Math 145. Problem solving. Fall 2019.
Prerequisites: Math 111; EHD 50 (may be enrolled concurrently; or permission of instructor).
Hours: 3.
Content: Topics to be covered include: brief review of logic, types of proofs, and principle of mathematical induction;
Dirichlet's box principle (also called pigeonhole principle), number theory, case analysis, invariants, coloring proofs, working backwards, geometry, mathematical games
and algorithms, non-routine calculus problems, problems from math olympiads and journals. Other topics may also be discussed. Students will be asked to
present their solutions on the board. Working in groups is encouraged. The purpose of this class is to develop/improve problem solving skills, communicate your mathematical
ideas, and to practice writing proofs in a correct and complete way.
Catalog description: A study of formulation
of problems into mathematical form; analysis of methods of attack such as
specialization, generalization, analogy, induction, recursion, etc. applied to a
variety of non-routine problems. Topics will be handled through student presentation.
Course goals
Introduce students to a variety of problem solving techniques as well as challenging ("non-routine") problems accessible to 7-12 graders; practice communicating
mathematical ideas verbally and symbolically in the problem solving setting.
Learning objectives
Upon completion of this course, students should know/understand:
- Importance of a formal (rigorous) proof
- Meaning of logical operations (and, or, negation, implication) and quantifiers (universality and existence)
- Difference between different types of proof (direct, by contradiction, by contrapositive, by example/counterexample, etc.)
- Principle of Mathematical Induction
- Dirichlet's Box Principle
- Divisibility properties of integer numbers; congruences
- The concept of invariant
- Basic graph theory terminology (vertex, edge, degree of a vertex, etc.)
Primary learning outcomes
Upon completion of this course, students will be able to do:
- Translate word problems into mathematical equations and relations
- Recognize problems which can be solved using one of the following techniques:
- Dirichlet's Box Principle
- Mathematical Induction
- Divisibility and/or Congruences
- Case Analysis
- Patterns
- Invariants
- Coloring
- Symmetry
- Graphs
- Working backwards
- Equations
Grading policy:
Your grade for the course will be based mostly on your performance on exams and homework (more information on these below). You will also receive points for class presentations,
submitting solutions of problems to jounals, and participating in Math Department's problem solving activities outside (more information below).
The number of points awarded for these are shown below.
Test 1 | 60 points |
Test 2 | 60 points |
Test 3 | 60 points |
Final exam | 90 points |
Problem of the day presentations |
40 points |
Written Homework |
150 points |
Participation in departmental problem solving activities |
40 points |
Total | 500 points |
Points earned | grade |
437.5-500 (87.5-100%) | A |
375-437 (75-87.4%) | B |
312.5-374.5 (62.5-74.9%) | C |
250-312 (50-62.4%) | D |
0-249.5 (0-49.9%) | F |
Exams:
There will be three 50-minutes tests and one final comprehensive exam. Tests will be given in class. The final exam will be given during the final exam week. If for any reason
you can not take a test at the scheduled time, please let me know as soon as possible, and certainly before the test. See schedule for exam
dates. No books, notes, or calculators are allowed on any of the tests.
Class attendance is required. One "problem of the day" will be assigned for each class period. Students will be randomly
called to present their ideas/solutions. These presentations are part of your grade. If you are called and you are absent, you will receive zero points for your
presentation. (If you are sick or have another serious reason to miss a class, notify your instructor before the class to be missed. Up to three absences during the semester for
undocumented reasons are allowed. If you are unable to notify the instructor before class or exceed three absences, please provide doctor's note, etc.)
In addition to these presentations and new material/problems discussions, important course information will be given in class, sometimes handouts may be distributed.
There will be weekly homework due Wednesday at 1 pm, usually on the topic(s) of the preceding week. There will be 12 written
homework assignments. Each homework will consist of 3-5 problems and will be worth 15 points total. Two lowest homework scores will be dropped. Homework problems
will be new problems, sometimes somewhat similar to the problems discussed in class and sometimes quite different. You may have to spend a few hours on each
homework (remember, the main purpose of this class is to learn to solve unfamiliar, non-routine problems). If you have any questions, or need someone to listen to and
possibly comment on your ideas, please do not hesitate to ask your instructor (this is what her office hours are for!) Also, you are encouraged to work with your
classmates. Please bring your homework to class. We will often discuss homework in class, so no late papers will be accepted unless you are sick or have another serious
reason to miss the class (notify your instructor in advance, or provide doctor's note, etc.)
