# West Virginia State Standards for Mathematics: Grade 12

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WV.MA.S.A1. Algebra I Mathematics

MA.S.A1.2. Algebra: Through communication, representation, reasoning and proof, problem solving, and making connections within and beyond the field of mathematics, students will demonstrate understanding of patterns, relations and functions, represent and analyze mathematical situations and structures using algebraic symbols, use mathematical models to represent and understand quantitative relationships, and analyze change in various contexts.

MA.O.A1.2.1. Formulate algebraic expressions for use in equations and inequalities that require planning to accurately model real-world problems.

MA.O.A1.2.2. Create and solve multi-step linear equations, absolute value equations, and linear inequalities in one variable, (with and without technology); apply skills toward solving practical problems such as distance, mixtures or motion and judge the reasonableness of solutions.

MA.O.A1.2.3. Evaluate data provided, given a real-world situation, select an appropriate literal equation and solve for a needed variable.

MA.O.A1.2.4. Develop and test hypotheses to derive the laws of exponents and use them to perform operations on expressions with integral exponents.

MA.O.A1.2.5. Analyze a given set of data and prove the existence of a pattern numerically, algebraically and graphically, write equations from the patterns and make inferences and predictions based on observing the pattern.

MA.O.A1.2.6. Determine the slope of a line through a variety of strategies (e.g. given an equation or graph).

MA.O.A1.2.7. Analyze situations and solve problems by determining the equation of a line given a graph of a line, two points on the line, the slope and a point, or the slope and y intercept.

MA.O.A1.2.8. Identify a real life situation that involves a constant rate of change; pose a question; make a hypothesis as to the answer; develop, justify, and implement a method to collect, organize, and analyze related data; extend the nature of collected, discrete data to that of a continuous linear function that describes the known data set; generalize the results to make a conclusion; compare the hypothesis and the conclusion; present the project numerically, analytically, graphically and verbally using the predictive and analytic tools of algebra (with and without technology).

MA.O.A1.2.9. Create and solve systems of linear equations graphically and numerically using the elimination method and the substitution method, given a real-world situation.

MA.O.A1.2.10. Simplify and evaluate algebraic expressions add and subtract polynomials multiply and divide binomials by binomials or monomials.

MA.O.A1.2.11. Create polynomials to represent and solve problems from real-world situations while focusing on symbolic and graphical patterns.

MA.O.A1.2.12. Use area models and graphical representations to develop and explain appropriate methods of factoring.

MA.O.A1.2.13. Simplify radical expressions through: adding, subtracting, multiplying and dividing; exact and approximate forms.

MA.O.A1.2.14. Choose the most efficient method to solve quadratic equations by graphing (with and without technology), factoring quadratic formula and draw reasonable conclusions about a situation being modeled.

MA.O.A1.2.15. Describe real life situations involving exponential growth and decay equations including y=2 to the x and y=(1/2) to the x; compare the equation with attributes of an associated table and graph to demonstrate an understanding of their interrelationship.

MA.O.A1.2.16. Simplify and evaluate rational expressions add, subtract, multiply and divide determine when an expression is undefined.

MA.O.A1.2.17. Perform a linear regression (with and without technology): compare and evaluate methods of fitting lines to data, identify the equation for the line of regression, examine the correlation coefficient to determine how well the line fits the data, use the equation to predict specific values of a variable.

MA.O.A1.2.18. Compute and interpret the expected value of random variables in simple cases using simulations and rules of probability (with and without technology).

MA.O.A1.2.19. Gather data to create histograms, box plots, scatter plots and normal distribution curves and use them to draw and support conclusions about the data.

MA.O.A1.2.20. Design experiments to model and solve problems using the concepts of sample space and probability distribution.

MA.O.A1.2.21. Use multiple representations, such as words, graphs, tables of values and equations, to solve practical problems; describe advantages and disadvantages of the use of each representation.

WV.MA.S.G. Geometry Mathematics

MA.S.G.3. Geometry: Through communication, representation, reasoning and proof, problem solving, and making connections within and beyond the field of mathematics, students will analyze characteristics and properties of two- and three-dimensional geometric shapes and develop mathematical arguments about geometric relationships, specify locations and describe spatial relationships using coordinate geometry and other representational systems, apply transformations and use symmetry to analyze mathematical situations, and solve problems using visualization, spatial reasoning, and geometric modeling.

