West Virginia Code of State Rules › Agency 126 - Education › Title 126 - LEGISLATIVE RULE BOARD OF EDUCATION › Series 126-044B - 21st Century Mathematics Content Standards and Objectives for West Virginia Schools (2520.2) › Section 126-44B-5 - Severability

Current through Register Vol. XLI, No. 28, July 12, 2024
5.1. If any
				provision of this rule or the application thereof to any person or circumstance
				is held invalid, such invalidity shall not affect other provisions or
				applications of this rule.
Foreword
A 21st century mathematics
				  curriculum is an increasingly important aspect of developing learners prepared
				  for success in the 21st century. Thus, the West
				  Virginia Board of Education and the West Virginia Department of Education are
				  pleased to present Policy 2520.2, 21st Century
				  Mathematics Content Standards and Objectives for West Virginia Schools. The
				  West Virginia Mathematics Standards for 21st Century
				  Learning includes 21st century
content
standards and objectives as well as
				  21st century standards and objectives for
learning skills
and
technology tools
. This
				  broadened scope of mathematics curriculum is built on the firm belief that
				  quality engaging instruction must be built on a curriculum that triangulates
				  rigorous 21st century content,
				  21st century learning skills and the use of
				  21st century technology tools.
Committees of educators from across the state convened
				  to revise the content standards and objectives. The overarching goal was to
				  build a rigorous, relevant and challenging mathematics curriculum that would
				  prepare students for the 21st century. West Virginia
				  educators, including regular classroom teachers, special education teachers,
				  and teachers representing higher education institutions played a key role in
				  shaping the content standards to align with national standards, rigorous
				  national assessments and research and best practice in the field of mathematics
				  education. The contribution of these professionals was critical in creating a
				  policy that is meaningful to classroom teachers and appears in a format that
				  can easily be used and understood.
Policy 2520.2 is organized around the three major
				  components of a standards-based curriculum: learning standards, instructional
				  objectives and performance descriptors. The learning standards are the
broad descriptions
of what
all
students must
				  know and be able to do at the conclusion of the instructional sequence. The
				  accompanying grade-level objectives are specific descriptors of knowledge,
				  skills and attitudes that when mastered will enable the student to attain the
				  standard. The instructional objectives guide instructional
planning
and provide a basis for determining appropriate
assessments, instructional strategies and resources
. The
				  performance descriptors provide the basis for
assessing
overall student competence of grade level standards. The performance
				  descriptors define the five student performance levels ranging from novice to
				  distinguished. With the ultimate goal of "learning for all," these descriptors
				  allow the teacher, students and parents to judge the
level
of
				  student proficiency in each 21st century learning
				  standard.
In combination, the use of learning standards,
				  instructional objectives and performance descriptors become a comprehensive
				  guide for delivering a rigorous and relevant mathematics curriculum to all West
				  Virginia students. These elements, when used to guide the instructional process
				  and when delivered with the creativity and instructional expertise of West
				  Virginia teachers, will become a powerful resource for preparing students to
				  meet the challenges of the 21st
				  century.
Explanation of Terms
Content Standards
are broad descriptions of what
				  students should know and be able to do in a content area. Content standards
				  describe what students' knowledge and skills should be at the end of a 9-12
				  sequence of study.
Objectives
are incremental steps toward
				  accomplishment of content standards. Objectives are listed by grade level and
				  are organized around the content standards. Objectives build across grade
				  levels as students advance in their knowledge and skills.
Performance Descriptors
describe in narrative
				  format how students demonstrate achievement of the content standards. Line
				  breaks within the narrative format indicate clusters of concepts and skills.
				  West Virginia has designed five performance levels: distinguished, above
				  mastery, mastery, partial mastery and novice. Performance Descriptors serve two
				  functions. Instructionally, they give teachers more information about the level
				  of knowledge and skills students need to acquire. Performance levels and
				  descriptors are also used to categorize and explain student performance on
				  statewide assessment instruments.
Distinguished:
A student at this level has
				  demonstrated exemplary performance. The work shows a distinctive and
				  sophisticated application of knowledge and skills in real world situations that
				  go beyond course or grade level applications.
Above Mastery:
A student at this level has
				  demonstrated effective performance and exceeds the standard. The work shows a
				  thorough and effective application of knowledge and skills in real world
				  situations within the subject matter and grade level.
Mastery:
A student at this level has
				  demonstrated competency over challenging subject matter, including knowledge
				  and skills that are appropriate to the subject matter and grade level. The work
				  is accurate, complete and addresses real world applications. The work shows
				  solid academic performance at the course or grade level.
Partial Mastery:
A student at this level has
				  demonstrated limited knowledge and skills toward meeting the standard. The work
				  shows basic but inconsistent application of knowledge and skills characterized
				  by errors and/or omissions. Performance needs further development.
Novice:
A student at this level has demonstrated
				  minimal fundamental knowledge and skills needed to meet the standard.
				  Performance at this level is fragmented and/or incomplete and needs
				  considerable development.
Abbreviations
Content Area
M Mathematics
High School Courses
Mathematics
A1 Algebra
A2 Algebra II
A3 Algebra III
C Calculus
CM Conceptual Mathematics
G Geometry
PC Pre-calculus
PS Probability and Statistics
T Trigonometry
Other Abbreviations
O Objective
D Performance Descriptors
S Standard (Content Standard)
MATHEMATICS - POLICY 2520.2
The six principles for school mathematics, as articulated in
Principles and Standards for School Mathematics,
address six
				  overarching themes to be considered when focused on the continuous improvement
				  of mathematics education:
1.
Equity.
High expectations and strong support for all student
2.
Curriculum.
Coherent focus on
				important mathematics that is well-articulated across the grades
3.
Teaching.
Understanding what
				students know and need to learn and then challenging and supporting them to
				learn it well
4.
Learning.
Students must learn mathematics with understanding,
				actively building new knowledge from experience and prior knowledge
5.
Assessment.
Assessment should
				support the learning of important mathematics and provide useful information to
				both teachers and students.
6.
Technology.
Technology is essential in teaching and learning
				mathematics; it influences the mathematics that is taught and enhances
				students' learning.
The standards, objectives and performance descriptors
				  presented in this policy are designed to provide clear, consistent priorities
				  and focus, as well as depth of knowledge. The standards describe what each
				  student of mathematics should be able to accomplish in grades 9-12. The
				  objectives spiral upward through the grade levels, eliminating repetition of
				  content and increasing in rigor and depth of knowledge throughout the student's
				  academic career. It is important that all students value mathematics and see
				  themselves as mathematical problem solvers who can communicate mathematically
				  and make connections to other content areas and the real-world application of
				  mathematics.
The vision of the West Virginia Board of Education and the
				  West Virginia Department of Education includes the triangulation of mathematics
				  content, learning skills and technology tools standards within each classroom
				  so that students will be able to think critically, analyze information,
				  comprehend new ideas, communicate, collaborate, solve problems and make
				  decisions. All West Virginia mathematics teachers are responsible for the
				  integration of Policy 2520.14 21st Century Learning
				  Skills and Technology Tools in their classroom instruction.
It is important that teachers of mathematics become familiar
				  with the performance descriptors at each grade level. The Mastery level
				  performance descriptor, supported by the accompanying standard and objectives,
				  describes student proficiency at that grade level. An understanding of the
				  performance descriptors, standards and objectives provides a clear picture of
				  what every student should know, understand and be able to do at each grade
				  level. Teachers are encouraged to become familiar with the performance
				  descriptors and objectives at the previous and subsequent grade level to
				  support a well-articulated curriculum. The abbreviation
e.g.
is used to indicate examples for teaching the objectives.
Policy 2510 states that "students in the professional pathway
				  and college bound students in the skilled pathway, who do not achieve the State
				  assessment college readiness benchmarks for mathematics, shall be required to
				  take a college transition mathematics course during their senior year." In
				  keeping with this policy, representatives from the West Virginia Department of
				  Education and the Higher Education Policy commission assembled classroom
				  teachers and professors of mathematics to establish the college readiness
				  benchmarks for mathematics. An additional collaborative effort from classroom
				  teachers and mathematics professors resulted in identification of a set of
				  objectives from Policy 2520.2 courses in Algebra I, Geometry, Algebra II and
				  Trigonometry that align to those benchmarks. The educational program for any
				  student placed in a college transition mathematics course will be aligned to
				  those objectives identified for Transition Mathematics. Therefore the college
				  transition mathematics course is an individualized course relating to a
				  student's identified skill deficiencies as related to previously approved
				  objectives. Consequently, there is not an identified set of standards and
				  objectives for the college transitions mathematics course required by Policy
				  2510.
Mathematics Content Standards 9-12
Standard 1: Number and Operations
Through communication, representation, reasoning and proof,
				  problem solving, and making connections within and beyond the field of
				  mathematics, students will demonstrate an understanding of numbers, ways of
				  representing numbers, and relationships among numbers and number systems,
				  demonstrate meanings of operations and how they relate to one another, and
				  compute fluently and make reasonable estimates.
Standard 2: Algebra
Through communication, representation, reasoning and proof,
				  problem solving, and making connections within and beyond the field of
				  mathematics, students will demonstrate understanding of patterns, relations and
				  functions, represent and analyze mathematical situations and structures using
				  algebraic symbols, use mathematical models to represent and understand
				  quantitative relationships, and analyze change in various contexts.
Standard 3: Geometry
Through communication, representation, reasoning and proof,
				  problem solving, and making connections within and beyond the field of
				  mathematics, students will analyze characteristics and properties of two- and
				  three-dimensional geometric shapes and develop mathematical arguments about
				  geometric relationships, specify locations and describe spatial relationships
				  using coordinate geometry and other representational systems, apply
				  transformations and use symmetry to analyze mathematical situations, and solve
				  problems using visualization, spatial reasoning, and geometric modeling.
Standard 4: Measurement
Through communication, representation, reasoning and proof,
				  problem solving, and making connections within and beyond the field of
				  mathematics, students will demonstrate understanding of measurable attributes
				  of objects and the units, systems, and processes of measurement, and apply
				  appropriate techniques, tools and formulas to determine measurements.
Standard 5: Data Analysis and Probability
Through communication, representation, reasoning and proof,
				  problem solving, and making connections within and beyond the field of
				  mathematics, students will formulate questions that can be addressed with data
				  and collect, organize, and display relevant data to answer them,
select and use appropriate statistical methods to analyze
				  data, develop and evaluate inferences and predictions that are based on models,
				  and apply and demonstrate an understanding of basic concepts of
				  probability.
ALGEBRA I CONTENT STANDARDS AND
				  OBJECTIVES
Algebra I objectives provide the gateway to all higher
				  mathematics courses. An emphasis on conceptual development and multiple
				  representations will be used to draw generalizations and to serve as a tool for
				  solving real-world problems. Algeblocks may be used to bridge the gap from the
				  concrete to the abstract. Available technology such as calculators, computers,
				  and interactive utilities are to be used as tools to enhance learning. The West
				  Virginia Standards for 21st Century Learning include
				  the following components: 21st Century Content
				  Standards and Objectives and 21st Century Learning
				  Skills and Technology Tools. All West Virginia teachers are responsible for
				  classroom instruction that integrates learning skills, technology tools and
				  content standards and objectives.
