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AlgebraIICurriculum PASSAICCOUNTYTECHNICALINSTITUTE 45ReinhardtRoad Wayne,NewJersey REVISED2016 AlgebraIICurriculum AlgebraIIOverview–Buildingontheunderstandingoflinear,quadraticandexponentialfunctions fromAlgebraI,thiscoursewillextendfunctionconceptstoincludepolynomial,rational,andradical functions.Thestandardsinthiscoursecontinuetheworkofmodelingsituationsandsolving equations. Unit1standardswillbuildonthestudents’previousknowledgeoffunctions,trigonometricratiosandcirclesin geometrytoextendtrigonometrytomodelperiodicphenomena.Standardwillcontinuewithanalyzationof trigonometricidentities. Unit2willextendstudent’salgebraknowledgeofpolynomials.Standardwillcontinuewithfactoring,solvingand graphingpolynomials,focusingontransformationsofgraphs. Unit3willextendstudent’salgebraknowledgeofrational,radical,andexponentialfunctions.Standardwill continuewithlogarithmicfunctions.Thegraphsofallthefunctionsinunit3willbemodeledandtheir transformationswillbeanalyzed. Unit4willinvestigateinversefunctions,usingcompositions.Theunitwillcontinuewithrelationandvisual displaysandsummarystatisticslearnedinpriorcoursestodifferenttypesofdataandtoprobabilitydistributions. Samples,surveys,experimentsandsimulationswillbeusedasmethodstocollectdata. PCTIMATHEMATICSDEPARTMENT COURSE:AlgebraII UNIT:1 UNITNAME:TRIGONOMETRY TOPICS Green Book Blue Book Obj. # STUDENTLEARNINGOBJECTIVES CCSScode 0.SummerPacketReview(10days) 10days 1 Reviewkeyconceptsfromalgebra1withinthesummerpacket. I. TRIGONOMETRICRATIOSANDFUNCTIONS(35days) 7days 9.2 10.3 1 Understandtheradianmeasureofanangleasthelengthofthearcontheunitcircle subtendedbytheangle. F.TF.1 7days 9.2;9.3 10.3 2 Explainhowtheunitcircleinthecoordinateplaneenablestheextensionof trigonometricfunctionstoallrealnumbers. F.TF.2 7days 10.1 11.1; 11.2 3 Graphtrigonometricfunctions,showingperiod,midline,andamplitude. F.IF.4,F.IF.7E, F.BF.3 7days 10.2 11.1; 11.2 4 Choosetrigonometricfunctionstomodelperiodicphenomenawithspecified amplitude,frequency,andmidline.** F.TF.5 7days 10.3 11.3 5 VerifyreciprocaltrigonometricandPythagoreanidentities. F.TF.8 **TaughtinApps/NotConceptsorResource TECHNOLOGYSTANDARDS I.1.Understandtheradianmeasureofanangleasthelengthofthearcontheunitcirclesubtendedbytheangle. https://www.youtube.com/watch?v=cwL3uktPLEs https://www.youtube.com/watch?v=sw5QNMLpep0 13.2–DefineGeneralAnglesandUseRadianMeasure.ppt http://www.youtube.com/watch?v=vlSb74Gk8yc http://www.youtube.com/watch?v=N6Lsjb4s5Ak http://ceemrr.com/Geometry2/GeneralAngles/GeneralAngles_print.html 13_2Notes.doc-MiraCostaHighSchool I.2.Explainhowtheunitcircleinthecoordinateplaneenablestheextensionoftrigonometricfunctionstoallreal numbers. http://www.youtube.com/watch?v=cNjzynK5QqE http://cims.nyu.edu/~kiryl/Precalculus/Section_6.3Trigonometric%20Functions%20of%20Angles/Trigonometric%20Functions%20of%20Angles.pdf http://www.youtube.com/watch?v=Cg70E506maw http://www.youtube.com/watch?v=zFvxHQtUHrQ http://www.youtube.com/watch?v=sb9oZZeBNhg I.3.Graphtrigonometricfunctions,showingperiod,midline,andamplitude. http://www.mathsisfun.com/algebra/trig-sin-cos-tan-graphs.html http://www.analyzemath.com/unitcircle/unit_circle_applet.html http://www.youtube.com/watch?v=pxFZCTAl9G8 http://exchange.smarttech.com/search.html?q=%22Sine%20%20Cosine%22 http://youtu.be/yR7y8hyOpDU http://youtu.be/c1VD_LEs5ZY I.4.Choosetrigonometricfunctionstomodelperiodicphenomenawithspecifiedamplitude,frequency,andmidline.