MATHEMATICS CURRICULUM Grade 4

MIDDLETOWN PUBLIC SCHOOLS MATHEMATICS CURRICULUM Grade 4 Middle Schl REVISED June 2014

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T 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

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 6-8 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

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 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 Items Cmmn Unit Assessment Cmmn Units/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 (PARCC Perfrmance Level Descriptrs) 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

FOR Textbks Supplementary Classrm Instructin That Wrks, Marzan Exemplars (grade 6) NECAP, MCAS, NAEP Released Tasks NWEA MAP Assessments PARCC released Tasks 2014 Technlgy Calculatr Cmputers ELMO Graphing Calculatr Interactive bards LCD prjectrs Overhead calculatr Smart bard TI Navigatr Websites http://illuminatins.nctm.rg/ http://ww.center.k12.m.us/edtech/everydaymath.htm http://www.achieve.rg/http://my.hrw.cm http://www.discveryeducatin.cm/ http://www.de.state.h.us/gd/templates/pages/ode/odedefaultpage.aspx?page=1 www.pearsnsuccess.net www.cmmncresheets.cm www.mathdrills.cm http://www.parccnline.rg/parcc-cntent-framewrks http://www.parccnline.rg/sites/parcc/files/parcc_draft_mdelcntentframewrksfrmathematics0.pdf www.cmmncre.rg/maps www.crestandards.rg www.csme.cm www.explrelearning.cm (Gizm ) www.fasttmath.cm www.glence.cm www.khanacademy.cm www.mathfrum.rg www.phschl.cm www.ride.ri.gv www.studyisland www.successnet.cm Materials Clred chips Cmpasses Dice EDM Templates Exp markers Fractin/decimal tiles Number line Pattern blcks Prtractrs Rulers Student white bards Task Type Task Type 8/20/2014 5

Task Type I. Tasks assessing cncepts, skills and prcedures 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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DOMAINS UNIT Students use the fur peratins with whle numbers t slve prblems. OPERATIONS AND ALGEBRAIC THINKING (4.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 4.OA.1 Interpret a multiplicatin equatin as a cmparisn, e.g., interpret 35 = 5 7 as a statement that 35 is 5 times as many as 7 and 7 times as many as 5. Represent verbal statements f multiplicative cmparisns as multiplicatin equatins. Majr cntent A multiplicatin equatin can be interpreted as multiplicative cmparisn. Multiplicative cmparisn answers the questin, hw many times as much r hw many times as many? A multiplicative cmparisn is a situatin in which ne quantity is multiplied by a specified number t get anther quantity (e.g., a is n times as much as b ). Students shuld be able t identify and verbalize which quantity is being multiplied and which number tells hw many times. 5 x 8 = 40. Sally is five years ld. Her mm is eight times lder. Hw ld is Sally s Mm? 5 x 5 = 25 Sally has five times as many pencils as Mary. If Sally has 5 pencils, hw many des Mary have? PARCC Clarificatin EOY - 4.OA.1-1 Interpret a multiplicatin equatin as a cmparisn, e.g., interpret 35 = 5 7 as a statement that 35 is 5 times as many as 7 and 7 times as many as 5. Tasks have thin cntext r n cntext. 4.OA.1-2 Represent verbal statements f multiplicative cmparisns as multiplicatin equatins. Tasks have thin cntext r n cntext. Sub Claim A, Task Type I (PBA & EOY) Sub Claim D, Task Type III (PBA) http://www.parccnline.rg/sites/parcc/files/grade4- ThreeFriends%27Beads.pdf multiplicatin equatin multiplicative cmparisns 2. Reasn abstractly and quantitatively 4. Mdel with mathematics TEACHER NOTES See instructinal strategies in the intrductin Students represent an unknwn number in a wrd prblem with a symbl. Wrd prblems which require multiplicatin r divisin are slved by using drawings and equatins. Present multistep wrd prblems with whle numbers and whle-number answers using the fur peratins. Students shuld knw which peratins are needed t slve the prblem. Drawing pictures r using mdels will help students understand what the prblem is asking. They shuld check the reasnableness f their answer using mental cmputatin and estimatin strategies. ODE Emply mathematics best practice strategies e.g. 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 RESOURCE NOTES See resurces in the intrductin Supplementary Classrm Instructin That Wrks, Marzan PARCC Released Tasks NWEA MAP Assessments Technlgy Cmputers ELMO Graphing calculatr Interactive bards LCD prjectrs MIMIO Overhead scientific calculatr Scientific calculatr Smart bard Websites Live Binder http://www.livebinder s.cm/play/play/1171 650 http://illuminatins.nc tm.rg/ http://www.achieve. http://www.parccnlin e.rg/parcc-cntentframewrks http://www.parccnlin e.rg/sites/parcc/files/ PARCC_Draft_MdelC ntentframewrksfr Mathematics0.pdf www.cmmncre.rg /maps www.crestandards.r g www.csme.cm www.explrelearning.c m (Gizm ) www.fasttmath.cm 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 8/20/2014 8 Exhibits Interviews Graphic rganizers Jurnals Mathematical Practices Mdeling Multiple Intelligences assessments, e.g. Rle playing - bdily kinesthetic Graphic rganizing - visual Cllabratin - interpersnal Oral presentatins Prblem/Perfrmanc e based/cmmn tasks

M 4.OA.2 Multiply r divide t slve wrd prblems invlving multiplicative cmparisn, e.g., by using drawings and equatins with a symbl fr the unknwn number t represent the prblem, distinguishing multiplicative cmparisn frm additive cmparisn. (see belw) Majr cntent Additive cmparisn answers the questin, hw many mre? T determine what the remainder represents yu must understand the cntext f the prblem. Students need many pprtunities t slve cntextual prblems. Table 2 includes the fllwing multiplicatin prblem: A blue hat csts $6. A red hat csts 3 times as much as the blue hat. Hw much des the red hat cst? In slving this prblem, the student shuld identify $6 as the quantity that is being multiplied by 3. The student shuld write the prblem using a symbl t represent the unknwn. additive cmparisn 1. Make sense f prblems and persevere in slving them 4. Mdel with Mathematics 5. Use apprpriate tls strategically www.glence.cm www.khanacademy.c m www.mathfrum.rg www.phschl.cm www.ride.ri.gv www.studyisland www.successnet.cm Materials Research Rubrics/checklists PARCC Perfrmance Level Descriptrs District Tests and quizzes Technlgy Think-aluds Table 2 includes the fllwing divisin prblem: A red hat csts $18 and a blue hat csts $6. Hw many times as much des the red hat cst as the blue hat? In slving this prblem, the student shuld identify $18 as the quantity being divided int shares f $6. The student shuld write the prblem using a symbl t represent the unknwn. ($18 $6 = ) When distinguishing multiplicative cmparisn frm additive cmparisn, students shuld nte that additive cmparisns fcus n the difference between tw quantities (e.g., Deb has 3 apples and Karen has 5 apples. 8/20/2014 9

Hw many mre apples des Karen have?). A simple way t remember this is, Hw many mre? multiplicative cmparisns fcus n cmparing tw quantities by shwing that ne quantity is a specified number f times larger r smaller than the ther (e.g., Deb ran 3 miles. Karen ran 5 times as many miles as Deb. Hw many miles did Karen run?). A simple way t remember this is Hw many times as much? r Hw many times as many? PARCC Clarificatin EOY - See the Prgressin fr Operatins and Algebraic Thinking, especially p. 29 and Table 3 n p. 23. Tasks sample equally the situatins in the third rw f Table 2, p. 89 f CCSS. Sub Claim A, Task Type I (PBA & EOY) Sub Claim D, Task Type III (PBA) http://www.parccnline.rg/sites/parcc/files/grade4- ThreeFriends%27Beads.pdf 8/20/2014 10

