Content Outline and Competencies:

I. Whole Numbers 
  A. Identify place value of the digits in a whole number; read whole numbers; write whole numbers; compare the size of whole numbers using inequality symbols. 
  B. Round whole numbers; estimate results using rounding. 
  C. Add, subtract, multiply, and divide whole numbers. 
  D. Evaluate expressions written in exponential form. 
  E. Apply the rules of order of operation to whole numbers. 
  F. Use the tests for divisibility for 2, 3, 5, and 10. 
  G. Determine if a number is prime or composite; determine the prime factorization of whole numbers; find least common multiple and greatest common factor. 
  H. Solve applications using whole numbers. 

II. Fraction Notation 
  A. Identify the numerator and denominator of a fraction; identify whether a fraction is proper or improper; identify mixed numbers; change mixed numbers to improper fractions; change an improper fraction to a mixed number; compare the size of fractions using inequality symbols. 
  B. Simplify fractions; identify equivalent fractions; change a fraction to an equivalent fraction with a different denominator. 
  C. Add, subtract, multiply, and divide fractions and mixed numbers. 
  D. Apply the rules of order of operation to fractions and mixed numbers. 
  E. Solve applications using fractions. 

III. Ratios and Proportions 
  A. Define a ratio; create a ratio. 
  B. Determine if a proportion is true.
  C. Solve proportions. 
  D. Solve applications using proportions. 

IV. Decimal Notation 
  A. Identify place value of the digits in a decimal number; read decimal numbers; write decimal numbers. 
  B. Round decimal numbers; estimate results using rounding. 
  C. Add, subtract, multiply, and divide decimals. 
  D. Compare the size of decimals using inequality symbols. 
  E. Apply the rules of order of operation to decimals. 
  F. Change from fraction notation to decimal notation; change from decimal notation to fraction notation. 
  G. Solve applications using decimals. 

V. Percent Notation 
  A. Define percent; change from percent notation to fraction notation; change from percent notation to decimal notation; change from fraction notation to percent notation; change from decimal notation to percent notation. 
  B. Solve percent problems by using proportions and/or by using basic equations. 
  C. Solve applications using percents. 

VI. Measurement 
  A. Convert from one unit of measurement to another in the U.S.. System. 
  B. Convert from one unit of measurement to another in the Metric System. 
  C. Convert between U.S.. and metric measurements. 
  D. Solve applications using U.S. and metric measurements. 

VII. Geometry 
  A. Classify angles; identify supplementary and complementary angles. 
  B. Identify rectangles, squares, parallelograms, triangles and trapezoids. 
  C. Calculate the perimeter of polygons; calculate the circumference of circles; calculate the perimeter and circumference of composite figures. 
  D. Calculate the area of a rectangle, a square, a parallelogram, a triangle, and a trapezoid; calculate the area of composite figures. 
  E. Calculate the volume of rectangular solids, circular cylinders, spheres, and circular cones.
  F. Define similar figures; determine if two triangles are similar; use a proportion to find a missing side of similar triangles. 
  G. Use the Pythagorean theorem to determine the missing side of a right triangle. 
  H. Solve applications using geometry. 

VIII. Signed Numbers 
  A. Compare the size of signed numbers using inequality symbols. 
  B. Evaluate absolute value expressions.
  C. Add, subtract, multiply, and divide signed numbers. 
  D. Apply the rules of order of operation to signed numbers. 
  E. Solve applications using signed numbers. 

IX. Introduction to Algebra Topics 
  A. Identify constants, variables, terms, and coefficients in algebraic expressions. 
  B. Identify like terms; add and subtract like terms. 
  C. Apply the distributive property to algebraic expressions. 
  D. Simplify algebraic expressions; evaluate algebraic expressions; translate words into algebraic expressions. 
  E. Solve simple linear equations using the addition and multiplication properties of equality.

X. Statistics 
  A. Calculate the mean, median, and mode of a set of numbers.
  B. Read tables, circle graphs, line graphs and bar graphs. C. Interpret tables, circle graphs, line graphs and bar graphs.

Content Outline and Competencies:

I. Arithmetic and Algebraic Manipulation
   A. Evaluate arithmetic expressions using the order of operations
   B. Evaluate algebraic expressions 
   C.  Apply the laws of exponents to simplify expressions containing
integer exponents
   D.  Express numbers in scientific notation
   E. Perform addition, subtraction, multiplication, and division on
polynomial expressions
   F. Factor expressions with common factors, expressions that require
grouping, quadratic expressions, and difference of square expressions
   G. Perform addition, subtraction, multiplication, and division on
rational expressions
   H. Calculate radicals, approximating those that are irrational
   I. Simplify radicals of any order using the product and quotient rules

II. Equations and Inequalities
   A. Solve linear equations in one variable
   B. Solve proportion equations
   C. Solve linear inequalities in one variable showing solutions on a
number line
   D. Solve literal equations that do not require factoring  
   E. Solve quadratic equations by factoring
   F. Solve quadratic equations by using the square root property
   G. Solve equations developed from number, geometry, proportion, and
percent applications

III. Graphs on a coordinate plane
   A. Graph linear equations by plotting points
   B. Graph linear equations using intercepts
   C. Graph linear equations using the y-intercept and slope 

IV. Analysis of equations and graphs
   A. Identify the x-intercept, y-intercept, and slope of a line given its
graph
   B. Construct an equation of a line given its graph
   C. Construct an equation of a line given its slope and y-intercept
   D. Construct an equation of a line given its slope and a point
   E. Construct an equation of a horizontal line
   F. Construct an equation of a vertical line
   G. Determine whether or not an equation is linear
   H. Calculate the slope of a line passing through two given points

Content Outline and Competencies:

I. Arithmetic and Algebraic Manipulation
   A. Factor quadratic in form expressions, sum of cubes expressions,
and difference of cubes expressions
   B. Perform addition, subtraction, multiplication, and division on
rational expressions
   C. Simplify complex fractions
   D. Apply the laws of exponents to simplify expressions containing
rational exponents
   E. Apply the laws of radicals to perform addition, subtraction, and
multiplication
   F. Rationalize denominators containing radicals
   G. Simplify radicals containing negative radicands 
   H. Perform addition, subtraction, multiplication, and division on
complex numbers
   I. Evaluate functions 

II. Equations and Inequalities
   A. Solve linear inequalities in one variable showing solutions on a
number line and in interval notation
   B. Solve literal equations that require factoring  
   C. Solve systems of linear equations in two variables
   D. Solve quadratic equations by completing the square
   E. Solve quadratic equations by using the quadratic formula
   F. Solve equations that are quadratic in form
   G. Solve quadratic inequalities in one variable showing solutions on a
number line and in interval notation
   H. Solve equations containing rational expressions
   I. Solve equations containing radicals
   J. Solve absolute value equations in one variable 
   K. Solve equations developed from mixture, motion, work, and geometry
applications

III. Graphs on a coordinate plane
   A. Graph linear inequalities
   B. Graph quadratic equations
   C. Graph circles

IV. Analysis of equations and graphs
   A. Construct an equation of a line given two points
   B. Construct an equation of a line perpendicular to a given line
through a specific point
   C. Construct an equation of a line parallel to a given line through a
specific point
   D. Construct an equation of a circle given its center and radius
   E. Calculate the distance between two points
   F. Determine the center and radius of a circle by completing the
square
   G. Determine the midpoint between two points
   H. Distinguish between functions and non-functions using the vertical
line test 
   I. Identify the domain and range of a function given its graph

Content Outline and Competencies:

I. Geometric Shapes
   A. Describe points, lines, and planes.
   B. Compute the distance between two points on a number line.
   C. Define line segments, rays, and angles.
   D. Identify types of angles.
   E. Determine measures of angles with a protractor.
   F. Distinguish between complementary and supplementary angles.
   G. Define triangles and list their types.
   H. Define polygons and list their types.
   I. Define quadrilaterals and list their types.
   J. Identify lines of symmetry in a polygon.
   K. Find the angle measures in a polygon.
   L.I dentify prisms, pyramids, cylinders, cones, and spheres.
   M. Identify regular polyhedra and list their types.
   N. Convert linear dimensions using dimensional analysis.