** TranslateandReflectTrigonometricGraphs-MiraCostaHighSchool 10-2TranslateandReflectTrigonometricGraphs http://youtu.be/CuvO9-Zk2Xc http://youtu.be/0KxVA7NWZiM I.5.Verifyreciprocaltrigonometricandpythagoreanidentities. ProvingTrigonometricIdentities-Purplemath Verifyingtrigonometricidentities-SlideShare http://youtu.be/vA6451TpSig http://youtu.be/TCdhf9iVkYc http://youtu.be/q8k-sS7qRts http://youtu.be/mAnw4ImaPK0 http://youtu.be/a70-dYvDJZY KEY VOCABULARY • • • • • • • • • • • • • • • • • • • • • • • • • • Sine Cosine Tangent Cosecant Secant Cotangent Initialside Terminalside Standardposition Coterminal Radian Sector Centralangle Unitcircle Quadrantalangle Referenceangle Amplitude Periodicfunction Cycle Period Frequency Translation Reflection Amplitude Period Trigonometric identity SelectedOpportunitiesforConnectionsto MathematicalPractices 1. 2. 3. 4. 5. 6. 7. 8. Makesenseofproblemsandpersevereinsolvingthem. Reasonabstractlyandquantitatively. Constructviableargumentsandcritiquethereasoningofothers. Modelwithmathematics. Useappropriatetoolsstrategically. Attendtoprecision. Lookforandmakeuseofstructure. Lookforandexpressregularityinrepeatedreasoning. Unit1-LinkstoOpenEndedProblems RelatedtotheStandards F.TF.2 F.TF.3 F.IF.7 F.TF.5 F.TF.8 CCSS Code F.TF.1 F.TF.2 F.IF.7.E F.BF.3 F.TF.5 F.TF.8 CCSSCodeDESRCIPTION Understandradianmeasureofanangleasthelengthofthearcontheunitcirclesubtendedbytheangle. Explainhowtheunitcircleinthecoordinateplaneenablestheextensionoftrigonometricfunctionstoallrealnumbers,interpretedasradian measuresofanglestraversedcounterclockwisearoundtheunitcircle. Graphexponentialandlogarithmicfunctions,showinginterceptsandendbehavior,andtrigonometricfunctions,showingperiod,midline,and amplitude. Identifytheeffectonthegraphofreplacingf(x)byf(x)+k,kf(x),f(kx),andf(x+k)forspecificvaluesofk(bothpositiveandnegative);findthe valueofkgiventhegraphs.Experimentwithcasesandillustrateanexplanationoftheeffectsonthegraphusingtechnology.Includerecognizing evenandoddfunctionsfromtheirgraphsandalgebraicexpressionsforthem. Choosetrigonometricfunctionstomodelperiodicphenomenawithspecifiedamplitude,frequency,andmidline.* ProvethePythagoreanidentitysin$ 𝜃 + cos $ 𝜃 = 1anduseittofindsin(θ),cos(θ),ortan(θ)givensin(θ),cos(θ),ortan(θ)andthequadrantofthe angle. PCTIMATHEMATICSDEPARTMENT COURSE:AlgebraII UNIT:2 UNITNAME:POLYNOMIALS TOPICS Green Book II. Blue Book Obj. # STUDENTLEARNINGOBJECTIVES POLYNOMIALS(45days) 7days 1.7,1.8 2.3,2.4, 2.6 1 Solvequadraticequationswithrealcoefficientsthathavecomplexsolutions. A.REI.4.b 5days 1.6 2.9 2 UsePropertiesofoperationstoadd,subtract,andmultiplycomplexnumbers. N.CN.1,2,7 6days 1.2 2.1 3 GraphQuadraticFunctionsInVertexandInterceptForm. A.SSE.3a,F.IF.4, 7,F.BF.3 5days 2.3 3.1-3.2 4 Performingarithmeticoperationsonpolynomials. A.APR.4 5days 2.5 3.3,3.4 5 Usetheremainderandfactortheorems. A.APR.2 7days 2.4 3.4 6 Useanappropriatefactoringtechniquetofactorexpressionscompletely. A.SSE.2,A.APR.3 5days 2.7 3.6 7 KnowtheFundamentalTheoremofAlgebra;showthatitistrueforquadratic polynomials. N.CN.9 5days 2.2,2.8 3.7 8 Interpretkeyfeaturesofpolynomialgraphsintermsofthequantities,andsketch graphsshowingkeyfeaturesgivenaverbaldescriptionoftherelationship. F.IF.4,F.BF.3 **TaughtinApps/NotConceptsorResource CCSScode TECHNOLOGYSTANDARDS II.1.Solvequadraticequationswithrealcoefficientsthathavecomplexsolutions. http://www.webgraphing.com/quadraticequation_completingthesquare.jsp https://www.khanacademy.org/math/algebra/quadratics/completing_the_square/v/solving-quadratic-equations-bysquare-roots https://www.khanacademy.org/math/algebra/quadratics/quadratic_formula/v/using-the-quadratic-formula http://www.webgraphing.com/quadraticequation_quadraticformula.jsp http://www.mathplanet.com/education/algebra-1/exponents-and-exponential-functions/properties-of-exponents II.2.UsePropertiesofoperationstoadd,subtract,andmultiplycomplexnumbers. http://www.purplemath.com/modules/complex2.htm http://www.youtube.com/watch?v=MEuPzvh0roM