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Standards of Learning
Algebra I
The standards below outline the content for a oneyear course in
Algebra I. All students are expected to achieve the Algebra I
standards. When planning for instruction, consideration should be
given to the student's cognitive level and readiness for dealing
with abstract concepts. Students should be helped to make
connections and to build relationships between algebra and
arithmetic, geometry, and probability and statistics. Connections
also should be made to other subject areas through practical
applications. This approach to teaching algebra should help
students attach meaning to the abstract concepts of algebra.
These standards require students to use algebra as a tool for
representing and solving a variety of practical problems. Tables
and graphs will be used to interpret algebraic expressions,
equations, and inequalities and to analyze functions. Matrices
will be used to organize and manipulate data.
Calculators, computers, spreadsheets, and graphing utilities
(graphing calculators or computer graphing simulators) should be
used as tools to assist in problem solving. Graphing utilities
enhance the understanding of functions; they provide a powerful
tool for solving and verifying solutions to equations and
inequalities.
Throughout the course, students should be encouraged to talk about
mathematics, to use the language and symbols of mathematics to
communicate, to discuss problems and problem solving, and to
develop their confidence in mathematics.
A.1 The student will solve 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.2 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.3 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.
A.4 The student will use matrices to organize and manipulate
data, including matrix addition, subtraction, and scalar
multiplication. Data will arise from business, industrial,
and consumer situations.
A.5 The student will analyze a given set of data for the
existence of a pattern, represent the pattern algebraically
and graphically, if possible, and determine if the relation
is a function.
A.6 The student will select, justify, and apply an appropriate
technique to graph a linear function in two variables.
Techniques will include slopeintercept, x and yintercepts,
graphing by transformation, and the use of the graphing
calculator.
A.7 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.8 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.9 The student will solve systems of two linear equations in two
variables, both algebraically and graphically, and apply
these techniques to solve practical problems. Graphing
calculators will be used as both a primary tool of solution
and to confirm an algebraic solution.
A.10 The student will apply the laws of exponents to perform
operations on expressions with integral exponents, using
scientific notation when appropriate.
A.11 The student will add, subtract, and multiply polynomials and
divide polynomials with monomial divisors, using concrete
objects, pictorial representations, and algebraic
manipulations.
A.12 The student will factor completely first and seconddegree
binomials and trinomials in one or two variables. The
graphing calculator will be used as both a primary tool for
factoring and for confirming an algebraic factorization.
A.13 The student will estimate square roots to the nearest tenth
and use a calculator to compute decimal approximations of
radicals.
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.15 The student will determine the domain and range of a relation
given a graph or a set of ordered pairs and will identify the
relations that are functions.
A.16 The student will, given a rule, find the values of a function
for elements in its domain and locate the zeros of the
function both algebraically and with a graphing calculator.
The value of f(x) will be related to the ordinate on the
graph.
A.17 The student will, given a set of data points, write an
equation for a line of best fit, using the median fit method,
and use the equation to make predictions.
A.18 The student will compare multiple onevariable data sets,
using statistical techniques that include measures of central
tendency, range, stemandleaf plots, and boxandwhisker
graphs.
A.19 The student will analyze a relation to determine whether a
direct or inverse variation exists and represent it
algebraically and graphically, if possible.
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,
twocolumn 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
threedimensional reasoning skills, coordinate and transformational
geometry, and the use of geometric models to solve problems. A
variety of applications and some general problemsolving 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 threedimensional figure
from a twodimensional drawing and make a twodimensional
representation of a threedimensional 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
threedimensional 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
Algebra II
The standards below outline the content for a oneyear course in
Algebra II. Students enrolled in Algebra II are assumed to have
mastered those concepts outlined in the Algebra I standards. A
thorough treatment of advanced algebraic concepts is provided
through the study of functions, polynomials, rational expressions,
complex numbers, matrices, and sequences and series. Emphasis
should be placed on practical applications and modeling throughout
the course of study. Oral and written communication concerning the
language of algebra, logic of procedures, and interpretation of
results also should permeate the course.
These standards include a transformational approach to graphing
functions. Transformational graphing uses translation, reflection,
dilation, and rotation to generate a "family of graphs" from a
given graph and builds a strong connection between algebraic and
graphic representations of functions. Students will vary the
coefficients and constants of an equation, observe the changes in
the graph of the equation, and make generalizations that can be
applied to many graphs.
Graphing utilities (graphing calculators or computer graphing
simulators) and spreadsheets will be used by students and teachers.
Graphing utilities enhance the understanding of realistic
applications through mathematical modeling and aid in the
investigation and study of functions and their inverses. They also
provide an effective tool for solving/verifying equations and
inequalities. Any other available technology that will enhance
student learning should be used.
AII.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.2 The student will add, subtract, multiply, divide, and
simplify rational expressions, including complex fractions.
AII.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.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.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.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.7 The student will solve equations containing rational
expressions and equations containing radical expressions
algebraically and graphically. Graphing calculators will be
used for solving and confirming algebraic solutions.
AII.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.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.10 The student will investigate and describe the
relationships between the solution of an equation, zero of
a function, xintercept of a graph, and factors of a
polynomial expression through the use of graphs.
AII.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.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 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.14 The student will solve nonlinear systems of equations,
including linearquadratic and quadraticquadratic,
algebraically and graphically. The graphing calculator
will be used as a tool to visualize graphs and predict the
number of solutions.
AII.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.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.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 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
of conic sections, using transformations.
AII.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.20 The student will identify, create, and solve practical
problems involving a combination of direct and inverse
variations.
Mathematics
Standards of Learning
Trigonometry
The standards below outline the content for a onesemester 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 oneperiod 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, xintercept 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 linearquadratic and quadraticquadratic,
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 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 oneperiod 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.
Mathematics
Standards of Learning
Mathematical Analysis
The standards below outline the content for a oneyear course in
Mathematical Analysis. Mathematical Analysis is intended not only
to extend students' knowledge of function characteristics but also
to introduce them to another mode of mathematical reasoning.
Students enrolled in Mathematical Analysis are assumed to have
mastered Algebra II concepts and have some exposure to
trigonometry. The content of this course will serve as appropriate
preparation for a calculus course.
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 functions and
their inverses. They also provide a powerful tool for solving and
verifying equations and inequalities. Any other technology that
will enhance student learning should be used if available.
MA.1 The student will investigate and identify the characteristics
of polynomial and rational functions and use these to sketch
the graphs of the functions. This will include determining
zeros, upper and lower bounds, yintercepts, symmetry,
asymptotes, intervals for which the function is increasing or
decreasing, and maximum or minimum points. Graphing
utilities will be used to investigate and verify these
characteristics.
MA.2 The student will perform operations, including composition
and inversion of functions, and determine the domain and
range of results. Continuity of functions and special
functions such as absolute value, step functions, and
piecewise, will be included. Curve sketching and
transformations will be included. Graphing utilities will be
used to investigate and verify the graphs.
MA.3 The student will use graphs to investigate and describe the
continuity of functions. The functions will include
piecewisedefined and step functions.
MA.4 The student will expand binomials having positive integral
exponents through the use of the Binomial Theorem, the
formula for combinations, and Pascal's Triangle.
MA.5 The student will solve problems involving arithmetic and
geometric sequences and series. This will include finding
the sum (sigma notation included) of finite and infinite
convergent series that will lead to an intuitive approach to
a limit.
MA.6 The student will apply the method of mathematical induction
to prove formulas/statements.
MA.7 The student will find the limit of an algebraic function, if
it exists, as the variable approaches either a finite number
or infinity. A graphing utility will be used to verify
intuitive reasoning, algebraic methods, and numerical
substitution.
MA.8 The student will apply the techniques of translation and
rotation of axes in the coordinate plane to graphing
functions and conic sections. A graphing utility will be
used to investigate and verify the graphs. Matrices will be
used to represent transformations.
MA.9 The student will investigate and identify the characteristics
of exponential and logarithmic functions in order to graph
these functions and to solve equations and practical
problems. This will include the role of e, natural and
common logarithms, laws of exponents and logarithms, and the
solution of logarithmic and exponential equations. Graphing
utilities will be used to investigate and verify the graphs
and solutions.
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.11 The student will perform operations with vectors in the
coordinate plane and solve practical problems using vectors.
This will include the following topics: operations of
addition, subtraction, scalar multiplication, and inner (dot)
product; norm of a vector; unit vector; graphing; properties;
simple proofs; complex numbers (as vectors); and
perpendicular components.
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.
MA.13 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
Advanced Placement Calculus
This course is intended for students who have a thorough knowledge
of analytic geometry and elementary functions in addition to
college preparatory algebra, geometry, and trigonometry. The
purpose of the course is to prepare the student for advanced
placement in college calculus. These standards incorporate the
19951996 College Board Advanced Placement Course Description
Syllabus. Teachers should update course content as changes occur
in future College Board publications.
As mandated by The College Board, graphing calculators will be
required for this course. Computers should be used where feasible
by the student and by the teacher. Any technology that will
enhance student learning should be used if available.
Instructional activities that engage students in solving
application problems of varying complexities are encouraged.
APC.1 The student will define and apply the properties of
elementary functions, including algebraic, trigonometric,
exponential, and composite functions and their inverses, and
graph these functions using a graphing calculator.
Properties of functions will include domains, ranges,
combinations, odd, even, periodicity, symmetry, asymptotes,
zeros, upper and lower bounds, and intervals where the
function is increasing or decreasing.
APC.2 The student will define and apply the properties of limits
of functions. This will include limits of a constant, sum,
product, quotient, onesided limits, limits at infinity,
infinite limits, and nonexistent limits.
*AP Calculus BC will include the rigorous definitions of a
limit.
APC.3 The student will state the definition of continuity and
determine where a function is continuous or discontinuous.
This will include
* continuity at a point;
* continuity over a closed interval;
* application of the Intermediate Value Theorem; and
* graphical interpretation of continuity and discontinuity.
APC.4 The student will find the derivative of an algebraic
function by using the definition of a derivative. This will
include investigating and describing the relationship
between differentiability and continuity.
APC.5 The student will apply formulas to find the derivative of
algebraic, trigonometric, exponential, and logarithmic
functions and their inverses.
APC.6 The student will apply formulas to find the derivative of
the sum, product, quotient, inverse, and composite (chain
rule) of elementary functions.
APC.7 The student will find the derivative of an implicitly
defined function.
APC.8 The student will find the higher order derivatives of
algebraic, trigonometric, exponential, and logarithmic
functions.
APC.9 The student will use logarithmic differentiation as a
technique to differentiate nonlogarithmic functions.
APC.10 The student will state (without proof) the Mean Value
Theorem for derivatives and apply it both algebraically and
graphically.
APC.11 The student will use l'Hopital's rule to find the limit of
functions whose limits yield the indeterminate forms:
0/0 and infinity/infinity
* For AP Calculus BC, these functions will also include
functions whose limits yield the indeterminate forms:
0 to the 0th power
1 to the infinity power
infinity to the infinity power
infinity minus infinity
APC.12 The student will apply the derivative to solve problems,
including tangent and normal lines to a curve, curve
sketching, velocity, acceleration, related rates of change,
Newton's method, differentials and linear approximations,
and optimization problems.
APC.13 The student will find the indefinite integral of algebraic,
exponential, logarithmic, and trigonometric functions. The
special integration techniques of substitution (change of
variables) and integration by parts will be included.
*AP Calculus BC will also include integration by
trigonometric substitution and integration by partial
fractions (only linear factors in the denominator).
APC.14 The student will identify the properties of the definite
integral. This will include the Fundamental Theorem of
Calculus and the definite integral as an area and as a
limit of a sum as well as the fundamental theorem:
The integral from a to x of f(t)d(t) dt/dx = f(x)
*AP Calculus BC will include composite functions defined by
integrals, e.g.,
f(x) = the integral from 0 to x squared of
e to the t squared power d(t)
APC.15 The student will apply the definite integral to solve
problems. These problems will include finding distance
traveled on a line and velocity from acceleration with
initial conditions, growth and decay problems, solutions of
separable differential equations, the average value of a
function, area between curves, volumes of solids of
revolution about the axes or lines parallel to the axes
using disc/washer and shell methods, and volumes of solids
with known crosssectional areas.
* AP Calculus BC will also include areas bounded by polar
curves.
APC.16 The student will compute an approximate value for a definite
integral. This will include numerical calculations using
Riemann Sums and the Trapezoidal Rule.
*AP Calculus BC will also utilize Simpson's Rule.
*APC.17 The student will find the derivatives of
vector functions and parametrically defined
functions and use them to solve problems.
The problems will include tangent and normal
lines to parametrically defined curves,
velocity and acceleration, and velocity and
acceleration vectors for motion on a plane
curve.
*APC.18 The student will use integration to solve
problems. This will include areas bounded by
polar curves, length of a path (including
parametric curves), work (Hooke's law), and
improper integrals.
*APC.19 The student will define and test for
convergence of a series of real numbers and
of functions. This will include geometric
series, comparison (including limit
comparison), ratio, root, and integral tests,
absolute and conditional convergence,
alternating series and error approximation,
and pseries.
*APC.20 The student will define, restate, and apply
power series. This will include addition,
substitution, termbyterm differentiation
and integration, interval of convergence,
Taylor's series, Maclaurin series expansions,
and Taylor polynomials with remainder and
Lagrange error approximation.
* For those students who are enrolled in AP Calculus BC.
Mathematics
Standards of Learning
Computer Mathematics
This Computer Mathematics course is intended to provide students
with experiences in using the computer to solve problems which can
be set up as mathematical models. Students who successfully
complete the standards for this course may earn high school
mathematics credit. It is recognized that many students will gain
computer skills in other mathematics courses or in a separate
curriculum outside of mathematics and prior to high school. In
such cases, the standards indicated by an asterisk (*) should be
included in the student's course of study and treated as a review
for those students who enroll in Computer Mathematics.
