Numbers: rational, irrational, algebraic,

constructible, transcendental

Mth 496-02: Course Syllabus

Room: C-302 Wells Hall Time: MWF 3:00-3:50

Instructor: Curtis Bennett Office: D-216 Wells Hall

Tentative Office Hours: MWF 2:00-2:50 or by appt. Phone: 353-3833


Please use the first to contact me. The pilot address will get to me at home, but not at the office.

Course Description: This is a capstone course in mathematics. It is specifically designed for students with an interest in education. As a capstone course, it will integrate the topics of many undergraduate mathematics courses. Consequently, students must have taken all of the prerequisite courses. In this course, we will discuss various understandings of numbers and their strengths and weaknesses. We shall work towards a firm understanding of the intermediate value theorem and the definition of the real numbers. This is not a course on material from high school, but rather, a course in advanced mathematics. The topics discussed are first touched upon in the high school curriculum, but rarely discussed in any detail there. In this course, we will build up from our previous knowledge and the high school curriculum to more advanced topics.

Course Objectives:

    1. For students to see some of the questions that drove the development of mathematics answered in the context of the course. — Students shall learn proofs of many of the classical problems that are often discussed in secondary mathematics classes. In particular, we shall prove the irrationality of all square roots, e, and p; we shall prove the impossibility of trisecting the general angle and doubling the cube with straightedge and compass; and we shall prove that one can solve a general cubic equation with radicals.
    2. For students to learn the mathematical ideas behind the answers to these questions. — Students shall be able to use the concepts from the proofs in solving mathematical problems.
    3. For students to be able to look at advanced mathematics topics and see how they are reflected in the high school curriculum and how these advanced topics might inform their teaching of high school mathematics. — In particular, students will reflect on how learning to solve a cubic equation relates to learning to solve quadratic equations and on how geometric constructions influence understanding of number.
    4. For students to gain a deeper understanding of themes of mathematics and how various subjects in mathematics can relate to a common theme. — In particular, students should gain an understanding of the interconnectedness of mathematical disciplines in this course, and see the integration of geometric, algebraic, and analytic thinking.
    5. For students to successfully experience mathematical research on their own. — Students will complete individual research projects, exposing them to mathematics as practiced on the highest levels.

Course Policies:

Attendance: Attendance will be expected in this mathematics class and will be taken into account in the class participation grade. It will also be used in deciding borderline grades. You are responsible for attending the lectures; however, if you are unable to make a class, you should inform the instructor as to the reason as soon as possible, preferably prior to the lecture.

Missed Work: If a student should miss an assignment due to a documented University sponsored event or religious holiday, arrangements should be made with the instructor prior to the absence. You are expected to complete missed assignments and make arrangements to turn them in. I reserve the right to deduct points from assignments that are turned in late (half the points for each day late).

Group Work: Working in groups is strongly encouraged when working on homework. The insights of your peers will often lead to greater understandings, and explaining material to others will also help solidify it in your head. Some assignments are to be turned in as a group, but others will be turned in individually. On individual assignments, you may discuss the problem together, but you are expected to write up the assignment by yourself. No two individual assignments should be handed in looking too much alike.

Also, the examinations will probably be on an individual basis, and therefore, you should be sure that you understand all material being tested.

Academic Honesty: The standards of academic honesty at Michigan State University as listed in the Spartan Life Handbook will be enforced. The minimum penalty for a violation for submitting work other than your own will be failure on the examination or assignment. The one exception to this rule is if the assignment involved group participation and a single product is given by the group by which all members will be graded. Note: It is necessary when writing up mathematics to properly credit information obtained from other sources. This includes information from the world wide web, in text books, and from other individuals.

Technology: A graphing calculator is strongly recommended for this class. I will support TI-83 graphing calculators, but I do not know how to program other calculators. You will also find it beneficial to have both a spreadsheet program and Geometer’s sketchpad available.

Course Grades: The highest possible grade cutoff will be 93% of possible points for a 4.0, 88% of possible points for a 3.5, 82% for a 3.0, 78% for a 2.5, 70% for a 2.0, 65% for a 1.5, and 60% for a 1.0.


Tests: Two tests, (midterm and a final) will be given during the semester. The final will be comprehensive, and it is highly likely that the final will include an oral portion.

Midterm: Wednesday, October 18.

Final: Tuesday, Dec. 12, 3pm-5pm.

Homework: For each section covered in lecture there will be a homework assignment. The homework problems are designed to promote deeper understandings of important mathematical concepts (that you should learn). As a result, conferencing with other students in the class is permissible; however, copying of work from any source is not. Individuals may work as a group, but the responses to homework problems should be written up individually.

Homework assignments may generally be composed of both lower and higher level problems. Problems of greater difficulty will be worth double the points of the lower level ones.

Scoring of Homework: (low level problems — double value for high level problems).

5. Complete and correct with no irrelevancies. Written well, claims backed up with details and well-reasoned arguments.

4. Complete but containing minor computation error(s) or minor irrelevancies. Minor writing errors allowed, claims backed up with details and well-reasoned arguments.

3. Incomplete homework, which has reasonably and sufficiently progressed toward completion although containing a student, identified gap (i.e., "I can get to here, and if I could prove this then I could complete the proof in the following manner"). Some writing errors, but a clear argumentative line. Some claims may not be defended well.

2. Incomplete homework progressing toward a solution, but the path may be confused meandering or ill-advised. Writing is poor, some claims are defended well.

