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Pre-Calculus Instructional Guide 2011-2012 1 Subject: Pre-Calculus/Pre-Calculus Honors Benchmark Assessments and Instructional Guide Instructional Guides are provided as resource for Alliance classroom teachers. They identify high priority grade-level standards to be taught during each quarter of instruction in the context of proposed units with a suggested amount of time. High priority standards are assessed on quarterly benchmark exams. Pre-Calculus begins with a study of different number systems. The real and complex number systems are explored. Basic set theory is introduced and used to make logical arguments about number systems and their subsets. The concept of sets are connected to the mapping of set A, domain, to set B, range, through the use of functions. A library of basic functions is established and transformations and compositions are used to graph and analyze these functions. Math modeling is introduced and connected to equations in one and two variables, and functions. This leads to a discussion of the relationship between the function and its graph to include the ability to predict behavior. In contrast, an analysis of the general equation of the second degree leads to a thorough study of circles, parabolas, ellipses, and hyperbolas, which are not necessarily functions. Polynomial and rational functions, and their graphs, are then studied in depth. Exponential and logarithmic functions are further explored, including a study of the logistic growth function. The study of trigonometry is introduced through a review of right triangle trigonometry and applications of the law of sines and law of cosines in an applied context. Trigonometry functions are defined on the unit circle. Graphs of the trigonometric functions are investigated and plane transformations are applied. The model for harmonic motion is discussed as an application of trigonometric functions of a real number. Trigonometric equations are introduced, and methods of verifying trigonometric identities are explored. Trigonometric equations are solved and their solutions are connected to the graph of a function and the unit circle. Infinite series are reviewed, and summation notation is introduced to write partial sums. Methods of probability and statistics are reviewed. Pre-calculus ends with a discussion of vectors, polar equations, and parametric equations. Unit High Priority Standards & Learning Targets # CST Items* # Q1 Items Supporting Medium/Low Priority Standards & Learning Targets Algebra 2: 5.0 Students demonstrate knowledge of how real and complex numbers are related both arithmetically and graphically. In particular, they can plot complex numbers as points in the plane. # CST Items* Textbook Prentice Hall** Unit 1: Mathematical Terminology Algebra 2: A67 #13,19,27,39,41 & Notation 15.0 Students determine 1 1 whether a specific alPre-Calculus begins with a formal A45 #73,79A94 #53,81 gebraic statement indiscussion of terminology and notavolving rational extion that has been used in past A94 #53,81 pressions, radical exmathematics courses. This begins pressions, or logawith an introduction to the concept of A11 #9,11 rithmic or exponential a set and set theory. The history of functions is someset theory, specifically the influence A85-86 #57, 73,83 times true, always of Georg Cantor, is investigated, and true, or never true. the importance of set theory in mathA94 #71,77,83,85 Learning Targets ematics is highlighted, namely that 1F Use mathematical properties to the language of set theory (in its explain how to simplify rasimplest form) provides the foundational and radical expressions tion for studying all areas of mathe1G Use the conjugate to rationalize the numerator of a commatics. The basic notion of a set is plex radical expression. defined, as well as the union and * Learning Targets (LT) break the standard down into relevant chunks for teaching and benchmark assessments. The LT number refers to the unit i.e. 1a is Unit 1 learning target A. *CST Items: Pre Calculus CST test is a Summative Exam (complete list of standards see Unit 13) **Textbook: Enhanced with Graphing Utilities, 5/E by Sullivan and Sullivan, publisher: Prentice Hall Pre-Calculus Instructional Guide 2011-2012 High Priority Standards & Learning Targets Unit intersection of two sets. Commonly used sets are defined and given names, including the integers Z {...,2,1,0,1,2,...} , the rational numbers Q {x | x ab , where a,b Z} , and the irrational numbers I {x | x ab , where a,b Z} . The union of the rational and irrational numbers is defined as real numbers Q I . A purely imaginary number is defined, and the complex numbers are defined as C {z | z a bi where a,b }. Properties associated with these sets are discussed, and used to justify logical statements. Subsets of these numbers are explored, as well as complements and closure. This naturally leads to a discussion of the size (or cardinality) of a set, as well as the notions of countable and uncountable, which sets up a discussion of intervals of the real number line, and interval notation. Quantifier notation is also developed and used throughout the unit, including (for all, for every), (there exists), (therefore), (element of), (implies), and (if and only if). Honors Level: With the discussion of the irrational numbers, honors stu- dents of prove the irrationality 2 # CST Items* # Q1 Items Supporting Medium/Low Priority Standards & Learning Targets # CST Items* Textbook Prentice Hall** 1J Use the distributive property and exponent rules to simplify complex rational expressions. 25.0 Students use properties from number systems to justify steps in combining and simplifying functions. Learning Targets 1A Justify mathematical statements using the properties of real and complex numbers. 1B Write subsets of the Rational, Real, and Complex number systems using set notation. 1C Explain relationships between sets within the Complex Number System and evaluate the connections. 1D Explain how the elements in the real number system and imaginaries connect to form the standard form of a complex number. 1H Find the union and intersection of two sets and justify the solution. 1I Compare and contrast set notation and interval notation and write mathematical statements using quantifier notation. 1K Explain the closure property and how a set of numbers can be closed under addition or multiplication, using a counterexample to justify your reasoning Math Analysis: 2.0 Students are adept at the arithmetic of complex numbers. Learning Targets 1E Add, subtract, multiply, divide, * Learning Targets (LT) break the standard down into relevant chunks for teaching and benchmark assessments. The LT number refers to the unit i.e. 1a is Unit 1 learning target A. *CST Items: Pre Calculus CST test is a Summative Exam (complete list of standards see Unit 13) **Textbook: Enhanced with Graphing Utilities, 5/E by Sullivan and Sullivan, publisher: Prentice Hall 2 Pre-Calculus Instructional Guide 2011-2012 Unit High Priority Standards & Learning Targets # CST Items* # Q1 Items Supporting Medium/Low Priority Standards & Learning Targets # CST Items* Textbook Prentice Hall** simplify, and graph complex numbers. 3.0 Students can give proofs of various formulas by using the technique of mathematical induction. (honors only) Learning Targets 1L Prove the irrationality of the square root of 2 and explain each step. (Honors) * Learning Targets (LT) break the standard down into relevant chunks for teaching and benchmark assessments. The LT number refers to the unit i.e. 1a is Unit 1 learning target A. *CST Items: Pre Calculus CST test is a Summative Exam (complete list of standards see Unit 13) **Textbook: Enhanced with Graphing Utilities, 5/E by Sullivan and Sullivan, publisher: Prentice Hall 3 Pre-Calculus Instructional Guide 2011-2012 High Priority Standards & Learning Targets Unit Unit 2: Domain We begin this unit by studying the concept of a function and its properties. The definition of a function, and the domain and range of a function, is explored, and multiple representations of functions are used. The domain or pre-image of a function is the set of allowable inputs, and the range or image of a function is the set of outputs. The graphs of a variety of basic functions are studied including linear functions ( f (x) ax b ), power functions ( f (x) x ), root n f (x) n x ), reciprocal ( f (x) 1n ) , and expofunctions x functions ( nential & logarithmic functions x ( f (x) e and f (x) ln x ). This group of functions is often referred to as the parent functions, or the library of functions. Shifting techniques are then applied to graphs in the library of functions. Piecewise functions are discussed in greater depth, featuring the greatest integer function and the absolute value function. The domain of a piecewise function is carefully analyzed and the behavior of the graph in each interval of the domain is discussed. Shifting techniques are also applied to piecewise functions. Once Algebra 2: 1.0 Students solve equations and inequalities involving absolute value. Learning Targets 2J Explain the conditions for solving linear absolute value problems. 2K Solve and graph the solution sets of absolute value inequalities, and explain the solution set. 24.0 Students solve problems involving functional concepts, such as composition, defining the inverse function and performing arithmetic operations on functions. Learning Targets 2A Explain the definition of a function. 2B Represent a function in a theoretical and applied context and justify using a counterexample 2E Composite functions and then simplify; justify each step in the simplification process and connect to concept of domain 2G Reconstruct a function involving composition. 2H Write translations as function compositions. Calculus (foundation only): 4.0 Students demonstrate an understanding of the formal definition # CST Items* # Q1 Items 1 1 Supporting Medium/Low Priority Standards & Learning Targets Calculus (foundation only): 2.0 Students demonstrate knowledge of both the formal definition and the graphical interpretation of continuity of a function. Learning Targets 2C Analyze the domain of a function and explain the similarities and differences between domains that produce different images- some continuous and others not continuous 2D Find the domain for functions such as f(x)=Ö(polynomial, degree greater than or equal to 2) 2P Explain how the domain of a piecewise function is determined, and how this relates to evaluating the function. Support with an example. # CST Items* 4 Textbook Prentice Hall ** Notes-definition p.67-68 #15,19,33 p.68 #51,55p.75 #13,15p.200 #13 p.166 #9,13p.218 #21,27,33 p.253-254 #11,21,33,35 p.68-69 #39h,77,80 p.254 #53 Notes p.97 #9-16 A58 #53,57,63 A58 #53,57,63A86 #89,93,95 Libraryp.108 #7-18 p.109-110 #27,39,43,45,47,51,61p.152 #21 p.87 #21,33,39,43 p.24-25#21,33,35,51,59p.75 #13,15 p.97 #25,27 * Learning Targets (LT) break the standard down into relevant chunks for teaching and benchmark assessments. The LT number refers to the unit i.e. 1a is Unit 1 learning target A. *CST Items: Pre Calculus CST test is a Summative Exam (complete list of standards see Unit 13) **Textbook: Enhanced with Graphing Utilities, 5/E by Sullivan and Sullivan, publisher: Prentice Hall Pre-Calculus Instructional Guide 2011-2012 Unit functions are graphically analyzed, properties of functions are analyzed, including even and odd properties. The average rate of change of a function is defined and used to determine whether a function is increasing or decreasing at a number. Given functions are also evaluated for the difference quotient: f (x h) f (x) . Function operah tions, including compositions, are then explored graphically. Honors Level: When students are applying the difference quotient, they need to justify each step mathematically using a property. High Priority Standards & Learning Targets of the derivative of a function at a point and the notion of differentiability: # CST Items* # Q1 Items Supporting Medium/Low Priority Standards & Learning Targets # CST Items* Textbook Prentice Hall ** p.97 #31,35,37 p.68-69 #39h,77,80 Learning Targets 2F Evaluate and simplify functions involving the difference quotient, explaining the connection between domains and functions 2R When applying the difference quotient, justify each step mathematically using a property. (Honors) 9.0 Students use differentiation to sketch, by hand, graphs of functions. They can identify maxima, minima, inflection points, and intervals in which the function is increasing and decreasing. Learning Targets 2I Illustrate the library of functions from memory, identify the domain and range of each function, and explain the reasoning behind memorizing these particular functions. 2L Illustrate and describe all possible shifting techniques for one function in the library of functions. 2M Graph functions using transformations and describe shifting. 2N Determine whether a function is even, odd, or neither, and explain symmetry. 2O Describe and illustrate symmetry about the origin and give specific cases where it exists. * Learning Targets (LT) break the standard down into relevant chunks for teaching and benchmark assessments. The LT number refers to the unit i.e. 1a is Unit 1 learning target A. *CST Items: Pre Calculus CST test is a Summative Exam (complete list of standards see Unit 13) **Textbook: Enhanced with Graphing Utilities, 5/E by Sullivan and Sullivan, publisher: Prentice Hall 5 Pre-Calculus Instructional Guide 2011-2012 Unit High Priority Standards & Learning Targets # CST Items* # Q1 Items Supporting Medium/Low Priority Standards & Learning Targets # CST Items* Textbook Prentice Hall ** 2Q Illustrate a piecewise function involving shifting techniques, the greatest integer function, and the library of functions * Learning Targets (LT) break the standard down into relevant chunks for teaching and benchmark assessments. The LT number refers to the unit i.e. 1a is Unit 1 learning target A. *CST Items: Pre Calculus CST test is a Summative Exam (complete list of standards see Unit 13) **Textbook: Enhanced with Graphing Utilities, 5/E by Sullivan and Sullivan, publisher: Prentice Hall 6 Pre-Calculus Instructional Guide 2011-2012 Unit High Priority Standards Unit 3: Introduction to Mathematical Modeling After formally defining the number systems and establishing the foundation for functions, it is then natural to study how the numbers and properties and concept of input and output discussed in the previous units are used in a real world context. The focus of this unit is to introduce the concept of a mathematical model, which is studied throughout this course. Mathematical models are created using previous knowledge of a variety of different types of mathematical concepts, with a focus on linear systems. The discussion of modeling begins with linear models in two variables. Linear equations and inequalities are used to model real life situations and contrasted to models involving absolute value equations and inequalities. Linear systems in two and three variables are categorized by their solution sets, and interpreted graphically. Certain types of non-linear systems are dis- Algebra 2: 8.0 Students solve and graph quadratic equations by factoring, completing the square, or using the quadratic formula. Students apply these techniques in solving word problems. They also solve quadratic equations in the complex number system. 1 1 cussed (i.e. involving and ), and y x the u-v method of substitution is used to solve such systems. Another type of non-linear mathematical model is introduced and analyzed, namely a quadratic model. The quadratic formula is proven using completing the square, and used to solve a variety Learning Targets 3E Prove the quadratic formula using completing the square and explain the connection to domain and solutions Linear Algebra: 6.0 Students demonstrate an understanding that linear systems are inconsistent (have no solutions), have exactly one solution, or have infinitely many solutions. Learning Targets 3B Design a model using a system of linear equations & inequalities (mixture, linear programming, piecewise functions), justify each step within the analysis and explain the validity of the solution. 3C Categorize and connect the solutions to mathematical linear models to types of linear systems and explain the importance of slope # CST Items* 3 # Q1 Items 3 Supporting Medium/Low Priority Standards Calculus (foundation only): 2.0 Students demonstrate knowledge of both the formal definition and the graphical interpretation of continuity of a function. 14.0 Students apply the definition of the integral to model problems in physics, economics, and so forth, obtaining results in terms of integrals. Learning Targets 3F Design and use a model involving a quadratic function 3H Apply a quadratic function model (equations and inequalities) to projectile motion and explain the difference between solutions found theoretically versus within an applied context (includes inequalities) # CST Items* 7 Textbook Prentice Hall ** p.42-43 #119,120p. 134 #37,38A75-77 #7-53A86-87 #111- 119 p.718-720 #55-68,71-81p.734 #77-88p.788 #57-61, p.794795 #19-31 p.40-41 #11,17,23,33,37,45,59,65,71,7 7,91p.132 #13p.717 #19,25,29 p.717 #35,36,39,40 Notes, Handout: Derive p.152 #29,35,39,51 (10 steps)p.167 #33,34 A59 #81-86 17.0 Students compute, by hand, the integrals of a wide variety of functions by using techniques of integration, such as substitution, integration by parts, and trigonometric substitution. They can also combine these techniques when appropriate. p.161 #11,12 Learning Targets 3D Use u-v substitution to solve certain non-linear systems of equations and justify each step (i.e. equations with 1/x). 3G Solve quadratic type problems, Notes p.117 #25,26(a-c)p.160 #7,8(ab)p.778-779 #85,87A76 #31,32 Notes p.117 #25,26(d)p.160-162 #3,7c,8c, 9,10,11,13,17,27p.169 #39 * Learning Targets (LT) break the standard down into relevant chunks for teaching and benchmark assessments. The LT number refers to the unit i.e. 1a is Unit 1 learning target A. *CST Items: Pre Calculus CST test is a Summative Exam (complete list of standards see Unit 13) **Textbook: Enhanced with Graphing Utilities, 5/E by Sullivan and Sullivan, publisher: Prentice Hall Pre-Calculus Instructional Guide 2011-2012 Unit High Priority Standards of quadratic equations and quadratic type equations used in mathematical models, including projectile motion. Solutions found theoretically are analyzed within an applied context for quadratic models. Next models for “fence”, “box” , and “garden” problems and designed and used to establish the foundation for optimization problems in Calculus. Models of direct, indirect, and joint variation are discussed, and used to solve problems related to physics. Honors Level: Students solve applied minimum and maximum problems, focusing on the box, garden, and fence problems Calculus (foundation only): 11.0 Students use differentiation to solve optimization (maximumminimum problems) in a variety of pure and applied contexts. # CST Items* # Q1 Items Supporting Medium/Low Priority Standards # CST Items* Textbook Prentice Hall ** justify each step, and explain the purpose of u- substitution. Learning Targets 3A Explain how the four step process (given, want, know, analysis) for problem solving supports finding solutions to a variety of complex problems 3I Design and use a model for fence, garden, and box problems 3J Explain the similarities and differences in solving a "box" problem vs. a "garden" problem vs. a fence problem vs a projectile motion problem 3K Write a direct, indirect, or joint variation model, and explain the differences between the three different types of variation. 