QR Courses without Prerequisite

The following introductory-level courses without prerequisites enable students to fulfill the skills requirement in quantitative reasoning (QR) through study in a wide range of quantitative disciplines. In addition, there are a multitude of QR courses with prerequisites (such as single-variable calculus, introductory computer programming, or introductory work in a natural or social science) that can be fulfilled through introductory-level course work or advanced placement. Students may view the complete list of QR courses or search for QR courses in specific departments using the OCI (Online Course Information) Web site.

Each student will wish to select courses that match his or her interests and level of preparation. Information about expected preparation is available through links following each course’s brief description.

Course Number Course Title and Description

Applied Mathematics

AMTH 160b

The Structure of Networks
Network structures and network dynamics described through examples and applications ranging from marketing to epidemics and the world climate. Study of social and biological networks as well as networks in the humanities. Mathematical graphs provide a simple common language to describe the variety of networks and their properties.

Applied Physics

APHY 110b

The Technological World
See ENAS 110b


ARCH 161b

Introduction to Structures
Basic principles governing the behavior of building structures. Developments in structural form combined with the study of force systems, laws of statics, and mechanics of materials and members and their application to a variety of structural systems. Prerequisites: trigonometry and some knowledge of calculus. Enrollment limited to 20.


ASTR 110a

An Introduction to Stars and Planetary Systems
Topics include the solar system and extrasolar planets, planet and stellar formation, and the evolution of stars from birth to death. No prerequisite other than a working knowledge of elementary algebra.

ASTR 120b

Galaxies and the Universe
An introduction to stellar populations and the structure and evolution of the Milky Way galaxy; external galaxies, radio galaxies, and quasars; cosmology and the expanding universe. 

Further Course Information and Preparation

ASTR 160b

Frontiers and Controversies in Astrophysics
A detailed study of three fundamental areas in astrophysics that are currently subjects of intense research and debate: (1) planetary systems around stars other than the sun; (2) pulsars, black holes, and the relativistic effects associated with them; (3) the age and ultimate fate of the universe. No prerequisite other than a working knowledge of elementary algebra.

ASTR 170a

Introduction to Cosmology
An introduction to modern cosmological theories and observations. Topics include aspects of special and general relativity; curved space-time; the Big Bang; inflation; primordial element synthesis; the cosmic microwave background; the formation of galaxies; and large-scale structure. 

Further Course Information and Preparation

ASTR 210b

Stars and Their Evolution
An intensive introduction to stars. Nuclear processes and element production, stellar evolution, stellar deaths and supernova explosions, and stellar remnants including white dwarfs, neutron stars, and black holes.  A close look at our nearest star, the sun. Prerequisite: a strong background in high school mathematics and physics.

ASTR 220a

Galaxies and Cosmology
An intensive introduction to extragalactic astronomy. The structure and contents of galaxies, evolution of galaxies, observational cosmology, and the history of the universe. 

Further Course Information and Preparation

ASTR 255a

Research Methods in Astrophysics 
The acquisition and analysis of astrophysical data, including the design and use of ground- and space-based telescopes, computational manipulation of digitized images and spectra, and confrontation of data with theoretical models. Examples taken from current research at Yale and elsewhere. Use of the Python programming language. Includes an optional field trip during October recess to the Arecibo 300-meter radio telescope. No previous programming experience required.  


CHEM 112a

Chemistry with Problem Solving
For beginning students in chemistry or for those whose exposure to the subject has been moderate. Special emphasis on scientific problem-solving skills through an additional discussion section devoted to quantitative reasoning. Enrollment by placement only and limited to freshmen.

CHEM 114a or b

Comprehensive General Chemistry
A comprehensive survey of modern descriptive, inorganic, and physical chemistry for students with a good secondary school exposure to general chemistry. Enrollment by placement only.

CHEM 118a

Quantitative Foundations of General Chemistry 
An advanced course emphasizing conceptual aspects and physical principles in general chemistry. Fulfills the general chemistry prerequisite for organic chemistry. Attendance at a weekly discussion section required. Must be taken concurrently with CHEM 119L. Enrollment by placement only.  

Computer Science

CPSC 101b

Great Ideas in Computer Science 
An introduction for nonmajors to some of the most important ideas in computer science: what the computer is; how it works; what it can do and what it cannot do, now and in the future. Topics include algorithms, elementary programming, hardware, language interpretation, software engineering, complexity, models of computation, and artificial intelligence. No previous programming experience required.

CPSC 112a or b

Introduction to Programming
Development on the computer of programming skills, problem-solving methods, and selected applications. 

