MATH 401 Spring 1996 Grader: Imre Patyi Tuesday and Thursdays 9 - 10:15 UNIV 117 Tuesday and Thursdays 3 - 4:15 CL50 125 ======================== GRADING POLICY: The grading system is designed to foster habits which aid in problem solving. Suppose a problem is worth p points. If the student cannot solve the problem and states "I can't solve the problem because of .... ", he or she will be rewarded with at least p/2 points. False statements and incorrect answers can lower the grade to 0 . A completely correct solution is worth p points. If the student cannot solve the problem, but can explain why he cannot solve the problem, the score will vary from p/2 to p depending upon the reason given by the student. Students who solve the problem will be eligible for extra credit by reflecting profitably upon the problem. Students are encouraged to look up solutions or to discuss the problem with their friends if they wish. I only require that the solution be written in the student's own words, and that any substantial help be cited. These citations will not harm the student's grade. Giving due credit is part of the complete honesty necessary to carry out mathematics. =========================== EXAMS: There will be a FINAL EXAM. It will cover vector algebra for Euclidean Geometry as well as spherical geometry. Students who miss less than three classes in the semester are expempt from the exam. ==================== COMPUTERS: Again this spring, the Mathematics Department will host Math Open Hours in the Macintosh Lab in LAEB B275 on Sunday through Thursday evenings from 7:00 p.m. till 10:00 p.m. The machines run Matlab, Maple, Geometer's Sketchpad and Excel software for doing mathematics and there will be student assistants on hand to help with the software and, to some extent, the mathematics. During these hours, students working on mathematics have priority for use of the machines, so there should not be a problem with access. ==================== ANNOUNCEMENTS: %%%%%%%%%%%%%%%%%%%%% Assignment 1: Write a page essay detailing the mathematics you have studied and your strengths and weaknesses and enthusiasm for mathematics. ========================== Assignment 2: Counting and Arithmetic to the ancients. Chapter 1, problems 2, 3, 4, 10, 11, 12 . Each problem is worth 10 points. In problem 11, just do as many of the reciprocals as is necessary to answer the question in the second sentence of the problem. =================== Assignment 3: Base 10 Polya, p. 163. Let a set of numbers contain exactly one each of the ten digits 0,1,2,3,4,5,6,7,8,9. Can the set add up to 100 ? =================== Assignment 4: Katz, Chapter 1, #16, 17, 18, 19, 20 Propose a problem about the digits of a number. Hint: Use Theorem 401 Theorem 401: Let N be a number which is expressed in terms of digits between zero and nine. Let d be a number. Divide d into 1, 10, 100, 1000, 10000, and so on and take the remainders. These form a Sequence of integers beginning with 1 and continuing. Call this Sequence the "Decimal Remainder Sequence for d " Now multiply the unit digit of N by 1, the first number of the Decimal Remainder Sequence for d , and then multiply the tens digit of N by the second number of the Decimal Remainder Sequence for d , and multiply the 100's digit by the third number in the Sequence and so on until there are no more digits. Then add all these procucts up to get a number R . Then d divides N if and only if d divides R. Note that the conclusion is still true if any number in the Decimal Remainder Sequence for d is replaced by itself plus or minus a multiple of d . ===================== Assignment 5: 1. In Katz, page 174, Proposition XIII-1 : What does "a straight line cut in mean and extreme ratio" mean? a) Try to find out from the proof. b) Try to find the definition in Katz, or in another book. Describe your experience. 2. On page 753 of Katz try to figure out the statment "nth order differences are constant." Try various examples. Is it only true for integers? What are the nth order differences of x^2 + x and of x^3 + x^2 + x ? ===================== Assignment 6: Katz, Chapter 1, # 34, 36, 37, 38 =================== Assignment 7: 1. Suppose the second sequence is the first sequence with every number multiplied by c . Show that the nth order differences of the second sequence is c times the nth order differences of the first sequence. 2. Suppose the first and second sequences are added together to give the third sequence. Show the nth order differences of the third sequence is the sum of the nth order differences of the first two sequences. 3. Suppose f(x) = cx^n + (polynomial in x of degree less than n) . Consider the sequence f(h), f(2h), f(3h), ... . Show the nth order differences of this sequence form a constant sequence, where the constant is c h^n n! ================== Assignment 8: Katz, Chapter 2. P.90, # 4, 5, 6, 7, 8, 9 . ======================== Assignment 9: Give examples of the following types of statements. 1. An axiom 2. A proposition 3. A hypothesis 4. A conclusion 5. A converse 6. A contrapositive 7. An equivalence 8. A definition 9. An example of some statement 10. A counter example to a statement ========================== Assignment 10. Katz Chapter 2. # 12, 13, 14, 15, 17, 19, 20, 21, 24, 23, 22, 41, 46 Due Tuesday February 20. ============ Assignment 11. 