Loyola University Chicago

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 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 1: Relations and Functions
    1.1  Sets of Real Numbers and The Cartesian Coordinate Plane
    1.2  Relations
    1.3  Introduction to Functions
    1.4  Function Notation
    1.5  Function Arithmetic
    1.6  Graphs of Functions
    1.7  Transformations

Chapter 2: Linear and Quadratic Functions
    2.1  Linear Functions
    2.2  Absolute Value Functions
    2.3  Quadratic Functions

Chapter 3: Polynomial Functions [2 weeks]
    3.1  Graphs of Polynomials
    3.2  The Factor Theorem and The Remainder Theorem
    3.3  Real Zeros of Polynomials
    3.4  Complex Zeros and the Fundamental Theorem of Algebra

Chapter 4: Rational Functions
    4.1  Introduction to Rational Functions
    4.2  Graphs of Rational Functions

Chapter 5: Further Topics in Functions
    5.1  Function Composition
    5.2  Inverse Functions
    5.3  Other Algebraic Functions

Chapter 7: Hooked on Conics
    7.2  Circles

Chapter 8: Systems of Equations and Matrices
    8.1  Systems of Linear Equations: Gaussian Elimination
    8.2  Systems of Non-Linear Equations and Inequalities

Common Syllabus for MATH 118

Chapter 6: Exponential and Logarithmic Functions
    Review  Section 5.2: Inverse Functions
    6.1  Introduction to Exponential and Logarithmic Functions
    6.2  Properties of Logarithms
    6.3  Exponential Equations and Inequalities
    6.4  Logarithmic Equations and Inequalities
    6.5  Applications of Exponential and Logarithmic Functions

Chapter 10: Foundations of Trigonometry
    10.1  Angles and their Measure
    10.2  The Unit Circle: Cosine and Sine
    10.3  The Six Circular Functions and Fundamental Identities
    10.4  Trigonometric Identities
    Review Section 1.7: Transformations
    10.5  Graphs of the Trigonometric Functions
    10.6  The Inverse Trigonometric Functions
    10.7  Trigonometric Equations and Inequalities

Chapter 11: Applications of Trigonometry
    11.2  The Law of Sines
    11.3  The Law of Cosines
    11.4  Polar Coordinates
    11.7  Polar Form of Complex Numbers

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

  • 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 interpretation 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 of Chapters 5 & 6.  Definite and Indefinite Integrals [1.5 to 2 Weeks]
(Prerequisite Material from MATH 131)
.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 and Models [1 week]
  1.1 Four Ways to Represent a Function
  1.2 Mathematical Models: A Catalog of Essential Functions
  1.3 New Functions from Old Functions
  1.4 Exponential Functions and Logarithms
  1.5 Inverse Functions and Logarithms 
    Optional: Graphing with calculators, Mathematica, Wolfram Alpha (pp. xxiv-xxv)

Chapter 2: Limits and Derivatives [2.5 weeks]
  2.1 The Tangent and Velocity Problems
  2.2 The Limit of a Function
  2.3 Calculating Limits Using the Limit Laws
  2.4 The Precise Definition of a Limit
  2.5 Continuity
  2.6 Limits at Infinity; Horizontal Asymptotes
  2.7 Derivatives and Rates of Change
  2.8 The Derivative as a Function

Chapter 3: Differentiation Rules [3 weeks]
  3.1 Derivatives of Polynomials and Exponential Functions
  3.2  The Product and Quotient Rules
  3.3 Derivatives of Trigonometric Functions
  3.4 The Chain Rule
  3.5 Implicit Differentiation
  3.6 Derivatives of Logarithmic Functions
  3.7 Rates of Change in Natural and Social Sciences
  3.8 Exponential Growth and Decay
  3.9 Related Rates
  3.10 Linear Approximations and Differentials 
  3.11 Optional: Hyperbolic Functions

Chapter 4: Applications of Derivatives [3 weeks]
  4.1 Maximum and Minimum Values
  4.2 The Mean Value Theorem
  4.3 How Derivatives Affect the Shape of a Graph
  4.4 Indeterminate Forms and l'Hospital's Rule
  4.5 Summary of Curve Sketching
  4.6 Optional: Graphing with Calculus and Calculators 
  4.7 Optimization Problems
  4.8 Optional: Newton's Method
  4.9 Antiderivatives

Chapter 5: Integrals [2.5 weeks]
  5.1 Areas and Distances
  5.2 The Definite Integral
  5.3 The Fundamental Theory of Calculus
  5.4 Indefinite Integrals and the Net Change Theorem
  5.5 The Substitution Rule

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 Integration  [1.5 weeks; 2 weeks if Section 6.4 is covered] 
  6.1  Area Between Curves
    6.2  Volumes
    6.3  Volumes by Cylindrical Shells
    6.4  Optional: Work ( This Sections could be covered with Chapter 8)
    6.5  Average Value of a Function

Chapter 7: Techniques of Integration [3 weeks]
    7.1  Integration by Parts
    7.2  Trigonometric Integrals
    7.3  Trigonometric Substitution
    7.4  Integration of Rational Functions by Partial Fractions
    7.5  Strategy for Integration
    7.6  Integration Using Tables and Computer Algebra Systems
    7.7  Approximate Integration
    7.8  Improper Integrals

Chapter 11: Infinite Sequences and Series [4 weeks]
   11.1  Sequences
   11.2  Series
   11.3  The Integral Test and Estimates of Sums
   11.4  The Comparison Tests
   11.5  Alternating Series
   11.6  Absolute Convergence and the Ratio and Root Tests
   11.7  Strategy for Testing Series
   11.8  Power Series
   11.9  Representations of Functions as Power Series
   11.10  Taylor and Maclauren Series
   11.11  Optional: Applications of Taylor Polynomials

Chapter 10: Parametric Equations and Polar Coordinates [1-1.5 weeks]
   10.1  Curves Defined by Parametric Equations
   10.2  Optional: Calculus with Parametric Curves
   10.3  Polar Coordinates
   10.4  Optional: Areas and Lengths in Polar Coordinates
   10.5  Optional: Conic Sections
   10.6  Optional: COnic Sections in Polar Coordinates


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 to Data, Chapters 1 and 2 [1 week]

    • Types of Data, experiments, sampling

    • Presenting Data: Organizing and graphing data

  • Summaries of Center and Variation, Chapter 3 [1 week]

    • Mean, Median, Standard Deviation, 5 number summary, Percentiles

  • Probability, Chapter 5 [1 week]

    • Independence, Conditional Probability

  • Binomial and Normal Random Variables, Chapter 6 [1 week]

    • Areas as probability

    • Normal Approximation to Binomial

  • Estimation of Population Proportions, Chapter 7 [2 weeks]

    • Confidence intervals for proportions

    • Comparing two proportions

  • Hypothesis Testing for Proportions, Chapter 8 [2 weeks]

    • Ingredients and logic behind hypothesis testing

    • Testing one and two proportions

  • Inference for Population Means, Chapter 9 [2 weeks]

    • Testing one and two means

  • Linear Regression Analysis, Chapter 4 [1 week]

    • Scatter Plots

    • Correlation

  • Categorical variables, Chapter 10 (optional, time permitting) [1 week]

    • One and Two-way tables

    • χ2 tests

  • Inference for Regression, Chapter 14 (optional, time permitting) [1 week]

    • Hypothesis testing correlation

    • Prediction