Math Standards.indb

Common Core State StandardS for

mathematics

Common Core State StandardS for matHematICS

table of Contents

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Grade6

Grade 7

Grade8 52HighSchool—Introduction

HighSchool—NumberandQuantity 58HighSchool—Algebra 62HighSchool—Functions 67HighSchool—Modeling 72HighSchool—Geometry 74HighSchool—StatisticsandProbability 79

Glossary 85Sample of Works Consulted 91

Common Core State StandardS for matHematICSG

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mathematics | Grade 6InGrade6,instructionaltimeshouldfocusonfourcriticalareas:(1)

connectingratioandratetowholenumbermultiplicationanddivision

andusingconceptsofratioandratetosolveproblems;(2)completing

understandingofdivisionoffractionsandextendingthenotionofnumber

tothesystemofrationalnumbers,whichincludesnegativenumbers;

(3)writing,interpreting,andusingexpressionsandequations;and(4)

developingunderstandingofstatisticalthinking.

(1)Studentsusereasoningaboutmultiplicationanddivisiontosolve

ratioandrateproblemsaboutquantities.Byviewingequivalentratios

andratesasderivingfrom,andextending,pairsofrows(orcolumns)in

themultiplicationtable,andbyanalyzingsimpledrawingsthatindicate

therelativesizeofquantities,studentsconnecttheirunderstandingof

multiplicationanddivisionwithratiosandrates.Thusstudentsexpandthe

scopeofproblemsforwhichtheycanusemultiplicationanddivisionto

solveproblems,andtheyconnectratiosandfractions.Studentssolvea

widevarietyofproblemsinvolvingratiosandrates.

(2)Studentsusethemeaningoffractions,themeaningsofmultiplication

anddivision,andtherelationshipbetweenmultiplicationanddivisionto

understandandexplainwhytheproceduresfordividingfractionsmake

sense.Studentsusetheseoperationstosolveproblems.Studentsextend

theirpreviousunderstandingsofnumberandtheorderingofnumbers

tothefullsystemofrationalnumbers,whichincludesnegativerational

numbers,andinparticularnegativeintegers.Theyreasonabouttheorder

andabsolutevalueofrationalnumbersandaboutthelocationofpointsin

allfourquadrantsofthecoordinateplane.

(3)Studentsunderstandtheuseofvariablesinmathematicalexpressions.

Theywriteexpressionsandequationsthatcorrespondtogivensituations,

evaluateexpressions,anduseexpressionsandformulastosolveproblems.

Studentsunderstandthatexpressionsindifferentformscanbeequivalent,

andtheyusethepropertiesofoperationstorewriteexpressionsin

equivalentforms.Studentsknowthatthesolutionsofanequationarethe

valuesofthevariablesthatmaketheequationtrue.Studentsuseproperties

ofoperationsandtheideaofmaintainingtheequalityofbothsidesof

anequationtosolvesimpleone-stepequations.Studentsconstructand

analyzetables,suchastablesofquantitiesthatareinequivalentratios,

andtheyuseequations(suchas3x=y)todescriberelationshipsbetween

quantities.

(4)Buildingonandreinforcingtheirunderstandingofnumber,students

begintodeveloptheirabilitytothinkstatistically.Studentsrecognizethata

datadistributionmaynothaveadefinitecenterandthatdifferentwaysto

measurecenteryielddifferentvalues.Themedianmeasurescenterinthe

sensethatitisroughlythemiddlevalue.Themeanmeasurescenterinthe

sensethatitisthevaluethateachdatapointwouldtakeonifthetotalof

thedatavalueswereredistributedequally,andalsointhesensethatitisa

balancepoint.Studentsrecognizethatameasureofvariability(interquartile

rangeormeanabsolutedeviation)canalsobeusefulforsummarizing

databecausetwoverydifferentsetsofdatacanhavethesamemeanand


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medianyetbedistinguishedbytheirvariability.Studentslearntodescribe

andsummarizenumericaldatasets,identifyingclusters,peaks,gaps,and

symmetry,consideringthecontextinwhichthedatawerecollected.

StudentsinGrade6alsobuildontheirworkwithareainelementary

schoolbyreasoningaboutrelationshipsamongshapestodeterminearea,

surfacearea,andvolume.Theyfindareasofrighttriangles,othertriangles,

andspecialquadrilateralsbydecomposingtheseshapes,rearranging

orremovingpieces,andrelatingtheshapestorectangles.Usingthese

methods,studentsdiscuss,develop,andjustifyformulasforareasof

trianglesandparallelograms.Studentsfindareasofpolygonsandsurface

areasofprismsandpyramidsbydecomposingthemintopieceswhose

areatheycandetermine.Theyreasonaboutrightrectangularprisms

withfractionalsidelengthstoextendformulasforthevolumeofaright

rectangularprismtofractionalsidelengths.Theyprepareforworkon

scaledrawingsandconstructionsinGrade7bydrawingpolygonsinthe

coordinateplane.

Common Core State StandardS for matHematICS

ratios and Proportional relationships

• Understand ratio concepts and use ratio reasoning to solve problems.

the number System

• apply and extend previous understandings of multiplication and division to divide fractions by fractions.

• Compute fluently with multi-digit numbers and find common factors and multiples.

• apply and extend previous understandings of numbers to the system of rational numbers.

expressions and equations

• apply and extend previous understandings of arithmetic to algebraic expressions.

• reason about and solve one-variable equations and inequalities.

• represent and analyze quantitative relationships between dependent and independent variables.

Geometry

• Solve real-world and mathematical problems involving area, surface area, and volume.

Statistics and Probability

• develop understanding of statistical variability.

• Summarize and describe distributions.

mathematical Practices

1. Makesenseofproblemsandpersevereinsolvingthem.

2. Reasonabstractlyandquantitatively.

3. Constructviableargumentsandcritiquethereasoningofothers.

4. Modelwithmathematics.

5. Useappropriatetoolsstrategically.

6. Attendtoprecision.

7. Lookforandmakeuseofstructure.

8. Lookforandexpressregularityinrepeatedreasoning.

Grade 6 overview


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ratios and Proportional relationships 6.rP

Understand ratio concepts and use ratio reasoning to solve problems.

1. Understandtheconceptofaratioanduseratiolanguagetodescribearatiorelationshipbetweentwoquantities.For example, “The ratio of wings to beaks in the bird house at the zoo was 2:1, because for every 2 wings there was 1 beak.” “For every vote candidate A received, candidate C received nearly three votes.”

2. Understandtheconceptofaunitratea/bassociatedwitharatioa:bwithb≠0,anduseratelanguageinthecontextofaratiorelationship.For example, “This recipe has a ratio of 3 cups of flour to 4 cups of sugar, so there is 3/4 cup of flour for each cup of sugar.” “We paid $75 for 15 hamburgers, which is a rate of $5 per hamburger.”1

3. Useratioandratereasoningtosolvereal-worldandmathematicalproblems,e.g.,byreasoningabouttablesofequivalentratios,tapediagrams,doublenumberlinediagrams,orequations.

a. Maketablesofequivalentratiosrelatingquantitieswithwhole-numbermeasurements,findmissingvaluesinthetables,andplotthepairsofvaluesonthecoordinateplane.Usetablestocompareratios.

b. Solveunitrateproblemsincludingthoseinvolvingunitpricingandconstantspeed.For example, if it took 7 hours to mow 4 lawns, then at that rate, how many lawns could be mowed in 35 hours? At what rate were lawns being mowed?

c. Findapercentofaquantityasarateper100(e.g.,30%ofaquantitymeans30/100timesthequantity);solveproblemsinvolvingfindingthewhole,givenapartandthepercent.

d. Useratioreasoningtoconvertmeasurementunits;manipulateandtransformunitsappropriatelywhenmultiplyingordividingquantities.

the number System 6.nS

Apply and extend previous understandings of multiplication and division to divide fractions by fractions.

1. Interpretandcomputequotientsoffractions,andsolvewordproblemsinvolvingdivisionoffractionsbyfractions,e.g.,byusingvisualfractionmodelsandequationstorepresenttheproblem.For example, create a story context for (2/3) ÷ (3/4) and use a visual fraction model to show the quotient; use the relationship between multiplication and division to explain that (2/3) ÷ (3/4) = 8/9 because 3/4 of 8/9 is 2/3. (In general, (a/b) ÷ (c/d) = ad/bc.) How much chocolate will each person get if 3 people share 1/2 lb of chocolate equally? How many 3/4-cup servings are in 2/3 of a cup of yogurt? How wide is a rectangular strip of land with length 3/4 mi and area 1/2 square mi?

Compute fluently with multi-digit numbers and find common factors and multiples.

2. Fluentlydividemulti-digitnumbersusingthestandardalgorithm.

3. Fluentlyadd,subtract,multiply,anddividemulti-digitdecimalsusingthestandardalgorithmforeachoperation.

4. Findthegreatestcommonfactoroftwowholenumberslessthanorequalto100andtheleastcommonmultipleoftwowholenumberslessthanorequalto12.Usethedistributivepropertytoexpressasumoftwowholenumbers1–100withacommonfactorasamultipleofasumoftwowholenumberswithnocommonfactor.For example, express 36 + 8 as 4 (9 + 2).

1Expectationsforunitratesinthisgradearelimitedtonon-complexfractions.


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Apply and extend previous understandings of numbers to the system of rational numbers.

5. Understandthatpositiveandnegativenumbersareusedtogethertodescribequantitieshavingoppositedirectionsorvalues(e.g.,temperatureabove/belowzero,elevationabove/belowsealevel,credits/debits,positive/negativeelectriccharge);usepositiveandnegativenumberstorepresentquantitiesinreal-worldcontexts,explainingthemeaningof0ineachsituation.

6. Understandarationalnumberasapointonthenumberline.Extendnumberlinediagramsandcoordinateaxesfamiliarfrompreviousgradestorepresentpointsonthelineandintheplanewithnegativenumbercoordinates.

a. Recognizeoppositesignsofnumbersasindicatinglocationsonoppositesidesof0onthenumberline;recognizethattheoppositeoftheoppositeofanumberisthenumberitself,e.g.,–(–3)=3,andthat0isitsownopposite.

b. Understandsignsofnumbersinorderedpairsasindicatinglocationsinquadrantsofthecoordinateplane;recognizethatwhentwoorderedpairsdifferonlybysigns,thelocationsofthepointsarerelatedbyreflectionsacrossoneorbothaxes.

c. Findandpositionintegersandotherrationalnumbersonahorizontalorverticalnumberlinediagram;findandpositionpairsofintegersandotherrationalnumbersonacoordinateplane.

7. Understandorderingandabsolutevalueofrationalnumbers.

a. Interpretstatementsofinequalityasstatementsabouttherelativepositionoftwonumbersonanumberlinediagram.For example, interpret –3 > –7 as a statement that –3 is located to the right of –7 on a number line oriented from left to right.

b. Write,interpret,andexplainstatementsoforderforrationalnumbersinreal-worldcontexts.For example, write –3 oC > –7 oC to express the fact that –3 oC is warmer than –7 oC.

c. Understandtheabsolutevalueofarationalnumberasitsdistancefrom0onthenumberline;interpretabsolutevalueasmagnitudeforapositiveornegativequantityinareal-worldsituation.For example, for an account balance of –30 dollars, write |–30| = 30 to describe the size of the debt in dollars.

d. Distinguishcomparisonsofabsolutevaluefromstatementsaboutorder.For example, recognize that an account balance less than –30 dollars represents a debt greater than 30 dollars.

8. Solvereal-worldandmathematicalproblemsbygraphingpointsinallfourquadrantsofthecoordinateplane.Includeuseofcoordinatesandabsolutevaluetofinddistancesbetweenpointswiththesamefirstcoordinateorthesamesecondcoordinate.

expressions and equations 6.ee

Apply and extend previous understandings of arithmetic to algebraic expressions.

1. Writeandevaluatenumericalexpressionsinvolvingwhole-numberexponents.

2. Write,read,andevaluateexpressionsinwhichlettersstandfornumbers.

a. Writeexpressionsthatrecordoperationswithnumbersandwithlettersstandingfornumbers.For example, express the calculation “Subtract y from 5” as 5 – y.


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b. Identifypartsofanexpressionusingmathematicalterms(sum,term,product,factor,quotient,coefficient);viewoneormorepartsofanexpressionasasingleentity.For example, describe the expression 2 (8 + 7) as a product of two factors; view (8 + 7) as both a single entity and a sum of two terms.

c. Evaluateexpressionsatspecificvaluesoftheirvariables.Includeexpressionsthatarisefromformulasusedinreal-worldproblems.Performarithmeticoperations,includingthoseinvolvingwhole-numberexponents,intheconventionalorderwhentherearenoparenthesestospecifyaparticularorder(OrderofOperations).For example, use the formulas V = s3 and A = 6 s2 to find the volume and surface area of a cube with sides of length s = 1/2.

3. Applythepropertiesofoperationstogenerateequivalentexpressions. For example, apply the distributive property to the expression 3 (2 + x) to produce the equivalent expression 6 + 3x; apply the distributive property to the expression 24x + 18y to produce the equivalent expression 6 (4x + 3y); apply properties of operations to y + y + y to produce the equivalent expression 3y.

4. Identifywhentwoexpressionsareequivalent(i.e.,whenthetwoexpressionsnamethesamenumberregardlessofwhichvalueissubstitutedintothem).For example, the expressions y + y + y and 3y are equivalent because they name the same number regardless of which number y stands for.

Reason about and solve one-variable equations and inequalities.

5. Understandsolvinganequationorinequalityasaprocessofansweringaquestion:whichvaluesfromaspecifiedset,ifany,maketheequationorinequalitytrue?Usesubstitutiontodeterminewhetheragivennumberinaspecifiedsetmakesanequationorinequalitytrue.

6. Usevariablestorepresentnumbersandwriteexpressionswhensolvingareal-worldormathematicalproblem;understandthatavariablecanrepresentanunknownnumber,or,dependingonthepurposeathand,anynumberinaspecifiedset.

7. Solvereal-worldandmathematicalproblemsbywritingandsolvingequationsoftheformx+p=qandpx=qforcasesinwhichp,qandxareallnonnegativerationalnumbers.

8. Writeaninequalityoftheformx>corx <c torepresentaconstraintorconditioninareal-worldormathematicalproblem.Recognizethatinequalitiesoftheformx>corx<chaveinfinitelymanysolutions;representsolutionsofsuchinequalitiesonnumberlinediagrams.

Represent and analyze quantitative relationships between dependent and independent variables.

9. Usevariablestorepresenttwoquantitiesinareal-worldproblemthatchangeinrelationshiptooneanother;writeanequationtoexpressonequantity,thoughtofasthedependentvariable,intermsoftheotherquantity,thoughtofastheindependentvariable.Analyzetherelationshipbetweenthedependentandindependentvariablesusinggraphsandtables,andrelatethesetotheequation.For example, in a problem involving motion at constant speed, list and graph ordered pairs of distances and times, and write the equation d = 65t to represent the relationship between distance and time.

Geometry 6.G

Solve real-world and mathematical problems involving area, surface area, and volume.

1. Findtheareaofrighttriangles,othertriangles,specialquadrilaterals,andpolygonsbycomposingintorectanglesordecomposingintotrianglesandothershapes;applythesetechniquesinthecontextofsolvingreal-worldandmathematicalproblems.


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2. Findthevolumeofarightrectangularprismwithfractionaledgelengthsbypackingitwithunitcubesoftheappropriateunitfractionedgelengths,andshowthatthevolumeisthesameaswouldbefoundbymultiplyingtheedgelengthsoftheprism.ApplytheformulasV = l w h and V = b htofindvolumesofrightrectangularprismswithfractionaledgelengthsinthecontextofsolvingreal-worldandmathematicalproblems.

3. Drawpolygonsinthecoordinateplanegivencoordinatesforthevertices;usecoordinatestofindthelengthofasidejoiningpointswiththesamefirstcoordinateorthesamesecondcoordinate.Applythesetechniquesinthecontextofsolvingreal-worldandmathematicalproblems.

4. Representthree-dimensionalfiguresusingnetsmadeupofrectanglesandtriangles,andusethenetstofindthesurfaceareaofthesefigures.Applythesetechniquesinthecontextofsolvingreal-worldandmathematicalproblems.

Statistics and Probability 6.SP

Develop understanding of statistical variability.

1. Recognizeastatisticalquestionasonethatanticipatesvariabilityinthedatarelatedtothequestionandaccountsforitintheanswers.For example, “How old am I?” is not a statistical question, but “How old are the students in my school?” is a statistical question because one anticipates variability in students’ ages.

2. Understandthatasetofdatacollectedtoanswerastatisticalquestionhasadistributionwhichcanbedescribedbyitscenter,spread,andoverallshape.

3. Recognizethatameasureofcenterforanumericaldatasetsummarizesallofitsvalueswithasinglenumber,whileameasureofvariationdescribeshowitsvaluesvarywithasinglenumber.

Summarize and describe distributions.

