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Polymath Sample Syllabus - Mathematics
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Polymath Institute

Sample Syllabus (Core Curriculum)

Field Topic Units: Mathematics

One (1) Topic Unit =  One (1)  20- to 30- minute Lecture

So, a 3 credit hour class would be 9 Topic Units per week,

and a 16 week semester would consist of 144 Topic Units.

Depending on the nature of the class and subject matter, for some subjects, students may elect to take only some of the Topic Units for a class, or they may choose to take all of them while they learn at their own pace.  

Professors:  TBA  (with contact information)

Office Hours: TBA

Course Description: Topic Units are organized by subject.  A Field Topic Unit is an overarching subject that encompasses many more specialized subjects in the field.  For example, the Field Topic of Mathematics includes Advanced Algebra, Geometry, Trigonometry, Calculus, Statistics, and more.

Graduation Requirements for Field Topic Units:

Minimum For All Students:

30 Topic Units in Psychology

50 Topic Units in Mathematics

50 Topic Units in Sciences

50 Topic Units in English Language and Literature

30 Topic Units in History

Students with Professional Applications related to given subject area(s) are required to take more Topic Units in those areas (i.e. a student utilizing a significant amount of mathematical formulas and/or skills in his project must take at least 120 Topic Units in Mathematics).


After each Topic Unit, students will take a Topic Assessment.  Assessments are computer-based, closed-book exams that are tested to a statistical proficiency level rather than graded on a traditional A-F scale.  Students need a minimum of 90% correct answers on a Topic Assessment to pass.

Topic Unit #                 Lecture Title        

A01                         Advanced Algebra - Intro and Basics

        A02                          Polynomial Functions

        A03                          Exponential Functions

        A04                          Logarithms

        A05                          Data Analysis

        A06                         Roots and Radicals

        A07                         Quadratic Functions

        A08                         3-Variable Equation Systems

        A09                        Rational Equations

        A10                        Systems of Inequalities

        A11                        Factoring

        A12                        Real-World Application Part 1

        A13                        Real-World Application Part 2

        A14                        Abstract Algebra

        A15                        Universal Algebra

        A16                        Normed Linear Spaces

        A17                        Banach Spaces

        A18                        Hilbert Spaces

        A19                        Topological Groups

        A20                        Combinatorics


G01                          Geometry - Intro and Basics

        G02                        Creating Proofs

        G03                        Lines and Linear Equations

        G04                        Polygons

        G05                        Congruence and Similarity

        G06                        Determining Surface Area

        G07                        Determining Volume

        G08                        Determining Mass

        G09                        Properties of Triangles

        G10                        Types of Triangles

        G11                        Calculations With Triangles

        G12                        Real-World Applications With Triangles

        G13                        Concave Polygons

        G14                        Convex Polygons

        G15                        Circles and Spheres

        G16                        Calculations with Circles

        G17                        Calculations with Spheres

        G18                        Parallel Lines

        G19                        Perpendicular Lines

        G20                        Properties of Quadrilaterals

        G21                        Types of Quadrilaterals

        G22                        Calculations With Quadrilaterals

        G23                        Real-World Applications With Quadrilaterals


T01                        Trigonometry - Intro and Basics

        T02                        Intro to the Six Basic Trigonometric Functions

        T03                        The Sine

        T04                        The Cosine

        T05                        The Tangent

        T06                        The Secant

        T07                        The Cosecant

        T08                        The Cotangent

        T09                        Verifying Trigonometric Identities

        T10                           Graphing Trigonometric Functions

        T11                        Law of Sines

        T12                        Law of Cosines

        T13                        Formulas for Area of Triangles

        T14                        Vector Addition and Scalar Multiplication

        T15                        Rectangular and Polar Coordinates

        T16                        The Dot Product

        T17                        Complex Numbers

        T18                        Addition and Subtraction Formulas

        T19                        Multiple Angle Formulas

        T20                        Real-World Trigonometry Application Part 1

        T21                        Real-World Trigonometry Application Part 2


P01                        Pre-Calculus - Intro and Basics

        P02                        Linear Function Review

        P03                        Quadratic Equation Review

        P04                        Polynomial Review

        P05                        Rational Functions

        P06                        Dividing Polynomials

        P07                        Integer Exponents

        P08                        Rational and Real Exponents

        P09                        Logarithms and Inverses of Exponents

        P10                        Properties of Logarithms

        P11                        e the Natural Logarithm

        P12                        Approximating Area Under A Curve

        P13                        Unit Circles

        P14                        Negative Angles

        P15                        Computing Circular Arc Length

        P16                        Radians

        P17                        Converting Radians to Degrees

        P18                        Trigonometric Function Review Part 1

        P19                        Trigonometric Function Review Part 2

        P20                        Right Triangle Review

        P21                        Inverse Trigonometric Function Review Part 1

        P22                        Inverse Trigonometric Function Review Part 2

        P23                        Double-Angle Formulas

        P24                        Half-Angle Formulas

        P25                        Addition and Subtraction Formulas

        P26                        Polar Coordinate Review

        P27                        Real-World Precalculus Applications

C01                        Calculus - Intro and Basics

C02                        Basic Differentiation Rules - Intro

C03                        Basic Integration Formulas - Intro

C04                        Functions and their Graphs

C05                        FItting Models to Data Graphs

C06                        What Are Limits?

