Math 5020

Fundamentals of Mathematics for Teachers

Description: Number theory and combinatorics are branches of mathematics in which challenging problems can be explored that require a background with which most students are familiar. This course deals with topics in these two fundamental mathematical fields, including modular arithmetic, linear and quadratic diophantine equations, continued fractions, permutations and combinations, distributions and partitions, recurrence relations and generating functions. It is one of two required courses for the Mathematics for Teachers program. An emphasis in this course is placed on writing and explaining mathematics clearly.

Prerequisite: Permission of the instructor is required for students who are not in the Graduate Programme in Mathematics and Statistics.

Professor: Mike Zabrocki
Office: TEL 2028
Hours: Thursday 4-5:30pm
phone: 416-736-2100 x33980
e-mail: zabrocki at mathstat dot yorku dot ca
Class meets Thursday 6-9pm in Ross S105
Text : Number Theory by George Andrews

We have a FORUM for math 5020.  This is a web site that will allow you to post questions/comments/communications about the course.  I will try to keep track of postings and answer them on a regular basis.

Grades for the course will be based on the following criteria:

Class presentation
Un-exams (4 x)
Forum exercises Fall 15%+Winter 15%
Class participation (incl attendance) Fall 5%+Winter 5%

A list of homework problems for this class (you should be doing a lot, if not all of ALL of these homework problems). I will be grading your solutions only to the ones that you post on the forum though.

A list of topics for presentations. I will be updating the dates for these as we go along. Make sure you talk to me one or two weeks in advance of your presentation.


3/23/08- the last unexam
3/23/08- A partition identity explained with generating functions
3/23/08- Mario's presentation on binomial coefficients in power point in in pdf
3/6/08- Partition generating function problems- this one I will be assigning the Forum problems from
3/6/08- Partition generating function matching problems- worksheet for March 20
2/28/08- Fibonacci generating function exercises
2/21/08- a worksheet on taking coefficients in generating functions
2/21/08- the third unexam
2/7/08- Jen Woodcock's computation of some generating functions
2/7/08- Combinatorial problems with generating functions
1/31/08- A proof that the number of solutions to x1+x2+...+xk=n is k^n
1/24/08- Thomas' slides on Elliptic curves in PDF and PowerPoint formats.
1/24/08- a reference sheet of some generating function examples
1/24/08- Articles from the 80's and 90's about the first big breakthroughs of quadratic sieve algorithm
1/17/08- taking coefficients in a generating function
1/10/08- summary of RSA public key cryptography
1/10/08- presentation on Goldbach's conjecture
1/3/08- Sequences and sets of objects and the OLEIS
12/6 - Drill and Skill Number Theory Questions
11/29 - Multiplicative functions
11/08 - Chinese Remainder Theorem


May 9, 2008- I posted the grades for this class here. Sorry that it took me so long. I had to have every grade in place before I could submit them and there were a number of late assignments from the third unexam which I hoped would take less time to do than it did.

March 23, 2008- I have posted two files. One is the last unexam which I had wanted to get to you on Thursday but I decided needed to be edited before I gave it out. I have listed a due date as April 10. You can email or give me a hardcopy before that date. If you decide to give me an electronic copy then make sure that your file is either in PDF format or contains ONLY text. If you try to use other symbols then it will not show up on my computer. Please do not discuss these assignments except in class next week. The other file I posted is an example of a justification of a generating function identity which is an infinite sum on one side of an equation and an infinite product on another. I will talk about this in class next week also because I claim that a slight modification of my explanation will also show that

March 23, 2008-I posted Mario's power point presentation on binomial coefficients.

March 20, 2008- I listen to NPR podcasts 'most emailed stories' and in the last couple of days there were two stories about math education in U.S. schools. One is about a program called Singapore math, another is about how Massachusetts is trying to improve its math program. These stories are related to a report which is due to be released by a panel of experts about how to improve mathematics education in the U.S..

March 18, 2008- After reading your unexams I get an impression that some of you are behind in this class. If you skipped one or two of the worksheets that I gave you go back and do them. Just because I am not grading them does not mean that they are an optional part of this course. You should be capable of doing every problem on every worksheet. I hope that one thing that you take away from this class will be that math is cumulative and if you do not keep a steady momentum then you will fall behind. If you said a few months ago "I don't have to know this widgets and doodles c^%$, he is just making us do these assignments because he doesn't have anything better to do with his time," then I hope you are not saying that now.

March 18, 2008- I have only 3 people who have asked to check their answers on the matching worksheet that I assigned posted last week. The rest of you are going to have to do it sometime. If you want to get a head start on generating functions involving Durfee squares this diagram will help.

