PUMP Courses

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.

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.

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.

IMPORTANT NOTE: This summer the Faculty of Arts & 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 the Physical and Mathematical Sciences, Computer Science, Life Sciences, Social Sciences, Rotman Commerce admission streams, commencing studies in fall 2023. 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 incoming students can register through the Arrive Ready website

Course Level Offered Sessions Course Start Date Course End Date Course Schedule Registration Opens Registration
PUMP Level 1 
Summer 2024 Tuesday, July 9th, 2024 Thursday, August 22nd, 2024

Lectures: Tu & Th 11:00am - 1:00pm 
Tutorials: TBA

PUMP Level 2
Summer 2024 Monday, July 8th, 2024 Wednesday, August 21st, 2024

Lectures: Mon & Wed 3:00pm - 5:00pm 
Tutorials: TBA 




PUMP 1 optional textbook (Summer 2023):

Functions Modeling Change: A Preparation for Calculus, 6th Edition, ISBN: 978-1-119-49837-7

PUMP 2 optional textbooks (Summer 2023): 

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.

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
Summer 2023 PUMP Level 1 Friday, July 19th, 2024
Summer 2023 PUMP Level 2 Friday, July 19th, 2024


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.


The deadline to submit your bursary application is Friday, July 19th, 2024, and then can be submitted to 1styearmath.undergrad@utoronto.caLate 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.



Note: First year incoming students should register through the Arrive Ready website

Contact Information

For inquiries about our PUMP courses please email 1styearmath.undergrad@utoronto.ca with "PUMP inquiry" in the subject line.

Meet Your Course Instructors

PUMP I Instructor: Sebastian Gherghe

"My full name is Sebastian Tudor Gherghe and I was born and raised just outside of Toronto, ON. My parents are immigrants from Romania and I've spent quite a bit of time there as well (my parents made sure that I have dual citizenship). I did my undergrad at UofT in math and physics as well as a masters in math and I'm currently chasing down my PhD! Some of my hobbies include sports (I am a huge soccer fan, but I also enjoy bouldering, powerlifting, salsa dancing) and video games (I love Fromsoft games). Last fall I had a bad knee injury playing soccer and managed to tear my meniscus and ACL, so the last year has been pretty rough for me. I'm looking forward to playing sports again soon! 

Fun fact about me: Even though I was born in Canada, I started off school as ESL for half a year because my grandparents never learned English!"

PUMP II Instructor: TBD