MATH 1200
section B - Problems, Conjectures, Proofs
2012 - 2013
Professor:
Mike Zabrocki
email:
Office: TEL 2028
Office hours: Monday 12:30-2:30pm, Thursday 4-5pm
Textbook: Mathematical Proofs:
A transition to advanced mathematics, by Chartrand,
Polimeni, Zhang
As a alternate/optional textbook: Thinking
Mathematically, by Mason, Burton, Stacey
Calendar copy: Extended exploration of elementary problems
leading to conjectures, partial solutions, revisions, and
convincing reasoning, and hence to proofs. Emphasis on
problem solving, reasoning, and proving. Regular
participation is required. Prerequisite: 12U Advanced
Functions (MHF4U) or Advanced Functions and Introductory
Calculus (MCB4U). NCR note: Not open to any
student who is taking or has passed a MATH course at
the 3000 level or higher.
Most High School mathematics problems are solved using
algorithmic methods or via reference to model solutions. One
purpose of this course is to enable students to develop the
confidence and ability to attack richer and more demanding
problems. The attempt to check work and to explain one’s
discoveries to others leads naturally to the need for
explanation. Learning how to present convincing reasoning —
or proof — is one of the course outcomes.
With an emphasis on communication/convincing argument, there
is a critical contribution to be made by: group work,
reading a proposed 'proof' including other student's work,
presenting and discussing as a whole class. There is also
value in working through several different approaches to
solve a problem, and taking the time to understand an
alternative approach offered by a peer in the class. Doing
mathematics well includes talking and listening to
mathematics and there will be assignments that require
collaborative work with another student in the class, as
well as support for forming study groups.
The main goal of this course is to develop skills that lead
to understanding and communicating a convincing argument.
Support will be given for proof presentation, especially for
the kinds of proofs that students are expected to produce in
their second year and higher level courses. This includes
inductions, and arguments with counting and with
inequalities. Formal proof writing exercises will be
introduced in the second half of the course, once problem
solving and informal justification skills reach an
acceptable level.
Class and tutorial attendance is mandatory and active
participation is expected of all students.
(Sept 6, 2012) URGENT! This course has currently too many students.
We are looking for volunteers to move to a new section which has just opened
up for Thursdays from 8:30-10am. The first meeting of this section
will be on September 13 so please enquire if you would be willing.
Interested students should contact Janice Grant at the mathematics undergraduate
office (janiceg@mathstat.yorku.ca).
Evaluation:
The evaluation will be based on the
following criteria
Participation
based
on attendance and in class assignments
Assignments
roughly
one every 4 weeks
20%
Tutorial
writeups
see
below
25%
Quizzes
6
total, 3 per term, best 2 from each term
25%
Final
Examination
Winter
exam period
30%
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.
Participation: You are expected to show your commitment to
this course and your fellow students by sharing your mathematical
knowledge of the material. Attendance at the
weekly classes and the tutorials is obligatory and you will lose 2
points from your course grade for each class or tutorial that you
miss each term. Sometimes attendance at the weekly classes will be
measured by a short in class assignment. Non participation in
these assignments will result in a lowering of your participation
grade. Note that participation is not a percentage of your grade,
but non-participation and attendance can lower your overall score.
Assignments: There will be roughly one assignment 4-5
weeks. Most assignments will require explanation beyond a simple
one or two word/numerical answer. It is good practice to RECOPY
THE QUESTION EVERY SINGLE TIME when you do the assignment. This
makes it possible to understand what the assignment when it is
handed back to you and it attempts to reduce the error of
answering a different question than is on the assignment. Full
credit is given to papers which demonstrate deep understanding of
the problem by providing multiple solutions and considers
variations based on the original question when this is
appropriate. Your assignment should include complete sentences and
explanations and not just a few equations or numbers. A solution
will not receive full credit unless you explain what your answer
represents and where it came from. You may discuss the homework
with other students in the class, but please write your own
solutions.
Note: Late assignments will be penalized by 20% per day. This will
apply to any homework handed in after the class time in which it
is due. In addition, assignments which are handed in late are
unlikely to be marked in a timely manner.