4. Displaynumericaldatainplotsonanumberline,includingdotplots,histograms,andboxplots.

5. Summarizenumericaldatasetsinrelationtotheircontext,suchasby:

a. Reportingthenumberofobservations.

b. Describingthenatureoftheattributeunderinvestigation,includinghowitwasmeasuredanditsunitsofmeasurement.

c. Givingquantitativemeasuresofcenter(medianand/ormean)andvariability(interquartilerangeand/ormeanabsolutedeviation),aswellasdescribinganyoverallpatternandanystrikingdeviationsfromtheoverallpatternwithreferencetothecontextinwhichthedataweregathered.

d. Relatingthechoiceofmeasuresofcenterandvariabilitytotheshapeofthedatadistributionandthecontextinwhichthedataweregathered.


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mathematics | Grade 7InGrade7,instructionaltimeshouldfocusonfourcriticalareas:(1)

developingunderstandingofandapplyingproportionalrelationships;

(2)developingunderstandingofoperationswithrationalnumbersand

workingwithexpressionsandlinearequations;(3)solvingproblems

involvingscaledrawingsandinformalgeometricconstructions,and

workingwithtwo-andthree-dimensionalshapestosolveproblems

involvingarea,surfacearea,andvolume;and(4)drawinginferencesabout

populationsbasedonsamples.

(1)Studentsextendtheirunderstandingofratiosanddevelop

understandingofproportionalitytosolvesingle-andmulti-stepproblems.

Studentsusetheirunderstandingofratiosandproportionalitytosolve

awidevarietyofpercentproblems,includingthoseinvolvingdiscounts,

interest,taxes,tips,andpercentincreaseordecrease.Studentssolve

problemsaboutscaledrawingsbyrelatingcorrespondinglengthsbetween

theobjectsorbyusingthefactthatrelationshipsoflengthswithinan

objectarepreservedinsimilarobjects.Studentsgraphproportional

relationshipsandunderstandtheunitrateinformallyasameasureofthe

steepnessoftherelatedline,calledtheslope.Theydistinguishproportional

relationshipsfromotherrelationships.

(2)Studentsdevelopaunifiedunderstandingofnumber,recognizing

fractions,decimals(thathaveafiniteorarepeatingdecimal

representation),andpercentsasdifferentrepresentationsofrational

numbers.Studentsextendaddition,subtraction,multiplication,anddivision

toallrationalnumbers,maintainingthepropertiesofoperationsandthe

relationshipsbetweenadditionandsubtraction,andmultiplicationand

division.Byapplyingtheseproperties,andbyviewingnegativenumbers

intermsofeverydaycontexts(e.g.,amountsowedortemperaturesbelow

zero),studentsexplainandinterprettherulesforadding,subtracting,

multiplying,anddividingwithnegativenumbers.Theyusethearithmetic

ofrationalnumbersastheyformulateexpressionsandequationsinone

variableandusetheseequationstosolveproblems.

(3)StudentscontinuetheirworkwithareafromGrade6,solvingproblems

involvingtheareaandcircumferenceofacircleandsurfaceareaofthree-

dimensionalobjects.Inpreparationforworkoncongruenceandsimilarity

inGrade8theyreasonaboutrelationshipsamongtwo-dimensionalfigures

usingscaledrawingsandinformalgeometricconstructions,andtheygain

familiaritywiththerelationshipsbetweenanglesformedbyintersecting

lines.Studentsworkwiththree-dimensionalfigures,relatingthemtotwo-

dimensionalfiguresbyexaminingcross-sections.Theysolvereal-world

andmathematicalproblemsinvolvingarea,surfacearea,andvolumeof

two-andthree-dimensionalobjectscomposedoftriangles,quadrilaterals,

polygons,cubesandrightprisms.

(4)Studentsbuildontheirpreviousworkwithsingledatadistributionsto

comparetwodatadistributionsandaddressquestionsaboutdifferences

betweenpopulations.Theybegininformalworkwithrandomsampling

togeneratedatasetsandlearnabouttheimportanceofrepresentative

samplesfordrawinginferences.


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ratios and Proportional relationships

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

the number System

• apply and extend previous understandings of operations with fractions to add, subtract, multiply, and divide rational numbers.


• Use properties of operations to generate equivalent expressions.

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

Geometry

• 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.


• 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.










Grade 7 overview


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ratios and Proportional relationships 7.rP

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

1. Computeunitratesassociatedwithratiosoffractions,includingratiosoflengths,areasandotherquantitiesmeasuredinlikeordifferentunits.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.

2. Recognizeandrepresentproportionalrelationshipsbetweenquantities.

a. Decidewhethertwoquantitiesareinaproportionalrelationship,e.g.,bytestingforequivalentratiosinatableorgraphingonacoordinateplaneandobservingwhetherthegraphisastraightlinethroughtheorigin.

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,withspecialattentiontothepoints(0,0)and(1, r) where r istheunitrate.

3. Useproportionalrelationshipstosolvemultistepratioandpercentproblems.Examples: simple interest, tax, markups and markdowns, gratuities and commissions, fees, percent increase and decrease, percent error.


Apply and extend previous understandings of operations with fractions to add, subtract, multiply, and divide rational numbers.

1. Applyandextendpreviousunderstandingsofadditionandsubtractiontoaddandsubtractrationalnumbers;representadditionandsubtractiononahorizontalorverticalnumberlinediagram.

a. Describesituationsinwhichoppositequantitiescombinetomake0.For example, a hydrogen atom has 0 charge because its two constituents are oppositely charged.

b. Understandp+qasthenumberlocatedadistance|q|fromp,inthepositiveornegativedirectiondependingonwhetherqispositiveornegative.Showthatanumberanditsoppositehaveasumof0(areadditiveinverses).Interpretsumsofrationalnumbersbydescribingreal-worldcontexts.

c. Understandsubtractionofrationalnumbersasaddingtheadditiveinverse,p–q=p+(–q).Showthatthedistancebetweentworationalnumbersonthenumberlineistheabsolutevalueoftheirdifference,andapplythisprincipleinreal-worldcontexts.

d. Applypropertiesofoperationsasstrategiestoaddandsubtractrationalnumbers.

2. Applyandextendpreviousunderstandingsofmultiplicationanddivisionandoffractionstomultiplyanddividerationalnumbers.

a. Understandthatmultiplicationisextendedfromfractionstorationalnumbersbyrequiringthatoperationscontinuetosatisfythepropertiesofoperations,particularlythedistributiveproperty,leadingtoproductssuchas(–1)(–1)=1andtherulesformultiplyingsignednumbers.Interpretproductsofrationalnumbersbydescribingreal-worldcontexts.


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b. Understandthatintegerscanbedivided,providedthatthedivisorisnotzero,andeveryquotientofintegers(withnon-zerodivisor)isarationalnumber.Ifpandqareintegers,then–(p/q)=(–p)/q=p/(–q).Interpretquotientsofrationalnumbersbydescribingreal-worldcontexts.

c. Applypropertiesofoperationsasstrategiestomultiplyanddividerationalnumbers.

d. Convertarationalnumbertoadecimalusinglongdivision;knowthatthedecimalformofarationalnumberterminatesin0soreventuallyrepeats.

3. Solvereal-worldandmathematicalproblemsinvolvingthefouroperationswithrationalnumbers.1


Use properties of operations to generate equivalent expressions.

1. Applypropertiesofoperationsasstrategiestoadd,subtract,factor,andexpandlinearexpressionswithrationalcoefficients.

2. Understandthatrewritinganexpressionindifferentformsinaproblemcontextcanshedlightontheproblemandhowthequantitiesinitarerelated.For example, a + 0.05a = 1.05a means that “increase by 5%” is the same as “multiply by 1.05.”

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

3. Solvemulti-stepreal-lifeandmathematicalproblemsposedwithpositiveandnegativerationalnumbersinanyform(wholenumbers,fractions,anddecimals),usingtoolsstrategically.Applypropertiesofoperationstocalculatewithnumbersinanyform;convertbetweenformsasappropriate;andassessthereasonablenessofanswersusingmentalcomputationandestimationstrategies.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.

4. Usevariablestorepresentquantitiesinareal-worldormathematicalproblem,andconstructsimpleequationsandinequalitiestosolveproblemsbyreasoningaboutthequantities.

a. Solvewordproblemsleadingtoequationsoftheformpx+q=randp(x+q)=r,wherep,q,andrarespecificrationalnumbers.Solveequationsoftheseformsfluently.Compareanalgebraicsolutiontoanarithmeticsolution,identifyingthesequenceoftheoperationsusedineachapproach.For example, the perimeter of a rectangle is 54 cm. Its length is 6 cm. What is its width?

b. Solvewordproblemsleadingtoinequalitiesoftheformpx+q>rorpx+q<r,wherep,q,andrarespecificrationalnumbers.Graphthesolutionsetoftheinequalityandinterpretitinthecontextoftheproblem.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.

Geometry 7.G

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

1. Solveproblemsinvolvingscaledrawingsofgeometricfigures,includingcomputingactuallengthsandareasfromascaledrawingandreproducingascaledrawingatadifferentscale.

1Computationswithrationalnumbersextendtherulesformanipulatingfractionstocomplexfractions.


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2. Draw(freehand,withrulerandprotractor,andwithtechnology)geometricshapeswithgivenconditions.Focusonconstructingtrianglesfromthreemeasuresofanglesorsides,noticingwhentheconditionsdetermineauniquetriangle,morethanonetriangle,ornotriangle.

3. Describethetwo-dimensionalfiguresthatresultfromslicingthree-dimensionalfigures,asinplanesectionsofrightrectangularprismsandrightrectangularpyramids.

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

4. Knowtheformulasfortheareaandcircumferenceofacircleandusethemtosolveproblems;giveaninformalderivationoftherelationshipbetweenthecircumferenceandareaofacircle.

5. Usefactsaboutsupplementary,complementary,vertical,andadjacentanglesinamulti-stepproblemtowriteandsolvesimpleequationsforanunknownangleinafigure.

6. Solvereal-worldandmathematicalproblemsinvolvingarea,volumeandsurfaceareaoftwo-andthree-dimensionalobjectscomposedoftriangles,quadrilaterals,polygons,cubes,andrightprisms.


Use random sampling to draw inferences about a population.

1. Understandthatstatisticscanbeusedtogaininformationaboutapopulationbyexaminingasampleofthepopulation;generalizationsaboutapopulationfromasamplearevalidonlyifthesampleisrepresentativeofthatpopulation.Understandthatrandomsamplingtendstoproducerepresentativesamplesandsupportvalidinferences.

2. Usedatafromarandomsampletodrawinferencesaboutapopulationwithanunknowncharacteristicofinterest.Generatemultiplesamples(orsimulatedsamples)ofthesamesizetogaugethevariationinestimatesorpredictions.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.

Draw informal comparative inferences about two populations.

3. Informallyassessthedegreeofvisualoverlapoftwonumericaldatadistributionswithsimilarvariabilities,measuringthedifferencebetweenthecentersbyexpressingitasamultipleofameasureofvariability.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.

4. Usemeasuresofcenterandmeasuresofvariabilityfornumericaldatafromrandomsamplestodrawinformalcomparativeinferencesabouttwopopulations.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.

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

5. Understandthattheprobabilityofachanceeventisanumberbetween0and1thatexpressesthelikelihoodoftheeventoccurring.Largernumbersindicategreaterlikelihood.Aprobabilitynear0indicatesanunlikelyevent,aprobabilityaround1/2indicatesaneventthatisneitherunlikelynorlikely,andaprobabilitynear1indicatesalikelyevent.


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6. Approximatetheprobabilityofachanceeventbycollectingdataonthechanceprocessthatproducesitandobservingitslong-runrelativefrequency,andpredicttheapproximaterelativefrequencygiventheprobability.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. Developaprobabilitymodelanduseittofindprobabilitiesofevents.Compareprobabilitiesfromamodeltoobservedfrequencies;iftheagreementisnotgood,explainpossiblesourcesofthediscrepancy.

a. Developauniformprobabilitymodelbyassigningequalprobabilitytoalloutcomes,andusethemodeltodetermineprobabilitiesofevents.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?

8. Findprobabilitiesofcompoundeventsusingorganizedlists,tables,treediagrams,andsimulation.

a. Understandthat,justaswithsimpleevents,theprobabilityofacompoundeventisthefractionofoutcomesinthesamplespaceforwhichthecompoundeventoccurs.

b. Representsamplespacesforcompoundeventsusingmethodssuchasorganizedlists,tablesandtreediagrams.Foraneventdescribedineverydaylanguage(e.g.,“rollingdoublesixes”),identifytheoutcomesinthesamplespacewhichcomposetheevent.

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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mathematics | Grade 8InGrade8,instructionaltimeshouldfocusonthreecriticalareas:(1)formulating

andreasoningaboutexpressionsandequations,includingmodelinganassociation

inbivariatedatawithalinearequation,andsolvinglinearequationsandsystems

oflinearequations;(2)graspingtheconceptofafunctionandusingfunctions

todescribequantitativerelationships;(3)analyzingtwo-andthree-dimensional

spaceandfiguresusingdistance,angle,similarity,andcongruence,and

understandingandapplyingthePythagoreanTheorem.

(1)Studentsuselinearequationsandsystemsoflinearequationstorepresent,

analyze,andsolveavarietyofproblems.Studentsrecognizeequationsfor

proportions(y/x=mory=mx)asspeciallinearequations(y=mx+b),

understandingthattheconstantofproportionality(m)istheslope,andthegraphs

arelinesthroughtheorigin.Theyunderstandthattheslope(m)ofalineisa

constantrateofchange,sothatiftheinputorx-coordinatechangesbyanamount

A,theoutputory-coordinatechangesbytheamountm·A.Studentsalsousealinear

equationtodescribetheassociationbetweentwoquantitiesinbivariatedata(such

asarmspanvs.heightforstudentsinaclassroom).Atthisgrade,fittingthemodel,

andassessingitsfittothedataaredoneinformally.Interpretingthemodelinthe

contextofthedatarequiresstudentstoexpressarelationshipbetweenthetwo

quantitiesinquestionandtointerpretcomponentsoftherelationship(suchasslope

andy-intercept)intermsofthesituation.

Studentsstrategicallychooseandefficientlyimplementprocedurestosolvelinear

equationsinonevariable,understandingthatwhentheyusethepropertiesof

equalityandtheconceptoflogicalequivalence,theymaintainthesolutionsofthe

originalequation.Studentssolvesystemsoftwolinearequationsintwovariables

andrelatethesystemstopairsoflinesintheplane;theseintersect,areparallel,or

arethesameline.Studentsuselinearequations,systemsoflinearequations,linear

functions,andtheirunderstandingofslopeofalinetoanalyzesituationsandsolve

problems.

(2)Studentsgrasptheconceptofafunctionasarulethatassignstoeachinput

exactlyoneoutput.Theyunderstandthatfunctionsdescribesituationswhereone

quantitydeterminesanother.Theycantranslateamongrepresentationsandpartial

representationsoffunctions(notingthattabularandgraphicalrepresentations

maybepartialrepresentations),andtheydescribehowaspectsofthefunctionare

reflectedinthedifferentrepresentations.

(3)Studentsuseideasaboutdistanceandangles,howtheybehaveunder

translations,rotations,reflections,anddilations,andideasaboutcongruenceand

similaritytodescribeandanalyzetwo-dimensionalfiguresandtosolveproblems.

Studentsshowthatthesumoftheanglesinatriangleistheangleformedbya

straightline,andthatvariousconfigurationsoflinesgiverisetosimilartriangles

becauseoftheanglescreatedwhenatransversalcutsparallellines.Students

understandthestatementofthePythagoreanTheoremanditsconverse,andcan

explainwhythePythagoreanTheoremholds,forexample,bydecomposinga

squareintwodifferentways.TheyapplythePythagoreanTheoremtofinddistances

betweenpointsonthecoordinateplane,tofindlengths,andtoanalyzepolygons.

Studentscompletetheirworkonvolumebysolvingproblemsinvolvingcones,

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the number System

• Know that there are numbers that are not rational, and approximate them by rational numbers.


• Work with radicals and integer exponents.

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

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

functions

• define, evaluate, and compare functions.

• Use functions to model relationships between quantities.

Geometry

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

• Understand and apply the Pythagorean theorem.

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


• Investigate patterns of association in bivariate data.










Grade 8 overview


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Know that there are numbers that are not rational, and approximate them by rational numbers.

1. Knowthatnumbersthatarenotrationalarecalledirrational.Understandinformallythateverynumberhasadecimalexpansion;forrationalnumbersshowthatthedecimalexpansionrepeatseventually,andconvertadecimalexpansionwhichrepeatseventuallyintoarationalnumber.

2. Userationalapproximationsofirrationalnumberstocomparethesizeofirrationalnumbers,locatethemapproximatelyonanumberlinediagram,andestimatethevalueofexpressions(e.g.,π2).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.


Work with radicals and integer exponents.

1. Knowandapplythepropertiesofintegerexponentstogenerateequivalentnumericalexpressions.For example, 32 × 3–5 = 3–3 = 1/33 = 1/27.