3L Design and use a model to maximize/minimize solutions to “projectile”, “box”, “garden” problems. 3M Describe the purpose of mathematical modeling and give examples. * Learning Targets (LT) break the standard down into relevant chunks for teaching and benchmark assessments. The LT number refers to the unit i.e. 1a is Unit 1 learning target A. *CST Items: Pre Calculus CST test is a Summative Exam (complete list of standards see Unit 13) **Textbook: Enhanced with Graphing Utilities, 5/E by Sullivan and Sullivan, publisher: Prentice Hall 8 Pre-Calculus Instructional Guide 2011-2012 Unit Unit 4: The General Equation of the Second Degree in Two Variables The history of the conic sections begins in ancient Greece with the mathematician Apollonius. His role in mathematics and the study of conic sections is investigated. The general quadratic equation of second degree in two variables is then defined as Ax Bxy Cy Dx Ey F 0 2 2 . The coefficient of xy represents the rotation of the graph of the equation, which cannot be studied without trigonometry. Therefore, by setting B 0 , the following conditions are developed for A and C to determine which conic sections are produced from the equation: parabolas [ A 0 or C 0 , not both], ellipses [ AC 0 ], circles [ A C ], and hyperbolas [ AC 0 ]. Using completing the square, each type of conic section is written in its standard form, and key information about the graph is identified. Using this information, graphs of each conic section are drawn from the standard forms and equations in standard form are written given certain information. Continuing the theme of mathematical modeling, conic sections are used to solve problems in a real world context, including those about High Priority Standards Math Analysis: 5.0 Students are familiar with conic sections, both analytically and geometrically: Learning Targets 3A Know the general equation of the second degree in two variables. 3B Describe the conditions of the coefficients of the general second-degree equation that produce each conic section (circle, parabola, ellipse, hyperbola). 3C Describe how each conic section is obtained from the intersection of a plane and a cone, including the degenerate forms. 3D Describe how completing the square is involved in the study of conic sections. 3N Design and use a mathematical model involving conic sections. # CST Items* # Q1 Items Supporting Medium/Low Priority Standards Math Analysis: 5.2 Students can take a geometric description of a conic section - for example, the locus of points whose sum of its distances from (1, 0) and (-1, 0) is 6 - and derive a quadratic equation representing it. # CST Items* Textbook Prentice Hall ** notes notes Learning Targets 3K Given the focus and directrix of a parabola find the standard form 3L Write equations for conic sections given key information (foci, vertices, etc.) 3M Write equations for conic sections from a graph. 5.1 Students can take a quadratic equation in two variables; put it in standard form by completing the square and using rotations and translations, if necessary; determine what type of conic section the equation represents; and determine its geometric components (foci, asymptotes, and so forth). * Learning Targets (LT) break the standard down into relevant chunks for teaching and benchmark assessments. The LT number refers to the unit i.e. 1a is Unit 1 learning target A. *CST Items: Pre Calculus CST test is a Summative Exam (complete list of standards see Unit 13) **Textbook: Enhanced with Graphing Utilities, 5/E by Sullivan and Sullivan, publisher: Prentice Hall 9 Pre-Calculus Instructional Guide 2011-2012 Unit High Priority Standards physics and astronomy. Honors Level: While the discussing the history of the conic sections, honors students investigate the intersection of a plane and a doublenapped cone, including the degenerate forms. Learning Targets 3E Rewrite a circle from the general form into the standard form and identify the center, radius, and sketch 3F Rewrite a parabola from the general form into the standard form and identify the vertex, focus, directrix, and sketch 3G Rewrite an ellipse from the general form into the standard form and identify the center, vertices, foci, co-vertices, eccentricity, and sketch 3J Rewrite a hyperbola from the general form into the standard form and identify the center, vertices, foci, asymptotes, and sketch # CST Items* # Q1 Items Supporting Medium/Low Priority Standards # CST Items* 10 Textbook Prentice Hall ** * Learning Targets (LT) break the standard down into relevant chunks for teaching and benchmark assessments. The LT number refers to the unit i.e. 1a is Unit 1 learning target A. *CST Items: Pre Calculus CST test is a Summative Exam (complete list of standards see Unit 13) **Textbook: Enhanced with Graphing Utilities, 5/E by Sullivan and Sullivan, publisher: Prentice Hall Pre-Calculus Instructional Guide 2011-2012 Unit High Priority Standards & Learning Targets Unit 5: Polynomial and Rational Functions The focus of the next few units is to use the relationships developed in the previous unit between functions and graphs, to analyze specific types of functions, and then to compare and contrast the different types of functions studied. This discussion begins with polynomial functions. Polynomial functions in standard form are defined, and the graphs of polynomial functions are studied, as well as the domain (pre-image) and range (image). Graphs of polynomials and non-polynomials are compared and contrasted in order to develop an intuitive understanding of the smooth and continuous nature of the graph of polynomial functions. The end behavior of graphs of polynomial functions is then explored. Math Analysis: 4.0 Students know the statement of, and can apply, the fundamental theorem of algebra. The focus of the unit shifts from the graphs of polynomial functions to the roots/zeros/solutions of the function. The roots/zeros/solutions of a polynomial function are defined, and the values of any real roots/zeros/solutions are estimated from the graph of the function. Connections are then made between the Binomial Theorem and roots/zeros/solutions of multiplicity, and this information is used to review the binomial expansion to a certain 8.0 Students are familiar with the notion of the limit of a sequence and the limit of a function as the independent variable approaches a number or infinity. They determine whether certain sequences converge or diverge. Learning Targets 5J Explain how the division algorithm, the remainder theorem, and the factor theorem are related. 5L Explain how to find the equation of a polynomial if given the degree of the polynomial and one of the conjugate pairs: , as well as information about multiplicity. 5M Describe the behavior of the graph of a polynomial function near a root of multiplicity. 5N Classify the real roots of a polynomial function using Descartes Rule of Signs and FTA. 5O Find the roots of a polynomial function using the Rational Roots Theorem Learning Targets 5U Explain how rational functions behave close to vertical as- # CST Items* # Q2 Items Supporting Medium/Low Priority Standards & Learning Targets Math Analysis: 6.0 Students find the roots and poles of a rational function and can graph the function and locate its asymptotes. Algebra 2: 3.0 Students are adept at operations on polynomials, including long division. # CST Items* 11 Textbook Prentice Hall ** 1 Learning Targets 5H Use the division algorithm, remainder theorem, and factor theorem. 4.0 Students factor polynomials representing the difference of squares, perfect square trinomials, and the sum and difference of two cubes. 1 Learning Targets G Describe the relationship between roots and factors. 5P Compare and contrast linear factors and quadratic factors. 7.0 Students add, subtract, multiply, divide, reduce, and evaluate rational expressions with monomial and polynomial denominators and simplify complicated rational expressions, including those with negative exponents in the denominator. 