Monroe Township Schools
Monroe Township Schools
[pic]
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 O’Donnell, 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:
Precalculus
Scope and Sequence
|Quarter I |
|Big Idea: Functions |Big Idea: Trigonometric Functions |
|Transformations of functions |Introduction to Trigonometric Functions |
|a. Reflection in the x-axis, y-axis, and line y=x (inverse functions) |a. Degree and radian measures of angles |
|b. Symmetry in the x-axis, y-axis, and origin |b. Arc length and area of a sector |
|c. Periodic Functions |c. Evaluating trigonometric expressions |
|d. Translations of y=f(x) to y-k=f(x-h) |d. Graphs of trigonometric functions |
|e. Vertical (y=cf(x)) and horizontal (y=f(cx)) stretching or shrinking of y=f(x). | |
|Big 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 equations | |
|Quarter II |
|Big Idea: Triangle Trigonometry |Big Idea: Trigonometric Addition Formulas |
|Triangle Trigonometry |Trigonometric Addition Formulas |
|a. Measurements in right triangles |a. Sum and difference formulas for sine, cosine, and tangent |
|b. Area of a triangle |b. Double angle formulas |
|c. Law of Sines |c. Trigonometric equations |
|d. Law of Cosines | |
| | |
|Quarter III |
|Big Idea: Polar Coordinates and Complex Numbers |Big Idea: Sequences and Series |
|Polar Coordinates and Complex Numbers |Sequences and series |
|a. Graphing polar coordinates |a. nth term of an arithmetic sequence |
|b. Conversions of rectangular and polar coordinates |b. nth term of a geometric sequence |
|c. Graphs of polar functions |c. Recursive definitions |
|d. Conversions of complex numbers between rectangular and polar form |d. Sum of finite arithmetic and geometric series |
|e. Product of two complex numbers in polar form |e. Sum of infinite geometric series |
|f. De Moivre’s theorem |f. Sigma notation |
|g. Roots of complex numbers | |
| | |
|Quarter IV |
|Big Idea: Limits |Big Idea: Exponents and Logarithms |
|Limits |Exponents and Logarithms |
|a. Limits of functions that approach infinity or negative infinity |a. Simplify numeric and algebraic expressions with integral and rational exponents |
|b. Limits of functions that approach a real number |b. Compound interest formula and formula for interest compounded continually |
|c. Graphs of rational functions |c. Evaluate logarithmic expressions (change of base formula) |
| |d. Expand and condense logarithmic expressions |
| |e. Logarithmic and exponential equations |
| | |
|Suggested |Curriculum Management System |Big Idea: Functions |
|blocks of |Grade Level/Subject: | |
|Instructio|Grade 11/Precalculus | |
|n | | |
| | |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. |
| |Objectives / Cluster Concepts / |Essential Questions |Instructional Tools / Materials / Technology / Resources / Learning |
| |Cumulative Progress Indicators (CPI's) |Sample Conceptual Understandings |Activities / Interdisciplinary Activities / Assessment Model |
| | | | |
| |The student will be able to: | | |
| |To sketch the reflection of a graph in the |Essential Questions: |NOTE: The assessment models provided in this document are suggestions |
| |[pic]-axis and[pic]-axis. (4.2.12B.1; 4.2.12B.3) |What changes in an equation produces the reflection of its graph in the |for the teacher. If the teacher chooses to develop his/her own model, |
| |To write the inverse of an equation and sketch |[pic]-axis, the [pic]-axis, and the line [pic]? |it must be of equal or better quality and at the same or higher |
| |the graph of the reflection in the line [pic]. |How do you tell whether the graph of an equation has symmetry in the the |cognitive levels (as noted in parentheses). |
| |(4.2.12B.1; 4.3.12B.3) |[pic]-axis, the [pic]-axis, the line [pic], and the origin? |Depending upon the needs of the class, the assessment questions may be |
| |To determine if the graph of an equation has |Given the graph of [pic], what effect does [pic]have on the graph of [pic]and |answered in the form of essays, quizzes, mobiles, PowerPoint, oral |
| |symmetry in the [pic]-axis, the [pic]-axis, the |[pic]? |reports, booklets, or other formats of measurement used by the teacher.|
| |line [pic], and the origin. (4.2.12B.1; |Given the graph of [pic], what effect does [pic]and [pic]have on the graph of | |
| |4.3.12B.4) |[pic]? |Resources: |
| |To determine if a function is periodic. |How can the vertical-line test be used to justify the horizontal-line test? |Precalculus with Limits A Graphing Approach, Fifth Edition, Larson et |
| |(4.3.12B.1) | |al; Houghton Mifflin, 2008 |
| |To evaluate a function using the fundamental |Enduring Understandings: | |
| |period. (4.3.12B.2) |If the equation [pic]is changed to: | |
| |To determine the period and amplitude of a |a. [pic], then the graph of [pic]is reflected in the [pic]-axis. |Learning Activity: |
| |periodic function. (4.3.12B.2) |b. [pic], then the graph of [pic]is unchanged when [pic]and reflected in the |In the following activity, how does a change in the equation result in |
| |To understand the effect of [pic]and sketch the |[pic]-axis when [pic]. |the reflection of its graph in some line? (analysis) |
| |graph of [pic]by vertically stretching or |c. [pic], then the graph of [pic]is reflected in the [pic]-axis. |1. Graph [pic] and [pic]. Graph [pic] and [pic]. How are the graphs of |
| |shrinking the graph of [pic]. (4.2.12B.1; |d. [pic], then the graph of [pic]is reflected in the line [pic]. |[pic]and [pic]related? |
