FocusingLens(es) / RepresentationModeling / StandardsandGradeLevelExpectationsAddressedinthisUnit / MA10-GR.HS-S.2-GLE.1
InquiryQuestions(Engaging-Debatable): /
- Arethereanyexamplesofsomethingthatcan’tbemodeledbyafunction?(MA10-GR.HS-S.2-GLE.1-EO.b)
UnitStrands / Functions:InterpretingFunctions
Concepts / Function,input,output,domain,range,context/situation,visualization,coordinateplane,keyfeaturesoffunctions,maxima,minima,intercepts,averagerateofchange,functionalrepresentations,comparisons,interpretation
Generalizations
MystudentswillUnderstandthat…
GuidingQuestions
FactualConceptual
Functionsdescribesituationswhereeachinputdeterminesexactlyoneoutput.(MA10-GR.HS-S.2-GLE.1-EO.a) / Givenaninputandoutput,howdoyoudeterminearule?
Whatnotationisusedtowriteafunction?Whatdoesy=f(x)denote? / Whyisonlyoneoutputpermissibleforeveryinputinafunction?
Whyispossibletohavetwoinputsforoneoutput?
Whyarefunctionsanimportanttoolinmathematicalmodeling?
Mathematicianslimitdomainsofafunctiontoensureboththedomainandrangemakesenseinagivencontext. (MA10-GR.HS-S.2-GLE.1-EO.b) / Whatisanothernamefortheinputsofafunction?
Outputs?
Whatistherelationshipbetweendomainandrangeofafunction?
Howcanyouquantifytherelationshipbetweentwovariables? / Howdoyoudetermineareasonabledomainandrangeforacontext?
Whyisitnecessarytoconstrainthedomainandrangeofafunctionmodel?
Visualizingavarietyoffunctionsonacoordinateplanehelpstointerpretkeyfeatures,suchasdomain,range,maxima,minima,intercepts,symmetry,endbehaviorandaveragerateofchange.(MA10-GR.HS-S.2-GLE.1-EO.b.i) / Whatareimportantcharacteristicsofafunctionthatcanbeseenonagraph?
Whatdothegraphsoflinear,exponential,squareroot,cuberoot,stepandabsolutevaluefunctionslooklike?
Whatistherelationshipbetweenanaveragerateofchangeofanyfunctionandtheslopeofalinearfunction? / Whyaremultipletypesoffunctionsneededtomodelrealworldphenomena?
Howdoesvisualizingafunctionhelpinterprettherelationshipbetweentwovariables?
Howisthegraphofanequationrelatedtoitssolutions?(A.REI.10)
Mathematicianuseavarietyoffunctionalrepresentationstocompareandinterpretthedifferencesandsimilaritiesoffunctions.(MA10-GR.HS-S.2-GLE.1-EO.c.v.3) / Whataredifferenttypesoffunctionalrepresentations?
Howdographs,equations,andtablesshowthesimilarityanddifferencesoffunctions? / Whyisitimportanttointerpretthedifferencesandsimilaritiesoffunctionsthroughmultiplerepresentations?
KeyKnowledgeandSkills:
Mystudentswill…
Whatstudentswillknowandbeabletodoaresocloselylinkedintheconcept-baseddisciplineofmathematics.Therefore,inthemathematics
sampleswhatstudentsshouldknowanddoarecombined.
- Understandafunctionfromoneset(calledthedomain)toanotherset(calledtherange)assignstoeachelementofthedomainexactlyoneelementoftherange;iffisafunctionandxisanelementofitsdomain,thenf(x)denotestheoutputoffcorrespondingtotheinputx.Thegraphoffisthegraphoftheequationy=f(x).
- Understand theconceptofa function
- Determine whethera relationbetweentwoquantitiescan bedescribed byafunction
- Identifythe domainand rangeofafunction
- Usefunctionnotationandevaluatefunctionsforinputsintheirdomains.(MA10-GR.HS-S.2-GLE.1-EO.a.ii)
- Use functionnotation
- Evaluate functions forgiven inputs
- Interpretstatementsthat use functionnotation in areal-world context
- Interpretstatementsthatusefunctionnotationintermsofacontext.(MA10-GR.HS-S.2-GLE.1-EO.a.ii)
- Use functionnotation
- Evaluate functions forgiven inputs
- Interpretstatementsthat use functionnotation in areal-world context
- Interpretkeyfeaturesofgraphsandtables,forafunctionthatmodelsarelationshipbetweentwoquantities,intermsofthequantitiesandsketchgraphsshowingkeyfeaturesgivenaverbaldescriptionoftherelationship;keyfeaturesinclude:intercepts;intervalswherethefunctionisincreasing,decreasing,positive,ornegative;relativemaximumsandminimums;symmetries;endbehavior;andperiodicity.(MA10-GR.HS-S.2-GLE.1-EO.b.i)
- Interpretkeyfeatures ofgraphs oflinear andexponentialfunctions
- Intercepts
- Intervalswhere thefunction is increasingor decreasing
- Positiveor negative slopes
- Endbehavior
- Sketchgraphs offunctions showingkeyfeaturesgivena verbaldescription ofthe relationshipsbetweentwoquantities
- Relatethedomainofafunctiontoitsgraphand,whereapplicable,tothequantitativerelationshipitdescribes.(MA10-GR.HS-S.2-GLE.1-EO.b.ii)
- Understand therelationshipbetweendomainandindependentvariable
- Relate the domain ofafunction to its graph
- Interpretdomainofafunction in a real-world context
- Calculateandinterprettheaveragerateofchangeofafunction(presentedsymbolicallyorasatable)overaspecifiedintervalandestimatetherateofchangefromagraph.(MA10-GR.HS-S.2-GLE.1-EO.b.iii)
- Interpret theaveragerateof change ofalinearorexponentialfunctionover a specifiedinterval
- Estimate the rate ofchangefrom agraph
- Comparepropertiesoftwofunctionseachrepresentedinadifferentway(algebraically,graphically,numericallyintables,orbyverbaldescriptions);limitfunctionstolinear,squareroot,cuberoot,piece-wise,step,absolutevalueandexponentialwithintegerdomain.(MA10-GR.HS-S.2-GLE.1-EO.c.v.3)
- Compare twolinearfunctions eachrepresented in a different way(algebraically,graphically, in tables, orbyverbaldescription)
- Compare twoexponentialfunctionseach represented in a different way(algebraically,graphically,intables, orbyverbaldescription)
EXAMPLE:AstudentinLanguageArtscandemonstratetheabilitytoapplyandcomprehendcriticallanguagethroughthefollowingstatement:“MarkTwainexposesthehypocrisyofslaverythroughtheuseofsatire.”
Astudent incandemonstratetheabilitytoapplyandcomprehendcriticallanguagethroughthefollowingstatement(s): / Whenusingfunctionstomodelrealworldphenomenaitisimportanttoconstrainthedomainandrangetoinputsandoutputsthatmakesensewithinthecontextofthemodel.
AcademicVocabulary: / Interpret,transform,represent,relationships,compare,distinguish,increasing,decreasing,symmetry,input,output,visualization,coordinateplane,
TechnicalVocabulary: / Function,domain,range,functionalrepresentations,piecewisefunction,squareroot,cuberoot,linear,exponential,squareroot,cuberoot,andpiecewise-definedfunctions,stepfunctions,absolutevalues,keyfeaturesoffunctions,maxima,minima,averagerateofchange,endbehavior,intercepts,