Participation in departmental problem solving activities is expected and counts toward your grade in the course. Below are the
activities supported by the Mathematics Department. You are welcome to participate in any of these. Choose anything that interests you and fits your schedule.
- Journal Problems. During the semester, the instructor will bring a few journals with problems for you to try to solve. (Problems from some journals have already
been posted on the schedule page.) When you have a solution, write it down and show to the instructor. If your solution is correct, depending on its completeness/clarity and
difficulty of the problem, you will get at least 15 points (possibly more) for your solution. If it is correct and complete, you are encouraged to send your solution to the
journal (and your name and/or solution may get published!) You can also present your solution in class. You will get 5 additional points for sending your solution to the journal,
and 5 more points for your presentation in class.
- Problem Solving Playground. In the past Dr. Markin held problem solving sessions called PSP. If such sessions are held this semester, meeting info will be posted here
and announced in class when it becomes available. One of the goals of the PSP is to practice for the Putnam Exam (see below) as well as the Fresno State problem solving contest
(also see below). You will get 5 points for attending each session (typically 1 hour long).
- Fresno Math Circle. This is a math enrichment program for children in grades 3-12. We are focusing on
problem solving and deep understanding of mathematics concepts. You are invited to participate/help run the sessions, which are held every other Saturday starting
September 14 from 9:45-11:30 AM for grades 4-9 and every other Thursday starting September 12 from 5:30-7:30 PM for grades 10-12. No prior experience is necessary.
Contact your instructor if you would like to participate/help. More precisely, your choices are:
- Atttend and participate (as a student) in Thursday sessions for high school grades. No preparation is necessary. Some of the topics will be overlapping
with those in this class. Space is limited. To sign up, contact your instructor. You will get 10 points for each session (you must attend for 2 hours).
- Help with Saturday sessions for grades 4-5 (lead by Math Circle Leaders who are Fresno State students) or grades 6-7 (lead by your instructor).
To prepare for each session, we will meet for about an hour earlier in the week. Space is limited. If you would like to help, contact your instructor
at least one week before the session. You will get 15 points for each session.
- Problem Solving Contest. The contest is usually held the week of the DMD
(in November), and is open to all Fresno State students. Good ways to practice are to attend the Problem Solving Playground meetings (see above) and/or Putnam Prep Sessions (see below). You will get 10 points
for participating in this contest, and additional points if you place in the top 10 (10 additional points for 1st place, 9 points for 2nd place, and so on). There is no limit on the number of participants, but advance registration will be required.
- Putnam Competition. This is an annual nation-wide competition in mathematics problem solving
for undergraduate students. All undergraduate students enrolled at Fresno State are welcome to participate. It is held on the first Saturday in December (that's December 7 this year),
however, advance registration is required. The registration date is October 14, email Dr. Stefaan Delcroix (sdelcroix@csufresno.edu) to register. Good ways to practice are
to attend the Putnam Prep Sessions (see below) and/or Problem Solving Playground meetings (see above). You will get 20 points for participating in this competition
(regardless of how you do, however, you must stay for the whole time period and try your best).
- Putnam Prep Sessions. Dr. Delcroix is hosting Putnam Prep (problem solving) sessions on Fridays from 2-3:50 pm in S2 206. Various problem solving techniques will be discussed, and problems from previous Putnam Competitions,
as well as possibly other problems, will be discussed. You will get 10 points for attending each session (provided you stay for the whole session).
- GRE Prep Workshops. If you consider going to
a graduate school in mathematics, you will need to take GRE in Mathematics. These sessions focus on review of and problem solving on Calculus and Linear Algebra,
and perhaps a bit of other undergraduate material. Dr. Vega is hosting these sessions on Fridays starting August 30 from 1:30-3 (however, you will only
be able to attend from 2-3 since our class meets until 1:50) in PB 136. You will get 5 points for attending (1 hour of) each workshop.
- Department of Mathematics Day.
This annual department event is usually held the last Friday before Thanksgiving (which is November 22 this year). You may choose to give a talk,
e.g. about your solution to a journal problem, or participate in the Math Jeopardy game. The old schedule is given in the
flyer. The info for this year will be announced as soon as it becomes available.
