Alliance for College-Ready Public Schools



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 |# CST Items*|# Q1 Items |Supporting Medium/Low Priority Standards |# CST Items* |Textbook |

| |& Learning Targets | | |& Learning Targets | |Prentice Hall** |

|Unit 1: Mathematical Terminology & Notation |Algebra 2: | | |Algebra 2: | |A67 #13,19,27,39,41 |

|Pre-Calculus begins with a formal discussion of |15.0 Students determine whether a |1 |1 |5.0 Students demonstrate knowledge of how | | |

|terminology and notation that has been used in past |specific algebraic statement involving | | |real and complex numbers are related both | |A45 #73,79A94 #53,81 |

|mathematics courses. This begins with an |rational expressions, radical | | |arithmetically and graphically. In | | |

|introduction to the concept of a set and set theory.|expressions, or logarithmic or | | |particular, they can plot complex numbers | |A94 #53,81 |

|The history of set theory, specifically the |exponential functions is sometimes | | |as points in the plane. | | |

|influence of Georg Cantor, is investigated, and the |true, always true, or never true. | | | | |A11 #9,11 |

|importance of set theory in mathematics is |Learning Targets | | | | | |

|highlighted, namely that the language of set theory |1F Use mathematical properties to | | | | |A85-86 #57, 73,83 |

|(in its simplest form) provides the foundation for |explain how to simplify rational and | | | | | |

|studying all areas of mathematics. The basic |radical expressions | | | | |A94 #71,77,83,85 |

|notion of a set is defined, as well as the union and|1G Use the conjugate to rationalize the| | | | | |

|intersection of two sets. Commonly used sets are |numerator of a complex radical | | | | | |

|defined and given names, including the integers |expression. | | | | | |

|[pic], the rational numbers [pic], and the |1J Use the distributive property and | | | | | |

|irrational numbers [pic]. The union of the |exponent rules to simplify complex | | | | | |

|rational and irrational numbers is defined as real |rational expressions. | | | | | |

|numbers [pic]. A purely imaginary number is |25.0 Students use properties from | | | | | |

|defined, and the complex numbers are defined as |number systems to justify steps in | | | | | |

|[pic]. Properties associated with these sets are |combining and simplifying functions. | | | | | |

|discussed, and used to justify logical statements. |Learning Targets | | | | | |

|Subsets of these numbers are explored, as well as |1A Justify mathematical statements | | | | | |

|complements and closure. This naturally leads to a|using the properties of real and | | | | | |

|discussion of the size (or cardinality) of a set, as|complex numbers. | | | | | |

|well as the notions of countable and uncountable, |1B Write subsets of the Rational, Real,| | | | | |

|which sets up a discussion of intervals of the real |and Complex number systems using set | | | | | |

|number line, and interval notation. Quantifier |notation. | | | | | |

|notation is also developed and used throughout the |1C Explain relationships between sets | | | | | |

|unit, including [pic] (for all, for every), [pic] |within the Complex Number System and | | | | | |

|(there exists), [pic] (therefore), [pic] (element |evaluate the connections. | | | | | |

|of), [pic] (implies), and [pic] (if and only if). |1D Explain how the elements in the real| | | | | |

|Honors Level: With the discussion of the irrational |number system and imaginaries connect | | | | | |

|numbers, honors students prove the irrationality of |to form the standard form of a complex | | | | | |

|[pic] |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, | | | | | |

| |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) | | | | | |

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|Unit |High Priority Standards |# CST Items*|# Q1 Items |Supporting Medium/Low Priority Standards |# CST Items* |Textbook |

| |& Learning Targets | | |& Learning Targets | |Prentice Hall ** |

|Unit 2: Domain |Algebra 2: | | |Calculus (foundation only): | |Notes-definition |

|We begin this unit by studying the concept of a |1.0 Students solve equations and |1 |1 |2.0 Students demonstrate knowledge of both | | |

|function and its properties. The definition of a |inequalities involving absolute value. | | |the formal definition and the graphical | |p.67-68 #15,19,33 |

|function, and the domain and range of a function, is|Learning Targets | | |interpretation of continuity of a function.| | |

|explored, and multiple representations of functions |2J Explain the conditions for solving | | | | |p.68 #51,55p.75 #13,15p.200 #13 |

|are used. The domain or pre-image of a function is |linear absolute value problems. | | |Learning Targets | | |

|the set of allowable inputs, and the range or image |2K Solve and graph the solution sets of| | |2C Analyze the domain of a function and | |p.166 #9,13p.218 #21,27,33 |

|of a function is the set of outputs. The graphs of |absolute value inequalities, and | | |explain the similarities and differences | | |

|a variety of basic functions are studied including |explain the solution set. | | |between domains that produce different | |p.253-254 #11,21,33,35 |

|linear functions ([pic]), power functions ([pic]), |24.0 Students solve problems involving | | |images- some continuous and others not | | |

|root functions ([pic]), reciprocal functions [pic], |functional concepts, such as | | |continuous | |p.68-69 #39h,77,80 |

|and exponential & logarithmic functions ([pic] and |composition, defining the inverse | | |2D Find the domain for functions such as | | |

|[pic]). This group of functions is often referred |function and performing arithmetic | | |f(x)=Ö(polynomial, degree greater than or | |p.254 #53 |

|to as the parent functions, or the library of |operations on functions. | | |equal to 2) | | |

|functions. Shifting techniques are then applied to |Learning Targets | | |2P Explain how the domain of a piecewise | |Notes |

|graphs in the library of functions. |2A Explain the definition of a | | |function is determined, and how this | | |

| |function. | | |relates to evaluating the function. Support| |p.97 #9-16 |

|Piecewise functions are discussed in greater depth, |2B Represent a function in a | | |with an example. | | |

|featuring the greatest integer function and the |theoretical and applied context and | | | | |A58 #53,57,63 |

|absolute value function. The domain of a piecewise |justify using a counterexample | | | | | |

|function is carefully analyzed and the behavior of |2E Composite functions and then | | | | |A58 #53,57,63A86 #89,93,95 |