2 Learning Targets 5K Demonstrate how to use long division to find the roots of a poly- * Learning Targets (LT) break the standard down into relevant chunks for teaching and benchmark assessments. The LT number refers to the unit i.e. 1a is Unit 1 learning target A. *CST Items: Pre Calculus CST test is a Summative Exam (complete list of standards see Unit 13) **Textbook: Enhanced with Graphing Utilities, 5/E by Sullivan and Sullivan, publisher: Prentice Hall Pre-Calculus Instructional Guide 2011-2012 Unit whole number power. The remainder and factor theorems are reviewed, and used to find roots/zeros/solutions of polynomial functions given certain information. The connection between a root/zero/solution of a polynomial function and a factor of the function is emphasized. The Fundamental Theorem of Algebra is reviewed, as well as the Complex Conjugate Theorem. Descartes Rule of Signs is used to classify the roots/zeros/solutions of a polynomial function and the Rational Roots Theorem is used to find the rational roots/zeros/solutions of a polynomial function. The focus of the unit shifts to the study of rational functions, with an emphasis on the domain (pre-image) of a rational function. Asymptotic behavior is reviewed, and the concept of a limit is discussed using limit notation. Conditions for the existence of different asymptotes of a rational function are discussed. The process of long division is introduced in order to find the oblique asymptotes of a rational function. Rational functions are analyzed (find domain/range, intercepts, asymptotes, symmetry) and graphed. Both polynomial and rational functions are used to develop mathematical models representing a real world situa- High Priority Standards & Learning Targets ymptotes and find the vertical asymptotes if they exist 5V Explain how rational functions behave close to horizontal asymptotes and find the horizontal asymptotes if they exist using the limit of the function as x approaches infinity # CST Items* # Q2 Items Supporting Medium/Low Priority Standards & Learning Targets # CST Items* 12 Textbook Prentice Hall ** nomial function. Calculus (foundation only): 9.0 Students use differentiation to sketch, by hand, graphs of functions. They can identify maxima, minima, inflection points, and intervals in which the function is increasing and decreasing. Learning Targets 5A Describe a polynomial function in standard form and classify special cases of polynomials 5B Apply the Fundamental Theorem of Algebra. 5C Determine the end behavior of a graph a polynomial function. 5D Explain how symmetry affects the behavior of the polynomial graph 5E Explain how symmetry affects the behavior of the graph 5F Find the maximum(s) and minimum(s) of a function. 5Q Determine whether a polynomial function is increasing or decreasing over an interval. 5R Analyze a polynomial that involves multiplicity and nonreal roots 5S Explain the difference between a polynomial function and rational function * Learning Targets (LT) break the standard down into relevant chunks for teaching and benchmark assessments. The LT number refers to the unit i.e. 1a is Unit 1 learning target A. *CST Items: Pre Calculus CST test is a Summative Exam (complete list of standards see Unit 13) **Textbook: Enhanced with Graphing Utilities, 5/E by Sullivan and Sullivan, publisher: Prentice Hall Pre-Calculus Instructional Guide 2011-2012 Unit tion. Honors Level: Honors students use limit notation to describe the end behavior of the graphs of polynomial functions.. High Priority Standards & Learning Targets # CST Items* # Q2 Items Supporting Medium/Low Priority Standards & Learning Targets # CST Items* 13 Textbook Prentice Hall ** 5X Graph rational functions. 5Y Defend the purpose for analyzing functions. * Learning Targets (LT) break the standard down into relevant chunks for teaching and benchmark assessments. The LT number refers to the unit i.e. 1a is Unit 1 learning target A. *CST Items: Pre Calculus CST test is a Summative Exam (complete list of standards see Unit 13) **Textbook: Enhanced with Graphing Utilities, 5/E by Sullivan and Sullivan, publisher: Prentice Hall Pre-Calculus Instructional Guide 2011-2012 Unit High Priority Standards Unit 6: Functions, Graphs, and their Inverses Composite functions are re-visited within the context of solving radical equations, rational equations and defining an inverse. Namely, f and g are inverses if and only if f (g(x)) x and g( f (x)) x . The concept of a one to one function is then explored algebraically and graphically (via symmetry), and an alternative definition is developed for 1 an inverse function: f and f are inverses if and only if f is a one-to- Algebra 2: 24.0 Students solve problems involving functional concepts, such as composition, defining the inverse function and performing arithmetic operations on functions. 1 one function and f ( f (x)) x . The method for finding inverses by is used to find switching variables inverse functions. Finally, the action of taking an inverse is linked with the concept of reflection about the line y x and connected to odd functions Honors Level: Honors students analyze the proof of solving a depressed general cubic equation, to include the history behind the solution to the depressed cubic. Learning Targets 6C Explain how to justify that a function is one-to-one. 6D Explain how composition is used to determine if two functions are inverses 6E Describe the connection between symmetry and one-toone functions. 6F Find inverse functions for algebraic and transcendental functions # CST Items* # Q2 Items 1 1 Supporting Medium/Low Priority Standards Algebra 2: 15.0 Students determine whether a specific algebraic statement involving rational expressions, radical expressions, or logarithmic or exponential functions is sometimes true, always true, or never true. # CST Items* 14 Textbook Prentice Hall ** 1 Learning Targets 6A Simplify radical expressions and solve radical equations and explain extraneous solutions 6B Solve rational equations, justify each step, and connect to analyzing rational functions Trigonometry (foundational): 8.0 Students know the definitions of the inverse trigonometric functions and can graph the functions. Learning Targets 6F Find inverse functions for algebraic and transcendental functions 6G Explain the relationship between domain and range for inverses. * Learning Targets (LT) break the standard down into relevant chunks for teaching and benchmark assessments. The LT number refers to the unit i.e. 1a is Unit 1 learning target A. *CST Items: Pre Calculus CST test is a Summative Exam (complete list of standards see Unit 13) **Textbook: Enhanced with Graphing Utilities, 5/E by Sullivan and Sullivan, publisher: Prentice Hall Pre-Calculus Instructional Guide 2011-2012 Unit High Priority Standards Unit 7: The Relationship Between Exponential and Logarithmic Functions The focus of the previous unit was the study of certain algebraic functions. Algebraic functions are functions that can be produced using basic operations. The focus of this unit, on the other hand, is to study certain type of non-algebraic, or transcendental, functions. The discussion begins with analyzing exponential and logarithmic functions and their graphs. These functions are analyzed and graphed in base Algebra 2: 11.1 Students understand the inverse relationship between exponents and logarithms and use this relationship to solve problems involving logarithms and exponents. 1 n e lim 1 2.71828459 and n n arbitrary a 1 , and key information Learning Targets 7A Explain the relationship between exponents and logarithms. 7C Explain the definition of a logarithm and use to simplify and solve. 11.2 Students judge the validity of an argument according to whether the properties of real numbers, exponents, and logarithms have been applied correctly at each step. # CST Items* # Q2 Items 1 1 Supporting Medium/Low Priority Standards # CST Items* 15 Textbook Prentice Hall ** Calculus (foundational): 4.2 Students demonstrate an understanding of the interpretation of the derivative as an instantaneous rate of change. Students can use derivatives to solve a variety of problems from physics, chemistry, economics, and so forth that involve the rate of change of a function. Learning Targets 7H Graph and analyze logistics growth functions. 7J Solve mathematical modeling problems using exponential functions involving compound interest, growth and decay. and logarithmic functions to include growth and decay and justify each step within the analysis. 7K Solve mathematical modeling problems using logarithmic functions about these functions is identified. This analysis includes the study of logistics growth functions. The unit begins with an emphasis on the in9.0 Students use differentiaverse relationship between exponention to sketch, by hand, tial and logarithmic functions. ExpoLearning Targets graphs of functions. nential equations and functions are 7I Solve exponential and logarithThey can identify maximic equations, justifying each analyzed both algebraically and ma, minima, inflection step. graphically, followed by an analysis points, and intervals in 12.0 Students know the 2 2 of logarithmic equations and funcwhich the function is inlaws of fractional extions. The properties of logarithms creasing and decreasponents, understand are proven and used to expand or ing. exponential functions, condense expressions containing Learning Targets and use these funclogarithms. Methods of estimating 7B Analyze an exponential function, tions in problems inincluding finding the domain, the value of a logarithm are disvolving exponential range, horizontal asymptote, incussed, including mental estimation tercepts, and inverse- sketch growth and decay. and approximating using the change 7G Analyze a logarithmic function, 14.0 Students understand 1 1 of base formulas. The focus of the including finding the domain, * Learning Targets (LT) break the standard down into relevant chunks for teaching and benchmark assessments. The LT number refers to the unit i.e. 1a is Unit 1 learning target A. *CST Items: Pre Calculus CST test is a Summative Exam (complete list of standards see Unit 13) **Textbook: Enhanced with Graphing Utilities, 5/E by Sullivan and Sullivan, publisher: Prentice Hall Pre-Calculus Instructional Guide 2011-2012 Unit unit then shifts to explore mathematical modeling and equations involving exponentials and logarithms. The methods of solving certain [accessible] equations that involve exponentials and logarithms are explored, and are applied to solve problems involving exponential growth & decay, logistic growth, and logarithmic models. Honors Level: The “rule of 72” is analyzed in terms of logarithmic and exponential functions. Furthermore, honors students develop more complex mathematical models involving growth and decay, including logistic growth. High Priority Standards and use the properties of logarithms to simplify logarithmic numeric expressions and to identify their approximate values. # CST Items* # Q2 Items Supporting Medium/Low Priority Standards # CST Items* 16 Textbook Prentice Hall ** range, vertical asymptote, intercepts, and inverse- sketch Learning Targets 7D Prove the multiplication and division property of logarithms. 