Grade 9-12
Mathematics: Algebra I
Standard 2
Algebra
M.S.A1.2
Through communication, representation, reasoning and
							 proof, problem solving, and making connections within and beyond the field of
							 mathematics, students will
· demonstrate understanding of patterns,
							 relations and functions,
· represent and analyze mathematical
							 situations and structures using algebraic symbols,
· use mathematical models to represent and
							 understand quantitative relationships, and
· analyze change in various contexts.
Performance Descriptors (M.PD.A1.2)
Distinguished
Above Mastery
Mastery
Partial Mastery
Novice
Algebra I students at the distinguished level
							 will:
formulate and simplify algebraic expressions for use
							 in equations and inequalities, developing and justifying each step, derive and
							 use the laws of integral exponents;
create, solve, and concisely and clearly interpret
							 solutions for multi-step equations; and solve literal equations;
identify a real life situation with experiments to
							 collect, organize, and analyze related data in a clear concise manner for
							 display in multiple representations; formulate a conclusion; present the
							 project with clarity and conciseness;
model real-life situations involving exponential
							 growth and decay equations and summarize the relationship in a clear, concise
							 manner;
develop and explain operations with and factoring of
							 higher order polynomials, rational and radical expressions. Use intercepts on a
							 graph in problem solving;
use simulations and rules of probability to design
							 experiments to solve problems justifying the reasonableness of the approach in
							 a clear, concise manner.
Algebra I students at the above mastery level
							 will:
formulate and simplify algebraic expressions for use
							 in equations and inequalities, derive and use the laws of integral
							 exponents;
create, solve, and interpret solutions for multi-step
							 equations; and solve literal equations;
identify a real life situation and collect, organize,
							 and analyze related data in a clear concise manner for display in multiple
							 representations; formulate a conclusion; present the project;
model real-life situations involving exponential
							 growth and decay equations;
develop and explain operations with and factoring of
							 polynomials, rational and radical expressions. Use intercepts on a graph in
							 problem solving;
use simulations and rules of probability to design
							 and interpret experiments to solve problems.
Algebra I students at the mastery level will:
formulate and simplify algebraic expressions for use
							 in equations and inequalities, derive and use the laws of integral
							 exponents;
create, solve, and interpret solutions for multi-step
							 equations; and solve literal equations;
identify a real life situation; collect, organize,
							 and analyze related data for display in multiple representations; make a
							 conclusion; present the project;
describe real-life situations involving exponential
							 growth and decay equations;
develop and explain operations with and factoring of
							 polynomials, rational and radical expressions;
use simulations and rules of probability to design
							 experiments to solve problems.
Algebra I students at the partial mastery level
							 will:
formulate and simplify algebraic expressions with
							 integer coefficients for use in equations and inequalities, and use the laws of
							 integral exponents;
create, solve, and interpret solutions for multi-step
							 equations that contain only integral coefficients; and solve literal
							 equations;
identify a real life situation; collect and organize
							 related data for display in multiple representations; make a conclusion;
							 present the project;
identify real-life situations involving exponential
							 growth and decay equations;
model and explain operations with and factoring of
							 polynomials, rational and radical expressions;
use simulations and rules of probability to conduct
							 and interpret experiments to solve problems.
Algebra I students at the novice level will:
formulate and simplify algebraic expressions with
							 whole number coefficients for use in equations and inequalities, and use
							 integral exponents;
create, solve, and interpret solutions for multi-step
							 equations that contain only whole number coefficients; and solve literal
							 equations;
identify a real life situation; collect and organize
							 related data for display in multiple representations; make a conclusion; prove
							 the existence of a pattern;
identify real-life situations involving exponential
							 growth;
model operations with and factoring of polynomials,
							 rational and radical expressions;
use simulations and rules of probability to conduct
							 experiments to solve problems.
Objectives
Students will
M.O.A1.2.1
formulate algebraic expressions for use in equations
							 and inequalities that require planning to accurately model real-world
							 problems.
M.O.A1.2.2
create and solve multi-step linear equations,
							 absolute value equations, and linear inequalities in one variable, (with and
							 without technology); apply skills toward solving practical problems such as
							 distance, mixtures or motion and judge the reasonableness of solutions.
M.O.A1.2.3
evaluate data provided, given a real-world situation,
							 select an appropriate literal equation and solve for a needed variable.
M.O.A1.2.4
develop and test hypotheses to derive the laws of
							 exponents and use them to perform operations on expressions with integral
							 exponents.
M.O.A1.2.5
analyze a given set of data and prove the existence
							 of a pattern numerically, algebraically and graphically, write equations from
							 the patterns and make inferences and predictions based on observing the
							 pattern.
M.O.A.1.2.6
determine the slope of a line through a variety of
							 strategies (e.g. given an equation or graph).
M.O.A1.2.7
analyze situations and solve problems by determining
							 the equation of a line given a graph of a line, two points on the line, the
							 slope and a point, or the slope and y intercept.
M.O.A1.2.8
identify a real life situation that involves a
							 constant rate of change; pose a question; make a hypothesis as to the answer;
							 develop, justify, and implement a method to collect, organize, and analyze
							 related data; extend the nature of collected, discrete data to that of a
							 continuous linear function that describes the known data set; generalize the
							 results to make a conclusion; compare the hypothesis and the conclusion;
							 present the project numerically, analytically, graphically and verbally using
							 the predictive and analytic tools of algebra (with and without
							 technology).
M.O.A1.2.9
create and solve systems of linear equations
							 graphically and numerically using the elimination method and the substitution
							 method, given a real-world situation.
M.O.A1.2.10
simplify and evaluate algebraic expressions
· add and subtract polynomials
· multiply and divide binomials by binomials
							 or monomials
M.O.A1.2.11
create polynomials to represent and solve problems
							 from real-world situations while focusing on symbolic and graphical
							 patterns.
M.O.A1.2.12
use area models and graphical representations to
							 develop and explain appropriate methods of factoring.
M.O.A1.2.13
simplify radical expressions
· through adding, subtracting, multiplying and
							 dividing
· exact and approximate forms
M.O.A1.2.14
choose the most efficient method to solve quadratic
							 equations by
· graphing (with and without
							 technology),
· factoring
· quadratic formula and draw reasonable
							 conclusions about a situation being modeled.
M.O.A1.2.15
describe real life situations involving exponential
							 growth and decay equations including y=2x and
							 y=(½)x; compare the equation with attributes
							 of an associated table and graph to demonstrate an understanding of their
							 interrelationship.
M.O.A1.2.16
simplify and evaluate rational expressions
· add, subtract, multiply and divide
· determine when an expression is
							 undefined.
M.O.A1.2.17
perform a linear regression (with and without
							 technology),
· compare and evaluate methods of fitting
							 lines to data.
· identify the equation for the line of
							 regression,
· examine the correlation coefficient to
							 determine how well the line fits the data
· use the equation to predict specific values
							 of a variable.
M.O.A1.2.18
compute and interpret the expected value of random
							 variables in simple cases using simulations and rules of probability (with and
							 without technology).
M.O.A1.2.19
gather data to create histograms, box plots, scatter
							 plots and normal distribution curves and use them to draw and support
							 conclusions about the data.
M.O.A1.2.20
design experiments to model and solve problems using
							 the concepts of sample space and probability distribution.
M.O.A1.2.21
use multiple representations, such as words, graphs,
							 tables of values and equations, to solve practical problems; describe
							 advantages and disadvantages of the use of each representation.
Geometry Content Standards and Objectives
Geometry objectives are designed for students who have
				  completed the objectives for Algebra I. Study includes experiences and
				  activities that foster in students a feeling for the value of geometry in their
				  lives. Emphasis is placed on development of conjectures by inductive processes
				  using manipulatives and computer software. Cooperative learning groups are
				  particularly effective in allowing students to become proficient in analyzing
				  conjectures and in formulating both formal and informal proofs. Emphasis should
				  be placed on connections to other branches of mathematics and other
				  disciplines, and on workplace applications. The West Virginia Standards for
				  21st Century Learning include the following
				  components: 21st Century Content Standards and
				  Objectives and 21st Century Learning Skills and
				  Technology Tools. All West Virginia teachers are responsible for classroom
				  instruction that integrates learning skills, technology tools and content
				  standards and objectives.
Grade 9-12
Mathematics: Geometry and Applied
							 Geometry
Standard 3
Geometry
M.S.G.3
Through communication, representation, reasoning and
							 proof, problem solving, and making connections within and beyond the field of
							 mathematics, students will
· analyze characteristics and properties of
							 two- and three-dimensional geometric shapes and develop mathematical arguments
							 about geometric relationships,
· specify locations and describe spatial
							 relationships using coordinate geometry and other representational
							 systems,
· apply transformations and use symmetry to
							 analyze mathematical situations, and
· solve problems using visualization, spatial
							 reasoning, and geometric modeling.
Performance Descriptors
							 (M.PD.G.3)
Distinguished
Above Mastery
Mastery
Partial Mastery
Novice
Geometry students at the distinguished level:
investigate, create arguments, justify, compare and
							 contrast, make conjectures, critique arguments and apply relationships
							 involving the properties of lines, polygons, measures of angles, circles,
							 Pythagorean Theorem, transformational geometry, tessellating figures,
							 trigonometric ratios, and the properties of Euclidean geometry with other
							 geometries concepts of analytical geometry;
construct the parts of a triangle and develop and
							 justify logical concepts to be used in solving real-world problems;
draw and justify conclusions in real-world settings
							 and construct proofs, counterexamples, and logical arguments;
identify a real life situation involving similarity;
							 pose a question; make a hypothesis; collect, organize, and analyze related
							 data; make a conclusion; compare the hypothesis and the conclusion; and present
							 the project.
Geometry students at the above mastery level:
investigate, justify, make conjectures, compare and
							 contrast, critique arguments and apply relationships involving the properties
							 of lines, polygons, measures of angles, circles, Pythagorean Theorem,
							 transformational geometry, tessellating figures, trigonometric ratios, the
							 properties of Euclidean geometry with other geometries, and concepts of
							 analytical geometry;
construct the parts of a triangle and develop and
							 justify logical concepts to be used in solving real-world problems;
draw and justify conclusions in real-world settings
							 and construct proofs, counterexamples, and logical arguments;
identify a real life situation involving similarity;
							 pose a question; make a hypothesis; collect, organize, and analyze related
							 data; make a conclusion; compare the hypothesis and the conclusion; and present
							 the project.
Geometry students at the mastery level:
investigate, make conjectures, compare and contrast,
							 and/or apply relationships involving the properties of lines, polygons,
							 measures of angles, circles, Pythagorean Theorem, transformational geometry,
							 tessellating figures, trigonometric ratios, the properties of Euclidean
							 geometry with other geometries, and concepts of analytical geometry;
construct the parts of a triangle and develop logical
							 concepts to be used in solving real-world problems;
draw and justify conclusions in real-world settings
							 and construct proofs and logical arguments;
identify a real life situation involving similarity;
							 pose a question; make a hypothesis; collect, organize, and analyze related
							 data; make a conclusion; compare the hypothesis and the conclusion; and present
							 the project.