4 ships A, B, C, D, are sailing in a fog with constant and different speeds along different straight line courses. Five of the six possible simple near collisions are known to have occurred. Explain why the sixth near collision must occur. ============ Assignment 12. 1. Are all numbers of the form abcabc divisible by 7 ? 2. Katz, Chapter 2, # 30 3. Do assignment 11 again 4. A man walked from the sea to the top of a mountain on the first day, The next day he walked down to the sea by a different route. Show there is a time on the clock at which he was at the same altitude each day. ================ Assignment 13. Katz, Chapter 3. # 1, 2, 5, 12 . =================== Assignment 14. Katz Chapter 4, # 10, 22 Chapter 5, #22 ================= Assignment 15. 1. Show that the medians of a triangle meet in a single point. 2. Given four space ships in intergalactic space travelling in straight lines with different constant velocities, if there are five near misses involving each time only two space craft, show that there are in fact six near misses. =================== Assignment 16: Katz, Chapter 5; # 8, 9, 11, 12, 15, 20, 21, 22, 23 ================= Assignment 17: 1. Show that the medians of a triangle meet in a point. Show that the point cuts the line into two parts in ratio 2 : 1 . 2. Show there is a triangle whose sides are parallel and equal in length to the medians of an arbitrary triangle ============== Assignment 18: Katz: Chapter 6; # 2, 4, 6, 32, 33, 34, 39 ============ Assignment 19: Katz: Chapter 7; # 6, 7a, 20, 22a,b, 28 ============= Assignment 20: Katz: Chapter 8; # 1, 2, 3, 4, 5, 6, 7, 8, 9, 12, 13, 21, 24 ==================== Assignment 21: Katz: Chapter 9; # 26, 28, 32 ================== Assignment 22: Katz: Chapter 9; # 1, 2, 3, 4, 5, 6, 7, 14, 15, 16, 17 ================ Assignment 23: Davis and Snider, P. 6 , # 1, 2, 3, 4, 5, 6, 7 Write 1.25252525... as a common fraction =============== Assignment 24: Problem sheet from Vectors: 1, 2, 3, 4, 5, 6 ================ Assignment 25: Problem sheet from Vectors: 7, 8 ================ Assignment 26: Problem sheet from Vectors: 9, 10, 11 ============== Assignment 27: Katz, Chapter 9: 18, 19, 21, 22, 43 %%%%%%%%%%%%%%%%%%%%%%%% Bibliography: Required Text: Victor J. Katz, A History of Mathematics. Recommended: G. Polya, How to Solve it, 2nd edition. John Mason, Thinking Mathematically. Used in class: Lecture 1: Richard P. Feynman, "What do YOU Care What Other People Think" pp. 54 - 59. Polya, problem 1, p. 234 Lecture 3: Mason, Page 5 . Lecture 8: Howard Eves, An Introduction to the History of Mathematics, 6th edition, Saunders p. 84 Lecture 10: Journey Though Genius, William Dunham pp. 29 - 32 The Elements, Euclid, Translated by Sir Thomas Heath Call number 513 Eu 2t2 1956 Proposition 5 p. 251, Pappus' proof p.254, pons Asinorum p.415 Lecture 13: Mathematics in Western Culture, Morris Kline Lecture 16: Introduction to Vector Analysis, Davis, Snider Lecture 17: My written notes on Vectors. Lecture 21: Journey Though Genius, William Dunham pp. 134-142. Lecture 23: H. F. Davis and A. D. Snider, Introduction to Vector Analysis, Sixth ed. %%%%%%%%%%%%%%%%%%%%%%%%%%%%% Report on Math 401 (problem solving) in Spring 1996 Daniel Henry Gottlieb I think I have had some success with this course. The students were education majors, training to teach math in High School or Junior High School. I gave them a brief summary of Polya's book, "How to Solve it". The grade depended upon the homework assignment given every class period. Those who came to class all but for one week were exempted from the final exam. The grading of the homework was 50% for no false statements with deductions to zero for FALSE statements, and extra credit for good questions or explainations as to why the student couldn't slve the problem, or generalizations or similar problems. I said they could use any means to solve the problems as long as they referenced the source of help. Most of the problems were taken from V. Katz' History of Mathematics. I went from the Babylonians to the Renaissance. I quoted from numerous texts. Occasionally, I would follow a topic: Telling divisors from the digits of a number; difference series for polynomials; problems leading to the idea of space-time; vector algebra for solving statics, velocity and elementary geometry problems. I wanted to talk about spherical geometry and functions, but I ran out of time. I am convinced that pedigogical strategies of teaching various courses have the unintended consequences of giving the students an ethic which which argues against looking in books or asking friends for help. I see by the final essays the students wrote that many of them were surprised that there were more than one way to solve a problem. Some students liked the history. A few were very enthusiastic. I would say that most of the students worked hard and only a few coasted. So my lenient way of grading helped students find their own way, and did not appreciably permit sloughing off. Dan Gottlieb