Content Outline and Competencies:

I. Whole Numbers

A. Identify place value of the digits in a whole number.

B. Read whole numbers.

C. Write whole numbers.

D. Compare the size of whole numbers using inequality symbols.

E. Round whole numbers.

F. Estimate results using rounding.

G. Add, subtract, multiply, and divide whole numbers.

H. Evaluate expressions written in exponential form.

I. Calculate square roots.

J. Apply the rules of order of operation to whole numbers.

K. Use the tests for divisibility for 2, 3, 5, and 10.

L. Determine if a number is prime or composite.

M. Determine the prime factorization of whole numbers.

N. Calculate the least common multiple and greatest common factor.

O. Solve applications using whole numbers.

II. Fraction Notation

A. Identify the numerator and denominator of a fraction.

B. Identify mixed numbers.

C. Convert between mixed numbers and improper fractions.

D. Compare the size of fractions and mixed numbers using inequality symbols.

E. Simplify fractions.

F. Identify equivalent fractions.

G. Change a fraction to an equivalent fraction with a different denominator.

H. Add, subtract, multiply, and divide fractions and mixed numbers.

I. Apply the rules of order of operation to fractions and mixed numbers.

J. Solve applications using fractions and mixed numbers.

III. Ratios and Proportions

A. Define a ratio.

B. Create a ratio.

C. Determine if a proportion is true.

D. Solve proportions.

E. Solve applications using proportions.

IV. Decimal Notation

A. Identify place value of the digits in a decimal number.

B. Read and write decimal numbers.

C. Round decimal numbers.

D. Estimate results using rounding.

E. Add, subtract, multiply, and divide decimal numbers.

F. Compare the size of decimal numbers using inequality symbols.

G. Apply the rules of order of operation to decimal numbers.

H. Convert between fraction notation and decimal notation.

I. Solve applications using decimal numbers.

V. Percent Notation

A. Define percent.

B. Convert between percent notation and decimal notation.

C. Convert between percent notation and fraction notation.

D. Solve percent problems using proportions.

E. Solve percent problems using basic equations.

F. 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. Convert between Fahrenheit and Celsius temperature measurements.

E. Solve applications using U.S. and metric measurements.

VII. Geometry

A. Classify angles.

B. Identify supplementary and complementary angles.

C. Identify rectangles, squares, parallelograms, triangles and trapezoids.

D. Calculate the perimeter of polygons.

E. Calculate the circumference of circles.

F. Calculate the area of a rectangle, a square, a parallelogram, a triangle, a trapezoid, and a circle.

G. Calculate the perimeter, circumference, and area of composite figures.

H. Calculate the volume of rectangular solids, circular cylinders, spheres, and circular cones.

I. Define similar figures.

J. Determine if two triangles are similar.

K. Use a proportion to find a missing side of similar triangles.

L. Use the Pythagorean theorem to determine the missing side of a right triangle.

M. 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. Statistics

A. Calculate the mean, median, and mode of a set of numbers.

B. Interpret tables, circle graphs, line graphs and bar graphs.

C. Solve applications using tables, circle graphs, line graphs, and bar graphs.

Content Outline and Competencies:

I. Arithmetic and Algebraic Manipulation

A. Evaluate algebraic expressions using the order of operations.

B. Apply the laws of exponents to simplify expressions containing integer and rational exponents.

C. Simplify complex fractions.

D. Perform addition, subtraction, multiplication, and division on polynomial and rational expressions.

E. Factor expressions with common factors, expressions that require grouping, quadratic expressions, quadratic in form expressions, difference of square expressions, and sum and difference of cubes expressions.

F. Apply the laws of radicals to perform addition, subtraction and multiplication.

G. Rationalize denominators containing radicals.

H. Simplify radicals containing negative radicands.

I. Calculate radicals, approximating those that are irrational.

J. Simplify radicals of any order using the product and quotient rules.

K. Perform addition, subtraction, multiplication, and division on complex numbers.

L. Evaluate functions.

M. Calculate the distance between two points using the distance formula.

N. Determine the midpoint between two points.

II. Equations and Inequalities

A. Solve linear equations in one variable.

B. Solve proportion equations.

C. Solve linear inequalities in one variable showing solutions using interval notation, set builder notation and on a number line.