7E Expand and condense logarithmic expressions using properties of logarithms. * Learning Targets (LT) break the standard down into relevant chunks for teaching and benchmark assessments. The LT number refers to the unit i.e. 1a is Unit 1 learning target A. *CST Items: Pre Calculus CST test is a Summative Exam (complete list of standards see Unit 13) **Textbook: Enhanced with Graphing Utilities, 5/E by Sullivan and Sullivan, publisher: Prentice Hall Pre-Calculus Instructional Guide 2011-2012 Unit Unit 8: Applied Trigonometry The study of non-algebraic functions continues as the trigonometric functions and their properties are developed. The trigonometric ratios of a right triangle, learned in previous courses, are reviewed and used to solve right triangles for all sides and angles. Applications of right triangle trigonometry are studied, including problems related to indirect measurement. Up to this point, the focus of the study of trigonometry has been strictly related to right triangle trigonometry. Concepts of right triangle trigonometry are the foundation for studying the trigonometry of acute or obtuse triangles. The Law of Sines and Law of Cosines are introduced and proven in order to establish trigonometric relationships of these triangles. Any triangle (acute, right, or obtuse) is solved using these laws and the inverse trigonometric functions, and the different conditions for which each law is used are explored. Applications of the Law of Sines and Law of Cosines are explored, including problems related to navigation and finding area. Honor’s Level: While discussing the Law of Sines, honors students explore the SSA ambiguous case for the Law of Sines and the reasons why two there are two cases. Fur- High Priority Standards Trigonometry: 13.0 Students know the law of sines and the law of cosines and apply those laws to solve problems. Learning Targets 8C Derive the Law of Sines and the different forms of the Law of Cosines. 8D Solve triangles using the Law of Sines 8E Explain why the SSA case yields more than one triangle. (challenge) 8F Describe conditions when an SSA case yields two solutions, one solution, and no solutions. (challenge) 8G Solve triangles using the Law of Cosines 19.0 Students are adept at using trigonometry in a variety of applications and word problems. # CST Items* # Q2 Items Supporting Medium/Low Priority Standards # CST Items* 17 Textbook Prentice Hall ** Trigonometry: 12.0 Students use trigonometry to determine unknown sides or angles in right triangles. Learning Targets 8A Define the six trigonometric ratios of a right triangle, and solve a right triangle for all sides and angles 8D Solve triangles using the Law of Sines 8G Solve triangles using the Law of Cosines 14.0 Students determine the area of a triangle, given one angle and the two adjacent sides. Learning Targets 8J Find the area of any triangle 8K Write a formal proof of Heron's formula using the Law of Cosines. (challenge) 8L Prove Heron's formula for finding the area of a triangle. (challenge) Learning Targets 8B Use right triangle trigonometry to solve problems of indirect measurement. 8H Explain the difference between and angle of depression and angle of elevation. 8I Design and solve problems related to navigation using right triangle trigonometry, the Law of Sines and the Law of Cosines and explain the analysis leading to the solutions * Learning Targets (LT) break the standard down into relevant chunks for teaching and benchmark assessments. The LT number refers to the unit i.e. 1a is Unit 1 learning target A. *CST Items: Pre Calculus CST test is a Summative Exam (complete list of standards see Unit 13) **Textbook: Enhanced with Graphing Utilities, 5/E by Sullivan and Sullivan, publisher: Prentice Hall Pre-Calculus Instructional Guide 2011-2012 Unit High Priority Standards # CST Items* # Q2 Items Supporting Medium/Low Priority Standards # CST Items* 18 Textbook Prentice Hall ** thermore, honors students study formulas for the area of a triangle, and prove Heron’s formula using either Law of Sines or Law of Cosines. * Learning Targets (LT) break the standard down into relevant chunks for teaching and benchmark assessments. The LT number refers to the unit i.e. 1a is Unit 1 learning target A. *CST Items: Pre Calculus CST test is a Summative Exam (complete list of standards see Unit 13) **Textbook: Enhanced with Graphing Utilities, 5/E by Sullivan and Sullivan, publisher: Prentice Hall Pre-Calculus Instructional Guide 2011-2012 Unit High Priority Standards Unit 9: Analyzing Trigonometric Functions This unit begins with a discussion of how angles are measured. Radian measure is defined in terms of arc length on the unit circle. Trigonometric functions of angles are defined in terms of the unit circle. This definition is used to determine which quadrants the trigonometric functions are positive and negative. The concept of a reference angle is explored and references angles are then used to find the value of the trigonometric functions of any angle. Trigonometry 4.0 Students graph functions of the form f(t) = A sin (Bt + C) or f(t) = A cos (Bt + C) and interpret A, B, and C in terms of amplitude, frequency, period, and phase shift. The unit circle is then discussed in more detail. Terminal points on a unit circle are defined, as well as reference numbers. The trigonometric functions of a real number are then defined, using terminal points of the unit circle. Comparisons are made between the concept of the trigonometric functions as functions of an angle, and functions of a real number. Special values of the trigonometric functions are derived. The domain and range of the trigonometric functions are discussed and the graphs of the sine and cosine are built. Connections between the sine and cosine graphs and the unit circle are illustrated, including properties of periodicity. Transformations are applied to the graphs of the sine and cosine, and the general forms of Learning Targets 9J Explain the relationship between the graph of a sine and cosine function, and the unit circle. Learning Targets 9K Use transformations to graph functions of the form or , and identify key information about the function (i.e. period, amplitude, phase shift, etc.). 5.0 Students know the definitions of the tangent and cotangent functions and can graph them. Learning Targets 9L Graph the tangent, cotangent, secant, cosecant functions, and identify the domain, range, and period for each. 8.0 Students know the definitions of the inverse trigonometric functions and can graph the functions. Learning Targets 9M Compare the domain and range of the trigonometric functions and their inverses and explain the relationship between the two. # CST Items* # Q3 Items Supporting Medium/Low Priority Standards # CST Items* 19 Textbook Prentice Hall ** Trigonometry 1.0 Students understand the notion of angle and how to measure it, in both degrees and radians. They can convert between degrees and radians. Learning Targets 9A Explain how to find arc length, and how this concept is used in the definition of radian measure. 9B Identify angles in degree and radian measure, and convert between the measures. 9C Find exact values of trig functions 2.0 Students know the definition of sine and cosine as y-and x-coordinates of points on the unit circle and are familiar with the graphs of the sine and cosine functions. Learning Targets 9D Define a reference angle and describe the reference angle in each quadrant. 9E Describe a method of finding reference angles. 9F Calculate the exact trig function values and explain the purpose of the angle 9G Use the unit circle to evaluate trig expressions and explain the process 19.0 Students are adept at using trigonometry in a variety of applications and word problems. Learning Targets 9N Describe simple harmonic motion * Learning Targets (LT) break the standard down into relevant chunks for teaching and benchmark assessments. The LT number refers to the unit i.e. 1a is Unit 1 learning target A. *CST Items: Pre Calculus CST test is a Summative Exam (complete list of standards see Unit 13) **Textbook: Enhanced with Graphing Utilities, 5/E by Sullivan and Sullivan, publisher: Prentice Hall Pre-Calculus Instructional Guide 2011-2012 Unit High Priority Standards these functions are built in the following order: f (x) sin( x c) , f (x) sin( bx c) , f (x) asin( bx c), and f (x) asin( bx c) d . The same order is used for the cosine function. Other graphs of trigonometric functions are studied including the tangent, the reciprocal functions, and the inverse functions. The difference between reciprocal functions and inverse functions is emphasized. The domain and range of trigonometric functions and their inverses are also compared. 9.0 Students compute, by hand, the values of the trigonometric functions and the inverse trigonometric functions at various standard points. # CST Items* # Q3 Items Supporting Medium/Low Priority Standards # CST Items* 20 Textbook Prentice Hall ** and identify the amplitude, frequency, and period of harmonic motion model, and explain their real world significance. 9O Describe damped harmonic motion verbally and algebraically. 9P Compare and contrast simple and damped harmonic motion. Learning Targets Use the concept of inverse functions and the unit circle to find exact values 9I Explain how to use the unit circle to evaluate trig functions as well as find locations of ordered pairs (detail how to “work” all aspects of the unit circle). Trigonometric functions are then used as mathematical models, specifically modeling harmonic motion. The general form of the sine and cosine is applied to simple harmonic motion, and used to find amplitude, frequency, and period. These components of the graph of a harmonic motion model are all interpreted in a real world context, and used to solve problems within that context. Honors Level: Honors students extend the study of harmonic motion, to include damped harmonic motion. * Learning Targets (LT) break the standard down into relevant chunks for teaching and benchmark assessments. The LT number refers to the unit i.e. 1a is Unit 1 learning target A. *CST Items: Pre Calculus CST test is a Summative Exam (complete list of standards see Unit 13) **Textbook: Enhanced with Graphing Utilities, 5/E by Sullivan and Sullivan, publisher: Prentice Hall Pre-Calculus Instructional Guide 2011-2012 Unit High Priority Standards Unit 10: Trigonometric Equations The trigonometric functions have now been study in depth, as well as their application. The focus of this unit is on trigonometric equations. The discussion begins with a review of basic identities (e.g. x 2 x 2 ). The fundamental trigonometric identities are introduced and illustrated. These include the reciprocal identities, the Pythagorean identities, the even and odd identities, and the cofunction identities. Basic trigonometric identities are used to simplify trigonometric expressions, and different methods of simplifying these expressions are explored. Trigonometry: 3.1 Students prove that this identity is equivalent to the Pythagorean theorem (i.e., students can prove this identity by using the Pythagorean theorem and, conversely, they can prove the Pythagorean theorem as a consequence of this identity). The basic trigonometric identities are then used to prove more complex trigonometric identities. A variety of methods of proving trigonometric identities are used. More advanced identities are introduced, including the following: addition and subtraction identities, double angle and half angle identities, and product and sum identities. All of these are used to prove more complex trigonometric identities. # CST Items* # Q3 Items Supporting Medium/Low Priority Standards # CST Items* 21 Textbook Prentice Hall ** Trigonometry: 3.0 Students know the iden2 2 tity cos (x) + sin (x) = 1: 10C Know the Pythagorean identities Learning Targets 10B Derive the Pythagorean Trig. identities. 10L Verify more complex trigonometric identities using the basic trigonometric identities, Pythagorean identities, cofunction identities, odd/even identities, sum & difference identities, double angle identities, half-angle identities and justify each step in the verification process. 3.2 Students prove other trigonometric identities and simplify others by using the iden2 2 tity cos (x) + sin (x) = 1. For example, students use this identity 2 to prove that sec (x) The focus of the unit shifts to a study of solving trigonometric equations. The inverse trigonometric functions are used to solve basic trigonometric equations on a restricted domain 2 = tan (x) + 1. Learning Targets 10A Know the basic trigonometric identities, including the reciprocal identities * Learning Targets (LT) break the standard down into relevant chunks for teaching and benchmark assessments. The LT number refers to the unit i.e. 1a is Unit 1 learning target A. *CST Items: Pre Calculus CST test is a Summative Exam (complete list of standards see Unit 13) **Textbook: Enhanced with Graphing Utilities, 5/E by Sullivan and Sullivan, publisher: Prentice Hall Pre-Calculus Instructional Guide 2011-2012 Unit (e.g. sin x 1 on [0,2 ) ). Once this domain restriction is removed, the periodic nature of the solution sets of trigonometric equations is explored. Connections are made between trigonometric equations, the unit circle, and graphs of trigonometric functions. Various methods of solving trigonometric equations are studied, and used to solve mathematical models involving the trigonometric functions. Honors Level: While solving trigonometric equations, honors students manipulate the known trigonometric identities before finding the solution set. Honors students also prove the multiple angle identities, and the addition and subtraction identities. High Priority Standards # CST Items* # Q3 Items Supporting Medium/Low Priority Standards # CST Items* 22 Textbook Prentice Hall ** 10D Simplify trigonometric expressions using the basic trigonometric identities. 10E Know the co-function identities using the subtraction identities. 10F Know and explain the odd/even identities for sine & cosine 10G Know the sum and difference identities 10I Prove the co-function identities using the subtraction identities. 9.0 Students compute, by hand, the values of the trigonometric functions and the inverse trigonometric functions at various standard points. Learning Targets 10O Explain the role of the inverse trigonometric functions in solving trigonometric expressions and equations. 10.0 Students demonstrate an understanding of the addition formulas for sines and cosines and their proofs and can use those formulas to prove and/or simplify other trigonometric identities. Learning Targets 0H Use the sum and difference identities to find the exact value of trigonometric functions at a given number. 10L Verify more complex trigonometric identities using the * Learning Targets (LT) break the standard down into relevant chunks for teaching and benchmark assessments. The LT number refers to the unit i.e. 1a is Unit 1 learning target A. *CST Items: Pre Calculus CST test is a Summative Exam (complete list of standards see Unit 13) **Textbook: Enhanced with Graphing Utilities, 5/E by Sullivan and Sullivan, publisher: Prentice Hall Pre-Calculus Instructional Guide 2011-2012 Unit High Priority Standards # CST Items* # Q3 Items Supporting Medium/Low Priority Standards # CST Items* 23 Textbook Prentice Hall ** basic trigonometric identities, Pythagorean identities, cofunction identities, odd/even identities, sum & difference identities, double angle identities, half-angle identities and justify each step in the verification process. 10M Solve trigonometric equations on a restricted domain and justify each step 10N Use the periodicity of the trigonometric functions to find the general solutions of a trigonometric equation. 11.0 Students demonstrate an understanding of half-angle and double-angle formulas for sines and cosines and can use those formulas to prove and/or simplify other trigonometric identities. Learning Targets 10J Know the double angle identities 10K Know the half-angle identities 10L Verify more complex trigonometric identities using the basic trigonometric identities, Pythagorean identities, cofunction identities, odd/even identities, sum & difference identities, double angle identities, half-angle identities and justify each step in the verification process. 10M Solve trigonometric equations on a restricted domain and justify each step 10N Use the periodicity of the trigonometric functions to find the general solutions of a * Learning Targets (LT) break the standard down into relevant chunks for teaching and benchmark assessments. The LT number refers to the unit i.e. 1a is Unit 1 learning target A. *CST Items: Pre Calculus CST test is a Summative Exam (complete list of standards see Unit 13) **Textbook: Enhanced with Graphing Utilities, 5/E by Sullivan and Sullivan, publisher: Prentice Hall Pre-Calculus Instructional Guide 2011-2012 Unit High Priority Standards # CST Items* # Q3 Items Supporting Medium/Low Priority Standards # CST Items* 24 Textbook Prentice Hall ** trigonometric equation. * Learning Targets (LT) break the standard down into relevant chunks for teaching and benchmark assessments. The LT number refers to the unit i.e. 1a is Unit 1 learning target A. *CST Items: Pre Calculus CST test is a Summative Exam (complete list of standards see Unit 13) **Textbook: Enhanced with Graphing Utilities, 5/E by Sullivan and Sullivan, publisher: Prentice Hall Pre-Calculus Instructional Guide 2011-2012 Unit High Priority Standards Unit 11: Statistics and Probability This unit begins with a discussion of how most distributions encountered in statistics give a normal (“bell” curve) distribution, which corresponds to the graph of the function 2 x2 f (x) e 1 . In this graph, 2 we note that 67% of all values lie within one standard deviation of the mean, 95% of all values within two standard deviations of the mean, and 99.7% of all values within three standard deviations of the mean. Examples from standardized tests such as the SAT are given and explored. The binomial distribution and its relation to Pascal’s triangle are discussed, as well as how the binomial distribution approximates the normal distribution. The definition of the mean using sigma notan tion, x i1 n Probability and Statistics 7.0 Students compute the variance and the standard deviation of a distribution of data. Learning Targets 11C Use formulas to find variation and standard deviation of a distribution of data. # CST Items* # Q3 Items Supporting Medium/Low Priority Standards # CST Items* 25 Textbook Prentice Hall ** Probability and Statistics 4.0 Students are familiar with the standard distributions (normal, binomial, and exponential) and can use them to solve for events in problems in which the distribution belongs to those families. Learning Targets 11A Identify and explain the normal distribution. 