HIGH SCHOOL MATHEMATICS EXIT PERFORMANCE STANDARD 1:

Student will apply number sense and order relations in problem solving situations to perform estimations and/or calculations with equations, matrices, and sequences involving complex numbers (counting numbers, whole numbers, integers, rational numbers, irrational numbers, real numbers, etc.) with and without calculators and will communicate the reasoning used in solving these problems.

Based on Kentucky’s Core Content for Mathematics Assessment, the Kentucky Program of Studies, and Academic Expectations: 1.5-1.9 Mathematical Communication & Reasoning, 1.16 Technology, 2.7 Number, 2.8 Procedures, 2.11 Change, 2.12 Structure; Goal 5 Think & Solve Problems; & Goal 6 Connect & Integrate Knowledge

Algebra II: Number/Computation Standards

NC/A2.1 Show number sense by interpreting, modeling and using appropriate mathematical notation and operations (%,!,/, pi, square roots-taking a root, scientific notation, absolute value, exponents-raising to a fixed power, matrix, opposite, reciprocal, factorial, permutation, combination, logarithm) for complex numbers and by comparing and contrasting various subsets of the complex number system (counting, whole, integers, rational, irrational, real, and imaginary).

NC/A2.2 Select and apply appropriate concrete, pictorial, and abstract models and strategies to simplify and solve quadratic, cubic, rational, and exponential equations that contain complex numbers, radicals and absolute values including applications involving direct, inverse, combined and joint variation; and solve systems of linear equations and inequalities with two or three variables simultaneously using linear combinations (elimination), substitution, graphing, and matrices for practical situations.

NC/A2.3 Select and apply appropriate concrete, pictorial, and abstract models and strategies to apply the laws of exponents, perform operations on expressions with integral exponents including complex algebraic fractions with negative exponents in the numerator and denominator, and expand powers of binomials using Pascal’s Triangle for coefficients or by using the Binomial Theorem.

NC/A2.4 Recognize, generate, find and defend the general term, find the sums and derive the summation formulas for arithmetic & geometric series, and communicate the concept of limit (fractals).

