MATHEMATICS CURRICULUM Grade 3

Transcription

1 MIDDLETOWN PUBLIC SCHOOLS MATHEMATICS CURRICULUM Grade 3 Elementary Schl Curriculum Writers: Mary Alice Chrabascz, Mary Claneri, Danielle Laurie, Laurie Oliveira, Cathy Palkvic, Kim Pearce, Jen Pesare, and Karen Weikert REVISED June 2014

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3 T MATHEMATICS CURRICULUM Grade 3 he Mathematics Curriculum fr grades K 12 was revised June 2014 by a K 12 team f teachers. The team, identified as the Mathematics Task Frce and Mathematics Curriculum Writers referenced extensive resurces t design the dcument that included: Cmmn Cre State Standards fr Mathematics Cmmn Cre State Standards fr Mathematics, Appendix A Understanding Cmmn Cre State Standards, Kendall PARCC Mdel Cntent Framewrks Numerus state curriculum Cmmn Cre framewrks,, e.g. Ohi, Arizna, Nrth Carlina, and New Jersey High Schl Traditinal Plus Mdel Curse Sequence, Achieve, Inc. Grade Level and Grade Span Expectatins (GLEs/GSEs) fr Mathematics Third Internatinal Mathematics and Science Test (TIMSS) Best Practice, New Standards fr Teaching and Learning in America s Schls; Differentiated Instructinal Strategies Instructinal Strategies That Wrk, Marzan Gals fr the district Missin Statement Our missin is t prvide a sequential and cmprehensive K 12 mathematics curriculum in a cllabrative student centered learning envirnment that develps critical thinkers, skillful prblem slvers, and effective cmmunicatrs f mathematics. The Mathematics Curriculum identifies what students shuld knw and be able t d in mathematics. Each grade r curse includes Cmmn Cre State Standards (CCSS), Grade Level Expectatins (GLEs), Grade Span Expectatins (GSEs), grade level supprtive tasks, teacher ntes, best practice instructinal strategies, resurces, a map (r suggested timeline), rubrics, checklists, and cmmn frmative and summative assessments. COMMON CORE STATE STANDARDS The Cmmn Cre State Standards (CCSS): Are fewer, higher, deeper, and clearer. Are aligned with cllege and wrkfrce expectatins. Include rigrus cntent and applicatins f knwledge thrugh high rder skills. Build upn strengths and lessns f current state standards (GLEs and GSEs). Are internatinally benchmarked, s that all students are prepared fr succeeding in ur glbal ecnmy and sciety. Are research and evidence based. Cmmn Cre State Standards cmpnents include: Standards fr Mathematical Practice (K 12) Standards fr Mathematical Cntent: Categries (high schl nly): e.g. numbers, algebra, functins, data Dmains: larger grups f related standards Clusters: grups f related standards Standards: define what students shuld understand and are able t d The Cmmn Cre Mathematics Curriculum prvides all students with a sequential cmprehensive educatin in mathematics thrugh the study f: Standards fr Mathematical Practice (K 12) Make sense f prblems and persevere in slving them Reasn abstractly and quantitatively Cnstruct viable arguments and critique the reasning f thers Mdel with mathematics* Use apprpriate tls strategically Attend t precisin Lk fr and make use f structure Lk fr and express regularity in repeated reasning Standards fr Mathematical Cntent: 8/20/2014 2

4 K 5 Grade Level Dmains f Cunting and Cardinality Operatins and Algebraic Thinking Number and Operatins in Base Ten Number and Operatins Fractins Measurement and Data Gemetry MATHEMATICS CURRICULUM Grade Grade Level Dmains f Ratis and Prprtinal Relatinships The Number System Expressins and Equatins Functins Gemetry 9 12 Grade Level Cnceptual Categries f Number and Quantity Algebra Functins Mdeling Gemetry Statistics and Prbability RESEARCH BASED The Cmmn Cre Mathematics Curriculum prvides a list f research based best practice instructinal strategies that the teacher may mdel and/r facilitate. It is suggested the teacher: Use frmative assessment t guide instructin Prvide pprtunities fr independent, partner and cllabrative grup wrk Differentiate instructin by varying the cntent, prcess, and prduct and prviding pprtunities fr: anchring cubing jig sawing pre/pst assessments tiered assignments Address multiple intelligences instructinal strategies, e.g. visual, bdily kinesthetic, interpersnal Prvide pprtunities fr higher level thinking: Webb s Depth f Knwledge, 2,3,4, skill/cnceptual understanding, strategic reasning, extended reasning Facilitate the integratin f in all cntent areas f mathematics Facilitate integratin f the Applied Learning Standards (SCANS): cmmunicatin critical thinking prblem slving reflectin/evaluatin research Emply strategies f best practice (student centered, experiential, hlistic, authentic, expressive, reflective, scial, cllabrative, demcratic, cgnitive, develpmental, cnstructivist/heuristic, and challenging) Prvide rubrics and mdels Address multiple intelligences and brain dminance (spatial, bdily kinesthetic, musical, linguistic, intrapersnal, interpersnal, mathematical/lgical, and naturalist) Emply mathematics best practice strategies e.g. 8/20/2014 3

5 using manipulatives facilitating cperative grup wrk discussing mathematics questining and making cnjectures justifying f thinking writing abut mathematics facilitating prblem slving apprach t instructin integrating cntent using calculatrs and cmputers facilitating learning using assessment t mdify instructin MATHEMATICS CURRICULUM Grade 3 COMMON The Cmmn Cre Mathematics Curriculum includes cmmn assessments. Required (red ink) indicates the assessment is required f all students e.g. cmmn tasks/perfrmance based tasks, standardized mid term exam, standardized final exam. Required Assessments PARCC Released Test Prblems Cmmn Unit Assessment Cmmn Tasks NWEA Test Perfrmance Level Descriptrs (PARCC) Cmmn Instructinal Assessments (I) used by teachers and students during the instructin f CCSS. Cmmn Frmative Assessments (F) used t measure hw well students are mastering the cntent standards befre taking state assessments teacher and student use t make decisins abut what actins t take t prmte further learning n ging, dynamic prcess that invlves far mre frequent testing serves as a practice fr students Cmmn Summative Assessment (S) used t measure the level f student, schl, r prgram success make sme srt f judgment, e.g. what grade prgram effectiveness e.g. state assessments (AYP), mid year and final exams Additinal assessments include: Anecdtal recrds Oral presentatins Cnferencing Prblem/Perfrmance based/cmmn tasks Exhibits Rubrics/checklists (mathematical practice, mdeling) Interviews Tests and quizzes Graphic rganizers Technlgy Jurnals Think aluds Mdeling Multiple Intelligences assessments, e.g. Rle playing bdily kinesthetic Graphic rganizing visual Cllabratin interpersnal 8/20/2014 4

6 FOR MATHEMATICS CURRICULUM Grade 3 MATHEMATICS CURRICULUM Grade 3 Textbks Supplementary Classrm Instructin That Wrks, Marzan Engage NY EnVisin Grade 3 EnVisin Online Cmpnent Exemplars (grade 4) PARCC Released Items NWEA MAP Assessments Prblem Slver Technlgy Calculatr Cmputers ELMO GIZMO Graphing Calculatr Interactive bards LCD prjectrs Overhead calculatr Smart bard TI Navigatr Websites cntent framewrks atics0.pdf (Gizm ) Materials Clred chips Dice Everyday Templates Exp markers Fractin sticks Number line Pattern blcks Prtractrs Rulers Student white bards 8/20/2014 5

7 Task Type I. Tasks assessing cncepts, skills and prcedures MATHEMATICS CURRICULUM Grade 3 Descriptin f Task Type Balance f cnceptual understanding, fluency, and applicatin Can invlve any r all mathematical practice standards Machine screable including innvative, cmputer based frmats Will appear n the End f Year and Perfrmance Based Assessment cmpnents Sub claims A, B and E II. Tasks assessing expressing mathematical reasning III. Tasks assessing mdeling / applicatins Each task calls fr written arguments / justificatins, critique f reasning, r precisin in mathematical statements (MP.3, 6). Can invlve ther mathematical practice standards May include a mix f machine scred and hand scred respnses Included n the Perfrmance Based Assessment cmpnent Sub claim C Each task calls fr mdeling/applicatin in a real wrld cntext r scenari (MP.4) Can invlve ther mathematical practice standards May include a mix f machine scred and hand scred respnses Included n the Perfrmance Based Assessment cmpnent Sub claim D 8/20/2014 6

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9 OPERATIONS AND ALGEBRAIC THINKING (3.OA) Use Mathematical Practices t 1. Make sense f prblems and persevere in slving them 2. Reasn abstractly and quantitatively 3. Cnstruct viable arguments and critique the reasning f thers 4. Mdel with mathematics 5. Use apprpriate tls strategically 6. Attend t precisin 7. Lk fr and make use f structure 8. Lk fr and express regularity in repeated reasning M Students represent and slve prblems invlving multiplicatin and divisin. 3.OA.1 Interpret prducts f whle numbers, e.g., interpret 5 7 as the ttal number f bjects in 5 grups f 7 bjects each. Majr cntent This standard interprets prducts f whle numbers. Students recgnize multiplicatin as a means t determine the ttal number f bjects when there are a specific number f grups with the same number f bjects in each grup r f an equal amunt f bjects were added r cllected numerus times.. Multiplicatin requires students t think in terms f grups f things rather than individual things. students learn that the multiplicatin symbl x means grups f and prblems such as 5 x 7 refer t 5 grups f 7. Students recgnize multiplicatin as a means t determine the ttal number f bjects when there are a specific number f grups with the same number f bjects in each grup. Multiplicatin requires students t think in terms f grups f things rather than individual things. Students learn that the multiplicatin symbl x means grups f and prblems such as 5 x 7 refer t 5 grups f 7 Fr example, describe a cntext in which a ttal number f bjects can be expressed as 5 7. Jim purchased 5 packages f muffins. Each package cntained 3 muffins. Hw many muffins did Jim purchase? 5 grups f 3, 5 x 3 = 15. Describe anther situatin where there wuld be 5 grups f 3 r 5 x 3. Snya earns $7 a week pulling weeds. After 5 weeks f wrk, hw much has Snya wrked? Write an equatin and find the answer. Describe anther situatin that wuld match 7x5. PARCC Clarificatin EOY Tasks invlve interpreting prducts in terms f equal grups, arrays, area, and/r measurement quantities. See CCSS Table 2, p. 89 Tasks d nt require students t interpret prducts in terms f repeated additin, skip cunting, r jumps n the number line. The italicized example refers t describing a cntext. But describing a cntext is nt the nly way t meet the standard. Fr example, anther way t meet the standard wuld be t identify cntexts in which a ttal can be expressed as a specified prduct. Sub Claim A, Task Type I (PBA & EOY) Sub Claim D, Task Type III (PBA) Area Array Multiplicatin Prduct 2. Reasn abstractly and quantitatively 4. Mdel with mathematics TEACHER NOTES See instructinal strategies in the intrductin RESOURCE NOTES See resurces in the intrductin Refer t Algebra I Live Binder m/play/play/ fr mid year evidence statements and clarificatin ASSESSMENT NOTES REQUIRED PARCC Released Test Prblems Cmmn Unit Assessment Cmmn Tasks NWEA Test Perfrmance Level Descriptrs (PARCC See assessments in the intrductin SUGGESTED Anecdtal recrds Cnferencing 8/20/ Exhibits Interviews Graphic rganizers Jurnals Mathematical Practices Mdeling Multiple Intelligences assessments, e.g. Rle playing bdily kinesthetic Graphic rganizing visual Cllabratin interpersnal Oral presentatins

