Emphases in Cmmn Cre Standards fr Mathematical Cntent Kindergarten High Schl Cntent Emphases by Cluster March 12, 2012 Describes cntent emphases in the standards at the cluster level fr each grade. These are prvided because curriculum, instructin and assessment at each grade must reflect the fcus and emphasis f the standards. Nt all f the cntent in a given grade is emphasized equally in the standards. The list f cntent standards fr each grade is nt a flat, ne dimensinal checklist; this is by design. There are smetimes strng differences f emphasis even within a single dmain. Sme clusters require greater emphasis than the thers based n the depth f the ideas, the time that they take t master, and/r their imprtance t future mathematics r the demands f cllege and career readiness. In additin, an intense fcus n the mst critical material at each grade allws depth in learning, which is carried ut thrugh the Standards fr Mathematical Practice. Withut such fcus, attentin t the practices wuld be difficult and unrealistic, as wuld best practices like frmative assessment. Therefre, t make relative emphases in the standards mre transparent and useful, the NY Math Emphases designate clusters as Majr, Additinal and Supprting fr the grade in questin. T say that sme things have greater emphasis is nt t say that anything in the standards can safely be neglected in instructin. Neglecting material will leave gaps in student skill and understanding and may leave students unprepared fr the challenges f a later grade. The assessments will mirrr the message that is cmmunicated here: Majr Clusters will be a majrity f the assessment, Supprting Clusters will be assessed thrugh their success at supprting the Majr Clusters and Additinal Clusters will be assessed as well. The assessments will strngly fcus where the standards strngly fcus. In additin t identifying the Majr, Additinal and Supprting Clusters fr each grade, suggestins are given in each grade fr ways t cnnect the Supprting Clusters t the Majr Clusters f the grade. Thus, rather than suggesting even inadvertently that sme material nt be taught, there is direct advice fr teaching it in ways that fster greater fcus and cherence. Finally, the fllwing are sme recmmendatins fr using the cluster level emphases: D Use the guidance t infrm instructinal decisins regarding time and ther resurces spent n clusters f varying degrees f emphasis. Allw the fcus n the majr wrk f the grade t pen up the time and space t bring the Standards fr Mathematical Practice t life in mathematics instructin thrugh sense making, reasning, arguing and critiquing, mdeling, etc.
Evaluate instructinal materials taking the cluster level emphases int accunt. The majr wrk f the grade must be presented with the highest pssible quality; the supprting wrk f the grade shuld indeed supprt the majr fcus, nt detract frm it. Set pririties fr ther implementatin effrts taking the emphases int accunt, such as staff develpment; new curriculum develpment; r revisin f existing frmative r summative testing at the state, district r schl level. Dn t Neglect any material in the standards. (Instead, use the infrmatin prvided t cnnect Supprting Clusters t the ther wrk f the grade.) Srt clusters frm Majr t Supprting, and then teach them in that rder. T d s wuld strip the cherence f the mathematical ideas and miss the pprtunity t enhance the majr wrk f the grade with the supprting clusters. Use the cluster headings as a replacement fr the standards. All features f the standards matter frm the practices t surrunding text t the particular wrding f individual cntent standards. Guidance is given at the cluster level as a way t talk abut the cntent with the necessary specificity yet withut ging s far int detail as t cmprmise the cherence f the standards. Explanatins f terms used: Majr clusters areas f intensive fcus, where students need fluent understanding and applicatin f the cre cncepts (apprximately 70%). Supprting clusters rethinking and linking; areas where sme material is being cvered, but in a way that applies cre understandings (apprximately 20%). Additinal Clusters expse students t ther subjects, thugh at a distinct, level f depth and intensity (apprximately 10%).
Cunting and Cardinality Kindergarten Majr Supprting Additinal Knw number names and cunt sequence. Cunt t tell the number f bjects. Cmpare numbers. Operatins and Algebraic Thinking Understand additin as putting tgether and adding t, and understand subtractin as taking apart and taking frm. Number and Operatins in Base Ten Wrk with numbers 11 19 t grain fundatins fr place value. Depth Opprtunities: CC 4, 5, 6; OA 2, 4 Gemetry Identify and describe shapes. Analyze, cmpare, create, and cmpse shapes. Describe and cmpare measurable attributes. Classify bjects in categries.