2. Usesquarerootandcuberootsymbolstorepresentsolutionstoequationsoftheformx2=pandx3=p,wherepisapositiverationalnumber.Evaluatesquarerootsofsmallperfectsquaresandcuberootsofsmallperfectcubes.Knowthat√2isirrational.

3. Usenumbersexpressedintheformofasingledigittimesanintegerpowerof10toestimateverylargeorverysmallquantities,andtoexpresshowmanytimesasmuchoneisthantheother.For example, estimate the population of the United States as 3 × 108 and the population of the world as 7 × 109, and determine that the world population is more than 20 times larger.

4. Performoperationswithnumbersexpressedinscientificnotation,includingproblemswherebothdecimalandscientificnotationareused.Usescientificnotationandchooseunitsofappropriatesizeformeasurementsofverylargeorverysmallquantities(e.g.,usemillimetersperyearforseafloorspreading).Interpretscientificnotationthathasbeengeneratedbytechnology.

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

5. Graphproportionalrelationships,interpretingtheunitrateastheslopeofthegraph.Comparetwodifferentproportionalrelationshipsrepresentedindifferentways.For example, compare a distance-time graph to a distance-time equation to determine which of two moving objects has greater speed.

6. Usesimilartrianglestoexplainwhytheslopemisthesamebetweenanytwodistinctpointsonanon-verticallineinthecoordinateplane;derivetheequationy=mxforalinethroughtheoriginandtheequationy=mx+bforalineinterceptingtheverticalaxisatb.

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

7. Solvelinearequationsinonevariable.

a. Giveexamplesoflinearequationsinonevariablewithonesolution,infinitelymanysolutions,ornosolutions.Showwhichofthesepossibilitiesisthecasebysuccessivelytransformingthegivenequationintosimplerforms,untilanequivalentequationoftheformx=a,a=a,ora=bresults(whereaandbaredifferentnumbers).

b. Solvelinearequationswithrationalnumbercoefficients,includingequationswhosesolutionsrequireexpandingexpressionsusingthedistributivepropertyandcollectingliketerms.


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8. Analyzeandsolvepairsofsimultaneouslinearequations.

a. Understandthatsolutionstoasystemoftwolinearequationsintwovariablescorrespondtopointsofintersectionoftheirgraphs,becausepointsofintersectionsatisfybothequationssimultaneously.

b. Solvesystemsoftwolinearequationsintwovariablesalgebraically,andestimatesolutionsbygraphingtheequations.Solvesimplecasesbyinspection.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.

functions 8.f

Define, evaluate, and compare functions.

1. Understandthatafunctionisarulethatassignstoeachinputexactlyoneoutput.Thegraphofafunctionisthesetoforderedpairsconsistingofaninputandthecorrespondingoutput.1

2. Comparepropertiesoftwofunctionseachrepresentedinadifferentway(algebraically,graphically,numericallyintables,orbyverbaldescriptions).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.

3. Interprettheequationy=mx+basdefiningalinearfunction,whosegraphisastraightline;giveexamplesoffunctionsthatarenotlinear.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.

Use functions to model relationships between quantities.

4. Constructafunctiontomodelalinearrelationshipbetweentwoquantities.Determinetherateofchangeandinitialvalueofthefunctionfromadescriptionofarelationshiporfromtwo(x,y)values,includingreadingthesefromatableorfromagraph.Interprettherateofchangeandinitialvalueofalinearfunctionintermsofthesituationitmodels,andintermsofitsgraphoratableofvalues.

5. Describequalitativelythefunctionalrelationshipbetweentwoquantitiesbyanalyzingagraph(e.g.,wherethefunctionisincreasingordecreasing,linearornonlinear).Sketchagraphthatexhibitsthequalitativefeaturesofafunctionthathasbeendescribedverbally.

Geometry 8.G

Understand congruence and similarity using physical models, trans-parencies, or geometry software.

1. Verifyexperimentallythepropertiesofrotations,reflections,andtranslations:

a. Linesaretakentolines,andlinesegmentstolinesegmentsofthesamelength.

b. Anglesaretakentoanglesofthesamemeasure.

c. Parallellinesaretakentoparallellines.

2. Understandthatatwo-dimensionalfigureiscongruenttoanotherifthesecondcanbeobtainedfromthefirstbyasequenceofrotations,reflections,andtranslations;giventwocongruentfigures,describeasequencethatexhibitsthecongruencebetweenthem.

1FunctionnotationisnotrequiredinGrade8.


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3. Describetheeffectofdilations,translations,rotations,andreflectionsontwo-dimensionalfiguresusingcoordinates.

4. Understandthatatwo-dimensionalfigureissimilartoanotherifthesecondcanbeobtainedfromthefirstbyasequenceofrotations,reflections,translations,anddilations;giventwosimilartwo-dimensionalfigures,describeasequencethatexhibitsthesimilaritybetweenthem.

5. Useinformalargumentstoestablishfactsabouttheanglesumandexteriorangleoftriangles,abouttheanglescreatedwhenparallellinesarecutbyatransversal,andtheangle-anglecriterionforsimilarityoftriangles.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.

Understand and apply the Pythagorean Theorem.

6. ExplainaproofofthePythagoreanTheoremanditsconverse.

7. ApplythePythagoreanTheoremtodetermineunknownsidelengthsinrighttrianglesinreal-worldandmathematicalproblemsintwoandthreedimensions.

8. ApplythePythagoreanTheoremtofindthedistancebetweentwopointsinacoordinatesystem.

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

9. Knowtheformulasforthevolumesofcones,cylinders,andspheresandusethemtosolvereal-worldandmathematicalproblems.


Investigate patterns of association in bivariate data.

1. Constructandinterpretscatterplotsforbivariatemeasurementdatatoinvestigatepatternsofassociationbetweentwoquantities.Describepatternssuchasclustering,outliers,positiveornegativeassociation,linearassociation,andnonlinearassociation.

2. Knowthatstraightlinesarewidelyusedtomodelrelationshipsbetweentwoquantitativevariables.Forscatterplotsthatsuggestalinearassociation,informallyfitastraightline,andinformallyassessthemodelfitbyjudgingtheclosenessofthedatapointstotheline.

3. Usetheequationofalinearmodeltosolveproblemsinthecontextofbivariatemeasurementdata,interpretingtheslopeandintercept.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.

4. Understandthatpatternsofassociationcanalsobeseeninbivariatecategoricaldatabydisplayingfrequenciesandrelativefrequenciesinatwo-waytable.Constructandinterpretatwo-waytablesummarizingdataontwocategoricalvariablescollectedfromthesamesubjects.Userelativefrequenciescalculatedforrowsorcolumnstodescribepossibleassociationbetweenthetwovariables.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?

Common Core State StandardS for matHematICSH

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mathematics Standards for High SchoolThehighschoolstandardsspecifythemathematicsthatallstudentsshould

studyinordertobecollegeandcareerready.Additionalmathematicsthat

studentsshouldlearninordertotakeadvancedcoursessuchascalculus,

advancedstatistics,ordiscretemathematicsisindicatedby(+),asinthis

example:

(+)Representcomplexnumbersonthecomplexplaneinrectangular

andpolarform(includingrealandimaginarynumbers).

Allstandardswithouta(+)symbolshouldbeinthecommonmathematics

curriculumforallcollegeandcareerreadystudents.Standardswitha(+)

symbolmayalsoappearincoursesintendedforallstudents.

Thehighschoolstandardsarelistedinconceptualcategories:

• NumberandQuantity

• Algebra

• Functions

• Modeling

• Geometry

• StatisticsandProbability

Conceptualcategoriesportrayacoherentviewofhighschool

mathematics;astudent’sworkwithfunctions,forexample,crossesa

numberoftraditionalcourseboundaries,potentiallyupthroughand

includingcalculus.

Modelingisbestinterpretednotasacollectionofisolatedtopicsbutin

relationtootherstandards.MakingmathematicalmodelsisaStandardfor

MathematicalPractice,andspecificmodelingstandardsappearthroughout

thehighschoolstandardsindicatedbyastarsymbol(★).Thestarsymbol

sometimesappearsontheheadingforagroupofstandards;inthatcase,it

shouldbeunderstoodtoapplytoallstandardsinthatgroup.


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mathematics | High School—number and QuantityNumbers and Number Systems.Duringtheyearsfromkindergartentoeighth

grade,studentsmustrepeatedlyextendtheirconceptionofnumber.Atfirst,

“number”means“countingnumber”:1,2,3...Soonafterthat,0isusedtorepresent

“none”andthewholenumbersareformedbythecountingnumberstogether

withzero.Thenextextensionisfractions.Atfirst,fractionsarebarelynumbers

andtiedstronglytopictorialrepresentations.Yetbythetimestudentsunderstand

divisionoffractions,theyhaveastrongconceptoffractionsasnumbersandhave

connectedthem,viatheirdecimalrepresentations,withthebase-tensystemused

torepresentthewholenumbers.Duringmiddleschool,fractionsareaugmentedby

negativefractionstoformtherationalnumbers.InGrade8,studentsextendthis

systemoncemore,augmentingtherationalnumberswiththeirrationalnumbers

toformtherealnumbers.Inhighschool,studentswillbeexposedtoyetanother

extensionofnumber,whentherealnumbersareaugmentedbytheimaginary

numberstoformthecomplexnumbers.

Witheachextensionofnumber,themeaningsofaddition,subtraction,

multiplication,anddivisionareextended.Ineachnewnumbersystem—integers,

rationalnumbers,realnumbers,andcomplexnumbers—thefouroperationsstay

thesameintwoimportantways:Theyhavethecommutative,associative,and

distributivepropertiesandtheirnewmeaningsareconsistentwiththeirprevious

meanings.

Extendingthepropertiesofwhole-numberexponentsleadstonewandproductive

notation.Forexample,propertiesofwhole-numberexponentssuggestthat(51/3)3

shouldbe5(1/3)3=51=5andthat51/3shouldbethecuberootof5.

Calculators,spreadsheets,andcomputeralgebrasystemscanprovidewaysfor

studentstobecomebetteracquaintedwiththesenewnumbersystemsandtheir

notation.Theycanbeusedtogeneratedatafornumericalexperiments,tohelp

understandtheworkingsofmatrix,vector,andcomplexnumberalgebra,andto

experimentwithnon-integerexponents.

Quantities.Inrealworldproblems,theanswersareusuallynotnumbersbut

quantities:numberswithunits,whichinvolvesmeasurement.Intheirworkin

measurementupthroughGrade8,studentsprimarilymeasurecommonlyused

attributessuchaslength,area,andvolume.Inhighschool,studentsencountera

widervarietyofunitsinmodeling,e.g.,acceleration,currencyconversions,derived

quantitiessuchasperson-hoursandheatingdegreedays,socialscienceratessuch

asper-capitaincome,andratesineverydaylifesuchaspointsscoredpergameor

battingaverages.Theyalsoencounternovelsituationsinwhichtheythemselves

mustconceivetheattributesofinterest.Forexample,tofindagoodmeasureof

overallhighwaysafety,theymightproposemeasuressuchasfatalitiesperyear,

fatalitiesperyearperdriver,orfatalitiespervehicle-miletraveled.Suchaconceptual

processissometimescalledquantification.Quantificationisimportantforscience,

aswhensurfaceareasuddenly“standsout”asanimportantvariableinevaporation.

Quantificationisalsoimportantforcompanies,whichmustconceptualizerelevant

attributesandcreateorchoosesuitablemeasuresforthem.


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The Real Number System

• extend the properties of exponents to rational exponents

• Use properties of rational and irrational numbers.

Quantities

• reason quantitatively and use units to solve problems

The Complex Number System

• Perform arithmetic operations with complex numbers

• represent complex numbers and their operations on the complex plane

• Use complex numbers in polynomial identities and equations

Vector and Matrix Quantities

• represent and model with vector quantities.

• Perform operations on vectors.

• Perform operations on matrices and use matrices in applications.










number and Quantity overview


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the real number System n-rn

Extend the properties of exponents to rational exponents.

1. Explainhowthedefinitionofthemeaningofrationalexponentsfollowsfromextendingthepropertiesofintegerexponentstothosevalues,allowingforanotationforradicalsintermsofrationalexponents.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.

2. Rewriteexpressionsinvolvingradicalsandrationalexponentsusingthepropertiesofexponents.

Use properties of rational and irrational numbers.

3. Explainwhythesumorproductoftworationalnumbersisrational;thatthesumofarationalnumberandanirrationalnumberisirrational;andthattheproductofanonzerorationalnumberandanirrationalnumberisirrational.

Quantities★ n-Q

Reason quantitatively and use units to solve problems.

1. Useunitsasawaytounderstandproblemsandtoguidethesolutionofmulti-stepproblems;chooseandinterpretunitsconsistentlyinformulas;chooseandinterpretthescaleandtheoriginingraphsanddatadisplays.

2. Defineappropriatequantitiesforthepurposeofdescriptivemodeling.

3. Choosealevelofaccuracyappropriatetolimitationsonmeasurementwhenreportingquantities.

the Complex number System n-Cn

Perform arithmetic operations with complex numbers.

1. Knowthereisacomplexnumberisuchthati2=–1,andeverycomplexnumberhastheforma +bi withaandbreal.

2. Usetherelationi2=–1andthecommutative,associative,anddistributivepropertiestoadd,subtract,andmultiplycomplexnumbers.

3. (+)Findtheconjugateofacomplexnumber;useconjugatestofindmoduliandquotientsofcomplexnumbers.

Represent complex numbers and their operations on the complex plane.

4. (+)Representcomplexnumbersonthecomplexplaneinrectangularandpolarform(includingrealandimaginarynumbers),andexplainwhytherectangularandpolarformsofagivencomplexnumberrepresentthesamenumber.

5. (+)Representaddition,subtraction,multiplication,andconjugationofcomplexnumbersgeometricallyonthecomplexplane;usepropertiesofthisrepresentationforcomputation.For example, (–1+√3i)3=8because(–1+√3i)has modulus2and argument120°.

6. (+)Calculatethedistancebetweennumbersinthecomplexplaneasthemodulusofthedifference,andthemidpointofasegmentastheaverageofthenumbersatitsendpoints.

Use complex numbers in polynomial identities and equations.

7. Solvequadraticequationswithrealcoefficientsthathavecomplexsolutions.

8. (+)Extendpolynomialidentitiestothecomplexnumbers.For example, rewrite x2+4as(x+2i)(x–2i).

9. (+)KnowtheFundamentalTheoremofAlgebra;showthatitistrueforquadraticpolynomials.


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Vector and matrix Quantities n-Vm

Represent and model with vector quantities.

1. (+)Recognizevectorquantitiesashavingbothmagnitudeanddirection.Representvectorquantitiesbydirectedlinesegments,anduseappropriatesymbolsforvectorsandtheirmagnitudes(e.g.,v,|v|,||v||,v).

2. (+)Findthecomponentsofavectorbysubtractingthecoordinatesofaninitialpointfromthecoordinatesofaterminalpoint.

3. (+)Solveproblemsinvolvingvelocityandotherquantitiesthatcanberepresentedbyvectors.

Perform operations on vectors.

4. (+)Addandsubtractvectors.

a. Addvectorsend-to-end,component-wise,andbytheparallelogramrule.Understandthatthemagnitudeofasumoftwovectorsistypicallynotthesumofthemagnitudes.

b. Giventwovectorsinmagnitudeanddirectionform,determinethemagnitudeanddirectionoftheirsum.

c. Understandvectorsubtractionv–wasv+(–w),where–wistheadditiveinverseofw,withthesamemagnitudeaswandpointingintheoppositedirection.Representvectorsubtractiongraphicallybyconnectingthetipsintheappropriateorder,andperformvectorsubtractioncomponent-wise.

5. (+)Multiplyavectorbyascalar.

a. Representscalarmultiplicationgraphicallybyscalingvectorsandpossiblyreversingtheirdirection;performscalarmultiplicationcomponent-wise,e.g.,asc(v

x,v

y)=(cv

x,cv

y).

b. Computethemagnitudeofascalarmultiplecvusing||cv||=|c|v.Computethedirectionofcvknowingthatwhen|c|v≠0,thedirectionofcviseitheralongv(forc>0)oragainstv(forc<0).

Perform operations on matrices and use matrices in applications.

6. (+)Usematricestorepresentandmanipulatedata,e.g.,torepresentpayoffsorincidencerelationshipsinanetwork.

7. (+)Multiplymatricesbyscalarstoproducenewmatrices,e.g.,aswhenallofthepayoffsinagamearedoubled.

8. (+)Add,subtract,andmultiplymatricesofappropriatedimensions.

9. (+)Understandthat,unlikemultiplicationofnumbers,matrixmultiplicationforsquarematricesisnotacommutativeoperation,butstillsatisfiestheassociativeanddistributiveproperties.

10. (+)Understandthatthezeroandidentitymatricesplayaroleinmatrixadditionandmultiplicationsimilartotheroleof0and1intherealnumbers.Thedeterminantofasquarematrixisnonzeroifandonlyifthematrixhasamultiplicativeinverse.