II. Perimeter, Area, and Volume
   A. Find the perimeter of a polygon.
   B. Find the circumference of a circle.
   C. Find the area of rectangles, triangles, parallelograms, and
trapezoids.
   D. Find the area of a regular polygon.
   E. Find the area of a circle.
   F. Find the length of a missing side in a right triangle.
   G. Describe the relationship of sides of a 45°-45°-90° and of a
30°-60°-90° triangle.
   H. Find the surface area of prisms, pyramids, cylinders, cones, and
spheres.
   I. Find the volume of prisms, pyramids, cylinders, cones, and spheres.
   J. Convert area and volume measures using dimensional analysis.

III. Reasoning
   A. Draw conclusions.
   B. Apply conditional statements.
   C. Apply equivalent (biconditional) statements.
   D. Recognize valid and invalid deductions.
   E. Write the converse of a statement.
   F. Write a deductive proof.

IV. Triangles
   A. Identify vertical angles and exterior angles.
   B. List the triangle congruence theorems.
   C. Prove two triangles are congruent.
   D. Prove corresponding parts of congruent triangles are congruent.
   E. Define medians and perpendicular bisectors
   F. Use theorems related to isosceles and equilateral triangles.
   G. Use theorems related to triangle inequalities.
   H. Construct segments and angles using a compass and straightedge.
   I. Construct perpendicular lines using a compass and straightedge.

V. Parallel Lines and Quadrilaterals
   A. Identify alternate interior, alternate exterior, and corresponding
angles.
   B. Use theorems derived from the Parallel Postulate.
   C. Prove the Angle Sum in a Triangle Theorem.
   D. Use the Exterior Angle Theorem.
   E. Use theorems related to quadrilaterals to determine side lengths and
angle measures.
   F. Prove theorems related to quadrilaterals.
   G. Construct parallel lines using a compass and straightedge.
   H. Subdivide line segments using a compass and straightedge.

VI. Similarity
   A. Solve problems related to ratio and proportion.
   B. Use triangle similarity theorems to prove two triangles are
similar.
   C. Find the missing part of similar polygons.
   D. Use theorems related to the mean proportional in a right triangle.
   E. Use the Side Splitting Theorem

VII. Circles
   A. Define arc, central angle, and inscribed angle.
   B. Determine measures of angles and arcs of a circle.
   C. Find areas of sectors and measures of arc length.
   D. Define chords and tangents.
   E. Determine measures of segments and angles formed by chords.
   F. Prove theorems related to circles.
   G. Construct tangent lines using a compass and straightedge.
   H. Find the center of a circle using a compass and straightedge.

VIII. Coordinate Geometry
   A. Plot points in a coordinate system.
   B. Find the distance between two points in the plane.
   C. Determine if three points in the plane are collinear.
   D. Find the midpoint of a line segment.
   E. Find the slope of a line.
   F. Find the slopes of parallel and perpendicular lines.
   G. Find the equation of a line.
   H. Find the center and radius of a circle from its equation.
   I. Find the equation of a circle.
   J. Graph lines and circles from their equations.
   K. Prove theorems using coordinates.

Content Outline and Competencies:

I. The Mathematics of Percents.
   A. Solve for base, rate, or part in a basic percent problem.
   B. Solve for the old or new value in a percent increase problem.
   C. Solve for the old or new value in a percent decrease problem.

II. The Mathematics of Payroll.
   A. Given an hourly wage with an overtime policy, find the gross pay.
   B. Given a commission structure or a piecework rate, find the gross
pay.
   C. Use the Percentage Method to calculate federal and state withholding
tax.
   D. Calculate FICA taxes; explain the purpose of FICA taxes.
   E. Calculate federal and state unemployment taxes.
   F. Calculate an employee’s net pay.
   G. Calculate the cost of employment; identify the employment taxes an
employer must pay.
   H. Use a computer to analyze the effect of taxes on gross pay.

III. The Mathematics of Retailing.
   A. Read and analyze an invoice; explain key abbreviations.
   B. Calculate trade, series, and cash discounts.
   C. Calculate markup based on cost.
   D. Calculate markup based on selling price.
   E. Calculate markdowns.
   F. Calculate the adjusted cost when shrinkage is present.
   G. Calculate the net profit.
   H. Calculate operating loss and absolute loss; explain the meaning of
each.
   I. Calculate the amount of operating expenses from the percent.
   J. Calculate the breakeven point.

IV. The Mathematics of Finance and the Simple Interest Formula
   A. Find the interest earned using the simple interest formula.
   B. Solve for principal, rate, or time in a simple interest problem.
   C. Calculate the interest, the proceeds, and the maturity value on a
simple interest note or a simple discount note.
   D. Find the bank discount on a note; explain what discounting a note
means in terms of a problem.
   E. Determine the effective rate (APR) of a note.
   F. Compute the payoff amount on a loan or note using the U.S. Rule.

V. The Mathematics of Finance and Time Value of Money Problems
   A. Use the financial calculator to find the future value, initial
value, time, or rate of a lump sum deposit problem.
   B. Use the financial calculator to find the future value, initial
value, periodic payment, time, or rate of a savings plan.
   C. Use the financial calculator to find the future balance, present
balance, periodic payment, time, or rate of a loan.
   D. Calculate the total interest earned in a time value of money
problem.
   E. Calculate the present value of an annuity; explain the meaning of
present value.
   F. Use a computer to solve time value of money applications; analyze
computer results of time value of money applications.

VI. The Valuation of Assets.
    A. Determine the value of ending inventory using Average Cost
(Weighted Average), FIFO, and LIFO.
    B. Use the straight line depreciation method to find the value of an
asset.
    C. Use the double declining balance depreciation method to find the
value of an asset.
    D. Use the units of production depreciation method to find the value
of an asset.

Content Outline and Competencies:

I. Numeric Expressions
   A. Describe the properties of the real number system.
   B. Use the order of operations for the real number system to simplify
mathematical expressions.
   C. Simplify expressions involving exponents and radicals.
   D. Determine the number of significant digits in a number.
   E. Use scientific notation and rounding appropriately in computation.
   F. Convert between scientific notation and standard notation.

II. Algebraic Expressions
   A. Add and subtract polynomials.
   B. Multiply polynomials.
   C. Divide polynomials.
   D. Evaluate algebraic expressions.

III. Linear Equations
   A. Solve linear equations.
   B. Solve formulas for a particular variable, including those used in
technical work.
   C. Define ratio and proportion.
   D. Solve a proportion for a missing term.
   E. Solve a system of two or three equations by algebraic methods.

IV. Basic Geometry Skills
   A. Define parallel lines and angles formed by a transversal; use these
concepts to determine unknown angles.
   B. Classify triangles.
   C. Calculate area and perimeter of polygons.
   D. Calculate area and circumference of circles.
   E. Calculate volume and surface area of geometric solids.

V. Graphing
   A. Plot points on the rectangular coordinate system.
   B. Graph straight lines.
   C. Define the slope of a line; use the slope to graph a line.
   D. Solve a system of equations by graphing.

Content Outline and Competencies:

I. Algebraic Skills
   A. Factor polynomials.
      1. Factor using greatest common factor.
      2. Factor using grouping.
      3. Factor trinomials.
      4. Factor using the difference of squares.
   B. Simplify rational and radical expressions.
   C. Perform operations with rational expressions, radical expressions
and complex numbers.
   D. Convert between exponential and logarithmic notation.
   E. Expand and condense logarithmic expressions using properties of
logarithms.