Geometry students at the partial mastery
							 level:
investigate and apply relationships involving the
							 properties of lines, polygons, measures of angles, circles, Pythagorean
							 Theorem, transformational geometry, tessellating figures, trigonometric ratios,
							 the properties of Euclidean geometry with other geometries, and concepts of
							 analytical geometry;
construct the parts of a triangle and use logical
							 concepts to solve real-world problems;
draw and justify conclusions in real-world settings
							 and construct informal proofs;
identify a real life situation involving similarity;
							 pose a question; make a hypothesis; collect and organize data; make a
							 conclusion; compare the hypothesis and the conclusion.
Geometry students at the novice level:
investigate relationships involving the properties of
							 lines, polygons, measures of angles, circles, Pythagorean Theorem,
							 transformational geometry, tessellating figures, concepts of analytical
							 geometry trigonometric ratios, and use analytical geometry to apply
							 formulas;
identify corresponding parts of similar
							 triangles;
draw conclusions in real-world settings and construct
							 informal proof;
identify a real life situation involving similarity;
							 pose a question; make a hypothesis; collect and organize data; make a
							 conclusion; compare the hypothesis and the conclusion.
Objectives
Students will
M.O.G.3.1
represent geometric figures, such as points, lines,
							 planes, segments, rays, and angles pictorially with proper identification and
							 distinguish between undefined and defined terms.
M.O.G.3.2
differentiate and apply inductive and deductive
							 reasoning, justify conclusions in real-world settings.
M.O.G.3.3
use the basic concepts of symbolic logic including
							 identifying the converse, inverse, and contrapositive of a conditional
							 statement and test the validity of conclusions with methods that include Venn
							 Diagrams.
M.O.G.3.4
validate conclusions by constructing logical
							 arguments using both formal and informal methods with direct and indirect
							 reasoning.
M.O.G.3.5
construct formal and informal proofs by applying
							 definitions, theorems, and postulates related to such topics as
· complementary,
· supplementary,
· vertical angles,
· angles formed by perpendicular lines, and
							 justify the steps.
M.O.G.3.6
compare and contrast the relationships between angles
							 formed by two lines cut by a transversal when lines are parallel and when they
							 are not parallel, and use the results to develop concepts that will justify
							 parallelism.
M.O.G.3.7
make conjectures and justify congruence relationships
							 with an emphasis on triangles and employ these relationships to solve
							 problems.
M.O.G.3.8
identify general properties of and compare and
							 contrast the properties of convex and concave quadrilaterals
· parallelograms
· rectangles
· rhombuses
· squares
· trapezoids
M.O.G.3.9
identify a real life situation that involves
							 similarity in two or three dimensions; pose a question; make a hypothesis as to
							 the answer, develop, justify, and implement a method to collect, organize, and
							 analyze related data; generalize the results to make a conclusion; compare the
							 hypothesis and the conclusion; present the project numerically, analytically,
							 graphically and verbally using the predictive and analytic tools of algebra and
							 geometry (with and without technology).
M.O.G.3.10
investigate measures of angles and lengths of
							 segments to determine the existence of a triangle (triangle inequality) and to
							 establish the relationship between the measures of the angles and the length of
							 the sides (with and without technology).
M.O.G.3.11
verify and justify the basis for the trigonometric
							 ratios by applying properties of similar triangles and use the results to find
							 inaccessible heights and distances. Using the ratios of similar triangles to
							 find unknown side lengths and angle measures, construct a physical model that
							 illustrates the use of a scale drawing in a real-world situation.
M.O.G.3.12
apply the Pythagorean Theorem and its converse to
							 solve real-world problems and derive the special right triangle relationships
							 (i.e. 30-60-90, 45-45-90).
M.O.G.3.13
investigate measures of angles formed by chords,
							 tangents, and secants of a circle and draw conclusions for the relationship to
							 its arcs.
M.O.G.3.14
find angle measures of interior and exterior angles;
							 given a polygon, find the length of sides from given data; and use properties
							 of regular polygons to find any unknown measurements of sides or angles.
M.O.G.3.15
develop properties of tessellating figures and use
							 those properties to tessellate the plane.
M.O.G.3.16
derive and justify formulas for area, perimeter,
							 surface area, and volume using nets and apply them to solve real-world
							 problems.
M.O.G.3.17
apply concepts of analytical geometry such as
							 formulas for distance, slope, and midpoint and apply these to finding
							 dimensions of polygons on the coordinate plane.
M.O.G.3.18
construct a triangle's medians, altitudes, angle and
							 perpendicular bisectors using various methods; and develop logical concepts
							 about their relationships to be used in solving real-world problems.
M.O.G.3.19
create and apply concepts using transformational
							 geometry and laws of symmetry, of a
· reflection,
· translation,
· rotation,
· glide reflection,
· dilation of a figure, and develop logical
							 arguments for congruency and similarity.
M.O.G.3.20
compare and contrast Euclidean geometry to other
							 geometries (i.e. spherical, elliptic) using various forms of communication such
							 as development of physical models, oral or written reports.
M.O.G.3.21
approximate the area of irregularly shaped regions
							 based on the approximations and the attributes of the related region, develop a
							 formula for finding the area of irregularly shaped regions. Plan, organize and
							 present results by justifying conclusions.
Algebra II Content Standards and
				  Objectives
Algebra II objectives emphasize the use of
				  investigation to more advanced functions, using them to solve real-world
				  problems. Focus is on multiple representations to develop conjectures, testing
				  and justifying validity. Calculators, computers, and interactive utilities are
				  an integral part of instruction. The West Virginia Standards for
				  21st Century Learning include the following
				  components: 21st Century Content Standards and
				  Objectives and 21st Century Learning Skills and
				  Technology Tools. All West Virginia teachers are responsible for classroom
				  instruction that integrates learning skills, technology tools and content
				  standards and objectives.
Grade 9-12
Mathematics: Algebra II
Standard 2
Algebra
M.S.A2.2
Through communication, representation, reasoning and
							 proof, problem solving, and making connections within and beyond the field of
							 mathematics, students will
· demonstrate understanding of patterns,
							 relations and functions,
· represent and analyze mathematical
							 situations and structures using algebraic symbols,
· use mathematical models to represent and
							 understand quantitative relationships, and
· analyze change in various contexts.
Performance Descriptors
							 (M.PD.A2.2)
Distinguished
Above Mastery
Mastery
Partial Mastery
Novice
Algebra II students at the distinguished
							 level:
develop and analyze practical situations to
							 determine, graph and solve various types of equations, inequalities, and
							 systems and express answers using various formats;
extend the techniques of factoring polynomials and
							 explain their application;
convert between the graphs and equations of functions
							 and conic sections using an analysis of their properties and graphing
							 techniques and describe their characteristics;
justify properties used to simplify and expand
							 expressions and convert between appropriate forms;
generate quadratic regressions to make predictions
							 and present analysis of results;
identify a real world situation that models
							 quadratics, pose a question, collect and analyze data, and present and justify
							 their results.
Algebra II students at the above mastery
							 level:
analyze practical situations to determine, graph and
							 solve various types of equations, inequalities, and systems and express answers
							 using various formats;
extend the techniques of factoring
							 polynomials;
convert between the graphs and equations of functions
							 and conic sections using an analysis of their properties and graphing
							 techniques and describe their characteristics;
apply properties to simplify and expand expressions
							 and convert between appropriate forms;
generate quadratic regressions to make predictions
							 and analyze results;
identify a real world situation that models
							 quadratics, pose a question, collect and analyze data, and present their
							 results.
Algebra II students at the mastery level:
determine, graph and solve various types of
							 equations, inequalities, and systems and express answers using various
							 formats;
apply the appropriate method to factor
							 polynomials;
convert between the graphs and equations of functions
							 and conic sections using an analysis of their properties and graphing
							 techniques;
simplify and expand expressions and convert between
							 appropriate forms;
generate quadratic regressions to make
							 predictions;
identify a real world situation that models
							 quadratics, pose a question, collect and analyze data.
Algebra II students at partial mastery level:
graph and solve various types of equations,
							 inequalities, and systems and express answers using various formats;
factor polynomials when given the appropriate
							 method;
convert between the graphs and equations of functions
							 and conic sections;
simplify and expand expressions;
make predictions given a quadratic regression;
identify a real world situation that models
							 quadratics and pose a question.
Algebra II students at the novice level:
graph and solve various types of equations,
							 inequalities, and systems;
factor most polynomials when given the appropriate
							 method;
graph functions and conic sections from the given
							 equation;
simplify and expand most expressions;
recognize quadratic regressions;
identify a real world situation that models
							 quadratics.
Objectives
Students will
M.O.A2.2.1
determine equations of lines including parallel,
							 perpendicular, vertical and horizontal lines, and compare and contrast the
							 properties of these equations.
M.O.A2.2.2
factor higher order polynomials by applying various
							 methods including factoring by grouping and the sum and difference of two
							 cubes; analyze and describe the relationship between the factored form and the
							 graphical representation.
M.O.A2.2.3
define complex numbers, simplify powers of 'i',
							 perform basic operations with complex numbers, and give answers as complex
							 numbers in simplest form.
M.O.A2.2. 4
simplify expressions involving radicals and
							 fractional exponents, convert between the two forms, and solve equations
							 containing radicals and exponents.
M.O.A2.2. 5
solve quadratic equations over the set of complex
							 numbers: apply the techniques of factoring, completing the square, and the
							 quadratic formula; use the discriminate to determine the number and nature of
							 the roots; identify the maxima and minima; use words, graphs, tables, and
							 equations to generate and analyze solutions to practical problems.
M.O.A2.2.6
develop and use the appropriate field properties of
							 matrices by adding, subtracting, and multiplying; solve a system of linear
							 equations using matrices; and apply skills toward solving practical
							 problems.
M.O.A2.2.7
define a function and find its zeros; express the
							 domain and range using interval notation; find the inverse of a function; find
							 the value of a function for a given element in its domain; and perform basic
							 operations on functions including composition of functions.
M.O.A2.2.8
analyze families of functions and their
							 transformations; recognize linear, quadratic, radical, absolute value, step,
							 piece-wise, and exponential functions; analyze connections among words, graphs,
							 tables and equations when solving practical problems with and without
							 technology.
M.O.A2.2.9
solve quadratic inequalities, graph their solution
							 sets, and express solutions using interval notation.
M.O.A2.2.10
solve and graph the solution set of systems of linear
							 inequalities in two variables by finding the maximum or minimum values of a
							 function over the feasible region using linear programming techniques.
M.O.A2.2.11
solve practical problems involving direct, inverse
							 and joint variation.
M.O.A2.2.12
analyze the conic sections; identify and sketch the
							 graphs of a parabola, circle, ellipse, and hyperbola and convert between graphs
							 and equations.
M.O.A2.2.13
solve absolute value inequalities graphically,
							 numerically and algebraically and express the solution set in interval
							 notation.
M.O.A2.2.14
define a logarithmic function, transform between
							 exponential and logarithmic forms, and apply the basic properties of logarithms
							 to simplify or expand an expression.
