Algebra II Curriculum Guide Tier 1 & 2

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Algebra II Curriculum GuideTier 1 & 2Unit 2: Polynomial Function and EquationsSeptember 10 – November 28ORANGE PUBLIC SCHOOLS 2018 - 2019OFFICE OF CURRICULUM AND INSTRUCTIONOFFICE OF MATHEMATICS

Algebra II Unit 1ContentsUnit Overview . 2Common Core State Standards . Error! Bookmark not defined.Essential Learning Goals for Unit 2 6Calendar . 9Scope and Sequence . 12Assessment Framework .13Transition lessons 14Agile mind Topic overviews .49Ideal Math Block . 62Sample Lesson Plan (Agile Mind) . 64Supplement Materials. 66Multiple Representations . 67PARCC practice Question .701

Algebra II Unit 1Unit OverviewUnit 2: Polynomial Function and EquationsOverviewThis course uses Agile Mind as its primary resource, which can be accessed at the following URL: www.orange.agilemind.comEach unit consists of 1-3 topics. Within each topic, there are “Exploring” lessons with accompanyingactivity sheets, practice, and assessments. The curriculum guide provides an analysis of teach topic,detailing the standards, objectives, skills, and concepts to be covered. In addition, it will providesuggestions for pacing, sequence, and emphasis of the content provided.Essential Questions What is polynomial function? How do you perform arithmetic operation on polynomials? How do you interpret key features of graphs and tables in terms of the quantities? How do you identify odd and even function based on the symmetry? What is a rational expression? How do you simplify rational expressions? How do you re-write rational expressions? How are the degrees of polynomials related to its’ zeroes? How can you analyze functions using different representation? How do you sketch graphs showing key features given a verbal description of the relationship? What is the difference between absolute values and relative values? What is a short-term behavior? What is a long-term behavior? How can you analyze functions using different representation? What is polynomial equation? What is a complex number? How do you solve polynomial equation? How does discriminant help you make prediction about roots of quadratic equations? What is the fundamental theorem of Algebra? What is remainder theorem?Enduring Understandings Polynomial functions take the form f(x) anxn an - 1xn 1 . a1x a0, where n is anonnegative integer and an 0. Understand that polynomials form a system analogous to the integers, namely, they are closedunder the operations of addition, subtraction, and multiplication; add, subtract, and multiplypolynomials. Understand the Key features of graphs such as; intercepts, intervals where the function isincreasing, decreasing, positive, or negative; relative maximums and minimums; symmetries andpoint of inflections. Understand that a function that has line symmetry with respect to y axis is called even function Understand that function that has point symmetry with respect to the origin is called odd function A rational expression is the quotient of two polynomial expressions, expressed as a ratio. Rational expression can be simplified through factoring Understand how to use long division to Rewrite simple rational expressions in different forms;write a(x)/b(x) in the form q(x) r(x)/b(x), where a(x), b(x), q(x), and r(x) are polynomials with thedegree of r(x) less than the degree of b(x).2

