ࡱ> 'Os L ] [ .bjbj ΐΐz&a###Dggg<#%|go&<+,[X^XXX&&&&&&&(M+&#5&XX|.)&___"8X#X&_&__. X-g. J%?&0o&$?,נ?,?,#0dY_W#d&&o&?, : Monroe Township Schools  Curriculum Management System Precalculus Grade 11 June 2009 * For adoption by all regular education programs Board Approved: July 2009 as specified and for adoption or adaptation by all Special Education Programs in accordance with Board of Education Policy # 2220. Table of Contents Monroe Township Schools Administration and Board of Education Members Page 3 Acknowledgments Page 4 District Mission Statement and Goals Page 5 Introduction/Philosophy/Educational Goals Pages 6 National and State Standards Page 7 Scope and Sequence Page 8-9 Goals/Essential Questions/Objectives/Instructional Tools/Activities Pages 10-35 Benchmarks Page 36 Addendum Pages 37 - MONROE TOWNSHIP SCHOOL DISTRICT ADMINISTRATION Dr. Kenneth Hamilton, Superintendent Mr. Jeff Gorman, Assistant Superintendent BOARD OF EDUCATION Ms. Amy Antelis, President Ms. Kathy Kolupanowich, Vice President Mr. Marvin Braverman Mr. Ken Chiarella Mr. Lew Kaufman Mr. Mark Klein Mr. John Leary Ms. Kathy Leonard Mr. Ira Tessler JAMESBURG REPRESENTATIVE Ms. Patrice Faraone Student Board Members Ms. Nidhi Bhatt Ms. Reena Dholakia Acknowledgments The following individuals are acknowledged for their assistance in the preparation of this Curriculum Management System: Writers Names: Beth Goldstein Supervisor Name: Robert ODonnell, Supervisor of Mathematics & Educational Technology Technology Staff: Al Pulsinelli Reggie Washington Secretarial Staff: Debbie Gialanella Geri Manfre Gail Nemeth Monroe Township Schools Mission and Goals Mission The mission of the Monroe Township School District, a unique multi-generational community, is to collaboratively develop and facilitate programs that pursue educational excellence and foster character, responsibility, and life-long learning in a safe, stimulating, and challenging environment to empower all individuals to become productive citizens of a dynamic, global society. Goals To have an environment that is conducive to learning for all individuals. To have learning opportunities that are challenging and comprehensive in order to stimulate the intellectual, physical, social and emotional development of the learner. To procure and manage a variety of resources to meet the needs of all learners. To have inviting up-to-date, multifunctional facilities that both accommodate the community and are utilized to maximum potential. To have a system of communication that will effectively connect all facets of the community with the Monroe Township School District. To have a staff that is highly qualified, motivated, and stable and that is held accountable to deliver a safe, outstanding, and superior education to all individuals. INTRODUCTION, PHILOSOPHY OF EDUCATION, AND EDUCATIONAL GOALS Philosophy Monroe Township Schools are committed to providing all students with a quality education resulting in life-long learners who can succeed in a global society. The mathematics program, grades K - 12, is predicated on that belief and is guided by the following six principles as stated by the National Council of Teachers of Mathematics (NCTM) in the Principles and Standards for School Mathematics, 2000. First, a mathematics education requires equity. All students will be given worthwhile opportunities and strong support to meet high mathematical expectations. Second, a coherent mathematics curriculum will effectively organize, integrate, and articulate important mathematical ideas across the grades. Third, effective mathematics teaching requires the following: a) knowing and understanding mathematics, students as learners, and pedagogical strategies, b) having a challenging and supportive classroom environment and c) continually reflecting on and refining instructional practice. Fourth, students must learn mathematics with understanding. A student's prior experiences and knowledge will actively build new knowledge. Fifth, assessment should support the learning of important mathematics and provide useful information to both teachers and students. Lastly, technology enhances mathematics learning, supports effective mathematics teaching, and influences what mathematics is taught. As students begin their mathematics education in Monroe Township, classroom instruction will reflect the best thinking of the day. Children will engage in a wide variety of learning activities designed to develop their ability to reason and solve complex problems. Calculators, computers, manipulatives, technology, and the Internet will be used as tools to enhance learning and assist in problem solving. Group work, projects, literature, and interdisciplinary activities will make mathematics more meaningful and aid understanding. Classroom instruction will be designed to meet the learning needs of all children and will reflect a variety of learning styles. In this changing world those who have a good understanding of mathematics will have many opportunities and doors open to them throughout their lives. Mathematics is not for the select few but rather is for everyone. Monroe Township Schools are committed to providing all students with the opportunity and the support necessary to learn significant mathematics with depth and understanding. This curriculum guide is designed to be a resource for staff members and to provide guidance in the planning, delivery, and assessment of mathematics instruction. Educational Goals Precalculus is the fourth course in the regular college preparatory sequence. This course applies the skills obtained in Algebra I, Geometry and Algebra II. The topics covered include exponents, logarithms, trigonometric functions and identities, solving trigonometric equations, applications involving triangles, inverse trigonometric functions, trigonometric addition formulas, advanced graphing techniques, polar coordinates and complex numbers, sequences and series, and limits. This course is intended to prepare students for a post secondary education. It emphasizes higher-level mathematical thinking necessary to pursue the study of calculus.  New Jersey State Department of Education Core Curriculum Content Standards A note about Mathematics Standards and Cumulative Progress Indicators. The New Jersey Core Curriculum Content Standards for Mathematics were revised in 2004. The Cumulative Progress Indicators (CPI's) referenced in this curriculum guide refer to these new standards and may be found in the Curriculum folder on the district servers. A complete copy of the new Core Curriculum Content Standards for Mathematics may also be found at:  HYPERLINK "http://www.nj.gov/njded/cccs/s4_math.htm" http://www.nj.gov/njded/cccs/s4_math.htm Precalculus Scope and Sequence Quarter IBig Idea: Functions Transformations of functions a. Reflection in the x-axis, y-axis, and line y=x (inverse functions) b. Symmetry in the x-axis, y-axis, and origin c. Periodic Functions d. Translations of y=f(x) to y-k=f(x-h) e. Vertical (y=cf(x)) and horizontal (y=f(cx)) stretching or shrinking of y=f(x).Big Idea: Trigonometric Functions Introduction to Trigonometric Functions a. Degree and radian measures of angles b. Arc length and area of a sector c. Evaluating trigonometric expressions d. Graphs of trigonometric functionsBig Idea: Trigonometric Equations and Applications Trigonometric Equations and Applications a. Translation of sine and cosine graphs b. Vertical and horizontal stretching and shrinking of sine and cosine functions c. Simplifying trigonometric expressions and proving trigonometric identities d. Trigonometric equationsQuarter IIBig Idea: Triangle Trigonometry Triangle Trigonometry a. Measurements in right triangles b. Area of a triangle c. Law of Sines d. Law of CosinesBig Idea: Trigonometric Addition Formulas Trigonometric Addition Formulas a. Sum and difference formulas for sine, cosine, and tangent b. Double angle formulas c. Trigonometric equations Quarter IIIBig Idea: Polar Coordinates and Complex Numbers Polar Coordinates and Complex Numbers a. Graphing polar coordinates b. Conversions of rectangular and polar coordinates c. Graphs of polar functions d. Conversions of complex numbers between rectangular and polar form e. Product of two complex numbers in polar form f. De Moivres theorem g. Roots of complex numbersBig Idea: Sequences and Series Sequences and series a. nth term of an arithmetic sequence b. nth term of a geometric sequence c. Recursive definitions d. Sum of finite arithmetic and geometric series e. Sum of infinite geometric series f. Sigma notation Quarter IVBig Idea: Limits Limits a. Limits of functions that approach infinity or negative infinity b. Limits of functions that approach a real number c. Graphs of rational functionsBig Idea: Exponents and Logarithms Exponents and Logarithms a. Simplify numeric and algebraic expressions with integral and rational exponents b. Compound interest formula and formula for interest compounded