Skills, Concepts & Relationships

• Translate real world data into matrices (2x2) and demonstrate matrix addition, matrix subtraction, scalar multiplication, and matrix multiplication using multiplicative inverse and identity properties of matrices (1-3) [Appendix A]
• Review using graphs and/or properties of equalities to solve linear equations (2-1) [2-1]
• Review recognizing absolute value as a measure of distance, finding the absolute value of a given number, and solving one-variable equations and inequalities involving absolute values symbolically and graphically (boundary point, compound inequality) (2-3) [1-6]
• Graph & solve linear and absolute value inequalities with two variables that describe restrictions in real relationships (test points to locate graphs, use boundary lines) (2-3) [4-10]
• Solve systems of two or more linear equations simultaneously using substitution, linear combinations (elimination), and matrices (inverse matrices), and interpret the solution for real life situations (3-1) [4-5, 4-9]
• Solve a 3x3 system of linear equations with a graphing calculator (3-1) [4-8]
• Recognize and generate arithmetic sequences (to model patterns-nature) and find partial sums of arithmetic series (common difference) (4-1) [11-2, 11-3]
• Represent arithmetic sequences and series using sigma notation and explicit or nth term formula (4-1) [11-4]
• Recognize and generate geometric sequences (to model patterns-real data) and find partial sums of finite geometric series (common ratio) (4-1) [11-2]
• Represent geometric sequences and series using sigma notation and explicit or nth term formula (4-1) [11-4]
• Introduce the convergence of infinite geometric series and finding the sum of an infinite geometric series (communicate the concept of limit) (4-1) [11-4, 11-5]
• Recognizes arithmetic and geometric sequences as forms of linear and exponential functions, respectively (4-1) [11-1, 11-7]
• Identify, compare and contrast various subsets of the complex number system (counting numbers, whole numbers, integers, rational numbers, irrational numbers, real numbers, and imaginary numbers) (4-3) [1-1, 5-4, 10-2, 10-3]
• Simplify radical expressions (4-3) [1-2]
• Define and use imaginary and complex numbers to find square roots of negative numbers and simplify complex expressions (complex conjugate) (4-3) [10-2,10-3]
• Solve quadratic equations by using a table of values and by factoring to solve for exact roots (Principle of Zero Products) (5-2) [5-2]
• Review using Pascal’s Triangle to determine theoretical probability of binomial situations (6-2) [12-7]
• Relate counting Combinations, the Binomial Theorem and Pascal’s Triangle (6-2) [11-9, 11-10]
• Apply the properties of exponents to simplify expressions and use scientific notation (Product of Powers, Power of Powers, Quotient of Powers, and Power of Quotients) (7-1) [6-3 to 6-6]
• Use the properties of real and complex numbers to simplify radical expressions and solve related equations (simplify and evaluate expressions with rational exponents, solve simple radical equations with extraneous roots, and solve equations containing two or more radical expressions) (7-2) [8-4]
• Combine functions through addition, subtraction, multiplication and composition, and use a composite function to classify real world situations (find and evaluate functions of functions) (7-3) [4-4]
• Identify, and find inverse functions algebraically and graphicaally (use an inverse of a relation or function) (7-3) [6-12]
• Recognize and use terms and definitions associated with polynomial expressions and functions (Expressions: monomial-1term, binomial-2terms, trinomial-3terms; Functions: linear-degree1, quadratic-degree2, cubic-degree3) (8-1) [1-4]
• Find the x-intercepts, maximums, minimums and zeros of polynomial functions of degree 3 or more graphically (continuous, real root, zero) and apply the functions to real world contexts (8-1) [10-4 to 10-6, 7-6]
• Review adding, subtracting, multiplying and dividing polynomials using concrete (algebra tiles) and abstract models in the context of practical applications (8-2) [1-3, 1-4, 7-5]
• Find all solutions of polynomial equations by and recognize the nature of the solutions, and analyze and interpret the results (8-2) [10-4]
• Use factoring and division to solve for roots of polynomial equations (8-2 & supplement synthetic division) [10-3,10-4]
• Use Rational Roots Theorem to identify possible rational solutions and use the Fundamental Theorem of Algebra to identify number of roots (8-2) [7-6]
• Use the properties of real numbers to simplify rational expressions and solve related equations (write, evaluate, multiply, divide & simplify rational expressions and simplify sums and differences of rational expressions - LCD) (8-3) [7-7 to 7-10]
• Understand properties of logarithms to simplify logarithmic numeric expressions and identify their approximate values (9-2) [6-3]

03-SFAW-Focus on Advanced Algebra in ( ) / Advanced -Forester Algebra and Trig in [ ]

HIGH SCHOOL MATHEMATICS EXIT PERFORMANCE STANDARD 2:

Student will apply properties of measurement (ratio measures including slope, rate, indirect measurement, similarity; surface area and volume of prisms, pyramids, cylinders, cones, and spheres, etc.) and will use geometric concepts, properties and relationships (prove, use and apply theorems/conjectures involving lines, angles, triangles, quadrilaterals, regular, and non-regular polygons, circles, and transformations, etc.) in problem solving situations and communicate the inductive and simple deductive reasoning used in solving these problems.

Algebra II: Geometry/Measurement Standards

GM/A2.1 Visualize objects, paths and regions in space, including intersections and cross sections of three dimensional figures and describe these using geometric language [Demonstrate and explain how the graph of a conic section (asymptotes, foci) depends on the coefficients of the quadratic equation representing it (analytic geometry)].

GM/A2.2 Represent graphs of functions in standard coordinate systems.

Skills, Concepts & Relationships

• Explore generating fractal images (4-2) [11-7]
• Use circle and parabola curves to model real world phenomena and recognize & explain why they are known as conic sections (5-3) [9-6]
• Given a quadratric equation indentify as a parabola or circle (5-3) [9-2,9-5]
• Introduce ellipse and hyperbola curves as conic sections (5-3) [*master ellipse, hyperbola 9-3,9-4]

03-SFAW-Focus on Advanced Algebra in ( ) / Advanced -Forester Algebra and Trig in [ ]

HIGH SCHOOL MATHEMATICS EXIT PERFORMANCE STANDARD 3:

Student will use data collection & analysis, graphing of single-variable and two-variable data (line, bar & circle graphs, histogram, stem and leaf plots, box and whisker plots, scatterplot, linear regression & curve fitting), statistics (mean, median, mode, range, outliers, quartiles), and designing probability experiments & simulations to test theories about real world problems and communicate the reasoning used in solving these problems.

