Courses

Topical Outline with Specific Course Objectives

1. Number Work and Simplifying Skills
   1. Use the properties of radicals, complex numbers, and exponents within the context of solving equations and graphing.
   2. Decide when an exact answer is needed and when an estimate is acceptable. Estimate to a reasonable degree of accuracy.
   3. Define, use and recognize absolute value as distance on the coordinate line.
   4. Discuss the historical and scientific basis for the number “e”.
2. Functions and Relations
   1. Recognize and model linear, quadratic, exponential, logarithmic, absolute value, radical, rational and polynomial functions in various settings including applications.
   2. Know the properties, graphs, and equations of linear, quadratic, exponential, logarithmic, absolute value, radical, rational and polynomial functions.
   3. Identify the domain, range, and inverse for functions or relations expressed graphically or symbolically.
   4. Analyze situations, functions and data from charts, tables and graphs for the purpose of describing behavior (e.g., increasing, decreasing), identifying specific information (e.g., maximum, minimum values), and making predictions (e.g., predicting future trends).
   5. Determine the major characteristics of graphs such as intercepts and asymptotes.
   6. Deduce the possible shape of the graph from a given equation and vice-versa.
   7. Use the algebra of functions and be able to relate it to the graphical representations.
   8. State and illustrate the fundamental properties of inverse functions.  Discuss the concept of inverses in a broad context and give examples.
   9. Recognize, use, and discuss the inter-relationships among roots of polynomial functions, factors of polynomials, and solutions of polynomial equations.
   10. Discuss the historical and scientific basis for the logarithm.
   11. Use the properties of exponential and logarithmic functions to solve equations, including equations which arise in applied settings.
3. Equations and Systems of Equations and Inequalities
   1. Choose and apply appropriate methods for solving systems of equations and inequalities, and describe the situation graphically where possible. Justify choice of method.
   2. Describe the relationships among the discriminant, solutions of quadratic equations, and intercepts of parabolas.
   3. Apply appropriate methods for solving quadratic equations, and discuss the advantages and disadvantages of each method.

Department Policies

• Graphing Calculator Required
• Group Activities and Projects Required
• Common Comprehensive Final

Topical Outline with Specific Course Objectives

1. Trigonometric functions, their properties and graphs
   1. Demonstrate an understanding of the trigonometric functions as functions of real numbers.
   2. Analyze and interpret trigonometric functions using graphs, tables and equations.
   3. Discuss the scientific basis for radian measure.
2. Inverse trigonometric functions, their properties and graphs
   1. Construct the inverse of a trigonometric function, both graphically and algebraically when feasible.
3. Trigonometric identities
   1. Verify and apply trigonometric identities.
4. Conditional trigonometric equations
   1. Solve trigonometric equations numerically, graphically, and algebraically.
5. Solutions of triangles
   1. Demonstrate the ability to find unknown angles and side lengths using trigonometry
   2. Solve applications using solutions of triangles.
6. Vector algebra
   1. Demonstrate an understanding of vectors, their graphical representation and vector algebra.
7. Parametric equations
   1. Sketch and identify graphs using parametric equations.
   2. Convert rectangular equations to parametric form and vice versa.
8. Polar coordinates
   1. Sketch and identify graphs in polar coordinates.
   2. Convert rectangular equations to polar form and vice versa.
9. Applications
   1. Model real-world applications using the trigonometric functions.
   2. Apply radian measure to arc length and area.

Department Policies

• Graphing Calculator not allowed on proctored tests or the final exam
• Comprehensive Final
• The TI-30XSII2 scientific calculator may be allowed in this course on some assignments, please ask your instructor

Topical Outline with Specific Course Objectives

1. Introduction of Functions, the Properties, Graphs, and Inverses
   1. Analyze and interpret functions using graphs, tables, and equations.
   2. Determine domain and range both algebraically and graphically.
   3. Demonstrate an understanding of both composition and decomposition of functions.
   4. Construct the inverse of a function, both graphically and algebraically when feasible.
   5. Demonstrate an intuitive understanding of functions including limits, continuity, increasing, decreasing, maximum and minimum values, symmetry and concavity.
   6. Demonstrate an understanding of transformations graphically, symbolically, and numerically.
   7. Graph and write equations for piecewise-defined functions.
2. Polynomial and Rational Functions
   1. Identify multiplicities of zeros and their effect on the graph.
   2. Create graphs and write equations given sufficient information.
   3. Use formal limit definitions and limit notation to define the end behavior and vertical and horizontal asymptotes for functions given in numerical, graphical, and symbolic forms.
   4. Determine when a rational function has a hole and the point at which it occurs.
3. Exponential and Logarithmic Functions
   
   1. Create graphs and write equations for exponential and logarithmic functions given sufficient information.
   2. Use formal limit definitions and limit notation to define the end behavior and vertical and horizontal asymptotes for functions given in numerical, graphical, and symbolic forms.
   3. Demonstrate mastery of the logarithmic properties by applying them in a variety of settings.
   4. Understand the inverse relationship and use it to solve exponential and logarithmic equations.
   5. Use exponential, logarithmic, and related functions in science based applications.
4. Solving Equations and Inequalities
   1. Solve non-linear equations and inequalities numerically, graphically, and symbolically.
   2. Recognize when a solution is approximate and have a rough idea of its accuracy.
   3. State exact solutions in proper mathematical form.
5. Conic Sections
   1. Identify and graph a conic section from its equation.
   2. Find the equation of a conic section from its description.
   3. Identify the attributes of a conic section.
6. Matrices
   
   1. Perform basic matrix algebra and demonstrate knowledge of the commutative properties.
   2. Use expansion by cofactors to compute the determinant of matrices up to dimension
   3. Construct the inverse of a matrix using row operations.
   4. Use matrices and determinants to solve linear systems of equations.
7. Sequences and Series
   1. Recognize arithmetic and geometric sequences and series in summation or iterative notation.
   2. Demonstrate an understanding of sequences and series notation by expanding a given formula and by creating a formula for a given expansion.
   3. Find the sum of a finite series.
8. Special Topics
   1. Apply the Binomial Theorem.
   2. Demonstrate an understanding of Proof by Induction.

Department Policies

• Graphing Calculator not allowed on proctored tests or final exam
• Comprehensive Final
• The TI-30XSII2 scientific calculator may be allowed in this course on some assignments, please ask your instructor