D. Solve literal equations for a given variable.

E. Solve quadratic equations by factoring, using the square root property, completing the square, and the quadratic formula.

F. Solve equations that are quadratic in form.

G. Solve equations containing rational expressions or radicals.

H. Solve systems of linear equations in two variables.

I. Solve applications modeled by equations and systems of equations. 

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.

D. Graph linear inequalities in two variables.

E. Graph quadratic equations emphasizing using x = -b/(2a) to find the vertex and symmetry.

IV. Analysis of Equations and Graphs

A. Identify the x-intercept, y-intercept and slope of a line given its graph.

B. Calculate the slope of a line passing through two given points.

C. Construct an equation of a line given its graph.

D. Construct an equation of a line given its slope and y-intercept.

E. Construct an equation of a line given its slope and a point.

F. Construct an equation of a line given two points.

G. Construct an equation of a line parallel or perpendicular to a given line through a specific point.

H. Construct an equation of a horizontal line.

I. Construct an equation of a vertical line.

J. Determine whether an equation is linear.

K. Distinguish between functions and non-functions using the vertical line test.

L. Identify the domain and range of a relation from a graph.

Content Outline and Competencies:

I. Arithmetic and Algebraic Manipulation

A. Evaluate algebraic expressions using the order of operations.

B.  Apply the laws of exponents to simplify expressions containing integer exponents.

C.  Express numbers in scientific notation.

D. Addition, subtract, multiply, and divide polynomial expressions.

E. Factor expressions with common factors, expressions that require grouping, quadratic expressions, and difference of square expressions.

F. Multiply and divide rational expressions.

G. Calculate square roots, approximating those that are irrational.

H. Simplify square roots using the product and quotient rules.

II. Equations and Inequalities in One Variable

A. Solve linear equations in one variable.

B. Solve proportion equations.

C. Solve linear inequalities in one variable showing solutions using interval notation, set-builder notation, and 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. Equations and Graphs in Two Variables

A. Graph linear equations by plotting points.

B. Graph linear equations using intercepts.

C. Graph linear equations using the y-intercept and slope. 

D. Identify the x-intercept, y-intercept, and slope of a line given its graph.

E. Construct an equation of a line given its graph.

F. Construct an equation of a line given its slope and y-intercept.

G. Construct an equation of a line given its slope and a point.

H. Construct an equation of a horizontal line.

I. Construct an equation of a vertical line.

J. Determine whether or not an equation is linear.

K. Calculate the slope of a line passing through two given points.

Content Outline and Competencies:

I.  Baseline Math Assessment and Expanded Algebra Knowledge

A.  Complete an assessment to determine a student’s knowledge and understanding of prerequisite College Algebra topics.

B. Complete the individualized study plan for algebraic and arithmetic topics based on the results of the assessment.

C.  Demonstrate understanding of underlying connections between mathematical concepts.

D.  Apply learned algebra topics in a variety of situations showing the ability to generalize mathematical knowledge.

II.  Learning Strategies

A.  Develop effective study habits.

B.  Apply individualized study habits to concurrent coursework.

C.  Establish productive homework strategies.

D.   Analyze various test taking skills.

III.  Mathematical Mindset

A.  Investigate campus resources available to assist students’ needs.

B.  Identify math anxieties and phobias.

C.  Establish positive math-related attitudes.

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. Multiply and divide rational expressions containing sum or difference of cubes.

C. Add and subtract rational expressions.

D. Simplify complex fractions.

E. Apply the laws of exponents to simplify expressions containing rational exponents.

F. Apply the laws of radicals to perform addition, subtraction, and multiplication.

G. Rationalize denominators containing radicals.

H. Simplify radicals containing negative radicands.

I. Add, subtract, multiply, and divide complex numbers.

J. Evaluate functions. 

II. Equations and Inequalities

A. Solve literal equations that require factoring. 

B. Solve systems of linear equations in two variables.

C. Solve quadratic equations by completing the square.

D. Solve quadratic equations by using the quadratic formula.

E. Solve equations that are quadratic in form.

F. Solve equations containing rational expressions.

G. Solve equations containing radicals.

H. Solve equations developed from mixture, motion, work, and geometry applications.

I. Graph linear inequalities in two variables.

III. 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. Graph quadratic equations by emphasizing finding the vertex using x = - b/(2a) and symmetry.