| |4.3.12B.3) |e. [pic], then the graph of [pic]is stretched vertically. |2. Graph [pic] and [pic]. Graph [pic] and [pic]. How are the graphs of|
| |To understand the effect of [pic]and sketch the |f. [pic], then the graph of [pic]is |[pic]and [pic]related? |
| |graph of [pic]by horizontally stretching or |shrunk vertically. |3. Graph [pic] and [pic]. Graph [pic] and [pic]. How are the graphs of|
| |shrinking the graph of [pic]. (4.2.12B.1; |g. [pic], then the graph of [pic]is |[pic] and [pic]related? |
| |4.3.12B.3) |shrunk horizontally. |4. Graph [pic] and [pic]. Graph [pic] and [pic]. How is the graph of |
| |To understand the effect of [pic]and [pic]and |h. [pic], then the graph of [pic]is |an equation affected when you interchange the variables in the |
| |sketch the graph of the equation [pic]by |stretched horizontally. |equation? |
| |translating the graph of [pic] horizontally |i. [pic], then the graph of [pic]is translated [pic]units horizontally and |Given the graph of [pic], sketch the graphs of [pic], [pic], and [pic] |
| |[pic]units and vertically [pic]units. |[pic]units vertically. |using different colored crayons. |
| |To determine if two functions are inverse |A graph is symmetric in the [pic]-axis if [pic]is on the graph whenever [pic]is. | |
| |functions by applying the definition. |An equation of a graph is symmetric in the [pic]-axis if an equivalent equation | |
| |To apply the Horizontal Line Test to determine if|results after substituting [pic]for [pic]. | |
| |a function has an inverse. |A graph is symmetric in the [pic]-axis if [pic]is on the graph whenever [pic]is. | |
| | |An equation of a graph is symmetric in the [pic]-axis if an equivalent equation | |
| | |results after substituting [pic]for [pic]. | |
| | |A graph is symmetric in the line [pic] if [pic]is on the graph whenever [pic]is. | |
| | |An equation of a graph is symmetric in the line [pic] if an equivalent equation | |
| | |results after interchanging [pic] and [pic]. | |
| | |A graph is symmetric in the origin if [pic]is on the graph whenever [pic]is. An | |
| | |equation of a graph is symmetric in the origin if an equivalent equation results | |
| | |after substituting [pic]for [pic]and [pic]for [pic]. | |
| | |Two functions [pic]and [pic]are inverse functions if: | |
| | |1. [pic]for all [pic]in the domain of [pic], and | |
| | |2. [pic]for all [pic]in the domain of [pic]. | |
| | |The Horizontal Line Test: If the graph of the function [pic]is such that no | |
| | |horizontal line intersects the graph in more than one point, then [pic]is | |
| | |one-to-one and has an inverse. | |
|Suggested |Curriculum Management System |Big Idea: Trigonometric Functions |
|blocks of |Grade Level/Subject: | |
|Instructio|Grade 11/Precalculus | |
|n | | |
| | |The student will be able to evaluate and graph trigonometric functions. |
| |Objectives / Cluster Concepts / |Essential Questions |Instructional Tools / Materials / Technology / Resources / Learning |
| |Cumulative Progress Indicators (CPI's) |Sample Conceptual Understandings |Activities / Interdisciplinary Activities / Assessment Model |
| | | | |
| |The student will be able to: | | |
| |To convert degree measures of angles to radians. |Essential Questions: |Sample Assessment Questions: |
| |To convert radian measures of angles to degrees. |What are radians and how are they related to degrees? |Convert each angle to radians in terms of [pic]. |
| |To determine coterminal angles. |Explain the process of evaluating a trigonometric function using reference angles|[pic] |
| |To determine the arc length and area of a sector |and the unit circle. |[pic] |
| |of a circle with central angles in either degrees|How do the values on the unit circle correlate to the rectangular graph of a |[pic] |
| |or radians. |trigonometric function? |[pic] |
| |To use the definitions of sine and cosine to |Why is it necessary to restrict the domain in order to discuss inverse |Convert each angle to degrees. |
| |evaluate these functions. (4.3.12D.1) |trigonometric functions? |[pic] |
| |To use reference angles, calculators or tables, | |[pic] |
| |and special angles to evaluate sine and cosine |Enduring Understandings: |[pic] |
| |functions. (4.3.12D.1) |[pic], where [pic] is the measure of the central angle, in radians, [pic]is the |[pic] |
| |To sketch the graph of sine and cosine functions.|arc length, and [pic]is the length of the radius. |Give one positive and one negative coterminal angle for each angle |
| |(4.3.12B.2) |To convert each degree measure to radians, multiply by [pic]. |below. Use the given form of the angle. |
| |To use reference angles, calculators or tables, |To convert each radian measure to degrees, multiply by [pic]. |[pic] |
| |and special angles to evaluate tangent, |The following formulas are used for the arc length [pic]and area [pic]of a sector|[pic]. |
| |cotangent, secant, and cosecant functions. |with central angle [pic]: |A sector of a circle has central angle 1.2 radians and radius 6cm. |
| |(4.3.12D.1) |a. If [pic]is in degrees, then [pic]and [pic]. |Find its arc length. |
| |To sketch the graphs of tangent, cotangent, |b. If [pic]is in radians, then [pic]and [pic]. |Find its area. |
| |secant, and cosecant functions. (4.3.12B.2) |[pic] |Find the value of each expression leave answers in simplest radical |
| |To sketch the graph of the inverse of the sine, |[pic] |form. Show reference angle statement when necessary. |
| |cosine, and tangent functions, and determine the |[pic], [pic] | |