11. (+)Multiplyavector(regardedasamatrixwithonecolumn)byamatrixofsuitabledimensionstoproduceanothervector.Workwithmatricesastransformationsofvectors.

12. (+)Workwith2×2matricesastransformationsoftheplane,andinterprettheabsolutevalueofthedeterminantintermsofarea.


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mathematics | High School—algebraExpressions.Anexpressionisarecordofacomputationwithnumbers,symbolsthatrepresentnumbers,arithmeticoperations,exponentiation,and,atmoreadvancedlevels,theoperationofevaluatingafunction.Conventionsabouttheuseofparenthesesandtheorderofoperationsassurethateachexpressionisunambiguous.Creatinganexpressionthatdescribesacomputationinvolvingageneralquantityrequirestheabilitytoexpressthecomputationingeneralterms,abstractingfromspecificinstances.

Readinganexpressionwithcomprehensioninvolvesanalysisofitsunderlyingstructure.Thismaysuggestadifferentbutequivalentwayofwritingtheexpressionthatexhibitssomedifferentaspectofitsmeaning.Forexample,p+0.05pcanbeinterpretedastheadditionofa5%taxtoapricep.Rewritingp+0.05pas1.05pshowsthataddingataxisthesameasmultiplyingthepricebyaconstantfactor.

Algebraicmanipulationsaregovernedbythepropertiesofoperationsandexponents,andtheconventionsofalgebraicnotation.Attimes,anexpressionistheresultofapplyingoperationstosimplerexpressions.Forexample,p+0.05pisthesumofthesimplerexpressionspand0.05p.Viewinganexpressionastheresultofoperationonsimplerexpressionscansometimesclarifyitsunderlyingstructure.

Aspreadsheetoracomputeralgebrasystem(CAS)canbeusedtoexperimentwithalgebraicexpressions,performcomplicatedalgebraicmanipulations,andunderstandhowalgebraicmanipulationsbehave.

Equations and inequalities.Anequationisastatementofequalitybetweentwoexpressions,oftenviewedasaquestionaskingforwhichvaluesofthevariablestheexpressionsoneithersideareinfactequal.Thesevaluesarethesolutionstotheequation.Anidentity,incontrast,istrueforallvaluesofthevariables;identitiesareoftendevelopedbyrewritinganexpressioninanequivalentform.

Thesolutionsofanequationinonevariableformasetofnumbers;thesolutionsofanequationintwovariablesformasetoforderedpairsofnumbers,whichcanbeplottedinthecoordinateplane.Twoormoreequationsand/orinequalitiesformasystem.Asolutionforsuchasystemmustsatisfyeveryequationandinequalityinthesystem.

Anequationcanoftenbesolvedbysuccessivelydeducingfromitoneormoresimplerequations.Forexample,onecanaddthesameconstanttobothsideswithoutchangingthesolutions,butsquaringbothsidesmightleadtoextraneoussolutions.Strategiccompetenceinsolvingincludeslookingaheadforproductivemanipulationsandanticipatingthenatureandnumberofsolutions.

Someequationshavenosolutionsinagivennumbersystem,buthaveasolutioninalargersystem.Forexample,thesolutionofx+1=0isaninteger,notawholenumber;thesolutionof2x+1=0isarationalnumber,notaninteger;thesolutionsofx2–2=0arerealnumbers,notrationalnumbers;andthesolutionsofx2+2=0arecomplexnumbers,notrealnumbers.

Thesamesolutiontechniquesusedtosolveequationscanbeusedtorearrangeformulas.Forexample,theformulafortheareaofatrapezoid,A=((b

1+b

2)/2)h,can

besolvedforhusingthesamedeductiveprocess.

Inequalitiescanbesolvedbyreasoningaboutthepropertiesofinequality.Many,butnotall,ofthepropertiesofequalitycontinuetoholdforinequalitiesandcanbeusefulinsolvingthem.

Connections to Functions and Modeling. Expressionscandefinefunctions,andequivalentexpressionsdefinethesamefunction.Askingwhentwofunctionshavethesamevalueforthesameinputleadstoanequation;graphingthetwofunctionsallowsforfindingapproximatesolutionsoftheequation.Convertingaverbaldescriptiontoanequation,inequality,orsystemoftheseisanessentialskillinmodeling.


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Seeing Structure in Expressions

• Interpret the structure of expressions

• Write expressions in equivalent forms to solve problems

Arithmetic with Polynomials and Rational Expressions

• Perform arithmetic operations on polynomials

• Understand the relationship between zeros and factors of polynomials

• Use polynomial identities to solve problems

• rewrite rational expressions

Creating Equations

• Create equations that describe numbers or relationships

Reasoning with Equations and Inequalities

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

• Solve equations and inequalities in one variable

• Solve systems of equations

• represent and solve equations and inequalities graphically










algebra overview


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Seeing Structure in expressions a-SSe

Interpret the structure of expressions

1. Interpretexpressionsthatrepresentaquantityintermsofitscontext.★

a. Interpretpartsofanexpression,suchasterms,factors,andcoefficients.

b. Interpretcomplicatedexpressionsbyviewingoneormoreoftheirpartsasasingleentity.For example, interpretP(1+r)nas the product of P and a factor not depending on P.

2. Usethestructureofanexpressiontoidentifywaystorewriteit.For example, see x4–y4as(x2)2–(y2)2,thus recognizing it as a difference of squares that can be factored as(x2–y2)(x2+y2).

Write expressions in equivalent forms to solve problems

3. Chooseandproduceanequivalentformofanexpressiontorevealandexplainpropertiesofthequantityrepresentedbytheexpression.★

a. Factoraquadraticexpressiontorevealthezerosofthefunctionitdefines.

b. Completethesquareinaquadraticexpressiontorevealthemaximumorminimumvalueofthefunctionitdefines.

c. Usethepropertiesofexponentstotransformexpressionsforexponentialfunctions. For example the expression1.15tcan be rewritten as(1.151/12)12t≈1.01212tto reveal the approximate equivalent monthly interest rate if the annual rate is 15%.

4. Derivetheformulaforthesumofafinitegeometricseries(whenthecommonratioisnot1),andusetheformulatosolveproblems.For example, calculate mortgage payments.★

arithmetic with Polynomials and rational expressions a-aPr

Perform arithmetic operations on polynomials

1. Understandthatpolynomialsformasystemanalogoustotheintegers,namely,theyareclosedundertheoperationsofaddition,subtraction,andmultiplication;add,subtract,andmultiplypolynomials.

Understand the relationship between zeros and factors of polynomials

2. KnowandapplytheRemainderTheorem:Forapolynomialp(x)andanumbera,theremainderondivisionbyx–aisp(a),sop(a)=0ifandonlyif(x–a)isafactorofp(x).

3. Identifyzerosofpolynomialswhensuitablefactorizationsareavailable,andusethezerostoconstructaroughgraphofthefunctiondefinedbythepolynomial.

Use polynomial identities to solve problems

4. Provepolynomialidentitiesandusethemtodescribenumericalrelationships.For example, the polynomial identity(x2+y2)2=(x2–y2)2+(2xy)2can be used to generate Pythagorean triples.

5. (+)KnowandapplytheBinomialTheoremfortheexpansionof(x+y)ninpowersofxandyforapositiveintegern,wherexandyareanynumbers,withcoefficientsdeterminedforexamplebyPascal’sTriangle.1

1TheBinomialTheoremcanbeprovedbymathematicalinductionorbyacom-binatorialargument.


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Rewrite rational expressions

6. Rewritesimplerationalexpressionsindifferentforms;writea(x)/b(x)intheformq(x)+r(x)/b(x),wherea(x),b(x),q(x),andr(x)arepolynomialswiththedegreeofr(x)lessthanthedegreeofb(x),usinginspection,longdivision,or,forthemorecomplicatedexamples,acomputeralgebrasystem.

7. (+)Understandthatrationalexpressionsformasystemanalogoustotherationalnumbers,closedunderaddition,subtraction,multiplication,anddivisionbyanonzerorationalexpression;add,subtract,multiply,anddividerationalexpressions.

Creating equations★ a-Ced

Create equations that describe numbers or relationships

1. Createequationsandinequalitiesinonevariableandusethemtosolveproblems.Include equations arising from linear and quadratic functions, and simple rational and exponential functions.

2. Createequationsintwoormorevariablestorepresentrelationshipsbetweenquantities;graphequationsoncoordinateaxeswithlabelsandscales.

3. Representconstraintsbyequationsorinequalities,andbysystemsofequationsand/orinequalities,andinterpretsolutionsasviableornon-viableoptionsinamodelingcontext.For example, represent inequalities describing nutritional and cost constraints on combinations of different foods.

4. Rearrangeformulastohighlightaquantityofinterest,usingthesamereasoningasinsolvingequations.For example, rearrange Ohm’s law V = IR to highlight resistance R.

reasoning with equations and Inequalities a-reI

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

1. Explaineachstepinsolvingasimpleequationasfollowingfromtheequalityofnumbersassertedatthepreviousstep,startingfromtheassumptionthattheoriginalequationhasasolution.Constructaviableargumenttojustifyasolutionmethod.

2. Solvesimplerationalandradicalequationsinonevariable,andgiveexamplesshowinghowextraneoussolutionsmayarise.

Solve equations and inequalities in one variable

3. Solvelinearequationsandinequalitiesinonevariable,includingequationswithcoefficientsrepresentedbyletters.

4. Solvequadraticequationsinonevariable.

a. Usethemethodofcompletingthesquaretotransformanyquadraticequationinxintoanequationoftheform(x–p)2=qthathasthesamesolutions.Derivethequadraticformulafromthisform.

b. Solvequadraticequationsbyinspection(e.g.,forx2=49),takingsquareroots,completingthesquare,thequadraticformulaandfactoring,asappropriatetotheinitialformoftheequation.Recognizewhenthequadraticformulagivescomplexsolutionsandwritethemasa±biforrealnumbersaandb.

Solve systems of equations

5. Provethat,givenasystemoftwoequationsintwovariables,replacingoneequationbythesumofthatequationandamultipleoftheotherproducesasystemwiththesamesolutions.


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

7. Solveasimplesystemconsistingofalinearequationandaquadraticequationintwovariablesalgebraicallyandgraphically.For example, find the points of intersection between the line y=–3xandthecirclex2+y2=3.

8. (+)Representasystemoflinearequationsasasinglematrixequationinavectorvariable.

9. (+)Findtheinverseofamatrixifitexistsanduseittosolvesystemsoflinearequations(usingtechnologyformatricesofdimension3×3orgreater).

Represent and solve equations and inequalities graphically

10. Understandthatthegraphofanequationintwovariablesisthesetofallitssolutionsplottedinthecoordinateplane,oftenformingacurve(whichcouldbealine).

11. Explainwhythex-coordinatesofthepointswherethegraphsoftheequationsy=f(x)andy=g(x)intersectarethesolutionsoftheequationf(x)=g(x);findthesolutionsapproximately,e.g.,usingtechnologytographthefunctions,maketablesofvalues,orfindsuccessiveapproximations.Includecaseswheref(x)and/org(x)arelinear,polynomial,rational,absolutevalue,exponential,andlogarithmicfunctions.★

12. Graphthesolutionstoalinearinequalityintwovariablesasahalf-plane(excludingtheboundaryinthecaseofastrictinequality),andgraphthesolutionsettoasystemoflinearinequalitiesintwovariablesastheintersectionofthecorrespondinghalf-planes.


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mathematics | High School—functionsFunctionsdescribesituationswhereonequantitydeterminesanother.Forexample,thereturnon$10,000investedatanannualizedpercentagerateof4.25%isafunctionofthelengthoftimethemoneyisinvested.Becausewecontinuallymaketheoriesaboutdependenciesbetweenquantitiesinnatureandsociety,functionsareimportanttoolsintheconstructionofmathematicalmodels.

Inschoolmathematics,functionsusuallyhavenumericalinputsandoutputsandareoftendefinedbyanalgebraicexpression.Forexample,thetimeinhoursittakesforacartodrive100milesisafunctionofthecar’sspeedinmilesperhour,v;theruleT(v)=100/vexpressesthisrelationshipalgebraicallyanddefinesafunctionwhosenameisT.

Thesetofinputstoafunctioniscalleditsdomain.Weofteninferthedomaintobeallinputsforwhichtheexpressiondefiningafunctionhasavalue,orforwhichthefunctionmakessenseinagivencontext.

Afunctioncanbedescribedinvariousways,suchasbyagraph(e.g.,thetraceofaseismograph);byaverbalrule,asin,“I’llgiveyouastate,yougivemethecapitalcity;”byanalgebraicexpressionlikef(x)=a+bx;orbyarecursiverule.Thegraphofafunctionisoftenausefulwayofvisualizingtherelationshipofthefunctionmodels,andmanipulatingamathematicalexpressionforafunctioncanthrowlightonthefunction’sproperties.

Functionspresentedasexpressionscanmodelmanyimportantphenomena.Twoimportantfamiliesoffunctionscharacterizedbylawsofgrowtharelinearfunctions,whichgrowataconstantrate,andexponentialfunctions,whichgrowataconstantpercentrate.Linearfunctionswithaconstanttermofzerodescribeproportionalrelationships.

Agraphingutilityoracomputeralgebrasystemcanbeusedtoexperimentwithpropertiesofthesefunctionsandtheirgraphsandtobuildcomputationalmodelsoffunctions,includingrecursivelydefinedfunctions.

Connections to Expressions, Equations, Modeling, and Coordinates.

Determininganoutputvalueforaparticularinputinvolvesevaluatinganexpression;findinginputsthatyieldagivenoutputinvolvessolvinganequation.Questionsaboutwhentwofunctionshavethesamevalueforthesameinputleadtoequations,whosesolutionscanbevisualizedfromtheintersectionoftheirgraphs.Becausefunctionsdescriberelationshipsbetweenquantities,theyarefrequentlyusedinmodeling.Sometimesfunctionsaredefinedbyarecursiveprocess,whichcanbedisplayedeffectivelyusingaspreadsheetorothertechnology.


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Interpreting Functions

• Understand the concept of a function and use function notation

• Interpret functions that arise in applications in terms of the context

• analyze functions using different representations

Building Functions

• Build a function that models a relationship between two quantities

• Build new functions from existing functions

Linear, Quadratic, and Exponential Models

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

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

Trigonometric Functions

• extend the domain of trigonometric functions using the unit circle

• model periodic phenomena with trigonometric functions

• Prove and apply trigonometric identities










functions overview


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Interpreting functions f-If

Understand the concept of a function and use function notation

1. Understandthatafunctionfromoneset(calledthedomain)toanotherset(calledtherange)assignstoeachelementofthedomainexactlyoneelementoftherange.Iffisafunctionandxisanelementofitsdomain,thenf(x)denotestheoutputoffcorrespondingtotheinputx.Thegraphoffisthegraphoftheequationy=f(x).

2. Usefunctionnotation,evaluatefunctionsforinputsintheirdomains,andinterpretstatementsthatusefunctionnotationintermsofacontext.

3. Recognizethatsequencesarefunctions,sometimesdefinedrecursively,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.

Interpret functions that arise in applications in terms of the context

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

5. Relatethedomainofafunctiontoitsgraphand,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.★

6. Calculateandinterprettheaveragerateofchangeofafunction(presentedsymbolicallyorasatable)overaspecifiedinterval.Estimatetherateofchangefromagraph.★

Analyze functions using different representations

7. Graphfunctionsexpressedsymbolicallyandshowkeyfeaturesofthegraph,byhandinsimplecasesandusingtechnologyformorecomplicatedcases.★

a. Graphlinearandquadraticfunctionsandshowintercepts,maxima,andminima.

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

c. Graphpolynomialfunctions,identifyingzeroswhensuitablefactorizationsareavailable,andshowingendbehavior.

d. (+)Graphrationalfunctions,identifyingzerosandasymptoteswhensuitablefactorizationsareavailable,andshowingendbehavior.

e. Graphexponentialandlogarithmicfunctions,showinginterceptsandendbehavior,andtrigonometricfunctions,showingperiod,midline,andamplitude.

8. Writeafunctiondefinedbyanexpressionindifferentbutequivalentformstorevealandexplaindifferentpropertiesofthefunction.

a. Usetheprocessoffactoringandcompletingthesquareinaquadraticfunctiontoshowzeros,extremevalues,andsymmetryofthegraph,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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9. Comparepropertiesoftwofunctionseachrepresentedinadifferentway(algebraically,graphically,numericallyintables,orbyverbaldescriptions).For example, given a graph of one quadratic function and an algebraic expression for another, say which has the larger maximum.

Building functions f-Bf

Build a function that models a relationship between two quantities

1. Writeafunctionthatdescribesarelationshipbetweentwoquantities.★

a. Determineanexplicitexpression,arecursiveprocess,orstepsforcalculationfromacontext.

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.

c. (+)Composefunctions. For example, if T(y) is the temperature in the atmosphere as a function of height, and h(t) is the height of a weather balloon as a function of time, then T(h(t)) is the temperature at the location of the weather balloon as a function of time.

