Math 110 - Fall 2009
Discovering in Mathematics

Professor: Erika L.C. King
Email: eking@hws.edu
Office: Lansing 304
Phone: (315) 781-3355
Home Page

Office Hours: M 1:30-3:30pm, W 2:30-4:00pm, Th 11:30am-12:30pm, F 1:30-2:30pm, and by appointment
Class: TTh 1:30-2:55PM in Napier 201
Course Syllabus
Course Grade Scale

READING/EXAM WEEK

Review Session: Sunday, December 13th 2:00pm-3:00pm in Napier 201

Office Hours:

• Sunday, December 13th: 3:15pm-4:30pm
• Monday, December 14th: 3:30pm-5:30pm
• Tuesday, December 15th: 11:00am-1:00pm and 4:00pm-5:30pm
• Wednesday, December 16th: 10:30am-12:30pm and 4:00pm-5:00pm
• Thursday, December 17th: 11:00am-noon
• By appointment

Final Exam: Thursday, December 17th 1:30pm-4:30pm in Napier 201

Remember your journal (including the last assignment below) is due at the final exam!

Last Journal Homework:

• Read Section 5.3 in the textbook (pages 359-367).
• Work Mindscapes 6, 9, 10, 12, 13, 15 and 26 from Section 5.3 (pages 368-371) in your journal.

WEEK 15

Journal Homework for class Tuesday, December 8:

• Minimal spanning trees have been used in areas such as biomedical image analysis, pattern recognition, weather data interpretation, fungal spore pattern analysis, and the study of particle interactions in turbulent fluid flows. Research some past or current applications of minimal spanning trees online. Then, in your journal, jot down a report (roughly the equivalent of one typed page) that describes an application of interest to you.
• In your journal, draw a graph on eight vertices with twelve edges. (a) How many edges must you eliminate to form a tree? Explain. (b) Draw at least three spanning trees of your graph. (c) How many edges does the complement of your graph have? Explain. (d) Draw the complement of your graph.

Journal Homework for class Thursday, December 10:

• Review the worksheet from class. Make sure you understand all the concepts and bring any questions you have to class on Thursday.
• In your journal, consider the following. Let G be a graph on nine vertices and let v be a vertex of degree four. What is the degree of v in the complement of G? Explain.
• In your journal, consider the following. Let G be a graph and let v be a vertex such that the degree of v is six in G and seven in the complement of G. How many vertices are in G? Explain.
• In your journal, consider the following. Let G be a graph on 12 vertices with 24 edges. How many edges are in the complement of G? Explain.

Collected Homework (Due Friday, December 11, 3:00PM):

• Complete this worksheet.
• (a) Draw a graph on seven vertices with 10 edges. (b) How many edges will be in the complement of your graph. Explain. (c) Draw the complement of your graph.
• BONUS (2 points): Is it possible to have a graph on five vertices whose complement is the same as the original graph? Why or why not?
• Although you are encouraged to discuss the homework with others, write-ups should be done individually. Remember that your solutions should be typed or neatly written, stapled if more than one page, and if you discuss the homework with anyone else, they should be noted. Remember that for a solution to be complete it must include full sentences.

WEEK 14

IMPORTANT: Office hours on Friday of this week will be cancelled, as I will need to leave early to head out of town for a memorial service. It is possible that I may be able to meet with you earlier in the day, but try to see me earlier in the week to address any questions that you have.

Journal Homework for class Tuesday, December 1:

• If you missed class last Tuesday, be sure to get the notes from someone before class so that you know what is going on. The material that we are covering right now is not in our textbook and so it is vitally important that you are taking good notes in class.
• In your journal, draw all distinct trees on four vertices. Then draw all distinct forests on four vertices. Be careful that your trees and forests are truly different; remember that it does not matter how you draw the graphs, but rather what the connections between edges and vertices actually are.
• In your journal, draw all distinct trees on six vertices.
• In your journal, carefully explain how many edges a tree with 37 vertices has. Then explain how many vertices a tree with 59 edges has.

Journal Homework for class Thursday, December 3:

• Continue to work on the worksheet from class. Come up with at least one algorithm for how to find a cheapest network. Make sure your procedure is precise. If you followed it word for word, would the result be the networks you found in part 1?
• In your journal, prove that if every vertex of a connected graph G has degree at least two, then G contains a cycle.
• In your journal, show that a graph on n vertices with n-1 edges need not be a tree. (You probably have examples of this on your secret handout.)

