Virginia Standards of Learning: Secondary

A.01 The student will solve multistep linear equations and inequalities in one variable, solve literal equations (formulas) for a given variable, and apply these skills to solve practical problems. Graphing calculators will be used to confirm algebraic solutions.

A.02 The student will represent verbal quantitative situations algebraically and evaluate these expressions for given replacement values of the variables. Students will choose an appropriate computational technique, such as mental mathematics, calculator, or paper and pencil.

A.03 The student will justify steps used in simplifying expressions and solving equations and inequalities. Justifications will include the use of concrete objects; pictorial representations; and the properties of real numbers, equality, and inequality.

A.05 The student will create and use tabular, symbolic, graphical, verbal, and physical representations to analyze a given set of data for the existence of a pattern, determine the domain and range of relations, and identify the relations that are functions.

A.07 The student will determine the slope of a line when given an equation of the line, the graph of the line, or two points on the line. Slope will be described as rate of change and will be positive, negative, zero, or undefined. The graphing calculator will be used to investigate the effect of changes in the slope on the graph of the line.

A.08 The student will write an equation of a line when given the graph of the line, two points on the line, or the slope and a point on the line.

A.14 The student will solve quadratic equations in one variable both algebraically and graphically. Graphing calculators will be used both as a primary tool in solving problems and to verify algebraic solutions.

A.18 The student will analyze a relation to determine whether a direct variation exists and represent it algebraically and graphically, if possible.

AII.01 The student will identify field properties, axioms of equality and inequality, and properties of order that are valid for the set of real numbers and its subsets, complex numbers, and matrices.

AII.02 The student will add, subtract, multiply, divide, and simplify rational expressions, including complex fractions.

AII.03a The student will add, subtract, multiply, divide, and simplify radical expressions containing positive rational numbers and variables and expressions containing rational exponents.

AII.03b The student will write radical expressions as expressions containing rational exponents and vice versa.

AII.04 The student will solve absolute value equations and inequalities graphically and algebraically. Graphing calculators will be used as a primary method of solution and to verify algebraic solutions.

AII.05 The student will identify and factor completely polynomials representing the difference of squares, perfect square trinomials, the sum and difference of cubes, and general trinomials.

AII.06 The student will select, justify, and apply a technique to solve a quadratic equation over the set of complex numbers. Graphing calculators will be used for solving and for confirming the algebraic solutions.

AII.07 The student will solve equations containing rational expressions and equations containing radical expressions algebraically and graphically. Graphing calculators will be used for solving and for confirming the algebraic solutions.

AII.08 The student will recognize multiple representations of functions (linear, quadratic, absolute value, step, and exponential functions) and convert between a graph, a table, and symbolic form. A transformational approach to graphing will be employed through the use of graphing calculators.

AII.09 The student will find the domain, range, zeros, and inverse of a function; the value of a function for a given element in its domain; and the composition of multiple functions. Functions will include exponential, logarithmic, and those that have domains and ranges that are limited and/or discontinuous. The graphing calculator will be used as a tool to assist in investigation of functions.

AII.1

AII.10

AII.10 The student will investigate and describe through the use of graphs the relationships between the solution of an equation, zero of a function, x-intercept of a graph, and factors of a polynomial expression.

AII.12

AII.12 The student will represent problem situations with a system of linear equations and solve the system, using the inverse matrix method. Graphing calculators or computer programs with matrix capability will be used to perform computations.

AII.13

AII.13 The student will solve practical problems, using systems of linear inequalities and linear programming, and describe the results both orally and in writing. A graphing calculator will be used to facilitate solutions to linear programming problems.

AII.14

AII.14 The student will solve nonlinear systems of equations, including linear-quadratic and quadratic-quadratic, algebraically and graphically. The graphing calculator will be used as a tool to visualize graphs and predict the number of solutions.

AII.15

AII.15 The student will recognize the general shape of polynomial, exponential, and logarithmic functions. The graphing calculator will be used as a tool to investigate the shape and behavior of these functions.

AII.16

AII.16 The student will investigate and apply the properties of arithmetic and geometric sequences and series to solve practical problems, including writing the first n terms, finding the nth term, and evaluating summation formulas. Notation will include Σ and an.

AII.17

AII.17 The student will perform operations on complex numbers and express the results in simplest form. Simplifying results will involve using patterns of the powers of i.

AII.18

AII.18 The student will identify conic sections (circle, ellipse, parabola, and hyperbola) from his/her equations. Given the equations in (h, k) form, the student will sketch graphs of conic sections, using transformations.

AII.19

AII.19 The student will collect and analyze data to make predictions and solve practical problems. Graphing calculators will be used to investigate scatterplots and to determine the equation for a curve of best fit. Models will include linear, quadratic, exponential, and logarithmic functions.

AII.2

AII.20

AII.20 The student will identify, create, and solve practical problems involving inverse variation and a combination of direct and inverse variations.

AII.3.a

AII.3.b

AII.4

AII.5

AII.6

AII.7

AII.8

AII.9

AII/T.14 The student will solve nonlinear systems of equations, including linear-quadratic and quadratic-quadratic, algebraically and graphically. The graphing calculator will be used as a tool to visualize graphs and predict the number of solutions.

AII/T.15 The student will recognize the general shape of polynomial, exponential, and logarithmic functions. The graphing calculator will be used as a tool to investigate the shape and behavior of these functions.

AII/T.26 The student, given one of the six trigonometric functions in standard form [e.g., y = A sin (Bx + C) + D, where A, B, C, and D are real numbers], will state the domain and the range of the function; determine the amplitude, period, phase shift, and vertical shift; and, sketch the graph of the function by using transformations for at least a one-period interval. The graphing calculator will be used to investigate the effect of changing A, B, C, and D on the graph of a trigonometric function.

