common core state standards for mathematics appendix a - … · 2020-01-27 · common core state...

common core state stanDarDs For

mathematics

appendix a:

Designing High school mathematics courses Based on the common core state standards

JLong

Typewritten Text

Attachment 2 Item 12.C. May 3-4, 2012 Mathematics SMC

Common Core State StandardS for matHematICS

overview TheCommonCoreStateStandards(CCSS)forMathematicsareorganizedbygradelevelinGradesK–8.Atthehigh schoollevel,thestandardsareorganizedbyconceptualcategory(numberandquantity,algebra,functions,geometry, modelingandprobabilityandstatistics),showingthebodyofknowledgestudentsshouldlearnineachcategoryto becollegeandcareerready,andtobepreparedtostudymoreadvancedmathematics.Asstatesconsiderhowto implementthehighschoolstandards,animportantconsiderationishowthehighschoolCCSSmightbeorganized intocoursesthatprovideastrongfoundationforpost-secondarysuccess.Toaddressthisneed,Achieve(inpartnershipwiththeCommonCorewritingteam)hasconvenedagroupofexperts,includingstatemathematicsexperts, teachers,mathematicsfacultyfromtwoandfouryearinstitutions,mathematicsteachereducators,andworkforce representativestodevelopModelCoursePathwaysinMathematicsbasedontheCommonCoreStateStandards.

Inconsideringthisdocument,therearefourthingsimportanttonote:

1. Thepathwaysandcoursesaremodels,notmandates.Theyillustratepossibleapproachestoorganizingthe contentoftheCCSSintocoherentandrigorouscoursesthatleadtocollegeandcareerreadiness.Statesand districtsarenotexpectedtoadoptthesecoursesasis;rather,theyareencouragedtousethesepathwaysand coursesasastartingpointfordevelopingtheirown.

2. Allcollegeandcareerreadystandards(thosewithouta+)arefoundineachpathway.Afew(+)standardsare includedtoincreasecoherencebutarenotnecessarilyexpectedtobeaddressedonhighstakesassessments.

3. Thecoursedescriptionsdelineatethemathematicsstandardstobecoveredinacourse;theyarenotprescriptionsforcurriculumorpedagogy.Additionalworkwillbeneededtocreatecoherentinstructionalprogramsthat helpstudentsachievethesestandards.

4. Unitswithineachcourseareintendedtosuggestapossiblegroupingofthestandardsintocoherentblocks;in thisway,unitsmayalsobeconsidered“criticalareas”or“bigideas”,andthesetermsareusedinterchangeably throughoutthedocument.Theorderingoftheclusterswithinaunitfollowstheorderofthestandardsdocument inmostcases,nottheorderinwhichtheymightbetaught.Attentiontoorderingcontentwithinaunitwillbe neededasinstructionalprogramsaredeveloped.

5. Whilecoursesaregivennamesfororganizationalpurposes,statesanddistrictsareencouragedtocarefullyconsiderthecontentineachcourseandusenamesthattheyfeelaremostappropriate.Similarly,unittitlesmaybe adjustedbystatesanddistricts.

WhilethefocusofthisdocumentisonorganizingtheStandardsforMathematicalContentintomodelpathways tocollegeandcareerreadiness,thecontentstandardsmustalsobeconnectedtotheStandardsforMathematical Practicetoensurethattheskillsneededforlatersuccessaredeveloped.Inparticular,Modeling(definedbya*inthe CCSS)isdefinedasbotha conceptual category forhighschoolmathematicsanda mathematical practice andisan importantavenueformotivatingstudentstostudymathematics,forbuildingtheirunderstandingofmathematics, andforpreparingthemforfuturesuccess.Developmentofthepathwaysintoinstructionalprogramswillrequire carefulattentiontomodelingandthemathematicalpractices.Assessmentsbasedonthesepathwaysshouldreflect boththecontentandmathematicalpracticesstandards.

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thePathways Fourmodelcoursepathwaysareincluded:

1. AnapproachtypicallyseenintheU.S.(Traditional)thatconsistsoftwoalgebracoursesandageometrycourse, withsomedata,probabilityandstatisticsincludedineachcourse;

2. Anapproachtypicallyseeninternationally(Integrated)thatconsistsofasequenceofthreecourses,eachof whichincludesnumber,algebra,geometry,probabilityandstatistics;

3. A “compacted” version of the Traditional pathway where no content is omitted, in which students would complete the content of 7th grade, 8th grade, and the High School Algebra I course in grades 7 (Compacted 7th Grade) and 8 (8th Grade Algebra I), which will enable them to reach Calculus or other college level courses by their senior year. While the K-7 CCSS effectively prepare students for algebra in 8th grade, some standards from 8th grade have been placed in the Accelerated 7th Grade course to make the 8th Grade Algebra I course more manageable;

4. A “compacted” version of the Integrated pathway where no content is omitted, in which students would complete the content of 7th grade, 8th grade, and the Mathematics I course in grades 7 (Compacted 7th Grade) and 8 (8th Grade Mathematics I), which will enable them to reach Calculus or other college level courses by their senior year. While the K-7 CCSS effectively prepare students for algebra in 8th grade, some standards from 8th grade have been placed in the Accelerated 7th Grade course to make the 8th Grade Mathematics I course more manageable;

5. Ultimately,allofthesepathwaysareintendedtosignificantlyincreasethecoherenceofhighschoolmathematics.

Thenon-compacted,orregular,pathwaysassumemathematicsineachyearofhighschoolandleaddirectlytopreparednessforcollegeandcareerreadiness.InadditiontothethreeyearsofstudydescribedintheTraditionaland Integratedpathways,studentsshouldcontinuetotakemathematicscoursesthroughouttheirhighschoolcareerto keeptheirmathematicalunderstandingandskillsfreshforuseintrainingorcourseworkafterhighschool.Avariety ofcoursesshouldbeavailabletostudentsreflectingarangeofpossibleinterests;possibleoptionsarelistedinthe followingchart.Basedonavarietyofinputsandfactors,somestudentsmaydecideatanearlyagethattheywant totakeCalculusorothercollegelevelcoursesinhighschool.Thesestudentswouldneedtobeginthestudyofhigh schoolcontentinthemiddleschool,whichwouldleadtoPrecalculusorAdvancedStatisticsasajuniorandCalculus, AdvancedStatisticsorothercollegeleveloptionsasasenior.

Strategicuseoftechnologyisexpectedinallwork.Thismayincludeemployingtechnologicaltoolstoassiststudents informingandtestingconjectures,creatinggraphsanddatadisplaysanddeterminingandassessinglinesoffitfor data.Geometricconstructionsmayalsobeperformedusinggeometricsoftwareaswellasclassicaltoolsandtechnologymayaidthree-dimensionalvisualization.Testingwithandwithouttechnologicaltoolsisrecommended.

Ashasoftenoccurredinschoolsanddistrictsacrossthestates,greaterresourceshavebeenallocatedtoaccelerated pathways,suchasmoreexperiencedteachersandnewermaterials.TheAchievePathwaysGroupmembersstrongly believethateachpathwayshouldgetthesameattentiontoqualityandresourcesincludingclasssizes,teacher assignments,professionaldevelopment,andmaterials.Indeed,theseandotherpathwaysshouldbeavenuesforstudentstopursueinterestsandaspirations.Thefollowingflowchartshowshowthecoursesinthetworegularpathwaysaresequenced(the*inthechartonthefollowingpagemeansthatCalculusfollowsPrecalculusandisafifth course,inmostcases).Moreinformationaboutthecompactedpathwayscanbefoundlaterinthisappendix.

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Someteachersandschoolsareeffectivelygettingstudentstobecollegeandcareerready.Wecanlooktothese teachersandschoolstoseewhatkindsofcoursesaregettingresults,andtocomparepathwayscoursestothemathematicstaughtineffectiveclassrooms.

AstudydonebyACTandTheEducationTrustgivesevidencetosupportthesepathways.ThestudylookedathighpovertyschoolswhereahighpercentageofstudentswerereachingandexceedingACT’scollege-readinessbenchmarks.Fromtheseschools,themosteffectiveteachersdescribedtheircoursesandopeneduptheirclassroomsfor observation.Thecommonalityofmathematicstopicsintheircoursesgivesapictureofwhatittakestogetstudents tosucceed,andalsoprovidesagroundingforthepathways.(Therewereothercommonalities.Formoredetailed informationaboutthisstudy,searchforthereportOnCourseforSuccessatwww.act.org.)1

Implementationconsiderations:

Asstates,districtsandschoolstakeontheworkofimplementingtheCommonCoreStateStandards,theModel CoursePathwaysinMathematicscanbeausefulfoundationfordiscussinghowbesttoorganizethehighschoolstandardsintocourses.ThePathwayshavebeendesignedtobemodularinnature,wherethemodulesorcriticalareas (units)areidenticalinnearlyeverymannerbetweenthetwopathways,butarearrangedindifferentorderstoaccommodatedifferentorganizationalofferings.Assessmentdevelopersmayconsiderthecreationofassessmentmodules inasimilarfashion.Curriculumdesignersmaycreatealternativemodelpathwayswithaltogetherdifferentorganizationsofthestandards.Someofthisworkisalreadyunderway.Inshort,thisdocumentisintendedtocontributeto theconversationsaroundassessmentandcurriculumdesign,ratherthanendthem.Effectivelyimplementingthese standardswillrequirealong-termcommitmenttounderstandingwhatbestsupportsstudentlearningandattainment ofcollegeandcareerreadinessskillsbytheendofhighschool,aswellasregularrevisionofpathwaysasstudent learningdatabecomesavailable.

supportingstudents

One of the hallmarks of the Common Core State Standards for Mathematics is the specification of content that all students must study in order to be college and career ready. This “college and career ready line” is a minimum for all students. However, this does not mean that all students should progress uniformly to that goal. Some students progress

1The study provides evidence that the pathways’ High School Algebra I, Geometry, Algebra II sequence is a reasonable and rigorous option for preparing students for college and career. Topics aligned almost completely between the CCSS topics and topics taught in the study classrooms. The starting point for the pathways’ High School Algebra I course is slightly beyond the starting point for the study Algebra I courses due to the existence of many typical Algebra I topics in the 8th grade CCSS, therefore some of the study Algebra II topics are a part of the pathways’ High School Algebra I course, specifically, using the quadratic formula; a bit more with exponential functions including comparing and contrasting linear and exponential growth; and the inclusion of the spread of data sets. The pathways’ Geometry course is very similar to what was done in the study Geometry courses, with the addition of the laws of sines and cosines and the work with conditional probability, plus applications involving completing the square because that topic was part of the pathways’ High School Algebra I course. The pathways’ Algebra II course then matches well with what was done in the study Algebra II courses and continues a bit into what was done in the study Precalculus classrooms, including inverse functions, the behavior of logarithmic and trigonometric functions, and in statistics with the normal distribution, margin of error, and the differences among sample surveys, experiments, and observational studies. All in all, the topics and the order of topics is very comparable between the pathways’ High School Algebra I, Geometry, Algebra II sequence and the sequence found in the study courses.

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more slowly than others. These students will require additional support, and the following strategies, consistent with Response to Intervention practices, may be helpful:

• Creatingaschool-widecommunityofsupportforstudents;

• Providingstudentsa“mathsupport”classduringtheschoolday;

• After-schooltutoring;

• Extendedclasstime(orblockingofclasses)inmathematics;and

• Additionalinstructionduringthesummer.

Watered-downcourseswhichleavestudentsuninspiredtolearn,unabletocatchuptotheirpeersandunreadyfor successinpostsecondarycoursesorforentryintomanyskilledprofessionsupongraduationfromhighschoolare neithernecessarynordesirable.Theresultsofnotprovidingstudentsthenecessarysupportstheyneedtosucceedin highschoolarewell-documented.Toooften,aftergraduation,suchstudentsattempttocontinuetheireducationat2- or4-yearpostsecondaryinstitutionsonlytofindtheymusttakeremedialcourses,spendingtimeandmoneymasteringhighschoollevelskillsthattheyshouldhavealreadyacquired.This,inturn,hasbeendocumentedtoindicatea greaterchanceofthesestudentsnotmeetingtheirpostsecondarygoals,whetheracertificateprogram,two-orfouryeardegree.Asaresult,intheworkplace,manycareerpathwaysandadvancementmaybedeniedtothem.Toensure studentsgraduatefullyprepared,thosewhoenterhighschoolunderpreparedforhighschoolmathematicscourses mustreceivethesupporttheyneedtogetbackoncourseandgraduatereadyforlifeafterhighschool.

Furthermore,researchshowsthatallowinglow-achievingstudentstotakelow-levelcoursesisnotarecipeforacademic success (Kifer, 1993). The research strongly suggests that the goal for districts should not be to stretch the high schoolmathematicsstandardsoverallfouryears.Rather,thegoalshouldbetoprovidesupportsothatallstudents canreachthecollegeandcareerreadylinebytheendoftheeleventhgrade,endingtheirhighschoolcareerwithone ofseveralhigh-qualitymathematicalcoursesthatallowsstudentstheopportunitytodeepentheirunderstandingof thecollege-andcareer-readystandards.

WiththeCommonCoreStateStandardsInitiativecomesanunprecedentedabilityforschools,districts,andstatesto collaborate.Whilethisiscertainlythecasewithrespecttoassessmentsandprofessionaldevelopmentprograms,itis alsotrueforstrategiestosupportstrugglingandacceleratedstudents.TheModelCoursePathwaysinMathematics areintendedtolaunchtheconversation,andgiveencouragementtoalleducatorstocollaborateforthebenefitofour states’children.

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HowtoreadthePathways:

Eachpathwayconsistsoftwoparts.Thefirstisachartthatshowsanoverviewofthepathway.Organizedbycourse andbyconceptualcategory(algebra,functions,geometry,etc…),thesechartsshowwhichclustersandstandards appearinwhichcourse(seepage5oftheCCSSfordefinitionsofclustersandstandards).Forexample,inthechart below,thethreestandards(N.Q.1,2,3)associatedwiththecluster“Reasonquantitativelyanduseunitstosolve problems,”arefoundinCourse1.Thisclusterisfoundunderthedomain“Quantities”inthe“NumberandQuantity” conceptualcategory.AllhighschoolstandardsintheCCSSarelocatedinatleastoneofthecoursesinthischart.

courses

Dom ain

clusters, notes,and standards

cecon ptual gorycate

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Thesecondpartofthepathwaysshowstheclustersandstandardsastheyappearinthecourses.Eachcourse containsthefollowingcomponents:

• Anintroductiontothecourseandalistoftheunitsinthecourse

• Unittitlesandunitoverviews(seebelow)

• Unitsthatshowtheclustertitles,associatedstandards,andinstructionalnotes(below)

Itisimportanttonotethattheunits(orcriticalareas)areintendedtoconveycoherentgroupingsofcontent.The clustersandstandardswithinunitsareorderedastheyareintheCommonCoreStateStandards,andarenot intendedtoconveyaninstructionalorder.Considerationsregardingconstraints,extensions,andconnectionsare foundintheinstructionalnotes.Theinstructionalnotesareacriticalattributeofthecoursesandshouldnotbe overlooked.Forexample,onewillseethatstandardssuchasA.CED.1andA.CED.2arerepeatedinmultiplecourses, yettheiremphaseschangefromonecoursetothenext.Thesechangesareseenonlyintheinstructionalnotes, makingthenotesanindispensablecomponentofthepathways.

cluster

standards associated with

cluster

Unit titleand overview

Instructionalnote

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overviewofthetraditionalPathwayfor thecommoncorestatemathematicsstandards ThistableshowsthedomainsandclustersineachcourseintheTraditionalPathway.Thestandardsfromeachclusterincluded inthatcoursearelistedbeloweachcluster.Foreachcourse,limitsandfocusfortheclustersareshowninitalics.

Domains High School Algebra I Geometry AlgebraII Fourth Courses *

Nu

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TheReal Number System

Quantities

TheComplex Number System

•Extendtheproperties ofexponentsto rationalexponents.

N.RN.1,2

•Usepropertiesof rationalandirrational numbers.

N.RN.3

•Reasonquantitatively anduseunitstosolve problems.

Foundation for work with expressions,

equations and functions

N.Q.1,2,3

•Performarithmetic operationswith complexnumbers.

N.CN.1,2

•Usecomplexnumbers in polynomial identities andequations.

Polynomials with real


(+) N.CN.3

•Representcomplex numbersandtheir operationsonthe complexplane.

Vector Quantitiesand Matrices

coefficients

N.CN.7,(+)8,(+)9

(+) N.CN.4,5,6

•Representandmodel withvectorquantities.

(+) N.VM.1,2,3

• Performoperationson vectors.

(+) N.VM.4a,4b,4c,5a, 5b

• Performoperations onmatricesand usematricesin applications.

(+) N.VM.6,7,8,9, 10, 11, 12

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*The(+)standardsinthiscolumnarethoseintheCommonCoreStateStandardsthatarenotincludedinanyoftheTraditionalPathwaycourses. TheywouldbeusedinadditionalcoursesdevelopedtofollowAlgebraII.

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Domains High School Algebra I Geometry AlgebraII Fourth Courses

Alg

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Seeing Structurein Expressions

Arithmetic with

•Interpretthestructure ofexpressions.

Linear, exponential, quadratic

A.SSE.1a,1b,2

•Writeexpressionsin equivalentformsto solveproblems.

Quadratic and exponential

A.SSE.3a,3b,3c

•Performarithmetic operationson polynomials.

Linear and quadratic

A.APR.1


Polynomial and rational

A.SSE.1a, 1b, 2


A.SSE.4


Beyond quadratic

A.APR.1

•Understandthe relationshipbetween zerosandfactorsof polynomials.

A.APR.2,3 Polynomials andRational Expressions

Creating Equations

•Createequationsthat describenumbersor relationships.

Linear, quadratic, and exponential (integer

inputs only); for A.CED.3 linear only

A.CED.1,2,3,4

•Usepolynomial identitiestosolve problems.

A.APR.4, (+) 5

•Rewriterational expressions.

Linear and quadratic denominators

A.APR.6,(+)7


Equations using all available types of

expressions, including simple root functions

A.CED.1,2,3,4

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Alg

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Interpreting Functions

•Understandsolving equationsasaprocess ofreasoningand explainthereasoning.

Master linear; learn as general principle

A.REI.1

•Solveequationsand inequalitiesinone variable.

Linear inequalities; literal that are linear

in the variables being solved for; quadratics

with real solutions

A.REI.3,4a,4b

•Solvesystemsof equations.

Linear-linear and linear-quadratic

A.REI.5,6,7

•Representand solveequationsand inequalities graphically.

Linear and exponential; learn as general

principle

A.REI.10,11,12

•Understandthe conceptofafunction andusefunction notation.

Learn as general principle; focus on

linear and exponential and on arithmetic and geometric sequences

F.IF.1,2,3

•Interpret functions that ariseinapplicationsin termsofacontext.

Linear, exponential, and quadratic


Simple radical and rational

A.REI.2


Combine polynomial, rational, radical,

absolute value, and exponential functions

A.REI.11

•Interpret functions that ariseinapplicationsin termsofacontext.

Emphasize selection of appropriate models

F.IF.4,5,6

•Analyzefunctions usingdifferent representations.

Focus on using key features to guide

selection of appropriate type of model function

F.IF.7b,7c,7e,8,9


(+)A.REI.8,9


Logarithmic and trigonometric functions

(+)F.IF.7d

Fu

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F.IF.4,5,6


Linear, exponential, quadratic, absolute

value, step, piecewise-defined

F.IF.7a,7b,7e,8a,8b,9

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11 Common Core State StandardS for matHematICS


Building Functions

•Buildafunctionthat modelsarelationship betweentwo quantities.

For F.BF.1, 2, linear, exponential, and

quadratic

F.BF.1a,1b,2

•Buildnewfunctions fromexisting


Include all types of functions studied

F.BF.1b

•Buildnewfunctions fromexisting functions.


(+)F.BF.1c


(+) F.BF.4b,4c,4d,5

Fu

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Linear, Quadratic,and Exponential Models

functions.

Linear, exponential, quadratic, and absolute value; for F.BF.4a, linear

only

F.BF.3,4a

•Constructand comparelinear, quadratic,and exponentialmodels andsolveproblems.

F.LE.1a,1b,1c,2,3

•Interpretexpressions forfunctionsinterms

Include simple radical, rational, and

exponential functions; emphasize common

effect of each transformation across

function types

F.BF.3,4a


Logarithms as solutions for exponentials

F.LE.4

Trigonometric Functions

ofthesituationthey model.

Linear and exponential of form f(x)=bx+k

F.LE.5

•Extendthedomain oftrigonometric functionsusingthe unitcircle.

F.TF.1,2

•Modelperiodic phenomenawith trigonometric functions.

F.TF.5

•Proveandapply trigonometric identities.

F.TF.8


(+)F.TF.3,4


(+)F.TF.6,7


(+)F.TF.9

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Ge

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Congruence

•Experimentwith transformationsinthe plane.

G.CO.1,2,3,4,5

•Understand congruence in terms of rigidmotions.

Build on rigid motions as a familiar starting

point for development of concept of geometric

proof

G.CO.6,7,8

•Provegeometric theorems.

Focus on validity of underlying reasoning while using variety of ways of writing proofs

G.CO.9,10,11

•Makegeometric constructions.

Similarity, Right Triangles,and Trigonometry

Formalize and explain processes

G.CO.12,13

•Understandsimilarity intermsofsimilarity transformations.

G.SRT.1a,1b,2,3

•Provetheorems involvingsimilarity.

G.SRT.4,5

•Definetrigonometric ratiosandsolve problemsinvolving righttriangles.

G.SRT.6,7,8

•Applytrigonometryto generaltriangles.

G.SRT.9.10,11

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Ge

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Circles

Expressing Geometric Properties withEquations

Geometric Measurement andDimension

Modelingwith Geometry

Interpreting Categorical and Quantitative Data

•Summarize,represent, andinterpretdata onasinglecountor measurementvariable.

S.ID.1,2,3

•Summarize,represent, andinterpretdataon twocategoricaland quantitativevariables.

Linear focus, discuss general principle

S.ID.5,6a,6b,6c

•Interpretlinearmodels

S.ID.7,8,9

•Understandand applytheoremsabout circles.

G.C.1,2,3,(+)4

•Findarclengthsand areasofsectorsof circles.

Radian introduced only as unit of measure

G.C.5

•Translatebetweenthe geometricdescription andtheequationfora conicsection.

G.GPE.1,2

•Usecoordinates toprovesimple geometrictheorems algebraically.

Include distance formula; relate to

Pythagorean theorem

G.GPE.4,5,6,7

•Explainvolume formulasandusethem tosolveproblems.

G.GMD.1, 3

•Visualizetherelation betweentwodimensionalandthreedimensionalobjects.

G.GMD.4

•Applygeometric conceptsinmodeling situations.

G.MG.1,2,3


S.ID.4


(+)G.GPE.3


(+)G.GMD.2

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Sta

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an

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Making Inferences andJustifying Conclusions

Conditional Probability andtheRules ofProbability

•Understand independenceand conditionalprobability andusethemto interpretdata.

Link to data from simulations or experiments

S.CP.1,2,3,4,5

•Usetherulesof probabilitytocompute

•Understandand evaluaterandom processesunderlying statisticalexperiments.

S.IC.1,2

•Makeinferencesand justifyconclusions fromsamplesurveys, experimentsand observationalstudies.

S.IC.3,4,5,6

Using Probability toMake Decisions

probabilitiesof compoundeventsin auniformprobability model.

S.CP.6,7, (+) 8,(+)9

•Useprobabilityto evaluateoutcomesof decisions.

Introductory; apply counting rules

(+) S.MD.6,7


Include more complex situations

(+) S.MD.6,7

•Calculateexpected valuesandusethemto solveproblems.

(+)S.MD.1,2,3,4

•Useprobabilityto evaluateoutcomesof decisions..

(+)S.MD.5a,5b


traditionalPathway:HighschoolalgebraI Thefundamentalpurposeofthiscourseistoformalizeandextendthemathematicsthatstudentslearnedinthe middlegrades.Becauseitisbuiltonthemiddlegradesstandards,thisisamoreambitiousversionofAlgebraI thanhasgenerallybeenoffered.Thecriticalareas,calledunits,deepenandextendunderstandingoflinearand exponentialrelationshipsbycontrastingthemwitheachotherandbyapplyinglinearmodelstodatathatexhibita lineartrend,andstudentsengageinmethodsforanalyzing,solving,andusingquadraticfunctions.TheMathematical PracticeStandardsapplythroughouteachcourseand,togetherwiththecontentstandards,prescribethatstudents experiencemathematicsasacoherent,useful,andlogicalsubjectthatmakesuseoftheirabilitytomakesenseof problemsituations.

CriticalArea1:Bytheendofeighthgrade,studentshavelearnedtosolvelinearequationsinonevariableandhave appliedgraphicalandalgebraicmethodstoanalyzeandsolvesystemsoflinearequationsintwovariables.Now, studentsanalyzeandexplaintheprocessofsolvinganequation.Studentsdevelopfluencywriting,interpreting,and translatingbetweenvariousformsoflinearequationsandinequalities,andusingthemtosolveproblems.Theymaster thesolutionoflinearequationsandapplyrelatedsolutiontechniquesandthelawsofexponentstothecreationand solutionofsimpleexponentialequations.

CriticalArea2:Inearliergrades,studentsdefine,evaluate,andcomparefunctions,andusethemtomodel relationshipsbetweenquantities.Inthisunit,studentswilllearnfunctionnotationanddeveloptheconceptsof domainandrange.Theyexploremanyexamplesoffunctions,includingsequences;theyinterpretfunctionsgiven graphically,numerically,symbolically,andverbally,translatebetweenrepresentations,andunderstandthelimitations ofvariousrepresentations.Studentsbuildonandinformallyextendtheirunderstandingofintegerexponents toconsiderexponentialfunctions.Theycompareandcontrastlinearandexponentialfunctions,distinguishing betweenadditiveandmultiplicativechange.Studentsexploresystemsofequationsandinequalities,andtheyfind andinterprettheirsolutions.Theyinterpretarithmeticsequencesaslinearfunctionsandgeometricsequencesas exponentialfunctions.

CriticalArea3:Thisunitbuildsuponpriorstudents’priorexperienceswithdata,providingstudentswithmore formalmeansofassessinghowamodelfitsdata.Studentsuseregressiontechniquestodescribeapproximately linearrelationshipsbetweenquantities.Theyusegraphicalrepresentationsandknowledgeofthecontexttomake judgmentsabouttheappropriatenessoflinearmodels.Withlinearmodels,theylookatresidualstoanalyzethe goodnessoffit.

CriticalArea4:Inthisunit,studentsbuildontheirknowledgefromunit2,wheretheyextendedthelawsofexponents torationalexponents.Studentsapplythisnewunderstandingofnumberandstrengthentheirabilitytoseestructure inandcreatequadraticandexponentialexpressions.Theycreateandsolveequations,inequalities,andsystemsof equationsinvolvingquadraticexpressions.

CriticalArea5:Inthisunit,studentsconsiderquadraticfunctions,comparingthekeycharacteristicsofquadratic functionstothoseoflinearandexponentialfunctions.Theyselectfromamongthesefunctionstomodelphenomena. Studentslearntoanticipatethegraphofaquadraticfunctionbyinterpretingvariousformsofquadraticexpressions. Inparticular,theyidentifytherealsolutionsofaquadraticequationasthezerosofarelatedquadraticfunction. Studentsexpandtheirexperiencewithfunctionstoincludemorespecializedfunctions—absolutevalue,step,and thosethatarepiecewise-defined.

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Units IncludesStandardClusters* MathematicalPractice

Standards

Unit1

Relationships BetweenQuantities andReasoningwith

Equations

• Reason quantitatively and use units to solve problems.

• Interpret the structure of expressions.

• Create equations that describe numbers or relationships.

• Understand solving equations as a process of reasoning and explain the reasoning.

• Solve equations and inequalities in one variable.

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Unit2

LinearandExponential Relationships

• Extend the properties of exponents to rational exponents.

• Solve systems of equations.

• Represent and solve equations and inequalities graphically.

• Understand the concept of a function and use function notation.

• Interpret functions that arise in applications in terms of a context.

• Analyze functions using different representations.

• Build a function that models a relationship between two quantities.

• Build new functions from existing functions.

• Construct and compare linear, quadratic, and exponential models and solve problems.

• Interpret expressions for functions in terms of the situation they model.

Makesenseofproblems andpersevereinsolving them.

Reasonabstractlyand quantitatively.

Constructviable argumentsandcritiquethe reasoningofothers.

Modelwithmathematics.

Unit3

DescriptiveStatistics

Unit4

Expressionsand Equations

• Summarize, represent, and interpret data on a single count or measurement variable.

• Summarize, represent, and interpret data on two categorical and quantitative variables.

• Interpret linear models.


• Write expressions in equivalent forms to solve problems.

• Perform arithmetic operations on polynomials.




Useappropriatetools strategically.

Attendtoprecision.

Lookforandmakeuseof structure.

Lookforandexpress regularityinrepeated reasoning.

Unit5

Quadratic Functions and Modeling

• Use properties of rational and irrational numbers.






*Insomecasesclustersappearinmorethanoneunitwithinacourseorinmorethanonecourse.Instructionalnoteswillindicatehow thesestandardsgrowovertime.Insomecasesonlycertainstandardswithinaclusterareincludedinaunit.


Unit1:relationshipsBetweenQuantitiesandreasoningwithequations

Bytheendofeighthgradestudentshavelearnedtosolvelinearequationsinonevariableandhaveappliedgraphical andalgebraicmethodstoanalyzeandsolvesystemsoflinearequationsintwovariables.Thisunitbuildsontheseearlierexperiencesbyaskingstudentstoanalyzeandexplaintheprocessofsolvinganequation.Studentsdevelopfluencywriting,interpreting,andtranslatingbetweenvariousformsoflinearequationsandinequalities,andusingthem tosolveproblems.Theymasterthesolutionoflinearequationsandapplyrelatedsolutiontechniquesandthelawsof exponentstothecreationandsolutionofsimpleexponentialequations.Allofthisworkisgroundedonunderstandingquantitiesandonrelationshipsbetweenthem.

Unit1:RelationshipsbetweenQuantitiesandReasoningwithEquations

ClusterswithInstructionalNotes CommonCoreStateStandards

N.Q.1Useunitsasawaytounderstandproblemsandtoguidethe solutionofmulti-stepproblems;chooseandinterpretunitsconsistently informulas;chooseandinterpretthescaleandtheoriginingraphsand datadisplays.

N.Q.2Defineappropriatequantitiesforthepurposeofdescriptive modeling.

N.Q.3Choosealevelofaccuracyappropriatetolimitationson measurementwhenreportingquantities.

SKILLSTOMAINTAIN

Reinforce understanding of the properties of integer exponents. The initial experience with exponential expressions, equations, and functions involves integer exponents and builds on this understanding.*

• Reasonquantitativelyanduseunitsto solveproblems.

Working with quantities and the relationships between them provides grounding for work with expressions, equations, and functions.

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• Interpretthestructureofexpressions.

Limit to linear expressions and to exponential expressions with integer exponents.

A.SSE.1Interpretexpressionsthatrepresentaquantityintermsofits context.★

a. Interpretpartsofanexpression,suchasterms,factors,andcoefficients.

b. Interpretcomplicatedexpressionsbyviewingoneormoreof theirpartsasasingleentity. For example, interpret P(1+r)n as the product of P and a factor not depending on P.

• Createequationsthatdescribenumbersorrelationships.

Limit A.CED.1 and A.CED.2 to linear and exponential equations, and, in the case of exponential equations, limit to situations requiring evaluation of exponential functions at integer inputs. Limit A.CED.3 to linear equations and inequalities. Limit A.CED.4 to formulas which are linear in the variable of interest.

A.CED.1Createequationsandinequalitiesinonevariableandusethem tosolveproblems. Include equations arising from linear and quadratic functions, and simple rational and exponential functions.

A.CED.2Createequationsintwoormorevariablestorepresent relationshipsbetweenquantities;graphequationsoncoordinateaxes withlabelsandscales.

A.CED.3Representconstraintsbyequationsorinequalities,andby systemsofequationsand/orinequalities,andinterpretsolutionsas viableornon-viableoptionsinamodelingcontext. For example, represent inequalities describing nutritional and cost constraints on combinations of different foods.

A.CED.4Rearrangeformulastohighlightaquantityofinterest,usingthe samereasoningasinsolvingequations. For example, rearrange Ohm’s law V = IR to highlight resistance R.

*Instructionalsuggestionswillbefoundinitalicsinthiscolumnthroughoutthedocument.




• Understandsolvingequationsasa processofreasoningandexplainthe reasoning.

Students should focus on and master A.REI.1 for linear equations and be able to extend and apply their reasoning to other types of equations in future courses. Students will solve exponential equations with logarithms in Algebra II.

A.REI.1Explaineachstepinsolvingasimpleequationasfollowingfrom theequalityofnumbersassertedatthepreviousstep,startingfromthe assumptionthattheoriginalequationhasasolution.Constructaviable argumenttojustifyasolutionmethod.

• Solveequationsandinequalitiesin A.REI.3Solvelinearequationsandinequalitiesinonevariable,including onevariable. equationswithcoefficientsrepresentedbyletters.

Extend earlier work with solving linear equations to solving linear inequalities in one variable and to solving literal equations that are linear in the variable being solved for. Include simple exponential equations that rely only on application of the laws of exponents, such as 5x=125or2x=1/16.

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Unit2:Linearandexponentialrelationships

Inearliergrades,studentsdefine,evaluate,andcomparefunctions,andusethemtomodelrelationshipsbetween quantities.Inthisunit,studentswilllearnfunctionnotationanddeveloptheconceptsofdomainandrange.They movebeyondviewingfunctionsasprocessesthattakeinputsandyieldoutputsandstartviewingfunctionsasobjects intheirownright.Theyexploremanyexamplesoffunctions,includingsequences;theyinterpretfunctionsgiven graphically,numerically,symbolically,andverbally,translatebetweenrepresentations,andunderstandthelimitations ofvariousrepresentations.Theyworkwithfunctionsgivenbygraphsandtables,keepinginmindthat,depending uponthecontext,theserepresentationsarelikelytobeapproximateandincomplete.Theirworkincludesfunctions thatcanbedescribedorapproximatedbyformulasaswellasthosethatcannot.Whenfunctionsdescriberelationshipsbetweenquantitiesarisingfromacontext,studentsreasonwiththeunitsinwhichthosequantitiesaremeasured.Studentsexploresystemsofequationsandinequalities,andtheyfindandinterprettheirsolutions.Students buildonandinformallyextendtheirunderstandingofintegerexponentstoconsiderexponentialfunctions.They compareandcontrastlinearandexponentialfunctions,distinguishingbetweenadditiveandmultiplicativechange. Theyinterpretarithmeticsequencesaslinearfunctionsandgeometricsequencesasexponentialfunctions.

Unit2:LinearandExponentialRelationships


• Extendthepropertiesofexponentsto rationalexponents.

In implementing the standards in curriculum, these standards should occur before discussing exponential functions with continuous domains.

• Solvesystemsofequations.

Build on student experiences graphing and solving systems of linear equations from middle school to focus on justification of the methods used. Include cases where the two equations describe the same line (yielding infinitely many solutions) and cases where two equations describe parallel lines (yielding no solution); connect to GPE.5 when it is taught in Geometry, which requires students to prove the slope criteria for parallel lines.

