PUMP courses

About PUMP Level 1 | About PUMP Level 2 | Current/Upcoming Sessions | Textbooks | Finances | RegistrationAccommodations | Contact

Preparing for University Math Program (PUMP) offers two non credit courses run by the Department of Mathematics on the St. George campus of the University of Toronto.

What are the advantages of PUMP?

PUMP provides the necessary foundation upon which the greater generality and abstraction of university mathematics is based.

PUMP is a non-credit course.  Because it does not count towards a degree, the stress associated with course marks is reduced and students can better concentrate on their learning.

PUMP participants are given an inside view of what to expect in their future university math courses, how to cope with the demands of courses, and what their prospects are.  In short, successful PUMP students will have better insight into which first-year math course(s) to choose and will be generally better prepared for university.

PUMP has regularly-scheduled assessments that reflect university standards.  Students will have the opportunity to measure their own progress throughout the course.

PUMP is now available at two levels: PUMP Level 1 and PUMP Level 2.

       PUMP Level 1 is ideal for:
  • High school students who will be starting a university level education in an upcoming semester and who need grounding in pre-calculus topics.
  • University level students engaged in studies outside the field of mathematics who desire to become functionally literate in mathematics.
  • People who wish to learn (or re-learn) high school pre-Calculus
  • Any student who intends to register for MAT135H1 or MAT133Y1 or MAT186H1 (or their analogues).
   PUMP Level 2 is ideal for:
  • People who want a quick review of high school pre-Calculus followed by an introduction to logic, problem solving and methods of proof.
  • Any student who intends to register for MAT137Y1 and MAT157Y1, MAT223H1, or MAT194H1 (or their analogues).
Important Note about U of T Programs that Require Calculus before Admissions

Ontario high school students applying on the 101 application form are expected to present MCV4U - Calculus and Vectors for admission to all science, math, and commerce programs in the Faculty of Arts and Science, St. George Campus. PUMP Level I and Level 2 cannot be used as a replacement for this admission requirement.

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About PUMP Level 1:

A certain background knowledge is strongly recommended for PUMP Level 1. In general terms, a student must have a working knowledge of basic high school algebra, linear and quadratic functions, and elementary analytic geometry.  Students who do not have this background may still enrol and can do well if they work hard.

What does PUMP Level 1 cover?

This course focuses on the mathematical background needed for entry-level university science and math courses, expanding and developing relevant skills and techniques of reasoning. Topics to be covered include:

  • Fundamental algebraic background: sets, operations and properties; Numbers, fractions, exponents and rational expressions; Factorization and reduction; Completing the square; Binomial expansion; Operations with general algebraic expressions.
  • Equations, inequalities and systems: equations in quadratic form; Absolute value, rational and radical equations; General equation solving; Polynomial, rational and absolute value inequalities; General solving and graphing of algebraic inequalities; Systems of linear equations; Gaussian elimination; Non-linear systems and systems containing inequalities; General system solving; Setting up equations; inequalities and systems; Working with word problems; Applications.
  • Basic trigonometry: The number p; Radians and degrees; Trigonometric functions and their graphs; Generalizations and inverse trigonometric functions; The algebra of trigonometric identities and equations; Right triangle trigonometry; The law of sines and the law of cosines; Applications.
  • Exponential and logarithmic functions: the number e; Exponential functions and their graphs; The inverse of an exponential function; Logarithmic functions and their graphs; Properties of the logarithms; Exponential and logarithmic equations and inequalities; Exponential growth and decay.
  • Analytic geometry and introductory calculus: rectangular co-ordinates; Basic formulas, equations and graphing; Lines, parabolas, and circles; The tangent line problem. Limits.
  • Additional topics (if time permits): Sequences and series; Complex numbers and vectors; Mathematical induction; Matrices and determinants.
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About PUMP Level 2:

PUMP Level 2 focuses on reading and understanding definitions and theorems, and writing basic mathematical proofs.  Within each topic, students will see definitions, theorems, ways to use each, and basic proofs.  There will be plenty of opportunities to write proofs in the style of each topic, explore the topic at a higher level of rigour than before, and receive feedback.