continually c. Evaluate logarithmic expressions (change of base formula) d. Expand and condense logarithmic expressions e. Logarithmic and exponential equations  Suggested blocks of InstructionCurriculum Management System Grade Level/Subject: Grade 11/PrecalculusBig Idea: FunctionsThe student will be able to stretch, shrink, reflect, or translate the graph of a function, and determine the inverse of a function, if it exists.Objectives / Cluster Concepts / Cumulative Progress Indicators (CPI's) The student will be able to:Essential Questions Sample Conceptual UnderstandingsInstructional Tools / Materials / Technology / Resources / Learning Activities / Interdisciplinary Activities / Assessment ModelTo sketch the reflection of a graph in the  EMBED Equation.DSMT4 -axis and EMBED Equation.DSMT4 -axis. (4.2.12B.1; 4.2.12B.3) To write the inverse of an equation and sketch the graph of the reflection in the line  EMBED Equation.DSMT4 . (4.2.12B.1; 4.3.12B.3) To determine if the graph of an equation has symmetry in the  EMBED Equation.DSMT4 -axis, the  EMBED Equation.DSMT4 -axis, the line  EMBED Equation.DSMT4 , and the origin. (4.2.12B.1; 4.3.12B.4) To determine if a function is periodic. (4.3.12B.1) To evaluate a function using the fundamental period. (4.3.12B.2) To determine the period and amplitude of a periodic function. (4.3.12B.2) To understand the effect of  EMBED Equation.DSMT4 and sketch the graph of  EMBED Equation.DSMT4 by vertically stretching or shrinking the graph of  EMBED Equation.DSMT4 . (4.2.12B.1; 4.3.12B.3) To understand the effect of  EMBED Equation.DSMT4 and sketch the graph of  EMBED Equation.DSMT4 by horizontally stretching or shrinking the graph of  EMBED Equation.DSMT4 . (4.2.12B.1; 4.3.12B.3) To understand the effect of  EMBED Equation.DSMT4 and  EMBED Equation.DSMT4 and sketch the graph of the equation  EMBED Equation.DSMT4 by translating the graph of  EMBED Equation.DSMT4  horizontally  EMBED Equation.DSMT4 units and vertically  EMBED Equation.DSMT4 units. To determine if two functions are inverse functions by applying the definition. To apply the Horizontal Line Test to determine if a function has an inverse.Essential Questions: What changes in an equation produces the reflection of its graph in the  EMBED Equation.DSMT4 -axis, the  EMBED Equation.DSMT4 -axis, and the line  EMBED Equation.DSMT4 ? How do you tell whether the graph of an equation has symmetry in the the  EMBED Equation.DSMT4 -axis, the  EMBED Equation.DSMT4 -axis, the line  EMBED Equation.DSMT4 , and the origin? Given the graph of  EMBED Equation.DSMT4 , what effect does  EMBED Equation.DSMT4 have on the graph of  EMBED Equation.DSMT4 and  EMBED Equation.DSMT4 ? Given the graph of  EMBED Equation.DSMT4 , what effect does  EMBED Equation.DSMT4 and  EMBED Equation.DSMT4 have on the graph of  EMBED Equation.DSMT4 ? How can the vertical-line test be used to justify the horizontal-line test? Enduring Understandings: If the equation  EMBED Equation.DSMT4 is changed to: a.  EMBED Equation.DSMT4 , then the graph of  EMBED Equation.DSMT4 is reflected in the  EMBED Equation.DSMT4 -axis. b.  EMBED Equation.DSMT4 , then the graph of  EMBED Equation.DSMT4 is unchanged when  EMBED Equation.DSMT4 and reflected in the  EMBED Equation.DSMT4 -axis when  EMBED Equation.DSMT4 . c.  EMBED Equation.DSMT4 , then the graph of  EMBED Equation.DSMT4 is reflected in the  EMBED Equation.DSMT4 -axis. d.  EMBED Equation.DSMT4 , then the graph of  EMBED Equation.DSMT4 is reflected in the line  EMBED Equation.DSMT4 . e.  EMBED Equation.DSMT4 , then the graph of  EMBED Equation.DSMT4 is stretched vertically. f.  EMBED Equation.DSMT4 , then the graph of  EMBED Equation.DSMT4 is shrunk vertically. g.  EMBED Equation.DSMT4 , then the graph of  EMBED Equation.DSMT4 is shrunk horizontally. h.  EMBED Equation.DSMT4 , then the graph of  EMBED Equation.DSMT4 is stretched horizontally. i.  EMBED Equation.DSMT4 , then the graph of  EMBED Equation.DSMT4 is translated  EMBED Equation.DSMT4 units horizontally and  EMBED Equation.DSMT4 units vertically. A graph is symmetric in the  EMBED Equation.DSMT4 -axis if  EMBED Equation.DSMT4 is on the graph whenever  EMBED Equation.DSMT4 is. An equation of a graph is symmetric in the  EMBED Equation.DSMT4 -axis if an equivalent equation results after substituting  EMBED Equation.DSMT4 for  EMBED Equation.DSMT4 . A graph is symmetric in the  EMBED Equation.DSMT4 -axis if  EMBED Equation.DSMT4 is on the graph whenever  EMBED Equation.DSMT4 is. An equation of a graph is symmetric in the  EMBED Equation.DSMT4 -axis if an equivalent equation results after substituting  EMBED Equation.DSMT4 for  EMBED Equation.DSMT4 . A graph is symmetric in the line  EMBED Equation.DSMT4  if  EMBED Equation.DSMT4 is on the graph whenever  EMBED Equation.DSMT4 is. An equation of a graph is symmetric in the line  EMBED Equation.DSMT4  if an equivalent equation results after interchanging  EMBED Equation.DSMT4  and  EMBED Equation.DSMT4 . A graph is symmetric in the origin if  EMBED Equation.DSMT4 is on the graph whenever  EMBED Equation.DSMT4 is. An equation of a graph is symmetric in the origin if an equivalent equation results after substituting  EMBED Equation.DSMT4 for  EMBED Equation.DSMT4 and  EMBED Equation.DSMT4 for  EMBED Equation.DSMT4 . Two functions  EMBED Equation.DSMT4 and  EMBED Equation.DSMT4 are inverse functions if: 1.  EMBED Equation.DSMT4 for all  EMBED Equation.DSMT4 in the domain of  EMBED Equation.DSMT4 , and 2.  EMBED Equation.DSMT4 for all  EMBED Equation.DSMT4 in the domain of  EMBED Equation.DSMT4 . The Horizontal Line Test: If the graph of the function  EMBED Equation.DSMT4 is such that no horizontal line intersects the graph in more than one point, then  EMBED Equation.DSMT4 is one-to-one and has an inverse. NOTE: The assessment models provided in this document are suggestions for the teacher. If the teacher chooses to develop his/her own model, it must be of equal or better quality and at the same or higher cognitive levels (as noted in parentheses). Depending upon the needs of the class, the assessment questions may be answered in the form of essays, quizzes, mobiles, PowerPoint, oral reports, booklets, or other formats of measurement used by the teacher. Resources: Precalculus with Limits A Graphing Approach, Fifth Edition, Larson et al; Houghton Mifflin, 2008 Learning Activity: In the following activity, how does a change in the equation result in the reflection of its graph in some line? (analysis) 1. Graph  EMBED Equation.DSMT4  and  EMBED Equation.DSMT4 . Graph  EMBED Equation.DSMT4  and  EMBED Equation.DSMT4 . How are the graphs of  EMBED Equation.DSMT4 and  EMBED Equation.DSMT4 related? 2. Graph  EMBED Equation.DSMT4  and  EMBED Equation.DSMT4 . Graph  EMBED Equation.DSMT4  and  EMBED Equation.DSMT4 . How are the graphs of  EMBED Equation.DSMT4 and  EMBED Equation.DSMT4 related? 3. Graph  EMBED Equation.DSMT4  and  EMBED Equation.DSMT4 . Graph  EMBED Equation.DSMT4  and  EMBED Equation.DSMT4 . How are the graphs of  EMBED Equation.DSMT4  and  EMBED Equation.DSMT4 related? 4. Graph  EMBED Equation.DSMT4  and  EMBED Equation.DSMT4 . Graph  EMBED Equation.DSMT4  and  EMBED Equation.DSMT4 . How is the graph of an equation affected when you interchange the variables in the equation? Given the graph of  EMBED Equation.DSMT4 , sketch the graphs of  EMBED Equation.DSMT4 ,  EMBED Equation.DSMT4 , and  EMBED Equation.DSMT4  using different colored crayons.  Suggested blocks of InstructionCurriculum Management System Grade Level/Subject: Grade 11/PrecalculusBig Idea: Trigonometric FunctionsThe student will be able to evaluate and graph trigonometric functions.Objectives / Cluster Concepts / Cumulative Progress Indicators (CPI's) The student will be able to:Essential Questions Sample Conceptual UnderstandingsInstructional Tools / Materials / Technology / Resources / Learning Activities / Interdisciplinary Activities / Assessment ModelTo convert degree measures of angles to radians. To convert radian measures of angles to degrees. To determine coterminal angles. To determine the arc length and area of a sector of a circle with central angles in either degrees or radians. To use the definitions of sine and cosine to evaluate these functions. (4.3.12D.1) To use reference angles, calculators or tables, and special angles to evaluate sine and cosine functions. (4.3.12D.1) To sketch the graph of sine and cosine functions. (4.3.12B.2) To use reference angles, calculators or tables, and special angles to evaluate tangent, cotangent, secant, and cosecant functions. (4.3.12D.1) To sketch the graphs of tangent, cotangent, secant, and cosecant functions. (4.3.12B.2) To sketch the graph of the inverse of the sine, cosine, and tangent functions, and determine the domain and range. (4.3.12B.2) To evaluate the inverse of sine, cosine, and tangent functions with and without a calculator or table. (4.3.12D.1)Essential Questions: What are radians and how are they related to degrees? Explain the process of evaluating a trigonometric function using reference angles and the unit circle. How do the values on the unit circle correlate to the rectangular graph of a trigonometric function? Why is it necessary to restrict the domain in order to discuss inverse trigonometric functions? Enduring Understandings:  EMBED Equation.DSMT4 , where  EMBED Equation.DSMT4  is the measure of the central angle, in radians,  EMBED Equation.DSMT4 is the arc length, and  EMBED Equation.DSMT4 is the length of the radius. To convert each degree measure to radians, multiply by  EMBED Equation.DSMT4 . To convert each radian measure to degrees, multiply by  EMBED Equation.DSMT4 . The following formulas are used for the arc length  EMBED Equation.DSMT4 and area  EMBED Equation.DSMT4 of a sector with central angle  EMBED Equation.DSMT4 : a. If  EMBED Equation.DSMT4 is in degrees, then  EMBED Equation.DSMT4 and  EMBED Equation.DSMT4 . b. If  EMBED Equation.DSMT4 is in radians, then  EMBED Equation.DSMT4 and  EMBED Equation.DSMT4 .  