10 M 3.OA.2 Interpret whle number qutients f whle numbers, e.g., interpret 56 8 as the number f bjects in each share when 56 bjects are partitined equally int 8 shares, r as a number f shares when 56 bjects are partitined int equal shares f 8 bjects each. Majr cntent This standard fcuses n tw distinct mdels f divisin: partitin mdels and measurement (repeated subtractin) mdels. Fr example, describe a cntext in which a number f shares r a number f grups can be expressed as Partitin mdels prvide students with a ttal number and the number f grups. These mdels fcus n the questin, Hw many bjects are in each grup s that the grups are equal? A cntext fr partitin mdels wuld be: There are 12 ckies n the cunter. If yu are sharing the ckies equally amng three bags, hw many ckies will g in each bag? Measurement (repeated subtractin) mdels prvide students with a ttal number and the number f bjects in each grup. These mdels fcus n the questin, Hw many equal grups can yu make? A cntext fr measurement mdels wuld be: There are 12 ckies n the cunter. If yu put 3 ckies in each bag, hw many bags will yu fill? Divisin Equal grups Grup size Partitined equally Qutients 2. Reasn abstractly and quantitatively 4. Mdel with mathematics Prblem/Perfrma nce based/cmmn tasks Research Rubrics/checklists PARCC Perfrmance Level Descriptrs District Tests and quizzes Technlgy Think aluds PARCC Clarificatin EOY Tasks invlve interpreting qutients in terms f equal grups, arrays, area, and/r measurement quantities. See CCSS Table 2, p. 89. Tasks d nt require students t interpret qutients in terms f repeated subtractin, skip cunting, r jumps n the number line. The italicized example refers t describing a cntext. But describing a cntext is nt the nly way t meet the standard. Fr example, anther way t meet the standard wuld be t identify cntexts in which a number f bjects can be expressed as a specified qutient. 50% f tasks require interpreting qutients as a number f bjects in each share. 50% f tasks require interpreting qutients as a number f equal shares. Sub Claim A, Task Type I (PBA & EOY) 8/20/2014 9

11 Sub Claim D, Task Type III (PBA) M 3.OA.3 Use multiplicatin and divisin within 100 t slve wrd prblems in situatins invlving equal grups, arrays, and measurement quantities, e.g., by using drawings and equatins with a symbl fr the unknwn number t represent the prblem. (see table belw). Majr cntent This standard references varius prblem slving cntext and strategies that students are expected t use while slving wrd prblems invlving multiplicatin & divisin. Students shuld use a variety f representatins fr creating and slving ne step wrd prblems, such as: If yu divide 4 packs f 9 brwnies amng 6 peple, hw many ckies des each persn receive? (4 x 9 = 36, 36 6 = 6). Glssary page 89, Table 2 (table als included at the end f this dcument fr yur cnvenience) gives examples f a variety f prblem slving cntexts, in which students need t find the prduct, the grup size, r the number f grups. Students shuld be given ample experiences t explre all f the different prblem structures. f multiplicatin: There are 24 desks in the classrm. If the teacher puts 6 desks in each rw, hw many rws are there? This task can be slved by drawing an array by putting 6 desks in each rw. This is an array mdel Arrays Equal grups/equal shares Equatins Unknwn 1. Make sense f prblems and persevere in slving them 4. Mdel with mathematics This task can als be slved by drawing pictures f equal grups. 4 grups f 6 equals 24 bjects A student can als reasn thrugh the prblem mentally r verbally, I knw 6 and 6 are and 12 are 24. Therefre, there are 4 grups f 6 giving a ttal f 24 desks in the classrm. A number line culd als be used t shw equal jumps. Students in third grade shuld use a variety f pictures, such as 8/20/

12 stars, bxes, flwers t represent unknwn numbers (variables). Letters are als intrduced t represent unknwns in third grade. f Divisin: There are sme students at recess. The teacher divides the class int 4 lines with 6 students in each line. Write a divisin equatin fr this stry and determine hw many students are in the class ( 4 = 6. There are 24 students in the class). Determining the number f bjects in each share (partitin mdel f divisin, where the size f the grups is unknwn): Example: The bag has 92 hair clips, and Laura and her three friends want t share them equally. Hw many hair clips will each persn receive? Determining the number f shares (measurement divisin, where the number f grups is unknwn) Example: Max the mnkey lves bananas. Mlly, his trainer, has 24 bananas. If she gives Max 4 bananas each day, hw many days will the bananas last? Slutin: The bananas will last fr 6 days. PARCC Clarificatin EOY 3.OA.3 1 Use multiplicatin within 100 (bth factrs less than r equal t 10) t slve wrd prblems in situatins invlving equal grups, arrays, r area, e.g., by using drawings and equatins with a symbl fr the unknwn number t represent the prblem. All prducts cme frm the harder three quadrants f the 8/20/

13 times table (a x b where a>5 and/r b>5). 50% f tasks invlve multiplying t find the ttal number (equal grups, arrays); 50% invlve multiplying t find the area. Fr mre infrmatin see CCSS Table 2, p. 89 and the Prgressin dcument fr Operatins and Algebraic Thinking 3.OA.3 2 Use multiplicatin within 100 (bth factrs less than r equal t 10) t slve wrd prblems in situatins invlving measurement quantities ther than area, e.g., by using drawings and equatins with a symbl fr the unknwn number t represent the prblem. All prducts cme frm the harder three quadrants f the times table ( where and/r ). Tasks invlve multiplying t find a ttal measure (ther than area). Fr mre infrmatin see CCSS Table 2, p. 89 and the Prgressin dcument fr Operatins and Algebraic Thinking 3.OA.3 3 Use divisin within 100 (qutients related t prducts having bth factrs less than r equal t 10) t slve wrd prblems in situatins invlving equal grups, arrays r area, e.g. by using drawings and equatins with a symbl fr the unknwn number t represent the prblem. All qutients are related t prducts frm the harder three quadrants f the times table ( where and/r ). A third f tasks invlve dividing t find the number in each equal grup r in each equal rw/clumn f an array; a third f tasks invlve dividing t find the number f equal grups r the number f equal rws/clumns f an array; a third f tasks invlve dividing an area by a side length t find an unknwn side length. Fr mre infrmatin see CCSS Table 2, p. 89 and the Prgressin dcument fr Operatins and Algebraic Thinking 3.OA.3 3 Use divisin within 100 (qutients related t prducts having bth factrs less than r equal t 10) t slve wrd prblems in situatins invlving measurement quantities ther than area, e.g., by using drawings and equatins with a symbl fr the unknwn number t represent the prblem. All qutients are related t prducts frm the harder three quadrants f the times table ( a x b where a > b and/r b> 5). 50% f tasks invlve finding the number f equal pieces; 50% invlve finding the measure f each piece. Fr mre infrmatin see CCSS Table 2, p. 89 and the Prgressin dcument fr Operatins and Algebraic Thinking Sub Claim A, Task Type I (PBA & EOY) Sub Claim D, Task Type III (PBA) 8/20/

14 (Great fr EOY) M 3.OA.4 Determine the unknwn whle number in a multiplicatin r divisin equatin relating three whle numbers. Majr cntent This standard is strngly cnnected t 3.OA.3 where students slve prblems and determine unknwns in equatins. Students shuld als experience creating stry prblems fr given equatins. When crafting stry prblems, they shuld carefully cnsider the questin(s) t be asked and answered t write an apprpriate equatin. Students may apprach the same stry prblem differently and write either a multiplicatin equatin r divisin equatin. Fr example, determine the unknwn number that makes the equatin true in each f the equatins 8? = 48, 5 = 3, 6 6 =?. Students apply their understanding f the meaning f the equal sign as the same as t interpret an equatin with an 8/20/

15 unknwn. When given 4 x? = 40, they might think: 4 grups f sme number is the same as 40 4 times sme number is the same as 40 I knw that 4 grups f 10 is 40 s the unknwn number is 10 The missing factr is 10 because 4 times 10 equals 40. Equatins in the frm f a x b = c and c = a x b shuld be used interchangeably, with the unknwn in different psitins. : Slve the equatins belw: 24 =? x 6 72 = 9 Rachel has 3 bags. There are 4 marbles in each bag. Hw many marbles des Rachel have altgether? 3 x 4 = m Students may use interactive whitebards t create digital mdels t explain and justify their thinking PARCC Clarificatin EOY Tasks d nt have a cntext. Only the answer is required (methds, representatins, etc. are nt assessed here). All prducts and related qutients are frm the harder three quadrants f the times table( a x b where a > 5 and/r b > 5). Sub Claim A, Task Type I (PBA & EOY) Sub Claim D, Task Type III (PBA) OPERATIONS AND ALGEBRAIC THINKING (3.OA) Use Mathematical Practices t 1. Make sense f prblems and persevere in slving them 2. Reasn abstractly and quantitatively 3. Cnstruct viable arguments and critique the reasning f thers 4. Mdel with mathematics 5. Use apprpriate tls strategically 6. Attend t precisin 7. Lk fr and make use f structure 8. Lk fr and express regularity in repeated reasning M Students understand prperties f multiplicatin and the relatinship between multiplicatin and divisin. 3. OA.5 Apply prperties f peratins as strategies t multiply and divide. Majr cntent Students represent expressins using varius bjects, pictures, wrds and symbls in rder t develp their understanding f prperties. They multiply by 1 and 0 and divide by 1. They change the rder f numbers t determine that the rder f numbers factrs des nt make a difference in multiplicatin (but des make a difference in divisin). Given three factrs, they investigate changing the rder f hw they multiply the numbers t determine that changing the rder f the factrs des nt change the prduct. They Divide Factr Multiply Operatin Prperties (cmmunative, assciative, distributive) TEACHER NOTES See instructinal strategies in the intrductin Students need nt use frmal terms fr these prperties Students need t apply prperties f peratins (cmmutative, assciative and distributive) as strategies t multiply and divide. Applying the cncept invlved is mre imprtant than students knwing the name f the prperty. Understanding the cmmutative prperty f RESOURCE NOTES See resurces in the intrductin Refer t Algebra I Live Binder m/play/play/ fr mid year evidence statements and clarificatin ASSESSMENT NOTES REQUIRED PARCC Released Test Prblems Cmmn Unit Assessment Cmmn Tasks NWEA Test Perfrmance Level Descriptrs (PARCC See assessments in the intrductin 8/20/