Topical Outline with Specific Course Objectives

1. Introduction of Functions, the Properties, Graphs, and Inverses
   1. Analyze and interpret functions using graphs, tables, and equations.
   2. Determine domain and range both algebraically and graphically.
   3. Demonstrate an understanding of both composition and decomposition of functions.
   4. Construct the inverse of a function, including trigonometric functions, both graphically and algebraically when feasible.
   5. Demonstrate an intuitive understanding of functions including limits, continuity, increasing, decreasing, maximum and minimum values, and concavity.
   6. Demonstrate an understanding of transformations graphically, symbolically, and numerically.
   7. Graph and write equations for piecewise-defined functions.
2. Polynomial and Rational Functions
   1. Identify multiplicities of zeros and their effect on the graph.
   2. Use formal limit definitions and limit notation to define the end behavior and vertical and horizontal asymptotes for functions given in numerical, graphical, and symbolic forms.
   3. Demonstrate mastery of the logarithmic properties by applying them in a variety of settings.
   4. Determine when a rational function has a hole and the point at which it occurs.
3. Exponential and Logarithmic Functions
   1. Create graphs and write equations for exponential and logarithmic functions given sufficient information.
   2. Use formal limit definitions and limit notation to define the end behavior and vertical and horizontal asymptotes for functions given in numerical, graphical, and symbolic forms.
   3. Demonstrate mastery of the logarithmic properties by applying them in a variety of settings.
   4. Understand the inverse relationship and use it to solve exponential and logarithmic equations.
   5. Use exponential, logarithmic, and related functions in science based applications.
4. Trigonometric Functions
   1. Demonstrate an understanding of the trigonometric functions as functions of real numbers.
   2. Analyze and interpret trigonometric functions using graphs, tables and equations.
   3. Discuss the scientific basis for radian measure.
   4. Model real-world applications using trigonometric functions.
   5. Apply radian measure to arc length and area.
5. Solving Equations and Inequalities
   1. Solve non-linear equations and inequalities numerically, graphically, and symbolically.
   2. Recognize when a solution is approximate and have a rough idea of its accuracy. .
   3. State exact solutions in proper mathematical form.
6. Conic Sections
   
   1. Identify and graph a conic section from its equation.
   2. Find the equation of a conic section from its description.
   3. Identify the attributes of a conic section.
7. Matrices
   1. Perform basic matrix algebra and demonstrate knowledge of the commutative properties.
   2. Use expansion by cofactors to compute the determinant of matrices up to dimension 4x4.
   3. Construct the inverse of a matrix using row operations.
   4. Use matrices and determinants to solve linear systems of equations.
8. Sequences and Series
   1. Recognize arithmetic and geometric sequences and series in summation or iterative notation.
   2. Demonstrate an understanding of sequences and series notation by expanding a given formula and by creating a formula for a given expansion.
   3. Find the sum of a finite series.
9. Additional Topics
   
   1. Apply the Binomial Theorem.
   2. Demonstrate an understanding of Proof by Induction.
   3. Demonstrate the ability to find unknown angles and side lengths of right and non-right triangles using trigonometry.
   4. Solve applications using solutions of triangles.
10. Trigonometric identities and conditional trigonometric equations
    
    1. Verify and apply trigonometric identities.
    2. Solve trigonometric equations numerically, graphically, and algebraically.
11. Vector algebra and polar coordinates
    
    1. Demonstrate an understanding of vectors, their graphical representation and vector algebra.
    2. Sketch and identify graphs in polar coordinates.
    3. Convert rectangular equations to polar form and vice versa.
12. Parametric equations
    1. Sketch and identify graphs using parametric equations.
    2. Convert rectangular equations to parametric form and vice versa.

Evaluation: Each instructor will determine the specific criteria for determining the final course grade. These criteria will be delineated in the first day handout provided to each student. Each instructor will give a comprehensive final exam during the assigned final exam period.

Commonality: All instructors will use the same textbook and cover all topics in the topical outline. All students will participate in at least two technology based lab activities. A computer lab with mathematical software is provided to facilitate collaboration and the use of technology.

Department Policies

• Graphing Calculator not allowed on proctored tests or final exam
• Comprehensive Final
• The TI-30XSII2 scientific calculator may be allowed in this course on some assignments, please ask your instructor

Topical Outline with Specific Course Objectives

1. Review of Basic functions and Graphs
   1. Recognize algebraic functions in various forms.
   2. Understand and use function notation in any setting.
2. Limits and Continuity
   
   1. Find limits graphically and numerically.
   2. Use limits to describe asymptotic behavior.
   3. Determine the continuity of functions graphically
3. The Derivative
   1. Define, discuss, and interpret the concept of the derivative algebraically, verbally, numerically, and graphically.
   2. Find the derivative of a function using differentiation rules.
   3. Identify the major characteristics of graphs and relate them to first and second derivatives.
   4. Apply approximating techniques for derivatives.
4. The Integral
   1. Demonstrate an understanding of the concept of the definite integral and describe the difference between the definite and indefinite integral.
   2. Find the indefinite integral of a function using antidifferentiation rules.
   3. Find and interpret the definite integral numerically, algebraically, and graphically.
   4. State and apply the Fundamental Theorem of Calculus.
   5. Use definite integrals to find areas.
   6. Apply approximating techniques for integrals.

Department Policies

• Graphing Calculator Required
• Comprehensive Final

Topical Outline with Specific Course Objectives

1. The Integral
   1. Find the indefinite integral of a function using antidifferentiation rules.
   2. Apply the Fundamental Theorem of Calculus.
   3. Use definite integrals to find the area between two curves.
   4. Apply a variety of numerical methods for approximating definite integrals such as the Trapezoidal Rule and Simpson's Rule
   5. Apply antidifferentiation techniques, such as integration by substitution and integration by parts, to find indefinite and definite integrals.
   6. Use definite integrals to model applied problems such as producer surplus, consumer surplus, present value and future value.
   7. Recognize an improper integral and determine whether it is convergent or divergent.
2. Functions of Several Variables
   1. Graph functions of several variables.
   2. Use functions of several variables to model applied problems, including Cobb-Douglas production functions.
   3. Find partial derivatives numerically and symbolically.
   4. Give graphical interpretations of partial derivatives.
   5. Find maxima and minima of functions of several variables.
   6. Use the method of Least Squares to find the line of best fit.
   7. Use the Method of Lagrange Multipliers to solve constrained optimization problems.
   8. Find the total differential of a function of several variables and use it to approximate incremental change in the function.
   9. Evaluate double integrals and use them to model applied problems such as areas of regions, volumes of solids, population density and average values of real-life models.
3. Differential Equations
   1. Verify general solutions of differential equations and find particular solutions, given the general solution.
   2. Use differential equations to model applied problems such as population and mixture problems.
   3. Use separation of variables to solve differential equations.
   4. Use separable differential equations to model applied problems.
   5. Use Euler’s method to approximate solutions to initial value problems.
4. Probability Distributions of Random Variables
   1. Verify that a function is a continuous probability density function and use continuous probability density functions to find probabilities.
   2. Find the cumulative distribution function for a random variable with a given probability density function.
   3. Find the expected value and standard deviation of continuous probability density functions.
   4. Model applied problems using special probability density functions such as the uniform probability density function, the exponential probability density function, and the normal probability density function.
5. Sequences and Series
   1. Find Taylor polynomials for functions.
   2. Use Taylor polynomials to determine the maximum errors of approximations and to approximate definite integrals.
   3. Determine the convergence or divergence of sequences and find the limits of convergent sequences.
   4. Define the sequence of partial sums and determine the convergence or divergence of infinite series (focus on telescoping series and geometric series).
   5. Model applied problems using geometric series.
6. Systems of Linear Equations and Matrices
   1. Model applied problems using systems of linear equations.
   2. Use Gauss-Jordan elimination to solve systems of linear equations.
   3. Multiply matrices and find the inverse of a square matrix.
   4. Model applied problems using matrices, including Leontief Input-Output Models.
7. Linear Inequalities and Linear Programming Problems
   1. Graph systems of linear inequalities in two variables.
   2. Model applied problems using linear programming.
   3. Find graphical solutions for linear programming problems.
   4. Use the Simplex Method to solve linear programming problems.

