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Advanced Algebra/PrecalculusCurriculum GuideOverview and Scope & SequenceLoudoun County Public Schools2014-2015(Additional curriculum information and resources for teachers can be accessed through CMS and VISION)Advanced Algebra/PreCalculus Semester Overview1st Semester2nd SemesterUnit 1 – MatricesAA.1AA.2Unit 2 – Conic SectionsAA.3AA.4Unit 3 – FunctionsAII.2 (reference)AII.4AII.7Unit 4 – Trigonometry (Continued into Semester 2)AA.5AA.7AA.5AA.6AA.5AA.8AA.9AA.11AA.12AA.8AA.10Unit 4 – Trigonometry (Continued from Semester 1)AA.11AA.13Unit 5 – Polynomial and Rational Functions ReviewAII.6Unit 6 – Exponential and Logarithmic FunctionsAA.14Unit 7 – Sequences and SeriesAA.15AA.16AA.17Unit 8 – Topics from Calculus: Limits44 blocks46 blocksThe student will use problem solving, mathematical communication, mathematical reasoning, connections, and representations as they engage in mathematics activities throughout the year.Advanced Algebra/PreCalculus Scope & SequenceSemester 1: 44 blocks total Number of blocks: 3StandardTopicUnit 1 – Matrices1AA.1AA.2Writing a MatrixDimensionMatrix OperationsAdditionSubtractionScalar MultiplicationMatrix MultiplicationLarger than a 3 x 3 on a calculator onlyRecommended that matrix multiplication for matrices smaller than 3 x 3 are done without the assistance of a calculator1DeterminantsMeaning of a determinant of zeroLarger than a 2 x 2 on a calculator only, recommending that all others are calculated without the aid of a calculatorArea of a triangle (Application)InversesLarger than a 2 x 2 on a calculator only, recommending that all others are calculated without the aid of a calculatorWhen an inverse does not existSolving Systems of EquationsWrite the system as a matrixSolve using an inverse matrixMatrices larger than a 2 x 2 on a calculator only, but write the matrix equationApplications Involving MatricesDetermine if points are collinearCryptographyEquation of Parabola given 3 points1 block – Suggested assessment, enrichment, and remediation for this unit.Semester 1: 44 blocks total (continued)Number of blocks: 6StandardTopicUnit 2 – Conic Sections1AA.3AA.4ParabolaDefine a parabola using the focus and the directrixIdentify whether the parabola is horizontal or vertical and how it opensGraph a parabola, identify the vertex, focus, directrix, and latus rectumWrite the equation of a graph given the vertex, focus, directrix, or latus rectum1CircleGraph the standard formWrite the equation of a circle given information about its center, radius, or diameterGraph a circle from general formFind the x- and y- intercepts of the circleConvert between standard form and general formSatellite dishes (parabola)1EllipseDefine an ellipse using the fociIdentify whether the ellipse is horizontal or verticalGraph the ellipse, identify the major/minor axes, the center, the vertices, and the fociWrite the equation of the ellipse given the center, foci, vertices, or axesFind the x- and y- intercepts of the ellipseConvert between standard form and general formOrbits, whispering galleries (ellipses)1AA.3AA.4HyperbolaDefine a hyperbola using the fociIdentify whether the hyperbola is horizontal or verticalGraph the hyperbola, identify the center, vertices, foci, and the equations of the asymptotesWrite the equation of the hyperbola given information about the center, vertices, foci, or the equations of the asymptotesFind the x- and andy- intercepts of the hyperbolaConvert between standard form and general formLOng RAnge Navigation system (LORAN) (hyperboa) (optional)Semester 1: 44 blocks total (continued)1Applications of ConicsSolving systems of conicsIdentify all conic sections1 block – Suggested assessment, enrichment, and remediation for this unit.Number of blocks: 6StandardTopicUnit 3 – Functions5AII.2AII.4AII.7One-to-one functions (new)Inverse FunctionsComposition of FunctionsFamilies of FunctionsTransformations (connect to graphs and equations)Domain and range (interval notation)Even and odd functions (new)FactoringPiecewise FunctionsConverting between standard form and vertex form (h, k) form1 block – Suggested assessment, enrichment, and remediation for this unit.Semester 1: 44 blocks total (continued)Number of blocks: 26StandardTopicUnit 4 – Trigonometry1AA.5Introduction to Trigonometry: AnglesDefine angle, including its initial and terminal sidesIdentify the direction of rotation for angles to be positive and negativeDefine standard position of an angle, direction of rotation, quadrant in which an angle lies, and quadrantal anglesDraw angles of varying measures1AA.7AA.6Angle MeasurementsDefine radianUse the arc length and area of a sector to solve problemsConvert from radians to degrees and backUse unit analysis to solve problems involving linear speed, rotational speed, and angular speed6AA.5AA.8AA.9AA.11AA.12Unit CircleUse the concepts of the special right triangles to create the unit circleReproduce the unit circle without the aid of a calculatorDefine the six trigonometric functionsDetermine the signs of the trigonometric functions in each quadrantFind the six trigonometric functionsIn terms of the coordinates of any point on the unit circleUsing a calculatorIn terms of the coordinates of any point on a circle of radius, r Given the coordinates of any point on the terminal side of an angleWrite the following fundamental identities:Reciprocal identitiesQuotient identitiesPythagorean identitiesSolve problems by finding values of the six trigonometric functionsUsing even-odd properties to find the exact values of the trigonometric functionsUsing fundamental trigonometric identitiesGiven one of the functions and the quadrant of the angle1 block – Suggested assessment, enrichment, and remediation for this unit.Semester 1: 44 blocks