16 als decmpse numbers t build fluency with multiplicatin., if 6 4 = 24 is knwn, then 4 6 = 24 is als knwn. (Cmmutative prperty f multiplicatin.) can be fund by 3 5 = 15, then 15 2 = 30, r by 5 2 = 10, then 3 10 = 30. (Assciative prperty f multiplicatin.) Knwing that 8 5 = 40 and 8 2 = 16, ne can find 8 7 as 8 (5 + 2) = (8 5) + (8 2) = = 56. (Distributive prperty.) Mdels help build understanding f the cmmutative prperty: Example: 3 x 6 = 6 x 3 In the fllwing diagram it may nt be bvius that 3 grups f 6 is the same as 6 grups f 3. A student may need t cunt t verify this. is the same quantity as Example: 4 x 3 = 3 x 4 An array explicitly demnstrates the cncept f the cmmutative prperty. multiplicatin is develped thrugh the use f mdels as basic multiplicatin facts are learned. Fr example, the result f multiplying 3 x 5 (15) is the same as the result f multiplying 5 x 3 (15). T find the prduct f three numbers, students can use what they knw abut the prduct f tw f the factrs and multiply this by the third factr. Fr example, t multiply 5 x 7 x 2, students knw that 5 x 2 is 10. Then, they can use mental math t find the prduct f 10 x 7 (70). Allw students t use their wn strategies and share with the class when applying the assciative prperty f multiplicatin. Splitting arrays can help students understand the distributive prperty. They can use a knwn fact t learn ther facts that may cause difficulty. Fr example, students can split a 6 x 9 array int 6 grups f 5 and 6 grups f 4; then, add the sums f the grups 4 rws f 3, r 4 x 3 3 rws f 4, r 3 x 4 Students are intrduced t the distributive prperty f multiplicatin ver additin as a strategy fr using prducts they knw t slve prducts they dn t knw. Fr example, if students are asked t find the prduct f 7 x 8, they might decmpse 7 int 5 and 2 and then multiply 5 x 8 and 2 x 8 t arrive at r 56. Students shuld learn that they can decmpse either f the factrs. It is imprtant t nte that the students may recrd their thinking in different ways 5 x 8 = 40 2 x 8 = 16 The 6 grups f 5 is 30 and the 6 grups f 4 is 24. Students can write 6 x 9 as 6 x x 4. Students understanding f the part/whle relatinships is critical in understanding the cnnectin between multiplicatin and divisin. ODE 8/20/

17 56 7 x 4 = 28 7 x 4 = T further develp understanding f prperties related t multiplicatin and divisin, students use different representatins and their understanding f the relatinship between multiplicatin and divisin t determine if the fllwing types f equatins are true r false. 0 x 7 = 7 x 0 = 0 (Zer Prperty f Multiplicatin) 1 x 9 = 9 x 1 = 9 (Multiplicative Identity Prperty f 1) 3 x 6 = 6 x 3 (Cmmutative Prperty) 8 2 = 2 8 (Students are nly t determine that these are nt equal) 2 x 3 x 5 = 6 x 5 10 x 2 < 5 x 2 x 2 2 x 3 x 5 = 10 x 3 0 x 6 > 3 x 0 x 2 PARCC Clarificatin EOY NONE Sub Claim, Task Type (EOY) Sub Claim C, Task Type II (PBA) M 3. OA.6 Understand divisin as an unknwn factr prblem. Majr cntent Multiplicatin and divisin are inverse peratins and that understanding can be used t find the unknwn. Fact family triangles demnstrate the inverse peratins f multiplicatin and divisin by shwing the tw factrs and hw thse factrs relate t the prduct and/r qutient. Inverse Unknwn factr Fr example, find 32 8 by finding the number that makes 32 when multiplied by 8. 8/20/

18 : 3 x 5 = 15 5 x 3 = = = 3 Students use their understanding f the meaning f the equal sign as the same as t interpret an equatin with an unknwn. When given 32 = 4, students may think: 4 grups f sme number is the same as 32 4 times sme number is the same as 32 I knw that 4 grups f 8 is 32 s the unknwn number is 8 The missing factr is 8 because 4 times 8 is 32. Equatins in the frm f a b = c and c = a b need t be used interchangeably, with the unknwn in different psitins. PARCC Clarificatin EOY Sub Claim A, Task Type I (PBA & EOY) Sub Claim C, Task Type II (PBA) Sub Claim D, Task Type III (PBA) ms_mathematics_g3vansfrfieldtrip_081513_final.pdf OPERATIONS AND ALGEBRAIC THINKING (3.OA) Use Mathematical Practices t 1. Make sense f prblems and persevere in slving them 2. Reasn abstractly and quantitatively 3. Cnstruct viable arguments and critique the reasning f thers 4. Mdel with mathematics 5. Use apprpriate tls strategically 6. Attend t precisin 7. Lk fr and make use f structure 8. Lk fr and express regularity in repeated reasning M Students multiply and divide within OA.7 Fluently multiply and divide within 100, using strategies such as the relatinship between multiplicatin and divisin. Majr cntent By the end f Grade 3, knw frm memry all prducts f tw ne digit numbers.. By studying patterns and relatinships in multiplicatin facts Strategies and relating multiplicatin and divisin, students build a fundatin fr fluency with multiplicatin and divisin facts. Students demnstrate fluency with multiplicatin facts thrugh _ 10 and the related divisin facts. Multiplying and dividing fluently refers t knwledge f prcedures, knwledge f when and hw t use them apprpriately, and skill in perfrming them flexibly, accurately, and efficiently. Fr example, knwing that 8 5 = 40, ne knws 40 5 = 8) r prperties f peratins. T EACHER NOTES See instructinal strategies in the intrductin Students need t understand the part/whle relatinships in rder t understand the cnnectin between multiplicatin and divisin. They need t develp efficient strategies that lead t the big ideas f multiplicatin and divisin. These big ideas include understanding the prperties f peratins, such as the cmmutative and assciative prperties f multiplicatin and the distributive prperty. The naming f the prperty is nt RESOURCE NOTES See resurces in the intrductin Refer t Algebra I Live Binder m/play/play/ fr mid year evidence statements and clarificatin ASSESSMENT NOTES REQUIRED PARCC Released Test Prblems Cmmn Unit Assessment Cmmn Tasks NWEA Test Perfrmance Level Descriptrs (PARCC See assessments in the intrductin 8/20/

19 Strategies students may use t attain fluency include: Multiplicatin by zers and nes Dubles (2s facts), Dubling twice (4s), Dubling three times (8s) Tens facts (relating t place value, 5 x 10 is 5 tens r 50) Five facts (half f tens) Skip cunting (cunting grups f and knwing hw many grups have been cunted) Square numbers (ex: 3 x 3) Nines (10 grups less ne grup, e.g., 9 x 3 is 10 grups f 3 minus ne grup f 3) Decmpsing int knwn facts (6 x 7 is 6 x 6 plus ne mre grup f 6) Turn arund facts (Cmmutative Prperty) Fact families (Ex: 6 x 4 = 24; 24 6 = 4; 24 4 = 6; 4 x 6 = 24) Missing factrs General Nte: Students shuld have expsure t multiplicatin and divisin prblems presented in bth vertical and hrizntal frms. Knw frm memry shuld nt fcus nly n timed tests and repetitive practice, but ample experiences wrking with manipulatives, pictures, arrays, wrd prblems, and numbers t internalize the basic facts. PARCC Clarificatin EOY Tasks d nt have a cntext. Only the answer is required (strategies, representatins, etc., are nt assessed here). Tasks require fluent (fast and accurate) finding f prducts and related qutients. Fr example, each ne pint task might require fur r mre cmputatins, tw r mre multiplicatin and tw r mre divisin. 75% f tasks are frm the harder three quadrants f the times table (a x b where a > 5 and/r b >5) Sub Claim, Task Type (PBA & EOY) 20Sample%20Prblems_GR3FluencyV2.pdf necessary at this stage f learning. Instructinal Strategies In Grade 2, students fund the ttal number f bjects using rectangular arrays, such as a 5 x 5, and wrte equatins t represent the sum. This is called unitizing, and it requires students t cunt grups, nt just bjects. They see the whle as a number f grups f a number f bjects. This strategy is a fundatin fr multiplicatin in that students shuld make a cnnectin between repeated additin and multiplicatin. As students create arrays fr multiplicatin using bjects r drawing n graph paper, they may discver that three grups f fur and fur grups f three yield the same results. They shuld bserve that the arrays stay the same, althugh hw they are viewed changes. Prvide numerus situatins fr students t develp this understanding. T develp an understanding f the distributive prperty, students need decmpse the whle int grups. Arrays can be used t develp this understanding. T find the prduct f 3 9, students can decmpse 9 int the sum f 4 and 5 and find 3 (4 + 5). The distributive prperty is the basis fr the standard multiplicatin algrithm that students can use t fluently 8/20/

20 multiply multi digit whle numbers in Grade 5. Once students have an understanding f multiplicatin using efficient strategies, they shuld make the cnnectin t divisin. Using varius strategies t slve different cntextual prblems that use the same tw ne digit whle numbers requiring multiplicatin allws fr students t cmmit t memry all prducts f tw ne digit numbers. ODE OPERATIONS AND ALGEBRAIC THINKING (3.OA) Use Mathematical Practices t 1. Make sense f prblems and persevere in slving them 2. Reasn abstractly and quantitatively 3. Cnstruct viable arguments and critique the reasning f thers 4. Mdel with mathematics 5. Use apprpriate tls strategically 6. Attend t precisin 7. Lk fr and make use f structure 8. Lk fr and express regularity in repeated reasning M Students slve prblems invlving the fur peratins, and identify and explain patterns in arithmetic. 3. OA.8 Slve tw step wrd prblems using the fur peratins. Represent these prblems using equatins with a letter standing fr the unknwn quantity. Assess the reasnableness f answers using mental cmputatin and estimatin strategies including runding. Majr cntent Students shuld be expsed t multiple prblem slving strategies (using any cmbinatin f wrds, numbers, diagrams, physical bjects r symbls) and be able t chse which nes t use. Jerry earned 231 pints at schl last week. This week he earned 79 pints. If he uses 60 pints t earn free time n a cmputer, hw many pints will he have left? A student may use the number line abve t describe his/her thinking, = 240 s nw I need t add 70 mre. 240, 250 (10 mre), 260 (20 mre), 270, 280, 290, 300, 310 (70 mre). Nw I need t cunt back , 300 (back 10), 290 (back 20), 280, 270, 260, 250 (back 60). Equatins Estimatin Mental cmputatin Runding Unknwn quantity 1. Make sense f prblems and persevere in slving them 4. Mdel with mathematics TEACHER NOTES See instructinal strategies in the intrductin This standard is limited t prblems psed with whle numbers and having whle number answers; students shuld knw hw t perfrm peratins in the cnventinal rder when there are n parentheses t specify a particular rder (Order f Operatins). Students gain a full understanding f which peratin t use in any given situatin thrugh cntextual prblems. Number skills and cncepts are develped as students slve prblems. Prblems shuld be presented n a regular basis as students wrk with numbers and cmputatins. Researchers and mathematics educatrs advise against prviding key wrds fr students t lk RESOURCE NOTES See resurces in the intrductin Refer t Algebra I Live Binder m/play/play/ fr mid year evidence statements and clarificatin ASSESSMENT NOTES REQUIRED PARCC Released Test Prblems Cmmn Unit Assessment Cmmn Tasks NWEA Test Perfrmance Level Descriptrs (PARCC See assessments in the intrductin 8/20/

