Common Syllabi for Mathematics Courses
All sections for any course offered in the calculus & pre-calculus sequence share a common syllabus, listed below.
Common Syllabus for MATH 100
Chapter 1: Basic Concepts [1.5 Weeks]
• Real numbers
• Order of operations
• Scientific notation
Chapter 2: Equations and Inequalities [2.5 Weeks]
• Solving linear equations
• Story problems including (rate)(time)=distance and mixture problems
• Solving linear inequalities
• Solving equations and inequalities containing absolute values
Chapter 3: Graphs and Functions [2.5 Weeks]
• Graphs, Functions
• Linear Functions, Graphs and story problems
• Slope-intercept form of a linear equation
• Point-slope form of a linear equation
• Algebra of functions
Chapter 4: Systems of Equations and Inequalities [1 Week]
• Solving systems of linear equations in two variables by substitution and by elimination
• Story problems
Chapter 5: Polynomials and Polynomial and Functions [3 Weeks]
• Addition, subtraction and multiplication of polynomials
• Division of polynomials (not including synthetic division)
• Remainder theorem
• Factoring methods (as time permits)
Chapter 6: Rational Expressions and Equations [1.5 Weeks]
• Domains, addition, subtraction, multiplication and division of rational expressions
• Work and rate story problems
Chapter 7: Radicals and Complex Numbers [2 Weeks]
• Roots and radicals, rational exponents
• Simplifying radicals
• Adding, subtracting and multiplying radicals (as time permits)
• Rationalizing denominators
Sample Syllabus for MATH 108
Part I - Management Science
Chapter 1: Urban Services [0.5 Weeks]
Euler Circuits, Finding Euler Circuits, Circuits with Reused Edges
Chapter 2: Business Efficiency [1 Week]
Hamiltonian Circuits, Fundamental Principle of Counting, Traveling Salesman Problem, Strategies for Solution, Nearest-Neighbor Algorithm, Sorted-Edges Algorithm, Minimum-Cost Spanning Trees, Kruskal's Algorithm
Chapter 3: Planning and Scheduling [1 Week]
Scheduling Tasks, Assumptions and Goals, List-Processing Algorithm, When is a Schedule Optimal?, Strange Happenings, Critical-Path Schedules, Independent Tasks, Decreasing-Time Lists
Chapter 4: Linear Programming [1 Week]
Mixture Problems, Mixture Problems Having One Resource, One Product and One Resource: Making Skateboards, Common Features of Mixture Problems, Two Products and One Resource: Skateboards and Dolls, Mixture Charts, Resource Constraints, Graphing the Constraints to Form the Feasible Region, Finding the Optimal Production Policy, General Shape of Feasible Regions, The Role of the Profit Formula: Skateboards and Dolls, Setting Minimum Quantities for Products: Skateboards and Dolls, Drawing a Feasible Region When There are Nonzero Minimum Constraints, Finding Corner Points of a Feasible Region Having Nonzero Minimums, Evaluating the Profit Formula at the Corners of a Feasible Region with Nonzero Minimums, Summary of the Pictorial Method, Mixture Problems Having Two Resources, Two Products and Two Resources: Skateboards and Dolls, The Corner Point Principle, Linear Programming: The Wider Picture, Characteristics of Linear Programming Algorithms, The Simplex Method, An Alternative to the Simplex Method
Part III - Voting and Social Choice
Chapter 9: Social Choice: The Impossible Dream [1.5 Weeks]
Elections with Only Two Alternatives, Elections with Three or More Alternatives: Procedures and Problems, Plurality Voting and the Condorcet Winner Criterion, The Borda Count and Independence of Irrelevant Alternatives, Sequential Pairwise Voting and the Pareto Condition, the Hare System and Monotonicity, Insurmountable Difficulties: From Paradox to Impossibility, The Voting Paradox of Condorcet, Impossibility, A Better Approach? Approval Voting
Chapter 11: Weighted Voting Systems [2 Weeks]
How Weighted Voting Works, Notation for Weighted Voting, The Banzhaf Power Index, How to Count Combinations, Equivalent Voting Systems, The Shapley-Shubik Power Index, How to Compute the Shapley-Shubik Power Index, Comparing the Banzhaf and Shapley-Shubik Models
Part IV - Fairness and Game Theory
Chapter 13: Fair Division [1.5 Weeks]
The Adjusted Winner Procedure, The Knaster Inheritance Procedure, Divide-and-Choose, Cake-Division Procedures: Proportionality, Cake-Division Procedures: The Problem of Envy
Chapter 14: Apportionment [1.5 Weeks]
The Apportionment Problem, The Hamilton Method, Paradoxes of the Hamilton Method, Divisor Methods, The Jefferson Method, Critical Divisors, The Webster Method, The Hill-Huntington Method, Which Divisor Method is the Best?
