Advanced Algebra/Precalculus

ADVANCED ALGEBRA/PRECALCULUS CURRICULUM GUIDE Overview and Scope & Sequence

Loudoun County Public Schools 2016-2017

(Additional curriculum information and resources for teachers can be accessed through CMS and VISION) Loudoun County Public Schools Advanced Algebra/Pre-Calculus Curriculum Guide 2016-2017 1 Advanced Algebra/Pre-calculus Semester Overview

The student will use problem solving, mathematical communication, mathematical reasoning, connections, and representations as they engage in mathematics activities throughout the year.

1st Semester 2nd Semester Unit 1 – Matrices Unit 4 – Trigonometric Identities, Equations and AA.1 Applications (Continued from Semester 1) AA.2 AA.11 MA.14 AA.13 T.5 Unit 2 – Conic Sections T.8 AA.3 MA.13 (T. 9) AA.4 MA.8 Unit 5 – Polynomial and Rational Functions Review AII.6 MA.1 Unit 3 – Introduction to Trigonometry, Unit Circle, Graphing Trigonometric Functions , Inverse Trigonometric Unit 6 – Exponential and Logarithmic Functions Functions (Continued in Semester 2) AA.14 AA.5 MA.9 AA.6 AA.7 Unit 7 – Sequences and Series AA.8 AA.15 AA.9 AA.16 AA.10 AA.17 AA.11 MA.5 AA.12 MA.4 T.3 T.1 Unit 8 – Topics from Calculus: Limits (if time allows) T.2 MA.7 T.4 T.6 T.7

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Number Topics and Essential Questions Standard(s) of Learning Additional Instructional Loudoun County Public Schools Advanced Algebra/Pre-Calculus Curriculum Guide 2016-2017 2 Essential Knowledge and Skills of Blocks Resources Essential Understandings Unit 1 Matrices MA.14 The student will use Writing a matrix matrices to organize data and will Websites –  Dimensions add and subtract matrices, multiply www.kutasoftware.com for free 5-7 matrices, multiply matrices by a worksheets Operations scalar, and use matrices to solve  Addition systems of equations. www.edhelper.com for free  Subtraction worksheets  Scalar Multiplication www.math.about.com for free Matrix Multiplication worksheets  Larger than a 2 X 2 on calculator only, all others without http://people.richland.edu/james/lec the aid of a calculator ture/m116/matrices/applications.ht Determinants ml for applications  Larger than a 2 X 2 on calculator only, all others without the aid of a calculator  Meaning of a determinant of zero Solving systems of equations  Write the system as a matrix  Solve using an inverse matrix  Larger than a 2 X 2 on calculator only, but write the matrix equation Applications involving matrices  Area of a triangle  Determine if points are collinear  Cryptography  Equation of Parabola given 3 points

Loudoun County Public Schools Advanced Algebra/Pre-Calculus Curriculum Guide 2016-2017 3 Standard(s) of Learning Number Additional Instructional Topic and Essential Questions Essential Knowledge and Skills of Blocks Resources Essential Understandings Unit 2 Conic Sections MA.8 The student will Websites – Circle investigate and identify the http://math2.org/math/algebra/co  Graph a circle given standard or characteristics of conic nics.htm to see pictures of how 8-9 general form section equations in (h, k) the conic sections are formed  Write the equation of a circle given and standard forms. when a plane cuts through a information about its center, radius, Transformations in the or diameter coordinate plane will be used cone. This website will also to graph conic sections. have the equations for the conic  Convert between Standard Form and sections. General Form To see some awesome pictures  Recognize the equation of a semi- of conics and to read about the circle centered at the origin history of conics , visit Ellipse http://xahlee.org/SpecialPlaneCu  Define an ellipse using the foci rves_dir/ConicSections_dir/conic  Identify whether the ellipse is Sections.html horizontal or vertical This website has links to real life  Graph the ellipse – identify the applications of conic sections major/minor axes, the center, the including engineering, physics, vertices, and the foci navigation, and roller coaster  Write the equation of the ellipse design. given the center, foci, vertices, or http://www.k12.hi.us/~mathappl/ axes MAch3Curves.html  Convert between Standard Form and General Form Visit this website for further lessons on how to work conic Hyperbola section problems. There are also  Define a hyperbola using the foci some real life applications to be  Identify whether the hyperbola is found here. horizontal or vertical http://jwilson.coe.uga.edu/emt66  Graph the hyperbola –identify 9/student.Folders/Jones.June/con the center, vertices, foci and the ics/conics.html equations of the asymptotes  Write the equation of the Sullivan Sec 1.2 hyperbola given information about Sullivan Sec 9.2 the center, vertices, foci or the Sullivan Sec 9.3 Loudoun County Public Schools Advanced Algebra/Pre-Calculus Curriculum Guide 2016-2017 4 equations of the asymptotes Sullivan Sec 9.4  Convert between Standard Form and General Form

