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 2025. 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. 

Textbooks

PUMP 1 optional textbook (Summer 2025): TBA

PUMP 2 optional textbooks (Summer 2025): TBA

Financial Information

What is the cost?

The fee to register is $600.00 per course and is payable by Visa or Mastercard only. 
Note: If you are a first-year incoming student at the St. George campus, please do not register through this website and do not pay the fee. Instead, register through the Arrive Ready website.

If you are unsure whether you are eligible to register with Arrive Ready, please contact their team at arriveready@utoronto.ca.

Refunds

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.

Bursaries

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.

BURSARY FORM - PUMP 2025

The deadline to submit your bursary application is Friday, July 18th, 2025, and then can be submitted to 1styearmath.undergrad@utoronto.ca. 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.

Registration

PUMP REGISTRATION - SUMMER 2024

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.