21 A student writes the equatin, = m and uses runding ( ) t estimate. A student writes the equatin, = m and calculates = 19 and then calculates = m. The sccer club is ging n a trip t the water park. The cst f attending the trip is $63. Included in that price is $13 fr lunch and the cst f 2 wristbands, ne fr the mrning and ne fr the afternn. Write an equatin representing the cst f the field trip and determine the price f ne wristband. W W The abve diagram helps the student write the equatin, w + w + 13 = 63. Using the diagram, a student might think, I knw that the tw wristbands cst $50 ($63 $13) s ne wristband csts $25. T check fr reasnableness, a student might use frnt end estimatin and say = 50 and 50 2 = 25. When students slve wrd prblems, they use varius estimatin skills which include identifying when estimatin is apprpriate, determining the level f accuracy needed, selecting the apprpriate methd f estimatin, and verifying slutins r determining the reasnableness f slutins. Estimatin strategies include, but are nt limited t: using benchmark numbers that are easy t cmpute frnt end estimatin with adjusting (using the highest place value and estimating frm the frnt end making adjustments t the estimate by taking int accunt the remaining amunts) runding and adjusting (students rund dwn r rund up and then adjust their estimate depending n hw much the runding changed the riginal values) fr in prblem situatins because they can be misleading. Students shuld use varius strategies t slve prblems. Students shuld analyze the structure f the prblem t make sense f it. They shuld think thrugh the prblem and the meaning f the answer befre attempting t slve it. Encurage students t represent the prblem situatin in a drawing r with cunters r blcks. Students shuld determine the reasnableness f the slutin t all prblems using mental cmputatins and estimatin strategies. Students can use base ten blcks n centimeter grid paper t cnstruct rectangular arrays t represent prblems. Students are t identify arithmetic patterns and explain them using prperties f peratins. They can explre patterns by determining likenesses, differences and changes. Use patterns in additin and multiplicatin tables. ODE PARCC Clarificatin EOY Only the answer is required (methds, representatins, etc., are nt assessed here).ii) Additin, subtractin, multiplicatin and divisin situatins in these prblems may invlve any f the basic situatin types with unknwns in varius psitins (see CCSS Tables 1 2, p. 88 and the Prgressin dcument fr Operatins and Algebraic Thinking.) If scafflded, ne f the 2 parts must require 2 steps. The ther part many cnsist f 1 step. Cnversins shuld be part f the 2 steps and shuld nt be a 8/20/

22 step n its wn. If the item is 2 pints, the item shuld be a 2 pint, unscafflded item but the rubric shuld allw fr pints. Sub Claim A, Task Type I (PBA & EOY) Sub Claim C, Task Type II (PBA) Sub Claim D, Task Type III (PBA) gulararray_0.pdf M 3. OA.9 Identify arithmetic patterns (including patterns in the additin table r multiplicatin table), and explain them using prperties f peratins. Majr cntent This standard calls fr students t examine arithmetic patterns invlving bth additin and multiplicatin. Arithmetic patterns are patterns that change by the same rate, such as adding the same number. Fr example, the series 2, 4, 6, 8, 10 is an arithmetic pattern that increases by 2 between each term. Fr example, bserve that 4 times a number is always even, and explain why 4 times a number can be decmpsed int tw equal addends. This standards als mentins identifying patterns related t the prperties f peratins. : Even numbers are always divisible by 2. Even numbers can always be decmpsed int 2 equal addends (14 = 7 + 7). Multiples f even numbers (2, 4, 6, and 8) are always even numbers. On a multiplicatin chart, the prducts in each rw and clumn increase by the same amunt (skip cunting). On an additin chart, the sums in each rw and clumn increase by the same amunt. What d yu ntice abut the numbers highlighted in pink in the multiplicatin table? Explain a pattern using prperties f peratins. When (cmmutative prperty) ne changes the rder f the factrs they will still gets the same prduct, example Patterns Prperties f peratin 1. Make sense f prblems and persevere in slving them 2. Reasn abstractly and quantitatively 4. Mdel with mathematics 7. Lk fr and make use f structure 8/20/

23 6 x 5 = 30 and 5 x 6 = 30. Teacher: What pattern d yu ntice when 2, 4, 6, 8, r 10 are multiplied by any number (even r dd)? Student: The prduct will always be an even number. Teacher: Why? What patterns d yu ntice in this additin table? Explain why the pattern wrks this way? 8/20/

24 Students need ample pprtunities t bserve and identify imprtant numerical patterns related t peratins. They shuld build n their previus experiences with prperties related t additin and subtractin. Students investigate additin and multiplicatin tables in search f patterns and explain why these patterns make sense mathematically. Example: Any sum f tw even numbers is even. Any sum f tw dd numbers is even. Any sum f an even number and an dd number is dd. The multiples f 4, 6, 8, and 10 are all even because they can all be decmpsed int tw equal grups. The dubles (2 addends the same) in an additin table fall n a diagnal while the dubles (multiples f 2) in a multiplicatin table fall n hrizntal and vertical lines. The multiples f any number fall n a hrizntal and a vertical line due t the cmmutative prperty. All the multiples f 5 end in a 0 r 5 while all the multiples f 10 end with 0. Every ther multiple f 5 is a multiple f 10. Students als investigate a hundreds chart in search f additin and subtractin patterns. They recrd and rganize all the different pssible sums f a number and explain why the pattern makes sense. PARCC Clarificatin EOY Sub Claim, Task Type (EOY) Sub Claim C, Task Type II (PBA) 8/20/

25 NUMBER AND OPERATIONS IN BASE TEN (3.NBT) Use Mathematical Practices t 1. Make sense f prblems and persevere in slving them 2. Reasn abstractly and quantitatively 3. Cnstruct viable arguments and critique the reasning f thers 4. Mdel with mathematics 5. Use apprpriate tls strategically 6. Attend t precisin 7. Lk fr and make use f structure 8. Lk fr and express regularity in repeated reasning A A Students use place value understanding and prperties f peratins t perfrm multi digit arithmetic. 3.NBT.1 Use place value understanding t rund whle numbers t the nearest 10 r Additinal cntent There are patterns when multiplying by multiples f 10. Students learn when and why t rund numbers. They identify pssible answers and halfway pints. Then they narrw where the given number falls between the pssible answers and halfway pints. They als understand that by cnventin if a number is exactly at the halfway pint f the tw pssible answers, the number is runded up. Example: Rund 178 t the nearest 10. Step 1: The answer is either 170 r 180. Step 2: The halfway pint is 175. Step 3: 178 is between 175 and 180. Step 4: Therefre, the runded number is 180. PARCC Clarificatin EOY Nne Sub Claim, Task Type (EOY) Place value Rund Whle number Mathematical Practices 3.NBT.2 Fluently add and subtract within 1000 using strategies and algrithms based n place value, prperties f peratins, and/r the relatinship between additin and subtractin. Additinal cntent Prblems shuld include bth vertical and hrizntal frms, including pprtunities fr students t apply the cmmutative and assciative prperties. Adding and subtracting fluently refers t knwledge f prcedures, Additin Algrithm Place value TEACHER NOTES See instructinal strategies in the intrductin A range f algrithms may be used. Prir t implementing rules fr runding students need t have pprtunities t investigate place value. A strng understanding f place value is essential fr the develped number sense and the subsequent wrk that invlves runding numbers. Instructinal Strategies Building n previus understandings f the place value f digits in multi digit numbers, place value is used t rund whle numbers. Dependence n learning rules can be eliminated with strategies such as the use f a number line t determine which multiple f 10 r f100, a number is nearest (5 r mre runds up, less than 5 runds dwn). As students understanding f place value increases, the strategies fr runding are valuable fr estimating, justifying and predicting the reasnableness f slutins in prblemslving. Strategies used t add and subtract tw digit numbers are nw applied t fluently add and subtract whle numbers within These strategies shuld be discussed s that students can make cmparisns and mve tward efficient methds. Number sense and cmputatinal understanding RESOURCE NOTES See resurces in the intrductin Refer t Algebra I Live Binder m/play/play/ fr mid year evidence statements and clarificatin ASSESSMENT NOTES REQUIRED PARCC Released Test Prblems Cmmn Unit Assessment Cmmn Tasks NWEA Test Perfrmance Level Descriptrs (PARCC See assessments in the intrductin 8/20/

26 knwledge f when and hw t use them apprpriately, and skill in perfrming them flexibly, accurately, and efficiently. Students explain their thinking and shw their wrk by using strategies and algrithms, and verify that their answer is reasnable. An interactive whitebard r dcument camera may be used t shw and share student thinking. Mary read 573 pages during her summer reading challenge. She was nly required t read 399 pages. Hw many extra pages did Mary read beynd the challenge requirements? Students may use several appraches t slve the prblem including the traditinal algrithm. f ther methds students may use are listed belw: = 400, = 500, = 573, therefre = 174 pages (Adding up strategy) is 500; is 573; is 173 plus 1 (fr 399, t 400) is 174 (Cmpensating strategy) Take away 73 frm 573 t get t 500, take away 100 t get t 400, and take away 1 t get t 399. Then = 174 (Subtracting t cunt dwn strategy) is 400, 500 (that s 100 mre). 510, 520, 530, 540, 550, 560, 570, (that s 70 mre), 571, 572, 573 (that s 3 mre) s the ttal is = 174 (Adding by tens r hundreds strategy) PARCC Clarificatin EOY Tasks have n cntext. Sub Claim, Task Type (EOY) Prperties f peratin Subtractin Mathematical Practices is built n a firm understanding f place value. Understanding what each number in a multiplicatin expressin represents is imprtant. Multiplicatin prblems need t be mdeled with pictures, diagrams r cncrete materials t help students understand what the factrs and prducts represent. The effect f multiplying numbers needs t be examined and understd. The use f area mdels is imprtant in understanding the prperties f peratins f multiplicatin and the relatinship f the factrs and its prduct. Cmpsing and decmpsing area mdels is useful in the develpment and understanding f the distributive prperty in multiplicatin. Cntinue t use manipulative like hundreds charts and place value charts. Have students use a number line r a rller caster example t blck ff the numbers in different clrs. Fr example this chart shw what numbers will rund t the tens place. A 3.NBT.3 Multiply ne digit whle numbers by multiples f 10 in the range (e.g., 9 80, 5 60) using strategies based n place value and prperties f peratins. Additinal cntent There are patterns when multiplying by multiples f 10. The special rle f 10 in the base ten system is imprtant in understanding multiplicatin f ne digit numbers with multiples f 10. Fr example, the prduct 3 x 50 can be represented as 3 grups f 5 tens, which is 15 tens, which is Multiples f ten Multiply Place value Prperties f peratin Whle numbers 8/20/