II.3.GraphQuadraticFunctionsInVertexandInterceptForm. https://www.desmos.com/calculator http://www.youtube.com/watch?v=y99lNRqLjBA http://vimeo.com/47178923 II.4.Performingarithmeticoperationsonpolynomials. http://www.virtualnerd.com/algebra-1/polynomials-and-factoring/add-subtract/ http://www.virtualnerd.com/algebra-1/polynomials-and-factoring/multiply-divide/ II.5.Usetheremainderandfactortheorems. http://www.virtualnerd.com/algebra-1/polynomials-and-factoring/factoring-strategy-solving-equations/choosingstrategy-grouping/ http://www.youtube.com/watch?v=_IPqCaspZOs II.6.Useanappropriatefactoringtechniquetofactorexpressionscompletely. http://www.virtualnerd.com/algebra-1/polynomials-and-factoring/factoring-strategy-solving-equations/solvingequations/ http://www.virtualnerd.com/algebra-1/polynomials-and-factoring/factoring-strategy-solving-equations/choosingstrategy-grouping/ II.7.KnowtheFundamentalTheoremofAlgebra;showthatitistrueforquadraticpolynomials. http://www.mathsisfun.com/algebra/fundamental-theorem-algebra.html http://www.youtube.com/watch?v=ox3zyPKRnlM II.8.Interpretkeyfeaturesofpolynomialgraphsintermsofthequantities,andsketchgraphsshowingkeyfeatures givenaverbaldescriptionoftherelationship. http://www.youtube.com/watch?v=Vl4pBa_XroE http://www.youtube.com/watch?v=-LJ5Bt8UwCo KEYVOCABULARY • • • • • • • • • • • • • • • • • • • • • • • • • • • • • Complexconjugates Constantterm Factorbygrouping Factoredcompletely Irrationalconjugates Leadingcoefficient Liketerms Localmaximum Localminimum Polynomiallongdivision Quadraticform Repeatedsolution ScientificNotation Syntheticdivision Zeroofafunction Complexconjugates Constantterm Factorbygrouping Factoredcompletely Irrationalconjugates Leadingcoefficient Liketerms Localmaximum Localminimum Polynomiallongdivision Quadraticform Repeatedsolution ScientificNotation Syntheticdivision Zeroofafunction SelectedOpportunitiesforConnectionsto MathematicalPractices 1. 2. 3. 4. 5. 6. 7. 8. Makesenseofproblemsandpersevereinsolvingthem. Reasonabstractlyandquantitatively. Constructviableargumentsandcritiquethereasoningofothers. Modelwithmathematics. Useappropriatetoolsstrategically. Attendtoprecision. Lookforandmakeuseofstructure. Lookforandexpressregularityinrepeatedreasoning. CCSS Code Unit2-LinkstoOpenEndedProblems RelatedtotheStandards A.REI.4 N.CN.1 N.CN.2 N.CN.7 A.SSE.3a F.IF.4 F.IF.7 A.APR.1 A.APR.2 A.APR.3 A.APR.4 A.SSE.1 A.SSE.2 A.SSE.4 CCSSCodeDESRCIPTION A.REI.4 Solve quadratic equations in one variable. Use the method of completing the square to transform any quadratic equation in x into an equation of the form 𝑥 − 𝑝 $ = 𝑞 that has the same solutions. A.REI.4.A Derive the quadratic formula from this form. Solve quadratic equations by inspection (e.g., for 𝑥 $ = 49), taking square roots, completing the square, the quadratic formula and factoring, as appropriate A.REI.4.B to the initial form of the equation. Recognize when the quadratic formula gives complex solutions and write them as a ± bi for real numbers a and b. N.CN.1 Know there is a complex number i such that 𝑖 $ = −1, and every complex number has the form a + bi with a and b real. N.CN.2 Use the relation 𝑖 $ = −1 and the commutative, associative, and distributive properties to add, subtract, and multiply complex numbers. N.CN.7 Solve quadratic equations with real coefficients that have complex solutions. A.SSE.3.A Factor a quadratic expression to reveal the zeros of the function it defines. For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. Key features include: intercepts; intervals where the function is increasing, decreasing, F.IF.4 positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity.* Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases.* F.IF.7 F.IF.7.A Graph linear and quadratic functions and show intercepts, maxima, and minima. F.IF.7.B Graph square root, cube root, and piecewise-defined functions, including step functions and absolute value functions. Understand that polynomials form a system analogous to the integers, namely, they are closed under the operations of addition, subtraction, and multiplication; add, subtract, and multiply polynomials. Know and apply the Remainder Theorem: For a polynomial p(x) and a number a, the remainder on division by x - a is p(a), so p(a) = 0 if and only if (x - a) is a factor of p(x). Identify zeros of polynomials when suitable factorizations are available, and use the zeros to construct a rough graph of the function defined by the polynomial. Prove polynomial identities and use them to describe numerical relationships. For example, the polynomial identity 𝑥 $ + 𝑦 $ $ = 𝑥 $ − 𝑦 $ $ + 2𝑥𝑦 $ can be used to generate Pythagorean triples. Interpret expressions that represent a quantity in terms of its context.* A.APR.1 A.APR.2 A.APR.3 A.APR.4 A.SSE.1 A.SSE.1.A Interpret parts of an expression, such as terms, factors, and coefficients. Identify zeros of polynomials when suitable factorizations are available, and use the zeros to construct a rough graph of the function defined by the A.SSE.2 polynomial. A.SSE.4 Derive the formula for the sum of a finite geometric series (when the common ratio is not 1), and use the formula to solve problems. For example, calculate mortgage payments.* PCTIMATHEMATICSDEPARTMENT COURSE:AlgebraII UNIT:3 UNITNAME:RATIONAL,RADICAL,EXPONENTIAL,ANDLOGARITHMICFUNCTIONS TOPICS Green Book III. Blue Book Obj. # PerformarithmeticoperationswithRationalExpressions. Rewritesimplerationalexpressionsindifferentformsusinginspection,longdivision, or,forthemorecomplicatedexamples,acomputeralgebrasystem.** A.APR.7 3 SolveRationalEquations. A.REI.1, A.REI.2 5.4 4 Interpretkeyfeaturesofsimpleandgeneralrationalgraphsintermsofthequantities, andsketchgraphsshowingkeyfeaturesgivenaverbaldescriptionoftherelationship. F.IF.4,F.BF.3 3.1,3.2 5.6 5 Usepropertiesofintegerexponentstoexplainandconvertbetweenexpressions involvingradicalsandrationalexponents,usingcorrectnotation. N.RN.1,N.RN.2 3.6 5.8 6 SolveRadicalEquations. A.REI.1, A.REI.2 7 InterpretkeyfeaturesofSquareRootFunctionandCubeRootgraphsintermsofthe quantities,andsketchgraphsshowingkeyfeaturesgivenaverbaldescriptionofthe relationship. F.IF.4,F.BF.3 5.4,5.5 5.2,5.3 1 4days 5.2Ext 2 5days 5.6 5.5 3days 5.2,5.3 2days 5days IV. CCSS code RATIONALANDRADICALFUNCTIONS(27days) 5days 3days STUDENTLEARNINGOBJECTIVES 3.5 5.7 A.APR.6 EXPONENTIALANDLOGARITHMICFUNCTIONSRATIONALANDRADICALFUNCTIONS(18days) 2days 2.1 1 Chooseandproduceequivalentexpressionsforexponentialfunctionsusingthe propertiesofexponents. A.SSE.3 2days 4.1,4.2 4.1 2 Usepropertiesofexponentstorewriteafunctioninanequivalentformtorevealand explaindifferentpropertiesoftheexponentialfunction. F.IF.8b 2days 4.1,4.2 4.1 3 Graphexponentialfunctionsexpressedsymbolicallyorverballyandshowkey featuresofthegraph. F.IF.4,F.BF.3 2days 4.3 4.6 4 UseFunctionsInvolvinge. F.IF.4,7,8, F.BF.3 3days 2days 3days 4.4 4.5 4.6 4.3 4.4 4.5-4.6 5 6 7 RewritingExponentialEquationsinLogarithmicForm. ApplyPropertiesofLogarithms. SolveExponentialandLogarithmicEquations. InterpretkeyfeaturesofLogarithmicFunctionsgraphsintermsofthequantities,and 2days 4.4 4.7 8 sketchgraphsshowingkeyfeaturesgivenaverbaldescriptionoftherelationship. **TaughtinApps/NotConceptsorResource F.IF.4,7,8 F.LE.4 F.LE.4 F.IF.4,F.BF.3 TECHNOLOGYSTANDARDS III.1.PerformarithmeticoperationswithRationalExpressions. http://www.youtube.com/watch?v=z4Hssd_oiDQ http://www.youtube.com/watch?v=WbvLjmK4Kmc http://www.youtube.com/watch?v=d3xr5a4cln0 http://www.youtube.com/watch?v=r_mkzOlfQOo http://www.youtube.com/watch?v=8_2x1kktEPc http://www.youtube.com/watch?v=SyVq0-ObQdA III.2.Rewritesimplerationalexpressionsindifferentformsusinginspection,longdivision,or,forthemorecomplicated examples,acomputeralgebrasystem.