total (continued)Number of blocksStandardTopicUnit 4 – Trigonometry (continued)4AA.10Graphing Trigonometric FunctionsState the domain, range, and periodicity of the trigonometric functionsGraph the sine, cosine and tangent functions as parent functionsIdentify the properties of the sine, cosine, and tangent graphsUse the sine, cosine, and tangent graphs to develop and explain the graphs of the remaining trigonometric functions (i.e. sec, csc, cot)Use knowledge of transformations to graph general form of trigonometric equations in the form Describe how changes in amplitude, period, phase shifts, and vertical shifts affect sine and cosine graphs from a practical application standpointDetermine the amplitude, period, phase shift, and vertical shift of trigonometric functions and their graphsWrite a sine or cosine function that satisfies a given data set allowing for amplitude, period, phase shift, and vertical shift.2 blocks – Suggested assessment, enrichment, and remediation for this unit.4AA.11Inverse Trigonometric FunctionsDefine the inverse of each of the six trigonometric functions, to include restricted domain and rangeFind the exact value of the inverse trigonometric functions usingGraphing calculatorsUnit circleComposite trigonometric functions1 block – Suggested assessment, enrichment, and remediation for this unit.Semester 1: 44 blocks total (continued)6AA.11Trig EquationsEstablish/verify trigonometric identitiesSolve trigonometric equations usingSum and difference formulasDouble and half angle formulasGraphing Factoring2 blocks – Suggested assessment, enrichment, and remediation for this unit.Note: The sum of blocks is less than the number of blocks in the first semester. This allowance is to take into account exam preparation, exam days and variance in emphasis you may need as you teach your students.End Semester 1Begin Semester 2Semester 2: 46 blocks total Number of blocks: 4StandardTopicUnit 4 – Trigonometry (continued)3AA.13Applications of Trigonometric FunctionsSolve right trianglesSolve oblique triangles usingLaw of SinesLaw of CosinesFind the area of triangles usingHeron’s formulaTwo sides and the sine of the included angle1 block – Suggested assessment, enrichment, and remediation for this unit.Number of blocks: 10StandardTopicUnit 5 – Polynomial and Rational Functions8AII.6Rational functions (include horizontal, vertical, and slant asymptotes, zeros, graphing & end behavior)Polynomial functions (include zeros and graphing)Inequalities2 blocks – Suggested assessment, enrichment, and remediation for this plex FractionsFactoring (sum/difference cubes. Sum/difference squares, grouping, hard trinomials)Semester 2: 46 blocks total (continued)Number of blocks: 10StandardTopicUnit 6 – Exponential and Logarithmic Functions3AA.14Rewrite exponential expressions as logarithmic expressions and vice versaEvaluate exponential and logarithmic expressionsApply properties of logarithms to express in expanded/condensed formUse the change of base formula to evaluate logarithms whose base in neither 10 nor e5AA.14Solving Logarithmic EquationsComparing arguments when the bases matchRewriting as an exponential equationUsing the change of base formulaSolve literal logarithmic equations(Emphasis is to be given to checking solutions for extraneous solutions in the domain)Using the graphing calculatorUse a graphing calculator to approximate the value of non-rational solutions to exponential and logarithmic equationsUse a graphing utility to fit data to an exponential functionUse a graphing utility to fit data to a logarithmic functionApplicationsDetermine the future value or present value of a lump sum of moneyDetermine the time required for a lump sum of money to double or tripleFind equations of populations that obey the law of decay/growthUse Newton’s Law of CoolingLogistic Growth Models2 blocks – Suggested assessment, enrichment, and remediation for this unit.Semester 2: 46 blocks total (continued)Number of blocks: 8StandardTopicUnit 7 – Sequences and Series6AA.15AA.16AA.17Sigma NotationArithmetic, Geometric, Infinite GeometricExplicit vs Recursive FormulasApplicationsMathematical InductionThe Binomial TheoremStress binomials expansion to the 2nd and 3rd power. (Pascal’s Triangle etc.)2 blocks – Suggested assessment, enrichment, and remediation for this unit.Number of blocks: 8StandardTopicUnit 8 – Limits5Finding Limits by InspectionFind a limit numerically (using a table of values)Find a limit graphicallyFind the limit of a function2 block – Suggested assessment, enrichment, and remediation for this unit.If Time Permits: the following topics should be covered.Finding Limits AnalyticallyFind a limit by simplifying rational expressions, adding fractions, multiplying numerator and denominator by a conjugateFind one-sided limits of a function (numerically, graphically, analytically)Find limits at infinity (numerically, graphically, analytically)Application of LimitsUse limits to determine if a function is continuous at a pointFind the limit of an average rate of changeWrite the equation of the tangent line to a graph at a pointUsing Graphing UtilitiesUse a graphing utility to find a limit numerically and graphicallyUse a graphing utility to verify the equation of the tangent line to a graph at a point. 1 block – Suggested assessment, enrichment, and remediation for EOY AssessmentNote: The sum of blocks is less than the number of blocks in the second semester. This allowance is to take into account missed classes due to spring testing, exam preparation, exam days and variance in emphasis you may need as you teach your students.Additional information about math vocabulary can be found in theVDOE Vocabulary Word Wall Cards(click link above) ................
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