| |domain and range. (4.3.12B.2) |[pic], [pic] |sin[pic] |
| |To evaluate the inverse of sine, cosine, and |[pic], [pic] |cos [pic] |
| |tangent functions with and without a calculator |[pic], [pic] |tan [pic] |
| |or table. (4.3.12D.1) |The signs of the trigonometric functions (sine and cosecant, cosine and secant, |sec[pic] |
| | |and tangent and cotangent) in the four quadrants can be summarized by the |cos ([pic]) |
| | |following phrase: All Students Take Calculus. | |
| | | |sin 3[pic] |
| | | |csc [pic] |
| | | |cot [pic] |
| | | |sec [pic] |
| | | |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 |
| | |In order to evaluate a trigonometric expression, |angles [pic]. |
| | |1. Determine the quadrant of the terminal ray of the angle. |Graph each of the 6 trigonometric |
| | |2. Determine if it is positive, negative, or zero. |functions on a separate graph. Include an accurate scale and asymptotes|
| | |3. Determine the reference angle. |where appropriate. |
| | |4. Determine the exact value, if possible, in simplest radical form. | |
| | | | |
|Suggested |Curriculum Management System |Big Idea: Trigonometric Equations and Applications |
|blocks of |Grade Level/Subject: | |
|Instructio|Grade 11/Precalculus | |
|n | | |
| | |The student will be able to stretch, shrink, and translate sine and cosine functions, simplify trigonometric expressions, and solve trigonometric |
| | |equations. |
| |Objectives / Cluster Concepts / |Essential Questions |Instructional Tools / Materials / Technology / Resources / Learning |
| |Cumulative Progress Indicators (CPI's) |Sample Conceptual Understandings |Activities / Interdisciplinary Activities / Assessment Model |
| | | | |
| |The student will be able to: | | |
| |To solve and apply simple trigonometric |Essential Questions: |Learning Activities: |
| |equations. (4.3.12D.2) |How does a change in amplitude or period affect the graph of a Sine or Cosine |Translating Graphs of Trigonometric Functions (See Addendum) |
| |To determine the slope and equation of a line |curve? |Students will work in pairs to complete the graphing calculator |
| |given the angle of inclination and the |Explain the effect of A, B, h, and k on the graph of a sine or Cosine curve |activity on translating graphs of trigonometric functions. |
| |coordinates of a point on the line, and determine|using the equations [pic] | |
| |the angle of inclination given the equation of a |Explain how to find all possible solutions to simple trigonometric equations over|Deriving Basic Trigonometric Identities |
| |line or information about the line. (4.2.12C.1) |a given domain. |(See Addendum) |
| |To stretch and shrink the graphs of sine and |How can the graph of a trigonometric function be used to anticipate the number of|Students will learn about negative angle relationships, Pythagorean |
| |cosine functions. (4.2.12B.1) |solutions to a trigonometric equation? |relationships, and reciprocal relationships among trigonometric |
| |To determine the period and amplitude of [pic] |Could a single curve be described using both a Sine function and a Cosine |functions. |
| |and [pic]. (4.3.12B.2) |function? Why? | |
| |To determine the amplitude and period, and write | |Sample Assessment Questions: |
| |the equation of sine and cosine curves. |Enduring Understandings: |Solve [pic] for [pic]. |
| |(4.3.12B.2) |For any line with slope m and inclination [pic], [pic] if [pic]. If [pic], then |Solve [pic] for [pic]. |
| |To solve equations of the form [pic] and [pic]. |the line has no slope. (The line is vertical.) |Solve [pic] for [pic]without using tables or a calculator. |
| |(4.3.12D.2) |For functions [pic] and [pic][pic] and [pic]: amplitude = [pic] and period = |To the nearest degree, find the inclination of the line [pic]. |
| | |[pic]. |Find the slope and equation of a line with an inclination of [pic]and |
| | | |contains (2,3). |
| | | | |
| | | | |
| |To determine the amplitude, period, axis of wave,| | |
| |and sketch the graph of translated sine and |If the graphs of [pic]and [pic]are translated horizontally h units and vertically|Give the amplitude and period of the function [pic]. Then sketch at |
| |cosine functions. (4.3.12B.3; 4.3.12B.4) |k units, then the resulting graphs have equations [pic] and [pic]. The amplitude |least one cycle of its graph. |
| |To determine the amplitude, period, axis of wave,|is [pic]. To find the period, p = horizontal distance between successive |Solve [pic] for [pic]. |
| |and write the equation of the graph of translated|maximums. Use the formula [pic]. |For each function: |
| |sine and cosine functions. (4.2.12B.1; 4.3.12B.2)|Relationships with negatives: |State the amplitude of the curve. |
| |To simplify trigonometric expressions using the |[pic] and [pic] |State the period of the curve. |
| |reciprocal relationships, relationships with |[pic]and [pic] |Describe any vertical or horizontal translations of the curve. |
| |negatives, Pythagorean relationships, and |[pic] and [pic] |Sketch the graph by hand. |
| |cofunction relationships of trigonometric |Pythagorean Relationships: |Confirm your sketch using a graphing calculator. |
| |functions. (4.3.12D.1) |[pic] |[pic] |
| |To prove trigonometric identities using the |[pic] |[pic] |
| |reciprocal relationships, relationships with |[pic] |[pic] |
| |negatives, Pythagorean relationships, and |Cofunction Relationships |[pic] |
| |cofunction relationships of trigonometric |[pic] and [pic] |Simplify [pic]. |
| |functions. (4.2.12A.4) |[pic] and [pic] |Prove: [pic]. |