11D Explain the binomial distribution, and its relationship to the normal distribution. 5.0 Students determine the mean and the standard deviation of a normally distributed random variable. Learning Targets 11B Explain the concept of standard deviation and connect to the concept of mean i is explored. The dif- ference between a subscript and a value is emphasized. This leads into the equation for the standard devia- n tion (x i ) 2 i1 n . These concepts are then explored through examples and problems. * Learning Targets (LT) break the standard down into relevant chunks for teaching and benchmark assessments. The LT number refers to the unit i.e. 1a is Unit 1 learning target A. *CST Items: Pre Calculus CST test is a Summative Exam (complete list of standards see Unit 13) **Textbook: Enhanced with Graphing Utilities, 5/E by Sullivan and Sullivan, publisher: Prentice Hall Pre-Calculus Instructional Guide 2011-2012 Unit Unit 12: Sequences and Series Properties of sequences and series are reviewed, with an emphasis on the algebraic derivation of the summation formulas for the arithmetic and geometric series. For the arithmetic series, mean of the terms is used to find the sum (the pairing method: pair the 1st and nth terms, the 2nd and (n 1) st terms, etc.). Geometric series and the factorization High Priority Standards Algebra 2: 20.0 Students know the binomial theorem and use it to expand binomial expressions that are raised to positive integer powers. Learning Targets 12D Explain the relationship between Pascal's triangle and the Binomial theorem. Calculus (foundational): 13.0 Students know the n 1 2 n 1 x (1 x x ... x )(1 x) definition of the definite integral by using discussed. Summation notation are Riemann sums. They is then used to simplify summation use this definition to calculations and do slightly more approximate intecomplex sums. The relationship begrals. tween Pascal’s triangle, the binomial Learning Targets coefficients, and the binomial theo12A Know the summation formulas for k^1, k^2, and k^3 rem is also discussed and the binomial theorem and Pascal’s triangle is used to expand powers of binomials. Honors Level: The derivation of the summation properties using Pascal’s theorem is discussed. This includes the “hockey stick” formula: n k n 1 k k 1. m 0 m 0 n # CST Items* # Q3 Items Supporting Medium/Low Priority Standards Algebra 2: 4.0 Students are familiar with the standard distributions (normal, binomial, and exponential) and can use them to solve for events in problems in which the distribution belongs to those families. 5.0 Students determine the mean and the standard deviation of a normally distributed random variable. 22.0 Students find the general term and the sums of arithmetic series and of both finite and infinite geometric series. # CST Items* 26 Textbook Prentice Hall ** 1 1 Learning Targets 12B Explain the derivations of the summation formulas for the arithmetic and geometric series. 12C Find the sum of an arithmetic and geometric series. 23.0 Students derive the summation formulas for arithmetic series and for both finite and infinite geometric series. * Learning Targets (LT) break the standard down into relevant chunks for teaching and benchmark assessments. The LT number refers to the unit i.e. 1a is Unit 1 learning target A. *CST Items: Pre Calculus CST test is a Summative Exam (complete list of standards see Unit 13) **Textbook: Enhanced with Graphing Utilities, 5/E by Sullivan and Sullivan, publisher: Prentice Hall Pre-Calculus Instructional Guide 2011-2012 Unit High Priority Standards Unit 13: CST Review The CST unit in Pre-Calculus is a review of essential topics from Algebra I, Geometry, and Algebra II in preparation for students to take the summative mathematics exam. The essential standards below are from the blueprint for the STAR Summative exam. Algebra 2: 1.0 Students solve equations and inequalities involving absolute value. 2.0 Students solve systems of linear equations and inequalities (in two or three variables) by substitution, with graphs, or with matrices. 3.0 Students are adept at operations on polynomials, including long division. 4.0 Students factor polynomials representing the difference of squares, perfect square trinomials, and the sum and difference of two cubes. 6.0 Students add, subtract, multiply, and divide complex numbers. 7.0 Students add, subtract, multiply, divide, reduce, and evaluate rational expressions with monomial and polynomial denominators and simplify complicated rational expressions, including those with nega- # CST Items* # Q3 Items 1 1 3 3 1 1 1 1 1 1 2 2 Supporting Medium/Low Priority Standards # CST Items* 27 Textbook Prentice Hall ** * Learning Targets (LT) break the standard down into relevant chunks for teaching and benchmark assessments. The LT number refers to the unit i.e. 1a is Unit 1 learning target A. *CST Items: Pre Calculus CST test is a Summative Exam (complete list of standards see Unit 13) **Textbook: Enhanced with Graphing Utilities, 5/E by Sullivan and Sullivan, publisher: Prentice Hall Pre-Calculus Instructional Guide 2011-2012 Unit High Priority Standards tive exponents in the denominator. 8.0 Students solve and graph quadratic equations by factoring, completing the square, or using the quadratic formula. Students apply these techniques in solving word problems. They also solve quadratic equations in the complex number system. 10.0 Students graph quadratic functions and determine the maxima, minima, and zeros of the function. 11.1 Students understand the inverse relationship between exponents and logarithms and use this relationship to solve problems involving logarithms and exponents. 12.0 Students know the laws of fractional exponents, understand exponential functions, and use these functions in problems involving exponential growth and decay. 14.0 Students understand and use the proper- # CST Items* # Q3 Items 3 3 2 2 1 1 2 2 1 1 Supporting Medium/Low Priority Standards # CST Items* 28 Textbook Prentice Hall ** * Learning Targets (LT) break the standard down into relevant chunks for teaching and benchmark assessments. The LT number refers to the unit i.e. 1a is Unit 1 learning target A. *CST Items: Pre Calculus CST test is a Summative Exam (complete list of standards see Unit 13) **Textbook: Enhanced with Graphing Utilities, 5/E by Sullivan and Sullivan, publisher: Prentice Hall Pre-Calculus Instructional Guide 2011-2012 Unit High Priority Standards ties of logarithms to simplify logarithmic numeric expressions and to identify their approximate values. 15.0 Students determine whether a specific algebraic statement involving rational expressions, radical expressions, or logarithmic or exponential functions is sometimes true, always true, or never true. 18.0 Students use fundamental counting principles to compute combinations and permutations. 19.0 Students use combinations and permutations to compute probabilities. 22.0 Students find the general term and the sums of arithmetic series and of both finite and infinite geometric series. 24.0 Students solve problems involving functional concepts, such as composition, defining the inverse function and performing arithmetic operations on functions. # CST Items* # Q3 Items 1 1 1 1 1 1 1 1 1 1 Supporting Medium/Low Priority Standards # CST Items* 29 Textbook Prentice Hall ** * Learning Targets (LT) break the standard down into relevant chunks for teaching and benchmark assessments. The LT number refers to the unit i.e. 1a is Unit 1 learning target A. *CST Items: Pre Calculus CST test is a Summative Exam (complete list of standards see Unit 13) **Textbook: Enhanced with Graphing Utilities, 5/E by Sullivan and Sullivan, publisher: Prentice Hall Pre-Calculus Instructional Guide 2011-2012 Unit High Priority Standards Geometry 3.0 Students construct and judge the validity of a logical argument and give counterexamples to disprove a statement. 4.0 Students prove basic theorems involving congruence and similarity. 5.0 Students prove that triangles are congruent or similar, and they are able to use the concept of corresponding parts of congruent triangles. 7.0 Students prove and use theorems involving the properties of parallel lines cut by a transversal, the properties of quadrilaterals, and the properties of circles. 8.0 Students know, derive, and solve problems involving the perimeter, circumference, area, volume, lateral area, and surface area of common geometric figures. 9.0 Students compute the volumes and surface areas of prisms, pyr- # CST Items* # Q3 Items 1 1 3 3 2 2 2 2 1 1 1 1 Supporting Medium/Low Priority Standards # CST Items* 30 Textbook Prentice Hall ** * Learning Targets (LT) break the standard down into relevant chunks for teaching and benchmark assessments. The LT number refers to the unit i.e. 1a is Unit 1 learning target A. *CST Items: Pre Calculus CST test is a Summative Exam (complete list of standards see Unit 13) **Textbook: Enhanced with Graphing Utilities, 5/E by Sullivan and Sullivan, publisher: Prentice Hall Pre-Calculus Instructional Guide 2011-2012 Unit High Priority Standards amids, cylinders, cones, and spheres; and students commit to memory the formulas for prisms, pyramids, and cylinders. 