Tutorial Assignments: You are expected to continue working
on the problems discussed in class and in the tutorials and to
keep a running record of the problems from those exercises (these
will be listed on the web page) as well as your progress and the
development of a solution for them. Here is a breakdown of some
aspects that I plan to evaluate your solutions from the tutorial:
(1) The discussion begins with an explanation of the problem
(2) The explanation should convince the reader that the meaning of
the question is understood (e.g. small examples, a clearly labeled
table of data, and/or a discussion of the meaning of the question)
(3) diagrams, tables or images that are drawn to aid the reader in
understanding the problem are well labeled and explained
(4) Clear statements are made of conjectures that are believed to
be true
(5) Explanations of why those conjectures are true are included
(6) An explanation of how the problem solving process proceeded is
clear from the explanation
(7) The entries consist of writing which is clear and
grammatically correct
(8) A conclusion about the solution to the problem is reached
On both your journals and assignments, I will be looking for
evidence of your solutions demonstrating one of the following 4
levels of understanding: Level 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: 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: Incomplete understanding of the problem. Limited
progress to solution or a solution marred by major errors. Level 1: Minimal understanding of the problem. Work
submitted shows little progress toward solution.
Note that to receive full credit you must go beyond simply solving
the problem as posed. Learn to think of your solutions as a
starting point.
Quizzes and Final Examination: There will be 3 quizzes per
term (dates listed below). A final examination will be scheduled
for the April exam period and the date announced in
late-February/early March.
(Sept 6, 2012) URGENT! This course has currently too many students.
We are looking for volunteers to move to a new section which has just opened
up for Thursdays from 8:30-10am. The first meeting of this section
will be on September 13 so please enquire if you would be willing.
Interested students should contact Janice Grant at the mathematics undergraduate
office (janiceg@mathstat.yorku.ca).
(Sept 10, 2012) There may be several
editions of the textbook floating around. Any of them should
suffice. We will primarily be following along with Mathematical
Proofs: A transition to advanced mathematics. Having the textbook
is not critically necessary but is recommended. You may also want
to find Thinking Mathematically as an optional textbook. It gives
techniques that are helpful for solving problems as well as sample
problems.
(Sept 10, 2012) Those that would like to review some concepts that
you are expected to have from high school, there are a few online
references I can recommend to start: (1) There is a quite
extensive set of algebra tutorials, covering a wide range of
topics, maintained by West Texas A&M University. The URL for
the main page of this resource is http://www.wtamu.edu/academic/anns/mps/math/mathlab/col_algebra
though a quicker way to get to this page is via the link on the
Bethune College Math Help page, http://www.yorku.ca/bethune/math
(2) There is also a detailed online course in Trig basics, which
starts right from the beginning, the URL for the main page of the
course being http://www.yorku.ca/bethune/math/trig.html
(as you can guess from this URL, the Bethune Math Help page has a
link which takes you to the Trig course main page). Other
potentially useful information is also listed on the Bethune Math
Help web page, and further online resources will be added there as
they become available.
(Sept 17, 2012) There has not been significant movement on the enrolment numbers
for this class. At the moment the only people who are getting into 1200 are
those that are signing up for the Thursday morning section.
(October 1, 2012) Just to give you an idea of what we are aiming towards in this
class, here are some copies of old exams (2011)
(2010).
(October 10, 2012) I will have to cancel my office hours October 11, 2012 because I will
be out of town. If you need to meet with me I will be available Monday 12:30-2:30 or
it is possible to make an appointment to meet with me.
(October 15, 2012) I have opened up a moodle
for this course which will allow you to track your grades. This is an internet tool
which will allow me to post interactive course material. You should log in with
your passport York account. Currently your quiz grade as well as your
homework grade should be online.
(October 22, 2012) I said at one point in one of the classes that I would try to put
up a better explanation of why the number of subsets of an $n$ element set is equal
to $2 \times$ the number of subsets of an $n-1$ element set because my first one
was awkward when I wrote it on the board in class.
It doesn't matter what the $n$ element set or the $n-1$ set that we are working with
is, so let us consider the set $\{1,2,3,\ldots, n-1\}$ as our set with $n-1$ elements.
Every subset of this set is also a subset $\{1,2,3,\ldots, n\}$. Moreover, every
subset of $\{1,2,3,\ldots, n-1\}$ with the element $n$ added in the set is also
a subset of $\{1,2,3,\ldots, n\}$. Therefore, we have produced $2 \times$ the
number of subsets of $\{1,2,\ldots,n-1\}$ which are subsets of $\{1,2,3,\ldots, n\}$.
Since every subset of $\{1,2,3,\ldots, n\}$ either does not contain $n$ (and hence is one
of the first type of subsets) or it contains $n$ (and is one of the second type of
subsets) then we have accounted for all of them.
(October 26, 2012) I was reading a newsletter and the following advice was written by
a mathematician Audrey Terras
(who was one of my instructors when I went to school).
"1. Don't (EVER) give up. Mathematics is hard to do.
Progress can be slow. Patience is necessary. It is important to keep working.