| |To use trigonometric identities or graphing |[pic] and [pic] |Solve [pic] for [pic]. |
| |calculator to solve more difficult trigonometric | |Solve [pic]for [pic]. |
| |equations. (4.3.12D.2) | |Solve [pic] for [pic]. |
| | | |Solve [pic] for [pic]. |
|Suggested |Curriculum Management System |Big Idea: Triangle Trigonometry |
|blocks of |Grade Level/Subject: | |
|Instructio|Grade 11/Precalculus | |
|n | | |
| | |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. |
| |Objectives / Cluster Concepts / |Essential Questions |Instructional Tools / Materials / Technology / Resources / Learning |
| |Cumulative Progress Indicators (CPI's) |Sample Conceptual Understandings |Activities / Interdisciplinary Activities / Assessment Model |
| | | | |
| |The student will be able to: | | |
| |To use trigonometry to find the lengths of |Essential Questions: |Learning Activities: |
| |unknown sides or measures of unknown angles of a |What does the acronym SOH-CAH-TOA stand for? |Perform the following activity to illustrate that when given the |
| |right triangle. (4.2.12A.1; 4.2.12E.1) |Given the lengths of two sides of a right triangle, or the length of one side and|lengths of two sides of a triangle and the measure of a non-included |
| |To determine the area of a triangle given the |the measure of one acute angle, how can you find the measures of the remaining |angle (SSA), it may be possible to construct no triangle, one triangle,|
| |lengths of two sides of a triangle and the |sides and angles using the trigonometric functions? |or two triangles. Draw [pic]with measure [pic]. Along one ray of [pic],|
| |measure of the included angle. (4.2.12E.2) |How can the area of a triangle be determined given the lengths of two sides and |locate point [pic]10 cm from point [pic]. For each of the following |
| |To use the Law of Sines to find unknown parts of |the measure of the included angle? |compass settings, draw a large arc. Then determine if the arc |
| |a triangle. (4.2.12E.1) |When given the lengths of two sides and the measure of a non-included angle of a |intersects the other ray of [pic] and, if so, in how many points. |
| |To use the Law of Cosines to find unknown parts |triangle, how many measurements are possible for the unknown angles and why? |a. Compass at [pic] and opened to 4 cm. (0) |
| |of a triangle. (4.2.12E.1) |For which of the following situation is the Law of Cosines used? SAS, SSS, ASA, |b. Compass at [pic] and opened to 5 cm. (1) |
| |To use trigonometry to solve navigation and |AAS, or SSA |c. Compass at [pic] and opened to 6 cm. (2) |
| |surveying problems. (4.2.12E.1) |For which of the following situations is the Law of Sines used? SAS, SSS, ASA, | |
| | |AAS, or SSA |Sample Assessment Questions: |
| | |How is measuring an angle in standard form different from measuring an angle from|For right triangle [pic] with right angle [pic], [pic]and [pic]. Find |
| | |a compass bearing? |[pic]. |
| | | |The safety instructions for a 20 ft. ladder indicate that the ladder |
| | |Enduring Understandings: |should not be inclined at more than a [pic]angle with the ground. |
| | |In [pic]with right angle [pic] |Suppose the ladder is leaned against a house with a [pic]angle with the|
| | |[pic] |ground. Find (a) the distance x from the base of the house to the foot |
| | |[pic] |of the ladder and (b) the height y reached by the ladder. |
| | |[pic] |The highest tower in the world is in Toronto, Canada, and is 553 m |
| | |[pic] |high. An observer at point [pic], 100 m from the center of the tower’s |
| | |[pic] |base, sights the top of the tower. The angle of elevation is [pic]. |
| | |[pic] |Find the measure of this angle. |
| | |The area [pic]of [pic]is given by: [pic]. In other words, the area of any |A triangle has sides of length 8, 8, and 4. Find the measures of the |
| | |triangle is [pic](one side) [pic](another side)[pic](sine of the included angle).|angles of the triangle. |
| | |Derive the formula for the area of a triangle by rewriting the height of the |Two sides of a triangle have lengths 7cm and 4cm. The angle between the|
| | |triangle in terms of an angle and a side (that doesn’t intersect the height of |sides measures [pic]. Find the area of the triangle. If the angle |
| | |the triangle). |between the sides is changed to [pic], what is the area of the new |
| | |Derive the Law of Sines using the three different ways of writing the area of a |triangle? Why are the two areas the same? |
| | |triangle (using each of the three angles). |The area of [pic] is 15. If [pic] and [pic], find all possible measures|
| | |The Law of Sines: In [pic], [pic]. |of [pic]. Why are there two answers? |
| | |The Law of Cosines: in [pic], [pic]. In other words, the square of the side |In [pic], [pic], [pic], and [pic]. Find [pic]and [pic]. |
| | |opposite an angle is equal to the square of one side of the angle plus the square|In [pic], [pic], [pic], and [pic]. Determine whether [pic] exists, and,|
| | |of the other side of the angle minus twice the product of the two sides of the |if so, find all possible measures of [pic]. |
| | |angle times the cosine of the angle. |Two sides of a triangle have lengths 3 cm and 7 cm, and the included |
| | |The Law of Cosines (alternate form): [pic]. |angle has a measure of [pic]. Find the length of the third side. |
| | |Given SAS, use the Law of Cosines to find the measure of the third side and then |The lengths of the sides of a triangle are 5, 6, and 7. Solve the |