Operatins and Algebraic Thinking Grade 1 Majr Supprting Additinal Represent and slve prblems invlving additin and subtractin. Understand and apply prperties f peratins and the relatinship between additin and subtractin. Add and subtract within 20. Wrk with additin and subtractin equatins. Number and Operatins in Base Ten Extend the cunting sequence. Understand place value. Use place value understanding and prperties f peratins t add and subtract. Measure lengths indirectly and by iterating length units. Depth Opprtunities: OA 1, 6; NBT 2, 4; MD 2 Gemetry Reasn with shapes and their attributes. Tell and write time. Represent and interpret data.
Operatins and Algebraic Thinking Grade 2 Majr Supprting Additinal Represent and slve prblems invlving additin and subtractin. Add and subtract within 20. Wrk with equal grups f bjects t gain fundatins fr multiplicatin. Number and Operatins in Base Ten Understand place value. Use place value understanding and prperties f peratins t add and subtract. Measure and estimate lengths in standard units. Relate additin and subtractin t length. Depth Opprtunities: OA 1, 2; NBT 1, 7; MD 5 Gemetry Reasn with shapes and their attributes. Wrk with time and mney. Represent and interpret data.
Operatins and Algebraic Thinking Grade 3 Majr Supprting Additinal Represent and slve prblems invlving multiplicatin and divisin. Understand the prperties f multiplicatin and the relatinship between multiplicatin and divisin. Multiply and divide within 100. Slve prblems invlving the fur peratins, and identify and explain patterns in arithmetic. Number and Operatins Fractins Develp understanding f fractins as numbers. Slve prblems invlving measurement and estimatin f intervals f time, liquid vlumes, and masses f bjects. Gemetric measurement: understand cncepts f area and relate area t multiplicatin and t additin. Gemetry Reasn with shapes and their attributes. 1 Represent and interpret data. 2 Number and Operatins in Base Ten Use place value understanding and prperties f peratins t perfrm multi digit arithmetic. Gemetric measurement: recgnize perimeter as an attribute f plane figures and distinguish between linear and area measures. Depth Opprtunities: OA 3, 6; NF 3; MD 2, 7 1 Wrk shuld be psitined in supprt f area measurement and understanding f fractins. 2 Students multiple and divide t slve prblems using infrmatin presented in scaled bar graphs. Pictgraphs and scaled bar graphs are a visually appealing cntext fr ne and tw step wrd prblems.
Operatins and Algebraic Thinking Grade 4 Majr Supprting Additinal Use the fur peratins with whle numbers t slve prblems. Number and Operatins in Base Ten Generalize place value understanding fr multi digit whle numbers. Use place value understanding and prperties f peratins t perfrm multi digit arithmetic. Number and Operatins Fractins Extend understanding f fractin equivalence and rdering. Build fractins frm unit fractins by applying and extending previus understandings f peratins n whle numbers. Understand decimal ntatin fr fractins, and cmpare decimal fractins. Depth Opprtunities: NBT 5, 6; NF 1, 3, 4 Operatins and Algebraic Thinking Gain familiarity with factrs and multiples. 3 Slve prblems invlving measurement and cnversin f measurements frm a larger unit t a smaller unit. Represent and interpret data. 4 Operatins and Algebraic Thinking Generate and analyze patterns. Gemetric measurement: understand cncepts f angles and measure angles. Gemetry Draw and identify lines and angles, and classify shapes by prperties f their lines and angles. 3 Wrk in this cluster supprts students wrk with multi digit arithmetic as well as their wrk with fractin equivalence. 4 The standard in this cluster requires students t use a line plt t display measuresments in fractins f a unit and t slve prblems invlving additin and subtractin f fractins, cnnecting it directly t the Number and Operatins Fractins clusters.
Grade 5 Majr Supprting Additinal Number and Operatins in Base Ten Understand the place value system. Perfrm peratins with multi digit whle numbers and with decimals t hundredths. Number and Operatins Fractins Use equivalent fractins as a strategy t add and subtract fractins. Apply and extend previus understandings f multiplicatin and divisin t multiply and divide fractins. Gemetric measurement: understand cncepts f vlume and relate vlume t multiplicatin and t additin. Represent and interpret data. 5 Cnvert like measurement units within a given measurement system. 6 Operatins and Algebraic Thinking Write and interpret numerical expressins. Analyze patterns and relatinships. Gemetry Graph pints n the crdinate plane t slve real wrld and mathematical prblems. Classify tw dimensinal figures int categries based n their prperties. Depth Opprtunities: NBT 1, 6; NF 2, 4; MD 5 5 The standard in this cluster prvides an pprtunity fr slving real wrld prblems with peratins n fractins, cnnecting directly t bth number and Operatins Fractins clusters. 6 Wrk in these standards supprts cmputatin with decimals. Fr example, cnverting 5 cm t.05 m invlves cmputatin with decimals t hundredths.