Chapter 15: Game Theory: The Mathematics of Competition [as time permits]
Two-Person Total-Conflict Games: Pure Strategies, Two-Person Total-Conflict Games: Mixed Strategies, A Flawed Approach, A Better Idea, Partial-Conflict Games, Larger Games, Using Game Theory, Solving Games, Practical Applications
Part V - The Digital Revolution
Chapter 16: Identification Numbers [1 Week]
Check digits, the Zip Code, Bar Codes, Encoding Personal Data
Chapter 17: Transmitting Information [1.5 Weeks]
Binary Codes, Encoding with Parity-Check Sums, Data Compression, Cryptography
Note: Instructors may vary the topics covered, and length of time devoted to each. Material shall be selected from:
- Management Science (Street Networks, Visiting Vertices, Planning and Scheduling, Linear Programming)
- Coding Information (Identification Numbers, Transmitting Information)
- Social Choice and Decision Making (Social Choice: The Impossible Dream, Weighted Voting Systems, Fair Division, Apportionment, Game Theory: The Mathematics of Competition)
- On Size and Shape (Growth and Form, Symmetry and Patterns, Tilings), Modeling in Mathematics (Logic and Modeling, Consumer Finance Models)
Common Syllabus for MATH 117
Chapter 2. Combining Algebra and Geometry [1.5 Weeks]
2.1 - (Cover very quickly.) The coordinate plane: coordinates, graphs of equations, distance formula, length, perimeter, circumference.
2.2 – (Cover very quickly.) Lines: slope, equation of a line, parallel/perpendicular lines, midpoints.
2.3 – Quadratic Expressions and Conic Sections: completing the square, quadratic formula, circles, ellipses, hyperbolas. Foci for ellipses/hyperbolas optional.
2.4 – (Cover very quickly.) Area: squares, rectangles, parallelograms, triangles, trapezoids, stretching, circles and ellipses.
Chapter 3. Functions and their Graphs [2.5 Weeks]
3.1 – Functions: definition, graphs, domain, range, tables.
3.2 – Function transformations and graphs: vertical/horizontal shifts, stretches, flips, combinations of transformations, even/odd functions.
3.3 – Composition of Functions: definition, importance of order, decomposing functions, composing 3 or more functions, transformations as functions.
3.4 – Inverse Functions: definition, one-to-one functions, domain/range of inverse functions, composition of a function and its inverse, importance of notation.
3.5 – A graphical approach to inverse functions: graph of inverse functions, horizontal line test, increasing/decreasing function, inverses via tables.
Chapter 4. Polynomial and Rational Functions [2.5 Weeks]
4.1 – Integer exponents: positive integer exponents, properties of exponents, negative integer exponents.
4.2 – Polynomials: degree, algebra of polynomials, zeros, factorization, behavior as x approaches positive/negative infinity, graphs.
4.3 – Rational functions: definition, algebra of rational functions, polynomial division, behavior as x approaches positive/negative infinity, graphs.
4.4 – Complex numbers: complex number system, arithmetic/algebra of complex numbers, conjugates, division, relation to zeros and factorization.
Chapter 5. Exponents and Logarithms [3 Weeks]?
5.1 – Exponents and exponential functions: roots, rational exponents, real exponents, exponential functions.
5.2 – Logarithms as inverses of exponential functions: logarithms with arbitrary base, common logarithms, number of digits, logarithm of a power, decay/half-life problems.
5.3 – Applications of Logarithms: logarithm of products/quotients, change of base, richter scale, decibels. Apparent magnitude and sound intensity optional.
5.4 – Exponential growth: functions with exponential growth, population growth, compound interest.