Further Course Information and Preparation

CPSC 202a

Mathematical Tools for Computer Science 
Introduction to formal methods for reasoning and to mathematical techniques basic to computer science. Topics include propositional logic, discrete mathematics, and linear algebra. Emphasis on applications to computer science: recurrences, sorting, graph traversal, Gaussian elimination.


ECON 108a or b

Quantitative Foundations of Microeconomics
Introductory microeconomics with a special emphasis on quantitative methods and examples. Intended for students with limited or no prior exposure to calculus.
Preference to freshmen. Permission of Economics DUS required. 

Further Course Information and Preparation

ECON 110a

Introduction to Economic Analysis: Microeconomics

An introduction to microeconomics, taught as a lecture discussion, that is an alternative to ECON 115a or b.
Limited to freshmen. Preregistration for ECON 110a is required by signing the preregistration sheet posted on the bulletin board at 28 Hillhouse Avenue on the first Monday before classes. 

Further Course Information and Preparation

ECON 115a or b

Introductory Economics: Microeconomics
An introduction that stresses the basic tools of micro economics and the problem solving involved in economic policy analysis. 

Further Course Information and Preparation

Electrical Engineering

EENG 201b

Introduction to Computer Engineering
Introduction to the theoretical principles underlying the design and programming of simple processors that can perform algorithmic computational tasks. Topics include data representation in digital form, combinational logic design and Boolean algebra, sequential logic design and finite state machines, and basic computer architecture principles. Hands-on laboratory involving the active design, construction, and programming of a simple processor. 

Further Course Information and Preparation

Engineering and Applied Science

ENAS 110b

The Technological World
An exploration of modern technologies that play a role in everyday life, including the underlying science, current applications, and future prospects. Examples include solar cells, light-emitting diodes (LEDs), computer displays, the global positioning system, fiber-optic communication systems, and the application of technological advances to medicine. For students not committed to a major in science or engineering; no college-level science or mathematics required.

ENAS 120b

Introduction to Environmental Engineering
Introduction to engineering principles related to the environment, with emphasis on causes of problems and technologies for abatement. Topics include air and water pollution, global climate change, hazardous chemical and emerging environmental technologies. Prerequisites: high school calculus and chemistry or CHEM 114, 115 (may be taken concurrently) or permission of instructor.

Environmental Studies

EVST 201a

Atmosphere, Ocean, and Environmental Change
Physical processes that control Earth’s atmosphere, ocean, and climate. Quantitative methods for constructing energy and water budgets. Topics include clouds, rain, severe storms, regional climate, the ozone layer, air pollution, ocean currents and productivity, the seasons, El Niño, the history of Earth’s climate, global warming, energy, and water resources. Must be taken concurrently with EVST 202La.

Global Affairs

GLBL 121a

Applied Quantitative Analysis
Mathematical fundamentals that underlie analytical approaches in public policy and the social sciences. Development of mathematical skills in areas such as linear functions, single and multiple variable differentiation, exponential functions, and optimization. Statistical approaches include descriptive statistics, principles of sampling, hypothesis tests, simple linear regression, multiple regression, and models for analyzing categorical outcomes.

Geology and Geophysics

G&G 140a

Atmosphere, Ocean, and Environmental Change
(see EVST 201a)


LING 224a

Formal Foundations of Linguistic Theories

Study of formal systems that play an important role in the scientific study of language. Exploration of a range of mathematical structures and techniques; demonstrations of their application in theories of grammatical competence and performance including set theory, graphs and discrete structures, algebras, formal language and automata theory. Evaluation of strengths and weaknesses of existing formal theories of linguistic knowledge.

LING 263a

Introduction to truth-conditional compositional semantics. Set theory, first- and higher-order logic, and the lambda calculus as they relate to the study of natural language meaning. Some attention to analyzing the meanings of tense/aspect markers, adverbs, and modals.


MATH 101b

Geometry of Nature
Geometric patterns in nature, including classical models of spirals in seashells and sunflowers, symmetry of honeycombs and snowflakes, and the curvature of soap films; the shape of the universe; ways to visualize the fourth dimension; and a brief introduction to fractal geometry. Enrollment limited to freshmen and sophomores who have not previously taken a high school or college calculus course.

MATH 107a

Mathematics in the Real World
The use of mathematics to address real-world problems. Applications of exponential functions to  population growth and radiocarbon dating; geometric series in mortgage payments, amortization of loans, future value of money; applications of basic probability theory and Bayes’s law in disease detection and drug testing; elements of logic; elements of cryptography. No knowledge of calculus required. Enrollment limited to students who have not previously taken a high school or college calculus or statistics course. 