II.  Equations
   A. Solve quadratic equations by factoring and by the quadratic
formula.
   B. Solve problems involving direct, inverse, and joint variation.
   C. Solve equations involving rational expressions.
   D. Solve equations involving radical expressions.
   E. Solve exponential equations.
   F. Solve logarithmic equations.

III. Trigonometry
   A. Define trigonometric functions using both a unit circle and a right
triangle.
   B. Define angle measurement in both radians and degrees.
   C. Evaluate trigonometric functions of any angle.
   D. Define inverse trigonometric functions.
   E. Solve right triangles for missing parts.
   F. Solve application problems using right triangle trigonometry.
   G. Define vectors.
   H. Perform operations with vectors.
   I. Solve oblique triangles for missing parts using the Law of Sines and
the Law of Cosines.
   J. Solve application problems using the Law of Sines and the Law of
Cosines.
   K. Graph transformations of the sine and cosine functions.

Content Outline and Competencies:

I. Set Theory
   A. Model set problems with Venn Diagrams.
   B. Use set operations to calculate union, intersection, complement, and
cardinality of sets.
   C. Identify subsets.

II. Logic and Deductive Reasoning
   A. Define conjunction, disjunction, and negation of statements.
   B. Determine the truth value of simple and compound statements using
truth tables.
   C. Write the converse, inverse, and contrapositive of a conditional
statement.
   D. Determine the validity of an argument using direct, indirect, and
transitive reasoning.

III. Probability
   A. Use probability notation including the "or" condition and the "and"
condition.
   B. Determine whether or not two events are mutually exclusive.
   C. Calculate mathematical expectation.
   D. Calculate conditional probabilities.
   E. Use counting formulas to determine permutations and combinations.

IV. Mathematics of Finance
   A. Calculate simple and compound interest.
   B. Calculate the value of an annunity.
   C. Calculate the annual percentage rate.
   D. Analyze an amortization schedule.
   E. Calculate loan and credit card payments.

V. Systems of Equations and Matrix Algebra
   A. Use linear functions to model applications.
   B. Solve systems of linear equations in two or more variables.
   C. Identify the dimension of a matrix.
   D. Use matrix operations (addition, multiplication, and scalar
multiplication) to calculate resultant matrices.
   E. Use Gaussian elimination to solve an augmented matrix.

VI. Linear Programming
   A. Graph systems of inequalities with all points of intersection.
   B. Determine the constraints and the objective function of a linear
programming problem.
   C. Solve a linear programming problem graphically.
   D. Use the Simplex Matrix to solve linear programming
applications.

Content Outline and Competencies:

I. Analysis and graphing of functions and non-functions
   A. Use function notation. 
   B. Recognize equations of functions and non-functions. 
   C. Use concepts of symmetry, intercepts, left to right behavior,
asymptotes, and transformations to sketch graphs of functions (constant,
linear, quadratic, piecewise-defined, absolute value, square root, cubic, polynomial,
rational, exponential, and  logarithmic) and non-functions (circles). 
   D. Determine the domain and range of a function. 
   E. Write the equation of a function (constant, linear, quadratic,
absolute value, square root, cubic, polynomial, rational, exponential,
and logarithmic) or non-function (circle) given its description. 
   F. Use graphs of functions for analysis. 
   G. Find combinations and composites of functions. 
   H. Find inverses of functions. 

II. Solutions of equations and inequalities
   A. Solve quadratic equations. 
   B. Solve equations involving rational expressions, radicals, and absolute 
value expressions.
   C. Solve polynomial equations. 
   D. Solve exponential equations. 
   E. Solve logarithmic equations. 
   F. Solve polynomial, absolute value, and rational inequalities. 
   G. Solve systems of linear equations in three variables by various 
methods including matrices. 
   H. Solve systems of non-linear equations.
   I. Graph systems of linear inequalities. 
        
III. Applications to Real-World Situations
   A. Apply equations to real-world situations, including but not limited 
to depreciation, growth and decay, and max/min problems. 
   B. Examine and analyze data, make predictions and interpretations, and 
do basic modeling.
   C. Use systems of linear equations in three variables in various applications.

Content Outline and Competencies:

I. The Six Trigonometric Functions
   A. Determine the six trigonometric functions of an angle in standard
position given a  point on its terminal side.
   B. Determine the remaining trigonometric functions of an angle given
the value of one trigonometric function and the quadrant in which the
angle lies.
   C. Determine the six trigonometric functions of an angle in a triangle
given two sides.
   D. Calculate the six trigonometric functions of an angle using a
calculator.
   E. Determine exact values of the six trigonometric functions given a
30°, 45°, 60°, or quadrantal angle.
   F. Determine the six trigonometric functions of an angle in standard
position using reference angles.

II. Applications
   A. Solve real world-problems involving angles of elevation and
depression.
   B. Calculate the length of an arc and the area of a sector in a
circle.

III.  Graphs of Trigonometric Functions
   A. Graph the six basic trigonometric functions of the form y = k +
Afunc(Bx + C).
   B. Determine the equation of a function given its graph.
   C. Graph combinations of functions.

IV. Trigonometric Identities
   A. Simplify trigonometric expressions using trigonometric identities.
   B. Prove trigonometric identities.
   C. Evaluate function values using sum, difference, double, and half
angles identities.
 
V. Inverse Trigonometric Functions
   A. Evaluate inverse trigonometric functions with and without a
calculator.
   B. Evaluate expressions involving inverse trigonometric functions.
 
VI. Trigonometric Equations
   A. Solve trigonometric equations on a given interval in degrees or
radians.
   B. Solve trigonometric equations for all angle solutions.

VII. Triangles
   A. Solve triangles using trigonometric ratios, the Law of Sines, or the
Law of Cosines, where appropriate.
   B. Find areas of triangles.
   C Apply the solutions of triangles to real-world problems.

VIII. Vectors
   A. Determine the magnitude and direction of vectors. 
   B. Resolve vectors into components.
   C. Find the sum, difference, dot product, magnitude, and angle between
two vectors in algebraic form.
   D. Apply vectors to real-world problems.

IX. Complex Numbers
   A. Graph in the complex plane.
   B. Convert complex numbers from standard form to trigonometric form and
vice versa.
   C. Find products, quotients, powers, and roots of complex numbers.

X. Polar Coordinates
   A. Convert from rectangular to polar coordinates and vice versa.
   B. Convert equations from rectangular to polar form and vice versa.
   C. Graph polar equations.

Content Outline and Competencies:

I. Concepts of Functions in Context
   A. Interpret a problem solution in context of a situation.
   B. Discuss appropriate notation for different contexts.
   C. Label answers including units and possible description in context.
   D. Calculate and interpret the average rate of change for a given
problem situation.
   E. Discuss the relationship between a function and its inverse.
   F. Interpret an inverse function and inverse function values in
context.

II. Elementary Functions
   A. Determine whether a relation represented using words, a table of
values, a graph, or a formula is a function.
   B. Convert from one representation of a function to another.
   C. Use function notation to symbolically represent functions.
   D. Graph parent functions including linear, power, polynomial,
rational, radical, exponential, logarithmic, trigonometric, and absolute
value functions.
   E. Apply the concepts of symmetry, intercepts, long-run behavior,
periodic behavior, asymptotes, transformations, and maximums and minimums
to analyze and graph functions.
   F. Use a graphing calculator or computer-generated graphs of functions
for analysis.

III. Domain and Range
   A. State the domains and ranges of parent functions including linear,
power, polynomial, rational, radical, exponential, logarithmic,
trigonometric, inverse trigonometric, and absolute value functions.
   B. Use the domains and ranges of the parent functions and the ideas of
transformations, symmetry, periodic behavior, asymptotes, and pattern
recognition to determine the domains and ranges of general functions.
   C. Connect the domains and ranges of functions with combinations,
compositions, and inverse functions.

