A Guide to First-Year Mathematics

A guide to First Year Calculus Courses offered through the Faculty of Arts & Science, Department of Mathematics.

Which calculus course is right for you? How to get ready for it?

We offer four different first-year calculus courses for students in the Faculty of Arts & Science. This page will help you understand the differences, choose the one that is right for you, and get ready for it.

At a glance:

  • MAT133Y1 - ‘Calculus and linear algebra for commerce’.
  • MAT135H1 + MAT136H1 - ‘Calculus 1 + Calculus 2’, the default course for most science students.
  • MAT137Y1 - ‘Calculus with proofs’, including an introduction to proofs and abstraction
  • MAT157Y1 - ‘Analysis I’, intended for math specialists.

Are you prepared for calculus?

If you want to succeed in university calculus, make sure your precalculus background is very strong. How much high-school calculus you have already studied is nowhere near as important. This applies to all our calculus courses.

To help you assess your readiness, we have prepared this website: Entry Level Math Preparation
We recommend you complete the self-assessment quizzes on these topics:

These quizzes cover what we assume you know before the first day of your calculus course. If you don’t, it is your responsibility to make up for it before the course begins. The website also contains tutorials and practice questions in case you need them

High-school math vs university math

You may notice a big difference between your high-school and your university math courses. Many high-school math courses (even Advanced Placement and other “university-level” courses) focus on memorization and computations. They teach you techniques and show you examples that you memorize and imitate. By contrast, university courses focus on understanding why methods work and what the concepts mean. It may take a while to get used to this approach, but we invite you to embrace it.

We emphasize understanding over memorization because we want to sharpen your problem solving skills. We want you to be comfortable solving novel problems that are different from everything we have taught you before. That is the most useful skill you will develop in your math courses.


MAT133Y1 is designed for commerce students. By the end of this course, students will understand and apply concepts from differential and integral calculus, of one and two variables, to solve practical business problems. If you are more interested in theory, proofs, or in a larger family of applications, you may wish to consider the other calculus courses.

Please note that MAT133Y1 does not serve as a prerequisite for second-year mathematics courses with a calculus prerequisite. If you would like to take further mathematics courses, you may want to consider a different option, even if you are a commerce student.

If you are interested in a Statistics program, please note that any of our calculus courses MAT133Y1, MAT135H1+MAT136H1, MAT137Y1, or MAT157Y1 may be applied toward the Minor in Statistics. However, if you take MAT133Y1, you must earn a score of at least 70% in MAT133Y1 in order to enrol in a Minor in Statistics. MAT133Y1 may not be applied toward a Major in Statistics. Please consult the calendar for further details.

MAT135H1 + MAT136H1

MAT135H1 + MAT136H1 is the default calculus sequence for most science students. It is a traditional calculus course with an emphasis on modelling and applications to science and social science. We designed these courses after consulting with other departments in the Faculty of Arts & Science to make sure we teach you calculus in a way that will be useful to you.

There are various second-year courses that you can take after completing MAT135H1 and MAT136H1, either because your program requires them or out of interest.

MAT135H1 + MAT136H1 will teach you to understand concepts, but will not include proofs. If you would like to develop a more theoretical foundation in mathematics, you should consider a different course. Otherwise, this is an excellent option for most science students. In addition, this sequence serves as the beginning of a math minor or a math major.


In addition to teaching calculus, MAT137Y1 serves as an introduction to logic, proofs, abstraction, and mathematical rigour. We learn how to write precise mathematical statements, and how to read, critique, and write mathematical proofs.

You will learn this slowly and from scratch. Nevertheless, most students find this leap difficult, perhaps because it is very different from everything they have done before. While this is not necessary, if you would like to, you can Get a head start on Proofs.

MAT137Y1 serves as the starting point to four specialists programs

  • Mathematical applications in economics and finance
  • Mathematics and its applications (physical science)
  • Mathematics and its applications (probability/statistics)
  • Mathematics and its applications (teaching)

However, most students in MAT137Y1 are not in one of these programs. They are students who have an interest in rigorous mathematics, mostly in computer science, physics, statistics, economics, or actuarial science. These programs require or (more often) simply recommend MAT137Y1 because it helps with analytical thinking.


MAT157Y1 is an analysis course that helps lay a theoretical, rigorous foundation to your mathematics. Like MAT137Y1 it introduces proofs, but the complexity of the proofs and the depth of the abstraction are higher. It is intended for students who are already comfortable with mathematical rigour or who will pick it up quickly on the fly, and who have an interest in abstract mathematics.

MAT157Y1 serves as the starting point to four specialists programs

  • Mathematics
  • Applied mathematics
  • Mathematics and physics
  • Mathematics and philosophy


MAT223H1 is a first-year course on Linear Algebra, which is about linear transformations, finite dimentional vector spaces, and the underlying mathematical ideas that help us understand how linear transformations work by studying their associated matrix representations.  This course does not require prior background knowledge in Linear Algebra, and we design and teach the course assuming students are learning Linear Algebra for the first time.  MAT223H1 is not a proof-based course, however, we do emphasize conceptual understanding and developing early proof methods for some concepts. 

MAT223H1 is offers every semester. After taking MAT223H1, students can take MAT224H1 as a follow-up course. MAT224H1 is a second course in Linear Algebra and emphasizes proofs more than MAT223.

Which course should you choose?

If you are trying to decide between two courses, your best plan is to attend both of them during the first two weeks (even if you are not enrolled). Do not just attend: do all the work. That will give you the best idea of which one is the right fit for you.

Some students choose a course that is more abstract than the one required by their program. As a general rule, any program that requires MAT135H1+MAT136H1 will also accept MAT137Y1 or MAT157Y1 instead, and any program that requires MAT137Y1 will also accept MAT157Y1 instead.

A good reason to take MAT135H1 and MAT136H1 is that you want to learn more about modelling and applications to science, rather than theory. A good reason to take MAT137Y1 or MAT157Y1 is that you are personally interested in abstract mathematics. Remember that an applied course is simply less theoretical or abstract, not necessarily easier.