27 150. This reasning relies n the assciative prperty f multiplicatin: 3 x 50 = 3 x (5 x 10) = (3 x 5) x 10 = 15 x 10 = 150. It is an example f hw t explain an instance f a calculatin pattern fr these prducts: calculate the prduct f the nn zer digits, and then shift the prduct ne place t the left t make the result ten times as large 7. Lk fr and make use f structure PARCC Clarificatin EOY Tasks have n cntext. Sub Claim B, Task Type I (EOY) NUMBER AND OPERATIONS FRACTIONS (3.NF) Use Mathematical Practices t 1. Make sense f prblems and persevere in slving them 2. Reasn abstractly and quantitatively 3. Cnstruct viable arguments and critique the reasning f thers 4. Mdel with mathematics 5. Use apprpriate tls strategically 6. Attend t precisin 7. Lk fr and make use f structure 8. Lk fr and express regularity in repeated M Students develp understanding f fractins as numbers 3.NF.1 Understand a fractin 1/b as the quantity frmed by 1 part when a whle is partitined int b equal parts; understand a fractin a/b as the quantity frmed by a parts f size 1/b. Majr cntent. This standard refers t the sharing f a whle being partitined. Fractin mdels in third grade include nly area (parts f a whle) mdels (circles, rectangles, squares) and number lines. Set mdels (parts f a grup) are nt addressed in Third Grade. In 3.NF.1 students start with unit fractins (fractins with numeratr 1), which are frmed by partitining a whle int Denminatr Equals parts Fractin Numeratr Partitined Quantity Whle TEACHER NOTES This is the initial experience students will have with fractins and is best dne ver time. Students need many pprtunities t discuss fractinal parts using cncrete mdels t develp familiarity and understanding f fractins. Expectatins in this dmain are limited t fractins with denminatrs 2, 3, 4, 6 and 8. Instructinal Strategies Understanding that a fractin is a quantity frmed by part RESOURCE NOTES See resurces in the intrductin Refer t Algebra I Live Binder m/play/play/ fr mid year evidence statements and clarificatin ASSESSMENT NOTES REQUIRED PARCC Released Test Prblems Cmmn Unit Assessment Cmmn Tasks NWEA Test Perfrmance Level Descriptrs (PARCC See assessments in the intrductin 8/20/

28 reasning equal parts and reasning abut ne part f the whle, e.g., if a whle is partitined int 4 equal parts then each part is. f the whle, and 4 cpies f that part make the whle. Next, students build fractins frm unit fractins, seeing the numeratr 3 f. as saying that. is the quantity yu get by putting 3 f the. s tgether. There is n need t intrduce imprper fractins" initially. Sme imprtant cncepts related t develping understanding f fractins include: Understand fractinal parts must be equal sized. Example Nn example These are thirds These are NOT thirds 2. Reasn abstractly and quantitatively f a whle is essential t number sense with fractins. Fractinal parts are the building blcks fr all fractin cncepts. Students need t relate dividing a shape int equal parts and representing this relatinship n a number line, where the equal parts are between tw whle numbers. Help students plt fractins n a number line, by using the meaning f the fractin. Fr example, t plt 4/5 n a number line, there are 5 equal parts with 4 cpies f the 5 equal parts. As students cunted with whle numbers, they shuld als cunt with fractins. Cunting equal sized parts helps students determine the number f parts it takes t make a whle and recgnize fractins that are equivalent t whle numbers. The number f equal parts tell hw many make a whle. As the number f equal pieces in the whle increases, the size f the fractinal pieces decreases. The size f the fractinal part is relative t the whle. One half f a small pizza is relatively smaller than ne half f a large pizza. When a whle is cut int equal parts, the denminatr represents the number f equal parts. The numeratr f a fractin is the cunt f the number f equal parts. 3 means that there are 3 ne furths. 4 Students can cunt ne furth, tw furths, three furths. Students express fractins as fair sharing r, parts f a whle. They use varius cntexts (candy bars, fruit, and cakes) and a variety f mdels (circles, squares, rectangles, fractin bars, and number lines) t develp understanding f fractins and represent fractins. Students need many pprtunities t slve Students need t knw hw big a particular fractin is and can easily recgnize which f tw fractins is larger. The fractins must refer t parts f the same whle. Benchmarks such as 1/2 and 1 are als useful in cmparing fractins. Equivalent fractins can be recgnized and generated using fractin mdels. Students shuld use different mdels and decide when t use a particular mdel. Make transparencies t shw hw equivalent fractins measure up n the number line. 8/20/

29 wrd prblems that require them t create and reasn abut fair share. Initially, students can use an intuitive ntin f same size and same shape (cngruence) t explain why the parts are equal, e.g., when they divide a square int fur equal squares r fur equal rectangles. Students cme t understand a mre precise meaning fr equal parts as parts with equal measurements. Fr example, when a ruler is partitined int halves r quarters f an inch, they see that each subdivisin has the same length. In area mdels they reasn abut the area f a shaded regin t decide what fractin f the whle it represents. Venn diagrams are useful in helping students rganize and cmpare fractins t determine the relative size f the fractins, such as mre than 1/2, exactly 1/2 r less than 1/2. Fractin bars shwing the same sized whle can als be used as mdels t cmpare fractins. Students are t write the results f the cmparisns with the symbls >, =, r <, and justify the cnclusins with a mdel. M PARCC Clarificatin EOY Tasks d nt invlve the number line. Sub Claim A, Task Type I (PBA & EOY) Sub Claim D, Task Type III (PBA) 20Sample%20Prblems_GR3_The%20Field_PartAV2.pdf 3.NF.2 Understand a fractin as a number n the number line; represent fractins n a number line diagram. Majr cntent a. Represent a fractin 1/b n a number line diagram by defining the interval frm 0 t 1 as the whle and partitining it int b equal parts. Recgnize that each part has size 1/b and that the endpint f the part based at 0 lcates the number 1/b n the number line. 3.NF.2a Students transfer their understanding f parts f a whle t partitin a number line int equal parts. In the number line diagram belw, the space between 0 and 1 is divided (partitined) int 4 equal regins. The distance frm 0 t the first segment is 1 f the 4 segments frm 0 t 1 r. (3.NF.2a). Similarly, the distance frm 0 t the third segment is 3 segments that are each ne End pint Equal parts Interval (equal distance) Number line Partitin Whle part The idea that the smaller the denminatr, the smaller the piece r part f the set, r the larger the denminatr, the larger the piece r part f the set, is based n the cmparisn that in whle numbers, the smaller a number, the less it is, r the larger a number, the mre it is. The use f different mdels, such as fractin bars and number lines, allws students t cmpare unit fractins t reasn abut their sizes. Cmmn Miscnceptins Students think all shapes can be divided the same way. Present shapes ther than circles, squares r rectangles t prevent students frm vergeneralizing that all shapes can be divided the same way. Fr example, have students fld a triangle int eighths. Prvide ral directins fr flding the triangle: 1. Fld the triangle int half by flding the left vertex (at the base f the triangle) ver t meet the right vertex. 8/20/

30 furth lng. Therefre, the distance f 3 segments frm 0 is the fractin. (3.NF.2b). 5. Use apprpriate tls strategically 2. Fld in this manner tw mre times. 3. Have students label each eighth using fractinal ntatin. Then, have students cunt the fractinal parts in the triangle (ne eighth, tweighths, three eighths, and s n). ODE (Prgressins fr the CCSSM, Number and Operatin Fractins, CCSS Writing Team, August 2011, page 3) PARCC Clarificatin EOY Fractins may be greater than 1. Fractins equal whle numbers in 20% f these tasks. Tasks have thin cntext r n cntext. Tasks are limited t fractins with denminatrs 2, 3, 4, 6, and 8. (See ftnte CCSS p 24) Sub Claim A, Task Type I (PBA & EOY) Sub Claim C, Task Type II (PBA) Sub Claim D, Task Type III (PBA) b. Represent a fractin a/b n a number line diagram by marking ff a lengths 1/b frm 0. Recgnize that the resulting interval has size a/b and that its endpint lcates the number a/b n the number line. 3.NF.2b Students transfer their understanding f parts f a whle t partitin a number line int equal parts. See abve 3.NF.2a End pint Equal parts Interval (equal distance) Number line Whle part PARCC Clarificatin EOY Fractins may be greater than 1. Fractins equal whle numbers in 20% f these tasks. 8/20/

31 Tasks have thin cntext r n cntext. Tasks are limited t fractins with denminatrs 2, 3, 4, 6, and 8. (See ftnte CCSS p 24) Sub Claim A, Task Type I (PBA & EOY) Sub Claim C, Task Type II (PBA) Sub Claim D, Task Type III (PBA) 20Sample%20Prblems_GR3_Frac Num LineV2.pdf 5. Use apprpriate tls strategically M 3.NF.3 Explain equivalence f fractins in special cases, and cmpare fractins by reasning abut their size. Majr cntent a. Understand tw fractins as equivalent (equal) if they are the same size, r the same pint n a number line. 3.NF.3a An imprtant cncept when cmparing fractins is t lk at the size f the parts and the number f the parts. Fr example, 1 is smaller than 1 because when 1 whle is cut 8 2 int 8 pieces, the pieces are much smaller than when1 whle is cut int 2 pieces. These standards call fr students t use visual fractin mdels (area mdels) and number lines t explre the idea f equivalent fractins. Students shuld nly explre equivalent fractins using mdels, rather than using algrithms r prcedures. Cmpare Equal Equivalence Equivalent Fractin Pint 5. Use apprpriate tls strategically This standard includes writing whle numbers as fractins. The cncept relates t fractins as divisin prblems, where the fractin 3/1 is 3 whles divided int ne grup. This standard is the building blck fr later wrk where students divide a set f bjects int a specific number f grups. Students must understand the meaning f a/1. Example: If 6 brwnies are shared between 2 peple, hw many brwnies wuld each persn get? This standard invlves cmparing fractins with r withut visual fractin mdels including number lines. Experiences shuld encurage students t reasn abut the size f pieces, the fact that 1/3 f a cake is larger than. f the same cake. Since the same cake (the whle) is split int equal pieces, 8/20/

32 thirds are larger than furths. In this standard, students shuld als reasn that cmparisns are nly valid if the whles are identical. Fr example, ½ f a large pizza is a different amunt than. f a small pizza. Students shuld be given pprtunities t discuss and reasn abut which. is larger. Previusly, in secnd grade, students cmpared lengths using a standard measurement unit. In third grade they build n this idea t cmpare fractins with the same denminatr. They see that fr fractins that have the same denminatr, the underlying unit fractins are the same size, s the fractin with the greater numeratr is greater because it is made f mre unit fractins. Fr example, segment frm 0 t. is shrter than the segment frm 0 t 5/4 because it measures 3 units f. as ppsed t 5 units f., therefre. < 5/4. PARCC Clarificatin EOY 3.NF.3a 1a. Understand tw fractins as equivalent (equal) if they are the same size Tasks d nt invlve the number line. Tasks are limited t fractins with denminatrs 2, 3, 4, 6 and 8. (See ftnte CCSS p 24) The explanatin aspect f 3.NF.3 is nt assessed here. 3.NF.3a 2a. Understand tw fractins as equivalent (equal) if they are the same pint n a number line. Tasks are limited t fractins with denminatrs 2, 3, 4, 6, and 8. (See ftnte CCSS p 24) The explanatin aspect f 3.NF.3 is nt assessed here. Sub Claim A, Task Type I (PBA & EOY) Sub Claim D, Task Type III (PBA) b. Recgnize and generate simple equivalent fractins, e.g., 1/2 = 2/4, 4/6 = 2/3). Explain why the fractins are equivalent, e.g., by using a visual fractin mdel. 3.NF.3b An imprtant cncept when cmparing fractins is t lk at the size f the parts and the number f the parts. Fr example, 1 is smaller than 1 because when 1 whle is cut 8 2 int 8 pieces, the pieces are much smaller than when1 whle is cut int 2 pieces. Cmpare Equal Equivalence Equivalent Fractin Pint These standards call fr students t use visual fractin mdels 8/20/