Topics include:

  • Symbolic Logic and Understanding Theorems: Basics of first-order logic, well formed formulas, and quantifiers; Translating between mathematical theorems as logical statements; Implications, the contrapositive, converse, and inverse; Basic proof techniques including direct proof, proof by contradiction, and proving the contrapositive.
  • Polynomials and Other Functions: Factoring polynomials of degree two, three, and higher; Factor Theorem; Simplifying rational expressions; Domain/Range; Injective/Surjective properties; Graphing and basic transformations.
  • Planar and 3-D Geometry: Equations of lines and planes; Normal and tangent lines; Triangles, circles, rectangles; Pythagorean Theorem, distance formula between points, a point and a line, planes; Adding/Subtracting vectors algebraically and geometrically.
  • Trigonometry: Definition of radians; Converting between degrees and radians - how to do it and motivation; Trigonometric rations; inverse trigonometric rations and right-angle triangle geometry; The unit circle; Sine and Cosine laws; Techniques for proving trigonometric identities; Proofs of the double/half angle formulas and adding/subtracting formulas; Graphs of trigonometirc functions and their tranformations; Applications.
  • Exponential and Logarithmic Functions: Definitions of "e", "In", and "log"; Exponential growth, decay, and modelling real-world problems; Inverse functions and solving exponential/logarithmic equations; Properities of logarithmic and exponential functions and their graphs.
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Current/Upcoming Sessions

We hope to offer PUMP courses in the upcoming academic year.  Please note for the Winter session, a minimum enrollment of 25 students will be needed in order for the Department to run this session.

For the Summer 2020, both levels of PUMP will be offered:

PUMP Level 1 will be offered in one session.

  • This session will run at a faster speed for 7 weeks with 6 hours of lecture per week (July-August).

PUMP Level 2 will be offered in one session.

  • This session will run at a faster speed for 7 weeks with 6 hours of lecture per week (July-August).  Week 1 and Week 6 will have 3 hours of lecture due to the Canada Day Holiday on July 1 and the Civic Holiday on August 5 respectively.

There will be a session of PUMP Level 1 in the Winter 2020 session.

Course Level OfferedSessionsCourse Start DateCourse End DateCourse ScheduleRegistration OpensRegistration
PUMP Level 1 - Regular Speed Winter 2020 Monday January 13, 2020 Wednesday April1, 2020 Lectures: Mo & We 6-8pm

Tutorials: W 5-6pm
October 7, 2019 January 10, 2020
PUMP Level 1 - Faster Speed Summer 2020 TBA   Lectures: Tu & Th 11am-2pm

Tutorials: Tu & Th: 3-5pm
PUMP Level 2 - Faster Speed Summer 2020 TBA   Lectures: Mo & We 11am-2pm

Tutorials: Mo & We 3-5pm


The location of lectures in the Summer 2019 session will be emailed to you.  In addition to lecture hours, there will be one hour of tutorial per week and possibly math aid centre hours.  The tutorial and math aid centre schedule will be announced in the lecture.





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Financial Information

What is the cost?

The fee is $600.00 and is payable by Visa or Mastercard only.


If you withdraw from the PUMP sessions on/or before the partial refund deadline, $500.00 will be refunded.  You must inform the office of your plans to withdraw as soon as you decide. Non-attendance is not accepted as proof of withdrawal.

No refunds of any kind will be made after the refund deadline dates.

Session Partial Refund Deadline Date
Winter 2020 PUMP Level 1 - Regular Speed Wednesday January 22, 2020 before 5:00pm EST
Summer 2020 PUMP Level 1 - Faster Speed TBA
Summer 2020 PUMP Level 2 - Faster Speed TBA
Winter 2021 PUMP Level 1 - Regular Speed TBA



Bursary assistance is available for current University of Toronto students based on financial need. 

Criteria to apply for bursaries:

-Bursary applicants must be returning or newly-admitted students to U of T (non-U of T students are NOT eligible).

-Returning students must be eligible to register (must not be suspended).

-Newly-admitted students must have accepted an offer of admission and must have registered in the Fall/Winter session (enrolled in courses and paid/deferred fees for the Fall/Winter).

-The student must demonstrate financial need.

-The student must present valid reasons for taking the course.

-The student must not have fees owing to U of T.

To apply for bursary assistance, please pick up a form at the main office BA 6290 - Bahen Centre, 40 St. George Street, Room 6290.  Late applications will not be accepted.  Please note that assistance is not guarantee.



To register for PUMP sessions, please click here.

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Are you coming from out-of-town?  Do you need accommodations?

If you require accommodation for the duration of PUMP, please contact the University of Toronto Housing Service at:

Koffler Student Services Centre
214 College St., Room 203
Toronto, ON M5T 2Z9
(416) 978-8045
(416) 978-1616 (fax)

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Contact Information

Academic Content and General Program Inquires

Sarah Cook
Undergraduate Assistant
Department of Mathematics
40 St. George Street, room BA 6290
(416) 978-5082
***We strongly encourage you to contact us by e-mail. Please put "PUMP inquiry" in the subject line

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Last Updated: June 21, 2019