Chapter 6. e and the Natural Logarithm [2 Weeks]
6.1 – Defining e and ln: estimating area using rectangles, definitions of e and ln, properties of ln.
6.2 (Optional) – Approximations of e and ln, and an area formula.
6.3 – Exponential growth revisited: continuous compounding of interest, continuous growth rates, doubling time.
Chapter 7. Systems of Equations [1 Week]
7.1 – Equations and systems of equations: solving an equation, solving a system graphically, solving a system by substitution.
7.2 – Solving systems of linear equations: linear equations and number of solutions, systems of linear equations, Gaussian elimination.
Common Syllabus for MATH 118
Review. Prerequisite Material from MATH 117 [1.5 Weeks]
Quick review of algebra, lines, circles, quadratic expressions, and functions, followed by a more comprehensive review of the definitions and properties of exponential functions and logarithms. Exponential growth modeling can be covered lightly.
Chapter 8. Sequences, Series, and Limits [2 Weeks]
8.1 – Sequences: definition of sequence, arithmetic/geometric sequences, recursively defined sequences.
8.2 – Series: sums of sequences, definition of series, arithmetic/geometric series. Emphasize: summation notation. Binomial theorem is optional.
8.3 – Limits: introduction to limits, infinite series, decimals as series, special series.
Chapter 9. Trigonometric Functions [3 Weeks]
9.1 – The unit circle: equation of unit circle, angles, negative angles, angles greater than 360 degrees, arc length, special points on unit circle.
9.2 – Radians: motivation of radians, radius corresponding to an angle, arc length revisited, area of slices, special points on unit circle revisited.
9.3 – Cosine and sine: definition of cosine and sine, signs of cosine and sine, pythagorean identity, graphs of cosine and sine.
9.4 – More trigonometric functions: tangent, sign of tangent, connections between cosine, sine, and tangent, graph of tangent, definitions of cotangent, secant, cosecant.
9.5 – Trigonometry in right triangles: definition of trigonometric functions via right triangles, two sides of a right triangle, one side and one angle of a right triangle
9.6 – Trigonometric identities: relationship between cosine, sine, tangent, identities for negative angles, identities involving pi/2, identities involving multiples of pi
Chapter 10. Trigonometric Algebra and Geometry [3 Weeks]
10.1 – Inverse trigonometric functions: arccosine, arcsine, and arctangent functions.
10.2 (Optional) – Inverse trigonometric identities, graphical and algebraic approach to evaluation at –t
10.3 – Using trigonometry to compute area: area of triangle/parallelogram via trigonometry, ambiguous angles, areas of polygons, trigonometric approximations.
10.4 – Law of Sines and Law of Cosines: statement and uses of laws of sines/cosines, when to use which law.
10.5 – Double-Angle and Half-Angle Formulas: sine/cosine double-angle and half-angle formulas. The corresponding formulas for tangent are optional.
10.6 – Addition and subtraction formulas: sine/cosine/ sum and difference formulas. The corresponding formulas for tangent are optional.
Chapter 11. Applications of Trigonometry [2.5 Weeks]
Suggestion: If time is short, 11.1 is optional. Focus on 11.2, 11.3, and 11.5.
Suggestion: Quickly review Chapter 3, Section 2 before covering 11.2
11.1 (Optional) – Parametric curves: curves in the plane, inverse functions as parametric curves, shifts/flips of parametric curves. Stretches of parametric curves is optional.
11.2 – Transformations of trigonometric functions: amplitude, period, phase shift, modeling periodic phenomena, modeling with data.
11.3 Polar Coordinates: Definition of polar coordinates, conversion between polar/rectangular coordinates, graphs of circles and rays. Other polar graphs are optional.
11.4 (Optional) – Algebraic and geometric introduction to vectors, addition and subtraction, scalar multiplication, dot product.
11.5 – The complex plane: complex numbers as points in the plane, geometric interpretation of multiplication/division of complex numbers, De Moivre's theorem, finding complex roots.
Common Syllabus for MATH 131
1.1 – Functions and Change
1.2 – Exponential Functions
1.3 – New Functions from Old
- Skip: the subsection on Shifts and Stretches and Odd and Even Symmetry.
- Review of composite and inverse functions, framed in a practical, not merely algebraic, setting.
