ADVANCED ALGEBRA/PRECALCULUS CURRICULUM GUIDE

Transcription

1 ADVANCED ALGEBRA/PRECALCULUS CURRICULUM GUIDE Loudoun County Public Schools Complete scope, sequence, pacing and resources are available on the CD and will be available on the LCPS Intranet. 1

2 INTRODUCTION TO LOUDOUN COUNTY S MATHEMATICS CURRICULUM GUIDE This CURRICULUM GUIDE is a merger of the Virginia Standards of Learning (SOL) and the Mathematics Achievement Standards for Loudoun County Public Schools. The CURRICULUM GUIDE includes excerpts from documents published by the Virginia Department of Education. Other statements, such as suggestions on the incorporation of technology and essential questions, represent the professional consensus of Loudoun s teachers concerning the implementation of these standards. In many instances the local expectations for achievement exceed state requirements. The GUIDE is the lead document for planning, assessment and curriculum work. It is a summarized reference to the entire program that remains relatively unchanged over several student generations. Other documents, called RESOURCES, are updated more frequently. These are published separately but teachers can combine them with the GUIDE for ease in lesson planning. Mathematics Internet Safety Procedures 1. Teachers should review all Internet sites and links prior to using it in the classroom. During this review, teachers need to ensure the appropriateness of the content on the site, checking for broken links, and paying attention to any inappropriate pop-ups or solicitation of information. 2. Teachers should circulate throughout the classroom while students are on the internet checking to make sure the students are on the appropriate site and are not minimizing other inappropriate sites. Teachers should periodically check and update any web addresses that they have on their LCPS web pages. 3. Teachers should assure that the use of websites correlate with the objectives of lesson and provide students with the appropriate challenge. 4. Teachers should assure that the use of websites correlate with the objectives of the lesson and provide students with the appropriate challenge. 2

3 Advanced Algebra/Precalculus Nine Weeks Overview 1 st Quarter 2 nd Quarter 3 rd Quarter 4 th Quarter AA.1 AA.2 AA.3 AA.4 AA.5 38 days AA.5 AA.7 AA.5 AA.6 AA.5 AA.8 AA.9 AA.11 AA.12 AA.8 AA.10 AA.11 AA.13 AA days Topics from Calculus 47 days 48 days 3

4 Quarter1: Academic Year Advanced Algebra/Precalculus Quarter 1: Matrices Writing a matrix Dimensions Operations Addition Subtraction Scalar Multiplication Matrix Multiplication Larger than a 3 X 3 on calculator only, all others without the aid of a calculator Determinants Larger than a 3 X 3 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 3 X 3 on calculator only, but write the matrix equation Applications involving matrices Area of a triangle Determine if points are collinear Cryptography AA.1 The student will use matrices to organize data and will add and subtract matrices, multiply matrices, and multiply matrices by a scalar. AA.2 The student will represent real world problems with a system of linear equations and solve the system, using the inverse matrix method, and be able to find the determinant of a matrix. Websites for free worksheets for free worksheets for free worksheets ture/m116/matrices/applications.ht ml for applications 4

5 Quarter1: Academic Year Advanced Algebra/Precalculus Conic Sections Circle Graph the Standard Form Write the equation of a circle given information about its center, radius, or diameter Graph a circle from General Form Find the x and y intercepts of the circle 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 Ellipse Define an ellipse using the foci Identify whether the ellipse is horizontal or vertical Graph the ellipse identify the major/minor axes, the center, the vertices, and the foci Write the equation of the ellipse given the center, foci, vertices, or axes Find the x and y intercepts of the ellipse Convert between Standard Form and General Form AA.3 The student will identify conic sections (circle, ellipse, parabola, and hyperbola) from his/her equations. Given the equations in any form, the student will sketch graphs of conic sections, using transformations. Websites nics.htm to see pictures of how the conic sections are formed when a plane cuts through a cone. This website will also have the equations for the conic sections. To see some awesome pictures of conics and to read about the history of conics, visit rves_dir/conicsections_dir/conic Sections.html This website has links to real life applications of conic sections including engineering, physics, navigation, and roller coaster design. MAch3Curves.html Visit this website for further lessons on how to work conic section problems. There are also some real life applications to be found here. 9/student.Folders/Jones.June/con ics/conics.html 5