| | |one of the remaining angles. |triangle. |
| | |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.) | |
|Suggested |Curriculum Management System |Big Idea: Trigonometric Addition Formulas |
|blocks of |Grade Level/Subject: | |
|Instructio|Grade 11/Precalculus | |
|n | | |
| | |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. |
| |Objectives / Cluster Concepts / |Essential Questions |Instructional Tools / Materials / Technology / Resources / Learning |
| |Cumulative Progress Indicators (CPI's) |Sample Conceptual Understandings |Activities / Interdisciplinary Activities / Assessment Model |
| | | | |
| |The student will be able to: | | |
| |To derive and apply the formulas for [pic] and |Essential Questions: |Sample Assessment Questions: |
| |for [pic]. (4.3.12D.1) |For what angle measurements can the sum and difference formulas be used? |Find the exact value of [pic]. |
| |To derive and apply the formulas for [pic]. |How can the double-angle formulas be derived from the sum and difference formulas|Find the exact value of : |
| |(4.3.12D.1) |for sine, cosine, and tangent? |a. [pic] |
| |To derive and apply the double-angle formulas. | |b. [pic]. |
| |(4.3.12D.1) |Enduring Understandings: |If [pic]and [pic], where [pic]and [pic], find [pic]. |
| |To use the double-angle formulas to solve |Use the Law of Cosines or the distance formula to derive the formulas for the |If [pic], find [pic]. |
| |trigonometric equations. (4.3.12D.2) |difference (and then the sum) of two angles for cosines. Then use the cofunction |Solve [pic]for [pic]using a graphing calculator and algebraically. |
| | |relationship to derive the formula for the sum (and then the difference) of two |Solve [pic]for [pic]. |
| | |angles for sine. (See pages 369-370.) |Solve [pic] for [pic]. |
| | |The sum and difference formulas for sine, cosine, and tangent are as follows: | |
| | |[pic] | |
| | |[pic] | |
| | |[pic] | |
| | |[pic] | |
| | |[pic] | |
| | |[pic] | |
| | |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: | |
| | |[pic] | |
| | |[pic] | |
| | |[pic] | |
| | |A quick review of factoring polynomials may be helpful before solving | |
| | |trigonometric equations using the double-angle formulas. | |
|Suggested |Curriculum Management System |Big Idea: Polar Coordinates and Complex Numbers |
|blocks of |Grade Level/Subject: | |
|Instructio|Grade 11/Precalculus | |
|n | | |
| | |The student will be able to represent points in rectangular and polar coordinates, and multiply and find powers of complex numbers. |
| |Objectives / Cluster Concepts / |Essential Questions |Instructional Tools / Materials / Technology / Resources / Learning |
| |Cumulative Progress Indicators (CPI's) |Sample Conceptual Understandings |Activities / Interdisciplinary Activities / Assessment Model |
| | | | |
| |The student will be able to: | | |
| |To graph points given polar coordinates. |Essential Questions: |Graphing Project: |
| |(4.2.12C.1) |How many ways can a point be represented using polar coordinates? |(See Addendum) |
| |To state two additional polar coordinates for the|How are rectangular coordinates converted to polar coordinates? |Polar vs. Rectangular Graphs |
| |same point. (4.2.12C.1) |How are polar coordinates converted to rectangular coordinates? |In this activity may help students see the connections between |
| |To convert from rectangular (Cartesian) to polar |What is the advantage of using De Moivre’s theorem? |rectangular and polar graphs. |
| |coordinates. | | |
| |To convert from polar to rectangular coordinates.|Enduring Understandings: | |
| |To graph polar equations. (4.3.12B.1) |The polar coordinates of a point [pic]are [pic], where [pic]is the directed | |
| |Explore special polar graphs including Cardiod, |distance from the pole to [pic]and [pic] is the polar angle measured from the | |
| |Limacon, and Rose curves. (4.3.12B.1; 4.3.12B.4) |polar axis to the ray [pic]. Although a point has only one pair of rectangles | |
| |To convert complex numbers in rectangular form to|coordinates, it has many pairs of polar coordinates. For example, [pic]all | |
| |polar form. |represent the same point. | |
| |To express complex numbers in polar form to |Formulas for converting from polar to rectangular coordinates: [pic]. | |
| |rectangular form. |Formulas for converting from rectangular to polar coordinates: [pic]. |Sample Assessment Questions: |
| |To determine the product of two complex numbers |The complex plane can be represented by an Argand diagram. In this diagram, the |Express [pic]in rectangular form. |
| |in polar form. (4.3.12D.3) |complex number [pic]is represented by the point [pic]or by an arrow from the |Express [pic]in polar form. |
| |To use De Moivre’s theorem to determine powers of|origin to [pic]. |If [pic]: |
| |complex numbers. (4.3.12D.3) |The absolute value of a complex number [pic]is [pic]. Thus, [pic]. |a. find [pic]in rectangular form by multiplying [pic]and [pic]. |
| |To determine roots of complex numbers. |To multiply two complex numbers in polar form: |b. find [pic], [pic], and [pic]in polar form. |
| |(4.3.12D.2) |1. Multiply their absolute values. |c. show that [pic]in polar form agree with [pic]in rectangular form. |
| | |2. Add their polar angles. |If [pic], find [pic]and [pic]. Plot these on an Argand diagram. |
| | |In other words, if [pic]and [pic], then [pic]. |Find the cube roots of [pic]. |
| | |De Moivre’s Theorem: If [pic], then [pic]. |Find the four fourth roots of -16. |
| | |The [pic]nth roots of [pic] are:[pic] for [pic]. | |