EMBED Equation.DSMT4   EMBED Equation.DSMT4   EMBED Equation.DSMT4 ,  EMBED Equation.DSMT4   EMBED Equation.DSMT4 ,  EMBED Equation.DSMT4   EMBED Equation.DSMT4 ,  EMBED Equation.DSMT4   EMBED Equation.DSMT4 ,  EMBED Equation.DSMT4  The signs of the trigonometric functions (sine and cosecant, cosine and secant, and tangent and cotangent) in the four quadrants can be summarized by the following phrase: All Students Take Calculus. In order to evaluate a trigonometric expression, 1. Determine the quadrant of the terminal ray of the angle. 2. Determine if it is positive, negative, or zero. 3. Determine the reference angle. 4. Determine the exact value, if possible, in simplest radical form. Sample Assessment Questions: Convert each angle to radians in terms of  EMBED Equation.DSMT4 .  EMBED Equation.DSMT4   EMBED Equation.DSMT4   EMBED Equation.DSMT4   EMBED Equation.DSMT4  Convert each angle to degrees.  EMBED Equation.DSMT4   EMBED Equation.DSMT4   EMBED Equation.DSMT4   EMBED Equation.DSMT4  Give one positive and one negative coterminal angle for each angle below. Use the given form of the angle.  EMBED Equation.DSMT4   EMBED Equation.DSMT4 . A sector of a circle has central angle 1.2 radians and radius 6cm. Find its arc length. Find its area. Find the value of each expression leave answers in simplest radical form. Show reference angle statement when necessary. sin EMBED Equation.DSMT4  cos  EMBED Equation.DSMT4  tan  EMBED Equation.DSMT4  sec EMBED Equation.DSMT4  cos ( EMBED Equation.DSMT4 ) sin 3 EMBED Equation.DSMT4  csc  EMBED Equation.DSMT4  cot  EMBED Equation.DSMT4  sec  EMBED Equation.DSMT4  Graphing Project (See Addendum) Students will complete the following tasks and present their graphs in a neat and accurate presentation. Complete a table of exact values for all special angles and quadrantal angles  EMBED Equation.DSMT4 . Graph each of the 6 trigonometric functions on a separate graph. Include an accurate scale and asymptotes where appropriate. Suggested blocks of InstructionCurriculum Management System Grade Level/Subject: Grade 11/PrecalculusBig Idea: Trigonometric Equations and ApplicationsThe student will be able to stretch, shrink, and translate sine and cosine functions, simplify trigonometric expressions, and solve trigonometric equations.Objectives / Cluster Concepts / Cumulative Progress Indicators (CPI's) The student will be able to:Essential Questions Sample Conceptual UnderstandingsInstructional Tools / Materials / Technology / Resources / Learning Activities / Interdisciplinary Activities / Assessment ModelTo solve and apply simple trigonometric equations. (4.3.12D.2) To determine the slope and equation of a line given the angle of inclination and the coordinates of a point on the line, and determine the angle of inclination given the equation of a line or information about the line. (4.2.12C.1) To stretch and shrink the graphs of sine and cosine functions. (4.2.12B.1) To determine the period and amplitude of  EMBED Equation.DSMT4  and  EMBED Equation.DSMT4 . (4.3.12B.2) To determine the amplitude and period, and write the equation of sine and cosine curves. (4.3.12B.2) To solve equations of the form  EMBED Equation.DSMT4  and  EMBED Equation.DSMT4 . (4.3.12D.2) To determine the amplitude, period, axis of wave, and sketch the graph of translated sine and cosine functions. (4.3.12B.3; 4.3.12B.4) To determine the amplitude, period, axis of wave, and write the equation of the graph of translated sine and cosine functions. (4.2.12B.1; 4.3.12B.2) To simplify trigonometric expressions using the reciprocal relationships, relationships with negatives, Pythagorean relationships, and cofunction relationships of trigonometric functions. (4.3.12D.1) To prove trigonometric identities using the reciprocal relationships, relationships with negatives, Pythagorean relationships, and cofunction relationships of trigonometric functions. (4.2.12A.4) To use trigonometric identities or graphing calculator to solve more difficult trigonometric equations. (4.3.12D.2) Essential Questions: How does a change in amplitude or period affect the graph of a Sine or Cosine curve? Explain the effect of A, B, h, and k on the graph of a sine or Cosine curve using the equations  EMBED Equation.DSMT4  Explain how to find all possible solutions to simple trigonometric equations over a given domain. How can the graph of a trigonometric function be used to anticipate the number of solutions to a trigonometric equation? Could a single curve be described using both a Sine function and a Cosine function? Why? Enduring Understandings: For any line with slope m and inclination  EMBED Equation.DSMT4 ,  EMBED Equation.DSMT4  if  EMBED Equation.DSMT4 . If  EMBED Equation.DSMT4 , then the line has no slope. (The line is vertical.) For functions  EMBED Equation.DSMT4  and  EMBED Equation.DSMT4  EMBED Equation.DSMT4  and  EMBED Equation.DSMT4 : amplitude =  EMBED Equation.DSMT4  and period =  EMBED Equation.DSMT4 . If the graphs of  EMBED Equation.DSMT4 and  EMBED Equation.DSMT4 are translated horizontally h units and vertically k units, then the resulting graphs have equations  EMBED Equation.DSMT4  and  EMBED Equation.DSMT4 . The amplitude is  EMBED Equation.DSMT4 . To find the period, p = horizontal distance between successive maximums. Use the formula  EMBED Equation.DSMT4 . Relationships with negatives:  EMBED Equation.DSMT4  and  EMBED Equation.DSMT4   EMBED Equation.DSMT4 and  EMBED Equation.DSMT4   EMBED Equation.DSMT4  and  EMBED Equation.DSMT4  Pythagorean Relationships:  EMBED Equation.DSMT4   EMBED Equation.DSMT4   EMBED Equation.DSMT4  Cofunction Relationships  EMBED Equation.DSMT4  and  EMBED Equation.DSMT4   EMBED Equation.DSMT4  and  EMBED Equation.DSMT4   EMBED Equation.DSMT4  and  EMBED Equation.DSMT4  Learning Activities: Translating Graphs of Trigonometric Functions (See Addendum) Students will work in pairs to complete the graphing calculator activity on translating graphs of trigonometric functions. Deriving Basic Trigonometric Identities (See Addendum) Students will learn about negative angle relationships, Pythagorean relationships, and reciprocal relationships among trigonometric functions. Sample Assessment Questions: Solve  EMBED Equation.DSMT4  for  EMBED Equation.DSMT4 . Solve  EMBED Equation.DSMT4  for  EMBED Equation.DSMT4 . Solve  EMBED Equation.DSMT4  for  EMBED Equation.DSMT4 without using tables or a calculator. To the nearest degree, find the inclination of the line  EMBED Equation.DSMT4 . Find the slope and equation of a line with an inclination of  EMBED Equation.DSMT4 and contains (2,3). Give the amplitude and period of the function  EMBED Equation.DSMT4 . Then sketch at least one cycle of its graph. Solve  EMBED Equation.DSMT4  for  EMBED Equation.DSMT4 . For each function: State the amplitude of the curve. State the period of the curve. Describe any vertical or horizontal translations of the curve. Sketch the graph by hand. Confirm your sketch using a graphing calculator.  