ewspapers, magazies, ad the Iteret. Draw 2-D shapes or 3-D objects that have edges, faces, ad sides that are parallel, itersectig, perpedicular, vertical, or horizotal. Describe the faces ad edges of a give 3-D object usig terms such as parallel, itersectig, perpedicular, vertical, or horizotal. Shape ad Space (3-D Objects ad 2-D Shapes) 3

246 PRIOR KNOWLEDGE Studets should be able to do the followig: (2.SS.7 ad 4.SS.5) Idetify cubes, spheres, coes, cyliders, pyramids, triagular prisms, ad rectagular prisms (2.SS.8) Idetify triagles, squares, rectagles, ad circles (3.SS.6) Idetify the faces, edges, ad vertices of 3-D objects (3.SS.7) Sort regular ad irregular polygos icludig triagles, quadrilaterals, petagos, hexagos, ad octagos accordig to the umber of sides RELATED KNOWLEDGE Studets should be able to do the followig: (5.SS.6) Idetify ad sort quadrilaterals BACKGROUND INFORMATION Poits, lies, ad plaes are the buildig blocks of geometry. These cocepts are udefied ad, like umber, they are abstractios that caot be see or touched. Studets uderstadig of these cocepts evolves from their experieces with physical objects (e.g., the tip of a pecil, the corer of a table or block, ad the dot draw o a piece of paper suggest the idea of a poit to studets). Lies are sometimes described as a set of poits extedig edlessly i two directios. They have legth but o other dimesio. Physical models, such as a rope stretched out, a wire held taut, ad the cetre lie o a highway, ca help studets develop a uderstadig of this cocept. A lie segmet is part of a lie. It cosists of two edpoits ad all the poits betwee them. Examples of lie segmets iclude the rugs of a ladder, the edges of a box, ad the bars i a grill. A plae is two-dimesioal. Ay smooth, flat surface, such as a tabletop, a floor, or a ceilig, ca be thought of as a plae. However, each of these models is oly a part of a plae because a plae exteds ifiitely i two directios. Two lies i a plae ca itersect or be parallel to each other. Itersectig lies have oe poit i commo; parallel lies have o poits i commo. The distace betwee them is the same everywhere. Sometimes lies itersect at right agles. These lies are perpedicular. Because studets are ot itroduced to agles util Grade 6, perpedicular lies are described as two lies that form square corers. I additio, lies ca be either horizotal or vertical. A horizotal lie is a lie that is parallel to the horizo. A vertical lie is a lie that is at right agles to the horizo. Studets usually describe horizotal lies as goig across, ad vertical lies as goig up ad dow. However, their perceptio of whether a lie is horizotal or vertical might differ accordig to their perspective. 4 Grade 5 Mathematics: Support Documet for Teachers

247 Whe describig the edges of prisms, some studets may thik that ay two lie segmets that do ot itersect are parallel. For example, cosider the cube show below: Some studets may thik the two dark edges are parallel sice they do ot itersect. However, these edges lie i differet plaes ad therefore are ot parallel. MATHEMATICAL LANGUAGE Coe Cube Cylider Edge Face Horizotal lie Itersectig lie Lie Lie segmet Parallel lies Perpedicular lies Pyramid Rectagular prism Sphere Triagular prism Vertex (Vertices) Vertical lie Shape ad Space (3-D Objects ad 2-D Shapes) 5

248 LEARNING EXPERIENCES Assessig Prior Kowledge Materials: A set of 3-D objects that icludes a coe, sphere, cylider, a pyramid, a cube, a rectagular prism, ad a triagular prism Orgaizatio: Whole class/idividual a) Put the 3-D objects i a place where all studets ca see them. Tell the studets that you will be askig them some questios about the shapes to fid out what they already kow about them. b) Tell studets they ca look at the shapes to help them idetify the 3-D objects or parts of objects that fit the followig clues: 1. I have six faces all the same size ad shape. (cube) 2. I am formed by the itersectio of two faces. (edge) 3. Two of my faces are circular. (cylider) 4. I am the poit where three or more edges meet. (vertex) 5. I have six rectagular faces. (rectagular prism) 6. I have o flat faces. (sphere) 7. My shape is foud o every pyramid. (triagle) 8. We are the faces foud o a triagular prism. (triagle ad rectagle) 9. I am oe face of a coe. (circle) Observatio Checklist Use studets resposes to the questios to determie whether further review o the idetificatio ad characteristics of 3-D objects ad 2-D shapes is eeded. 6 Grade 5 Mathematics: Support Documet for Teachers

249 Provide examples from the eviromet that show parallel, itersectig, perpedicular, vertical, ad horizotal lie segmets. Materials: Copies of the cocept descriptio sheet (BLM 5 8.2). Orgaizatio: Idividual/Whole class a) Tell studets that i the ext few lessos they will be learig about lies ad today they will be discussig what a lie is. Before begiig the discussio, you wat them to write dow what they already kow about lies. b) Have studets complete the cocept descriptio sheet. Let studets kow that it is alright if they caot thik of aythig to put i a sectio. They will have aother opportuity to complete the sheet whe they lear more about lies. c) Whe studets fiish, begi a discussio by askig, What is a lie? What are some examples of lies? As the discussio progresses, clear up ay miscoceptios studets may have about lies ad make sure they see a variety of examples ad o-examples. d) Have studets add to their cocept descriptio. Observatio Checklist Moitor studets resposes to determie whether they ca do the followig: describe the characteristics of a lie (e.g., it cotiues idefiitely i two directios) idetify examples of a lie idetify o-examples of a lie Shape ad Space (3-D Objects ad 2-D Shapes) 7

250 Material: A log rope or a skei of yar Orgaizatio: Whole class Note: This activity could be doe i the gym or outside. a) Take studets outside to the playgroud. Stretch the rope across the playgroud. Have studets hold oto the rope ad hold it taut. Tell studets that the rope represets a lie that keeps goig forever (e.g., if they were to tie aother piece o the ed ad pull it taut, the lie would cotiue). Have studets discuss what thigs the lie would go through as it goes beyod each ed of the rope. Discuss how they thik they could show that the rope/yar would cotiue o. b) Tell studets that everyoe holdig oto the lie is a poit ad the space betwee each pair of them is a lie segmet or part of a lie. Name the lie segmets usig the studets ames (e.g., lie segmet Jack ad Josie). Call out the ames of several lie segmets. Each time you call out a lie segmet, have the amed studets hold o to the rope ad the studets betwee them let go to show how log the lie segmet is. Have studets take turs amig lie segmets. c) Ask studets to remember who is stadig o either side of them. Retur to the classroom ad draw a arrow o the chalkboard. Put a arrow o either ed to idicate that the lie goes o forever. Idicate poits o the lie ad write the ames of the studets uder them. d) Have studets discuss the differeces betwee their experieces outdoors ad the ideas represeted by the lie o the board. For example, studets should ote that the arrows o the lie idicate that the lie goes o forever, ad the poits where studets ames appear show that a lie segmet has defiite eds. e) Explai that i math we use a double arrow to idicate a lie ad capital letters istead of studets ames to idicate poits o the lie. Draw aother lie o the board. Name several lie segmets ad have studets idetify where they are o the lie. Observatio Checklist Observe studets resposes to determie whether they ca do the followig: recogize that a lie exteds ifiitely i two directios recogize that a lie segmet is part of a lie ame lie segmets correctly 8 Grade 5 Mathematics: Support Documet for Teachers

251 Idetify parallel, itersectig, perpedicular, vertical, ad horizotal sides o 2-D shapes. Provide examples from the eviromet that show parallel, itersectig, perpedicular, vertical, ad horizotal lie segmets. Draw 2-D shapes or 3-D objects that have edges, faces, ad sides that are parallel, itersectig, perpedicular, vertical, or horizotal. Describe the faces ad edges of a 3-D object usig terms such as parallel, itersectig, perpedicular, vertical, or horizotal. Materials: Stir sticks or toothpicks, a mat or a rug Orgaizatio: Whole class/small group a) Have two or three studets come to the frot of the class. Ask them to lie dow o the floor. Explai that whe the studets are lyig o the floor they are horizotal. Now ask the studets to stad up. Explai that whe the studets are stadig up they are vertical. b) Ask differet studets to either lie dow or stad up. Have the other studets idicate whether the studets are horizotal or vertical. c) Draw a horizotal ad a vertical lie o the board. Explai that, i math, lies ca be horizotal or vertical. Lies that are lyig dow are horizotal ad those that are stadig up straight (go up ad dow o their paper) are vertical. d) Draw several lies o the board like the oes show below: Ask, Are these lies horizotal? Why or why ot? Are the lie segmets vertical? Why or why ot? Shape ad Space (3-D Objects ad 2-D Shapes) 9

252 e) Ask studets to use the stir sticks to show the followig: A vertical lie segmet A horizotal lie segmet A lie segmet that is either horizotal or vertical A vertical lie segmet that crosses a horizotal lie segmet A horizotal lie segmet that crosses a lie segmet that is ot vertical A horizotal lie segmet that crosses three vertical lie segmets A vertical lie segmet that crosses two lie segmets that are either horizotal or vertical Four horizotal lie segmets Two vertical lie segmets ad three horizotal lie segmets A vertical lie segmet ad two horizotal lie segmets ad a lie segmet that is either vertical or horizotal f) Have studets idetify whether the edge of their desk is a horizotal or a vertical lie segmet. Have them idetify two or three other horizotal or vertical lie segmets i the room. g) Ask each group to make a list of horizotal ad vertical lie segmets that they see i the school. Have them share their lists with the other members of the class. Observatio Checklist Moitor studets resposes to determie whether they ca do the followig: use the terms horizotal ad vertical correctly make vertical ad horizotal lie segmets idetify examples of horizotal ad vertical lie segmets i the eviromet. idetify lie segmets that are either horizotal or vertical 10 Grade 5 Mathematics: Support Documet for Teachers

253 Idetify parallel, itersectig, perpedicular, vertical, ad horizotal edges ad faces o 3-D objects. Idetify parallel, itersectig, perpedicular, vertical, ad horizotal sides o 2-D shapes. Draw 2-D shapes or 3-D objects that have edges, faces, ad sides that are parallel, itersectig, perpedicular, vertical, or horizotal. Describe faces ad edges of a 3-D object usig terms such as parallel, itersectig, perpedicular, vertical, or horizotal. Describe the sides of 2-D shapes usig terms such as parallel, itersectig, perpedicular, vertical, or horizotal. Materials: 20 stir sticks or toothpicks for each pair of studets, cubes, rectagular prisms, pyramids, ad triagular prisms Orgaizatio: Pairs/Whole class a) Show studets a square made out of stir sticks. Have them idetify the shape, as well as the horizotal ad vertical lie segmets that form it. b) Have studets make as may shapes as they ca with their stir sticks. Explai that a shape ca be made with ay umber of stir sticks, but it must have oly horizotal ad vertical lies. Ask studets to draw a picture of each shape they make. Tell studets that they should write the ame of each shape uder it, ad label the horizotal ad vertical lie segmets. c) Give a 3-D object to each pair of studets. Ask studets to discuss their shape with their parter, ad the write a descriptio of their shape i their math joural. Explai that they should write the ame of the shape, ad the use words ad pictures to explai which edges ad faces of their shape are horizotal ad which are vertical. Observatio Checklist Check studets work to determie whether they ca do the followig: costruct 2-D shapes that have vertical ad horizotal lies draw 2-D shapes that have vertical ad horizotal lies idetify the horizotal ad vertical lie segmets o a 2-D shape idetify the ames of 2-D shapes idetify the horizotal ad vertical edges ad faces of a 3-D object draw 3-D objects that have vertical ad horizotal edges ad faces Shape ad Space (3-D Objects ad 2-D Shapes) 11