|Suggested |Curriculum Management System |Big Idea: Sequences and Series |
|blocks of |Grade Level/Subject: | |
|Instructio|Grade 11/Precalculus | |
|n | | |
| | |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. |
| |Objectives / Cluster Concepts / |Essential Questions |Instructional Tools / Materials / Technology / Resources / Learning |
| |Cumulative Progress Indicators (CPI's) |Sample Conceptual Understandings |Activities / Interdisciplinary Activities / Assessment Model |
| | | | |
| |The student will be able to: | | |
| |To identify an arithmetic and geometric sequence.|Essential Questions: |Sample Assessment Questions: |
| | |What makes a sequence arithmetic? What is the graph of an arithmetic sequence? |State the next term in each sequence. Then write a rule for the nth |
| |To write a formula for the nth term of an |What makes a sequence geometric? What is the graph of a geometric sequence? |term. Identify is the sequence is arithmetic, geometric, or neither. |
| |arithmetic sequence. (4.3.12A.1) |What is the difference between an explicit definition and a recursive definition |1. [pic] |
| |To write a formula for the nth term of a |of a sequence? |2. [pic] |
| |geometric sequence. (4.3.12A.1) |What is the difference between a sequence and a series? |3. [pic] |
| |To write the formula for the nth term of a |When does the sum of a series converge or diverge? |4. [pic] |
| |sequence that is neither arithmetic nor | |5. [pic] |
| |geometric. (4.3 12A.1; 4.3.12A.3) |Enduring Understandings: |6. [pic] (discuss[pic]) |
| |To use and write recursive definitions of |A sequence is arithmetic if the difference d of any two consecutive terms is |Write a recursive rule for the sequences #1, 2, and 5 above. |
| |sequences. (4.3.12A.1) |constant. |Write a rule for the nth term of the arithmetic sequence. Then find |
| |To determine the sum of the first n terms of |The formula for the nth term in an arithmetic sequence is: [pic]. |[pic]. |
| |arithmetic and geometric series. (4.3 12A.1) |A sequence is geometric if the ratio r of any two consecutive terms is constant. |1. [pic] |
| |To find or estimate the limit of an infinite |The formula for the nth term in a geometric sequence is: [pic]. |2. [pic] |
| |sequence or to determine that the limit does not |A recursive definition consists of two parts: |3. [pic] |
| |exist. (4.2.12B.4; 4.3.12A.1; 4.3.12A.2) |1. An initial condition that states the first term of the sequence. |Write a rule for the nth term of the geometric sequence. Then find |
| |To determine the sum of an infinite geometric |2. A recursive equation (or recursive formula) that states how any term in the |[pic]. |
| |series. (4.2.12B.4; 4.3.12A.1) |sequence is related to the preceding term. |[pic][pic] |
| |To expand series written in sigma notation. |A series is an indicated sum of terms of a sequence. |One term of a geometric sequence is [pic]. The common ratio is [pic]. |
| |(4.3.12A.1) |The sum of the first n terms in an arithmetic series is [pic]. |Write a rule for the nth term. |
| |To condense series into sigma notation. |The sum of the first n terms of a geometric series is [pic], where r is the |Two terms of a geometric sequence are [pic] and [pic]. Find a rule for |
| |(4.3.12A.1) |common ratio and [pic]. |the nth term. |
| | |The sum of an infinite geometric series when [pic]is: [pic]. If [pic] and [pic], |State the first five terms of the sequence. |
| | |then the series diverges. |1. [pic] |
| | | |[pic] |
| | | |2. [pic] |
| | | |[pic] |
| | | |3. [pic] |
| | | |[pic] |
| | | |4. [pic] |
| | | |[pic] |
| | | |For the arithmetic series [pic] |
| | | |1. Determine the sum of the first 30 terms. |
| | | |2. Find n such that [pic]. |
| | | |For the geometric series |
| | | |[pic] |
| | | |1. Determine the sum of the first 10 terms. |
| | | |2. Find n such that [pic]. |
| | | |Find the sum of [pic] |
| | | |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. [pic] |
| | | |2. [pic] |
| | | |3. [pic] |
| | | |4. [pic] |
| | | |Write in sigma notation. |
| | | |1. [pic] |
| | | |2. [pic] |
|Suggested |Curriculum Management System |Big Idea: Limits |
|blocks of |Grade Level/Subject: | |
|Instructio|Grade 11/Precalculus | |
|n | | |
| | |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 / |Essential Questions |Instructional Tools / Materials / Technology / Resources / Learning |
| |Cumulative Progress Indicators (CPI's) |Sample Conceptual Understandings |Activities / Interdisciplinary Activities / Assessment Model |
| |The student will be able to: | | |
| |To determine the limit of a function or the |Essential Questions: |NOTE: The assessment models provided in this document are suggestions |
| |quotient of two functions as x approaches [pic] |How do we find the limit of a function as [pic]approaches [pic]or [pic]? |for the teacher. If the teacher chooses to develop his/her own model, |
| |or [pic]. (4.3.12A.2) |How do we find the limit of a function as [pic]approaches a real number [pic]? |it must be of equal or better quality and at the same or higher |
| |To determine the limit of a function or the |How do we find the limit of a quotient of two functions? |cognitive levels (as noted in parentheses). |
| |quotient of two functions as x approaches a real|What is an inverse variation and how does it compare to other rational functions?|Depending upon the needs of the class, the assessment questions may be |
| |number. (4.3.12A.2) |What is significant about the graph of a rational function? |answered in the form of essays, quizzes, mobiles, PowerPoint, oral |