M.O.A2.2.15
identify a real life situation that exhibits
							 characteristics of change that can be modeled by a quadratic equations; pose a
							 questions; make a hypothesis as to the answer; develop, justify, and implement
							 a method to collect, organize and analyze related data; extend the nature of
							 collected, discrete data to that of a continuous function that describes the
							 known data set; generalize the results to make a conclusion; compare the
							 hypothesis and the conclusion; present the project numerically, analytically,
							 graphically and verbally using the predictive and analytic tools of algebra
							 (with and without technology).
M.O.A2.2.16
describe and illustrate how patterns and sequences
							 are used to develop recursive and closed form equations; analyze and describe
							 characteristics of each form.
Conceptual Mathematics Content Standards and
				  Objectives
Conceptual Mathematics objectives include major topics
				  from algebra and geometry and extend these ideas to practical usage. Basic
				  ideas of probability and statistics and the mathematics of finance are
				  included. These big ideas are to be presented in the context of their
				  historical development. Full integration of calculators, computers, and
				  interactive utilities are essential for mastery. The West Virginia Standards
				  for 21st Century Learning include the following
				  components: 21st Century Content Standards and
				  Objectives and 21st Century Learning Skills and
				  Technology Tools. All West Virginia teachers are responsible for classroom
				  instruction that integrates learning skills, technology tools and content
				  standards and objectives.
Grade 9-12
Mathematics: Conceptual
							 Mathematics
Standard 2
Algebra
M.S.CM.2
Through communication, representation, reasoning and
							 proof, problem solving, and making connections within and beyond the field of
							 mathematics, students will
· demonstrate understanding of patterns,
							 relations and functions,
· represent and analyze mathematical
							 situations and structures using algebraic symbols,
· use mathematical models to represent and
							 understand quantitative relationships, and
· analyze change in various contexts.
Performance Descriptors
							 (M.PD.CM.2)
Distinguished
Above Mastery
Mastery
Partial Mastery
Novice
Conceptual Mathematics students at the distinguished
							 level:
research, create, apply, and compare a variety of
							 problem-solving strategies to solve real-world problems and justify the
							 reasonableness of the solutions;
develop and solve applications problems involving
							 functions and assess their usefulness in the real world;
pose questions, make hypotheses, and implement
							 appropriate methods to collect, organize, analyze, draw conclusions, and
							 present data (with and without technology);
research and investigate real-world personal finance
							 situations and differentiate between the types of personal finance functions in
							 order to solve and make recommendations to optimize the situation.
Conceptual Mathematics students at the above mastery
							 level:
create, apply, and compare a variety of
							 problem-solving strategies to solve real-world problems and justify the
							 reasonableness of the solutions;
develop and solve application problems involving
							 functions and interpret and analyze their graphs;
pose questions, make hypotheses, and implement
							 appropriate methods to collect, organize, analyze, draw conclusions, and
							 present data (with and without technology);
research and investigate real-world personal finance
							 situations and differentiate between the types of personal finance functions in
							 order to solve the problem.
Conceptual Mathematics students at the mastery
							 level:
apply and compare a variety of problem-solving
							 strategies to solve real-world problems and justify the reasonableness of the
							 solutions;
solve application problems involving functions and
							 interpret and analyze the graphs;
pose questions, make hypotheses, and implement
							 appropriate methods to collect, organize, analyze, draw conclusions, and
							 present data (with and without technology);
differentiate between the types of personal finance
							 functions in order to solve real-world problems.
Conceptual Mathematics students at the partial
							 mastery level:
apply problem-solving strategies to solve real-world
							 problems;
solve application problems involving functions and
							 read the graphs;
implement appropriate methods to collect, organize,
							 and present data (with and without technology);
solve real-world problems when given the appropriate
							 personal finance function.
Conceptual Mathematics students at the novice
							 level:
apply problem-solving strategies to solve
							 problems;
recognize problems involving functions;
collect, organize, and present data (with and without
							 technology);
compute using personal finance formulas.
Objectives
Students will
M.O.CM.2.1
use a variety of problem solving strategies (e.g.,
							 draw a diagram, look for a pattern, work backwards) to solve real-world
							 problems.
M.O.CM.2.2
interpret graphs of functions including linear,
							 quadratic, and exponential.
M.O.CM.2.3
solve application problems using linear, quadratic
							 and exponential functions with emphasis on data collection and analysis.
M.O.CM.2.4
choose the appropriate formulas to solve workplace
							 problems and judge the reasonableness of the solutions.
M.O.CM.2.5
describe and illustrate how calculating costs, simple
							 and compound interest, finance charge, loan payment and tax functions are used
							 to solve real-world problems.
M.O.CM.2.6
identify a real life situation that involves
							 investing money over time; pose a question; make a hypothesis as to the answer;
							 develop, justify, and implement a method to collect, organize, and analyze
							 related data; generalize the results to make a conclusion; compare the
							 hypothesis and the conclusion; present the project numerically, analytically,
							 graphically and verbally using words, graphs, models, or tables (with and
							 without technology).
Grade 9-12
Mathematics: Conceptual
							 Mathematics
Standard 3
Geometry
M.S.CM.3
Through communication, representation, reasoning and
							 proof, problem solving, and making connections within and beyond the field of
							 mathematics, students will
· analyze characteristics and properties of
							 two- and three-dimensional geometric shapes and develop mathematical arguments
							 about geometric relationships,
· specify locations and describe spatial
							 relationships using coordinate geometry and other representational
							 systems,
· apply transformations and use symmetry to
							 analyze mathematical situations, and
· solve problems using visualization, spatial
							 reasoning, and geometric modeling.
Performance Descriptors
							 (M.PD.CM.3)
Distinguished
Above Mastery
Mastery
Partial Mastery
Novice
Conceptual Mathematics students at the distinguished
							 level:
design and implement a project which applies concepts
							 of geometry to compute measures and analyze connections between geometric
							 shapes and their real-world applications.
Conceptual Mathematics students at the above mastery
							 level:
determine and apply concepts of geometry to compute
							 measures and analyze connections between geometric shapes and their real-world
							 applications.
Conceptual Mathematics students at the mastery
							 level:
apply concepts of geometry to compute measures and
							 analyze connections between geometric shapes and their real-world
							 applications.
Conceptual Mathematics students at the partial
							 mastery level:
use concepts of geometry to compute measures and
							 model connections between geometric shapes and their real-world
							 applications.
Conceptual Mathematics students at the novice
							 level:
recognize concepts of geometry to compute measures
							 and describe connections between geometric shapes and their real-world
							 applications.
Objectives
Students will
M.O.CM.3.1
apply concepts of geometry including the Pythagorean
							 Theorem, similar triangles, and right triangle trigonometry.
M.O.CM.3.2
compute measures to solve real-world problems, using
							 relationships involving perimeter, area, surface area and volume of geometric
							 figures.
M.O.CM.3.3
analyze the connections of various geometric shapes
							 and patterns to art, architecture, and nature.
Grade 9-12
Mathematics: Conceptual
							 Mathematics
Standard 3
Data Analysis and Probability
M.S.CM.5
Through communication, representation, reasoning and
							 proof, problem solving, and making connections within and beyond the field of
							 mathematics, students will
· formulate questions that can be addressed
							 with data and collect, organize, and display relevant data to answer
							 them,
· select and use appropriate statistical
							 methods to analyze data,
· develop and evaluate inferences and
							 predictions that are based on models, and
· apply and demonstrate an understanding of
							 basic concepts of probability.
Performance Descriptors (M.PD
							 CM.5)
Distinguished
Above Mastery
Mastery
Partial Mastery
Novice
Conceptual Mathematics students at distinguished
							 level:
research and relate mathematical content to its
							 historical development and connect to other disciplines;
design and conduct probability investigations using
							 counting techniques, and determine, analyze and communicate the results and
							 develop rules of probability;
compare and contrast more than one set of data that
							 they collect, summarize, and interpret numerically and graphically.
Conceptual Mathematics students at above mastery
							 level:
research and relate mathematical content to its
							 historical development and integrate other disciplines into the study of
							 mathematics;
design and conduct probability investigations using
							 counting techniques, and determine, analyze and communicate the results and
							 develop rules of probability;
collect, summarize, and interpret data numerically
							 and graphically to make predictions.
Conceptual Mathematics students at mastery
							 level:
relate mathematical content to its historical
							 development and integrate other disciplines into the study of
							 mathematics;
design and conduct probability investigations using
							 counting techniques, and determine, analyze and communicate the results;
collect, summarize, and interpret data numerically
							 and graphically to make predictions.
Conceptual Mathematics students at partial mastery
							 level:
recognize mathematical content as it relates to its
							 historical development and relate how other disciplines are integrated into the
							 study of mathematics;
conduct probability investigations using counting
							 techniques, and determine, analyze and communicate the results;
collect and summarize data numerically and
							 graphically.
Conceptual Mathematics students at novice
							 level:
recognize that mathematical content is related to its
							 historical development and see how other disciplines are integrated into the
							 study of mathematics;
conduct probability investigations using counting
							 techniques and communicate results;
collect and summarize data numerically and
							 graphically.
Objectives
Students will
M.O.CM.5.1
relate mathematical content to its historical
							 development.
M.O.CM.5.2
integrate other disciplines into the study of
							 mathematics through simulations, research, and projects.
M.O.CM.5.3
determine possible outcomes using tree diagrams and
							 the counting principles of permutations and combinations, develop conclusions
							 and offer solutions for new situations, using real-world data.
M.O.CM.5.4
design and conduct probability investigations and
							 then determine, analyze, and communicate the results.
M.O.CM.5.5
collect and interpret data using various methods of
							 displaying numerical data, including frequency distributions, graphs,
							 histograms, stem-and-leaf plots, and box-and-whiskers plots, using technology
							 when appropriate.
M.O.CM.5.6
relate the measures of central tendency and the
							 measures of dispersion to a normal distribution.
M.O.CM.5.7
apply the measures of central tendency and the
							 measures of dispersion to workplace situations.
M.O.CM.5.8
use statistical tools for workplace applications such
							 as quality control, marketing and predicting trends.
Algebra III Content Standards and
				  Objectives
Algebra III is intended for students who have mastered
				  the concepts of Algebra I, Geometry, and Algebra II. Algebra III objectives
				  develop and extend properties of higher degree polynomial functions, rational
				  functions, exponential functions and logarithmic functions using the common
				  concepts and language of algebraic, graphical, and tabular representations. The
				  use of analytic geometry for sense making, conceptual understanding of abstract
				  ideas and modeling real world applications is stressed, making use of
				  calculators, computers, and interactive activities. The West Virginia Standards
				  for 21st Century Learning include the following components:
				  21st Century Content Standards and Objectives and
				  21st Century Learning Skills and Technology Tools.
				  All West Virginia teachers are responsible for classroom instruction that
				  integrates learning skills, technology tools and content standards and
				  objectives.
Grade 9-12
Mathematics: Algebra III
Standard 2
Algebra
M.S.A3.2
Through communication, representation, reasoning and
							 proof, problem solving, and making connections within and beyond the field of
							 mathematics, students will
· demonstrate understanding of patterns,
							 relations and functions,
· represent and analyze mathematical
							 situations and structures using algebraic symbols,
· use mathematical models to represent and
							 understand quantitative relationships, and
· analyze change in various contexts.