|the graph in each interval of the domain is |simplify; justify each step in the | | | | | |

|discussed. Shifting techniques are also applied to |simplification process and connect to | | | | |Libraryp.108 #7-18 |

|piecewise functions. Once functions are graphically|concept of domain | | | | | |

|analyzed, properties of functions are analyzed, |2G Reconstruct a function involving | | | | |p.109-110 #27,39,43,45,47,51,61p.152 #21 |

|including even and odd properties. The average rate|composition. | | | | | |

|of change of a function is defined and used to |2H Write translations as function | | | | |p.87 #21,33,39,43 |

|determine whether a function is increasing or |compositions. | | | | | |

|decreasing at a number. Given functions are also |Calculus (foundation only): | | | | |p.24-25#21,33,35,51,59p.75 #13,15 |

|evaluated for the difference quotient: [pic]. |4.0 Students demonstrate an | | | | | |

|Function operations, including compositions, are |understanding of the formal definition | | | | |p.97 #25,27 |

|then explored graphically. |of the derivative of a function at a | | | | | |

|Honors Level: When students are applying the |point and the notion of | | | | |p.97 #31,35,37 |

|difference quotient, they need to justify each step |differentiability: | | | | | |

|mathematically using a property. |Learning Targets | | | | |p.68-69 #39h,77,80 |

| |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. | | | | | |

| |2Q Illustrate a piecewise function | | | | | |

| |involving shifting techniques, the | | | | | |

| |greatest integer function, and the | | | | | |

| |library of functions | | | | | |

| | | | | | | |

|Unit |High Priority Standards |# CST Items* |# Q1 Items |Supporting Medium/Low Priority Standards |# CST Items* |Textbook |

| | | | | | |Prentice Hall ** |

|Unit 3: Introduction to Mathematical Modeling |Algebra 2: | | |Calculus (foundation only): | |p.42-43 #119,120p. 134 #37,38A75-77 |

|After formally defining the number systems and |8.0 Students solve and graph quadratic | | |2.0 Students demonstrate knowledge of both | |#7-53A86-87 #111- 119 |

|establishing the foundation for functions, it is |equations by factoring, completing the |3 |3 |the formal definition and the graphical | | |

|then natural to study how the numbers and properties|square, or using the quadratic formula.| | |interpretation of continuity of a function.| |p.718-720 #55-68,71-81p.734 #77-88p.788 |

|and concept of input and output discussed in the |Students apply these techniques in | | | | |#57-61, p.794-795 #19-31 |

|previous units are used in a real world context. |solving word problems. They also solve | | |14.0 Students apply the definition of the | | |

|The focus of this unit is to introduce the concept |quadratic equations in the complex | | |integral to model problems in physics, | |p.40-41 |

|of a mathematical model, which is studied throughout|number system. | | |economics, and so forth, obtaining results | |#11,17,23,33,37,45,59,65,71,77,91p.132 |

|this course. Mathematical models are created using |Learning Targets | | |in terms of integrals. | |#13p.717 #19,25,29 |

|previous knowledge of a variety of different types |3E Prove the quadratic formula using | | |Learning Targets | | |

|of mathematical concepts, with a focus on linear |completing the square and explain the | | |3F Design and use a model involving a | |p.717 #35,36,39,40 |

|systems. The discussion of modeling begins with |connection to domain and solutions | | |quadratic function | | |

|linear models in two variables. Linear equations |Linear Algebra: | | |3H Apply a quadratic function model | |Notes, Handout: Derive |

|and inequalities are used to model real life |6.0 Students demonstrate an | | |(equations and inequalities) to projectile | | |

|situations and contrasted to models involving |understanding that linear systems are | | |motion and explain the difference between | |p.152 #29,35,39,51 (10 steps)p.167 #33,34 |

|absolute value equations and inequalities. Linear |inconsistent (have no solutions), have | | |solutions found theoretically versus within| | |

|systems in two and three variables are categorized |exactly one solution, or have | | |an applied context (includes inequalities) | |A59 #81-86 |

|by their solution sets, and interpreted graphically.|infinitely many solutions. | | |17.0 Students compute, by hand, the | | |

|Certain types of non-linear systems are discussed |Learning Targets | | |integrals of a wide variety of functions by| |p.161 #11,12 |

|(i.e. involving [pic] and [pic]), and the u-v method|3B Design a model using a system of | | |using techniques of integration, such as | | |

|of substitution is used to solve such systems. |linear equations & inequalities | | |substitution, integration by parts, and | |p.117 #25,26(a-c)p.160 #7,8(a-b)p.778-779 |

|Another type of non-linear mathematical model is |(mixture, linear programming, piecewise| | |trigonometric substitution. They can also | |#85,87A76 #31,32 |

|introduced and analyzed, namely a quadratic model. |functions), justify each step within | | |combine these techniques when appropriate. | | |

|The quadratic formula is proven using completing the|the analysis and explain the validity | | |Learning Targets | |Notes |

|square, and used to solve a variety of quadratic |of the solution. | | |3D Use u-v substitution to solve certain | | |

|equations and quadratic type equations used in |3C Categorize and connect the solutions| | |non-linear systems of equations and justify| |p.117 #25,26(d)p.160-162 #3,7c,8c, |

|mathematical models, including projectile motion. |to mathematical linear models to types | | |each step (i.e. equations with 1/x). | |9,10,11,13,17,27p.169 #39 |

|Solutions found theoretically are analyzed within an|of linear systems and explain the | | |3G Solve quadratic type problems, justify | | |

|applied context for quadratic models. Next models |importance of slope | | |each step, and explain the purpose of u- | |Notes |

|for “fence”, “box” , and “garden” problems and |Calculus (foundation only): | | |substitution. | | |

|designed and used to establish the foundation for |11.0 Students use differentiation to | | | | | |

|optimization problems in Calculus. Models of direct,|solve optimization (maximum-minimum | | | | | |

|indirect, and joint variation are discussed, and |problems) in a variety of pure and | | | | | |