| |To determine if a function is continuous. |How can the vertical and horizontal asymptotes of a rational function be |reports, booklets, or other formats of measurement used by the teacher.|
| |(4.3.12B.1; 4.3.12B.2) |identified analytically? | |
| |To sketch the graph of a rational function by |Explain the procedure to sketch a possible graph of a rational function. |Sample Assessment Questions: |
| |determining the x-intercepts, vertical and | | |
| |horizontal asymptotes, performing a sign |Enduring Understandings: |Evaluate: |
| |analysis, and use limits. |If [pic], then [pic].. |a) [pic] |
| |(4.3.12B.1; 4.3.12B.2) |The Quotient Theorem for Limits: If [pic]and [pic]both exist, and [pic], then |b) [pic] |
| | |[pic]. |c) [pic] |
| | |Techniques for evaluating [pic]: |d) [pic] |
| | |1. If possible, use the quotient theorem for limits. |e) [pic] |
| | |2. If [pic]and [pic], try the following techniques: |f) [pic] |
| | |a) Factor [pic]and [pic], and reduce [pic]to lowest terms. |g) [pic] |
| | |b) If [pic]or [pic]involves a square root, try multiplying both [pic]and [pic]by|h) [pic] |
| | |the conjugate of the square root expression. |i) [pic] |
| | |3. If [pic]and [pic], then either statement (a) or (b) is true: |j) [pic] |
| | |a) [pic]does not exist |k) [pic] |
| | |b) [pic]or [pic] |l) [pic] |
| | |4. If [pic]is approaching infinity or negative infinity, divide the numerator |m) [pic] |
| | |and denominator by the highest power of [pic]in the denominator. |n) [pic] |
| | |5. Evaluate [pic]by evaluating [pic]for very large values of [pic], and evaluate|o) [pic] |
| | |[pic]by evaluating [pic] for [pic]-values very near [pic]. These limits can also |p) [pic] |
| | |be guessed by using a graphing calculator to examine the graph of [pic]for very | |
| | |large values of [pic], or for [pic]-values very near [pic]. (A graphing |For each rational function: |
| | |calculator might not show points of discontinuity.) |Find all horizontal and vertical asymptotes of the graph. (synthesis) |
| | |Given a rational function of the form [pic], find the horizontal asymptotes by |Identify any holes in the graph. (application) |
| | |the following rules: |Sketch a possible graph of the function using asymptotes, intercepts |
| | |a. If the degree of [pic]and [pic]are the same, then the horizontal asymptote is|and sign analysis. (synthesis) |
| | |the ratio of the leading coefficients of[pic]and [pic]. |Confirm your graph using a graphing calculator. (application) |
| | |b. If the degree of [pic]is greater then the degree of [pic], then the |State the domain and range of the function. (analysis) |
| | |horizontal asymptote is the [pic]axis. |[pic] |
| | |c. If the degree of [pic]is less than the degree of [pic], there is no |[pic] |
| | |horizontal asymptote. |[pic] |
| | |Given a rational function of the form [pic], a discontinuity (vertical asymptote |[pic] |
| | |or hole) will occur whenever [pic]. |[pic] |
| | |Given a rational function of the form [pic], the [pic]-intercepts will occur |The function [pic] relates atmospheric pressure, [pic], in inches of |
| | |when [pic](unless it is also a zero of [pic]). |mercury, to altitude, [pic], 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 |Curriculum Management System |Big Idea: Exponents and Logarithms |
|blocks of |Grade Level/Subject: | |
|Instructio|Grade 11/Precalculus | |
|n | | |
| | |The student will be able to simplify expressions and solve equations with exponents and logarithms. |
| |Objectives / Cluster Concepts / |Essential Questions |Instructional Tools / Materials / Technology / Resources / Learning |
| |Cumulative Progress Indicators (CPI's) |Sample Conceptual Understandings |Activities / Interdisciplinary Activities / Assessment Model |
| | | | |
| |The student will be able to: | | |
| |To apply the laws of exponents to simplify |Essential Questions: |Sample Assessment Questions: |
| |numeric and algebraic expressions with integral |For what values of b does [pic]represent exponential growth? Exponential decay? |Simplify [pic] |
| |exponents. (4.1.12B.4; 4.3.12D.1) | |Simplify [pic], where [pic]and [pic]. |
| |To apply the laws of exponents to simplify |Enduring Understandings: |Simplify [pic] and [pic], where [pic]. |
| |numeric and algebraic expressions with rational |Properties of Exponents: |A radioactive isotope decays so that the radioactivity present |
| |exponents. (4.1.12B.4; 4.3.12D.1) |1. [pic] |decreases by 15% per day. If 40 kg are present now, find the amount |
| |To solve real-world problems involving |2. [pic] |present (a) 6 days from now, and (b) 6 days ago. |
| |exponential growth and decay. (4.3.12C.1) |When multiplying expressions with the same base, add the exponents. [pic] |Find the balance after 10 years if $5000 is invested in a bank account |
| |To estimate the doubling time using the Rule of |When dividing expressions with the same base, subtract the exponents. [pic] |that pays 4.5% interest compounded |
| |72. (4.3.12C.1) |When raising a power to a power, multiply the exponents. [pic] |1. annually. |
| |To apply the formula for compounded interest. |[pic] |2. quarterly. |
| |(4.3.12C.1) |[pic] |3. monthly. |
| |To define and apply the natural exponential |If [pic], then [pic]if and only if [pic]. |4. daily. |
| |function. |An exponential function has the form [pic], where [pic]and[pic]. If [pic], it is |Find the balance after 10 years if $5000 is invested in a bank account |