Department policies

• Graphing Calculator Required
• Comprehensive Final

Topical Outline with Specific Course Objectives

1. Limits and Continuity
   1. Find limits algebraically, graphically, and numerically.
   2. Identify and find indeterminate limits, when they exist, using appropriate tools, e.g. L'Hopital's Rule.
   3. Use limits to describe asymptotic behavior.
   4. Determine the continuity of functions algebraically and graphically.
   5. Test the hypothesis and, where appropriate, apply the conclusion of the Intermediate Value Theorem for a function on a given interval.
2. The Derivative
   1. Define, discuss, and interpret the concept of the derivative algebraically, verbally, numerically, and graphically.
   2. Find the derivative of a function using differentiation rules.
   3. Find the derivatives of inverse functions.
   4. Identify the major characteristics of graphs and relate them to first and second derivatives.
   5. Find the differential of a function and use it to find a tangent line approximation to the function.
   6. Apply Newton's method to solve equations.
   7. Model related rates, optimization, and motion problems using the language of calculus. Find and discuss solutions.
   8. Test the hypothesis and, where appropriate, apply the conclusion of the Mean Value Theorem for a function on a given interval.
3. The Integral
   1. Understand the concept of Riemann Sums and apply the definition of Riemann sums to evaluate definite integrals.
   2. Demonstrate an understanding of the concept of the definite integral and describe the difference between the definite and indefinite and indefinite integral.
   3. Find the indefinite integral of a function using antidifferentiation rules.
   4. Find and interpret the definite integral numerically, algebraically, and graphically.
   5. State and apply the Fundamental Theorem of Calculus.
   6. Use definite integrals to find areas including areas between curves.
   7. Apply integrals to functions describing rectilinear motion, displacement, and distance traveled.
   8. Apply approximating techniques for integrals.
   9. Find definite and indefinite integrals using substitution.
   10. Use definite integrals to find areas under a curve and areas between curves.

Department Policies

• Graphing Calculator not allowed on proctored tests or final exam
• Comprehensive Final
• The TI-30XSII2 scientific calculator may be allowed in this course on some assignments, please ask your instructor

Topical Outline with Specific Course Objectives

1. Integration
   1. Apply antidifferentiation techniques, such as integration by parts, partial fractions, trigonometric substitution, and the use of a table of integrals.
   2. Develop an organized approach for classifying a given integrand and determining the appropriate technique of antidifferentiation.
   3. Model applied problems of area, volume, arc length, and work using integrals.
   4. Apply a variety of numerical methods for approximating definite integrals.
   5. Estimate and compare the errors associated with each approximation method for definite integrals.
   6. Recognize an improper integral and determine whether it is convergent or divergent.
   7. Find the value of a convergent improper integral symbolically when possible, otherwise estimate the value numerically.
2. Infinite Sequences and Series
   
   1. Verify the convergence or divergence of a sequence by employing appropriate tools and find or estimate the limit of a convergent sequence.
   2. Define the sequence of partial sums for an infinite series and relate the convergence of this sequence to the convergence of the series. Then find or estimate the sum.
   3. Exhibit knowledge of convergence tests, their usefulness, conditions, and limitations, and apply the tests to determine the convergence or divergence of a series.
   4. Develop and apply an organized approach for proving the convergence or divergence of a series.
   5. Find the interval and radius of convergence for a given power series.
   6. Find the Taylor and Maclaurin series representations of a function and determine the interval of convergence.
   7. Relate differentiation/integration of a given function to differentiation/integration of the corresponding power series representation.
   8. Use the sequence of partial sums of a power series, in its interval of convergence, as a useful approximation for a function.
   9. Calculate error terms for Alternating Series and Taylor Series.
3. Parametric Equations
   1. Represent a plane curve parametrically and determine its orientation.
   2. Use parametric equations to model and analyze physical processes such as curvilinear motion.
   3. Apply the techniques of Calculus to analyze parametric equations.
4. Polar Coordinates
   1. Apply calculus to examine the properties of curves represented in polar coordinates, e.g. area, tangent lines, and arc length.
   2. Convert between rectangular and polar coordinates.

Department Policies

• Graphing Calculator not allowed on proctored tests or final exam
• Comprehensive Final
• The TI-30XSII2 scientific calculator may be allowed in this course on some assignments, please ask your instructor

Topical Outline with Specific Course Objectives

1. Functions and the Geometry of Space
   1. Identify, describe, and visualize equations in 3-space.
   2. Use contour maps for functions of two or three variables to analyze the functions.
   3. Use the algebra of vectors to study geometry in 3-space.
2. Calculus of Vector-Valued Functions
   1. Use the calculus of vector-valued functions to analyze motions in 3-space.
   2. Find and interpret the unit tangent and unit normal vectors and curvature.
3. Calculus of Functions of Several Variables - Differentiation
   1. Prove a limit does or does not exist.
   2. Find partial derivatives numerically and symbolically and use them to analyze and interpret the way a function varies.
   3. Find and interpret the gradient and directional derivatives for a function at a given point.
   4. Find the total differential of a function of several variables and use it to approximate incremental change in the function.
   5. Analyze and solve optimization problems using the Second Derivative Test and Lagrange multipliers.
4. Calculus of Functions of Several Variables - Integration
   1. Explain the relationship between multiple and iterated integrals.
   2. Evaluate multiple integrals either by using iterated integrals or approximation methods.
   3. Evaluate multiple integrals using polar coordinates.
   4. Relate rectangular coordinates in 3-space to spherical and cylindrical coordinates, and use spherical and cylindrical coordinates as an aid in evaluating multiple integrals.
   5. Model applied problems using multiple integrals.
   6. Evaluate multiple integrals using a change of variables and the Jacobian.
5. Vector Analysis
   1. Define a line integral, and use it to find the total change in a function given its gradient field.
   2. Understand and apply Fundamental Theorem for line integrals.
   3. Understand and apply Green’s Theorem.
   4. Calculate and interpret curl and divergence for a vector field.
   5. Define a surface integral, and use it to find the flux of a vector field over a surface.
   6. Understand and apply Stoke’s Theorem.

Department Policies

• Graphing Calculator not allowed on proctored tests or final exam
• Comprehensive Final
• The TI-30XSII2 scientific calculator may be allowed in this course on some assignments, please ask your instructor

Topical Outline with Specific Course Objectives

1. First Order Differential Equations
   1. Understand the Existence and Uniqueness Theorem and its ramifications.
   2. Apply calculus techniques to first order differential equations to determine properties of solutions such as monotonicity, concavity, symmetry, and singularities.
   3. Use graphical techniques such as direction (slope) fields, phase diagrams and orbits to study the relationship between a first order differential equation and its solution.
   4. Investigate the sensitivity of models and their solutions to initial conditions and parameters.
   5. Use numerical methods to approximate solutions to initial value problems and provide error estimates.
   6. Solve first order differential equations which are exact, separable, homogeneous, linear or Bernoulli using symbolic methods.
   7. Construct and analyze models, interpret results, and make predictions within the original context.
2. Higher Order Linear Differential Equations
   1. Use series, graphical, and symbolic methods to generate and analyze solutions to differential equations or boundary value problems.
   2. Solve higher order linear differential equations with constant coefficients using symbolic methods.
   3. Apply the methods of undetermined coefficients and variation of parameters.
   4. Identify and solve Cauchy-Euler equations.
3. The Laplace Transform
   1. Use the Laplace Transform to solve differential equations.
   2. Construct and analyze models, interpret results, and make predictions within the original context.
   3. Use Laplace Transforms to solve systems of linear differential equations.