254 Idetify parallel, itersectig, perpedicular, vertical, ad horizotal edges ad faces o 3-D objects. Idetify parallel, itersectig, perpedicular, vertical, ad horizotal sides o 2-D shapes. Provide examples from the eviromet that show parallel, itersectig, perpedicular, vertical, ad horizotal lie segmets. Draw 2-D shapes or 3-D objects that have edges, faces, ad sides that are parallel, itersectig, perpedicular, vertical, or horizotal. Materials: Stir sticks ad orage patter block squares Orgaizatio: Small groups a) Draw the followig lie segmets o the board or overhead. Explai that these lies are called itersectig lies because they cross each other. Ask studets to use their stir sticks to show three differet pairs of itersectig lie segmets two lie segmets that do ot itersect A lie segmet itersected by more tha oe lie segmet b) Explai that sometimes lies itersect i a special way. Ask studets what is special about how these two lie segmets itersect. Explai that these lies are special because they form square corers. Demostrate this by placig the orage squares at the itersectio of the lies. Tell studets that these lies are perpedicular. Have studets make three differet pairs of perpedicular lie segmets with their stir sticks. Have them use the orage squares to show that each pair of lies forms a square corer. Ask studets to make a pair of lie segmets that are ot perpedicular ad explai why they are ot perpedicular. 12 Grade 5 Mathematics: Support Documet for Teachers

255 c) Ask studets what is special about this pair of lies (figure 1). Explai that lies that ever meet are parallel. Demostrate that the lies ever meet by placig orage squares betwee the two lies (figure 2) ad havig studets ote that the distace betwee the two lies is the same everywhere. Ask studets to make three differet pairs of parallel lie segmets with their stir sticks. Have them demostrate that the lie segmets are the same distace apart by usig the orage squares or their rulers. Have studets make a pair of lies that are ot parallel ad explai why they are ot parallel. d) Ask each group to idetify examples of parallel, itersectig, ad perpedicular lie segmets iside ad outside the classroom. Have them draw ad label a diagram of each lie segmet pair, ad list uder each diagram real-world examples of the lie segmet pair. e) Have each group share its examples of each type of lie segmet pair with the other members of the class. Observatio Checklist Observe studets resposes to determie whether they ca do the followig: costruct pairs of lie segmets that are parallel, perpedicular, ad itersectig idetify lie segmets that are ot parallel idetify lie segmets that are ot itersectig idetify lie segmets that are ot perpedicular idetify real-world examples of parallel, perpedicular, ad itersectig lie segmets Shape ad Space (3-D Objects ad 2-D Shapes) 13

256 Idetify parallel, itersectig, perpedicular, vertical, ad horizotal sides o 2-D shapes. Draw 2-D shapes or 3-D objects that have edges, faces, ad sides that are parallel, itersectig, perpedicular, vertical, or horizotal. Materials: Pait or watercolours, black marker, ad samples of Piet Modria artwork (ca be foud o the Iteret) Orgaizatio: Large group/idividual a) Show studets a picture of Piet Modria s artwork. Explai that Piet Modria was a Dutch paiter who was famous for paitigs that he called compositios. b) Ask studets to describe the picture. Ecourage them to discuss the types of lies ad shapes he used to create the picture. c) Have studets use black markers ad watercolours to create a picture i the style of Piet Modria. d) Display studets artwork o walls ad coduct a gallery walk so studets ca look at each other s work. Observatio Checklist Observe studets resposes to determie whether they ca do the followig: idetify parallel, itersectig, perpedicular, vertical, ad horizotal lies create a piece of artwork that is comprised of parallel, itersectig, perpedicular, vertical, ad horizotal lies 14 Grade 5 Mathematics: Support Documet for Teachers

257 Materials: Patter blocks Orgaizatio: Pairs a) Ask studets to use their patter blocks to complete the followig activity. Have them draw a sketch of each shape that they make. 1. Use two differet blocks to make a shape with exactly two pairs of parallel sides exactly oe pair of parallel sides o parallel sides 2. Use three differet blocks to make a shape with exactly three pairs of parallel sides exactly two pairs of parallel sides exactly oe pair of parallel sides o parallel sides 3. What is the largest umber of pairs of parallel sides of a shape you ca make with two pieces? three pieces? four pieces? 4. Ca you use six differet patter blocks to make a shape with o parallel sides? b) Have studets share their shapes with the other members of the class. Ecourage studets to idetify lies that are parallel, perpedicular, itersectig, vertical, ad horizotal. Observatio Checklist Moitor studets resposes to determie whether they ca do the followig: idetify parallel, itersectig, perpedicular, horizotal, ad vertical lie segmets o 2-D shapes draw 2-D shapes with parallel, itersectig, perpedicular, horizotal, ad vertical lie segmets Shape ad Space (3-D Objects ad 2-D Shapes) 15

258 Idetify parallel, itersectig, perpedicular, vertical, ad horizotal edges ad faces o 3-D objects. Describe the faces ad edges of a 3-D object usig terms such as parallel, itersectig, perpedicular, vertical, or horizotal. Materials: Cubes, rectagular prisms, square-based ad triagular-based prisms, ad triagular prisms, cards with the ames of the 3-D objects o them, oe ame per card (e.g., cube, rectagular prism, etc.), stir sticks or toothpicks, ad plasticie Orgaizatio: Small groups a) Show studets a triagular-based prism ad ask them to describe it. Ecourage studets to idetify the faces ad edges that are parallel, itersectig, perpedicular, vertical, ad horizotal. b) Give each group a set of the 3-D objects. Have studets take turs describig oe of the shapes to the other members of their group. Ecourage studets to poit out the faces ad edges that are parallel, itersectig, perpedicular, vertical, ad horizotal. c) Give each studet a card. Tell studets you will be describig the characteristics of 3-D objects. If the shape o their card has that characteristic, they should stad up ad show their card to the other members of the class. For example, if I say, I am a 3-D object that has three pairs of parallel faces, the studets who have cards with cube ad rectagular prism writte o them should stad up. The other members of the class have to check the cards to make sure that the right shapes have bee idetified. d) Show studets how to use toothpicks ad plasticie to build a 3-D object. Have studets use the materials to build a 3-D object that fits each set of characteristics. Each edge is perpedicular to four other edges. The edges are ot all the same legth. There are six edges. No edges are perpedicular. The side edges are perpedicular to the bottom edges. There are three side edges. e) Have studets discuss questios like the followig: Why are the roofs of most houses ot parallel to the groud? Why are the shelves of a bookcase parallel to the floor? Observatio Checklist Observe studets resposes to determie whether they ca do the followig: idetify faces of 3-D objects that are parallel, itersectig, perpedicular, horizotal, ad vertical idetify the edges of 3-D objects that are parallel, itersectig, perpedicular, horizotal, ad vertical 16 Grade 5 Mathematics: Support Documet for Teachers

259 Grade 5: Shape ad Space (3-D Objects ad 2-D Shapes) Edurig Uderstadigs: Shapes are distiguished by their properties. Geeral Outcome: Describe the characteristics of 3-D objects ad 2-D shapes, ad aalyze the relatioship betwee them. SPECIFIC LEARNING OUTCOME(S): ACHIEVEMENT INDICATORS: 5.SS.6 Idetify ad sort quadrilaterals accordig to their attributes, icludig rectagles squares trapezoids parallelograms rhombuses [C, R, V] Idetify ad describe the characteristics of a pre-sorted set of quadrilaterals. Sort a set of quadrilaterals ad explai the sortig rule. Sort a set of quadrilaterals accordig to the legths of the sides. Sort a set of quadrilaterals accordig to whether or ot opposite sides are parallel. PRIOR KNOWLEDGE Studets should be able to do the followig: (3.SS.7) Idetify quadrilaterals (4.PR.4) Idetify ad explai mathematical relatioships usig a Ve diagram RELATED KNOWLEDGE Studets should be able to do the followig: (5.SS.5) Idetify parallel, perpedicular, horizotal, ad itersectig lie segmets Shape ad Space (3-D Objects ad 2-D Shapes) 17

260 BACKGROUND INFORMATION A simple closed curve is a curve that does ot cross itself ad ca be draw by startig ad stoppig at the same poit (e.g., i the diagram below, figures (a) ad (b) are simple closed curves while (c) ad (d) are curves that are ot closed). a) b) c) d) Polygos are simple closed curves formed by the uio of lie segmets. I the diagram above, (b) is the oly polygo sice it is both a simple closed curve ad made up of lie segmets. The lie segmets that form the polygo are the sides of the polygo. A poit where two sides meet is a vertex of the polygo. Polygos are classified accordig to the umber of sides they have. The most commo classificatios are: triagle (three sides), quadrilateral (four sides), petago (five sides), hexago (six sides), heptago (7 sides), octago (8 sides), oago (9 sides), decago (10 sides), ad dodecago (12 sides). Other polygos are commoly referred to as -gos, where is the umber of sides. For example, a eleve-sided polygo ca be referred to as a 11-go ad a 14-sided polygo ca be referred to as a 14-go. Quadrilaterals ca be classified accordig to the umber of parallel sides that they have. The defiitio of each type of quadrilateral is give below. Trapezium A quadrilateral with o pairs of parallel sides. Trapezoid A quadrilateral with at least oe pair of parallel sides. Parallelogram A quadrilateral i which each pair of opposite sides is parallel. The opposite sides of parallelograms are also cogruet (same legth). Rectagle A parallelogram that has four right agles. Rhombus A parallelogram that has four cogruet sides. Square A parallelogram that has four cogruet sides ad four right agles. Some texts defie a trapezoid as a quadrilateral with exactly oe pair of parallel sides. If the support material you are usig defies a quadrilateral i this way, studets should be show both defiitios. This ca help them uderstad that mathematics is ot a rigid subject ad that mathematicias do ot always agree o the defiitio of a cocept. Studets ca also be asked to examie how the differet defiitios affect their solutios to problems ivolvig trapezoids. Moreover, eve though the learig experieces focus o quadrilaterals that have parallel sides, some of the activities iclude quadrilaterals that have o parallel sides. This has bee doe to avoid givig studets a flawed cocept of a quadrilateral. 18 Grade 5 Mathematics: Support Documet for Teachers