|used to solve problems related to physics. |applied contexts. | | | | | |

|Honors Level: Students solve applied minimum and |Learning Targets | | | | | |

|maximum problems, focusing on the box, garden, and |3A Explain how the four step process | | | | | |

|fence problems |(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. | | | | | |

|Unit |High Priority Standards |# CST Items*|# Q1 Items |Supporting Medium/Low Priority Standards |# CST Items* |Textbook |

| | | | | | |Prentice Hall ** |

|Unit 4: The General Equation of the Second Degree in|Math Analysis: | | |Math Analysis: | |notes |

|Two Variables |5.0 Students are familiar with conic | | |5.2 Students can take a geometric | | |

|The history of the conic sections begins in ancient |sections, both analytically and | | |description of a conic section - for | |notes |

|Greece with the mathematician Apollonius. His role |geometrically: | | |example, the locus of points whose sum of | | |

|in mathematics and the study of conic sections is |Learning Targets | | |its distances from (1, 0) and (-1, 0) is 6 | | |

|investigated. The general quadratic equation of |3A Know the general equation of the | | |- and derive a quadratic equation | | |

|second degree in two variables is then defined as |second degree in two variables. | | |representing it. | | |

|[pic]. The coefficient of [pic] represents the |3B Describe the conditions of the | | |Learning Targets | | |

|rotation of the graph of the equation, which cannot |coefficients of the general | | |3K Given the focus and directrix of a | | |

|be studied without trigonometry. Therefore, by |second-degree equation that produce | | |parabola find the standard form | | |

|setting[pic], the following conditions are developed|each conic section (circle, parabola, | | |3L Write equations for conic sections given| | |

|for [pic]and [pic] to determine which conic sections|ellipse, hyperbola). | | |key information (foci, vertices, etc.) | | |

|are produced from the equation: parabolas [[pic] or |3C Describe how each conic section is | | |3M Write equations for conic sections from | | |

|[pic], not both], ellipses [[pic]], circles [[pic]],|obtained from the intersection of a | | |a graph. | | |

|and hyperbolas [[pic]]. Using completing the |plane and a cone, including the | | | | | |

|square, each type of conic section is written in its|degenerate forms. | | | | | |

|standard form, and key information about the graph |3D Describe how completing the square | | | | | |

|is identified. Using this information, graphs of |is involved in the study of conic | | | | | |

|each conic section are drawn from the standard forms|sections. | | | | | |

|and equations in standard form are written given |3N Design and use a mathematical model | | | | | |

|certain information. Continuing the theme of |involving conic sections. | | | | | |

|mathematical modeling, conic sections are used to |5.1 Students can take a quadratic | | | | | |

|solve problems in a real world context, including |equation in two variables; put it in | | | | | |

|those about physics and astronomy. |standard form by completing the square | | | | | |

|Honors Level: While the discussing the history of |and using rotations and translations, | | | | | |

|the conic sections, honors students investigate the |if necessary; determine what type of | | | | | |

|intersection of a plane and a double-napped cone, |conic section the equation represents; | | | | | |

|including the degenerate forms. |and determine its geometric components | | | | | |

| |(foci, asymptotes, and so forth). | | | | | |

| |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 | | | | | |

|Unit |High Priority Standards |# CST Items*|# Q2 Items |Supporting Medium/Low Priority Standards |# CST Items* |Textbook |

| |& Learning Targets | | |& Learning Targets | |Prentice Hall ** |

|Unit 5: Polynomial and Rational Functions |Math Analysis: | | |Math Analysis: | | |

|The focus of the next few units is to use the |4.0 Students know the statement of, and| | |6.0 Students find the roots and poles of a | | |

|relationships developed in the previous unit between|can apply, the fundamental theorem of | | |rational function and can graph the | | |

|functions and graphs, to analyze specific types of |algebra. | | |function and locate its asymptotes. | | |

|functions, and then to compare and contrast the |Learning Targets | | |Algebra 2: | | |

|different types of functions studied. This |5J Explain how the division algorithm, | | |3.0 Students are adept at operations on | | |

|discussion begins with polynomial functions. |the remainder theorem, and the factor | | |polynomials, including long division. | | |

|Polynomial functions in standard form are defined, |theorem are related. | | |Learning Targets |1 | |

|and the graphs of polynomial functions are studied, |5L Explain how to find the equation of | | |5H Use the division algorithm, remainder | | |

|as well as the domain (pre-image) and range (image).|a polynomial if given the degree of the| | |theorem, and factor theorem. | | |

|Graphs of polynomials and non-polynomials are |polynomial and one of the conjugate | | |4.0 Students factor polynomials | | |

|compared and contrasted in order to develop an |pairs: , as well as information about | | |representing the difference of squares, | | |

|intuitive understanding of the smooth and continuous|multiplicity. | | |perfect square trinomials, and the sum and | | |

|nature of the graph of polynomial functions. The |5M Describe the behavior of the graph | | |difference of two cubes. | | |

|end behavior of graphs of polynomial functions is |of a polynomial function near a root of| | |Learning Targets |1 | |

|then explored. |multiplicity. | | |G Describe the relationship between roots | | |

| |5N Classify the real roots of a | | |and factors. | | |

|The focus of the unit shifts from the graphs of |polynomial function using Descartes | | |5P Compare and contrast linear factors and | | |

|polynomial functions to the roots/zeros/solutions of|Rule of Signs and FTA. | | |quadratic factors. | | |

|the function. The roots/zeros/solutions of a |5O Find the roots of a polynomial | | |7.0 Students add, subtract, multiply, | | |

|polynomial function are defined, and the values of |function using the Rational Roots | | |divide, reduce, and evaluate rational | | |

|any real roots/zeros/solutions are estimated from |Theorem | | |expressions with monomial and polynomial | | |

|the graph of the function. Connections are then |8.0 Students are familiar with the | | |denominators and simplify complicated | | |

|made between the Binomial Theorem and |notion of the limit of a sequence and | | |rational expressions, including those with | | |

|roots/zeros/solutions of multiplicity, and this |the limit of a function as the | | |negative exponents in the denominator. |2 | |