Collected Homework (Due Friday, December 4, 3:00PM):

• Draw all distinct trees on seven vertices. (Hint: There are more than seven and less than fifteen.)
• Draw all distinct forests on five vertices. (Note: A forest may have only one tree.)
• Six comic book collectors, Alex, Bailey, Carla, Joe, Leya and Tina, met to trade comic books with each other. Each trade that took place was between only two people. After the meeting, each of the six was asked how many people he or she had traded with. The answers were 5, 4, 2, 1, 3 and 2, respectively. Model this situation with graph theory by describing what would represent each vertex and what would represent each edge in the resulting graph. You need not actually draw a graph. Then prove that at least one person is mistaken.
• Show that in a room with at least two people, there are at least two people who have the same number of friends in the room. (Assume that if A is B's friend, then B is also A's friend.) Hint: Make a graph model as in the previous question. Then rewrite the question in graph theoretic terms.
• BONUS (5 points): Suppose T is a tree on 13 vertices such that each of its vertices has degree 1, 2 or 5. If T has exactly three vertices of degree two, how many vertices of degree one does it have? Carefully justify your argument with complete sentences using definitions and theorems where appropriate.
• Although you are encouraged to discuss the homework with others, write-ups should be done individually. Remember that your solutions should be typed or neatly written, stapled if more than one page, and if you discuss the homework with anyone else, they should be noted. Remember that for a solution to be complete it must include full sentences.
• As I will be out of town at 3PM on Friday, I will grant an extension on this assignment for anyone who wishes to use it. The extended deadline is 10AM on Monday, December 7th. You may drop off your homework at any time before Monday at 10AM; just slip it under my door if I am not in my office.

WEEK 13

There will be no collected homework due this week.

Journal Homework for class Tuesday, November 24:

• Continue working on your worksheet from class. Come up with at least two conjectures for question number 3. Then try to prove or disprove your conjectures. Use the graphs that you came up with in questions 1 and 2 to give you ideas for these conjectures.
• Come up with a conjecture for question 6 from the November 17th worksheet. That is, what can you say about the sum of the degrees of the vertices of a graph? Can you prove or disprove your idea? What information does this give you about the number of vertices of odd degree in a graph?
• In your journal, work Mindscapes 6 and 7 from the pdf, first as asked (they are looking for Eulerian circuits), and then with Hamiltonian cycles in mind.
• In your journal, prove or disprove the conjecture: If a graph is Eulerian, then it is Hamiltonian.
• In your journal, prove or disprove the conjecture: If a graph has no cut-vertex and all its vertices are of even degree, then it is Hamiltonian.
• Be prepared to share your results in class on Tuesday.

Have a great Thanksgiving Break!

WEEK 12

Journal Homework for class Tuesday, November 17:

• Read Section 3.5 in the textbook (pages 190-201). Check out the cardinality of the points in the square versus the cardinality of points on a line segment. Wow!
• Work Mindscapes 3, 4, 5, 9 and 20 from Section 3.5 (pages 201-204) in your journal.

Journal Homework for class Thursday, November 19:

• Read through the comments I wrote in your journal. Rework any of the Mindscapes for which you did not earn full credit. Let me know if you have any questions.
• Continue working on the handout from class. Particularly work on coming up with conjectures that sound something like, "If a graph has this property, then it contains a Hamiltonian cycle." or "If a graph contains a Hamiltonian cycle, then it has this property." Can you prove or disprove any of your conjectures? (This is question 5 from the groupwork sheet.)
• The third edition of our text was just published. It has a new section on Eulerian graphs. Click here for the Mindscapes from that section (sorry it isn't the best scan). Work Mindscapes 1, 2 (make up your own letters), 3, 8, 11 and 19 from this pdf in your journal.

Collected Homework (Due Friday, November 20, 3:00PM):

• Write up solutions to Mindscapes 7 and 10 at the end of section 3.5 (pages 202-203). The solution for Mindscape 7 should include both a clear diagram and an explanation with full sentences. For Mindscape 10 you need only include one brief sentence describing your procedure, which should be visualized in a clear diagram.
• Write up solutions to Mindscapes 10 (if you need to add edges, justify you have used the fewest possible), 12 (they are asking you about Eulerian cycles; note that just putting arrows on your graph is not clear enough - you need to write out the order in which you traverse the edges if you can), and 20 (this is about Eulerian paths) from the pdf. Use theorems and definitions to justify your answers.
• Prove or disprove the following conjecture that was made by a student: "If a graph has all vertices of odd degree, then it is Hamiltonian." Give a detailed description of your answer. If you are having trouble, work some examples and give details into your explorations for the answer.
• Although you are encouraged to discuss the Mindscapes with others, write-ups should be done individually. Remember that your solutions should be typed or neatly written, stapled if more than one page, and if you discuss the Mindscapes with anyone else, they should be noted. Remember that for a solution to be complete it must include full sentences.