Department Policies

• Graphing Calculator not allowed on proctored tests or final exam
• Comprehensive Final
• The TI-30XSII2 scientific calculator may be allowed in this course on some assignments, please ask your instructor

Topical Outline with Specific Course Objectives

1. Systems of Linear Equations
   1. Recognize a linear equation in n variables.
   2. Find a parametric representation of a solution set of a system of linear equations.
   3. Write a system of linear equations in matrix form.
   4. Solve a system of linear equations by substitution, graphing, using a computer or graphing calculator, Gaussian elimination, Gauss-Jordan elimination, LU-factorization, Cramer’s Rule.
   5. Determine whether a system of linear equations is consistent or inconsistent.
   6. Write a given system of linear equations in the form Ax=b and use it to solve for x.
   7. Find a general solution of a consistent system.
2. Matrices
   1. Determine the size, transpose, inverse, rank, and LU-factorization of a matrix.
   2. Write an augmented or coefficient matrix from a system of linear equations.
   3. Use elementary row operations to write a matrix in row-echelon form and reduced row-echelon form.
   4. Perform matrix operations and solve matrix equations.
   5. Factor a given matrix into a product of elementary matrices.
   6. Find the adjoint of a matrix and use it to find the inverse of the matrix. Know and use equivalent conditions for an invertible matrix.
3. Determinants
   1. Find the determinant, minors and cofactors of a given matrix.
   2. Use the determinant to decide whether a given matrix is singular or nonsingular.
   3. Use properties of determinants.
4. Vectors and Vector Spaces
   1. Perform vector operations for vectors in Rn.
   2. Determine whether a given set with two operations is a vector space.
   3. Recognize standard examples of vector spaces: nth dimensional Euclidean space, the set of all m x n matrices, the set of all polynomials, the set of all polynomials of degree ≤ n, the set of all continuous functions defined on the real line, the set of all continuous functions defined on a closed interval [a,b].
   4. Determine whether a given subset of a vector space is a subspace .
   5. Recognize subspaces of R2 and R3 and understand their geometric interpretations.
   6. Determine whether a vector is a linear combination of a given finite set of vectors in a vector space and be able to write this linear combination.
   7. Determine whether a given set of vectors in a vector space is a spanning set for that vector space.
   8. Determine whether a given finite set of vectors in a vector space is linearly independent.
   9. Determine whether a given set of vectors in a vector space forms a basis for that vector space and recognize standard bases in the vector spaces nth dimensional Euclidean space, the set of all m x n matrices, and the set of all polynomials of degree ≤ n.
   10. Find the dimension of a subspace.
   11. Find a basis for the column or row space of a matrix.
   12. Find a basis for and the dimension of the nullspace of a matrix.
   13. Find the coordinate matrix for a vector relative to a basis.
   14. Find the transition matrix from one basis to another basis.
   15. For a given vector v, find its length, a unit vector in the same or opposite direction, all vectors that are orthogonal to v, and the projection of v onto a given vector or vector space.
   16. Find the distance, the dot product, the inner product, the cross product and the angle between any two given vectors in a Euclidean space.
   17. Verify and use the Cauchy-Schwarz Inequality, the Triangle Inequality and the Pythagorean Theorem for vectors.
   18. Determine whether any two given vectors are orthogonal, parallel, or neither.
   19. Determine whether a given set of vectors is orthogonal, orthonormal, or neither.
   20. Determine whether given subspaces are orthogonal.
   21. Find the orthogonal complement of a given subspace.
   22. Apply the Gram-Schmidt orthonormalization process to find an orthonormal basis for a given basis, subspace, or inner product space.
   23. Find an orthonormal basis for the solution space of a homogeneous system of linear equations.
5. Linear Transformations
   1. Find the image and preimage of a given function.
   2. Determine whether a function from one vector space to another is a linear transformation.
   3. For a given linear transformation, find the kernel and range, find the basis for the kernel and range, and determine the nullity and rank.
   4. Determine whether a given linear transformation is one-to-one or onto.
   5. Determine whether two vector spaces are isomorphic.
   6. Find the standard matrix for a given linear transformation and the composition of linear transformations.
   7. Determine whether a given linear transformation is invertible and find its inverse if it exists.
   8. Know and use the properties of similar matrices.
6. Eigenvalues and Eigenvectors
   1. Verify an eigenvalue and an eigenvector of a given matrix.
   2. Understand the geometrical interpretation of the eigenvalue and eigenvector of a given matrix.
   3. Find the characteristic equation and the eigenvalues and corresponding eigenvectors of a given matrix.
   4. Determine whether a given matrix is diagonalizable, symmetric, or orthogonal.
   5. Find (if possible) a nonsingular matrix P for a given matrix A such that P-1AP is diagonal.
   6. Find a basis B (if possible) for the domain of a linear transformation T such that the matrix of T relative to B is diagonal.
   7. Find the eigenvalues of a given symmetric matrix and determine the dimension of the corresponding eigen space.
   8. Find an orthogonal matrix that diagonalizes a given matrix.
7. Applications
   1. Applications of the above objectives to geometry and other topics which could include differential equations, systems of linear differential equations, least squares approximations, Fourier approximations, population growth or models in dynamical systems.

Topical Outline with Specific Course Objectives

1. Linear and Nonlinear Diophantine Equations
   1. Solve Linear Diophantine equation in two variables.
   2. Represent a Primitive Pythagorean Triples with a unique pair of relatively prime integers.
   3. Investigate the historical background of Fermat's Last Theorem.
2. Primes and Greatest Common Divisions
   1. Investigate the distribution of prime numbers.
   2. Represent integers in different bases.
   3. Find the greatest common factor using the Euclidean Algorithm.
   4. Investigate different factorization methods, such as the sieve of Eratosthenes and Fermat factorization.
   5. Investigate the proof of the Fundamental Theorem of Arithmetic.
3. Congruence
   1. Solve systems of linear congruences.
   2. Solve systems of linear congruences with different moduli using the Chinese Remainder Theorem.
   3. Be able to factor using the Pollard Rho Method.
   4. Use Wilson's Theorem and Fermat's Little Theorem as the basis for primality tests and factoring algorithms.
   5. Investigate Pseudo-primes.
   6. Investigate Carmichael numbers.
   7. Develop divisibility tests.
   8. Describe how congruences are used to detect errors in strings of digits.
4. Multiplicative Functions
   1. Determine if a function is multiplicative using the Euler Phi-function.
   2. Find the value of the Euler-Phi function for integers.
   3. Investigate perfect numbers and Mersenne prime numbers and their connection.
   4. Explore the use of arithmetical functions, the Mobius function, and the Euler totient function.
   5. Investigate the Dirichlet product of arithmetical functions.
5. Cryptography
   1. Learn to encrypt and decrypt a message using character ciphers.
   2. Learn to encrypt and decrypt a message using Public-Key cryptology.