MA.O.G.3.1. Represent geometric figures, such as points, lines, planes, segments, rays, and angles pictorially with proper identification and distinguish between undefined and defined terms.

MA.O.G.3.2. Differentiate and apply inductive and deductive reasoning, justify conclusions in real-world settings.

MA.O.G.3.3. Use the basic concepts of symbolic logic including identifying the converse, inverse, and contrapositive of a conditional statement and test the validity of conclusions with methods that include Venn Diagrams.

MA.O.G.3.4. Validate conclusions by constructing logical arguments using both formal and informal methods with direct and indirect reasoning.

MA.O.G.3.5. Construct formal and informal proofs by applying definitions, theorems, and postulates related to such topics as complementary, supplementary, vertical angles, angles formed by perpendicular lines, and justify the steps.

MA.O.G.3.6. Compare and contrast the relationships between angles formed by two lines cut by a transversal when lines are parallel and when they are not parallel, and use the results to develop concepts that will justify parallelism.

MA.O.G.3.7. Make conjectures and justify congruence relationships with an emphasis on triangles and employ these relationships to solve problems.

MA.O.G.3.8. Identify general properties of and compare and contrast the properties of convex and concave quadrilaterals: parallelograms, rectangles, rhombuses, squares, trapezoids

MA.O.G.3.9. Identify a real life situation that involves similarity in two or three dimensions; pose a question; make a hypothesis as to the answer, develop, justify, and implement a method to collect, organize, and analyze related data; generalize the results to make a conclusion; compare the hypothesis and the conclusion; present the project numerically, analytically, graphically and verbally using the predictive and analytic tools of algebra and geometry (with and without technology).

MA.O.G.3.10. Investigate measures of angles and lengths of segments to determine the existence of a triangle (triangle inequality) and to establish the relationship between the measures of the angles and the length of the sides (with and without technology).

MA.O.G.3.11. Verify and justify the basis for the trigonometric ratios by applying properties of similar triangles and use the results to find inaccessible heights and distances. Using the ratios of similar triangles to find unknown side lengths and angle measures, construct a physical model that illustrates the use of a scale drawing in a real-world situation.

MA.O.G.3.12. Apply the Pythagorean Theorem and its converse to solve real-world problems and derive the special right triangle relationships (i.e. 30-60-90, 45-45-90).

MA.O.G.3.13. Investigate measures of angles formed by chords, tangents, and secants of a circle and draw conclusions for the relationship to its arcs.

MA.O.G.3.14. Find angle measures of interior and exterior angles; given a polygon, find the length of sides from given data; and use properties of regular polygons to find any unknown measurements of sides or angles.

MA.O.G.3.15. Develop properties of tessellating figures and use those properties to tessellate the plane.

MA.O.G.3.16. Derive and justify formulas for area, perimeter, surface area, and volume using nets and apply them to solve real-world problems.

MA.O.G.3.17. Apply concepts of analytical geometry such as formulas for distance, slope, and midpoint and apply these to finding dimensions of polygons on the coordinate plane.

MA.O.G.3.18. Construct a triangle's medians, altitudes, angle and perpendicular bisectors using various methods; and develop logical concepts about their relationships to be used in solving real-world problems.

MA.O.G.3.19. Create and apply concepts using transformational geometry and laws of symmetry, of a reflection, translation, rotation, glide reflection, dilation of a figure, and develop logical arguments for congruency and similarity.

MA.O.G.3.20. Compare and contrast Euclidean geometry to other geometries (i.e. spherical, elliptic) using various forms of communication such as development of physical models, oral or written reports.

MA.O.G.3.21. Approximate the area of irregularly shaped regions based on the approximations and the attributes of the related region, develop a formula for finding the area of irregularly shaped regions. Plan, organize and present results by justifying conclusions.