EMBED Equation.DSMT4   EMBED Equation.DSMT4   EMBED Equation.DSMT4   EMBED Equation.DSMT4  Simplify  EMBED Equation.DSMT4 . Prove:  EMBED Equation.DSMT4 . Solve  EMBED Equation.DSMT4  for  EMBED Equation.DSMT4 . Solve  EMBED Equation.DSMT4 for  EMBED Equation.DSMT4 . Solve  EMBED Equation.DSMT4  for  EMBED Equation.DSMT4 . Solve  EMBED Equation.DSMT4  for  EMBED Equation.DSMT4 . Suggested blocks of InstructionCurriculum Management System Grade Level/Subject: Grade 11/PrecalculusBig Idea: Triangle TrigonometryThe student will be able to apply the trigonometric definitions, Law of Sines, and Law of Cosines to determine the lengths of unknown sides or measures of unknown angles in triangles.Objectives / Cluster Concepts / Cumulative Progress Indicators (CPI's) The student will be able to:Essential Questions Sample Conceptual UnderstandingsInstructional Tools / Materials / Technology / Resources / Learning Activities / Interdisciplinary Activities / Assessment ModelTo use trigonometry to find the lengths of unknown sides or measures of unknown angles of a right triangle. (4.2.12A.1; 4.2.12E.1) To determine the area of a triangle given the lengths of two sides of a triangle and the measure of the included angle. (4.2.12E.2) To use the Law of Sines to find unknown parts of a triangle. (4.2.12E.1) To use the Law of Cosines to find unknown parts of a triangle. (4.2.12E.1) To use trigonometry to solve navigation and surveying problems. (4.2.12E.1) Essential Questions: What does the acronym SOH-CAH-TOA stand for? Given the lengths of two sides of a right triangle, or the length of one side and the measure of one acute angle, how can you find the measures of the remaining sides and angles using the trigonometric functions? How can the area of a triangle be determined given the lengths of two sides and the measure of the included angle? When given the lengths of two sides and the measure of a non-included angle of a triangle, how many measurements are possible for the unknown angles and why? For which of the following situation is the Law of Cosines used? SAS, SSS, ASA, AAS, or SSA For which of the following situations is the Law of Sines used? SAS, SSS, ASA, AAS, or SSA How is measuring an angle in standard form different from measuring an angle from a compass bearing? Enduring Understandings: In  EMBED Equation.DSMT4 with right angle  EMBED Equation.DSMT4   EMBED Equation.DSMT4   EMBED Equation.DSMT4   EMBED Equation.DSMT4   EMBED Equation.DSMT4   EMBED Equation.DSMT4   EMBED Equation.DSMT4  The area  EMBED Equation.DSMT4 of  EMBED Equation.DSMT4 is given by:  EMBED Equation.DSMT4 . In other words, the area of any triangle is  EMBED Equation.DSMT4 (one side)  EMBED Equation.DSMT4 (another side) EMBED Equation.DSMT4 (sine of the included angle). Derive the formula for the area of a triangle by rewriting the height of the triangle in terms of an angle and a side (that doesnt intersect the height of the triangle). Derive the Law of Sines using the three different ways of writing the area of a triangle (using each of the three angles). The Law of Sines: In  EMBED Equation.DSMT4 ,  EMBED Equation.DSMT4 . The Law of Cosines: in  EMBED Equation.DSMT4 ,  EMBED Equation.DSMT4 . In other words, the square of the side opposite an angle is equal to the square of one side of the angle plus the square of the other side of the angle minus twice the product of the two sides of the angle times the cosine of the angle. The Law of Cosines (alternate form):  EMBED Equation.DSMT4 . Given SAS, use the Law of Cosines to find the measure of the third side and then one of the remaining angles. Given SSS, use the Law of Cosines to find the measures of any two angles. Given ASA or AAS, use the Law of Sines to find the measures of the remaining sides. Given SSA, use the Law of Sines to find an angle opposite a given side and then the third side. (Note that 0, 1, or 2 triangles are possible.)Learning Activities: Perform the following activity to illustrate that when given the lengths of two sides of a triangle and the measure of a non-included angle (SSA), it may be possible to construct no triangle, one triangle, or two triangles. Draw  EMBED Equation.DSMT4 with measure  EMBED Equation.DSMT4 . Along one ray of  EMBED Equation.DSMT4 , locate point  EMBED Equation.DSMT4 10 cm from point  EMBED Equation.DSMT4 . For each of the following compass settings, draw a large arc. Then determine if the arc intersects the other ray of  EMBED Equation.DSMT4  and, if so, in how many points. a. Compass at  EMBED Equation.DSMT4  and opened to 4 cm. (0) b. Compass at  EMBED Equation.DSMT4  and opened to 5 cm. (1) c. Compass at  EMBED Equation.DSMT4  and opened to 6 cm. (2) Sample Assessment Questions: For right triangle  EMBED Equation.DSMT4  with right angle  EMBED Equation.DSMT4 ,  EMBED Equation.DSMT4 and  EMBED Equation.DSMT4 . Find  EMBED Equation.DSMT4 . The safety instructions for a 20 ft. ladder indicate that the ladder should not be inclined at more than a  EMBED Equation.DSMT4 angle with the ground. Suppose the ladder is leaned against a house with a  EMBED Equation.DSMT4 angle with the ground. Find (a) the distance x from the base of the house to the foot of the ladder and (b) the height y reached by the ladder. The highest tower in the world is in Toronto, Canada, and is 553 m high. An observer at point  EMBED Equation.DSMT4 , 100 m from the center of the towers base, sights the top of the tower. The angle of elevation is  EMBED Equation.DSMT4 . Find the measure of this angle. A triangle has sides of length 8, 8, and 4. Find the measures of the angles of the triangle. Two sides of a triangle have lengths 7cm and 4cm. The angle between the sides measures  EMBED Equation.DSMT4 . Find the area of the triangle. If the angle between the sides is changed to  EMBED Equation.DSMT4 , what is the area of the new triangle? Why are the two areas the same? The area of  EMBED Equation.DSMT4  is 15. If  EMBED Equation.DSMT4  and  EMBED Equation.DSMT4 , find all possible measures of  EMBED Equation.DSMT4 . Why are there two answers? In  EMBED Equation.DSMT4 ,  EMBED Equation.DSMT4 ,  EMBED Equation.DSMT4 , and  EMBED Equation.DSMT4 . Find  EMBED Equation.DSMT4 and  EMBED Equation.DSMT4 . In  EMBED Equation.DSMT4 ,  EMBED Equation.DSMT4 ,  EMBED Equation.DSMT4 , and  EMBED Equation.DSMT4 . Determine whether  EMBED Equation.DSMT4  exists, and, if so, find all possible measures of  EMBED Equation.DSMT4 . Two sides of a triangle have lengths 3 cm and 7 cm, and the included angle has a measure of  EMBED Equation.DSMT4 . Find the length of the third side. The lengths of the sides of a triangle are 5, 6, and 7. Solve the triangle. Suggested blocks of InstructionCurriculum Management System Grade Level/Subject: Grade 11/PrecalculusBig Idea: Trigonometric Addition FormulasThe student will be able to derive and apply the sum and difference formulas and the double-angle formulas for sine, cosine, and tangent, and apply these formulas to solve trigonometric equations.Objectives / Cluster Concepts / Cumulative Progress Indicators (CPI's) The student will be able to:Essential Questions Sample Conceptual UnderstandingsInstructional Tools / Materials / Technology / Resources / Learning Activities / Interdisciplinary Activities / Assessment ModelTo derive and apply the formulas for  EMBED Equation.DSMT4  and for  EMBED Equation.DSMT4 . (4.3.12D.1) To derive and apply the formulas for  EMBED Equation.DSMT4 . (4.3.12D.1) To derive and apply the double-angle formulas. (4.3.12D.1) To use the double-angle formulas to solve trigonometric equations. (4.3.12D.2) Essential Questions: For what angle measurements can the sum and difference formulas be used? How can the double-angle formulas be derived from the sum and difference formulas for sine, cosine, and tangent? Enduring Understandings: Use the Law of Cosines or the distance formula to derive the formulas for the difference (and then the sum) of two angles for cosines. Then use the cofunction relationship to derive the formula for the sum (and then the difference) of two angles for sine. (See pages 369-370.) The sum and difference formulas for sine, cosine, and tangent are as follows:  EMBED Equation.DSMT4   EMBED Equation.DSMT4   EMBED Equation.DSMT4   EMBED Equation.DSMT4   EMBED Equation.DSMT4   EMBED Equation.DSMT4  Use the sum and difference formulas for sine, cosine, and tangent, and the Pythagorean relationships to derive the double angle formulas. The double angle formulas are as follows:  EMBED Equation.DSMT4   EMBED Equation.DSMT4   EMBED Equation.DSMT4  A quick review of factoring polynomials may be helpful before solving trigonometric equations using the double-angle formulas. Sample Assessment Questions: Find the exact value of  EMBED Equation.DSMT4 . Find the exact value of : a.  EMBED Equation.DSMT4  b.  