N.RN.1Explainhowthedefinitionofthemeaningofrationalexponents followsfromextendingthepropertiesofintegerexponentsto thosevalues,allowingforanotationforradicalsintermsofrational exponents. For example, we define 51/3 to be the cube root of 5 because we want (51/3)3 = 5(1/3)3 to hold, so (51/3)3 must equal 5.

N.RN.2Rewriteexpressionsinvolvingradicalsandrationalexponents usingthepropertiesofexponents.

A.REI.5Provethat,givenasystemoftwoequationsintwovariables, replacingoneequationbythesumofthatequationandamultipleof theotherproducesasystemwiththesamesolutions.

A.REI.6Solvesystemsoflinearequationsexactlyandapproximately (e.g.,withgraphs),focusingonpairsoflinearequationsintwovariables.

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• Representandsolveequationsand inequalitiesgraphically.

For A.REI.10, focus on linear and exponential equations and be able to adapt and apply that learning to other types of equations in future courses. For A.REI.11, focus on cases where f(x) and g(x) are linear or exponential.

A.REI.10Understandthatthegraphofanequationintwovariablesis thesetofallitssolutionsplottedinthecoordinateplane,oftenforming acurve(whichcouldbealine).

A.REI.11Explainwhythex-coordinatesofthepointswherethegraphs oftheequations y = f(x) andy=g(x)intersectarethesolutionsof theequationf(x)=g(x);findthesolutionsapproximately,e.g.,using technologytographthefunctions,maketablesofvalues,orfind successiveapproximations.Includecaseswhere f(x) and/or g(x) are linear,polynomial,rational,absolutevalue,exponential,andlogarithmic functions.★

A.REI.12Graphthesolutionstoalinearinequalityintwovariablesasa half-plane(excludingtheboundaryinthecaseofastrictinequality), andgraphthesolutionsettoasystemoflinearinequalitiesintwo variablesastheintersectionofthecorrespondinghalf-planes.




• Understandtheconceptofafunction andusefunctionnotation.

Students should experience a variety of types of situations modeled by functions. Detailed analysis of any particular class of functions at this stage is not advised. Students should apply these concepts throughout their future mathematics courses.

Draw examples from linear and exponential functions. In F.IF.3, draw connection to F.BF.2, which requires students to write arithmetic and geometric sequences. Emphasize arithmetic and geometric sequences as examples of linear and exponential functions.

• Interpretfunctionsthatariseinapplicationsintermsofacontext.

For F.IF.4 and 5, focus on linear and exponential functions. For F.IF.6, focus on linear functions and exponential functions whose domain is a subset of the integers. Unit 5 in this course and the Algebra II course address other types of functions.

F.IF.1Understandthatafunctionfromoneset(calledthedomain)to anotherset(calledtherange)assignstoeachelementofthedomain exactlyoneelementoftherange.If f isafunctionand x isanelementof itsdomain,then f(x) denotestheoutputof f correspondingtotheinput x.Thegraphof f isthegraphoftheequation y = f(x).

F.IF.2Usefunctionnotation,evaluatefunctionsforinputsintheir domains,andinterpretstatementsthatusefunctionnotationintermsof acontext.

F.IF.3Recognizethatsequencesarefunctions,sometimesdefined recursively,whosedomainisasubsetoftheintegers. For example, the Fibonacci sequence is defined recursively by f(0) = f(1) = 1, f(n+1) = f(n) + f(n-1) for n ≥ 1.

F.IF.4Forafunctionthatmodelsarelationshipbetweentwoquantities, interpretkeyfeaturesofgraphsandtablesintermsofthequantities, andsketchgraphsshowingkeyfeaturesgivenaverbaldescription oftherelationship. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity.★

F.IF.5Relatethedomainofafunctiontoitsgraphand,whereapplicable, tothequantitativerelationshipitdescribes. For example, if the function h(n) gives the number of person-hours it takes to assemble n engines in a factory, then the positive integers would be an appropriate domain for the function.★

F.IF.6Calculateandinterprettheaveragerateofchangeofafunction (presentedsymbolicallyorasatable)overaspecifiedinterval.Estimate therateofchangefromagraph.★

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• Analyzefunctionsusingdifferentrepresentations.

For F.IF.7a, 7e, and 9 focus on linear and exponentials functions. Include comparisons of two functions presented algebraically. For example, compare the growth of two linear functions, or two exponential functions such as y=3n and y=1002

F.IF.7Graphfunctionsexpressedsymbolicallyandshowkeyfeatures ofthegraph,byhandinsimplecasesandusingtechnologyformore complicatedcases.★

a. Graphlinearandquadraticfunctionsandshowintercepts, maxima,andminima.

e.Graphexponentialandlogarithmicfunctions,showingintercepts andendbehavior,andtrigonometricfunctions,showingperiod, midline,andamplitude.

F.IF.9Comparepropertiesoftwofunctionseachrepresentedina differentway(algebraically,graphically,numericallyintables,orby verbaldescriptions). For example, given a graph of one quadratic function and an algebraic expression for another, say which has the larger maximum.




• Buildafunctionthatmodelsarelationshipbetweentwoquantities.

Limit to F.BF.1a, 1b, and 2 to linear and exponential functions. In F.BF.2, connect arithmetic sequences to linear functions and geometric sequences to exponential functions.

F.BF.1Writeafunctionthatdescribesarelationshipbetweentwo quantities.★

a. Determineanexplicitexpression,arecursiveprocess,orstepsfor calculationfromacontext.

b. Combinestandardfunctiontypesusingarithmeticoperations. For example, build a function that models the temperature of a cooling body by adding a constant function to a decaying exponential, and relate these functions to the model.

F.BF.2Writearithmeticandgeometricsequencesbothrecursivelyand withanexplicitformula,usethemtomodelsituations,andtranslate betweenthetwoforms.★

• Buildnewfunctionsfromexistingfunctions.

Focus on vertical translations of graphs of linear and exponential functions. Relate the vertical translation of a linear function to its y-intercept.

While applying other transformations to a linear graph is appropriate at this level, it may be difficult for students to identify or distinguish between the effects of the other transformations included in this standard.

F.BF.3Identifytheeffectonthegraphofreplacing f(x) by f(x) + k, k f(x), f(kx), and f(x + k) forspecificvaluesof k (bothpositiveand negative);findthevalueof k giventhegraphs.Experimentwith casesandillustrateanexplanationoftheeffectsonthegraphusing technology. Include recognizing even and odd functions from their graphs and algebraic expressions for them.

• Constructandcomparelinear,quadratic,andexponentialmodelsandsolve problems.

For F.LE.3, limit to comparisons between linear and exponential models. In constructing linear functions in F.LE.2, draw on and consolidate previous work in Grade 8 on finding equations for lines and linear functions (8.EE.6, 8.F.4).

F.LE.1Distinguishbetweensituationsthatcanbemodeledwithlinear functionsandwithexponentialfunctions.

a. Provethatlinearfunctionsgrowbyequaldifferencesoverequal intervals;andthatexponentialfunctionsgrowbyequalfactors overequalintervals.

b. Recognizesituationsinwhichonequantitychangesataconstant rateperunitintervalrelativetoanother.

c. Recognizesituationsinwhichaquantitygrowsordecaysbya constantpercentrateperunitintervalrelativetoanother.

F.LE.2Constructlinearandexponentialfunctions,includingarithmetic andgeometricsequences,givenagraph,adescriptionofarelationship, ortwoinput-outputpairs(includereadingthesefromatable).

F.LE.3Observeusinggraphsandtablesthataquantityincreasing exponentiallyeventuallyexceedsaquantityincreasinglinearly, quadratically,or(moregenerally)asapolynomialfunction.

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• Interpretexpressionsforfunctionsin F.LE.5Interprettheparametersinalinearorexponentialfunctionin termsofthesituationtheymodel. termsofacontext.

Limit exponential functions to those of the form f(x) = bx + k.


Unit3:Descriptivestatistics

ExperiencewithdescriptivestatisticsbeganasearlyasGrade6.Studentswereexpectedtodisplaynumericaldata andsummarizeitusingmeasuresofcenterandvariability.Bytheendofmiddleschooltheywerecreatingscatterplotsandrecognizinglineartrendsindata.Thisunitbuildsuponthatpriorexperience,providingstudentswithmore formalmeansofassessinghowamodelfitsdata.Studentsuseregressiontechniquestodescribeapproximatelylinear relationshipsbetweenquantities.Theyusegraphicalrepresentationsandknowledgeofthecontexttomakejudgmentsabouttheappropriatenessoflinearmodels.Withlinearmodels,theylookatresidualstoanalyzethegoodness offit.

Unit3:DescriptiveStatistics


• Summarize,represent,andinterpret dataonasinglecountormeasurement variable.

In grades 6 – 8, students describe center and spread in a data distribution. Here they choose a summary statistic appropriate to the characteristics of the data distribution, such as the shape of the distribution or the existence of extreme data points.

S.ID.1Representdatawithplotsontherealnumberline(dotplots, histograms,andboxplots).

S.ID.2Usestatisticsappropriatetotheshapeofthedatadistribution tocomparecenter(median,mean)andspread(interquartilerange, standarddeviation)oftwoormoredifferentdatasets.

S.ID.3Interpretdifferencesinshape,center,andspreadinthecontext ofthedatasets,accountingforpossibleeffectsofextremedatapoints (outliers).

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• Summarize,represent,andinterpret dataontwocategoricalandquantitativevariables.

Students take a more sophisticated look at using a linear function to model the relationship between two numerical variables. In addition to fitting a line to data, students assess how well the model fits by analyzing residuals.

S.ID.6b should be focused on linear models, but may be used to preview quadratic functions in Unit 5 of this course.

S.ID.5Summarizecategoricaldatafortwocategoriesintwo-way frequencytables.Interpretrelativefrequenciesinthecontextofthe data(includingjoint,marginal,andconditionalrelativefrequencies). Recognizepossibleassociationsandtrendsinthedata.

S.ID.6Representdataontwoquantitativevariablesonascatterplot, anddescribehowthevariablesarerelated.

a. Fitafunctiontothedata;usefunctionsfittedtodatatosolve problemsinthecontextofthedata. Use given functions or choose a function suggested by the context. Emphasize linear and exponential models.

b. Informallyassessthefitofafunctionbyplottingandanalyzing residuals.

c. Fitalinearfunctionforascatterplotthatsuggestsalinearassociation.

• Interpretlinearmodels.

Build on students’ work with linear relationships in eighth grade and introduce the correlation coefficient. The focus here is on the computation and interpretation of the correlation coefficient as a measure of how well the data fit the relationship. The important distinction between a statistical relationship and a causeand-effect relationship arises in S.ID.9.

S.ID.7Interprettheslope(rateofchange)andtheintercept(constant term)ofalinearmodelinthecontextofthedata.

S.ID.8Compute(usingtechnology)andinterpretthecorrelation coefficientofalinearfit.

S.ID.9Distinguishbetweencorrelationandcausation.


Unit4:expressionsandequations

Inthisunit,studentsbuildontheirknowledgefromunit2,wheretheyextendedthelawsofexponentstorational exponents.Studentsapplythisnewunderstandingofnumberandstrengthentheirabilitytoseestructureinandcreatequadraticandexponentialexpressions.Theycreateandsolveequations,inequalities,andsystemsofequations involvingquadraticexpressions.

Unit4:ExpressionsandEquations



Focus on quadratic and exponential expressions. For A.SSE.1b, exponents are extended from the integer exponents found in Unit 1 to rational exponents focusing on those that represent square or cube roots.




A.SSE.2Usethestructureofanexpressiontoidentifywaystorewrite it. For example, see x4 – y4 as (x2)2 – (y2)2, thus recognizing it as a difference of squares that can be factored as (x2 – y2)(x2 + y2).

• Writeexpressionsinequivalentforms tosolveproblems.

It is important to balance conceptual understanding and procedural fluency in work with equivalent expressions. For example, development of skill in factoring and completing the square goes hand-in-hand with understanding what different forms of a quadratic expression reveal.

A.SSE.3Chooseandproduceanequivalentformofanexpression torevealandexplainpropertiesofthequantityrepresentedbythe expression.★

a. Factoraquadraticexpressiontorevealthezerosofthefunction itdefines.

b. Complete the square in a quadratic expression to reveal the maximumorminimumvalueofthefunctionitdefines.

c. Usethepropertiesofexponentstotransformexpressionsfor exponentialfunctions. For example the expression 1.15t can be rewritten as (1.151/12)12t ≈ 1.01212t to reveal the approximate equivalent monthly interest rate if the annual rate is 15%.

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• Performarithmeticoperationson A.APR.1Understandthatpolynomialsformasystemanalogoustothe polynomials. integers,namely,theyareclosedundertheoperationsofaddition,

subtraction,andmultiplication;add,subtract,andmultiplypolynomials.

Focus on polynomial expressions that simplify to forms that are linear or quadratic in a positive integer power of x.


Extend work on linear and exponential equations in Unit 1 to quadratic equations. Extend A.CED.4 to formulas involving squared variables.




• Solveequationsandinequalitiesinone variable.

Students should learn of the existence of the complex number system, but will not solve quadratics with complex solutions until Algebra II.

A.REI.4Solvequadraticequationsinonevariable.

a. Usethemethodofcompletingthesquaretotransformanyquadraticequationin x intoanequationoftheform (x – p)2 = q that hasthesamesolutions.Derivethequadraticformulafromthis form.

b. Solvequadraticequationsbyinspection(e.g.,for x2 = 49), taking squareroots,completingthesquare,thequadraticformulaand factoring,asappropriatetotheinitialformoftheequation.Recognizewhenthequadraticformulagivescomplexsolutionsand writethemas a ± bi forrealnumbers a and b.





Include systems consisting of one linear and one quadratic equation. Include systems that lead to work with fractions. For example, finding the intersections between x2+y2=1 and y = (x+1)/2 leads to the point (3/5, 4/5) on the unit circle, corresponding to the Pythagorean triple 32+42=52.

A.REI.7Solveasimplesystemconsistingofalinearequationanda quadraticequationintwovariablesalgebraicallyandgraphically. For example, find the points of intersection between the line y = –3x and the circle x2 + y2 = 3.

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Unit5:QuadraticFunctionsandmodeling

Inpreparationforworkwithquadraticrelationshipsstudentsexploredistinctionsbetweenrationalandirrationalnumbers.Theyconsiderquadraticfunctions,comparingthekeycharacteristicsofquadraticfunctionstothoseoflinear and exponential functions. They select from among these functions to model phenomena. Students learn to anticipate thegraphofaquadraticfunctionbyinterpretingvariousformsofquadraticexpressions.Inparticular,theyidentify therealsolutionsofaquadraticequationasthezerosofarelatedquadraticfunction.Studentslearnthatwhenquadraticequationsdonothaverealsolutionsthenumbersystemmustbeextendedsothatsolutionsexist,analogous tothewayinwhichextendingthewholenumberstothenegativenumbersallowsx+1=0tohaveasolution.Formal workwithcomplexnumberscomesinAlgebraII.Studentsexpandtheirexperiencewithfunctionstoincludemore specializedfunctions—absolutevalue,step,andthosethatarepiecewise-defined.

Unit5:QuadraticFunctionsandModeling


• Usepropertiesofrationalandirrational N.RN.3Explainwhythesumorproductoftworationalnumbersis numbers. rational;thatthesumofarationalnumberandanirrationalnumberis

irrational;andthattheproductofanonzerorationalnumberandan irrationalnumberisirrational. Connect N.RN.3 to physical situations,

e.g., finding the perimeter of a square of area 2.


Focus on quadratic functions; compare with linear and exponential functions studied in Unit 2.


For F.IF.7b, compare and contrast absolute value, step and piecewise-defined functions with linear, quadratic, and exponential functions. Highlight issues of domain, range, and usefulness when examining piecewise-defined functions. Note that this unit, and in particular in F.IF.8b, extends the work begun in Unit 2 on exponential functions with integer exponents. For F.IF.9, focus on expanding the types of functions considered to include, linear, exponential, and quadratic.

Extend work with quadratics to include the relationship between coefficients and roots, and that once roots are known, a quadratic equation can be factored.






b. Graphsquareroot,cuberoot,andpiecewise-definedfunctions, includingstepfunctionsandabsolutevaluefunctions.

F.IF.8Writeafunctiondefinedbyanexpressionindifferentbut equivalentformstorevealandexplaindifferentpropertiesofthe function.

a. Usetheprocessoffactoringandcompletingthesquareinaquadraticfunctiontoshowzeros,extremevalues,andsymmetryof thegraph,andinterprettheseintermsofacontext.

b. Usethepropertiesofexponentstointerpretexpressionsforexponentialfunctions. For example, identify percent rate of change in functions such as y = (1.02)t, y = (0.97)t, y = (1.01)12t, y = (1.2)t/10, and classify them as representing exponential growth or decay.


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Focus on situations that exhibit a quadratic relationship.





For F.BF.3, focus on quadratic functions, and consider including absolute value functions. For F.BF.4a, focus on linear functions but consider simple situations where the domain of the function must be restricted in order for the inverse to exist, such as

f(x) = x2, x>0.


Compare linear and exponential growth to quadratic growth.


F.BF.4Findinversefunctions.

a. Solveanequationoftheform f(x) = c forasimplefunction f thathasaninverseandwriteanexpressionfortheinverse. For example, f(x) = 2 x3 or f(x) = (x+1)/(x-1) for x ≠ 1.


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traditionalPathway:Geometry TheThefundamentalpurposeofthecourseinGeometryistoformalizeandextendstudents’geometricexperiences fromthemiddlegrades.Studentsexploremorecomplexgeometricsituationsanddeepentheirexplanationsofgeometricrelationships,movingtowardsformalmathematicalarguments.ImportantdifferencesexistbetweenthisGeometrycourseandthehistoricalapproachtakeninGeometryclasses.Forexample,transformationsareemphasized earlyinthiscourse.CloseattentionshouldbepaidtotheintroductorycontentfortheGeometryconceptualcategory foundinthehighschoolCCSS.TheMathematicalPracticeStandardsapplythroughouteachcourseand,together withthecontentstandards,prescribethatstudentsexperiencemathematicsasacoherent,useful,andlogicalsubject thatmakesuseoftheirabilitytomakesenseofproblemsituations.Thecriticalareas,organizedintosixunitsareas follows.

CriticalArea1:Inpreviousgrades,studentswereaskedtodrawtrianglesbasedongivenmeasurements.Theyalso havepriorexperiencewithrigidmotions:translations,reflections,androtationsandhaveusedthesetodevelopnotions about what it means for two objects to be congruent. In this unit, students establish triangle congruence criteria, basedonanalysesofrigidmotionsandformalconstructions.Theyusetrianglecongruenceasafamiliarfoundation forthedevelopmentofformalproof.Studentsprovetheorems—usingavarietyofformats—andsolveproblemsabout triangles,quadrilaterals,andotherpolygons.Theyapplyreasoningtocompletegeometricconstructionsandexplain whytheywork.

CriticalArea2:Studentsapplytheirearlierexperiencewithdilationsandproportionalreasoningtobuildaformal understandingofsimilarity.Theyidentifycriteriaforsimilarityoftriangles,usesimilaritytosolveproblems,andapply similarityinrighttrianglestounderstandrighttriangletrigonometry,withparticularattentiontospecialrighttrianglesandthePythagoreantheorem.StudentsdeveloptheLawsofSinesandCosinesinordertofindmissingmeasuresofgeneral(notnecessarilyright)triangles,buildingonstudents’workwithquadraticequationsdoneinthefirst course.Theyareabletodistinguishwhetherthreegivenmeasures(anglesorsides)define0,1,2,orinfinitelymany triangles.

CriticalArea3:Students’experiencewithtwo-dimensionalandthree-dimensionalobjectsisextendedtoinclude informalexplanationsofcircumference,areaandvolumeformulas.Additionally,studentsapplytheirknowledgeof two-dimensionalshapestoconsidertheshapesofcross-sectionsandtheresultofrotatingatwo-dimensionalobject aboutaline.

CriticalArea4:BuildingontheirworkwiththePythagoreantheoremin8th gradetofinddistances,studentsusea rectangularcoordinatesystemtoverifygeometricrelationships,includingpropertiesofspecialtrianglesandquadrilateralsandslopesofparallelandperpendicularlines,whichrelatesbacktoworkdoneinthefirstcourse.Students continuetheirstudyofquadraticsbyconnectingthegeometricandalgebraicdefinitionsoftheparabola.

CriticalArea5:Inthisunitstudentsprovebasictheoremsaboutcircles,suchasatangentlineisperpendiculartoa radius,inscribedangletheorem,andtheoremsaboutchords,secants,andtangentsdealingwithsegmentlengths andanglemeasures.Theystudyrelationshipsamongsegmentsonchords,secants,andtangentsasanapplicationof similarity.IntheCartesiancoordinatesystem,studentsusethedistanceformulatowritetheequationofacirclewhen given the radius and the coordinates of its center. Given an equation of a circle, they draw the graph in the coordinate plane,andapplytechniquesforsolvingquadraticequations,whichrelatesbacktoworkdoneinthefirstcourse,to determineintersectionsbetweenlinesandcirclesorparabolasandbetweentwocircles.

CriticalArea6:Buildingonprobabilityconceptsthatbeganinthemiddlegrades,studentsusethelanguagesofset theorytoexpandtheirabilitytocomputeandinterprettheoreticalandexperimentalprobabilitiesforcompound events,attendingtomutuallyexclusiveevents,independentevents,andconditionalprobability.Studentsshouldmake useofgeometricprobabilitymodelswhereverpossible.Theyuseprobabilitytomakeinformeddecisions.

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Standards

• Experiment with transformations in the plane. Unit1 • Understand congruence in terms of rigid motions.

Congruence,Proof,and • Prove geometric theorems. Constructions • Make geometric constructions.

Unit2

Similarity,Proof,and Trigonometry

• Understand similarity in terms of similarity transformations.

• Prove theorems involving similarity.

• Define trigonometric ratios and solve problems involving right triangles.

• Apply geometric concepts in modeling situations.

• Apply trigonometry to general triangles.



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• Explain volume formulas and use them to solve Constructviable problems. argumentsandcritiquethe

reasoningofothers. Unit3

• Visualize the relation between two-dimensional Dimensions

ExtendingtoThree and three-dimensional objects.

• Apply geometric concepts in modeling situations. Modelwithmathematics.

• Use coordinates to prove simple geometric theorems algebraically.

Unit4

ConnectingAlgebra andGeometrythrough • Translate between the geometric description and

Coordinates the equation for a conic section.

Unit5

CirclesWithand WithoutCoordinates

• Understand and apply theorems about circles.

• Find arc lengths and areas of sectors of circles.

• Translate between the geometric description and the equation for a conic section.

• Use coordinates to prove simple geometric theorem algebraically.



Attendtoprecision.



Unit6

Applicationsof Probability

• Understand independence and conditional probability and use them to interpret data.

• Use the rules of probability to compute probabilities of compound events in a uniform probability model.

• Use probability to evaluate outcomes of decisions.



Unit1:congruence,Proof,andconstructions

Inpreviousgrades,studentswereaskedtodrawtrianglesbasedongivenmeasurements.Theyalsohavepriorexperiencewithrigidmotions:translations,reflections,androtationsandhaveusedthesetodevelopnotionsaboutwhatit meansfortwoobjectstobecongruent.Inthisunit,studentsestablishtrianglecongruencecriteria,basedonanalyses ofrigidmotionsandformalconstructions.Theyusetrianglecongruenceasafamiliarfoundationforthedevelopment offormalproof.Studentsprovetheorems—usingavarietyofformats—andsolveproblemsabouttriangles,quadrilaterals,andotherpolygons.Theyapplyreasoningtocompletegeometricconstructionsandexplainwhytheywork.

Unit1:Congruence,Proof,andConstructions

ClustersandInstructionalNotes CommonCoreStateStandards

• Experimentwithtransformationsinthe plane.

Build on student experience with rigid motions from earlier grades. Point out the basis of rigid motions in geometric concepts, e.g., translations move points a specified distance along a line parallel to a specified line; rotations move objects along a circular arc with a specified center through a specified angle.

• Understandcongruenceintermsof rigidmotions.

Rigid motions are at the foundation of the definition of congruence. Students reason from the basic properties of rigid motions (that they preserve distance and angle), which are assumed without proof. Rigid motions and their assumed properties can be used to establish the usual triangle congruence criteria, which can then be used to prove other theorems.

G.CO.1Knowprecisedefinitionsofangle,circle,perpendicularline, parallelline,andlinesegment,basedontheundefinednotionsofpoint, line,distancealongaline,anddistancearoundacirculararc.

G.CO.2Representtransformationsintheplaneusing,e.g., transparenciesandgeometrysoftware;describetransformationsas functionsthattakepointsintheplaneasinputsandgiveotherpoints asoutputs.Comparetransformationsthatpreservedistanceandangle tothosethatdonot(e.g.,translationversushorizontalstretch).

G.CO.3Givenarectangle,parallelogram,trapezoid,orregularpolygon, describetherotationsandreflectionsthatcarryitontoitself.

G.CO.4Developdefinitionsofrotations,reflections,andtranslations intermsofangles,circles,perpendicularlines,parallellines,andline segments.

G.CO.5Givenageometricfigureandarotation,reflection,or translation, draw the transformed figure using, e.g., graph paper, tracing paper,orgeometrysoftware.Specifyasequenceoftransformations thatwillcarryagivenfigureontoanother.

G.CO.6Usegeometricdescriptionsofrigidmotionstotransform figuresandtopredicttheeffectofagivenrigidmotiononagiven figure;giventwofigures,usethedefinitionofcongruenceintermsof rigidmotionstodecideiftheyarecongruent.

G.CO.7Usethedefinitionofcongruenceintermsofrigidmotionsto showthattwotrianglesarecongruentifandonlyifcorresponding pairsofsidesandcorrespondingpairsofanglesarecongruent.

G.CO.8Explainhowthecriteriafortrianglecongruence(ASA,SAS,and SSS)followfromthedefinitionofcongruenceintermsofrigidmotions.

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• Provegeometrictheorems.

Encourage multiple ways of writing proofs, such as in narrative paragraphs, using flow diagrams, in two-column format, and using diagrams without words. Students should be encouraged to focus on the validity of the underlying reasoning while exploring a variety of formats for expressing that reasoning. Implementation of G.CO.10 may be extended to include concurrence of perpendicular bisectors and angle bisectors as preparation for G.C.3 in Unit 5.

G.CO.9Provetheoremsaboutlinesandangles. Theorems include: vertical angles are congruent; when a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent; points on a perpendicular bisector of a line segment are exactly those equidistant from the segment’s endpoints.

G.CO.10Provetheoremsabouttriangles. Theorems include: measures of interior angles of a triangle sum to 180°; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point.

G.CO.11Provetheoremsaboutparallelograms. Theorems include: opposite sides are congruent, opposite angles are congruent, the diagonals of a parallelogram bisect each other, and conversely, rectangles are parallelograms with congruent diagonals.




• Makegeometricconstructions.

Build on prior student experience with simple constructions. Emphasize the ability to formalize and explain how these constructions result in the desired objects.

Some of these constructions are closely related to previous standards and can be introduced in conjunction with them.

G.CO.12Makeformalgeometricconstructionswithavarietyoftools andmethods(compassandstraightedge,string,reflectivedevices, paperfolding,dynamicgeometricsoftware,etc.). Copying a segment; copying an angle; bisecting a segment; bisecting an angle; constructing perpendicular lines, including the perpendicular bisector of a line segment; and constructing a line parallel to a given line through a point not on the line.

G.CO.13Constructanequilateraltriangle,asquare,andaregular hexagoninscribedinacircle.

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Unit2:similarity,Proof,andtrigonometry

Studentsapplytheirearlierexperiencewithdilationsandproportionalreasoningtobuildaformalunderstanding ofsimilarity.Theyidentifycriteriaforsimilarityoftriangles,usesimilaritytosolveproblems,andapplysimilarityin righttrianglestounderstandrighttriangletrigonometry,withparticularattentiontospecialrighttrianglesandthe Pythagoreantheorem.StudentsdeveloptheLawsofSinesandCosinesinordertofindmissingmeasuresofgeneral (notnecessarilyright)triangles.Theyareabletodistinguishwhetherthreegivenmeasures(anglesorsides)define0, 1,2,orinfinitelymanytriangles.

Unit2:Similarity,Proof,andTrigonometry


• Understandsimilarityintermsofsimilaritytransformations.

G.SRT.1Verifyexperimentallythepropertiesofdilationsgivenbya centerandascalefactor.

a. Adilationtakesalinenotpassingthroughthecenterofthedilationtoaparallelline,andleavesalinepassingthroughthecenter unchanged.

b. Thedilationofalinesegmentislongerorshorterintheratio givenbythescalefactor.

G.SRT.2Giventwofigures,usethedefinitionofsimilarityintermsof similaritytransformationstodecideiftheyaresimilar;explainusing similaritytransformationsthemeaningofsimilarityfortrianglesasthe equalityofallcorrespondingpairsofanglesandtheproportionalityof allcorrespondingpairsofsides.

G.SRT.3Usethepropertiesofsimilaritytransformationstoestablishthe AAcriterionfortwotrianglestobesimilar.

• Provetheoremsinvolvingsimilarity. G.SRT.4Provetheoremsabouttriangles. Theorems include: a line parallel to one side of a triangle divides the other two proportionally, and conversely; the Pythagorean Theorem proved using triangle similarity.

G.SRT.5Usecongruenceandsimilaritycriteriafortrianglestosolve problemsandtoproverelationshipsingeometricfigures.

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• Definetrigonometricratiosandsolve problemsinvolvingrighttriangles.

G.SRT.6Understandthatbysimilarity,sideratiosinrighttriangles arepropertiesoftheanglesinthetriangle,leadingtodefinitionsof trigonometricratiosforacuteangles.

G.SRT.7Explainandusetherelationshipbetweenthesineandcosineof complementaryangles.

G.SRT.8UsetrigonometricratiosandthePythagoreanTheoremtosolve righttrianglesinappliedproblems.★

• Applygeometricconceptsinmodeling situations.

Focus on situations well modeled by trigonometric ratios for acute angles.

G.MG.1Usegeometricshapes,theirmeasures,andtheirpropertiesto describeobjects(e.g.,modelingatreetrunkorahumantorsoasa cylinder).*

G.MG.2Applyconceptsofdensitybasedonareaandvolumein modelingsituations(e.g.,personspersquaremile,BTUspercubic foot).*

G.MG.3Applygeometricmethodstosolvedesignproblems(e.g., designinganobjectorstructuretosatisfyphysicalconstraintsor minimizecost;workingwithtypographicgridsystemsbasedonratios).*

• Applytrigonometrytogeneraltriangles.

With respect to the general case of the Laws of Sines and Cosines, the definitions of sine and cosine must be extended to obtuse angles.

G.SRT.9 (+) Derivetheformula A = 1/2 absin(C)fortheareaofatriangle bydrawinganauxiliarylinefromavertexperpendiculartotheopposite side.

G.SRT.10 (+) ProvetheLawsofSinesandCosinesandusethemtosolve problems.

G.SRT.11 (+) UnderstandandapplytheLawofSinesandtheLawof Cosinestofindunknownmeasurementsinrightandnon-righttriangles (e.g.,surveyingproblems,resultantforces).


Unit3:extendingtothreeDimensions

Students’experiencewithtwo-dimensionalandthree-dimensionalobjectsisextendedtoincludeinformalexplanationsofcircumference,areaandvolumeformulas.Additionally,studentsapplytheirknowledgeoftwo-dimensional shapestoconsidertheshapesofcross-sectionsandtheresultofrotatingatwo-dimensionalobjectaboutaline.

Unit3:ExtendingtoThreeDimensions


• Explainvolumeformulasandusethem tosolveproblems.

Informal arguments for area and volume formulas can make use of the way in which area and volume scale under similarity transformations: when one figure in the plane results from another by applying a similarity transformation with scale factor k, its area is k2 times the area of the first. Similarly, volumes of solid figures scale by k3 under a similarity transformation with scale factor k.

G.GMD.1Giveaninformalargumentfortheformulasforthe circumferenceofacircle,areaofacircle,volumeofacylinder,pyramid, andcone. Use dissection arguments, Cavalieri’s principle, and informal limit arguments.

G.GMD.3Usevolumeformulasforcylinders,pyramids,cones,and spherestosolveproblems.★

• Visualizetherelationbetweentwo G.GMD.4 Identifytheshapesoftwo-dimensionalcross-sectionsofthreedimensionalandthree-dimensional dimensionalobjects,andidentifythree-dimensionalobjectsgenerated objects. byrotationsoftwo-dimensionalobjects.

• Applygeometricconceptsinmodeling G.MG.1Usegeometricshapes,theirmeasures,andtheirpropertiesto situations. describeobjects(e.g.,modelingatreetrunkorahumantorsoasa

cylinder).*

Focus on situations that require relating two- and three-dimensional objects, determining and using volume, and the trigonometry of general triangles.

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Unit4:connectingalgebraandGeometrythroughcoordinates

BuildingontheirworkwiththePythagoreantheoremin8th gradetofinddistances,studentsusearectangularcoordinatesystemtoverifygeometricrelationships,includingpropertiesofspecialtrianglesandquadrilateralsandslopes ofparallelandperpendicularlines.Studentscontinuetheirstudyofquadraticsbyconnectingthegeometricand algebraicdefinitionsoftheparabola.

Unit4:ConnectingAlgebraandGeometryThroughCoordinates


• Usecoordinatestoprovesimplegeometrictheoremsalgebraically.

This unit has a close connection with the next unit. For example, a curriculum might merge G.GPE.1 and the Unit 5 treatment of G.GPE.4 with the standards in this unit. Reasoning with triangles in this unit is limited to right triangles; e.g., derive the equation for a line through two points using similar right triangles.

Relate work on parallel lines in G.GPE.5 to work on A.REI.5 in High School Algebra I involving systems of equations having no solution or infinitely many solutions.

G.GPE.7 provides practice with the distance formula and its connection with the Pythagorean theorem.

• Translatebetweenthegeometricdescriptionandtheequationforaconic section.

The directrix should be parallel to a coordinate axis.

G.GPE.4Usecoordinatestoprovesimplegeometrictheorems algebraically. For example, prove or disprove that a figure defined by four given points in the coordinate plane is a rectangle; prove or disprove that the point (1, √3) lies on the circle centered at the origin and containing the point (0, 2).

G.GPE.5Provetheslopecriteriaforparallelandperpendicularlinesand usesthemtosolvegeometricproblems(e.g.,findtheequationofaline parallelorperpendiculartoagivenlinethatpassesthroughagiven point).

G.GPE.6Findthepointonadirectedlinesegmentbetweentwogiven pointsthatpartitionsthesegmentinagivenratio.

G.GPE.7Usecoordinatestocomputeperimetersofpolygonsandareas oftrianglesandrectangles,e.g.,usingthedistanceformula.★

G.GPE.2Derivetheequationofaparabolagivenafocusanddirectrix.

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Unit5:circlesWithandWithoutcoordinates

Inthisunit,studentsprovebasictheoremsaboutcircles,withparticularattentiontoperpendicularityandinscribed angles,inordertoseesymmetryincirclesandasanapplicationoftrianglecongruencecriteria.Theystudyrelationshipsamongsegmentsonchords,secants,andtangentsasanapplicationofsimilarity.IntheCartesiancoordinate system,studentsusethedistanceformulatowritetheequationofacirclewhengiventheradiusandthecoordinates ofitscenter.Givenanequationofacircle,theydrawthegraphinthecoordinateplane,andapplytechniquesforsolvingquadraticequationstodetermineintersectionsbetweenlinesandcirclesorparabolasandbetweentwocircles.