Algebra II Unit 1 Identify zeros of polynomials when suitable factorizations are available. Use the zeros of a function, critical points (relative max, min) and intervals for increasing anddecreasing function, end behavior, and symmetries to construct a rough graph of the functiondefined by the polynomial. Understand that for any absolute values graph reaches the highest or lowest point then decreasesor increases over an interval Understand that for any local values graph reaches a high point then a low point and then it keepincreasing or decreasing and there is not absolute values The behavior of a function over small intervals is called the short-term behavior, or localbehavior, of a function Long-term behavior is the same as end behavior, of the polynomial. End behavior of thefunction is defined as the behavior of the values of f(x) as x approaches negative infinityand as x approaches positive infinity. Compare properties of two functions each represented in a different way (algebraically,graphically, numerically in tables, or by verbal descriptions). A polynomial equation is any equation that can be written in the form(anxn an - 1xn 1 . a1x a0 0. Know there is a complex number i such that i2 –1, and every complex number has the form a biwith a and b real. When you know one of the roots you can find other factor by dividing the polynomial by linearexpression. You can solve polynomial through factoring. If it is quadratic equation then you can also solve bycompleting the square or by using the quadratic equation If the discriminant is positive, there are two distinct real roots. If the discriminant is zero,there is one distinct real root. If the discriminant is negative, there are two distinct non-realcomplex roots. According to the Fundamental Theorem of Algebra, any polynomial with real coefficients ofdegree n has at least one complex root. For a polynomial p(x) and a number a, the remainder on division by x –a is p(a)NJSLS/CCSS1) A-SSE.1: Interpret expressions that represent a quantity in terms of its context.a. Interpret parts of an expression, such as terms, factors, and coefficients.b. Interpret complicated expressions by viewing one or more of their parts as a single entity. Forexample, interpret P(1 r)n as the product of P and a factor not depending on P.2) A.SSE.2: Use the structure of an expression to identify ways to rewrite it. For example, seex4 – y 4 as (x2 )2 – (y2 )2 , thus recognizing it as a difference of squares that can be factored as(x2 – y 2 )(x2 y2 )3) A.SSE.3: Choose and produce an equivalent form of an expression to reveal and explain propertiesof the quantity represented by the expression.a. Factor a quadratic expression to reveal the zeros of the function it defines.4) A-APR.1: Understand that polynomials form a system analogous to the integers, namely, they areclosed under the operations of addition, subtraction, and multiplication; add, subtract, andmultiply polynomials.5) A-APR.2: Know and apply the Remainder Theorem: For a polynomial p(x) and a number a, theremainder on division by x – a is p(a), so p(a) 0 if and only if (x – a) is a factor of p(x).3

Algebra II Unit 16) A-APR.3: Identify zeros of polynomials when suitable factorizations are available, and use the zerosto construct a rough graph of the function defined by the polynomial.7) A-APR.4: Prove polynomial identities and use them to describe numerical relationships. Forexample, the polynomial identity (x2 y2) 2 (x2 – y 2) 2 (2xy)2 can be used to generatePythagorean triples.8) A-APR.6. Rewrite simple rational expressions in different forms; write a(x)/b(x) in the form q(x) r(x)/b(x), where a(x), b(x), q(x), and r(x) are polynomials with the degree of r(x) less than thedegree of b(x), using inspection, long division, or, for the more complicated examples, a computeralgebra system.9) A-REI.4: Solve quadratic equations in one variable.b. Solve quadratic equations by inspection (e.g., for x 2 49), taking square roots, completing thesquare, the quadratic formula and factoring, as appropriate to the initial form of the equation.Recognize when the quadratic formula gives complex solutions and write them as a bi for realNumbers a and bd. Represent and solve equations and inequalities graphically10) A-REI.11: Explain why the x-coordinates of the points where the graphs of the equations y f(x)and y g(x) intersect are the solutions of the equation f(x) g(x); find the solutions approximately,e.g., using technology to graph the functions, make tables of values, or find successiveapproximations. Include cases where f(x) and/or g(x) are linear, polynomial, rational, absolutevalue, exponential, and logarithmic functions.11) F-IF.4: For a function that models a relationship between two quantities, interpret key features ofgraphs and tables in terms of the quantities, and sketch graphs showing key features given a verbaldescription of the relationship. Key features include: intercepts; intervals where the function isincreasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; endbehavior; and periodicity.12) F-IF.5: Relate the domain of a function to its graph and, where applicable, to the quantitativerelationship it describes. For example, if the function h(n) gives the number of person-hours ittakes to assemble n engines in a factory, then the positive integers would be an appropriatedomain for the function13) F-IF.7: Graph functions expressed symbolically and show key features of the graph, by hand insimple cases and using technology for more complicated cases.c. Graph polynomial functions, identifying zeros when suitable factorizations are available, andshowing end behavior.14) F-IF.8: Write a function defined by an expression in different but equivalent forms to reveal andexplain different properties of the function.a. Use the process of factoring and completing the square in a quadratic function to show zeros,extreme values, and symmetry of the graph, and interpret these in terms of a context15) F-IF.9: Compare properties of two functions each represented in a different way (algebraically,graphically, numerically in tables, or by verbal descriptions). For example, given a graph of onequadratic function and an algebraic expression for another, say which has the larger maximum16) F-BF.1: Write a function that describes a relationship between two quantities.b. Combine standard function types using arithmetic operations. For example, build a function thatmodels the temperature of a cooling body by adding a constant function to a decayingexponential, and relate these functions to the model.4

