First year student resources

Welcome to math at Yale! 

We are glad you are considering studying math with us at Yale. Whether you plan to take one course, several, or are considering a mathematics major, we are here to support you. This page will help you get started. 

You can also take a look at our orientation video, which provides basic information about our courses - and it has a puzzle game at the end!

Math introductory courses 

Differential calculus (comparable to AP Calculus AB)

Math 112 focuses on differential calculus, where the goal is to measure a function’s instantaneous rate of change (the derivative). First, we define limits, which allow us to talk rigorously about instantaneous changes. Then, we define the derivative and find many rules that simplify its computation. Next, we apply the derivative to better understand function behavior (allowing us to minimize cost or maximize profit in economic models). Finally, we develop strategies for reversing the derivative process to find antiderivative functions. These antiderivatives have surprising applications, like computing areas and modeling population growth.

Math 112 requires the placement exam. The necessary background is pre-calculus. 

Another option is the Math 110/111 sequence, a two-semester course that integrates pre-calculus and differential calculus topics. Successful completion of Math 110 and 111 is equivalent to Math 112 in that it satisfies the same major and professional-school (e.g. pre-med) requirements; however a student completing Math 110 and 111 receives two course credits and two QR credits.  

Math 110 requires the placement exam. Some prior knowledge of pre-calculus is helpful. 

Integral calculus + sequences and series (comparable to AP Calculus BC)

Math 115 investigates the mathematics of accumulation. With integral calculus, we use information about instantaneous rates (like the velocity of a moving object) to compute net changes (like change in position). Series convergence tests and Taylor series help us understand how calculators work, and various other topics (solids of revolution, parametric equations, polar coordinates) allow us to apply the tools of single variable calculus in broader contexts. Throughout the semester, we consider both exact and approximate solutions to problems and investigate the role of error.

Math 115 and 116 both cover integral calculus, and both can be used as pre-requisites for any course that requires Math 115. Math 116 places emphasis on applications to biology, and is particularly suitable for biology and pre-medical students.

Math 115 and 116 require the placement exam, or completion of Math 111 or 112 at Yale. The necessary background is differential calculus (equivalent to Math 112 or AP Calculus AB). 

Multivariable calculus

Math 120 extends skills and knowledge you gain in single variable calculus to two and three variables. We will study how to graph surfaces and solids in three dimensions, differentiate and integrate functions of several variables, optimize functions of several variables (for example when maximizing profit in economics),  we will learn how to integrate along curves and surfaces (for example when calculating how much energy a solar panel will generate while the sun is shining on it), and many other topics.

Math 120 and 121 both cover multivariable calculus, and both can be used as pre-requisites for any course that requires Math 120. Math 121 places emphasis on applications to biology, and is particularly suitable for biology and pre-medical students.

Math 120 and 121 require the placement exam or completion of Math 115 or 116 at Yale. The necessary background is integral calculus (such as Math 115 or AP Calculus BC). 

Linear algebra and introduction to proofs

Math 225 is a linear algebra course, and a part of our introductory sequence into the math major. In linear algebra, you will learn key language and concepts used throughout pure mathematics as well as in a wide variety of applications.  Linear algebra starts by studying systems of equations with many variables, and it builds a detailed understanding of how to work in abstract n-dimensional space. This version of linear algebra focuses on concepts, and provides an introduction to writing mathematical proofs.

We also offer an intensive version of this course, Math 226, for students who are looking for an extra challenge. Math 226 will teach the same main topics as Math 225, but it may go into more depth, ask more challenging problems on homework and exams, or cover optional side topics.

Math 222 is a linear algebra course best suited for students who wish to focus on applications and practical solving problem practice, rather than abstract mathematics and mathematical proofs. It is often taken by students majoring in engineering, technology, science, social sciences, and economics. Mathematics majors need to complete Math 225 or 226, rather than 222. 

Math 222, 225, and 226 do not use the placement exam or preference selection. 

Real analysis and introduction to proofs

Math 255 is an introduction to real analysis, and a part of our introductory sequence into the math major. The topics are similar to differential and integral calculus, but it’s done in a rigorous way that focuses on proofs and details of the concepts that happen behind the scenes in calculus.  Said differently, our calculus sequence teaches you how to use the tools of limits, derivatives, and integrals to solve a variety of problems, whereas this course teaches you how to build all of those tools from scratch.

We also offer an intensive version of this course, Math 256, for students who are looking for an extra challenge. Math 256 will teach the same main topics as Math 255, but it may go into more depth, ask more challenging problems on homework and exams, or cover some optional side topics.

We recommend taking linear algebra first, because it is more universally applicable, and because it is a more gentle introduction to proofs – but it is completely fine to take analysis first.

Math 255 and 256 do not use the placement exam or preference selection. 

Multivariable analysis

Math 302 is a rigorous and exciting course that explores some of the beautiful connections between linear algebra and real analysis. For this reason, it is intended to be taken after completing 225/226 and 255/256. 

This course is an introduction to manifolds, which are low-dimensional shapes living in a higher dimensional universe (e.g., a one-dimensional coat hanger or a two-dimensional eggshell each living in a three-dimensional world), and it teaches how to do calculus in this abstract setting.  Although the focus of this course will be on abstract mathematical objects and their properties, the material in this course also has applications to a broad range of topics including theoretical physics and machine learning.

Many students who have not completed multivariable calculus in high school would benefit from taking Math 120. However, prospective math majors, and students excited about rigorous proof-based math should consider taking Math 225/226, Math 255/256 and Math 302 instead.  Math 302 is also appropriate for students who have already completed Math 120 and are curious to learn more.