When stuck on a problem, start writing up the proofs of the things you have
already done. Heisenberg said: 'You just have to be able to drill in very hard wood
... and keep thinking beyond the point where thinking begins to hurt.'" If you would
like to read the rest of her comments,
I found them here.
(October 29, 2012) Since classes aren't being held on Thursday, I won't be holding
my office hours Thursday afternoon either. If you need to see me then, please
either schedule an appointment for another time (I can make Tuesday at 4pm this
week).
(November 4, 2012) I ran across an article
Why Writers Should Learn Math. His case says that a writer should understand
the beauty and ideas that one finds in mathematics.
I would go much further than this author does because
I often see how mathematical logic (as an example of a recent topic of discussion in this class)
is imbedded deep within our language and how
this logic is then lost by many who are in a position to exposit political
arguments, philosophical questions and descriptions of scientific reasoning. Sigh.
(November 17, 2012) I found a video that I really liked. Starting off in math
and want to tell people why?
(November 19, 2012) I ran over a little tonight. I showed that if $2|m$ and $3|m$,
then $6|m$. Next, I asked you to show that for $a$ and $b$ are positive integers:
if $a|m$ and $b|m$, then $(ab)|m$. Try and come up with an explanation of this
for next time.
(March 3, 2013) Here is an application of symbolic logic.
(March 9, 2013) I finished grading the quizzes only yesterday. They took a long time to
read. The solution to the quiz is posted on the moodle
under quiz #5. Please read it carefully and make sure that you understand it. I posted
the following instructions at the end of the description.
If you had a score $\leq$ 24 on this quiz I would like you to do homework #5
problem 1 and hand it in for Monday, March 18
If you had a score $\leq$ 16 on this quiz I would like you to do all of homework #5
and hand it in for Monday March 18
(I am extending the due date because it took me several days to read
these assignments)
(March 9, 2013) As discussed in class, you should do the following 80 problems as practice for the final.
Find some finite sets $A, B, C$ (each with just a few elements)
and functions $f : B \rightarrow C$ and $g : A \rightarrow B$ such that
1-8. $f$ is $X$, $g$ is $Y$ and $f \circ g$ is $Z$
where $X, Y, Z \in \{$ onto, not onto $\}$ or show that such functions do not exist.
9-16. $f$ is $X$, $g$ is $Y$, $f \circ g$ is $Z$
where $X, Y, Z \in \{$ 1-1, not 1-1 $\}$ or show that such functions do not exist.
17-80. $f$ is $X_0$ and $X_1$, $g$ is $Y_0$ and $Y_1$, $f \circ g$ is $Z_0$ and $Z_1$
where $X_0, Y_0, Z_0 \in \{$ onto, not onto $\}$
and $X_1, Y_1, Z_1 \in \{$ 1-1, not 1-1 $\}$ or show that such functions do not exist.
You should also do the practice for the final
that I posted and be prepared to ask questions in tutorial.
(March 9, 2013) You should be practicing
for the final exam now. It will take place April 18 at 7pm.
(March 18, 2013) I am not going to collect the 6th homework assignment since I can't really
use it to give you feedback before the last quiz and final. But like all of the homework
assignments it is necessary practice for your last quiz (compare the quizzes with
the homeworks to see how they are related, the same will be true of this one) and
for the final exam.
(March 18, 2013) I received the following message about course evaluations:
The Winter 2013 and Fall/Winter 2012-2013 Course evaluations will be available on-line tomorrow (March 19, 2013).
The URL link is http://courseevaluations.yorku.ca.
Students should go on-line and complete the course evaluations.
They have from March 19, 2013 to April 4, 2013 to submit the evaluations.
Please encourage and remind your students to go on-line and complete the course evaluations.
(March 18, 2013) Here are pictures of working and after the work is over.
I tried, but the photo doesn't quite seem readable.
(April 1, 2013) I wasn't joking about my bucket list for university:
1. Learn a second (or third) language
2. Learn to program
3. Don't read - do problems, problems problems
4. Get help from professors, TAs, mathlab, Club Infinity, friends, classmates
5. Do a research project, study abroad, or do something else that will make you happy while you are here (join clubs, sports, hobbys...)
(April 1, 2013) I will not have office hours the week of April 8, but I will have office
hours April 15, 3-5pm and April 17, 10am-12pm. The exam is April 18 at 7pm in LAS C
(Lassonde Building LSA (formerly Computer Science & Engineering Building)). NB: I had to change my office hours from 2-4 to 3-5 on April 15,
because I had a meeting scheduled 12-3
