Mathematics

   Standards of Learning

   Geometry

This course is designed for students who have successfully

completed the standards for Algebra I.  The course, among other

things, includes the deductive axiomatic method of proof to justify

theorems and to tell whether conclusions are valid.  Methods of

justification will include paragraph proofs, flow charts,

two-column proofs, indirect proofs, coordinate proofs, and verbal

arguments.  A gradual development of formal proof is encouraged. 

Inductive and intuitive approaches also should be used.

This set of standards includes emphasis on two- and

three-dimensional reasoning skills, coordinate and transformational

geometry, and the use of geometric models to solve problems.  A

variety of applications and some general problem-solving techniques

should be used to implement these standards, including algebraic

skills.  Calculators, computers, and graphing utilities (graphing

calculators or computer graphing simulators) should be used by the

student where feasible.  Any technology that will enhance student

learning should be used.

G.1   The student will construct and judge the validity of a

      logical argument consisting of a set of premises and a

      conclusion.  This will include

      * identifying the converse, inverse, and contrapositive of a

        conditional statement;

      * translating a short verbal argument into symbolic form;

      * diagramming arguments involving quantifiers (all, no,

        none, some), using Venn diagrams; and

      * using valid forms of deductive reasoning, including the

        law of syllogism.

G.2   The student will use pictorial representations, including

      computer software and coordinate methods to solve problems

      involving symmetry and transformation.  This will include

      * using formulas for finding distance, midpoint, and slope;

      * investigating 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.3   The student will solve practical problems involving

      complementary, supplementary, and congruent angles that

      include vertical angles, angles formed when parallel lines

      are cut by a transversal, and angles in polygons.

G.4   The student will use the relationships between angles formed

      by two lines cut by a transversal to determine if two lines

      are parallel and verify, using algebraic and coordinate

      methods as well as deductive proofs.

G.5   The student will

      * investigate and identify congruence and similarity

        relationships between triangles; and

      * prove two triangles are congruent or similar given

        information in the form of a figure or statement, using

        algebraic and coordinate as well as deductive proofs.

G.6   The student, given information concerning the lengths of

      sides and/or measures of angles, will apply the triangle

      inequality properties to determine whether a triangle exists

      and to order sides and angles.  These concepts will be

      considered in the context of practical situations.

G.7   The student will solve practical problems involving right

      triangles by using the Pythagorean Theorem and its converse,

      properties of special right triangles, and right triangle

      trigonometry.  Calculators will be used to solve problems and

      find decimal approximations for the solutions.

G.8   The student will

      * investigate and identify properties of quadrilaterals

        involving opposite sides and angles, consecutive sides and

        angles, and diagonals;

      * prove these properties of quadrilaterals using algebraic

        and coordinate as well as deductive proofs; and

      * use properties of quadrilaterals to solve practical

        problems.

G.9   The student will use measures of interior and exterior angles

      of polygons to solve problems.  Tessellations and tiling

      problems will be used to make connections to art,

      construction, and nature.

G.10  The student will investigate and use the properties of

      angles, arcs, chords, tangents, and secants to solve problems

      involving circles.  Problems will include finding the area of

      a sector and applications of architecture, art, and

      construction.

G.11  The student will construct, using a compass and straightedge,

      a line segment congruent to a given line segment, the

      bisector of a line segment, a perpendicular to a given line

      from a point not on the line, a perpendicular to a given line

      at a point on the line, the bisector of a given angle, and an

      angle congruent to a given angle.

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.

G.14  The student, given similar geometric objects, will use

      proportional reasoning to solve practical problems;

      investigate relationships between linear, square, and cubic

      measures; and describe how changes in one of the measures of

      the object affect the others.

G.15  The student will

      * draw a system of vectors and find the resultant

        graphically, write the components of a vector as a column

        matrix, and find the resultant by matrix addition; and

      * solve practical problems using a system of vectors.

  Mathematics

  Standards of Learning

  Trigonometry

The standards below outline the content for a one-semester course

in trigonometry.  A thorough treatment of trigonometry is provided

through the study of trigonometric definitions, applications,

graphing, and solving trigonometric equations and inequalities. 

Emphasis should be placed on using connections between right

triangle ratios, trigonometric functions, and circular functions. 

In addition, applications and modeling should be included

throughout the course of study.  Emphasis should be placed on oral

and written communication concerning the language of mathematics,

logic of procedure, and interpretation of results.  Students

enrolled in trigonometry are assumed to have mastered those

concepts outlined in the Algebra II standards.

Graphing utilities (graphing calculators or computer graphing

simulators) will be used by students and teachers.  Graphing

utilities enhance the understanding of realistic applications

through modeling and aid in the investigation of trigonometric

functions and their inverses.  They also provide a powerful tool

for solving/verifying trigonometric equations and inequalities. 

Any other technology that will enhance student learning should be

used if available.

T.1   The student will use the definitions of the six trigonometric

      functions to find the sine, cosine, tangent, cotangent,

      secant, and cosecant of an angle in standard position, given

      a point, other than the origin, on the terminal side of the

      angle.  Circular function definitions will be connected with

      trigonometric function definitions.

T.2   The student, given the value of one trigonometric function,

      will find the values of the other trigonometric functions. 

      Properties of the unit circle and definitions of circular

      functions will be applied.

T.3   The student will find the values of the trigonometric

      functions of the special angles and their related angles as

      found in the unit circle without the aid of a calculating

      utility.  This will include converting radians to degrees and

      vice versa.

T.4   The student will use a calculator to find the value of any

      trigonometric function and inverse trigonometric function.