EMBED Equation.DSMT4 . If  EMBED Equation.DSMT4 and  EMBED Equation.DSMT4 , where  EMBED Equation.DSMT4 and  EMBED Equation.DSMT4 , find  EMBED Equation.DSMT4 . If  EMBED Equation.DSMT4 , find  EMBED Equation.DSMT4 . Solve  EMBED Equation.DSMT4 for  EMBED Equation.DSMT4 using a graphing calculator and algebraically. Solve  EMBED Equation.DSMT4 for  EMBED Equation.DSMT4 . Solve  EMBED Equation.DSMT4  for  EMBED Equation.DSMT4 . Suggested blocks of InstructionCurriculum Management System Grade Level/Subject: Grade 11/PrecalculusBig Idea: Polar Coordinates and Complex NumbersThe student will be able to represent points in rectangular and polar coordinates, and multiply and find powers of complex numbers. Objectives / Cluster Concepts / Cumulative Progress Indicators (CPI's) The student will be able to:Essential Questions Sample Conceptual UnderstandingsInstructional Tools / Materials / Technology / Resources / Learning Activities / Interdisciplinary Activities / Assessment ModelTo graph points given polar coordinates. (4.2.12C.1) To state two additional polar coordinates for the same point. (4.2.12C.1) To convert from rectangular (Cartesian) to polar coordinates. To convert from polar to rectangular coordinates. To graph polar equations. (4.3.12B.1) Explore special polar graphs including Cardiod, Limacon, and Rose curves. (4.3.12B.1; 4.3.12B.4) To convert complex numbers in rectangular form to polar form. To express complex numbers in polar form to rectangular form. To determine the product of two complex numbers in polar form. (4.3.12D.3) To use De Moivres theorem to determine powers of complex numbers. (4.3.12D.3) To determine roots of complex numbers. (4.3.12D.2) Essential Questions: How many ways can a point be represented using polar coordinates? How are rectangular coordinates converted to polar coordinates? How are polar coordinates converted to rectangular coordinates? What is the advantage of using De Moivres theorem? Enduring Understandings: The polar coordinates of a point  EMBED Equation.DSMT4 are  EMBED Equation.DSMT4 , where  EMBED Equation.DSMT4 is the directed distance from the pole to  EMBED Equation.DSMT4 and  EMBED Equation.DSMT4  is the polar angle measured from the polar axis to the ray  EMBED Equation.DSMT4 . Although a point has only one pair of rectangles coordinates, it has many pairs of polar coordinates. For example,  EMBED Equation.DSMT4 all represent the same point. Formulas for converting from polar to rectangular coordinates:  EMBED Equation.DSMT4 . Formulas for converting from rectangular to polar coordinates:  EMBED Equation.DSMT4 . The complex plane can be represented by an Argand diagram. In this diagram, the complex number  EMBED Equation.DSMT4 is represented by the point  EMBED Equation.DSMT4 or by an arrow from the origin to  EMBED Equation.DSMT4 . The absolute value of a complex number  EMBED Equation.DSMT4 is  EMBED Equation.DSMT4 . Thus,  EMBED Equation.DSMT4 . To multiply two complex numbers in polar form: 1. Multiply their absolute values. 2. Add their polar angles. In other words, if  EMBED Equation.DSMT4 and  EMBED Equation.DSMT4 , then  EMBED Equation.DSMT4 . De Moivres Theorem: If  EMBED Equation.DSMT4 , then  EMBED Equation.DSMT4 . The  EMBED Equation.DSMT4 nth roots of  EMBED Equation.DSMT4  are: EMBED Equation.DSMT4  for  EMBED Equation.DSMT4 .Graphing Project: (See Addendum) Polar vs. Rectangular Graphs In this activity may help students see the connections between rectangular and polar graphs. Sample Assessment Questions: Express  EMBED Equation.DSMT4 in rectangular form. Express  EMBED Equation.DSMT4 in polar form. If  EMBED Equation.DSMT4 : a. find  EMBED Equation.DSMT4 in rectangular form by multiplying  EMBED Equation.DSMT4 and  EMBED Equation.DSMT4 . b. find  EMBED Equation.DSMT4 ,  EMBED Equation.DSMT4 , and  EMBED Equation.DSMT4 in polar form. c. show that  EMBED Equation.DSMT4 in polar form agree with  EMBED Equation.DSMT4 in rectangular form. If  EMBED Equation.DSMT4 , find  EMBED Equation.DSMT4 and  EMBED Equation.DSMT4 . Plot these on an Argand diagram. Find the cube roots of  EMBED Equation.DSMT4 . Find the four fourth roots of -16. Suggested blocks of InstructionCurriculum Management System Grade Level/Subject: Grade 11/PrecalculusBig Idea: Sequences and SeriesThe student will be able to identify arithmetic and geometric sequences and series, write a formula for the nth term of sequences and series using an explicit or recursive definition, apply sigma notation, and find the sum of finite arithmetic, and finite and infinite geometric series.Objectives / Cluster Concepts / Cumulative Progress Indicators (CPI's) The student will be able to:Essential Questions Sample Conceptual UnderstandingsInstructional Tools / Materials / Technology / Resources / Learning Activities / Interdisciplinary Activities / Assessment ModelTo identify an arithmetic and geometric sequence. To write a formula for the nth term of an arithmetic sequence. (4.3.12A.1) To write a formula for the nth term of a geometric sequence. (4.3.12A.1) To write the formula for the nth term of a sequence that is neither arithmetic nor geometric. (4.3 12A.1; 4.3.12A.3) To use and write recursive definitions of sequences. (4.3.12A.1) To determine the sum of the first n terms of arithmetic and geometric series. (4.3 12A.1) To find or estimate the limit of an infinite sequence or to determine that the limit does not exist. (4.2.12B.4; 4.3.12A.1; 4.3.12A.2) To determine the sum of an infinite geometric series. (4.2.12B.4; 4.3.12A.1) To expand series written in sigma notation. (4.3.12A.1) To condense series into sigma notation. (4.3.12A.1) Essential Questions: What makes a sequence arithmetic? What is the graph of an arithmetic sequence? What makes a sequence geometric? What is the graph of a geometric sequence? What is the difference between an explicit definition and a recursive definition of a sequence? What is the difference between a sequence and a series? When does the sum of a series converge or diverge? Enduring Understandings: A sequence is arithmetic if the difference d of any two consecutive terms is constant. The formula for the nth term in an arithmetic sequence is:  EMBED Equation.DSMT4 . A sequence is geometric if the ratio r of any two consecutive terms is constant. The formula for the nth term in a geometric sequence is:  EMBED Equation.DSMT4 . A recursive definition consists of two parts: 1. An initial condition that states the first term of the sequence. 2. A recursive equation (or recursive formula) that states how any term in the sequence is related to the preceding term. A series is an indicated sum of terms of a sequence. The sum of the first n terms in an arithmetic series is  EMBED Equation.DSMT4 . The sum of the first n terms of a geometric series is  EMBED Equation.DSMT4 , where r is the common ratio and  EMBED Equation.DSMT4 . The sum of an infinite geometric series when  EMBED Equation.DSMT4 is:  EMBED Equation.DSMT4 . If  EMBED Equation.DSMT4  and  EMBED Equation.DSMT4 , then the series diverges. Sample Assessment Questions: State the next term in each sequence. Then write a rule for the nth term. Identify is the sequence is arithmetic, geometric, or neither. 1.  EMBED Equation.DSMT4  2.  EMBED Equation.DSMT4  3.  EMBED Equation.DSMT4  4.  EMBED Equation.DSMT4  5.  EMBED Equation.DSMT4  6.  EMBED Equation.DSMT4  (discuss EMBED Equation.DSMT4 ) Write a recursive rule for the sequences #1, 2, and 5 above. Write a rule for the nth term of the arithmetic sequence. Then find  EMBED Equation.DSMT4 . 1.  EMBED Equation.DSMT4  2.  EMBED Equation.DSMT4  3.  EMBED Equation.DSMT4  Write a rule for the nth term of the geometric sequence. Then find  EMBED Equation.DSMT4 .  EMBED Equation.DSMT4  EMBED Equation.DSMT4  One term of a geometric sequence is  EMBED Equation.DSMT4 . The common ratio is  EMBED Equation.DSMT4 . Write a rule for the nth term. Two terms of a geometric sequence are  EMBED Equation.DSMT4  and  EMBED Equation.DSMT4 . Find a rule for the nth term. State the first five terms of the sequence. 1.  EMBED Equation.DSMT4   EMBED Equation.DSMT4  2.  EMBED Equation.DSMT4   EMBED Equation.DSMT4  3.  EMBED Equation.DSMT4   EMBED Equation.DSMT4  4.  EMBED Equation.DSMT4   EMBED Equation.DSMT4  For the arithmetic series  EMBED Equation.DSMT4  1. Determine the sum of the first 30 terms. 2. Find n such that  EMBED Equation.DSMT4 . For the geometric series  EMBED Equation.DSMT4  1. Determine the sum of the first 10 terms. 2. Find n such that  EMBED Equation.DSMT4 . Find the sum of  EMBED Equation.DSMT4  A ball is dropped from a height of 5 feet. Each time it hits the ground, it bounces one half of its previous height. Find the total distance traveled by the ball. Find the sum of the series. 1.  EMBED Equation.DSMT4  2.  EMBED Equation.DSMT4  3.  EMBED Equation.DSMT4  4.  EMBED Equation.DSMT4  Write in sigma notation. 1.  EMBED Equation.DSMT4  2.  EMBED Equation.DSMT4   Suggested blocks of InstructionCurriculum Management System Grade Level/Subject: Grade 11/PrecalculusBig Idea: Limits The student will be able to determine the limit of a function or the quotient of two functions, and sketch the graph of a rational function using limits. Objectives / Cluster Concepts / Cumulative Progress Indicators (CPI's) The student will be able to:Essential Questions Sample Conceptual UnderstandingsInstructional Tools / Materials / Technology / Resources / Learning Activities / Interdisciplinary Activities / Assessment ModelTo determine the limit of a function or the quotient of two functions as x approaches  EMBED Equation.DSMT4  or  EMBED Equation.DSMT4 . (4.3.12A.2) To determine the limit of a function or the quotient of two functions as x approaches a real number. (4.3.12A.2) To determine if a function is continuous. (4.3.12B.1; 4.3.12B.2) To sketch the graph of a rational function by determining the x-intercepts, vertical and horizontal asymptotes, performing a sign analysis, and use limits. (4.3.12B.1; 4.3.12B.2)Essential Questions: How do we find the limit of a function as  EMBED Equation.DSMT4 approaches  EMBED Equation.DSMT4 or  EMBED Equation.DSMT4 ? How do we find the limit of a function as  EMBED Equation.DSMT4 approaches a real number  EMBED Equation.DSMT4 ? How do we find the limit of a quotient of two functions? What is an inverse variation and how does it compare to other rational functions? What is significant about the graph of a rational function? How can the vertical and horizontal asymptotes of a rational function be identified analytically? Explain the procedure to sketch a possible graph of a rational function. Enduring Understandings: If  EMBED Equation.DSMT4 , then  EMBED Equation.DSMT4 .. The Quotient Theorem for Limits: If  EMBED Equation.DSMT4 and  EMBED Equation.DSMT4 both exist, and  EMBED Equation.DSMT4 , then  EMBED Equation.DSMT4 . Techniques for evaluating  EMBED Equation.DSMT4 : 1. If possible, use the quotient theorem for limits. 