** https://www.youtube.com/watch?v=-DUMVzvu7-g III.3.SolveRationalEquations http://www.youtube.com/watch?v=JkhePJajfEc http://www.youtube.com/watch?v=f9QBU4v0X2U http://www.youtube.com/watch?v=r9LYqfqssS0 III.4.Interpretkeyfeaturesofsimpleandgeneralrationalgraphsintermsofthequantities,andsketchgraphsshowing keyfeaturesgivenaverbaldescriptionoftherelationship. https://www.youtube.com/watch?v=-DUMVzvu7-g http://www.purplemath.com/modules/grphrtnl.htm III.5.Usepropertiesofintegerexponentstoexplainandconvertbetweenexpressionsinvolvingradicalsandrational exponents,usingcorrectnotation. http://www.youtube.com/watch?v=PpuKD2_OfFk http://www.youtube.com/watch?v=NCEB8x-lYBM http://www.youtube.com/watch?v=pafRsMuHNOQ http://www.youtube.com/watch?v=-T1punCdxas III.6.SolveRadicalEquations. http://www.youtube.com/watch?v=hkF8ej72NAg http://www.youtube.com/watch?v=8EOGx68kdwo http://www.youtube.com/watch?v=Ef2gOQbDv7M http://www.youtube.com/watch?v=QCnPnV4mE8g III.7.InterpretkeyfeaturesofSquareRootFunctionandCubeRootgraphsintermsofthequantities,andsketchgraphs showingkeyfeaturesgivenaverbaldescriptionoftherelationship. http://www.youtube.com/watch?v=xQ6DTxjUA7w http://www.youtube.com/watch?v=17rEN-fVPvE IV.1.Chooseandproduceequivalentexpressionsforexponentialfunctionsusingthepropertiesofexponents. http://www.mathplanet.com/education/algebra-1/exponents-and-exponential-functions/properties-of-exponents IV.2.Usepropertiesofexponentstorewriteafunctioninanequivalentformtorevealandexplaindifferentpropertiesof theexponentialfunction. http://www.youtube.com/watch?v=c6pcRR5Uy6w http://www.youtube.com/watch?v=4vmR8XMvQ_I http://www.youtube.com/watch?v=f-3QDswUc8o http://www.youtube.com/watch?v=sf8KdpN67Sk http://www.youtube.com/watch?v=XJlawwoGhxQ IV.3.Graphexponentialfunctionsexpressedsymbolicallyorverballyandshowkeyfeaturesofthegraph. http://www.youtube.com/watch?v=c6pcRR5Uy6w KEYVOCABULARY • • • • • • • • • • • • • • • • • • • • • • • • • • • • • nthrootofa Indexofaradical Simplestformofa radical Likeradicals PowerFunction Composition Inverserelation InverseFunction Radicalequation Extraneous solution Exponential Function Exponentialgrowth GrowthFactor Asymptote RationalFunction Domain Range Asymptote EndBehavior RationalFunction Simplifiedformofa rationalexpression Reciprocal ComplexFraction Crossmultiplying Extraneous solution Increasing Decreasing Oddfunction Evenfunction http://www.youtube.com/watch?v=4vmR8XMvQ_I http://www.youtube.com/watch?v=f-3QDswUc8o http://www.youtube.com/watch?v=sf8KdpN67Sk http://www.youtube.com/watch?v=XJlawwoGhxQ IV.4.UseFunctionsInvolvinge. http://www.youtube.com/watch?v=SNZgbj3UaRE http://www.youtube.com/watch?v=gvwXYc7Qa9E http://www.youtube.com/watch?v=Yo-UN392NDc IV.5.RewritingExponentialEquationsinLogarithmicForm. https://www.youtube.com/watch?v=fjBZCEZpISQ http://www.youtube.com/user/learnmathtutorials?v=wfYsiJcVWy0 http://www.youtube.com/watch?v=REPqXHu7gXc&feature=c4-overview-vl&list=PLBFAE01DF8EA66352 http://www.youtube.com/watch?v=mUfXkwTQB8o&feature=c4-overview-vl&list=PLBFAE01DF8EA66352 http://www.youtube.com/watch?v=Fo0pciptGWs IV.6.ApplyPropertiesofLogarithms. http://www.youtube.com/watch?v=eLapHtvQbFo http://www.youtube.com/watch?v=kXYbrOjm9D8 http://www.youtube.com/watch?v=dLJAuL_dveQ IV.7.SolveExponentialandLogarithmicEquations. http://www.youtube.com/watch?v=CqYDBfoiwOc http://www.youtube.com/watch?v=QMesv6cJxtI http://www.youtube.com/watch?v=Fo0pciptGWs IV.8.InterpretkeyfeaturesofLogarithmicFunctionsgraphsintermsofthequantities,andsketchgraphsshowingkey featuresgivenaverbaldescriptionoftherelationship. https://www.youtube.com/watch?v=fjBZCEZpISQ http://www.youtube.com/user/learnmathtutorials?v=wfYsiJcVWy0 http://www.youtube.com/watch?v=REPqXHu7gXc&feature=c4-overview-vl&list=PLBFAE01DF8EA66352 http://www.youtube.com/watch?v=mUfXkwTQB8o&feature=c4-overview-vl&list=PLBFAE01DF8EA66352 http://www.youtube.com/watch?v=Fo0pciptGWs SelectedOpportunitiesforConnectionsto MathematicalPractices 1. 