WV.MA.S.A2. Algebra II Mathematics

MA.S.A2.2. Algebra: Through communication, representation, reasoning and proof, problem solving, and making connections within and beyond the field of mathematics, students will demonstrate understanding of patterns, relations and functions, represent and analyze mathematical situations and structures using algebraic symbols, use mathematical models to represent and understand quantitative relationships, and analyze change in various contexts.

MA.O.A2.2.1. Determine equations of lines including parallel, perpendicular, vertical and horizontal lines, and compare and contrast the properties of these equations.

MA.O.A2.2.2. Factor higher order polynomials by applying various methods including factoring by grouping and the sum and difference of two cubes; analyze and describe the relationship between the factored form and the graphical representation.

MA.O.A2.2.3. Define complex numbers, simplify powers of ''i', perform basic operations with complex numbers, and give answers as complex numbers in simplest form.

MA.O.A2.2.4. Simplify expressions involving radicals and fractional exponents, convert between the two forms, and solve equations containing radicals and exponents.

MA.O.A2.2.5. Solve quadratic equations over the set of complex numbers: apply the techniques of factoring, completing the square, and the quadratic formula; use the discriminate to determine the number and nature of the roots; identify the maxima and minima; use words, graphs, tables, and equations to generate and analyze solutions to practical problems.

MA.O.A2.2.6. Develop and use the appropriate field properties of matrices by adding, subtracting, and multiplying; solve a system of linear equations using matrices; and apply skills toward solving practical problems.

MA.O.A2.2.7. Define a function and find its zeros; express the domain and range using interval notation; find the inverse of a function; find the value of a function for a given element in its domain; and perform basic operations on functions including composition of functions.

MA.O.A2.2.8. Analyze families of functions and their transformations; recognize linear, quadratic, radical, absolute value, step, piece-wise, and exponential functions; analyze connections among words, graphs, tables and equations when solving practical problems with and without technology.

MA.O.A2.2.9. Solve quadratic inequalities, graph their solution sets, and express solutions using interval notation.

MA.O.A2.2.10. Solve and graph the solution set of systems of linear inequalities in two variables by finding the maximum or minimum values of a function over the feasible region using linear programming techniques.

MA.O.A2.2.11. Solve practical problems involving direct, inverse and joint variation.

MA.O.A2.2.12. Analyze the conic sections; identify and sketch the graphs of a parabola, circle, ellipse, and hyperbola and convert between graphs and equations.

MA.O.A2.2.13. Solve absolute value inequalities graphically, numerically and algebraically and express the solution set in interval notation.

MA.O.A2.2.14. Define a logarithmic function, transform between exponential and logarithmic forms, and apply the basic properties of logarithms to simplify or expand an expression.

MA.O.A2.2.15. Identify a real life situation that exhibits characteristics of change that can be modeled by a quadratic equations; pose a questions; make a hypothesis as to the answer; develop, justify, and implement a method to collect, organize and analyze related data; extend the nature of collected, discrete data to that of a continuous function that describes the known data set; generalize the results to make a conclusion; compare the hypothesis and the conclusion; present the project numerically, analytically, graphically and verbally using the predictive and analytic tools of algebra (with and without technology).

MA.O.A2.2.16. Describe and illustrate how patterns and sequences are used to develop recursive and closed form equations; analyze and describe characteristics of each form.

WV.MA.S.CM. Conceptual Mathematics

MA.S.CM.2. Algebra: Through communication, representation, reasoning and proof, problem solving, and making connections within and beyond the field of mathematics, students will demonstrate understanding of patterns, relations and functions, represent and analyze mathematical situations and structures using algebraic symbols, use mathematical models to represent and understand quantitative relationships, and analyze change in various contexts.

MA.O.CM.2.1. Use a variety of problem solving strategies (e.g., draw a diagram, look for a pattern, work backwards) to solve real-world problems.

MA.O.CM.2.2. Interpret graphs of functions including linear, quadratic, and exponential.

MA.O.CM.2.3. Solve application problems using linear, quadratic and exponential functions with emphasis on data collection and analysis.

MA.O.CM.2.4. Choose the appropriate formulas to solve workplace problems and judge the reasonableness of the solutions.