Topical Outline with Specific Course Objectives

1. Focus on Math Study Skills
   1. Develop awareness of preferred learning styles and learn to make adaptations for other instructional styles.
   2. Develop mathematical study skills that can be utilized in college-level math courses.
   3. Utilize support systems (faculty office hours, labs, SI, advising, etc.) to maximize learning opportunities.
   4. Utilize critical thinking skills to solve problems and determine reasonableness of solutions.
2. Arithmetic
   1. Classify sets of numbers
   2. Identify and apply the properties of real numbers
   3. Write the prime factorization of a number
   4. Perform operations with integers (with applications)
   5. Perform operations with fractions (with applications)
   6. Perform operations with decimals (with applications)
   7. Perform operations on whole number (with applications, including area and perimeter)
   8. Convert among percents, fractions, and decimals
   9. Identify place value and round decimals
   10. Identify place value and round whole numbers
   11. Simplify fractions
   12. Perform order of operations including absolute values
   13. Evaluate exponents with whole numbers
   14. Evaluate exponents with integers
   15. Compare magnitude of real numbers
3. Expressions and Equations
   1. Solve percent equations with applications
   2. Evaluate absolute value expressions
   3. Solve application problems involving geometry (circumference of circle, perimeter of polygons, area of triangles, parallelograms, circles)
   4. Solve formulas with given values
   5. Set up and solve ratios and proportions with simple algebraic expressions
   6. Covert units of measurements with same measurement system
   7. Define variables and write an expression to represent a quantity in a problem
   8. Evaluate algebraic expressions
   9. Simplify algebraic expressions involving one variable
   10. Solve linear equations involving the addition and multiplication property of equalities
   11. Graph an inequality on a number line

Department Policies

• Comprehensive Common Final
• No calculator of any kind

Topical Outline with Specific Course Objectives

1. Focus on Math Study Skills
   1. Develop awareness of preferred learning styles and learn to make adaptations for other instructional styles.
   2. Develop mathematical study skills that can be utilized in college-level math courses.
   3. Utilize support systems (faculty office hours, labs, SI, advising, etc.) to maximize learning opportunities.
   4. Utilize critical thinking skills to solve problems and determine reasonableness of solutions.
2. Arithmetic
   1. Classify sets of numbers
   2. Identify and apply the properties of real numbers
   3. Write the prime factorization of a number
   4. Perform operations with integers (with applications)
   5. Perform operations with fractions (with applications)
   6. Perform operations with decimals (with applications)
   7. Perform operations on whole number (with applications, including area and perimeter)
   8. Convert among percents, fractions, and decimals
   9. Identify place value and round decimals
   10. Identify place value and round whole numbers
   11. Simplify fractions
   12. Perform order of operations including absolute values
   13. Evaluate exponents with whole numbers
   14. Evaluate exponents with integers
   15. Compare magnitude of real numbers
   16. Convert between scientific notation and standard notation
3. Expressions and Equations
   1. Solve percent equations with applications
   2. Evaluate absolute value expressions
   3. Solve application problems involving geometry (circumference of circle, perimeter of polygons, area of triangles, parallelograms, circles)
   4. Solve formulas with given values
   5. Set up and solve ratios and proportions with simple algebraic expressions
   6. Covert units of measurements with same measurement system
   7. Define variables and write an expression to represent a quantity in a problem
   8. Evaluate algebraic expressions
   9. Simplify algebraic expressions involving one variable
   10. Solve linear equations involving the addition and multiplication property of equalities
   11. Solve multi-step problems involving fractions and percentages (include situations such as simple interest, tax, markups/markdowns, gratuities and commissions, fees, percent increase and decrease, percent error, expressing rent as a percentage of take home pay)
   12. Recognize proportional relationships and solve problems involving rates and ratios
   13. Apply the order of operations to evaluate algebraic expressions, including those with parentheses and exponents
   14. Solve application problems involving geometry (Pythagorean Theorem)
   15. Solve application problems involving geometry (perimeter and area with algebraic expressions)
   16. Convert units of measurements across measurement systems
   17. Solve literal equations for a given variable with applications (geometry, motion [d=rt], simple interest [i=prt]
   18. Solve linear equations in one variable using manipulations guided by the rules of arithmetic and the properties of equality
   19. Simplify an expression with integer exponents
   20. Simplify radical expressions – square roots only
4. Graphing Linear Equations and Inequalities
   1. Graph an inequality on a number line
   2. Identify the slope of a line (from slope formula, graph, and equation)
   3. Solve linear inequalities in one variable and graph the solution set on a number line
   4. Graph linear equations using table of values, intercepts, slope intercept form
   5. Identify the intercepts of a linear equation
5. Polynomials and Rational Expressions
   1. Add, subtract, multiply, and divide polynomials. Division by monomials only
   2. Add, subtract, and multiply square roots of monomials
   3. Factor polynomial expressions (GCF, grouping, trinomials, difference of squares)
   4. Solve quadratic equations in one variable by factoring
   5. Rationalize the denominator (monomials only)
   6. Simplify, multiply, and divide rational expressions
   7. Add and subtract rational expressions with monomial denominators

Department Policies

• Comprehensive common final
• Four function basic calculator allowed

Topical Outline with Specific Course Objectives

1. Focus on Math Study Skills
   1. Develop awareness of preferred learning styles and learn to make adaptations for other instructional styles.
   2. Develop mathematical study skills that can be utilized in college-level math courses.
   3. Utilize support systems (faculty office hours, labs, SI, advising, etc.) to maximize learning opportunities.
   4. Utilize critical thinking skills to solve problems and determine reasonableness of solutions.
2. Arithmetic, Expressions and Equations
   1. Convert between scientific notation and standard notation
   2. Convert units of measurements across measurement systems
   3. Solve multi-step problems involving fractions and percentages (include situations such as simple interest, tax, markups/markdowns, gratuities and commissions, fees, percent increase and decrease, percent error, expressing rent as a percentage of take home pay)
   4. Recognize proportional relationships and solve problems involving rates and ratios
   5. Apply the order of operations to evaluate algebraic expressions, including those with parentheses and exponents
   6. Solve application problems involving geometry (Pythagorean Theorem)
   7. Solve application problems involving geometry (perimeter and area with algebraic expressions)
   8. Solve literal equations for a given variable with applications (geometry, motion [d=rt], simple interest [i=prt]
   9. Solve linear equations in one variable using manipulations guided by the rules of arithmetic and the properties of equality
   10. Simplify an expression with integer exponents
   11. Simplify radical expressions – square roots only
3. Graphing Linear Equations and Inequalities
   1. Identify the slope of a line (from slope formula, graph, and equation)
   2. Solve linear inequalities in one variable and graph the solution set on a number line
   3. Graph linear equations using table of values, intercepts, slope intercept form
   4. Identify the intercepts of a linear equation
4. Polynomials and Rational Expressions
   1. Add, subtract, multiply, and divide polynomials. Division by monomials only
   2. Add, subtract, and multiply square roots of monomials
   3. Factor polynomial expressions (GCF, grouping, trinomials, difference of squares)
   4. Solve quadratic equations in one variable by factoring
   5. Rationalize the denominator (monomials only)
   6. Simplify, multiply, and divide rational expressions
   7. Add and subtract rational expressions with monomial denominators