6 Quarter1: Academic Year Advanced Algebra/Precalculus Hyperbola Define a hyperbola using the foci Identify whether the hyperbola is horizontal or vertical Graph the hyperbola identify the center, vertices, foci and the equations of the asymptotes Write the equation of the hyperbola given information about the center, vertices, foci or the equations of the asymptotes Find the x and y intercepts of the hyperbola Convert between Standard Form and General Form Sullivan Sec 1.2 Sullivan Sec 9.2 Sullivan Sec 9.3 Sullivan Sec 9.4 Applications of conics Solving systems of conics Satellite dishes (parabola) Orbits, whispering galleries (ellipses) LOng RAnge Navigation system (LORAN) (hyperbola) AA.4 The student will apply their knowledge of the conic sections to analyze real world situations. 6

7 Quarter1: Academic Year Advanced Algebra/Precalculus Introduction to Trigonometry Angles Define angle, including its initial and terminal sides Identify the direction of rotation for angles to be positive and negative Define standard position of an angle, direction of rotation, quadrant in which an angle lies, and quadrantal angles Draw angles of varying measures Assessment, Enrichment, and Remediation AA.5 The student will find, without the use of a calculator, the values of the trigonometric functions of the special angles and their related angles as found in the unit circle. This will include converting angles from radians to degrees and vice versa. Sullivan Sec 5.1 Sullivan Sec 5.1 Sullivan Sec 5.1 Sullivan Sec 5.2 Sullivan Sec 5.3 Sullivan Sec

8 Quarter 2 Academic Year Advanced Algebra/Precalculus Quarter 2: 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 Use the concepts of the special right triangles to create the Unit Circle a. Reproduce the Unit Circle without aid of a calculator Define the six trigonometric functions Find the six trigonometric functions a. in terms of the coordinates of any point on the Unit Circle b. using a calculator c. in terms of the coordinates of any point on a circle of radius r d. given the coordinates of any point on the terminal 8 AA.5 The student will find, without the use of a calculator, the values of the trigonometric functions of the special angles and their related angles as found in the unit circle. This will include converting angles from radians to degrees and vice versa. AA.7 The student will apply their knowledge of length of an arc and the area of a sector to analyze real world situations. AA.6 The student will apply their knowledge of linear and angular speed to analyze real world situations. AA.5 The student will find, without the use of a calculator, the values of the trigonometric functions of the special angles and their related angles as found in the unit circle. This will include converting angles from radians to degrees and vice versa. AA.8 The student, given a point other than the origin on the terminal side of an angle, will use the definitions of the six trigonometric functions to find the sine, cosine, tangent, cotangent, secant, and cosecant of Websites etry gonometry

9 Quarter 2 Academic Year Advanced Algebra/Precalculus side of an angle Write the following fundamental identities: a. reciprocal identities b. quotient identities c. Pythagorean identities Solve problems by finding values of the six trigonometric functions a. using even-odd properties to find the exact values of the trigonometric functions b. using fundamental trigonometric identities c. given one of the functions and the quadrant of the angle Determine the signs of the trigonometric functions in each quadrant Graphing Trigonometric Functions State the domain, range, and periodicity of the trigonometric functions Graph the sine and cosine functions as parent functions a. identify the properties of the sine and cosine graphs b. use the sine and cosine graphs to develop and explain the graphs of the remaining trigonometric functions Use knowledge of transformations to graph general form of trigonometric equations in the form y = Asin( Bx C) + D a. describe how changes in amplitude, period, phase shifts, and vertical shifts affect sine and cosine graphs from a practical application standpoint the angle in standard position. Trigonometric functions defined on the unit circle will be related to trigonometric functions defined in right triangles. AA.9 The student, given the value of one trigonometric function, will find the values of the other trigonometric functions, using the definitions and properties of the trigonometric functions. AA.11 11The student will verify basic trigonometric identities and make substitutions, using the basic identities, AA.12 The student will solve trigonometric equations and trigonometric inequalities. AA.10 The student will use the transformational approach to graphing the six trigonometric functions and the inverse functions. Attention will be paid to the restricted domains. 9

