Math 1200 C Course Information
(Mar 15) The tenth homework assignment is available here. The class notes for March 11 are available here.
(Mar 6) The handout for the eleventh tutorial is available here.
(Feb 25) The handout for the tenth tutorial is available here. The ninth homework assignment is available
(Feb 11) A summary of what we have done to date with Pascal's
triangle is available here.
(Feb 2) The handout for the ninth tutorial is available here. The eighth homework assignment is available
(Jan 18) The handout for the eighth tutorial is available here.
(Jan 13) Click for an excerpt from Stillwell, Mathematics and its History and for the seventh homework assignment.
(Jan 6) The handout for the seventh tutorial is available here.
(Dec 17) The sixth homework assigment is available here. It is due on January 14.
(Nov 25) The handout for the sixth tutorial is available here.
(Nov 18) The fifth homework assignment is available here. It is due on December 3.
(Nov 11) The handout for the fifth tutorial is available here.
(Nov 5) Homework problems from Chapter 6 of the text are 6.4 (1), 6.5, 6.6, 6.9, 6.38, 6.40, 6.41. You will be expected to be able to discuss these problems and present solutions to them next class.
(Nov 4) The fourth homework assignment is available here. It is due on November 19.
(Oct 28 ) The handout for
the fourth tutorial is available here.
(Oct 21) The next homework assignment will be handed out next week. Use your time to prepare for the quiz and to
work on your
submission for the investigation project. Some exercises to help you prepare for the quiz are available here. (Oct 2) The handout for the
third tutorial is available here. Students
registered in Tutorial 2 who are interested in starting work on this
problem should consider attending the tutorial on October 9 (10:30 a.m. in
Ross South 101 A). (Sep 30) Additional questions to consider from
Chapter 2 of the text are 2.27, 2.29, 2.32, 2.33, 2.34, 2.36, 2.38, 2.41.
(Sep 29) The third homework assignment is available here. (Sep 29) The quiz on Thursday will consist
of one question in three parts. It is designed to see how well you
understand and can apply what you did to complete the first homework
(Sep 21) The handout for
the second tutorial is available here. Followup to Tutorial 1 is available here.
(Sep 14) The second homework assignment is available here. It is due on October 1. The explanation as to why
your answers are
correct involves material from sections 2.1 -- 2.4 of
Chartrand, Polimeni, Zhang. Sections 2.1 -- 2.5 will be discussed in
class. Suggested problems to consider are 2.3, 2.5, 2.11, 2.13,
(Sep 4) The handout for the first tutorial is available here. The first homework assignment is available here.
Instructor: Eli Brettler
Office: South 508 Ross
Telephone: 736-2100 Extension 66321
Normal Office hours: By appointment. I am normally available Tuesday between 4:30 and 6:30 p.m., Thursday between noon and 2:00 p.m., Friday between noon and 2:00 p.m.
Schedule of Classes and Tutorials (with some due dates shown):
FULL CLASS MEETINGS, Thursdays 10:00 - 11:30, CLH 110
Sep 10, 17, 24,
Oct 1 (Quiz 1), 8, 22, 29 (Quiz 2),
Nov 5, 12, 19, 26 (Quiz 3),
Jan 7, 14, 21, 28 (Quiz 4),
Feb 4, 11, 25 (Quiz 5),
Mar 4, 11, 18, 25 (Quiz 6),
TUTORIAL 01, Fridays 10:30 - 11:30, RS 101A
Sep 11, 25,
Oct 9, 30,
Feb 13, 27,
Jan 8, 22,
Feb 5, 26,
Mar 12, 26.
TUTORIAL 02, Fridays 10:30 - 11:30, RS 125
Oct 2, 23,
Nov 6, 20,
Jan 15, 29
Mar 5, 19,
Apr 5 (Monday)
Tutor: Dorota Mazur
Text: Chartrand, Polimeni, Zhang:
Mathematical Proofs, A Transition to Advanced Mathematics, Second
Edition. We will cover Chapters 2 - 7.
Supplementary Text: John Mason with Leone Burton and Kaye Stacey,
Thinking Mathematically. This book gives an approach to problem solving and the problem solving experience. It is also a source for rich and varied problems.
Statement of Purpose:
This is a critical skills course. Here are some questions to consider.
Just what are the objects which you consider when you do mathematics?
What is meant by the fraction one half? How does it represent a ratio? How does it represent a quantity? Are these conceptions different? Can you reconcile them?
What is meant by a proof?
How would you describe a triangle to someone (for example a blind person) who has never seen one.
How would you describe a circle?
What conventional conceptions do you have which inform your own thinking about these and other mathematical objects?
How you convince yourself, and how do you convince others that an answer is correct? What are the conventions for presenting concise mathematical proofs? How well does the presentation reflect the means by which a particular mathematical discovery was made? What does it mean for an ordinary language argument (mathematical or otherwise) to be valid? What is a counterexample? How does one make conjectures and how does one go about trying to assess whether they are correct?
Can you learn problem solving?
It is pretty easy to convince oneself or others of the correctness of answers which seem intuitively correct. What is much harder is to convince when answers while correct are counterintuitive. An example some of you may have seen is the "Monty Hall Problem".
Most of the problems you solved in High School were done mechanically or by mimicking solutions to similar problems in the textbook? What means are available to deal with problems which are genuinely novel?
Are there techniques which extend your problem solving and proving capabilities?
The text, "Thinking Mathematically" by John Mason has a rich selection of problems for consideration. Most require minimal technical background but almost all require hard thinking. Mason suggests a way of working strongly grounded in self awareness both in terms of what you are doing, and how you feel while doing it.