E. Calculate the distance between two points using the distance formula.

F. Determine the midpoint between two points.

G. Distinguish between functions and non-functions using the vertical line test.

H. Identify the domain and range of a relation.

Content Outline and Competencies:

I. Geometric Shapes

A. Define points, lines, and planes.

B. Define line segments, rays, and angles.

C. Distinguish between different types of angles.

D. Measure angles with a protractor.

E. Apply properties of vertical, complementary, and supplementary angles.

F. Define triangles and list their types.

G. Define polygons and list their types.

H. Define quadrilaterals and list their types.

I. Compute the angle measures in a polygon.

J. Identify prisms, pyramids, cylinders, cones, and spheres.

K. Identify regular polyhedra and list their types.

II. Perimeter, Area, and Volume

A. Compute the perimeter of a polygon.

B. Compute the circumference of a circle.

C. Compute the area of rectangles, triangles, parallelograms, and trapezoids.

D. Compute the area of a regular polygon.

E. Compute the area of a circle.

F. Compute 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. Compute the surface area of prisms, pyramids, cylinders, cones, and spheres.

I. Compute the volume of prisms, pyramids, cylinders, cones, and spheres.

III. Reasoning

A. Formulate conclusions.

B. Apply conditional statements.

C. Apply equivalent (biconditional) statements.

D. Distinguish between valid and invalid deductions.

E. Write the converse, inverse, and contrapositive of a conditional statement.

F. Write a deductive proof.

IV. Triangles

A. Apply the triangle congruence theorems.

B. Prove two triangles are congruent.

C. Prove corresponding parts of congruent triangles are congruent.

D. Define medians and perpendicular bisectors

E. Apply theorems related to isosceles and equilateral triangles.

F. Apply theorems related to triangle inequalities.

V. Parallel Lines and Quadrilaterals

A. Apply properties of alternate interior, alternate exterior, and corresponding angles.

B. Apply theorems derived from the Parallel Postulate.

C. Prove the Angle Sum in a Triangle Theorem.

D. Apply the Exterior Angle Theorem.

E. Apply theorems related to quadrilaterals to determine side lengths and angle measures.

F. Prove theorems related to quadrilaterals.

VI. Similarity

A. Solve problems related to ratio and proportion.

B. Apply triangle similarity theorems to prove two triangles are similar.

C. Compute the missing part of similar polygons.

D. Apply theorems related to the geometric mean in a right triangle.

E. Apply the Side-Splitting Theorem

VII. Circles

A. Define arc, central angle and inscribed angle.

B. Compute measures of angles and arcs of a circle.

C. Compute areas of sectors and measures of arc length.

D. Define chords and tangents.

E. Compute measures of segments and angles formed by chords.

F. Prove theorems related to circles.

VIII. Constructions

A. Construct segments and angles using a compass and straightedge.

B. Construct perpendicular lines using a compass and straightedge.

C. Construct parallel lines using a compass and straightedge.

D. Subdivide line segments using a compass and straightedge.

E. Construct the center of a circle using a compass and straightedge.

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.

E. Calculate federal and state unemployment taxes.

F. Calculate an employee’s net pay.

G. Calculate the cost of employment to an employer.

H. Use a computer to analyze the effect of taxes on gross pay.

III. The Mathematics of Retailing

A. Analyze an invoice with 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.

I. Calculate the amount of operating expenses from the percent.

J. Calculate the break-even 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. Determine the effective rate (APR) of a note.

E. Compute the payoff amount on a loan or note using the actuarial method.

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.

F. Use a computer to solve and analyze time value of money applications for both savings plans and loans.

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. Convert numbers in standard notation to and from scientific notation and engineering notation.

F. Define ratio and proportion.

G. Use conversion factors to convert various units of measure.

H. Measure using a ruler.

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. Solve a proportion for a missing term.

D. 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.

VI. Logic

A. Understand Venn diagrams.

B. Construct truth tables.

C. Evaluate using Boolean algebra.

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 conditional probabilities.