WEEK 11

Exam 2 will be in class on Thursday, November 12th covering Sections 2.7 and 3.1-3.4 of the text.

Due to the exam, there will be no collected homework due this week.

Journal Homework for class Tuesday, November 10:

• Catch up on all your journal homework. Remember you need to turn in your journal on Thursday at the beginning of the exam. Review the journal guidelines in your syllabus to make sure that your journal conforms to these standards.
• Review for the exam; read class notes, homeworks and the text.
• Read Section 3.4 in the textbook (pages 173-185).
• Work Mindscapes 8, 19 and 22 from Section 3.4 (pages 187-189) in your journal.

Preparing for the exam Thursday, November 12:

• Work through the review sheet and practice problems (these do not need to go in your journal). Review past journal work and reread all comments on past homeworks.
• Remember to bring your journal to the exam to be turned in.

WEEK 10

My office hours on Friday, November 6th will be moved later to 3:00-4:00.

Journal Homework for class Tuesday, November 3:

• Read Section 3.3 in the textbook (pages 162-168).
• Work Mindscapes 5, 11, 17 and 20 from Section 3.3 (pages 169-171) in your journal.

Remember Exam 2 is next week! Make sure that you are up to date on your journal work! Use my office hours if you have questions!

Journal Homework for class Thursday, November 5:

• Review the ideas we were discussing at the end of class about how the subset T is formed. Work on your groups one-to-one correspondence.
• Start reading Section 3.4 in the textbook (especially pages 173-178).
• Work Mindscapes 1, 4, 6 and 10 from Section 3.4 (pages 185-187) in your journal.

Collected Homework (Due Friday, November 6, 4:00PM - note special time extension due to my moved office hours):

• Write up a solution to Mindscape 13 at the end of section 3.2 (page 157). Remember that complete solutions include explanations with full sentences.
• Write up solutions to Mindscapes 4, 10 and 14 at the end of section 3.3 (pages 169-170). Remember that complete solutions include explanations with full sentences.
• Write up a solution to Mindscape 13 at the end of section 3.4 (page 187). Be sure to briefly explain the process, including at least a few of the specific steps as applied to this correspondence.
• Remember that your solutions should be typed or neatly written, stapled if more than one page, and if you discuss the Mindscapes with _anyone_ else, they should be noted. Remember that for a solution to be complete it must include full sentences.

WEEK 9

The group project is due Friday, October 30th at 3:00pm. There is no collected homework this week.

Journal Homework for class Tuesday, October 27:

• Work on our question of whether or not we can make a "list" of the rational numbers. Can we?
• Work Mindscapes 3, 4, 6, 7, 9 and 14 from Section 3.2 (pages 156-157) in your journal.

Journal Homework for class Thursday, October 29:

• Read Section 3.2 in the textbook (pages 145-155). Welcome to the wild and wonderful world of infinity!
• Work Mindscapes 16, 30, 31, 32 and 36 from Section 3.2 (pages 157-160) in your journal.

WEEK 8

Get started on part I of your project. Remember that the project is due October 30th. Your group assignments have been posted here.

Journal Homework for class Tuesday, October 20:

• Read Section 2.7 in the textbook (pages 121-131).
• Work Mindscapes 2, 7, 13, 19 and 24 (given what we did in class, you should have an intuition of what this last one is, but be sure that you prove it) from Section 2.7 (pages 131-133) in your journal.

Journal Homework for class Thursday, October 22:

• Read Section 3.1 in the textbook (pages 138-140).
• Work Mindscapes 4, 7, 10, and 12 from Section 3.1 (pages 141-142) in your journal.

Collected Homework (Due Friday, October 23, 3:00PM):

• Write up solutions to Mindscapes 10 (use the hint if you need to, but explain why it works), 20 and 22 at the end of section 2.7 (pages 132-133). Be sure to explain your answers carefully!
• Write up solutions to Mindscapes 18 at the end of section 3.1 (page 143). Be sure to explain your answer carefully!
• Remember that your solutions should be typed or neatly written, stapled if more than one page, and if you discuss the Mindscapes with _anyone_ else, they should be noted. Remember that for a solution to be complete it must include full sentences.