261 MATHEMATICAL LANGUAGE Cogruet Polygo Parallel Parallelogram Perpedicular Quadrilateral Rectagle Rhombus (Rhombuses or Rhombi) Set Side Square Square corer Trapezoid Vertex (Vertices) LEARNING EXPERIENCES Assessig Prior Kowledge Materials: Cocept descriptio sheet (BLM 5 8.2). Orgaizatio: Idividual/Whole class a) Tell studets that they will be learig about a family of shapes called quadrilaterals, but before they begi you eed to fid out what they already kow about this shape. b) Have studets complete the cocept descriptio sheet. Let studets kow that it is all right if they caot thik of aythig to put i a sectio. They will have aother opportuity to complete the sheet after they have leared more about the shape. c) Whe studets complete the sheet, begi a discussio of their resposes by askig, What is a quadrilateral? What does it look like? As the discussio progresses, make sure studets see a variety of examples ad o-examples. I particular, studets should see examples of quadrilaterals that have o parallel sides Observatio Checklist Whe the discussio eds, have studets add to the cocept descriptio sheet to determie whether they ca do the followig: recogize that a quadrilateral is a four-sided polygo give appropriate examples ad o-examples of quadrilaterals Shape ad Space (3-D Objects ad 2-D Shapes) 19

262 Idetify ad describe the characteristics of a pre-sorted set of quadrilaterals. Materials: Five evelopes (oe labelled trapezoid, oe labelled square, oe labelled rectagle, oe labelled parallelogram, ad oe labelled rhombus), three differet cutouts of each quadrilateral, oe large sheet of paper, ad oe marker for each group Orgaizatio: Small group/large group a) Place the cut-outs ito the appropriate evelopes ad the divide the class ito five groups. Give each group a large sheet of paper, a evelope, ad a marker. b) Tell studets that each group has a differet type of quadrilateral ad that their task is to teach the other groups about their quadrilateral. To do this, they eed to look at the examples of the quadrilateral i their evelopes ad determie its characteristics. Let studets kow that they should pay particular attetio to the sides ad corers of their quadrilateral. Tell studets that they should record the ame of their quadrilateral ad its characteristics o the large sheet of paper. c) Have each group post their sheet of paper i the frot of the room ad tell the other members of the class about their quadrilateral. Help studets idetify ay characteristics they might have missed. d) Have studets make a graphic orgaizer to help them lear the characteristics of the differet quadrilaterals ad their relatioship to each other (e.g., studets ca make a chart like the oe show below). Quadrilateral Diagram At least oe pair of parallel sides Two pairs of parallel sides All sides cogruet Opposite sides cogruet All square corers Parallelogram Square Rectagle Trapezoid 4 Rhombus Grade 5 Mathematics: Support Documet for Teachers

263 Observatio Checklist Observe studets resposes to make sure that for each quadrilateral they have correctly idetified the characteristics show i the chart. Materials: Stir sticks or straws Orgaizatio: Whole class a) Have the studets use the stir sticks to make a quadrilateral whose opposite sides are cogruet that is ot a rhombus that has at least oe pair of parallel sides that has o parallel sides that has four square corers that is either a square or a rectagle ad has two pairs of parallel sides that has oe square corer b) Whe studets fiish makig each shape, ask them: What shape did you make? How do you kow that it is a quadrilateral? Is there aother shape that you could have made? What is it? How does it differ from the shape you made? What other characteristics does your shape have? Observatio Checklist Observe studets resposes to determie whether they ca do the followig: costruct ad idetify a quadrilateral with the give characteristic(s) describe the characteristics of each quadrilateral that they make idetify other quadrilaterals that have the same characteristics as the oe that was give describe how squares, rectagles, parallelograms, trapezoids, ad rhombuses differ from each other recogize that there are other quadrilaterals besides squares, rectagles, trapezoids, parallelograms, ad rhombuses Shape ad Space (3-D Objects ad 2-D Shapes) 21

264 Materials: Tagrams Orgaizatio: Idividual a) Have the studets use the tagram pieces to make two differet rectagles parallelograms trapezoids squares rhombuses Let studets kow that they ca use two or more of the tagram pieces to make each shape. b) Have studets place each shape they make o a piece of paper ad trace aroud it. Ask them to write the ame of the shape udereath their drawig ad write a setece statig which tagram pieces they used to make the shape. Observatio Checklist Check studets work to see whether they ca do the followig: make two differet rectagles, parallelograms, trapezoids, ad squares correctly idetify each quadrilateral that they made ad the tagram pieces that they used to make it spell the ames of the quadrilaterals correctly Materials: Quadrilateral cards (BLM 5.SS.6.1) Orgaizatio: Whole group a) Give each studet a card with a ame of a quadrilateral o it. b) Tell studets that they are goig to play a game called Name that Quadrilateral. Explai that you will be describig a characteristic of a quadrilateral. Studets who have a card with the ame of a quadrilateral with that characteristic o it should stad up ad show their card to the rest of the class (e.g., if I say, I am a quadrilateral whose sides are all cogruet, the studets who have square or rhombus writte o their card should stad up. The rest of the class checks to see whether studets have idetified the right quadrilaterals.). 22 Grade 5 Mathematics: Support Documet for Teachers

265 c) Vary the game by selectig five studets to be pael members. Give each pael member a card with a ame of a quadrilateral o it ad tell him or her to keep it hidde from the rest of the class. Tell studets that everyoe will have a chace to ask a pael member a questio about his or her quadrilateral ad the oly questio studets ca t ask is: What is your quadrilateral? The game is over whe every studet has had a opportuity to ask a questio. The perso who correctly idetifies the quadrilateral o each pael member s card is the wier. Observatio Checklist Observe studets resposes to determie whether they idetify ad describe the characteristics of the differet quadrilaterals as show i the chart. Sort a set of quadrilaterals ad explai the sortig rule. Sort a set of quadrilaterals accordig to the legths of the sides. Sort a set of quadrilaterals accordig to whether or ot opposite sides are parallel. Materials: A set of quadrilateral cards ad a set of rule cards for each pair of studets (BLM 5.SS.6.2); two loops of strig or yar for each pair of studets Orgaizatio: Pairs a) Have the studets lay the strig loops ad the label cards at least oe square corer ad opposite sides cogruet o their workspace, as show below. b) Have studets sort their quadrilaterals ito the appropriate sets. Whe studets fiish sortig the shapes, have them describe their solutios ad explai how they kew where to place each quadrilateral. Shape ad Space (3-D Objects ad 2-D Shapes) 23

266 c) Repeat the activity. Have the studets sort the quadrilaterals ito sets with at least oe pair of parallel sides/ all sides cogruet all square corers/two pairs of parallel sides o parallel sides/at least oe pair of parallel sides Have studets make up their ow rules for sortig the quadrilaterals. d) Vary the activity by showig studets pre-sorted sets ad askig them to describe the rules that were used to sort the quadrilaterals. For example, show studets the followig set ad ask them to idetify the rule that was used to sort the quadrilaterals. Observatio Checklist Observe studets resposes to determie whether they ca do the followig: sort the set of quadrilaterals accordig to the stated rule explai how they kew where each quadrilateral beloged recogize the relatioships amog the quadrilaterals, such as all rectagles are squares all parallelograms are trapezoids all rectagles, squares, ad rhombuses are parallelograms all squares are rhombuses 24 Grade 5 Mathematics: Support Documet for Teachers

267 Sort a set of quadrilaterals ad explai the sortig rule. Materials: Quadrilateral activity sheet (BLM 5.SS.6.3) Orgaizatio: Whole class/idividual a) Ask studets to complete the activity. b) Have studets share their resposes with the other members of the class. Observatio Checklist Check studets resposes to the questios to determie whether they ca do the followig: recogize the characteristics of rectagles, squares, trapezoids, rhombuses, ad parallelograms recogize the relatioships amog quadrilaterals, such as the followig: All squares are rectagles All squares are rhombuses All rectagles are parallelograms All squares are parallelograms All parallelograms, rectagles, squares, ad rhombuses are trapezoids All rhombuses are parallelograms Shape ad Space (3-D Objects ad 2-D Shapes) 25

268 PUTTING THE PIECES TOGETHER A Parallel World Purpose: The purpose of this activity is to have studets recogize real-world examples of lies ad quadrilaterals. I particular, the ivestigatio is desiged to eable studets to idetify real-world examples of faces ad edges of 3-D objects ad sides of 2-D shapes that are examples of parallel, itersectig, perpedicular, vertical, ad horizotal lies (5.SS.5) rectagles, squares, trapezoids, parallelograms, ad rhombuses (5.SS.6) I additio, the ivestigatio is desiged to ehace studets ability to commuicate mathematically use techology coect mathematical cocepts to each other ad the real world Materials/Resources: Digital camera or video camera* Computer Orgaizatio: Large group/small groups a) Tell studets that they will be creatig a digital scrapbook (or a video recordig). Explai that each group is resposible for takig pictures of examples of lies ad quadrilaterals that they fid either iside or outside of school. Whe they fiish takig their pictures, they will create a digital scrapbook. Each picture i their scrapbook must iclude a descriptio of the types of lies ad quadrilaterals foud i the picture. b) Help studets determie the guidelies they should follow whe takig their pictures (e.g., studets eed to cosider the amout of time eeded to fid ad take the pictures, their coduct as they move withi ad outside the school, ad the resposibilities of each group member). c) Have studets take their pictures ad create their scrapbooks. d) Have studets choose a picture to preset to the class. Ask them to explai why they chose the picture ad where i the picture they see lies ad quadrilaterals. Ecourage them to idetify the types of lies ad quadrilaterals foud i their picture. * If digital cameras or computers are ot available, have studets fid pictures of lies ad quadrilaterals i magazies ad ewspapers. 26 Grade 5 Mathematics: Support Documet for Teachers

269 Observatio Checklist Use the followig rubric to assess studet mastery of learig outcomes for ad of learig (durig ad at the completio of the activity) Scrapbook icludes: a example of each type of lie a example of at least 3 differet quadrilaterals All lies are correctly idetified. All quadrilaterals are correctly idetified. Writte descriptio is clear. Mathematical terms are used correctly. Scrapbook icludes: a example of 3 or 4 types of lies a example of at least 2 differet quadrilaterals Some lies are correctly idetified. Some quadrilaterals are correctly idetified. Writte descriptio is clear. Some mathematical terms are used correctly. Scrapbook icludes: examples of 1 or 2 types of lies a example of 1 type of quadrilateral Not all lies are correctly idetified. Quadrilateral is icorrectly idetified. Writte descriptio is ot clear. Some mathematical terms are used correctly. Shape ad Space (3-D Objects ad 2-D Shapes) 27