Unit5:CirclesWithandWithoutCoordinates


• Understandandapplytheoremsabout circles.

G.C.1Provethatallcirclesaresimilar.

G.C.2Identifyanddescriberelationshipsamonginscribedangles,radii, andchords. Include the relationship between central, inscribed, and circumscribed angles; inscribed angles on a diameter are right angles; the radius of a circle is perpendicular to the tangent where the radius intersects the circle.

G.C.3Constructtheinscribedandcircumscribedcirclesofatriangle, andprovepropertiesofanglesforaquadrilateralinscribedinacircle.

G.C.4 (+) Constructatangentlinefromapointoutsideagivencircleto thecircle.

• Findarclengthsandareasofsectors ofcircles.

Emphasize the similarity of all circles. Note that by similarity of sectors with the same central angle, arc lengths are proportional to the radius. Use this as a basis for introducing radian as a unit of measure. It is not intended that it be applied to the development of circular trigonometry in this course.

G.C.5Deriveusingsimilaritythefactthatthelengthofthearc interceptedbyanangleisproportionaltotheradius,anddefinethe radianmeasureoftheangleastheconstantofproportionality;derive theformulafortheareaofasector.

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• Translatebetweenthegeometricde G.GPE.1Derivetheequationofacircle ofgivencenterandradiususing scriptionandtheequationforaconic thePythagoreanTheorem;completethesquaretofindthecenterand section. radiusofacirclegivenbyanequation.

• Usecoordinatestoprovesimplegeo G.GPE.4Usecoordinatestoprovesimplegeometrictheorems metrictheoremsalgebraically. algebraically. For example, prove or disprove that a figure defined

by four given points in the coordinate plane is a rectangle; prove or Include simple proofs involving circles. disprove that the point (1, √3) lies on the circle centered at the origin

and containing the point (0, 2).

• Applygeometricconceptsinmodeling G.MG.1Usegeometricshapes,theirmeasures,andtheirpropertiesto situations. describeobjects(e.g.,modelingatreetrunkorahumantorsoasa

cylinder).*

Focus on situations in which the analysis of circles is required.


Unit6:applicationsofProbability

Buildingonprobabilityconceptsthatbeganinthemiddlegrades,studentsusethelanguagesofsettheorytoexpand theirabilitytocomputeandinterprettheoreticalandexperimentalprobabilitiesforcompoundevents,attendingto mutuallyexclusiveevents,independentevents,andconditionalprobability.Studentsshouldmakeuseofgeometric probabilitymodelswhereverpossible.Theyuseprobabilitytomakeinformeddecisions.

Unit6:ApplicationsofProbability


• Understandindependenceandconditionalprobabilityandusethemto interpretdata.

Build on work with two-way tables from Algebra I Unit 3 (S.ID.5) to develop understanding of conditional probability and independence.

• Usetherulesofprobabilitytocompute probabilitiesofcompoundeventsina uniformprobabilitymodel.

S.CP.1Describeeventsassubsetsofasamplespace(thesetof outcomes)usingcharacteristics(orcategories)oftheoutcomes,or asunions,intersections,orcomplementsofotherevents(“or,”“and,” “not”).

S.CP.2Understandthattwoevents A and B areindependentifthe probabilityof A and B occurringtogetheristheproductoftheir probabilities,andusethischaracterizationtodetermineiftheyare independent.

S.CP.3Understandtheconditionalprobabilityof A given B as P(A and B)/P(B),andinterpretindependenceof A and B assayingthat theconditionalprobabilityof A given B isthesameastheprobability of A,andtheconditionalprobabilityof B given A isthesameasthe probabilityof B.

S.CP.4Constructandinterprettwo-wayfrequencytablesofdatawhen twocategoriesareassociatedwitheachobjectbeingclassified.Usethe two-waytableasasamplespacetodecideifeventsareindependent andtoapproximateconditionalprobabilities. For example, collect data from a random sample of students in your school on their favorite subject among math, science, and English. Estimate the probability that a randomly selected student from your school will favor science given that the student is in tenth grade. Do the same for other subjects and compare the results.

S.CP.5Recognizeandexplaintheconceptsofconditionalprobability andindependenceineverydaylanguageandeverydaysituations. For example, compare the chance of having lung cancer if you are a smoker with the chance of being a smoker if you have lung cancer.

S.CP.6Findtheconditionalprobabilityof A given B asthefractionof B’s outcomesthatalsobelongto A,andinterprettheanswerintermsofthe model.

S.CP.7ApplytheAdditionRule, P(A or B) = P(A) + P(B) – P(A and B), andinterprettheanswerintermsofthemodel.

S.CP.8 (+) ApplythegeneralMultiplicationRuleinauniformprobability model, P(A and B) = P(A)P(B|A) = P(B)P(A|B),andinterprettheanswer intermsofthemodel.

S.CP.9 (+) Usepermutationsandcombinationstocomputeprobabilities ofcompoundeventsandsolveproblems.

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• Useprobabilitytoevaluateoutcomes ofdecisions.

This unit sets the stage for work in Algebra II, where the ideas of statistical inference are introduced. Evaluating the risks associated with conclusions drawn from sample data (i.e. incomplete information) requires an understanding of probability concepts.

S.MD.6 (+) Useprobabilitiestomakefairdecisions(e.g.,drawingbylots, usingarandomnumbergenerator).

S.MD.7 (+) Analyzedecisionsandstrategiesusingprobabilityconcepts (e.g.,producttesting,medicaltesting,pullingahockeygoalieattheend ofagame).


traditionalPathway:algebraII Buildingontheirworkwithlinear,quadratic,andexponentialfunctions,studentsextendtheirrepertoireoffunctions toincludepolynomial,rational,andradicalfunctions.2 Studentsworkcloselywiththeexpressionsthatdefinethe functions,andcontinuetoexpandandhonetheirabilitiestomodelsituationsandtosolveequations,including solvingquadraticequationsoverthesetofcomplexnumbersandsolvingexponentialequationsusingtheproperties oflogarithms.TheMathematicalPracticeStandardsapplythroughouteachcourseand,togetherwiththecontent standards,prescribethatstudentsexperiencemathematicsasacoherent,useful,andlogicalsubjectthatmakesuse oftheirabilitytomakesenseofproblemsituations.Thecriticalareasforthiscourse,organizedintofourunits,areas follows:

CriticalArea1:Thisunitdevelopsthestructuralsimilaritiesbetweenthesystemofpolynomialsandthesystem ofintegers.Studentsdrawonanalogiesbetweenpolynomialarithmeticandbase-tencomputation,focusingon propertiesofoperations,particularlythedistributiveproperty.Studentsconnectmultiplicationofpolynomialswith multiplicationofmulti-digitintegers,anddivisionofpolynomialswithlongdivisionofintegers.Studentsidentify zerosofpolynomials,includingcomplexzerosofquadraticpolynomials,andmakeconnectionsbetweenzerosof polynomialsandsolutionsofpolynomialequations.Theunitculminateswiththefundamentaltheoremofalgebra.A centralthemeofthisunitisthatthearithmeticofrationalexpressionsisgovernedbythesamerulesasthearithmetic ofrationalnumbers.

CriticalArea2:Buildingontheirpreviousworkwithfunctions,andontheirworkwithtrigonometricratiosandcircles inGeometry,studentsnowusethecoordinateplanetoextendtrigonometrytomodelperiodicphenomena.

CriticalArea3:Inthisunitstudentssynthesizeandgeneralizewhattheyhavelearnedaboutavarietyoffunction families.Theyextendtheirworkwithexponentialfunctionstoincludesolvingexponentialequationswithlogarithms. Theyexploretheeffectsoftransformationsongraphsofdiversefunctions,includingfunctionsarisinginan application,inordertoabstractthegeneralprinciplethattransformationsonagraphalwayshavethesameeffect regardlessofthetypeoftheunderlyingfunction.Theyidentifyappropriatetypesoffunctionstomodelasituation, theyadjustparameterstoimprovethemodel,andtheycomparemodelsbyanalyzingappropriatenessoffitand makingjudgmentsaboutthedomainoverwhichamodelisagoodfit.Thedescriptionofmodelingas“the process of choosing and using mathematics and statistics to analyze empirical situations, to understand them better, and to make decisions”isattheheartofthisunit.Thenarrativediscussionanddiagramofthemodelingcycleshouldbe consideredwhenknowledgeoffunctions,statistics,andgeometryisappliedinamodelingcontext.

CriticalArea4:Inthisunit,studentsseehowthevisualdisplaysandsummarystatisticstheylearnedinearliergrades relatetodifferenttypesofdataandtoprobabilitydistributions.Theyidentifydifferentwaysofcollectingdata— includingsamplesurveys,experiments,andsimulations—andtherolethatrandomnessandcarefuldesignplayinthe conclusionsthatcanbedrawn.

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2Inthiscourserationalfunctionsarelimitedtothosewhosenumeratorsareofdegreeatmost1anddenominatorsofdegreeatmost2; radicalfunctionsarelimitedtosquarerootsorcuberootsofatmostquadraticpolynomials.



Standards

Unit1

Polynomial,Rational, andRadical

Relationships

Unit2

TrigonometricFunctions

Unit3

ModelingwithFunctions

• Perform arithmetic operations with complex numbers.

• Use complex numbers in polynomial identities and equations.




• Understand the relationship between zeros and factors of polynomials.

• Use polynomial identities to solve problems.

• Rewrite rational expressions.




• Extend the domain of trigonometric functions using the unit circle.

• Model periodic phenomena with trigonometric function.

• Prove and apply trigonometric identites.












Attendtoprecision.



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Unit4

Inferencesand ConclusionsfromData

• Summarize, represent, and interpret data on single count or measurement variable.

• Understand and evaluate random processes underlying statistical experiments.

• Make inferences and justify conclusions from sample surveys, experiments and observational studies.




Unit1:Polynomial,rational,andradicalrelationships

Thisunitdevelopsthestructuralsimilaritiesbetweenthesystemofpolynomialsandthesystemofintegers.Students drawonanalogiesbetweenpolynomialarithmeticandbase-tencomputation,focusingonpropertiesofoperations, particularlythedistributiveproperty.Studentsconnectmultiplicationofpolynomialswithmultiplicationofmulti-digit integers,anddivisionofpolynomialswithlongdivisionofintegers.Studentsidentifyzerosofpolynomials,including complexzerosofquadraticpolynomials,andmakeconnectionsbetweenzerosofpolynomialsandsolutionsofpolynomialequations.Theunitculminateswiththefundamentaltheoremofalgebra.Rationalnumbersextendthearithmeticofintegersbyallowingdivisionbyallnumbersexcept0.Similarly,rationalexpressionsextendthearithmeticof polynomialsbyallowingdivisionbyallpolynomialsexceptthezeropolynomial.Acentralthemeofthisunitisthatthe arithmeticofrationalexpressionsisgovernedbythesamerulesasthearithmeticofrationalnumbers.

Unit1:Polynomial,Rational,andRadicalRelationships


• Performarithmeticoperationswith N.CN.1Knowthereisacomplexnumber i suchthat i2 =−1,andevery complexnumbers. complexnumberhastheform a + bi with a and b real.

N.CN.2Usetherelation i2 =–1andthecommutative,associative,and distributivepropertiestoadd,subtract,andmultiplycomplexnumbers.

• Usecomplexnumbersinpolynomial N.CN.7Solvequadraticequationswithrealcoefficientsthathave identitiesandequations. complexsolutions.

N.CN.8 (+) Extendpolynomialidentitiestothecomplexnumbers. For Limit to polynomials with real example, rewrite x2 + 4 as (x + 2i)(x – 2i). coefficients.

N.CN.9 (+) KnowtheFundamentalTheoremofAlgebra;showthatitis trueforquadraticpolynomials.


Extend to polynomial and rational expressions.





• Writeexpressionsinequivalentforms A.SSE.4Derivetheformulaforthesumofafinitegeometricseries tosolveproblems. (whenthecommonratioisnot1),andusetheformulatosolve

problems. For example, calculate mortgage payments.★

Consider extending A.SSE.4 to infinite geometric series in curricular implementations of this course description.



Extend beyond the quadratic polynomials found in Algebra I.

• Understandtherelationshipbetween A.APR.2KnowandapplytheRemainderTheorem:Forapolynomial zerosandfactorsofpolynomials. p(x) andanumber a,theremainderondivision by x – a is p(a),so p(a) =

0ifandonly if (x – a) isafactorofp(x).

A.APR.3Identifyzerosofpolynomialswhensuitablefactorizationsare available,andusethezerostoconstructaroughgraphofthefunction definedbythepolynomial.

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Unit1:Polynomial,Rational,andRadicalRelationships


• Usepolynomialidentitiestosolve problems.

This cluster has many possibilities for optional enrichment, such as relating the example in A.APR.4 to the solution of the system u2+v2=1, v = t(u+1), relating the Pascal triangle property of binomial coefficients to (x+y)n+1 = (x+y)(x+y)n, deriving explicit formulas for the coefficients, or proving the binomial theorem by induction.

A.APR.4Provepolynomialidentitiesandusethemtodescribe numericalrelationships.Forexample,thepolynomialidentity (x2 + y2)2

= (x2 – y2)2 + (2xy)2 canbeusedtogeneratePythagoreantriples.

A.APR.5 (+) KnowandapplytheBinomialTheoremfortheexpansion of (x + y)n inpowersof x and y forapositiveinteger n,where x and y areanynumbers,withcoefficientsdeterminedforexamplebyPascal’s Triangle.

• Rewriterationalexpressions

The limitations on rational functions apply to the rational expressions in A.APR.6. A.APR.7 requires the general division algorithm for polynomials.

A.APR.6Rewritesimplerationalexpressionsindifferentforms;write a(x)/b(x) intheform q(x) + r(x)/b(x),where a(x), b(x), q(x), and r(x) arepolynomialswiththedegreeof r(x) lessthanthedegreeof b(x), usinginspection,longdivision,or,forthemorecomplicatedexamples,a computeralgebrasystem.

A.APR.7 (+) Understandthatrationalexpressionsformasystem analogoustotherationalnumbers,closedunderaddition,subtraction, multiplication,anddivisionbyanonzerorationalexpression;add, subtract,multiply,anddividerationalexpressions.

• Understandsolvingequationsasa A.REI.2Solvesimplerationalandradicalequationsinonevariable,and processofreasoningandexplainthe giveexamplesshowinghowextraneoussolutionsmayarise. reasoning.

Extend to simple rational and radical equations.


Include combinations of linear, polynomial, rational, radical, absolute value, and exponential functions.

A.REI.11Explainwhythe x-coordinatesofthepointswherethegraphs oftheequations y = f(x) and y = g(x) intersectarethesolutionsof theequation f(x) = g(x);findthesolutionsapproximately,e.g.,using technologytographthefunctions,maketablesofvalues,orfind successiveapproximations.Includecaseswhere f(x) and/or g(x) are linear,polynomial,rational,absolutevalue,exponential,andlogarithmic functions.★

• Analyzefunctionsusingdifferentrep F.IF.7Graphfunctionsexpressedsymbolicallyandshowkeyfeatures resentations. ofthegraph,byhandinsimplecasesandusingtechnologyformore

complicatedcases.★

Relate F.IF.7c to the relationship c.Graphpolynomialfunctions,identifyingzeroswhensuitablefacbetween zeros of quadratic functions torizationsareavailable,andshowingendbehavior. and their factored forms

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Unit2:trigonometricFunctions

Buildingontheirpreviousworkwithfunctions,andontheirworkwithtrigonometricratiosandcirclesinGeometry, studentsnowusethecoordinateplanetoextendtrigonometrytomodelperiodicphenomena.

Unit2:TrigonometricFunctions


• Extendthedomainoftrigonometric F.TF.1Understandradianmeasureofanangleasthelengthofthearcon functionsusingtheunitcircle. theunitcirclesubtendedbytheangle.

F.TF.2Explainhowtheunitcircleinthecoordinateplaneenablesthe extensionoftrigonometricfunctionstoallrealnumbers,interpretedas radianmeasuresofanglestraversedcounterclockwisearoundtheunit circle.

• Modelperiodicphenomenawithtrigo F.TF.5Choosetrigonometricfunctionstomodelperiodicphenomena nometricfunctions. withspecifiedamplitude,frequency,andmidline.★

• Proveandapplytrigonometricidentities.

An Algebra II course with an additional focus on trigonometry could include the (+) standard F.TF.9: Prove the addition and subtraction formulas for sine, cosine, and tangent and use them to solve problems. This could be limited to acute angles in Algebra II.

F.TF.8ProvethePythagoreanidentitysin2(θ)+cos2(θ)=1anduseitto findsin(θ),cos(θ),ortan(θ),givensin(θ),cos(θ),ortan(θ),andthe quadrantoftheangle.

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Unit3:modelingwithFunctions

Inthisunitstudentssynthesizeandgeneralizewhattheyhavelearnedaboutavarietyoffunctionfamilies.They extendtheirworkwithexponentialfunctionstoincludesolvingexponentialequationswithlogarithms.Theyexplore theeffectsoftransformationsongraphsofdiversefunctions,includingfunctionsarisinginanapplication,inorderto abstractthegeneralprinciplethattransformationsonagraphalwayshavethesameeffectregardlessofthetypeof theunderlyingfunction.Theyidentifyappropriatetypesoffunctionstomodelasituation,theyadjustparametersto improvethemodel,andtheycomparemodelsbyanalyzingappropriatenessoffitandmakingjudgmentsaboutthe domainoverwhichamodelisagoodfit. The description of modeling as “the process of choosing and using mathematics and statistics to analyze empirical situations, to understand them better, and to make decisions” is at the heart of this unit. The narrative discussion and diagram of the modeling cycle should be considered when knowledge of functions, statistics, and geometry is applied in a modeling context.

Unit3:ModelingwithFunctions



For A.CED.1, use all available types of functions to create such equations, including root functions, but constrain to simple cases. While functions used in A.CED.2, 3, and 4 will often be linear, exponential, or quadratic the types of problems should draw from more complex situations than those addressed in Algebra I. For example, finding the equation of a line through a given point perpendicular to another line allows one to find the distance from a point to a line. Note that the example given for A.CED.4 applies to earlier instances of this standard, not to the current course.


Emphasize the selection of a model function based on behavior of data and context.








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Unit3:ModelingwithFunctions



Focus on applications and how key features relate to characteristics of a situation, making selection of a particular type of function model appropriate.


b.Graphsquareroot,cuberoot,andpiecewise-definedfunctions, includingstepfunctionsandabsolutevaluefunctions.



F.IF.9Comparepropertiesoftwofunctionseachrepresentedina differentway(algebraically,graphically,numericallyintables,orby verbaldescriptions).Forexample,givenagraphofonequadratic functionandanalgebraicexpressionforanother,saywhichhasthe largermaximum.


Develop models for more complex or sophisticated situations than in previous courses.

F.BF.1Writeafunctionthatdescribesarelationshipbetweentwo quantities.*

b.Combinestandardfunctiontypesusingarithmeticoperations. For example, build a function that models the temperature of a cooling body by adding a constant function to a decaying exponential, and relate these functions to the model..

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Use transformations of functions to find models as students consider increasingly more complex situations.

For F.BF.3, note the effect of multiple transformations on a single graph and the common effect of each transformation across function types.

Extend F.BF.4a to simple rational, simple radical, and simple exponential functions; connect F.BF.4a to F.LE.4.

F.BF.3Identifytheeffectonthegraphofreplacing f(x) by f(x) + k, k f(x), f(kx),and f(x + k) forspecificvaluesof k (bothpositiveand negative);findthevalueof k giventhegraphs.Experimentwith casesandillustrateanexplanationoftheeffectsonthegraphusing technology.Includerecognizingevenandoddfunctionsfromtheir graphsandalgebraicexpressionsforthem.



• Constructandcomparelinear,quadrat F.LE.4Forexponentialmodels,expressasalogarithmthesolutionto ic,andexponentialmodelsandsolve a bct = d where a, c, and d arenumbersandthebase b is2,10,or e; problems. evaluatethelogarithmusingtechnology.

Consider extending this unit to include the relationship between properties of logarithms and properties of exponents, such as the connection between the properties of exponents and the basic logarithm property that log xy = log x +log y.


Unit4:InferencesandconclusionsfromData

Inthisunit,studentsseehowthevisualdisplaysandsummarystatisticstheylearnedinearliergradesrelatetodifferenttypesofdataandtoprobabilitydistributions.Theyidentifydifferentwaysofcollectingdata—includingsample surveys,experiments,andsimulations—andtherolethatrandomnessandcarefuldesignplayintheconclusionsthat canbedrawn.

Unit4:InferencesandConclusionsfromData


• Summarize,represent,andinterpretdataonasingle countormeasurementvariable.

While students may have heard of the normal distribution, it is unlikely that they will have prior experience using it to make specific estimates. Build on students’ understanding of data distributions to help them see how the normal distribution uses area to make estimates of frequencies (which can be expressed as probabilities). Emphasize that only some data are well described by a normal distribution.

S.ID.4Usethemeanandstandarddeviationofadata settofitittoanormaldistributionandtoestimate populationpercentages.Recognizethattherearedata setsforwhichsuchaprocedureisnotappropriate.Use calculators,spreadsheets,andtablestoestimateareas underthenormalcurve.

• Understandandevaluaterandomprocessesunderlying statisticalexperiments.

For S.IC.2, include comparing theoretical and empirical results to evaluate the effectiveness of a treatment.

S.IC.1Understandstatisticsasaprocessformaking inferencesaboutpopulationparametersbasedona randomsamplefromthatpopulation.

S.IC.2Decideifaspecifiedmodelisconsistentwith resultsfromagivendata-generatingprocess,e.g., usingsimulation. For example, a model says a spinning coin falls heads up with probability 0.5. Would a result of 5 tails in a row cause you to question the model?

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• Makeinferencesandjustifyconclusionsfromsample surveys,experiments,andobservationalstudies.

In earlier grades, students are introduced to different ways of collecting data and use graphical displays and summary statistics to make comparisons. These ideas are revisited with a focus on how the way in which data is collected determines the scope and nature of the conclusions that can be drawn from that data. The concept of statistical significance is developed informally through simulation as meaning a result that is unlikely to have occurred solely as a result of random selection in sampling or random assignment in an experiment.

For S.IC.4 and 5, focus on the variability of results from experiments—that is, focus on statistics as a way of dealing with, not eliminating, inherent randomness.

S.IC.3Recognizethepurposesofanddifferences amongsamplesurveys,experiments,andobservational studies;explainhowrandomizationrelatestoeach.

S.IC.4Usedatafromasamplesurveytoestimatea populationmeanorproportion;developamarginof errorthroughtheuseofsimulationmodelsforrandom sampling.

S.IC.5Usedatafromarandomizedexperimentto comparetwotreatments;usesimulationstodecideif differencesbetweenparametersaresignificant.

S.IC.6Evaluatereportsbasedondata.

• Useprobabilitytoevaluateoutcomesofdecisions.

Extend to more complex probability models. Include situations such as those involving quality control, or diagnostic tests that yield both false positive and false negative results.

S.MD.6 (+) Use probabilities to make fair decisions (e.g., drawingbylots,usingarandomnumbergenerator).

S.MD.7(+)Analyzedecisionsandstrategiesusing probabilityconcepts(e.g.,producttesting,medical testing,pullingahockeygoalieattheendofagame).


overviewoftheIntegratedPathwayforthe commoncorestatemathematicsstandards ThistableshowsthedomainsandclustersineachcourseintheIntegratedPathway.Thestandardsfromeachclusterincluded inthatcoursearelistedbeloweachcluster.Foreachcourse,limitsandfocusfortheclustersareshowninitalics.

Domains MathematicsI MathematicsII MathematicsIII Fourth Courses *

Nu

mb

er

an

dQ

uan

tity


Quantities


•Reasonquantitatively anduseunitstosolve problems.



N.Q.1,2,3

•Extendtheproperties ofexponentsto rationalexponents.

N.RN.1,2

•Usepropertiesof rationalandirrational numbers.

N.RN.3


i2 as highest power of i

N.CN.1,2


Quadratics with real


Polynomials with real coefficients; apply

N.CN.9 to higher degree polynomials

(+)N.CN.8,9


(+) N.CN.3

•Representcomplex numbersandtheir operationsonthe complexplane.

(+) N.CN.4,5,6

Vector Quantitiesand Matrices

coefficients

N.CN.7,(+)8,(+)9

•Representandmodel withvectorquantities.

(+) N.VM.1,2,3

•Performoperationson vectors.

(+) N.VM.4a,4b,4c,5a, 5b

•Performoperations onmatricesand usematricesin applications.

(+) N.VM.6,7,8,9, 10, 11, 12

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*The(+)standardsinthiscolumnarethoseintheCommonCoreStateStandardsthatarenotincludedinanyoftheIntegratedPathwaycourses. TheywouldbeusedinadditionalcoursesdevelopedtofollowMathematicsIII.

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Domains MathematicsI MathematicsII MathematicsIII Fourth Courses



Linear expressions and exponential expressions with integer exponents

A.SSE.1a,1b



A.SSE.1a,1b,2

•Writeexpressionsin equivalentformsto



A.SSE.1a,1b,2


Alg

eb

ra

Arithmetic with Polynomials

solveproblems.


A.SSE.3a,3b,3c


Polynomials that simplify to quadratics

A.APR.1

A.SSE.4


Beyond quadratic

A.APR.1

•Understandthe relationshipbetween zerosandfactorsof polynomials.

A.APR.2,3

andRational Expressions

Creating Equations


Linear, and exponential (integer inputs only);

for A.CED.3, linear only

A.CED.1,2,3,4


In A.CED.4, include formulas involving quadratic terms

A.CED.1,2,4


A.APR.4, (+) 5


Linear and quadratic denominators

A.APR.6,(+)7


Equations using all available types of

expressions including simple root functions

A.CED.1,2,3,4

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Alg

eb

ra Reasoning with Equationsand Inequalities



Master linear, learn as general principle

A.REI.1


Linear inequalities; literal that are linear

in the variables being solved for;, exponential

of a form, such as 2x = 1/16

A.REI.3


Linear systems

A.REI.5, 6


Linear and exponential; learn as general

principle

A.REI.10, 11, 12

•Understandthe conceptofafunction andusefunction notation.

Learn as general principle. Focus on

linear and exponential (integer domains)

and on arithmetic and geometric sequences

F.IF.1,2,3

•Interpretfunctions thatarisein applicationsinterms ofacontext.


Quadratics with real coefficients

A.REI.4a,4b


Linear-quadratic systems

A.REI.7


Quadratic

F.IF.4,5,6




F.IF.7a,7b,8a,8b,9



A.REI.2


Combine polynomial, rational, radical,

absolute value, and exponential functions

A.REI.11


Include rational, square root and cube root;

emphasize selection of appropriate models

F.IF.4,5,6


Include rational and radical; focus on using key features to guide


(+)A.REI.8,9


Logarithmic and trigonometric functions

(+)F.IF.7d

Fu

ncti

on

s

Linear and exponential, (linear domain)

F.IF.4,5,6


Linear and exponential

F.IF.7a,7e,9

selection of appropriate type of model function

F.IF.7b,7c,7e,8,9

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Building Functions


For F.BF.1, 2, linear and exponential (integer

inputs)

F.BF.1a,1b,2

•Buildnewfunctions fromexisting



F.BF.1a,1b




F.BF.1b



(+)F.BF.1c


(+) F.BF.4b,4c,4d,5

Fu

ncti

on

s

Linear, Quadratic,and

functions.

Linear and exponential; focus on vertical translations for

exponential

F.BF.3



F.LE.1a,1b,1c,2,3

Quadratic, absolute value

F.BF.3,4a


Include quadratic

F.LE.3

Include simple radical, rational, and

exponential functions; emphasize common

effect of each transformation across

function types

F.BF.3,4a



F.LE.4 Exponential Models

Trigonometric Functions

•Interpretexpressions forfunctionsinterms ofthesituationthey model.

Linear and exponential of form

f(x) = bx + k

F.LE.5


F.TF.8


F.TF.1,2


F.TF.5


(+)F.TF.3,4


(+)F.TF.6,7


(+)F.TF.9

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Ge

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Congruence

•Experimentwith transformationsinthe plane.

G.CO.1,2,3,4,5

•Understand congruenceinterms ofrigidmotions.

Build on rigid motions as a familiar starting

point for development of concept of geometric

proof

G.CO.6,7,8



G.CO.12,13


Focus on validity of underlying reasoning while using variety of ways of writing proofs

G.CO.9,10,11

•Understandsimilarity intermsofsimilarity transformations.

G.SRT.1a,1b,2,3

•Provetheorems involvingsimilarity.

•Applytrigonometryto generaltriangles.

(+)G.SRT.9.10,11

Similarity, Right Triangles,and Trigonometry

Circles

Focus on validity of underlying reasoning while using variety of

formats

G.SRT.4,5

•Definetrigonometric ratiosandsolve problemsinvolving righttriangles.

G.SRT.6,7,8

•Understandand applytheoremsabout circles.

G.C.1,2,3,(+)4

•Findarclengthsand areasofsectorsof circles.

Radian introduced only as unit of measure

G.C.5

a

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Ge

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Geometric Measurement andDimension

Modelingwith Geometry



Include distance formula; relate to

Pythagorean theorem

G.GPE.4,5,7


S.ID.1,2,3

•Summarize,represent, andinterpretdataon twocategoricaland quantitativevariables.

Linear focus; discuss general principle

S.ID.5,6a,6b,6c

•Interpretlinear models.

S.ID.7,8,9


G.GPE.1,2


For G.GPE.4 include simple circle theorems

G.GPE.4


G.GMD.1, 3

•Visualizetherelation betweentwodimensionalandthreedimensionalobjects.

G.GMD.4

•Applygeometric conceptsinmodeling situations.

G.MG.1,2,3


S.ID.4

•Understandand evaluaterandom

*


(+)G.GPE.3


(+)G.GMD.2

Sta

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Making Inferences andJustifying Conclusions

processesunderlying statisticalexperiments.

S.IC.1,2

•Makeinferencesand justifyconclusions fromsamplesurveys, experimentsand observationalstudies.

S.IC.3,4,5,6

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Conditional Probability andtheRules ofProbability

•Understand independenceand conditionalprobability andusethemto interpretdata.


S.CP.1,2,3,4,5

•Usetherulesof probabilitytocompute probabilitiesof compoundeventsin auniformprobability model.

S.CP.6,7, (+) 8,(+)9

•Useprobabilityto •Useprobabilityto •Calculateexpected


evaluateoutcomesof decisions.


(+) S.MD.6,7

evaluateoutcomesof decisions.


(+) S.MD.6,7

values and use them to solveproblems.

(+) S.MD.1,2,3,4


(+) S.MD. 5a, 5b


IntegratedPathway:mathematicsI ThefundamentalpurposeofMathematicsIistoformalizeandextendthemathematicsthatstudentslearnedinthe middlegrades.Thecriticalareas,organizedintounits,deepenandextendunderstandingoflinearrelationships,in partbycontrastingthemwithexponentialphenomena,andinpartbyapplyinglinearmodelstodatathatexhibita lineartrend.Mathematics1usespropertiesandtheoremsinvolvingcongruentfigurestodeepenandextendunderstandingofgeometricknowledgefrompriorgrades.Thefinalunitinthecoursetiestogetherthealgebraicand geometricideasstudied.TheMathematicalPracticeStandardsapplythroughouteachcourseand,togetherwiththe contentstandards,prescribethatstudentsexperiencemathematicsasacoherent,useful,andlogicalsubjectthat makesuseoftheirabilitytomakesenseofproblemsituations.

CriticalArea1:Bytheendofeighthgradestudentshavehadavarietyofexperiencesworkingwithexpressionsand creatingequations.Inthisfirstunit,studentscontinuethisworkbyusingquantitiestomodelandanalyzesituations, tointerpretexpressions,andbycreatingequationstodescribesituations.

CriticalArea2:Inearliergrades,studentsdefine,evaluate,andcomparefunctions,andusethemtomodelrelationshipsbetweenquantities.Inthisunit,studentswilllearnfunctionnotationanddeveloptheconceptsofdomain andrange.Theymovebeyondviewingfunctionsasprocessesthattakeinputsandyieldoutputsandstartviewing functionsasobjectsintheirownright.Theyexploremanyexamplesoffunctions,includingsequences;theyinterpret functionsgivengraphically,numerically,symbolically,andverbally,translatebetweenrepresentations,andunderstand thelimitationsofvariousrepresentations.Theyworkwithfunctionsgivenbygraphsandtables,keepinginmindthat, dependinguponthecontext,theserepresentationsarelikelytobeapproximateandincomplete.Theirworkincludes functionsthatcanbedescribedorapproximatedbyformulasaswellasthosethatcannot.Whenfunctionsdescribe relationshipsbetweenquantitiesarisingfromacontext,studentsreasonwiththeunitsinwhichthosequantitiesare measured.Studentsbuildonandinformallyextendtheirunderstandingofintegerexponentstoconsiderexponential functions.Theycompareandcontrastlinearandexponentialfunctions,distinguishingbetweenadditiveandmultiplicativechange.Theyinterpretarithmeticsequencesaslinearfunctionsandgeometricsequencesasexponential functions.

CriticalArea3:Bytheendofeighthgrade,studentshavelearnedtosolvelinearequationsinonevariableandhave appliedgraphicalandalgebraicmethodstoanalyzeandsolvesystemsoflinearequationsintwovariables.Thisunit buildsontheseearlierexperiencesbyaskingstudentstoanalyzeandexplaintheprocessofsolvinganequation andtojustifytheprocessusedinsolvingasystemofequations.Studentsdevelopfluencywriting,interpreting,and translatingbetweenvariousformsoflinearequationsandinequalities,andusingthemtosolveproblems.Theymaster thesolutionoflinearequationsandapplyrelatedsolutiontechniquesandthelawsofexponentstothecreationand solutionofsimpleexponentialequations.Studentsexploresystemsofequationsandinequalities,andtheyfindand interprettheirsolutions.Allofthisworkisgroundedonunderstandingquantitiesandonrelationshipsbetweenthem.

CriticalArea4:Thisunitbuildsuponpriorstudents’priorexperienceswithdata,providingstudentswithmoreformal meansofassessinghowamodelfitsdata.Studentsuseregressiontechniquestodescribeapproximatelylinearrelationshipsbetweenquantities.Theyusegraphicalrepresentationsandknowledgeofthecontexttomakejudgments abouttheappropriatenessoflinearmodels.Withlinearmodels,theylookatresidualstoanalyzethegoodnessoffit.

CriticalArea5:Inpreviousgrades,studentswereaskedtodrawtrianglesbasedongivenmeasurements.Theyalso havepriorexperiencewithrigidmotions:translations,reflections,androtationsandhaveusedthesetodevelopnotions about what it means for two objects to be congruent. In this unit, students establish triangle congruence criteria, basedonanalysesofrigidmotionsandformalconstructions.Theysolveproblemsabouttriangles,quadrilaterals,and otherpolygons.Theyapplyreasoningtocompletegeometricconstructionsandexplainwhytheywork.

CriticalArea6:BuildingontheirworkwiththePythagoreanTheoremin8th gradetofinddistances,studentsusea rectangularcoordinatesystemtoverifygeometricrelationships,includingpropertiesofspecialtrianglesandquadrilateralsandslopesofparallelandperpendicularlines.