Performance Descriptors
							 (M.PD.A3.2)
Distinguished
Above Mastery
Mastery
Partial Mastery
Novice
Algebra III students at the distinguished
							 level:
research practical solutions to choose appropriate
							 representations from the families of functions using characteristics of the
							 functions;
demonstrate, relate, and assess connections between
							 functions and their inverses, justify restricting the domain to guarantee an
							 inverse, and apply transformations, compositions, and operations;
use properties of analytic geometry to apply elements
							 of equations, interpret rates of change, convert between forms of equations,
							 develop and justify the use of the distance and midpoint formulas;
collaborate to choose a real world problem that can
							 be modeled using algebraic and graphical techniques, predict, justify, and
							 explain the model, and screen for extraneous solutions explaining their
							 existence.
Algebra III students at the above mastery
							 level:
analyze practical solutions to compare and apply
							 multiple representations of families of functions using characteristics of the
							 functions;
recognize, demonstrate, and relate connections
							 between functions and their inverses, appropriately restrict the domain to
							 guarantee an inverse and apply transformations, compositions, and
							 operations;
use properties of analytic geometry to determine
							 equations, interpret rates of change, convert between forms of equations, and
							 develop the distance and midpoint formulas;
create models of real world problems using algebraic
							 and graphical techniques, screen for extraneous solutions, and explain their
							 existence.
Algebra III students at the mastery level:
compare and apply multiple representations of
							 families of functions using characteristics of the functions;
recognize and demonstrate connections between
							 functions and their inverses and apply transformations, compositions and
							 operations;
use properties of analytic geometry to determine
							 equations, their components and relationships and apply the distance and
							 midpoint formulas;
model real world problems using algebraic and
							 graphical techniques and screen for extraneous solutions.
Algebra III students at the partial mastery
							 level:
explore multiple representations of families of
							 functions using characteristics of the function;
recognize connections between functions and their
							 inverses by performing transformations, compositions, and operations;
use properties from analytic geometry to determine
							 slope, equations of circles, and apply the distance and midpoint
							 formulas;
solve real world problems using algebraic and
							 graphing techniques and recognize extraneous solutions.
Algebra III students at the novice level:
recognize multiple representations of families of
							 functions using basic characteristics of the functions;
recognize inverse functions and perform compositions
							 and arithmetic operations;
recognize slope of a line, equations of circles, and
							 calculate distance and midpoint using formulas;
confirm solutions of real world problems using
							 algebraic and graphical techniques and recognize extraneous roots.
Objectives
Students will
M.O.A3.2.1
use properties of analytic geometry to justify and
							 use the distance and midpoint formulas and negative reciprocal criterion for
							 non-vertical perpendicular lines.
M.O.A3.2.2
factor higher order polynomials by using techniques
							 that can be applied to the factoring of second degree polynomials; relate
							 factored forms of polynomials to graphs, tables, and solutions to problems in
							 context.
M.O.A3.2.3
relate analytical attributes such as characteristics
							 of zeros, x- and y- intercepts, symmetry, asymptotes, end behavior, maximum and
							 minimum points, and domain and range, to graphical and algebraic
							 representations of polynomials and rational functions.
M.O.A3.2.4
analyze the discriminant to classify the roots of
							 quadratic equations with real coefficients, and relate the existence of
							 x-intercepts of the graph to information obtained from the discriminant.
M.O.A3.2.5
solve equations with extraneous roots; explain why
							 the extraneous roots are excluded from the solution set.
M.O.A3.2.6
compare and contrast the use of interval notation,
							 set notation, and number line representations to express the domain and range
							 of functions.
M.O.A3.2.7
compare and contrast the domain and range of a
							 modeling function with the restricted domain and range used in a real world
							 situation; justify the restricted domain and range choice for a problem in
							 context.
M.O.A3.2.8
differentiate between functions and relations;
							 evaluate, add, subtract, multiply, divide, rationalize, simplify, and compose
							 functions (including rational, radical and those with fractional exponents);
							 express domain and range of functions.
M.O.A3.2.9
convert between graphs and equations of circles
							 identifying important features from either representation; translate from
							 general form to standard form by completing the square and describe readily
							 usable characteristics of each form; represent a circle as two functions
							 graphically and algebraically.
M.O.A3.2.10
analyze a piecewise defined function in multiple
							 representations, to give its domain, intercepts, range, constituent pieces as
							 elementary functions, and end behavior; apply to real world data.
M.O.A3.2.11
determine the average rate of change of a function
							 between any two points on its graph and use this rate to find the equation of a
							 secant line; interpret the average rate of change to solve real world problems;
							 relate signs of average rate of change to the function increasing or
							 decreasing; and demonstrate a geometrical and conceptual understanding of the
							 difference quotient.
M.O.A3.2.12
use synthetic division to divide a polynomial, verify
							 a factor, and determine its roots; compare and contrast synthetic division to
							 long division.
M.O.A3.2.13
investigate how the multiplicity of zeros of
							 polynomial functions affects the graph; characterize a polynomial given the
							 zeros, the behavior of the graph at the zeros, and the end-behavior.
M.O.A3.2.14
given the characteristics of a transformation
							 involving polynomial, radical, absolute value, logarithmic, or exponential
							 functions, determine a representative function; unravel the effect of a series
							 of transformations using multiple representations.
M.O.A3.2.15
define and discuss one-to-one functions including the
							 role of the Vertical and Horizontal Line Tests; use multiple representations in
							 describing the relationship between a function and its inverse, including the
							 domain and range of each; identify and explain the need for appropriate
							 restrictions necessary to guarantee an inverse function; discuss the
							 symmetrical relationship associated with the line y=x between the function and
							 its inverse and explain the geometric reason the symmetry exists; demonstrate
							 how to algebraically verify that two functions are inverses of each
							 other.
M.O.A3.2.16
prioritize relevant techniques to graph a given
							 rational function, explaining the relevance of symmetry, end behavior, and
							 domain and range; use zeros of the denominator to differentiate between
							 vertical asymptotes and points of discontinuity; use long division to determine
							 end behavior and explain the role of quotient and remainder in the process;
							 explain how the factors of the numerator and denominator can be used to
							 analytically and graphically determine where the graph will fall above or below
							 the x-axis.
M.O.A3.2.17
restrict the possible rational zeros of a polynomial
							 function by using the Rational Zeros Theorem and Descartes' Rule of Signs;
							 confirm the real zeros of a polynomial function by using the Remainder and
							 Factor Theorems; approximate zeros of a polynomial or rational function using a
							 graphing utility and the Intermediate Value Theorem.
M.O.A3.2.18
analyze polynomial equations with real coefficients
							 and complex roots using factoring, the Conjugate Roots Theorem, the quadratic
							 formula, or root restricting theorems; confirm roots using numerical and
							 graphical methods; discuss and justify how the graph of a polynomial function
							 gives information about complex zeros.
M.O.A3.2.19
compare and contrast the cases when 0a1 and a1 for
							 the general exponential function f(x) =ax: graphs,
							 asymptotes, domain and range, and transformations. Interpret the number e as a
							 limit and use e to build exponential functions modeling real world
							 applications.
M.O.A3.2.20
use common and natural logarithms in the evaluation
							 of logarithmic functions whose base is neither 10 nor e. Incorporate the change
							 of base formula and properties of logarithms to simplify and expand algebraic
							 expressions and to solve logarithmic and exponential equations.
M.O.A3.2.21
through algebraic, graphical, numerical, and verbal
							 techniques, solve equations involving radical, exponential, and logarithmic
							 expressions. Formulate strategies to solve real life problems including
							 compound interest and exponential growth and decay.
M.O.A3.2.22
build on the skills of solving linear equations in
							 two variables using elimination, substitution, or matrix methods to solve
							 systems with three or more unknowns involving real world applications.
							 Categorize systems of equations as zero, one, or infinitely many solutions, by
							 both geometric and algebraic methods.
M.O.A3.2.23
work in groups to choose a real life situation that
							 could be modeled by a polynomial, rational, exponential, or logarithmic
							 function, and make a hypothesis, design an experiment, gather data, analyze
							 data, refine the hypothesis into an appropriate mathematical model, use the
							 model to make a prediction, test the prediction using the experimental setup,
							 and compare the results. Present the collaboration as a project using words,
							 graphs, tables, equations, and appropriate presentation tools.
Trigonometry Content Standards and
				  Objectives
Trigonometry objectives emphasize making connections
				  between right triangle trigonometry and circular functions. Calculators,
				  computers, and interactive utilities will be used to enhance student learning.
				  The West Virginia Standards for 21st Century
				  Learning include the following components: 21st
				  Century Content Standards and Objectives and 21st
				  Century Learning Skills and Technology Tools. All West Virginia teachers are
				  responsible for classroom instruction that integrates learning skills,
				  technology tools and content standards and objectives.
Grade 9-12
Mathematics: Trigonometry
Standard 3
Geometry
M.S.T.3
Through communication, representation, reasoning and
							 proof, problem solving, and making connections within and beyond the field of
							 mathematics, students will
· analyze characteristics and properties of
							 two- and three-dimensional geometric shapes and develop mathematical arguments
							 about geometric relationships,
· specify locations and describe spatial
							 relationships using coordinate geometry and other representational
							 systems,
· apply transformations and use symmetry to
							 analyze mathematical situations, and
· solve problems using visualization, spatial
							 reasoning, and geometric modeling.
Performance Descriptors
							 (M.PD.T.3)
Distinguished
Above Mastery
Mastery
Partial Mastery
Novice
Trigonometry students at the distinguished
							 level:
define and relate the six trigonometric functions in
							 right triangles and in circular functions; graph and evaluate them and their
							 inverse functions; solve trigonometric equations and apply them to real-world
							 problems;
convert from degrees to radians (and vice versa) and
							 test hypothesis or hypotheses to derive formulas to find applications of radian
							 measure;
determine the appropriate use for the Law of Sines
							 and the Law of Cosines; apply to real-world situations; solve triangles and
							 figures of multiple shapes;
perform graphical and algebraic addition of vectors;
							 convert complex numbers to polar form and graph in the polar coordinate plane
							 and compare the graph to real world situations; identify three-dimensional
							 vectors and use graphs, tables and equations to model periodic data sets and to
							 analyze real world problems;
verify the basic identities and use them to verify
							 and evaluate other identities.
Trigonometry students at the above mastery
							 level:
define and relate the six trigonometric functions in
							 right triangles and in circular functions; graph and evaluate them and their
							 inverse functions; solve trigonometric equations and apply them to real-world
							 problems;
convert from degrees to radians (and vice versa) and
							 test hypothesis to derive formulas to find applications of radian
							 measure;
determine the appropriate use for the Law of Sines
							 and the Law of Cosines and solve triangles and figures of multiple
							 shapes;
perform graphical and algebraic addition of vectors;
							 convert complex numbers to polar form and graph in the polar coordinate plane;
							 compare the graph to real-world situations;
verify the basic identities and use them to verify
							 and evaluate other identities.