WEEK 7

Have a great fall break!

Important: Your project assignment will be handed out soon. Write down on a piece of paper your name and your top three choices of who to partner with for your project. If there is anyone in the class with whom you do not think you would work well, you may record that as well. Bring this with you to class on Thursday to hand in.

Journal Homework for class Thursday, October 15:

• Review Section 2.6 in the textbook (pages 110-117).
• Work Mindscapes 16, 19, 22 and 29 from Section 2.6 (pages 90-93) in your journal. (Some of you do not have your journals, in which case you can work these on separate pages that may be stapled or taped neatly into your journal later.)

Collected Homework (Due Friday, October 16, 3:00PM):

• Write up solutions to Mindscapes 13, 18 and 26 at the end of section 2.6 (page 119). Be sure to explain your answers carefully!
• Remember that your solutions should be typed or neatly written, stapled if more than one page, and if you discuss the Mindscapes with _anyone_ else, they should be noted. Remember that for a solution to be complete it must include full sentences.

WEEK 6

Exam 1 will be in class on Thursday, October 8th covering Chapters 1 and 2 of the text.

There will be no collected homework due this week.

Journal Homework for class Tuesday, October 6:

• Catch up on all your journal homework. There are a bunch of new Mindscapes below, but many of them are shorter exercises. Remember you need to turn in your journal on Thursday at the beginning of the exam. Review the journal guidelines in your syllabus to make sure that your journal conforms to these standards.
• Review for the exam; read class notes, homeworks and the text.
• Read Section 2.5 in the textbook (pages 95-106). Carefully work through the details of how the public key coding scheme works. Try to explain it to a friend.
• Read pages 110-115 of Section 2.6 in the textbook. Review your groupwork of proving that the square root of three (and/or five) is irrational, and prepare to present it at the beginning of class.
• Work Mindscapes 1, 2, 4, 5, 7, 16 and 17 from Section 2.5 (pages 107-108) in your journal.
• Work Mindscapes 2, 3, 4, 6, 7, 8, 11 and 12 from Section 2.6 (pages 118-119) in your journal.

Preparing for the exam Thursday, October 8:

• Work through the review sheet and practice problems (these do not need to go in your journal).
• Reread Section 2.6 in the textbook, including the last two pages.
• You do not need to put these in your journal, but for some extra practice you can work Mindscapes 15, 16, 17 and 26 from Section 2.6 (page 119).

WEEK 5

Journal Homework for class Tuesday, September 29:

• Bring something to class that has a UPC symbol on it. You probably do this every day already, but feel free to get creative!
• Read Section 2.3 in the textbook (pages 64-77), and start reading Section 2.4 (pages 82-85).
• Work Mindscapes 16 and 27 from Section 2.3 (pages 77-80) in your journal.
• Work Mindscapes 2, 5, 7 and 8 from Section 2.4 (pages 89-90) in your journal.

Journal Homework for class Thursday, October 1:

• Read Section 2.4 in the textbook (pages 82-89). Review what we discussed in class. In your own words, explain why some errors in UPC readings can be detected and some cannot.
• Work Mindscapes 9, 13, 16 and 38 from Section 2.4 (pages 90-93) in your journal.

Collected Homework (Due Friday, October 2, 3:00PM):

• Write up a solution to Mindscape 28 at the end of section 2.3 (page 79). Be sure to explain your answers carefully!
• Write up solutions to Mindscapes 3 and 4 at the end of section 2.4 (page 89). Be sure to explain your answers carefully!
• Write up a solution to Mindscapes 15 (note that doing journal homework first might help!) and 24 at the end of section 2.4 (page 91). Be sure to explain your answers carefully!
• Remember that your solutions should be typed or neatly written, stapled if more than one page, and if you discuss the Mindscapes with _anyone_ else, they should be noted. Remember that for a solution to be complete it must include full sentences.

WEEK 4

Journal Homework for class Tuesday, September 22:

• Thoroughly read pages 64-69 of Section 2.3 in the textbook. You might want to take a peek at what is beyond as well. Pages 68-69 review the idea behind the proof of the infinitude of primes that we discussed in class on Thursday. Work through your class notes and this reading carefully and see if it makes sense. Try to explain it to someone else not in the class. Write down any questions you have about it so that we can discuss it at the beginning of class on Tuesday.
• Work Mindscapes 1, 2, 3, 8, 24 and 35 from Section 2.3 (pages 77-80) in your journal.