Ratis and Prprtinal Relatinships Grade 6 Majr Supprting Additinal Understand rati cncepts and use rati reasning t slve prblems. The Number System Apply and extend previus understandings f numbers t the system f ratinal numbers. Apply and extend previus understandings f multiplicatin and divisin t divide fractins by fractins. Expressins and Equatins Apply and extend previus understandings f arithmetic t algebraic expressins. Reasn abut and slve nevariable equatins and inequalities. Represent and analyze quantitative relatinships between dependent and independent variables. Depth Opprtunities: RP 3; NS 1; NS 8; EE 3, 7 Gemetry Slve real wrld and mathematical prblems invlving area, surface area, and vlume. 7 Statistics and Prbability Develp understanding f statistical variability. Summarize and describe distributins. The Number System Cmpute fluently with multidigit numbers and find cmmn factrs and multiples. 7 In this cluster, students wrk n prblems with areas f triangles and vlumes f right rectangular prisms, which cnnects t wrk in the Expressins and Equatins dmain. In additin, anther standard within this cluster asks students t draw plygns in the crdinate plane, which supprts wrk with the crdinate plane in the Number System dmain.
Ratis and Prprtinal Relatinships Grade 7 Majr Supprting Additinal Analyze prprtinal relatinships and use them t slve real wrld and mathematical prblems. The Number System Apply and extend previus understandings f peratins with fractins t add, subtract, multiply, and divide ratinal numbers. Expressins and Equatins Use prperties f peratins t generate equivalent expressins. Slve real life and mathematical prblems using numerical and algebraic expressins and equatins. Depth Opprtunities: RP 2; NS 3; EE 3, 4; G 6 Statistics and Prbability Use randm sampling t draw inferences abut a ppulatin. 8 Investigate chance prcesses and develp, use, and evaluate prbability mdels. 9 Statistics and Prbability Draw infrmal cmparative inferences abut tw ppulatins. Gemtetry Slve real life and mathematical prblems invlving angle measure, area, surface area, and vlume. Draw, cnstruct and describe gemetrical figures and describe the relatinships between them. 8 The standards in this cluster represent pprtunities t apply percentages and prprtinal reasning. In rder t make inferences abut a ppulatin, ne needs t apply such reasning t the sample and the entire ppulatin. 9 Prbability mdels draw n prprtinal reasning and shuld be cnnected t the majr wrk in thse standards.
Grade 8 Majr Supprting Additinal Expressins and Equatins Wrk with radicals and integer expnents. Understand the cnnectins between prprtinal relatinships, lines, and linear equatins. Analyze and slve linear equatins and pairs f simultaneus linear equatins. Functins Define, evaluate, and cmpare functins. Gemetry Understand and apply the Pythagrean Therem. Understand cngruence and similarity using physical mdels, transparencies, r gemetry sftware. Depth Opprtunities: EE 5, 7, 8; F 2; G 7 The Number System Knw that there are numbers that are nt ratinal, and apprximate them by ratinal numbers. 10 Functins Use functins t mdel relatinships between quantities. 11 Statistics and Prbability Investigate patterns f assciatin in bivariate data. 12 Gemetry Slve real wrld and mathematical prblems invlving vlume f cylinders, cnes, and spheres. 10 Wrk with the number system in this grade is intimately related t wrk with radicals, and bth f these may be cnnected t the Pythagrean Therem as well as t vlume prblems, e.g., in which a cube has knwn vlume but unknwn edge lengths. 11 The wrk in this cluster invlves functins fr mdeling linear relatinships and a rate f change/initial value, which supprts wrk with prprtinal relatinships and setting up linear equatins. 12 Lking fr patterns in scatterplts and using linear mdels t describe data are directly cnnected t the wrk in the Expressins and Equatins clusters. Tgether, these represent a cnnectin t the Standard fr Mathematical Practice Mdel with mathematics.