2. Writearithmeticandgeometricsequencesbothrecursivelyandwithanexplicitformula,usethemtomodelsituations,andtranslatebetweenthetwoforms.★

Build new functions from existing functions

3. Identifytheeffectonthegraphofreplacingf(x)byf(x)+k,kf(x),f(kx),andf(x+k)forspecificvaluesofk(bothpositiveandnegative);findthevalueofkgiventhegraphs.Experimentwithcasesandillustrateanexplanationoftheeffectsonthegraphusingtechnology.Include recognizing even and odd functions from their graphs and algebraic expressions for them.

4. Findinversefunctions.

a. Solveanequationoftheformf(x)=cforasimplefunctionfthathasaninverseandwriteanexpressionfortheinverse.For example, f(x) =2 x3 or f(x) = (x+1)/(x–1) for x ≠ 1.

b. (+)Verifybycompositionthatonefunctionistheinverseofanother.

c. (+)Readvaluesofaninversefunctionfromagraphoratable,giventhatthefunctionhasaninverse.

d. (+)Produceaninvertiblefunctionfromanon-invertiblefunctionbyrestrictingthedomain.

5. (+)Understandtheinverserelationshipbetweenexponentsandlogarithmsandusethisrelationshiptosolveproblemsinvolvinglogarithmsandexponents.

Linear, Quadratic, and exponential models★ f-Le

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

1. Distinguishbetweensituationsthatcanbemodeledwithlinearfunctionsandwithexponentialfunctions.

a. Provethatlinearfunctionsgrowbyequaldifferencesoverequalintervals,andthatexponentialfunctionsgrowbyequalfactorsoverequalintervals.

b. Recognizesituationsinwhichonequantitychangesataconstantrateperunitintervalrelativetoanother.

c. Recognizesituationsinwhichaquantitygrowsordecaysbyaconstantpercentrateperunitintervalrelativetoanother.


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2. Constructlinearandexponentialfunctions,includingarithmeticandgeometricsequences,givenagraph,adescriptionofarelationship,ortwoinput-outputpairs(includereadingthesefromatable).

3. Observeusinggraphsandtablesthataquantityincreasingexponentiallyeventuallyexceedsaquantityincreasinglinearly,quadratically,or(moregenerally)asapolynomialfunction.

4. Forexponentialmodels,expressasalogarithmthesolutiontoabct=dwherea,c,anddarenumbersandthebasebis2,10,ore;evaluatethelogarithmusingtechnology.

Interpret expressions for functions in terms of the situation they model

5. Interprettheparametersinalinearorexponentialfunctionintermsofacontext.

trigonometric functions f-tf

Extend the domain of trigonometric functions using the unit circle

1. Understandradianmeasureofanangleasthelengthofthearcontheunitcirclesubtendedbytheangle.

2. Explainhowtheunitcircleinthecoordinateplaneenablestheextensionoftrigonometricfunctionstoallrealnumbers,interpretedasradianmeasuresofanglestraversedcounterclockwisearoundtheunitcircle.

3. (+)Usespecialtrianglestodeterminegeometricallythevaluesofsine,cosine,tangentforπ/3,π/4andπ/6,andusetheunitcircletoexpressthevaluesofsine,cosine,andtangentforπ–x,π+x,and2π–xintermsoftheirvaluesforx,wherexisanyrealnumber.

4. (+)Usetheunitcircletoexplainsymmetry(oddandeven)andperiodicityoftrigonometricfunctions.

Model periodic phenomena with trigonometric functions

5. Choosetrigonometricfunctionstomodelperiodicphenomenawithspecifiedamplitude,frequency,andmidline.★

6. (+)Understandthatrestrictingatrigonometricfunctiontoadomainonwhichitisalwaysincreasingoralwaysdecreasingallowsitsinversetobeconstructed.

7. (+)Useinversefunctionstosolvetrigonometricequationsthatariseinmodelingcontexts;evaluatethesolutionsusingtechnology,andinterpretthemintermsofthecontext.★

Prove and apply trigonometric identities

8. ProvethePythagoreanidentitysin2(θ)+cos2(θ)=1anduseittofindsin(θ),cos(θ),ortan(θ)givensin(θ),cos(θ),ortan(θ)andthequadrantoftheangle.

9. (+)Provetheadditionandsubtractionformulasforsine,cosine,andtangentandusethemtosolveproblems.


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mathematics | High School—modelingModelinglinksclassroommathematicsandstatisticstoeverydaylife,work,anddecision-making.Modelingistheprocessofchoosingandusingappropriatemathematicsandstatisticstoanalyzeempiricalsituations,tounderstandthembetter,andtoimprovedecisions.Quantitiesandtheirrelationshipsinphysical,economic,publicpolicy,social,andeverydaysituationscanbemodeledusingmathematicalandstatisticalmethods.Whenmakingmathematicalmodels,technologyisvaluableforvaryingassumptions,exploringconsequences,andcomparingpredictionswithdata.

Amodelcanbeverysimple,suchaswritingtotalcostasaproductofunitpriceandnumberbought,orusingageometricshapetodescribeaphysicalobjectlikeacoin.Evensuchsimplemodelsinvolvemakingchoices.Itisuptouswhethertomodelacoinasathree-dimensionalcylinder,orwhetheratwo-dimensionaldiskworkswellenoughforourpurposes.Othersituations—modelingadeliveryroute,aproductionschedule,oracomparisonofloanamortizations—needmoreelaboratemodelsthatuseothertoolsfromthemathematicalsciences.Real-worldsituationsarenotorganizedandlabeledforanalysis;formulatingtractablemodels,representingsuchmodels,andanalyzingthemisappropriatelyacreativeprocess.Likeeverysuchprocess,thisdependsonacquiredexpertiseaswellascreativity.

Someexamplesofsuchsituationsmightinclude:

• Estimatinghowmuchwaterandfoodisneededforemergencyreliefinadevastatedcityof3millionpeople,andhowitmightbedistributed.

• Planningatabletennistournamentfor7playersataclubwith4tables,whereeachplayerplaysagainsteachotherplayer.

• Designingthelayoutofthestallsinaschoolfairsoastoraiseasmuchmoneyaspossible.

• Analyzingstoppingdistanceforacar.

• Modelingsavingsaccountbalance,bacterialcolonygrowth,orinvestmentgrowth.

• Engagingincriticalpathanalysis,e.g.,appliedtoturnaroundofanaircraftatanairport.

• Analyzingriskinsituationssuchasextremesports,pandemics,andterrorism.

• Relatingpopulationstatisticstoindividualpredictions.

Insituationslikethese,themodelsdeviseddependonanumberoffactors:Howpreciseananswerdowewantorneed?Whataspectsofthesituationdowemostneedtounderstand,control,oroptimize?Whatresourcesoftimeandtoolsdowehave?Therangeofmodelsthatwecancreateandanalyzeisalsoconstrainedbythelimitationsofourmathematical,statistical,andtechnicalskills,andourabilitytorecognizesignificantvariablesandrelationshipsamongthem.Diagramsofvariouskinds,spreadsheetsandothertechnology,andalgebraarepowerfultoolsforunderstandingandsolvingproblemsdrawnfromdifferenttypesofreal-worldsituations.

Oneoftheinsightsprovidedbymathematicalmodelingisthatessentiallythesamemathematicalorstatisticalstructurecansometimesmodelseeminglydifferentsituations.Modelscanalsoshedlightonthemathematicalstructuresthemselves,forexample,aswhenamodelofbacterialgrowthmakesmorevividtheexplosivegrowthoftheexponentialfunction.

Thebasicmodelingcycleissummarizedinthediagram.Itinvolves(1)identifyingvariablesinthesituationandselectingthosethatrepresentessentialfeatures,(2)formulatingamodelbycreatingandselectinggeometric,graphical,tabular,algebraic,orstatisticalrepresentationsthatdescriberelationshipsbetweenthevariables,(3)analyzingandperformingoperationsontheserelationshipstodrawconclusions,(4)interpretingtheresultsofthemathematicsintermsoftheoriginalsituation,(5)validatingtheconclusionsbycomparingthemwiththesituation,andtheneitherimprovingthemodelor,ifit


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isacceptable,(6)reportingontheconclusionsandthereasoningbehindthem.Choices,assumptions,andapproximationsarepresentthroughoutthiscycle.

Indescriptivemodeling,amodelsimplydescribesthephenomenaorsummarizestheminacompactform.Graphsofobservationsareafamiliardescriptivemodel—forexample,graphsofglobaltemperatureandatmosphericCO

2overtime.

Analyticmodelingseekstoexplaindataonthebasisofdeepertheoreticalideas,albeitwithparametersthatareempiricallybased;forexample,exponentialgrowthofbacterialcolonies(untilcut-offmechanismssuchaspollutionorstarvationintervene)followsfromaconstantreproductionrate.Functionsareanimportanttoolforanalyzingsuchproblems.

Graphingutilities,spreadsheets,computeralgebrasystems,anddynamicgeometrysoftwarearepowerfultoolsthatcanbeusedtomodelpurelymathematicalphenomena(e.g.,thebehaviorofpolynomials)aswellasphysicalphenomena.

modeling Standards Modeling is best interpreted not as a collection of isolated topics but rather in relation to other standards. Making mathematical models is a Standard for Mathematical Practice, and specific modeling standards appear throughout the high school standards indicated by a star symbol (★).


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mathematics | High School—GeometryAnunderstandingoftheattributesandrelationshipsofgeometricobjectscanbeappliedindiversecontexts—interpretingaschematicdrawing,estimatingtheamountofwoodneededtoframeaslopingroof,renderingcomputergraphics,ordesigningasewingpatternforthemostefficientuseofmaterial.

Althoughtherearemanytypesofgeometry,schoolmathematicsisdevotedprimarilytoplaneEuclideangeometry,studiedbothsynthetically(withoutcoordinates)andanalytically(withcoordinates).EuclideangeometryischaracterizedmostimportantlybytheParallelPostulate,thatthroughapointnotonagivenlinethereisexactlyoneparallelline.(Sphericalgeometry,incontrast,hasnoparallellines.)

Duringhighschool,studentsbegintoformalizetheirgeometryexperiencesfromelementaryandmiddleschool,usingmoreprecisedefinitionsanddevelopingcarefulproofs.LaterincollegesomestudentsdevelopEuclideanandothergeometriescarefullyfromasmallsetofaxioms.

Theconceptsofcongruence,similarity,andsymmetrycanbeunderstoodfromtheperspectiveofgeometrictransformation.Fundamentalaretherigidmotions:translations,rotations,reflections,andcombinationsofthese,allofwhicharehereassumedtopreservedistanceandangles(andthereforeshapesgenerally).Reflectionsandrotationseachexplainaparticulartypeofsymmetry,andthesymmetriesofanobjectofferinsightintoitsattributes—aswhenthereflectivesymmetryofanisoscelestriangleassuresthatitsbaseanglesarecongruent.

Intheapproachtakenhere,twogeometricfiguresaredefinedtobecongruentifthereisasequenceofrigidmotionsthatcarriesoneontotheother.Thisistheprincipleofsuperposition.Fortriangles,congruencemeanstheequalityofallcorrespondingpairsofsidesandallcorrespondingpairsofangles.Duringthemiddlegrades,throughexperiencesdrawingtrianglesfromgivenconditions,studentsnoticewaystospecifyenoughmeasuresinatriangletoensurethatalltrianglesdrawnwiththosemeasuresarecongruent.Oncethesetrianglecongruencecriteria(ASA,SAS,andSSS)areestablishedusingrigidmotions,theycanbeusedtoprovetheoremsabouttriangles,quadrilaterals,andothergeometricfigures.

Similaritytransformations(rigidmotionsfollowedbydilations)definesimilarityinthesamewaythatrigidmotionsdefinecongruence,therebyformalizingthesimilarityideasof"sameshape"and"scalefactor"developedinthemiddlegrades.Thesetransformationsleadtothecriterionfortrianglesimilaritythattwopairsofcorrespondinganglesarecongruent.

Thedefinitionsofsine,cosine,andtangentforacuteanglesarefoundedonrighttrianglesandsimilarity,and,withthePythagoreanTheorem,arefundamentalinmanyreal-worldandtheoreticalsituations.ThePythagoreanTheoremisgeneralizedtonon-righttrianglesbytheLawofCosines.Together,theLawsofSinesandCosinesembodythetrianglecongruencecriteriaforthecaseswherethreepiecesofinformationsufficetocompletelysolveatriangle.Furthermore,theselawsyieldtwopossiblesolutionsintheambiguouscase,illustratingthatSide-Side-Angleisnotacongruencecriterion.

Analyticgeometryconnectsalgebraandgeometry,resultinginpowerfulmethodsofanalysisandproblemsolving.Justasthenumberlineassociatesnumberswithlocationsinonedimension,apairofperpendicularaxesassociatespairsofnumberswithlocationsintwodimensions.Thiscorrespondencebetweennumericalcoordinatesandgeometricpointsallowsmethodsfromalgebratobeappliedtogeometryandviceversa.Thesolutionsetofanequationbecomesageometriccurve,makingvisualizationatoolfordoingandunderstandingalgebra.Geometricshapescanbedescribedbyequations,makingalgebraicmanipulationintoatoolforgeometricunderstanding,modeling,andproof.Geometrictransformationsofthegraphsofequationscorrespondtoalgebraicchangesintheirequations.

Dynamicgeometryenvironmentsprovidestudentswithexperimentalandmodelingtoolsthatallowthemtoinvestigategeometricphenomenainmuchthesamewayascomputeralgebrasystemsallowthemtoexperimentwithalgebraicphenomena.

Connections to Equations. Thecorrespondencebetweennumericalcoordinatesandgeometricpointsallowsmethodsfromalgebratobeappliedtogeometryandviceversa.Thesolutionsetofanequationbecomesageometriccurve,makingvisualizationatoolfordoingandunderstandingalgebra.Geometricshapescanbedescribedbyequations,makingalgebraicmanipulationintoatoolforgeometricunderstanding,modeling,andproof.


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Congruence

• experiment with transformations in the plane

• Understand congruence in terms of rigid motions

• Prove geometric theorems

• make geometric constructions

Similarity, Right Triangles, and Trigonometry

• Understand similarity in terms of similarity transformations

• Prove theorems involving similarity

• define trigonometric ratios and solve problems involving right triangles

• apply trigonometry to general triangles

Circles

• Understand and apply theorems about circles

• find arc lengths and areas of sectors of circles

Expressing Geometric Properties with Equations

• translate between the geometric description and the equation for a conic section

• Use coordinates to prove simple geometric theorems algebraically

Geometric Measurement and Dimension

• explain volume formulas and use them to solve problems

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

Modeling with Geometry

• apply geometric concepts in modeling situations










Geometry overview


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Congruence G-Co

Experiment with transformations in the plane

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

2. Representtransformationsintheplaneusing,e.g.,transparenciesandgeometrysoftware;describetransformationsasfunctionsthattakepointsintheplaneasinputsandgiveotherpointsasoutputs.Comparetransformationsthatpreservedistanceandangletothosethatdonot(e.g.,translationversushorizontalstretch).

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

4. Developdefinitionsofrotations,reflections,andtranslationsintermsofangles,circles,perpendicularlines,parallellines,andlinesegments.

5. Givenageometricfigureandarotation,reflection,ortranslation,drawthetransformedfigureusing,e.g.,graphpaper,tracingpaper,orgeometrysoftware.Specifyasequenceoftransformationsthatwillcarryagivenfigureontoanother.

Understand congruence in terms of rigid motions

6. Usegeometricdescriptionsofrigidmotionstotransformfiguresandtopredicttheeffectofagivenrigidmotiononagivenfigure;giventwofigures,usethedefinitionofcongruenceintermsofrigidmotionstodecideiftheyarecongruent.

7. Usethedefinitionofcongruenceintermsofrigidmotionstoshowthattwotrianglesarecongruentifandonlyifcorrespondingpairsofsidesandcorrespondingpairsofanglesarecongruent.

8. Explainhowthecriteriafortrianglecongruence(ASA,SAS,andSSS)followfromthedefinitionofcongruenceintermsofrigidmotions.

Prove geometric theorems

9. Provetheoremsaboutlinesandangles. 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.

10. Provetheoremsabouttriangles.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.

11. Provetheoremsaboutparallelograms.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.

Make geometric constructions

12. Makeformalgeometricconstructionswithavarietyoftoolsandmethods(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.

13. Constructanequilateraltriangle,asquare,andaregularhexagoninscribedinacircle.


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Similarity, right triangles, and trigonometry G-Srt

Understand similarity in terms of similarity transformations

1. Verifyexperimentallythepropertiesofdilationsgivenbyacenterandascalefactor:

a. Adilationtakesalinenotpassingthroughthecenterofthedilationtoaparallelline,andleavesalinepassingthroughthecenterunchanged.

b. Thedilationofalinesegmentislongerorshorterintheratiogivenbythescalefactor.