You will learn about combinatorial proofs which are arguments based on the analysis of situations rather the manipulation of formulas.
What are expected outcomes of this course?
You will learn about recursive methods and mathematical induction as a tool in calculations and in proofs.
You will learn to use representations from other branches of mathematics (for example, geometric models to solve probability problems) to help obtain answers.
You will learn to present proofs and explanations which are concise and logically correct.
You will learn to take risks as you engage with learning new mathematics and doing mathematical problem solving.
You will learn to express mathematical ideas with precision and clarity.
You will learn to ask questions whose consideration can lead to deeper understanding.
|Individual Investigation and Writing Assignments||One assignment to be handed in every other week||25%|
|Investigation Projects ||See below||20%|
|Quizzes||3 Fall, 3 Winter||15%|
|Final Examination||Winter examination period||30%|
Participation: Participation is how you show your commitment to the course and to the other students taking the course with you. You are expected to share both of your mathematical knowledge and the feelings you have as you engage in doing mathematics.
Attendance at the weekly classes and at the tutorials is obligatory. You will lose 2 points from your course grade for each class or tutorial in excess of two which you miss each term.
The tutorial handout, which includes the problem or problems for consideration, will be posted prior to the tutorial meeting. This is to give you an opportunity to review it and consider (in the case of a problem) what might be involved in its solution. You are expected to actively participate in small group and whole tutorial group discussion.
Individual Investigation and Proof Assignments: Questions for investigation and solution will be assigned biweekly. Solutions are to be handed in. You may be asked to include a journal style discussion of how your solution or solutions were discovered. The following grading rubric will be used.
Homework will be graded from 4 points. Grades will be assigned as follows:
Note that to receive full credit (4 points from 4) you must go beyond simply solving the problem as posed. Learn to think of your solutions as a starting point.
- Level 4: (4 points from 4) Deep understanding of the problem.
Complete solution carefully presented. Provides multiple alternative solutions where
possible. Considers variations based on the original question (with or without solutions).
- Level 3: (3 points from 4) Good understanding of the problem.
Problem solved or a solution provided which can easily be completed, for example,
one with a minor error which would be simple to correct. No evidence of engagement
beyond finding an answer to the problem as posed.
- Level 2: (2 points from 4) Incomplete understanding of the problem.
Limited progress to solution or a solution marred by major errors.
- Level 1: (1 point from 4) Minimal understanding of the problem.
Work submitted shows little progress toward solution.
Do your own work. Don't look for a solution on the web or take one from another student's work unless you already have found your own solution and intend to review another to make a comparison. Work that is not original will be graded accordingly. Presenting someone else's work as your own without proper citation is academic dishonesty. You must cite any internet sources which you have consulted. You will be required to take the York University Academic Integrity Tutorial.
The lowest assignment grade each term will be dropped. Some assignments may be designed so that they can be handed in a second time with corrections.
Investigation Projects: After each tutorial, you are expected to continue working on the problems discussed. Each project will consist of the results of deep and sustained investigations of your choice of three (of the six or more) tutorial problems considered each term. As with the homework, you should consider multiple solution methods, extensions of the problems, relationships with other related problems. You should include a report on your experience (how you felt) during the process of investigation. Each term you will be required to hand in early (as an indication of your progress) your report for one (of the three) problems. Due dates for the first problem are Monday, November 2 (Fall) and Monday, February 23 (Winter). Due dates for the full reports are December 8 (Fall) and April 5 (Winter). The grading breakdown is 3 points for the each of the single problem reports and 7 for each of the three problem reports.
These replace the Final Group Investigation Project which was assigned last year. Click here for an example of an A+ project submitted last year.
There will be 6 in class quizzes, 3 per term.
Here are some sample quiz question types:
The grade will be obtained by taking the average of the best 2 quiz grades from each of the terms. There will be no makeups for missed quizzes.
- Given a problem and a sketch of a solution, formulate a more complete solution and present it with justification.
- Given a proof of some result, find any errors and correct them.
- Given various conjectures, find counterexamples if false, proofs if true.
- Provide simple proofs including direct proofs, indirect proofs, proofs by mathematical induction.
Final Examination: This will be a conventional timed, closed book exam, scheduled during the University Final Examination period. Question types would be similar to those examples given for the quizzes. Click here to view last year's examination.
Handouts from class:
Tutorial 1 (September 11 and 18)
Assignment due September 17
A summary of some material discussed in the first two class meetings.
Assignment due October 1
Tutorial 2 (September 25 and October 2)
A summary of discussion, September 11 and 18 (Tutorial 1)
Problem from Krantz, Techniques of Problem Solving, September 24
Assignment due October 22
Tutorial 3 (October 9 and October 23)
Sample "Proof" Exercises.
Tutorial 4 (October 30 and November 6)
Assignment due November 19
Tutorial 5 (November 13 and November 20)
Assignment due December 3
Tutorial 6 (November 27 and December 4)
Proof from Krantz (December 3) and Proof Reading Details Handout
Assignment due January 14
Tutorial 7 (January 8 and January 15)
Excerpt from Stillwell, Mathematics and its History
Assignment due January 28
Tutorial 8 (January 22 and January 29)
Tutorial 9 (February 5 and February 12)
Assignment due January 28
Assignment due February 25
Assignment due March 11
Tutorial 10 (February 26 and March 5)
Tutorial 11 (March 12 and March 19)
Assignment due April 1
Class notes for March 11
Tutorial 12 (March 26 and April 5)
To read files in pdf format you can use the
the free Acrobat reader.
Note: The last date to drop the course without
academic penalty is Feb 6.