33 (area mdels) and number lines t explre the idea f equivalent fractins. Students shuld nly explre equivalent fractins using mdels, rather than using algrithms r prcedures. This standard includes writing whle numbers as fractins. The cncept relates t fractins as divisin prblems, where the fractin 3/1 is 3 whles divided int ne grup. This standard is the building blck fr later wrk where students divide a set f bjects int a specific number f grups. Students must understand the meaning f a/1. Example: If 6 brwnies are shared between 2 peple, hw many brwnies wuld each persn get? This standard invlves cmparing fractins with r withut visual fractin mdels including number lines. Experiences shuld encurage students t reasn abut the size f pieces, the fact that 1/3 f a cake is larger than. f the same cake. Since the same cake (the whle) is split int equal pieces, thirds are larger than furths. In this standard, students shuld als reasn that cmparisns are nly valid if the whles are identical. Fr example, ½ f a large pizza is a different amunt than. f a small pizza. Students shuld be given pprtunities t discuss and reasn abut which. is larger. Previusly, in secnd grade, students cmpared lengths using a standard measurement unit. In third grade they build n this idea t cmpare fractins with the same denminatr. They see that fr fractins that have the same denminatr, the underlying unit fractins are the same size, s the fractin with the greater numeratr is greater because it is made f mre unit fractins. Fr example, segment frm 0 t. is shrter than the segment frm 0 t 5/4 because it measures 3 units f. as ppsed t 5 units f., therefre. < 5/4. Students als see that fr unit fractins, the ne with the larger denminatr is smaller, by reasning, fr example, that in rder fr mre (identical) pieces t make the same whle, the pieces must be smaller. Frm this they reasn that fr fractins that have the same numeratr, the fractin with the smaller denminatr is greater. Fr example, 2/5 > 2/7, because 1/7 < 1/5, s 2 lengths f 1/7 is less than 2 lengths f 1/5. As with equivalence f fractins, it is imprtant in cmparing fractins t make sure that each fractin refers t the same whle. 7. Lk fr and make use f structure 8/20/

34 (Prgressins fr the CCSSM, Number and Operatin Fractins, CCSS Writing Team, August 2011, page 4) (Prgressins fr the CCSSM, Number and Operatin Fractins, CCSS Writing Team, August 2011, page 4) PARCC Clarificatin EOY Tasks are limited t fractins with denminatrs 2, 3, 4, 6, and 8 (See ftnte, CCSS p 24). The explanatin aspect f 3.NF.3 is nt assessed here. Sub Claim A, Task Type I (PBA & EOY) Sub Claim C, Task Type II (PBA) Sub Claim D, Task Type III (PBA) c. Express whle numbers as fractins, and recgnize fractins that are equivalent t whle numbers. See abve 3.NF.3b : Express 3 in the frm 3 = 3/1; recgnize that 6/1 = 6; lcate 4/4 and 1 at the same pint f a number line diagram. 3.NF.3c See abve 3.NF.3b Cmpare Equal Equivalence Equivalent Fractin Pint PARCC Clarificatin EOY Tasks are limited t fractins with denminatrs 2, 3, 4, 6, and 8. (See ftnte CCSS p 24) The explanatin aspect f 3.NF.3 is nt assessed here. 3. Cnstruct viable arguments and critique the 8/20/

35 Sub Claim A, Task Type I (PBA & EOY) reasning f thers 5. Use apprpriate tls strategically 7. Lk fr and make use f structure d. Cmpare tw fractins with the same numeratr r the same denminatr by reasning abut their size. Recgnize that cmparisns are valid nly when the tw fractins refer t the same whle. Recrd the results f cmparisns with the symbls >, =, r <, and justify the cnclusins, e.g., by using a visual fractin mdel. 3.NF.3d See abve 3.NF.3b See abve 3.NF.3b PARCC Clarificatin EOY.Tasks are limited t fractins with denminatrs 2, 3, 4, 6, and 8. (See ftnte CCSS p 24) Justifying is nt assessed here. Prmpts d nt prvide visual fractin mdels; students may at their discretin draw visual fractin mdels as a strategy. Sub Claim A, Task Type I (EOY) Sub Claim C, Task Type II (PBA) Cmpare Denminatr Equal Equivalence Equivalent Fractin Justify Numeratr Pint 7. Lk fr and make use f structure MEASUREMENT AND DATA (3.MD) Students slve prblems invlving measurement and estimatin f intervals f time, liquid vlumes, and masses f bjects. TEACHER NOTES See instructinal strategies in the intrductin RESOURCE NOTES ASSESSMENT NOTES See resurces in the REQUIRED Use Mathematical M 3.MD.1 Tell and write time t the nearest minute and measure time intervals in minutes. intrductin PARCC Released Practices t Slve wrd prblems invlving additin and subtractin f time intervals in minutes, Test Prblems 1. Make sense f prblems and e.g., by representing the prblem n a number line diagram. Majr cntent Excludes cmpund units persevere in slving them Cmmn Unit 2. Reasn abstractly and such as cm 3 and finding the Refer t Algebra I Assessment quantitatively gemetric vlume f a Live Binder 3. Cnstruct viable arguments This standard calls fr students t slve elapsed time, Minute cntainer. Cmmn Tasks and critique the reasning f thers including wrd prblems. Students culd use clck mdels r Number line diagram m/play/play/ NWEA Test 4. Mdel with mathematics number lines t slve. On the number line, students shuld Time Excludes multiplicative fr mid year evidence Perfrmance Level 5. Use apprpriate tls be given the pprtunities t determine the intervals and size Time intervals cmparisn prblems statements and Descriptrs 8/20/

36 strategically 6. Attend t precisin 7. Lk fr and make use f structure 8. Lk fr and express regularity in repeated reasning M f jumps n their number line. Students culd use pre d Tnya wakes up at 6:45 a.m. It takes her 5 minutes t shwer, 15 minutes t get dressed, and 15 minutes t eat breakfast. What time will she be ready fr schl? PARCC Clarificatin EOY 3.MD.1 1 Tell and write time t the nearest minute and measure time intervals in minutes. Time intervals are limited t 60 minutes N mre than 20% f items require determining a time interval frm clck readings having different hur values Acceptable intervals: ex. Start time 1:20, end time 2:10 time interval is50 minutes. Unacceptable intervals: ex. Start time 1:20, end time 2:30 time interval exceeds 60 minutes. 3.MD.1 2 Slve wrd prblems invlving additin and subtractin f time intervals in minutes, e.g., by representing the prblem n a number line diagram.\. Only the answer is required (methds, representatins, etc. are nt assessed here). Tasks d nt invlve reading start/stp times frm a clck nr calculating elapsed time. Sub Claim A, Task Type I (PBA & EOY) Sub Claim C, Task Type II (PBA) Sub Claim D, Task Type III (PBA) Patricia%27sReadingTime.pdf 3.MD Make sense f prblems and persevere in slving them 2. Reasn abstractly and quantitatively 4. Mdel with mathematics 5. Use apprpriate tls strategically 3.MD.2 Measure and estimate liquid vlumes and masses f bjects using standard units f grams (g), kilgrams (kg), and liters (l). Majr cntent Add, subtract, multiply, r divide t slve ne step wrd prblems invlving masses r vlumes that are given in the same units, e.g., by using drawings (such as a beaker with a measurement scale) t represent the prblem. Students need multiple pprtunities weighing classrm bjects and filling cntainers t help them develp a basic understanding f the size and weight f a liter, a gram, and a Beaker Estimate Gram (g) (prblems invlving ntins f times as much (see table in 3.OA.3) A clck is a cmmn instrument fr measuring time. Learning t tell time has much t d with learning t read a dial type instrument and little with time measurement. Students have experience in telling and writing time frm analg and digital clcks t the hur and half hur in Grade 1 and t the nearest five minutes, using a.m. and p.m. in Grade 2. Nw students will tell and write time t the nearest minute and measure time intervals in minutes. Prvide analg clcks that allw students t mve the minute hand. Students need experience representing time frm a digital clck t an analg clck and vice versa. Prvide wrd prblems invlving additin and subtractin f time intervals in minutes. Have students represent the prblem n a number line. Student shuld relate using the number line with subtractin frm Grade 2. Prvide pprtunities fr students t use apprpriate tls t measure and estimate liquid vlumes in liters nly and masses f bjects in grams and kilgrams. Students need practice in reading the scales n measuring tls since the markings may nt always be in intervals f ne. The scales clarificatin 8/20/ (PARCC See assessments in the intrductin

37 kilgram. Milliliters may als be used t shw amunts that are less than a liter. Students identify 5 things that weigh abut ne gram. They recrd their findings with wrds and pictures. (Students can repeat this fr 5 grams and 10 grams.) This activity helps develp gram benchmarks. One large paperclip weighs abut ne gram. A paper clip weighs abut a) a gram, b) 10 grams, c) 100 grams? Explain why. PARCC Clarificatin EOY 3.MD.2 1 Measure and estimate liquid vlumes and masses f bjects using standard units f grams (g), kilgrams (kg), and liters (l). 3.MD.2 2 Add, subtract, multiply r divide t slve ne step wrd prblems invlving masses r vlumes that are given in the same units, e.g., by using drawings (such as a beaker with a measurement scale) t represent the prblem. Only the answer is required (methds, representatins, etc. are nt assessed here). Sub Claim A, Task Type I (PBA & EOY) Sub Claim D, Task Type III (PBA) Kilgram (g) Liquid vlume Liter (l) Mass Measure Metric Standard unit 1. Make sense f prblems and persevere in slving them 2. Reasn abstractly and quantitatively 4. Mdel with mathematics 5. Use apprpriate tls strategically 6. Attend t precisin may be marked in intervals f tw, five r ten. Allw students t hld gram and kilgram weights in their hand t use as a benchmark. Use water clred with fd clring s that the water can be seen in a beaker. Students shuld estimate vlumes and masses befre actually finding the measuring. Shw students a grup cntaining the same kind f bjects. Then, shw them ne f the bjects and tell them its weight. Fill a cntainer with mre bjects and ask students t estimate the weight f the bjects. Use similar strategies with liquid measures. Be sure that students have pprtunities t pur liquids int different size cntainers t see hw much liquid will be in certain whle liters. Shw students cntainers and ask, Hw many liters d yu think will fill the cntainer? ODE MEASUREMENT AND DATA (3.MD) Use Mathematical Practices t 1. Make sense f prblems and persevere in slving them 2. Reasn abstractly and S Students represent and interpret data. 3.MD.3 Draw a scaled picture graph and a scaled bar graph t represent a data set with several categries. Slve ne and tw step hw many mre and hw many less prblems using infrmatin presented in scaled bar graphs. Supprting cntent TEACHER NOTES See instructinal strategies in the intrductin Representatin f a data set is extended frm picture RESOURCE NOTES See resurces in the intrductin ASSESSMENT NOTES REQUIRED PARCC Released Test Prblems Cmmn Unit 8/20/