Algebra II Unit 117) N-CN.1: Know there is a complex number i such that i 2 –1, and every complex number has theform a bi with a and b real18) N-CN-2: Use the relation i 2 –1 and the commutative, associative, and distributive properties toadd, subtract, and multiply complex numbers.19) N-CN.7: Solve quadratic equations with real coefficients that have complex solutions20) N-CN.8: ) Extend polynomial identities to the complex numbers. For example, rewrite x 2 4as (x 2i)(x – 2i).21) N-CN.9: ( ) Know the Fundamental Theorem of Algebra; show that it is true for quadraticpolynomials.Major ContentSupporting ContentAdditional ContentParts of standard not contained in this unitAlgebra I Content21st Century Career Ready Practice5

Essential Learning Goals for Algebra 2 Unit 1CCSSF.IF.8a:Use the process of factoring andcompleting the square in a quadraticfunction to show zeros, extreme values,and symmetry of the graph, andinterpret these in terms of a context.A-REI.4: Solve quadratic equations inone variable.A. Use the method of completing thesquare to transform any quadraticequation in xinto an equation of theform (x -p)2 q that has the samesolutions. Derive the quadratic formulafrom this form.B. Solve quadratic equations byinspection (e.g., for x 2 49), takingsquare roots, completing thesquare, the quadratic formula andfactoring, as appropriate to the initialform of the equation. Recognize whenthe quadratic formula gives complexsolutions and write them as a bi forreal numbers a and bA-REI.11: Explain why the xcoordinates of the points where thegraphs of the equations y f(x) andy g(x) intersect are the solutionsof the equation f(x) g(x); find thesolutions approximately, e.g., usingtechnology to graph the functions,make tables of values, or findSuccessive approximations. Includecases where f(x) and/or g(x) areLinear, polynomial, rational, absolutevalue, exponential, and logarithmicfunctions.F-IF.4: For a function that models arelationship between two quantities,interpret key features of graphs andtables in terms of the quantities, andsketch graphs showing key featuresgiven a verbal description of therelationship. Key features include:intercepts; intervals where thefunction is increasing, decreasing,positive, or negative; relativemaximums and minimums;symmetries; end behavior; mplexnumbersLesson ObjectiveNotes1.1a F.IF.7Give a quadratic graph and its function in standardform, students will identify key features of graph, and justifyzeros algebraically.Give a quadratic function in standard from, studentswill sketch the graph showing y-intercept & endbehavior.(Transitionlesson ifneeded) –1.1 b F.IF.7 , F.IF.4Students will: Identify key features of quadraticfunctions in factored form and standardform and sketch showing key features Using a graphing calculator to graph aquadratic function, and use the graph tore-write the standard form into factoredform(Transitionlesson ifneeded) –1.2a F.IF.8a, ASSE.3bStudents will: Re-write standard form in to factored formusing area model, or any factoredstrategies1.2b - A.REI.4, F.IF.7, A.REI.11Students will Solve quadratic equations in standard formby factoring or graphing and sketch thegraph to show key features.1.2 c - F.IF.4 F.IF.7Given a vertex form of the quadratic functionstudents will Identify key features of the vertex Sketch the graph of the quadraticequation in vertex form without thecalculator(Transitionlesson ifneeded) –1.2d: – A.RE.I4, F.IF.7Given a vertex form of the quadratic functionstudents will Solve simple quadratic equations(eg. X2 49, (x 2)2 49, x2 – 2 2)and sketch the graph to show keyfeatures. Solve quadratic equations in vertex formby taking square roots(Transitionlesson ifneeded) –(Transitionlesson ifneeded) –