270 N OTES 28 Grade 5 Mathematics: Support Documet for Teachers

271 G RADE 5 MATHEMATICS Shape ad Space (Trasformatios)

272

273 Grade 5: Shape ad Space (Trasformatios) Edurig Uderstadigs: The positio of shapes ca be chaged by traslatig, rotatig, or reflectig them. Geeral Outcome: Describe ad aalyze positio ad motio of objects ad shapes. SPECIFIC LEARNING OUTCOME(S): ACHIEVEMENT INDICATORS: 5.SS.7 Perform a sigle trasformatio (traslatio, rotatio, or reflectio) of a 2-D shape ad draw ad describe the image. [C, CN, T, V] 5.SS.8 Idetify a sigle trasformatio (traslatio, rotatio, or reflectio) of 2-D shapes. [C, CN, T, V] Traslate a 2-D shape horizotally, vertically, or diagoally, ad describe the positio ad orietatio of the image. Rotate a 2-D shape about a poit, ad describe the positio ad orietatio of the image. Reflect a 2-D shape i a lie of reflectio, ad describe the positio ad orietatio of the image. Perform a trasformatio of a 2-D shape by followig istructios. Draw a 2-D shape, traslate the shape, ad record the traslatio by describig the directio ad magitude of the movemet. Draw a 2-D shape, rotate the shape, ad describe the directio of the tur (clockwise or couter-clockwise), the fractio of the tur, ad poit of rotatio. Draw a 2-D shape, reflect the shape, ad idetify the lie of reflectio ad the distace of the image from the lie of reflectio. Predict the result of a sigle trasformatio of a 2-D shape ad verify the predictio. Provide a example of a traslatio, a rotatio, ad a reflectio. Idetify a sigle trasformatio as a traslatio, rotatio, or reflectio. Describe a rotatio by the directio of the tur (clockwise or couter-clockwise). Shape ad Space (Trasformatios) 3

274 PRIOR KNOWLEDGE Studets should be able to do the followig: (2.SS.8, 3.SS.7) Idetify triagles, quadrilaterals, petagos, hexagos, octagos, ad circles RELATED KNOWLEDGE Studets should be able to do the followig: (5.SS.5) Idetify vertical ad horizotal lies (5.SS.6) Idetify rectagles, squares, trapezoids, rhombuses, ad parallelograms (5.SS.2) Measure the legths of lies to the earest cm or mm (5.SS.7) Idetify equivalet fractios BACKGROUND INFORMATION Trasformatios play a importat role i the mathematics curriculum. I the Middle Years, the study of trasformatio ca support studets work i patterig, algebra, problem solvig, geometry, ad statistics. I high school ad beyod, the study of trasformatios helps studets recogize the coectios betwee algebra ad geometry ad ehaces their uderstadig of other topics such as matrices, scalig, ad complex umbers. A trasformatio ca be thought of as a chage i the positio, size, or shape of a figure. I the learig activities that follow, studets are itroduced to three trasformatios that chage the positio of a figure. Iformally, these trasformatios are referred to as slides, flips, ad turs. Formally, they are kow as traslatios, reflectios, ad rotatios. Studets should kow both the formal ad iformal termiology. A traslatio slides a figure a fixed distace i a give directio. The figure ad its traslatio are cogruet (same size ad shape) ad face i the same directio. I the diagram show below, square ABCD has bee traslated to a ew positio represeted by square A B C D. Note that Square A B C D, which is called the image of Square ABCD, is cogruet to Square ABCD ad faces i the same directio. The arrow idicates the distace ad the directio of the traslatio. 4 Grade 5 Mathematics: Support Documet for Teachers

275 A rotatio turs a figure ay umber of degrees aroud a fixed poit called the cetre of rotatio. The cetre of rotatio may be ay poit withi or outside the figure. The figure ad its image (the result of the trasformatio) are cogruet but they may face i opposite directios (e.g., i the diagram below, the arrow ABCDE has bee rotated 90 couter-clockwise about its midpoit). The image arrow A B C D E is cogruet to Arrow ABCDE but faces i a differet directio. A reflectio flips the figure over a lie, creatig a mirror image. The figure ad its image are cogruet but have differet orietatios. The lie the figure is flipped over is called the lie of reflectio ad it is the same distace from the figure as its image (e.g., i the diagram below, petago ABCDE has bee flipped over lie k). Note that Petago A B C D E is cogruet to Petago ABCDE but faces i the opposite directio. Lie k, the lie of reflectio, is equidistat from the two petagos. MATHEMATICAL LANGUAGE Clockwise Couter-clockwise Cogruet Diagoally Horizotal Image Lie of reflectio Polygo Reflectio (flip) Rotatio (tur) Trasformatio Traslatio (slide) Vertical lie Shape ad Space (Trasformatios) 5

276 LEARNING EXPERIENCES Traslate a 2-D shape horizotally, vertically, or diagoally, ad describe the positio ad orietatio of the image. Rotate a 2-D shape about a poit, ad describe the positio ad orietatio of the image. Reflect a 2-D shape i a lie of reflectio, ad describe the positio ad orietatio of the image. Perform a trasformatio of a 2-D shape by followig istructios. Materials: Carpeted area or floor mats Orgaizatio: Whole class a) Have studets lie dow o a carpet or mat. Ask them to slide a short distace i oe directio. Have them repeat the movemet several times by askig them to slide up, dow, ad sideways. After each slide, ask, What chaged? What remaied the same? Emphasize that whe a slide is made, the directio i which a object is poitig does ot chage. b) Have studets demostrate flips. At the ed of a flip, studets should have chaged from stomach to back or back to stomach. Discuss the differet ways flips ca be completed. For example, studets may roll to the left or to the right, or over the feet or head. Have studets try these differet ways. After each flip, ask, What chaged? What remaied the same? Emphasize that whe a object is flipped, its orietatio chages. Ask studets how this is differet from lookig at their reflectio i the mirror. Emphasize that a true reflectio of oeself would have exactly the same image, just i a differet orietatio. c) Have studets demostrate turs. To perform a tur, studets must keep either their feet or their heads (or belly butto) at the same locatio for the duratio of the tur. If the feet are the poit (cetre) of rotatio, the the arms ad head are used to move the body. If the head is the poit (cetre) of rotatio, the feet are used to make the move. Have studets tur all the way aroud or partway aroud. Have them tur i either a clockwise or couter-clockwise directio. After each tur, ask studets, What chaged? What remaied the same? Discuss the fact that after a tur, the directio i which the head poits is differet, except whe a complete tur is made. d) Iform studets that i the ext few lessos they will be learig more about slides, flips, ad turs. 6 Grade 5 Mathematics: Support Documet for Teachers

277 Observatio Checklist Observe studets resposes to determie whether they are able to perform a slide, flip, ad a tur. Traslate a 2-D shape horizotally, vertically, or diagoally, ad describe the positio ad orietatio of the image. Perform a trasformatio of a 2-D shape by followig istructios. Materials: Grid paper ad a overhead trasparecy of a grid Orgaizatio: Whole class/parters a) Remid studets of the first activity by askig them to describe a slide. Discuss other objects i the eviromet that slide, such as drawers, slidig doors, ad swigs. Explai that to perform a slide, we eed to kow the distace ad directio of the move. b) Tell studets that they are goig to perform a umber of slides. Have them stad i a large ope area or take them to the gym. Ask studets to slide: oe step to the right; oe step up ad two steps to the left; ad three steps back ad two steps to the right. After each slide, ask studets, What chaged? What remaied the same? c) Tur the activity ito a game by playig Simo Says. Tell studets that if they move the wrog way or slide whe Simo does t tell them to, they must take their seats. The last perso stadig is the wier. d) Explai that aother ame for a slide is a traslatio. Draw a triagle o the overhead grid ad show it to studets. Tell studets that to traslate or slide the triagle, we eed to kow the distace ad the directio of the move. Draw a arrow ad explai that the arrow idicates the directio of the traslatio ad its legth describes the distace. Draw the image of the shape ad explai that the origial triagle has bee traslated horizotally four uits to the right. Explai that the traslated shape is called the image of the origial shape. Ask, How are the shape ad its image alike? How do they differ? Shape ad Space (Trasformatios) 7

278 e) Do two or three more examples. Make sure you iclude a example where the shape is traslated diagoally (e.g., i the figure show below, the petago has bee traslated diagoally [o a slat] three uits dow ad three uits across). f) Ask studets to draw a shape o their grid paper ad a slide arrow. Have them exchage papers with their parter. The parter must traslate the shape accordig to the directio ad legth of the arrow. Have studets repeat the activity several times. 8 Grade 5 Mathematics: Support Documet for Teachers

279 g) Vary the activity by havig studets draw a shape ad its image o grid paper, but ot the slide arrow. Have them exchage papers with their parter. The parter must draw the slide arrow that correspods to the traslatio. Have studets repeat the activity several times. Observatio Checklist Observe studets work to determie whether they ca do the followig: provide real-world examples of slides perform a trasformatio by followig istructios traslate a 2-D shape horizotally, vertically, ad diagoally, ad describe its positio ad orietatio determie the directio ad distace of a traslatio recogize that a shape ad its image are cogruet ad face i the same directio draw a 2-D shape ad traslate the shape, ad describe the distace ad magitude of the traslatio Traslate a 2-D shape horizotally, vertically, or diagoally, ad describe the positio ad orietatio of the image. Reflect a 2-D shape i a lie of reflectio, ad describe the positio ad orietatio of the image. Materials: Pait, paper, ad black markers or crayos Orgaizatio: Whole class/idividual a) Remid studets of the first activity by askig them to describe a flip. Explai that a flip is a mirror image. Tell studets that they will be doig a activity that ivolves creatig mirror images with their parters. b) Have studets stad up ad face their parter. Ask studets to: raise their right arm above their heads; bed their kees as if they were sittig; tur aroud ad put their backs together; hold both arms straight out; ad hop oce o their left foot. Let oe of the studets be the leader ad, without talkig, make motios for the other studets to follow. The let aother studet be the leader. c) Egage studets i a discussio about their movemets by askig them to describe how their movemets were similar to ad differet from their parters. Explai that a mirror image or a flip is also called a reflectio. Have studets describe real-world occurreces of reflectios, such as seeig oe s reflectio i a pool of water. Shape ad Space (Trasformatios) 9

280 d) Give studets a piece of paper ad ask them to fold it i half vertically. Have studets put dabs of differet coloured pait o oe side of the paper. Before the pait dries, have studets fold the other half of the paper over the paited side ad smooth it out. Have studets ope the picture to see the desig. e) Tell studets that the fold i their paper is a lie of reflectio because it acts like a mirror. Have studets draw a lie segmet alog the fold lie with a black marker or crayo ad label the segmet reflectio lie. f) Have studets show their desigs ad their reflectio to the etire class. Ecourage studets to describe why their pictures illustrate reflectios. Display the desigs i the classroom. Observatio Checklist Observe studets to determie whether they ca do the followig: provide real-world examples of reflectios perform a trasformatio by followig istructios reflect a 2-D shape over a lie of reflectio ad describe its positio ad orietatio Reflect a 2-D shape i a lie of reflectio, ad describe the positio ad orietatio of the image. Perform a trasformatio of a 2-D shape by followig istructios. Draw a 2-D shape, reflect the shape, ad idetify the lie of reflectio ad the distace of the image from the lie of reflectio. Materials: Miras or reflective plastic, copies of the activity sheet (BLM 5.SS.7&8.1), ad rulers Orgaizatio: Idividual/Pairs a) Show studets how to use a Mira. Explai that the top ad bottom of the Mira are differet. The bottom of the Mira has a beveled drawig edge. Whe usig the Mira to draw a figure, the beveled edge should always be facig the drawer. 10 Grade 5 Mathematics: Support Documet for Teachers