Parabola  Define a parabola using the focus and the directrix  Identify whether the parabola is horizontal or vertical and how it opens  Graph a parabola – identify the vertex, focus, directrix and latus rectum  Write the equation of a graph given the vertex, focus, directrix or latus rectum  Convert between Standard Form and General Form Applications of conics  Solving systems of conics  Satellite dishes (parabola)  Orbits, whispering galleries (ellipses)  LOng RAnge Navigation system (LORAN) (hyperbola)

Number Standard(s) of Learning Additional Instructional of Blocks Topics and Essential Questions Essential Knowledge and Skills Resources Essential Understandings

Loudoun County Public Schools Advanced Algebra/Pre-Calculus Curriculum Guide 2016-2017 5 Unit 3 Introduction to Trigonometry T.3 The student will find, without Sullivan Sec 5.1—5.4, 5.6 Angles the use of a calculator, the values of the trigonometric functions of Websites –  Define angle, including its the special angles and their www.funmaths.com/worksheets 2-3 initial and terminal sides related angles as found in the unit www.analyzemath.com/ trigonom  Identify the direction of circle. This will include etry rotation for angles to be converting angles from radians to www.onlinemathlearning.com/ tri positive and negative degrees and vice versa. gonometry  Define standard position of an www.kutasoftware.com angle, direction of rotation, www.ti.com quadrant in which an angle lies, and quadrantal angles  Draw angles of varying measures Angle Measurements  Define radian  Use the arc length and area of a sector to solve problems  Convert from radians to degrees and back  Use unit analysis to solve problems involving linear speed, rotational speed and angular speed Unit Circle T.1 The student, given a  Use the concepts of the special point other than the origin on right triangles to create the unit the terminal side of the 6 circle angle, will use the definitions  Reproduce the unit circle of the six trigonometric without aid of a calculator functions to find the sine,  Define the six trigonometric cosine, tangent, cotangent, functions secant, and cosecant of the angle in standard position.  Find the six trigonometric Trigonometric functions functions defined on the unit circle will

Loudoun County Public Schools Advanced Algebra/Pre-Calculus Curriculum Guide 2016-2017 6 a. in terms of the be related to trigonometric coordinates of any point on functions defined in right the Unit Circle triangles. b. using a calculator c. in terms of the T.2 The student, given the coordinates of any point on value of one trigonometric a circle of radius r function, will find the values d. given the of the other trigonometric coordinates of any point on functions, using the the terminal side of an definitions and properties of angle the trigonometric functions.  Write the following fundamental identities: T.3 The student will find, without the use of a calculator, the values a. reciprocal identities of the trigonometric functions of b. quotient identities the special angles and their c. Pythagorean related angles as found in the unit identities circle. This will include  Solve problems by finding converting angles from radians to values of the six trigonometric degrees and vice versa. functions a. using even-odd T.4 The student will find, properties to find the exact with the aid of a calculator, values of the trigonometric the value of any functions trigonometric function and b. using fundamental inverse trigonometric trigonometric identities function. c. given one of the functions and the quadrant of the angle  Determine the signs of the trigonometric functions in each quadrant

Loudoun County Public Schools Advanced Algebra/Pre-Calculus Curriculum Guide 2016-2017 7 Suggested placement of Assessment, Enrichment, and Remediation

Graphing Trigonometric T. 6 The student, given one of the Functions six trigonometric functions in  State the domain, range, and standard form, will periodicity of the trigonometric a) state the domain and the functions range of the function; b) determine the amplitude,  Graph the sine and cosine period, phase shift, vertical shift, functions as parent functions and asymptotes; a. identify the c)sketch the graph of the function properties of the sine and by using transformations for at cosine graphs least a two-period interval; and 6 -7 d) investigate the effect of b. use the sine and changing the parameters in a cosine graphs to develop trigonometric function on the and explain the graphs of graph of the function. the remaining trigonometric functions  Use knowledge of transformations to graph general form of trigonometric equations in the form a. describe how changes in amplitude, period, phase shifts, and vertical shifts affect sine and cosine graphs from a practical application standpoint b. determine the amplitude, period, phase shift, and vertical shift of trigonometric functions and their graphs