Department Policies

• Comprehensive common final
• Four function basic calculator allowed

Topical Outline with Specific Course Objectives

1. Focus on Math Study Skills
2. Review
   1. Demonstrate an ability to factor algebraic expressions into primes using techniques of removing common factors, and factoring the difference of squares and trinomials.
   2. Use the properties of inequalities and equivalent inequalities to solve linear inequalities in one variable and express the solutions graphically or in interval notation.
3. Linear Equations and Inequalities in Two Variables
   1. Use tables and graphs as tools to interpret expressions, equations, and inequalities.
   2. Locate the x and y intercepts graphically and algebraically and interpret them in the context of the problem
   3. Explain and determine the slope of a line as the ratio of change in the dependent variable with respect to change in the independent variable.
4. Systems of Linear Inequalities and their Graphs
   1. Connect the solution set of a system of two linear inequalities in two variables with the graphs of the two equations.
   2. Graph the solution set of a system of two linear inequalities in two variables.
5. Introduction to Functions
   1. Recognize functions in table, graph, equation or verbal form.
   2. Understand that for a function one input value results in one output value.
   3. Determine the acceptability of a value to be used for the independent variable in an equation that defines a function.
   4. Determine the domain and range of a relation from a graph.
   5. Use and understand functional notation.
6. Linear Functions and Their Applications
   1. Express linear functions in table, graph, equation, or verbal form.
   2. Make connections between the parameters of a function and the behavior of the function.
   3. Recognize that a variety of problem situations can be modeled by the same type of function.
   4. Use patterns and functions to represent and solve problems.
   5. Extract and interpret information presented in a graph.

Department Policies

• Graphing Calculator Required
• Comprehensive Common Final

Topical Outline with Specific Course Objectives

1. Review
   1. Demonstrate an ability to factor algebraic expressions into primes using techniques of removing common factors, and factoring the difference of squares and trinomials.
   2. Use the properties of inequalities and equivalent inequalities to solve linear inequalities in one variable and express the solutions graphically or in interval notation.
2. Rational Expressions and Equations
   1. Evaluate rational expressions, and use prime factorization to reduce simple rational expressions (decreased emphasis).
   2. Use the properties of equalities and equivalent equalities to solve rational equations; apply to word problems involving ratios and proportions.
3. Radicals and Rational Exponents
   1. Demonstrate the relationship between exponents and radicals.
   2. Use the properties of radicals to simplify simple radicals.
   3. Use the properties of equality to solve equations involving one radical expression.
4. Quadratic Equations
   1. Recognize a quadratic equation; choose and apply the most efficient method to solve it.
   2. Apply skills to word problems involving quadratic equations.
5. Linear Equations and Inequalities in Two Variables
   1. Use tables and graphs as tools to interpret expressions, equations, and inequalities.
   2. Locate the x and y intercepts graphically and algebraically and interpret them in the context of the problem.
   3. Explain and determine the slope of a line as the ratio of change in the dependent variable with respect to change in the independent variable.
6. Systems of Linear Equations and Inequalities and their Graphs
   1. Connect the solution set of a system of two linear equations in two variables with the graphs of the two equations.
   2. Connect the solution set of a system of two linear inequalities in two variables with the graphs of the two inequalities.
7. Introduction to Functions
   1. Recognize functions in table, graph, equation or verbal form.
   2. Understand that for a function one input value results in one output value.
   3. Determine the acceptability of a value to be used for the independent variable in an equation that defines a function.
   4. Determine the domain and range of a relation from a graph.
   5. Use and understand functional notation.
8. Linear Functions and Their Applications
   1. Express linear and quadratic functions in table, graph, equation, or verbal form.
   2. Make connections between the parameters of a function and the behavior of the function.
   3. Recognize that a variety of problem situations can be modeled by the same type of function.
   4. Use patterns and functions to represent and solve problems.
   5. Extract and interpret information presented in a graph.

Department Policies

• Graphing Calculator Required
• Comprehensive Common Final

Topical Outline with Specific Course Objectives

1. Review of MAT 1032 material
2. Focus on math study skills
3. Systems of Linear Equations and their Graphs
   1. Connect the solution set of a system of two linear equations in two variables with the graphs of the two equations.
   2. Graph the solution set of a system of two linear equations in two variables
   3. Find the solution to a system of two linear equations algebraically and graphically
4. Rational Expressions and Equations
   1. Evaluate rational expressions, and use prime factorization to reduce simple rational expressions (decreased emphasis).
   2. Use the properties of equalities and equivalent equalities to solve rational equations; apply to word problems involving ratios and proportions.
5. Radicals and Rational Exponents
   1. Demonstrate the relationship between exponents and radicals.
   2. Use the properties of radicals to simplify simple radicals.
   3. Use the properties of equality to solve equations involving one radical expression.
6. Quadratic Equations
   1. Recognize a quadratic equation; choose and apply the most efficient method to solve it.
   2. Apply skills to word problems involving quadratic equations.
   3. Express quadratic functions in table, graph, equation, or verbal form.
   4. Make connections between the parameters of a function and the behavior of the function.
   5. Recognize that a variety of problem situations can be modeled by the same type of function.
   6. Use patterns and functions to represent and solve problems.
   7. Extract and interpret information presented in a graph.

Department Policies

• Graphing Calculator Required
• Comprehensive Common Final

Topical Outline with Specific Course Objectives

1. Study Strategies and Techniques
   1. Develop critical thinking skills and techniques to become problem-solvers and logical thinkers.
   2. Use number sense to determine if a proposed answer to a mathematics problem is appropriate in that context.
   3. Check that a proposed answer to a problem is correct.
   4. Take class notes in an efficient manner.
2. Arithmetic
   1. Demonstrate an understanding of which numbers are in the natural numbers, whole numbers, integers, rational numbers, irrational numbers, and real numbers.
   2. Round numbers to the specified place.
   3. Demonstrate understanding of the relationship between fractions, decimals, and percentages.
      1. Convert fractions to decimals and percentages.
      2. Convert decimals to percentage and fractions.
      3. Convert percentages to decimals and fractions.
   4. Correctly perform the following operations with signed real numbers.
      1. Addition
      2. Subtraction
      3. Multiplication
      4. Division
      5. Use whole number exponents
      6. Any combination of the above operations
      7. Extra focus on arithmetic with fractions
   5. Demonstrate appropriate use of order of operations in the following areas.
      1. Solving linear equations
      2. Arithmetic with signed numbers
      3. Simplifying algebraic expressions
      4. Graph inequalities on a number line.
3. Linear Patterns and Their Connection to a Graph
   1. Find a pattern in a given table and fill in blanks in a table as well as write a formula representing the relation in the table.
   2. Plot points given in a table on the cartesian plane. Draw a smooth curve through those points.
   3. Select reasonable input values for a formula and calculate their output values.
4. Linear Functions and Equations
   1. Distinguish between algebraic expressions and equations.
   2. Simplify algebraic expressions by combining like terms.
   3. Determine if an equation is linear.
   4. Calculate the slope of a linear equation in two variables.
      1. Including horizontal and vertical lines
      2. Solve linear equations in one variable.
   5. Graph linear equations (diagonal, horizontal, and vertical) in two variables on the cartesian plane in the following ways.
      1. Plotting points
      2. Plotting the intercepts
      3. Plotting by using a point and the slope
   6. Find an equation of a line given a graph.
   7. Make connections between the parameters of a function and the behavior of the function.
   8. Recognize that a variety of problem situations can be modeled by the same type of function.
   9. Apply skills to word problems involving linear equations.
5. Linear Inequalities in Two Variables
   1. Graph a linear inequality on the cartesian plane.
   2. Connect the solution set of a linear inequality with the graph of the linear inequality.
6. Systems of Linear Equations and their Graphs
   1. Recognize the three cases for systems of two linear equations in two variables.
   2. Connect the solution set of a system of two linear equations in two variables with the graphs of the two equations.
   3. Solve a system of two linear equations in two variables using a graph.