1.4 – Logarithmic Functions
1.5 – Trigonometric Functions
1.6 – Powers, Polynomials, and Rational Functions
- Skip: the subsection on rational functions (pg 49-50).
- Power functions, graph properties for both positive and negative exponents.
- Comparison of long-run behavior of exponentials and polynomials.
1.7 – Introduction to Continuity
- Skip: the majority of the section.
- Understand the graphical viewpoint of continuity. (are there holes, breaks, or jumps in the graph?)
1.8 – Limits
- Skip: the subsection “Definition of Limit” (bottom of pg 58 – 59).
- Skip: the subsection “Definition of Continuity” (see comment on 1.7 above).
- Understand the concept, notation, and properties of limits and one-sided limites at a point and limits at infinity.
Chapter 2 – Key Concept – The Derivative (1.5 weeks)
2.1 – How do we measure speed?
2.2 – The Derivative at a Point
- Emphasis on observing what happens to the value of the average rate of change as the interval gets smaller and smaller.
- Emphasis placed on visualizing the derivative as the slope of a tangent.
2.3 – The Derivative Function
- Focus on the conceptual and practical understanding of the derivative.
- Sketch the graph of f ’ given the graph of f.
2.4 – Interpretation of the Derivative
2.5 – The Second Derivative
Chapter 3 – Shortcuts to Differentiation (2.5 weeks)
3.1 – Powers and Polynomials
3.2 – The Exponential Function
- Emphasis on graphical, not epsilon-delta, definition of derivative.
- Add: differentiation rule for y=ln(x), from section 3.6.
3.3 – Product and Quotient Rules
3.4 – The Chain Rule
3.5 – The Trigonometric Functions
Chapter 4 – Using the Derivative (3 weeks)
4.1 – Using First and Second Derivatives
4.2 – Optimization
4.3 – Optimization and Modeling
4.4 – Families of Functions and Modeling
4.5 – Applications to Marginality
4.7 – L’Hopital’s Rule, Growth, and Dominance
Chapter 5 – Key Concept – The Definite Integral (2 weeks)
5.1 – How Do We Measure Distance Traveled?
- Skip: accuracy of estimates (pg 277)
5.2 – The Definite Integral
- Approximation using area and interpretion as accumulated change.
5.3 – The Fundamental Theorem and Interpretations
5.4 – Theorems About Definite Integrals
- Finding area between curves; using the definite integral to find an average.
- Skip: the subsection “Comparing Integrals”
Chapter 6 – Constructing Antiderivatives (1 week)
6.1 – Antiderivatives Graphically and Numerically
6.2 – Constructing Antiderivatives Analytically
Common Syllabus for MATH 132
Review. Prerequisite Material from MATH 131 [1 Week]
Derivatives and Definite Integrals.
Chapter 7. Antiderivatives [2 Weeks]
7.1 – Constructing antiderivatives analytically.
7.2 – Integration by substitution.
7.3 – Using the fundamental theorem to find definite integrals.
7.4 – Integration by parts.
7.5 – Analyzing Antiderivatives graphically and numerically.
Chapter 6. Using the Definite Integral [2 Weeks]
6.1 – Average value.
6.2 – Consumer and producer surplus.
6.3 – Present and future value.
6.4 – Integrating relative growth rates.
Chapter 8. Probability [1 Week]
8.1 – Density functions.
8.2 – Cumulative distribution functions and probability.
8.3 – The median and the mean.
Chapter 9. Functions of Several Variables [3 Weeks]
9.1 – Understanding functions of two variables.
9.2 – Contour diagrams.
9.3 – Partial derivatives.
9.4 – Computing partial derivatives.
9.5 – Critical points and optimization.
9.6 – Constrained optimization.
Chapter 10. Mathematical Modeling Using Differential Equations [3 Weeks]
10.1 – Mathematical modeling: Setting up a differential equation.
10.2 – Solutions to differential equations.
10.3 – Slope fields.
10.4 – Exponential growth and decay.
10.5 – Applications to modeling.
10.6 – Modeling the interactions of two populations.
10.7 – Modeling the spread of a disease.
Common Syllabus for MATH 161
Chapter 1. Functions [1 week]
Functions and their graphs: identifying functions, mathematical models.