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Standards

• Reason quantitatively and use units to solve problems. Unit1

• Interpret the structure of expressions. Quantities

RelationshipsBetween


Unit2














Useappropriatetools • Understand solving equations as a process of Unit3 strategically. reasoning and explain the reasoning.

Reasoningwith • Solve equations and inequalities in one variable. Equations

Attendtoprecision. • Solve systems of equations.

• Summarize, represent, and interpret data on a Lookforandmakeuseof single count or measurement variable. structure. Unit4

• Summarize, represent, and interpret data on two DescriptiveStatistics categorical and quantitative variables.

Lookforandexpress • Interpret linear models. regularityinrepeated reasoning. • Experiment with transformations in the plane. Unit5

• Understand congruence in terms of rigid motions. Congruence,Proof,and Constructions • Make geometric constructions.

Unit6 • Use coordinates to prove simple geometric theorems algebraically. ConnectingAlgebra

andGeometrythrough Coordinates


†Notethatsolvingequationsandsystemsofequationsfollowsastudyoffunctionsinthiscourse.Toexamineequationsbeforefunctions,thisunitcouldbemergedwithUnit1.

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Unit1:relationshipsBetweenQuantities

Bytheendofeighthgradestudentshavehadavarietyofexperiencesworkingwithexpressionsandcreatingequations.Inthisfirstunit,studentscontinuethisworkbyusingquantitiestomodelandanalyzesituations,tointerpret expressions,andbycreatingequationstodescribesituations.




Unit1:RelationshipsbetweenQuantities


SKILLSTOMAINTAIN

Reinforce understanding of the properties of integer exponents. The initial experience with exponential expressions, equations, and functions involves integer exponents and builds on this understanding.



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Limit A.CED.1 and A.CED.2 to linear and exponential equations, and, in the case of exponential equations, limit to situations requiring evaluation of exponential functions at integer inputs. Limit A.CED.3 to linear equations and inequalities. Limit A.CED.4 to formulas with a linear focus.






Unit2:Linearandexponentialrelationships

Inearliergrades,studentsdefine,evaluate,andcomparefunctions,andusethemtomodelrelationshipsbetween quantities.Inthisunit,studentswilllearnfunctionnotationanddeveloptheconceptsofdomainandrange.They movebeyondviewingfunctionsasprocessesthattakeinputsandyieldoutputsandstartviewingfunctionsasobjects intheirownright.Theyexploremanyexamplesoffunctions,includingsequences;theyinterpretfunctionsgiven graphically,numerically,symbolically,andverbally,translatebetweenrepresentations,andunderstandthelimitations ofvariousrepresentations.Theyworkwithfunctionsgivenbygraphsandtables,keepinginmindthat,depending uponthecontext,theserepresentationsarelikelytobeapproximateandincomplete.Theirworkincludesfunctions thatcanbedescribedorapproximatedbyformulasaswellasthosethatcannot.Whenfunctionsdescriberelationshipsbetweenquantitiesarisingfromacontext,studentsreasonwiththeunitsinwhichthosequantitiesaremeasured.Studentsbuildonandinformallyextendtheirunderstandingofintegerexponentstoconsiderexponential functions.Theycompareandcontrastlinearandexponentialfunctions,distinguishingbetweenadditiveandmultiplicativechange.Theyinterpretarithmeticsequencesaslinearfunctionsandgeometricsequencesasexponential functions.




For A.REI.10 focus on linear and exponential equations and be able to adapt and apply that learning to other types of equations in future courses. For A.REI.11, focus on cases where f(x) and g(x) are linear or exponential.




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Students should experience a variety of types of situations modeled by functions. Detailed analysis of any particular class of function at this stage is not advised. Students should apply these concepts throughout their future mathematics courses.

Draw examples from linear and exponential functions. In F.IF.3, draw connection to F.BF.2, which requires students to write arithmetic and geometric sequences. Emphasize arithmetic and geometric sequences as examples of linear and exponential functions.








For F.IF.4 and 5, focus on linear and exponential functions. For F.IF.6, focus on linear functions and intervals for exponential functions whose domain is a subset of the integers. Mathematics II and III will address other function types.

N.RN.1 and N.RN. 2 will need to be referenced here before discussing exponential models with continuous domains.





For F.IF.7a, 7e, and 9 focus on linear and exponential functions. Include comparisons of two functions presented algebraically. For example, compare the growth of two linear functions, or two exponential functions such as y=3n and y=100·2n.





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Limit F.BF.1a, 1b, and 2 to linear and exponential functions. In F.BF.2, connect arithmetic sequences to linear functions and geometric sequences to exponential functions.













For F.LE.3, limit to comparisons between exponential and linear models.


a. Provethatlinearfunctionsgrowbyequaldifferencesoverequal intervals;exponentialfunctionsgrowbyequalfactorsoverequal intervals.






Limit exponential functions to those of the form f(x) = bx + k .

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Unit3:reasoningwithequations

Bytheendofeighthgrade,studentshavelearnedtosolvelinearequationsinonevariableandhaveappliedgraphicalandalgebraicmethodstoanalyzeandsolvesystemsoflinearequationsintwovariables.Thisunitbuildsonthese earlierexperiencesbyaskingstudentstoanalyzeandexplaintheprocessofsolvinganequationandtojustifythe process used in solving a system of equations. Students develop fluency writing, interpreting, and translating between variousformsoflinearequationsandinequalities,andusingthemtosolveproblems.Theymasterthesolutionoflinearequationsandapplyrelatedsolutiontechniquesandthelawsofexponentstothecreationandsolutionofsimple exponentialequations.Studentsexploresystemsofequationsandinequalities,andtheyfindandinterprettheirsolutions.Allofthisworkisgroundedonunderstandingquantitiesandonrelationshipsbetweenthem.

Unit3:ReasoningwithEquations



Students should focus on and master A.REI.1 for linear equations and be able to extend and apply their reasoning to other types of equations in future courses. Students will solve exponential equations with logarithms in Mathematics III.



Extend earlier work with solving linear equations to solving linear inequalities in one variable and to solving literal equations that are linear in the variable being solved for. Include simple exponential equations that rely only on application of the laws of exponents, such as 5x = 125 or 2x = 1/16 .


Build on student experiences graphing and solving systems of linear equations from middle school to focus on justification of the methods used. Include cases where the two equations describe the same line (yielding infinitely many solutions) and cases where two equations describe parallel lines (yielding no solution); connect to GPE.5, which requires students to prove the slope criteria for parallel lines.

A.REI.5Provethat,givenasystemoftwoequationsintwovariables, replacingoneequationbythesumofthatequationandamultiple of theotherproducesasystemwiththesamesolutions.


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ExperiencewithdescriptivestatisticsbeganasearlyasGrade6.Studentswereexpectedtodisplaynumerical dataandsummarizeitusingmeasuresofcenterandvariability.Bytheendofmiddleschooltheywerecreating scatterplotsandrecognizinglineartrendsindata.Thisunitbuildsuponthatpriorexperience,providingstudentswith moreformalmeansofassessinghowamodelfitsdata.Studentsuseregressiontechniquestodescribeapproximately linearrelationshipsbetweenquantities.Theyusegraphicalrepresentationsandknowledgeofthecontexttomake judgmentsabouttheappropriatenessoflinearmodels.Withlinearmodels,theylookatresidualstoanalyzethe goodnessoffit.








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S.ID.6b should be focused on situations for which linear models are appropriate.





c. Fit a linear function for scatter plots that suggest a linear association.


Build on students’ work with linear relationships in eighth grade and introduce the correlation coefficient. The focus here is on the computation and interpretation of the correlation coefficient as a measure of how well the data fit the relationship. The important distinction between a statistical relationship and a causeand-effect relationship arises in S.ID.9.






Inpreviousgrades,studentswereaskedtodrawtrianglesbasedongivenmeasurements.Theyalsohavepriorexperiencewithrigidmotions:translations,reflections,androtationsandhaveusedthesetodevelopnotionsaboutwhatit meansfortwoobjectstobecongruent.Inthisunit,studentsestablishtrianglecongruencecriteria,basedonanalyses ofrigidmotionsandformalconstructions.Theysolveproblemsabouttriangles,quadrilaterals,andotherpolygons. Theyapplyreasoningtocompletegeometricconstructionsandexplainwhytheywork.








G.CO.2 Represent transformations in the plane using, e.g., transparencies andgeometrysoftware;describetransformationsasfunctionsthattake pointsintheplaneasinputsandgiveotherpointsasoutputs.Compare transformationsthatpreservedistanceandangletothosethatdonot (e.g.,translationversushorizontalstretch).



G.CO.5Givenageometricfigureandarotation,reflection,ortranslation, drawthetransformedfigureusing,e.g.,graphpaper,tracingpaper,or geometrysoftware.Specifyasequenceoftransformationsthatwill carryagivenfigureontoanother.

*

G.CO.6Usegeometricdescriptionsofrigidmotionstotransformfigures andtopredicttheeffectofagivenrigidmotiononagivenfigure;given twofigures,usethedefinitionofcongruenceintermsofrigidmotions todecideiftheyarecongruent.

G.CO.7Usethedefinitionofcongruenceintermsofrigidmotionsto showthattwotrianglesarecongruentifandonlyifcorrespondingpairs ofsidesandcorrespondingpairsofanglesarecongruent.



Build on prior student experience with simple constructions. Emphasize the ability to formalize and defend how these constructions result in the desired objects.




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BuildingontheirworkwiththePythagoreanTheoremin8th gradetofinddistances,studentsusearectangularcoordinatesystemtoverifygeometricrelationships,includingpropertiesofspecialtrianglesandquadrilateralsandslopes ofparallelandperpendicularlines.

Unit6:ConnectingAlgebraandGeometryThroughCoordinates



This unit has a close connection with the next unit. For example, a curriculum might merge G.GPE.1 and the Unit 5 treatment of G.GPE.4 with the standards in this unit. Reasoning with triangles in this unit is limited to right triangles; e.g., derive the equation for a line through two points using similar right triangles.

Relate work on parallel lines in G.GPE.5 to work on A.REI.5 in Mathematics I involving systems of equations having no solution or infinitely many solutions.


G.GPE.4Usecoordinatestoprovesimplegeometrictheorems algebraically. For example, prove or disprove that a figure defined by four given points in the coordinate plane is a rectangle; prove or disprove that the point (1, √3) lies on the circle centered at the origin and containing the point (0, 2).

G.GPE.5Provetheslopecriteriaforparallelandperpendicularlines; usethemtosolvegeometricproblems(e.g.,findtheequationofaline parallelorperpendiculartoagivenlinethatpassesthroughagiven point).


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IntegratedPathway:mathematicsII ThefocusofMathematicsIIisonquadraticexpressions,equations,andfunctions;comparingtheircharacteristics andbehaviortothoseoflinearandexponentialrelationshipsfromMathematicsIasorganizedinto6criticalareas, orunits.Theneedforextendingthesetofrationalnumbersarisesandrealandcomplexnumbersareintroducedso thatallquadraticequationscanbesolved.Thelinkbetweenprobabilityanddataisexploredthroughconditional probabilityandcountingmethods,includingtheiruseinmakingandevaluatingdecisions.Thestudyofsimilarity leadstoanunderstandingofrighttriangletrigonometryandconnectstoquadraticsthroughPythagorean relationships.Circles,withtheirquadraticalgebraicrepresentations,roundoutthecourse.TheMathematical PracticeStandardsapplythroughouteachcourseand,togetherwiththecontentstandards,prescribethatstudents experiencemathematicsasacoherent,useful,andlogicalsubjectthatmakesuseoftheirabilitytomakesenseof problemsituations.

CriticalArea1:Studentsextendthelawsofexponentstorationalexponentsandexploredistinctionsbetweenrational andirrationalnumbersbyconsideringtheirdecimalrepresentations.InUnit3,studentslearnthatwhenquadratic equationsdonothaverealsolutionsthenumbersystemmustbeextendedsothatsolutionsexist,analogoustothe wayinwhichextendingthewholenumberstothenegativenumbersallowsx+1=0tohaveasolution.Students explorerelationshipsbetweennumbersystems:wholenumbers,integers,rationalnumbers,realnumbers,and complexnumbers.Theguidingprincipleisthatequationswithnosolutionsinonenumbersystemmayhavesolutions inalargernumbersystem.

CriticalArea2:Studentsconsiderquadraticfunctions,comparingthekeycharacteristicsofquadraticfunctionsto thoseoflinearandexponentialfunctions.Theyselectfromamongthesefunctionstomodelphenomena.Students learntoanticipatethegraphofaquadraticfunctionbyinterpretingvariousformsofquadraticexpressions.In particular,theyidentifytherealsolutionsofaquadraticequationasthezerosofarelatedquadraticfunction. Whenquadraticequationsdonothaverealsolutions,studentslearnthatthatthegraphoftherelatedquadratic functiondoesnotcrossthehorizontalaxis.Theyexpandtheirexperiencewithfunctionstoincludemorespecialized functions—absolutevalue,step,andthosethatarepiecewise-defined.

CriticalArea3:Studentsbeginthisunitbyfocusingonthestructureofexpressions,rewritingexpressionstoclarify andrevealaspectsoftherelationshiptheyrepresent.Theycreateandsolveequations,inequalities,andsystemsof equationsinvolvingexponentialandquadraticexpressions.

CriticalArea4:Buildingonprobabilityconceptsthatbeganinthemiddlegrades,studentsusethelanguagesofset theorytoexpandtheirabilitytocomputeandinterprettheoreticalandexperimentalprobabilitiesforcompound events,attendingtomutuallyexclusiveevents,independentevents,andconditionalprobability.Studentsshouldmake useofgeometricprobabilitymodelswhereverpossible.Theyuseprobabilitytomakeinformeddecisions.

CriticalArea5:Studentsapplytheirearlierexperiencewithdilationsandproportionalreasoningtobuildaformal understandingofsimilarity.Theyidentifycriteriaforsimilarityoftriangles,usesimilaritytosolveproblems,and applysimilarityinrighttrianglestounderstandrighttriangletrigonometry,withparticularattentiontospecialright trianglesandthePythagoreanTheorem.Itisinthisunitthatstudentsdevelopfacilitywithgeometricproof.They usewhattheyknowaboutcongruenceandsimilaritytoprovetheoremsinvolvinglines,angles,triangles,andother polygons.Theyexploreavarietyofformatsforwritingproofs.

CriticalArea6:Inthisunitstudentsprovebasictheoremsaboutcircles,suchasatangentlineisperpendiculartoa radius,inscribedangletheorem,andtheoremsaboutchords,secants,andtangentsdealingwithsegmentlengths andanglemeasures.IntheCartesiancoordinatesystem,studentsusethedistanceformulatowritetheequationofa circlewhengiventheradiusandthecoordinatesofitscenter,andtheequationofaparabolawithverticalaxiswhen givenanequationofitsdirectrixandthecoordinatesofitsfocus.Givenanequationofacircle,theydrawthegraphin thecoordinateplane,andapplytechniquesforsolvingquadraticequationstodetermineintersectionsbetweenlines andcirclesoraparabolaandbetweentwocircles.Studentsdevelopinformalargumentsjustifyingcommonformulas forcircumference,area,andvolumeofgeometricobjects,especiallythoserelatedtocircles.

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Standards


Unit1 • Use properties of rational and irrational numbers. ExtendingtheNumber

• Perform arithmetic operations with complex System numbers.


Unit2

QuadraticFunctions andModeling








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Unit3


Unit4

Applicationsof Probability

Unit5

Similarity,RightTriangle Trigonometry,andProof

Unit6

CirclesWithand WithoutCoordinates







• Understand independence and conditional probability and use them to interpret data.

• Use the rules of probability to compute probabilities of compound events in a uniform probability model.


• Understand similarity in terms of similarity transformations.

• Prove geometric theorems.

• Prove theorems involving similarity.

• Use coordinates to prove simple geometric theorems algebraically.

• Define trigonometric ratios and solve problems involving right triangles.

• Prove and apply trigonometric identities.

• Understand and apply theorems about circles.

• Find arc lengths and areas of sectors of circles.

• Translate between the geometric description and the equation for a conic section.

• Use coordinates to prove simple geometric theorem algebraically.

• Explain volume formulas and use them to solve problems.




Attendtoprecision.




†Notethatsolvingequationsfollowsastudyoffunctionsinthiscourse.Toexamineequationsbeforefunctions,thisunitcouldcome beforeUnit2.


Unit1:extendingthenumbersystem

Studentsextendthelawsofexponentstorationalexponentsandexploredistinctionsbetweenrationalandirrational numbersbyconsideringtheirdecimalrepresentations.InUnit2,studentslearnthatwhenquadraticequationsdonot haverealsolutionsthenumbersystemmustbeextendedsothatsolutionsexist,analogoustothewayinwhichextendingthewholenumberstothenegativenumbersallowsx+1=0tohaveasolution.Studentsexplorerelationships betweennumbersystems:wholenumbers,integers,rationalnumbers,realnumbers,andcomplexnumbers.Theguidingprincipleisthatequationswithnosolutionsinonenumbersystemmayhavesolutionsinalargernumbersystem.

Unit1:ExtendingtheNumberSystem





• Usepropertiesofrationalandirrational N.RN.3Explainwhysumsandproductsofrationalnumbersarerational, numbers. thatthesumofarationalnumberandanirrationalnumberisirrational,

andthattheproductofanonzerorationalnumberandanirrational numberisirrational. Connect N.RN.3 to physical situations,


• Performarithmeticoperationswith N.CN.1Knowthereisacomplexnumber i suchthat i2 =−1,andevery complexnumbers. complexnumberhastheform a + bi with a and b real.

N.CN.2Usetherelation i2 =–1andthecommutative,associative,and Limit to multiplications that involve i2

distributivepropertiestoadd,subtract,andmultiplycomplexnumbers. as the highest power of i.




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Studentsconsiderquadraticfunctions,comparingthekeycharacteristicsofquadraticfunctionstothoseoflinearand exponentialfunctions.Theyselectfromamongthesefunctionstomodelphenomena.Studentslearntoanticipatethe graphofaquadraticfunctionbyinterpretingvariousformsofquadraticexpressions.Inparticular,theyidentifythe realsolutionsofaquadraticequationasthezerosofarelatedquadraticfunction.Whenquadraticequationsdonot haverealsolutions,studentslearnthatthatthegraphoftherelatedquadraticfunctiondoesnotcrossthehorizontal axis.Theyexpandtheirexperiencewithfunctionstoincludemorespecializedfunctions—absolutevalue,step,and thosethatarepiecewise-defined.




Focus on quadratic functions; compare with linear and exponential functions studied in Mathematics I.





For F.IF.7b, compare and contrast absolute value, step and piecewise-defined functions with linear, quadratic, and exponential functions. Highlight issues of domain, range and usefulness when examining piecewise-defined functions. Note that this unit, and in particular in F.IF.8b, extends the work begun in Mathematics I on exponential functions with integer exponents. For F.IF.9, focus on expanding the types of functions considered to include, linear, exponential, and quadratic.





F.IF.8Writeafunction definedbyanexpressionindifferentbut equivalentformstorevealandexplaindifferentpropertiesofthe function.


b. Usethepropertiesofexponentstointerpretexpressionsfor exponentialfunctions. For example, identify percent rate of change in functions such as y = (1.02)t, y = (0.97)t, y = (1.01)12t, y = (1.2)t/10, and classify them as representing exponential growth or decay.


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Focus on situations that exhibit a quadratic or exponential relationship.








For F.BF.3, focus on quadratic functions and consider including absolute value functions.. For F.BF.4a, focus on linear functions but consider simple situations where the domain of the function must be restricted in order for the inverse to exist, such as f(x) = x2, x>0.

F.BF.3Identifytheeffectonthegraphofreplacing f(x) by f(x) + k, k f(x), f(kx),and f(x + k) forspecificvaluesof k (bothpositiveand negative);findthevalueof k giventhegraphs.Experimentwith casesandillustrateanexplanationoftheeffectsonthegraphusing technology. Include recognizing even and odd functions from their graphs and algebraic expressions for them.





Compare linear and exponential growth studied in Mathematics I to quadratic growth.

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Studentsbeginthisunitbyfocusingonthestructureofexpressions,rewritingexpressionstoclarifyandreveal aspectsoftherelationshiptheyrepresent.Theycreateandsolveequations,inequalities,andsystemsofequations involvingexponentialandquadraticexpressions.




Focus on quadratic and exponential expressions. For A.SSE.1b, exponents are extended from the integer exponents found in Mathematics I to rational exponents focusing on those that represent square or cube roots.




A.SSE.2Usethestructureofanexpressiontoidentifywaystorewrite it.Forexample,see x4 – y4 as (x2)2 – (y2)2,thusrecognizingitasa differenceofsquaresthatcanbefactoredas (x2 – y2)(x2 + y2).


It is important to balance conceptual understanding and procedural fluency in work with equivalent expressions. For example, development of skill in factoring and completing the square goes hand-in-hand with understanding what different forms of a quadratic expression reveal.





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Extend work on linear and exponential equations in Mathematics I to quadratic equations. Extend A.CED.4 to formulas involving squared variables.





Extend to solving any quadratic equation with real coefficients, including those with complex solutions.




• Usecomplexnumbersinpolynomial N.CN.7Solvequadraticequationswithrealcoefficientsthathave identitiesandequations. complexsolutions.

N.CN.8 (+) Extendpolynomialidentitiestothecomplexnumbers. For Limit to quadratics with real example, rewrite x2 + 4 as (x + 2i)(x – 2i).coefficients.

N.CN.9 (+) KnowtheFundamentalTheoremofAlgebra;showthatitis trueforquadraticpolynomials.





Include systems consisting of one linear and one quadratic equation. Include systems that lead to work with fractions. For example, finding the intersections between x2 + y2 = 1 and y = (x+1)/2 leads to the point (3/5, 4/5) on the unit circle, corresponding to the Pythagorean triple 32 + 42 = 52.

A.REI.7Solveasimplesystemconsistingofalinearequationanda quadraticequationintwovariablesalgebraicallyandgraphically. For example, find the points of intersection between the line y = –3x and the circle x2 + y2 = 3.

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Unit4:applicationsofProbability

Buildingonprobabilityconceptsthatbeganinthemiddlegrades,studentsusethelanguagesofsettheorytoexpand theirabilitytocomputeandinterprettheoreticalandexperimentalprobabilitiesforcompoundevents,attendingto mutuallyexclusiveevents,independentevents,andconditionalprobability.Studentsshouldmakeuseofgeometric probabilitymodelswhereverpossible.Theyuseprobabilitytomakeinformeddecisions.

Unit4:ApplicationsofProbability


• Understandindependenceandconditionalprobabilityandusethemto interpretdata.

Build on work with two-way tables from Mathematics I Unit 4 (S.ID.5) to develop understanding of conditional probability and independence.

• Usetherulesofprobabilitytocompute probabilitiesofcompoundeventsina uniformprobabilitymodel.

S.CP.1Describeeventsassubsetsofasamplespace(thesetof outcomes)usingcharacteristics(orcategories)oftheoutcomes,or asunions,intersections,orcomplementsofotherevents(“or,”“and,” “not”).

S.CP.2Understandthattwoevents A and B areindependentifthe probabilityof A and B occurringtogetheristheproductoftheir probabilities,andusethischaracterizationtodetermineiftheyare independent.

S.CP.3Understandtheconditionalprobabilityof A given B as P(A and B)/P(B),andinterpretindependenceof A and B assayingthat theconditionalprobabilityof A given B isthesameastheprobability of A,andtheconditionalprobabilityof B given A isthesameasthe probabilityof B.

S.CP.4Constructandinterprettwo-wayfrequencytablesofdatawhen twocategoriesareassociatedwitheachobjectbeingclassified.Usethe two-waytableasasamplespacetodecideifeventsareindependent andtoapproximateconditionalprobabilities. For example, collect data from a random sample of students in your school on their favorite subject among math, science, and English. Estimate the probability that a randomly selected student from your school will favor science given that the student is in tenth grade. Do the same for other subjects and compare the results.

S.CP.5Recognizeandexplaintheconceptsofconditionalprobability andindependenceineverydaylanguageandeverydaysituations. For example, compare the chance of having lung cancer if you are a smoker with the chance of being a smoker if you have lung cancer.

S.CP.6Findtheconditionalprobabilityof A given B asthefractionof B’s outcomesthatalsobelongto A,andinterprettheanswerintermsofthe model.

S.CP.7ApplytheAdditionRule, P(A or B) = P(A) + P(B) – P(A and B), andinterprettheanswerintermsofthemodel.

S.CP.8 (+) ApplythegeneralMultiplicationRuleinauniformprobability model, P(A and B) = P(A)P(B|A) = P(B)P(A|B),andinterprettheanswer intermsofthemodel.

S.CP.9 (+) Usepermutationsandcombinationstocomputeprobabilities ofcompoundeventsandsolveproblems.

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• Useprobabilitytoevaluateoutcomes ofdecisions.

This unit sets the stage for work in Mathematics III, where the ideas of statistical inference are introduced. Evaluating the risks associated with conclusions drawn from sample data (i.e. incomplete information) requires an understanding of probability concepts.

S.MD.6 (+) Useprobabilitiestomakefairdecisions(e.g.,drawingbylots, usingarandomnumbergenerator).

S.MD.7 (+) Analyzedecisionsandstrategiesusingprobabilityconcepts (e.g.,producttesting,medicaltesting,pullingahockeygoalieattheend ofagame).


Unit5:similarity,righttriangletrigonometry,andProof

Studentsapplytheirearlierexperiencewithdilationsandproportionalreasoningtobuildaformalunderstanding ofsimilarity.Theyidentifycriteriaforsimilarityoftriangles,usesimilaritytosolveproblems,andapplysimilarityin righttrianglestounderstandrighttriangletrigonometry,withparticularattentiontospecialrighttrianglesandthe Pythagoreantheorem.

Itisinthisunitthatstudentsdevelopfacilitywithgeometricproof.Theyusewhattheyknowaboutcongruenceand similaritytoprovetheoremsinvolvinglines,angles,triangles,andotherpolygons.Theyexploreavarietyofformats forwritingproofs.

Unit5:Similarity,RightTriangleTrigonometry,andProof


• Understandsimilarityintermsofsimilaritytransformations.

G.SRT.1Verifyexperimentallythepropertiesofdilationsgivenbya centerandascalefactor.

a. Adilationtakesalinenotpassingthroughthecenterofthedilationtoaparallelline,andleavesalinepassingthroughthecenter unchanged.

b. Thedilationofalinesegmentislongerorshorterintheratio givenbythescalefactor.

G.SRT.2Giventwofigures,usethedefinitionofsimilarityintermsof similaritytransformationstodecideiftheyaresimilar;explainusing similaritytransformationsthemeaningofsimilarityfortrianglesasthe equalityofallcorrespondingpairsofanglesandtheproportionalityof allcorrespondingpairsofsides.

G.SRT.3Usethepropertiesofsimilaritytransformationstoestablishthe AAcriterionfortwotrianglestobesimilar.

• Provegeometrictheorems.

Encourage multiple ways of writing proofs, such as in narrative paragraphs, using flow diagrams, in two-column format, and using diagrams without words. Students should be encouraged to focus on the validity of the underlying reasoning while exploring a variety of formats for expressing that reasoning. Implementation of G.CO.10 may be extended to include concurrence of perpendicular bisectors and angle bisectors as preparation for G.C.3 in Unit 6.

G.CO.9Provetheoremsaboutlinesandangles. Theorems include: vertical angles are congruent; when a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent; points on a perpendicular bisector of a line segment are exactly those equidistant from the segment’s endpoints.

G.CO.10Provetheoremsabouttriangles. Theorems include: measures of interior angles of a triangle sum to 180°; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point.

G.CO.11Provetheoremsaboutparallelograms. Theorems include: opposite sides are congruent, opposite angles are congruent, the diagonals of a parallelogram bisect each other, and conversely, rectangles are parallelograms with congruent diagonals.

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• Provetheoremsinvolvingsimilarity. G.SRT.4Provetheoremsabouttriangles. Theorems include: a line parallel to one side of a triangle divides the other two proportionally, and conversely; the Pythagorean Theorem proved using triangle similarity.

G.SRT.5Usecongruenceandsimilaritycriteriafortrianglestosolve problemsandtoproverelationshipsingeometricfigures.

• Usecoordinatestoprovesimplegeo G.GPE.6Findthepointonadirectedlinesegmentbetweentwogiven metrictheoremsalgebraically. pointsthatpartitionsthesegmentinagivenratio.

• Definetrigonometricratiosandsolve problemsinvolvingrighttriangles.

G.SRT.6Understandthatbysimilarity,sideratiosinrighttriangles arepropertiesoftheanglesinthetriangle,leadingtodefinitionsof trigonometricratiosforacuteangles.

G.SRT.7Explainandusetherelationshipbetweenthesineandcosineof complementaryangles.

G.SRT.8UsetrigonometricratiosandthePythagoreanTheoremtosolve righttrianglesinappliedproblems.


Unit5:Similarity,RightTriangleTrigonometry,andProof


• Proveandapplytrigonometricidentities.

In this course, limit θ to angles between 0 and 90 degrees. Connect with the Pythagorean theorem and the distance formula. A course with a greater focus on trigonometry could include the (+) standard F.TF.9: Prove the addition and subtraction formulas for sine, cosine, and tangent and use them to solve problems. This could continue to be limited to acute angles in Mathematics II.

Extension of trigonometric functions to other angles through the unit circle is included in Mathematics III.

F.TF.8ProvethePythagoreanidentitysin2(θ)+cos2(θ)=1anduseitto findsin(θ),cos(θ),ortan(θ),givensin(θ),cos(θ),ortan(θ),andthe quadrantoftheangle.

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Unit6:circlesWithandWithoutcoordinates

Inthisunitstudentsprovebasictheoremsaboutcircles,suchasatangentlineisperpendiculartoaradius,inscribed angletheorem,andtheoremsaboutchords,secants,andtangentsdealingwithsegmentlengthsandanglemeasures.Theystudyrelationshipsamongsegmentsonchords,secants,andtangentsasanapplicationofsimilarity.In theCartesiancoordinatesystem,studentsusethedistanceformulatowritetheequationofacirclewhengiventhe radiusandthecoordinatesofitscenter,andtheequationofaparabolawithverticalaxiswhengivenanequation ofitsdirectrixandthecoordinatesofitsfocus.Givenanequationofacircle,theydrawthegraphinthecoordinate plane,andapplytechniquesforsolvingquadraticequationstodetermineintersectionsbetweenlinesandcirclesora parabolaandbetweentwocircles.Studentsdevelopinformalargumentsjustifyingcommonformulasforcircumference,area,andvolumeofgeometricobjects,especiallythoserelatedtocircles.

Unit6:CirclesWithanWithoutCoordinates


• Understandandapplytheoremsabout circles.

G.C.1Provethatallcirclesaresimilar.

G.C.2Identifyanddescriberelationshipsamonginscribedangles,radii, andchords. Include the relationship between central, inscribed, and circumscribed angles; inscribed angles on a diameter are right angles; the radius of a circle is perpendicular to the tangent where the radius intersects the circle.

G.C.3Constructtheinscribedandcircumscribedcirclesofatriangle, andprovepropertiesofanglesforaquadrilateralinscribedinacircle.

G.C.4 (+) Constructatangentlinefromapointoutsideagivencircleto thecircle.

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• Findarclengthsandareasofsectors ofcircles.

Emphasize the similarity of all circles. Note that by similarity of sectors with the same central angle, arc lengths are proportional to the radius. Use this as a basis for introducing radian as a unit of measure. It is not intended that it be applied to the development of circular trigonometry in this course.

G.C.5Deriveusingsimilaritythefactthatthelengthofthearc interceptedbyanangleisproportionaltotheradius,anddefinethe radianmeasureoftheangleastheconstantofproportionality;derive theformulafortheareaofasector.

• Translatebetweenthegeometricdescriptionandtheequationforaconic section.

Connect the equations of circles and parabolas to prior work with quadratic equations. The directrix should be parallel to a coordinate axis.

G.GPE.1Derivetheequationofacircle ofgivencenterandradiususing thePythagoreanTheorem;completethesquaretofindthecenterand radiusofacirclegivenbyanequation.

G.GPE.2Derivetheequationofaparabolagivenafocusanddirectrix.

• Usecoordinatestoprovesimplegeo G.GPE.4Usecoordinatestoprovesimplegeometrictheorems metrictheoremsalgebraically. algebraically. For example, prove or disprove that a figure defined

by four given points in the coordinate plane is a rectangle; prove or Include simple proofs involving circles. disprove that the point (1, √3) lies on the circle centered at the origin

and containing the point (0, 2).

• Explainvolumeformulasandusethem tosolveproblems.

Informal arguments for area and volume formulas can make use of the way in which area and volume scale under similarity transformations: when one figure in the plane results from another by applying a similarity transformation with scale factor k, its area is k2 times the area of the first. Similarly, volumes of solid figures scale by k3 under a similarity transformation with scale factor k.

G.GMD.1Giveaninformalargumentfortheformulasforthe circumferenceofacircle,areaofacircle,volumeofacylinder,pyramid, andcone. Use dissection arguments, Cavalieri’s principle, and informal limit arguments.

G.GMD.3Usevolumeformulasforcylinders,pyramids,cones,and spherestosolveproblems.★


IntegratedPathway:mathematicsIII ItisinMathematicsIIIthatstudentspulltogetherandapplytheaccumulationoflearningthattheyhavefromtheir previouscourses,withcontentgroupedintofourcriticalareas,organizedintounits.Theyapplymethodsfrom probabilityandstatisticstodrawinferencesandconclusionsfromdata.Studentsexpandtheirrepertoireoffunctions toincludepolynomial,rational,andradicalfunctions.3 Theyexpandtheirstudyofrighttriangletrigonometryto includegeneraltriangles.And,finally,studentsbringtogetheralloftheirexperiencewithfunctionsandgeometryto createmodelsandsolvecontextualproblems.TheMathematicalPracticeStandardsapplythroughouteachcourse and,togetherwiththecontentstandards,prescribethatstudentsexperiencemathematicsasacoherent,useful,and logicalsubjectthatmakesuseoftheirabilitytomakesenseofproblemsituations.

CriticalArea1:Inthisunit,studentsseehowthevisualdisplaysandsummarystatisticstheylearnedinearliergrades relatetodifferenttypesofdataandtoprobabilitydistributions.Theyidentifydifferentwaysofcollectingdata— includingsamplesurveys,experiments,andsimulations—andtherolethatrandomnessandcarefuldesignplayinthe conclusionsthatcanbedrawn.

CriticalArea2:Thisunitdevelopsthestructuralsimilaritiesbetweenthesystemofpolynomialsandthesystem ofintegers.Studentsdrawonanalogiesbetweenpolynomialarithmeticandbase-tencomputation,focusingon propertiesofoperations,particularlythedistributiveproperty.Studentsconnectmultiplicationofpolynomialswith multiplicationofmulti-digitintegers,anddivisionofpolynomialswithlongdivisionofintegers.Studentsidentify zerosofpolynomialsandmakeconnectionsbetweenzerosofpolynomialsandsolutionsofpolynomialequations. Theunitculminateswiththefundamentaltheoremofalgebra.Rationalnumbersextendthearithmeticofintegers byallowingdivisionbyallnumbersexcept0.Similarly,rationalexpressionsextendthearithmeticofpolynomialsby allowingdivisionbyallpolynomialsexceptthezeropolynomial.Acentralthemeofthisunitisthatthearithmeticof rationalexpressionsisgovernedbythesamerulesasthearithmeticofrationalnumbers.