38 quantitatively 3. Cnstruct viable arguments and critique the reasning f thers 4. Mdel with mathematics 5. Use apprpriate tls strategically 6. Attend t precisin 7. Lk fr and make use f structure 8. Lk fr and express regularity in repeated reasning Students shuld have pprtunities reading and slving prblems using scaled graphs befre being asked t draw ne. The fllwing graphs all use five as the scale interval, but students shuld experience different intervals t further develp their understanding f scale graphs and number facts. Fr example, draw a bar graph in which each square in the bar graph might represent 5 pets. Pictgraphs: Scaled pictgraphs include symbls that represent multiple units. Belw is an example f a pictgraph with symbls that represent multiple units. Graphs shuld include a title, categries, categry label, key, and data. Hw many mre bks did Juan read than Nancy? Single Bar Graphs: Students use bth hrizntal and vertical bar graphs. Bar graphs include a title, scale, scale label, categries, categry label, and data. PARCC Clarificatin EOY 3.MD.3 1 Draw a scaled picture graph and a scaled bar graph Categries Data set Hw many less Hw many mre Multi step Put tgether prblems Scale Scaled bar graph Scaled picture graph 2. Reasn abstractly and quantitatively 4. Mdel with mathematics graphs and bar graphs with single unit scales t scaled picture graphs and scaled bar graphs. Intervals fr the graphs shuld relate t multiplicatin and divisin with 100 (prduct is 100 r less and numbers used in divisin are 100 r less). In picture graphs, use values fr the icns in which students are having difficulty with multiplicatin facts. Fr example, represents 7 peple. If there are three, students shuld use knwn facts t determine that the three icns represents 21 peple. The intervals n the vertical scale in bar graphs shuld nt exceed 100. Instructinal Strategies Students are t draw picture graphs in which a symbl r picture represents mre than ne bject. Bar graphs are drawn with intervals greater than ne. Ask questins that require students t cmpare quantities and use mathematical cncepts and skills. Use symbls n picture graphs that student can easily represent half f, r knw hw many half f the symbl represents. Students are t measure lengths using rulers marked with halves and furths f an inch and recrd the data n a line plt. The hrizntal scale f the line plt is marked ff in whle numbers, halves r furths. Students can create rulers with apprpriate markings and use the ruler t create the line plts. ODE Refer t Algebra I Live Binder m/play/play/ fr mid year evidence statements and clarificatin Frm the Natinal Cuncil f Teachers f Mathematics, Illuminatins: Instructinal Resurces/Tls Bar Grapher This is a NCTM site that cntains a bar graph tl t create bar graphs. Frm the Natinal Cuncil f Teachers f Mathematics, Illuminatins: All Abut Multiplicatin Explring equal sets Students listen t the cunting stry, What Cmes in 2's, 3's, & 4's, and then use cunters t set up multiple sets f equal size. They fill in a table listing the number f sets, the number f bjects in each set, and the ttal number in all. They study the table t find examples f the rder (cmmutative) prperty. Finally, they apply the equal sets mdel f multiplicatin by creating pictgraphs in which each icn represents several data pints. Frm the Natinal Cuncil f Teachers f Mathematics, Assessment Cmmn Tasks NWEA Test Perfrmance Level Descriptrs (PARCC See assessments in the intrductin 8/20/

39 t represent a data set with several categries. Fr example, draw a bar graph in which each square in the bar graph might represent 5 pets. Tasks invlve n mre than 10 items in 2 5 categries. Tasks d nt require students t create the entire graph, but might ask students t cmplete a graph r therwise demnstrate knwledge f its creatin. 3.MD.3 3 Slve a put tgether prblem using infrmatin presented in a scaled bar graph, then use the result t answer a hw many mre r hw many less prblem using infrmatin presented in the scaled bar graph. See 3.MD.3 Be careful that tasks d nt require cmputatins beynd the grade 3 expectatins. See 4.NBT fr cmputatins expected nly at the next grade. Sub Claim B, Task Type I (EOY) Cmmn Miscnceptins Althugh intervals n a bar graph are nt in single units, students cunt each square as ne. T avid this errr, have students include tick marks between each interval. Students shuld begin each scale with 0. They shuld think f skipcunting when determining the value f a bar since the scale is nt in single units. ODE Illuminatins: What s in a Name? Creating Pictgraphs. Students create pictgraphs and answer questins abut the data set. S 3.MD.4 Generate measurement data by measuring lengths using rulers marked with halves and furths f an inch. Shw the data by making a line plt, where the hrizntal scale is marked ff in apprpriate units whle numbers, halves, r quarters. Supprting cntent Students in secnd grade measured length in whle units using bth metric and U.S. custmary systems. It s imprtant t review with students hw t read and use a standard ruler including details abut halves and quarter marks n the ruler. Students shuld cnnect their understanding f fractins t measuring t ne half and ne quarter inch. Third graders need many pprtunities measuring the length f varius bjects in their envirnment. Sme imprtant ideas related t measuring with a ruler are: The starting pint f where ne places a ruler t begin measuring Measuring is apprximate. Items that students measure will nt always measure exactly ¼, ½ r ne whle inch. Students will need t decide n an apprpriate estimate length. Making paper rulers and flding t find the half and quarter marks will help students develp a strnger understanding f measuring length Students generate data by measuring and create a line plt t display their findings. An example f a line plt is shwn belw: Furths Halves Hrizntal scale Inch Length Line plt Quarters Unit Whle number 2. Reasn abstractly and quantitatively 5. Use apprpriate tls strategically 8/20/

40 PARCC Clarificatin EOY NONE Sub Claim B, Task Type I (EOY) 5cctask.ncdpi.wikispaces.net/3.MD.3 3.MD.4 (Use 3.MD.4 Task 2.dc) MEASUREMENT AND DATA (3.MD) Use Mathematical Practices t 1. Make sense f prblems and persevere in slving them 2. Reasn abstractly and quantitatively 3. Cnstruct viable arguments and critique the reasning f thers 4. Mdel with mathematics 5. Use apprpriate tls strategically 6. Attend t precisin 7. Lk fr and make use f structure 8. Lk fr and express regularity in repeated reasning M Students understand cncepts f area and relate area t multiplicatin and t additin (gemetric measurement). 3.MD.5 Recgnize area as an attribute f plane figures and understand cncepts f area measurement. Majr cntent a. A square with side length 1 unit, called a unit square, is said t have ne square unit f area, and can be used t measure area. 3.MD.5a These standards call fr students t explre the cncept f cvering a regin with unit squares, which culd include square tiles r shading n grid r graph paper. Based n students develpment, they shuld have ample experiences filling a regin with square tiles befre transitining t pictrial representatins n graph paper. Area Attribute Measure Plane figure Square unit 7. Lk fr and make use f structure TEACHER NOTES See instructinal strategies in the intrductin RESOURCE NOTES See resurces in the intrductin Refer t Algebra I Live Binder m/play/play/ fr mid year evidence statements and clarificatin ASSESSMENT NOTES REQUIRED PARCC Released Test Prblems Cmmn Unit Assessment Cmmn Tasks NWEA Test Perfrmance Level Descriptrs (PARCC See assessments in the intrductin 8/20/

41 (Prgressins fr the CCSSM, Gemetric Measurement, CCSS Writing Team, June 2012, page 16) PARCC Clarificatin EOY NONE Sub Claim A, Task Type I (PBA & EOY) Sub Claim C, Task Type II (PBA) Sub Claim D, Task Type III (PBA) 5cctask.ncdpi.wikispaces.net/3.MD.3 3.MD.4 (Use 3.MD.4 Task 2.dc) b. A plane figure which can be cvered withut gaps r verlaps by n unit squares is said t have an area f n square units. 3.MD.5b These standards call fr students t explre the cncept f cvering a regin with unit squares, which culd include square tiles r shading n grid r graph paper. Based n students develpment, they shuld have ample experiences filling a regin with square tiles befre transitining t pictrial representatins n graph paper. See abve 3.MD.5a PARCC Clarificatin EOY Area Attribute Measure Plane figure Regin Square unit Sub Claim A, Task Type I (PBA & EOY) Sub Claim C, Task Type II (PBA) Sub Claim D, Task Type III (PBA) 8/20/

42 M 3.MD.6 Measure areas by cunting unit squares (square cm, square m, square in, square ft., and imprvised units). Majr cntent Students shuld be cunting the square units t find the area culd be dne in metric, custmary, r nn standard square units. Using different sized graph paper, students can explre the areas measured in square centimeters and square inches. The task shwn abve wuld prvides a great experience fr students t tile a regin and cunt the number f square units Area Gap Measure Overlap Square centimeter Square feet Square inch Square meter Unit square 7. Lk fr and make use f structure (Prgressins fr the CCSSM, Gemetric Measurement, CCSS Writing Team, June 2012, page 16) PARCC Clarificatin EOY NONE Sub Claim A, Task Type I (PBA & EOY) Sub Claim C, Task Type II (PBA) Sub Claim D, Task Type III (PBA) M 3.MD.7 Relate area t the peratins f multiplicatin and additin. Majr cntent a. Find the area f a rectangle with whle number side lengths by tiling it, and shw that the area is the same as wuld be fund by multiplying the side lengths. 3.MD.7a Students tile areas f rectangles, determine the area, recrd the length and width f the rectangle, investigate the patterns in the numbers, and discver that the area is the length times the width. Additive Area Decmpsing Distributive 8/20/

43 Many activities that invlve seeing and making arrays f squares t frm a rectangle might be needed t build rbust cnceptins f a rectangular area structured int squares. Gap Nn verlap Overlapping Prducts Prperty Rectilinear figures Side length Tiling Students shuld understand and explain why multiplying the side lengths f a rectangle yields the same measurement f area as cunting the number f tiles (with the same unit length) that fill the rectangle s interir Fr example, students might explain that ne length tells hw many unit squares in a rw and the ther length tells hw many rws there are. (Prgressins fr the CCSSM, Gemetric Measurement, CCSS Writing Team, June 2012, page 17) Students shuld tile rectangle then multiply the side lengths t shw it is the same. T find the area ne culd cunt the squares r multiply 3 x 4 = 12. PARCC Clarificatin EOY Sub Claim A, Task Type I (EOY) Sub Claim C, Task Type II (PBA) 8/20/

44 b. Multiply side lengths t find areas f rectangles with whle number side lengths in the cntext f slving real wrld and mathematical prblems, and represent whle number prducts as rectangular areas in mathematical reasning. 3.MD.7b Students tile areas f rectangles, determine the area, recrd the length and width f the rectangle, investigate the patterns in the numbers, and discver that the area is the length times the width See abve 3.MD.7a Students shuld slve real wrld and mathematical prblems Drew wants t tile the bathrm flr using 1 ft tiles. Hw many square ft tiles will he need? Additive Area Decmpsing Distributive Gap Nn verlap Overlapping Prperty Rectilinear figures Side length Tiling Students might slve prblems such as finding all the rectangular regins with whle number side lengths that have an area f 12 area units, ding this fr larger rectangles (e.g., enclsing 24, 48, 72 area units), making sketches rather than drawing each square. Students learn t justify their belief they have fund all pssible slutins. (Prgressins fr the CCSSM, Gemetric Measurement, CCSS Writing Team, June 2012, page 18) 4. Mdel with Mathematics 7. Lk fr and make use f structure This standard extends students wrk with the distributive prperty. Fr example, in the picture belw the area f a 7 x 6 figure can be determined by finding the area f a 5 x 6 and 2 x 6 and adding the tw sums. Using cncrete bjects r drawings students build cmpetence with cmpsitin and decmpsitin f shapes, spatial structuring, and additin f area measurements, students learn t investigate arithmetic prperties using area mdels. Fr example, they learn t rtate rectangular arrays physically and mentally, understanding that their areas are preserved under rtatin, and thus, fr example, 4 x 7 = 7 x 4, illustrating the cmmutative prperty f multiplicatin. Students als learn t understand and explain that the area 8/20/