D. Use counting formulas to determine permutations and combinations.

IV. Statistics

A. Calculate mean, median, mode and standard deviation of a data set.

B. Calculate expected value.

C. Interpret statistical graphs.

V. Mathematics of Finance

A. Calculate simple and compound interest.

B. Calculate the value of an annuity.

C. Calculate the annual percentage rate.

D. Analyze an amortization schedule.

VI. 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.

VII. 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 application graphically.

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) given its description.

F. Construct an equation of a circle in standard form using:

1. The center and radius.

2. The endpoints of the diameter.

3. The method of completing the square.

G. Use graphs of functions for analysis.

H. Find combinations and composites of functions.

I. 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 linear inequalities by graphing.

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.

Content Outline and Competencies:

I. The Six Trigonometric Functions

A. Determine the six trigonometric functions of an angle in standard position.

B. Determine the six trigonometric functions of an angle in a right triangle.

C. Determine the six trigonometric functions of an angle using the unit circle.

D. Calculate the six trigonometric functions of an angle using a calculator.

E. Determine exact values of the six trigonometric functions given an angle which is a multiple of  30°, 45°, 60°, or 90°.

II. Graphs

A. Analyze the graphs of the six basic trigonometric functions.

B. Graph transformations of trigonometric functions using the concepts of period, phase shift, amplitude and displacement.

C. Determine the equation of a function given its graph.

D. Convert from rectangular to polar coordinates and vice versa.

E. Convert equations from rectangular to polar form and vice versa.

F. Graph polar equations.

III. Inverse Trigonometric Functions

A. Evaluate inverse trigonometric functions with and without a calculator.

B. Evaluate expressions involving inverse trigonometric functions.

C. Graph inverse trigonometric functions, stating the domain and range.

IV. Trigonometric Identities

A. Simplify trigonometric expressions using trigonometric identities.

B. Verify trigonometric identities.

C. Evaluate function values using sum, difference, double, and half angle identities.

V. Trigonometric Equations

A. Solve trigonometric equations on a given interval in degrees or radians.

B. Solve trigonometric equations for all angle solutions.

VI. Solving Triangles

A. Solve triangles using trigonometric ratios, the Law of Sines, or the Law of Cosines.

B. Apply the solutions of triangles to real-world problems.

VII. Complex Numbers in Trigonometric Form

A. Graph in the complex plane.

B. Convert complex numbers from standard form to trigonometric form and vice versa.

C. Calculate products, quotients, and powers of complex numbers in trigonometric form.

VIII. Applications

A. Solve applications involving angles of elevation and depression.

B. Calculate the length of an arc and the area of a sector of a circle.

C. Determine the magnitude and direction of vectors.

D. Calculate the sum, difference, dot product, and angle between two vectors.

E. Apply vectors to real-world problems.

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. Describe 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.

F. Use inverse function notation.

G. Use function composition to verify an inverse function relationship.

H. 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, the law of sines, and the law of 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. Graph circles.

D. Sketch simple polar graphs.

E. Use technology to approximate trigonometric function values.

F. Perform binomial expansions through pattern recognition and use of the binomial theorem.

G. Investigate concavity as a rate of change.

H. Define a radian and use radian measures.

I. Generate sequences and associated sums using simple patterns and determine their formulas.

J. Find general terms of sequences with emphasis on appropriate notation.

K. Use sigma notation to express series.

L. Graph the inverse sine, inverse cosine and inverse tangent functions.

M. Graph solution sets for systems of non-linear inequalities.

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 the Banzhaf power distribution.

G. Define apportionment.

H. Calculate a solution to apportionment problems.

I. Discuss the effect of apportionment methods in the U.S. Congress.

III. Sequences and Recursion

A. Determine the terms of a sequence, both explicitly and recursively.

B. Determine the terms of the Fibonacci sequence.

C. Determine the sums of sequences.

D. Describe populations using linear and exponential growth models.

E. Apply linear and exponential growth models to financial applications.

F. Define fractals and the Mandelbrot Set.

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 the graph of the feasible region.

Content Outline and Competencies:

I. Descriptive Statistics

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

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

C. Construct appropriate graphical displays of qualitative (categorical) and quantitative (numerical) data such as dotplots, histograms, box plots, bar charts, pie charts or scatterplots.