10 Quarter 2 Academic Year Advanced Algebra/Precalculus b. determine the amplitude, period, phase shift, and vertical shift of trigonometric functions and their graphs c. write a sine or cosine function that satisfies a given data set allowing for amplitude, period, phase shift, and vertical shift Assessment, Enrichment, and Remediation 10

11 Quarter 3 Academic Year Advanced Algebra/Precalculus Quarter 3: Inverse Trigonometric Functions Define the inverse of each of the six trigonometric functions, to include restricted domain and range Find the exact value of the inverse trigonometric functions using AA.11 The student will verify basic trigonometric identities and make substitutions, using the basic identities, graphing calculators unit circle composite trigonometric functions Establish/verify trigonometric identities Solve trigonometric equations using a. Sum and Difference Formulas b. Double and half Angle Formulas c. Graphing on calculators d. Factoring Applications of Trigonometric Functions Solve right triangles Solve oblique triangles using a. Law of Sines b. Law of Cosines Find the area of triangles using a. Heron s formula Two sides and the sine of the included angle AA.13 The student will identify, create, and solve real-world problems involving triangles. Techniques will include using the trigonometric functions, the Pythagorean Theorem, the Law of Sines, the Law of Cosines, and the formulas to calculate the areas of triangles. 11

12 Quarter 3 Academic Year Advanced Algebra/Precalculus Exponential and Logarithmic Functions Rewrite exponential expressions as logarithmic expressions and vice versa Evaluate exponential and logarithmic expressions Apply properties of logarithms to express in expanded/condensed form Use the change of base formula to evaluate logarithms whose base in neither 10 nor e AA.14 The student will be able to identify, graph, and write exponential and logarithmic functions and to apply the concepts of exponential and logarithmic functions to real world models. Websites nential-logarithmic.html Sullivan Sec 4.2 Sullivan Sec 4.6 Sullivan Sec 4.7 Sullivan Sec

13 Quarter 3 Academic Year Advanced Algebra/Precalculus Solving Logarithmic Equations Comparing arguments when the bases match Rewriting as an exponential equation Using the change of base formula Solve literal logarithmic equations (Emphasis is to be given to checking solutions for extraneous solutions in the domain) Using the graphing calculator Use a graphing calculator to approximate the value of non-rational solutions to exponential and logarithmic equations Use a graphing utility to fit data to an exponential function Use a graphing utility to fit data to a logarithmic function Applications Determine the future value or present value of a lump sum of money Determine the time required 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 AA.14 The student will be able to identify, graph, and write exponential and logarithmic functions and to apply the concepts of exponential and logarithmic functions to real world models. 13

14 Quarter 3 Academic Year Advanced Algebra/Precalculus Sequences and Series: Sigma Notation Applications *Mathematical Induction *The Binomial Theorem (*May be moved until after Trigonometry) AA.15 The student will find the sum (sigma notation included) of finite and infinite convergent series, which will lead to the intuitive approach to a limit. AA.16 The student will use mathematical induction to prove formulas and mathematical statements. AA.17 The student will expand binomials having positive integral exponents through the use of the Binomial Theorem, the formula for combinations, and Pascal s Triangle. Sullivan Sec 11.1 Sullivan Sec Sullivan Sec 11.4 Sullivan Sec 11.5 Assessment, Enrichment, and Remediation 14

15 Quarter 4 Academic Year Advanced Algebra/Precalculus LIMITS Finding Limits by Inspection Find a limit numerically (using a table of values) Find a limit graphically Find the limit of a polynomial Find the limit of a power or root Finding Limits Analytically Find a limit by simplifying rational expressions, adding fractions, multiplying numerator and denominator by a conjugate Find one-sided limits of a function (numerically, graphically, analytically) Find limits at infinity (numerically, graphically, analytically) Websites culus/limits/introduction.html culus_worksheets/limits/limit_1.p df alc/limits.htm Application of Limits 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 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 /limcon/limcon04/limcon04.html 15