March 18, 2008- Rant: How is it possible that 1/2 of the class did not write the question on the last unexam assignment that they handed in? Many of you answered a question that did not have to do with calvinball. It is almost as if I asked for an essay about the Mexican-American war and I got one back about Teddy Rosevelt. In order for me to grade these I am looking for a clear explanation which takes me from the question to the answer. It is impossible to grade this assignment if you don't start your explanation at the question!

March 12, 2008- I assigned forum problems from the worksheet on partition generating functions. These problems are similar to the examples that I did in class, but I haven't had the chance to do more advanced examples yet. I put two examples on the list of problems page.

March 12, 2008- We only have two classes left so I expect you to come prepared next week with questions. The unexam should have been a piece of cake yet I still found errors or people did not even get the right answer. "Why?" I need to know what you don't understand so STOP me in class if you are confused. I think I am using just addition and multiplication for everything so that everyone should be able to do these.

March 6, 2008- There will be no class March 14 for the March break week that Ontario schools have off.

February 29, 2008- Take a look at the GIMPS website. This is just one of several massively parallel processing projects which are interesting applications of mathematics and computers. I would like to know if you decide to try out the computational software.

February 29, 2008- Please note that the following instructions still hold for the third unexam which is due next week: Please do not work together on this assignment. I would like to see original compositions.

February 29, 2008- I asked you to do as homework to find a formula for the Lucas numbers in terms of phi^n and phibar^n. Violeta has agreed to post her solution on the forum. I am looking for volunteers for posting a solution to problems from the Fibonacci generating function worksheet on the forum. If you are willing and able let me know.

February 25, 2008- I said in class that in SAGE the command for computing the taylor series for n terms is "series( expr, q, n)". I was mistaken and the correct command should be "taylor( expr, x, 0, n)". For example:
sage: taylor( x^2/(1-x), x, 0, 10 )
this command should compute the taylor series for q^2/(1-q) for 10 or so terms. If you want to use another variable like q, you should execute a command like:
sage: q = var('q')
sage: taylor( q^2/(1-q)/(1-q^2), q, 0, 10 )

February 21, 2008- A link to the computer algebra package SAGE. You also might want to use your computer browswer for computing with SAGE, (although I haven't gotten this to work yet) and you should consider exploring this link here.

February 21, 2008- A link to the BBC's website for the Simon Singh's radio program on the number 0.

February 21, 2008- I posted the next unexam. Please come to class tonight and ask questions. It is due in two weeks.

February 19, 2008- Catherine found a website about the historical calculation of Mersenne primes during class on Feb 7. I forgot to add this link here. FYI, what is not clear on that page is that the 40th through the 44th are not know for sure to be the only ones. There still might be a few missing Mersenne primes. Try to find a pattern on the first 39. Watch the newspapers when the next one is found because there is usually a short news story about it when they are found.

February 18, 2008- What do you think? (this is from

February 18, 2008- The education department is thinking of switching to a 5:30pm class schedule (rather than 6-9 it would run 5:30-8:30). I've been asked to get feedback about what our students think about the MA for Teachers program also switching to an earlier time. Personally, I hate being at York until 9pm or later but I don't want to move future classes earlier if there are too many people who will be late (we already have people trickling in later than 6pm). I will try to remember to ask you about on Thursday.

February 7, 2008- Reminder: no class on Feb 14. Please work on your forum problems on the week off. I won't be able to respond until after I get back, but I plan to give you the next unexam on the 21st and it will be similar.

February 6, 2008- I looked in some of the textbooks that people leant me towards the beginning of the school year. There is one titled "Geometry and Discrete Mathematics" which is published by Addison-Wesley. On page 380 there is a page titled "Focus on ... The Divider" which is essentially special cases of the second unexam. You are expected to explain this to your students, you should be able to explain it to me.

February 5, 2008- The summer course "Math 5300: Computation for Teachers" will start Monday, May 5, 2008 and will be offered 6-9pm Monday and Wednesday evenings. I will be teaching the first half from May 5 until June 9. Hongmei Zhu will be teaching the second half. The topic will be on digital images and computer graphics. The reason we chose these topics is that there are great applications of high school level mathematics built into computer software such as Photoshop and similar programs, and there is a lot of mathematics to learn by looking at them more closely.
For example: How do you change an image from color to black and white or sepia? Why is the number of megapixels on your digital camera important? How do you remove red eye from a digital photo? What do any of these questions have to do with mathematics?

Course description: This course will explore the relationship between mathematics and the use of computers. The topic of digital images and computer graphics will be a main focus this term and the mathematics behind popular image manipulation software. Students will learn how computers represent images and then how subjects learned in high school such as calculus, linear algebra and discrete mathematics relate to computer graphics and digital image processing.