| |To derive and apply the formula for interest |an exponential growth function, and if [pic], then it is an exponential growth |that pays 4.5% annual interest compounded continuously. |
| |compounded continuously. (4.3.12C.1) |function. |In about how many years will it take for $1000 to double in value with |
| |To define and evaluate logarithms. (4.3.12D.1) |Compound Interest Formula: |a 6% annual interest rate compounded continuously? (Discuss the rule of|
| |To solve logarithmic equations without and with a|The amount [pic]in an account earning interest compounded [pic]times per year for|72) |
| |calculator. (4.3.12D.2) |[pic] years is: [pic] , where [pic] is the principal and [pic]is the annual |Evaluate without a calculator: |
| |To apply the laws of logarithms to expand and |interest rate expressed as a decimal. |1. [pic] |
| |condense logarithmic expressions. (4.3.12D.1) |The rule of 72 provides an approximation of the doubling time for exponential |2. [pic] |
| |To solve exponential equations by rewriting both |growth. If a quantity is growing at r% per year, then the doubling time [pic]. |3. [pic] |
| |sides of the equation with the same base. |The number [pic]is defined as: [pic]. |4. [pic] |
| |(4.3.12D.2) |Continuously Compounded Interest Formula: [pic]. |5. [pic] |
| |To solve exponential equations by using the |The logarithm of x to the base b [pic] is the exponent a such that [pic]. Thus, |6. [pic] |
| |definition of logarithms to rewrite as a |[pic]if and only if [pic]. |Expand the expression. |
| |logarithmic equation. (4.3.12D.2) |Laws of Logarithms: |1. [pic] |
| |To evaluate logarithmic expressions using the |1. [pic] |2. [pic] |
| |Change of Base formula. (4.3.12D.1) |2. [pic] |Condense the expression. |
| | |3. [pic] |1. [pic] |
| | |4. [pic]if and only if [pic] |2. [pic] |
| | |The change of base formula enables you to write logarithms in any given base in |Use the change of base formula to evaluate |
| | |terms in any other base. [pic] |1. [pic] |
| | | |2. [pic] |
| | | |Solve. Check for extraneous solutions. |
| | | |1. [pic] |
| | | |2. [pic] |
| | | |3. [pic] |
| | | |4. [pic] |
| | | |5. [pic] |
| | | |6. [pic] |
Precalculus
COURSE BENCHMARKS
1. 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.
2. The student will be able to evaluate and graph trigonometric functions.
3. The student will be able to stretch, shrink, and translate sine and cosine functions, simplify trigonometric expressions, and solve trigonometric equations.
4. 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.
5. 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.
6. The student will be able to represent points in rectangular and polar coordinates, and multiply and find powers of complex numbers.
7. 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.
8. 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.
9. 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
|[pic]in degrees |[pic]in radians |
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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 [pic] 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 [pic].
B represents the number of cycles in [pic]units. B is related to the period P as follows:
[pic] or [pic].
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 [pic] which means that
4 marks = [pic].
8. Sketch each graph and state the amplitude and period.
[pic] [pic] [pic]
A=_______ P=_______ A=_______ P=_______ A=________ P=_______
9. Match each equation with its graph.
Label each graph [pic]or [pic].
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.
[pic] [pic] [pic]
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 = [pic]
x-scale = [pic]
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
[pic].
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)[pic] b) [pic] c) [pic]
[pic] [pic] [pic]
d) [pic] e) [pic] f) [pic]
[pic] [pic] [pic]
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) [pic] b) [pic] c) [pic]
[pic] [pic] [pic]
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
[pic] and [pic].
1 y
[pic] Use the Pythagorean Theorem to state
x the relationship between x, y and 1.
Substitute [pic] and [pic] 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 Geometer’s 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.
1. Show a change in amplitude of a sine or cosine curve.
2. Show a change in period of sine or cosine function. Be sure to state the period on this graph.
3. Show a reflection of sine or cosine over the horizontal axis.
4. Show a horizontal translation of sine or cosine. Make notations on the graph to illustrate the translation of a particular point on the curve.
5. Show a vertical translation of sine or cosine. Make notations on the graph to illustrate the translation of a particular point on the curve.
6. 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 S32°E for 260 feet, then along a bearing of S68°W 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 Geometer’s 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([pic])=sin[pic]
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 [pic], then graph.
You will be graded on the following:
• a complete and correct table of values
• reasonable scales, clearly labeled
• neat and accurate curves (hint: there are no line segments on these graphs!)
• all points must be plotted!
• BE CREATIVE!
Sample problems:
1. [pic]
2. [pic]
Sample graph and table:
EQUATION:_______________________________
| [pic] |[pic] |
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