AFDA.03 The student will collect data and generate an equation for the curve (linear, quadratic, exponential, and logarithmic) of best fit to model real- world problems or applications. Students will use the best fit equation to interpolate function values, make decisions, and justify conclusions with algebraic and/or graphical models.

APC.04 The student will investigate asymptotic and unbounded behavior in functions. This will include describing and understanding asymptotes in terms of graphical behavior and limits involving infinity; and, comparing relative magnitudes of functions and their rates of change.

APC.05 The student will investigate derivatives presented in graphic, numerical, and analytic contexts and the relationship between continuity and differentiability. The derivative will be defined as the limit of the difference quotient and interpreted as an instantaneous rate of change.

APC.06 The student will investigate the derivative at a point on a curve. This will include finding the slope of a curve at a point, including points at which the tangent is vertical and points at which there are no tangents; using local linear approximation to find the slope of a tangent line to a curve at the point; defining instantaneous rate of change as the limit of average rate of change; and, approximating rate of change from graphs and tables of values.

APC.08 The student will apply the derivative to solve problems. This will include analysis of curves and the ideas of concavity and monotonicity; optimization involving global and local extrema; modeling of rates of change and related rates; use of implicit differentiation to find the derivative of an inverse function; interpretation of the derivative as a rate of change in applied contexts, including velocity, speed, and acceleration; and, differentiation of nonlogarithmic functions, using the technique of logarithmic differentiation. * * AP Calculus BC will also apply the derivative to solve problems. This will include analysis of planar curves given in parametric form, polar form, and vector form, including velocity and acceleration vectors; numerical solution of differential equations, using Euler's method; l'Hopital's Rule to test the convergence of improper integrals and series; and, geometric interpretation of differential equations via slope fields and the relationship between slope fields and the solution curves for the differential equations.

APC.09 The student will apply formulas to find derivatives. This will include derivatives of algebraic, trigonometric, exponential, logarithmic, and inverse trigonometric functions; derivations of sums, products, quotients, inverses, and composites (chain rule) of elementary functions; derivatives of implicitly defined functions; and, higher order derivatives of algebraic, trigonometric, exponential, and logarithmic, functions. * * AP Calculus BC will also include finding derivatives of parametric, polar, and vector functions.

APC.15 The student will use integration techniques and appropriate integrals to model physical, biological, and economic situations. The emphasis will be on using the integral of a rate of change to give accumulated change or on using the method of setting up an approximating Riemann sum and representing its limit as a definite integral. Specific applications will include the area of a region; the volume of a solid with known cross-section; the average value of a function; and, the distance traveled by a particle along a line. * * AP Calculus BC will include finding the area of a region (including a region bounded by polar curves) and finding the length of a curve (including a curve given in parametric form).

G.02 The student will use pictorial representations, including computer software, constructions, and coordinate methods, to solve problems involving symmetry and transformation. This will include investigating and using formulas for finding distance, midpoint, and slope; investigating symmetry and determining whether a figure is symmetric with respect to a line or a point; and, determining whether a figure has been translated, reflected, or rotated.

G.12 The student will make a model of a three-dimensional figure from a two-dimensional drawing and make a two-dimensional representation of a three-dimensional object. Models and representations will include scale drawings, perspective drawings, blueprints, or computer simulations.

G.13 The student will use formulas for surface area and volume of three-dimensional objects to solve practical problems. Calculators will be used to find decimal approximations for results.

MA.08 The student will investigate and identify the characteristics of conic section equations in (h, k) and standard forms. The techniques of translation and rotation of axes in the coordinate plane will be used to graph conic sections.

MA.10 The student will investigate and identify the characteristics of the graphs of polar equations, using graphing utilities. This will include classification of polar equations, the effects of changes in the parameters in polar equations, conversion of complex numbers from rectangular form to polar form and vice versa, and the intersection of the graphs of polar equations.

MA.12 The student will use parametric equations to model and solve application problems. Graphing utilities will be used to develop an understanding of the graph of parametric equations.

PS.01 The student will analyze graphical displays of data, including dotplots, stemplots, and histograms, to identify and describe patterns and departures from patterns, using central tendency, spread, clusters, gaps, and outliers. Appropriate technology will be used to create graphical displays.

PS.02 The student will analyze numerical characteristics of univariate data sets to describe patterns and departure from patterns, using mean, median, mode, variance, standard deviation, interquartile range, range, and outliers. Appropriate technology will be used to calculate statistics.

PS.03 The student will compare distributions of two or more univariate data sets, analyzing center and spread (within group and between group variations), clusters and gaps, shapes, outliers, or other unusual features. Appropriate technology will be used to generate graphical displays.

PS.05 The student will find and interpret linear correlation, use the method of least squares regression to model the linear relationship between two variables, and use the residual plots to assess linearity. Appropriate technology will be used to compute correlation coefficients and residual plots.

PS.12 The student will identify and describe two or more events as complementary, dependent, independent, and/or mutually exclusive.

PS.13 The student will find probabilities (relative frequency and theoretical), including conditional probabilities for events that are either dependent or independent, by applying the "law of large numbers" concept, the addition rule, and the multiplication rule.

PS.5 The student will find and interpret linear correlation, use the method of least squares regression to model the linear relationship between two variables, and use the residual plots to assess linearity. Appropriate technology will be used to compute

T.06 The student, given one of the six trigonometric functions in standard form [e.g., y = A sin (Bx + C) + D, where A, B, C, and D are real numbers], will state the domain and the range of the function; determine the amplitude, period, phase shift, and vertical shift; and sketch the graph of the function by using transformations for at least a one-period interval. The graphing calculator will be used to investigate the effect of changing A, B, C, and D on the graph of a trigonometric function.