IV. Function Relationships and Operations
   A. Find combinations and compositions of functions represented using a
description, a table of values, a graph, or a formula.
   B. Use and create piecewise defined functions for a given problem
situation.
   C. Represent an absolute value function as a piecewise defined
function.
   D. Relate the concept of an inverse function to function composition,
one-to-one functions, and domain/range relationships.
   E. Find inverse functions and use inverse function notation.
   F. Use function composition to verify an inverse function
relationship.
   G. Graph an inverse function.

V. Real World Applications through Modeling
   A. Determine a valid domain and range for situations represented in
words, a table of values, a graph, or a formula.
   B. Use a given mathematical model to analyze a problem.
   C. Write a mathematical model for a problem situation represented as a
description, a table of values, or a graph.
   D. Use a mathematical model to make predictions.
   E. Compare and contrast linear, exponential and logarithmic growth
models.
   F. Use right triangle trigonometry and the Laws of Sines and Cosines to
solve problems.

VI. Problem Solving
   A. Develop proficiency in solving equations both by hand and with
appropriate technology using pattern recognition, inverse function
relations, or other algebraic techniques.
   B. Recognize techniques that potentially introduce extraneous
solutions.
   C. Solve polynomial equations emphasizing the root-factor relationship
and the number of expected solutions based on the Fundamental Theorem of
Algebra.
   D. Solve exponential and logarithmic equations.
   E. Solve trigonometric equations using methods including algebraic
techniques, inverse trigonometric functions, memorization of basic
trigonometric values, trigonometric identities, and the use of
technology.
   F. Analyze graphs of non-linear systems of equations to determine the
number of solutions.

VII. Calculus Transition Topics
   A. Evaluate the difference quotient and interpret it in the context of
a problem situation.
   B. Verify trigonometric identities, including reciprocal identities,
Pythagorean identities, sum, difference, double and half-angle
identities.
   C. Write the trigonometric form of a complex number given in standard
form and convert between the two forms.
   D. Sketch simple polar graphs.
   E. Use technology to approximate trigonometric function values.
   F. Determine the magnitude and direction of vectors, add and subtract
vectors geometrically and algebraically, resolve vectors into components
and evaluate dot products.
   G. Perform binomial expansions through pattern recognition and use of
The Binomial Theorem.
   H. Investigate concavity as a rate of change.
   I. Define a radian and use radian measures.
   J. Generate sequences and associated sums using simple patterns and
determine their formulas.
   K. Find general terms of sequences with emphasis on appropriate
notation.
   L. Use sigma notation to express series.
   M. Evaluate finite arithmetic and geometric series, recognizing linear
and exponential connections.
   N. Graph the inverse sine, inverse cosine, and inverse tangent
functions.

Content Outline and Competencies:

I. Graphs and Trees
   A. Define Euler paths and circuits.
   B. Identify Euler paths and circuits
   C. Define Hamilton paths and circuits.
   D. Identify Hamilton paths and circuits.
   E. Determine optimal solutions to routing problems.
   F. Define network.
   G. Define spanning tree.
   H. Determine optimal solutions to network problems.
   I. Discuss Euler’s and Hamilton’s contributions to graph theory

II.  Social Choice
   A. Introduce mathematically-oriented voting methods
   B. Define ballot counting.
   C. Define voting methods.
   D. Determine election winners and rankings.
   E..Define voting power.
   F.  Calculate power distributions.
   G. Define fair-division problems.
   H. Calculate a solution to fair-division problems.
   I. Define apportionment.
   J. Calculate a solution to apportionment problems.
   K. Discuss the effect of apportionment methods in the U.S. Congress

III. Recursion and Difference Equations
   A. Introduce sequences and functions in the context of recursion.
   B. Define recursive growth problems.
   C. Calculate recursive growth sequences.
   D. Introduce Fractals

IV. Matrices
   A. Define matrix operations.
   B. Utilize matrices to solve systems of equations.
   C. Define a linear programming problem.
   D. Calculate solutions to linear programming problems using augmented
matrices and an appropriate calculator.
   E. Calculate solutions to linear programming problems using the simplex
method and an appropriate calculator.

Content Outline and Competencies:

I. Descriptive Statistics

A. Define and distinguish between categorical (qualitative) and numerical (quantitative) data.

B. Distinguish between data from an observational study and data from a designed experiment.

C. Organize data in frequency tables and contingency tables.

D. Construct appropriate graphical displays of qualitative and quantitative data such as dotplots, histograms, stem-and-leaf diagrams, boxplots, bar charts, pie charts or scatterplots.

E. Describe the general shape of data such as skewed left, skewed right, normal or other symmetric.

F. Calculate the measures of center including mean, median and mode.

G. Calculate the measures of spread including range, standard deviation and interquartile range.

H. Explain the meaning of spread as it relates to a problem.

I. Use a statistical package on a graphics calculator or a computer to enter data and analyze results.

J. Determine potential outliers in a distribution.

K. Measure the position of a data point by computing a percentile.

L. Recognize features of misleading graphs.

II. Introduction to Probability

A. Solve basic probability problems.

B. Use probability notation including the “or” condition and the “and” condition.

C. Determine whether or not two events are mutually exclusive.

D. Determine whether or not two events are independent.

E. Calculate the probability of compound events.

F. Calculate conditional probabilities using conditional notation.

G. Explain the meaning of conditional probabilities.

III. Random Variables

A. Distinguish between discrete and continuous random variables.

B. Find and interpret the mean and the standard deviation of a discrete probability distribution.

C. Determine probabilities for a discrete random variable.

D. Recognize Bernoulli populations.

E. Solve probability problems of independent events with two outcomes for random variables that are binomially distributed.

F. Use the normal distribution to solve percent problems for normally distributed populations.

G. Use the normal distribution to solve probability problems for normally distributed random variables.

IV. Random Sampling and Sampling Theory

A. Calculate the mean for a distribution of sample means and/or proportions.

B. Calculate the standard deviation for a distribution of sample means and/or proportions.

C. Use the shape of a plot of a sample to decide if the sampling distribution appears to be normal.

D. Demonstrate use of the Central Limit Theorem for means and/or proportions.

V. Estimating the Mean

A. Construct a confidence interval for a population mean and a difference of two population means and interpret them in context.

B. Construct a confidence interval for a population proportion and a difference of two population proportions and interpret them in context.

VI. Hypothesis Tests

A. Perform a hypothesis test for a population mean and a difference of two population means and interpret results.

B. Perform a hypothesis test for a population proportion and a difference of two population proportions and interpret results.

C. Perform a hypothesis test with more than two categories for procedures using the Chi-square distribution.

D. Explain Type I error, Type II error, p-value, significance level and power of a test in context.

VII. Linear Regression

A. Draw a scatterplot in order to observe any association between the two variables.

B. Calculate a linear regression equation.

1. Interpret the regression equation parameters in terms of the problem.

2. Test significance of the slope.

3. Use the linear regression equation to make predictions.

4. Calculate the linear correlation coefficient and interpret its meaning.

5. Calculate the coefficient of determination and interpret its meaning.

Content Outline and Competencies:

I. Basic Math
   A. Write the equations of circles, lines, planes and spheres.
   B. Determine if two circles or two spheres are intersecting.
   C. Use trigonometry to determine the components of a vector and the
angle produced by a vector.
   D. Analyze a trigonometric function for amplitude and period.
   E. Convert between polar and rectangular coordinates.
   F. Convert units of measurement.
   G. Construct code that will detect collisions between circles, lines,
planes and spheres.
 
II. Vectors
   A. Compare the concepts of scalar and vector.
   B. Add and subtract vectors.
   C. Multiply vectors by scalars.
   D. Normalize vectors.
   E. Find dot products and cross products of vectors.
   F. Find the angle between two vectors.
   G. Find the normal vector to a surface.
   H. Construct code that will perform vector arithmetic and
normalization.
 
III. Matrices
   A. Add, subtract, and multiply matrices.
   B. Multiply matrices by scalars.
   C. Describe translations using matrices and homogeneous coordinates.
   D. Describe scalings using matrices and homogeneous coordinates.
   E. Describe rotations using matrices and homogeneous coordinates.
   F. Construct code that will perform scaling, rotation, and translations
on vectors and geometric objects.
 