MA.O.CM.2.5. Describe and illustrate how calculating costs, simple and compound interest, finance charge, loan payment and tax functions are used to solve real-world problems.

MA.O.CM.2.6. Identify a real life situation that involves investing money over time; pose a question; make a hypothesis as to the answer; develop, justify, and implement a method to collect, organize, and analyze related data; generalize the results to make a conclusion; compare the hypothesis and the conclusion; present the project numerically, analytically, graphically and verbally using words, graphs, models, or tables (with and without technology).

MA.S.CM.3. Geometry: Through communication, representation, reasoning and proof, problem solving, and making connections within and beyond the field of mathematics, students will analyze characteristics and properties of two- and three-dimensional geometric shapes and develop mathematical arguments about geometric relationships, specify locations and describe spatial relationships using coordinate geometry and other representational systems, apply transformations and use symmetry to analyze mathematical situations, and solve problems using visualization, spatial reasoning, and geometric modeling.

MA.O.CM.3.1. Apply concepts of geometry including the Pythagorean Theorem, similar triangles, and right triangle trigonometry.

MA.O.CM.3.2. Compute measures to solve real-world problems, using relationships involving perimeter, area, surface area and volume of geometric figures.

MA.O.CM.3.3. Analyze the connections of various geometric shapes and patterns to art, architecture, and nature.

MA.S.CM.5. Data Analysis and Probability: Through communication, representation, reasoning and proof, problem solving, and making connections within and beyond the field of mathematics, students will formulate questions that can be addressed with data and collect, organize, and display relevant data to answer them, select and use appropriate statistical methods to analyze data, develop and evaluate inferences and predictions that are based on models, and apply and demonstrate an understanding of basic concepts of probability.

MA.O.CM.5.1. Relate mathematical content to its historical development.

MA.O.CM.5.2. Integrate other disciplines into the study of mathematics through simulations, research, and projects.

MA.O.CM.5.3. Determine possible outcomes using tree diagrams and the counting principles of permutations and combinations, develop conclusions and offer solutions for new situations, using real-world data.

MA.O.CM.5.4. Design and conduct probability investigations and then determine, analyze, and communicate the results.

MA.O.CM.5.5. Collect and interpret data using various methods of displaying numerical data, including frequency distributions, graphs, histograms, stem-and-leaf plots, and box-and-whiskers plots, using technology when appropriate.

MA.O.CM.5.6. Relate the measures of central tendency and the measures of dispersion to a normal distribution.

MA.O.CM.5.7. Apply the measures of central tendency and the measures of dispersion to workplace situations.

MA.O.CM.5.8. Use statistical tools for workplace applications such as quality control, marketing and predicting trends.

WV.MA.S.T. Trigonometry Mathematics

MA.S.T.3. Geometry: Through communication, representation, reasoning and proof, problem solving, and making connections within and beyond the field of mathematics, students will analyze characteristics and properties of two- and three-dimensional geometric shapes and develop mathematical arguments about geometric relationships, specify locations and describe spatial relationships using coordinate geometry and other representational systems, apply transformations and use symmetry to analyze mathematical situations, and solve problems using visualization, spatial reasoning, and geometric modeling.

MA.O.T.3.1. Apply the right triangle definition of the six trigonometric functions of an angle to determine the values of the function values of an angle in standard position given a point on the terminal side of the angle.

MA.O.T.3.1.1. Determine the value of the other trigonometric functions given the value of one of the trigonometric functions and verify these values with technology.

MA.O.T.3.1.2. Using geometric principles and the Pythagorean Theorem, determine the six function values for the special angles and the quadrantal angles and use them in real-world problems.

MA.O.T.3.1.3. Compare circular functions and the trigonometric function values to draw inferences about coterminal angles and co-functions.

MA.O.T.3.2. Convert angle measures from degrees to radians (and vice versa) and apply this concept to

MA.O.T.3.2.1. Create a data set, analyze, and formulate a hypothesis to test and develop formulas for the arc length, area of a sector, and angular velocity and use the formula for application in the real-world.

MA.O.T.3.2.2. Compare and contrast the concepts of angular velocity and linear velocity and demonstrate by graphical or algebraic means relationship between them and apply to real-world problems.