2. Giventwofigures,usethedefinitionofsimilarityintermsofsimilaritytransformations todecideiftheyaresimilar;explainusingsimilaritytransformationsthemeaningofsimilarityfortrianglesastheequalityofallcorrespondingpairsofanglesandtheproportionalityofallcorrespondingpairsofsides.

3. UsethepropertiesofsimilaritytransformationstoestablishtheAAcriterionfortwotrianglestobesimilar.

Prove theorems involving similarity

4. Provetheoremsabouttriangles.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.

5. Usecongruenceandsimilaritycriteriafortrianglestosolveproblemsandtoproverelationshipsingeometricfigures.

Define trigonometric ratios and solve problems involving right triangles

6. Understandthatbysimilarity,sideratiosinrighttrianglesarepropertiesoftheanglesinthetriangle,leadingtodefinitionsoftrigonometricratiosforacuteangles.

7. Explainandusetherelationshipbetweenthesineandcosineofcomplementaryangles.

8. UsetrigonometricratiosandthePythagoreanTheoremtosolverighttrianglesinappliedproblems.★

Apply trigonometry to general triangles

9. (+)DerivetheformulaA=1/2absin(C)fortheareaofatrianglebydrawinganauxiliarylinefromavertexperpendiculartotheoppositeside.

10. (+)ProvetheLawsofSinesandCosinesandusethemtosolveproblems.

11. (+)UnderstandandapplytheLawofSinesandtheLawofCosinestofindunknownmeasurementsinrightandnon-righttriangles(e.g.,surveyingproblems,resultantforces).

Circles G-C

Understand and apply theorems about circles

1. Provethatallcirclesaresimilar.

2. Identifyanddescriberelationshipsamonginscribedangles,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.

3. Constructtheinscribedandcircumscribedcirclesofatriangle,andprovepropertiesofanglesforaquadrilateralinscribedinacircle.

4. (+)Constructatangentlinefromapointoutsideagivencircletothecircle.


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Find arc lengths and areas of sectors of circles

5. Deriveusingsimilaritythefactthatthelengthofthearcinterceptedbyanangleisproportionaltotheradius,anddefinetheradianmeasureoftheangleastheconstantofproportionality;derivetheformulafortheareaofasector.

expressing Geometric Properties with equations G-GPe

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

1. DerivetheequationofacircleofgivencenterandradiususingthePythagoreanTheorem;completethesquaretofindthecenterandradiusofacirclegivenbyanequation.

2. Derivetheequationofaparabolagivenafocusanddirectrix.

3. (+)Derivetheequationsofellipsesandhyperbolasgiventhefoci,usingthefactthatthesumordifferenceofdistancesfromthefociisconstant.

Use coordinates to prove simple geometric theorems algebraically

4. Usecoordinatestoprovesimplegeometrictheoremsalgebraically.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).

5. Provetheslopecriteriaforparallelandperpendicularlinesandusethemtosolvegeometricproblems(e.g.,findtheequationofalineparallelorperpendiculartoagivenlinethatpassesthroughagivenpoint).

6. Findthepointonadirectedlinesegmentbetweentwogivenpointsthatpartitionsthesegmentinagivenratio.

7. Usecoordinatestocomputeperimetersofpolygonsandareasoftrianglesandrectangles,e.g.,usingthedistanceformula.★

Geometric measurement and dimension G-Gmd

Explain volume formulas and use them to solve problems

1. Giveaninformalargumentfortheformulasforthecircumferenceofacircle,areaofacircle,volumeofacylinder,pyramid,andcone.Use dissection arguments, Cavalieri’s principle, and informal limit arguments.

2. (+)GiveaninformalargumentusingCavalieri’sprinciplefortheformulasforthevolumeofasphereandothersolidfigures.

3. Usevolumeformulasforcylinders,pyramids,cones,andspherestosolveproblems.★

Visualize relationships between two-dimensional and three-dimensional objects

4. Identifytheshapesoftwo-dimensionalcross-sectionsofthree-dimensionalobjects,andidentifythree-dimensionalobjectsgeneratedbyrotationsoftwo-dimensionalobjects.

modeling with Geometry G-mG

Apply geometric concepts in modeling situations

1. Usegeometricshapes,theirmeasures,andtheirpropertiestodescribeobjects(e.g.,modelingatreetrunkorahumantorsoasacylinder).★

2. Applyconceptsofdensitybasedonareaandvolumeinmodelingsituations(e.g.,personspersquaremile,BTUspercubicfoot).★

3. Applygeometricmethodstosolvedesignproblems(e.g.,designinganobjectorstructuretosatisfyphysicalconstraintsorminimizecost;workingwithtypographicgridsystemsbasedonratios).★


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mathematics | High School—Statistics and Probability★

Decisionsorpredictionsareoftenbasedondata—numbersincontext.Thesedecisionsorpredictionswouldbeeasyifthedataalwayssentaclearmessage,butthemessageisoftenobscuredbyvariability.Statisticsprovidestoolsfordescribingvariabilityindataandformakinginformeddecisionsthattakeitintoaccount.

Dataaregathered,displayed,summarized,examined,andinterpretedtodiscoverpatternsanddeviationsfrompatterns.Quantitativedatacanbedescribedintermsofkeycharacteristics:measuresofshape,center,andspread.Theshapeofadatadistributionmightbedescribedassymmetric,skewed,flat,orbellshaped,anditmightbesummarizedbyastatisticmeasuringcenter(suchasmeanormedian)andastatisticmeasuringspread(suchasstandarddeviationorinterquartilerange).Differentdistributionscanbecomparednumericallyusingthesestatisticsorcomparedvisuallyusingplots.Knowledgeofcenterandspreadarenotenoughtodescribeadistribution.Whichstatisticstocompare,whichplotstouse,andwhattheresultsofacomparisonmightmean,dependonthequestiontobeinvestigatedandthereal-lifeactionstobetaken.

Randomizationhastwoimportantusesindrawingstatisticalconclusions.First,collectingdatafromarandomsampleofapopulationmakesitpossibletodrawvalidconclusionsaboutthewholepopulation,takingvariabilityintoaccount.Second,randomlyassigningindividualstodifferenttreatmentsallowsafaircomparisonoftheeffectivenessofthosetreatments.Astatisticallysignificantoutcomeisonethatisunlikelytobeduetochancealone,andthiscanbeevaluatedonlyundertheconditionofrandomness.Theconditionsunderwhichdataarecollectedareimportantindrawingconclusionsfromthedata;incriticallyreviewingusesofstatisticsinpublicmediaandotherreports,itisimportanttoconsiderthestudydesign,howthedataweregathered,andtheanalysesemployedaswellasthedatasummariesandtheconclusionsdrawn.

Randomprocessescanbedescribedmathematicallybyusingaprobabilitymodel:alistordescriptionofthepossibleoutcomes(thesamplespace),eachofwhichisassignedaprobability.Insituationssuchasflippingacoin,rollinganumbercube,ordrawingacard,itmightbereasonabletoassumevariousoutcomesareequallylikely.Inaprobabilitymodel,samplepointsrepresentoutcomesandcombinetomakeupevents;probabilitiesofeventscanbecomputedbyapplyingtheAdditionandMultiplicationRules.Interpretingtheseprobabilitiesreliesonanunderstandingofindependenceandconditionalprobability,whichcanbeapproachedthroughtheanalysisoftwo-waytables.

Technologyplaysanimportantroleinstatisticsandprobabilitybymakingitpossibletogenerateplots,regressionfunctions,andcorrelationcoefficients,andtosimulatemanypossibleoutcomesinashortamountoftime.

Connections to Functions and Modeling.Functionsmaybeusedtodescribedata;ifthedatasuggestalinearrelationship,therelationshipcanbemodeledwitharegressionline,anditsstrengthanddirectioncanbeexpressedthroughacorrelationcoefficient.


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Interpreting Categorical and Quantitative Data

• 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

Making Inferences and Justifying Conclusions

• Understand and evaluate random processes underlying statistical experiments

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

Conditional Probability and the Rules of Prob-ability

• 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

Using Probability to Make Decisions

• Calculate expected values and use them to solve problems

• Use probability to evaluate outcomes of decisions










Statistics and Probability overview


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Interpreting Categorical and Quantitative data S-Id

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

1. Representdatawithplotsontherealnumberline(dotplots,histograms,andboxplots).

2. Usestatisticsappropriatetotheshapeofthedatadistributiontocomparecenter(median,mean)andspread(interquartilerange,standarddeviation)oftwoormoredifferentdatasets.

3. Interpretdifferencesinshape,center,andspreadinthecontextofthedatasets,accountingforpossibleeffectsofextremedatapoints(outliers).

4. Usethemeanandstandarddeviationofadatasettofitittoanormaldistributionandtoestimatepopulationpercentages.Recognizethattherearedatasetsforwhichsuchaprocedureisnotappropriate.Usecalculators,spreadsheets,andtablestoestimateareasunderthenormalcurve.

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

5. Summarizecategoricaldatafortwocategoriesintwo-wayfrequencytables.Interpretrelativefrequenciesinthecontextofthedata(includingjoint,marginal,andconditionalrelativefrequencies).Recognizepossibleassociationsandtrendsinthedata.

6. Representdataontwoquantitativevariablesonascatterplot,anddescribehowthevariablesarerelated.

a. Fitafunctiontothedata;usefunctionsfittedtodatatosolveproblemsinthecontextofthedata.Use given functions or choose a function suggested by the context. Emphasize linear, quadratic, and exponential models.

b. Informallyassessthefitofafunctionbyplottingandanalyzingresiduals.

c. Fitalinearfunctionforascatterplotthatsuggestsalinearassociation.

Interpret linear models

7. Interprettheslope(rateofchange)andtheintercept(constantterm)ofalinearmodelinthecontextofthedata.

8. Compute(usingtechnology)andinterpretthecorrelationcoefficientofalinearfit.

9. Distinguishbetweencorrelationandcausation.

making Inferences and Justifying Conclusions S-IC

Understand and evaluate random processes underlying statistical experiments

1. Understandstatisticsasaprocessformakinginferencesaboutpopulationparametersbasedonarandomsamplefromthatpopulation.

2. Decideifaspecifiedmodelisconsistentwithresultsfromagivendata-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?

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

3. Recognizethepurposesofanddifferencesamongsamplesurveys,experiments,andobservationalstudies;explainhowrandomizationrelatestoeach.


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4. Usedatafromasamplesurveytoestimateapopulationmeanorproportion;developamarginoferrorthroughtheuseofsimulationmodelsforrandomsampling.

5. Usedatafromarandomizedexperimenttocomparetwotreatments;usesimulationstodecideifdifferencesbetweenparametersaresignificant.

6. Evaluatereportsbasedondata.

Conditional Probability and the rules of Probability S-CP

Understand independence and conditional probability and use them to interpret data

1. Describeeventsassubsetsofasamplespace(thesetofoutcomes)usingcharacteristics(orcategories)oftheoutcomes,orasunions,intersections,orcomplementsofotherevents(“or,”“and,”“not”).

2. UnderstandthattwoeventsAandBareindependentiftheprobabilityofAandBoccurringtogetheristheproductoftheirprobabilities,andusethischaracterizationtodetermineiftheyareindependent.

3. UnderstandtheconditionalprobabilityofAgivenBas P(AandB)/P(B),andinterpretindependenceofAandBassayingthattheconditionalprobabilityof Agiven BisthesameastheprobabilityofA,andtheconditionalprobabilityofBgivenAisthesameastheprobabilityofB.

4. Constructandinterprettwo-wayfrequencytablesofdatawhentwocategoriesareassociatedwitheachobjectbeingclassified.Usethetwo-waytableasasamplespacetodecideifeventsareindependentandtoapproximateconditionalprobabilities.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.

5. Recognizeandexplaintheconceptsofconditionalprobabilityandindependenceineverydaylanguageandeverydaysituations.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.

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

6. FindtheconditionalprobabilityofAgivenBasthefractionofB’soutcomesthatalsobelongtoA,andinterprettheanswerintermsofthemodel.

7. ApplytheAdditionRule,P(AorB)=P(A)+P(B)–P(AandB),andinterprettheanswerintermsofthemodel.

8. (+)ApplythegeneralMultiplicationRuleinauniformprobabilitymodel,P(AandB)=P(A)P(B|A)=P(B)P(A|B),andinterprettheanswerintermsofthemodel.

9. (+)Usepermutationsandcombinationstocomputeprobabilitiesofcompoundeventsandsolveproblems.

Using Probability to make decisions S-md

Calculate expected values and use them to solve problems

1. (+)Definearandomvariableforaquantityofinterestbyassigninganumericalvaluetoeacheventinasamplespace;graphthecorrespondingprobabilitydistributionusingthesamegraphicaldisplaysasfordatadistributions.

2. (+)Calculatetheexpectedvalueofarandomvariable;interpretitasthemeanoftheprobabilitydistribution.


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3. (+)Developaprobabilitydistributionforarandomvariabledefinedforasamplespaceinwhichtheoreticalprobabilitiescanbecalculated;findtheexpectedvalue.For example, find the theoretical probability distribution for the number of correct answers obtained by guessing on all five questions of a multiple-choice test where each question has four choices, and find the expected grade under various grading schemes.

4. (+)Developaprobabilitydistributionforarandomvariabledefinedforasamplespaceinwhichprobabilitiesareassignedempirically;findtheexpectedvalue.For example, find a current data distribution on the number of TV sets per household in the United States, and calculate the expected number of sets per household. How many TV sets would you expect to find in 100 randomly selected households?

Use probability to evaluate outcomes of decisions

5. (+)Weighthepossibleoutcomesofadecisionbyassigningprobabilitiestopayoffvaluesandfindingexpectedvalues.

a. Findtheexpectedpayoffforagameofchance.For example, find the expected winnings from a state lottery ticket or a game at a fast-food restaurant.

b. Evaluateandcomparestrategiesonthebasisofexpectedvalues.For example, compare a high-deductible versus a low-deductible automobile insurance policy using various, but reasonable, chances of having a minor or a major accident.

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

7. (+)Analyzedecisionsandstrategiesusingprobabilityconcepts(e.g.,producttesting,medicaltesting,pullingahockeygoalieattheendofagame).

Common Core State StandardS for matHematICS8

4

note on courses and transitions

ThehighschoolportionoftheStandardsforMathematicalContentspecifiesthemathematicsallstudentsshouldstudyforcollegeandcareerreadiness.Thesestandardsdonotmandatethesequenceofhighschoolcourses.However,theorganizationofhighschoolcoursesisacriticalcomponenttoimplementationofthestandards.Tothatend,samplehighschoolpathwaysformathematics–inbothatraditionalcoursesequence(AlgebraI,Geometry,andAlgebraII)aswellasanintegratedcoursesequence(Mathematics1,Mathematics2,Mathematics3)–willbemadeavailableshortlyafterthereleaseofthefinalCommonCoreStateStandards.Itisexpectedthatadditionalmodelpathwaysbasedonthesestandardswillbecomeavailableaswell.

Thestandardsthemselvesdonotdictatecurriculum,pedagogy,ordeliveryofcontent.Inparticular,statesmayhandlethetransitiontohighschoolindifferentways.Forexample,manystudentsintheU.S.todaytakeAlgebraIinthe8thgrade,andinsomestatesthisisarequirement.TheK-7standardscontaintheprerequisitestopreparestudentsforAlgebraIby8thgrade,andthestandardsaredesignedtopermitstatestocontinueexistingpoliciesconcerningAlgebraIin8thgrade.

Asecondmajortransitionisthetransitionfromhighschooltopost-secondaryeducationforcollegeandcareers.Theevidenceconcerningcollegeandcareerreadinessshowsclearlythattheknowledge,skills,andpracticesimportantforreadinessincludeagreatdealofmathematicspriortotheboundarydefinedby(+)symbolsinthesestandards.Indeed,someofthehighestprioritycontentforcollegeandcareerreadinesscomesfromGrades6-8.Thisbodyofmaterialincludespowerfullyusefulproficienciessuchasapplyingratioreasoninginreal-worldandmathematicalproblems,computingfluentlywithpositiveandnegativefractionsanddecimals,andsolvingreal-worldandmathematicalproblemsinvolvinganglemeasure,area,surfacearea,andvolume.Becauseimportantstandardsforcollegeandcareerreadinessaredistributedacrossgradesandcourses,systemsforevaluatingcollegeandcareerreadinessshouldreachasfarbackinthestandardsasGrades6-8.Itisimportanttonoteaswellthatcutscoresorotherinformationgeneratedbyassessmentsystemsforcollegeandcareerreadinessshouldbedevelopedincollaborationwithrepresentativesfromhighereducationandworkforcedevelopmentprograms,andshouldbevalidatedbysubsequentperformanceofstudentsincollegeandtheworkforce.


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Addition and subtraction within 5, 10, 20, 100, or 1000.Additionorsubtractionoftwowholenumberswithwholenumberanswers,andwithsumorminuendintherange0-5,0-10,0-20,or0-100,respectively.Example:8+2=10isanadditionwithin10,14–5=9isasubtractionwithin20,and55–18=37isasubtractionwithin100.