Algebra II: Probability/Statistics Standards

PS/A2.1 Analyze sets of data using dispersion (range, outliers, quartiles, standard deviation, variance), arithmetic & geometric means, and use curve fitting (linear, exponential, & power regression equations on the graphing calculator) to model real data and make predictions about trends.

PS/A2.2 Analyze sets of data using normal curve distributions.

PS/A2.3 Formulate and test theories using a variety of quantified information, sampling techniques, probability simulations and discrete probability distributions (fundamental addition and multiplication counting principles, combinations, permutations, conditional probabilities).

Skills, Concepts & Relationships

• Review using scatterplots to determine whether two quantities are related and to look for patterns and trends in order to make predictions about their values (negative, positive, or no association) (1-1) [supplement]
• Review measuring the probability of an event with theoretical probability (formula, tree diagram, geometric probability model) or with experimental probability (simulation, random number charts, etc.) and differentiating probability and "odds" (1-2, supplement odds) [12-2, 12-3]
• Review using scatterplots, trend lines and linear functions as tools for creating a linear model for interpreting data that do not fall neatly on a line in practical real world situations (make predictions using trend line, its equation, and interpret reasonableness of predictions) (2-2) [3-5]
• Find a trend line of best fit graphically using a graphing calculator (2-2) [supplement]
• Count the number of ways a series of events can occur using tree diagrams or counting principles (Addition Counting Principle for mutually exclusive events, Multiplication Fundamental Counting Principle) (6-1) [12-3]
• Use permutations to count the number of ways that items in a set can be arranged when all of the items are different, when all of the items are the same, and to find probabilities in complex situations (6-1) [12-4,11-8]
• Use combinations to count the number of ways items in a set can be arranged without regard to order to find probabilities (6-1) [12-5]
• Use real number exponents to model data with a power regression curve and determine the practical use of rational exponents and radicals in the real world (7-2) [6-14]
• Use exponential regression to find the equation of a curve to model data that shows exponential growth or decay in practical contexts (9-1) [6-14]
• Calculate and distinguish among probabilities of complementary, compound, dependent (conditional with and without replacement), and independent events to solve real world problems (11-1) [12-6]
• Review differences between experimental probability (actual), theoretical probability (expected), and odds (supplement) [supplement]
• Review constructing, interpreting, and analyzing: histograms, line plots, stem and leaf plots, circle graphs, box and whisker plots, scatterplots, etc. (supplement) [supplement]
• Recognize methods of random sampling (11-2) [12-2]
• Formulate a hypothesis to investigate a problem about a statistical theory, and design an experiment (supplement) [supplement]
• Recognize normally distributed data and estimate the probabilities that normally distributed data lies within one or two standard deviations of the mean (11-2) [12-9]
• Determine concepts of variance and dispersion and compute for a distribution of data: standard deviation and variance (quartile, interquartile, range, outliers) (supplement) [12.9 and supplement]

03-SFAW-Focus on Advanced Algebra in ( ) / Advanced -Forester Algebra and Trig in [ ]

HIGH SCHOOL MATHEMATICS EXIT PERFORMANCE STANDARD 4:

Student will model, analyze, compare and apply linear & nonlinear algebraic functions (quadratic, polynomial, exponential, etc.) using tables, graphs in the coordinate plane, variables, expressions, equations, formulas and inequalities in practical situations and communicate the reasoning used in solving these problems.

Algebra II: Algebraic Ideas Standards

AI/A2.1 Describe real world phenomena as functions by translating among graphic, algebraic, numeric and verbal mathematical models.

AI/A2.2 Model relationships between real world quantities and find solutions for quadratic functions by graphing, factoring, completing the square, and using the quadratic formula (including equations whose roots are complex numbers); interpret the maximum and minimum values and intercepts in the context of the problem; and recognize the function as a circle or parabola.