IV. Linear Motion
   A. Compute distance, displacement, velocity, speed, and acceleration
for one-dimensional motion.
   B. Use vectors to describe displacements, velocities, and accelerations
in two and three dimensions.
   C. Write equations which model the motion of projectiles.
   D. Use Newton's Laws to determine the effect of forces on the motion of
an object.
   E. Solve for the motion of an object 
   F. Calculate the work done by a force on an object.
   G. Calculate the kinetic energy, potential energy, and momentum of an
object.
   H. Use conservation of energy and conservation of momentum to model the
collision of objects.
   I. Construct code that can simulate the motion of a projectile.
   J. Construct code that can simulate the motion of an object according
to Newton’s Laws of Motion.
   K. Construct code that can simulate the collision between two objects.
 
V. Rotational Motion
   A. Compute angular displacement, angular velocity, and angular
acceleration.
   B. Determine the angular motion caused by a torque on an object.
   C. Find the kinetic energy and angular momentum of a rotating object.
   D. Construct code that can model the three-dimensional motion of a
rigid body incorporating the concepts of the conservation of energy and
momentum, and Newton’s Laws of Motion.

Content Outline and Competencies:

I. Set Theory
   A. Determine if two sets are equal or equivalent.
   B. Write the subsets and proper subsets of a given set.
   C. Find the intersection of two sets.
   D. Find the union of two sets.
   E. Find the complement of a set.
   F. Use Venn diagrams to determine the validity of a set statement.
   G. Solve applications using Venn Diagrams.

II. Whole Numbers
   A. Use the definitions of whole number operations to explain the
elementary processes of addition, subtraction, multiplication, and
division.
   B. Identify the properties of whole number addition and multiplication,
such as closure, commutativity, associativity, distributive property, and
identity element.
   C. Model situations that involve whole numbers using objects, pictures,
and symbols.
   D. Estimate the results of whole number calculations.
   E. Use a variety of problem solving strategies and critical thinking
skills with applications involving whole numbers.
   F. Apply the definition and properties of exponents to whole numbers.
   G. Use the correct order of operations when evaluating expressions.

III. Integers
   A. Use the definitions of integer operations to explain the processes
of addition, subtraction, multiplication, and division.
   B. Identify the properties of integer addition such as closure,
commutativity, associativity, identity element, and additive inverse.
   C. Identify the properties of integer multiplication such as closure,
commutativity, associativity, distributive property, zero multiplication,
and identity element.
   D. Demonstrate operations on integers using a variety of
manipulatives.
   E. Use a variety of problem solving strategies and critical thinking
skills with applications involving integers.

IV. Number Theory
   A. Use divisibility tests to determine if a given number is divisible
by another.
   B. Classify numbers as prime or composite or neither.
   C. Apply the Fundamental Theorem of Arithmetic.
   D. Use the Sieve of Eratosthenes to find the primes less than 100.
   E. Find the greatest common divisor and the least common multiple of
numbers using a variety of methods.

V. Rational Numbers
   A. Write fractions in simplest form.
   B. Determine if fractions are equivalent.
   C. Arrange a set of fractions in numerical order.
   D. Use the definitions and properties of addition, subtraction,
multiplication, and division of fractions.
   E. Express ratios as fractions in simplest forom.
   F. Solve for the missing term of a proportion.
   G. Demonstrate operations on fractions using a variety of
manipulatives.
   H. Use a variety of problem solving strategies and critical thinking
skills with applications involving fractions.

VI. Decimal Numbers
   A. Write decimal numbers in expanded form.
   B. Convert from standard decimal notation to scientific notation and
vice versa.
   C. Arrange decimal numbers in numerical order.
   D. Use standard algorithms to add, subtract, multiply, and divide
decimal numbers.
   E. Round decimal numbers to a specified place value.
   F. Convert a fraction to a decimal number.
   G. Convert repeating decimal numbers and terminating decimal numbers to
fractions.
   H. Convert numbers in percent form to fractional or decimal form and
vice versa.
   I. Use a variety of problem solving strategies and critical thinking
skills with decimal and percent applications.

VII. Numeration Systems
   A. Convert Roman numerals to Hindu-Arabic numerals and vice versa.
   B. Convert Egyptian numerals to Hindu-Arabic numerals and vice versa.
   C. Convert Babylonian numerals to Hindu-Arabic numerals and vice
versa.
   D. Convert Mayan numerals to Hindu-Arabic numerals and vice versa.
   E. Write Hindu-Arabic numerals in expanded form.
   F. Convert numerals from one base system to another.
   G. Trace the historical development of the Hindu-Arabic number system.

VIII. Functions and Relations
   A. Identify relations as functions.
   B. Identify the domain and range of functions.

Content Outline and Competencies:

I. Probability
   A. Determine the probability of outcomes of simple experiments.
   B. Determine the probability of outcomes of complex experiments.
   C. Determine the probability of events involving conditional
probability.
   D. Determine the odds in favor of and against different events.
   E. Determine the expected value of an experiment.
   F. Solve problems involving the concepts of permutations and
combinations.
   G. Distinguish between theoretical probability and experimental
probability.

II. Statistics
   A. Interpret statistical graphs such as line graphs, stem-and-leaf
plots, pictographs, histograms, circle graphs, and box plots.
   B. Draw statistical graphs such as line graphs, stem-and-leaf plots,
pictographs, histograms, circle graphs, and box plots.
   C. Determine measures of central tendency such as mean, median, mode,
and their applications to box plots and percentiles.
   D. Determine measures of dispersion such as range, variance, and
standard deviation.
   E. Describe the attributes of the normal distribution.
   F. Recognize misleading statistics.

III. Measurement
   A. Calculate temperature conversions for Celsius and Fahrenheit.
   B. Measure items using both English and metric units.
   C. Convert measurements within the English system.
   D. Convert measurements within the metric system.
   E. Use dimensional analysis to convert measurements between the English
system and the metric system.
   F. Calculate the perimeter and the area of polygons.
   G. Calculate the circumference and area of circles.
   H. Apply the Pythagorean Theorem to right triangles to find
measurements.
   I. Calculate the surface area and volume of figures such as prisms,
cylinders, pyramids, cones, and spheres.

IV. Geometric Shapes
   A. Describe the attributes of different 2- and 3-dimensional shapes.
   B. Describe the attributes of different types of angles.
   C. Draw basic 2- and 3-dimensional shapes.
   D. Explain the relationships between angles formed by parallel lines
intersected by transversals.
   E. Solve problems involving angle measurement.
   F. Describe the attributes of the Platonic solids.

V. Relationships between Geometric Shapes
   A. Apply the concept of congruent triangles to constructions.
   B. Construct parallel lines, perpendicular bisectors and angle
bisectors.
   C. Inscribe a regular polygon in a circle.
   D. Solve problems involving congruency properties.
   E. Solve problems involving similar figures.

VI. Motion Geometry
   A. Find the image of geometrical shapes under various translations and
rotations.
   B. Find the image of geometrical shapes under various reflections.
   C. Find the image of geometrical shapes under various size
transformations with a given scale factor.
   D.Analyze the properties of tessellations.

Content Outline and Competencies:

I. Teaching as a Career

A. Determine suitability/interest in teaching as a career through thoughtful self-reflection.

B. Explore pathways to a career in teaching.

C. Identify personal learning styles and discuss their implications for classroom interactions.

II. Strategies for Practical Lesson Design

A. Design and deliver inquiry-based hands-on lessons.

B. Write performance objectives for each lesson, including mathematics and/or science connections, and appropriate assessments for those objectives.