CriticalArea3:StudentsdeveloptheLawsofSinesandCosinesinordertofindmissingmeasuresofgeneral(not necessarilyright)triangles.Theyareabletodistinguishwhetherthreegivenmeasures(anglesorsides)define0,1,2, orinfinitelymanytriangles.Thisdiscussionofgeneraltrianglesopenuptheideaoftrigonometryappliedbeyondthe righttriangle—thatis,atleasttoobtuseangles.Studentsbuildonthisideatodevelopthenotionofradianmeasure foranglesandextendthedomainofthetrigonometricfunctionstoallrealnumbers.Theyapplythisknowledgeto modelsimpleperiodicphenomena.

CriticalArea4:Inthisunitstudentssynthesizeandgeneralizewhattheyhavelearnedaboutavarietyoffunction families.Theyextendtheirworkwithexponentialfunctionstoincludesolvingexponentialequationswithlogarithms. Theyexploretheeffectsoftransformationsongraphsofdiversefunctions,includingfunctionsarisinginan application,inordertoabstractthegeneralprinciplethattransformationsonagraphalwayshavethesameeffect regardlessofthetypeoftheunderlyingfunctions.Theyidentifyappropriatetypesoffunctionstomodelasituation, theyadjustparameterstoimprovethemodel,andtheycomparemodelsbyanalyzingappropriatenessoffitand makingjudgmentsaboutthedomainoverwhichamodelisagoodfit.Thedescriptionofmodelingas“theprocess ofchoosingandusingmathematicsandstatisticstoanalyzeempiricalsituations,tounderstandthembetter,andto makedecisions”isattheheartofthisunit.Thenarrativediscussionanddiagramofthemodelingcycleshouldbe consideredwhenknowledgeoffunctions,statistics,andgeometryisappliedinamodelingcontext.

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3Inthiscourserationalfunctionsarelimitedtothosewhosenumeratorsareofdegreeatmost1anddenominatorsofdegreeatmost2; radicalfunctionsarelimitedtosquarerootsorcuberootsofatmostquadraticpolynomials.



Standards

Unit1

Inferencesand ConclusionsfromData

• Summarize, represent, and interpret data on single count or measurement variable.

• Understand and evaluate random processes underlying statistical experiments.

• Make inferences and justify conclusions from sample surveys, experiments, and observational studies.







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Unit2

Polynomial,Rational, andRadical

Relationships.





• Understand the relationship between zeros and factors of polynomials.

• Use polynomial identities to solve problems.

• Rewrite rational expressions.




• Apply trigonometry to general triangles. Unit3

• Extend the domain of trigonometric functions Attendtoprecision. Trigonometryof using the unit circle.

GeneralTrianglesand TrigonometricFunctions • Model periodic phenomena with trigonometric

Lookforandmakeuseof function. structure.

Unit4

MathematicalModeling

• Create equations that describe numbers or relationships. Lookforandexpress

regularityinrepeated • Interpret functions that arise in applications in reasoning. terms of a context.





• Visualize relationships between two-dimensional and three-dimensional objects.




Unit1:InferencesandconclusionsfromData

Inthisunit,studentsseehowthevisualdisplaysandsummarystatisticstheylearnedinearliergradesrelatetodifferenttypesofdataandtoprobabilitydistributions.Theyidentifydifferentwaysofcollectingdata—includingsample surveys,experiments,andsimulations—andtherolethatrandomnessandcarefuldesignplayintheconclusionsthat canbedrawn.

Unit1:InferencesandConclusionsfromData


• Summarize,represent,andinterpretdataonasingle countormeasurementvariable.

While students may have heard of the normal distribution, it is unlikely that they will have prior experience using it to make specific estimates. Build on students’ understanding of data distributions to help them see how the normal distribution uses area to make estimates of frequencies (which can be expressed as probabilities). Emphasize that only some data are well described by a normal distribution.

S.ID.4Usethemeanandstandarddeviationofadata settofitittoanormaldistributionandtoestimate populationpercentages.Recognizethattherearedata setsforwhichsuchaprocedureisnotappropriate.Use calculators,spreadsheets,andtablestoestimateareas underthenormalcurve.

• Understandandevaluaterandomprocessesunderlying statisticalexperiments.

For S.IC.2, include comparing theoretical and empirical results to evaluate the effectiveness of a treatment.

S.IC.1Understandthatstatisticsallowsinferencestobe madeaboutpopulationparametersbasedonarandom samplefromthatpopulation.

S.IC.2Decideifaspecifiedmodelisconsistentwith resultsfromagivendata-generatingprocess,e.g., usingsimulation. For example, a model says a spinning coin falls heads up with probability 0.5. Would a result of 5 tails in a row cause you to question the model?

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• Makeinferencesandjustifyconclusionsfromsample surveys,experiments,andobservationalstudies.

In earlier grades, students are introduced to different ways of collecting data and use graphical displays and summary statistics to make comparisons., These ideas are revisited with a focus on how the way in which data is collected determines the scope and nature of the conclusions that can be drawn from that data. The concept of statistical significance is developed informally through simulation as meaning a result that is unlikely to have occurred solely as a result of random selection in sampling or random assignment in an experiment.

For S.IC.4 and 5, focus on the variability of results from experiments—that is, focus on statistics as a way of dealing with, not eliminating, inherent randomness.

S.IC.3Recognizethepurposesofanddifferences amongsamplesurveys,experiments,andobservational studies;explainhowrandomizationrelatestoeach.

S.IC.4Usedatafromasamplesurveytoestimatea populationmeanorproportion;developamarginof errorthroughtheuseofsimulationmodelsforrandom sampling.

S.IC.5Usedatafromarandomizedexperimentto comparetwotreatments;usesimulationstodecideif differencesbetweenparametersaresignificant.

S.IC.6Evaluatereportsbasedondata.

• Useprobabilitytoevaluateoutcomesofdecisions.

Extend to more complex probability models. Include situations such as those involving quality control or diagnostic tests that yields both false positive and false negative results.

S.MD.6 (+) Use probabilities to make fair decisions (e.g., drawingbylots,usingarandomnumbergenerator).

S.MD.7(+)Analyzedecisionsandstrategiesusing probabilityconcepts(e.g.,producttesting,medical testing,pullingahockeygoalieattheendofagame).


Unit2:Polynomials,rational,andradicalrelationships

Thisunitdevelopsthestructuralsimilaritiesbetweenthesystemofpolynomialsandthesystemofintegers.Students drawonanalogiesbetweenpolynomialarithmeticandbase-tencomputation,focusingonpropertiesofoperations, particularlythedistributiveproperty.Studentsconnectmultiplicationofpolynomialswithmultiplicationofmultidigitintegers,anddivisionofpolynomialswithlongdivisionofintegers.Studentsidentifyzerosofpolynomialsand makeconnectionsbetweenzerosofpolynomialsandsolutionsofpolynomialequations.Theunitculminateswith thefundamentaltheoremofalgebra.Rationalnumbersextendthearithmeticofintegersbyallowingdivisionbyall numbersexcept0.Similarly,rationalexpressionsextendthearithmeticofpolynomialsbyallowingdivisionbyall polynomialsexceptthezeropolynomial.Acentralthemeofthisunitisthatthearithmeticofrationalexpressionsis governedbythesamerulesasthearithmeticofrationalnumbers.

Unit2:Polynomials,Rational,andRadicalRelationships


• Usecomplexnumbersinpolynomial N.CN.8 (+) Extendpolynomialidentitiestothecomplexnumbers. For identitiesandequations. example, rewrite x2 + 4 as (x + 2i)(x – 2i).

N.CN.9(+)KnowtheFundamentalTheoremofAlgebra;showthatitis Build on work with quadratics trueforquadraticpolynomials. equations in Mathematics II. Limit to polynomials with real coefficients.


Extend to polynomial and rational expressions.




A.SSE.2Usethestructureofanexpressiontoidentifywaystorewrite it.Forexample,see x4 – y4 as (x2)2 – (y2)2,thusrecognizingitasa differenceofsquaresthatcanbefactoredas (x2 – y2)(x2 + y2).

• Writeexpressionsinequivalentforms A.SSE.4Derivetheformulaforthesumofageometricseries(when tosolveproblems. thecommonratioisnot1),andusetheformulatosolveproblems. For

example, calculate mortgage payments. ★

Consider extending A.SSE.4 to infinite geometric series in curricular implementations of this course description.



Extend beyond the quadratic polynomials found in Mathematics II.

• Understandtherelationshipbetween A.APR.2KnowandapplytheRemainderTheorem:Forapolynomial p(x) zerosandfactorsofpolynomials. andanumber a,theremainderondivisionby x – a is p(a), so p(a) =0if

andonlyif (x – a) isafactorof p(x).

A.APR.3Identifyzerosofpolynomialswhensuitablefactorizationsare available,andusethezerostoconstructaroughgraphofthefunction definedbythepolynomial.

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Unit2:Polynomials,Rational,andRadicalRelationships


• Usepolynomialidentitiestosolve problems.

This cluster has many possibilities for optional enrichment, such as relating the example in A.APR.4 to the solution of the system u2+v2=1, v = t(u+1), relating the Pascal triangle property of binomial coefficients to (x+y)n+1 = (x+y)(x+y)n, deriving explicit formulas for the coefficients, or proving the binomial theorem by induction.

A.APR.4 Prove polynomial identities and use them to describe numerical relationships. For example, the polynomial identity (x2 + y2)2 = (x2 – y2)2 + (2xy)2 can be used to generate Pythagorean triples.

A.APR.5 (+) KnowandapplytheBinomialTheoremfortheexpansion of(x + y)n inpowersof x and y forapositiveinteger n,where x and y areanynumbers,withcoefficientsdeterminedforexamplebyPascal’s Triangle.

• Rewriterationalexpressions

The limitations on rational functions apply to the rational expressions in A.APR.6. A.APR.7 requires the genera division algorithm for polynomials.

A.APR.6Rewritesimplerationalexpressionsindifferentforms;write a(x)/b(x)intheform q(x) + r(x)/b(x), where a(x), b(x), q(x), and r(x) arepolynomialswiththedegreeof r(x) lessthanthedegreeof b(x), usinginspection,longdivision,or,forthemorecomplicatedexamples,a computeralgebrasystem.

A.APR.7 (+) Understandthatrationalexpressionsformasystem analogoustotherationalnumbers,closedunderaddition,subtraction, multiplication,anddivisionbyanonzerorationalexpression;add, subtract,multiply,anddividerationalexpressions.

• Understandsolvingequationsasa A.REI.2Solvesimplerationalandradicalequationsinonevariable,and processofreasoningandexplainthe giveexamplesshowinghowextraneoussolutionsmayarise. reasoning.

Extend to simple rational and radical equations.


Include combinations of linear, polynomial, rational, radical, absolute value, and exponential functions.


• Analyzefunctionsusingdifferentrep F.IF.7Graphfunctionsexpressedsymbolicallyandshowkeyfeatures resentations. ofthegraph,byhandinsimplecasesandusingtechnologyformore

complicatedcases.★

Relate F.IF.7c to the relationship c.Graphpolynomialfunctions,identifyingzeroswhensuitablefacbetween zeros of quadratic functions torizationsareavailable,andshowingendbehavior. and their factored forms.

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Unit3:trigonometryofGeneraltrianglesandtrigonometricFunctions

StudentsdeveloptheLawsofSinesandCosinesinordertofindmissingmeasuresofgeneral(notnecessarilyright) triangles.Theyareabletodistinguishwhetherthreegivenmeasures(anglesorsides)define0,1,2,orinfinitelymany triangles.Thisdiscussionofgeneraltrianglesopenuptheideaoftrigonometryappliedbeyondtherighttriangle— thatis,atleasttoobtuseangles.Studentsbuildonthisideatodevelopthenotionofradianmeasureforanglesand extendthedomainofthetrigonometricfunctionstoallrealnumbers.Theyapplythisknowledgetomodelsimple periodicphenomena.

Unit3:TrigonometryofGeneralTrianglesandTrigonometricFunctions


• Applytrigonometrytogeneraltriangles.

With respect to the general case of the Laws of Sines and Cosines, the definitions of sine and cosine must be extended to obtuse angles.

G.SRT.9 (+) Derivetheformula A = 1/2 ab sin(C)fortheareaofatriangle bydrawinganauxiliarylinefromavertexperpendiculartotheopposite side.

G.SRT.10 (+) ProvetheLawsofSinesandCosinesandusethemtosolve problems.

G.SRT.11 (+) UnderstandandapplytheLawofSinesandtheLawof Cosinestofindunknownmeasurementsinrightandnon-righttriangles (e.g.,surveyingproblems,resultantforces).

• Extendthedomainoftrigonometric F.TF.1Understandradianmeasureofanangleasthelengthofthearcon functionsusingtheunitcircle. theunitcirclesubtendedbytheangle.

F.TF.2Explainhowtheunitcircleinthecoordinateplaneenablesthe extensionoftrigonometricfunctionstoallrealnumbers,interpretedas radianmeasuresofanglestraversedcounterclockwisearoundtheunit circle.

• Modelperiodicphenomenawithtrigo F.TF.5Choosetrigonometricfunctionstomodelperiodicphenomena nometricfunctions. withspecifiedamplitude,frequency,andmidline.★

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Unit4:mathematicalmodeling

Inthisunitstudentssynthesizeandgeneralizewhattheyhavelearnedaboutavarietyoffunctionfamilies.They extendtheirworkwithexponentialfunctionstoincludesolvingexponentialequationswithlogarithms.Theyexplore theeffectsoftransformationsongraphsofdiversefunctions,includingfunctionsarisinginanapplication,inorderto abstractthegeneralprinciplethattransformationsonagraphalwayshavethesameeffectregardlessofthetypeof theunderlyingfunctions.Theyidentifyappropriatetypesoffunctionstomodelasituation,theyadjustparameters toimprovethemodel,andtheycomparemodelsbyanalyzingappropriatenessoffitandmakingjudgmentsabout thedomainoverwhichamodelisagoodfit.Thedescriptionofmodelingas“the process of choosing and using mathematics and statistics to analyze empirical situations, to understand them better, and to make decisions” isatthe heartofthisunit.Thenarrativediscussionanddiagramofthemodelingcycleshouldbeconsideredwhenknowledge offunctions,statistics,andgeometryisappliedinamodelingcontext.

Unit4:MathematicalModeling



For A.CED.1, use all available types of functions to create such equations, including root functions, but constrain to simple cases. While functions used in A.CED.2, 3, and 4 will often be linear, exponential, or quadratic the types of problems should draw from more complex situations than those addressed in Mathematics I. For example, finding the equation of a line through a given point perpendicular to another line allows one to find the distance from a point to a line. Note that the example given for A.CED.4 applies to earlier instances of this standard, not to the current course.


Emphasize the selection of a model function based on behavior of data and context.








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Focus on applications and how key features relate to characteristics of a situation, making selection of a particular type of function model appropriate.


b.Graphsquareroot,cuberoot,andpiecewise-definedfunctions, includingstepfunctionsandabsolutevaluefunctions.


F.IF.8 Write a function defined by an expression in different but equivalent forms to reveal and explain different properties of the function.

F.IF.9 Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions). For example, given a graph of one quadratic function and an algebraic expression for another, say which has the larger maximum.


Unit4:MathematicalModeling



Develop models for more complex or sophisticated situations than in previous courses.

F.BF.1Writeafunctionthatdescribesarelationshipbetweentwo quantities.*

b.Combinestandardfunctiontypesusingarithmeticoperations. For example, build a function that models the temperature of a cooling body by adding a constant function to a decaying exponential, and relate these functions to the model.


Use transformations of functions to find more optimum models as students consider increasingly more complex situations.

For F.BF.3, note the effect of multiple transformations on a single function and the common effect of each transformation across function types. Include functions defined only by a graph.

Extend F.BF.4a to simple rational, simple radical, and simple exponential functions; connect F.BF.4a to F.LE.4.




• Constructandcomparelinear,quadrat F.LE.4Forexponentialmodels,expressasalogarithmthesolutionto ic,andexponentialmodelsandsolve a bct = d where a, c,and d arenumbersandthebase b is2,10,or e; problems. evaluatethelogarithmusingtechnology.

Consider extending this unit to include the relationship between properties of logarithms and properties of exponents, such as the connection between the properties of exponents and the basic logarithm property that log xy = log x + log y.

• Visualizerelationshipsbetweentwo G.GMD.4 Identifytheshapesoftwo-dimensionalcross-sectionsofthreedimensionalandthree-dimensional dimensionalobjects,andidentifythree-dimensionalobjectsgenerated objects. byrotationsoftwo-dimensionalobjects.

• Applygeometricconceptsinmodeling situations.

G.MG.1Usegeometricshapes,theirmeasures,andtheirpropertiesto describeobjects(e.g.,modelingatreetrunkorahumantorsoasa cylinder).★

G.MG.2Applyconceptsofdensitybasedonareaandvolumein modelingsituations(e.g.,personspersquaremile,BTUspercubic foot).★

G.MG.3Applygeometricmethodstosolvedesignproblems(e.g., designinganobjectorstructuretosatisfyphysicalconstraintsor minimizecost;workingwithtypographicgridsystemsbasedonratios).★

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Highschoolmathematicsinmiddleschool4

Therearesomestudentswhoareabletomovethroughthemathematicsquickly.Thesestudentsmaychoosetotake highschoolmathematicsbeginningineighthgrade5 orearliersotheycantakecollege-levelmathematicsinhigh school.6 Studentswhoarecapableofmovingmorequicklydeservethoughtfulattention,bothtoensurethattheyare challengedandthattheyaremasteringthefullrangeofmathematicalcontentandskills—withoutomittingcritical conceptsandtopics.Caremustbetakentoensurethatstudentsmasterandfullyunderstandallimportanttopics inthemathematicscurriculum,andthatthecontinuityofthemathematicslearningprogressionisnotdisrupted.In particular,theStandardsforMathematicalPracticeoughttocontinuetobeemphasizedinthesecases.

Thenumberofstudentstakinghighschoolmathematicsineighthgradehasincreasedsteadilyforyears.Partofthis trendistheresultofaconcertedefforttogetmorestudentstotakeCalculusandothercollege-levelmathematics coursesinhighschool.EnrollmentinbothAPStatisticsandAPCalculus,forexample,haveessentiallydoubledover thelastdecade(CollegeBoard,2009).Thereisalsopowerfulresearchshowingthatamongacademicfactors,the strongestpredictorofwhetherastudentwillearnabachelor’sdegreeisthehighestlevelofmathematicstakenin highschool(Adelman,1999).ArecentstudycompletedbyTheCollegeBoardconfirmsthis.Usingdatafrom65,000 studentsenrolledin110colleges,students’highschoolcourseworkwasevaluatedtodeterminewhichcourses werecloselyassociatedwithstudents’successfulperformanceincollege.Thestudyconfirmedtheimportanceof arigorouscurriculumthroughoutastudents’highschoolcareer.Amongotherconclusions,thestudyfoundthat studentswhotookmoreadvancedcourses,suchasPre-Calculusinthe11thgradeorCalculusin12thgrade,were moresuccessfulincollege.StudentswhotookAPCalculusatanytimeduringtheirhighschoolcareersweremost successful(Wyatt&Wiley,2010).Andevenasmorestudentsareenrolledinmoredemandingcourses,itdoesnot necessarilyfollowthattheremustbeacorrespondingdecreaseinengagementandsuccess(Cooney&Bottoms, 2009,p.2).

Atthesametime,therearecautionarytalesofpushingunderpreparedstudentsintothefirstcourseofhighschool mathematicsintheeighthgrade.TheBrookingsInstitute’s2009BrownCenterReportonAmericanEducationfound thattheNAEPscoresofstudentstakingAlgebraIintheeighthgradevariedwidely,withthebottomtenpercent scoringfarbelowgradelevel.AndareportfromtheSouthernRegionalEducationBoard,whichsupportsincreasing thenumberofmiddlestudentstakingAlgebraI,foundthatamongstudentsinthelowestquartileonachievement tests,thoseenrolledinhigher-levelmathematicshadaslightlyhigherfailureratethanthoseenrolledinlower-level mathematics(Cooney&Bottoms,2009,p.2).Inallotherquartiles,studentsscoringsimilarlyonachievementtests werelesslikelytofailiftheywereenrolledinmoredemandingcourses.Thesetworeportsareremindersthat,rather thanskippingorrushingthroughcontent,studentsshouldhaveappropriateprogressionsoffoundationalcontentto maximizetheirlikelihoodsofsuccessinhighschoolmathematics.

Itisalsoimportanttonotethatnotionsofwhatconstitutesacoursecalled“AlgebraI”or“MathematicsI”vary widely.IntheCCSS,studentsbeginpreparingforalgebrainKindergarten,astheystartlearningabouttheproperties ofoperations.Furthermore,muchofthecontentcentraltotypicalAlgebraIcourses—namelylinearequations, inequalities,andfunctions—isfoundinthe8th gradeCCSS.TheAlgebraIcoursedescribedhere(“HighSchoolAlgebra I”),however,isthefirstformalalgebracourseintheTraditionalPathway(conceptsfromthisAlgebraIcourseare developedacrossthefirsttwocoursesoftheintegratedpathway).Enrollinganeighth-gradestudentinawatered downversionofeithertheAlgebraIcourseorMathematicsIcoursedescribedheremayinfactdostudentsa disservice,asmasteryofalgebraincludingattentiontotheStandardsforMathematicalPracticeisfundamentalfor successinfurthermathematicsandoncollegeentranceexaminations.Asmentionedabove,skippingmaterialto getstudentstoaparticularpointinthecurriculumwilllikelycreategapsinthestudents’mathematicalbackground, whichmaycreateadditionalproblemslater,becausestudentsmaybedeniedtheopportunityforarigorousAlgebraI orMathematicsIcourseandmaymissimportantcontentfromeighth-grademathematics.

middleschoolacceleration

Takingtheaboveconsiderationsintoaccount,aswellastherecognitionthatthereareothermethodsfor accomplishingthesegoals,theAchievePathwaysGroupendorsesthenotionthatallstudentswhoarereadyfor rigoroushighschoolmathematicsineighthgradeshouldtakesuchcourses(AlgebraIorMathematicsI),andthatall middleschoolsshouldofferthisopportunitytotheirstudents.Topreparestudentsforhighschoolmathematicsin eighthgrade,districtsareencouragedtohaveawell-craftedsequenceof compactedcourses.Theterm“compacted” meanstocompresscontent,whichrequiresafasterpacetocomplete,asopposedtoskippingcontent.TheAchieve PathwaysGrouphasdevelopedtwocompactedcoursesequences,onedesignedfordistrictsusingatraditional AlgebraI–Geometry–AlgebraIIhighschoolsequence,andtheotherfordistrictsusinganintegratedsequence, whichiscommonlyfoundinternationally.Botharebasedontheideathatcontentshouldcompact3yearsofcontent into2years,atmost.Inotherwords,compactingcontentfrom2yearsinto1yearwouldbetoochallenging,and compacting4yearsofcontentinto3yearsstartingingrade7runstheriskofcompactingacrossmiddleandhigh schools.Assuch,grades7,8,and9werecompactedintogrades7and8(a3:2compaction).Asaresult,some8th

gradecontentisinthe7th gradecourses,andhighschoolcontentisin8th grade.

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4Thissectionreferstomathematicscontent,nothighschoolcredit.Thedeterminationforhighschoolcreditispresumedtobemade bystateandlocaleducationagencies. 5Either8thGradeAlgebraIorAcceleratedMathematicsI. 6SuchasCalculusorAdvancedStatistics.


Thecompactedtraditionalsequence,or,“AcceleratedTraditional,”compactsgrades7,8,andHighSchoolAlgebra Iintotwoyears:“Accelerated7th Grade”and“8th GradeAlgebraI.”Uponsuccessfullycompletionofthispathway, studentswillbereadyforGeometryinhighschool.Thecompactedintegratedsequence,or,“AcceleratedIntegrated,” compactsgrades7,8,andMathematicsIintotwoyears:“Accelerated7th Grade”and“8th GradeMathematicsI.”At theendof8th grade,thesestudentswillbereadyforMathematicsIIinhighschool.WhiletheK-7CCSSeffectively preparestudentsforalgebrain8th grade,somestandardsfrom8th gradehavebeenplacedintheAccelerated7th

Gradecoursetomakethe8th Gradecoursesmoremanageable.

TheAchievePathwaysGrouphasfollowedasetofguidelines7 forthedevelopmentofthesecompactedcourses.

1. CompactedcoursesshouldincludethesameCommonCoreStateStandardsasthenon-compactedcourses. Itisrecommendedtocompactthreeyearsofmaterialintotwoyears,ratherthancompactingtwoyearsinto one.Therationaleisthatmathematicalconceptsarelikelytobeomittedwhentryingtosqueezetwoyearsof materialintoone.Thisistobeavoided,asthestandardshavebeencarefullydevelopedtodefineclearlearning progressionsthroughthemajormathematicaldomains.Moreover,thecompactedcoursesshouldnotsacrifice attentiontotheMathematicalPracticesStandard.

2. DecisionstoacceleratestudentsintotheCommonCoreStateStandardsforhighschoolmathematicsbefore ninthgradeshouldnotberushed. Placingstudentsintotrackstooearlyshouldbeavoidedatallcosts.Itisnot recommendedtocompactthestandardsbeforegradeseven.Inthisdocument,compactionbeginsinseventh gradeforboththetraditionalandintegrated(international)sequences.

3. Decisionstoacceleratestudentsintohighschoolmathematicsbeforeninthgradeshouldbebasedonsolid evidenceofstudentlearning. Researchhasshowndiscrepanciesintheplacementofstudentsinto“advanced” classesbyrace/ethnicityandsocioeconomicbackground.Whilesuchdecisionstoacceleratearealmostalways ajointdecisionbetweentheschoolandthefamily,seriouseffortsmustbemadetoconsidersolidevidenceof studentlearninginordertoavoidunwittinglydisadvantagingtheopportunitiesofparticulargroupsofstudents.

4. AmenuofchallengingoptionsshouldbeavailableforstudentsaftertheirthirdyearofmathematicsÑan d allstudentsshouldbestronglyencouragedtotakemathematicsinallyearsofhighschool. Traditionally, studentstakinghighschoolmathematicsintheeighthgradeareexpectedtotakePrecalculusintheirjunior yearsandthenCalculusintheirsenioryears.Thisisagoodandworthygoal,butitshouldnotbetheonlyoption forstudents.AdvancedcoursescouldalsoincludeStatistics,DiscreteMathematics,orMathematicalDecision Making.Anarrayofchallengingoptionswillkeepmathematicsrelevantforstudents,andgivethemanewsetof toolsfortheirfuturesincollegeandcareer(seeFourthCoursessectionofthispaperforfurtherdetail).

otherWaystoacceleratestudents

Justascareshouldbetakennottorushthedecisiontoacceleratestudents,careshouldalsobetakentoprovide morethanoneopportunityforacceleration.Somestudentsmaynothavethepreparationtoentera“Compacted Pathway”butmaystilldevelopaninterestintakingadvancedmathematics,suchasAPCalculusorAPStatisticsin theirsenioryear.Additionalopportunitiesforaccelerationmayinclude:

• Allowingstudentstotaketwomathematicscoursessimultaneously(suchasGeometryandAlgebraII,or PrecalculusandStatistics).

• Allowingstudentsinschoolswithblockschedulingtotakeamathematicscourseinbothsemestersofthesame academicyear.

• Offering summer courses that are designed to provide the equivalent experience of a full course in all regards, including attention to the Mathematical Practices.8

• Creating different compaction ratios, including four years of high school content into three years beginning in 9th grade.

• Creating a hybrid Algebra II-Precalculus course that allows students to go straight to Calculus.

A combination of these methods and our suggested compacted sequences would allow for the most mathematically-inclined students to take advanced mathematics courses during their high school career. The compacted sequences begin here:

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7BasedonworkpublishedbyWashingtonOfficeoftheSuperintendentofPublicSchools,2008 8Aswithothermethodsofacceleratingstudents,enrollingstudentsinsummercoursesshouldbehandledwithcare,asthepaceofthe courseslikelybeenormouslyfast.


overviewoftheacceleratedtraditionalPathwayfor thecommoncorestatemathematicsstandards ThistableshowsthedomainsandclustersineachcourseintheAcceleratedTraditionalPathway.Thestandardsfromeach clusterincludedinthatcoursearelistedbeloweachcluster.Foreachcourse,limitsandfocusfortheclustersareshowninitalics.Fororganizationalpurposes,clustersfrom7th Gradeand8th Gradehavebeensituatedinthematrixwithinthehighschool domains.

Domains Accelerated7th

Grade 8th Grade AlgebraI

Geometry AlgebraII Fourth Courses *

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•Applyand extendprevious understandings ofoperations withfractionsto add,subtract, multiply,and dividerational numbers.

7.NS.1a,1b,1c,1d, 2a,2b,2c,2d,3

•Knowthatthere arenumbersthat arenotrational, andapproximate thembyrational numbers.

8.NS.1,2

•Workwith radicals andinteger

•Extendthe propertiesof exponents torational exponents.

N.RN.1,2

•Useproperties ofrational andirrational numbers.

N.RN.3.

Quantities

exponents.

8.EE.1,2,3,4

•Analyze proportional relationshipsand usethemtosolve real-worldand mathematical problems.

7.RP.1,2a,2b,2c, 2d,3

•Reason quantitatively and useunitstosolve problems.



N.Q.1,2,3

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*The(+)standardsinthiscolumnarethoseintheCommonCoreStateStandardsthatarenotincludedinanyoftheAccelerated TraditionalPathwaycourses.TheywouldbeusedinadditionalcoursesdevelopedtofollowAlgebraII.

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Geometry AlgebraII Fourth Courses

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tity

The Complex Number System

Vector Quantities andMatrices


•Useproperties ofoperations togenerate equivalent expressions.

7.EE.1,2

•Solvereal-lifeand mathematical problemsusing numerical andalgebraic

•Interpretthe structureof expressions.

Linear, exponential, quadratic

A.SSE.1a,1b,2

•Writeexpressions inequivalent formstosolve problems.

•Perform arithmetic operations withcomplex numbers.

N.CN.1,2

•Usecomplex numbersin polynomial identitiesand equations.

Polynomials with real coefficients

N.CN.7,(+)8,(+)9



A.SSE.1a,1b,2



(+) N.CN.3

•Represent complexnumbers andtheir operationsonthe complexplane.

(+) N.CN.4,5,6

•Representand modelwith vectorquantities.

(+) N.VM.1,2,3

• Perform operationson vectors.

(+) N.VM.4a,4b, 4c,5a,5b

•Perform operationson matricesand usematricesin applications.

(+) N.VM.6,7,8,9, 10, 11, 12

Alg

eb

ra

expressionsand equations..

7.EE.3,4a,4b


A.SSE.3a,3b,3c

A.SSE.4

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Arithmetic with

•Perform arithmetic operationson polynomials.


A.APR.1


Beyond quadratic

A.APR.1

•Understand therelationship betweenzeros andfactorsof polynomials.

Alg

eb

ra

Polynomials andRational Expressions

A.APR.2,3


A.APR.4, (+) 5

Creating Equations

•Createequations thatdescribe numbersor relationships.

Linear, quadratic, and exponential (integer inputs

only) for A.CED.3, linear only

A.CED.1,2,3,4



denominators

A.APR.6,(+)7


Equations using all available types

of expressions, including simple root functions

A.CED.1,2,3,4

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•Understandthe connections between proportional relationships, lines,andlinear equations.

8.EE.5,6

•Analyzeandsolve linearequations andpairsof simultaneous linearequations.

8.EE.7a,7b

•Understand solvingequations asaprocess ofreasoning andexplainthe reasoning.


A.REI.1

•Solveequations andinequalitiesin onevariable.

Linear inequalities; literal equations



A.REI.2

•Representand solveequations andinequalities graphically.

Combine


(+)A.REI.8,9

Alg

eb

ra

Reasoning with Equations and

that are linear in the variables

being solved for; quadratics with real solutions

A.REI.3,4a,4b

•Analyzeandsolve

polynomial, rational, radical, absolute value, and exponential

functions

A.REI.11

Inequalities linearequations andpairsof simultaneous linearequations.

8.EE.8a,8b,8c


Linear-linear and linear-quadratic

A.REI.5, 6, 7


Linear and exponential; learn

as general principle

A.REI.10, 11, 12

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•Define,evaluate, andcompare functions.

8.F.1,2,3

•Understandthe

•Interpret functions thatarisein applications intermsofa context.


Logarithmic and trigonometric

functions

Fu

ncti

on

s


conceptofa functionanduse functionnotation.

Learn as general principle; focus on linear and

exponential and on arithmetic

and geometric sequences

F.IF.1,2,3

•Usefunctions tomodel relationships between quantities.

8.F.4,5

Emphasize selection of appropriate

models

F.IF.4,5,6


Focus on using key features to

guide selection of appropriate type of model function

F.IF.7b,7c,7e,8,9

(+)F.IF.7d


Linear, exponential, and quadratic

F.IF.4,5,6




F.IF.7a,7b,7e,8a, 8b,9

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•Buildafunction thatmodels arelationship betweentwo quantities.

For F.BF.1, 2, linear, exponential, and

quadratic

F.BF.1a,1b,2



F.BF.1b

•Buildnew


(+)F.BF.1c

•Buildnew functionsfrom existingfunctions.

Building Functions


functionsfrom existingfunctions.

Include simple

(+) F.BF.4b,4c, 4d,5

Fu

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Linear, Quadratic, and Exponential Models

Linear, exponential, quadratic, and

absolute value; for F.BF.4a, linear only

F.BF.3,4a

•Constructand comparelinear, quadratic,and exponential modelsandsolve problems.

F.LE.1a,1b,1c,2,3

•Interpret expressionsfor functionsinterms ofthesituation

radical, rational, and exponential

functions; emphasize common

effect of each transformation across function

types

F.BF.3,4a



F.LE.4


theymodel.

Linear and exponential of

form f(x) = bx + k

F.LE.5

•Extendthe domainof trigonometric functionsusing theunitcircle.

F.TF.1,2


F.TF.5


F.TF.8


(+)F.TF.3,4


(+)F.TF.6,7


(+)F.TF.9

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Congruence

•Draw,construct, anddescribe geometrical figuresand describethe relationships betweenthem.

Focus on constructing

triangles

7.G.2

•Understand congruenceand similarityusing physicalmodels, transparencies, orgeometric software.

8.G.1a,1b,1c,2,5

•For 8.G.5, informal argumentsto establishangle sumandexterior angletheorems fortriangles

•Experimentwith transformationsin theplane.

G.CO.1,2,3,4,5

•Understand congruencein termsofrigid motions.

Build on rigid motions as a

familiar starting point for

development of concept of

geometric proof

G.CO.6,7,8


Focus on validity of underlying

reasoning while using variety of ways of writing

proofs

Ge

om

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y andangles

relationships whenparallel linesarecutbya transversal

G.CO.9,10,11


Formalize and

Similarity, RightTriangles,and Trigonometry


Scale drawings

7.G.1


8.G.3,4,5

•For 8.G.5, informal argumentsto establishthe angle-angle criterionfor similartriangles

explain processes

G.CO.12,13

•Understand similarityinterms ofsimilarity transformations.

G.SRT.1a,1b,2,3

•Provetheorems involving similarity.

G.SRT.4,5

•Define trigonometric ratiosand solveproblems involvingright triangles.

G.SRT.6,7,8

•Apply trigonometryto generaltriangles.

G.SRT.9.10,11

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Ge

om

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y

Circles


•Draw,construct, anddescribe geometrical

•Understand andapplythe Pythagorean

•Understandand applytheorems aboutcircles.