1. A reasonable attempt is made at a solution. Writing has some structure, and student has tried to answer the question.

0. No reasonable attempt made toward solution.

Project: A project accounting for 30% of your final grade will be assigned. The paper describing your project will be due on Monday, Dec. 4. Each week of the term, however, I will expect a progress report in writing, or via e-mail (see attached list of questions to be answered in each progress report). While it is acceptable (although not recommended) if once in a while you report having done little work on the project, you should be working on the project every week. It is expected (and highly recommended) that you will meet with me occasionally during the term to discuss the project.

The problem you choose is a starting point. As is the case in mathematics research, there is no set stopping point for your project, although there are minimum expectations. For each problem, I have a vision of what you will accomplish during the term, but this is only a vague outline. I fully expect the best projects to stray (sometimes wildly) from this outline, and I will work very hard to help you decide where you should proceed after completing the first part of the project.

There is no set guideline for the number of pages for your project. In mathematics, some of the best papers have been fewer than 10 pages long. Of course, one of the most famous papers in algebra took up an entire journal. In your paper, you should discuss some of the background of the project and the mathematical information in the project in addition to writing up complete proofs of the theorems with which you end up. At some stage, I may give you more background information on your project, although this will only happen after you have completed the first portion of it. You should also either think about how the project could be used in a classroom or about the connections of your problem with high school mathematics.

Projects will be completed either individually or in teams of at most three students. In the case of a group, a single write-up is necessary addressing the indicated question(s) asked and any additional questions that you raise, or that are raised by Dr. Bennett. Projects will be graded based upon (a) communication, (b) visual representation, (c) computations, (d) proofs, (e) decision making, (f) interpretation of results, (g) conclusions drawn and supported, (h) overall presentation, and (i) overall impression. Each member of the team will also be rated for their contribution to the project by other team members.

Writing Assignments: From time to time in the semester, writing assignments will be given requiring reflection upon material presented in class or assigned for reading. These will be graded and scored in with the homework. The point values of individual writing assignments will vary depending on the amount of work required.

Class Participation and Discussion: Occasionally, the class will work on problems in groups, and frequently there will be in class discussions. A participation and discussion grade will be assigned based on students contributing in a meaningful way to group work and discussions. (Such contribution need not be confined to time during class. Comments delivered to the instructor outside of class often contribute valuably to what happens in class.)

Grades will be computed in the following manner:

Midterm 10%

Project Paper 30%

Homework/Writing Assignments 35%

Participation and Discussion 5%

Final 20%

Total 100%

Import Dates:

Friday September 1: Close of Computer Registration.

Friday September 8: Last day to Add or change sections.

Thursday September 21: Last day to drop with 100% refund.


Tentative Sequence of Topics:

  1. The rational and irrational numbers.
  2. Numbers as lengths: constructible numbers, the impossibility of trisecting a general angle or doubling the cube with straightedge and compass.
  3. Numbers as sums, products, quotients, and differences of radicals: Solving quadratic and cubic equations.
  4. Algebraic versus transcendental numbers.
  5. Dedekind Cuts and the real number line.
  6. Desert topics (as time permits).


Evaluating Projects


5) Students use language and symbols correctly and communicates their ideas clearly.

4) Students use language and symbols correctly most of the time (only a few mistakes) and communicate their ideas relatively clearly.

3) Students use language and symbols correctly some of the time (many mistakes) and communication of ideas is vague.

2) Students’ paper is difficult to read and the reader must work hard to understand what the paper is saying.

1) Students’ paper is unreadable.

Visual Representation (if appropriate)

5. Tables and/or graphs are correctly constructed, labeled, and presented.

3. Tables and/or graphs have some errors but are mostly correct.

1. Errors in constructing or labeling tables and/or graphs.


5. Computations are correctly done.

3. Some correct, some incorrect computations

1. Errors in computations lead to answers that are unreasonable, or a formula is used incorrectly.


5. Proofs clearly indicate the theorem and the line of reasoning is complete and correct.

3. Proof has some errors but is mostly correct.

1. Failure to prove what is claimed due to errors arising from unreasonable assumptions.

Decision Making

5. All decisions appear to be appropriate.

3. Some decisions appear to be inappropriate, whereas others are appropriate.

1. Decisions do not appear to be based upon obvious clues in the problem.

Interpretation of Results

5. Good interpretation of results using all appropriate information as supporting material.

3. Interpretation is too brief, the student fails to interpret some of the information, or the interpretation is only partly correct.

1. Poor interpretation of results, tables, graphs, and/or computations.

Conclusions Drawn

5. Conclusions are made on the basis of analysis and readings. Comments are made about the study as it relates to the teaching of mathematics at the secondary level and/or the problem is naturally extended.

3. Weak conclusions are made, but some attempt is made to look beyond the problem.

1. Students fail to draw conclusions, draws unsubstantiated conclusions, or fails to notice inconsistencies of conclusions made.

Overall Presentation

5. Presentation is appropriate to the profession of a classroom teacher.

3. Presentation is neat, clean, orderly, but inconsistencies are in the document (e.g., fonts change inappropriately).

1. Presentation is sloppy; elements of write-up are missing or unreadable.

Overall Impression

5. Paper gives a strong overall impression to the instructor.

3. Paper gives mediocre overall impression to the instructor.

1. Paper seems put together at the last minute.

Member Review

Project members will all receive the same grade unless other team members report that a student did not contribute or contributed very little to the project. In this case, the instructor will meet with the project team members individually to discuss the grading of the project. In mathematics, it is often hard to judge the contributions of co-authors, consequently, the authors in all joint mathematics papers are listed alphabetically (unlike other scientific disciplines). Consequently, project members should only be considered to not have contributed if there is malfeasance on their part (i.e., they miss meetings, don’t complete promised work or do so shoddily, etc.).