PHASE 1 CURRICULUM MAP M. Fellmeth Course/Subject: Honors Precalculus Grade: 11 th Teacher: M. Hart

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1 Month: September 1. How to describe angles using different units of measure and how to find the lengths associated with those angles A Select and use appropriate units and tools to measure to the degree of accuracy required in particular measurement B Measure and compare angles in degrees and radians C Demonstrate the ability to produce measures with specified levels of precision A Select and use appropriate mathematical concepts and techniques from different areas of mathematics and apply them to nonroutine and multi-step problems A Use graphing calculators to display periodic and circular functions; describe properties of the graphs. Trigonometric Functions 1. Given an angle in degrees, what is the measure in radians? 2. What is the area and arc length of a given sector? 3. How do you find co-terminal angles? 4. What is the approximate apparent size of an object? Find measure of angle in degree/radius Find coterminal angles. Solve problems involving angles. Find arc length and area of a sector of a circle Apply formulas to solve problems involving apparent size. June 2011

2 Month: September (continued) 1. How to use the definitions of sine, cosine and tangent and their inverses to find values of these functions and solve simple trigonometric equations. 2. How to graph and apply the graphs of the trig functions to find values of these functions. 1. What is the value of the sine, cosine, and tangent of a given angle? 2. How do you calculate the exact value of a given trig function? C Demonstrate the ability to produce measures with specified levels of precision A Select and use appropriate mathematical concepts and techniques from different areas of mathematics and apply them to nonroutine and multi-step problems F Use the properties of angles, arcs, chords, tangents and secants to solve problems involving circles A Use graphing calculators to display periodic and circular functions; describe properties of the graphs B Identify, create, and solve practical problems involving right triangles using the trigonometric functions and the Pythagorean Theorem. Trigonometric Functions Inverse Trigonometric Functions Define sine, cosine, tangent and their inverses, find values and solve simple trigonometry functions. Graph sine, cosine, tangent and their inverses. Find values of the inverse trigonometric functions. Recognize/graph the inverse functions. Solve simple trigonometric equations.

3 Month: October 1. How to find equations of different sine and cosine curves and to apply these equations. 2. To use trigonometric functions to model periodic behavior A Develop and use computation concepts, operations and procedures with real numbers in problem- situations A Use graphing calculators to display periodic and circular functions; describe properties of the graphs A Determine maximum and minimum values of a function over a specified interval B Interpret maximum and minimum values in problem situations. Equations of Sine and Cosine Functions Identities 1. What is the amplitude, period, and translations of a sine and cosine equation and/or graph? 2. How do you sketch the graph of a simple sine or cosine equations and identify the solutions? 3. How do you simplify trigonometric expressions and prove trigonometric identities? Identify equations of different sine/cosine curves. Use trigonometric functions to model periodic behavior. Sketch trigonometric functions. Transform sin/cos functions. Apply electronics, physics and music formulas to sin/cos equations. Memorize basic identifies Simplify trigonometric expressions. District Assessment: Honors Precalculus, Performance Assessment Daily Homework

4 Month: November 1. Trigonometric functions can be used to solve for missing lengths and angles in right triangles. 2. Using the sine function one can derive a formula for the area of a triangle given the lengths of two sides and the measure of the included angle. 3. The law of sines and the law of cosines are used to solve for missing angles and lengths in all triangles. 4. Trigonometry is used to solve problems in the field of navigation and surveying. 1. How do you find the unknown sides or angles of a right triangle using trigonometry? 2. Given two sides and the included angle of a triangle, how do you calculate the area? 3. How do you use the law of sines and the law of cosines to solve for the unknown parts of a triangle? 4. In problems involving navigation and surveying, how do you apply trigonometry to solve for unknown angles or lengths? D Describe and explain the amount of error that may exist in a computation using estimates C Demonstrate the ability to produce measures with specified levels of precision A Use direct proofs, indirect proofs, or proof by contradiction to validate conjectures B Construct valid arguments from stated facts. Identities and Equations Triangle Trigonometry Prove trigonometric identities Solve more difficult trigonometric equations. Find unknown sides/angles of a right triangle. Find area of a triangle given lengths of 2 sides and measure of included angle. Solve triangles using law of sines/cosines Apply triangle trigonometry to solve word problems. with 10%

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6 Month: December 1. To derive and apply formulas for the sine or cosine of the sum or difference of two angles. 2. To derive and apply formulas for the tangent of the sum or difference of two angles. 3. To derive and apply double-angle and half-angle formulas. 4. To use identities to solve trigonometric equations. 1. How do you use the sum and difference identities to simplify a trigonometric expression? 2. How do you find the exact value of an angle that is equal to the sum or difference of two special angles? 3. How do you use the difference formula for the tangent to calculate the angle between two intersecting lines? Trigonometric Addition Formulas Derive and apply formulas for sin, cos and tan (a+b). Polar Coordinates Derive and apply double/half angle formulas. Solve trigonometric equations. Graph polar coordinates

7 Month: January 1. Real numbers are plotted on the x-y coordinate plane, complex numbers are plotted on the complex plane, and polar numbers on the polar plane. 2. Equations can be converted from rectangular to polar form and graphed on the polar plane. 3. All polar numbers can be expressed, operated, and plotted in complex form. 1. How do you graph polar and complex polar ordered pairs and polar equations? 2. How can you convert an ordered pair from polar to rectangular and vice versa? 3. How do you write complex numbers in polar form and find the product, power, and roots of two complex numbers? Polar Coordinates Graph polar equations Convert polar coordinates to rectangular form. Write complex numbers in polar form. Products, powers and roots of complex numbers. Represent products, powers and roots of complex numbers on an Argand Diagram Polar Graph Project