Trigonometry students at the mastery level:
define and relate the six trigonometric functions in
							 right triangles and in circular functions; graph and evaluate them and their
							 inverse functions; solve trigonometric equations and apply them to real-world
							 problems;
convert from degrees to radians (and vice versa) and
							 develop formulas to find applications of radian measure;
determine the appropriate use for the Law of Sines
							 and the Law of Cosines and solve triangles;
perform graphical and algebraic addition of vectors;
							 convert complex numbers to polar form and graph in the polar coordinate
							 plane;
verify the basic identities and use them to verify
							 other identities.
Trigonometry students at the partial mastery
							 level:
define and relate the six trigonometric functions in
							 right triangles and in circular functions; graph and evaluate them and their
							 inverse functions; solve trigonometric equations;
convert from degrees to radians and develop formulas
							 to find applications of radian measure;
use for the Law of Sines and the Law of Cosines and
							 solve triangles;
perform graphical and algebraic addition of vectors;
							 convert complex numbers to polar form and identify graphs in the polar
							 coordinate plane;
identify the basic identities and use them to verify
							 other identities.
Trigonometry students at the novice level:
recognize the six trigonometric functions in right
							 triangles and in circular functions, recognize the graphs and evaluate the
							 functions and their inverses; solve trigonometric equations;
recognize radians and formulas to convert from
							 degrees to radians and recognize formulas to find applications of radian
							 measure;
recognize the Law of Sines and the Law of Cosines and
							 find the area of triangles;
recognize graphical and algebraic addition of
							 vectors; recognize the conversion of complex numbers to polar form and identify
							 graphs in the polar coordinate plane;
recognize the formulas for the basic
							 identities.
Objectives
Students will
M.O.T.3.1
apply the right triangle definition of the six
							 trigonometric functions of an angle to determine the values of the function
							 values of an angle in standard position given a point on the terminal side of
							 the angle.
· determine the value of the other
							 trigonometric functions given the value of one of the trigonometric functions
							 and verify these values with technology.
· using geometric principles and the
							 Pythagorean Theorem, determine the six function values for the special angles
							 and the quadrantal angles and use them in real-world problems.
· compare circular functions and the
							 trigonometric function values to draw inferences about coterminal angles and
							 co-functions.
M.O.T.3.2
convert angle measures from degrees to radians (and
							 vice versa) and apply this concept to
· create a data set, analyze, and formulate a
							 hypotheses to test and develop formulas for the arclength, area of a sector,
							 and angular velocity and use the formula for application in the
							 real-world.
· compare and contrast the concepts of angular
							 velocity and linear velocity and demonstrate by graphical or algebraic means
							 relationship between them and apply to real-world problems.
M.O.T.3.3
using various methods, basic identities and graphical
							 representation
· verify trigonometric identities
· prove the sum and difference to two angles,
							 double-angles, and half-angle identities
M.O.T.3.4
justify and present the solutions of trigonometric
							 equations that include both infinite and finite (over a restricted domain)
							 solutions.
M.O.T.3.5
find the value of the inverse trigonometric functions
							 using special angle trigonometric function values and technology.
· draw inferences of restricted domain to
							 recognize and produce a graph of the inverse trigonometric functions.
· prove conjectures made about the solution of
							 the equations such as x = sin (arcsin y), x = sin (arcos y) being sure to
							 consider restrictions of the domain.
M.O.T.3.6
identify a real life problem utilizing graphs of
							 trigonometric functions and/or the inverse functions; make a hypothesis as to
							 the outcome; develop, justify, and implement a method to collect, organize, and
							 analyze data; generalize the results to make a conclusion; compare the
							 hypothesis and the conclusion; present the project using words, graphs,
							 drawings, models, or tables.
M.O.T.3.7
model periodic data sets using graphs, tables, and
							 equations and use them to analyze real-world problems such as electricity and
							 harmonic motion.
M.O.T.3.8
investigate real-world problems within a project
							 based investigation involving triangles using the trigonometric functions, the
							 law of sines and the law of cosines, justify and present results.
M.O.T.3.9
develop and test a hypothesis to find the area of a
							 triangle given the measures of two sides and the included angle or the measures
							 of three sides (Heron's formula) and use these formulas to find total area of
							 figures constructed of multiple shapes.
M.O.T.3.10
express complex numbers in polar form:
· perform operations including adding,
							 subtracting, multiplying, and dividing;
· evaluate powers and roots of complex numbers
							 using De Moivre's Theorem; and graph complex numbers.
· graph complex numbers in the polar
							 coordinate plane and make conjectures about some polar graphs and real-world
							 situations such as the paths that the planets travel.
M.O.T.3.11
create graphical and algebraic representations for
							 performing vector operations and analyze these to solve real-world problems
							 such as force analysis and navigation.
Probability and Statistics Content Standards and
				  Objectives
Probability and Statistics is one of the most important
				  branches of the mathematical sciences. Knowledge of these topics is critical to
				  decision-making and to the analysis of data. Using concepts of probability and
				  statistics, individuals are able to predict the likelihood of an event
				  occurring, organize and evaluate data, and identify the significance of
				  statements. Connections between content and applications to the real-world will
				  be emphasized. Graphing utilities such as calculators and computers will be
				  used to enhance student learning and to aid in the solution of practical
				  problems. Prerequisites for this course are successful completion of Algebra II
				  and Geometry. The West Virginia Standards for 21st
				  Century Learning include the following components:
				  21st Century Content Standards and Objectives and
				  21st Century Learning Skills and Technology Tools.
				  All West Virginia teachers are responsible for classroom instruction that
				  integrates learning skills, technology tools and content standards and
				  objectives.
Grade 9-12
Mathematics: Probability and
							 Statistics
Standard 5
Data Analysis and Probability
M.S.PS.5
Through communication, representation, reasoning and
							 proof, problem solving, and making connections within and beyond the field of
							 mathematics, students will
· formulate questions that can be addressed
							 with data and collect, organize, and display relevant data to answer
							 them,
· select and use appropriate statistical
							 methods to analyze data,
· develop and evaluate inferences and
							 predictions that are based on models, and
· apply and demonstrate an understanding of
							 basic concepts of probability.
Performance Descriptors
							 (M.PD.PS.3)
Distinguished
Above Mastery
Mastery
Partial Mastery
Novice
Probability and Statistics students at the
							 distinguished level:
distinguish between, justify and investigate types of
							 probability using multiple counting principles and distributions;
use proper sampling techniques to compare and
							 contrast more than one set of data that they collect, summarize, and interpret
							 numerically and graphically in both one-variable and two-variable
							 situations;
test the validity of a hypothesis in real-world
							 situations by determining the appropriate inference technique to make a
							 conclusion about the population of interest
Probability and Statistics students at the above
							 mastery level:
distinguish between, justify and investigate types of
							 probability using multiple counting principles and distributions;
use proper sampling techniques to collect, summarize,
							 and interpret data numerically and graphically in both one-variable and
							 two-variable situations;
test the validity of a hypothesis in real-world
							 situations by determining the appropriate inference technique to make a
							 conclusion about the population of interest
Probability and Statistics students at the mastery
							 level:
distinguish between types of probability using
							 multiple counting principles and distributions;
use proper sampling techniques to collect, summarize,
							 and interpret data numerically and graphically in both one-variable and
							 two-variable situations;
test the validity of a hypothesis in real-world
							 situations by determining the appropriate inference technique to make a
							 conclusion about the population of interest.
Probability and Statistics students at the partial
							 mastery level:
calculate probabilities given the type using multiple
							 counting principles and distributions;
use proper sampling techniques to collect and
							 summarize data numerically and graphically in both one-variable and
							 two-variable situations;
test the validity of a hypothesis in real-world
							 situations using the provided inference technique to make a conclusion about
							 the population of interest.
Probability and Statistics students at the novice
							 level:
recognize the types of probability using multiple
							 counting principles and distributions;
collect and summarize data numerically and
							 graphically in both one-variable and two-variable situation;
identify a hypothesis in real-world situations to
							 recognize that an inference technique needs to be used in order to make a
							 conclusion about the population of interest.
Objectives
Students will
M.O.PS.5.1
distinguish between experimental and theoretical
							 probability.
M.O.PS.5.2
using a real-world problem solving investigation,
							 create and interpret data using various methods of displaying circle graphs,
							 histograms, and frequency curves, make predictions, include information
							 concerning outliers, present and justify results.
M.O.PS.5.3
determine possible outcomes using tree diagrams and
							 the counting principles of permutations and combinations.
M.O.PS.5.4
express the chances of events occurring either in
							 terms of a probability or odds.
M.O.PS.5.5
use the normal distribution and the binomial
							 distribution including Pascal's triangle, to determine probability of
							 events.
M.O.PS.5.6
analyze measures of central tendency (mean, median,
							 and mode) from data presented in a variety of forms such as charts, tables, and
							 graphs or from data created through experimentation.
M.O.PS.5.7
interpret and calculate measures of dispersions
							 (range and standard deviation) from data presented in a variety of forms such
							 as charts, tables and graphs or from data created through
							 experimentation.
M.O.PS.5.8
analyze individual performances in terms of
							 percentiles, z-scores, and t- scores.
M.O.PS.5.9
analyze the role of sampling, randomness, bias, and
							 sample size in data collection and interpretation.
M.O.PS.5.10
identify a real life situation that involves
							 statistical concepts including a t-test, make a hypothesis as to the outcome;
							 develop, justify, and implement a method to collect, organize and analyze data;
							 generalize the results to make a conclusion, compare the hypothesis and the
							 conclusion; present the project using predictive and analytic tools (with and
							 without technology).
M.O.PS.5.11
determine the correlation values for given data or
							 for data generated by students and use the results to describe the association
							 of the variables within the given data. Identify whether this association is
							 systematic or predictable.
M.O.PS.5.12
calculate the Chi-Square values for a given
							 population.
M.O.PS.5.13
perform a regression analysis on a set of data,
							 either given or created through experimentation, and use the results to predict
							 specific values of a variable. Identify the regression equation.
M.O.PS.5.14
perform an analysis of variance (ANOVA) and interpret
							 the results.
Pre-Calculus Content Standards and
				  Objectives
Pre-Calculus objectives extend students' knowledge of
				  functions and equations (e.g., higher-order functions, exponential, and
				  logarithmic) as well as provide preparation for a calculus course. Available
				  technology will be used by students and teachers to enhance learning. Graphing
				  utilities are powerful tools for solving and verifying equations and
				  inequalities. They also aid in investigating functions, and their inverses. The
				  West Virginia Standards for 21st Century Learning
				  include the following components: 21st Century
				  Content Standards and Objectives and 21st Century
				  Learning Skills and Technology Tools. All West Virginia teachers are
				  responsible for classroom instruction that integrates learning skills,
				  technology tools, and content standards and objectives.
Grade 9-12
Mathematics: Pre-Calculus
Standard 2
Algebra
M.S.PC.2
Through communication, representation, reasoning and
							 proof, problem solving, and making connections within and beyond the field of
							 mathematics, students will
· demonstrate understanding of patterns,
							 relations, and functions,
· represent and analyze mathematical
							 situations and structures using algebraic symbols,
· use mathematical models to represent and
							 understand quantitative relationships, and
· analyze change in various contexts.