C. Use technology and the Internet to enhance classroom lessons, collaborate, and communicate.

III. Concepts and Components of Teaching Theory

A. Identify instructional strategies that meet the needs of diverse learners.

B. Distinguish between learner-centered and teacher-centered instructional strategies.

C. Discuss state and national science and mathematics standards and their implications for curriculum decisions.

D. Identify current issues in the theory and practice of teaching.

IV. Components of a Field Experience

A. Observe a math-science lesson taught by a cooperating teacher.

B. Interact with a population of diverse student learners in a school setting while teaching a lesson in an elementary school classroom.

C. Receive and synthesize feedback from a cooperating teacher as a peer and mentoring colleague in order to improve techniques.

Content Outline and Competencies:

I. Data collection
   A. Describe characteristics of populations and samples.
   B. Examine methods of sampling, explaining sources of bias in these
methods.
   C. Examine methods of experimentation, identifying sources of
confounding in experiments.

II. Descriptive statistics
   A. Describe data with graphical methods including histograms, box
plots, scatterplots and regression lines.
   B. Describe data with numerical methods including measures of central
tendency and measures of spread.
   C. Analyze data using a statistical package on a graphics calculator or
a computer.

III. Probability and inferential statistics
   A. Describe sample spaces for random experiments.
   B. Apply counting techniques to determine the number of possible
outcomes of an experiment.
   C. Calculate the probability of a given event.
   D. Interpret features of the normal distribution including mean,
standard deviation, and confidence intervals.

IV. Patterns and symmetry
   A. Use geometric similarity to assess and interpret the physical
limitations of size.
   B. Identify scaling factors relating to geometrically similar objects.
   C. Convert between different units of measure in U.S. Customary and
Metric system.
   D. Recognize the golden ratio in geometric applications.
   E. Identify types of rigid motion such as rotation, reflection, and
translation.
   F. Identify terms of the Fibonacci sequence from patterns found in
nature.
   G. Apply concepts of arithmetic and geometric growth to financial and
biological population models.

Content Outline and Competencies:

I. Demonstrate a knowledge of limits
   A. Evaluating limits.
      1. Evaluate a limit at a point using algebraic techniques or a
table.
      2. Evaluate a limit of a function at infinity using algebraic
techniques or a table.
      3. Evaluate a limit using a graph.
      4. Evaluate left and right hand limits using algebraic techniques or
a graph.
   B. Use of limits
      1. Use of the limit to determine continuity of a function at a
point.
      2. Use a limit to determine differentiability of a function.
      3. Use the limit definition of the derivative to determine
differentiability of a function and to find the derivative of a function.

II. Demonstrate a knowledge of derivatives
   A. Finding and estimating derivatives.
      1. Find the derivatives of algebraic functions using the power rule,
product rule, quotient rule and chain rule.
      2. Find the derivatives of natural exponential and logarithmic
functions.
      3. Find the derivatives using implicit differentiation.
      4. Use a graph to estimate the intervals over which the first
derivative is positive or negative.
      5. Use a graph to estimate the intervals over which the second
derivative is positive or negative.
      6. Find the equation of a tangent line to a curve at a given point.

III. Using derivatives
   A. Apply derivative techniques to curve sketching
      1. Using the first derivative, find critical points.
      2. Determine the behavior of a function using the first derivative.
      3. Using the second derivative, find inflection points.
      4. Determine the concavity of a function using the second
derivative.
      5. Sketch a function using information gathered from the first and
second derivatives.

   B. Apply derivative techniques to physical, economics and social
sciences problems
      1. Use derivatives to predict rates of change.
      2. Use derivatives to determine the maxima/minima (optimization).
      3. Use derivatives to determine outcomes in related rates problems.
      4. Use derivatives to find and explain rates of change for position
functions, including the relationship between position, velocity and
acceleration.
      5. Find extrema of functions with restricted domains.

IV. Finding integrals
   A. Identify the antiderivative for a given function using elementary
techniques.
   B. Identify the antiderivative for a given function using
u-substitution.
   C. Apply the Fundamental Theorem of Calculus.
      1. Evaluate a definite integral using elementary techniques.
      2. Evaluate a definite integral using u-substition.
      3. Calculate the area between two curves.

Content Outline and Competencies:

I. Additional Integration Techniques
   A. Calculate definite and indefinite integrals using integration by
parts.
   B. Calculate definite and indefinite integrals using the Integration
Tables.
   C. Determine value of definite integrals using numerical integration.
   D. Determine whether improper integrals converge.
   E. Apply l’Hôpital’s Rule to find limits.

II. Multivariable Calculus
   A. Discover the need for and the use of functions of several
variables.
   B. Discover how to compute the first and second partial derivatives of
functions of several variables.
   C. Compute the value of double integrals.
   D. Locate the coordinates of any relative extrema of a function of two
variables.
   E. Utilize Lagrange Multipliers to compute the maximum or minimum of a
function subject to constraints.
   F. Find the best fitting line through three points using the method of
least squares. (Optional)
   G. Use total differentials to obtain an approximation of an expression.
(Optional)

III. Differential Equations
   A. Identify differential equations.
   B. Solve differential equations using the method of separation of
variables.
   C. Calculate approximate solutions to differential equations using
Euler’s method. (Optional)
   D. Determine the qualitative behavior of solutions to differential
equations. (Optional)
   E. Apply differential equations to problems, e.g. logistic growth.

IV. Calculus and Trigonometric Functions
   A. Review the sine and cosine functions.
   B. Discover the derivatives of the sine and cosine functions.
   C. Discover the integrals of the sine and cosine functions.
   D. Extend the derivatives and integrals of the sine and cosine
functions to the other trigonometric functions.

V. Calculus and Probability Theory
   A. Define discrete probability
   B. Identify continuous probability density functions.
   C. Compute expected value and variance of a continuous random variable
and compare to discrete probability.
   D. Convert a normal distribution function to a standard normal
distribution function.
   E. Use the standard normal distribution function to calculate the
probabilities of a random variable.

Content Outline and Competencies:

I. Using Limits
   A. Evaluation of limits
      1. Evaluate the limit of a function at a point both algebraically
and graphically.
      2. Evaluate the limit of a function at infinity both algebraically
and graphically.
      3. Use the definition of a limit to verify a value of the limit of a
function.
   B. Use of limits
      1. Use the limit to determine the continuity of a function. (KCOG)
      2. Use the limit to determine differentiability of a function. (KCOG)
      3. Apply the Intermediate Value Theorem. (KCOG)
   C. Limiting process
      1. Use the limiting process to find the derivative of a function. (KCOG)
 
II. Finding Derivatives
   A. Find derivatives involving powers, exponents, and sums. (KCOG)
   B. Find derivatives involving products and quotients. (KCOG)
   C. Find derivatives involving the chain rule. (KCOG)
   D. Find derivatives involving exponential and logarithmic functions. (KCOG)
   E. Find derivatives involving trigonometric (KCOG) and inverse trigonometric
functions.
   F. Find derivatives involving implicit differentiation. (KCOG)   
    
III. Using Derivatives
   A. Curve sketching
      1. Use the first derivative to find critical points. (KCOG)
      2. Apply the Mean-Value Theorem for derivatives. (KCOG)
      3. Determine the behavior of a function using the first derivative. (KCOG)
      4. Use the second derivative to find inflection points. (KCOG)
      5. Determine the concavity of a function using the second
derivative. (KCOG)
      6. Sketch the graph of the function using information gathered from
the first and second derivatives. (KCOG)
      7. Interpret graphs of functions. (KCOG)
   B. Applications of the derivative
      1.  Use the derivative to find velocity, acceleration, and other rates of change. (KCOG)
      2.  Use the derivative to find a tangent line to a curve at a given point. (KCOG)
      3. Solve related rates problems. (KCOG)
      4. Use optimization techniques in economics, the physical sciences,
and geometry. (KCOG)
      5. Use differentials to estimate change. (KCOG)
      6. Use Newton’s Method. (KCOG)
      
IV. Finding Integrals
   A. Find area using Riemann sums. (KCOG)
   B. Express the limit of a Riemann sum as a definite integral. (KCOG)
   C. Evaluate the definite integral using geometry. (KCOG)
   D. Approximate definite integral using the Trapezoid Rule and Simpson’s Rule. (KCOG)
   E. Evaluate definite integrals using the Fundamental Theorem of
Calculus. (KCOG)
   F. Integrate algebraic, natural exponential, natural logarithm,
trigonometric, (KCOG) and inverse trigonometric functions. 
   G. Integrate indefinite integrals. (KCOG)
   H. Integrate by substitution. (KCOG)
   I. Integrate by parts.
   