Math 302 does not use the placement exam or preference selection. 

Other courses at the 100 level

Math 118 teaches a combination of multivariable calculus and linear algebra. It is suited for students majoring in Economics or Social sciences who wish to learn the most relevant mathematics in one semester.  Students intending to take further courses in mathematics should take multivariable calculus (Math 120) and linear algebra (Math 222, 225, or 226) instead. Pre-requisite for Math 118 is differential calculus (such as Math 112 or AP Calculus AB). Knowledge of integral calculus is recommended. 

The math department offers three Quantitative Reasoning (QR) courses that assume no calculus experience:

Math 106 (The shape of space) provides an introduction to mathematical thinking through ideas in geometry and graph theory. The course follows a historical narrative starting from antiquity, and builds mathematical and geometric intuition through problem solving and other activities. 

In Math 107 (Mathematics in the real world), students use mathematical ideas to solve real world problems. Topics include compound interest, population growth, probability and its applications to games of chance, mortgage payments, false positives in drug testing, computer security, and other questions. 

Math 108 (Estimation and error) leads students through a problem-based investigation of basic mathematical principles and techniques that help make sense of the world. Applications include geology, ecology, finance, and other fields. 

Other courses at the 200 level

Math 241 and 242: probability and statistics teach us how to collect, analyze, and interpret data. 

Math 244: discrete mathematics includes many topics of interest in computer science.

Math 246: differential equations can be used for modeling complex systems in biology, physics, economics, and other fields. 

Math 270: set theory is at the foundation of mathematics and often taken by philosophy students.

Choosing your first math course at Yale

Many incoming students enroll in one of our calculus courses (Math 110, 112, 115, 116, 120, or 121). If you are considering this option, you will need to  complete our placement exam, so that we can help you determine the best place to start.

Notes about the placement exam

  • For detailed information about the test, visit our Calculus and placement exam FAQ
  • The Calculus Placement Exam is open July 1 - July 31, 2021.
  • All incoming first-year students are automatically enrolled on the Math Placement Exam Canvas course at the end of June. Rising sophomores, juniors, and seniors should contact j.barnes@yale.edu to be added to the site. 
  • Results of the test are available immediately upon submission, by clicking on the “placement results” tab. (If the tab does not load, try a different browser.) 
  • If you would like to take a course that differs from your placement exam result, please attend our calculus advising session in August. The advisers will be happy to help you. Note that you must take the test first - advisers cannot help you without it.
  • After receiving placement to take your first calculus course, pre-register for the class through Yale’s preference selection website. Preference selection is typically open in August (for the Fall term), in December (for the Spring term), and again in April (for the Fall term, continuing students). 
  • The placement exam is NOT required for any math course level 200 or above. In particular, you do not need it in order to enroll in Math 225, 226, 255, or 256.

Checklist for enrolling in calculus

  1. Take the placement exam on Canvas.
  2. If you have questions about the calculus courses or placement exam, check the calculus and placement exam FAQ.
  3. If you have questions about your placement, attend our calculus advising session in August. 
  4. Pre-register for your first calculus course via Preference selection. Note that you must have placement in the course in order to enroll, and you may not enter preference selection for more than one calculus course at the same time.
  5. If you need help with your placement and missed the advising session, write to the faculty member listed below as a contact for the course into which you were placed. 

Students who completed multivariable calculus during high school often start at Yale with linear algebra:  Math 222, 225, or 226. For details about how this can be done, please see the corresponding question in our Math major FAQ. 

If you completed linear algebra during high school, please take a look at the options listed in the corresponding Math major FAQ question

If you completed real analysis, abstract algebra, or other advanced courses during high school, please consult the math DUS for advice about math courses. We are happy to help. 

Overview of the math major introductory sequence

Pre-requisite for the mathematics major is integral calculus (for example Math 115, or AP Calculus BC). 

Following the pre-requisite, the introductory sequence includes:

  • Linear algebra (Math 225 or 226),
  • Real analysis (Math 255 or 256), and
  • Multivariable analysis or calculus (Math 302 or 120). 

Most students complete multivariable calculus before enrolling in Math 225 or 226.

Prospective mathematics majors and students with interest in abstract mathematics may consider enrolling in Math 225 directly after integral calculus (e.g. Math 115 or AP Caclulus BC), and completing their vector analysis/multivariable calculus requirement with Math 302. 

If you have completed some part of the sequence during high school, please visit the Math Major FAQ for information about the options. You should then consult the DUS, who will evaluate your previous coursework, see if an exception could be granted from the requirements, and help you select the best class to continue. 

For advice about the mathematics major or courses, visit our Math Major FAQ, attend our Academic Fair in August, or contact the Math DUS. 

If you are thinking about the math major, we encourage you to sign up for our Math DUS mailing list, to get news from the math department. You can also take a look at our Math community page!

If you did not take calculus in high school and wish to take a math course other than calculus, you may enjoy one of our non-calculus level 100 courses described above. 

Contact information

With additional questions, please visit the 

If your question is not addressed there, please contact the faculty member designated in the following table. We would be glad to help!

question type email (to be followed by @yale.edu)
For questions about your placement, please attend our advising session in August.*
Other placement exam questions j.barnes
Math 110-111 robert.j.mcdonald
Math 112 ian.adelstein
Math 115 brett.c.smith
Math 116 john.hall
Math 120 john.hall
Math 225 patrick.devlin
Math major / advanced courses / other questions miki.havlickova

* If you missed the advising session, write to the faculty member who is listed as a contact for the course into which you were placed. Please be sure to write at least three days before preference selection closes, or the instructor might not be able to assist you in time.