281 b) Have studets place a Mira o a piece of paper ad draw a lie alog the drawig edge. Next, have studets draw a triagle like the oe show below o the beveled side of the Mira. Have them look through the Mira ad ask, Do you see a reflectio of the triagle you just drew? Now have studets reach aroud the Mira ad draw the image of the triagle. Have studets remove the Mira. Ask, Does your drawig show a reflectio? How do you kow? c) Help studets recogize that a object ad its image are the same distace from the lie of reflectio by askig studets to measure the distaces from the vertices of the triagle (ad its image) to the lie of the reflectio. d) Have studets repeat parts (b) ad (c) several times usig differet shapes. e) Give studets a copy of the activity sheet (BLM 5.SS.7&8.1) ad tell them to use their Miras to draw the lie of reflectio for each shape. After they draw the lie of reflectio, they should look through the Mira ad draw the image of the shape. f) Have studets discuss the reflectios. Ecourage studets to describe the distace the shape ad its image are from the lie of reflectio ad how the orietatio of the image differs from the shape s orietatio. Observatio Checklist Moitor studets resposes to determie whether they ca do the followig: draw a 2-D shape, reflect the shape, ad idetify the lie of reflectio ad its distace from the shape ad its image describe the orietatio of the image recogize that the shape ad its image are cogruet perform a traslatio by followig istructios Shape ad Space (Trasformatios) 11

282 Reflect a 2-D shape i a lie of reflectio, ad describe the positio ad orietatio of the image. Perform a trasformatio of a 2-D shape by followig istructios. Materials: Cetimetre grid paper (BLM 5 8.9), Cuiseaire rods, mirrors, ad black markers Orgaizatio: Pairs or groups of four (if groups of four are used, divide each group ito two teams of two) a) Ask studets to draw a black lie horizotally across the middle of the grid paper. Tell them that the black lie is the lie of reflectio. b) Have oe studet i each pair arrage the Cuiseaire rods o the grid paper i some way o oe side of the black lie. c) Have the other studet i each pair build the reflectio of the arragemet o the other side of the black lie, ad the use a Mira to check whether his or her arragemet is correct. d) Have studets cotiuig creatig reflectios, chagig who builds ad who reflects. Observatio Checklist Observe studets resposes to determie whether they ca do the followig: recogize that a object ad its reflectio are the same distace from the lie of reflectio recogize that the orietatio of the image is differet from the orietatio of the origial shape recogize that the shape ad its image are cogruet traslate the shape over a lie of reflectio, ad describe its positio ad orietatio 12 Grade 5 Mathematics: Support Documet for Teachers

283 Perform a trasformatio of a 2-D shape by followig istructios. Materials: Noe Orgaizatio: Whole class a) Remid studets of the first activity by askig them to describe a tur. Ecourage studets to idetify objects i their eviromet that tur, such as doorkobs, tires, ad the hads o aalog clocks. Explai that to tur a shape, we eed to kow how far ad i what directio to tur it. b) Have studets stad up ad face the frot of the room. Have them tur aroud slowly util they see the frot of the room agai. Explai that they just made a full tur. c) Ask studets what they thik they will see if they make a half-tur. Ask, Will you be able to see me? What part of you will I be able to see? Have studets make a half-tur. d) Ask studets what they will see if they make a quarter-tur. Explai that what they see depeds o the directio of their tur (e.g., if they tur couter-clockwise, they might see widows, ad if they tur clockwise, they might see a bulleti board). Have studets make: a quarter-tur clockwise; a three-quarter-tur couterclockwise; a quarter-tur couter-clockwise, ad a three-quarter-tur clockwise. Ask questios such as, What are two ¼-turs clockwise equivalet to? What are three quarter-turs couter-clockwise equivalet to? Have studets perform the differet turs to check their resposes. e) Have studets practice makig full-, quarter-, three-quarter-, ad half-turs. Tur the activity ito a game by playig Simo Says. Tell studets that if they tur the wrog way or tur whe Simo does t tell them to, they must take their seats. The last perso stadig is the wier. Observatio Checklist Observe studets to determie whether they ca do the followig: uderstad the meaig of the terms clockwise ad couter-clockwise make quarter-, half-, three-quarter-, ad full turs i a clockwise directio make quarter-, half-, three-quarter-, ad full turs i a couterclockwise directio perform a trasformatio by followig directios Shape ad Space (Trasformatios) 13

284 Rotate a 2-D shape about a poit, ad describe the positio ad orietatio of the image. Materials: Scissors, paper ad pecil Orgaizatio: Idividual a) Ask studets to draw a rectagle o a piece of paper. Have studets label the vertices of their rectagle A, B, C, ad D. Have them cut out aother rectagle that is the same size as the rectagle that they drew o their paper, ad label the vertices A, B, C, ad D. Have studets place the cut-out o their drawig so that the vertices of the two rectagles match. b) Tell studets aother ame for a tur is a rotatio. Explai that to rotate the shape, we ot oly eed to kow the directio ad how far to tur the shape, we also eed to kow the poit (cetre) of rotatio. Ay poit o or off the shape ca be used as the poit (cetre) of rotatio. c) Have studets place the tip of a pecil o the cetre of their rectagles ad the have them rotate their rectagles a quarter-tur clockwise. Have studets describe the positio ad orietatio of the image. Ask studets to rotate their rectagles: ½-tur couter-clockwise; a ¾-tur clockwise, ¼-tur couter-clockwise; a ¾-tur couter-clockwise; ad a ½-tur clockwise. After each rotatio, have studets describe the positio ad orietatio of the rectagle. Have studets discuss relatioships, such as two ¼-turs is the same as a ½-tur; three ¼-turs is the same as a ¾-tur or a ½-tur plus a ¼-tur; ad four ¼-turs is the same as a full tur. d) Vary the activity by chagig the poit of rotatio (e.g., let oe of the vertices be the poit of rotatio, or select a poit that is ot o the rectagle as the poit of rotatio). e) Repeat the activity usig differet shapes. 14 Grade 5 Mathematics: Support Documet for Teachers

285 Observatio Checklist Observe studets resposes to determie whether they ca do the followig: uderstad the terms clockwise ad couter-clockwise rotate a 2-D shape ¼-tur, ½-tur, ¾-tur, ad a full tur, both clockwise ad couter-clockwise, aroud ay poit i the iterior or o the shape rotate a 2-D shape ¼-tur, ½- tur, ¾-tur, or a full tur from a poit i the exterior of the shape describe the positio ad orietatio of a 2-D shape after it has bee rotated recogize that the shape ad its image are cogruet Rotate a 2-D shape about a poit, ad describe the positio ad orietatio of the image. Perform a trasformatio of a 2-D shape by followig istructios. Materials: Math jourals Orgaizatio: Idividual a) Have studets draw i their math jourals a square like the oe show below. b) Ask studets to draw a picture of how the square would look after it has bee rotated aroud its cetre poit ¼-tur clockwise ½-tur clockwise ¾-tur clockwise c) Ask studets to write a paragraph describig the positio ad orietatio of the image after each rotatio. Shape ad Space (Trasformatios) 15

286 Observatio Checklist Check to determie whether studets: kow the term clockwise ca perform a trasformatio by followig istructios ca rotate a 2-D shape ¼-tur, ½-tur, ad ¾-tur ca describe the directio ad the orietatio of the image of a rotatio Traslate a 2-D shape horizotally, vertically, or diagoally, ad describe the positio ad orietatio of the image. Rotate a 2-D shape about a poit, ad describe the positio ad orietatio of the image. Reflect a 2-D shape i a lie of reflectio, ad describe the positio ad orietatio of the image. Materials: Patter blocks, math jourals Orgaizatio: Pairs/Idividual a) Preset studets with the followig situatio: Michaela wats to use the same patter block to show her cousi how to rotate, traslate, ad reflect a shape. She is ot sure which patter block she should use to show her cousi all three moves. b) Tell studets it is their job to help Michaela decide which block she should use. Explai that if a block looks the same after it has bee trasformed, it is ot a good model. Help them uderstad what you mea by this by demostratig why a circle is ot a good model for illustratig a reflectio. c) Have studets traslate, rotate, ad reflect each patter block. Ecourage them to record their fidigs i a table like the oe show below: Shape Traslatio Rotatio Reflectio Orage Square Blue Rhombus Red Trapezoid Yellow Hexago Gree Triagle Ta Rhombus d) Have studets write a paragraph i the math joural explaiig which shape Michaela should use ad the reasos for their choice. 16 Grade 5 Mathematics: Support Documet for Teachers

287 Observatio Checklist Check studets resposes to determie whether they ca do the followig: traslate a give 2-D shape ad describe the positio ad orietatio of the trasformed shape rotate a give 2-D shape ad describe the positio ad orietatio of the trasformed shape reflect a give 2-D shape ad describe the positio ad orietatio of the trasformed shape Materials: Geoboards, elastics, dot paper, a set of cards (BLM 5.SS.7&8.2) for each group Orgaizatio: Groups of four a) Tell studets that they will be doig a activity ivolvig traslatios, reflectios, ad rotatios. Explai that each group will get four cards. The cards should be placed face dow o the work area ad each studet should draw oe. The studet with the card that says Origial Positio makes a shape o his or her geoboard. The studet who has the card statig Rotatio creates a rotatio of the origial shape o his or her geoboard. The studet who has the card statig Reflectio creates a reflectio of the origial shape o his or her geoboard, ad the oe who has the card statig Traslatio creates a traslatio of the origial shape o his or her geoboard. Studets should record the origial shape o their dot paper ad the trasformatio that they performed. Udereath the trasformatio, they should describe how the shape was trasformed (e.g., vertical traslatio two uits up) ad how the positio ad orietatio of the shape chaged. b) Tell studets that they have to fid a way to verify that they have successfully traslated, rotated, ad reflected the origial shape. Whe they have verified their trasformatios, they should exchage cards ad do the activity agai. Each studet should have the opportuity to create, traslate, rotate, ad reflect a shape. Observatio Checklist Observe studets resposes to determie whether they ca do the followig: traslate a give shape horizotally, vertically, or diagoally, ad describe the positio ad orietatio of the image rotate a shape aroud a poit ad describe the positio ad orietatio of the image reflect a shape across a lie of reflectio ad describe the positio ad orietatio of the image Shape ad Space (Trasformatios) 17

288 Traslate a 2-D shape horizotally, vertically, or diagoally, ad describe the positio ad orietatio of the image. Rotate a 2-D shape about a poit, ad describe the positio ad orietatio of the image. Reflect a 2-D shape i a lie of reflectio, ad describe the positio ad orietatio of the image. Predict the result of a sigle trasformatio of a 2-D shape ad verify the predictio. Idetify a sigle trasformatio as a traslatio, rotatio, or reflectio. Describe a rotatio by the directio of the tur (clockwise or couterclockwise). Materials: A red trapezoid from the set of patter blocks, a die, ad math jourals Orgaizatio: Whole class a) Tell studets that they will be playig a game. Explai that you will be placig a trapezoid o the overhead ad studets will be takig turs rollig a die. After the die has bee rolled, the overhead will be tured off ad the studet who rolled the die will trasform the trapezoid accordig to the followig rules: If a 1 or 2 is rolled, the studet will traslate the trapezoid. If a 3 or 4 is rolled, the studet will reflect the trapezoid. If a 5 or 6 is rolled, the studet will rotate the trapezoid either clockwise or couter-clockwise. After the studet has chaged the positio of the trapezoid, the overhead will be tured back o ad the other members of the class must record i their math jourals the trasformatio they thik the studet performed o the trapezoid. b) Demostrate the procedure for playig the game ad aswer ay questios studets may have. c) Have studets share their respose to each roud of the game. Ecourage studets to explai how the positio ad orietatio of the trapezoid chaged ad why the trasformatio was a traslatio, rotatio, or reflectio. 18 Grade 5 Mathematics: Support Documet for Teachers