|information is used to review the binomial expansion|independent variable approaches a | | |Learning Targets | | |

|to a certain whole number power. The remainder and |number or infinity. They determine | | |5K Demonstrate how to use long division to | | |

|factor theorems are reviewed, and used to find |whether certain sequences converge or | | |find the roots of a polynomial function. | | |

|roots/zeros/solutions of polynomial functions given |diverge. | | | | | |

|certain information. The connection between a |Learning Targets | | | | | |

|root/zero/solution of a polynomial function and a |5U Explain how rational functions | | | | | |

|factor of the function is emphasized. The |behave close to vertical asymptotes and| | | | | |

|Fundamental Theorem of Algebra is reviewed, as well |find the vertical asymptotes if they | | | | | |

|as the Complex Conjugate Theorem. Descartes Rule of|exist | | | | | |

|Signs is used to classify the roots/zeros/solutions |5V Explain how rational functions | | | | | |

|of a polynomial function and the Rational Roots |behave close to horizontal asymptotes | | | | | |

|Theorem is used to find the rational |and find the horizontal asymptotes if | | | | | |

|roots/zeros/solutions of a polynomial function. |they exist using the limit of the | | | | | |

| |function as x approaches infinity | | | | | |

|The focus of the unit shifts to the study of |Calculus (foundation only): | | | | | |

|rational functions, with an emphasis on the domain |9.0 Students use differentiation to | | | | | |

|(pre-image) of a rational function. Asymptotic |sketch, by hand, graphs of functions. | | | | | |

|behavior is reviewed, and the concept of a limit is |They can identify maxima, minima, | | | | | |

|discussed using limit notation. Conditions for the |inflection points, and intervals in | | | | | |

|existence of different asymptotes of a rational |which the function is increasing and | | | | | |

|function are discussed. The process of long |decreasing. | | | | | |

|division is introduced in order to find the oblique |Learning Targets | | | | | |

|asymptotes of a rational function. Rational |5A Describe a polynomial function in | | | | | |

|functions are analyzed (find domain/range, |standard form and classify special | | | | | |

|intercepts, asymptotes, symmetry) and graphed. Both|cases of polynomials | | | | | |

|polynomial and rational functions are used to |5B Apply the Fundamental Theorem of | | | | | |

|develop mathematical models representing a real |Algebra. | | | | | |

|world situation. |5C Determine the end behavior of a | | | | | |

|Honors Level: Honors students use limit notation to |graph a polynomial function. | | | | | |

|describe the end behavior of the graphs of |5D Explain how symmetry affects the | | | | | |

|polynomial functions.. |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 non-real roots | | | | | |

| |5S Explain the difference between a | | | | | |

| |polynomial function and rational | | | | | |

| |function | | | | | |

| |5X Graph rational functions. | | | | | |

| |5Y Defend the purpose for analyzing | | | | | |

| |functions. | | | | | |

|Unit |High Priority Standards |# CST Items*|# Q2 Items |Supporting Medium/Low Priority Standards |# CST Items* |Textbook |

| | | | | | |Prentice Hall ** |

|Unit 6: Functions, Graphs, and their Inverses |Algebra 2: | | |Algebra 2: | | |

|Composite functions are re-visited within the |24.0 Students solve problems involving |1 |1 |15.0 Students determine whether a specific |1 | |

|context of solving radical equations, rational |functional concepts, such as | | |algebraic statement involving rational | | |

|equations and defining an inverse. Namely, f and g |composition, defining the inverse | | |expressions, radical expressions, or | | |

|are inverses if and only if [pic] and [pic]. The |function and performing arithmetic | | |logarithmic or exponential functions is | | |

|concept of a one to one function is then explored |operations on functions. | | |sometimes true, always true, or never true.| | |

|algebraically and graphically (via symmetry), and an|Learning Targets | | | | | |

|alternative definition is developed for an inverse |6C Explain how to justify that a | | |Learning Targets | | |

|function: [pic] and [pic] are inverses if and only |function is one-to-one. | | |6A Simplify radical expressions and solve | | |

|if [pic] is a one-to-one function and [pic]. The |6D Explain how composition is used to | | |radical equations and explain extraneous | | |

|method for finding inverses by switching variables |determine if two functions are inverses| | |solutions | | |

|is used to find inverse functions. Finally, the | | | |6B Solve rational equations, justify each | | |

|action of taking an inverse is linked with the |6E Describe the connection between | | |step, and connect to analyzing rational | | |

|concept of reflection about the line[pic] and |symmetry and one-to-one functions. | | |functions | | |

|connected to odd functions |6F Find inverse functions for algebraic| | | | | |

|Honors Level: Honors students analyze the proof of |and transcendental functions | | | | | |

|solving a depressed general cubic equation, to |Trigonometry (foundational): | | | | | |

|include the history behind the solution to the |8.0 Students know the definitions of | | | | | |

|depressed cubic. |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. | | | | | |

|Unit |High Priority Standards |# CST Items* |# Q2 Items |Supporting Medium/Low Priority Standards |# CST Items* |Textbook |

| | | | | | |Prentice Hall ** |

|Unit 7: The Relationship Between Exponential and |Algebra 2: | | |Calculus (foundational): | | |

|Logarithmic Functions |11.1 Students understand the inverse |1 |1 |4.2 Students demonstrate an understanding | | |

|The focus of the previous unit was the study of |relationship between exponents and | | |of the interpretation of the derivative as | | |

|certain algebraic functions. Algebraic functions |logarithms and use this relationship to| | |an instantaneous rate of change. Students | | |

|are functions that can be produced using basic |solve problems involving logarithms and| | |can use derivatives to solve a variety of | | |

|operations. The focus of this unit, on the other |exponents. | | |problems from physics, chemistry, | | |

|hand, is to study certain type of non-algebraic, or |Learning Targets | | |economics, and so forth that involve the | | |

|transcendental, functions. The discussion begins |7A Explain the relationship between | | |rate of change of a function. | | |

|with analyzing exponential and logarithmic functions|exponents and logarithms. | | |Learning Targets | | |