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

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

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

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

H. Determine potential outliers in a distribution.

I. Use the General Normal Distribution to calculate probabilities and percentiles.

II. Introduction to Probability

A. Solve basic probability problems.

B. Determine whether or not two events are independent.

C. Calculate and explain, in context, a conditional probability.

III. General Normal Distribution

A. Use the General Normal Distribution to solve problems involving random variables.

B. Use the normal distribution to solve percentile or measurement value problems.

IV. Random Sampling and Sampling Theory

A. Describe the shape, center, and spread for the distribution of the samples of a single variable.

B. Verify that the conditions are met to use the Central Limit Theorem.

V. Confidence Intervals

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

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

VI. Hypothesis Tests

A. Perform hypothesis tests for a population mean and the difference of two population means, and interpret the results.

B. Perform hypothesis tests for a population proportion and the difference of two population proportions, and interpret the results.

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

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

VII. Linear Regression

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

B. Calculate a linear regression equation.

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

2. Use the linear regression equation to make predictions.

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

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

Content Outline and Competencies:

I. Vector Algebra and Transformations

A. Use trigonometry to determine the components and direction angles of a vector.

B. Compare the concepts of scalar and vector.

C. Compute vector arithmetic graphically and numerically.

D. Compute the angle between two vectors.

E. Normalize vectors.

F. Compute the normal vector to a surface.

G. Construct code that will perform vector arithmetic and normalization.

H. Convert between polar and rectangular coordinates.

I. Convert units of measurement.

J. Compute matrix arithmetic graphically and numerically.

K. Describe scaling using matrices and homogeneous coordinates.

L. Construct code that will perform scaling on vectors and geometric objects using matrices.

II. 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. Use Newton's Laws to determine the effect of forces on the motion of an object.

D. Derive and solve the equations of motion of an object.

E. Calculate the work done by a force on an object.

F. Calculate the kinetic energy, potential energy, and momentum of an object.

G. Compute the force vector acting on an object resulting from a scalar potential energy field.

H. Describe the Forward Euler and Velocity Verlet integration methods.

I. Compare the advantages and disadvantages of the Forward Euler and Velocity Verlet integration methods.

J. Construct code that can simulate the motion of an object according to Newton’s Laws of Motion.

K. Describe translations using matrices and homogeneous coordinates.

L. Construct code that will perform translation on vectors and geometric objects using matrices.

III. Collision Detection and Resolution

A. Determine the distance between an object and a line or plane.

B. Construct code that will compute the distance between an object and a line or plane.

C. Determine if two circles or two spheres are intersecting.

D. Calculate the point of intersection of two line segments.

E. Determine if two axially-aligned bounding boxes are intersecting.

F. Construct code that will detect collisions between circles, spheres, axially aligned bounding boxes and line segments.

G. Use conservation of energy and conservation of momentum to model the collision of objects.

H. Construct code that can simulate the collision between two objects.

IV. Rotational Motion

A. Describe rotations using matrices and homogeneous coordinates.

B. Construct code that will rotate an object using matrices.

C. Compute angular displacement, angular velocity and angular acceleration.

D. Determine the angular motion caused by a torque on an object.

E. Calculate the rotational kinetic energy and angular momentum of a rotating object.

F. Construct code that can model the two-dimensional motion of a rigid body incorporating the concepts of the conservation of energy and momentum and Newton’s Laws of Motion.

G. Compute quaternion arithmetic numerically.

H. Construct code that will rotate an object using quaternions.

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 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 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 form.

F. Solve for the missing term of a proportion.

G. Demonstrate operations on fractions using 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 and their applications to box plots and percentiles.

D. Determine measures of dispersion.

E. Describe the attributes of the normal distribution.

F. Recognize misleading graphs.

III. Measurement

A. Calculate temperature conversions for Celsius and Fahrenheit.

B. Measure items using both the U.S. measurement systems and the metric system.

C. Convert measurements within the U.S. measurement system.

D. Convert measurements within the metric system.

E. Use dimensional analysis to convert measurements between the U.S. measurement 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. Limits

A. Evaluate a limit, including one-sided limits, at a point using graphs, tables and algebraic techniques.

B. Evaluate a limit involving infinity using graphs, tables and algebraic techniques.

C. Determine continuity of a function at a point using the three conditions of continuity.

D. Determine differentiability of a function at a point using limits and graphs.

II. Derivatives

A. Find derivatives of a function using the limit definition.

B. Find the derivatives of algebraic, natural exponential and logarithmic functions using the power rule, product rule, quotient rule and the chain rule.