Loudoun County Public Schools Advanced Algebra/Pre-Calculus Curriculum Guide 2016-2017 8 c. Write a sine or cosine function that satisfies a given data set allowing for amplitude, period, phase shift, and vertical shift Suggested placement of Assessment, Enrichment, and Remediation

Loudoun County Public Schools Advanced Algebra/Pre-Calculus Curriculum Guide 2016-2017 9 Number Standard(s) of Learning Additional Instructional of Blocks Topics and Essential Questions Essential Knowledge and Skills Resources Essential Understandings Inverse Trigonometric Functions T.4 The student will find, with the  Define the inverse of each 5 aid of a calculator, the value of any of the six trigonometric trigonometric function and inverse functions, to include restricted trigonometric function. domain and range  Find the exact value of the T.7 The student will identify the inverse trigonometric functions domain and range of the inverse using trigonometric functions and  graphing calculators recognize the graphs of these  unit circle functions. Restrictions on the domains of the inverse  composite trigonometric trigonometric functions will be functions included.

Suggested placement of Assessment, Enrichment, and Remediation

Number Standard(s) of Learning Additional Instructional of Blocks Topics and Essential Questions Essential Knowledge and Skills Resources Essential Understandings Unit 4 Identities T.5 The student will verify basic Loudoun County Public Schools Advanced Algebra/Pre-Calculus Curriculum Guide 2016-2017 10 3 - 4  Establish/verify trigonometric identities and make trigonometric identities substitutions, using the basic identities.

Solve trigonometric equations T.8 The student will solve using 5 trigonometric equations that include Include finite and infinite number both infinite solutions and restricted of solutions domain solutions and solve basic a. Sum and difference trigonometric inequalities. formulas b. Double and half angle formulas c. Graphing d. Factoring Applications of Trigonometric Functions  Solve right triangles  Solve oblique triangles MA.13 (T. 9) The student will using identify, create, and solve real- a. Law of Sines world problems involving triangles. 3-4 (Include ambiguous case) Techniques will include using the b. Law of Cosines trigonometric functions, the Pythagorean Theorem, the Law of  Find the area of triangles Sines, and the Law of Cosines. using a. Heron’s formula  Two sides and the sine of the included angle Suggested placement of Assessment, Enrichment, and Remediation

Loudoun County Public Schools Advanced Algebra/Pre-Calculus Curriculum Guide 2016-2017 11 Standard(s) of Learning Number Additional Instructional Topics and Essential Questions Essential Knowledge and Skills of Blocks Resources Essential Understandings Unit 5 Polynomial and Rational MA.1 The student will Functions investigate and identify the  Polynomial functions characteristics of polynomial a. Identify the domain, and rational functions and range, zeros of a function, use these to sketch the graphs the x and y intercepts, of the functions. This will intervals where the function include determining zeros, is increasing and decreasing, upper and lower bounds, y- intercepts, symmetry, and end behavior, asymptotes, intervals for graphically and which the function is algebraically. Express increasing or decreasing, and characteristics in interval maximum or minimum notation. points. Graphing utilities will b. Explore why be used to investigate and discontinuities do not occur verify these characteristics. 5-7 in polynomial functions. c. Describe relationships among solutions, zeros, x-intercepts, and factors of higher order polynomial functions. d. Review factoring (sum/difference cubes, Sum/difference squares, grouping, trinomials with lead coefficient ≠ 1) e. Solve polynomials graphically and algebraically using a variety of methods including factoring, long Loudoun County Public Schools Advanced Algebra/Pre-Calculus Curriculum Guide 2016-2017 12 division, and synthetic division. f. Write equations of polynomials given various characteristics of the function.

 Rational functions a. Investigate and analyze rational functions graphically first, then algebraically. b. Identify the domain, range, zeros of a function, the x and y intercepts, intervals where the function is increasing and decreasing, asymptotes (horizontal, vertical, & slant or oblique) and other discontinuities, and end behavior, graphically and algebraically. Express characteristics in interval notation. c. Review simplifying complex fractions and operations as necessary. d. Solve rational equations and inequalities graphically and algebraically using a variety of methods. e. Write equations of Loudoun County Public Schools Advanced Algebra/Pre-Calculus Curriculum Guide 2016-2017 13 rational functions given various characteristics of the function. Suggested placement of Assessment, Enrichment, and Remediation