2. If  EMBED Equation.DSMT4 and  EMBED Equation.DSMT4 , try the following techniques: a) Factor  EMBED Equation.DSMT4 and  EMBED Equation.DSMT4 , and reduce  EMBED Equation.DSMT4 to lowest terms. b) If  EMBED Equation.DSMT4 or  EMBED Equation.DSMT4 involves a square root, try multiplying both  EMBED Equation.DSMT4 and  EMBED Equation.DSMT4 by the conjugate of the square root expression. 3. If  EMBED Equation.DSMT4 and  EMBED Equation.DSMT4 , then either statement (a) or (b) is true: a)  EMBED Equation.DSMT4 does not exist b)  EMBED Equation.DSMT4 or  EMBED Equation.DSMT4  4. If  EMBED Equation.DSMT4 is approaching infinity or negative infinity, divide the numerator and denominator by the highest power of  EMBED Equation.DSMT4 in the denominator. 5. Evaluate  EMBED Equation.DSMT4 by evaluating  EMBED Equation.DSMT4 for very large values of  EMBED Equation.DSMT4 , and evaluate  EMBED Equation.DSMT4 by evaluating  EMBED Equation.DSMT4  for  EMBED Equation.DSMT4 -values very near  EMBED Equation.DSMT4 . These limits can also be guessed by using a graphing calculator to examine the graph of  EMBED Equation.DSMT4 for very large values of  EMBED Equation.DSMT4 , or for  EMBED Equation.DSMT4 -values very near  EMBED Equation.DSMT4 . (A graphing calculator might not show points of discontinuity.) Given a rational function of the form  EMBED Equation.DSMT4 , find the horizontal asymptotes by the following rules: a. If the degree of  EMBED Equation.DSMT4 and  EMBED Equation.DSMT4 are the same, then the horizontal asymptote is the ratio of the leading coefficients of EMBED Equation.DSMT4 and  EMBED Equation.DSMT4 . b. If the degree of  EMBED Equation.DSMT4 is greater then the degree of  EMBED Equation.DSMT4 , then the horizontal asymptote is the  EMBED Equation.DSMT4 axis. c. If the degree of  EMBED Equation.DSMT4 is less than the degree of  EMBED Equation.DSMT4 , there is no horizontal asymptote. Given a rational function of the form  EMBED Equation.DSMT4 , a discontinuity (vertical asymptote or hole) will occur whenever  EMBED Equation.DSMT4 . Given a rational function of the form  EMBED Equation.DSMT4 , the  EMBED Equation.DSMT4 -intercepts will occur when  EMBED Equation.DSMT4 (unless it is also a zero of  EMBED Equation.DSMT4 ).NOTE: The assessment models provided in this document are suggestions for the teacher. If the teacher chooses to develop his/her own model, it must be of equal or better quality and at the same or higher cognitive levels (as noted in parentheses). Depending upon the needs of the class, the assessment questions may be answered in the form of essays, quizzes, mobiles, PowerPoint, oral reports, booklets, or other formats of measurement used by the teacher. Sample Assessment Questions: Evaluate: a)  EMBED Equation.DSMT4  b)  EMBED Equation.DSMT4  c)  EMBED Equation.DSMT4  d)  EMBED Equation.DSMT4  e)  EMBED Equation.DSMT4  f)  EMBED Equation.DSMT4  g)  EMBED Equation.DSMT4  h)  EMBED Equation.DSMT4  i)  EMBED Equation.DSMT4  j)  EMBED Equation.DSMT4  k)  EMBED Equation.DSMT4  l)  EMBED Equation.DSMT4  m)  EMBED Equation.DSMT4  n)  EMBED Equation.DSMT4  o)  EMBED Equation.DSMT4  p)  EMBED Equation.DSMT4  For each rational function: Find all horizontal and vertical asymptotes of the graph. (synthesis) Identify any holes in the graph. (application) Sketch a possible graph of the function using asymptotes, intercepts and sign analysis. (synthesis) Confirm your graph using a graphing calculator. (application) State the domain and range of the function. (analysis)  EMBED Equation.DSMT4   EMBED Equation.DSMT4   EMBED Equation.DSMT4   EMBED Equation.DSMT4   EMBED Equation.DSMT4  The function  EMBED Equation.DSMT4  relates atmospheric pressure,  EMBED Equation.DSMT4 , in inches of mercury, to altitude,  EMBED Equation.DSMT4 , in miles. Graph the function. (application) Find the atmospheric pressure at Mt. Kilimanjaro with altitude 19,340 ft. (application) Is there an altitude at which the atmospheric pressure is 0 inches of mercury? Use your graph to justify your answer. (synthesis/evaluation  Suggested blocks of InstructionCurriculum Management System Grade Level/Subject: Grade 11/PrecalculusBig Idea: Exponents and LogarithmsThe student will be able to simplify expressions and solve equations with exponents and logarithms.Objectives / Cluster Concepts / Cumulative Progress Indicators (CPI's) The student will be able to:Essential Questions Sample Conceptual UnderstandingsInstructional Tools / Materials / Technology / Resources / Learning Activities / Interdisciplinary Activities / Assessment ModelTo apply the laws of exponents to simplify numeric and algebraic expressions with integral exponents. (4.1.12B.4; 4.3.12D.1) To apply the laws of exponents to simplify numeric and algebraic expressions with rational exponents. (4.1.12B.4; 4.3.12D.1) To solve real-world problems involving exponential growth and decay. (4.3.12C.1) To estimate the doubling time using the Rule of 72. (4.3.12C.1) To apply the formula for compounded interest. (4.3.12C.1) To define and apply the natural exponential function. To derive and apply the formula for interest compounded continuously. (4.3.12C.1) To define and evaluate logarithms. (4.3.12D.1) To solve logarithmic equations without and with a calculator. (4.3.12D.2) To apply the laws of logarithms to expand and condense logarithmic expressions. (4.3.12D.1) To solve exponential equations by rewriting both sides of the equation with the same base. (4.3.12D.2) To solve exponential equations by using the definition of logarithms to rewrite as a logarithmic equation. (4.3.12D.2) To evaluate logarithmic expressions using the Change of Base formula. (4.3.12D.1) Essential Questions: For what values of b does  EMBED Equation.DSMT4 represent exponential growth? Exponential decay? Enduring Understandings: Properties of Exponents: 1.  EMBED Equation.DSMT4  2.  EMBED Equation.DSMT4  When multiplying expressions with the same base, add the exponents.  EMBED Equation.DSMT4  When dividing expressions with the same base, subtract the exponents.  EMBED Equation.DSMT4  When raising a power to a power, multiply the exponents.  EMBED Equation.DSMT4   EMBED Equation.DSMT4   EMBED Equation.DSMT4  If  EMBED Equation.DSMT4 , then  EMBED Equation.DSMT4 if and only if  EMBED Equation.DSMT4 . An exponential function has the form  EMBED Equation.DSMT4 , where  EMBED Equation.DSMT4 and EMBED Equation.DSMT4 . If  EMBED Equation.DSMT4 , it is an exponential growth function, and if  EMBED Equation.DSMT4 , then it is an exponential growth function. Compound Interest Formula: The amount  EMBED Equation.DSMT4 in an account earning interest compounded  EMBED Equation.DSMT4 times per year for  EMBED Equation.DSMT4  years is:  EMBED Equation.DSMT4  , where  EMBED Equation.DSMT4  is the principal and  EMBED Equation.DSMT4 is the annual interest rate expressed as a decimal. The rule of 72 provides an approximation of the doubling time for exponential growth. If a quantity is growing at r% per year, then the doubling time  EMBED Equation.DSMT4 . The number  EMBED Equation.DSMT4 is defined as:  EMBED Equation.DSMT4 . Continuously Compounded Interest Formula:  EMBED Equation.DSMT4 . The logarithm of x to the base b  EMBED Equation.DSMT4  is the exponent a such that  EMBED Equation.DSMT4 . Thus,  EMBED Equation.DSMT4 if and only if  EMBED Equation.DSMT4 . Laws of Logarithms: 1.  EMBED Equation.DSMT4  2.  EMBED Equation.DSMT4  3.  EMBED Equation.DSMT4  4.  EMBED Equation.DSMT4 if and only if  EMBED Equation.DSMT4  The change of base formula enables you to write logarithms in any given base in terms in any other base.  EMBED Equation.DSMT4 Sample Assessment Questions: Simplify  EMBED Equation.DSMT4  Simplify  EMBED Equation.DSMT4 , where  EMBED Equation.DSMT4 and  EMBED Equation.DSMT4 . Simplify  EMBED Equation.DSMT4  and  EMBED Equation.DSMT4 , where  EMBED Equation.DSMT4 . A radioactive isotope decays so that the radioactivity present decreases by 15% per day. If 40 kg are present now, find the amount present (a) 6 days from now, and (b) 6 days ago. Find the balance after 10 years if $5000 is invested in a bank account that pays 4.5% interest compounded 1. annually. 