Further Course Information and Preparation

MATH 112a or b

Calculus of Functions of One Variable I
Limits and their properties. Definitions and some techniques of differentiation and the evaluation of definite integrals, with applications. Graphical, symbolic, and numerical methods use the mathematical software package Mathematica. 

Further Course Information and Preparation

MATH 190a

Fractal Geometry
A visual introduction to the geometry of fractals and the dynamics of chaos, accessible to students not majoring in science. Study of mathematical patterns repeating on many levels and expressions of these patterns in nature, art, music, and literature. 

Further Course Information and Preparation


MUSI 175b

The Mathematics of Music 
An introduction to applied mathematics in the context of music theory and analysis. Concepts from algebra, modular arithmetic, set theory, geometry, and elementary topology are applied to the study of musical rhythms, melodies, and chords across a wide repertoire of classical, atonal, and popular musics. Prerequisite: ability to read music.


PHIL 115a

First Order Logic
An introduction to formal logic. Study of the formal deductive systems and semantics for both propositional and predicate logic. Some discussion of metatheory. 

Further Course Information and Preparation

PHIL 281b

The idea of infinity. Traditional and contemporary versions of the paradoxes of space, time, and motion, as well as the paradoxes of classes, chances, and truth. Some elementary arithmetic, geometry, probability theory, and set theory.


PHYS 060a

Energy Technology and Society
The technology and use of energy. Impacts on the environment, climate, security, and economy. Application of scientific reasoning and quantitative analysis. Intended for students not committed to a major in science or engineering; no college-level science or mathematics required. Enrollment limited to freshmen. Preregistration required; see under Freshman Seminar Program.

PHYS 101a or b

Movie Physics
A critical evaluation of Hollywood action movies using the laws of physics and back of the envelope estimates to distinguish between fictional and real movie physics. Enrollment limited to freshmen and sophomores. Intended for students with little or no prior exposure to calculus and statistics.

PHYS 115a

The Physics of Dance
Critical investigation of introductory concepts in physics through the lens of dance. Topics in physics include the normal force, friction, Newton’s laws, projectile motion, potential and kinetic energy, and conservation of energy. Topics in dance include aspects of dance history, contemporary artists who engage with science, and the development of movement studies. Class meetings include movement exercises. Prerequisite: basic trigonometry and algebra. Prior dance experience is not required.

PHYS 295a

Research Methods in Astrophysics 
(see ASTR 255a)

Political Science

PLSC 452a

Introduction to Statistics: Political Science
(see STAT 102a)

PLSC 453a

Introduction to Statistics: Social Sciences
(see STAT 103a)


PSYC 200b

Measures of central tendency, variability, association, and the application of probability concepts in determining the significance of research findings.


SOCY 162a

Methods in Quantitative Sociology 
Introduction to methods for reading and conducting quantitative sociological research. Data description and graphical approaches to data analysis; elementary probability theory; assumptions and properties of bivariate and multivariate linear regression; regression diagnostics. 

Further Course Information and Preparation


STAT 100b

Introductory Statistics
An introduction to statistical reasoning. Topics include numerical and graphical summaries of data, data acquisition and experimental design, probability, hypothesis testing, confidence intervals, correlation and regression. Application of statistical concepts to data; analysis of real-world problems.

STAT 101a

Introduction to Statistics: Life Sciences
Statistical and probabilistic analysis of biological problems presented with a unified foundation in basic statistical theory. Problems are drawn from genetics, ecology, epidemiology, and bioinformatics. 

Further Course Information and Preparation

STAT 102a

Introduction to Statistics: Political Science
Statistical analysis of politics, elections, and political psychology. Problems presented with reference to a wide array of examples: public opinion, campaign finance, racially motivated crime, and public policy.

Further Course Information and Preparation

STAT 103a

Introduction to Statistics: Social Sciences
Descriptive and inferential statistics applied to analysis of data from the social sciences. Introduction of concepts and skills for understanding and conducting quantitative research. 

Further Course Information and Preparation

STAT 105a

Introduction to Statistics: Medicine
Statistical methods relied upon in medicine and medical research. Practice in reading medical literature competently and critically, as well as practical experience performing statistical analysis of medical data.

Further Course Information and Preparation

STAT 230b

Introductory Data Analysis 
Survey of statistical methods: plots, transformations, regression, analysis of variance, clustering, principal components, contingency tables, and time series analysis. The R computing language and Web data sources are used.  
EP&E: Intro Statistics