289 Observatio Checklist Observe studets resposes to determie whether they ca do the followig: traslate, rotate, ad reflect a 2-D shape idetify a trasformatio as a rotatio, reflectio, or traslatio describe a rotatio by the directio of the tur (either clockwise or couter-clockwise) describe the positio ad orietatio of a traslated shape, a rotated shape, ad a reflected shape Shape ad Space (Trasformatios) 19

290 N OTES 20 Grade 5 Mathematics: Support Documet for Teachers

291 G RADE 5 MATHEMATICS Statistics ad Probability (Data Aalysis)

292

293 Grade 5: Statistics ad Probability (Data Aalysis) Edurig Uderstadigs: Graphs are a way of orgaizig, represetig, ad commuicatig iformatio. Geeral Outcome: Collect, display, ad aalyze data to solve problems. SPECIFIC LEARNING OUTCOME(S): ACHIEVEMENT INDICATORS: 5.SP.1 Differetiate betwee first-had ad secod-had data. [C, R, T, V] 5.SP.2 Costruct ad iterpret double bar graphs to draw coclusios. [C, PS, R, T, V] Explai the differece betwee first-had ad secod-had data. Formulate a questio that ca best be aswered usig first-had data ad explai why. Formulate a questio that ca best be aswered usig secod-had data ad explai why. Fid examples of secod-had data i prit ad electroic media, such as ewspapers, magazies, ad the Iteret. Determie the attributes (title, axes, itervals, ad leged) of double bar graphs by comparig a set of double bar graphs. Represet a set of data by creatig a double bar graph, label the title ad axes, ad create a leged with or without the use of techology. Draw coclusios from a give double bar graph to aswer questios. Provide examples of double bar graphs used i a variety of prit ad electroic media, such as ewspapers, magazies, ad the Iteret. Solve a problem by costructig ad iterpretig a double bar graph. Statistics ad Probability (Data Aalysis) 3

294 PRIOR KNOWLEDGE Studets should be able to do the followig: (3.SP.2) Collect first-had data ad orgaize it usig tally marks, lie plots, charts, ad lists (4.SP.2) Costruct ad iterpret bar graphs ad pictographs ivolvig may-to-oe correspodece to draw coclusios (various outcomes) Classify objects or items ito groups (various outcomes) Make reasoable estimates of quatities (3.SS.4) Demostrate a uderstadig of kilograms ad grams (4.N.9) Describe ad represet decimals to hudredths RELATED KNOWLEDGE Studets should be able to do the followig: (5.N.1) Represet ad describe whole umbers to BACKGROUND INFORMATION Graphs are visual displays of data that provide a overall picture of the iformatio that has bee collected. There are may types of graphs, the most commo of which are bar graphs, lie graphs, circle graphs, ad pictographs. Before studets ca decide o which graph to use, they eed to kow what type of data have bee collected. More specifically, the data that they collect ca be either discrete or cotiuous. Discrete data ivolve observatios that are separate ad distict ad that ca be couted. Shoe sizes, the umber of studets assiged to each classroom i a school, ad the aimals o a farm are all examples of discrete data. Cotiuous data ivolve observatios that ca take o may values withi a fiite or ifiite iterval. Height, temperature, ad age are examples of cotiuous data. Bar graphs, circle graphs, ad pictographs are used to illustrate discrete data. Lie graphs are used to illustrate cotiuous data. I additio, studets eed to be aware of the advatages ad disadvatages of usig each type of graph. This will eable them to select a appropriate graph for their data ad to defed their choices. The data that studets collect ad display ca be either first-had or secod-had. Firsthad data provide iformatio that a idividual obtais directly by askig questios, measurig, observig, or experimetig. Askig Grade 6 studets what their favourite CD is, or observig the umber of times the copy machie is used i a day, are examples of first-had data. Secod-had data provide iformatio that is readily available ad oly eeds to be extracted from sources such as ewspapers, electroic media, magazies, almaacs, ad jourals. Usig a compay s records to obtai iformatio about the umber of sales over a five-year period, ad usig cesus data to fid iformatio about Caadia households, are examples of secod-had data. 4 Grade 5 Mathematics: Support Documet for Teachers

295 All graphs should have a title, labels, ad eat ad cocise etry of data. Whe graphs show icremets, those icremets should start at 0, be of equal size, ad be umbered. The axes the horizotal ad vertical lie segmets that divide the coordiate plae ito quadrats must also be labelled. Bar graphs compare the frequecy of discrete data. Data are displayed usig a umber of rectagles (bars) that are the same width. Each bar represets oe of the categories that the data have bee sorted ito. The bars are displayed either horizotally or vertically with a space betwee them. The height (or legth) of a bar represets the umber of observatios i that category. The umbers o the y-axis of a vertical bar graph or the x-axis of horizotal bar graph are called the scale. Sometimes a bar graph will have a squiggle i its scale. This meas that part of the scale has bee omitted. However, the use of a squiggle teds to give a misleadig visual picture (see Graph 2 below). Statistics ad Probability (Data Aalysis) 5

296 Double bar graphs are used to make comparisos betwee ad amog sets of data (e.g., the double bar graph show below compares the desserts favoured by boys with the desserts favoured by girls). I additio to kowig the differet types of graphs ad how to costruct them, studets eed to kow how to read ad iterpret them. Curcio (2001) idetifies three levels of graph comprehesio: readig the data, readig betwee the data, ad readig beyod the data. Note: The Statistics Caada Website at < is a excellet source of secod-had data. Readig the Data This level of comprehesio requires a literal readig of the graph. The reader simply lifts data explicitly stated i the graph, or the iformatio foud i the graph title ad axes labels, from the graph. There is o iterpretatio at this level. Readig that requires this type of comprehesio is a very low-level cogitive task. Readig betwee the Data This level of comprehesio icludes the iterpretatio ad itegratio of the data i the graph. It requires the ability to compare quatities (e.g., greater tha, tallest, smallest) ad the use of other mathematical cocepts ad skills (e.g., additio, subtractio, multiplicatio, divisio) that allow the reader to combie ad itegrate data ad idetify the mathematical relatioships expressed i the graph. Readig beyod the Data This level of comprehesio requires the reader to predict or ifer from the data by tappig existig schemata (i.e., backgroud kowledge, kowledge i memory) for iformatio that is either explicitly or implicitly stated i the graph. Whereas readig betwee the data might require the reader to make a iferece that is based o the data preseted i the graph, readig beyod the data requires that the iferece be made o the basis of iformatio i the reader s head, ot i the graph. 6 Grade 5 Mathematics: Support Documet for Teachers

297 MATHEMATICAL LANGUAGE Bar graph Data Double bar graph Estimate First-had data Horizotal axis Iterval Leged Scale Secod-had data Vertical axis LEARNING EXPERIENCES Assessig Prior Kowledge Materials: Graph paper, BLM 5.SP.1&2.1, markers or crayos, straight edge Orgaizatio: Idividual/Whole class a) Have studets complete the activity show below (BLM 5.SP.1&2). Let them kow that the purpose of the activity is to fid out what they kow about bar graphs. 1. Louis is doig a project o aimals. He foud that aimals have differet heart rates (e.g., he foud that a cow has a heart rate of 60 to 80 beats per miute). Here are some of the heart rates he foud. Cow 60 to 80 Cat 110 to 140 Horse 30 to 40 Chicke 300 to 350 Rabbit 140 to 160 Sheep 70 to 140 Make a bar graph of the highest heart rate of each aimal. 2. Study your graph. List three coclusios you ca draw from it. 3. If you added the heart rate of a mouse to your graph, which aimals do you thik would have a slower heart rate tha the mouse? Explai your aswer. b) Have studets share their graph ad coclusios with the rest of the class. Statistics ad Probability (Data Aalysis) 7

298 Observatio Checklist Use the followig checklist to assess studets kowledge of bar graphs. Makes a Bar Graph that Icludes Yes No Commets A title Category labels A label for each axis Bars whose legths correctly represet the umber of observatios i each category Bars the same width Spaces betwee bars A appropriate scale with equal icremets that starts at zero A iterpretatio of bar graphs Readig betwee the data (Questio 2) Readig beyod the data (Questio 3) Explai the differece betwee first-had ad secod-had data. Formulate a questio that ca best be aswered usig first-had data ad explai why. Formulate a questio that ca best be aswered usig secod-had data ad explai why. Represet a set of data by creatig a double bar graph, label the title ad axes, ad create a leged with or without the use of techology. Materials: BLM 5.SP.1&2.2, two bar graphs, oe illustratig first-had data ad oe illustratig secod-had data, math joural Orgaizatio: Pairs/Whole class a) Tell studets that i the ext few lessos they will be doig some graphig. Explai that graphs are a quick ad easy way to illustrate iformatio. Ask studets what a bar graph is ad where they have see oe. 8 Grade 5 Mathematics: Support Documet for Teachers

299 b) Show studets the two bar graphs ad have them discuss how they are alike ad how they differ. Ecourage studets to thik about how the iformatio i the graphs was obtaied. c) Tell studets that each of the followig statemets represets iformatio that ca be graphed. Have them sort the statemets ito two groups i as may ways as they ca ad to record their fidigs (BLM 5.SP.1&2.2). Names of the highest moutais i North America Studets i the class who have pets The wo/lost record of the Wiipeg Goldeyes The makes of cars parked i a shoppig cetre parkig lot The sum of the umbers rolled whe two dice are throw The populatio of Caadia provices The umber of people who have immigrated to Maitoba How far studets i Grade 7 ca throw a softball The umber of cm a plat grows over a five-week period The umber of houses sold i Wiipeg last year c) Have studets share their classificatios with the other members of the class. d) Explai that iformatio we collect ourselves is called first-had data, ad iformatio that we get from other sources, such as ewspapers, magazies or the Iteret, is called secod-had data. Have studets decide which statemets are examples of first-had data ad which are secod-had data. e) Itroduce the ever-edig graphig project. Tell studets that they will be makig a book of graphs. Each week, they will be makig a graph that orgaizes iformatio they have collected about themselves, their commuity, or ay other topic of iterest. Ask studets what they would like to kow about their classmates ad their commuity. Make a list of their questios ad have studets determie which questios are best aswered with first-had data ad which are best aswered with secod-had data. Note: Questios ca be added to the list throughout the year. Each time a ew questio is added, studets should discuss whether the questio is best aswered with first- or secod-had data. The graphig activities ca be itegrated ito other subject areas (e.g., i laguage arts, studets could make a graph illustratig the umber of books they have read, ad i sciece they ca make a graph comparig distaces differet-sized balloos travel i a jet expulsio experimet. Also, the activities should be structured so studets have opportuities to practice creatig ad iterpretig differet types of graphs [pictographs, bar graphs, ad double bar graphs].). Statistics ad Probability (Data Aalysis) 9