2. 3. 4. 5. 6. 7. 8. Makesenseofproblemsandpersevereinsolvingthem. Reasonabstractlyandquantitatively. Constructviableargumentsandcritiquethereasoningofothers. Modelwithmathematics. Useappropriatetoolsstrategically. Attendtoprecision. Lookforandmakeuseofstructure. Lookforandexpressregularityinrepeatedreasoning. Unit3-LinkstoOpenEndedProblems RelatedtotheStandards A.APR.6 A.APR.7 A.REI.1 A.REI.2 A.REI.6 A.REI.7 F.IF.4 F.BF.3 N.RN.1 N.RN.2 A.SSE.3 F.IF.7 F.IF.8 F.LE.4 CCSS Code A.APR.6 A.APR.7 A.REI.1 A.REI.2 A.REI.6 A.REI.7 F.IF.4 F.BF.3 N.RN.1 CCSSCodeDESRCIPTION Rewritesimplerationalexpressionsindifferentforms;writea(x)/b(x)intheformq(x)+r(x)/b(x),wherea(x),b(x),q(x),andr(x)arepolynomialswith thedegreeofr(x)lessthanthedegreeofb(x),usinginspection,longdivision,or,forthemorecomplicatedexamples,acomputeralgebrasystem. (+)Understandthatrationalexpressionsformasystemanalogoustotherationalnumbers,closedunderaddition,subtraction,multiplication,and divisionbyanonzerorationalexpression;add,subtract,multiply,anddividerationalexpressions. Explaineachstepinsolvingasimpleequationasfollowingfromtheequalityofnumbersassertedatthepreviousstep,startingfromtheassumption thattheoriginalequationhasasolution.Constructaviableargumenttojustifyasolutionmethod. Solvesimplerationalandradicalequationsinonevariable,andgiveexamplesshowinghowextraneoussolutionsmayarise. Solvesystemsoflinearequationsexactlyandapproximately(e.g.,withgraphs),focusingonpairsoflinearequationsintwovariables. Solveasimplesystemconsistingofalinearequationandaquadraticequationintwovariablesalgebraicallyandgraphically.Forexample,findthe pointsofintersectionbetweentheliney=-3xandthecircle𝑥 $ + 𝑦 $ = 3. Forafunctionthatmodelsarelationshipbetweentwoquantities,interpretkeyfeaturesofgraphsandtablesintermsofthequantities,andsketch graphsshowingkeyfeaturesgivenaverbaldescriptionoftherelationship.Keyfeaturesinclude:intercepts;intervalswherethefunctionisincreasing, decreasing,positive,ornegative;relativemaximumsandminimums;symmetries;endbehavior;andperiodicity.* Identifytheeffectonthegraphofreplacingf(x)byf(x)+k,kf(x),f(kx),andf(x+k)forspecificvaluesofk(bothpositiveandnegative);findthevalue ofkgiventhegraphs.Experimentwithcasesandillustrateanexplanationoftheeffectsonthegraphusingtechnology.Includerecognizingevenand oddfunctionsfromtheirgraphsandalgebraicexpressionsforthem. Explainhowthedefinitionofthemeaningofrationalexponentsfollowsfromextendingthepropertiesofintegerexponentstothosevalues,allowing foranotationforradicalsintermsofrationalexponents.Forexample,wedefine5 so 5 7 7 7 tobethecuberootof5becausewewant 5 6 7 7 =5 6 7 8 tohold, mustequal5. N.RN.2 Rewriteexpressionsinvolvingradicalsandrationalexponentsusingthepropertiesofexponents. A.SSE.3 Chooseandproduceanequivalentformofanexpressiontorevealandexplainpropertiesofthequantityrepresentedbytheexpression.* Graphfunctionsexpressedsymbolicallyandshowkeyfeaturesofthegraph,byhandinsimplecasesandusingtechnologyformorecomplicated cases.