|and their graphs. These functions are analyzed and |7C Explain the definition of a | | |7H Graph and analyze logistics growth | | |

|graphed in base [pic] and arbitrary [pic], and key |logarithm and use to simplify and | | |functions. | | |

|information about these functions is identified. |solve. | | |7J Solve mathematical modeling problems | | |

|This analysis includes the study of logistics growth|11.2 Students judge the validity of an | | |using exponential functions involving | | |

|functions. The unit begins with an emphasis on the |argument according to whether the | | |compound interest, growth and decay. and | | |

|inverse relationship between exponential and |properties of real numbers, exponents, | | |logarithmic functions to include growth and| | |

|logarithmic functions. Exponential equations and |and logarithms have been applied | | |decay and justify each step within the | | |

|functions are analyzed both algebraically and |correctly at each step. | | |analysis. | | |

|graphically, followed by an analysis of logarithmic |Learning Targets | | |7K Solve mathematical modeling problems | | |

|equations and functions. The properties of |7I Solve exponential and logarithmic | | |using logarithmic functions | | |

|logarithms are proven and used to expand or condense|equations, justifying each step. | | |9.0 Students use differentiation to sketch,| | |

|expressions containing logarithms. Methods of |12.0 Students know the laws of | | |by hand, graphs of functions. They can | | |

|estimating the value of a logarithm are discussed, |fractional exponents, understand | | |identify maxima, minima, inflection points,| | |

|including mental estimation and approximating using |exponential functions, and use these | | |and intervals in which the function is | | |

|the change of base formulas. The focus of the unit |functions in problems involving | | |increasing and decreasing. | | |

|then shifts to explore mathematical modeling and |exponential growth and decay. | | |Learning Targets | | |

|equations involving exponentials and logarithms. The|14.0 Students understand and use the | | |7B Analyze an exponential function, | | |

|methods of solving certain [accessible] equations |properties of logarithms to simplify | | |including finding the domain, range, | | |

|that involve exponentials and logarithms are |logarithmic numeric expressions and to | | |horizontal asymptote, intercepts, and | | |

|explored, and are applied to solve problems |identify their approximate values. |2 |2 |inverse- sketch | | |

|involving exponential growth & decay, logistic |Learning Targets | | |7G Analyze a logarithmic function, | | |

|growth, and logarithmic models. |7D Prove the multiplication and | | |including finding the domain, range, | | |

|Honors Level: The “rule of 72” is analyzed in terms |division property of logarithms. | | |vertical asymptote, intercepts, and | | |

|of logarithmic and exponential functions. |7E Expand and condense logarithmic | | |inverse- sketch | | |

|Furthermore, honors students develop more complex |expressions using properties of | | | | | |

|mathematical models involving growth and decay, |logarithms. | | | | | |

|including logistic growth. | | | | | | |

| | |1 |1 | | | |

|Unit |High Priority Standards |# CST Items* |# Q2 Items |Supporting Medium/Low Priority Standards |# CST Items* |Textbook |

| | | | | | |Prentice Hall ** |

|Unit 8: Applied Trigonometry |Trigonometry: | | |Trigonometry: | | |

|The study of non-algebraic functions continues as |13.0 Students know the law of sines and| | |12.0 Students use trigonometry to determine| | |

|the trigonometric functions and their properties are|the law of cosines and apply those laws| | |unknown sides or angles in right triangles.| | |

|developed. The trigonometric ratios of a right |to solve problems. | | | | | |

|triangle, learned in previous courses, are reviewed |Learning Targets | | |Learning Targets | | |

|and used to solve right triangles for all sides and |8C Derive the Law of Sines and the | | |8A Define the six trigonometric ratios of a| | |

|angles. Applications of right triangle trigonometry|different forms of the Law of Cosines. | | |right triangle, and solve a right triangle | | |

|are studied, including problems related to indirect |8D Solve triangles using the Law of | | |for all sides and angles | | |

|measurement. |Sines | | |8D Solve triangles using the Law of Sines | | |

| |8E Explain why the SSA case yields more| | |8G Solve triangles using the Law of Cosines| | |

|Up to this point, the focus of the study of |than one triangle. (challenge) | | | | | |

|trigonometry has been strictly related to right |8F Describe conditions when an SSA case| | |14.0 Students determine the area of a | | |

|triangle trigonometry. Concepts of right triangle |yields two solutions, one solution, and| | |triangle, given one angle and the two | | |

|trigonometry are the foundation for studying the |no solutions. (challenge) | | |adjacent sides. | | |

|trigonometry of acute or obtuse triangles. The Law |8G Solve triangles using the Law of | | |Learning Targets | | |

|of Sines and Law of Cosines are introduced and |Cosines | | |8J Find the area of any triangle | | |

|proven in order to establish trigonometric |19.0 Students are adept at using | | |8K Write a formal proof of Heron's formula | | |

|relationships of these triangles. Any triangle |trigonometry in a variety of | | |using the Law of Cosines. (challenge) | | |

|(acute, right, or obtuse) is solved using these laws|applications and word problems. | | |8L Prove Heron's formula for finding the | | |

|and the inverse trigonometric functions, and the |Learning Targets | | |area of a triangle. (challenge) | | |

|different conditions for which each law is used are |8B Use right triangle trigonometry to | | | | | |

|explored. Applications of the Law of Sines and Law |solve problems of indirect measurement.| | | | | |

|of Cosines are explored, including problems related |8H Explain the difference between and | | | | | |

|to navigation and finding area. |angle of depression and angle of | | | | | |

|Honor’s Level: While discussing the Law of Sines, |elevation. | | | | | |

|honors students explore the SSA ambiguous case for |8I Design and solve problems related to| | | | | |

|the Law of Sines and the reasons why two there are |navigation using right triangle | | | | | |

|two cases. Furthermore, honors students study |trigonometry, the Law of Sines and the | | | | | |

|formulas for the area of a triangle, and prove |Law of Cosines and explain the analysis| | | | | |

|Heron’s formula using either Law of Sines or Law of |leading to the solutions | | | | | |