300 f) Have studets aswer the followig questios i their math jourals. Ecourage studets to give examples that have ot bee discussed i class. What are two examples of first-had data What are two examples of secod-had data? What is the differece betwee first- ad secod-had data? Observatio Checklist Examie studets resposes to determie whether they ca do the followig: formulate a questio that is best aswered with first-had data ad explai why formulate a questio that is best aswered with secod-had data ad explai why provide examples of first-had data provide examples of secod-had data explai the differece betwee first- ad secod-had data 10 Grade 5 Mathematics: Support Documet for Teachers

301 Explai the differece betwee first-had ad secod-had data. Represet a set of data by creatig a double bar graph, label the title ad axes, ad create a leged with or without the use of techology. Solve a problem by costructig ad iterpretig a double bar graph. Materials: Graph paper, markers, pik ad yellow sticky otes Orgaizatio: Whole class a) Select oe questio that studets have about their classmates (e.g., What is your favourite pizza toppig? ). Have studets idetify three differet toppigs (e.g., pepperoi, mushrooms, ad sausage). b) Have the girls record their favourite toppig o the yellow sticky otes ad the boys o the pik sticky otes. Write pepperoi, mushrooms, ad sausage o the board ad have studets place their sticky otes uder the appropriate headig ad fid the total umber of studets i each category. c) Have studets make a bar graph illustratig their favourite pizza toppig ad write a paragraph explaiig what the graph meas to them. Whe studets fiish writig their paragraphs, have them share their paragraph with a parter. The graph ad paragraph ca be the studets first etry ito their graphig book (itroduced i Part E of the previous learig experiece). d) Ask studets how they ca show how may girls like each toppig ad how may boys like each toppig. Have studets sort the sticky otes uder each headig ito two groups ad fid the total umber i each category. e) Show studets how to make a double bar graph that compares girls favourite pizza toppigs with boys favourite pizza toppigs. Have them make a double bar graph alog with you. Help studets iterpret their graph by askig questios such as the followig: What toppig do girls like the most? The least? What toppig do the boys like the most? How may more girls like tha boys? If you order pizza for the class ad ca oly pick two toppigs, which two would you pick? Why? Does the graph illustrate first-had or secod-had data? Explai. f) Have studets compare the bar graph of favourite pizza toppigs with the double bar graph of favourite pizza toppigs. Ask, How are they alike? How do they differ? Statistics ad Probability (Data Aalysis) 11

302 Observatio Checklist For Part C, use the followig checklist to determie whether studets ca create ad iterpret a appropriate bar graph. Makes a Bar Graph that Icludes Yes No Commets A title Category labels A label for each axis Bars whose legths correctly represet the umber of observatios i each category Bars the same width Spaces betwee bars A appropriate scale with equal icremets that starts at zero A iterpretatio of bar graphs Readig betwee the data (Questio 2) Readig beyod the data (Questio 3) Observatio Checklist Observe studets resposes to Parts E ad F to determie whether they ca do the followig: make the double bar graph correctly iterpret the double bar graph correctly recogize that both bar graphs ad double bar graphs iclude a title, labelled axes, ad a scale with equal icremets that begi at 0 recogize that double bar graphs eed a leged recogize that bar graphs illustrate oe set of data ad double bar graphs compare two sets of data 12 Grade 5 Mathematics: Support Documet for Teachers

303 Determie the attributes (title, axes, itervals, ad leged) of double bar graphs by comparig a set of double bar graphs. Represet a set of data by creatig a double bar graph, label the title ad axes, ad create a leged with or without the use of techology. Draw coclusios from a double bar graph to aswer questios. Solve problems by costructig ad iterpretig a double bar graph. Materials: Two packages of assorted bags of cady for each group (e.g., two bags of jelly beas, two bags of gum drops, two bags of skittles), large graphig paper, markers, BLM 5.SP.1&2.3, BLM 5 8.4, paper plates for sortig the cady, self-assessmet sheet Orgaizatio: Groups of three or four a) Tell studets that they will be ivestigatig what is iside a package of assorted cady. Explai that each group will get two packages of the same type of cady. There are differet colours of cady i each package, ad it is their job to determie whether the umber of pieces of each colour i the package is the same or whether oe colour appears more ofte tha aother. Whe they fiish their ivestigatio, they should be able to aswer the questio, How assorted is a package of cadies? b) Have each studet complete the recordig sheet (BLM 5.SP.1&2.3). Explai that the members of each group will eed to work together to make a double bar graph that they will preset to the rest of the class. c) Have each group preset its graph ad its fidigs to the rest of the class. d) Have the studets compare all the double graphs. Ask them how the graphs are alike ad how they differ ad if there are ay commo patters ad relatioships amog them. Observatio Checklist Moitor studets resposes to determie whether they ca do the followig: make reasoable estimates of quatities idetify the attributes (title, axes, itervals, ad leged) of double bar graphs represet sets of data by creatig a double bar graph that icludes a title, labelled axes, appropriate itervals, ad a leged without the use of techology draw coclusios from a double bar graph that illustrate that they ca read betwee ad beyod the data solve a problem usig double bar graphs Statistics ad Probability (Data Aalysis) 13

304 Self-Assessmet Have studets do a self-assessmet (BLM 5 8.4) of how they work i a group. Explai the differece betwee first-had ad secod-had data. Draw coclusios from a double bar graph to aswer questios. Materials: Copies of immigratio map, math jourals, calculators Orgaizatio: Idividual/Large group a) Provide studets with the followig graph to aalyze. b) Ask studets to record their aswers to these questios i their math jourals. Tell studets that they ca use a calculator to help them iterpret the graph. 1. What is the graph about? 2. Does the graph illustrate first-had or secod-had data? Explai. 3. What coclusios ca you draw from the data? 4. What reasos ca you give for the results? 5. If you add Otario to the graph, do you thik more people would have immigrated to Maitoba or to Otario? Explai. 14 Grade 5 Mathematics: Support Documet for Teachers

305 c) Have the studets share their resposes with the other members of the class. Ecourage studets to provide reasos for why more people immigrate to oe provice tha aother. d) Ask studets what questios they have as a result of aalyzig the graph. Make a list of their questios ad have studets devise a pla for aswerig them. Extesio: Visit the Statistics Caada Website at < to see if the data have chaged sice Observatio Checklist Check studets aswers to determie whether they ca do the followig: differetiate betwee first-had ad secod-data draw coclusios from a double bar graph that illustrate they ca read betwee the data (e.g., A total of approximately people immigrated to Maitoba i the two years) read beyod the data (e.g., More people would immigrate to Otario tha Maitoba) provide appropriate reasos for their coclusios (e.g., More people might have immigrated to B.C. because there were more jobs available) Explai the differece betwee first-had ad secod-had data. Represet a set of data by creatig a double bar graph, labellig the title ad axes, ad creatig a leged with or without the use of techology. Draw coclusios from a double bar graph to aswer questios. Solve a problem by costructig ad iterpretig a double bar graph. Materials: Empty cereal boxes, large graph paper, ad markers Orgaizatio: Idividual or pairs a) Show studets a empty cereal box. Poit out that there are may differet types of cereal. Ask studets why some cereals are cosidered to be better for them tha others. b) Explai that each box of cereal has a list of utriets. Have studets fid the list of utriets o oe of their boxes. Ask them to fid the umber of grams of fibre i oe servig of their cereal. Explai that cereals that are high i fibre ad low i sugar cotet are cosidered healthier tha those that are low i fibre ad high i sugar. Statistics ad Probability (Data Aalysis) 15

306 c) Tell studets that their task is to determie which cereal is better for them. Explai that they eed to fid the grams of fibre ad sugar i three differet cereals ad represet this iformatio o a double bar graph. Have studets discuss whether they will be graphig first- or secod-had data, ad how they kow. d) Have studets write a paragraph describig their graph ad all the coclusios that they ca draw from it. Explai that they eed to iclude a statemet i their paragraph idicatig which cereal is better for them ad the reasos for their coclusio. e) Have studets share their graphs with the other members of the class, ad explai which cereal they thik is better for them. Ecourage studets to provide reasos for their coclusios. Observatio Checklist Checks studets graphs ad their coclusios about them to determie whether they ca do the followig: represet data o a double bar graph that icludes a title, labelled axes, appropriate itervals, ad a leged with or without the use of techology draw valid coclusios from the data that illustrate they ca read betwee data solve a problem usig a double bar graph Represet a set of data by creatig a double bar graph, labellig the title ad axes, ad creatig a leged with or without the use of techology. Draw coclusios from a double bar graph to aswer questios. Solve a problem by costructig ad iterpretig a double bar graph. Materials: A computer with a spreadsheet program, paper, ad pecil Orgaizatio: Idividual or pairs a) Ask, Which city do you thik is colder: Fli Flo or Motreal? Why? Explai that oe way to determie which city is colder is to compare the umber of days the temperature i each city is less tha or equal to zero. b) Tell studets they are goig to fid out which cities i Caada are the coldest. Have studets go to < ad select the statistics lik. Tell them to pick ay two cities i Caada ad fid the umber of days i each moth of the year that each city s temperature is less tha or equal to zero. 16 Grade 5 Mathematics: Support Documet for Teachers

307 c) Have studets make a double bar graph usig a computer spreadsheet program that shows the umber of days i each moth that each city s temperature is less tha or equal to zero. d) Have studets make a list of questios about their graph. Ask them to prepare a poster cosistig of the origial data, the graph, ad the questios that they developed. e) Display the posters aroud the room ad coduct a gallery walk. Have studets aswer the questios o each poster. Have studets compare all the graphs to determie which city is the coldest. Observatio Checklist Observe studets resposes to determie whether they ca do the followig: collect secod-had data from a electroic database create a double bar graph usig techology draw valid coclusios from the data that illustrate they ca read betwee ad beyod the data ask appropriate questios about the data represeted o a graph Statistics ad Probability (Data Aalysis) 17

308 N OTES 18 Grade 5 Mathematics: Support Documet for Teachers

309 G RADE 5 MATHEMATICS Statistics ad Probability (Chace ad Ucertaity)

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311 Grade 5: Statistics ad Probability (Chace ad Ucertaity) Edurig Uderstadigs: Chace is a elemet of may aspects of our lives. The chace that a evet will occur varies from impossible to certai. Geeral Outcome: Use experimetal or theoretical probabilities to represet ad solve problems ivolvig ucertaity. SPECIFIC LEARNING OUTCOME(S): ACHIEVEMENT INDICATORS: 5.SP.3 Describe the likelihood of a sigle outcome occurrig, usig words such as impossible possible certai [C, CN, PS, R] 5.SP.4 Compare the likelihood of two possible outcomes occurrig, usig words such as less likely equally likely more likely [C, CN, PS, R] Provide examples of evets that are impossible, possible, or certai from persoal cotexts. Classify the likelihood of a sigle evet occurrig i a probability experimet as impossible, possible, or certai. Desig ad coduct a probability experimet i which the likelihood of a sigle outcome occurrig is impossible, possible, or certai. Coduct a probability experimet a umber of times, record the outcomes, ad explai the results. Idetify outcomes from a probability experimet that are less likely, equally likely, or more likely to occur tha other outcomes. Desig ad coduct a probability experimet i which oe outcome is less likely to occur tha the other outcome. Desig ad coduct a probability experimet i which oe outcome is equally likely to occur as the other outcome. Desig ad coduct a probability experimet i which oe outcome is more likely to occur tha the other outcome. Statistics ad Probability (Chace ad Ucertaity) 3