Combining functions; shifting and scaling graphs.
Graphing with calculators and computers (introduction to Mathematica).
Inverse functions and logarithms.
Optional: Hyperbolic functions.
Chapter 2. Limits and Continuity [1.5 weeks]
Rates of change and tangents to curves.
Calculating limits using the limit laws.
The precise definition of a limit.
One-sided limits, continuity.
Limits involving infinity: limits at infinity, infinite limits; asymptotes of graphs.
Chapter 3. Differentiation [4 weeks]
Tangents and the derivative at a point.
The derivative as a function.
Differentiation rules: for polynomials and exponentials; for products and quotients.
The derivative as a rate of change.
Derivatives of trigonometric functions.
The chain rule.
Derivatives of inverse functions and logarithms.
Inverse trigonometric functions.
Linearization and differentials.
Additional material (§§11.1, 11.2): parametric equations and their derivatives.
Derivatives of Hyperbolic Functions.
Chapter 4. Applications of Derivatives [3 weeks]
Extreme values of functions.
Rolle’s theorem and the mean value theorem.
Monotonic functions and the first derivative test.
Concavity and curve sketching.
Applied optimization problems.
Indeterminate forms and l’Hopital’s rule.
Chapter 5. Integration [4 weeks]
Area estimates with finite sums.
Sigma notation and limits of finite sums.
The definite integral.
The fundamental theorem of calculus.
Indefinite integrals and the substitution method.
Substitution and area between curves.
Common Syllabus for MATH 162
Review. Prerequisite Material from MATH 161 [1 Week]
Rapid review of differentiation rules.
More detailed review of integration (Ch. 5): areas and distances; the definite integral; the fundamental theorem of calculus.
Chapter 6. Applications of Definite Integrals [2-3 weeks]
Using integrals to calculate volumes of solids, length of curves, surface areas of solids of revolution.
Applications to physics (instructor to select from moments and center of mass, work, fluid pressures and forces).
Chapter 7. Integrals of Transcendental Functions [1 week]
The logarithm defined as an integral.
Exponential functions and hyperbolic functions.
Chapter 8. Techniques of Integration [3 weeks]
Basic integration formulas; integration by parts.
Integration of rational functions by partial fractions; trigonometric integrals; trigonometric substitution.
Integration using tables and computer algebra systems; approximate integration.
Improper integrals of Type I and Type II; comparison tests for convergence of improper integrals.
Chapter 10. Infinite Sequences and Series [4 weeks]
Numerical sequences and series; integral test and estimates of sums; comparison tests.
The ratio and root tests.
Alternating series; absolute and conditional convergence.
Strategy for testing convergence of series; the rearrangement theorem for absolutely convergent series.
Power series; representations of functions as power series.
Taylor and Maclaurin series.
Binomial series; applications of Taylor polynomials; complex numbers and Euler's identity.
Chapter 11. Conic Sections and Polar Coordinates [1-2 weeks]
Rotations, polar coordinates; arc length in polar coordinates.
Conic sections and quadratic equations.
Chapter 9. First Order Differential Equations [as time permits]
Optional: Selected topics from §§ 1 & 2. Solutions, slope fields, Euler's method, first-order linear equations.
Common Syllabi for Statistics Courses
The statistics courses that share a common syllabus across all sections are listed below.
Common Syllabus for STAT 103
Introduction (Chapters 1 and 2)
• Controlled Experiments
• Observational Studies
Descriptive Statistics (Chapters 3–6)
• Mean and Standard Deviation
• Normal Approximation for Data
• Measurement Error
Chance Variability (Chapters 16–18)
• Law of Averages
• Expected Value and Standard Error
• Normal Approximation for Probability Histograms
Sampling (Chapters 19–21, 23)
• Sample Surveys
• Chance Errors in Sampling
• Accuracy of Percentages
• Accuracy of Averages
Tests of Significance (Chapters 26–27)
• Hypothesis Tests
• More Tests for Averages
Correlation and Regression (Chapters 8–12)
• More Correlation
• R.M.S. Error for Regression
• Regression Line
Supplemental topics chosen from the following, if time allows:
Probability (Chapters 13–14)
More Hypothesis Tests (Chapters 28–29)