2. quarterly. 3. monthly. 4. daily. Find the balance after 10 years if $5000 is invested in a bank account that pays 4.5% annual interest compounded continuously. In about how many years will it take for $1000 to double in value with a 6% annual interest rate compounded continuously? (Discuss the rule of 72) Evaluate without a calculator: 1.  EMBED Equation.DSMT4  2.  EMBED Equation.DSMT4  3.  EMBED Equation.DSMT4  4.  EMBED Equation.DSMT4  5.  EMBED Equation.DSMT4  6.  EMBED Equation.DSMT4  Expand the expression. 1.  EMBED Equation.DSMT4  2.  EMBED Equation.DSMT4  Condense the expression. 1.  EMBED Equation.DSMT4  2.  EMBED Equation.DSMT4  Use the change of base formula to evaluate 1.  EMBED Equation.DSMT4  2.  EMBED Equation.DSMT4  Solve. Check for extraneous solutions. 1.  EMBED Equation.DSMT4  2.  EMBED Equation.DSMT4  3.  EMBED Equation.DSMT4  4.  EMBED Equation.DSMT4  5.  EMBED Equation.DSMT4  6.  EMBED Equation.DSMT4   Precalculus COURSE BENCHMARKS The student will be able to stretch, shrink, reflect, or translate the graph of a function, and determine the inverse of a function, if it exists. The student will be able to evaluate and graph trigonometric functions. The student will be able to stretch, shrink, and translate sine and cosine functions, simplify trigonometric expressions, and solve trigonometric equations. The student will be able to apply the trigonometric definitions, Law of Sines, and Law of Cosines to determine the lengths of unknown sides or measures of unknown angles in triangles. The student will be able to derive and apply the sum and difference formulas and the double-angle formulas for sine, cosine, and tangent, and apply these formulas to solve trigonometric equations. The student will be able to represent points in rectangular and polar coordinates, and multiply and find powers of complex numbers. The student will be able to identify arithmetic and geometric sequences and series, write a formula for the nth term of sequences and series using an explicit or recursive definition, apply sigma notation, and find the sum of finite arithmetic, and finite and infinite geometric series. The student will be able to determine the limit of a function or the quotient of two functions, and sketch the graph of a rational function using limits. The student will be able to simplify expressions and solve equations with exponents and logarithms. Addendum Goal 2: Graphing Project GRAPHING TRIGONOMETRIC FUNCTIONS DIRECTIONS: Fill in all values on the table below. Use the values to graph each trig function on a separate sheet of graph paper. You will be graded on the following: A complete and correct table of values in simplest radical form. Six separate graphs including : title, scale, points plotted accurately using decimal approximations, asymptotes where necessary and smooth continuous curves  EMBED Equation.DSMT4 in degrees EMBED Equation.DSMT4 in radians EMBED Equation.DSMT4  EMBED Equation.DSMT4  EMBED Equation.DSMT4  EMBED Equation.DSMT4  EMBED Equation.DSMT4  EMBED Equation.DSMT4 -360 EMBED Equation.DSMT4 -330 EMBED Equation.DSMT4 -315 EMBED Equation.DSMT4 -300 EMBED Equation.DSMT4 -270 EMBED Equation.DSMT4 -240 EMBED Equation.DSMT4 -225 EMBED Equation.DSMT4 -210 EMBED Equation.DSMT4 -180 EMBED Equation.DSMT4 -150 EMBED Equation.DSMT4 -135 EMBED Equation.DSMT4 -120 EMBED Equation.DSMT4 -90 EMBED Equation.DSMT4 -60 EMBED Equation.DSMT4 -45 EMBED Equation.DSMT4 -30 EMBED Equation.DSMT4 0 EMBED Equation.DSMT4  EMBED Equation.DSMT4 in degrees EMBED Equation.DSMT4 in radians EMBED Equation.DSMT4  EMBED Equation.DSMT4  EMBED Equation.DSMT4  EMBED Equation.DSMT4  EMBED Equation.DSMT4  EMBED Equation.DSMT4 0 EMBED Equation.DSMT4 30 EMBED Equation.DSMT4 45 EMBED Equation.DSMT4 60 EMBED Equation.DSMT4 90 EMBED Equation.DSMT4 120 EMBED Equation.DSMT4  135 EMBED Equation.DSMT4 150 EMBED Equation.DSMT4 180 EMBED Equation.DSMT4 210 EMBED Equation.DSMT4 225 EMBED Equation.DSMT4 240 EMBED Equation.DSMT4 270 EMBED Equation.DSMT4 300 EMBED Equation.DSMT4 315 EMBED Equation.DSMT4 330 EMBED Equation.DSMT4 360 EMBED Equation.DSMT4  (The grid below can be used instead of traditional graph paper to help students create a reasonable scale on the horizontal axis) TITLE OF GRAPH:     Goal 3 Learning Activity-Translating Graphs of Trigonometric Functions Students may work in pairs or alone to complete the exercises and discover the relationship between translations of trigonometric functions and their equations. TRANSLATING GRAPHS OF TRIGONOMETRIC FUNCTIONS Complete each task in the space provided using a graphing calculator. 1. Set the MODE on your calculator to agree with the window below. 2. Set you WINDOW to agree with the window below. type in  EMBED Equation.DSMT4  for the first three values. 3. Graph the following equations in the window provided.   4. Using the graphs above and your class notes, describe the effect that a change in A has on the graph of y = A sin(x). Use the proper vocabulary word for A in your explanation. Will the effect be the same for y = A cos(x)? Test your conclusion using the calculator. 5. Sketch each graph in the space provided.  6. Using the graphs above, describe the effects that a change in B has on the graph of y = sin(Bx) or y = cos(Bx). 7. Use the information below to answer questions about the exercises that follow. Given y = Asin(Bx) or y = Acos(Bx) A represents amplitude  EMBED Equation.DSMT4 . B represents the number of cycles in  EMBED Equation.DSMT4 units. B is related to the period P as follows:  EMBED Equation.DSMT4  or  EMBED Equation.DSMT4 . Try the following exercises without the calculator. You may check your work with the calculator when you are done. NOTE: use the same window that you set up in the beginning of this packet.  Remember: each mark on the x-axis represents  EMBED Equation.DSMT4  which means that 4 marks =  EMBED Equation.DSMT4 . 8. Sketch each graph and state the amplitude and period.  EMBED Equation.DSMT4   EMBED Equation.DSMT4   EMBED Equation.DSMT4   A=_______ P=_______ A=_______ P=_______ A=________ P=_______ 9. Match each equation with its graph.  Label each graph  EMBED Equation.DSMT4 or  EMBED Equation.DSMT4 .  10. Reset your window using the sample below:  NOTE: Now each mark on the x-axis is equal to 1.0 decimal radians. 11. Sketch each graph in the window provided.  12. Using the graphs above, describe the effect of adding or subtracting a number on the outside of the function had on the graph. 13. Sketch the graph of each function below:  14. Using the graphs above, describe the effect that adding or subtracting a number on the inside of the function has on its graph. (You may want to graph the y = sin(x) or y = cos(x) on the same axis to compare. 15. Sketch each graph below:  16. Try to sketch each graph without the calculator, then check your answer.  EMBED Equation.DSMT4   EMBED Equation.DSMT4   EMBED Equation.DSMT4   17. Use the descriptions in #7 to find the amplitude and period for each graph in #16. 18. Match each equation with is graph. Note the scale on each graph.  x-scale =  EMBED Equation.DSMT4   x-scale =  EMBED Equation.DSMT4   x-scale = 1.0  x-scale = 1.0 19. Summarize the effects of A,B,H, and K on the graphs of sine and cosine using the equations  EMBED Equation.DSMT4 . Goal 3: Learning Activity-Deriving Basic Trigonometric Identities. The following exercises can be used to help students discover the basic trigonometric identities. After students have completed the exercises, a formal discussion of trigonometric identities should follow. DERIVING TRIGONOMETRIC IDENTITIES COMPETE EACH SET OF EXERCISES AND DRAW CONCLUSIONS IN THE SPACE PROVIDED. 1. Give the exact value for each expression in simplest radical form. a) EMBED Equation.DSMT4  b)  EMBED Equation.DSMT4  c)  EMBED Equation.DSMT4   EMBED Equation.DSMT4   EMBED Equation.DSMT4   EMBED Equation.DSMT4  d)  EMBED Equation.DSMT4  e)  EMBED Equation.DSMT4  f)  EMBED Equation.DSMT4   EMBED Equation.DSMT4   EMBED Equation.DSMT4   EMBED Equation.DSMT4  2. Which functions changed signs when the angle was changed from a positive to a negative? Will that happen every time? Explain. 3. Give the value of each expression to the nearest hundredth. a)  EMBED Equation.DSMT4  b)  EMBED Equation.DSMT4  c)  EMBED Equation.DSMT4   EMBED Equation.DSMT4   EMBED Equation.DSMT4   EMBED Equation.DSMT4  4. What is the relationship between the angles in each pair of functions? What can you conclude about the cofunctions?  5. For the given unit circle, we know that  EMBED Equation.DSMT4  and  EMBED Equation.DSMT4 . 