312 PRIOR KNOWLEDGE Although this is the first time that studets should formally ecouter the study of chace ad ucertaity, studets should (4.N.8) Have a uderstadig of fractios BACKGROUND INFORMATION Probability is a measure of the chace that a evet will occur. The formal study of probability begis i Grade 5 with the developmet of the laguage of probability. Kowledge of the terms associated with probability facilitates studets uderstadig of the role chace plays i their lives ad their awareess that some evets are more probable tha others. The learig experieces i this substrad are divided ito two parts. The first part focuses o the developmet of the terms impossible, possible, certai, more likely, less likely, ad equally likely ad their applicatio to real-life evets. The learig experieces i the secod part have studets applyig the laguage of probability to predict ad explai the outcomes (the results) of probability experimets. The research o probability idicates that studets ofte have miscoceptios about the outcomes of evets. For example, if a coi is tossed three times ad lads heads up each time, may studets believe the ext time the coi is tossed it is boud to lad tails up because thigs will eve out. Cosequetly, the learig activities i this sectio egage studets i geeratig ad aalyzig data that will help them overcome their miscoceptios ad pave the way for a deeper uderstadig of theoretical probability. MATHEMATICAL LANGUAGE Impossible Possible Certai Less likely More likely Equally likely 4 Grade 5 Mathematics: Support Documet for Teachers

313 LEARNING EXPERIENCES Provide examples of evets that are impossible, possible, or certai from persoal cotexts. Idetify outcomes from a probability experimet that are less likely, equally likely, or more likely to occur tha other outcomes. Materials: It s Probably Pey by Loree Leedy (ISBN-13: / ISBN-10: ) Orgaizatio: Whole class a) Read the book It s Probably Pey to the class. b) Discuss the story. Begi the discussio by askig, What was Pey s homework assigmet? What evet did Pey thik was possible? What are some other evets that are possible? What does possible mea? c) Create a class list of evets that are impossible possible likely certai Have studets explai why they thik each evet they ame is impossible, possible, likely, or certai to occur. Observatio Checklist Observe the studets to determie whether they ca do the followig: idetify evets that are impossible, possible, certai, or ulikely provide valid reasos for why a evet is impossible, possible, ulikely, or certai Statistics ad Probability (Chace ad Ucertaity) 5

314 Classify the likelihood of a sigle outcome occurrig i a probability experimet as impossible, possible, or certai. Idetify outcomes from a probability experimet that are less likely, equally likely, or more likely to occur tha other outcomes. Materials: Evet cards ad label cards BLM 5.SP.3&4.1, loops made of strig or yar Orgaizatio: Pairs a) Have the studets place the strig loops o their workspace ad place a label card beside each set. b) Ask studets to place each of the give statemets ito oe of the sets. c) Whe studets are fiished sortig the statemets, have them share their aswers ad explai their reasos for placig a evet i a give set. Observatio Checklist Moitor studets to determie whether they have valid reasos for categorizig a give evet as impossible, ulikely, possible, or certai, ad whether they uderstad that if a evet is impossible, it will ot occur if a evet is certai, it will occur if a evet is possible, it may or may ot occur some evets are more possible tha others 6 Grade 5 Mathematics: Support Documet for Teachers

315 Material: Copies of the learig experiece (BLM 5.SP.3&4.2) Orgaizatio: Idividual a) Have studets complete the followig activity sheet (BLM 5.SP.3&4.2). b) Whe studets have fiished aswerig the questios, have them share their aswers ad their reasos for them. Observatio Checklist Observe studets resposes to determie whether they uderstad the followig: the more possibilities there are, the more likely a evet will occur if a evet is impossible, it will ot occur if a outcome is certai, it is the oly oe if evets are equally likely, they have the same chace of occurrig Statistics ad Probability (Chace ad Ucertaity) 7

316 Provide examples from evets that are impossible, possible, or certai from persoal cotexts. Materials: Joural/Learig log Orgaizatio: Idividual Have the studets complete the followig statemets i their math jourals: 1. I am less likely to.. 2. I am more likely to I am certai to. 4. It is impossible for me to.. 5. It is equally likely that.. 6. It is possible that I will Observatio Checklist Examie studets resposes to determie whether they ca do the followig: idetify evets other tha oes discussed i class that are certai, impossible, possible, more likely, less likely, ad equally likely to occur uderstad the meaig of the terms certai, impossible, possible, more likely, less likely, ad equally likely 8 Grade 5 Mathematics: Support Documet for Teachers

317 Coduct a probability experimet a umber of times, record the outcomes, ad explai the results. Idetify outcomes from a probability experimet that are less likely, equally likely, or more likely to occur tha other outcomes. Materials: A set of cards (BLM 5 8.5) Orgaizatio: Whole class Show studets the cards ad the spread them out face dow o a table. Have oe studet select a card ad show it to the rest of the class. Ask: Is it possible to pick a umber that is greater tha the oe just selected? Why or why ot? Is it possible to pick a umber that is less tha the oe just selected? Why or why ot? Is the umber more likely or less likely to be greater (less) tha the oe just selected? Why do you thik so? Have a studet pick aother card so the class ca check their predictio ad discuss why their predictio may or may ot have bee correct. Repeat the activity several times. Observatio Checklist Observe studets resposes to determie whether they are doig the followig: basig their predictios o the relatioships betwee the umbers usig the terms more likely ad less likely correctly providig valid reasos for their predictios Statistics ad Probability (Chace ad Ucertaity) 9

318 Coduct a probability experimet a umber of times, record the outcomes, ad explai the results. Idetify outcomes from a probability experimet that are less likely, equally likely, or more likely to occur tha other outcomes. Material: Coloured tiles or blocks, paper bag Orgaizatio: Whole class a) Ask studets, If you put four blue tiles i a bag ad the take oe out without lookig, ca you be sure what colour tile you will get? Why or why ot? b) Next, ask, If you put a red tile i the bag with the blue tiles ad take oe out without lookig, ca you be sure what coloured tile you will get? Why or why ot? Are you more likely or less likely to get a red tile? Why? Are you more likely or less likely to get a blue tile? Why? Have studets test their aswer by placig four blue tiles ad oe red tile i a bag, ad the askig several studets to draw a tile from the bag ad show it to the rest of the class. Have them put the tile back i the bag before askig aother studet to draw a tile. Have studets keep a record of the umber of times a blue tile is draw ad the umber of times a red tile is draw. Discuss the results. c) Fially, ask, If you put three red tiles i a bag with seve blue tiles ad take oe out without lookig, is oe coloured tile more likely to be draw from the bag tha the other? Why or why ot? Are you more likely or less likely to get a red tile? Have studets test their aswer by placig three red ad seve blue tiles i the bag. Ask several studets to draw a tile from the bag ad show it to the rest of the class. Have them put the tile back ito the bag before askig aother studet to draw a tile. Have studets keep a record of the umber of times a red tile is draw ad the umber of times a blue tile is draw. Discuss the results. Observatio Checklist Moitor studets resposes to the questios to determie whether they are doig the followig: basig their predictios o the umber of red ad blue tiles usig the terms more likely ad less likely correctly providig valid reasos for their predictios explaiig the results of the experimets 10 Grade 5 Mathematics: Support Documet for Teachers

319 Idetify outcomes from a probability experimet that are less likely, equally likely, or more likely to occur tha other outcomes. Materials: BLM 5.SP.3&4.3 Orgaizatio: Pairs a) Examie the followig statemets about the spier ad decide whether the statemets are true or false. Check your predictios by spiig the spier 20 times ad recordig the result of each spi. Statemets: It is more likely to lad o 4 tha o 1, 2, or 3. It is less likely to lad o 1 tha 3. It is equally likely to lad o 1 as 2. b) Whe all pairs have checked the validity of these statemets, discuss their results as a class. Observatio Checklist Observe studets resposes to determie whether they ca do the followig: uderstad that the spier is more likely to stop o the sectio that is the greatest fractio of the whole uderstad the spier is less likely to stop o the sectio that is the least fractio of the whole uderstad that sectios of the spier that are equivalet fractios of the whole are equally likely to occur provide valid reasos for their predictios explai the results of their experimet ad why their results may differ from others Statistics ad Probability (Chace ad Ucertaity) 11

320 Idetify outcomes from a probability experimet that are less likely, equally likely, or more likely to occur tha other outcomes. Materials: Two-coloured chips, paper cups, copies of the istructio for the experimet (BLM 5.SP.3&4.4) Orgaizatio: Pairs Have studets complete activity sheet (BLM 5.SP.3&4.4). Observatio Checklist Examie studets resposes to determie whether they are doig the followig: usig the terms more likely ad less likely correctly explaiig the results of their experimet Desig ad coduct a probability experimet i which the likelihood of a sigle outcome occurrig is impossible, possible, or certai. Coduct a probability experimet a umber of times, record the outcomes, ad explai the results. Desig ad coduct a probability experimet i which oe outcome is less likely to occur tha the other outcome. Desig ad coduct a probability experimet i which oe outcome is equally likely to occur as the other outcome. Desig ad coduct a probability experimet i which oe outcome is more likely to occur tha the other outcome. Materials: Copies of the Mystery Spier activity sheet (BLM 5.SP.3&4.5) Orgaizatio: Idividual Have studets complete the activity sheet (BLM 5.SP.3&4.5). 12 Grade 5 Mathematics: Support Documet for Teachers

321 Observatio Checklist Moitor studets resposes to the spiig activities to determie whether: their spiers meet the coditios that were specified they use the terms more likely, less likely, impossible, certai, ad equally likely correctly they refer to the fractio of the whole that each part of their spier occupies to explai why the coditios that were specified are met PUTTING THE PIECES TOGETHER Picturig Probability Purpose: The itet of this ivestigatio is to have studets demostrate their uderstadig of the laguage of probability by depictig situatios that illustrate differet degrees of chace. I particular, the ivestigatio is desiged to reiforce the meaig ad use of the terms impossible, possible, ad certai (5.SP.3) less likely, equally likely, ad more likely (5.SP.4) The ivestigatio is also desiged to exted studets ability to commuicate mathematically use techology make coectios to other subject areas (LwICT ad ELA) make coectios to real-world situatios Materials/Resources: Digital cameras Card readers (optioal) Microphoes Computer projectors Computers Microsoft Photo Story Statistics ad Probability (Chace ad Ucertaity) 13

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