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Mathematics and Statistics

Course Syllabi

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
   • Exponents
   • 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:

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

Chapter 1 – A Library of Functions (1.5–2 weeks):

1.1 – Functions and Change
1.2 – Exponential Functions
1.3 – New Functions from Old

1.4 – Logarithmic Functions
1.5 – Trigonometric Functions
1.6 – Powers, Polynomials, and Rational Functions

1.7 – Introduction to Continuity

1.8 – Limits

Chapter 2 – Key Concept – The Derivative (1.5 weeks)

2.1 – How do we measure speed?
2.2 – The Derivative at a Point

2.3 – The Derivative Function

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

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?

5.2 – The Definite Integral

5.3 – The Fundamental Theorem and Interpretations
5.4 – Theorems About Definite Integrals


Chapter 6 – Constructing Antiderivatives  (1 week)

6.1 – Antiderivatives Graphically and Numerically
6.2 – Constructing Antiderivatives Analytically

Common Syllabus for MATH 132

Review of Chapters 5 & 6.  Definite and Indefinite Integrals [1.5 to 2 Weeks]
(Prerequisite Material from MATH 131)
   5
.1 – How Do We Measure Distance Traveled?
      Skip: accuracy of estimates (pg 277)
   
5.2 – The Definite Integral
   5.3 – The Fundamental Theorem and Interpretations
   5.4 – Theorems About Definite Integrals
      properties of definite integrals; area between curves; using the definite integral to find an average
      Skip: the subsection “Comparing Integrals”

   6.1 – Antiderivatives Graphically and Numerically
   6.2 – Constructing Antiderivatives Analytically
   Skip: Sections 6.3-6.4

Chapter 7.  Integration [1.5 to 2 Weeks]
   7.1 – Integration by Substitution
   7.2 – Integration by Parts
   7.6 – Improper Integrals
      consider only those where limits of integral are infinite
      Skip: cases where value of integrand becomes infinite (pp. 398–400)
   Skip: Sections 7.3, 7.4, 7.5 & 7.7

Chapter 8.  Using the Definite Integral [2 Weeks]

   8.6 – Applications to Economics
   8.7 – Distribution Functions
   8.8 – Probability, Mean, and Median
   Skip: Sections 8.1–8.5

Chapter 9.  Functions of Several Variables [2.5 to 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 11.  Differential Equations [3.5 Weeks]
   11.1 – What is a Differential Equation?
   11.2 – Slope Fields
   11.3 – Euler's Method
   11.4 – Separation of Variables
   11.5 – Growth and Decay
   11.6 – Applications and Modeling
   11.7 – The Logistic Model
   11.8 – Systems of Differential Equations
   11.9 – Analyzing the Phase Plane (time permitting)

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). 
    Exponential functions.
    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.
    Implicit Differentiation. 
    Derivatives of inverse functions and logarithms. 
    Inverse trigonometric functions. 
    Related Rates. 
    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. 
    Newton’s Method. 
    Antiderivatives.

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)
   • Histograms 
   • 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)
   • Correlation
   • More Correlation
   • Regression
   • 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)

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