February 4, 2008- Remember that I gave as homework : Show that
(q+q^2)/(1-q)^3 = sum_{n\geq0} n^2 q^n
and (q+4q^2+q^3)/(1-q)^4 = sum_{n\geq0} n^3 q^n.
You already know that
1/(1-q)^2 = \sum_{n\geq0} (n+1) q^n,
1/(1-q)^3 = \sum_{n\geq0} (n+1)(n+2)/2 q^n
and 1/(1-q)^4 = \sum_{n\geq0} (n+1)(n+2)(n+3)/6 q^n
so all you need to do is use the equations for the generating functions you know to find the ones that you want a formula for.

February 4, 2008- I said that I would give a 'meaning' to the expression 1/(1-q A(q)) and we didn't have time/I forgot last week.
The coefficient of q^n in 1/(1-q A(q)) is

This is the generating function for tuples of widgets (of any length) such that the sum of the sizes of the widgets plus the number of widgets is n.

If you have any questions about this generating function please ask me about it during class this week.

February 1, 2008- There will be no class on February 14th because this is our reading week and I am planning to not be there. I will give you an assignment to work on the week before.

January 28, 2008- Thomas gave a problem in class on Thursday. If you stack cannonballs in a pile with 1^2 cannonballs on the first level, 2^2 cannonballs on the second level, 3^2 cannonballs on the third level, etc. How many levels (greater than 1) do you need to have so that when the stack is laid out flat that it can be made into a square?
(a) write an elliptic equation which describes the cannonball problem
(b) use elliptic equation addition to solve the problem (hint: (0,0) and (1,1) should both be points on your curve and by adding them together you will get a third).

January 23, 2008- Betty posted a message on the forum about a book review of "The Indian Clerk" about the relationship between Hardy, Littlewood and Ramanujan. The book review was in The Toronto Star. For reference.

January 5, 2008- I asked everyone to do one problem from the latest worksheet that I posted (your choice). If you want a really challenging ones, try 21-25. I don't want to assign these as forum problems because they are not exactly appropriate but I will be assigning forum problems next week from other related worksheets.

January 5, 2008- We had a low turnout on January 3. I thought it was a fun class because with only about 8 people it was easy to get a lot done. We used the discrete log to solve equations and talked about its use in cryptography, we talked about sequences and sets of objects and did part of one problem on the worksheet, and we talked about the unexam. The second unexam is due January 17.
I say it should be done in one page but it looks to me that it should be longer than the last one and the shortest of those ranged from 1/2 page to 1 page so my estimate is off.

January 3, 2008- A sneak peak for those of you who look at the website before class: the second unexam.

January 3, 2008- I will be handing back the unexams tonight. Here is a list of scores that I have for work you have turned in. Note that the unexams were out of 5 (5 means that I had no issues with your explanation, 4 means I had only minor comments, 3 and 2 mean that I had several comments on your explanation), and the forum problems were out of 3 each (3 means you finished the problem, 2 means mostly finished, 1 means some progress, 0 means I didn't see it posted).

January 2, 2008- Welcome back everyone. First day of class is January 3. See you there.

December 7, 2007- I am working on answers for the worksheets that we did last night. I will continue to plug away at the grading for the forum whenever I have a chance. For the questions on widgets and doodles, don't just give me a guess! Once you have a combinatorial description, apply the addition and multiplication princples and explain why it is right. Then I won't have to come back to you and say 'no, do it again.' I am sticking to the Dec 17 deadline.

December 7, 2007 - Here is a copy of the worksheet on number theory problems from last night.

November 30, 2007- I have finished assigning 3 homework problems to everyone in the class. Please check yours on the list. If you have any questions (especially about the last ones that we didn't get a lot of class time on, please feel free to contact me).

November 29, 2007- I announced at the beginning of class, Thursday December 6 is our last class for the term and because this is during the final exam time I will not be able to get us the same room. I have reserved the Math and Statistics lab in S525 for that evening. Please let everyone know!

November 28, 2007- On Thursday we will be doing more on sequences represented by sets of combintorial objects. I expect to give a bunch of homework problems associated to this. So about the homework problems I would like a FINAL version of all of them finished by December 17th. If you post the first draft of your solution on December 17th and I say 'no, do it again' you are s*#% out of luck because I am not able to read your postings every day (I seem to have time about twice a week). Solution: do it now!

November 26, 2007- I posted a summary of what we did in class on November 22. Please read it, and do the worksheet if you were not there that night.

November 23, 2007- We talked about the unexam last night. About half the class wasn't there. I will post a summary for what we did do (because the last hour was spent doing something new) but it is clear from my schedule that I won't be able to do this for the next few days. I also am not sure that I will be able to respond to forum postings until Monday.