M 4.OA.3 Slve multistep wrd prblems psed with whle numbers and having whle-number answers using the fur peratins, including prblems in which remainders must be interpreted. Majr cntent 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. T determine what the remainder represents yu must understand the cntext f the prblem. Kim is making candy bags. There will be 5 pieces f candy in each bag. She had 53 pieces f candy. She ate 14 pieces f candy. Hw many candy bags can Kim make nw? (7 bags with 4 leftver) Kim has 28 ckies. She wants t share them equally between herself and 3 friends. Hw many ckies will each persn get? (7 ckies each) 28 4 = a Estimatin skills 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 situatins using varius estimatin strategies. Estimatin strategies include, but are nt limited t: 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), clustering arund an average (when the values are clse tgether an average value is selected and multiplied by the number f values t determine an estimate), runding and adjusting (students rund dwn r rund up and then adjust their estimate depending n hw much the runding affected the riginal values), using friendly r cmpatible numbers such as factrs (students seek t fit numbers tgether - e.g., runding t factrs and gruping numbers tgether that have rund sums like 100 r 1000), using benchmark numbers that are easy t cmpute (students select clse whle numbers fr fractins r Academic vcabulary Remainders Interpreted unknwn quantity Reasnableness Estimatin runding Mathematical Practices 1. Make sense f prblems and persevere in slving them 4. Mdel with Mathematics 5. Use apprpriate tls strategically 8/20/2014 11

decimals t determine an estimate). Yur class is cllecting bttled water fr a service prject. The gal is t cllect 300 bttles f water. On the first day, Max brings in 3 packs with 6 bttles in each cntainer. Sarah wheels in 6 packs with 6 bttles in each cntainer. Abut hw many bttles f water still need t be cllected? Student 1 First, I multiplied 3 and 6 which equals 18. Then I multiplied 6 and 6 which is 36. I knw 18 plus 36 is abut 50. I m trying t get t 300. 50 plus anther 50 is 100. Then I need 2 mre hundreds. S we still need 250 bttles. Student 2 First, I multiplied 3 and 6 which equals 18. Then I multiplied 6 and 6 which is 36. I knw 18 is abut 20 and 36 is abut 40. 40+20=60. 300-60 = 240, s we need abut 240 mre bttles On a vacatin, yur family travels 267 miles n the first day, 194 miles n the secnd day and 34 miles n the third day. Hw many miles did they travel ttal? Sme typical estimatin strategies fr this prblem: Student 1 I first thught abut 267 and 34. I nticed that their sum is abut 300. Then I knew that 194 is clse t 200. When I put 300 and 200 tgether, I get 500. Student 2 I first thught abut 194. It is really clse t 200. I als have 2 hundreds in 267. That gives me a ttal f 4 hundreds. Then I have 67 in 267 and the 34. When I put 67 and 34 tgether that is really clse t 100. When I add that hundred t the 4 hundreds that I already had, I end up with 500. Student 3 I runded 267 t 300. I runded 194 t 200. I runded 34 t 30.When I added 300, 200 and 30, I knw my answer will be abut 530. PARCC Clarificatin EOY - 4.OA.3.1 Slve multistep wrd prblems psed with whle numbers and having whle-number answers using the fur peratins. Assessing reasnableness f answer is nt assessed here. Tasks d nt invlve interpreting remainders. 8/20/2014 12

4.OA.3.2 Slve multistep wrd prblems psed with whle numbers and having whle-number answers using the fur peratins, in which remainders must be interpreted. Assessing reasnableness f answer is nt assessed here. Tasks invlve interpreting reminders. See page 30 f the Prgressin fr Operatins and Algebraic Thinking. Sub Claim A, Task Type I (PBA& EOY) Sub Claim C, Task Type II (PBA) Sub Claim D, Task Type III (PBA) http://www.parccnline.rg/sites/parcc/files/grade4- ThreeFriends%27Beads.pdf http://www.recity.k12.r.us/files/4th_grade_assessment_4 _8_Divisin_0.pdf OPERATIONS AND ALGEBRAIC THINKING (4.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 S Students gain familiarity with factrs and multiples. 4. OA.4 Find all factr pairs fr a whle number in the range 1 100. Supprting cntent Recgnize that a whle number is a multiple f each f its factrs. Determine whether a given whle number in the range 1 100 is a multiple f a given ne-digit number. Determine whether a given whle number in the range 1 100 is prime r cmpsite. A whle number is a multiple f each f its factrs. Factrs and multiples are interrelated. Sme numbers are prime because they nly have ne factr pair; sme numbers are cmpsite because they have mre than ne factr pair. Multiples are the result f skip cunting by each factr. Students shuld understand the prcess f finding factr pairs s they can d this fr any number 1-100, Example: Factr pairs fr 96: 1 and 96, 2 and 48, 3 and 32, 4 and 24, 6 and 16, 8 and 12. Multiples can be thught f as the result f skip cunting by each f the factrs. When skip cunting, students shuld be able t identify the number f factrs cunted e.g., 5, 10, 15, 20 (there are 4 fives in 20). Cmpsite Factr pairs Factrs Multiple Prime TEACHER NOTES See instructinal strategies in the intrductin Students need t develp an understanding f the cncepts f number thery such as prime numbers and cmpsite numbers. This includes the relatinship f factrs and multiples. Multiplicatin and divisin are used t develp cncepts f factrs and multiples. Divisin prblems resulting in remainders are used as cunter-examples f factrs. Review vcabulary s that students have an understanding f terms such as factr, prduct, multiples, and dd and even numbers. Students need t develp strategies fr determining if a number is prime r cmpsite, in ther wrds, if a number has a whle number factr that is nt ne RESOURCE NOTES See resurces in the intrductin Refer t Algebra I PBA/MYA @ Live Binder http://www.livebinders.c m/play/play/1171650 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 8/20/2014 13

Example: Factrs f 24: 1, 2, 3, 4, 6, 8,12, 24 Multiples : 1,2,3,4,5 24 2,4,6,8,10,12,14,16,18,20,22,24 3,6,9,12,15,18,21,24 4,8,12,16,20,24 8,16,24 12,24 24 T determine if a number between1-100 is a multiple f a given ne-digit number, sme helpful hints include the fllwing: all even numbers are multiples f 2 all even numbers that can be halved twice (with a whle number result) are multiples f 4 all numbers ending in 0 r 5 are multiples f 5 r itself. ODE Prime vs. Cmpsite: A prime number is a number greater than 1 that has nly 2 factrs, 1 and itself. Cmpsite numbers have mre than 2 factrs. Students investigate whether numbers are prime r cmpsite by building rectangles (arrays) with the given area and finding which numbers have mre than tw rectangles (e.g. 7 can be made int nly 2 rectangles, 1 x 7 and 7 x 1, therefre it is a prime number) finding factrs f the number PARCC Clarificatin EOY - NONE Sub Claim B, Task Type I (EOY) Calculatr - http://www.parccnline.rg/sites/parcc/files/grade4- ThreeFriends%27Beads.pdf 8/20/2014 14

Students generate and analyze patterns. OPERATIONS AND ALGEBRAIC THINKING (4.OA) 1. Use Mathematical Practices t 2. Make sense f prblems and persevere in slving them 3. Reasn abstractly and quantitatively 4. Cnstruct viable arguments and critique the reasning f thers 5. Mdel with mathematics 6. Use apprpriate tls strategically 7. Attend t precisin 8. Lk fr and make use f structure 9. Lk fr and express regularity in repeated reasning A 4. OA.5 Generate a number r shape pattern that fllws a given rule. Identify apparent features f the pattern that were nt explicit in the rule itself. Additinal cntent A pattern is a sequence that repeats the same prcess ver and ver based n a rule. Given a pattern yu can generate a rule; given a rule yu can generate a pattern. Fr example, given the rule Add 3 and the starting number 1, generate terms in the resulting sequence and bserve that the terms appear t alternate between dd and even numbers. Explain infrmally why the numbers will cntinue t alternate in this way. PARCC Clarificatin EOY Tasks d nt require students t determine a rule ;the rule is given. 75% f patterns shuld be number patterns. Sub Claim B Task Type I (EOY) Number pattern Rule Shape pattern 8. Lk fr and make use f structure TEACHER NOTES See instructinal strategies in the intrductin Students shuld generate numerical r gemetric patterns that fllw a given rule. They shuld lk fr relatinships in the patterns and be able t describe and make generalizatins. ODE RESOURCE NOTES See resurces in the intrductin Refer t Algebra I PBA/MYA @ Live Binder http://www.livebinders.c m/play/play/1171650 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 NUMBER AND OPERATIONS IN BASE TEN (4.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 M Students generalize place value under-standing fr multi-digit whle numbers. 4.NBT.1 Recgnize that in a multi-digit whle number, a digit in ne place represents ten times what it represents in the place t its right. Majr cntent A digit in ne place is ten times mre than the same digit in a place t the right. Fr example, recgnize that 700 70 = 10 by applying cncepts f place value and divisin. This standard calls fr students t extend their understanding f place value related t multiplying and dividing by multiples f 10. In this standard, students Multi-digit Place value 7. Lk fr and make use f structure TEACHER NOTES See instructinal strategies in the intrductin Grade 4 expectatins in this dmain are limited t whle numbers less than r equal t 1,000,000. In Grade 4, runding is nt new, and students need t build n the Grade 3 skill f runding t the nearest 10 r 100 t include larger numbers and place value. What is new fr Grade 4 is runding t RESOURCE NOTES See resurces in the intrductin Refer t Algebra I PBA/MYA @ Live Binder http://www.livebinders.c m/play/play/1171650 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 8/20/2014 15