C07                         How To Find and Evaluate Limits

C08                        Types of Limits

C09                        Infinite Limits

C10                        Differentiation and Derivatives

C11                        Real-World Application of Derivatives

        P01                        Classical Mechanics - Intro and Basics

        P02                        Force

        P04                        Velocity

        P05                        Acceleration

C12                        The Product Rule

C13                        The Quotient Rule

C14                        Higher-Order Derivatives

C15                        The Chain Rule

C16                        Rolle’s Theorem

C17                        The Mean Value Theorem

C18                        Curve Sketching

C19                        Newton’s Method

C20                        Antiderivatives

C21                        Riemann Sums

C22                        The Fundamental Theorem of Calculus

C23                        Integration and Integrals

C24                        Real-World Application of Integrals

        CE01                        Civic Engineering - Intro and Basics

        P02                        Torque

C25                        e and the ln function

C26                        Differential Equations

C27                        Hyperbolic Functions

C28                        Inverse Functions

C29                        Integration and Volume

C30                        Integration and Area

C31                        Conics

C32                        Parametric Equations

C33                        Graphing Polar Coordinates

C34                        Working With Vectors - Intro

C35                        Vectors and Differentiation

C36                        Vectors and Integration

C37                        Tangent Vectors

C38                        Real-World Applications of Vectors

C39                        Working With Several Variables - Intro

C40                        Chain Rule

C41                        Partial Derivatives

C42                        Directional Derivatives

C43                        Gradients

C44                        Double and Triple Integrals

C45                        Divergence Theorem

C46                        Stokes’ Theorem

S01                        Statistics - Intro and Basics

        S02                        Statistical Formulas Part 1

        S03                        Statistical Formulas Part 2

        S04                        Data Relationships

        S05                        Data Inferences

        S06                        Probability Part 1

        S07                        Probability and Randomness

        S08                        Statistical Sampling

        S09                        Inference - Intro

        S10                        Statistical Regression

        S11                        Multiple Regression

        S12                        One-Way Variance Analysis

        S13                        Two-Way Variance Analysis

        S14                        Parametric Tests

        S15                        Nonparametric Tests

        S16                        Statistical Applications in Science

        S17                        Statistical Applications in Medicine

        S18                        Statistical Applications in Psychology

        S19                        Statistical Applications in Sociology

        S20                        Statistical Applications in Engineering

        S21                        Statistical Applications in Business

        S22                        Statistical Applications in Economics


Students may choose from our suggested resources in order to get a better grasp on their understanding of Topic Unit material in addition to listening to Lecture material, or they may also look for additional resources on their own.  This list is to be viewed as a starting point, not as comprehensive.

Related resources do not have a one-to-one Topic Unit correlation, although certain resources do tend to apply more to certain Topic Units.


Birkhoff, G. & MacLane, S. (1998). A Survey of Modern Algebra. London: A.K. Peters.

Lial, M., Hornsby, J., Schneider, D. II., & McGinnis,M. (2009). Essentials of College Algebra.  Boston: Addison-Wesley.


Devadoss, S. & O’Rourke, J. (2011). Discrete and Computational Geometry. Princeton University Press.

Jacobs, R. (2003). Geometry: Seeing, Doing, Understanding. New York: W. H. Freeman.

Rhoad, R., Milauskas, G., & Whipple, R. (1991). Geometry for Enjoyment and Challenge.  Boston: McDougal Littell/Houghton-Mifflin.


Larson, Ron & Hostetler, Robert P. (2010).  Trigonometry.  Independence, KY: Cengage Learning.

McKeague, Charles D. & Turner, Mark P. (2007). Trigonometry. Belmont, CA: Brooks-Cole.


Cohen, D., Lee, T., & Sklar, J. (2011). Precalculus (7th ed.).  Belmont, CA: Brooks-Cole.

Dugopolski, Mark. (2008). Precalculus: Functions and Graphs (3rd ed.). Boston: Addison-Wesley.

Simmons, George. (2003).  Precalculus Mathematics in a Nutshell: Geometry, Algebra, Trigonometry.  Eugene, OR: Wipf & Stock.


Larson, R. & Edwards, Bruce H. (2009). Calculus. Independence, KY: Cengage Learning.

Stewart, James. (2011). Calculus. Belmont, CA: Thomson Brooks-Cole.


Kokoska, Stephen. (2010). Introductory Statistics: A Problem-Solving Approach. New York: W.H. Freeman.

Triola, Mario F. (2010). Essentials of Statistics (4th ed.). Upper Saddle River, NJ: Pearson Education.

Online statistics book list: