Mathematics

   Standards of Learning

  Mathematical Analysis

The standards below outline the content for a one-year 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, y-intercepts, 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

      piece-wise, 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

      piece-wise-defined 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.

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   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

1995-1996 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, one-sided 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

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 p-series.

   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 concepts-Standards 1 through 21-and those

dealing with mathematical applications-Standards 22 and 24.  This

separation is not intended to suggest that they be treated

separately in the instructional program.  Programming concepts,

problem-solving 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 pre-conditions,

        post-conditions, 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 step-by-step 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 user-defined

        function, subroutines, and procedures.  Enrichment

        topics can include user-defined libraries (units) and

        object-oriented 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 built-in (library)

        functions in processing data, which include trigonometric

        functions, absolute value functions, random number

        functions, end of line, end of file, and string.

  Educational Technology Standards
  and Performance Indicators
   for All Teachers

Building on the NETS for Students, the ISTE NETS for Teachers (NETS•T), which focus on preservice teacher education, define the fundamental concepts, knowledge, skills, and attitudes for applying technology in educational settings. All candidates seeking certification or endorsements in teacher preparation should meet these educational technology standards. It is the responsibility of faculty across the university and at cooperating schools to provide opportunities for teacher candidates to meet these standards.

The six standards areas with performance indicators listed below are designed to be general enough to be customized to fit state, university, or district guidelines and yet specific enough to define the scope of the topic. Performance indicators for each standard provide specific outcomes to be measured when developing a set of assessment tools.The standards and the performance indicators also provide guidelines for teachers currently in the classroom .

I.                        TECHNOLOGY OPERATIONS AND CONCEPTS.
Teachers demonstrate a sound understanding of technology operations and concepts. Teachers:

A.     demonstrate introductory knowledge, skills, and understanding of concepts related to technology (as described in the ISTE National Education Technology Standards for Students)

B.     demonstrate continual growth in technology knowledge and skills to stay abreast of current and emerging technologies.

                             II.            PLANNING AND DESIGNING LEARNING ENVIRONMENTS AND EXPERIENCES.
Teachers plan and design effective learning environments and experiences supported by technology. Teachers:

 .        design developmentally appropriate learning opportunities that apply technology-enhanced instructional strategies to support the diverse needs of learners.

A.     apply current research on teaching and learning with technology when planning learning environments and experiences.

B.     identify and locate technology resources and evaluate them for accuracy and suitability.

C.     plan for the management of technology resources within the context of learning activities.

D.     plan strategies to manage student learning in a technology-enhanced environment.

                           III.            TEACHING, LEARNING, AND THE CURRICULUM.
Teachers implement curriculum plans, that include methods and strategies for applying technology to maximize student learning. Teachers:

 .        facilitate technology-enhanced experiences that address content standards and student technology standards.

A.     use technology to support learner-centered strategies that address the diverse needs of students.

B.     apply technology to develop students' higher order skills and creativity.

C.     manage student learning activities in a technology-enhanced environment.

                          IV.            ASSESSMENT AND EVALUATION.
Teachers apply technology to facilitate a variety of effective assessment and evaluation strategies. Teachers:

 .        apply technology in assessing student learning of subject matter using a variety of assessment techniques.

A.     use technology resources to collect and analyze data, interpret results, and communicate findings to improve instructional practice and maximize student learning.

B.     apply multiple methods of evaluation to determine students' appropriate use of technology resources for learning,communication,and productivity.

                             V.            PRODUCTIVITY AND PROFESSIONAL PRACTICE.
Teachers use technology to enhance their productivity and professional practice. Teachers:

 .        use technology resources to engage in ongoing professional development and lifelong learning.

A.     continually evaluate and reflect on professional practice to make informed decisions regarding the use of technology in support of student learning.

B.     apply technology to increase productivity.

C.     use technology to communicate and collaborate with peers, parents, and the larger community in order to nurture student learning.

                          VI.            SOCIAL, ETHICAL, LEGAL, AND HUMAN ISSUES.
Teachers understand the social,ethical,legal,and human issues surrounding the use of technology in PK-12 schools and apply those principles in practice. Teachers:

 .        model and teach legal and ethical practice related to technology use.

A.     apply technology resources to enable and empower learners with diverse backgrounds, characteristics, and abilities.

B.     identify and use technology resources that affirm diversity

C.     promote safe and healthy use of technology resources.

D.     facilitate equitable access to technology resources for all students.

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     Technology Foundation Standards
     for All Students

The technology foundation standards for students are divided into six broad categories. Standards within each category are to be introduced, reinforced, and mastered by students. These categories provide a framework for linking performance indicators within the Profiles for Technology Literate Students to the standards. Teachers can use these standards and profiles as guidelines for planning technology-based activities in which students achieve success in learning, communication, and life skills.

Technology Foundation Standards for Students

1. Basic operations and concepts

• Students demonstrate a sound understanding of the nature and operation of technology systems.
• Students are proficient in the use of technology.
  
   
1. Social, ethical, and human issues
   • Students understand the ethical, cultural, and societal issues related to technology.
   • Students practice responsible use of technology systems, information, and software.
   • Students develop positive attitudes toward technology uses that support lifelong learning, collaboration, personal pursuits, and productivity.
     
      
2. Technology productivity tools
   • Students use technology tools to enhance learning, increase productivity, and promote creativity.
   • Students use productivity tools to collaborate in constructing technology-enhanced models, prepare publications, and produce other creative works.
     
      
3. Technology communications tools
   • Students use telecommunications to collaborate, publish, and interact with peers, experts, and other audiences.
   • Students use a variety of media and formats to communicate information and ideas effectively to multiple audiences.
     
      
4. Technology research tools
   • Students use technology to locate, evaluate, and collect information from a variety of sources.
   • Students use technology tools to process data and report results.
   • Students evaluate and select new information resources and technological innovations based on the appropriateness for specific tasks.
     
      
5. Technology problem-solving and decision-making tools
   • Students use technology resources for solving problems and making informed decisions.
   • Students employ technology in the development of strategies for solving problems in the real world.

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