G.C.1,2,3,(+)4

•Findarclengths andareasof sectorsofcircles.

Radian introduced only as unit of

measure

G.C.5

•Translatebetween thegeometric descriptionand theequationfora conicsection.

G.GPE.1,2

•Usecoordinates toprovesimple geometric theorems algebraically.

Include distance formula; relate to Pythagorean

theorem

G.GPE.4,5,6,7

•Explainvolume formulasanduse themtosolve


(+)G.GPE.3

•Explainvolume formulasanduse themtosolve

figuresand describethe relationships betweenthem.

theorem.

Connect to radicals, rational exponents, and

problems.

G.GMD.1, 3

•Visualizethe

problems.

(+)G.GMD.2

Geometric Measure-mentand

Slicing 3-D figures

7.G.3

•Solvereal-lifeand mathematical problems involvingangle

irrational numbers

8.G.6,7,8

relationbetween two-dimensional andthreedimensional objects.

G.GMD.4

Dimension

Modeling with Geometry

measure,area, surfacearea,and volume.

7.G.4,5,6

•Solvereal-world andmathematical problems involvingvolume ofcylinders, cones,and spheres.

8.G.9

•Applygeometric conceptsin modeling situations.

G.MG.1,2,3

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•Summarize, represent,and interpretdataon asinglecount ormeasurement variable.

S.ID.1,2,3

•Investigate patternsof associationin bivariatedata.

8.SP.1,2,3,4

•Summarize, represent,and interpretdataon twocategorical andquantitative variables.

Linear focus; discuss general

principle

S.ID.5,6a,6b,6c



S.ID.4

Making Inferences and Justifying Conclusions

•Userandom samplingtodraw inferencesabout apopulation.

7.SP.1,2

•Drawinformal comparative inferencesabout twopopulations.

7.SP.3,4

S.ID.7,8,9

•Understand andevaluate random processes underlying statistical experiments.

S.IC.1,2

•Makeinferences andjustify conclusionsfrom samplesurveys, experimentsand observational studies.

S.IC.3,4,5,6

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Sta

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Conditional Probability andthe Rulesof Probability

•Investigate chanceprocesses anddevelop, use,andevaluate probability models.

7.SP.5,6,7a,7b,8a, 8b,8c

•Understand independence andconditional probabilityand usethemto interpretdata.


S.CP.1,2,3,4,5

•Usetherules ofprobability tocompute probabilitiesof compoundevents inauniform probabilitymodel.

S.CP.6,7, (+) 8, (+)9


•Useprobability toevaluate outcomesof decisions.


(+) S.MD.6,7



(+) S.MD.6,7

•Calculate expectedvalues andusethemto solveproblems.

(+) S.MD.1,2,3,4


(+) S.MD.5a,5b


acceleratedtraditionalPathway: accelerated7th Grade Thiscoursediffersfromthenon-accelerated7th Gradecourseinthatitcontainscontentfrom8th grade.While coherenceisretained,inthatitlogicallybuildsfromthe6th Grade,theadditionalcontentwhencomparedtothenonacceleratedcoursedemandsafasterpaceforinstructionandlearning.Contentisorganizedintofourcriticalareas,or units.TheMathematicalPracticeStandardsapplythroughouteachcourseand,togetherwiththecontentstandards, prescribethatstudentsexperiencemathematicsasacoherent,useful,andlogicalsubjectthatmakesuseoftheir abilitytomakesenseofproblemsituations.Thecriticalareasareasfollows:

CriticalArea1:Studentsdevelopaunifiedunderstandingofnumber,recognizingfractions,decimals(thathavea finiteorarepeatingdecimalrepresentation),andpercentsasdifferentrepresentationsofrationalnumbers.Students extendaddition,subtraction,multiplication,anddivisiontoallrationalnumbers,maintainingthepropertiesof operationsandtherelationshipsbetweenadditionandsubtraction,andmultiplicationanddivision.Byapplyingthese properties,andbyviewingnegativenumbersintermsofeverydaycontexts(e.g.,amountsowedortemperatures belowzero),studentsexplainandinterprettherulesforadding,subtracting,multiplying,anddividingwithnegative numbers.Theyusethearithmeticofrationalnumbersastheyformulateexpressionsandequationsinonevariable andusetheseequationstosolveproblems.Theyextendtheirmasteryofthepropertiesofoperationstodevelopan understandingofintegerexponents,andtoworkwithnumberswritteninscientificnotation.

CriticalArea2:Studentsuselinearequationsandsystemsoflinearequationstorepresent,analyze,andsolvea varietyofproblems.Studentsrecognizeequationsforproportions(y/x=mory=mx)asspeciallinearequations(y =mx+b),understandingthattheconstantofproportionality(m)istheslope,andthegraphsarelinesthroughthe origin.Theyunderstandthattheslope(m)ofalineisaconstantrateofchange,sothatiftheinputorx-coordinate changesbyanamountA,theoutputory-coordinatechangesbytheamountm×A.Studentsstrategicallychoose andefficientlyimplementprocedurestosolvelinearequationsinonevariable,understandingthatwhentheyusethe propertiesofequalityandtheconceptoflogicalequivalence,theymaintainthesolutionsoftheoriginalequation.

CriticalArea3:Studentsbuildontheirpreviousworkwithsingledatadistributionstocomparetwodatadistributions andaddressquestionsaboutdifferencesbetweenpopulations.Theybegininformalworkwithrandomsamplingto generatedatasetsandlearnabouttheimportanceofrepresentativesamplesfordrawinginferences.

CriticalArea4:StudentscontinuetheirworkwithareafromGrade6,solvingproblemsinvolvingtheareaand circumferenceofacircleandsurfaceareaofthree-dimensionalobjects.Inpreparationforworkoncongruence andsimilarity,theyreasonaboutrelationshipsamongtwo-dimensionalfiguresusingscaledrawingsandinformal geometricconstructions,andtheygainfamiliaritywiththerelationshipsbetweenanglesformedbyintersecting lines.Studentsworkwiththree-dimensionalfigures,relatingthemtotwo-dimensionalfiguresbyexaminingcrosssections.Theysolvereal-worldandmathematicalproblemsinvolvingarea,surfacearea,andvolumeoftwo-and three-dimensionalobjectscomposedoftriangles,quadrilaterals,polygons,cubesandrightprisms.Studentsuseideas aboutdistanceandangles,howtheybehaveundertranslations,rotations,reflections,anddilations,andideasabout congruenceandsimilaritytodescribeandanalyzetwo-dimensionalfiguresandtosolveproblems.Studentsshow thatthesumoftheanglesinatriangleistheangleformedbyastraightline,andthatvariousconfigurationsoflines giverisetosimilartrianglesbecauseoftheanglescreatedwhenatransversalcutsparallellines.Studentscomplete theirworkonvolumebysolvingproblemsinvolvingcones,cylinders,andspheres.

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Standards

• Apply and extend previous understandings of operations with fractions to add, subtract,

Unit1 multiply, and divide rational numbers. RationalNumbersand • Know that there are numbers that are not rational,

Exponents and approximate them by rational numbers. Makesenseofproblems • Work with radicals and integer exponents. andpersevereinsolving

Unit2

Proportionalityand LInearRelationships

• Analyze proportional relationships and use them to solve real-world and mathematical problems.

• Use properties of operations to generate equivalent expressions.

• Solve real-life and mathematical problems using numerical and algebraic expressions and equations.

• Understand the connections between proportional relationships, lines, and linear equations.

• Analyze and solve linear equations and pairs of simultaneous linear equations.

them.




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Unit3

Introductionto SamplingInference

Unit4

Creating,Comparing, andAnalyzing

GeometricFigures

• Use random sampling to draw inferences about a population.

• Draw informal comparative inferences about two populations.

• Investigate chance processes and develop, use, and evaluate probability models.

• Draw, construct and describe geometrical figures and describe the relationships between them.

• Solve real-life and mathematical problems involving angle measure, area, surface area, and volume.

• Understand congruence and similarity using physical models, transparencies, or geometry software.

• Solve real-world and mathematical problems involving volume of cylinders, cones and spheres.


Attendtoprecision.





Unit1:rationalnumbersandexponents

Studentsdevelopaunifiedunderstandingofnumber,recognizingfractions,decimals(thathaveafiniteorarepeatingdecimalrepresentation),andpercentsasdifferentrepresentationsofrationalnumbers.Studentsextendaddition, subtraction,multiplication,anddivisiontoallrationalnumbers,maintainingthepropertiesofoperationsandthe relationshipsbetweenadditionandsubtraction,andmultiplicationanddivision.Byapplyingtheseproperties,andby viewingnegativenumbersintermsofeverydaycontexts(e.g.,amountsowedortemperaturesbelowzero),students explainandinterprettherulesforadding,subtracting,multiplying,anddividingwithnegativenumbers.Theyusethe arithmeticofrationalnumbersastheyformulateexpressionsandequationsinonevariableandusetheseequations tosolveproblems.Theyextendtheirmasteryofthepropertiesofoperationstodevelopanunderstandingofinteger exponents,andtoworkwithnumberswritteninscientificnotation.

Unit1:RationalNumbersandExponents


• Applyandextendpreviousunderstandingsofoperationswithfractions toadd,subtract,multiply,anddivide rationalnumbers.

7.NS.1Applyandextendpreviousunderstandingsofadditionand subtractiontoaddandsubtractrationalnumbers;representaddition andsubtractiononahorizontalorverticalnumberlinediagram.

a. Describesituationsinwhichoppositequantitiescombineto make0. For example, a hydrogen atom has 0 charge because its two constituents are oppositely charged.

b. Understand p + q asthenumberlocatedadistance|q|from p, inthepositiveornegativedirectiondependingonwhether q is positiveornegative.Showthatanumberanditsoppositehavea sumof0(areadditiveinverses).Interpretsumsofrationalnumbersbydescribingreal-worldcontexts.

c. Understandsubtractionofrationalnumbersasaddingtheadditiveinverse, p – q = p +(–q).Showthatthedistancebetweentwo rationalnumbersonthenumberlineistheabsolutevalueoftheir difference,andapplythisprincipleinreal-worldcontexts.

d. Applypropertiesofoperationsasstrategiestoaddandsubtract rationalnumbers.

7.NS.2Applyandextendpreviousunderstandingsofmultiplicationand divisionandoffractionstomultiplyanddividerationalnumbers.

a. Understandthatmultiplicationisextendedfromfractionsto rationalnumbersbyrequiringthatoperationscontinuetosatisfy thepropertiesofoperations,particularlythedistributiveproperty,leadingtoproductssuchas(–1)(–1)=1andtherulesformultiplyingsignednumbers.Interpretproductsofrationalnumbers bydescribingreal-worldcontexts.

b. Understandthatintegerscanbedivided,providedthatthedivisorisnotzero,andeveryquotientofintegers(withnon-zero divisor)isarationalnumber.If p and q areintegers,then –(p/q) = (–p)/q = p/(–q).Interpretquotientsofrationalnumbersby describingreal-worldcontexts.

c. Applypropertiesofoperationsasstrategiestomultiplyand dividerationalnumbers.

d. Convertarationalnumbertoadecimalusinglongdivision;know thatthedecimalformofarationalnumberterminatesin0sor eventuallyrepeats.

7.NS.3Solvereal-worldandmathematicalproblemsinvolvingthefour operationswithrationalnumbers.*

*Computationswithrationalnumbersextendtherulesformanipulatingfractionstocomplexfractions.

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• Knowthattherearenumbersthatare notrational,andapproximatethemby rationalnumbers.

8.NS.1Knowthatnumbersthatarenotrationalarecalledirrational. Understandinformallythateverynumberhasadecimalexpansion;for rationalnumbersshowthatthedecimalexpansionrepeatseventually, andconvertadecimalexpansionwhichrepeatseventuallyintoa rationalnumber.

8.NS.2Userationalapproximationsofirrationalnumberstocomparethe sizeofirrationalnumbers,locatethemapproximatelyonanumberline diagram,andestimatethevalueofexpressions(e.g.,p2). For example, by truncating the decimal expansion of √2, show that √2 is between 1 and 2, then between 1.4 and 1.5, and explain how to continue on to get better approximations.

• Workwithradicalsandintegerexponents.

8.EE.1Knowandapplythepropertiesofintegerexponentstogenerate equivalentnumericalexpressions. For example, 32 x 3–5 = 3–3 = 1/33 = 1/27.

8.EE.2Usesquarerootandcuberootsymbolstorepresentsolutions toequationsoftheformx2 =pandx3 =p,wherepisapositiverational number.Evaluatesquarerootsofsmallperfectsquaresandcuberoots ofsmallperfectcubes.Knowthat √2isirrational.

8.EE.3Usenumbersexpressedintheformofasingledigittimesan integerpowerof10toestimateverylargeorverysmallquantities, andtoexpresshowmanytimesasmuchoneisthantheother. For example, estimate the population of the United States as 3 x 108 and the population of the world as 7 x 109, and determine that the world population is more than 20 times larger.

8.EE.4Performoperationswithnumbersexpressedinscientific notation,includingproblemswherebothdecimalandscientificnotation areused.Usescientificnotationandchooseunitsofappropriatesize formeasurementsofverylargeorverysmallquantities(e.g.,use millimetersperyearforseafloorspreading).Interpretscientificnotation thathasbeengeneratedbytechnology.

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Unit2:ProportionalityandLinearrelationships

Studentsuselinearequationsandsystemsoflinearequationstorepresent,analyze,andsolveavarietyofproblems. Studentsrecognizeequationsforproportions(y/x = m or y = mx)asspeciallinearequations(y = mx + b),understandingthattheconstantofproportionality(m)istheslope,andthegraphsarelinesthroughtheorigin.They understandthattheslope(m)ofalineisaconstantrateofchange,sothatiftheinputor x-coordinatechangesby anamount A,theoutputor y-coordinatechangesbytheamount m×A.Studentsstrategicallychooseandefficiently implementprocedurestosolvelinearequationsinonevariable,understandingthatwhentheyusethepropertiesof equalityandtheconceptoflogicalequivalence,theymaintainthesolutionsoftheoriginalequation.

Unit2:ProportionalityandLinearRelationships


• Analyzeproportionalrelationshipsand usethemtosolvereal-worldandmathematicalproblems.

• Usepropertiesofoperationstogenerateequivalentexpressions.

7.RP.1 Compute unit rates associated with ratios of fractions, including ratios of lengths, areas and other quantities measured in like or different units. For example, if a person walks 1/2 mile in each 1/4 hour, compute the unit rate as the complex fraction 1/2 / 1/4 miles per hour, equivalently 2 miles per hour.

7.RP.2 Recognize and represent proportional relationships between quantities.

a. Decidewhethertwoquantitiesareinaproportionalrelationship, e.g.,bytesting for equivalent ratios in a table or graphing on a coordinate plane and observing whether the graph is a straight line through the origin.

b. Identifytheconstantofproportionality(unitrate)intables, graphs,equations,diagrams,andverbaldescriptionsofproportionalrelationships.

c. Representproportionalrelationshipsbyequations. For example, if total cost t is proportional to the number n of items purchased at a constant price p, the relationship between the total cost and the number of items can be expressed as t = pn.

d. Explainwhatapoint(x,y)onthegraphofaproportionalrelationshipmeansintermsofthesituation,withspecialattentionto thepoints(0,0)and(1,r)whereristheunitrate.

7.RP.3Useproportionalrelationshipstosolvemultistepratioand percentproblems. Examples: simple interest, tax, markups and markdowns, gratuities and commissions, fees, percent increase and decrease, percent error.

7.EE.1Applypropertiesofoperationsasstrategiestoadd,subtract, factor,andexpandlinearexpressionswithrationalcoefficients.

7.EE.2Understandthatrewritinganexpressionindifferentformsina problemcontextcanshedlightontheproblemandhowthequantities initarerelated. For example, a + 0.05a = 1.05a means that “increase by 5%” is the same as “multiply by 1.05.”

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• Solvereal-lifeandmathematicalproblemsusingnumericalandalgebraic expressionsandequations.

• Understandtheconnectionsbetween proportionalrelationships,lines,and linearequations.

7.EE.3Solvemulti-stepreal-lifeandmathematicalproblemsposedwith positiveandnegativerationalnumbersinanyform(wholenumbers, fractions,anddecimals),usingtoolsstrategically.Applypropertiesof operationstocalculatewithnumbersinanyform;convertbetween formsasappropriate;andassessthereasonablenessofanswersusing mentalcomputationandestimationstrategies. For example: If a woman making $25 an hour gets a 10% raise, she will make an additional 1/10 of her salary an hour, or $2.50, for a new salary of $27.50. If you want to place a towel bar 9 3/4 inches long in the center of a door that is 27 1/2 inches wide, you will need to place the bar about 9 inches from each edge; this estimate can be used as a check on the exact computation.

7.EE.4 Usevariablestorepresentquantitiesinareal-worldor mathematicalproblem,andconstructsimpleequationsandinequalities tosolveproblemsbyreasoningaboutthequantities.

a. Solvewordproblemsleadingtoequationsoftheform px + q = r and p(x + q) = r,where p, q,and r arespecificrationalnumbers. Solveequationsoftheseformsfluently.Compareanalgebraic solutiontoanarithmeticsolution,identifyingthesequenceofthe operationsusedineachapproach. For example, the perimeter of a rectangle is 54 cm. Its length is 6 cm. What is its width?

b. Solvewordproblemsleadingtoinequalitiesoftheform px + q > r or px + q < r,where p, q,and r arespecificrationalnumbers. Graphthesolutionsetoftheinequalityandinterpretitinthe contextoftheproblem. For example: As a salesperson, you are paid $50 per week plus $3 per sale. This week you want your pay to be at least $100. Write an inequality for the number of sales you need to make, and describe the solutions.

8.EE.5Graphproportionalrelationships,interpretingtheunitrateasthe slopeofthegraph.Comparetwodifferentproportionalrelationships representedindifferentways. For example, compare a distance-time graph to a distance-time equation to determine which of two moving objects has greater speed.

8.EE.6Usesimilartrianglestoexplainwhytheslope m isthesame betweenanytwodistinctpointsonanon-verticallineinthecoordinate plane;derivetheequation y = mx foralinethroughtheoriginandthe equation y = mx + b foralineinterceptingtheverticalaxisat b.

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• Analyzeandsolvelinearequationsand pairsofsimultaneouslinearequations.

8.EE.7Solvelinearequationsinonevariable.

a. Giveexamplesoflinearequationsinonevariablewithonesolution,infinitelymanysolutions,ornosolutions.Showwhichof thesepossibilitiesisthecasebysuccessivelytransformingthe givenequationintosimplerforms,untilanequivalentequationof theform x = a, a = a,or a = b results(where a and b aredifferent numbers).

b. Solvelinearequationswithrationalnumbercoefficients,includingequationswhosesolutionsrequireexpandingexpressions usingthedistributivepropertyandcollectingliketerms.


Unit3:IntroductiontosamplingandInference

Studentsbuildontheirpreviousworkwithsingledatadistributionstocomparetwodatadistributionsandaddress questionsaboutdifferencesbetweenpopulations.Theybegininformalworkwithrandomsamplingtogeneratedata setsandlearnabouttheimportanceofrepresentativesamplesfordrawinginferences.

Unit3:IntroductiontoSamplingandInference


• Userandomsamplingtodrawinferencesaboutapopulation.

7.SP.1Understandthatstatisticscanbeusedtogaininformationabout apopulationbyexaminingasampleofthepopulation;generalizations aboutapopulationfromasamplearevalidonlyifthesampleis representativeofthatpopulation.Understandthatrandomsampling tendstoproducerepresentativesamplesandsupportvalidinferences.

7.SP.2Usedatafromarandomsampletodrawinferencesabouta populationwithanunknowncharacteristicofinterest.Generatemultiple samples(orsimulatedsamples)ofthesamesizetogaugethevariation inestimatesorpredictions. For example, estimate the mean word length in a book by randomly sampling words from the book; predict the winner of a school election based on randomly sampled survey data. Gauge how far off the estimate or prediction might be.

• Drawinformalcomparativeinferences abouttwopopulations.

7.SP.3Informallyassessthedegreeofvisualoverlapoftwonumerical datadistributionswithsimilarvariabilities,measuringthedifference betweenthecentersbyexpressingitasamultipleofameasureof variability. For example, the mean height of players on the basketball team is 10 cm greater than the mean height of players on the soccer team, about twice the variability (mean absolute deviation) on either team; on a dot plot, the separation between the two distributions of heights is noticeable.

7.SP.4Usemeasuresofcenterandmeasuresofvariabilityfornumerical datafromrandomsamplestodrawinformalcomparativeinferences abouttwopopulations. For example, decide whether the words in a chapter of a seventh-grade science book are generally longer than the words in a chapter of a fourth-grade science book.

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• Investigatechanceprocessesand develop,use,andevaluateprobability models.

7.SP.5Understandthattheprobabilityofachanceeventisanumber between0and1thatexpressesthelikelihoodoftheeventoccurring. Largernumbersindicategreaterlikelihood.Aprobabilitynear0 indicatesanunlikelyevent,aprobabilityaround 1/2 indicatesanevent thatisneitherunlikelynorlikely,andaprobabilitynear1indicatesa likelyevent.

7.SP.6Approximatetheprobabilityofachanceeventbycollecting dataonthechanceprocessthatproducesitandobservingitslong-run relativefrequency,andpredicttheapproximaterelativefrequencygiven theprobability. For example, when rolling a number cube 600 times, predict that a 3 or 6 would be rolled roughly 200 times, but probably not exactly 200 times.

7.SP.7Developaprobabilitymodelanduseittofindprobabilitiesof events.Compareprobabilitiesfromamodeltoobservedfrequencies;if theagreementisnotgood,explainpossiblesourcesofthediscrepancy.

a. Developauniformprobabilitymodelbyassigningequalprobabilitytoalloutcomes,andusethemodeltodetermineprobabilities ofevents. For example, if a student is selected at random from a class, find the probability that Jane will be selected and the probability that a girl will be selected.

b. Developaprobabilitymodel(whichmaynotbeuniform)byobservingfrequenciesindatageneratedfromachanceprocess. For example, find the approximate probability that a spinning penny will land heads up or that a tossed paper cup will land open-end down. Do the outcomes for the spinning penny appear to be equally likely based on the observed frequencies?

7.SP.8Findprobabilitiesofcompoundeventsusingorganizedlists, tables,treediagrams,andsimulation.

a. Understandthat,justaswithsimpleevents,theprobabilityofa compoundeventisthefractionofoutcomesinthesamplespace forwhichthecompoundeventoccurs.

b. Representsamplespacesforcompoundeventsusingmethods suchasorganizedlists,tablesandtreediagrams.Foranevent describedineverydaylanguage(e.g.,“rollingdoublesixes”), identifytheoutcomesinthesamplespacewhichcomposethe event.

c. Designanduseasimulationtogeneratefrequenciesforcompoundevents. For example, use random digits as a simulation tool to approximate the answer to the question: If 40% of donors have type A blood, what is the probability that it will take at least 4 donors to find one with type A blood?

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Unit4:creating,comparing,andanalyzingGeometricFigures

StudentscontinuetheirworkwithareafromGrade6,solvingproblemsinvolvingtheareaandcircumferenceofa circle and surface area of three-dimensional objects. In preparation for work on congruence and similarity, they reason aboutrelationshipsamongtwo-dimensionalfiguresusingscaledrawingsandinformalgeometricconstructions,and theygainfamiliaritywiththerelationshipsbetweenanglesformedbyintersectinglines.Studentsworkwiththreedimensionalfigures,relatingthemtotwo-dimensionalfiguresbyexaminingcross-sections.Theysolvereal-worldand mathematicalproblemsinvolvingarea,surfacearea,andvolumeoftwo-andthree-dimensionalobjectscomposedof triangles,quadrilaterals,polygons,cubesandrightprisms.Studentsuseideasaboutdistanceandangles,howthey behaveundertranslations,rotations,reflections,anddilations,andideasaboutcongruenceandsimilaritytodescribe andanalyzetwo-dimensionalfiguresandtosolveproblems.Studentsshowthatthesumoftheanglesinatriangleis theangleformedbyastraightline,andthatvariousconfigurationsoflinesgiverisetosimilartrianglesbecauseofthe anglescreatedwhenatransversalcutsparallellines.Studentscompletetheirworkonvolumebysolvingproblems involvingcones,cylinders,andspheres.

Unit4:Creating,Comparing,andAnalyzingGeometricFigures


• Draw,construct,anddescribegeometricalfiguresanddescribetherelationshipsbetweenthem.

7.G.1Solveproblemsinvolvingscaledrawingsofgeometricfigures, includingcomputingactuallengthsandareasfromascaledrawingand reproducingascaledrawingatadifferentscale.

7.G.2Draw(freehand,withrulerandprotractor,andwithtechnology) geometric shapes with given conditions. Focus on constructing triangles fromthreemeasuresofanglesorsides,noticingwhentheconditions determineauniquetriangle,morethanonetriangle,ornotriangle.

7.G.3Describethetwo-dimensionalfiguresthatresultfromslicingthreedimensionalfigures,asinplanesectionsofrightrectangularprismsand rightrectangularpyramids.

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• Solvereal-lifeandmathematicalproblemsinvolvinganglemeasure,area, surfacearea,andvolume.

7.G.4Knowtheformulasfortheareaandcircumferenceofacircle andusethemtosolveproblems;giveaninformalderivationofthe relationshipbetweenthecircumferenceandareaofacircle.

7.G.5Usefactsaboutsupplementary,complementary,vertical,and adjacentanglesinamulti-stepproblemtowriteandsolvesimple equationsforanunknownangleinafigure.

7.G.6Solvereal-worldandmathematicalproblemsinvolvingarea, volumeandsurfaceareaoftwo-andthree-dimensionalobjects composedoftriangles,quadrilaterals,polygons,cubes,andrightprisms.




• Understandcongruenceandsimilarity usingphysicalmodels,transparencies, orgeometrysoftware.

• Solvereal-worldandmathematical probleminvolvingvolumeofcylinders, cones,andspheres.

8.G.1Verifyexperimentallythepropertiesofrotations,reflections,and translations:

a. Linesaretakentolines,andlinesegmentstolinesegmentsof thesamelength.

b. Anglesaretakentoanglesofthesamemeasure.

c. Parallellinesaretakentoparallellines.

8.G.2Understandthatatwo-dimensionalfigureiscongruenttoanother ifthesecondcanbeobtainedfromthefirstbyasequenceofrotations, reflections,andtranslations;giventwocongruentfigures,describea sequencethatexhibitsthecongruencebetweenthem.

8.G.3Describetheeffectofdilations,translations,rotations,and reflectionsontwo-dimensionalfiguresusingcoordinates.

8.G.4Understandthatatwo-dimensionalfigureissimilartoanotherif thesecondcanbeobtainedfromthefirstbyasequenceofrotations, reflections,translations,anddilations;giventwosimilartwo-dimensional figures,describeasequencethatexhibitsthesimilaritybetweenthem.

8.G.5Useinformalargumentstoestablishfactsabouttheanglesum andexteriorangleoftriangles,abouttheanglescreatedwhenparallel linesarecutbyatransversal,andtheangle-anglecriterionforsimilarity oftriangles. For example, arrange three copies of the same triangle so that the sum of the three angles appears to form a line, and give an argument in terms of transversals why this is so.

8.G.9Knowtheformulasforthevolumesofcones,cylinders,and spheresandusethemtosolvereal-worldandmathematicalproblems.

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8th GradealgebraI Thefundamentalpurposeof8th GradeAlgebraIistoformalizeandextendthemathematicsthatstudentslearned throughtheendofseventhgrade.Thecriticalareas,calledunits,deepenandextendunderstandingoflinearand exponentialrelationshipsbycontrastingthemwitheachotherandbyapplyinglinearmodelstodatathatexhibita lineartrend,andstudentsengageinmethodsforanalyzing,solving,andusingquadraticfunctions.Inaddition,the unitswillintroducemethodsforanalyzingandusingquadraticfunctions,includingmanipulatingexpressionsfor them,andsolvingquadraticequations.StudentsunderstandandapplythePythagoreantheorem,andusequadratic functionstomodelandsolveproblems.TheMathematicalPracticeStandardsapplythroughouteachcourseand, togetherwiththecontentstandards,prescribethatstudentsexperiencemathematicsasacoherent,useful,and logicalsubjectthatmakesuseoftheirabilitytomakesenseofproblemsituations.

ThiscoursediffersfromHighSchoolAlgebraIinthatitcontainscontentfrom8th grade.Whilecoherenceisretained, inthatitlogicallybuildsfromtheAccelerated7th Grade,theadditionalcontentwhencomparedtothehighschool coursedemandsafasterpaceforinstructionandlearning.

CriticalArea1:Workwithquantitiesandrates,includingsimplelinearexpressionsandequationsformsthefoundation forthisunit.Studentsuseunitstorepresentproblemsalgebraicallyandgraphically,andtoguidethesolutionof problems.Studentexperiencewithquantityprovidesafoundationforthestudyofexpressions,equations,and functions.Thisunitbuildsonearlierexperienceswithequationsbyaskingstudentstoanalyzeandexplaintheprocess ofsolvinganequation.Studentsdevelopfluencywriting,interpreting,andtranslatingbetweenvariousformsoflinear equationsandinequalities,andusingthemtosolveproblems.Theymasterthesolutionoflinearequationsandapply relatedsolutiontechniquesandthelawsofexponentstothecreationandsolutionofsimpleexponentialequations.

CriticalArea2:Buildingonearlierworkwithlinearrelationships,studentslearnfunctionnotationandlanguagefor describingcharacteristicsoffunctions,includingtheconceptsofdomainandrange.Theyexploremanyexamples offunctions,includingsequences;theyinterpretfunctionsgivengraphically,numerically,symbolically,andverbally, translatebetweenrepresentations,andunderstandthelimitationsofvariousrepresentations.Theyworkwith functionsgivenbygraphsandtables,keepinginmindthatdependinguponthecontext,theserepresentations arelikelytobeapproximateandincomplete.Theirworkincludesfunctionsthatcanbedescribedorapproximated byformulasaswellasthosethatcannot.Whenfunctionsdescriberelationshipsbetweenquantitiesarisingfrom acontext,studentsreasonwiththeunitsinwhichthosequantitiesaremeasured.Studentsexploresystemsof equationsandinequalities,andtheyfindandinterprettheirsolutions.Studentsbuildonandinformallyextend theirunderstandingofintegralexponentstoconsiderexponentialfunctions.Theycompareandcontrastlinearand exponentialfunctions,distinguishingbetweenadditiveandmultiplicativechange.Theyinterpretarithmeticsequences aslinearfunctionsandgeometricsequencesasexponentialfunctions.

CriticalArea3:Studentsuseregressiontechniquestodescriberelationshipsbetweenquantities.Theyusegraphical representationsandknowledgeofthecontexttomakejudgmentsabouttheappropriatenessoflinearmodels.With linearmodels,theylookatresidualstoanalyzethegoodnessoffit.

CriticalArea4:Inthisunit,studentsbuildontheirknowledgefromunit2,wheretheyextendedthelawsofexponents torationalexponents.Studentsapplythisnewunderstandingofnumberandstrengthentheirabilitytoseestructure inandcreatequadraticandexponentialexpressions.Theycreateandsolveequations,inequalities,andsystemsof equationsinvolvingquadraticexpressions.

CriticalArea5:Inpreparationforworkwithquadraticrelationshipsstudentsexploredistinctionsbetweenrational andirrationalnumbers.Theyconsiderquadraticfunctions,comparingthekeycharacteristicsofquadraticfunctions tothoseoflinearandexponentialfunctions.Theyselectfromamongthesefunctionstomodelphenomena.Students learntoanticipatethegraphofaquadraticfunctionbyinterpretingvariousformsofquadraticexpressions.In particular,theyidentifytherealsolutionsofaquadraticequationasthezerosofarelatedquadraticfunction. Studentslearnthatwhenquadraticequationsdonothaverealsolutionsthenumbersystemmustbeextendedso thatsolutionsexist,analogoustothewayinwhichextendingthewholenumberstothenegativenumbersallowsx+1 =0tohaveasolution.FormalworkwithcomplexnumberscomesinAlgebraII.Studentsexpandtheirexperience withfunctionstoincludemorespecializedfunctions—absolutevalue,step,andthosethatarepiecewise-defined.

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Units IncludesStandardClusters MathematicalPractice

Standards *

Unit1

Relationships BetweenQuantities andReasoningwith

Equations






Attendtoprecision.



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Unit2


• Reason quantitatively and use units to solve problems.








• Represent and solve equations and inequalities graphically

• Define, evaluate, and compare functions.


• Use functions to model relationships between quantities.








• Investigate patterns of association in bivariate Unit3 data.

DescriptiveStatistics • Summarize, represent, and interpret data on two

categorical and quantitative variables.




Units IncludesStandardClusters MathematicalPractice

Standards

Unit4








Unit5

QuadraticsFuntions

andModeling

• Use properties of rational and irrational numbers.

• Understand and apply the Pythagorean theorem.





• Construct and compare linear, quadratic and exponential models and solve problems.

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Unit1:relationshipsbetweenQuantitiesandreasoningwithequations

Workwithquantitiesandrates,includingsimplelinearexpressionsandequationsformsthefoundationforthis unit.Studentsuseunitstorepresentproblemsalgebraicallyandgraphically,andtoguidethesolutionofproblems. Studentexperiencewithquantityprovidesafoundationforthestudyofexpressions,equations,andfunctions.This unitbuildsonearlierexperienceswithequationsbyaskingstudentstoanalyzeandexplaintheprocessofsolvingan equation.Studentsdevelopfluencywriting,interpreting,andtranslatingbetweenvariousformsoflinearequations andinequalities,andusingthemtosolveproblems.Theymasterthesolutionoflinearequationsandapplyrelated solutiontechniquesandthelawsofexponentstothecreationandsolutionofsimpleexponentialequations.




Working with quantities and the relationships between them provides grounding for work with expressions, equations, and functions.12









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Limit A.CED.1 and A.CED.2 to linear and exponential equations, and, in the case of exponential equations, limit to situations requiring evaluation of exponential functions at integer inputs. Limit A.CED.3 to linear equations and inequalities. Limit A.CED.4 to formulas which are linear in the variables of interest.






Students should focus on and master A.REI.1 for linear equations and be able to extend and apply their reasoning to other types of equations in future units and courses. Students will solve exponential equations in Algebra II.






Extend earlier work with solving linear equations to solving linear inequalities in one variable and to solving literal equations that are linear in the variable being solved for. Include simple exponential equations that rely only on application of the laws of exponents, such as 5x = 125 or 2x = 1/16 .

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Unit2:LinearandexponentialFunctions

Buildingonearlierworkwithlinearrelationships,studentslearnfunctionnotationandlanguagefordescribing characteristicsoffunctions,includingtheconceptsofdomainandrange.Theyexploremanyexamplesoffunctions, includingsequences;theyinterpretfunctionsgivengraphically,numerically,symbolically,andverbally,translate betweenrepresentations,andunderstandthelimitationsofvariousrepresentations.Theyworkwithfunctions givenbygraphsandtables,keepinginmindthatdependinguponthecontext,theserepresentationsarelikelyto beapproximateandincomplete.Theirworkincludesfunctionsthatcanbedescribedorapproximatedbyformulas aswellasthosethatcannot.Whenfunctionsdescriberelationshipsbetweenquantitiesarisingfromacontext, studentsreasonwiththeunitsinwhichthosequantitiesaremeasured.Studentsexploresystemsofequationsand inequalities,andtheyfindandinterprettheirsolutions.Studentsbuildonandinformallyextendtheirunderstanding ofintegralexponentstoconsiderexponentialfunctions.Theycompareandcontrastlinearandexponentialfunctions, distinguishingbetweenadditiveandmultiplicativechange.Theyinterpretarithmeticsequencesaslinearfunctions andgeometricsequencesasexponentialfunctions.