* Graphexponentialandlogarithmicfunctions,showinginterceptsandendbehavior,andtrigonometricfunctions,showingperiod,midline,and amplitude. Writeafunctiondefinedbyanexpressionindifferentbutequivalentformstorevealandexplaindifferentpropertiesofthefunction. Usethepropertiesofexponentstointerpretexpressionsforexponentialfunctions.Forexample,identifypercentrateofchangeinfunctionssuchasy =(1.02)ᵗ,y=(0.97)ᵗ,y=(1.01)12ᵗ,y=(1.2)ᵗ/10,andclassifythemasrepresentingexponentialgrowthordecay. Forexponentialmodels,expressasalogarithmthesolutiontoabct=dwherea,c,anddarenumbersandthebasebis2,10,ore;evaluatethe logarithmusingtechnology. F.IF.7 F.IF.7.E F.IF.8 F.IF.8.B F.LE.4 6 6 PCTIMATHEMATICSDEPARTMENT COURSE:AlgebraII UNIT:4 UNITNAME:INVERSEFUNCTIONS,SEQUENCES&SERIES,ANDDATA&STATISTICS TOPICS Green Book V. Blue Book Obj. # 3.3 6.5 1 5days 3.4 6.6 2 5days 8.7 12.7 3 5days 4 PerformFunctionComposition. Determinetheinversefunctionforasimplefunctionthathasaninverseandwritean expressionfortheinverse. Solvesystemsoflinearequationsandsimplesystemsconsistingofalinearanda quadraticequationintwovariables,algebraicallyandgraphically. Findapproximatesolutionsfortheintersectionsoffunctionsandexplainwhythexcoordinatesofthepointswherethegraphsoftheequationsy=f(x)andy=g(x) intersectarethesolutionsoftheequationf(x)=g(x)involvinglinear,polynomial, rational,absolutevalue,logarithmicandexponentialfunctions. 7.1-7.3, 7.5 9.1-9.4 1 7days 7.4 9.5 2 Writearithmeticandgeometricsequencesbothrecursivelyandwithanexplicit formula,usethemtomodelsituations,andtranslatebetweenthetwoforms. Derivetheformulaforthesumofafinitegeometricseries(whenthecommonratiois not1),andusetheformulatosolveproblems.Forexample,calculatemortgage payments. F.BF.4 A.REI.6,A.REI.7 A.REI.11 F.BF.2 A.SSE.4 ANALYZEDATAANDSTATISTICS(10days) Identifydifferentmethodsandpurposesforconductingsamplesurveys,experiments, andobservationalstudiesandexplainhowrandomizationrelatestoeach.** Usedatafromarandomizedexperimenttocomparetwotreatmentsanduse 5days 6.5EXT 8.3 2 simulationstodecideifdifferencesbetweenparametersaresignificant;evaluate reportsbasedondata.** **TaughtinApps/NotConceptsorResource 5days F.BF.4 SEQUENCEANDSERIES(14days) 7days VII. CCSScode INVERSEFUNCTIONSANDFUNCTIONSYSTEMS(21days) 6days VI. STUDENTLEARNINGOBJECTIVES 6.4-6.5 8.2 1 S.IC.3 S.IC.5,S.IC.6 TECHNOLOGYSTANDARDS V.1.PerformFunctionComposition. http://www.youtube.com/watch?v=LM_NyH4R6yw http://www.youtube.com/watch?v=Gl_0MhsuyO8 http://www.youtube.com/watch?v=v8JgEEimO7w V.2.Determinetheinversefunctionforasimplefunctionthathasaninverseandwriteanexpressionfortheinverse. http://www.youtube.com/watch?v=nSmFzOpxhbY http://www.youtube.com/watch?v=DsaJKV4M-vk V.3.Solvesystemsoflinearequationsandsimplesystemsconsistingofalinearandaquadraticequationintwovariables, algebraicallyandgraphically. http://www.youtube.com/watch?v=TSRhRbuL7E8 http://mathnmind.com/PDF%20Files/Algebra%202/chap09/section07/notetaking.pdf V.4.Findapproximatesolutionsfortheintersectionsoffunctionsandexplainwhythex-coordinatesofthepointswhere thegraphsoftheequationsy=f(x)andy=g(x)intersectarethesolutionsoftheequationf(x)=g(x)involvinglinear, polynomial,rational,absolutevalue,logarithmicandexponentialfunctions. http://www.youtube.com/watch?v=TSRhRbuL7E8 http://mathnmind.com/PDF%20Files/Algebra%202/chap09/section07/notetaking.pdf VI.1.Writearithmeticandgeometricsequencesbothrecursivelyandwithanexplicitformula,usethemtomodel situations,andtranslatebetweenthetwoforms. http://mathnmind.com/PDF%20Files/Algebra%202/chap10/section01/notetaking.pdf http://www.virtualnerd.com/algebra-2/sequences-series http://www.youtube.com/watch?v=oDQmXsXzNn0 http://www.youtube.com/watch?v=W561_exZn2k http://www.youtube.com/watch?v=lj_X9JVSF8k http://prezi.com/htshzbfuhzyi/122-analyze-arithmetic-sequences-and-series/ http://www.youtube.com/watch?v=rtsk8caxbr4 http://www.virtualnerd.com/algebra-2/sequences-series/geometric/geometric-sequences