November 23, 2007- On the web page for the program:
The final dates of Graduate classes are at the Professors' discretion but each half course should have at least 36 hours of classes.
So last day of class will be December 6, HOWEVER the first day of exam schedule is December 5 and so it is possible that an exam will be scheduled in our room. Details to be announced.

November 22, 2007- I am still unable to determine when the last day of graduate classes is if anyone knows where tofind this on York's website I would be grateful!

November 22, 2007 (4:17pm) As far as I can tell class will still be on tonight although the number of attendees is going down by the hour.

November 22, 2007- Several people have asked me if we are having class tonight. I will be there without fail because I take TTC home and I will be here anyway for my other classes. If classes are canceled at York I will post the announcement here. I understand that many of you are coming from a long distance and the roads are not safe. If you do decide to come please drive safely. I will be posting the unexam question here later.

November 17, 2007- Next week this class starts moving out of training mode. We will have the first (and only one this term) unexam. Please make progress on your forum problems. I will be applying the same grading criterion to the unexam as I do to the forum problems that I have been reading.

November 17, 2007- I announced during class last time that during the summer session of 08 we would be offering Math 5300: Mathematics of computation. At the moment it looks like Hongmei Zhu will be teaching one half (and I might be teaching the other). We are looking at a schedule of Math 5420: Algebra for Teachers (Paul Szeptycki), and Math 5400: History of Mathematics for Teachers (Stan Kochman) for the Fall/Winter terms in 08/09. For the Summer of 09 and Fall/Winter 08/09 we tentatively have scheduled to offer a new course on Mathematical Modeling by Jane Heffernan, the Math 5020: Fundamentals of Mathematics for Teachers, and possibly Math 5410: Analysis for teachers. In addition the education department is planning to offer:
EDUC 5215 3.0 research in math ed Winter 2008,
Math 5910/EDUC 5210 3.0 Summer 2008,
EDUC 5848 3.0 technology in Fall 2008,
EDUC 5845 3.0 early childhood development Winter 2009,
MATH 5900/EDUC 5841 3.0 Summer 2009.

November 3, 2007- I said that I would post the web address of the radio program Simon Singh's 5 numbers. The number seven is the one we listened to on the group of another 5 numbers. I have some of these in MP3 format if you would like them let me know and I can arrage to get them to you.

November 3, 2007- I updated the hw problem list (again). I am assigning problems now because the managed chaos of distributing problems to people became a bit uncontrolable. I will be adding more problems this week as I think of them. Z and AA came from the cryptography class that I am teaching. Remember, everyone should be doing a bunch of problems from this list. You should be able to do all of them if you work at them. I am holding you responsible just for the one that you post.

October 25, 2007- I updated the hw problem list. Please send me an email if you find it wrong. I may be missing a few of who signed up for what. I am hoping that people will get a variety of problems during the term. I don't want to see someone end up with three easy combinatorial problems. The type of writing involved in justifying a inductive proof is very different than the writing that is used in justifying a combintorial problem and this is different yet again than explaining other types of arguments.

October 22, 2007- I had a conversation with Nader last week that sounded a little like this comic:
This comic is from

October 22, 2007- I finally got the right program to update the topic list. Please look at the file and compare and make sure that you are listed as presented what I think you are supposed to present.

October 12, 2007- Here is one way of picturing the integers that I was thinking about and that I showed in class last night.
poset of the integers 1 through 30 ordered by division
Some neat properties of this graph: October 12, 2007- we still have a few people which do not have the book. I will try to post a few more pages, but please try to get it soon.

October 9, 2007-After a bunch of back and forth with account services at York they have activated AML accounts for everyone. This should be in effect by Oct 10.

October 7, 2007-I put a list of the topics that everyone signed up for on a page. I will continue to update it. There are a couple of conflicts and I will try to resolve them by email before Thursday. Also, keep watching the homework problems that I post on this page.

September 27, 2007- I mentioned that everyone would be doing a 20-30 minute presentation during the term. Here is a list of topics that I would like to cover during these presentations. The first half of them are closely linked to the material in the book that I would like to cover and we will try to time them so that they roughly correspond to our progress through the text. The second half of them may diverge from other topics in the class but are related to number theory and combinatorics.

September 26, 2007- It seems as though the book has not been ordered through the bookstore for this class. This means it will take longer on average to get this class started. In the meantime, here are 3 pages to get you started. I asked you to try to solve roughly 3-4 (maybe as many as 18) of the problems from section 1.1 to get started on proving results by induction.

August 23, 2007- Note that graduate classes for the Mathematics and Statistics program are not supposed to begin until the week of September 10, 2007. Thursday September 13th and Friday September 14th are Rosh Hashanah and classes are cancelled on that day (as well as the evening of Wednesday September 12th so please carefully check your class schedule). The first day of class will therefore be Thursday, September 20th, 2007.