Department Policies

• NO Calculators allowed
• Comprehensive Common Final

Topical Outline with Specific Course Objectives

1. Inductive and Deductive Reasoning
   1. Use inductive reasoning to reason to a general conclusion through observations of specific cases.
   2. Use deductive reasoning to reason to a specific conclusion from a general statement.
2. Sets and Venn Diagrams
   
   1. Indicate sets by description, roster form, and set-builder notation.
   2. Categorize equal and equivalent sets.
   3. Use symbols to illustrate the relationship between sets and to distinguish elements from subsets.
   4. Construct Venn diagrams to illustrate set relationships.
   5. Determine new sets from old using any combination of the set operations complement, intersection, and union. (The difference of two sets and the Cartesian product are optional.)
   6. Use Venn diagrams to determine if statements involving sets are equal.
   7. Solve practical problems using sets.
3. Logic
   1. Write negations of statements, including ones that contain quantifiers such as all, none, and some.
   2. Translate verbal compound statements (not, and, or, if-then, and if and only if) into symbolic form, and vice versa.
   3. Construct truth tables for compound statements.
   4. Determine if statements are logically equivalent.
   5. Use De Morgan’s laws to write equivalent statements.
   6. Use variations of the conditional (converse, inverse, and contrapositive) to write equivalent statements.
   7. Determine whether symbolic arguments are valid or invalid.
   8. Use Euler diagrams to determine whether syllogistic arguments are valid or invalid.
4. Geometry
   1. Classify lines (intersecting, parallel, perpendicular, skew) and angles (right, acute, obtuse, straight).
   2. State and apply the basic relationships between special pairs of angles such as complementary, supplementary, vertical, alternate interior, alternate exterior, and corresponding.
   3. Classify polygons including special types of triangles (acute, obtuse, right, isosceles, equilateral, scalene) and quadrilaterals (trapezoid, parallelogram, rhombus, rectangle, square).
   4. Use proportions to find missing lengths in similar figures.
   5. Calculate the perimeter and area of two-dimensional figures such as triangles, squares, rectangles, trapezoids, and circles.
   6. State and apply the Pythagorean theorem.
   7. Classify three-dimensional figures including special types of polyhedra (platonic solids, prisms, pyramids).
   8. Calculate the volume of three-dimensional figures such as rectangular solids, cubes, cylinders, cones, and spheres.
   9. Calculate the surface area of three-dimensional figures such as rectangular solids, cubes, cylinders, and spheres.
5. Probability
   1. Determine empirical probabilities.
   2. Determine theoretical probabilities using the definition.
   3. Use the fact that the sum of the probabilities of all possible outcomes of an experiment is 1.
   4. Write the sample space for an experiment.
   5. Use tree diagrams to determine probabilities.
   6. Find the probability of event A or event B.
   7. Find the probability of event A and event B.
   8. Solve probability problems by using combinations.
6. Counting Principles
   1. Use the formula n(A ∪ B) = n(A) + n(B) - n(A ∩ B).
   2. Count the number of items or events by listing them.
   3. State and apply the Fundamental Counting Principle.
7. Permutations and Combinations
   1. Evaluate factorials.
   2. Define the terms permutation and combination.
   3. Count permutations.
   4. Calculate a specified permutation.
   5. Count combinations
   6. Calculate a specified combination
8. Descriptive Statistics
   1. Organize and display data using tools such as frequency distributions, histograms, frequency polygons, and stem-and-leaf displays.
   2. Find and interpret measures of central tendency such as mean, median, and mode. Midrange is optional.
   3. Find and interpret measures of dispersion such as range and standard deviation.
   4. Apply the empirical rule which states the following. In any normal distribution,
      • Approximately 68% of all the data lies within one standard deviation of the mean (in both directions).
      • Approximately 95% of all the data lies within two standard deviations of the mean (in both directions).
      • Approximately 99.7% of all the data lies within three standard deviations of the mean (in both directions).
   5. Determine the relative relationships between the mean, median, and mode for various distributions.

Department Policies

• Comprehensive Final Assessment

Topical Outline with Specific Course Objectives

The exact topics to be covered will be determined by the three topics chosen by the instructor to cover in this course. Specific objectives are listed underneath the seven topics, with the numbers in parentheses indicating the topic(s) for which these objectives will be attained.

Three of these seven topics will be included in the course.

1. Exploratory data analysis
2. Sampling and decision making
3. Modeling - linear, quadratic and exponential
4. Financial mathematics
5. History of mathematics
6. Patterns, network and game theory
7. Mathematics in art, nature and daily life

For each topic chosen, the instructor will develop the ideas at a level appropriate to this class, and will emphasize the following concepts and skills. Concepts with numbers in parentheses will be covered under those topics; concepts without numbers illustrate the approach of the class and may or may not be specifically addressed:

• Analyze real data sets using tools such as histograms, timelines, boxplots, tables, and descriptive statistics. (1, 2)
• Prepare a report based on an analysis of a real data set. (1,2)
• Understand basic types of sampling in opinion polls and interpret sampling error. (1,2)
• Recognize abuses and manipulations of data that occur in polling. (1,2)
• Determine whether data can be modeled by functions such as linear, quadratic, or exponential. (3)
• Assess the model to determine its usefulness. (3)
• Use exponential functions to model data such as population growth and radioactive decay. (3)
• Make informed choices in areas such as credit card and long distance carrier offers. (4)
• Make informed choices in areas such as loans, mortgages, insurance, and investment plans. (4)
• Understand simple, compound, and effective interest rates; discounts, mark-ups and markdowns. (4)
• Understand and appreciate man's evolving understanding of mathematics. (5)
• Understand other systems of numeration and modulus arithmetic. (5)
• Recognize the patterns or mathematics involved in man-made or natural occurrences: Golden ratio, Fibonacci sequence, tessellation, fractals. (6,7)
• Recognize the patterns or mathematics involved in many games and recreations; network theory, game theory, magic squares, billiards. (6,7)
• Choose appropriate methods for solving problems: tables, graphs, equations, guess-and check, and working backwards.
• Make an educated “guess” as to the reasonableness of results.
• Take basic concepts and apply them in situations that have not previously been specifically demonstrated.
• Explain the process used in solving the problem.
• Explain what the answer means in the context of the problem.
• Recognize patterns or relationships and express them in clear sentences that will be meaningful to another student.
• Solve problems and recognize the relationship among information presented numerically, algebraically, and graphically.
• Express generalizations discovered through investigations.
• Ask clarifying and extending questions related to the mathematics at hand.