shuld reasn abut the magnitude f digits in a number. Students shuld be given pprtunities t reasn and analyze the relatinships f numbers that they are wrking with. In the base-ten system, the value f each place is 10 times the value f the place t the immediate right. Because f this, multiplying by 10 yields a prduct in which each digit f the multiplicand is shifted ne place t the left. digits ther than the leading digit, e.g., rund 23,960 t the nearest hundred. This requires greater sphisticatin than runding t the nearest ten thusand because the digit in the hundreds place represents 900 and when runded it becmes 1000, nt just zer. ODE Example: Hw is the 2 in the number 582 similar t and different frm the 2 in the number 528? PARCC Clarificatin EOY - NONE Sub Claim A, Task Type I (PBA & EOY) Sub Claim D, Task Type III (PBA) http://www.recity.k12.r.us/files/4th_grade_assessme nt_4_2_as_pv_2.pdf M 4.NBT.2 Read and write multi-digit whle numbers using base-ten numerals, number names, and expanded frm. Majr cntent Cmpare tw multi-digit numbers based n meanings f the digits in each place, using >, =, and < symbls t recrd the results f cmparisns. Numbers can be represented in many frms, including: >, =, and < symbls 8/20/2014 16

expanded ntatin, written frm, mdels/pictures, and standard ntatin. Expanded frm Number names Traditinal expanded frm is 285 = 200 + 80 + 5. Written frm r number name is tw hundred eighty-five. Hwever, students shuld have pprtunities t explre the idea that 285 culd als be 28 tens plus 5 nes r 1 hundred, 18 tens, and 5 nes. PARCC Clarificatin EOY - Tasks assess cnceptual understanding, e.g. by including a mixture (bth within and between items) f expanded frm, number names, and base ten numerals. Sub Claim A, Task Type I (PBA& EOY) Sub Claim D, Task Type III (PBA) 7. Lk fr and make use f structure Calculatr - http://www.recity.k12.r.us/files/4th_grade_assessment_ 4_2_AS_PV_2.pdf M 4.NBT.3 Use place value understanding t rund multi-digit whle numbers t any place. Majr cntent Place value understanding up t millins place When students are asked t rund large numbers, they first need t identify which digit is in the apprpriate place. Example: Rund 76,398 t the nearest 1000. Step 1: Since I need t rund t the nearest 1000, then the answer is either 76,000 r 77,000. Step 2: I knw that the halfway pint between these tw numbers is 76,500. Step 3: I see that 76,398 is between 76,000 and 76,500. Step 4: Therefre, the runded number wuld be 76,000. PARCC Clarificatin EOY - Grade 4 expectatins are limited t whle numbers less than r equal t 1,000,000 (CCSS ftnte, p. 29). Sub Claim A, Task Type I (EOY) Place value Rund 7. Lk fr and make use f structure http://www.recity.k12.r.us/files/4th_grade_assessment _4_2_AS_PV_2.pdf 8/20/2014 17

NUMBER AND OPERATIONS IN BASE TEN (4.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 M Students use place value understanding and prperties f peratins t perfrm multidigit arithmetic. 4.NBT.4 Fluently add and subtract multi-digit whle numbers using the standard algrithm. Majr cntent There is a relatinship between the fur peratins. Students build n their understanding f additin and subtractin, their use f place value and their flexibility with multiple strategies t make sense f the standard algrithm. They cntinue t use place value in describing and justifying the prcesses they use t add and subtract. When students begin using the standard algrithm their explanatin may be quite lengthy. After much practice with using place value t justify their steps, they will develp fluency with the algrithm. Students shuld be able t explain why the algrithm wrks. 3892 + 1567 Student explanatin fr this prblem: 1. Tw nes plus seven nes is nine nes. 2. Nine tens plus six tens is 15 tens. 3. I am ging t write dwn five tens and think f the10 tens as ne mre hundred.(ntates with a 1 abve the hundreds clumn) 4. Eight hundreds plus five hundreds plus the extra hundred frm adding the tens is 14 hundreds. 5. I am ging t write the fur hundreds and think f the 10 hundreds as ne mre 1000. (ntates with a 1 abve the thusands clumn) 6. Three thusands plus ne thusand plus the extra thusand frm the hundreds is five thusand. Student explanatin fr this prblem: 1. There are nt enugh nes t take 8 nes frm 6 nes s I have t use ne ten as 10 nes. Nw I have 3 tens and 16 nes. (Marks thrugh the 4 and ntates with a 3 abve the 4 and writes a 1 abve the nes clumn t be represented as 16 nes.) 2. Sixteen nes minus 8 nes is 8 nes. (Writes an 8 in the nes clumn f answer.) 3. Three tens minus 2 tens is ne ten. (Writes a 1 in the Algrithm 7. Lk fr and make use f structure TEACHER NOTES See instructinal strategies in the intrductin Grade 4 expectatins in this dmain are limited t whle numbers less than r equal t 1,000,000. Fr multi-digit additin and subtractin in Grade 4, the gal is fluency, which means students must be able t carry ut the calculatins efficiently and accurately. ODE Start with a student s understanding f a certain strategy, and then make intentinal, clear-cut cnnectins fr the student t the standard algrithm. This allws the student t gain understanding f the algrithm rather than just memrize certain steps t fllw. It is very imprtant fr sme students t talk thrugh their understanding f cnnectins between different strategies and standard additin and subtractins algrithms. Give students many pprtunities t talk with classmates abut hw they culd explain standard algrithms. Think- Pair-Share is a gd prtcl fr all students. ODE RESOURCE NOTES See resurces in the intrductin Refer t Algebra I PBA/MYA @ Live Binder http://www.livebinders.c m/play/play/1171650 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 8/20/2014 18

tens clumn f answer.) 4. There are nt enugh hundreds t take 9 hundreds frm 5 hundreds s I have t use ne thusand as 10 hundreds. (Marks thrugh the 3 and ntates with a 2 abve it. (Writes dwn a 1 abve the hundreds clumn.) Nw I have 2 thusand and 15 hundreds. 5. Fifteen hundreds minus 9 hundreds is 6 hundreds. (Writes a 6 in the hundreds clumn f the answer). 6. I have 2 thusands left since I did nt have t take away any thusands. (Writes 2 in the thusands place f answer.) Nte: Students shuld knw that it is mathematically pssible t subtract a larger number frm a smaller number but that their wrk with whle numbers des nt allw this as the difference wuld result in a negative number. PARCC Clarificatin EOY - Sub Claim, Task Type (EOY) 4.NBT.4-1 Fluently add multi-digit whle numbers using the standard algrithm. The given addends are such as t require an efficient/standard algrithm (e.g., 7263 + 4875). Addends in the task d nt suggest any bvius ad hc r mental strategy (as wuld be present fr example in a case such as 16,999 + 3,501). Tasks d nt have a cntext. Grade 4 expectatins in CCSS are limited t whle numbers less than r equal t 1,000,000; fr purpses f assessment, bth f the given numbers shuld have 4 digits. 4.NBT.4 Fluently subtract multi-digit whle numbers using the standard algrithm. The given subtrahend and minuend are such as t require an efficient/standard algrithm (e.g. 72634875 r 74064637). The subtrahend and minuend d nt suggest any bvius ad hc r mental strategy (as wuld be present fr example in a case such as 73006301. ). Tasks d nt have a cntext. Grade 4 expectatins in CCSS are limited t whle numbers less than r equal t 1,000,000; fr purpses f assessment, bth f the given numbers shuld have 4 digits. http://www.parccnline.rg/sites/parcc/files/parcc_sampl eitems_mathematics_g4subtractinfluency_081913_final. 8/20/2014 19

pdf M 4.NBT.5 Multiply a whle number f up t fur digits by a ne-digit whle number, and multiply tw tw-digit numbers, using strategies based n place value and the prperties f peratins. Illustrate and explain the calculatin by using equatins, rectangular arrays, and/r area mdels. Majr cntent Efficient prcedures and strategies t find prducts, qutients, sums, and differences, invlve the use f prperties f peratins (cmmutative, assciative, distributive and identity prperties), place value understanding, and/r flexibility with numbers. Students wh develp flexibility in breaking numbers apart have a better understanding f the imprtance f place value and the distributive prperty in multi-digit multiplicatin. Students use base ten blcks, area mdels, partitining, cmpensatin strategies, etc. when multiplying whle numbers and use wrds and diagrams t explain their thinking. They use the terms factr and prduct when cmmunicating their reasning. Multiple strategies enable students t develp fluency with multiplicatin and transfer that understanding t divisin. Use f the standard algrithm fr multiplicatin is an expectatin in the 5th grade. Area mdels Equatins Prperties f peratin Rectangular arrays 7. Lk fr and make use f structure Students may use digital tls t express their ideas. Use f place value and the distributive prperty are applied in the scaffld examples belw. T illustrate 154 x 6 students use base 10 blcks r use drawings t shw 154 six times. Seeing 154 six times will lead them t understand the distributive prperty, 154 X 6 = (100 + 50 + 4) x 6 = (100 x 6) + (50 X 6) + (4 X 6) = 600 + 300 + 24 = 924. The area mdel shws partial prducts. 14 x 16 = 224 Using the area mdel, students first verbalize their understanding: 10 x 10 is 100 4 x 10 is 40 10 x 6 is 60, and 8/20/2014 20