Algebra II Unit 1F-IF.7: Graph functions expressedsymbolically and show key features ofthe graph, by hand in simple casesand using technology for morecomplicated cases.7b: Graph polynomial functions,identifying zeros when suitablefactorizations are available, andshowing end behavior.A.SSE.31.3a A.SSE.3, A.REI.4After a mini lesson on expanding sum of squaresstudents will Solve quadratic equations by completingsquares1.3b - A.REI.4aBy completing squares students will Derive the quadratic formula Apply the quadratic formula to find realsolution identify the nature of the roots andnumber of real roots from graphs Andthe discriminate1.4 N.CN.7, N.CN.1By using the quadratic formula and the definition ofimaginary numbers students will Solve and graph Quadratic equationswith non-real solutions Derive the definition of complex numberComplete the square in aquadratic expression to reveal themaximum or minimum value ofthe function it defines.N-CN.1: Know there is a complexnumber i such that i 2 –1, and everycomplex number has the form a biwith a and b realN-CN-2: Use the relation i 2 –1 andthe commutative, associative, anddistributive properties to add,subtract, and multiply complexnumbers.1.5 N.CN.2,By using the definition of complex number studentswill Simply complex numbers Perform operation with complex numbersN-CN.7: Solve quadratic equationswith real coefficients that havecomplex solutionsA-APR.3: Identify zeros ofpolynomials when suitablefactorizations are available, and usethe zeros to construct a rough graphof the function defined by thepolynomial.A-APR.2: Know and apply theRemainder Theorem: For apolynomial p(x) and a number a, theremainder on division by x – a is p(a),so p(a) 0 if and only if (x – a) is afactor of p(x).Keyfeatures ofpolynomialandsketchingpolynomial2.1a – F.IF.7Objective: Given a polynomial functions in factoredform students will Identify zeroes and y intercept Plot them on the coordinate plane Develop strategies to find the end behavior Create a sketch of the cubic function Identify the end behavior of functions withpositive and negative leading coefficients1

Algebra II Unit 1A-APR.6. Rewrite simple rationalexpressions in different forms; writea(x)/b(x) in the form q(x) r(x)/b(x),where a(x), b(x), q(x), and r(x) arepolynomials with the degree of r(x)less than the degree of b(x), usinginspection, long division, or, for themoreA.SSE.2: Use the structure of anexpression to identify ways to rewriteit. For example, see x4 – y4 as (x2)2 –(y2)2, thus recognizing it as adifference of squares that can befactored as (x2-y2)(x2 y2)2.1b -F.IF.4, F.IF.6 A.CED.1Objective: Given a real life situation Students will Create a mathematical model (polynomialfunction) to represent the situation Sketch the graph Identify key features from graphs andequations; and interpret the key features interms of contextRemainderTheoremand LongDivision3.1a – A.APR.6,Using Area model students will Understand the concept of long division Perform long division on polynomial Rewrite simple rational expression indifferent form3.1b – A.APR.2Through long division and by evaluating thepolynomial for a given root students will Understand the remainder theorem Apply the remainder theorem to find theremainder of a polynomial and see theconnection between factor and theremainder.3.2 – A.SSE.2students will Re-write the sum and difference of cube asfactored form3.3a - A. APR.3Using Area model and or GCF students will Factor cubic polynomialsSolvingCubicEquation3.3b - A. APR.3By performing factoring by grouping or longdivision and quadratic formula students will Solve cubic equation Identify key features Create a rough sketch and show keyfeatures of the polynomial functon2

CalendarSeptember st Day forStudents1718192021222324252627282930

Algebra II Unit 1October usDay – ndowopensPDHalf Day282930311

Algebra II Unit 1November 6Tue2627WedThuFriSat891014NJEA ConvNo School15NJEA ConvNo School161721222324Half DayThanksGiving29Thanks giving28302

Algebra II Unit 1Assessment FrameworkAssessmentEstimated TimeDateFormatGrading WeightUnit 1 Readiness Assessment40 miutesBeforeStart theUnit 1IndividualGraded (weight : 0)Diagnostic Assessment1 – 1 ½ BlockWeek 2IndividualSoftware graded (baseline ofstudent growth)Performance Task50 minutesAfterLesson 2.4IndividualYesPerformance Task40 minuteAfterLesson 3.1Individual,YesBenchmark 1 Assessment1 BlockDistrict test IndividualwindowYesAssessment check points

Algebra II Unit 1 5 17) N-CN.1: Know there is a complex number i such that i 2 –1, and every complex number has the form a bi with a and b real 18) N-CN-2: Use the relation i 2 –1 and the commutative, associative, and distributive p

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