MATH 5520 S1

Faculty of Graduate Studies- MATH 5220 Problem Solving II, Summer 2015
Course Description:

This course aims to further develop the problem-solving techniques begun in 5210. These include exploring symmetries and generalizations, the pigeonhole principle, strong induction, recursion methods, inequalities, and applications of calculus. We will cover material ranging from combinatorics, elementary number theory, the summation of series, calculus of a real variable, basic complex numbers, geometry, and basic algebra.  Some emphasis will be on solving problems appearing in mathematical contests such as national and international mathematical Olympiads and the Putnam competition. Last but not least, this course will explore the design of a Canadian competition for mathematical modeling.

The marks will be base on in-class work, assignments and a group based project.

  • Course Times: Tuesdays and Thursdays, 6:00 p.m. to 9:00 p.m.
  • Dates: May 19, 2015, to June 25, 2015
  • Location: N638 Ross Building
  • Instructors website:
  • Course Moodle Site:
  • Prerequisites: The formal prerequisite for this course is MATH 5220 Problem Solving I.

Course Objectives:

  • Explain the different types of mathematical propositions.
  • Explain basic algebraic concepts of factorization, equations, and fundamentals structures such as groups, rings, and fields.
  • Apply number theory and combinatorics to solving arithmetic problems
  • Explain complex numbers.
  • Apply basic summation formulas.
  • Explore the continuous, differentiable, and integrable functions in R
  • Explore various types of inequalities and several techniques for solving them
  • Explain common techniques for solving problems in Euclidean geometry.
  • Design a mathematical modeling competition framework for Canadian high schools.

Expectations: You are expected to:

  • join in a group, work regularly in class and some group work outside of class
  • prepare and present some material in class, and in the written project

In addition, you are encouraged to use the resources of the Internet to track information and discussions about problem-solving --see the course website for a link to problem-solving sites. You may be required to sign onto one of these lists, for a few weeks, and comment on the potential of such lists as a resource.

TextLoren C. Larson, Problem-solving through problems, Springer 2006.

Reference Text: Newman, Donald J. A Problem Seminar. Problem books in mathematics. New York, NY: Springer-Verlag, 1982.

Other materials: We will make use of the following materials:

Attendance: attendance is required at all classes. Attendance will be taken at the beginning and end of each class.

Evaluation: There will be weekly homework assignment on each mathematical topic, due the week after it is assigned, mainly from the textbook, and a final project.  Late assignment will be penalized at 5% per class. Graded work will include

Assignments 50%
Participation 20%
Project 30%
Total 100%

Topic Outline:

Nr Date Topic Reading Assignment
1 May 19th, 2015 Combinatorics & Probability
2 May 21st, 2015 Pigeonhole Principle and Strong Induction Chapter 2 Assig1
3 May 26th, 2015 Pigeonhole Principle and Strong Induction Chapter 2
4 May 28th, 2015 Number theory Chapter 3 Assig2
5 June 2nd, 2015 Number theory Chapter 3
6 June 4th, 2015 Algebra Chapter 4 Assig3
7 June 9th, 2015 Summation of series Chapter 5 Project midpoint check
8 June 11th, 2015 Inequalities Chapter 7 Assig4
9 June 16th, 2016 Geometry Chapter 8
10 June 18th, 2015 Geometry Chapter 8 Assig5
11 June 23rd, 2015 Real Analyses Chapter 6 Assig6
12 June 25th, 2015 Project presentation: Mathematical Modeling Competition   Project Written Component: Due
Evaluation Standards for Graduate Student Work.
Level Standard to be achieved for performance at the specified level
  • Fully achieves the purpose of the task, while insightfully interpreting, extending beyond the task, or raising provocative original questions.
  • Demonstrates an in-depth understanding of concepts, content and reasoning,
  • Communicates effectively and clearly to the intended audience, using dynamic and diverse means, connecting pieces with convincing arguments.
  • Fully achieves the purpose of the task, with some insight.
  • Demonstrates an in-depth understanding of concepts, content and reasoning.
  • Communicates effectively and clearly to the intended audience.
  • Accomplishes the purposes of the task.
  • Displays understanding of major concepts, and basic connections.
  • Shows clear understanding of concepts, but may include extraneous information or some less important ideas may be missing.
  • Communicates effectively.
  • Substantially completes purposes of the task.
  • Displays a substantial understanding of major concepts, even though some less important ideas may be missing and includes extraneous information.
  • Communicates successfully, including reasoning connecting concepts.
  • Purpose of the task not fully achieved; needs elaboration; some strategies may be ineffectual or not appropriate; assumptions about the purposes may be flawed.
  • Gaps in conceptual understanding are evident.
  • Limits communication to some important ideas; results may be incomplete or not clearly presented.
  • Important purposes of the task not achieved; work may need redirection; approach to the task may lead away from its completion.
  • Presents a fragmented understanding of concepts; results may be incomplete or arguments may be logically invalid.

Each piece of written work is expected to include:

  • Your current problem-solving questions and/or ongoing dialog on your previous questions.
  • Failure to include this will result in a deduction of one level (point) in the grade for the assignment.

Note: Work below a B+ may be returned to be redone. At this stage (Graduate work), it is essential that students work to the corresponding standards of mathematical thinking and presentation. In professional mathematics, it is common for journals to require a rewrite of a mathematical paper.  It seems appropriate to require the same from graduate students!

Also – you will be using material from multiple sources: colleagues, the class, the internet.  It is essential that you both cite the sources and that you show your own understanding of the material you choose to present!  Otherwise, it is considered plagiarism.

Assignment Submissions, and Lateness Penalties

Assignments may be submitted in class (or no later than midnight) by email on their due dates. Assignments received later than the due date will be penalized 5% per class. Exceptions to the lateness penalty for valid reasons such as illness, compassionate grounds, etc., will be considered by the Course Director only when supported by written documentation (e.g., a doctor’s letter).

Academic Honesty

Students are responsible for familiarizing themselves with policies regarding academic honesty as set out by the Senate of York University. Please read the Senate Policy on Academic Honesty (which can be found in the University Policies and Regulations section of the York University Undergraduate Programs Calendar.

Cheating and/or impersonation are dealt with severely.

In-Class Behaviour: All cell phones and pagers are to be turned off during lectures.