4 x 6 is 24. They use different strategies t Students explain this strategy and the ne belw with base 10 blcks, drawings, r numbers. 25 x24 400 (20 x 20) 100 (20 x 5) 80 (4 x 20) 20 (4 x 5) 600 25 x24 500 (20 x 25) 100 (4 x 25) 600 Matrix mdel This mdel shuld be intrduced after students have facility with the strategies shwn abve. 20 5 20 400 100 500 4 80 20 100 480 + 120 600 8/20/2014 21

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) 4.NBT.5-1 Multiply a whle number f three r fur digits by a ne-digit whle number using strategies based n place value and the prperties f peratins. Tasks d nt have a cntext. The illustrative/explain aspect f 4.NBT.5 is nt assessed here. 4.NBT.5-2 Multiply tw tw-digit numbers, using strategies based n place value and the prperties f peratins. Tasks d nt have a cntext. The illustrative/explain aspect f 4.NBT.6 is nt assessed here. http://www.dadswrksheets.cm/v1/wrksheets/wrd%2 0Prblems.html#Mixed%20Additin%20and%20Subtractin %20Wrd M 4.NBT.6 Find whle-number qutients and remainders with up t fur-digit dividends and ne-digit divisrs, using strategies based n place value, the prperties f peratins, and/r the relatinship between multiplicatin and divisin. Majr cntent Illustrate and explain the calculatin by using equatins, rectangular arrays, and/r area mdels. Efficient prcedures and strategies t find prducts, Dividends qutients, sums, and differences, invlve the use f Divisrs prperties f peratins (cmmutative, assciative, Qutients 8/20/2014 22

distributive and identity prperties), place value understanding, and/r flexibility with numbers. In furth grade, students build n their third grade wrk with divisin within 100. Students need pprtunities t develp their understandings by using prblems in and ut f cntext. : A 4th grade teacher bught 4 new pencil bxes. She has 260 pencils. She wants t put the pencils in the bxes s that each bx has the same number f pencils. Hw many pencils will there be in each bx? Using Base 10 Blcks: Students build 260 with base 10 blcks and distribute them int 4 equal grups. Sme students may need t trade the 2 hundreds fr tens but thers may easily recgnize that 200 divided by 4 is 50. Using Place Value: 260 4 = (200 4) + (60 4) Using Multiplicatin: 4 x 50 = 200, 4 x 10 = 40, 4 x 5 = 20; 50 + 10 + 5 = 65; s 260 4 = 65 Students may use digital tls t express ideas. Using an Open Array r Area Mdel After develping an understanding f using arrays t divide, students begin t use a mre abstract mdel fr divisin. This mdel cnnects t a recrding prcess that will be frmalized in the 5th grade. Example: 150 6 Remainders 7. Lk fr and make use f structure 8. Lk fr and make use f structure Students make a rectangle and write 6 n ne f its sides. They express their understanding that they need t think f the rectangle as representing a ttal f 150. 1. Students think, 6 times what number is a number clse t 150? They recgnize that 6 x 10 is 60 s they recrd 10 as a factr and partitin the rectangle int 2 rectangles and label the area aligned t the factr f 10 with 60. They express that they have nly used 60 f the 150 s they have 90 left. 2. Recgnizing that there is anther 60 in what is left they repeat the prcess abve. They express that they 8/20/2014 23

have used 120 f the 150 s they have 30 left. 3. Knwing that 6 x 5 is 30. They write 30 in the bttm area f the rectangle and recrd 5 as a factr. 4. Students express their calculatins in varius ways: a. 150 150 6 = 10 + 10 + 5 = 25-60 (6 x 10) 90-60 (6 x 10) 30-30 (6 x 5) 0 b. 150 6 = (60 6) + (60 6) + (30 6) = 10 + 10 + 5 = 25 Example 2: 1917 9 A student s descriptin f his r her thinking may be: I need t find ut hw many 9s are in 1917. I knw that 200 x 9 is 1800. S if I use 1800 f the 1917, I have 117 left. I knw that 9 x 10 is 90. S if I have 10 mre 9s, I will have 27 left. I can make 3 mre 9s. I have 200 nines, 10 nines and 3 nines. S I made 213 nines. 1917 9 = 213. PARCC Clarificatin EOY 4.NBT.6-1 Find whle-number qutients and remainders with three-digit dividends and ne-digit divisrs, using strategies based n place value, the prperties f peratins, and/r the relatinship between multiplicatin and divisin Tasks d nt have a cntext. The illustrative/explain aspect f 4.NBT.6 is nt assessed here. 4.NBT.6-2 Find whle-number qutients and remainders with fur-digit dividends and ne-digit divisrs, using strategies based n place value, the prperties f peratins, and/r the relatinship between multiplicatin and divisin. Tasks d nt have a cntext. The illustrative/explain aspect f 4.NBT.6 is nt assessed here. 8/20/2014 24

Sub Claim A, Task Type I (PBA& EOY) Sub Claim C, Task Type II (PBA) Sub Claim D, Task Type III (PBA) http://www.recity.k12.r.us/files/4th_grade_assessme nt_4_8_divisin_0.pdf http://www.dadswrksheets.cm/v1/wrksheets/wrd %20Prblems.html#Mixed%20Additin%20and%20Subtr actin%20w NUMBER AND OPERATIONS FRACTIONS (4.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 reasning M Students extend understanding f fractin equivalence and rdering. 4.NF.1 Explain why a fractin a/b is equivalent t a fractin (n a)/(n b) by using visual fractin mdels, with attentin t hw the number and size f the parts differ even thugh the tw fractins themselves are the same size. Majr cntent Use this principle t recgnize and generate equivalent fractins. This standard refers t visual fractin mdels. This includes area mdels, number lines r it culd be a cllectin/set mdel. This standard extends the wrk in third grade by using additinal denminatrs. (5, 10, 12and 100) This standard addresses equivalent fractins by examining the idea that equivalent fractins can be created by multiplying bth the numeratr and denminatr by the same number r by dividing a shaded regin int varius parts. Students shuld begin t ntice cnnectins between the mdels and fractins in the way bth the parts and whles are cunted and begin t generate a rule fr writing equivalent fractins. 1/2 x 2/2 = 2/4. Visual fractin mdels Equivalent fractins 7. Lk fr and make use f structure TEACHER NOTES See instructinal strategies in the intrductin Grade 4 expectatins in this dmain are limited t fractins with denminatrs 2, 3, 4, 5, 6, 8, 10, 12, and 100. Students shuld use mdels t cmpare tw fractins with different denminatrs by creating cmmn denminatrs r numeratrs. The mdels shuld be the same (bth fractins shwn using fractin bars r bth fractins using circular mdels) s that the mdels represent the same whle. The mdels shuld be represented in drawings. Students shuld als use benchmark fractins such as t cmpare tw fractins. The result f the cmparisns shuld be recrded using, and = symbls. ODE RESOURCE NOTES See resurces in the intrductin Refer t Algebra I PBA/MYA @ Live Binder http://www.livebinders.c m/play/play/1171650 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 8/20/2014 25

PARCC Clarificatin EOY Use the principle a nxa = b nxb t recgnize and generate equivalent fractins The explanatin aspect f 4.NF.1 is nt assessed here. i) Tasks are limited t denminatrs 2, 3, 4, 5, 6, 8, 10, 12, and 100 (CCSS ftnte, p. 30). Tasks may include fractins that equal whle numbers. Sub Claim A, Task Type I (PBA& EOY) Sub Claim C, Task Type II (PBA) Sub Claim D, Task Type III (PBA) http://www.parccnline.rg/sites/parcc/files/grade4- FractinCmparisn.pdf M 4.NF.2 Cmpare tw fractins with different numeratrs and different denminatrs, e.g., by creating cmmn denminatrs r numeratrs, r by cmparing t a benchmark fractin such as 1/2. Majr cntent Recgnize that cmparisns are valid nly when the tw fractins refer t the same whle. Recrd the results f cmparisns with symbls >, =, r <, and justify the cnclusins, e.g., by using a visual fractin mdel. Fractins can be cmpared by using benchmark fractins, and by creating cmmn denminatrs r cmmn numeratrs. Cmparisns are nly valid when the tw fractins refer t the same whle. Benchmark fractins include cmmn fractins between 0 and 1 such as halves, thirds, furths, fifths, sixths, eighths, tenths, twelfths, and hundredths. There are tw cakes n the cunter that are the same size. The first cake has. f it left. The secnd cake has Benchmark fractin Denminatrs Numeratrs Visual fractin mdel 6. Attend t precisin 7. Lk fr and make use f structure 8/20/2014 26