Unit2:LinearandExponentialFunctions



In implementing the standards in curriculum, these standards should occur before discussing exponential models with continuous domains.



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While this content is likely subsumed by A.REI.3, 5, and 6, it could be used for scaffolding instruction to the more sophisticated content found there.

8.EE.8Analyzeandsolvepairsofsimultaneouslinearequations.

a. Understandthatsolutionstoasystemoftwolinearequationsin two variables correspond to points of intersection of their graphs, becausepointsofintersectionsatisfybothequationssimultaneously.

b. Solvesystemsoftwolinearequationsintwovariablesalgebra-ically,andestimatesolutionsbygraphingtheequations.Solve simplecasesbyinspection. For example, 3x + 2y = 5 and 3x + 2y = 6 have no solution because 3x + 2y cannot simultaneously be 5 and 6.

c. Solvereal-worldandmathematicalproblemsleadingtotwolinearequationsintwovariables. For example, given coordinates for two pairs of points, determine whether the line through the first pair of points intersects the line through the second pair.


Include cases where two equations describe the same line (yielding infinitely many solutions) and cases where two equations describe parallel lines (yielding no solution).

A.REI.5Provethat,givenasystemoftwoequationsintwovariables, replacingoneequationbythesumofthatequationandamultiple of theotherproducesasystemwiththesamesolutions.










• Define,evaluate,andcomparefunctions.

While this content is likely subsumed by F.IF.1-3 and F.IF.7a, it could be used for scaffolding instruction to the more sophisticated content found there.

8.F.1Understandthatafunctionisarulethatassignstoeachinput exactlyoneoutput.Thegraphofafunctionisthesetoforderedpairs consistingofaninputandthecorrespondingoutput.

8.F.2Comparepropertiesoftwofunctionseachrepresentedina differentway(algebraically,graphically,numericallyintables,orby verbaldescriptions). For example, given a linear function represented by a table of values and a linear function represented by an algebraic expression, determine which function has the greater rate of change.

8.F.3Interprettheequation y = mx + b asdefiningalinearfunction, whosegraphisastraightline;giveexamplesoffunctionsthatarenot linear. For example, the function A = s2 giving the area of a square as a function of its side length is not linear because its graph contains the points (1,1), (2,4) and (3,9), which are not on a straight line.

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Students should experience a variety of types of situations modeled by functions Detailed analysis of any particular class of function at this stage is not advised. Students should apply these concepts throughout their future mathematics courses.

Constrain examples to linear functions and exponential functions having integral domains. In F.IF.3, draw connection to F.BF.2, which requires students to write arithmetic and geometric sequences.

F.IF.1Understandthatafunctionfromoneset(calledthedomain)to anotherset(calledtherange)assignstoeachelementofthedomain exactlyoneelementoftherange.If f isafunctionandxisanelementof itsdomain,then f(x) denotestheoutputof f correspondingtotheinput x.Thegraphof f isthegraphoftheequation y = f(x).



• Usefunctionstomodelrelationships betweenquantities.

While this content is likely subsumed by F.IF.4 and F.BF.1a, it could be used for scaffolding instruction to the more sophisticated content found there.

8.F.4Constructafunctiontomodelalinearrelationshipbetweentwo quantities.Determinetherateofchangeandinitialvalueofthefunction fromadescriptionofarelationshiporfromtwo(x, y)values,including readingthesefromatableorfromagraph.Interprettherateofchange andinitialvalueofalinearfunctionintermsofthesituationitmodels, andintermsofitsgraphoratableofvalues.

8.F.5Describequalitativelythefunctionalrelationshipbetweentwo quantitiesbyanalyzingagraph(e.g.,wherethefunctionisincreasing ordecreasing,linearornonlinear).Sketchagraphthatexhibitsthe qualitativefeaturesofafunctionthathasbeendescribedverbally.





For F.IF.4 and 5, focus on linear and exponential functions. For F.IF.6, focus on linear functions and exponential functions whose domain is a subset of the integers. Unit 5 in this course and Algebra II course address other types of functions.





For F.IF.7a, 7e, and 9 focus on linear and exponentials functions. Include comparisons of two functions presented algebraically. For example, compare the growth of two linear functions, or two exponential functions such as y=3n and y=100·2n.




F.IF.9Comparepropertiesoftwofunctionseachrepresentedina differentway(algebraically,graphically,numericallyintables,orby verbaldescriptions).Forexample,givenagraphofonequadratic functionandanalgebraicexpressionforanother,saywhichhasthe largermaximum.

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Limit F.BF.1a, 1b, and 2 to linear and exponential functions. In F.BF.2, connect arithmetic sequences to linear functions and geometric sequences to exponential functions in F.BF.2.








F.BF.3Identifytheeffectonthegraphofreplacing f(x) byf(x) + k, k f(x), f(kx), and f(x + k) forspecificvaluesofk(bothpositiveand negative);findthevalueof k giventhegraphs.Experimentwith casesandillustrateanexplanationoftheeffectsonthegraphusing technology. Include recognizing even and odd functions from their graphs and algebraic expressions for them.





For F.LE.3, limit to comparisons between linear and exponential models.








Limit exponential functions to those of the form f(x) = bx + k .

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Studentsuseregressiontechniquestodescriberelationshipsbetweenquantities.Theyusegraphicalrepresentations andknowledgeofthecontexttomakejudgmentsabouttheappropriatenessoflinearmodels.Withlinearmodels, theylookatresidualstoanalyzethegoodnessoffit.








• Investigatepatternsofassociationin bivariatedata.

While this content is likely subsumed by S.ID.6-9, it could be used for scaffolding instruction to the more sophisticated content found there.

8.SP.1Constructandinterpretscatterplotsforbivariatemeasurement datatoinvestigatepatternsofassociationbetweentwoquantities. Describepatternssuchasclustering,outliers,positiveornegative association,linearassociation,andnonlinearassociation.

8.SP.2Knowthatstraightlinesarewidelyusedtomodelrelationships betweentwoquantitativevariables.Forscatterplotsthatsuggesta linearassociation,informallyfitastraightline,andinformallyassessthe modelfitbyjudgingtheclosenessofthedatapointstotheline.

8.SP.3Usetheequationofalinearmodeltosolveproblemsinthe contextofbivariatemeasurementdata,interpretingtheslopeand intercept. For example, in a linear model for a biology experiment, interpret a slope of 1.5 cm/hr as meaning that an additional hour of sunlight each day is associated with an additional 1.5 cm in mature plant height.

8.SP.4Understandthatpatternsofassociationcanalsobeseenin bivariatecategoricaldatabydisplayingfrequenciesandrelative frequenciesinatwo-waytable.Constructandinterpretatwo-waytable summarizingdataontwocategoricalvariablescollectedfromthesame subjects.Userelativefrequenciescalculatedforrowsorcolumnsto describepossibleassociationbetweenthetwovariables. For example, collect data from students in your class on whether or not they have a curfew on school nights and whether or not they have assigned chores at home. Is there evidence that those who have a curfew also tend to have chores?

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S.ID.6b should be focused on linear models, but may be used to preface quadratic functions in the Unit 6 of this course.










Build on students’ work with linear relationship and; introduce the correlation coefficient. The focus here is on the computation and interpretation of the correlation coefficient as a measure how well the data fit the relationship. The important distinction between a statistical relationship and a cause-and-effect relationship arises in S.ID.9.




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Inthisunit,studentsbuildontheirknowledgefromunit2,wheretheyextendedthelawsofexponentstorational exponents.Studentsapplythisnewunderstandingofnumberandstrengthentheirabilitytoseestructureinandcreatequadraticandexponentialexpressions.Theycreateandsolveequations,inequalities,andsystemsofequations involvingquadraticexpressions.




Focus on quadratic and exponential expressions. For A.SSE.1b, exponents are extended from integer found in Unit 1 to rational exponents focusing on those that represent square roots and cube roots.






Consider extending this unit to include the relationship between properties of logarithms and properties of exponents.





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Extend work on linear and exponential equations in Unit 1 to include quadratic equations. Extend A.CED.4 to formulas involving squared variables.








Students should learn of the existence of the complex number system, but will not solve quadratics with complex solutions until Algebra II.





Include systems consisting of one linear and one quadratic equation. Include systems that lead to work with fractions. For example, finding the intersections between x2+y2=1 and y = (x+1)/2 leads to the point (3/5, 4/5) on the unit circle, corresponding to the Pythagorean triple 32 + 42 = 52.

A.REI.7Solveasimplesystemconsistingofalinearequationanda quadraticequationintwovariablesalgebraicallyandgraphically. For example, find the points of intersection between the line y = –3x and the circle x2 + y2 = 3. a

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Inpreparationforworkwithquadraticrelationshipsstudentsexploredistinctionsbetweenrationalandirrationalnumbers.Theyconsiderquadraticfunctions,comparingthekeycharacteristicsofquadraticfunctionstothoseoflinear and exponential functions. They select from among these functions to model phenomena. Students learn to anticipate thegraphofaquadraticfunctionbyinterpretingvariousformsofquadraticexpressions.Inparticular,theyidentify therealsolutionsofaquadraticequationasthezerosofarelatedquadraticfunction.Studentslearnthatwhenquadraticequationsdonothaverealsolutionsthenumbersystemmustbeextendedsothatsolutionsexist,analogous tothewayinwhichextendingthewholenumberstothenegativenumbersallowsx+1=0tohaveasolution.Formal workwithcomplexnumberscomesinAlgebraII.Studentsexpandtheirexperiencewithfunctionstoincludemore specializedfunctions—absolutevalue,step,andthosethatarepiecewise-defined.



• Usepropertiesofrationalandirrational N.RN.3Explainwhythesumorproductoftworationalnumbersis numbers. rational;thatthesumofarationalnumberandanirrationalnumberis

irrational;andthattheproductofanonzerorationalnumberandan irrationalnumberisirrational. Connect N.RN.3 to physical situations,


• UnderstandandapplythePythagoreantheorem.

Discuss applications of the Pythagorean theorem and its connections to radicals, rational exponents, and irrational numbers.

8.G.6ExplainaproofofthePythagoreantheoremanditsconverse.

8.G.7ApplythePythagoreantheoremtodetermineunknownside lengthsinrighttrianglesinreal-worldandmathematicalproblemsin twoandthreedimensions.

8.G.8ApplythePythagoreantheoremtofindthedistancebetweentwo pointsinacoordinatesystem.

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Focus on quadratic functions; compare with linear and exponential functions studied in Unit 2.








For F.IF.7b, compare and contrast absolute value, step and piecewise-defined functions with linear, quadratic, and exponential functions. Highlight issues of domain, range, and usefulness when examining piecewise-defined functions. Note that this unit, and in particular in F.IF.8b, extends the work begun in Unit 2 on exponential functions with integral exponents. For F.IF.9, focus on expanding the types of functions considered to include, linear, exponential, and quadratic.







b. Usethepropertiesofexponentstointerpretexpressionsforexponentialfunctions. For example, identify percent rate of change in functions such as y = (1.02)t, y = (0.97)t, y = (1.01)12t, y = (1.2)t/10, and classify them as representing exponential growth or decay.


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Focus on situations that exhibit a quadratic relationship.





For F.BF.3, focus on quadratic functions, and consider including absolute value functions. For F.BF.4a, focus on linear functions but consider simple situations where the domain of the function must be restricted in order for the inverse to exist, such as f(x) = x2, x>0



a. Solveanequationoftheform f(x) = c forasimplefunction f thathasaninverseandwriteanexpressionfortheinverse. For example, f(x) =2 x3 for x > 0 or f(x) = (x+1)/(x-1) for x ≠ 1.

• Constructandcomparelinear,quadrat F.LE.3Observeusinggraphsandtablesthataquantityincreasing ic,andexponentialmodelsandsolve exponentiallyeventuallyexceedsaquantityincreasinglinearly, problems. quadratically,or(moregenerally)asapolynomialfunction.

Compare linear and exponential growth to growth of quadratic growth.


overviewoftheacceleratedIntegratedPathwayfor thecommoncorestatemathematicsstandards This table shows the domains and clusters in each course in the Accelerated Traditional Pathway. The standards from each cluster included in that course are listed below each cluster. For each course, limits and focus for the clusters are shown in italics. For organizational purposes, clusters from 7th Grade and 8th Grade have been situated in the matrix within the high school domains.


Grade 8th Grade

MathematicsI MathematicsII MathematicsIII FourthCourses *

Nu

mb

er

an

dQ

uan

tity


Quantities

•Applyand extendprevious understandingsof operationswith fractionstoadd, subtract,multiply, anddivide rationalnumbers.

7.NS.1a,1b,1c,1d, 2a,2b,2c,2d,3

•Knowthatthere arenumbersthat arenotrational, andapproximate thembyrational numbers.

8.NS.1,2

•Workwith radicals andinteger exponents.

8.EE.1,2,3,4

•Analyze proportional relationshipsand usethemtosolve real-worldand mathematical problems.

•Reason quantitativelyand useunitstosolve problems.


•Extendthe propertiesof exponents torational exponents.

N.RN.1,2

•Useproperties ofrational andirrational numbers.

N.RN.3.


7.RP.1,2a,2b,2c, 2d,3


N.Q.1,2,3


i2 as highest power of i

N.CN.1,2

•Usecomplex numbersin polynomial

•Usecomplex numbersin polynomial identitiesand equations.

Polynomials with real coefficients; apply N.CN.9 to higher degree polynomials

(+)N.CN.8,9


(+) N.CN.3

•Represent complexnumbers andtheir operationsonthe complexplane.

identitiesand equations.


N.CN.7,(+)8,(+)9

(+) N.CN.4,5,6

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117*The(+)standardsinthiscolumnarethoseintheCommonCoreStateStandardsthatarenotincludedinanyoftheAcceleratedIntegratedPath-waycourses.TheywouldbeusedinadditionalcoursesdevelopedtofollowMathematicsIII.

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Grade 8th Grade

MathematicsI MathematicsII MathematicsIII FourthCourses

Nu

mb

er

an

dQ

uan

tity

Vector Quantities andMatrices


•Useproperties ofoperations togenerate equivalent expressions.

7.EE.1,2

•Solvereal-lifeand mathematical problemsusing numerical andalgebraic


Linear expressions and exponential expressions with

integer exponents

A.SSE.1a,1b



A.SSE.1a,1b,2




A.SSE.1a,1b,2


•Representand modelwithvector quantities.

(+) N.VM.1,2,3

•Perform operationson vectors.

(+) N.VM.4a,4b, 4c,5a,5b

•Perform operationson matricesand usematricesin applications.

(+) N.VM.6,7,8,9, 10, 11, 12

Alg

eb

ra

expressionsand equations..

7.EE.3,4a,4b


A.SSE.3a,3b,3c


Polynomials that simplify to

quadratics

A.SSE.4


Beyond quadratic

A.APR.1

Arithmetic with Polynomials andRational Expressions

A.APR.1 •Understand therelationship betweenzeros andfactorsof polynomials.

A.APR.2,3


A.APR.4, (+) 5



denominators

A.APR.6,(+)7

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Grade 8th Grade


Creating Equations

•Understandthe connections between proportional relationships, lines,andlinear



(integer inputs only); for A.CED.3,

linear only.

A.CED.1,2,3,4



In A.CED.4 include formulas involving quadratic terms

A.CED.1, 2, 4




Equations using all available types

of expressions, including simple root functions

A.CED.1,2,3,4



(+)A.REI.8,9

equations.

8.EE.5,6

•Analyzeandsolve linearequations andpairsof simultaneous linearequations.

8.EE.7a,7b


A.REI.1


Linear inequalities; literal equations

A.REI.4a,4b


Linear-quadratic systems

A.REI.7


A.REI.2


Combine

Alg

eb

ra that are linear

in the variables being solved for; exponential of a

polynomial, rational, radical, absolute value, and exponential

Reasoning with Equations and Inequalities

form, such as 2x = 1/16

A.REI.3

•Analyzeandsolve linearequations andpairsof

functions.

A.REI.11

simultaneous linearequations

Systems of linear equations

8.EE.8a,8b,8c


Linear systems

A.REI.5, 6


Linear and exponential; learn

as general principle

A.REI.10, 11, 12

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Grade 8th Grade


•Define,evaluate, andcompare functions.

8.F.1,2,3

•Understandthe conceptofa functionanduse functionnotation.

Learn as general principle. Focus

on linear and exponential

(integer domains) and on arithmetic

and geometric sequences


Quadratic

F.IF.4,5,6





Include rational, square root

and cube root; emphasize selection of appropriate

models

F.IF.4,5,6

•Analyzefunctions usingdifferent


Logarithmic and trigonometric

functions

(+)F.IF.7d

Fu

ncti

on

s


F.IF.1,2,3

•Usefunctions tomodel relationships between quantities.

F.IF.7a, 7b, 8a, 8b, 9 representations.

Include rational and radical;

focus on using key features to

guide selection of appropriate type

8.F.4,5


Linear and exponential, (linear

domain)

F.IF.4,5,6



F.IF.7a,7e,9

of model function

F.IF.7b,7c,7e,8,9

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Grade 8th Grade




(integer inputs)

F.BF.1a,1b,2



F.BF.1a,1b

•Buildnew



F.BF.1b

•Buildnew


(+)F.BF.1c


Building Functions



Quadratic, all


Include simple

(+) F.BF.4b,4c, 4d,5

Fu

ncti

on

s

Linear, Quadratic,

For F.BF.1, 2, linear and exponential; focus on vertical translations for

exponential

F.BF.3



F.LE.1a,1b,1c,2,3

exponential, absolute value

F.BF.3,4a


Include quadratic

F.LE.3

radical, rational, and exponential

functions; emphasize common

effect of each transformation across function

types

F.BF.3,4a



and Exponential Models


•Interpret expressionsfor functionsinterms ofthesituation theymodel.

Linear and exponential of

form f(x) = bx = k

F.LE.5


F.TF.8

F.LE.4


F.TF.1,2


F.TF.5


(+)F.TF.3,4


(+)F.TF.6,7

Proveandapply trigonometric identities.

(+)F.TF.9

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Grade 8th Grade



•Experimentwith transformationsin theplane.

G.CO.1,2,3,4,5

•Understand congruencein



reasoning while using variety of ways of writing

Ge

om

etr

y

Congruence

Focus on constructing

triangles

7.G.2


8.G.1a,1b,1c,2,5

•For 8.G.5, informal argumentsto establishangle sumandexterior angletheorems fortriangles andangles relationships whenparallel linesarecutbya

termsofrigid motions.

Build on rigid motions as a

familiar starting point for

development of concept of

geometric proof

G.CO.6,7,8



G.CO.12,13

proofs

G.CO.9,10,11

transversal

•Draw,construct, anddescribe geometrical figuresand describethe relationships

•Understand similarityinterms ofsimilarity transformations.

G.SRT.1a,1b,2,3

•Apply trigonometryto generaltriangles.

(+) G.SRT.9.10,11

Similarity, RightTriangles,and Trigonometry

betweenthem.

Scale drawings

7.G.1


8.G.3,4,5

•For 8.G.5, informal argumentsto establishthe angle-angle criterionfor similartriangles

•Provetheorems involving similarity.


reasoning while using variety of

formats

G.SRT.4,5

•Define trigonometric ratios and solve problems involving right triangles.

G.SRT.6,7,8

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Grade 8th Grade


Ge

om

etr

y

Circles


•Draw,construct, anddescribe geometrical figuresand


Include distance formula; relate to Pythagorean

theorem

G.GPE.4,5,7

•Understand andapplythe Pythagorean theorem.

•Understandand applytheorems aboutcircles.

G.C.1,2,3,(+)4

•Findarclengths andareasof sectorsofcircles.

Radian introduced only as unit of

measure

G.C.5


G.GPE.1,2


For G.GPE.4 include simple circle theorems

G.GPE.4,6

•Explainvolume formulasanduse themtosolve problems.

•Visualizethe relationbetween two-dimensional andthree


(+)G.GPE.3

•Explainvolume formulasanduse themtosolve problems.

describethe relationships betweenthem.

Connect to radicals, rational exponents, and

G.GMD.1, 3 dimensional objects.

G.GMD.4

(+)G.GMD.2

Geometric Measure-mentand

Slicing 3-D figures

7.G.3

•Solvereal-lifeand mathematical problems involvingangle

irrational numbers

8.G.6,7,8

Dimension

Modeling with Geometry

measure,area, surfacearea,and volume.

7.G.4,5,6

•Solvereal-world andmathematical problems involvingvolume ofcylinders, cones,and spheres.

8.G.9

•Applygeometric conceptsin modeling situations.

G.MG.1,2,3

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Grade 8th Grade


Sta

tist

ics

an

dP

rob

ab

ilit

y

Interpreting Categorical andQuantitativeData


S.ID.1,2,3

•Investigate patternsof associationin bivariatedata.

8.SP.1,2,3,4

•Summarize, represent,and interpretdataon twocategorical andquantitative variables.

Linear focus; discuss general

principle

S.ID.5,6a,6b,6c



S.ID.4

MakingInferencesand Justifying Conclusions

•Userandom samplingtodraw inferencesabout apopulation.

7.SP.1,2

•Drawinformal comparative inferencesabout twopopulations.

7.SP.3,4

S.ID.7,8,9

•Understand andevaluate random processes underlying statistical experiments.

S.IC.1,2

•Makeinferences andjustify conclusionsfrom samplesurveys, experimentsand observational studies.

S.IC.3,4,5,6

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Grade 8th Grade


Sta

tist

ics

an

dP

rob

ab

ilit

y

Conditional Probabilityandthe Rulesof Probability

•Investigate chanceprocesses anddevelop, use,andevaluate probability models.

7.SP.5,6,7a,7b,8a, 8b,8c

•Understand independence andconditional probabilityand usethemto interpretdata.


S.CP.1,2,3,4,5

•Usetherules ofprobability tocompute probabilitiesof compoundevents inauniform probabilitymodel.

S.CP.6,7, (+) 8, (+)9

UsingProbabilityto MakeDecisions



(+) S.MD.6,7



(+) S.MD.6,7

•Calculate expectedvalues andusethemto solveproblems.

(+) S.MD.1,2,3,4


(+) S.MD.5a,5b


acceleratedIntegratedPathway: accelerated7th Grade Thiscoursediffersfromthenon-accelerated7th Gradecourseinthatitcontainscontentfrom8th grade.While coherenceisretained,inthatitlogicallybuildsfromthe6th Grade,theadditionalcontentwhencomparedtothenonacceleratedcoursedemandsafasterpaceforinstructionandlearning.Contentisorganizedintofourcriticalareas,or units.TheMathematicalPracticeStandardsapplythroughouteachcourseand,togetherwiththecontentstandards, prescribethatstudentsexperiencemathematicsasacoherent,useful,andlogicalsubjectthatmakesuseoftheir abilitytomakesenseofproblemsituations.Thecriticalareasareasfollows:

CriticalArea1:Studentsdevelopaunifiedunderstandingofnumber,recognizingfractions,decimals(thathavea finiteorarepeatingdecimalrepresentation),andpercentsasdifferentrepresentationsofrationalnumbers.Students extendaddition,subtraction,multiplication,anddivisiontoallrationalnumbers,maintainingthepropertiesof operationsandtherelationshipsbetweenadditionandsubtraction,andmultiplicationanddivision.Byapplyingthese properties,andbyviewingnegativenumbersintermsofeverydaycontexts(e.g.,amountsowedortemperatures belowzero),studentsexplainandinterprettherulesforadding,subtracting,multiplying,anddividingwithnegative numbers.Theyusethearithmeticofrationalnumbersastheyformulateexpressionsandequationsinonevariable andusetheseequationstosolveproblems.Theyextendtheirmasteryofthepropertiesofoperationstodevelopan understandingofintegerexponents,andtoworkwithnumberswritteninscientificnotation.

CriticalArea2:Studentsuselinearequationsandsystemsoflinearequationstorepresent,analyze,andsolvea varietyofproblems.Studentsrecognizeequationsforproportions(y/x=mory=mx)asspeciallinearequations(y =mx+b),understandingthattheconstantofproportionality(m)istheslope,andthegraphsarelinesthroughthe origin.Theyunderstandthattheslope(m)ofalineisaconstantrateofchange,sothatiftheinputorx-coordinate changesbyanamountA,theoutputory-coordinatechangesbytheamountm×A.Studentsstrategicallychoose andefficientlyimplementprocedurestosolvelinearequationsinonevariable,understandingthatwhentheyusethe propertiesofequalityandtheconceptoflogicalequivalence,theymaintainthesolutionsoftheoriginalequation.

CriticalArea3:Studentsbuildontheirpreviousworkwithsingledatadistributionstocomparetwodatadistributions andaddressquestionsaboutdifferencesbetweenpopulations.Theybegininformalworkwithrandomsamplingto generatedatasetsandlearnabouttheimportanceofrepresentativesamplesfordrawinginferences.

CriticalArea4:StudentscontinuetheirworkwithareafromGrade6,solvingproblemsinvolvingtheareaand circumferenceofacircleandsurfaceareaofthree-dimensionalobjects.Inpreparationforworkoncongruence andsimilarity,theyreasonaboutrelationshipsamongtwo-dimensionalfiguresusingscaledrawingsandinformal geometricconstructions,andtheygainfamiliaritywiththerelationshipsbetweenanglesformedbyintersecting lines.Studentsworkwiththree-dimensionalfigures,relatingthemtotwo-dimensionalfiguresbyexaminingcrosssections.Theysolvereal-worldandmathematicalproblemsinvolvingarea,surfacearea,andvolumeoftwo-and three-dimensionalobjectscomposedoftriangles,quadrilaterals,polygons,cubesandrightprisms.Studentsuseideas aboutdistanceandangles,howtheybehaveundertranslations,rotations,reflections,anddilations,andideasabout congruenceandsimilaritytodescribeandanalyzetwo-dimensionalfiguresandtosolveproblems.Studentsshow thatthesumoftheanglesinatriangleistheangleformedbyastraightline,andthatvariousconfigurationsoflines giverisetosimilartrianglesbecauseoftheanglescreatedwhenatransversalcutsparallellines.Studentscomplete theirworkonvolumebysolvingproblemsinvolvingcones,cylinders,andspheres.

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Standards

• Apply and extend previous understandings of operations with fractions to add, subtract,

Unit1 multiply, and divide rational numbers. RationalNumbersand • Know that there are numbers that are not rational,

Exponents and approximate them by rational numbers. Makesenseofproblems • Work with radicals and integer exponents. andpersevereinsolving

Unit2

Proportionalityand LinearRelationships

• Analyze proportional relationships and use them to solve real-world and mathematical problems.

• Use properties of operations to generate equivalent expressions.

• Solve real-life and mathematical problems using numerical and algebraic expressions and equations.

• Understand the connections between proportional relationships, lines, and linear equations.


them.




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Unit3

Introductionto Samplingand Interference

Unit4

Creating,Comparing, andAnalyzing

GeometricFigures

• Use random sampling to draw inferences about a population.

• Draw informal comparative inferences about two populations.

• Investigate chance processes and develop, use, and evaluate probability models.

• Draw, construct and describe geometrical figures and describe the relationships between them.

• Solve real-life and mathematical problems involving angle measure, area, surface area, and volume.

• Understand congruence and similarity using physical models, transparencies, or geometry software.

• Solve real-world and mathematical problems involving volume of cylinders, cones and spheres.


Attendtoprecision.





Unit1:rationalnumbersandexponents

Studentsdevelopaunifiedunderstandingofnumber,recognizingfractions,decimals(thathaveafiniteorarepeating decimalrepresentation),andpercentsasdifferentrepresentationsofrationalnumbers.Theyconvertbetweenafractionanddecimalformofanirrationalnumber.Studentsextendaddition,subtraction,multiplication,anddivisiontoall rationalnumbers,maintainingthepropertiesofoperationsandtherelationshipsbetweenadditionandsubtraction, andmultiplicationanddivision.Byapplyingtheseproperties,andbyviewingnegativenumbersintermsofeverydaycontexts(e.g.,amountsowedortemperaturesbelowzero),studentsexplainandinterprettherulesforadding, subtracting,multiplying,anddividingwithnegativenumbers.Theyusethearithmeticofrationalnumbersasthey formulateexpressionsandequationsinonevariableandusetheseequationstosolveproblems.Theyextendtheir masteryofthepropertiesofoperationstodevelopanunderstandingofintegerexponents,andtoworkwithnumbers writteninscientificnotation.



• Applyandextendpreviousunderstandingsofoperationswithfractions toadd,subtract,multiply,anddivide rationalnumbers.

7.NS.1Applyandextendpreviousunderstandingsofadditionand subtractiontoaddandsubtractrationalnumbers;representaddition andsubtractiononahorizontalorverticalnumberlinediagram.

a. Describesituationsinwhichoppositequantitiescombineto make0. For example, a hydrogen atom has 0 charge because its two constituents are oppositely charged.

b. Understand p + q asthenumberlocatedadistance|q|from p, inthepositiveornegativedirectiondependingonwhether q is positiveornegative.Showthatanumberanditsoppositehavea sumof0(areadditiveinverses).Interpretsumsofrationalnumbersbydescribingreal-worldcontexts.

c. Understandsubtractionofrationalnumbersasaddingtheadditiveinverse, p – q = p +(–q).Showthatthedistancebetweentwo rationalnumbersonthenumberlineistheabsolutevalueoftheir difference,andapplythisprincipleinreal-worldcontexts.

d. Applypropertiesofoperationsasstrategiestoaddandsubtract rationalnumbers.

7.NS.2Applyandextendpreviousunderstandingsofmultiplicationand divisionandoffractionstomultiplyanddividerationalnumbers.

a. Understandthatmultiplicationisextendedfromfractionsto rationalnumbersbyrequiringthatoperationscontinuetosatisfy thepropertiesofoperations,particularlythedistributiveproperty,leadingtoproductssuchas(–1)(–1)=1andtherulesformultiplyingsignednumbers.Interpretproductsofrationalnumbers bydescribingreal-worldcontexts.

b. Understandthatintegerscanbedivided,providedthatthedivisorisnotzero,andeveryquotientofintegers(withnon-zero divisor)isarationalnumber.If p and q areintegers,then –(p/q) = (–p)/q = p/(–q).Interpretquotientsofrationalnumbersby describingreal-worldcontexts.

c. Applypropertiesofoperationsasstrategiestomultiplyand dividerationalnumbers.

d. Convertarationalnumbertoadecimalusinglongdivision;know thatthedecimalformofarationalnumberterminatesin0sor eventuallyrepeats.

7.NS.3Solvereal-worldandmathematicalproblemsinvolvingthefour operationswithrationalnumbers.*

*Computationswithrationalnumbersextendtherulesformanipulatingfractionstocomplexfractions.

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• Knowthattherearenumbersthatare notrational,andapproximatethemby rationalnumbers.

8.NS.1Knowthatnumbersthatarenotrationalarecalledirrational. Understandinformallythateverynumberhasadecimalexpansion;for rationalnumbersshowthatthedecimalexpansionrepeatseventually, andconvertadecimalexpansionwhichrepeatseventuallyintoa rationalnumber.

8.NS.2Userationalapproximationsofirrationalnumberstocomparethe sizeofirrationalnumbers,locatethemapproximatelyonanumberline diagram,andestimatethevalueofexpressions(e.g., p2). For example, by truncating the decimal expansion of √2, show that √2 is between 1and 2, then between 1.4 and 1.5, and explain how to continue on to get better approximations.

• Workwithradicalsandintegerexponents.

8.EE.1Knowandapplythepropertiesofintegerexponentstogenerate equivalentnumericalexpressions. For example, 32 x 3–5 = 3–3 = 1/33 = 1/27.

8.EE.2Usesquarerootandcuberootsymbolstorepresentsolutions toequationsoftheformx2 =pandx3 =p,where p isapositiverational number.Evaluatesquarerootsofsmallperfectsquaresandcuberoots ofsmallperfectcubes.Knowthat √2isirrational.

8.EE.3Usenumbersexpressedintheformofasingledigittimesan integerpowerof10toestimateverylargeorverysmallquantities, andtoexpresshowmanytimesasmuchoneisthantheother. For example, estimate the population of the United States as 3 x 108 and the population of the world as 7 x 109, and determine that the world population is more than 20 times larger.

8.EE.4Performoperationswithnumbersexpressedinscientific notation,includingproblemswherebothdecimalandscientificnotation areused.Usescientificnotationandchooseunitsofappropriatesize formeasurementsofverylargeorverysmallquantities(e.g.,use millimetersperyearforseafloorspreading).Interpretscientificnotation thathasbeengeneratedbytechnology.

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Unit2:ProportionalityandLinearrelationships

Studentsuselinearequationsandsystemsoflinearequationstorepresent,analyze,andsolveavarietyofproblems. Studentsrecognizeequationsforproportions(y/x = m or y = mx)asspeciallinearequations(y = mx + b),understandingthattheconstantofproportionality(m)istheslope,andthegraphsarelinesthroughtheorigin.They understandthattheslope(m)ofalineisaconstantrateofchange,sothatiftheinputor x-coordinatechangesby anamount A,theoutputor y-coordinatechangesbytheamount m×A.Studentsstrategicallychooseandefficiently implementprocedurestosolvelinearequationsinonevariable,understandingthatwhentheyusethepropertiesof equalityandtheconceptoflogicalequivalence,theymaintainthesolutionsoftheoriginalequation.



• Analyzeproportionalrelationshipsand usethemtosolvereal-worldandmathematicalproblems.

• Usepropertiesofoperationstogenerateequivalentexpressions.

7.RP.1 Compute unit rates associated with ratios of fractions, including ratios of lengths, areas and other quantities measured in like or different units. For example, if a person walks 1/2 mile in each 1/4 hour, compute the unit rate as the complex fraction 1/2/1/4

miles per hour, equivalently 2 miles per hour.

7.RP.2 Recognize and represent proportional relationships between quantities.

a. Decidewhethertwoquantitiesareinaproportionalrelationship, e.g.,bytestingforequivalentratiosinatableorgraphingona coordinateplaneandobservingwhetherthegraphisastraight linethroughtheorigin.

b. Identifytheconstantofproportionality(unitrate)intables, graphs,equations,diagrams,andverbaldescriptionsofproportionalrelationships.

c. Representproportionalrelationshipsbyequations. For example, if total cost t is proportional to the number n of items purchased at a constant price p, the relationship between the total cost and the number of items can be expressed as t = pn.

d. Explainwhatapoint(x,y)onthegraphofaproportionalrelationshipmeansintermsofthesituation,withspecialattentionto thepoints(0,0)and(1,r)whereristheunitrate.

7.RP.3 Use proportional relationships to solve multi-step ratio and percent problems. Examples: simple interest, tax, markups and markdowns, gratuities and commissions, fees, percent increase and decrease, percent error.

7.EE.1Applypropertiesofoperationsasstrategiestoadd,subtract, factor,andexpandlinearexpressionswithrationalcoefficients.

7.EE.2Understandthatrewritinganexpressionindifferentformsina problemcontextcanshedlightontheproblemandhowthequantities initarerelated. For example, a + 0.05a = 1.05a means that “increase by 5%” is the same as “multiply by 1.05.”