V. Using the Integral
   A. Apply the Mean-Value Theorem for Integrals. (KCOG)
   B. Calculate the area between curves using integration.
   C. Calculate the volume of a solid of revolution by the disk method.
   D. Calculate the volume of a solid of revolution by the washer method.
   E. Calculate the volume of a solid of revolution by the cylindrical
shells method.
   F. Calculate the arc length and surface area of revolution using integration.

Content Outline and Competencies:

I. Inverse Trigonometric Functions 
   A. Find derivatives involving inverse trigonometric functions.
   B. Find antiderivatives involving inverse trigonometric functions.

II. Techniques of Integration
   A. Evaluate an integral using integration by parts.
   B. Evaluate an integral using the method of partial fractions.
   C. Evaluate an integral which involves powers of trigonometric
functions.
   D. Evaluate an integral by using trigonometric substitutions.
   E. Evaluate an integral by using integral tables.
   F. Apply L'Hopital's Rule to evaluate limits involving indeterminate
forms.
   G. Determine whether an improper integral converges or diverges.
   H. Test for the convergence or divergence of an improper integral using
an appropriate test.
   I. Evaluate a convergent improper integral.    

III. Infinite Sequences and Series
   A. Determine whether an infinite sequence converges or diverges.
   B. Find the limit of a convergent sequence.
   C. Identify geometric series, telescoping series, harmonic series,
p-series, alternating series, and power series. 
   D. Determine whether an infinite series converges or diverges. 
   E. Test for convergence or divergence of an infinite series using an
appropriate test. 
   F. Determine whether an infinite series converges absolutely or
conditionally.
   G. Find the radius and interval of convergence for a power series.
   H. Write a power series which represents a function.
   I. Write a Taylor polynomial which represents a function.
   J. Calculate the error in approximating a function with an infinite
series.
   
IV. Parametric and Polar Equations
   A. Differentiate parametric equations.
   B. Calculate the length of a smooth parametrized curve.
   C. Calculate the surface area from a parametrized curve.
   D. Graph polar equations.
   E. Convert from rectangular to polar coordinates and vice-versa.
   F. Differentiate polar equations.
   G. Integrate polar equations.
   H. Calculate the area in the plane using polar coordinates.
   I. Calculate the area between two curves using polar coordinates.
   J. Calculate the length of a polar curve.
   K. Calculate the surface area of a surface of revolution using polar
coordinates.

V. Applications of Definite Integrals
   A. Calculate area between curves using integration.
   B. Calculate the volume of a solid of revolution using appropriate methods,
including the disk, washer, or shell method.
   C. Calculate the arc length and surface area of revolution using integration. 
   D. Calculate work, moments and centers of mass.
   E. Calculate fluid pressures and forces.

Content Outline and Competencies:

I. Vectors in the Plane and in Space
   A. Determine the components of a vector in the plane and in space and
illustrate geometrically.
   B. Apply vector operations and properties and interpret them
geometrically.
   C. Calculate the dot product to determine the angle between two
vectors.
   D. Find the magnitude and direction of a resultant vector and
illustrate geometrically.
   E. Determine a vector passing through two points.
   F. Determine whether two vectors are orthogonal
   G. Determine whether two vectors are parallel.
   H. Determine the projection of one vector onto another vector.
   I. Calculate the cross product for two vectors and interpret
geometrically.
   J. Write the parametric equations for a line in space.
   K. Determine the equation for a plane.

II. Surfaces in Space
   A. Identify equations of right cylinders and sketch graphs of these
cylinders.
   B. Identify equations of quadric surfaces and sketch graphs of quadric
surfaces.

III. Vector-valued Functions
   A. Define vector-valued functions.
   B. Identify the space curve determined by a vector-valued function.
   C. Define differentiation and integration of vector-valued functions.
   D. Calculate derivatives of vector-valued functions.
   E. Calculate integrals of vector-valued functions.
   F. Identify intervals on which a parametrically defined curve is
smooth.
   G. For a given position vector, calculate the velocity and acceleration
vectors.
   H. Determine the principal unit tangent, unit normal and unit binormal
vectors to a given curve at a given point.
   I. Determine the tangential and normal components of an acceleration
vector.
   J. Calculate the arc length of a curve.
   K. Calculate the curvature of a curve.
   L. Calculate the torsion of a curve. 
   M. Use position, velocity, acceleration, speed, and/or tangential and
normal components of acceleration to analyze motion along a curve.

IV. Functions of Several Variables
   A. Determine the domain of functions of several variables.
   B. Graph functions of two variables.
   C. Sketch level curves and level surfaces of multivariable functions.
   D. Define limits and continuity for multivariable functions.
   E. Determine limits or show that the limit does not exist for
multivariable functions.
   F. Determine regions on which a multivariable function is continuous.
   G. Define partial derivatives.
   H. Calculate and interpret partial derivatives.
   I. Compute differentials.
   J. Define differentiability of multivariable functions.
   K. State the chain rule.
   L. Apply chain rule for multivariable functions.
   M. Compute and interpret directional derivatives.
   N. Calculate and interpret gradients.
   O. Find equations of tangent planes and normal lines to surfaces.
   P. Find critical points and classify as relative extrema or saddle
points for functions of two variables.
   Q. Find extrema of functions of two variables.
   R. Solve applied optimization problems using multivariable functions.
   S. Utilize Lagrange multipliers to solve constrained optimization
problems.

V. Multiple Integration
   A. Define double and triple integrals.
   B. Set up and evaluate iterated multiple integrals in rectangular
coordinates.
   C. Use polar coordinates to evaluate double integrals.
   D. Use cylindrical and spherical coordinates to evaluate triple
integrals. (Note:  Students’ prior familiarity with cylindrical and
spherical coordinates should not be assumed.) 
   E. Use double integrals to compute area in the plane.
   F. Use double and triple integrals to calculate volume.
   G. Use double integrals to compute surface area.
   H. Use double and triple integrals in applications including
calculation of mass, center of mass, and moments of inertia.
   I. Implement a change of variables to evaluate a double integral.

VI. Vector Analysis
   A. Define vector fields.
   B. Calculate divergence and curl for a vector field.
   C. Define conservative vector field.
   D. Determine whether a vector field is conservative or not.
   E. Calculate the potential function for a conservative vector field.
   F. Define line integrals.
   G. Evaluate a line integral over a given curve.
   H. Use line integrals to calculate work, circulation, and flow.
   I. Calculate flux of a vector field across a curve.
   J. Define path independence.
   K. Determine if a line integral is path independent.
   L. Apply the Fundamental Theorem of Line Integrals.
   M. Apply Green’s Theorem where appropriate.
   N. Define surface integrals.
   O. Evaluate surface integrals.
   P. Calculate flux of a vector field through a surface.
   Q. Parameterize a surface.
   R. Evaluate surface integrals over parameterized surfaces.
   S. Apply the Divergence Theorem where appropriate.
   T. Apply Stokes’s Theorem where appropriate.

Content Outline and Competencies:

I. Matrices, Vectors, and Systems of Linear Equations
   A. Define matrices and vectors. 
   B. Manipulate linear combinations, matrix-vector products, and special
matrices. 
   C. Solve systems of linear equations. 
   D. Perform Gaussian elimination.  
   E. Find the span of a set of vectors. 
   F. Determine linear dependence and independence. 