Additive inverses.Twonumberswhosesumis0areadditiveinversesofoneanother.Example:3/4and–3/4areadditiveinversesofoneanotherbecause3/4+(–3/4)=(–3/4)+3/4=0.

Associative property of addition.SeeTable3inthisGlossary.

Associative property of multiplication. SeeTable3inthisGlossary.

Bivariate data. Pairsoflinkednumericalobservations.Example:alistofheightsandweightsforeachplayeronafootballteam.

Box plot.Amethodofvisuallydisplayingadistributionofdatavaluesbyusingthemedian,quartiles,andextremesofthedataset.Aboxshowsthemiddle50%ofthedata.1

Commutative property.SeeTable3inthisGlossary.

Complex fraction.AfractionA/BwhereAand/orBarefractions(Bnonzero).

Computation algorithm.Asetofpredefinedstepsapplicabletoaclassofproblemsthatgivesthecorrectresultineverycasewhenthestepsarecarriedoutcorrectly.See also:computationstrategy.

Computation strategy.Purposefulmanipulationsthatmaybechosenforspecificproblems,maynothaveafixedorder,andmaybeaimedatconvertingoneproblemintoanother.See also:computationalgorithm.

Congruent.Twoplaneorsolidfiguresarecongruentifonecanbeobtainedfromtheotherbyrigidmotion(asequenceofrotations,reflections,andtranslations).

Counting on.Astrategyforfindingthenumberofobjectsinagroupwithouthavingtocounteverymemberofthegroup.Forexample,ifastackofbooksisknowntohave8booksand3morebooksareaddedtothetop,itisnotnecessarytocountthestackalloveragain.Onecanfindthetotalbycounting on—pointingtothetopbookandsaying“eight,”followingthiswith“nine,ten,eleven.Thereareelevenbooksnow.”

Dot plot. See: lineplot.

Dilation.Atransformationthatmoveseachpointalongtheraythroughthepointemanatingfromafixedcenter,andmultipliesdistancesfromthecenterbyacommonscalefactor.

Expanded form.Amulti-digitnumberisexpressedinexpandedformwhenitiswrittenasasumofsingle-digitmultiplesofpowersoften.Forexample,643=600+40+3.

Expected value. Forarandomvariable,theweightedaverageofitspossiblevalues,withweightsgivenbytheirrespectiveprobabilities.

First quartile. ForadatasetwithmedianM,thefirstquartileisthemedianofthedatavalueslessthanM.Example:Forthedataset{1,3,6,7,10,12,14,15,22,120},thefirstquartileis6.2See also:median,thirdquartile,interquartilerange.

Fraction.Anumberexpressibleintheforma/bwhereaisawholenumberandbisapositivewholenumber.(Thewordfractioninthesestandardsalwaysreferstoanon-negativenumber.)See also:rationalnumber.

Identity property of 0.SeeTable3inthisGlossary.

Independently combined probability models.Twoprobabilitymodelsaresaidtobecombinedindependentlyiftheprobabilityofeachorderedpairinthecombinedmodelequalstheproductoftheoriginalprobabilitiesofthetwoindividualoutcomesintheorderedpair.

1AdaptedfromWisconsinDepartmentofPublicInstruction,http://dpi.wi.gov/standards/mathglos.html,accessedMarch2,2010.2Manydifferentmethodsforcomputingquartilesareinuse.ThemethoddefinedhereissometimescalledtheMooreandMcCabemethod.SeeLangford,E.,“QuartilesinElementaryStatistics,”Journal of Statistics EducationVolume14,Number3(2006).

Glossary


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Integer.Anumberexpressibleintheformaor–aforsomewholenumbera.

Interquartile Range. Ameasureofvariationinasetofnumericaldata,theinterquartilerangeisthedistancebetweenthefirstandthirdquartilesofthedataset.Example:Forthedataset{1,3,6,7,10,12,14,15,22,120},theinterquartilerangeis15–6=9.See also:firstquartile,thirdquartile.

Line plot.Amethodofvisuallydisplayingadistributionofdatavalueswhereeachdatavalueisshownasadotormarkaboveanumberline.Alsoknownasadotplot.3

Mean.Ameasureofcenterinasetofnumericaldata,computedbyaddingthevaluesinalistandthendividingbythenumberofvaluesinthelist.4Example:Forthedataset{1,3,6,7,10,12,14,15,22,120},themeanis21.

Mean absolute deviation.Ameasureofvariationinasetofnumericaldata,computedbyaddingthedistancesbetweeneachdatavalueandthemean,thendividingbythenumberofdatavalues.Example:Forthedataset{2,3,6,7,10,12,14,15,22,120},themeanabsolutedeviationis20.

Median.Ameasureofcenterinasetofnumericaldata.Themedianofalistofvaluesisthevalueappearingatthecenterofasortedversionofthelist—orthemeanofthetwocentralvalues,ifthelistcontainsanevennumberofvalues.Example:Forthedataset{2,3,6,7,10,12,14,15,22,90},themedianis11.

Midline. Inthegraphofatrigonometricfunction,thehorizontallinehalfwaybetweenitsmaximumandminimumvalues.

Multiplication and division within 100.Multiplicationordivisionoftwowholenumberswithwholenumberanswers,andwithproductordividendintherange0-100.Example:72÷8=9.

Multiplicative inverses.Twonumberswhoseproductis1aremultiplicativeinversesofoneanother.Example:3/4and4/3aremultiplicativeinversesofoneanotherbecause3/4×4/3=4/3×3/4=1.

Number line diagram. Adiagramofthenumberlineusedtorepresentnumbersandsupportreasoningaboutthem.Inanumberlinediagramformeasurementquantities,theintervalfrom0to1onthediagramrepresentstheunitofmeasureforthequantity.

Percent rate of change.Arateofchangeexpressedasapercent.Example:ifapopulationgrowsfrom50to55inayear,itgrowsby5/50=10%peryear.

Probability distribution.Thesetofpossiblevaluesofarandomvariablewithaprobabilityassignedtoeach.

Properties of operations.SeeTable3inthisGlossary.

Properties of equality.SeeTable4inthisGlossary.

Properties of inequality.SeeTable5inthisGlossary.

Properties of operations.SeeTable3inthisGlossary.

Probability.Anumberbetween0and1usedtoquantifylikelihoodforprocessesthathaveuncertainoutcomes(suchastossingacoin,selectingapersonatrandomfromagroupofpeople,tossingaballatatarget,ortestingforamedicalcondition).

Probability model. Aprobabilitymodelisusedtoassignprobabilitiestooutcomesofachanceprocessbyexaminingthenatureoftheprocess.Thesetofalloutcomesiscalledthesamplespace,andtheirprobabilitiessumto1.See also: uniformprobabilitymodel.

Random variable. Anassignmentofanumericalvaluetoeachoutcomeinasamplespace.

Rational expression.Aquotientoftwopolynomialswithanon-zerodenominator.

Rational number.Anumberexpressibleintheforma/bor– a/bforsomefractiona/b.Therationalnumbersincludetheintegers.

Rectilinear figure. Apolygonallanglesofwhicharerightangles.

Rigid motion.Atransformationofpointsinspaceconsistingofasequenceof

3AdaptedfromWisconsinDepartmentofPublicInstruction,op. cit.4Tobemoreprecise,thisdefinesthearithmetic mean.


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oneormoretranslations,reflections,and/orrotations.Rigidmotionsarehereassumedtopreservedistancesandanglemeasures.

Repeating decimal.Thedecimalformofarationalnumber.See also:terminatingdecimal.

Sample space.Inaprobabilitymodelforarandomprocess,alistoftheindividualoutcomesthataretobeconsidered.

Scatter plot. Agraphinthecoordinateplanerepresentingasetofbivariatedata.Forexample,theheightsandweightsofagroupofpeoplecouldbedisplayedonascatterplot.5

Similarity transformation.Arigidmotionfollowedbyadilation.

Tape diagram.Adrawingthatlookslikeasegmentoftape,usedtoillustratenumberrelationships.Alsoknownasastripdiagram,barmodel,fractionstrip,orlengthmodel.

Terminating decimal. Adecimaliscalledterminatingifitsrepeatingdigitis0.

Third quartile.ForadatasetwithmedianM,thethirdquartileisthemedianofthedatavaluesgreaterthanM.Example:Forthedataset{2,3,6,7,10,12,14,15,22,120},thethirdquartileis15.See also:median,firstquartile,interquartilerange.

Transitivity principle for indirect measurement. IfthelengthofobjectAisgreaterthanthelengthofobjectB,andthelengthofobjectBisgreaterthanthelengthofobjectC,thenthelengthofobjectAisgreaterthanthelengthofobjectC.Thisprincipleappliestomeasurementofotherquantitiesaswell.

Uniform probability model.Aprobabilitymodelwhichassignsequalprobabilitytoalloutcomes.See also:probabilitymodel.

Vector. Aquantitywithmagnitudeanddirectionintheplaneorinspace,definedbyanorderedpairortripleofrealnumbers.

Visual fraction model. Atapediagram,numberlinediagram,orareamodel.

Whole numbers.Thenumbers0,1,2,3,….

5AdaptedfromWisconsinDepartmentofPublicInstruction,op. cit.


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Table 1.Commonadditionandsubtractionsituations.6

result Unknown Change Unknown Start Unknown

add to

Twobunniessatonthegrass.Threemorebunnieshoppedthere.Howmanybunniesareonthegrassnow?

2+3=?

Twobunniesweresittingonthegrass.Somemorebunnieshoppedthere.Thentherewerefivebunnies.Howmanybunnieshoppedovertothefirsttwo?

2+?=5

Somebunniesweresittingonthegrass.Threemorebunnieshoppedthere.Thentherewerefivebunnies.Howmanybunnieswereonthegrassbefore?

?+3=5

take from

Fiveappleswereonthetable.Iatetwoapples.Howmanyapplesareonthetablenow?

5–2=?

Fiveappleswereonthetable.Iatesomeapples.Thentherewerethreeapples.HowmanyapplesdidIeat?

5–?=3

Someappleswereonthetable.Iatetwoapples.Thentherewerethreeapples.Howmanyappleswereonthetablebefore?

?–2=3

total Unknown addend Unknown Both addends Unknown1

Put together/ take apart2

Threeredapplesandtwogreenapplesareonthetable.Howmanyapplesareonthetable?

3+2=?

Fiveapplesareonthetable.Threeareredandtherestaregreen.Howmanyapplesaregreen?

3+?=5,5–3=?

Grandmahasfiveflowers.Howmanycansheputinherredvaseandhowmanyinherbluevase?

5=0+5,5=5+0

5=1+4,5=4+1

5=2+3,5=3+2

difference Unknown Bigger Unknown Smaller Unknown

Compare3

(“Howmanymore?”version):

Lucyhastwoapples.Juliehasfiveapples.HowmanymoreapplesdoesJuliehavethanLucy?

(“Howmanyfewer?”version):

Lucyhastwoapples.Juliehasfiveapples.HowmanyfewerapplesdoesLucyhavethanJulie?

2+?=5,5–2=?

(Versionwith“more”):

JuliehasthreemoreapplesthanLucy.Lucyhastwoapples.HowmanyapplesdoesJuliehave?

(Versionwith“fewer”):

Lucyhas3fewerapplesthanJulie.Lucyhastwoapples.HowmanyapplesdoesJuliehave?

2+3=?,3+2=?

(Versionwith“more”):

JuliehasthreemoreapplesthanLucy.Juliehasfiveapples.HowmanyapplesdoesLucyhave?

(Versionwith“fewer”):

Lucyhas3fewerapplesthanJulie.Juliehasfiveapples.HowmanyapplesdoesLucyhave?

5–3=?,?+3=5

6AdaptedfromBox2-4ofMathematicsLearninginEarlyChildhood,NationalResearchCouncil(2009,pp.32,33).

1Thesetakeapartsituationscanbeusedtoshowallthedecompositionsofagivennumber.Theassociatedequations,whichhavethetotalontheleftoftheequalsign,helpchildrenunderstandthatthe=signdoesnotalwaysmeanmakesorresultsinbutalwaysdoesmeanisthesamenumberas.2Eitheraddendcanbeunknown,sotherearethreevariationsoftheseproblemsituations.BothAddendsUnknownisapro-ductiveextensionofthisbasicsituation,especiallyforsmallnumberslessthanorequalto10.3FortheBiggerUnknownorSmallerUnknownsituations,oneversiondirectsthecorrectoperation(theversionusingmoreforthebiggerunknownandusinglessforthesmallerunknown).Theotherversionsaremoredifficult.


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Table 2. Commonmultiplicationanddivisionsituations.7

Unknown ProductGroup Size Unknown

(“Howmanyineachgroup?”Division)

number of Groups Unknown(“Howmanygroups?”Division)

3 × 6 = ? 3 × ? = 18, and 18 ÷ 3 = ? ? × 6 = 18, and 18 ÷ 6 = ?

equal Groups

Thereare3bagswith6plumsineachbag.Howmanyplumsarethereinall?

Measurement example.Youneed3lengthsofstring,each6incheslong.Howmuchstringwillyouneedaltogether?

If18plumsaresharedequallyinto3bags,thenhowmanyplumswillbeineachbag?

Measurement example.Youhave18inchesofstring,whichyouwillcutinto3equalpieces.Howlongwilleachpieceofstringbe?

If18plumsaretobepacked6toabag,thenhowmanybagsareneeded?

Measurement example.Youhave18inchesofstring,whichyouwillcutintopiecesthatare6incheslong.Howmanypiecesofstringwillyouhave?

arrays,4 area5

Thereare3rowsofappleswith6applesineachrow.Howmanyapplesarethere?

Area example.Whatistheareaofa3cmby6cmrectangle?

If18applesarearrangedinto3equalrows,howmanyappleswillbeineachrow?

Area example.Arectanglehasarea18squarecentimeters.Ifonesideis3cmlong,howlongisasidenexttoit?

If18applesarearrangedintoequalrowsof6apples,howmanyrowswilltherebe?

Area example.Arectanglehasarea18squarecentimeters.Ifonesideis6cmlong,howlongisasidenexttoit?

Compare

Abluehatcosts$6.Aredhatcosts3timesasmuchasthebluehat.Howmuchdoestheredhatcost?

Measurement example.Arubberbandis6cmlong.Howlongwilltherubberbandbewhenitisstretchedtobe3timesaslong?

Aredhatcosts$18andthatis3timesasmuchasabluehatcosts.Howmuchdoesabluehatcost?

Measurement example.Arubberbandisstretchedtobe18cmlongandthatis3timesaslongasitwasatfirst.Howlongwastherubberbandatfirst?

Aredhatcosts$18andabluehatcosts$6.Howmanytimesasmuchdoestheredhatcostasthebluehat?

Measurement example.Arubberbandwas6cmlongatfirst.Nowitisstretchedtobe18cmlong.Howmanytimesaslongistherubberbandnowasitwasatfirst?

General a×b=? a×? =p,andp÷ a=? ?×b=p, and p÷ b =?

7Thefirstexamplesineachcellareexamplesofdiscretethings.Theseareeasierforstudentsandshouldbegivenbeforethemeasurementexamples.

4Thelanguageinthearrayexamplesshowstheeasiestformofarrayproblems.Aharderformistousethetermsrowsandcolumns:Theapplesinthegrocerywindowarein3rowsand6columns.Howmanyapplesareinthere?Bothformsarevaluable.5Areainvolvesarraysofsquaresthathavebeenpushedtogethersothattherearenogapsoroverlaps,soarrayproblemsincludetheseespeciallyimportantmeasurementsituations.


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Table 3.Thepropertiesofoperations.Herea,bandcstandforarbitrarynumbersinagivennumbersystem.Thepropertiesofoperationsapplytotherationalnumbersystem,therealnumbersystem,andthecomplexnumbersystem.

Associative property of addition

Commutative property of addition

Additive identity property of 0

Existence of additive inverses

Associative property of multiplication

Commutative property of multiplication

Multiplicative identity property of 1

Existence of multiplicative inverses

Distributive property of multiplication over addition

(a + b)+ c = a + (b + c)

a + b= b + a

a + 0 = 0+ a= a

Foreveryathereexists–asothata+(–a)= (–a)+a=0.

(a × b)× c = a × (b × c)

a × b= b × a

a × 1= 1×a= a

Foreverya≠0thereexists1/asothata×1/a=1/a× a=1.

a ×(b + c) = a × b + a × c

Table 4.Thepropertiesofequality.Herea,bandcstandforarbitrarynumbersintherational,real,orcomplexnumbersystems.

Reflexive property of equality

Symmetric property of equality

Transitive property of equality

Addition property of equality

Subtraction property of equality

Multiplication property of equality

Division property of equality

Substitution property of equality

a=a

If a = b,then b = a.

If a = b and b = c,then a = c.

If a = b,then a + c = b + c.

If a = b,then a – c = b – c.

If a = b,then a × c = b × c.

If a = b and c ≠ 0,then a ÷ c = b ÷ c.