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• Solvereal-lifeandmathematicalproblemsusingnumericalandalgebraic expressionsandequations.

• Understandtheconnectionsbetween proportionalrelationships,lines,and linearequations.

7.EE.3Solvemulti-stepreal-lifeandmathematicalproblemsposedwith positiveandnegativerationalnumbersinanyform(wholenumbers, fractions,anddecimals),usingtoolsstrategically.Applypropertiesof operationstocalculatewithnumbersinanyform;convertbetween formsasappropriate;andassessthereasonablenessofanswersusing mentalcomputationandestimationstrategies. For example: If a woman making $25 an hour gets a 10% raise, she will make an additional 1/10 of her salary an hour, or $2.50, for a new salary of $27.50. If you want to place a towel bar 9 3/4 inches long in the center of a door that is 27 1/2 inches wide, you will need to place the bar about 9 inches from each edge; this estimate can be used as a check on the exact computation.

7.EE.4 Usevariablestorepresentquantitiesinareal-worldor mathematicalproblem,andconstructsimpleequationsandinequalities tosolveproblemsbyreasoningaboutthequantities.

a. Solvewordproblemsleadingtoequationsoftheform px + q = r and p(x + q) = r,where p, q,and r arespecificrationalnumbers. Solveequationsoftheseformsfluently.Compareanalgebraic solutiontoanarithmeticsolution,identifyingthesequenceofthe operationsusedineachapproach. For example, the perimeter of a rectangle is 54 cm. Its length is 6 cm. What is its width?

b. Solvewordproblemsleadingtoinequalitiesoftheform px + q > r or px + q < r,where p, q,and r arespecificrationalnumbers. Graphthesolutionsetoftheinequalityandinterpretitinthe contextoftheproblem. For example: As a salesperson, you are paid $50 per week plus $3 per sale. This week you want your pay to be at least $100. Write an inequality for the number of sales you need to make, and describe the solutions.

8.EE.5Graphproportionalrelationships,interpretingtheunitrateasthe slopeofthegraph.Comparetwodifferentproportionalrelationships representedindifferentways. For example, compare a distance-time graph to a distance-time equation to determine which of two moving objects has greater speed.

8.EE.6Usesimilartrianglestoexplainwhytheslope m isthesame betweenanytwodistinctpointsonanon-verticallineinthecoordinate plane;derivetheequation y = mx foralinethroughtheoriginandthe equation y = mx + b foralineinterceptingtheverticalaxisat b.

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8.EE.7Solvelinearequationsinonevariable.

a. Giveexamplesoflinearequationsinonevariablewithonesolution,infinitelymanysolutions,ornosolutions.Showwhichof thesepossibilitiesisthecasebysuccessivelytransformingthe givenequationintosimplerforms,untilanequivalentequationof theform x = a, a = a,or a = b results(where a and b aredifferent numbers).

b. Solvelinearequationswithrationalnumbercoefficients,includingequationswhosesolutionsrequireexpandingexpressions usingthedistributivepropertyandcollectingliketerms.


Unit3:IntroductiontosamplingandInference

Studentsbuildontheirpreviousworkwithsingledatadistributionstocomparetwodatadistributionsandaddress questionsaboutdifferencesbetweenpopulations.Theybegininformalworkwithrandomsamplingtogeneratedata setsandlearnabouttheimportanceofrepresentativesamplesfordrawinginferences.



• Userandomsamplingtodrawinferencesaboutapopulation.

7.SP.1Understandthatstatisticscanbeusedtogaininformationabout apopulationbyexaminingasampleofthepopulation;generalizations aboutapopulationfromasamplearevalidonlyifthesampleis representativeofthatpopulation.Understandthatrandomsampling tendstoproducerepresentativesamplesandsupportvalidinferences.

7.SP.2Usedatafromarandomsampletodrawinferencesabouta populationwithanunknowncharacteristicofinterest.Generatemultiple samples(orsimulatedsamples)ofthesamesizetogaugethevariation inestimatesorpredictions. For example, estimate the mean word length in a book by randomly sampling words from the book; predict the winner of a school election based on randomly sampled survey data. Gauge how far off the estimate or prediction might be.

• Drawinformalcomparativeinferences abouttwopopulations.

7.SP.3Informallyassessthedegreeofvisualoverlapoftwonumerical datadistributionswithsimilarvariabilities,measuringthedifference betweenthecentersbyexpressingitasamultipleofameasureof variability. For example, the mean height of players on the basketball team is 10 cm greater than the mean height of players on the soccer team, about twice the variability (mean absolute deviation) on either team; on a dot plot, the separation between the two distributions of heights is noticeable.

7.SP.4Usemeasuresofcenterandmeasuresofvariabilityfornumerical datafromrandomsamplestodrawinformalcomparativeinferences abouttwopopulations. For example, decide whether the words in a chapter of a seventh-grade science book are generally longer than the words in a chapter of a fourth-grade science book.

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• Investigatechanceprocessesand develop,use,andevaluateprobability models.

7.SP.5Understandthattheprobabilityofachanceeventisanumber between0and1thatexpressesthelikelihoodoftheeventoccurring. Largernumbersindicategreaterlikelihood.Aprobabilitynear0 indicatesanunlikelyevent,aprobabilityaround 1/2 indicatesanevent thatisneitherunlikelynorlikely,andaprobabilitynear1indicatesa likelyevent.

7.SP.6Approximatetheprobabilityofachanceeventbycollecting dataonthechanceprocessthatproducesitandobservingitslong-run relativefrequency,andpredicttheapproximaterelativefrequencygiven theprobability. For example, when rolling a number cube 600 times, predict that a 3 or 6 would be rolled roughly 200 times, but probably not exactly 200 times.

7.SP.7Developaprobabilitymodelanduseittofindprobabilitiesof events.Compareprobabilitiesfromamodeltoobservedfrequencies;if theagreementisnotgood,explainpossiblesourcesofthediscrepancy.

a. Developauniformprobabilitymodelbyassigningequalprobabilitytoalloutcomes,andusethemodeltodetermineprobabilities ofevents. For example, if a student is selected at random from a class, find the probability that Jane will be selected and the probability that a girl will be selected.

b. Developaprobabilitymodel(whichmaynotbeuniform)byobservingfrequenciesindatageneratedfromachanceprocess. For example, find the approximate probability that a spinning penny will land heads up or that a tossed paper cup will land open-end down. Do the outcomes for the spinning penny appear to be equally likely based on the observed frequencies?

7.SP.8Findprobabilitiesofcompoundeventsusingorganizedlists, tables,treediagrams,andsimulation.

a. Understandthat,justaswithsimpleevents,theprobabilityofa compoundeventisthefractionofoutcomesinthesamplespace forwhichthecompoundeventoccurs.

b. Representsamplespacesforcompoundeventsusingmethods suchasorganizedlists,tablesandtreediagrams.Foranevent describedineverydaylanguage(e.g.,“rollingdoublesixes”), identifytheoutcomesinthesamplespacewhichcomposethe event.

c. Designanduseasimulationtogeneratefrequenciesforcompoundevents. For example, use random digits as a simulation tool to approximate the answer to the question: If 40% of donors have type A blood, what is the probability that it will take at least 4 donors to find one with type A blood?

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Unit4:creating,comparing,andanalyzingGeometricFigures

StudentscontinuetheirworkwithareafromGrade6,solvingproblemsinvolvingtheareaandcircumferenceofa circle and surface area of three-dimensional objects. In preparation for work on congruence and similarity, they reason aboutrelationshipsamongtwo-dimensionalfiguresusingscaledrawingsandinformalgeometricconstructions,and theygainfamiliaritywiththerelationshipsbetweenanglesformedbyintersectinglines.Studentsworkwiththreedimensionalfigures,relatingthemtotwo-dimensionalfiguresbyexaminingcross-sections.Theysolvereal-worldand mathematicalproblemsinvolvingarea,surfacearea,andvolumeoftwo-andthree-dimensionalobjectscomposedof triangles,quadrilaterals,polygons,cubesandrightprisms.Studentsuseideasaboutdistanceandangles,howthey behaveundertranslations,rotations,reflections,anddilations,andideasaboutcongruenceandsimilaritytodescribe andanalyzetwo-dimensionalfiguresandtosolveproblems.Studentsshowthatthesumoftheanglesinatriangleis theangleformedbyastraightline,andthatvariousconfigurationsoflinesgiverisetosimilartrianglesbecauseofthe anglescreatedwhenatransversalcutsparallellines.Studentscompletetheirworkonvolumebysolvingproblems involvingcones,cylinders,andspheres.



• Draw,construct,anddescribegeometricalfiguresanddescribetherelationshipsbetweenthem.

7.G.1Solveproblemsinvolvingscaledrawingsofgeometricfigures, includingcomputingactuallengthsandareasfromascaledrawingand reproducingascaledrawingatadifferentscale.

7.G.2Draw(freehand,withrulerandprotractor,andwithtechnology) geometric shapes with given conditions. Focus on constructing triangles fromthreemeasuresofanglesorsides,noticingwhentheconditions determineauniquetriangle,morethanonetriangle,ornotriangle.

7.G.3Describethetwo-dimensionalfiguresthatresultfromslicingthreedimensionalfigures,asinplanesectionsofrightrectangularprismsand rightrectangularpyramids.

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• Solvereal-lifeandmathematicalproblemsinvolvinganglemeasure,area, surfacearea,andvolume.

7.G.4Knowtheformulasfortheareaandcircumferenceofacircle andusethemtosolveproblems;giveaninformalderivationofthe relationshipbetweenthecircumferenceandareaofacircle.

7.G.5Usefactsaboutsupplementary,complementary,vertical,and adjacentanglesinamulti-stepproblemtowriteandsolvesimple equationsforanunknownangleinafigure.

7.G.6Solvereal-worldandmathematicalproblemsinvolvingarea, volumeandsurfaceareaoftwo-andthree-dimensionalobjects composedoftriangles,quadrilaterals,polygons,cubes,andrightprisms.




• Understandcongruenceandsimilarity usingphysicalmodels,transparencies, orgeometrysoftware.

• Solvereal-worldandmathematical probleminvolvingvolumeofcylinders, cones,andspheres.

8.G.1Verifyexperimentallythepropertiesofrotations,reflections,and translations:

a. Linesaretakentolines,andlinesegmentstolinesegmentsof thesamelength.

b. Anglesaretakentoanglesofthesamemeasure.

c. Parallellinesaretakentoparallellines.

8.G.2Understandthatatwo-dimensionalfigureiscongruenttoanother ifthesecondcanbeobtainedfromthefirstbyasequenceofrotations, reflections,andtranslations;giventwocongruentfigures,describea sequencethatexhibitsthecongruencebetweenthem.

8.G.3Describetheeffectofdilations,translations,rotations,and reflectionsontwo-dimensionalfiguresusingcoordinates.

8.G.4Understandthatatwo-dimensionalfigureissimilartoanotherif thesecondcanbeobtainedfromthefirstbyasequenceofrotations, reflections,translations,anddilations;giventwosimilartwo-dimensional figures,describeasequencethatexhibitsthesimilaritybetweenthem.

8.G.5Useinformalargumentstoestablishfactsabouttheanglesum andexteriorangleoftriangles,abouttheanglescreatedwhenparallel linesarecutbyatransversal,andtheangle-anglecriterionforsimilarity oftriangles. For example, arrange three copies of the same triangle so that the sum of the three angles appears to form a line, and give an argument in terms of transversals why this is so.

8.G.9Knowtheformulasforthevolumesofcones,cylinders,and spheresandusethemtosolvereal-worldandmathematicalproblems.

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8th GrademathematicsI Thefundamentalpurposeof8th GradeMathematicsIistoformalizeandextendthemathematicsthatstudents learnedthroughtheendofseventhgrade.Contentinthiscourseisgroupedintosixcriticalareas,orunits.The unitsofstudydeepenandextendunderstandingoflinearandexponentialrelationshipsbycontrastingthemwith eachotherandbyapplyinglinearmodelstodatathatexhibitalineartrend.8th GradeMathematics1includesan explorationoftheroleofrigidmotionsincongruenceandsimilarity.ThePythagoreantheoremisintroduced,and studentsexaminevolumerelationshipsofcones,cylinders,andspheres.8th GradeMathematics1usesproperties andtheoremsinvolvingcongruentfigurestodeepenandextendunderstandingofgeometricknowledgefrom priorgrades.Thefinalunitinthecoursetiestogetherthealgebraicandgeometricideasstudied.TheMathematical PracticeStandardsapplythroughouteachcourseand,togetherwiththecontentstandards,prescribethatstudents experiencemathematicsasacoherent,useful,andlogicalsubjectthatmakesuseoftheirabilitytomakesenseof problemsituations.

ThiscoursediffersfromMathematicsIinthatitcontainscontentfrom8th grade.Whilecoherenceisretained,in thatitlogicallybuildsfromAccelerated7th Grade,theadditionalcontentwhencomparedtothehighschoolcourse demandsafasterpaceforinstructionandlearning.

CriticalArea1:Workwithquantitiesandrates,includingsimplelinearexpressionsandequationsformsthefoundation forthisunit.Studentsuseunitstorepresentproblemsalgebraicallyandgraphically,andtoguidethesolutionof problems.Studentexperiencewithquantityprovidesafoundationforthestudyofexpressions,equations,and functions.

CriticalArea2:Buildingonearlierworkwithlinearrelationships,studentslearnfunctionnotationandlanguagefor describingcharacteristicsoffunctions,includingtheconceptsofdomainandrange.Theyexploremanyexamples offunctions,includingsequences;theyinterpretfunctionsgivengraphically,numerically,symbolically,andverbally, translatebetweenrepresentations,andunderstandthelimitationsofvariousrepresentations.Theyworkwith functionsgivenbygraphsandtables,keepinginmindthatdependinguponthecontext,theserepresentationsare likelytobeapproximateandincomplete.Theirworkincludesfunctionsthatcanbedescribedorapproximatedby formulasaswellasthosethatcannot.Whenfunctionsdescriberelationshipsbetweenquantitiesarisingfroma context,studentsreasonwiththeunitsinwhichthosequantitiesaremeasured.Studentsbuildonandinformally extendtheirunderstandingofintegralexponentstoconsiderexponentialfunctions.Theycompareandcontrast linearandexponentialfunctions,distinguishingbetweenadditiveandmultiplicativechange.Theyinterpretarithmetic sequencesaslinearfunctionsandgeometricsequencesasexponentialfunctions.

CriticalArea3:Thisunitbuildsonearlierexperiencesbyaskingstudentstoanalyzeandexplaintheprocessof solvinganequationandtojustifytheprocessusedinsolvingasystemofequations.Studentsdevelopfluency writing,interpreting,andtranslatingbetweenvariousformsoflinearequationsandinequalities,andusingthemto solveproblems.Theymasterthesolutionoflinearequationsandapplyrelatedsolutiontechniquesandthelawsof exponentstothecreationandsolutionofsimpleexponentialequations.Studentsexploresystemsofequationsand inequalities,andtheyfindandinterprettheirsolutions.

CriticalArea4:Thisunitbuildsuponpriorstudents’priorexperienceswithdata,providingstudentswithmore formalmeansofassessinghowamodelfitsdata.Studentsuseregressiontechniquestodescribeapproximately linearrelationshipsbetweenquantities.Theyusegraphicalrepresentationsandknowledgeofthecontexttomake judgmentsabouttheappropriatenessoflinearmodels.Withlinearmodels,theylookatresidualstoanalyzethe goodnessoffit.

CriticalArea5:Inpreviousgrades,studentswereaskedtodrawtrianglesbasedongivenmeasurements.Theyalso havepriorexperiencewithrigidmotions:translations,reflections,androtationsandhaveusedthesetodevelop notionsaboutwhatitmeansfortwoobjectstobecongruent.Inthisunit,studentsestablishtrianglecongruence criteria,basedonanalysesofrigidmotionsandformalconstructions.Theysolveproblemsabouttriangles, quadrilaterals,andotherpolygons.Theyapplyreasoningtocompletegeometricconstructionsandexplainwhythey work.

CriticalArea6:BuildingontheirworkwiththePythagoreanTheoremtofinddistances,studentsusearectangular coordinatesystemtoverifygeometricrelationships,includingpropertiesofspecialtrianglesandquadrilateralsand slopesofparallelandperpendicularlines.

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Standards

• Reason quantitatively and use units to solve problems. Unit1

• Interpret the structure of expressions. Quantities

RelationshipsBetween


Unit2



• Define, evaluate, and compare functions.


• Use functions to model relationships between quantities.











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Unit3

Reasoningwith Equations

Unit4

DescriptiveStatistics






• Investigate patterns of associate in bivariate data.

• Summarize, represent, and interpret data on two categorical and quantitative variables.



Attendtoprecision.



• Experiment with transformations in the plane. Unit5 • Understand congruence in terms of rigid motions.

Congruence,Proof,and • Make geometric constructions. Constructions • Understand and apply the Pythagorean theorem.

Unit6 • Use coordinates to prove simple geometric theorems algebraically. ConnectingAlgebra

andGeometrythrough Coordinates


†Notethatsolvingequationsandsystemsofequationsfollowsastudyoffunctionsinthiscourse.Toexamineequationsbeforefunctions,thisunitcouldbemergedwithUnit1.


Unit1:relationshipsBetweenQuantities

Workwithquantitiesandrates,includingsimplelinearexpressionsandequationsformsthefoundationforthisunit. Studentsuseunitstorepresentproblemsalgebraicallyandgraphically,andtoguidethesolutionofproblems.Studentexperiencewithquantityprovidesafoundationforthestudyofexpressions,equations,andfunctions.

Unit1:RelationshipsbetweenQuantities












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Limit A.CED.1 and A.CED.2 to linear and exponential equations, and, in the case of exponential equations, limit to situations requiring evaluation of exponential functions at integer inputs. Limit A.CED.3 to linear equations and inequalities. Limit A.CED.4 to formulas which are linear in the variables of interest.






Unit2:LinearandexponentialFunctions

Building on earlier work with linear relationships, students learn function notation and language for describing characteristicsoffunctions,includingtheconceptsofdomainandrange.Theyexploremanyexamplesoffunctions,includingsequences;theyinterpretfunctionsgivengraphically,numerically,symbolically,andverbally,translatebetween representations,andunderstandthelimitationsofvariousrepresentations.Theyworkwithfunctionsgivenbygraphs andtables,keepinginmindthatdependinguponthecontext,theserepresentationsarelikelytobeapproximateand incomplete.Theirworkincludesfunctionsthatcanbedescribedorapproximatedbyformulasaswellasthosethat cannot.Whenfunctionsdescriberelationshipsbetweenquantitiesarisingfromacontext,studentsreasonwiththe unitsinwhichthosequantitiesaremeasured.Studentsbuildonandinformallyextendtheirunderstandingofintegral exponentstoconsiderexponentialfunctions.Theycompareandcontrastlinearandexponentialfunctions,distinguishingbetweenadditiveandmultiplicativechange.Theyinterpretarithmeticsequencesaslinearfunctionsand geometricsequencesasexponentialfunctions.






A.REI.11Explainwhythex-coordinatesofthepointswherethegraphs oftheequations y = f(x) and y = g(x) intersectarethesolutionsof theequation f(x) = g(x);findthesolutionsapproximately,e.g.,using technologytographthefunctions,maketablesofvalues,orfind successiveapproximations.Includecaseswhere f(x) and/or g(x) are linear,polynomial,rational,absolutevalue,exponential,andlogarithmic functions.★


• Define,evaluate,andcomparefunctions.

While this content is likely subsumed by F.IF.1-3 and F.IF.7a, it could be used for scaffolding instruction to the more sophisticated content found there.

8.F.1Understandthatafunctionisarulethatassignstoeachinput exactlyoneoutput.Thegraphofafunctionisthesetoforderedpairs consistingofaninputandthecorrespondingoutput.

8.F.2Comparepropertiesoftwofunctionseachrepresentedina differentway(algebraically,graphically,numericallyintables,orby verbaldescriptions). For example, given a linear function represented by a table of values and a linear function represented by an algebraic expression, determine which function has the greater rate of change.

8.F.3Interprettheequation y = mx + b asdefiningalinearfunction, whosegraphisastraightline;giveexamplesoffunctionsthatarenot linear. For example, the function A = s2 giving the area of a square as a function of its side length is not linear because its graph contains the points (1,1), (2,4) and (3,9), which are not on a straight line.

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Students should experience a variety of types of situations modeled by functions. Detailed analysis of any particular class of function at this stage is not advised. Students should apply these concepts throughout their future mathematics courses.

Constrain examples to linear functions and exponential functions having integral domains. In F.IF.3, draw connection to F.BF.2, which requires students to write arithmetic and geometric sequences.







• Usefunctionstomodelrelationships betweenquantities.

While this content is likely subsumed by F.IF.4 and F.BF.1a, it could be used for scaffolding instruction to the more sophisticated content found there.

8.F.4Constructafunctiontomodelalinearrelationshipbetweentwo quantities.Determinetherateofchangeandinitialvalueofthefunction fromadescriptionofarelationshiporfromtwo(x, y)values,including readingthesefromatableorfromagraph.Interprettherateofchange andinitialvalueofalinearfunctionintermsofthesituationitmodels, andintermsofitsgraphoratableofvalues.

8.F.5Describequalitativelythefunctionalrelationshipbetweentwo quantitiesbyanalyzingagraph(e.g.,wherethefunctionisincreasing ordecreasing,linearornonlinear).Sketchagraphthatexhibitsthe qualitativefeaturesofafunctionthathasbeendescribedverbally.


For F.IF.4 and 5, focus on linear and exponential functions. For F.IF.6, focus on linear functions and exponential functions whose domain is a subset of the integers. Mathematics II and III will address other types of functions.

N.RN.1 and N.RN. 2 will need to be referenced here before discussing exponential functions with continuous domains.





For F.IF.7a, 7e, and 9 focus on linear and exponential functions. Include comparisons of two functions presented algebraically. For example, compare the growth of two linear functions, or two exponential functions such as y = 3n and y = 100 x 2n.





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Limit F.BF.1a, 1b, and 2 to linear and exponential functions. In F.BF.2, connect arithmetic sequences to linear functions and connect geometric sequences to exponential functions in F.BF.2.













For F.LE.3, limit to comparisons to those between exponential and linear models.







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Limit exponential, with exponential functions to those of the form f(x) = bx + k .


Unit3:reasoningwithequations

Thisunitbuildsonearlierexperiencesbyaskingstudentstoanalyzeandexplaintheprocessofsolvinganequation andtojustifytheprocessusedinsolvingasystemofequations.Studentsdevelopfluencywriting,interpreting,and translatingbetweenvariousformsoflinearequationsandinequalities,andusingthemtosolveproblems.Theymaster thesolutionoflinearequationsandapplyrelatedsolutiontechniquesandthelawsofexponentstothecreationand solutionofsimpleexponentialequations.Studentsexploresystemsofequationsandinequalities,andtheyfindand interprettheirsolutions.

Unit3:ReasoningwithEquations



Students should focus on and master A.REI.1 for linear equations and be able to extend and apply their reasoning to other types of equations in future courses. Students will solve exponential equations in Mathematics III.



Extend earlier work with solving linear equations to solving linear inequalities in one variable and to solving literal equations that are linear in the variable being solved for. Include simple exponential equations that rely only on application of the laws of exponents, such as 5x = 125 or 2x =1/16 .

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While this content is likely subsumed by A.REI.3, 5, and 6, it could be used for scaffolding instruction to the more sophisticated content found there.

8.EE.8Analyzeandsolvepairsofsimultaneouslinearequations.

a. Understandthatsolutionstoasystemoftwolinearequationsin two variables correspond to points of intersection of their graphs, becausepointsofintersectionsatisfybothequationssimultaneously.

b. Solvesystemsoftwolinearequationsintwovariablesalgebra-ically,andestimatesolutionsbygraphingtheequations.Solve simplecasesbyinspection. For example, 3x + 2y = 5 and 3x + 2y = 6 have no solution because 3x + 2y cannot simultaneously be 5 and 6.

c. Solvereal-worldandmathematicalproblemsleadingtotwolinearequationsintwovariables. For example, given coordinates for two pairs of points, determine whether the line through the first pair of points intersects the line through the second pair.


Include cases where two equations describe the same line (yielding infinitely many solutions) and cases where two equations describe parallel lines (yielding no solution).

A.REI.5Provethat,givenasystemoftwoequationsintwovariables, replacingoneequationbythesumofthatequationandamultipleof theotherproducesasystemwiththesamesolutions.




Studentsuseregressiontechniquestodescriberelationshipsbetweenquantities.Theyusegraphicalrepresentations andknowledgeofthecontexttomakejudgmentsabouttheappropriatenessoflinearmodels.Withlinearmodels, theylookatresidualstoanalyzethegoodnessoffit.








• Investigatepatternsofassociationin bivariatedata.

While this content is likely subsumed by S.ID.6-9, it could be used for scaffolding instruction to the more sophisticated content found there.

8.SP.1Constructandinterpretscatterplotsforbivariatemeasurement datatoinvestigatepatternsofassociationbetweentwoquantities. Describepatternssuchasclustering,outliers,positiveornegative association,linearassociation,andnonlinearassociation.

8.SP.2Knowthatstraightlinesarewidelyusedtomodelrelationships betweentwoquantitativevariables.Forscatterplotsthatsuggesta linearassociation,informallyfitastraightline,andinformallyassessthe modelfitbyjudgingtheclosenessofthedatapointstotheline.

8.SP.3Usetheequationofalinearmodeltosolveproblemsinthe contextofbivariatemeasurementdata,interpretingtheslopeand intercept. For example, in a linear model for a biology experiment, interpret a slope of 1.5 cm/hr as meaning that an additional hour of sunlight each day is associated with an additional 1.5 cm in mature plant height.

8.SP.4Understandthatpatternsofassociationcanalsobeseenin bivariatecategoricaldatabydisplayingfrequenciesandrelative frequenciesinatwo-waytable.Constructandinterpretatwo-waytable summarizingdataontwocategoricalvariablescollectedfromthesame subjects.Userelativefrequenciescalculatedforrowsorcolumnsto describepossibleassociationbetweenthetwovariables. For example, collect data from students in your class on whether or not they have a curfew on school nights and whether or not they have assigned chores at home. Is there evidence that those who have a curfew also tend to have chores?

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S.ID.6b should be focused on situations for which linear models are appropriate.










Build on students’ work with linear relationship and introduce the correlation coefficient. The focus here is on the computation and interpretation of the correlation coefficient as a measure how well the data fit the relationship. The important distinction between a statistical relationship and a cause-and-effect relationship arises in S.ID.9.




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Inpreviousgrades,studentswereaskedtodrawtrianglesbasedongivenmeasurements.Theyalsohavepriorexperiencewithrigidmotions:translations,reflections,androtationsandhaveusedthesetodevelopnotionsaboutwhatit meansfortwoobjectstobecongruent.Inthisunit,studentsestablishtrianglecongruencecriteria,basedonanalyses ofrigidmotionsandformalconstructions.Theysolveproblemsabouttriangles,quadrilaterals,andotherpolygons. Theyapplyreasoningtocompletegeometricconstructionsandexplainwhytheywork.

Unit5: Congruence,Proof,andConstructions





G.CO.2Representtransformationsintheplaneusing,e.g., transparenciesandgeometrysoftware;describetransformationsas functionsthattakepointsintheplaneasinputsandgiveotherpoints asoutputs.Comparetransformationsthatpreservedistanceandangle tothosethatdonot(e.g.,translationversushorizontalstretch).



G.CO.5Givenageometricfigureandarotation,reflection,or translation,drawthetransformedfigureusing,e.g.,graphpaper,tracing paper,orgeometrysoftware.Specifyasequenceoftransformations thatwillcarryagivenfigureontoanother.



G.CO.6Usegeometricdescriptionsofrigidmotionstotransform figuresandtopredicttheeffectofagivenrigidmotiononagiven figure;giventwofigures,usethedefinitionofcongruenceintermsof rigidmotionstodecideiftheyarecongruent.

G.CO.7Usethedefinitionofcongruenceintermsofrigidmotionsto show that two triangles are congruent if and only if corresponding pairs ofsidesandcorrespondingpairsofanglesarecongruent.


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Build on prior student experience with simple constructions. Emphasize the ability to formalize and defend how these constructions result in the desired objects.




• UnderstandandapplythePythagorean theorem.

Discuss applications of the Pythagorean theorem and its connections to radicals, rational exponents, and irrational numbers.

8.G.6ExplainaproofofthePythagoreantheoremanditsconverse.

8.G.7ApplythePythagoreantheoremtodetermineunknownside lengthsinrighttrianglesinreal-worldandmathematicalproblemsin twoandthreedimensions.

8.G.8ApplythePythagoreantheoremtofindthedistancebetweentwo pointsinacoordinatesystem.



BuildingontheirworkwiththePythagoreanTheoremtofinddistances,studentsusearectangularcoordinatesystem toverifygeometricrelationships,includingpropertiesofspecialtrianglesandquadrilateralsandslopesofparalleland perpendicularlines.

Unit6: ConnectingAlgebraandGeometryThroughCoordinates



Reasoning with triangles in this unit is limited to right triangles; e.g., derive the equation for a line through two points using similar right triangles.

Relate work on parallel lines in G.GPE.5 to work on A.REI.5 in Mathematics I involving systems of equations having no solution or infinitely many solutions.


G.GPE.4Usecoordinatestoprovesimplegeometrictheorems algebraically.Forexample,proveordisprovethatafiguredefined byfourgivenpointsinthecoordinateplaneisarectangle;proveor disprovethatthepoint(1, √3)liesonthecirclecenteredattheorigin andcontainingthepoint(0,2).

G.GPE.5Provetheslopecriteriaforparallelandperpendicularlines; usethemtosolvegeometricproblems(e.g.,findtheequationofaline parallelorperpendiculartoagivenlinethatpassesthroughagiven point).


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additionalmathematicscourses The“collegeandcareerready”linehasbeenbasedonevidencefromanumberofsources,includinginternational benchmarking,surveysofpostsecondaryfacultyandemployers,reviewofstatestandards,andexpertopinion. Studentsmeetingthesestandardsshouldbewell-preparedforintroductorymathematicscoursesin2-and4-year colleges.Still,therearepersuasivereasonsforstudentstocontinueontotakeafourthmathematicscourseinhigh school.

ResearchconsistentlyfindsthattakingmathematicsabovetheAlgebraIIlevelhighlycorrespondstomanymeasures ofstudentsuccess.InhisgroundbreakingreportAnswersintheToolbox,CliffordAdelmanfoundthatthestrongest predictorofpostsecondarysuccessisthehighestlevelofmathematicscompleted(ExecutiveSummary).ACThas foundthattakingmoremathematicscoursescorrelateswithgreatersuccessontheircollegeentranceexamination. Ofstudentstaking(AlgebraI,GeometryandAlgebraIIandnoothermathematicscourses),onlythirteenpercentof thosestudentsmetthebenchmarkforreadinessforcollegealgebra.Oneadditionalmathematicscoursegreatlyincreasedthelikelihoodthatastudentwouldreachthatbenchmark,andthree-fourthsofstudentstakingCalculusmet thebenchmark(ACTb13).

Studentsgoingthroughthepathwaysshouldbeencouragedtoselectfromarangeofhighqualitymathematics options.STEM-intendingstudentsshouldbestronglyencouragedtotakePrecalculusandCalculus(andperhapsa computersciencecourse).Astudentinterestedinpsychologymaybenefitgreatlyfromacourseindiscretemathematics,followedbyAPStatistics.Astudentinterestedinstartingabusinessafterhighschoolcoulduseknowledge andskillsgleanedfromacourseonmathematicaldecision-making.Mathematically-inclinedstudentscan,atthislevel, doubleuponcourses—astudenttakingcollegecalculusandcollegestatisticswouldbewell-preparedforalmostany postsecondarycareer.

Takentogether,thereiscompellingrationaleforurgingstudentstocontinuetheirmathematicaleducationthroughouthighschool,allowingstudentsseveralrichoptionsoncetheyhavedemonstratedmasteryofcorecontent.The Pathwaysdescribepossiblecoursesforthefirstthreeyearsofhighschool.OtherarrangementsoftheCommonCore StateStandardsforhighschoolarepossible.Standardsmarkedwitha(+)mayappeareitherincoursesrequiredfor allstudents,orinlatercourses.Inparticular,the(+)standardscanformthestartingpointforfourthyearcoursesin PrecalculusandinProbabilityandStatistics.Otherfourthyearcourses,forexampleCalculus,Modeling,orDiscrete Mathematicsarepossible.

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references Achieve. (2010). Closing the expectations gap 2010. Washington, DC: Achieve.

ACT & The Education Trust. (2004). On course for success: A close look at selected high school courses that prepare all students for college. Iowa City, IA: Authors.

ACT. (2008). The Forgotten middle: Ensuring that all students are on target for college and career readiness before high school. Iowa City, IA: ACT.

Adelman, C. (1999). Answers in the toolbox. Washington, DC: US Department of Education. Retrieved from U.S. De- partment of Education website: http://www2.ed.gov/pubs/Toolbox/index.html.

Burris, C., Heubert, J., & Levin, H. (2004). Math acceleration for all. Improving Achievement in Math and Science, 61(5), 68-71.

Retrieved from [Link removed Jun 2, 2017].

College Board. (2009). AP Examination volume changes (1999-2009). New York, NY: College Board. Retrieved from http://professionals.collegeboard.com/profdownload/exam-volume-change-09.pdf.

Cooney, S., & Bottoms, G. (2002). Middle grades to high school: Mending a weak link. Atlanta, GA: Southern Re- gional Education Board. Retrieved

from the Southern Regional Education Board website: http://publications.sreb.org/2002/02V08_Middle_Grades_To_HS.pdf

Ford, R. (2006, January). High school profiles of mathematically unprepared college freshmen. Paper present at the fourth annual International Conference on Education, Honolulu, HI. Retrieved from http://www.csuchico.edu/eapapp/ PDFfiles/CollegeReadiness3.htm.

Ginsburg, A., & Leinwand, S. (2009). Informing Grades 1-6 Mathematics Standards Development: What Can Be Learned From High-Performing Hong Kong, Korea and Singapore? Washington, DC: American Institutes for Research.

Kennelly, L., & Monrad, M. (Eds). (2007). Easing the Transition to High School: Research and Best Practices Designed to Support High School Learning. Washington, DC: National High School Center at the American Institutes for Re- search.

Schmidt, B., Houang, R., & Cogan, L. (2002). A coherent curriculum: The case of mathematics. American Educator, 26(2), 1-18.

Wallinger, Linda. (n.d.). Virginia’s algebra readiness initiative. Richmond, VA: Virginia Department of Education

Washington Office of the Superintendent of Public Instruction. (2008). Guidelines for accelerating students into high school mathematics in grade 8. Retrieved from http://www.k12.wa.us/mathematics /Standards/Compression.pdf

What Works Clearinghouse. (2008). Accelerated middle schools. Washington, DC: Institute of Education Sciences.

Wyatt, W.J. & Wiley, A. (2010). The development of an index of academic rigor for the SAT. (College Board Research Report). New York: The College Board.

Attachment 2

Item 12.C.

May 3-4,2012

Mathematics

SMC

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http://www2.ed.gov/pubs/Toolbox/index.html

http://professionals.collegeboard.com/profdownload/exam-volume-change-09.pdf

http://publications.sreb.org/2002/02V08_Middle_Grades_To_HS.pdf

http://www.csuchico.edu/eapapp

http://www.k12.wa.us/mathematics

http://www.csuchico.edu/eapapp/

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