1 y  EMBED Equation.DSMT4  Use the Pythagorean Theorem to state x the relationship between x, y and 1. Substitute  EMBED Equation.DSMT4  and  EMBED Equation.DSMT4  into the Pythagorean expression above. Goal 3: Graphing Assessment The following activity may be used as an alternative assessment for translating graphs of trigonometric functions. GRAPHING PROJECT: Translating Graphs of Trigonometric Functions For each situation listed below, write an original equation and use The Geometers Sketchpad to illustrate the translation. For 4-6, show the original sine or cosine curve and the translation on the same set of axes. Each graph should contain a text window with an explanation, in complete sentences, of the effects of the equation on the graph of sine or cosine. Include a text window on each graph with your name and date. Show a change in amplitude of a sine or cosine curve. Show a change in period of sine or cosine function. Be sure to state the period on this graph. Show a reflection of sine or cosine over the horizontal axis. Show a horizontal translation of sine or cosine. Make notations on the graph to illustrate the translation of a particular point on the curve. Show a vertical translation of sine or cosine. Make notations on the graph to illustrate the translation of a particular point on the curve. Show all of the above in a single graph. State this equation in terms of sine and cosine. NOTE: The equations to be used are your choice. Be sure to use sine and cosine equally throughout this project. Goal 4: Group Assessment-Surveying Poster Each group of students will be assigned a surveying problem. They should work together to solve the problem and then present a poster of the property that was surveyed. DIRECTIONS: Read the problem below. Working with the members of your group, make a rough diagram and find the area of the irregular quadrilateral. Document all work to show how you got the area. When all members of the group agree on the solution, draw a large scale diagram on the paper provided. Provide a NEAT copy of all work necessary to get the area. You will be graded on neatness and accuracy. All members of the group must present the solutions in their own notebook and be able to explain how they got their answers. Sample problem: Find the area of the plot of land described below. From a granite post, proceed 195 feet East along Tasker Hill Blvd, then along a bearing of S32E for 260 feet, then along a bearing of S68W for 385 feet, and finally along a line back to the granite post. Goal 6 Learning Activity-Polar vs. Rectangular Graphs The following activity may help students see the connections between rectangular and polar graphs. GRAPHING POLAR EQUATIONS Using Geometers Sketchpad or a graphing calculator, graph each of the equations below using rectangular coordinates and then polar coordinates. Sketch your graphs in the space provided. 1. Sketch f(x)=sin x  2. Sketch f( EMBED Equation.DSMT4 )=sin EMBED Equation.DSMT4   Describe the characteristics that the two graphs have in common Goal 11: Polar Graphing Project DIRECTIONS: For each equation below, complete the table of values for  EMBED Equation.DSMT4 , then graph. 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layt2D$IfK$L$q!vh5 5 5 5 5 5 5 5 5  5  5 5 5 5 55555555555#v #v :V l t065yt2`kd# $IfK$L$l*%Ee % E d =\{ t06dddd44 layt2D$IfK$L$q!vh5 5 5 5 5 5 5 5 5  5  5 5 5 5 55555555555#v #v :V l t065yt2`kd$IfK$L$l*%Ee % E d =\{ t06dddd44 layt2D$IfK$L$q!vh5 5 5 5 5 5 5 5 5  5  5 5 5 5 55555555555#v #v :V l t065yt2`kds$IfK$L$l*%Ee % E d =\{ t06dddd44 layt2D$IfK$L$q!vh5 5 5 5 5 5 5 5 5  5  5 5 5 5 55555555555#v #v :V l t065yt2`kd$IfK$L$l*%Ee % E d =\{ t06dddd44 layt2D$IfK$L$q!vh5 5 5 5 5 5 5 5 5  5  5 5 5 5 55555555555#v #v :V l t065yt2`kd#$IfK$L$l*%Ee % E d =\{ t06dddd44 layt2D$IfK$L$q!vh5 5 5 5 5 5 5 5 5  5  5 5 5 5 55555555555#v #v :V l t065yt2`kdk)$IfK$L$l*%Ee % E d =\{ t06dddd44 layt2D$IfK$L$q!vh5 5 5 5 5 5 5 5 5  5  5 5 5 5 55555555555#v #v  # !   :V l t065yt2`kd/$IfK$L$l*%Ee % E d =\{ t06dddd44 layt2D$IfK$L$q!vh5 5 5 5 5 5 5 5 5  5  5 5 5 5 55555555555#v #v :V l t065yt2`kd4$IfK$L$l*%Ee % E d =\{ t06dddd44 layt2D$IfK$L$q!vh5 5 5 5 5 5 5 5 5  5  5 5 5 5 55555555555#v #v :V l t065yt2`kdc:$IfK$L$l*%Ee % E d =\{ t06dddd44 layt2D$IfK$L$q!vh5 5 5 5 5 5 5 5 5  5  5 5 5 5 55555555555#v #v :V l t065yt2`kd @$IfK$L$l*%Ee % E d =\{ t06dddd44 layt2D$IfK$L$q!vh5 5 5 5 5 5 5 5 5  5  5 5 5 5 55555555555#v #v :V l t065yt2`kdE$IfK$L$l*%Ee % E d =\{ t06dddd44 layt2D$IfK$L$q!vh5 5 5 5 5 5 5 5 5  5  5 5 5 5 55555555555#v #v :V l t065yt2`kd[K$IfK$L$l*%Ee % E d =\{ t06dddd44 layt2D$IfK$L$q!vh5 5 5 5 5 5 5 5 5  5  5 5 5 5 55555555555#v #v :V l t065yt2`kdQ$IfK$L$l*%Ee % E d =\{ t06dddd44 layt2D$IfK$L$q!vh5 5 5 5 5 5 5 5 5  5  5 5 5 5 55555555555#v #v :V l t065yt2`kdV$IfK$L$l*%Ee % E d =\{ t06dddd44 layt2D$IfK$L$q!vh5 5 5 5 5 5 5 5 5  5  5 5 5 5 55555555555#v #v :V l t065yt2`kdS\$IfK$L$l*%Ee % E d =\{ t06dddd44 layt2D$IfK$L$q!vh5 5 5 5 5 5 5 5 5  5  5 5 5 5 55555555555#v #v :V l t065yt2`kda$IfK$L$l*%Ee % E d =\{ t06dddd44 layt2D$IfK$L$q!vh5 5 5 5 5 5 5 5 5  5  5 5 5 5 55555555555#v #v :V l t065yt2`kdg$IfK$L$l*%Ee % E d =\{ t06dddd44 layt2D$IfK$L$q!vh5 5 5 5 5 5 5 5 5  5  5 5 5 5 55555555555#v #v :V l t065yt2`kdKm$IfK$L$l*%Ee % E d =\{ t06dddd44 layt2D$IfK$L$q!vh5 5 5 5 5 5 5 5 5  5  5 5 5 5 55555555555#v #v :V l t065yt2`kdr$IfK$L$l*%Ee % E d =\{ t06dddd44 layt2D$IfK$L$q!vh5 5 5 5 5 5 5 5 5  5  5 5 5 5 55555555555#v #v :V l t065yt2`kdx$IfK$L$l*%Ee % E d =\{ t06dddd44 layt2D$IfK$L$q!vh5 5 5 5 5 5 5 5 5  5  5 5 5 5 55555555555#v #v :V l t065yt2`kdC~$IfK$L$l*%Ee % E d =\{ t06dddd44 layt2D$IfK$L$q!vh5 5 5 5 5 5 5 5 5  5  5 5 5 5 55555555555#v #v :V l t065yt2`kd$IfK$L$l*%Ee % E d =\{ t06dddd44 layt2D$IfK$L$q!vh5 5 5 5 5 5 5 5 5  5  5 5 5 5 55555555555#v #v :V l t065yt2`kd$IfK$L$l*%Ee % E d =\{ t06dddd44 layt2D$IfK$L$q!vh5 5 5 5 5 5 5 5 5  5  5 5 5 5 55555555555#v #v :V l t065yt2`kd;$IfK$L$l*%Ee % E d =\{ t06dddd44 layt2D$IfK$L$q!vh5 5 5 5 5 5 5 5 5  5  5 5 5 5 55555555555#v #v :V l t065yt2`kd$IfK$L$l*%Ee % E d =\{ t06dddd44 layt2D$IfK$L$q!vh5 5 5 5 5 5 5 5 5  5  5 5 5 5 55555555555#v #v :V l t065yt2`kd$IfK$L$l*%Ee % E d =\{ t06dddd44 layt2D$IfK$L$q!vh5 5 5 5 5 5 5 5 5  5  5 5 5 5 55555555555#v #v :V l t065yt2`kd3$IfK$L$l*%Ee % E d =\{ t06dddd44 layt2D$IfK$L$q!vh5 5 5 5 5 5 5 5 5  5  5 5 5 5 55555555555#v #v :V l t065yt2`kdۥ$IfK$L$l*%Ee % E d =\{ t06dddd44 layt2D$IfK$L$q!vh5 5 5 5 5 5 5 5 5  5  5 5 5 5 55555555555#v #v :V l t065yt2`kd$IfK$L$l*%Ee % E d =\{ t06dddd44 layt2$$If!vh5%5%#v%:V t065 4 yt2D$IfK$L$q!vh5 5 5 5 5 5 5 5 5  5  5 5 5 5 55555555555#v #v :V l t065yt2`kd$IfK$L$l*%Ee % E d =\{ t06dddd44 layt2D$IfK$L$q!vh5 5 5 5 5 5 5 5 5  5  5 5 5 5 55555555555#v #v :V l t065yt2`kd]$IfK$L$l*%Ee % E d =\{ t06dddd44 layt2D$IfK$L$q!vh5 5 5 5 5 5 5 5 5  5  5 5 5 5 55555555555#v #v :V l t065yt2`kd$IfK$L$l*%Ee % E d =\{ t06dddd44 layt2D$IfK$L$q!vh5 5 5 5 5 5 5 5 5  5  5 5 5 5 55555555555#v #v :V l t065yt2`kd$IfK$L$l*%Ee % E d =\{ t06dddd44 layt2D$IfK$L$q!vh5 5 5 5 5 5 5 5 5  5  5 5 5 5 55555555555#v #v :V l t065yt2`kdU$IfK$L$l*%Ee % E d =\{ t06dddd44 layt2D$IfK$L$q!vh5 5 5 5 5 5 5 5 5  5  5 5 5 5 55555555555#v #v :V l t065yt2`kd$IfK$L$l*%Ee % E d =\{ t06dddd44 layt2D$IfK$L$q!vh5 5 5 5 5 5 5 5 5  5  5 5 5 5 55555555555#v #v :V l t065yt2`kd$IfK$L$l*%Ee % E d =\{ t06dddd44 layt2D$IfK$L$q!vh5 5 5 5 5 5 5 5 5  5  5 5 5 5 55555555555#v #v :V l t065yt2`kdM$IfK$L$l*%Ee % E d =\{ t06dddd44 layt2D$IfK$L$q!vh5 5 5 5 5 5 5 5 5  5  5 5 5 5 55555555555#v #v :V l t065yt2`kd$IfK$L$l*%Ee % E d =\{ t06dddd44 layt2D$IfK$L$q!vh5 5 5 5 5 5 5 5 5  5  5 5 5 5 55555555555#v #v :V l t065yt2`kd$IfK$L$l*%Ee % E d =\{ t06dddd44 layt2D$IfK$L$q!vh5 5 5 5 5 5 5 5 5  5  5 5 5 5 55555555555#v #v :V l t065yt2`kdE$IfK$L$l*%Ee % E d =\{ t06dddd44 layt2D$IfK$L$q!vh5 5 5 5 5 5 5 5 5  5  5 5 5 5 55555555555#v #v :V l t065yt2`kd$IfK$L$l*%Ee % E d =\{ t06dddd44 layt2D$IfK$L$q!vh5 5 5 5 5 5 5 5 5  5  5 5 5 5 55555555555#v #v :V l t065yt2`kd$IfK$L$l*%Ee % E d =\{ t06dddd44 layt2D$IfK$L$q!vh5 5 5 5 5 5 5 5 5  5  5 5 5 5 55555555555#v #v :V l t065yt2`kd=$IfK$L$l*%Ee % E d =\{ t06dddd44 layt2D$IfK$L$q!vh5 5 5 5 5 5 5 5 5  5  5 5 5 5 55555555555#v #v :V l t065yt2`kd $IfK$L$l*%Ee % E d =\{ t06dddd44 layt2D$IfK$L$q!vh5 5 5 5 5 5 5 5 5  5  5 5 5 5 55555555555#v #v :V l t065yt2`kd $IfK$L$l*%Ee % E d =\{ t06dddd44 layt2D$IfK$L$q!vh5 5 5 5 5 5 5 5 5  5  5 5 5 5 55555555555#v #v :V l t065yt2`kd5 $IfK$L$l*%Ee % E d =\{ t06dddd44 layt2D$IfK$L$q!vh5 5 5 5 5 5 5 5 5  5  5 5 5 5 55555555555#v #v :V l t065yt2`kd $IfK$L$l*%Ee % E d =\{ t06dddd44 layt2D$IfK$L$q!vh5 5 5 5 5 5 5 5 5  5  5 5 5 5 55555555555#v #v :V l t065yt2`kd $IfK$L$l*%Ee % E d =\{ t06dddd44 layt2D$IfK$L$q!vh5 5 5 5 5 5 5 5 5  5  5 5 5 5 55555555555#v #v :V l t065yt2`kd- $IfK$L$l*%Ee % E d =\{ t06dddd44 layt2D$IfK$L$q!vh5 5 5 5 5 5 5 5 5  5  5 5 5 5 55555555555#v #v :V l t065yt2`kd" $IfK$L$l*%Ee % E d =\{ t06dddd44 layt2D$IfK$L$q!vh5 5 5 5 5 5 5 5 5  5  5 5 5 5 55555555555#v #v :V l t065yt2`kd}( $IfK$L$l*%Ee % E d =\{ t06dddd44 layt2D$IfK$L$q!vh5 5 5 5 5 5 5 5 5  5  5 5 5 5 55555555555#v #v :V l t065yt2`kd%. $IfK$L$l*%Ee % E d =\{ObjInfo Equation Native >_1267894030E FOle Times New 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