Math 145. Problem solving. Fall 2021.

Prerequisites: Math 111; EHD 50 (may be enrolled concurrently; or permission of instructor).

Hours: 3.

Content: Topics to be covered include: Principle of Mathematical Induction, Dirichlet's box principle (also called Pigeonhole principle), number theory, case analysis, working backwards, mathematical games and algorithms, invariants, coloring proofs, calculus. Other topics may also be discussed. Some problems will be from math olympiads and current journals. 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
• Principle of Mathematical Induction
• Dirichlet's Box Principle
• Divisibility properties of integer numbers; congruences
• Technique of working backwards
• Concept of invariant
• Technique of using coloring to obtain a contradiction
• 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
• Working backwards
• Patterns
• Invariants
• Coloring
• Symmetry
• Graphs
• 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 tests (50 minutes long each) and one final comprehensive exam (2 hours long). Tests will be given during class time, via Zoom, with camera on. 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.

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.

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. Your instructor will be present and take attendance at the Journal Problem Solving Group meetings and Math Circle, grades 8-9. For all other sessions, report your attendance via this Google form.

• Journal Problems. During the semester, the instructor will post a few collections of journal problems for you to try to solve. (Problems from some journals have already been posted on the schedule page.) You may work on these in any of the following two ways:
• Work individually, at your own schedule and pace. 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.
• Work in a group, on Fridays from 12-12:50 PM starting August 3, via Zoom using the class link. You are expected to be present for the whole time period and actively participate when you attend. Please turn your camera on. However, you may mute yourself when you are not talking. When the group makes any progress, start an overleaf file with solution(s). For each journal issue, designate one or two people responsible for proofreading, finalizing, and sending the file to the journal. The submission should indicate "Fresno State Problem Solving Group" or something of the sort rather than the names of just one or two individuals who were designated to send off the solution(s). You will get 5 points for attending each Tuesday or Friday session (for 50 minutes). The designated person(s) will get 5 additional points for sending the solution(s) to the journal. Any volunteer(s) may present the solution in class and get 5 more points for their presentation.
• Problem Solving Playground. Dr. Markin will hold problem solving sessions called PSP on Wednesdays starting October 6, from 4:00-5:00 PM. PSP is open to all Fresno State students interested in working on non-routine problems (not homework). Problems vary greatly in topic and difficulty, ranging from simple problems/puzzles to Putnam Competition problems. Dr. Markin welcomes any problems that the participants bring for the group to attempt/discuss. You are expected to be present for the whole hour and actively participate when you attend. Please turn your camera on. However, you may mute yourself when you are not talking. You will get 5 points for attending each session.
• 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 in/observe the sessions which are held via Zoom every other Wednesday starting September 1 from 5:30-7:30 PM for grades 9-12, and every other Saturday starting August 28 from 10:00-11:30 AM for grades 8-9. No prior experience is necessary. Contact your instructor if you would like to participate/observe. More precisely, your choices are:
• Atttend and participate (as a student) in Wednesday sessions for high school grades. No preparation is necessary. Topics sometimes overlap 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). You are expected to be present for the whole session and actively participate when you attend. Please turn your camera on. However, you may mute yourself when you are not talking.
• Observe Saturday sessions for grades 8-9. No preparation is necessary. Topics sometimes overlap with those in this class. Please turn your camera on but mute yourself. Space is limited. To sign up, contact your instructor. You will get 5 points for observing each session.
• Sabuwala Problem Solving Competition. This competition is usually held around mid-November and is open to all Fresno State students. Good ways to practice are to try problems from previous years (see the link above) and attend the Problem Solving Playground meetings (see above). This year, the competition will be held on Friday, November 12, from 5:25-7:35 pm, via Zoom. You will get 10 points for participating in this competition, and additional points if you place in the top 10 (at least 10 additional points for 1st place, 9 points for 2nd place, and so on; possibly more if there are many participants; bonus points may be awarded for exceptionally elegant solutions). There is no limit on the number of participants, but advance registration is required (see the link to the competition page above).
• Department of Mathematics Day. This annual department event is held the last Friday before Thanksgiving (which is November 19 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. 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. You are encouraged to attend/participate in the other components of this event, however, attendance will not be taken and no points are awarded. See more information in an announcement on Canvas
• 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 4 this year), however, advance registration is required. Email Dr. Stefaan Delcroix (sdelcroix@csufresno.edu) to register. You will also have to register with Putnam Competition directly. 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).
• 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 but not yet fixed mistakes is here. To be eligible for any extra credit, you must have your camera turned on every class.

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:

• Work with your classmates. Note: working on your homework together is encouraged, however, every group should write down their solutions on their own, in their own words. The best thing to do, after you have worked with other people/groups, 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.
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!

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.

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.