5/12 left. Which cake has mre left? Student 1 Area mdel: The first cake has mre left ver. The secnd cake has 5/12 left which is smaller than.. Student 2 Number Line mdel: Student 3 verbal explanatin: I knw that 6/12 equals.. Therefre, the secnd cake which has 5/12 left is less than.. When using the benchmark f 2 1 t cmpare 6 4 and 8 5, yu culd use diagrams such as these: 4 1 is larger than 1 5 1, while is 6 6 2 8 8 larger than 2 1. Since 6 1 is greater than 8 1, 6 4 is the greater fractin. 8/20/2014 27

PARCC Clarificatin EOY Only the answer is required (methds, representatin, justificatin, etc. are nt assessed here). Tasks require the student t chse the cmparisn strategy autnmusly. Tasks are limited t denminatrs 2, 3, 4, 5, 6, 8, 10, 12, and 100. (CCSS ftnte, p. 30). Tasks may include fractins that equal whle numbers. Sub Claim A, Task Type I (PBA & EOY) Sub Claim C, Task Type II (PBA) Sub Claim D, Task Type III (PBA) http://www.parccnline.rg/sites/parcc/files/grade4- FractinCmparisn.pdf NUMBER AND OPERATIONS FRACTIONS (4.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 reasning M Students build fractins frm unit fractins by applying and extending previus understandings f peratins n whle numbers. 4.NF.3 Understand a fractin a/b with a > 1 as a sum f fractins 1/b. Majr cntent a. Understand additin and subtractin f fractins as jining and parts referring t the same whle. 4.NF.3a Adding and subtracting fractins is the prcess f jining r separating parts that refer t the same whle. Fractins, with the exceptin f unit fractins, can be decmpsed int the sum f fractins with the same denminatr in mre than ne way. A fractin with a numeratr f ne is called a unit fractin. When students investigate fractins ther than unit fractins, such as 2/3, they shuld be able t decmpse the nn-unit fractin int a cmbinatin f several unit fractins. PARCC Clarificatin EOY Tasks are limited t denminatrs 2, 3, 4, 5, 6, 8, 10, 12, and 100. (CCSS ftnte, p. 30). Sub Claim A, Task Type I (PBA& EOY) Sub Claim C, Task Type II (PBA) Sub Claim D, Task Type III (PBA) Decmpse Unit fractin 2. Reasn abstractly and quantitatively 7. Lk fr and make use f structure 8. Lk fr and make use f structure TEACHER NOTES See instructinal strategies in the intrductin Grade 4 expectatins in this dmain are limited t fractins with denminatrs 2, 3, 4, 5, 6, 8, 10, 12, and 100. ODE Nw, they begin t represent a fractin by decmpsing the fractin as the sum f unit fractin and justify with a fractin mdel. Fr example, = + +. Students als represented whle numbers as fractins. They use this knwledge t add and subtract mixed numbers with like denminatrs using prperties f number and apprpriate fractin mdels. It is imprtant t stress that whichever mdel is used, it shuld be the same fr the RESOURCE NOTES See resurces in the intrductin Refer t Algebra I PBA/MYA @ Live Binder http://www.livebinders.c m/play/play/1171650 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 8/20/2014 28

http://www.pusmath.cm/cmmn-creclusters/4.nf.b-build-fractins-frm-unit-fractins b. Decmpse a fractin int a sum f fractins with the same denminatr in mre than ne way, recrding each decmpsitin by an equatin. Justify decmpsitins, e.g., by using a visual fractin mdel. : 3/8 = 1/8 + 1/8 + 1/8 ; 3/8 = 1/8 + 2/8 ; 2 1/8 = 1 + 1 + 1/8 = 8/8 + 8/8 + 1/8. 4.NF.3b same whle. Fr example, a circular mdel and a rectangular mdel shuld nt be used in the same prblem. ODE A mixed number is a whle number plus a fractin smaller than ne. Students shuld justify their breaking apart (decmpsing) f fractins using visual fractin mdels. The cncept f turning mixed numbers int imprper fractins needs t be emphasized using visual fractin mdels. Example: Justify decmpsitin Visual fractin mdel 7. Lk fr and make use f structure 8. Lk fr and make use f structure Similarly, cnverting an imprper fractin t a mixed number is a matter f decmpsing the fractin int a sum f a whle number and a number less than 1. Students can draw n their knwledge frm third grade f whle numbers as fractins. Example, knwing that 1 = 3/3, they see: (Prgressins fr the CCSSM, Number and Operatin 8/20/2014 29

Fractins, CCSS Writing Team, August 2011, page 8) PARCC Clarificatin EOY Understand a fractin a/b with a > 1 as a sum f fractins 1/b. Decmpse a fractin int a sum f fractins with the same denminatr in mre than ne way, recrding each decmpsitin by an equatin. : 3 1 1 1 3 1 2 = + + ; = + ; 8 8 8 8 8 8 8 1 1 8 8 2 = 1+ 1+ = + + 8 8 8 8 1 ; 8 Only the answer is required (methds, representatin, etc. are nt assessed here). Tasks are limited t denminatrs 2, 3, 4, 5, 6, 8, 10, 12, and 100. (CCSS ftnte, p. 30). Tasks may include fractins that equal whle numbers. Sub Claim A, Task Type I (PBA & EOY) Sub Claim C, Task Type II (PBA) Sub Claim D, Task Type III (PBA) c. Add and subtract mixed numbers with like denminatrs, e.g., by replacing each mixed number with an equivalent fractin, and/r by using prperties f peratins and the relatinship between additin and subtractin. 4.NF.3c Fractins with like denminatrs can be added and subtracted by using prperties f peratins and the relatinship between additin and subtractin. While slving the prblem, 3. + 2. students culd d the fllwing: Mixed numbers 7. Lk fr and make use f structure 8/20/2014 30

Student 1 3 + 2 = 5 and 3. + 1. = 1 s 5+ 1 = 6 4 4 Student 2 3 3. + 2 = 5 3. s 5 3. + 1. = 6 4 4 4 4 Student 3 3 3. = 15 and 2 1. = 9 s 15 + 9 = 24 = 6 4 4 4 4 4 4 4 PARCC Clarificatin EOY Understand a fractin /ab with 1a> as a sum f fractins 1/b. c. Add and subtract mixed numbers with like denminatrs, e.g., by replacing each mixed number with an equivalent fractin, and/r by using prperties f peratins and the relatinship between additin and subtractin. Tasks d nt have a cntext. Denminatrs are limited t grade 3 pssibilities (2, 3, 4, 6, 8) s as t keep cmputatinal difficulty lwer CCSS ftnte, p. 24). Sub Claim A, Task Type I (EOY) Sub Claim C, Task Type II (PBA) d. Slve wrd prblems invlving additin and subtractin f fractins referring t the same whle and having like denminatrs, e.g., by using visual fractin mdels and equatins t represent the prblem. 4.NF.3d. Susan and Maria need 8 3/8 feet f ribbn t package gift baskets. Susan has 3 1/8 feet f ribbn and Maria has 5 3/8 feet f ribbn. Hw much ribbn d they have altgether? Will it be enugh t cmplete the prject? Explain why r why nt. The student thinks: I can add the ribbn Susan has t the ribbn Maria has t find ut hw much ribbn they have 1. Make sense f prblems and persevere in slving them 4. Mdel with Mathematics 8/20/2014 31

altgether. Susan has 31/8 feet f ribbn and Maria has 5 3/8 feet f ribbn. I can write this as 31/8 + 53/8. I knw they have 8 feet f ribbn by adding the 3 and 5. They als have 1/8 and 3/8 which makes a ttal f 4/8 mre. Altgether they have 84/8 feet f ribbn. 84/8 is larger than 83/8 s they will have enugh ribbn t cmplete the prject. They will even have a little extra ribbn left, 1/8 ft. Trevr has 4 1/8 pizzas left ver frm his sccer party. After giving sme pizza t his friend, he has 2 4/8 f a pizza left. Hw much pizza did Trevr give t his friend? Slutin: Trevr had 4 1/8 pizzas t start. This is 33/8 f a pizza. The x s shw the pizza he has left which is 2 4/8 pizzas r 20/8 pizzas. The shaded rectangles withut the x s are the pizza he gave t his friend which is 13/8 r 1 5/8 pizzas. 5. Use apprpriate tls strategically A cake recipe calls fr yu t use. cup f milk,. cup f il, and 2/4 cup f water. Hw much liquid was needed t make the cake? 8/20/2014 32