|Cosines. | | | | | | |

|Unit |High Priority Standards |# CST Items* |# Q3 Items |Supporting Medium/Low Priority Standards |# CST Items* |Textbook |

| | | | | | |Prentice Hall ** |

|Unit 9: Analyzing Trigonometric Functions |Trigonometry | | |Trigonometry | | |

|This unit begins with a discussion of how angles are|4.0 Students graph functions of the | | |1.0 Students understand the notion of angle| | |

|measured. Radian measure is defined in terms of arc|form f(t) = A sin (Bt + C) or f(t) = A | | |and how to measure it, in both degrees and | | |

|length on the unit circle. Trigonometric functions |cos (Bt + C) and interpret A, B, and C | | |radians. They can convert between degrees | | |

|of angles are defined in terms of the unit circle. |in terms of amplitude, frequency, | | |and radians. | | |

|This definition is used to determine which quadrants|period, and phase shift. | | |Learning Targets | | |

|the trigonometric functions are positive and |Learning Targets | | |9A Explain how to find arc length, and how | | |

|negative. The concept of a reference angle is |9J Explain the relationship between the| | |this concept is used in the definition of | | |

|explored and references angles are then used to find|graph of a sine and cosine function, | | |radian measure. | | |

|the value of the trigonometric functions of any |and the unit circle. | | |9B Identify angles in degree and radian | | |

|angle. |Learning Targets | | |measure, and convert between the measures. | | |

| |9K Use transformations to graph | | |9C Find exact values of trig functions | | |

|The unit circle is then discussed in more detail. |functions of the form or , and | | |2.0 Students know the definition of sine | | |

|Terminal points on a unit circle are defined, as |identify key information about the | | |and cosine as y-and x-coordinates of points| | |

|well as reference numbers. The trigonometric |function (i.e. period, amplitude, phase| | |on the unit circle and are familiar with | | |

|functions of a real number are then defined, using |shift, etc.). | | |the graphs of the sine and cosine | | |

|terminal points of the unit circle. Comparisons are|5.0 Students know the definitions of | | |functions. | | |

|made between the concept of the trigonometric |the tangent and cotangent functions and| | |Learning Targets | | |

|functions as functions of an angle, and functions of|can graph them. | | |9D Define a reference angle and describe | | |

|a real number. Special values of the trigonometric |Learning Targets | | |the reference angle in each quadrant. | | |

|functions are derived. The domain and range of the |9L Graph the tangent, cotangent, | | |9E Describe a method of finding reference | | |

|trigonometric functions are discussed and the graphs|secant, cosecant functions, and | | |angles. | | |

|of the sine and cosine are built. Connections |identify the domain, range, and period | | |9F Calculate the exact trig function values| | |

|between the sine and cosine graphs and the unit |for each. | | |and explain the purpose of the angle | | |

|circle are illustrated, including properties of |8.0 Students know the definitions of | | |9G Use the unit circle to evaluate trig | | |

|periodicity. Transformations are applied to the |the inverse trigonometric functions and| | |expressions and explain the process | | |

|graphs of the sine and cosine, and the general forms|can graph the functions. | | |19.0 Students are adept at using | | |

|of these functions are built in the following order:|Learning Targets | | |trigonometry in a variety of applications | | |

|[pic], [pic], [pic], and [pic]. The same order is |9M Compare the domain and range of the | | |and word problems. | | |

|used for the cosine function. Other graphs of |trigonometric functions and their | | |Learning Targets | | |

|trigonometric functions are studied including the |inverses and explain the relationship | | |9N Describe simple harmonic motion and | | |

|tangent, the reciprocal functions, and the inverse |between the two. | | |identify the amplitude, frequency, and | | |

|functions. The difference between reciprocal |9.0 Students compute, by hand, the | | |period of harmonic motion model, and | | |

|functions and inverse functions is emphasized. The |values of the trigonometric functions | | |explain their real world significance. | | |

|domain and range of trigonometric functions and |and the inverse trigonometric functions| | |9O Describe damped harmonic motion verbally| | |

|their inverses are also compared. |at various standard points. | | |and algebraically. | | |

| |Learning Targets | | |9P Compare and contrast simple and damped | | |

|Trigonometric functions are then used as |Use the concept of inverse functions | | |harmonic motion. | | |

|mathematical models, specifically modeling harmonic |and the unit circle to find exact | | | | | |

|motion. The general form of the sine and cosine is |values | | | | | |

|applied to simple harmonic motion, and used to find |9I Explain how to use the unit circle | | | | | |

|amplitude, frequency, and period. These components |to evaluate trig functions as well as | | | | | |

|of the graph of a harmonic motion model are all |find locations of ordered pairs (detail| | | | | |

|interpreted in a real world context, and used to |how to “work” all aspects of the unit | | | | | |

|solve problems within that context. |circle). | | | | | |

|Honors Level: Honors students extend the study of | | | | | | |

|harmonic motion, to include damped harmonic motion. | | | | | | |

|Unit |High Priority Standards |# CST Items*|# Q3 Items |Supporting Medium/Low Priority Standards |# CST Items* |Textbook |

| | | | | | |Prentice Hall ** |

|Unit 10: Trigonometric Equations |Trigonometry: | | |Trigonometry: | | |

|The trigonometric functions have now been study in |3.1 Students prove that this identity | | |3.0 Students know the identity cos2 (x) + | | |

|depth, as well as their application. The focus of |is equivalent to the Pythagorean | | |sin2 (x) = 1: | | |

|this unit is on trigonometric equations. The |theorem (i.e., students can prove this | | |10C Know the Pythagorean identities | | |

|discussion begins with a review of basic identities |identity by using the Pythagorean | | | | | |

|(e.g. [pic]). The fundamental trigonometric |theorem and, conversely, they can prove| | | | | |

|identities are introduced and illustrated. These |the Pythagorean theorem as a | | | | | |

|include the reciprocal identities, the Pythagorean |consequence of this identity). | | | | | |