AI/A2.3 Model relationships between real world quantities and find solutions for polynomial functions by graphing and factoring; find zeros, intercepts, and approximate turning points; and recognize the degree of the function.

AI/A2.4 Model relationships between real world quantities (exponential growth and decay) and find solutions for exponential functions by graphing, substituting, and applying inverse relationships; and recognize exponential and logarithmic functions as inverses.

Skills, Concepts & Relationships

• Review representing functional situations with tables, graphs, and equations, recognizing the interrelationship of these and showing how each representation is useful in different situations (domain of independent variables and range of dependent variables of a function; difference between constant and variable quantities) (1-1) [2-3]
• Review that slope is the rate of change of one quantity relative to another quantity and describe slopes of vertical (undefined) and horizontal (zero) lines and oblique (2-1) [3-2]
• Review recognizing characteristics of linear (y=mx+b, constant change), nonlinear (not constant change) and proportional (y=ax, direct variation) functions and identify their equations (2-1) [3-1 to 3-3]
• Review using slopes and y-intercepts with linear graphs and forms of linear equalities to determine whether pairs of lines are parallel, intersecting, or perpendicular (find equation of a line through a point parallel or perpendicular to another line) (2-1) [3-4]
• Review solving and applying a system of two linear equations graphically (graph paper and graphing calculator) and interpreting the solution for real life situations (same line dependent infinite solutions, parallel lines inconsistent no solution) (3-1) [4-2,4-11]
• Sketch the region that represents the solution of a system of linear inequalities in problem solving situations (3-2) [4-10]
• Show how real world constraints of a linear programming problem can be modeled by systems of inequalities (feasible region) (3-2)
• Introduce solving linear programming problems as a tool for decision making (objective function, optimal solution, Vertex Theorem) (3-2 and supplement) [4-11]
• Review graphing quadratic functions of the form y=ax2 to model real relationships and recognize the curve is a parabola (axis of symmetry, vertex) (5-1) [5-2]
• Use translations to determine and explain how the graph of a parabola changes as a, h, & k vary in the equation y=a(x-h) 2+k ) (5-1) [9-5]
• Describe and analyze quadratic functions of the form y=ax2, y=ax2+c, and y=ax2+bx+c to help make decisions in practical situations (determine maximum, minimum, and zero values of the function by graphing) (5-1) [5-7]
• Use the method of completing the square to put an equation of the form ax2+by2+cx+dy+e=0 into standard form and recognize whether its graph is a parabola (vertex, axis of symmetry) or a circle (5-1) [ellipse, hyperbola 9-5, 9-6]
• Use graphs of quadratic functions to solve quadratic equations (y=ax2+bx+c standard form when y=0 which describes the x-axis, x-intercepts: zero,one, or two solutions) (5-2) [5-5]
• Derive, recognize and use the quadratic formula to solve quadratic equations in problems similar to those encountered by physicists, engineers, and business people (5-2) [5-3, 5-7]
• Use the discriminant of the quadratic formula to determine the nature of the roots of a quadratic equation (positive-two real roots, negative-two complex roots, zero-one real root -- double root) (5-2) [5-3]
• Recognize, graph, determine the standard form of the equation of a circle (center, radius) (5-3) [9-2]
• Introduce graphing, determining & recognizing the standard form of the equation of an ellipse and hyperbola (center, foci, major axis, minor axis, vertices) (5-3) [*master 9-3, 9-4]
• Classify relationships between variables by type of variation (direct: one increases while other increases-graph is line that contains origin; inverse: one increases while other decreases-graph is rectangular hyperbola; combined: relationship shows both inverse and direct variation; and joint: one variable varies directly with two or more variables but does not vary inversely with any other variable) and apply direct variation and scientific notation (7-1) [7-11]
• Use exponential functions of the form y=abx to model quantities increase or decrease over time at a given percentage, and to model exponential growth and decay (9-1) [6.14]
• Recognize the inverse relationship between exponential and logarithmic functions (9-2) [6.8]
• Use the definition of logarithms and the product formula for logs to translate between logs in any bases (9-2) [6.9]
• Use logarithmic functions to model real world problems (9-2) [6.14]

03-SFAW-Focus on Advanced Algebra in ( ) / Advanced -Forester Algebra and Trig in [ ]