PARCC Clarificatin EOY Understand a fractin with as a sum f fractins. /ab 1a> 1/b d. Slve wrd prblems invlving additin and subtractin f fractins referring t the same whle and having like denminatrs, e.g., by using visual fractin mdels and equatins t represent the prblem. Tasks are limited t denminatrs 2, 3, 4, 5, 6, 8, 10, 12, and 100 (CCSS ftnte, p. 30). Additin and subtractin situatins are limited t the darkr medium-shaded types in Table 2, p. 9 f the Prgressin fr Operatins and Algebraic Thinking; these situatins are sampled equally. 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 (PBA & EOY) Sub Claim C, Task Type II (PBA) Sub Claim D, Task Type III (PBA) http://www.pusmath.cm/cmmn-cre-clusters/4.nf.bbuild-fractins-frm-unit-fractins M 4.NF.4 Apply and extend previus understandings f multiplicatin t multiply a fractin by a whle number. Majr cntent a. Understand a fractin a/b as a multiple f 1/b. Fr example, use a visual fractin mdel t represent 5/4 as the prduct 5 (1/4), recrding the cnclusin by the equatin 5/4 = 5 (1/4). 4.NF.4a Multiplicatin f a fractin by a whle number can be mdeled as repeated additin f the unit fractin. This standard builds n students wrk f adding fractins and extending that wrk int multiplicatin. Example: 3/6 = 1/6 + 1/6 + 1/6 = 3 x (1/6) Multiple Prduct 5. Use apprpriate tls strategically 8/20/2014 33

Number line: 7. Lk fr and make use f structure Students shuld see a fractin as the numeratr times the unit fractin with the same denminatr. Example: (Prgressins fr the CCSSM, Number and Operatin Fractins, CCSS Writing Team, August 2011, page 8) PARCC Clarificatin Apply and extend previus understandings f multiplicatin t multiply a fractin by a whle number. a. Understand a fractin a/b as a multiple f 1/b. Fr example, use a visual fractin mdel t represent 5/4 as the prduct 1 5x 4 5 1 = 5x 4 4 equatin., recrding the cnclusin by the Tasks are limited t denminatrs 2, 3, 4, 5, 6, 8, 10, 12, and 100 (CCSS ftnte, p. 30). Sub Claim A, Task Type I (PBA& EOY) Sub Claim C, Task Type II (PBA) Sub Claim D, Task Type III (PBA) b. Understand a multiple f a/b as a multiple f 1/b, and use this understanding t multiply a fractin by a whle number. Fr example, use a visual fractin mdel t express 3 (2/5) as 6 (1/5), recgnizing this prduct as 6/5. (In general, n (a/b) = (n a)/b.) 4.NF.4b. 3 x (2/5) = 6 x (1/5) = 6/5 5. Use apprpriate tls strategically 7. Lk fr and make 8/20/2014 34

use f structure PARCC Clarificatin EOY 4.NF.4b-1 Apply and extend previus understandings f multiplicatin t multiply a fractin by a whle number. b. Use the understanding that a multiple f /ab is a multiple f 1/.b t multiply a fractin by a whle number. Fr example, use a visual fractin mdel t express 2 3x as 5 1 6x. 5 Tasks d nt have a cntext. Prmpts d nt prvide visual fractin mdels; students may at their discretin draw visual fractin mdels as a strategy. Tasks invlve expressing a multiple f /abas a fractin. Results may equal fractins greater than 1 (including thse equal t whle numbers). Tasks are limited t denminatrs 2, 3, 4, 5, 6, 8, 10, 12, and 100 (CCSS ftnte, p. 30). 4.NF.4b-2 Apply and extend previus understandings f multiplicatin t multiply a fractin by a whle number. b. Use the understanding that a multiple f /ab is a multiple f 1/.b t multiply a fractin by a whle number. Fr example, use a visual fractin mdel t express 2 3x as 6/5 (In general, 5 a n x a n x b b Tasks d nt have a cntext. Prmpts d nt prvide visual fractin mdels; students may at their discretin draw visual fractin mdels as a strategy. Tasks invlve expressing a multiple f a/b as a fractin. Results may equal fractins greater than 1 (including fractins equal t whle numbers). Tasks are limited t denminatrs 2, 3, 4, 5, 6, 8, 10, 12, 8/20/2014 35

and 100 (CCSS ftnte, p. 30). Sub Claim A, Task Type I (PBA& EOY) Sub Claim C, Task Type II (PBA) Sub Claim D, Task Type III (PBA) http://www.pusmath.cm/cmmn-cre-clusters/4.nf.bbuild-fractins-frm-unit-fractins c. Slve wrd prblems invlving multiplicatin f a fractin by a whle number, e.g., by using visual fractin mdels and equatins t represent the prblem. 5.NF.4c. Fr example, if each persn at a party will eat 3/8 f a pund f rast beef, and there will be 5 peple at the party, hw many punds f rast beef will be needed? Between what tw whle numbers des yur answer lie? Heather bught 12 plums and ate 1 f them. Paul bught 12 3 plums and ate 4 1 f them. Which statement is true? Draw a mdel t explain yur reasning. a. Heather and Paul ate the same number f plums. b. Heather ate 4 plums and Paul ate 3 plums. c. Heather ate 3 plums and Paul ate 4 plums. d. Heather had 9 plums remaining Wrd prblems 1. Make sense f prblems and persevere in slving them 4. Mdel with Mathematics 5. Use apprpriate tls strategically PARCC Clarificatin EOY Prmpts d nt prvide visual fractin mdels; students may at their discretin draw visual fractin mdels as a strategy. Situatins are limited t thse in which the prduct is unknwn (situatins d nt include thse with an unknwn factr). Situatins invlve a whle number f fractinal quantities, nt a fractin f a whle-number quantity. Results may equal fractins greater than 1 (including fractins equal t whle numbers). Tasks are limited t denminatrs 2, 3, 4, 5, 6, 8, 10, 12, and 100 (CCSS ftnte, p. 30). Sub Claim A, Task Type I (PBA & EOY) Sub Claim C, Task Type II (PBA) Sub Claim D, Task Type III (PBA) 8/20/2014 36

http://www.pusmath.cm/cmmn-cre-clusters/4.nf.bbuild-fractins-frm-unit-fractins NUMBER AND OPERATIONS FRACTIONS (4.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 reasning M M Students understand decimal ntatin fr fractins, and cmpare decimal fractins. 4.NF.5 Express a fractin with denminatr 10 as an equivalent fractin with denminatr 100, and use this technique t add tw fractins with respective denminatrs 10 and 100.4. Majr cntent Fractins with denminatrs f 10 can be expressed as equivalent fractins with the denminatr f 100, and can be used as a strategy fr adding decimal fractins. Fr example, express 3/10 as 30/100, and add 3/10 + 4/100 = 34/100. PARCC Clarificatin EOY Tasks d nt have a cntext. Sub Claim A, Task Type I (EOY) http://www.parccnline.rg/sites/parcc/files/parcc_sample Items_Mathematics_G4FractinMdel_081913_Final.pdf Decimal fractin 7. Lk fr and make use f structure 4.NF.6 Use decimal ntatin fr fractins with denminatrs 10 r 100. Majr cntent. Decimal fractins can be recrded with decimal ntatin and can be illustrated n a number line. Fr example, rewrite 0.62 as 62/100; describe a length as 0.62 meters; lcate 0.62 n a number line diagram Students make cnnectins between fractins with denminatrs f 10 and 100 and the place value chart. By reading fractin names, students say 32/100 as thirty-tw hundredths and rewrite this as 0.32 r represent it n a place value mdel as shwn belw. Decimal ntatin 7. Lk fr and make use f structure TEACHER NOTES See instructinal strategies in the intrductin Grade 4 expectatins in this dmain are limited t fractins with denminatrs 2, 3, 4, 5, 6, 8, 10, 12, and 100. Students wh can generate equivalent fractins can develp strategies fr adding fractins with unlike denminatrs in general. But additin and subtractin with unlike denminatrs in general is nt a requirement at this grade. Students need t make cnnectins between fractins and decimals. They shuld be able t write decimals fr fractins with denminatrs f 10 r 100. Have students say the fractin with denminatrs f 10 and 100 alud. Fr example 410 wuld fur tenths r wuld be twenty-seven hundredths. Students shuld be able t express decimals t the hundredths as the sum f tw decimals r fractins. In decimal numbers, the value f each place is 10 times the value f the place t its RESOURCE NOTES See resurces in the intrductin Refer t Algebra I PBA/MYA @ Live Binder http://www.livebinders.c m/play/play/1171650 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 8/20/2014 37

Hundre ds Tens Ones Tenths Hundre dths 3 2 Students use the representatins explred in 4.NF.5 t understand 32/100 can be expanded t 3/10 and 2/100. Students represent values such as 0.32 r 32/100 n a number line. 32/100 is mre than 30/100 (r 3/10) and less than 40/100 (r 4/10). It is clser t 30/100 s it wuld be placed n the number line near that value. immediate right. ODE PARCC Clarificatin EOY Measuring t the nearest mm r cm is equivalent t measuring n the number line. Sub Claim A, Task Type I (EOY) http://cmmncresheets.cm/srtedbygrade.php?srted= 4nf6 M 4.NF.7 Cmpare tw decimals t hundredths by reasning abut their size. Majr cntent Recgnize that cmparisns are valid nly when the tw decimals refer t the same whle. Recrd the results f cmparisns with the symbls >, =, r <, and justify the cnclusins, e.g., by using a visual mdel. Cmparisns are nly valid when the tw decimals refer t the same whle. Draw a mdel t shw that 0.3 < 0.5. (Students wuld sketch tw mdels f apprximately the same size t shw the area that represents three-tenths is smaller than the area that represents five-tenths. Hundredths Reasning 5. Use apprpriate tls strategically 7. Lk fr and make use f structure 8/20/2014 38