|identities, the even and odd identities, and the |Learning Targets | | | | | |

|cofunction identities. Basic trigonometric |10B Derive the Pythagorean Trig. | | | | | |

|identities are used to simplify trigonometric |identities. | | | | | |

|expressions, and different methods of simplifying |10L Verify more complex trigonometric | | | | | |

|these expressions are explored. |identities using the basic | | | | | |

| |trigonometric identities, Pythagorean | | | | | |

|The basic trigonometric identities are then used to |identities, co-function identities, | | | | | |

|prove more complex trigonometric identities. A |odd/even identities, sum & difference | | | | | |

|variety of methods of proving trigonometric |identities, double angle identities, | | | | | |

|identities are used. More advanced identities are |half-angle identities and justify each | | | | | |

|introduced, including the following: addition and |step in the verification process. | | | | | |

|subtraction identities, double angle and half angle |3.2 Students prove other trigonometric | | | | | |

|identities, and product and sum identities. All of |identities and simplify others by using| | | | | |

|these are used to prove more complex trigonometric |the identity cos2 (x) + sin2 (x) = 1. | | | | | |

|identities. |For example, students use this identity| | | | | |

| |to prove that sec2 (x) = tan2 (x) + 1. | | | | | |

|The focus of the unit shifts to a study of solving |Learning Targets | | | | | |

|trigonometric equations. The inverse trigonometric |10A Know the basic trigonometric | | | | | |

|functions are used to solve basic trigonometric |identities, including the reciprocal | | | | | |

|equations on a restricted domain (e.g. [pic] on |identities | | | | | |

|[pic]). Once this domain restriction is removed, |10D Simplify trigonometric expressions | | | | | |

|the periodic nature of the solution sets of |using the basic trigonometric | | | | | |

|trigonometric equations is explored. Connections |identities. | | | | | |

|are made between trigonometric equations, the unit |10E Know the co-function identities | | | | | |

|circle, and graphs of trigonometric functions. |using the subtraction identities. | | | | | |

|Various methods of solving trigonometric equations |10F Know and explain the odd/even | | | | | |

|are studied, and used to solve mathematical models |identities for sine & cosine | | | | | |

|involving the trigonometric functions. |10G Know the sum and difference | | | | | |

|Honors Level: While solving trigonometric equations,|identities | | | | | |

|honors students manipulate the known trigonometric |10I Prove the co-function identities | | | | | |

|identities before finding the solution set. Honors |using the subtraction identities. | | | | | |

|students also prove the multiple angle identities, |9.0 Students compute, by hand, the | | | | | |

|and the addition and subtraction identities. |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 basic | | | | | |

| |trigonometric identities, Pythagorean | | | | | |

| |identities, co-function 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, co-function 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. | | | | | |

|Unit |High Priority Standards |# CST Items* |# Q3 Items |Supporting Medium/Low Priority Standards |# CST Items*|Textbook |

| | | | | | |Prentice Hall ** |

|Unit 11: Statistics and Probability |Probability and Statistics | | |Probability and Statistics | | |

|This unit begins with a discussion of how most |7.0 Students compute the variance and | | |4.0 Students are familiar with the standard| | |

|distributions encountered in statistics give a |the standard deviation of a | | |distributions (normal, binomial, and | | |

|normal (“bell” curve) distribution, which |distribution of data. | | |exponential) and can use them to solve for | | |

|corresponds to the graph of the function [pic]. In |Learning Targets | | |events in problems in which the | | |

|this graph, we note that 67% of all values lie |11C Use formulas to find variation and | | |distribution belongs to those families. | | |

|within one standard deviation of the mean, 95% of |standard deviation of a distribution of| | |Learning Targets | | |

|all values within two standard deviations of the |data. | | |11A Identify and explain the normal | | |

|mean, and 99.7% of all values within three standard | | | |distribution. | | |

|deviations of the mean. Examples from standardized | | | |11D Explain the binomial distribution, and | | |

|tests such as the SAT are given and explored. The | | | |its relationship to the normal | | |

|binomial distribution and its relation to Pascal’s | | | |distribution. | | |

|triangle are discussed, as well as how the binomial | | | |5.0 Students determine the mean and the | | |

|distribution approximates the normal distribution. | | | |standard deviation of a normally | | |

|The definition of the mean using sigma notation, | | | |distributed random variable. | | |

|[pic] is explored. The difference between a | | | |Learning Targets | | |

|subscript and a value is emphasized. This leads | | | |11B Explain the concept of standard | | |

|into the equation for the standard deviation [pic]. | | | |deviation and connect to the concept of | | |

|These concepts are then explored through examples | | | |mean | | |

|and problems. | | | | | | |

|Unit |High Priority Standards |# CST Items* |# Q3 Items |Supporting Medium/Low Priority Standards |# CST Items* |Textbook |

| | | | | | |Prentice Hall ** |

|Unit 12: Sequences and Series |Algebra 2: | | |Algebra 2: | | |

|Properties of sequences and series are reviewed, |20.0 Students know the binomial theorem| | |4.0 Students are familiar with the standard|1 | |

|with an emphasis on the algebraic derivation of the |and use it to expand binomial | | |distributions (normal, binomial, and | | |

|summation formulas for the arithmetic and geometric |expressions that are raised to positive| | |exponential) and can use them to solve for | | |

|series. For the arithmetic series, mean of the |integer powers. | | |events in problems in which the | | |

|terms is used to find the sum (the pairing method: |Learning Targets | | |distribution belongs to those families. | | |

|pair the 1st and nth terms, the 2nd and [pic]st |12D Explain the relationship between | | |5.0 Students determine the mean and the | | |

|terms, etc.). Geometric series and the |Pascal's triangle and the Binomial | | |standard deviation of a normally | | |

|factorization [pic] are discussed. Summation |theorem. | | |distributed random variable. | | |

|notation is then used to simplify summation |Calculus (foundational): | | |22.0 Students find the general term and the| | |

|calculations and do slightly more complex sums. The|13.0 Students know the definition of | | |sums of arithmetic series and of both | | |

|relationship between Pascal’s triangle, the binomial|the definite integral by using Riemann | | |finite and infinite geometric series. | | |