You will get points for your participation in the above activities: at least 10 points for the talk if it is related to problem solving (possibly more depending on the topic
and time spent), and 10 points for participation in Math Jeopardy.
- If you earn more than 40 points participating in the above activities, you will receive extra credit (up to another 40 points).
Another opportunity for extra credit: to encourage you to read the book carefully (and to reward you for doing so) 1 point of extra credit will be given for reporting a
new mistake/typo in the textbook. The list of reported mistakes is here.
Extra Help: It is essential not to fall behind, because most
classes will use the material studied previously. If you have trouble with some material, seek help in the
following ways:
- Ask me, either in class or privately. My contact info is given on the course web page.
Don't be shy to ask questions. If you don't understand something, chances are very high that somebody else doesn't understand that either.
In particular, you are more than welcome to ask questions about solutions in the book. However, I can not tell you how to do the homework problems.
Those are for you to solve. Remember, the purpose of this class is for you to learn to solve non-routine problems. On the other hand, if anything is unclear then
questions will be answered, feedback on your ideas will certainly be given, and some hints may be given if needed, so please do not hesitate to ask questions.
You never loose anything by asking!
Also, if you'd like to know a solution to a homework problem after it's due, let me know, and I'll be very glad to discuss the problem.
-
Work with your classmates. Note: working on your homework together is allowed and even encouraged, however, every student should
write down their solutions on their own. The best thing to do, after you have worked with other people, is to put aside all the notes you made, and
write a complete solution from scratch, in your own words. Do not just copy a solution from someone else's or even joint papers.
Also, if you have worked with someone, please indicate that on your paper.
If you are having any difficulties, seek help immediately - don't wait
until it is too late to recover from falling behind, or failing to understand
a concept!
Computers:
At California State University, Fresno, computers and communications links to remote resources are recognized as being integral to the education and research
experience. Every student is required to have his/her own computer or have other personal access to a workstation (including a modem and a printer) with all the
recommended software. The minimum and recommended standards for the workstations and software, which may vary by academic major, are updated periodically and are
available from Information Technology Services or the University Bookstore. In the curriculum and class assignments, students are presumed to have 24-hour access to a
computer workstation and the necessary communication links to the University's information resources.
Students with disabilities:
upon identifying themselves to the University, students with disabilities will receive necessary accommodation for learning and
evaluation. For more information, see http://www.fresnostate.edu/studentaffairs/ssd/
Academic honesty: cheating and plagiarism will not be tolerated in this class. For information
on the University's policy, see the University Catalog (section Policies and Regulations).
Honor Code: Members of the CSU Fresno academic community adhere to principles of academic integrity and mutual respect
while engaged in university work and related activities. You should:
- understand or seek clarification about expectations for academic integrity in this course (including no cheating, plagiarism and inappropriate collaboration)
- neither give nor receive unauthorized aid on examinations or other course work that is used by the instructor as the basis of grading.
- take responsibility to monitor academic dishonesty in any form and to report it to the instructor or other appropriate official for action.
Disruptive Classroom Behavior: student conduct which disrupts the learning
process will not be tolerated and may lead to disciplinary action and/or removal from class.
Copyright policy: Copyright laws and fair use policies protect the rights of those who have produced the material. The copy
in this course has been provided for private study, scholarship, or research. Other uses may require permission from the copyright holder. The user of this work is
responsible for adhering to copyright law of the U.S. (Title 17, U.S. Code). Digital Campus course web sites contains material protected by copyrights held by the
instructor, other individuals or institutions. Such material is used for educational purposes in accord with copyright law and/or with permission given by the owners of
the original material. You may download one copy of the materials on any single computer for non-commercial, personal, or educational purposes only, provided that you
(1) do not modify it, (2) use it only for the duration of this course, and (3) include both this notice and any copyright notice originally included with the material.
Beyond this use, no material from the course web site may be copied, reproduced, re-published, uploaded, posted, transmitted, or distributed in any way without the
permission of the original copyright holder. The instructor assumes no responsibility for individuals who improperly use copyrighted material placed on the web site.
Subject to Change:
This syllabus and schedule are subject to change in the event of extenuating circumstances. If you are absent from
class, it is your responsibility to check on announcements made while you were absent.
Last revised on 20 August 2019.