Performance Descriptors
							 (M.PD.PC.2)
Distinguished
Above Mastery
Mastery
Partial Mastery
Novice
Pre-Calculus students at the distinguished
							 level:
investigate and formulate a solution to a real-world
							 problem involving higher-order polynomials, exponential and logarithmic
							 equations;
solve application problems involving sequences and
							 series, and formulate a hypothesis for evaluating limits;
consider and justify using induction to prove
							 formulas and statements;
differentiate between appropriate methods to expand
							 binomials in relation to real-world problems;
compare and contrast various graphs formulating a set
							 of rules that produce and support a solution to a real-world problem;
design and execute a method to solve a real-world
							 problem involving vectors.
Pre-Calculus students at the above mastery
							 level:
hypothesize and develop a plan to solve higher-order
							 polynomials, exponential and logarithmic equations;
solve application problems involving sequences and
							 series and evaluate limits;
devise a method for proving formulas and
							 statements;
expand binomials by applying appropriate methods and
							 relate the expansion to real-world situations;
identify and justify their solutions to real-world
							 problems which require various graphs;
recognize the application of vectors to practical
							 problems and perform operations on vectors to solve them.
Pre-Calculus students at the mastery level:
determine the reasonableness of the solutions of
							 higher-order polynomials, exponential and logarithmic equations;
evaluate sequences and series to find or estimate a
							 limit;
differentiate the process of proving formulas and
							 statements;
expand binomials by applying appropriate
							 methods;
interpret the techniques of curve sketching to graph
							 functions of real-world situations;
analyze and perform operations on vectors to solve
							 practical problems.
Pre-Calculus students at the partial mastery
							 level:
examine the solutions of higher-order polynomials,
							 exponential and logarithmic equations;
differentiate sequences and series, and find or
							 estimate a limit;
find formulas and statements by applying
							 induction;
identify the various methods for expanding
							 binomials;
graph various functions;
perform operations on vectors to solve practical
							 problems.
Pre-Calculus students at the novice level:
confirm the solutions of higher-order polynomials,
							 exponential and logarithmic equations;
recognize sequences and series, and find or estimate
							 a limit;
recognize induction as a process to prove statements
							 and formulas;
recognize the methods for expanding binomials;
identify the graphs of various functions;
perform operations on vectors.
Objectives
Students will
M.O.PC.2.1
investigate and sketch the graphs of polynomials and
							 rational functions by analyzing and using the characteristics of zeros, upper
							 and lower bounds, y-intercepts, symmetry, asymptotes and end behavior, maximum
							 and minimum points, and domain and range.
M.O.PC.2.2
solve higher order polynomial equations utilizing
							 techniques such as Descartes' Rule of Signs, upper and lower bounds, and the
							 Rational Root Theorem.
M.O.PC.2.3
relate Pascal's Triangle and the Binomial Theorem;
							 use both to expand binomials with positive integral exponents.
M.O.PC.2.4
establish and explain the inverse relationship
							 between exponential and logarithmic functions; graph related functions and
							 include their domain and range using interval notation.
M.O.PC.2.5
compare laws of exponents to properties of
							 logarithms; solve equations and practical problems involving exponential and
							 logarithmic expressions, including natural and common logarithms; confirm
							 solutions graphically and numerically.
M.O.PC.2.6
solve problems involving the sum of finite and
							 infinite sequences and series, including Sigma notation.
M.O.PC.2.7
use tables of values, graphs, conjectures, algebraic
							 methods, and numerical substitution to find or estimate the limit of a
							 function, a sequence or a series.
M.O.PC.2.8
analyze and describe the geometry of vectors, perform
							 mathematical operations with vectors and use vectors to solve practical
							 problems.
M.O.PC.2.9
apply the method of mathematical induction to prove
							 formulas and statements.
M.O.PC.2.10
apply parametric methods to represent motion of
							 objects.
M.O.PC.2.11
use multiple representations, such as words, graphs,
							 tables, and equations, to solve practical problems involving logarithmic,
							 exponential, polynomial, rational, and radical functions; explain how the
							 representations are related to each other, as well as to the problem.
Grade 9-12
Mathematics: Pre-Calculus
Standard 3
Geometry
M.S.PC.3
Through communication, representation, reasoning and
							 proof, problem solving, and making connections within and beyond the field of
							 mathematics, students will
· analyze characteristics and properties of
							 two- and three-dimensional geometric shapes and develop mathematical arguments
							 about geometric relationships,
· specify locations and describe spatial
							 relationships using coordinate geometry and other representational
							 systems,
· apply transformations and use symmetry to
							 analyze mathematical situations, and
· solve problems using visualization, spatial
							 reasoning, and geometric modeling.
Performance Descriptors
							 (M.PD.PD.3)
Distinguished
Above Mastery
Mastery
Partial Mastery
Novice
Pre-Calculus students at the distinguished
							 level:
hypothesize, organize, determine and explain the
							 justification for the solutions to real-world problems involving conic sections
							 and their transformations.
Pre-Calculus students at the above mastery
							 level:
analyze, interpret, and graph the conic sections
							 along with their transformations, and apply to real-world situations.
Pre-Calculus students at the mastery level:
analyze, interpret, and graph conic sections and
							 their transformations.
Pre-Calculus students at the partial mastery
							 level:
graph conic sections and their
							 transformations.
Pre-Calculus students at the novice level:
identify the graphs of conic sections and their
							 transformations.
Objectives
Students will
M.O.PC.3.1
graph functions and conic sections using
							 transformations.
M.O.PC.3.2
analyze and describe properties of conic sections;
							 explain the interrelationship among the properties; solve practical problems
							 involving conic sections.
Grade 9-12
Mathematics: Pre-Calculus
Standard 5
Data Analysis and Probability
M.S.PC.5
Through communication, representation, reasoning and
							 proof, problem solving, and making connections within and beyond the field of
							 mathematics, students will
· formulate questions that can be addressed
							 with data and collect, organize, and display relevant data to answer
							 them,
· select and use appropriate statistical
							 methods to analyze data,
· develop and evaluate inferences and
							 predictions that are based on models, and
· apply and demonstrate an understanding of
							 basic concepts of probability.
Performance Descriptors
							 (M.PD.PC.5)
Distinguished
Above Mastery
Mastery
Partial Mastery
Novice
Pre-Calculus students at the distinguished
							 level:
relate and defend a solution to a developed
							 real-world situation that involves use of regression equations.
Pre-Calculus students at the above mastery
							 level:
summarize the analysis of developed regression
							 equations.
Pre-Calculus students at the mastery level:
investigate, hypothesize, and develop a regression
							 equation.
Pre-Calculus students at the partial mastery
							 level:
investigate and hypothesize regarding a regression
							 equation.
Pre-Calculus students at the novice level:
investigate and select a regression equation.
Objectives
Students will
M.O.PC.5.1
identify a real life situation that exhibits
							 characteristics of exponential or logistic growth or decay; pose a question;
							 make a hypothesis as to the answer; develop, justify, and implement a method to
							 collect, organize, and analyze related data; extend the nature of collected,
							 discrete data to that of a continuous function that describes the known data
							 set; generalize the results to make a conclusion; compare the hypothesis and
							 the conclusion; present the project numerically, analytically, graphically and
							 verbally using the predictive and analytic tools of pre-calculus (with and
							 without technology).
Calculus Content Standards and Objectives
Calculus objectives are designed for students who have
				  completed Algebra I, Geometry, Algebra II, Trigonometry, and Pre-Calculus.
				  Study includes functions and continuity, limits, differentiation and
				  applications of derivatives, integration and its application to area, volume,
				  and displacement. The Rule of Four (Numerical, Analytical, Graphical and
				  Verbal) will be applied throughout the course. Available technology will be
				  used by students and teachers to enhance learning. Graphing utilities will be
				  used to investigate concepts and to evaluate derivatives and integrals. The
				  West Virginia Standards for 21st Century Learning
				  include the following components: 21st Century
				  Content Standards and Objectives and 21st Century
				  Learning Skills and Technology Tools. All West Virginia teachers are
				  responsible for classroom instruction that integrates learning skills,
				  technology tools, and content standards and objectives.
Grade 9-12
Mathematics: Calculus
Standard 2
Algebra
M.S.C.2
Through communication, representation, reasoning and
							 proof, problem solving, and making connections within and beyond the field of
							 mathematics, students will
· demonstrate understanding of patterns,
							 relations, and functions,
· represent and analyze mathematical
							 situations and structures using algebraic symbols,
· use mathematical models to represent and
							 understand quantitative relationships, and
· analyze change in various contexts.
Performance Descriptors
							 (M.PD.C.2)
Distinguished
Above Mastery
Mastery
Partial Mastery
Novice
Calculus students at the distinguished level:
closely connect all representations of a
							 function;
recognize real life situations that involve limits
							 and interpret these limits using multiple representations and evaluate them
							 using appropriate limit properties;
relate the Intermediate Value Theorem, continuity,
							 and root finding;
apply the various forms of the definition of the
							 derivative of a function at a point; interpreted as the slope of the tangent
							 line to the graph of the function at any x, and as the instantaneous rate of
							 change. They recognize the tangent line slope as a limit of the converging
							 secant line slopes, and apply the limit definition to find a general form for
							 f'(x);
compare the average rate of change and the
							 instantaneous rate of change in real-world applications. They prove that
							 differentiability implies continuity and give examples of continuous functions
							 that are not differentiable. They combine and apply the rules of
							 differentiation to various types of functions as appropriate. They use Rolle's
							 Theorem to derive the Mean Value Theorem;
use mathematical models to solve applied
							 problems;
efficiently calculate indefinite or definite
							 integrals. They calculate a definite integral of a polynomial function using an
							 infinite limit of a Riemann sum and apply the Fundamental Theorem of
							 Calculus.
Calculus students at the above mastery level:
explain connections among algebraic notation,
							 graphical analysis and tabular data;
explain limits using multiple representations and
							 evaluate limits using appropriate limit properties;
determine if a function is continuous at a point over
							 an interval;
apply the definition of the derivative of a function
							 at a point; interpret this as the slope of the tangent line and as the
							 instantaneous rate of change. They recognize the tangent line slope as a limit
							 of the converging secant line slopes and apply the limit definition to find a
							 general form for f'(x);
compare the average rate of change and the
							 instantaneous rate of change in real-world applications, demonstrate that
							 differentiability implies continuity, and give examples of continuous functions
							 that are not differentiable. They combine and apply the rules of
							 differentiation to various types of functions as appropriate. They recognize
							 when the hypotheses of Rolle's and the Mean Value Theorems are
							 satisfied;
construct and apply mathematical models to solve
							 applied problems;
find definite and indefinite integrals that may
							 involve multiple substitutions and change of limits and calculate a definite
							 integral of a polynomial function using an infinite limit of a Riemann sum.
							 They apply the Fundamental Theorem of Calculus.
Calculus students at the mastery level:
manipulate algebraic notation to study functions and
							 relate the results to graphs and tables;
determine limits algebraically, graphically and
							 numerically, using appropriate limit properties;
decide about continuity at a point and over an
							 interval;
interpret the derivative of a function as the slope
							 of the tangent line to the graph of the function at any x, or as the
							 instantaneous rate of change. They apply the limit definition to find the
							 derivative at a point;
investigate the average rate of change and
							 instantaneous rate of change in real-world applications. They relate
							 differentiability and continuity and combine and apply the algebraic rules of
							 differentiation and theoretical results;
construct and apply mathematical models to solve
							 applied problems;
calculate definite and indefinite integrals for
							 integrable elementary functions. They calculate definite integrals using both
							 Riemann sums and the Fundamental Theorem of Calculus.