http://www.youtube.com/watch?v=k-ygvxR47Fc http://www.youtube.com/watch?v=NEZuM5itx7o http://www.youtube.com/watch?v=y6QIro3Cdig http://www.youtube.com/watch?v=-XVIMMjtAmI VI.2.Derivetheformulaforthesumofafinitegeometricseries(whenthecommonratioisnot1),andusetheformulato solveproblems.Forexample,calculatemortgagepayments. http://www.youtube.com/watch?v=i6zeKlXYgE8 http://www.youtube.com/watch?v=mcnblnEsf98 http://www.youtube.com/watch?v=Ocd1iQyN-qk VII.1.Identifydifferentmethodsandpurposesforconductingsamplesurveys,experiments,andobservationalstudiesand explainhowrandomizationrelatestoeach.** http://www.slideserve.com/kiet/7-5-purple-select-and-draw-conclusions-from-samples Download6.4-SelectandDrawConclusions Lesson14.4ADrawingConclusionsfromSamples,Surveys-CORD... 7.5selectanddrawconclusionsfromsamples http://www.youtube.com/watch?v=wnNapKDGQH4 VII.2.Usedatafromarandomizedexperimenttocomparetwotreatmentsandusesimulationstodecideifdifferences betweenparametersaresignificant;evaluatereportsbasedondata.** http://www.slideserve.com/kiet/7-5-purple-select-and-draw-conclusions-from-samples KEYVOCABULARY • • • • • • • • • • • • • • • • • • • • • • • • • • • • • Composition Inverserelation InverseFunction Focus Directrix Parabola Vertex Quadraticsystem biasedquestions biasedsample controlgroup controlled experiment experiment marginoferror observationalstudy Population randomized comparative experiment sample treatmentgroup unbiasedsample Arithmeticsequence Commondifference Arithmeticseries Geometricsequence Commonratio Geometricseries Explicitrule Recursiverule Iteration Download6.4-SelectandDrawConclusions Lesson14.4ADrawingConclusionsfromSamples,Surveys-CORD... 7.5selectanddrawconclusionsfromsamples http://www.youtube.com/watch?v=wnNapKDGQH4 SelectedOpportunitiesforConnectionsto MathematicalPractices Unit4-LinkstoOpenEndedProblems 9. Makesenseofproblemsandpersevereinsolvingthem. 10. Reasonabstractlyandquantitatively. 11. Constructviableargumentsandcritiquethereasoningofothers. 12. Modelwithmathematics. 13. Useappropriatetoolsstrategically. 14. Attendtoprecision. 15. Lookforandmakeuseofstructure. 16. Lookforandexpressregularityinrepeatedreasoning. CCSS Code F.BF.4 RelatedtotheStandards F.BF.4 A.REI.6 A.REI.7 A.REI.11 F.BF.2 A.SSE.4 S.IC.3 S.IC.5 S.IC.6 CCSSCodeDESRCIPTION F.BF.4.B Findinversefunctions. Solveanequationoftheformf(x)=cforasimplefunctionfthathasaninverseandwriteanexpressionfortheinverse.Forexample,f(x)=2x3or f(x)=(x+1)/(x-1)forx≠1. (+)Verifybycompositionthatonefunctionistheinverseofanother. F.BF.4.C (+)Readvaluesofaninversefunctionfromagraphoratable,giventhatthefunctionhasaninverse. F.BF.4.D (+)Produceaninvertiblefunctionfromanon-invertiblefunctionbyrestrictingthedomain. F.BF.4 Findinversefunctions. A.REI.6 Solvesystemsoflinearequationsexactlyandapproximately(e.g.,withgraphs),focusingonpairsoflinearequationsintwovariables. F.BF.4.A A.REI.7 A.REI.11 F.BF.2 Solveasimplesystemconsistingofalinearequationandaquadraticequationintwovariablesalgebraicallyandgraphically.Forexample,find thepointsofintersectionbetweentheliney=-3xandthecircle𝑥 $ + 𝑦 $ = 3. 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.* Writearithmeticandgeometricsequencesbothrecursivelyandwithanexplicitformula,usethemtomodelsituations,andtranslatebetween thetwoforms.* A.SSE.4 Derivetheformulaforthesumofafinitegeometricseries(whenthecommonratioisnot1),andusetheformulatosolveproblems.For example,calculatemortgagepayments.* S.IC.3 Recognizethepurposesofanddifferencesamongsamplesurveys,experiments,andobservationalstudies;explainhowrandomizationrelatesto each. S.IC.5 Usedatafromarandomizedexperimenttocomparetwotreatments;usesimulationstodecideifdifferencesbetweenparametersare significant. S.IC.6 Evaluatereportsbasedondata.