PARCC Clarificatin EOY Tasks have thin cntext r n cntext. Justifying cnclusins 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) MEASUREMENT AND DATA (4.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 S http://www.kcps.rg/prtals/okcs/district/dcs/pacing%20 Guides/Furth%20Grade/4th%20Grade%20Pearsn%20Perf rmance%20tasks.pdf Students slve prblems invlving measurement and cnversin f measurements frm a larger unit t a smaller unit. 4.MD.1 Knw relative sizes f measurement units within ne system f units including km, m, cm; kg, g; lb, z.; l, ml; hr, min, sec. Supprting cntent Within a single system f measurement, express measurements in a larger unit in terms f a smaller unit. Recrd measurement equivalents in a tw clumn table. There are tw distinct systems f measurement with unique units f measure fr each ne, Metric and Custmary (smetimes referred t as U.S. Custmary). Fr example, knw that 1 ft is 12 times as lng as 1 in. Express the length f a 4 ft snake as 48 in. Generate a cnversin table fr feet and inches listing the number pairs (1, 12), (2, 24), (3, 36),... The units f measure that have nt been addressed in prir years are punds, unces, kilmeters, milliliters, and secnds. Students prir experiences were limited t measuring length, mass, liquid vlume, and elapsed time. Students did nt cnvert measurements. Students need ample pprtunities t becme familiar with these new units f measure. Students may use a tw-clumn chart t cnvert frm larger t smaller units and recrd equivalent measurements. They make statements such as, if ne ft is 12 inches, then 3 feet has t be 36 inches because there are 3 grups f 12. km, m, cm; kg, g; lb, z.; l, ml; hr, min, sec. equivalent Measurement 5. Use apprpriate tls strategically 8. 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 PBA/MYA @ Live Binder http://www.livebinders.c m/play/play/1171650 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 8/20/2014 39

kg g 1 1000 2 2000 3 3000 ft. in 1 12 2 24 3 36 lb z 1 16 2 32 3 48 PARCC Clarificatin EOY - NONE Sub Claim B, Task Type I (EOY) http://cmmncresheets.cm/math/measurement/cnv ersin%20tables/1.pdf S 4.MD.2 Use the fur peratins t slve wrd prblems invlving distances, intervals f time, liquid vlumes, masses f bjects, and mney, including prblems invlving simple fractins r decimals, and prblems that require expressing measurements given in a larger unit in terms f a smaller unit. Supprting cntent Represent measurement quantities using diagrams such as number line diagrams that feature a measurement scale. 8/20/2014 40

Units f measure can be expressed as whle numbers, decimals and fractins, (e.g., ne inch equals 1/12th f a ft, 1 gram is.01 kilgram). Additin: Masn ran fr an hur and 15 minutes n Mnday, 25 minutes n Tuesday, and 40 minutes n Wednesday. What was the ttal number f minutes Masn ran? Subtractin: A pund f apples csts $1.20. Rachel bught a pund and a half f apples. If she gave the clerk a $5.00 bill, hw much change will she get back? Multiplicatin: Mari and his 2 brthers are selling lemnade. Mari brught ne and a half liters, Javier brught 2 liters, and Ernest brught 450 milliliters. Hw many ttal milliliters f lemnade did the bys have? PARCC Clarificatin EOY 4.MD.2-1 Use the fur peratins t slve wrd prblems invlving distances, intervals f time, liquid vlumes, masses f bjects, and mney, in prblems that require expressing measurements given in a larger unit in terms f a smaller unit. Represent measurement quantities using diagrams such as number line diagrams that feature a measurement scale. Situatins invlve whle-number measurements and require expressing measurements given in a larger unit in terms f a smaller unit. Tasks may present number line diagrams featuring a measurement scale. Tasks may include measuring t the nearest cm r mm. 4.MD.2-2 Use the fur peratins t slve wrd prblems invlving distances, intervals f time, liquid vlumes, masses f bjects, and mney, in prblems invlving simple fractins r decimals. Represent measurement quantities using diagrams such as number line diagrams that feature a measurement scale. Situatins invlve tw measurements given in the same units, ne a whle-number measurement and the ther a nn-whle-number measurement (given as a fractin r a decimal). Tasks may present number line diagrams featuring a measurement scale. Tasks may include measuring distances t the nearest cm r mm. Diagrams Distances Intervals f time Liquid vlumes Mass f bjects Mney 4. Mdel with Mathematics 5. Use apprpriate tls strategically 8/20/2014 41

Sub Claim B, Task Type I (EOY) http://www.mathwrksheetsland.cm/4/24largerunits/ip. pdf 4.MD.3 Apply the area and perimeter frmulas fr rectangles in real wrld and mathematical prblems. Units f measure can be expressed as whle numbers, decimals and fractins, (e.g., ne inch equals 1/12th f a ft, 1 gram is.01 kilgram). The area f a rectangular can be calculated when the lengths f tw f the sides f the rectangle are knwn. Knwing the area and the length f ne side f a rectangle enables ne t determine the lengths f the ther three sides. Fr example, find the width f a rectangular rm given the area f the flring and the length, by viewing the area frmula as a multiplicatin equatin with an unknwn factr. Students learn t apply these understandings and frmulas t the slutin f real-wrld and mathematical prblems. Example: A rectangular garden has as an area f 80 square feet. It is 5 feet wide. Hw lng is the garden? Here, specifying the area and the width creates an unknwn factr prblem. Similarly, students culd slve perimeter prblems that give the perimeter and the length f ne side and ask the length f the adjacent side. Students shuld be challenged t slve multistep prblems. Example: A plan fr a huse includes rectangular rm with an area f 60 square meters and a perimeter f 32meters. What are the length and the width f the rm? Area Frmulas Length Linear units Perimeter Rectangles Square units Width 2. Reasn abstractly and quantitatively 5. Use apprpriate tls strategically PARCC Clarificatin EOY - NONE Sub Claim B, Task Type I (EOY) http://www.insidemathematics.rg/cmmn-cre-math- 8/20/2014 42

tasks/4th-grade/4-2009%20fair%20play.pdf http://www.recity.k12.r.us/files/4th_grade_assessme nt_4_5_areaper.pdf MEASUREMENT AND DATA (4.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 S Students represent and interpret data. 4.MD.4 Make a line plt t display a data set f measurements in fractins f a unit (1/2, 1/4, 1/8). Supprting cntent Slve prblems invlving additin and subtractin f fractins by using infrmatin presented in line plts. Data can be cllected and represented in many ways, including graphs r line plts. Data can be interpreted, analyzed and cmpared using graphs r line plts. The fundatin f a line plt is a number line. Data sets f measurements are recrded with an X abve the crrespnding value. Fr example, frm a line plt find and interpret the difference in length between the lngest and shrtest specimens in an insect cllectin. Ten students in Rm 31 measured their pencils at the end f the day. They recrded their results n the line plt belw. Data set Display Line plt 4. Mdel with Mathematics 5. Use apprpriate tls strategically TEACHER NOTES See instructinal strategies in the intrductin Data has been measured and represented n line plts in units f whle numbers, halves r quarters. Students have als represented fractins n number lines. Nw students are using line plts t display measurement data in fractin units and using the data t slve prblems invlving additin r subtractin f fractins. Instructinal Strategies Have students create line plts with fractins f a unit (,, ) and plt data shwing multiple data pints fr each fractin. ODE RESOURCE NOTES See resurces in the intrductin Refer t Algebra I PBA/MYA @ Live Binder http://www.livebinders.c m/play/play/1171650 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 X X X X X X X X X X 3 ½ 4 4 ¼ 5 1/8 5 1/2 Pssible questins: What is the difference in length frm the lngest t the shrtest pencil? If yu were t line up all the pencils, what wuld the ttal length be? If the 51/8 pencils are placed end t end, what wuld be their ttal length? Pse questins that students may answer, such as Hw many ne-eighths are shwn n the line plt? Expect tw ne-eighths as the answer. Then ask, What is the ttal f these tw neeighths? Encurage students t cunt the fractinal numbers as they wuld with whle-number cunting, but using the fractin name. What is the ttal number f inches fr insects 8/20/2014 43