C. Find derivatives using implicit differentiation.

D. Use the derivative to write the equation of a line tangent to a curve at a given point.

E. Determine the sign of the derivative to estimate the intervals over which the first and second derivative is positive or negative using a graph.

F. Use derivatives to find critical points, inflection points and determine concavity using the first and second derivative.

G. Apply derivative techniques to economics, business, and physical, social and life sciences applications.

H.  Predict, find, and explain rates of change for position functions, including the relationship between position, velocity and acceleration using derivatives.

I.  Use derivatives to determine relative and absolute extrema (optimization) including restricted domains.

J.   Determine outcomes in related rates problems using derivatives.

K.  Determine and interpret marginal analyses using derivatives.

III. Integrals

A. Antidifferentiate algebraic, exponential, and logarithmic functions using elementary techniques and u-substitution.

B. Apply the Fundamental Theorem of Calculus to evaluate a definite integral using elementary techniques and u- substitution.

C. Calculate the area under a curve.

D. Determine a function given the rate of change.

E. Determine total revenue, cost and profit from marginal functions and determine distance from velocity function.

F. Solve for the integration constant given initial conditions.

G. Determine the total accumulation between two fixed values for a given rate of change.

H. Determine the average value of a function.

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. Compute the first and second partial derivatives of functions of several variables.

B. Compute the value of double integrals.

C. Locate the coordinates of any relative extrema of a function of two variables.

D. Utilize Lagrange Multipliers to compute the maximum or minimum of a function subject to constraints.

E. Apply multivariable calculus to application problems.

III. Differential Equations

A. Identify differential equations.

B. Solve differential equations using the method of separation of variables.

C. Apply differential equations to problems involving limited, unlimited, and logistic growth.

IV. Calculus and Trigonometric Functions

A. Differentiate the sine and cosine functions.

B. Integrate the sine and cosine functions.

C. 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. Limits

A. Evaluate 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 limits.

1. Use the limit to determine the continuity of a function.

2. Use the limit to determine differentiability of a function.

3. Apply the Intermediate Value Theorem.

4. Use the limiting process to find the derivative of a function.

II. Derivatives

A. Find derivatives involving powers and sums.

B. Find derivatives involving products and quotients.

C. Find derivatives involving the chain rule.

D. Find derivatives involving exponential and logarithmic functions.

E. Find derivatives involving trigonometric and inverse trigonometric functions.

F. Find derivatives involving implicit differentiation.

III. Applications of Derivatives

A. Determine graphical properties of functions.

1. Use the first derivative to find critical points.

2. Apply the Mean-Value Theorem for derivatives.

3. Determine the behavior of a function using the first derivative.

4. Use the second derivative to find inflection points.

5. Determine the concavity of a function using the second derivative.

6. Sketch the graph of the function using information gathered from the first and second derivatives.

7. Interpret graphs of functions.

B. Apply the derivative.

1. Use the derivative to find velocity, acceleration and other rates of change.

2. Use the derivative to find a tangent line to a curve at a given point.

3. Solve related rates problems.

4. Use optimization techniques in economics, the physical sciences and geometry.

5. Use differentials to estimate change.

6. Use Newton’s Method.

IV. Integrals

A. Find area using Riemann sums.

B. Express the limit of a Riemann sum as a definite integral.

C. Evaluate the definite integral using geometry.

D. Approximate definite integrals using the Trapezoid Rule and Simpson’s Rule.

E. Evaluate definite integrals using the Fundamental Theorem of Calculus.

F. Integrate algebraic, natural exponential, natural logarithm, trigonometric and inverse trigonometric functions.

G. Integrate indefinite integrals.

H. Integrate by substitution.

I. Apply the Mean-Value Theorem for Integrals.

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 using trigonometric substitutions.