II. Matrices and Linear Transformations 
   A. Perform matrix multiplication.  
   B. Construct inverse matrices with elementary matrices. 
   C. Find the inverse of a matrix.  
   D. Describe the relationship between linear transformations and
matrices. 
   E. Determine the composition and invertibility of linear
transformations. 

III. Determinants 
   A. Perform cofactor expansions. 
   B. Use the properties of determinants.

IV. Subspaces and Their Properties 
   A. Define subspaces associated with matrices. 
   B. Construct a basis for a subspace.
   C. Determine the dimension of a subspace.
   D. Determine the dimension of subspace associated with a matrix. 
   E. Perform rotation of coordinate systems. 
   F. Construct matrix representations of linear operators. 

V. Eigenvalues, Eigenvectors, and Diagonalization 
   A. Determine eigenvalues and eigenvectors. 
   B. Construct the characteristic polynomial of a matrix. 
   C. Perform the diagonalization of matrices. 
   D. Perform the diagonalization of linear operators.

VI. Orthogonality 
   A. Describe vectors with geometry. 
   B.Construct orthonormal vectors. 
   C. Perform least-squares approximation.
   D. Produce orthogonal projection matrices. 
   E. Create orthogonal matrices and operators. 
   F. Formulate symmetric matrices.   

VII. Vector Spaces 
   A. Define vector spaces and their subspaces. 
   B. Determine dimension and isomorphism. 
   C. Develop linear transformations; find matrix representations. 
   D. Define inner product spaces. 

ADDITIONAL TOPICS OF INTEREST (time permitting)
Systems of linear equations in applications
Matrix multiplication in applications
LU decomposition of a matrix
Eigenvalues in applications
Singular value decomposition of a matrix
Rotations of R3 in computer graphics applications

Content Outline and Competencies:

I. Basic concepts
   A. Define ordinary versus partial derivatives.
   B. Discriminate between ordinary and partial differential equations.
   C. Determine the order of a differential equation.
   D. Discriminate between linear and nonlinear differential equations.
   E. Define a solution to a differential equation.
   F. Utilize direction fields in interpreting solutions of differential
equations.
 
II. First-Order Differential Equations
   A. Utilize the existence and uniqueness theorem for first-order
differential equations.
   B. Calculate solutions to linear first-order ordinary differential
equations.
   C. Utilize the existence and uniqueness theorem for first-order linear
differential equations.
   D. Calculate solutions to separable first-order ordinary differential
equations.
   E. Utilize first-order ordinary differential equations in modeling
activities.
 
 III. Numerical Methods
   A. Utilize Euler’s Method to approximate initial-value problem
solutions.
   B. Utilize improvements on Euler’s Method to approximate
initial-value problem solutions.
   C. Utilize the Runge-Kutta methods to approximate initial-value problem
solutions.

IV. Linear Algebra
   A. Perform elimination using pivots for a given system of equations.
   B. Define vector spaces and subspaces.
   C. Define linear independence and dependence.
   D. Define a basis for a vector space or subspace.
   E. Define the dimension for a vector space.
   F. Calculate the eigenvalues and associated eigenvectors for a given
matrix.

V. Systems of Differential Equations
   A. Define higher-order linear differential equations.
   B. Transform higher-order linear differential equations to systems of
first-order differential equations.
   C. Utilize the eigenvalue-eigenvector method in solving systems of
homogeneous linear differential equations with constant coefficients.
   D. Determine the linear dependence or independence of a set of
solutions.
   E. Calculate solutions given real distinct, real repeated, and complex
roots of the characteristic equation.
   F. Define fundamental solutions of linear homogeneous equations.
   G. Classify solutions of second-order linear homogeneous equations as
to underdamped, critically damped, or overdamped.
   H. Construct the general solution of nonhomogeneous linear systems
utilizing the method of variation of parameters.
   I. Analyze the stability of equilibrium solutions.
   J. Utilize the phase plane in solving systems.
   K. Analyze the solutions of almost linear systems.
   L. Utilize systems of differential equations in modeling activities.

VI. Series Solution Techniques
   A. Review power series.
   B. Construct series solutions near an ordinary point.
   C. Construct solutions for Euler equations

VII. The Laplace Transform
   A. Define the Laplace transform.
   B. Utilize Laplace transforms in solving initial-value problems.
   C. Solve initial-value problems involving step functions.
   D. Solve initial-value problems involving discontinuous forcing
functions.
   E. Solve initial-value problems involving impulse functions.
   F. Utilize the Convolution theorem

Content Outline and Competencies:

I. Basic Descriptive Statistics:  Organizing and describing data

   A. For a given set of data, draw a dotplot, histogram, stem-and-leaf

diagram, and a boxplot.

   B. Describe the general shape of data, skewed left, skewed right, or

symmetric.

   C. Calculate the measures of central tendency including mean, median,

and mode.

   D. Calculate the measures of dispersion including range, variance, and

standard deviation; explain the meaning of standard deviation as it

relates to a problem.

   E. Use a computer package to enter data and analyze results.

II. Introduction to Probability:  Finding the theoretical probability of

an event

   A. Construct the sample space for an experiment.

   B. Use the Fundamental Counting Rule to predict the number of things in

a sample space.

   C. Determine whether or not two events are mutually exclusive.

   D. Determine whether or not two events are independent.

   E. Calculate conditional probabilities.

III. Random Variables and Probability Density Functions (p.d.f.)

   A. List all possible values of a discrete random variable along with its

probabilities.

   B. Use calculus to generate a continuous cumulative density function (c.d.f.)

   C. Determine the expected value and the variance of a discrete p.d.f.

   D. Determine the expected value and the variance of a continuous

p.d.f.

   E. Determine the covariance and correlation of a discrete joint p.d.f.

IV. Special Probability Distributions:  Using statistics in the “real world”

   A. Use the Binomial distribution to solve problems with two outcomes and independent events.

   B. Use the Hypergeometric distribution to solve problems involving sampling without replacement.

   C. Use the Poisson distribution to solve problems involving occurrences of events that happen randomly over time, such as the number of customers entering a store.

   D. Use the Exponential distribution to solve problems describing the time between events such as the predicted time between customers.

   E. Use the Normal distribution to solve problems that are normally distributed.

V. Random Sampling and Sampling Theory: Generating distributions for

sample means

   A. Calculate the mean for a distribution of sample means.

   B. Calculate the standard deviation for a distribution of sample

means.

   C. Perform a normal probability plot; describe the shape of the

population based on the plot.

   D. Analyze the Central Limit Theorem.

VI. Estimating the Mean: Using statistics to determine averages and

deviations of a population

   A. Construct a confidence interval for a population mean; explain the meaning in terms of a problem.

   B. Construct a confidence interval for a population proportion;

explain the meaning in terms of a problem.

   C. Construct a confidence interval for a population median.

   D. Construct a confidence interval for a population standard deviation.

VII. Hypothesis Tests: Finding significance

   A. Perform a hypothesis test for a population mean using a sample mean.

   B. Perform a hypothesis test for a population proportion using a sample

proportion.

   C. Perform a hypothesis test for two population means.

   D. Perform a hypothesis test with more than two categories.

   E. Explain Type I and Type II errors with respect to a problem.

VIII. Linear Regression: Making predictions with linear data

   A. Calculate a linear regression equation; explain the equation in

terms of the problem.

   B. Use a linear regression equation to make predictions about data.

   C. Calculate the coefficient of determination for a linear regression

equation; use the coefficient of determination to explain the strength of

the regression equation.

IX. Quality Control:  Using statistics in the work setting

   A. Provide a brief history of quality control including a discussion of

W. Edwards Deming.

   B. Construct basic control charts using the distributions learned in

the course.

Content Outline and Competencies:

The content outline and competencies will vary, depending on the special topic
being offered. The Special Topics course competencies must follow standard
format for JCCC course outlines. The Special Topics proposer will submit
competencies to the Mathematics Division Curriculum Committee for review and
approval.