MA.O.T.3.3. Using various methods, basic identities and graphical representation

MA.O.T.3.3.1. Verify trigonometric identities

MA.O.T.3.3.2. Prove the sum and difference to two angles, double-angles, and half-angle identities

MA.O.T.3.4. Justify and present the solutions of trigonometric equations that include both infinite and finite (over a restricted domain) solutions.

MA.O.T.3.5. Find the value of the inverse trigonometric functions using special angle trigonometric function values and technology.

MA.O.T.3.5.1. Draw inferences of restricted domain to recognize and produce a graph of the inverse trigonometric functions.

MA.O.T.3.5.2. Prove conjectures made about the solution of the equations such as x = sin (arcsin y), x = sin (arcos y) being sure to consider restrictions of the domain.

MA.O.T.3.6. Identify a real life problem utilizing graphs of trigonometric functions and/or the inverse functions; make a hypothesis as to the outcome; develop, justify, and implement a method to collect, organize, and analyze data; generalize the results to make a conclusion; compare the hypothesis and the conclusion; present the project using words, graphs, drawings, models, or tables.

MA.O.T.3.7. Model periodic data sets using graphs, tables, and equations and use them to analyze real-world problems such as electricity and harmonic motion.

MA.O.T.3.8. Investigate real-world problems within a project based investigation involving triangles using the trigonometric functions, the law of sines and the law of cosines, justify and present results.

MA.O.T.3.9. Develop and test a hypothesis to find the area of a triangle given the measures of two sides and the included angle or the measures of three sides (Heron's formula) and use these formulas to find total area of figures constructed of multiple shapes.

MA.O.T.3.10. Express complex numbers in polar form:

MA.O.T.3.10.1. Perform operations including adding, subtracting, multiplying, and dividing;

MA.O.T.3.10.2. Evaluate powers and roots of complex numbers using De Moivre's Theorem; and graph complex numbers.

MA.O.T.3.10.3. Graph complex numbers in the polar coordinate plane and make conjectures about some polar graphs and real-world situations such as the paths that the planets travel.

MA.O.T.3.11. Create graphical and algebraic representations for performing vector operations and analyze these to solve real-world problems such as force analysis and navigation.

WV.MA.S.PS. Probability and Statistics Mathematics

MA.S.PS.5. Data Analysis and Probability: Through communication, representation, reasoning and proof, problem solving, and making connections within and beyond the field of mathematics, students will formulate questions that can be addressed with data and collect, organize, and display relevant data to answer them, select and use appropriate statistical methods to analyze data, develop and evaluate inferences and predictions that are based on models, and apply and demonstrate an understanding of basic concepts of probability.

MA.O.PS.5.1. Distinguish between experimental and theoretical probability.

MA.O.PS.5.2. Using a real-world problem solving investigation, create and interpret data using various methods of displaying circle graphs, histograms, and frequency curves, make predictions, include information concerning outliers, present and justify results.

MA.O.PS.5.3. Determine possible outcomes using tree diagrams and the counting principles of permutations and combinations.

MA.O.PS.5.4. Express the chances of events occurring either in terms of a probability or odds.

MA.O.PS.5.5. Use the normal distribution and the binomial distribution including Pascal's triangle, to determine probability of events.

MA.O.PS.5.6. Analyze measures of central tendency (mean, median, and mode) from data presented in a variety of forms such as charts, tables, and graphs or from data created through experimentation.

MA.O.PS.5.7. Interpret and calculate measures of dispersions (range and standard deviation) from data presented in a variety of forms such as charts, tables and graphs or from data created through experimentation.

MA.O.PS.5.8. Analyze individual performances in terms of percentiles, z-scores, and t- scores.

MA.O.PS.5.9. Analyze the role of sampling, randomness, bias, and sample size in data collection and interpretation.

MA.O.PS.5.10. Identify a real life situation that involves statistical concepts including a t-test, make a hypothesis as to the outcome; develop, justify, and implement a method to collect, organize and analyze data; generalize the results to make a conclusion, compare the hypothesis and the conclusion; present the project using predictive and analytic tools (with and without technology).