E. Identify appropriate techniques necessary for integration.

F. Evaluate an integral using integral tables and/or technology.

G. Apply L'Hopital's Rule to evaluate limits involving indeterminate forms.

H. Determine whether an improper integral converges or diverges by definition.

I. Determine the convergence or divergence of an improper integral using an appropriate test.

J. Evaluate a convergent improper integral.

III. 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 and shell methods.

C. Calculate the length of a curve defined by a Cartesian equation.

D. Calculate the surface area generated by revolving a Cartesian curve.

E. Calculate work.

IV. Parametric and Polar Equations

A. Graph parametric equations.

B. Convert between parametric and Cartesian equations.

C. Graph polar equations.

D. Convert from rectangular to polar coordinates and vice-versa.

E. Calculate the slope of a parametrized curve.

F. Calculate the length of a smooth parametrized curve.

G. Calculate the surface area generated by revolving a parametrized curve.

H. Differentiate polar equations.

I. Integrate polar equations.

J. Calculate area in the plane using polar coordinates.

K. Calculate the area between two curves using polar coordinates.

L. Calculate the length of a polar curve.

V. 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 by definition.

E. Determine 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. Construct a Taylor series which represents a function.

I. Estimate the error in approximating a function with a Taylor polynomial.

J. Approximate a definite integral, within a specified error magnitude, using a Taylor polynomial

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. Describe locations in three dimensions using rectangular, cylindrical, and spherical coordinates.

E. Convert between rectangular, cylindrical, and spherical coordinates.

F. Utilize rectangular, cylindrical, and spherical coordinates to describe surfaces.

G. Use cylindrical and spherical coordinates to evaluate triple integrals.

H. Use double integrals to compute area in the plane.

I. Use double and triple integrals to calculate volume.

J. Use double integrals to compute surface area.

K. Use double and triple integrals in applications including calculation of mass, center of mass, and moments of inertia.

L. 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. Apply the Divergence Theorem where appropriate.

R. 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. Determine 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. Compute 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. Calculate determinants using 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 a subspace associated with a matrix.

E. Calculate a rotation matrix for a given angle.

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. Diagonalize matrices.

D. Diagonalize linear operators.

VI. Orthogonality

A. Describe vectors with geometry.

B. Construct orthonormal vectors.

C. Compute the least-squares approximation.

D. Produce orthogonal projection matrices.

E. Create orthogonal matrices and operators.

F. Construct symmetric matrices.   

VII. Vector Spaces

A. Define vector spaces and their subspaces.

B. Determine dimension and isomorphism.

C. Express linear transformations in matrix representations.

D. Define inner product spaces.

Content Outline and Competencies:

I. Basic concepts

A. Discriminate between ordinary and partial differential equations.

B. Determine the order of a differential equation.

C. Discriminate between linear and nonlinear differential equations.

D. Define a solution to a differential equation.

E. 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.

F. Analyze the stability of equilibrium solutions.

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. Solve systems of equations using Gaussian elimination.

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 underdamped, critically damped or overdamped.

H. Construct the general solution of nonhomogeneous linear systems utilizing the method of variation of parameters.

I. Utilize the phase plane in solving systems.

J. Utilize systems of differential equations in modeling activities.

VI. Series Solution Techniques

A. Construct series solutions near an ordinary point.

B. 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. Descriptive Statistics

A. For a given set of data, draw a dotplot, histogram and/or a boxplot.

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

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

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

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

II. Probability

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

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

B. Determine the expected value and the variance of a discrete probability density function.

C. Use calculus to calculate probabilities, expected value and the variance for continuous probability density functions.

IV. Specific Probability Distributions

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.

D. Use the Exponential distribution to solve problems describing the time between events.

E. Use the Normal distribution and z-scores to solve problems.

V. Random Sampling and Sampling Theory

A. Calculate the mean and standard deviation for a distribution of sample means.

B. Construct a normal probability plot and describe the shape of the population based on the plot.

C. Verify that the conditions are met to use the Central Limit Theorem.

VI. Estimating a Population Parameter

A. Construct and interpret a confidence interval for a population mean.

B. Construct and interpret a confidence interval for a population proportion.

VII. Hypothesis Testing

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

A. Calculate a linear regression equation and 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.

D. Use the coefficient of determination to explain the strength of the regression equation.

IX. Introduction to Statistical Process Control

A. Provide a brief history of quality control.

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.