Ifa=b,thenbmaybesubstitutedfora

inanyexpressioncontaininga.

Table 5.Thepropertiesofinequality.Herea,bandcstandforarbitrarynumbersintherationalorrealnumbersystems.

Exactlyoneofthefollowingistrue:a<b,a=b,a>b.

Ifa>bandb>cthena>c.

Ifa>b,thenb<a.

Ifa>b,then–a<–b.

If a>b,thena±c>b±c.

Ifa>bandc>0,thena ×c>b ×c.

Ifa>bandc<0,thena ×c<b ×c.

Ifa>bandc>0,thena ÷c>b ÷c.

Ifa>bandc<0,thena ÷c<b ÷c.

Common Core State StandardS for matHematICSw

or

KS

Co

nS

Ult

ed

| 91

Existingstatestandardsdocuments.

ResearchsummariesandbriefsprovidedtotheWorkingGroupbyresearchers.

NationalAssessmentGoverningBoard,Mathematics Framework for the 2009 National Assessment of Educational Progress.U.S.DepartmentofEducation,2008.

NAEPValidityStudiesPanel,ValidityStudyoftheNAEPMathematicsAssessment:Grades4and8.Daroetal.,2007.

Mathematicsdocumentsfrom:Alberta,Canada;Belgium;China;ChineseTaipei;Denmark;England;Finland;HongKong;India;Ireland;Japan;Korea;NewZealand;Singapore;Victoria(BritishColumbia).

AddingitUp:HelpingChildrenLearnMathematics.NationalResearchCouncil,MathematicsLearningStudyCommittee,2001.

BenchmarkingforSuccess:EnsuringU.S.StudentsReceiveaWorld-ClassEducation.NationalGovernorsAssociation,CouncilofChiefStateSchoolOfficers,andAchieve,Inc.,2008.

Crossroads in Mathematics(1995)andBeyond Crossroads (2006).AmericanMathematicalAssociationofTwo-YearColleges(AMATYC).

Curriculum Focal Points for Prekindergarten through Grade 8 Mathematics: A Quest for Coherence.NationalCouncilofTeachersofMathematics,2006.

Focus in High School Mathematics: Reasoning and Sense Making.NationalCouncilofTeachersofMathematics.Reston,VA:NCTM.

Foundations for Success: The Final Report of the National Mathematics Advisory Panel. U.S.DepartmentofEducation:Washington,DC,2008.

Guidelines for Assessment and Instruction in Statistics Education (GAISE) Report: A PreK-12 Curriculum Framework.

How People Learn: Brain, Mind, Experience, and School. Bransford,J.D.,Brown,A.L.,andCocking,R.R.,eds.CommitteeonDevelopmentsintheScienceofLearning,CommissiononBehavioralandSocialSciencesandEducation,NationalResearchCouncil,1999.

Mathematics and Democracy, The Case for Quantitative Literacy,Steen,L.A.(ed.).NationalCouncilonEducationandtheDisciplines,2001.

Mathematics Learning in Early Childhood: Paths Toward Excellence and Equity. Cross,C.T.,Woods,T.A.,andSchweingruber,S.,eds.CommitteeonEarlyChildhoodMathematics,NationalResearchCouncil,2009.

The Opportunity Equation: Transforming Mathematics and Science Education for Citizenship and the Global Economy.TheCarnegieCorporationofNewYorkandtheInstituteforAdvancedStudy,2009.Online:http://www.opportunityequation.org/

Principles and Standards for School Mathematics.NationalCouncilofTeachersofMathematics,2000.

The Proficiency Illusion.Cronin,J.,Dahlin,M.,Adkins,D.,andKingsbury,G.G.;forewordbyC.E.Finn,Jr.,andM.J.Petrilli.ThomasB.FordhamInstitute,2007.

Ready or Not: Creating a High School Diploma That Counts.AmericanDiplomaProject,2004.

A Research Companion to Principles and Standards for School Mathematics.NationalCouncilofTeachersofMathematics,2003.

Sizing Up State Standards 2008.AmericanFederationofTeachers,2008.

A Splintered Vision: An Investigation of U.S. Science and Mathematics Education.Schmidt,W.H.,McKnight,C.C.,Raizen,S.A.,etal.U.S.NationalResearchCenterfortheThirdInternationalMathematicsandScienceStudy,MichiganStateUniversity,1997.

Stars By Which to Navigate? Scanning National and International Education Standards in 2009.Carmichael,S.B.,Wilson.W.S,Finn,Jr.,C.E.,Winkler,A.M.,andPalmieri,S.ThomasB.FordhamInstitute,2009.

Askey,R.,“KnowingandTeachingElementaryMathematics,”American Educator,Fall1999.

Aydogan,C.,Plummer,C.,Kang,S.J.,Bilbrey,C.,Farran,D.C.,&Lipsey,M.W.(2005).Aninvestigationofprekindergartencurricula:Influencesonclassroomcharacteristicsandchildengagement.PaperpresentedattheNAEYC.

Blum,W.,Galbraith,P.L.,Henn,H-W.andNiss,M.(Eds)Applications and Modeling in Mathematics Education,ICMIStudy14.Amsterdam:Springer.

Brosterman,N.(1997).Inventing kindergarten.NewYork:HarryN.Abrams.

Clements,D.H.,&Sarama,J.(2009).Learning and teaching early math: The learning trajectories approach.NewYork:Routledge.

Clements,D.H.,Sarama,J.,&DiBiase,A.-M.(2004).Clements,D.H.,Sarama,J.,&DiBiase,A.-M.(2004).Engaging young children in mathematics: Standards for early childhood mathematics education.Mahwah,NJ:LawrenceErlbaumAssociates.

CobbandMoore,“Mathematics,Statistics,andTeaching,”Amer. Math. Monthly104(9),pp.801-823,1997.

Confrey,J.,“TracingtheEvolutionofMathematicsContentStandardsintheUnitedStates:LookingBackandProjectingForward.”K12MathematicsCurriculumStandardsconferenceproceedings,February5-6,2007.

Conley,D.T.Knowledge and Skills for University Success,2008.

Conley,D.T.Toward a More Comprehensive Conception of College Readiness,2007.

Cuoco,A.,Goldenberg,E.P.,andMark,J.,“HabitsofMind:AnOrganizingPrincipleforaMathematicsCurriculum,”Journal of Mathematical Behavior,15(4),375-402,1996.

Carpenter,T.P.,Fennema,E.,Franke,M.L.,Levi,L.,&Empson,S.B.(1999).Children’s Mathematics: Cognitively Guided Instruction.Portsmouth,NH:Heinemann.

VandeWalle,J.A.,Karp,K.,&Bay-Williams,J.M.(2010).Elementary and Middle School Mathematics: Teaching Developmentally(Seventhed.).Boston:AllynandBacon.

Ginsburg,A.,Leinwand,S.,andDecker,K.,“InformingGrades1-6StandardsDevelopment:WhatCanBeLearnedfromHigh-PerformingHongKong,Korea,andSingapore?”AmericanInstitutesforResearch,2009.

Ginsburgetal.,“WhattheUnitedStatesCanLearnFromSingapore’sWorld-ClassMathematicsSystem(andwhatSingaporecanlearnfromtheUnitedStates),”AmericanInstitutesforResearch,2005.

Ginsburgetal.,“ReassessingU.S.InternationalMathematicsPerformance:NewFindingsfromthe2003TIMMSandPISA,”AmericanInstitutesforResearch,2005.

Ginsburg,H.P.,Lee,J.S.,&Stevenson-Boyd,J.(2008).Mathematicseducationforyoungchildren:Whatitisandhowtopromoteit.Social Policy Report,22(1),1-24.

SampleofWorksConsulted


or

KS

Co

nS

Ult

ed

| 92

Harel,G.,“WhatisMathematics?APedagogicalAnswertoaPhilosophicalQuestion,”inR.B.GoldandR.Simons(eds.),Current Issues in the Philosophy of Mathematics from the Perspective of Mathematicians.MathematicalAssociationofAmerica,2008.

Henry,V.J.,&Brown,R.S.(2008).First-gradebasicfacts:Aninvestigationintoteachingandlearningofanaccelerated,high-demandmemorizationstandard.Journal for Research in Mathematics Education,39,153-183.

Howe,R.,“FromArithmetictoAlgebra.”

Howe,R.,“StartingOffRightinArithmetic,”http://math.arizona.edu/~ime/2008-09/MIME/BegArith.pdf.

Jordan,N.C.,Kaplan,D.,Ramineni,C.,andLocuniak,M.N.,“Earlymathmatters:kindergartennumbercompetenceandlatermathematicsoutcomes,”Dev. Psychol.45,850–867,2009.

Kader,G.,“MeansandMADS,”Mathematics Teaching in the Middle School,4(6),1999,pp.398-403.

Kilpatrick,J.,Mesa,V.,andSloane,F.,“U.S.AlgebraPerformanceinanInternationalContext,”inLoveless(ed.),Lessons Learned: What International Assessments Tell Us About Math Achievement.Washington,D.C.:BrookingsInstitutionPress,2007.

Leinwand,S.,andGinsburg,A.,“MeasuringUp:HowtheHighestPerformingState(Massachusetts)ComparestotheHighestPerformingCountry(HongKong)inGrade3Mathematics,”AmericanInstitutesforResearch,2009.

Niss,M.,“QuantitativeLiteracyandMathematicalCompetencies,”inQuantitative Literacy: Why Numeracy Matters for Schools and Colleges,Madison,B.L.,andSteen,L.A.(eds.),NationalCouncilonEducationandtheDisciplines.ProceedingsoftheNationalForumonQuantitativeLiteracyheldattheNationalAcademyofSciencesinWashington,D.C.,December1-2,2001.

Pratt,C.(1948).Ilearnfromchildren.NewYork:SimonandSchuster.

Reys,B.(ed.),The Intended Mathematics Curriculum as Represented in State-Level Curriculum Standards: Consensus or Confusion? IAP-InformationAgePublishing,2006.

Sarama,J.,&Clements,D.H.(2009).Early childhood mathematics education research: Learning trajectories for young children.NewYork:Routledge.

Schmidt,W.,Houang,R.,andCogan,L.,“ACoherentCurriculum:TheCaseofMathematics,”American Educator,Summer2002,p.4.

Schmidt,W.H.,andHouang,R.T.,“LackofFocusintheIntendedMathematicsCurriculum:SymptomorCause?”inLoveless(ed.),Lessons Learned: What International Assessments Tell Us About Math Achievement.Washington,D.C.:BrookingsInstitutionPress,2007.

Steen,L.A.,“FacingFacts:AchievingBalanceinHighSchoolMathematics.”Mathematics Teacher,Vol.100.SpecialIssue.

Wu,H.,“Fractions,decimals,andrationalnumbers,”2007,http://math.berkeley.edu/~wu/(March19,2008).

Wu,H.,“LectureNotesforthe2009Pre-AlgebraInstitute,”September15,2009.

Wu,H.,“Preserviceprofessionaldevelopmentofmathematicsteachers,”http://math.berkeley.edu/~wu/pspd2.pdf.

MassachusettsDepartmentofEducation.ProgressReportoftheMathematicsCurriculumFrameworkRevisionPanel,MassachusettsDepartmentofElementaryandSecondaryEducation,2009.

www.doe.mass.edu/boe/docs/0509/item5_report.pdf.

ACTCollegeReadinessBenchmarks™

ACTCollegeReadinessStandards™

ACTNationalCurriculumSurvey™

Adelman,C.,The Toolbox Revisited: Paths to Degree Completion From High School Through College,2006.

Advanced Placement Calculus, Statistics and Computer Science Course Descriptions.May 2009, May 2010. CollegeBoard,2008.

Aligning Postsecondary Expectations and High School Practice: The Gap Defined(ACT:PolicyImplicationsoftheACTNationalCurriculumSurveyResults2005-2006).

Condition of Education, 2004: Indicator 30, Top 30 Postsecondary Courses, U.S.DepartmentofEducation,2004.

Condition of Education, 2007: High School Course-Taking.U.S.DepartmentofEducation,2007.

Crisis at the Core: Preparing All Students for College and Work,ACT.

Achieve,Inc.,FloridaPostsecondarySurvey,2008.

Golfin,Peggy,etal.CNACorporation.Strengthening Mathematics at the Postsecondary Level: Literature Review and Analysis,2005.

Camara,W.J.,Shaw,E.,andPatterson,B.(June13,2009).FirstYearEnglishandMathCollegeCoursework.CollegeBoard:NewYork,NY(Availablefromauthors).

CLEPPrecalculusCurriculumSurvey:SummaryofResults.TheCollegeBoard,2005.

CollegeBoardStandardsforCollegeSuccess:MathematicsandStatistics.CollegeBoard,2006.

Miller,G.E.,Twing,J.,andMeyers,J.“HigherEducationReadinessComponent(HERC)CorrelationStudy.”Austin,TX:Pearson.

On Course for Success: A Close Look at Selected High School Courses That Prepare All Students for College and Work,ACT.

Out of Many, One: Towards Rigorous Common Core Standards from the Ground Up.Achieve,2008.

Ready for College and Ready for Work: Same or Different?ACT.

RigoratRisk:ReaffirmingQualityintheHighSchoolCoreCurriculum,ACT.

The Forgotten Middle: Ensuring that All Students Are on Target for College and Career Readiness before High School,ACT.

Achieve,Inc.,VirginiaPostsecondarySurvey,2004.

ACTJobSkillComparisonCharts.

Achieve,MathematicsatWork,2008.

The American Diploma Project Workplace Study.NationalAllianceofBusinessStudy,2002.

Carnevale,AnthonyandDesrochers,Donna.Connecting Education Standards and Employment: Course-taking Patterns of Young Workers,2002.

ColoradoBusinessLeaders’TopSkills,2006.

Hawai’i Career Ready Study: access to living wage careers from high school,2007.

States’CareerClusterInitiative.Essential Knowledge and Skill Statements,2008.

ACTWorkKeysOccupationalProfiles™.

ProgramforInternationalStudentAssessment(PISA),2006.

TrendsinInternationalMathematicsandScienceStudy(TIMSS),2007.


or

KS

Co

nS

Ult

ed

| 93

InternationalBaccalaureate,MathematicsStandardLevel,2006.

UniversityofCambridgeInternationalExaminations:GeneralCertificateofSecondaryEducationinMathematics,2009.

EdExcel,GeneralCertificateofSecondaryEducation,Mathematics,2009.

Blachowicz,Camille,andFisher,Peter.“VocabularyInstruction.”InHandbook of Reading Research,VolumeIII,editedbyMichaelKamil,PeterMosenthal,P.DavidPearson,andRebeccaBarr,pp.503-523.Mahwah,NJ:LawrenceErlbaumAssociates,2000.

Gándara,Patricia,andContreras,Frances.The Latino Education Crisis: The Consequences of Failed Social Policies.Cambridge,Ma:HarvardUniversityPress,2009.

Moschkovich,JuditN.“SupportingtheParticipationofEnglishLanguageLearnersinMathematicalDiscussions.”For the Learning of Mathematics19(March1999):11-19.

Moschkovich,J.N.(inpress).Language,culture,andequityinsecondarymathematicsclassrooms.ToappearinF.Lester&J.Lobato(ed.),TeachingandLearningMathematics:TranslatingResearchtotheSecondaryClassroom, Reston,VA:NCTM.

Moschkovich,JuditN.“ExaminingMathematicalDiscoursePractices,”For the Learning of Mathematics 27 (March2007):24-30.

Moschkovich,JuditN.“UsingTwoLanguageswhenLearningMathematics:HowCanResearchHelpUsUnderstandMathematicsLearnersWhoUseTwoLanguages?”Research Brief and Clip,NationalCouncilofTeachersofMathematics,2009http://www.nctm.org/uploadedFiles/Research_News_and_Advocacy/Research/Clips_and_Briefs/Research_brief_12_Using_2.pdf.(accessedNovember25,2009).

Moschkovich,J.N.(2007)BilingualMathematicsLearners:Howviewsoflanguage,bilinguallearners,andmathematicalcommunicationimpactinstruction.InNasir,N.andCobb,P.(eds.),Diversity, Equity, and Access to Mathematical Ideas.NewYork:TeachersCollegePress,89-104.

Schleppegrell,M.J.(2007).Thelinguisticchallengesofmathematicsteachingandlearning:Aresearchreview.Reading & Writing Quarterly, 23:139-159.

IndividualswithDisabilitiesEducationAct(IDEA),34CFR§300.34(a).(2004).

IndividualswithDisabilitiesEducationAct(IDEA),34CFR§300.39(b)(3).(2004).

OfficeofSpecialEducationPrograms,U.S.DepartmentofEducation.“IDEARegulations:IdentificationofStudentswithSpecificLearningDisabilities,”2006.

Thompson,S.J.,Morse,A.B.,Sharpe,M.,andHall,S.,“AccommodationsManual:HowtoSelect,AdministerandEvaluateUseofAccommodationsandAssessmentforStudentswithDisabilities,”2ndEdition.CouncilofChiefStateSchoolOfficers,2005.

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