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9 Month: February 1. To identify a polynomial function and to determine its zeros. 2. How to graph a function and determine the domain, range, zeros and inverse. 1. How do you determine if a polynomial is a function? 2. What specific facts can be determined from a polynomial equation that will allow you to quickly sketch its graph? 3. What test do you perform to determine if a function has an inverse, how do you find the inverse equation, and sketch its graph? B Use symbols, mathematical terminology, standard notation, mathematical rules, graphing and other types of mathematical representations to communicate observations, predictions, concepts, procedures, generalizations, ideas and results O Determine the domain and range of a relation, given a graph or set of ordered pairs N Solve linear, quadratic and exponential equations both symbolically and graphically T Analyze and categorize functions by their characteristics. Functions Identify functions/graph functions Determine domain, range and zeros of a function. Graph parabolas using 3 b 1 different methods 2, ( hk a ), and intercept form. Perform operations on functions Reflect graphs and use symmetry to sketch graphs. Determine periodicity and amplitude from graphs. Find the inverse of a function, if the inverse exists.

10 Month: March 1. To define and apply integral and rational exponents. 2. To define and use exponential functions. 3. The definition of e. 4. To define and apply logarithms, and the laws of logarithms R Create and interpret functional models S Analyze properties and relationships of functions (e.g., linear, polynomial, rational, trigonometric, exponential, logarithmic) T Analyze and categorize functions by their characteristics A Determine maximum and minimum values of a function over a specified interval C Graph and interpret rates of growth/decay. Exponents: Logarithms 1. How are integral exponents different then rational exponents? 2. How do you identify and graph an exponential equation? 3. What is the definition of an exponential function and what is their application in the real world? 4. What is the number e? 5. How do you use the laws of logarithms to expand or simplify a logarithmic expression? Define and apply integral and rational exponents. Define and graph exponential functions. Apply exponential functions to growth/decay problems. Define e Define and apply logarithms Prove and apply laws of logs

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12 Month: April 1. To solve exponential equations and to change logarithms from one base to another. 2. Use coordinate geometry to prove that line segments are equal, lines are parallel, lines are perpendicular, line segments bisect each other, and that lines are concurrent. 3. Identify, sketch and/or write the equation of a conic those line of symmetry is vertical or horizontal. 4. Solve systems of equations that involve degree 1 and degree 2 equations A Use operations (e.g., opposite, reciprocal, absolute value, raising to a power, finding roots, finding logarithms) E Use equations to represent curves (e.g., lines, circles, ellipses, parabolas, hyperbolas) G Analyze and explain systems of equations, systems of inequalities and matrices N Solve linear, quadratic and exponential equations both symbolically and graphically I Model situations geometrically to formulate and solve problems J Analyze figures in terms of the kinds of symmetries they have. Logarithms Analytic Geometry 1. How do you solve an exponential equation? 2. What is the purpose of the logarithmic function and how do you apply it? 3. How can you prove a geometry theorem using coordinate methods? 4. How do you identify each conic equation and what essential information do you need to construct each conic? 5. How do you solve a degree 2 system of equations using the graphing method and the substitution method? Solve exponential equations Change logarithms from one base to another. Graph and reflect exponential and logarithmic functions. Prove theorems from geometry by using coordinates. Find, recognize and graph circles, ellipses, hyperbolas and parabolas. Solve systems of linear and quadratic equations.

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14 Month: May 1. How to identify and classify polynomials and use synthetic substitution to evaluate. 2. How to use synthetic substitution to apply the remainder and factor theorems. 3. Solve polynomial equations and determine max and min values of quadratic equations. 1. When using synthetic substitution, what information is derived from the process? 2. What information does the remainder and factor theorem tell you about a polynomial? 3. What methods can be applied to solve polynomial equations? 4. What essential information can you extract from a equation and how can you use this information to sketch its graph? J Demonstrate the connection between algebraic equations and inequalities and the geometry of relations in the coordinate plane N Solve linear, quadratic and exponential equations both symbolically and graphically R Create and interpret functional models S Analyze properties and relationships of functions (e.g., linear, polynomial, rational, trigonometric, exponential, logarithmic) T Analyze and categorize functions by their characteristics. Polynomial Functions Identify polynomial functions and evaluate using synthetic substitution. Use synthetic division to apply remainder and factor theorem. Graph and determine equations of polynomial functions. Apply quadratic functions to solve max/min problems. Solve polynomial equations by factoring, quadratic formula and rational root theorem. Apply general theorems re: polynomial equations.

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16 Month: June 1. How to solve and graph, one and two variable, polynomial inequalities. 1. How is graphing a polynomial equation different than graphing a polynomial inequality? 2. How do you graph and read the solutions of a system of polynomial inequalities on a graph? F Identify whether systems of equations and inequalities are consistent or inconsistent H Select and use an appropriate strategy to solve systems of equations and inequalities using graphing calculators, symbol manipulators, spreadsheets and other software J - Demonstrate the connection between algebraic equations and inequalities and the geometry of relations in the coordinate plane. Inequalities Review for Final Solve and graph linear inequalities in one variable. Solve and graph polynomial inequalities in one variable Graph polynomial inequalities in two variables. 10%

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