T.5   The student will verify basic trigonometric identities and

      make substitutions using the basic identities.

T.6   The student, given one of the six trigonometric functions in

      standard form (e.g.,  y = Asin (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.

T.7   The student will identify the domain and range of the inverse

      trigonometric functions and recognize the graph of these

      functions.  Restrictions on the domains of the inverse

      trigonometric functions will be included.

T.8   The student will solve trigonometric equations that include

      both infinite solutions and restricted domain solutions and

      solve basic trigonometric inequalities.  Graphing utilities

      will be used to solve equations, to check for reasonableness

      of results, and to verify algebraic solutions.

T.9   The student will identify, create, and solve practical

      problems involving triangles and vectors.  Techniques will

      include using the trigonometric functions, the Pythagorean

      Theorem, the Law of Sines, and the Law of Cosines.

  Mathematics

  Standards of Learning

  Algebra II and Trigonometry

The standards for this combined course in Algebra II and

Trigonometry include all of the standards listed for Algebra II and

Trigonometry.  This course is designed for advanced students who

are capable of a more rigorous course at an accelerated pace.  The

standards listed for this course provide the foundation for

students to pursue a sequence of advanced mathematical studies from

Mathematical Analysis to Advanced Placement Calculus.

AII/T.1    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/T.2    The student will add, subtract, multiply, divide, and

           simplify rational expressions, including complex

           fractions.

AII/T.3    The student will

           *  add, subtract, multiply, divide, and simplify radical

              expressions containing positive rational numbers and

              variables and expressions containing rational exponents;

              and

           *  write radical expressions as expressions containing

              rational exponents and vice versa.

AII/T.4    The student will solve absolute value equations and

           inequalities graphically and algebraically.  Graphing

           calculators will be used both as a primary method of

           solution and to verify algebraic solutions.

AII/T.5    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/T.6    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 confirming algebraic solutions.

AII/T.7    The student will solve equations containing rational

           expressions and equations containing radical expressions

           algebraically and graphically.  Graphing calculators

           will be used both as a primary tool for solving and

           confirming algebraic solutions.

AII/T.8    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/T.9    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 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,

           including exponential and logarithmic.

AII/T.10   The student will investigate and describe the

           relationships between the solution of an equation, zero

           of a function, x-intercept of a graph, and factors of a

           polynomial expression through the use of graphs.

AII/T.11   The student will use matrix multiplication to solve

           practical problems.  Graphing calculators or computer

           programs with matrix capabilities will be used to find

           the product.

AII/T.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/T.13   The student will solve systems of linear inequalities

           and linear programming problems and describe the results

           both orally and in writing.  A graphing calculator will

           be used to facilitate solutions to linear programming

           problems.

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 functions, locate the zeros, sketch the

           graphs, and verify graphical solutions algebraically. 

           The graphing calculator will be used as a tool to

           investigate the shape and behavior of polynomial

           functions.

AII/T.16   The student will investigate and apply the properties of

           arithmetic and geometric sequences and series to solve

           problems, including writing the first n terms, finding

           the nth term, and evaluating summation formulas. 

           Notation will include sigma and 'a sub n'.

AII/T.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/T.18   The student will identify conic sections  (circle,

           ellipse, parabola, and hyperbola) from his/her

           equations.  Given the equations in (h, k) form, students

           will sketch graphs, using transformations.

AII/T.19   The student will collect and analyze data to

AII/T.18   The student will identify conic sections  (circle,

           ellipse, parabola, and hyperbola) from his/her

           equations.  Given the equations in (h, k) form, students

           will sketch graphs, using transformations.

AII/T.19   The student will collect and analyze data to make

           predictions, write equations, and solve practical

           problems.  Graphing calculators will be used to

           investigate scatterplots to determine the equation for a

           curve of best fit.

AII/T.20   The student will solve practical problems involving a

           combination of direct and inverse variations.

AII/T.21   The student will use the definitions of the six

           trigonometric functions to find the sine, cosine,

           tangent, cotangent, secant, and cosecant of an angle in

           standard position, given a point, other than the origin,

           on the terminal side of the angle.  Circular function

           definitions will be connected with trigonometric

           function definitions.

AII/T.22   The student, given the value of one trigonometric

           function, will find the values of the other

           trigonometric functions.  Properties of the unit circle

           and definitions of circular functions will be applied.

AII/T.23   The student will find the values of the trigonometric

           functions of the special angles and their related angles

           as found in the unit circle without the aid of a

           calculating utility.  This will include converting

           radians to degrees and vice versa.

AII/T.24   The student will use a calculator to find the value of

           any trigonometric function and inverse trigonometric

           function.

AII/T.25   The student will verify basic trigonometric identities

           and make substitutions using the basic identities.

AII/T.26   The student, given one of the six trigonometric

           functions in standard form 

           (e.g.,  y = Asin (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.

AII/T.27   The student will identify the domain and range of the

           inverse trigonometric functions and recognize the graph

           of these functions.  Restrictions on the domains of the

           inverse trigonometric functions will be included.

AII/T.28   The student will solve trigonometric equations that

           include both infinite solutions as well as restricted

           domain solutions and solve basic trigonometric

           inequalities.  Graphing utilities will be used to solve

           equations, to check for reasonableness of results, and

           to verify algebraic solutions.

AII/T.29   The student will identify, create, and solve practical

           problems involving triangles and vectors.  Techniques

           will include using the trigonometric functions, the

           Pythagorean Theorem, the Law of Sines, and the Law of

           Cosines.