Vaughn Climenhaga

Math 35
A Survey of Mathematical Thought
Summer 2006

August 3:  Assignment 9 is posted and is due on Tuesday, August 8.  The final is next Friday; please note that the final is cumulative, so you may want to start dusting off those notes on probability and prime numbers.  There will be a review session sometime next Thursday (the free day between the last class and the final exam) - details to follow next week.

August 1:  Assignment 8 has been posted as a PDF file, and is due this Friday, August 4; there will also be a quiz in class on Friday.  The relevant concepts for the assignment which are not found in chapter 14 of the book are as follows:

Switching labels:

If you switch the labels of any two vertices of a tetrahedron, you reverse the orientation.  You can use this to see if two tetrahedra have the same orientation by seeing how many switches it takes to go from one to the other.

The 15-puzzle:

Chapter 14 in the book describes how to calculate B, the number of backwards pairs in a list of numbers, and how to apply this to the 15-puzzle.  Our invariant for the 15-puzzle is the number A.  To get A, colour the squares of the puzzle like a chess board.  If the empty square is shaded, then A is equal to B+1; if the empty square is not shaded, then A is just equal to B.  The value of A may change when we make a legal move, but the parity of A - whether it is even or odd - will not.

End behaviour of polynomials:

A polynomial is an expression like x² - 3x + 4 or -2x³ + x² - x + 10; the degree of a polynomial is the largest exponent it contains.  For example, the first polynomial given here has degree 2; the second has degree 3.  The parity of the degree determines the behaviour of the ends of the graph of the polynomial - if the degree is even, then the ends will go the same direction (either both up or both down).  If the degree is odd, they will go different directions (one up, one down).

July 31:  Assignment 7 has been posted as a PDF file, and is due this Wednesday, August 2.  These questions are not in the book, and will require the use of definitions and concepts which we will go over in class today, but which are not in the book.  The relevant ones are as follows:

Rules for combining even and odd numbers:

Orientation of triangles and tetrahedra:

If we label the vertices of a triangle with the numbers 1, 2, and 3, then the triangle has a clockwise orientation if following the labels from 1 to 2 to 3 moves us in a clockwise direction, and a counterclockwise orientation if it moves is in a counterclockwise direction.

If we label the vertices of a tetrahedron with the numbers 1, 2, 3, and 4, then the tetrahedron is right-handed if, when we look from the direction of the vertex labeled 1, the triangle labeled 2, 3, 4 has a counterclockwise orientation; the tetrahedron is left-handed if the triangle labeled 2, 3, 4 has a clockwise orientation.  This terminology is chosen so that if you put the appropriate hand inside the tetrahedron, and point your thumb towards the vertex labeled 1, then your other four fingers will curl in the direction that takes you from 2 to 3 to 4.

Two triangles are equivalent if they have the same orientation; this means that we can get from one to the other by means of the appropriate rotations.  Similarly, two tetrahedra are equivalent if they have the same orientation, and this has a similar implication.

Orientation as discussed here is one example of an invariant, which will be an important concept in this unit.

July 28:  Today we begin unit three, on parity - that is, even and odd numbers (and I hand back the second midterm).  This unit covers chapters 7 and 14 in the textbook, as well as some material not found in the textbook.  Assignment 7 will be posted on Monday, and will be due Wednesday, August 2.

Also, the final exam has been scheduled for 2:30-4:20 p.m. on Friday, August 11, in 109 Osmond.  Be sure to keep that afternoon free; you can do whatever you like in the evening.

July 18:  Assignment solutions and new assignments continue to be posted; also, there will be a quiz this Friday at the beginning of class, followed by an abbreviated lecture.

July 10:  I've put up the solutions from Friday's quiz, which will be handed back Tuesday or Wednesday in class.  The first midterm is this Thursday at 6:30, in 108 Henderson, and will cover everything we've done on probability.  There will be a review session on Wednesday evening, time and location to be announced in class.

July 5:  Solutions to Assignment 2 are up - you'll need Adobe Acrobat Reader to view them, since they're in PDF format.  If you don't have it yet (most computers do), you can get it here.  The problem set for Assignment 3, due next Tuesday, is also up.  Remember the quiz on Friday!

June 30:  Solutions to Assignment 1 have been posted - follow the link at right.  Assignment 2 is also available now, and is due next Wednesday.

There will be a quiz at the beginning of class next Friday, July 7 - it will cover everything we've done up to that point.  I will be out of town that day attending a wedding back in Canada; my replacement for the day will administer the quiz and give a brief lecture afterwards.

June 27:  Welcome to Math 35!  This course will be divided into three units, and we will spend about two weeks on each:

1. Probability, odds, expectations.  Applications to games of chance, etc.
   Chapter 13 in Mathematics: The Man-Made Universe, by Sherman K. Stein (1999 Dover edition)
2. Prime numbers, factoring integers, rational and irrational numbers.
   Chapters 2, 3, and 4
3. Parity (even and odd numbers).  Applications to various puzzles and problems.
   Chapters 7 and 14

The syllabus, which contains the same information as this web page, will be handed out at the first class.  It is available in printable format here as an MSWord document, or here as a PS file.

Textbook & materials:  The textbook for this course is the Stein book mentioned above.  You will need a calculator for this course, but only a very basic one capable of addition, subtraction, multiplication, and division.  Graphing calculators will not be required.

Lectures: This is a three credit hour course, with lectures every afternoon (Monday-Friday) in 173 Willard from 2:20-3:35pm.  The first class is Wednesday, June 28, and the last class is Wednesday, August 9.  There will be no class on Tuesday, July 4.

Office hours: Every weekday afternoon, Monday to Friday, from 1-2 p.m., in my office, 430 McAllister. Other times by appointment - email me (climenha AT math DOT psu DOT edu) or talk to me after class if you need to see me and cannot make the regular office hours.

Grading scheme: Your final grade (out of 100%) will be determined as follows:

The first midterm test will cover unit one (chapter 13), the second will cover unit two (chapters 2, 3, and 4), and the final exam will be cumulative, covering all three units.

There will be weekly assignments and occasional quizzes, which will be announced in class (and posted on this website), and which will total 30% of your final grade.  Because solutions will be posted here once the due date has passed, late assignments will not be accepted, nor make-up quizzes permitted.  However, your lowest quiz/assignment grade will be dropped.