MA.O.PS.5.11. Determine the correlation values for given data or for data generated by students and use the results to describe the association of the variables within the given data. Identify whether this association is systematic or predictable.

MA.O.PS.5.12. Calculate the Chi-Square values for a given population.

MA.O.PS.5.13. Perform a regression analysis on a set of data, either given or created through experimentation, and use the results to predict specific values of a variable. Identify the regression equation.

MA.O.PS.5.14. Perform an analysis of variance (ANOVA) and interpret the results.

WV.MA.S.PC. Pre-Calculus Mathematics

MA.S.PC.2. Algebra: Through communication, representation, reasoning and proof, problem solving, and making connections within and beyond the field of mathematics, students will demonstrate understanding of patterns, relations, and functions, represent and analyze mathematical situations and structures using algebraic symbols, use mathematical models to represent and understand quantitative relationships, and analyze change in various contexts.

MA.O.PC.2.1. Investigate and sketch the graphs of polynomials and rational functions by analyzing and using the characteristics of zeros, upper and lower bounds, y-intercepts, symmetry, asymptotes and end behavior, maximum and minimum points, and domain and range.

MA.O.PC.2.2. Solve higher order polynomial equations utilizing techniques such as Descartes' Rule of Signs, upper and lower bounds, and the Rational Root Theorem.

MA.O.PC.2.3. Relate Pascal's Triangle and the Binomial Theorem; use both to expand binomials with positive integral exponents.

MA.O.PC.2.4. Establish and explain the inverse relationship between exponential and logarithmic functions; graph related functions and include their domain and range using interval notation.

MA.O.PC.2.5. Compare laws of exponents to properties of logarithms; solve equations and practical problems involving exponential and logarithmic expressions, including natural and common logarithms; confirm solutions graphically and numerically.

MA.O.PC.2.6. Solve problems involving the sum of finite and infinite sequences and series, including Sigma notation.

MA.O.PC.2.7. Use tables of values, graphs, conjectures, algebraic methods, and numerical substitution to find or estimate the limit of a function, a sequence or a series.

MA.O.PC.2.8. Analyze and describe the geometry of vectors, perform mathematical operations with vectors and use vectors to solve practical problems.

MA.O.PC.2.9. Apply the method of mathematical induction to prove formulas and statements.

MA.O.PC.2.10. Apply parametric methods to represent motion of objects.

MA.O.PC.2.11. Use multiple representations, such as words, graphs, tables, and equations, to solve practical problems involving logarithmic, exponential, polynomial, rational, and radical functions; explain how the representations are related to each other, as well as to the problem.

MA.S.PC.3. Geometry: Through communication, representation, reasoning and proof, problem solving, and making connections within and beyond the field of mathematics, students will analyze characteristics and properties of two- and three-dimensional geometric shapes and develop mathematical arguments about geometric relationships, specify locations and describe spatial relationships using coordinate geometry and other representational systems, apply transformations and use symmetry to analyze mathematical situations, and solve problems using visualization, spatial reasoning, and geometric modeling.

MA.O.PC.3.1. Graph functions and conic sections using transformations.

MA.O.PC.3.2. Analyze and describe properties of conic sections; explain the interrelationship among the properties; solve practical problems involving conic sections.

MA.S.PC.5. Data Analysis and Probability: Through communication, representation, reasoning and proof, problem solving, and making connections within and beyond the field of mathematics, students will formulate questions that can be addressed with data and collect, organize, and display relevant data to answer them, select and use appropriate statistical methods to analyze data, develop and evaluate inferences and predictions that are based on models, and apply and demonstrate an understanding of basic concepts of probability.

MA.O.PC.5.1. Identify a real life situation that exhibits characteristics of exponential or logistic growth or decay; pose a question; make a hypothesis as to the answer; develop, justify, and implement a method to collect, organize, and analyze related data; extend the nature of collected, discrete data to that of a continuous function that describes the known data set; generalize the results to make a conclusion; compare the hypothesis and the conclusion; present the project numerically, analytically, graphically and verbally using the predictive and analytic tools of pre-calculus (with and without technology).

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