Calculus students at the partial mastery
							 level:
use algebraic notation for functions and confirm
							 results using graphs;
determine limits graphically or numerically and
							 evaluate limits using limit properties;
determine if a function is continuous at a point
							 numerically and graphically;
apply the definition of the derivative of a function
							 at a point to find the slope of the tangent line to the graph of the function,
							 interpreting the derivative as an instantaneous rate of change;
investigate the average rate of change and
							 instantaneous rate of change graphically. They recognize that differentiable
							 functions are also continuous. They apply the rules of differentiation to
							 various types of functions;
solve applied problems about motion, area, and
							 volume;
calculate definite and indefinite integrals for
							 integrable elementary functions. They apply the Fundamental Theorem of Calculus
							 to evaluate a definite integral.
Calculus students at the novice level:
recognize functions expressed algebraically and
							 graphically and use functional notation correctly;
given a graph or table, recognize a limit and
							 evaluate limits using limit properties;
determine graphically if a function is continuous at
							 a point;
construct the tangent line to a curve at a given
							 point and use derivatives to aid in graphing functions;
calculate the average rate of change and the
							 instantaneous rate of change;
solve simple optimization problems;
calculate definite and indefinite integrals for
							 polynomials. They apply the Fundamental Theorem of Calculus to evaluate a
							 definite integral.
Objectives
Students will
M.O.C.2.1
use abstract notation to apply properties of
							 algebraic, trigonometric, exponential, logarithmic and composite functions, as
							 well as their inverses, represented graphically, numerically, analytically, and
							 verbally; and demonstrate an understanding of the connections among these
							 representations.
M.O.C.2.2
demonstrate a conceptual understanding of the
							 definition of a limit via the analysis of continuous and discontinuous
							 functions represented using multiple representations (e.g. graphs and
							 tables).
M.O.C.2.3
Click Here To View Image
use
							 the properties of limits including addition, product, quotient, composition,
							 and squeeze/sandwich theorem to calculate the various forms of limits:
							 one-sided limits, limits at infinity, infinite limits, limits that do not
							 exist, and special limits such as
Click Here To View
							 Image
M.O.C.2.4
apply the definition of continuity to determine where
							 a function is continuous or discontinuous including continuity at a point,
							 continuity over an interval, application of the Intermediate Value Theorem, and
							 graphical interpretation of continuity and discontinuity.
M.O.C.2.5
investigate and apply the definition of the
							 derivative graphically, numerically, and analytically at a point, conceptually
							 interpreting the derivative as an instantaneous rate of change and the slope of
							 the tangent line.
M.O.C.2.6
discriminate between the average rate of change and
							 the instantaneous rate of change using real-world problems.
M.O.C.2.7
justify why differentiability implies continuity and
							 classify functional cases when continuity does not imply
							 differentiability.
M.O.C.2.8
recognize when the Extreme Value Theorem indicates
							 that function extrema exist.
M.O.C.2.9
quickly recall and apply rules of differentiation
							 including the constant multiple rule, sum rule, the difference rule, the
							 product rule, the quotient rule, the power rule, and the chain rule as applied
							 to algebraic, trigonometric, exponential, logarithmic, and inverse
							 trigonometric functions using techniques of both explicit and implicit
							 differentiation.
M.O.C.2.10
apply Rolle's Theorem and the Mean Value Theorem to
							 real-world problems.
M.O.C.2.11
construct and use mathematical models to solve
							 optimization, related-rates, velocity, and acceleration problems.
M.O.C.2.12
determine antiderivatives that follow from
							 derivatives of basic functions and apply substitution of variables.
M.O.C.2.13
calculate a definite integral using Riemann sums by
							 evaluating an infinite limit of a sum using summation notation and rules for
							 summation.
M.O.C.2.14
evaluate definite integrals using basic integration
							 properties such as addition, subtraction, constant multipliers, the power rule,
							 substitution, and change of limits.
M.O.C.2.15
characterize the definite integral as the total
							 change of a function over an interval and use this to solve real-world
							 problems.
M.O.C.2.16
apply the Fundamental Theorem of Calculus to evaluate
							 definite integrals and to formulate a cumulative area function and interpret
							 the function as it relates to the integrand.
M.O.C.2.17
use integration to solve problems that involve linear
							 displacement, total distance, position, velocity, acceleration and area between
							 curves by looking at both functions of x and functions of y; utilize units to
							 interpret the physical nature of the calculus process.
Grade 9-12
Mathematics: Calculus
Standard 3
Geometry
M.S.C.3
Through communication, representation, reasoning and
							 proof, problem solving, and making connections within and beyond the field of
							 mathematics, students will
· analyze characteristics and properties of
							 two- and three-dimensional geometric shapes and develop mathematical arguments
							 about geometric relationships,
· specify locations and describe spatial
							 relationships using coordinate geometry and other representational
							 systems,
· apply transformations and use symmetry to
							 analyze mathematical situations, and
· solve problems using visualization, spatial
							 reasoning, and geometric modeling.
Performance Descriptors
							 (M.PD.C.3)
Distinguished
Above Mastery
Mastery
Partial Mastery
Novice
Calculus students at the distinguished level:
apply the definition of continuity to categorize
							 discontinuities of functions presented algebraically and graphically;
use asymptotes to explain end behavior of functions,
							 and describe asymptotic behavior using multiple representations, develop
							 tangent lines as best linear approximations to functions near specific points
							 and apply this concept to Newton's Method;
investigate and explain the relationships among the
							 graphs of a function and its derivatives;
anticipate whether the left, right, or midpoint rule
							 will yield the best approximation to a definite integral using a Riemann Sum
							 with a finite number of sub-intervals. They propose better methods for
							 approximating the actual area.
Calculus students at the above mastery level:
recognize continuous and discontinuous functions
							 using limits;
use limits to find and justify the existence of
							 asymptotes of functions, develop tangent lines as best linear approximations to
							 functions near specific points, construct these tangent lines and apply this
							 concept to Newton's Method;
investigate and explain the relationships among the
							 graph of a function and its derivatives;
approximate the area under a curve using a Riemann
							 sum implementing left, right, or midpoint rules, and determine whether the left
							 hand and right hand approximations over-estimate or under-estimate the actual
							 area.
Calculus students at the mastery level:
recognize continuous and discontinuous functions
							 graphically;
apply limits to recognize asymptotes, use tangent
							 lines to locally approximate functions, and apply Newton's Method to
							 approximate zeroes of functions;
extract information about the graph of a function
							 from its derivative and limiting values;
approximate the area under a curve via a Riemann sum
							 using left, right, or midpoint rules.
Calculus students at the partial mastery
							 level:
distinguish between continuous and discontinuous
							 functions graphically;
apply limits to find asymptotes, use a tangent line
							 to approximate a function at a point and can apply Newton's Method to
							 approximate zeroes of functions;
use derivatives to aid in graphing functions;
approximate the area under a curve by constructing a
							 Riemann sum implementing left, right, or midpoint rules.
Calculus students at the novice level:
identify a discontinuous function graphically;
given a graph, identify the location of
							 asymptotes;
use information from derivatives to aid in graphing
							 functions;
approximate the area under a curve by applying a
							 finite Riemann sum implementing left, right, or midpoint rules, given the
							 subdivision.
Objectives
Students will
M.O.C.3.1
use limits to deduce asymptotic behavior of the graph
							 of a function.
M.O.C.3.2
compare and contrast the limit definition (not delta
							 epsilon) of continuity and the graphical interpretation of the continuity of a
							 function at a point; recognize different types of discontinuities.
M.O.C.3.3
develop tangent lines as best linear approximations
							 to functions near specific points; explain this conceptually; and construct
							 these tangent lines; and apply this concept to Newton's Method.
M.O.C.3.4
investigate and explain the relationships among the
							 graphs of a function, its derivative and its second derivative; construct the
							 graph of a function using the first and second derivatives including extrema,
							 points of inflection, and asymptotic behavior.
M.O.C.3.5
approximate areas under a curve using Riemann sums by
							 applying and comparing left, right, and midpoint methods for a finite number of
							 subintervals.
Grade 9-12
Mathematics: Calculus
Standard 5
Data Analysis and Probability
M.S.C.5
Through communication, representation, reasoning and
							 proof, problem solving, and making connections within and beyond the field of
							 mathematics, students will
· formulate questions that can be addressed
							 with data and collect, organize, and display relevant data to answer
							 them,
· select and use appropriate statistical
							 methods to analyze data,
· develop and evaluate inferences and
							 predictions that are based on models, and
· apply and demonstrate an understanding of
							 basic concepts of probability.
Performance Descriptors
							 (M.PD.C.5)
Distinguished
Above Mastery
Mastery
Partial Mastery
Novice
Calculus students at the distinguished level:
apportion individual tasks in small groups to
							 identify a real life situation that involves modeling change; pose a question;
							 make a hypothesis as to the answer; develop, justify, and implement a method to
							 collect, organize, and analyze related data; extend the nature of collected,
							 discrete data to that of a continuous function that describes the known data
							 set; generalize the results to make predictions to test their model; compare
							 the hypothesis and the conclusion. They present the project numerically,
							 analytically, graphically and verbally.
Calculus students at the above mastery level:
in small groups, identify a real life situation that
							 involves modeling change; pose a question; make a hypothesis as to the answer;
							 implement a method to collect, organize, and analyze related data; extend the
							 nature of collected, discrete data to that of a continuous function that
							 describes the known data set; generalize the results to make predictions to
							 test their model; compare the hypothesis and the conclusion. They collaborate
							 using concepts from calculus to present the project numerically, analytically,
							 graphically and verbally.
Calculus students at the mastery level:
working in small groups, identify a real life
							 situation that involves modeling change; pose a question; implement a method to
							 collect, organize, and analyze related data; find a continuous function that
							 describes the known data set; make predictions to test their model. They
							 collaborate using concepts from calculus to present the project numerically,
							 analytically, graphically and verbally.
Calculus students at the partial mastery
							 level:
working in teacher facilitated groups, solve a real
							 life problem using given data that involves modeling change. They organize and
							 analyze the data; find a continuous function that describes the known data set.
							 They collaborate using concepts from calculus to present the projects
							 numerically, analytically, graphically and verbally.
Calculus students at the novice level:
working in teacher facilitated groups, solve a real
							 life problem using provided data that involves modeling change. They extend
							 collected, discrete data to values of a continuous function that describes the
							 known data set. They present the projects numerically, analytically,
							 graphically and verbally.
Objectives
Students will
M.O.C.5.1
identify a real life situation that involves
							 quantities that change over time; pose a question; make a hypothesis as to the
							 answer; develop, justify, and implement a method to collect, organize, and
							 analyze related data; extend the nature of collected, discrete data to that of
							 a continuous function that describes the known data set; generalize the results
							 to make a conclusion; compare the hypothesis and the conclusion; present the
							 project numerically, analytically, graphically and verbally using the
							 predictive and analytic tools of calculus.