Quantities High Schl: Number and Quantity Majr Supprting Additinal Reasn quantitatively and use units t slve prblems. The Real Number System Extend the prperties f expnents t ratinal expnents. Depth Opprtunities: N NQ 1 The Cmplex Number System Perfrm arithmetic peratins with cmplex numbers. The Real Number System Use prperties f ratinal and irratinal numbers. The Cmplex Number System Represent cmplex numbers and their peratins n the cmplex plane. Use cmplex numbers in plynmial identities and equatins. Vectr and Matrix Quantities Represent and mdel with vectr quantities. Perfrm peratins n vectrs. Perfrm peratins n matrices and use matrices in applicatins.
Seeing the Structure in Expressins High Schl: Algebra Majr Supprting Additinal Interpret the structure f expressins. Write expressins in equivalent frms t slve prblems. Arithmetic with Plynmials and Ratinal Expressins Perfrm arithmetic peratins n plynmials. Understand the relatinship between zers and factrs f plynmials. Creating Equatins Create equatins that describe numbers r relatinships. Reasning with Equatins and Inequalities Understand slving equatins as a prcess f reasning and explain the reasning. Slve equatins and inequalities in ne variable. Slve systems f equatins. Depth Opprtunities: A SSE 2, 3; A APR 1; A CED 3; A REI 4 Arithmetic with Plynmials and Ratinal Expressins Rewrite ratinal expressins. Reasning with Equatins and Inequalities Represent and slve equatins and inequalities graphically. Arithmetic with Plynmials and Ratinal Expressins Use plynmial identities t slve prblems.
Interpreting Functins High Schl: Functins Majr Supprting Additinal Understand the cncept f a functin and understand functin ntatin. Interpret functins that arise in applicatins in terms f the cntext. Analyze functins using different representatins. Building Functins Build a functin that mdels a relatinship between tw quantities. Linear, Quadratic and Expnential Mdels Cnstruct and cmpare linear, quadratic, and expnential mdels and slve prblems. Interpret expressins fr functins in terms f the situatin they mdel. Depth Opprtunities: F IF 4, 8, 9; F LE 1 Building Functins Build new functins frm existing functins. Trignmetric Functins Extend the dmain f trignmetric functins using the unit circle. Mdel peridic phenmena with trignmetric functins. Prve and apply trignmetric identities.
Cngruence High Schl: Gemetry Majr Supprting Additinal Prve gemetric therems. Expressing Gemetric Prperties with Equatins Use crdinates t prve simple therems algebraically. Similarity, Right Triangles, and Trignmetry Define trignmetric ratis and slve prblems invlving right triangles. Mdeling with Gemetry Apply gemetric cncepts in mdeling situatins. Depth Opprtunities: GPE 1, 4, 7; G MG 2 Cngruence Experiment with transfrmatins in the plane. Understand cngruence in terms f rigid mtins. Make gemetric cnstructins. Circles Understand and apply therems abut circles. Find arc lengths and areas f sectrs f circles. Similarity, Right Triangles, and Trignmetry Understand similarity in terms f similarity transfrmatins. Similarity, Right Triangles, and Trignmetry Prve therems invlving similarity. Apply trignmetry t general triangles. Gemetric Measurement and Dimensin Explain vlume frmulas and use them t slve prblems. Visualize relatinships between tw dimensinal and three dimensinal bjects. Expressing Gemetric Prperties with Equatins Translate between the gemetric descriptin and the equatin fr a cnic sectin. (Here because f circles.)
High Schl: Statistics and Prbability Majr Supprting Additinal Interpreting Categrical and Quantitative Data Summarize, represent, and interpret data n a single cunt r measurement variable. Summarize, represent, and interpret data n tw categrical and quantitative variables. Making Inferences and Justifying Cnclusins Make inferences and justify cnclusins frm sample surveys, experiments, and bservatinal studies. Depth Opprtunities: S ID 3, 5, 6, 9; S IC 3 Making Inferences and Justifying Cnclusins Understand and evaluate randm prcesses underlying statistical experiments. Interpreting Categrical and Quantitative Data Interpret linear mdels. Cnditinal Prbability and the Rules f Prbability Understand independence and cnditinal prbability and use them t interpret data. Use the rules f prbability t cmpute prbabilities f cmpund events in a unifrm prbability mdel. Using Prbability t Make Decisins Calculate expected values and use them t slve prblems. Use prbability t evaluate utcmes f decisins.