PARCC Clarificatin EOY -Nne Sub Claim B, Task Type I (EOY) http://www.mathwrksheetsland.cm/4/26measfrac/ip.pdf measuring inches? Students can use skip cunting with fractin names t find the ttal, such as, three-eighths, six-eighths, nine-eighths. The last fractin names the ttal. Students shuld ntice that the denminatr did nt change when they were saying the fractin name. have them make a statement abut the result f adding fractins with the same denminatr. What is the ttal number f insects measuring inch r inches? Have students write number sentences t represent the prblem and slutin such as, + + = inches. Use visual fractin strips and fractin bars t represent prblems t slve prblems invlving additin and subtractin f fractins. ODE MEASUREMENT AND DATA (4.MD) Students understand cncepts f angle and measure angles (gemetric measurement). TEACHER NOTES RESOURCE NOTES ASSESSMENT NOTES 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 4.MD.5 Recgnize angles as gemetric shapes that are frmed wherever tw rays share a cmmn endpint, and understand cncepts f angle measurement. Additinal cntent a. An angle is measured with reference t a circle with its center at the cmmn endpint f the rays, by cnsidering the fractin f the circular arc between the pints where the tw rays intersect the circle. An angle that turns thrugh 1/360 f a circle is called a ne-degree angle, and can be used t measure angles. 4.MD.5a An angle is frmed when tw rays share a cmmn endpint. A 360 degree rtatin arund a pint makes a cmplete circle. Angles Center Circle Degree Endpint Intersect See instructinal strategies in the intrductin Students can als create an angle explrer (tw strips f cardbard attached with a brass fastener) t learn abut angles. They can use the angle explrer t get a feel f the See resurces in the intrductin Refer t Algebra I PBA/MYA @ Live Binder http://www.livebinders.c m/play/play/1171650 fr mid-year evidence statements and clarificatin REQUIRED PARCC Released Test Prblems Cmmn Unit Assessment Cmmn Tasks NWEA Test Perfrmance Level Descriptrs (PARCC See assessments in the intrductin 8/20/2014 44

Rays Mathematical Practices 2. Reasn abstractly and quantitatively relative size f angles as they rtate the cardbard strips arund. ODE (Prgressins fr the CCSSM, Gemetric Measurement, CCSS Writing Team, June 2012, page 23) PARCC Clarificatin EOY - NONE Sub Claim B, Task Type I (EOY) http://www.pusmath.cm/cmmn-cre-clusters/4.md.cangle-measurement b. An angle that turns thrugh n ne-degree angles is said t have an angle measure f n degrees. 4.MD.5b. 2. Reasn abstractly PARCC Clarificatin EOY - NONE and quantitatively 5. Use apprpriate tls strategically Sub Claim B, Task Type I (EOY) Calculatr - http://www.pusmath.cm/cmmn-cre-clusters/4.md.cangle-measurement A 4.MD.6 Measure angles in whle-number degrees using a prtractr. Sketch angles f specified measure. Additinal cntent Students shuld measure angles and sketch angles Benchmark angles include 45, 90, 180 and 360 degree angles. Angles can be classified and srted by the degrees f their angles (acute, btuse, right, and straight). Prtractr sketch 8/20/2014 45

Knwledge f benchmark angles can be used t find the measure f an unknwn angle. Befre students begin measuring angles with prtractrs, they need t have sme experiences with benchmark angles. They transfer their understanding that a 360º rtatin abut a pint makes a cmplete circle t recgnize and sketch angles that measure apprximately 90º and 180º. They extend this understanding and recgnize and sketch angles that measure apprximately 45º and 30º. They use apprpriate terminlgy (acute, right, and btuse) t describe angles and rays (perpendicular). PARCC Clarificatin EOY - NONE Sub Claim B, Task Type I (EOY) http://www.pusmath.cm/cmmn-cre-clusters/4.md.cangle-measurement 2. Reasn abstractly and quantitatively 5. Use apprpriate tls strategically A 4.MD.7 Recgnize angle measure as additive. When an angle is decmpsed int nn-verlapping parts, the angle measure f the whle is the sum f the angle measures f the parts. Additinal cntent Slve additin and subtractin prblems t find unknwn angles n a diagram in real wrld and mathematical prblems, e.g., by using an equatin with a symbl fr the unknwn angle measure.. Cmplementary angles add t 90 0 and supplementary angles add t 180 0 If the tw rays are perpendicular, what is the value f m? Additive Decmpse Nn-verlapping parts 1. Make sense f prblems and persevere in slving them 7. Lk fr and make use f structure PARCC Clarificatin EOY - NONE 8/20/2014 46

Sub Claim B, Task Type I (EOY) GEOMETRY (4.G) http://www.pusmath.cm/cmmn-cre-clusters/4.md.cangle-measurement Students draw and identify lines and angles, and classify shapes by prperties f their lines and angles. TEACHER NOTES RESOURCE NOTES ASSESSMENT NOTES 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 4.G.1 Draw pints, lines, line segments, rays, angles (right, acute, btuse), and perpendicular and parallel lines. Identify these in tw-dimensinal figures. Additinal cntent Tw-dimensinal gemetric figures can be analyzed, classified and cmpared based n their prperties (i.e., symmetry, parallel sides, particular angle measures, and perpendicular sides). f pints, line segments, lines, angles, parallelism, and perpendicularity can be seen daily. Students d nt easily identify lines and rays because they are mre abstract. Acute Angles Line segments Lines Obtuse Parallel lines Perpendicular Pints Rays Right Tw-dimensinal 5. Use apprpriate tls strategically See instructinal strategies in the intrductin Tw-dimensinal shapes are classified based n relatinships by the angles and sides. Students can determine if the sides are parallel r perpendicular, and classify accrdingly. Characteristics f rectangles (including squares) are used t develp the cncept f parallel and perpendicular lines. The characteristics and understanding f parallel and perpendicular lines are used t draw rectangles. Repeated experiences in cmparing and cntrasting shapes enable students t gain a deeper understanding abut shapes and their prperties. See resurces in the intrductin Refer t Algebra I PBA/MYA @ Live Binder http://www.livebinders.c m/play/play/1171650 fr mid-year evidence statements and clarificatin REQUIRED PARCC Released Test Prblems Cmmn Unit Assessment Cmmn Tasks NWEA Test Perfrmance Level Descriptrs (PARCC See assessments in the intrductin PARCC Clarificatin EOY - NONE http://www.pusmath.cm/cmmn-cre-clusters/4.g.alines-and-angles http://www.insidemathematics.rg/cmmn-cre-mathtasks/4th-grade/4-2008%20quilt%20making.pdf Infrmal understanding f the characteristics f triangles is develped thrugh angle measures and side length relatinships. Triangles are named accrding t their angle measures (right, acute r btuse) and side lengths (scalene, issceles r equilateral). These characteristics are used t draw triangles. ODE 8/20/2014 47

A 4.G.2 Classify tw-dimensinal figures based n the presence r absence f a parallel r perpendicular lines, r the presence r absence f angles f specified size. Recgnize right triangles as a categry, and identify right triangles. Additinal cntent Tw-dimensinal figures may be classified using different characteristics such as, parallel r perpendicular lines r by angle measurement Parallel r Perpendicular Lines: Students shuld becme familiar with the cncept f parallel and perpendicular lines. Tw lines are parallel if they never intersect and are always equidistant. Tw lines are perpendicular if they intersect in right angles (90º). Right triangles 7. Lk fr and make use f structure Which figure in the Venn diagram belw is in the wrng place, explain hw d yu knw? D yu agree with the label n each f the circles in the Venn diagram abve? Describe why sme shapes fall in the verlapping sectins f the circles. Draw and name a figure that has tw parallel sides and exactly 2 right angles. Fr each f the fllwing, sketch an example if it is pssible. If it is impssible, say s, and explain why r shw a cunter example. A parallelgram with exactly ne right angle. An issceles right triangle. A rectangle that is nt a parallelgram. (impssible) Every square is a quadrilateral. Every trapezid is a parallelgram. Identify which f these shapes have perpendicular r parallel sides and justify yur selectin. 8/20/2014 48

A pssible justificatin that students might give is: The square has perpendicular lines because the sides meet at a crner, frming right angles. Angle Measurement: This expectatin is clsely cnnected t 4.MD.5, 4.MD.6, and 4.G.1. Students experiences with drawing and identifying right, acute, and btuse angles supprt them in classifying tw-dimensinal figures based n specified angle measurements. They use the benchmark angles f 90, 180, and 360 t apprximate the measurement f angles. Right triangles can be a categry fr classificatin. A right triangle has ne right angle. There are different types f right triangles. An issceles right triangle has tw r mre cngruent sides and a scalene right triangle has n cngruent sides. 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) http://www.pusmath.cm/cmmn-cre-clusters/4.g.alines-and-angles http://www.insidemathematics.rg/cmmn-cre-mathtasks/4th-grade/4-2008%20quilt%20making.pdf A 4.G.3 Recgnize a line f symmetry fr a tw-dimensinal figure as a line acrss the figure such that the figure can be flded alng the line int matching parts Identify line-symmetric figures and draw lines f symmetry. Additinal cntent A line f symmetry is made when a shape r figure is Line-symmetrical 8/20/2014 49