Loudoun County Public Schools Advanced Algebra/Pre-Calculus Curriculum Guide 2016-2017 14 Standard(s) of Learning Number Additional Instructional Topics and Essential Questions Essential Knowledge and Skills of Blocks Resources Essential Understandings Unit 6 Exponential and Logarithmic MA.9 The student will Websites – Functions investigate and identify the www.kutasoftware.com  Rewrite exponential characteristics of exponential expressions as logarithmic and logarithmic functions in www.onlinemathlearning.com/ expo expressions and vice versa order to graph these nential - logarithmic .html 3  Evaluate exponential and functions and solve equations logarithmic expressions and real-world problems. www.analyzemath.com  Apply properties of logarithms This will include the role of to express in expanded/condensed e, natural and common www.ti.com form logarithms, laws of  Use the change of base formula exponents and logarithms, Sullivan Sec 4.2 to evaluate logarithms whose base and the solution of Sullivan Sec 4.6 in neither 10 nor e logarithmic and exponential Sullivan Sec 4.7 equations. Sullivan Sec 4.8 6 Solving Exponential and Logarithmic Equations  Comparing arguments when the bases match  Rewriting as an exponential or logarithmic equation  Using the change of base formula  Solve exponential and logarithmic equations using algebraic and graphic methods. (Emphasis is to be given to checking solutions for extraneous solutions in the domain) Applications  Determine the future value or present value of a lump sum of money  Determine the time required

Loudoun County Public Schools Advanced Algebra/Pre-Calculus Curriculum Guide 2016-2017 15 for a lump sum of money to double or triple  Find equations of populations that obey the law of decay/growth  Use Newton’s Law of Cooling  Logistic Growth Models  Use a graphing utility to fit data to an exponential function  Use a graphing utility to fit data to a logarithmic function Suggested placement of Assessment, Enrichment, and Remediation

Standard(s) of Learning Number Additional Instructional Topics and Essential Questions Essential Knowledge and Skills of Blocks Resources Essential Understandings Unit 7 Sequences and Series MA.5 The student will find Sullivan Sec 11.1  Sigma Notation the sum (sigma notation Sullivan Sec 11.1 – 11.3  Applications included) of finite and  The Binomial Theorem infinite convergent series, Sullivan Sec 11.4 5 (include Pascal’s Triangle) which will lead to an intuitive approach to a limit. Sullivan Sec 11.5

MA.4 The student will expand binomials having positive integral exponents through the use of the Binomial Theorem, the formula for combinations, and Pascal’s Triangle.

Suggested placement of Assessment, Enrichment, and Remediation

Loudoun County Public Schools Advanced Algebra/Pre-Calculus Curriculum Guide 2016-2017 16 Standard(s) of Learning Number Additional Instructional Topics and Essential Questions Essential Knowledge and Skills of Blocks Resources Essential Understandings Unit 8 LIMITS MA.7 The student will find Websites – If time allows Finding Limits by Inspection the limit of an algebraic  Find a limit numerically (using function, if it exists, as the http://www.coolmath.com/lesson- a table of values) variable approaches either a whats-a-limit-1.htm  Find a limit graphically 2 finite number or infinity. A  Find the limit of a polynomial graphing utility will be used http://www.analyzemath.com/calcu  Find the limit of a power or lus/limits/introduction.html root to verify intuitive reasoning, Finding Limits Analytically algebraic methods, and numerical substitution. http://www.analyzemath.com/calcu  Find a limit by simplifying lus_worksheets/limits/limit_1.pdf rational expressions, adding fractions, multiplying numerator http://www.themathpage.com/aCalc and denominator by a conjugate /limits.htm 2  Find one-sided limits of a function (numerically, graphically, analytically) http://www.sosmath.com/calculus/li  Find limits at infinity mcon/limcon04/limcon04.html (numerically, graphically, analytically) 2 Application of Limits

Loudoun County Public Schools Advanced Algebra/Pre-Calculus Curriculum Guide 2016-2017 17  Use limits to determine if a function is continuous at a point  Find the limit of an average rate of change  Write the equation of the tangent line to a graph at a point Using Graphing Utilities  Use a graphing utility to find a 2 limit numerically and graphically  Use a graphing utility to verify the equation of the tangent line to a graph at a point Assessment, Enrichment, and Remediation

Additional information about math vocabulary can be found in the

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Loudoun County Public Schools Advanced Algebra/Pre-Calculus Curriculum Guide 2016-2017 18