Even though computer ideas should be introduced in the context of
mathematical concepts, problem solving per se should be developed
in the most general sense, making the techniques applicable by
students in many other environments. Strategies include defining
the problem; developing, refining, and implementing a plan; and
testing and revising the solution. Programming, ranging from
simple programs involving only a few lines to complex programs
involving subprograms, should permeate the entire course.
These standards identify fundamental principles and concepts in the
field of computer science. Students will develop and refine skills
in logic, organization, and precise expression that will enhance
learning in other disciplines.
The standards that follow are separated into two groups: those
related to programming conceptsStandards 1 through 21and those
dealing with mathematical applicationsStandards 22 and 24. This
separation is not intended to suggest that they be treated
separately in the instructional program. Programming concepts,
problemsolving strategies, and mathematical applications should be
integrated throughout the course.
*COM.1 The student will describe the program development cycle:
defining the problem, planning a solution, carrying out
the plan, debugging the program, and providing program
documentation.
*COM.2 The student will write program specifications that
define the constraints of a given problem. These
specifications include descriptions of preconditions,
postconditions, the desired output, analysis of the
available input, and an indication as to whether or not
the program is solvable under the given conditions.
*COM.3 The student will design a stepbystep plan (algorithm)
to solve a given problem. The plan will be in the form
of a program flowchart, pseudo code, a hierarchy chart
and/or data flow diagram.
*COM.4 The student will use operating system commands, which
include creating a new file, opening an existing file,
saving a file, making a printed copy (hard copy) of the
file, and executing a program.
*COM.5 The student will divide a given problem into manageable
sections (modules) by task and implement the solution.
The modules will include an appropriate userdefined
function, subroutines, and procedures. Enrichment
topics can include userdefined libraries (units) and
objectoriented programming.
*COM.6 The student will design and implement the input phase of a
program, which will include designing screen layout and
getting information into the program by way of user
interaction, data statements (BASIC), and/or file input.
The input phase also will include methods of filtering out
invalid data (error trapping).
*COM.7 The student will design and implement the output phase of
a computer program, which will include designing output
layout, accessing a variety of output devices, using
output statements, and labeling results.
COM.8 The student will design and implement computer graphics,
which will include topics appropriate for the available
programming environment as well as student background.
Students will use graphics as an end in itself, as an
enhancement to other output, and as a vehicle for
reinforcing programming techniques.
COM.9 The student will define simple variable data types that
include integer, real (fixed and scientific notation),
character, string, and Boolean.
COM.10 The student will use appropriate variable data types,
including integer, real (fixed and scientific notation),
character, string, and Boolean. This will also include
variables representing structured data types.
*COM.11 The student will describe the way the computer stores,
accesses, and processes variables, including the following
topics: the use of variables versus constants, variables
addresses, pointers, parameter passing, scope of
variables, and local versus global variables. This will
also include use of terminology, including memory, CPU,
RAM, ROM, baud, byte, bits, floppy disc, and hard drive.
COM.12 The student will translate a mathematical expression into
a computer statement, which involves writing assignment
statements and using the order of operations.
COM.13 The student will select and implement builtin (library)
functions in processing data, which include trigonometric
functions, absolute value functions, random number
functions, end of line, end of file, and string.
COM.14 The student will implement conditional statements that
include if/then, if/then/else, case statements, and
Boolean logic.
COM.15 The student will implement a loop, including iterative
loops, pretest loops, and posttest loops. Other topics
will include single entry point, single exit point,
preconditions, postconditions and loop invariance.
*COM.16 The student will select and implement appropriate data
structures, including arrays (onedimensional and/or
multidimensional), files, and records. Implementation
will include creating the data structure, putting
information into the structure, and retrieving information
from the structure.
*COM.17 The student will implement preexisting algorithms,
including sort routines, search routines, and animation
routines.
COM.18 The student will test a program using an appropriate set
of data. The set of test data should be appropriate and
complete for the type of program being tested.
COM.19 The student will debug a program using appropriate
techniques (e.g., appropriately placed controlled breaks,
the printing of intermediate results, and other debugging
tools available in the programming environment), and
identify the difference between syntax errors and logic
errors.
COM.20 The student will properly document a program including the
preconditions and postconditions of program segments,
input/output specifications, the stepbystep plan, the
test data, a sample run, and the program listing with
appropriately placed comments.
COM.21 The student will design, write, test, debug, and document
a complete structured program which requires the synthesis
of many of the concepts contained in previous standards.
*COM.22 The student will solve practical consumer problems that
involve analyzing and interpreting graphs, charts, and/or
tables.
COM.23 The student will solve mathematical problems using
formulas, equations, and functions. Problems will include
those related to geometry, business, and leisure (e.g.,
sports and recreational activities).
COM.24 The student will solve probability, data analysis, and
statistical problems.
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