45 f a rectangular regin f, fr example, 12 length units by 5 length units can be fund either by multiplying 12 x 5, r by adding tw prducts, e.g., 10 x 5 and 2 x 5, illustrating the distributive prperty. Prgressins fr the CCSSM, Gemetric Measurement, CCSS Writing Team, June 2012, page 18) PARCC Clarificatin EOY 3.MD.7b 1 Relate area t the peratins f multiplicatin and additin. Multiply side lengths t find areas f rectangles with whlenumber side lengths in the cntext f slving real wrld and mathematical prblems. Prducts are limited t the 10x10 multiplicatin table Sub Claim A, Task Type I (EOY) Sub Claim C, Task Type II (PBA) c. Use tiling t shw in a cncrete case that the area f a rectangle with whlenumber side lengths a and b + c is the sum f a b and a c. Use area mdels t represent the distributive prperty in mathematical reasning. 3.MD.7c Students tile areas f rectangles, determine the area, recrd the length and width f the rectangle, investigate the patterns in the numbers, and discver that the area is the length times the width See abve 3.MD.7a, 3. MD.7b PARCC Clarificatin EOY Additive Area Decmpsing Distributive Gap Nn verlap Overlapping Prperty Rectilinear figures Side length Tiling Sub Claim A, Task Type I (EOY) Sub Claim C, Task Type II (PBA) d. Recgnize area as additive. Find areas f rectilinear figures by decmpsing them int nn verlapping rectangles and adding the areas f the nnverlapping parts, applying this technique t slve real wrld prblems. 3.MD.7d 8/20/

46 Students tile areas f rectangles, determine the area, recrd the length and width f the rectangle, investigate the patterns in the numbers, and discver that the area is the length times the width This standard uses the wrd rectilinear. A rectilinear figure is a plygn that has all right angles. Additive Area Decmpsing Distributive Gap Nn verlap Overlapping Prperty Rectilinear figures Side length Tiling Hw culd this figure be decmpsed t help find the area? 7. Lk fr and make use f structure Therefre the ttal area f this figure is 12 square units Example: A strage shed is pictured belw. What is the ttal area? Hw culd the figure be decmpsed t help find the area? Example: Students can decmpse a rectilinear figure int different rectangles. 8/20/

47 They find the area f the figure by adding the areas f each f the rectangles tgether. With strng and distinct cncepts f bth perimeter and area established, students can wrk n prblems t differentiate their measures. Fr example, they can find and sketch rectangles with the same perimeter and different areas r with the same area and different perimeters and justify their claims Differentiating perimeter frm area is facilitated by having students draw cngruent rectangles and measure, mark ff, and label the unit lengths all arund the perimeter n ne rectangle, then d the same n the ther rectangle but als draw the square units. This enables students t see the units invlved in length and area and find patterns in finding the lengths and areas f nn square and square rectangles. Students can cntinue t describe and shw the units invlved in perimeter and area after they n lnger need these. (Prgressins fr the CCSSM, Gemetric Measurement, CCSS Writing Team, June 2012, page 18) PARCC Clarificatin EOY NONE. Sub Claim A, Task Type I (EOY) Sub Claim C, Task Type II (PBA) MEASUREMENT AND DATA (3.MD) Students recgnize perimeter as an attribute f plane figures and distinguish between linear and area measures (gemetric measurement). TEACHER NOTES See instructinal strategies in the intrductin RESOURCE NOTES ASSESSMENT NOTES See resurces in the REQUIRED Use Mathematical M 3.MD.8 Slve real wrld and mathematical prblems invlving perimeters f plygns, intrductin PARCC Released Practices t including finding the perimeter given the side lengths, finding an unknwn side Students have created Test Prblems 1. Make sense f prblems and length, and exhibiting rectangles with the same perimeter and different areas r rectangles befre when persevere in slving them Cmmn Unit with the same area and different perimeters. Majr cntent 2. Reasn abstractly and finding the area f rectangles Refer t Algebra I Assessment quantitatively and cnnecting them t using Live Binder 3. Cnstruct viable arguments arrays in the multiplicatin f Cmmn Tasks and critique the reasning f 8/20/

48 thers 4. Mdel with mathematics 5. Use apprpriate tls strategically 6. Attend t precisin 7. Lk fr and make use f structure 8. Lk fr and express regularity in repeated reasning Students develp an understanding f the cncept f perimeter by walking arund the perimeter f a rm, using rubber bands t represent the perimeter f a plane figure n a gebard, r tracing arund a shape n an interactive whitebard. They find the perimeter f bjects; use additin t find perimeters; and recgnize the patterns that exist when finding the sum f the lengths and widths f rectangles. Students use gebards, tiles, graph paper, r technlgy t find all the pssible rectangles with a given area (e.g. find the rectangles that have an area f 12 square units.) They recrd all the pssibilities using dt r graph paper, cmpile the pssibilities int an rganized list r a table, and determine whether they have all the pssible rectangles. Students then investigate the perimeter f the rectangles with an area f 12. Area Length Width Perimeter 12 sq. in. 1 in. 12 in. 26 in. 12 sq. in. 2 in. 6 in. 16 in. 12 sq. in 3 in. 4 in. 14 in. 12 sq. in 4 in. 3 in. 14 in. 12 sq. in 6 in. 2 in. 16 in. 12 sq. in 12 in. 1 in. 26 in. The patterns in the chart allw the students t identify the factrs f 12, cnnect the results t the cmmutative prperty, and discuss the differences in perimeter within the same area. This chart can als be used t investigate rectangles with the same perimeter. It is imprtant t include squares in the investigatin. Area Attribute Linear Perimeter Plane figure Plygn Side length Unknwn side 2. Reasn abstractly and quantitatively 4. Mdel with mathematics 5. Use apprpriate tls strategically whle numbers. T explre finding the perimeter f a rectangle, have students use nnstretchy string. They shuld measure the string and create a rectangle befre cutting it int fur pieces. Then, have students use fur pieces f the nnstretchy string t make a rectangle. Tw pieces f the string shuld be f the same length and the ther tw pieces shuld have a different length that is the same. Students shuld be able t make the cnnectin that perimeter is the ttal distance arund the rectangle. Instructinal Strategies m/play/play/ fr mid year evidence statements and clarificatin NWEA Test Perfrmance Level Descriptrs (PARCC See assessments in the intrductin 8/20/

49 PARCC Clarificatin EOY NONE Sub Claim B, Task Type I (EOY) GEOMETRY (3.G) Use Mathematical Practices t 1. Make sense f prblems and persevere in slving them 2. Reasn abstractly and quantitatively 3. Cnstruct viable arguments and critique the reasning f thers 4. Mdel with mathematics 5. Use apprpriate tls strategically 6. Attend t precisin 7. Lk fr and make use f structure 8. Lk fr and express regularity in repeated reasning S Students reasn with shapes and their attributes. 3.G.1 Understand that shapes in different categries, e.g. rhmbuses, rectangles, and thers) may share attributes having fur sides, and that the shared attributes can define a larger categry quadrilaterals Recgnize rhmbuses, rectangles, and squares as examples f quadrilaterals, and draw examples f quadrilaterals that d nt belng t any f these subcategries. Supprting cntent In secnd grade, students identify and draw triangles, quadrilaterals, pentagns, and hexagns. Third graders build n this experience and further investigate quadrilaterals (technlgy may be used during this explratin). Students recgnize shapes that are and are nt quadrilaterals by Attribute Categry Parallagram Quadrilateral Rectangle TEACHER NOTES See instructinal strategies in the intrductin RESOURCE NOTES See resurces in the intrductin Refer t Algebra I Live Binder m/play/play/ fr mid year evidence statements and clarificatin ASSESSMENT NOTES REQUIRED PARCC Released Test Prblems Cmmn Unit Assessment Cmmn Tasks NWEA Test Perfrmance Level Descriptrs (PARCC See assessments in the intrductin 8/20/

50 examining the prperties f the gemetric figures. They cnceptualize that a quadrilateral must be a clsed figure with fur straight sides and begin t ntice characteristics f the angles and the relatinship between ppsite sides. Students shuld be encuraged t prvide details and use prper vcabulary when describing the prperties f quadrilaterals. They srt gemetric figures (see examples belw) and identify squares, rectangles, and rhmbuses as quadrilaterals. Rhmbus Shapes Side Subcategry Trapezid The standards d nt require the abve representatin be cnstructed by students, but they shuld represent be able t draw examples f quadrilaterals that are nt in the subcategries. (Prgressins fr the CCSSM, Gemetry, CCSS Writing Team, June 2012, page 13) 8/20/

51 Parallelgrams include: squares, rectangles, rhmbi, r ther shapes that have tw pairs f parallel sides. Als, the brad categry quadrilaterals include all types f parallelgrams, trapezids and ther fur sided figures. Example: Draw a picture f a quadrilateral. Draw a picture f a rhmbus. Hw are they alike? Hw are they different? Is a quadrilateral a rhmbus? Is a rhmbus a quadrilateral? Justify yur thinking. A kite is a quadrilateral whse fur sides can be gruped int tw pairs f equal length sides that are beside each ther. The ntin f cngruence ( same size and same shape ) may be part f classrm cnversatin but the cncepts f cngruence and similarity d nt appear until middle schl. PARCC Clarificatin EOY A trapezid is defined as A quadrilateral with at least ne pair f parallel sides. Sub Claim B, Task Type I (EOY) S 3.G.2 Partitin shapes int parts with equal areas. Express the area f each part as a unit fractin f the whle. Supprting cntent Given a shape, students partitin it int equal parts, recgnizing that these parts all have the same area. They identify the fractinal name f each part as ne f fur and ne furth, and are able t partitin a shape int parts with equal areas in several different ways. Equal areas Part Partitin Shape Unit fractin 8/20/

52 Fr example, partitin a shape int 4 parts with equal area, and describe the area f each part as 1/4 f the area f the shape. This figure was partitined/divided int fur equal parts. Each part is. f the ttal area f the figure. Whle Given a shape, students partitin it int equal parts, recgnizing that these parts all have the same area. They identify the fractinal name f each part and are able t partitin a shape int parts with equal areas in several different ways. PARCC Clarificatin EOY NONE Sub Claim B, Task Type I (EOY) NOTE In Additin the fllwing will be assessed frm grade 2 n the grade 3 PBA: Sub Claim D, Task Type III (PBA) 2.NBT, all grade 3 PBA Task Type I, 2.OA.A, 2.OA.B, 2.NBT, 2.MD.B 8/20/