PUMP Courses

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

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 MAT133Y1 or MAT135H1 (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 their analogues).

Important Note about U of T Programs that Require Calculus for Admission:

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. 

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

  • Four Representations of Functions: Representing functions algebraically (with an equation), verbally (in words), numerically (with a table of values), and graphically (with a graph); Converting between the four representations of a function; Domain and Range of a function; Describing how functions are changing. 
  • Inequalities & Systems of Equations: General solving and graphing of algebraic inequalities; Systems of linear equations; General system solving. 
  • Fundamental trigonometry: Radians and degrees; The unit circle; Trigonometric functions and their graphs from the unit circle; Right triangle trigonometry; Basic applications and four representations of functions.
  • Transformations of Functions: Shifts and stretches of functions, represented algebraically and graphically.
  • Inverse functions: Definition of inverse of a function using four representations; inverse trigonometric functions.
  • Exponential and logarithmic functions: The number e; Four representations of exponential functions; Four representations of logarithmic functions as the inverse of exponential functions; Properties of the logarithms; Exponential and logarithmic equations and inequalities; Exponential growth and decay.
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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 rigor 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.
  • Functions: Representing functions algebraically (with an equation), verbally (in words), numerically (with a table of values), and graphically (with a graph); Converting between the four representations of a function; Domain and Range of a function; Describing how functions are changing; Examples of functions, including exponential. logarithmic, polynomial, and rational functions; Injective/surjective properties.
  • 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 trigonometric functions and their transformations; Applications.
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Current/Upcoming Sessions

We will be offering both levels of PUMP in the summer term online from July-August. Math Learning Centre hours may also be available for students.

Due to COVID-19, these courses will be delivered online until it is safe to meet in person again. Both lectures and tutorials will be delivered synchronously, so students should expect to attend class virtually at the scheduled times.

** Important note ** This summer the Faculty of Arts and Science will again be collaborating with our Department so that first year incoming students may enrol in either PUMP level 1 or level 2 for free through the Arrive Ready Program. In order to be eligible for free registration through Arrive Ready, you must be accepted to a Bachelor Degree Program in the Faculty of Arts and Science, in either the Physical and Mathematical Sciences or Rotman admission streams, commencing studies in fall 2021. If you fall into this category, please DO NOT register through the Math Department, as you will be charged full price for the course. First year students will receive an email from the Faculty of Arts and Science later in June with information on how to register.

Course Level Offered Sessions Course Start Date Course End Date Course Schedule Registration Opens Registration
PUMP Level 1  Summer 2023 TBA TBA Lectures: Tu & Th 11am-1pm

Tutorials: Tu & Th: 3-5pm
PUMP Level 2  Summer 2023 TBA TBA Lectures: Mo & We 11am-1pm

Tutorials: Mo & We 3-5pm


PUMP 1 recommended textbook (Summer 2022):

Functions of Modelling Change: A Preparation for Calculus, 6th Edition, by Eric Connally, Deborah Hughes-Hallett, Andrew M. Gleason, ISBN: 978-1-119-49837-7 

PUMP 2 optional textbooks (Summer 2022): 

How to Read and Do Proofs: An Introduction to Mathematical Thought Processes, 6th Edition, by Daniel Solow, ISBN: 978-1-118-16402-0

Proofs and Fundamentals: A First Course in Abstract Math, 2nd Edition, by  Ethan D. Bloch, 978-1-4419-7127-2

Note: textbooks may change between terms. If you are planning on taking PUMP, please check with the course instructor to see which textbook (if any) will be used for the course.

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Financial Information

What is the cost?

The fee to register is $600.00 per course 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
Summer 2021 PUMP Level 1 - Faster Speed  Friday, July 22nd 
Summer 2021 PUMP Level 2 - Faster Speed  Friday, July 22nd


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.

Summer PUMP Bursary Form

The deadline to submit your bursary application is Friday, July 22, 2022, and then can be submitted to 1styear@math.toronto.edu. Late applications will not be accepted. Please note that assistance is not guaranteed. 

Successful candidates who receive a bursary will still be required to pay the full registration fee upfront. Then once the course has been completed with a passing grade, the student will then be reimbursed.

Bursaries are not available in the winter sessions due to the Faculty of Arts and Science funding model.



Registration for summer 2022: https://pump.math.toronto.edu/pump/welcome

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Contact Information

For inquiries about our PUMP courses please email 1styear@math.toronto.edu with "PUMP inquiry" in the subject line.

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Last Updated: June 7, 2022