|coefficients, and the binomial theorem is also |sums. They use this definition to | | |Learning Targets | | |

|discussed and the binomial theorem and Pascal’s |approximate integrals. | | |12B Explain the derivations of the | | |

|triangle is used to expand powers of binomials. |Learning Targets | | |summation formulas for the arithmetic and | | |

|Honors Level: The derivation of the summation |12A Know the summation formulas for | | |geometric series. |1 | |

|properties using Pascal’s theorem is discussed. |k^1, k^2, and k^3 | | |12C Find the sum of an arithmetic and | | |

|This includes the “hockey stick” formula: [pic]. | | | |geometric series. | | |

| | | | |23.0 Students derive the summation formulas| | |

| | | | |for arithmetic series and for both finite | | |

| | | | |and infinite geometric series. | | |

|Unit |High Priority Standards |# CST Items*|# Q3 Items |Supporting Medium/Low Priority Standards |# CST Items* |Textbook |

| | | | | | |Prentice Hall ** |

|Unit 13: CST Review |Algebra 2: | | | | | |

|The CST unit in Pre-Calculus is a review of |1.0 Students solve equations and |1 |1 | | | |

|essential topics from Algebra I, Geometry, and |inequalities involving absolute value. | | | | | |

|Algebra II in preparation for students to take the |2.0 Students solve systems of linear | | | | | |

|summative mathematics exam. The essential standards|equations and inequalities (in two or | | | | | |

|below are from the blueprint for the STAR Summative |three variables) by substitution, with |3 |3 | | | |

|exam. |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. |1 |1 | | | |

| |6.0 Students add, subtract, multiply, | | | | | |

| |and divide complex numbers. | | | | | |

| |7.0 Students add, subtract, multiply, | | | | | |

| |divide, reduce, and evaluate rational |1 |1 | | | |

| |expressions with monomial and | | | | | |

| |polynomial denominators and simplify | | | | | |

| |complicated rational expressions, | | | | | |

| |including those with negative exponents| | | | | |

| |in the denominator. | | | | | |

| |8.0 Students solve and graph quadratic | | | | | |

| |equations by factoring, completing the |1 |1 | | | |

| |square, or using the quadratic formula.| | | | | |

| |Students apply these techniques in | | | | | |

| |solving word problems. They also solve | | | | | |

| |quadratic equations in the complex |2 |2 | | | |

| |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 |3 |3 | | | |

| |functions in problems involving | | | | | |

| |exponential growth and decay. | | | | | |

| |14.0 Students understand and use the | | | | | |

| |properties 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. |2 |2 | | | |

| |18.0 Students use fundamental counting | | | | | |

| |principles to compute combinations and | | | | | |

| |permutations. | | | | | |

| |19.0 Students use combinations and | | | | | |

| |permutations to compute probabilities. |1 |1 | | | |

| |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. |2 |2 | | | |

| | | | | | | |

| |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. |1 |1 | | | |

| |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 |1 |1 | | | |

| |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, pyramids, | | | | | |

| |cylinders, cones, and spheres; and |1 |1 | | | |

| |students commit to memory the formulas | | | | | |

| |for prisms, pyramids, and cylinders. | | | | | |

| |10.0 Students compute areas of | | | | | |

| |polygons, including rectangles, scalene| | | | | |

| |triangles, equilateral triangles, |1 |1 | | | |

| |rhombi, parallelograms, and trapezoids.| | | | | |

| | | | | | | |

| |11.0 Students determine how changes in | | | | | |

| |dimensions affect the perimeter, area, |1 |1 | | | |

| |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. |1 |1 | | | |

| |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), (sin(x))2 + (cos(x)) 2 =| | | | | |

| |1. | | | | | |

| |19.0 Students use trigonometric | | | | | |

| |functions to solve for an unknown |1 |1 | | | |

| |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 |3 |3 | | | |

| |polygons of circles. | | | | | |

| |Algebra 1: | | | | | |

| |4.0 Students simplify expressions | | | | | |

| |before solving linear equations and |2 |2 | | | |

| |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. |2 |2 | | | |

| |6.0 Students graph a linear equation | | | | | |

| |and compute the x- and y-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 |1 |1 | | | |

| |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|1 |1 | | | |

| |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 to second- and simple | | | | | |

| |third-degree polynomials. These | | | | | |

| |techniques include finding a common |1 |1 | | | |

| |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 |1 |1 | | | |

| |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.|2 |2 | | | |

| | | | | | | |

| |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|2 |2 | | | |

| |to physical problems, such as the | | | | | |

| |motion of an object under the force of | | | | | |

| |gravity. | | | | | |

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|Unit |High Priority Standards |# CST Items*|# Q3 Items |Supporting Medium/Low Priority Standards |# CST Items* |Textbook |

| | | | | | |Prentice Hall ** |

|Unit 14: Introduction to Vectors, Polars, |Math Analysis: | | |Trigonometry: | | |

|Parametrics |1.0 Students are familiar with, and can| | |16.0 Students represent equations given in | | |

|This unit begins with the introduction of a vector |apply, polar coordinates and vectors in| | |rectangular coordinates in terms of polar | | |

|space. Vectors are introduced algebraically and |the plane. In particular, they can | | |coordinates. | | |

|geometrically in two and three dimensions. Vector |translate between polar and rectangular| | | | | |

|addition and scalar multiplication are introduced. |coordinates and can interpret polar | | | | | |

|The polar coordinate system is also introduced, and |coordinates and vectors graphically. | | | | | |

|connections are made between polar and rectangular |Learning Targets | | | | | |

|coordinates. Parametric equations are then |14A Explain the meaning of a vector | | | | | |

|introduced, and connected to the rectangular |within the context of vector space | | | | | |

|coordinate system by the method of eliminating the |14B Write and draw vectors in two and | | | | | |

|parameter. |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. | | | | | |

| |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 [pic], and Brahmagupta | | | | | | |

|theorem. | | | | | | |

Instruction Continues After CST- Q4

Math Culminating Projects

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