10.0 Students compute areas of polygons, including rectangles, scalene triangles, equilateral triangles, rhombi, parallelograms, and trapezoids. 11.0 Students determine how changes in dimensions affect the perimeter, area, and volume of common geometric figures and solids. 15.0 Students use the Pythagorean theorem to determine distance and find missing lengths of sides of right triangles. 18.0 Students know the definitions of the basic trigonometric functions defined by the angles of a right triangle. They also know and are able to use elementary relationships between them. For example, tan(x) = sin(x)/cos(x), # CST Items* # Q3 Items 1 1 1 1 2 2 2 2 Supporting Medium/Low Priority Standards # CST Items* 31 Textbook Prentice Hall ** * Learning Targets (LT) break the standard down into relevant chunks for teaching and benchmark assessments. The LT number refers to the unit i.e. 1a is Unit 1 learning target A. *CST Items: Pre Calculus CST test is a Summative Exam (complete list of standards see Unit 13) **Textbook: Enhanced with Graphing Utilities, 5/E by Sullivan and Sullivan, publisher: Prentice Hall Pre-Calculus Instructional Guide 2011-2012 Unit High Priority Standards (sin(x))2 + (cos(x)) 2 = 1. 19.0 Students use trigonometric functions to solve for an unknown length of a side of a right triangle, given an angle and a length of a side. 21.0 Students prove and solve problems regarding relationships among chords, secants, tangents, inscribed angles, and inscribed and circumscribed polygons of circles. Algebra 1: 4.0 Students simplify expressions before solving linear equations and inequalities in one variable, such as 3(2x-5) + 4(x-2) = 12. 5.0 Students solve multistep problems, including word problems, involving linear equations and linear inequalities in one variable and provide justification for each step. 6.0 Students graph a linear equation and compute the x- and y- # CST Items* # Q3 Items 1 1 2 2 1 1 3 3 2 2 Supporting Medium/Low Priority Standards # CST Items* 32 Textbook Prentice Hall ** * Learning Targets (LT) break the standard down into relevant chunks for teaching and benchmark assessments. The LT number refers to the unit i.e. 1a is Unit 1 learning target A. *CST Items: Pre Calculus CST test is a Summative Exam (complete list of standards see Unit 13) **Textbook: Enhanced with Graphing Utilities, 5/E by Sullivan and Sullivan, publisher: Prentice Hall Pre-Calculus Instructional Guide 2011-2012 Unit High Priority Standards intercepts (e.g., graph 2x + 6y = 4). They are also able to sketch the region defined by linear inequality (e.g., they sketch the region defined by 2x + 6y < 4). 7.0 Students verify that a point lies on a line, given an equation of the line. Students are able to derive linear equations by using the point-slope formula. 8.0 Students understand the concepts of parallel lines and perpendicular lines and how those slopes are related. Students are able to find the equation of a line perpendicular to a given line that passes through a given point. 10.0 Students add, subtract, multiply, and divide monomials and polynomials. Students solve multistep problems, including word problems, by using these techniques. 11.0 Students apply basic factoring techniques # CST Items* # Q3 Items 1 1 1 1 3 3 1 1 Supporting Medium/Low Priority Standards # CST Items* 33 Textbook Prentice Hall ** * Learning Targets (LT) break the standard down into relevant chunks for teaching and benchmark assessments. The LT number refers to the unit i.e. 1a is Unit 1 learning target A. *CST Items: Pre Calculus CST test is a Summative Exam (complete list of standards see Unit 13) **Textbook: Enhanced with Graphing Utilities, 5/E by Sullivan and Sullivan, publisher: Prentice Hall Pre-Calculus Instructional Guide 2011-2012 Unit High Priority Standards to second- and simple third-degree polynomials. These techniques include finding a common factor for all terms in a polynomial, recognizing the difference of two squares, and recognizing perfect squares of binomials. 12.0 Students simplify fractions with polynomials in the numerator and denominator by factoring both and reducing them to the lowest terms. 14.0 Students solve a quadratic equation by factoring or completing the square. 15.0 Students apply algebraic techniques to solve rate problems, work problems, and percent mixture problems. 20.0 Students use the quadratic formula to find the roots of a second-degree polynomial and to solve quadratic equations. 23.0 Students apply quadratic equations to physical problems, such as the motion of # CST Items* # Q3 Items 1 1 1 1 2 2 1 1 1 1 Supporting Medium/Low Priority Standards # CST Items* 34 Textbook Prentice Hall ** * Learning Targets (LT) break the standard down into relevant chunks for teaching and benchmark assessments. The LT number refers to the unit i.e. 1a is Unit 1 learning target A. *CST Items: Pre Calculus CST test is a Summative Exam (complete list of standards see Unit 13) **Textbook: Enhanced with Graphing Utilities, 5/E by Sullivan and Sullivan, publisher: Prentice Hall Pre-Calculus Instructional Guide 2011-2012 Unit High Priority Standards # CST Items* # Q3 Items Supporting Medium/Low Priority Standards # CST Items* 35 Textbook Prentice Hall ** an object under the force of gravity. * Learning Targets (LT) break the standard down into relevant chunks for teaching and benchmark assessments. The LT number refers to the unit i.e. 1a is Unit 1 learning target A. *CST Items: Pre Calculus CST test is a Summative Exam (complete list of standards see Unit 13) **Textbook: Enhanced with Graphing Utilities, 5/E by Sullivan and Sullivan, publisher: Prentice Hall Pre-Calculus Instructional Guide 2011-2012 Unit High Priority Standards Unit 14: Introduction to Vectors, Polars, Parametrics This unit begins with the introduction of a vector space. Vectors are introduced algebraically and geometrically in two and three dimensions. Vector addition and scalar multiplication are introduced. The polar coordinate system is also introduced, and connections are made between polar and rectangular coordinates. Parametric equations are then introduced, and connected to the rectangular coordinate system by the method of eliminating the parameter. Math Analysis: 1.0 Students are familiar with, and can apply, polar coordinates and vectors in the plane. In particular, they can translate between polar and rectangular coordinates and can interpret polar coordinates and vectors graphically. # CST Items* # Q3 Items Supporting Medium/Low Priority Standards # CST Items* 36 Textbook Prentice Hall ** Trigonometry: 16.0 Students represent equations given in rectangular coordinates in terms of polar coordinates. Learning Targets 14A Explain the meaning of a vector within the context of vector space 14B Write and draw vectors in two and three dimensions. 14C Perform vector addition and scalar multiplication in two dimensions, and explain their representation in R2. 14D Describe and categorize polar graphs 14E Convert between the polar coordinate system and the rectangular coordinate system. 7.0 Students demonstrate an understanding of functions and equations defined parametrically and can graph them. Learning Targets 14F Explain the parametric coordinate system, and eliminate the parameter in order to convert between the polar coordinate system and the rectangular coordinate system. * Learning Targets (LT) break the standard down into relevant chunks for teaching and benchmark assessments. The LT number refers to the unit i.e. 1a is Unit 1 learning target A. *CST Items: Pre Calculus CST test is a Summative Exam (complete list of standards see Unit 13) **Textbook: Enhanced with Graphing Utilities, 5/E by Sullivan and Sullivan, publisher: Prentice Hall Pre-Calculus Instructional Guide 2011-2012 Unit High Priority Standards # CST Items* # Q3 Items Supporting Medium/Low Priority Standards # CST Items* 37 Textbook Prentice Hall ** 14G Draw a graph of a parametric equation. Unit 15: Extension Project Honors Level: This final unit is in two parts. Individual students choose from a list of certain topics. Research is conducted on the chosen topic, and results are presented to the class. The following topics are used: proof by induction, DeMoivre’s theorem, the Chinese remainder theorem, modular arithmetic, Gaussian Elimination, introduction to group theory, and an overview of string theory. As a group, the concept of infinity is debated in the style of Kroneker and Cantor. Non-Honors Level: This final unit is in two parts. Individual students choose from a list of certain topics. Research is conducted on the chosen topic, and results are presented to the class. The following topics are used: Simpson’s paradox, Boolean algebra, chaos theory (fractals), angular and linear speed, history and mathematics of 0, history and mathematics of , and Brahmagupta theorem. Instruction Continues After CST- Q4 Math Culminating Projects * Learning Targets (LT) break the standard down into relevant chunks for teaching and benchmark assessments. The LT number refers to the unit i.e. 1a is Unit 1 learning target A. *CST Items: Pre Calculus CST test is a Summative Exam (complete list of standards see Unit 13) **Textbook: Enhanced with Graphing Utilities, 5/E by Sullivan and Sullivan, publisher: Prentice Hall