The following teaching techniques, which reflect the goals of the Math Department, will be especially encouraged in this course.

• Emphasize techniques that encourage the student to become an active, involved learner: less lecture, more group and cooperative learning, interesting and challenging problems, student presentations.
• Emphasize problem solving & mathematical modeling; less emphasis on manipulation for its own sake.
• Cover fewer topics but in deeper and more interesting ways.
• Use textbooks which reflect the NCTM and AMATYC standards.
• Use appropriate technology.

Department Policies

• Comprehensive Final

Topical Outline with Specific Course Objectives

1. Introduction to Basic Concepts of Logic
   1. Recognize an argument
      1. Determine the premises
      2. Determine the conclusion
   2. Recognize deductive arguments
      1. Valid
         1. Sound
         2. Unsound
      2. Invalid (and unsound)
   3. Recognize Inductive Arguments
      1. Strong
         1. Cogent
         2. Uncogent
      2. Weak (and uncogent)
2. Informal Fallacies
   1. Recognize instances of fallacies of relevance including:
      1. Argumentum ad baculum
      2. Argumentum ad hominem
      3. Argumentum ad populum
      4. Argumentum ad misericordiam
      5. Argumentum ad ignorantiam
      6. Argumentum ad verecundiam
      7. Accident
      8. Converse Accident
      9. False Cause
      10. Petitio Principii
      11. Complex Question
      12. False Dichotomy
   2. Recognize instances of fallacies of ambiguity, including:
      1. Equivocation
      2. Amphiboly
      3. Composition
      4. Division
3. Categorical Propositions
   1. Recognize and classify a standard-form categorical proposition as to:
      1. Form
      2. Quantifier
      3. Copula
      4. Subject term
      5. Predicate term
      6. Quantity
      7. Quality
      8. Distribution or non-distribution of terms by the proposition
   2. Use the traditional square of opposition to form valid inferences using the relationships of:
      1. Contrary
      2. Subcontrary
      3. Sub-alternation
      4. Contradictory
   3. Recognize and perform the immediate inference operations
      1. Conversion
      2. Obversion
      3. Contraposition
   4. Use the modern square of opposition
      1. Make inferences using the relationship of contradictory.
      2. Recognize the existential fallacy.
   5. Use 2-class Venn diagrams to:
      1. Visually represent the class relationship expressed in each categorical proposition.
      2. Validate the relationships found in the squares of opposition.
      3. Validate the inferences of conversion, obversion and contraposition.
4. Categorical Syllogisms
   1. Recognize a categorical syllogism and put it in standard form by:
      1. Recognizing the major, minor and middle terms.
      2. Recognizing the major and minor premises, and the conclusion.
   2. Classify a standard-form categorical syllogism as to:
      1. Mood
      2. Figure
      3. Test a given mood-figure form for validity
   3. Test a syllogism for validity by:
   4. 1. Venn diagram method
      2. Method of the syllogistic rules
      3. Numerical or other method
   5. Test the validity of:
   6. 1. Categorical arguments that can be reduced to logically equivalent standard-form syllogisms
      2. Categorical arguments that can be evaluated using diagrams representing four and five categories
      3. Sorites and/or enthymemes
5. Truth-Functional Logic
   1. Symbolize any combination of the five (5) types of compound statements (below) in propositional logic.
      1. Conjunction
      2. Disjunction
      3. Conditional
      4. Bi-conditional
      5. Negation
   2. Apply the definitional truth tables for each of the five (5) truth functional connectives in order to:
      1. Assess the truth-value of a statement based on the given values of its atomic components.
      2. Test the logical equivalence of two statement forms using truth tables.
      3. Classify statement forms as to contingent truths, logical truths or self-contradictions by means of truth tables.
      4. Test argument forms for validity by means of truth tables.
      5. Use indirect (shortcut) truth tables to test validity of argument forms.
      6. Test the consistency of sets of premises by the indirect truth table method.
6. Formal Deductive Proof
   1. Recognize instances of any of the following nine rules of inference:
      1. Modus ponens
      2. Modus tollens
      3. Disjunctive syllogism
      4. Hypothetical syllogism
      5. Constructive dilemma
      6. Simplification
      7. Addition
      8. Conjunction
      9. Absorption
   2. Recognize instances of the use of the rule of replacement with any of the following ten tautologous bi-conditionals:
      1. DeMorgan's theorems
      2. Commutation
      3. Association
      4. Distribution
      5. Double negation
      6. Transposition
      7. Material implication
      8. Material equivalence
      9. Exportation
      10. Tautology
   3. Recognize both the specific form and any non-specific forms of an argument:
      1. Given the argument in verbal form.
      2. Given the argument in symbolic form.
   4. Use the method of deduction and the 19 rules of deduction to formulate:
      1. Direct proofs
      2. Indirect proofs
      3. Conditional proofs
      4. Natural deduction of tautologies
      5. Proofs involving inconsistent premises

Department Policies

• Comprehensive Final

Topical Outline with Specific Course Objectives

1. Descriptive Statistics
   1. Calculate and interpret the various descriptive measures for centrality and dispersion.
   2. Determine potential outliers of data sets and understand how they affect the various numerical measures.
   3. Analyze and/or compare different sets of data using graphs, charts, tables, and numerical measures, and write about them in clear and precise sentences using statistical vocabulary.
   4. Demonstrate an understanding of the different types of distributions.
   5. Organize and display data by means of various tables, charts, and graphs.
   6. Define and use the basic terminology of statistics.
2. Simple Linear Regression and Correlation
   1. Find and interpret the sample correlation coefficient (r) to determine the strength and direction of the linear relationship between predictor and response variables.
   2. Use scatter plots to determine if outliers are present and if data can be represented by a simple linear regression model.
   3. Find the simple linear regression model and be able to interpret the slope and y-intercept.
   4. Use r-squared to determine if a simple linear regression model is a strong predictor.
   5. Predict values of “y” using the simple linear regression model
3. Normal Probability Distribution
   1. Calculate and interpret a z-score as a measure of relative standing and use it as it applies to the normal model.
   2. Understand the Normal Probability Distribution and be able to determine appropriate areas under a normal curve.
   3. Use the 68-95-99.7 Rule (Empirical Rule) to find probabilities on an approximately normal or bell-shaped distribution.
   4. Use a histogram or normal probability plot, determine if a sample comes from a normally distributed population.
4. Fundamentals of Probability
   1. Understand and apply basic rules of probability.
   2. Understand and apply the Binomial Probability Distribution.
   3. Identify the random variable involved in a statistical problem and distinguish between: categorical vs quantitative, discrete vs continuous, and binomial vs normal.
5. Inferential Statistics
   1. Demonstrate at least a rudimentary understanding of basic sample and experimental design (i.e. randomness, bias, etc. ).
   2. Understand and apply sampling distribution models for sample proportions and sample means
   3. Understand and apply the Central Limit Theorem.
   4. Estimate means and proportions using confidence intervals for one and two populations.
   5. Be able to perform hypothesis tests on means and proportions for one and two populations.
   6. Determine and interpret p-values.
6. Demonstrate competency in the use of technology, including graphing calculator and/or statistical computer software as it applies to topics I – V.

Department Policies

• Graphing Calculator Required
• Comprehensive Final