Journal Homework for class Thursday, September 24:

• Work on the groupwork handout from class today. You do not need to put this in your journal. Come to class with notes to share with your group. Find your group and start discussing when you get there.
• Read pages 64-75 of Section 2.3 in the textbook. Pay special attention to the story of Fermat's Last Theorem on the last three pages. A lesson to be learned: Have patience!
• Work Mindscapes 11, 19, 22 and 37 from Section 2.3 (pages 78-80) in your journal.

Collected Homework (Due Friday, September 25, 3:00PM):

• Write up solutions to Mindscapes 5 (there is more than one answer; you will probably use words to describe your list), 7 (How often do you think these numbers are prime? Explain.), 12 (Use an example as part of your explanation.), 23 (The Division Algorithm is helpful in your explanation here) and 25 at the end of section 2.3 (pages 77-80). Be sure to explain your answers carefully, including a short explanation of how you found them.
• Write up a solution to Mindscape 32 at the end of section 2.3 (page 80). Be sure to explain your answer carefully, including a short explanation of how you found it.
• BONUS (5 points): Mindscape 36 on page 80. A thorough and clear explanation is required for full credit.
• Remember that your solutions should be typed or neatly written, stapled if more than one page, and if you discuss the Mindscapes with _anyone_ else, they should be noted.

WEEK 3

Office hours are back to normal. Please use them!

Journal Homework for class Tuesday, September 15:

• Read pages 49-55 of Section 2.2 in the textbook.
• Check out this website about Fibonacci numbers in nature. It includes some nice pictures, as well as diagrams illustrating the different spirals. It also discusses how plant and tree growth exibit Fibonacci numbers through branching. Isn't this cool!!!
• Work Mindscapes 5, 6, 10 and 15 from Section 2.2 (pages 57-59) in your journal.

Journal Homework for class Thursday, September 17:

• Finish reading Section 2.2 in the textbook (pages 49-57).
• Work Mindscapes 17, 19, 22, and 24 from Section 2.2 (page 60) in your journal.

Collected Homework (Due Friday, September 18, 3:00PM):

• Write up solutions to Mindscapes 4, 8 and 37 at the end of section 2.2 (pages 57-62). Be sure to explain your answers carefully, including a short explanation of how you found them.
• Write up solutions to Mindscapes 20 and 23 at the end of section 2.2 (page 60). Be sure to explain your answers carefully, including a short explanation of how you found them.
• Remember that your solutions should be typed or neatly written, stapled if more than one page, and if you discuss the Mindscapes with anyone else, they should be noted.

WEEK 2

Note that due to individual appointments, some of my open office hours are shortened this week. The revised hours for this week are: Monday 1:30-3:00, Wednesday 2:30-3:30pm, Thursday 11:30am-Noon and Friday 1:30-2:00pm. If you cannot make these times and need to see me, please make an appointment.

Journal Homework for class Tuesday, September 8:

• Reread Sections 1.1-1.4 of the book (pages 4-28). Are there things that you picked up this time that you missed the first?
• Work Mindscapes 5, 10 and 12 (pages 29-31) in your journal.

Journal Homework for class Thursday, September 10:

• Buy a pineapple and bring it to class on Thursday. Both Wegmans and Tops have them in stock. If you would like to pair up with someone else in the class and bring one pineapple between the two of you, that will be sufficient, but at least half the class should have pineapples.
• Review the homework returned in class on Tuesday. Read my comments carefully, and rework any problems you missed the first time around.
• Read Section 2.1 (pages 40-45). Do you recognize the two mathematicians discussed? They were referred to in one of our stories. Work through the proof that "every natural number is interesting". Does it make sense?
• Work Mindscapes 8, 10 and 11 (pages 46-47) in your journal.

Collected Homework (Due Friday, September 11, 3:00PM):

• Write up solutions to Mindscapes 9 and 14 at the end of section 1.4 (pages 30-32). Be sure to explain your answers carefully, including a short explanation of how you found them. There are hints after the Mindscapes. Try not to use them (it is ok if you do, but try not to), but do read the Problem-solving techniques in blue (for ALL the problems) after you finish working on the problems if not before.
• Write up solutions to Mindscapes 4 and 14 at the end of section 2.1 (pages 46-47). Be sure to explain your answers carefully, including a short explanation of how you found them.
• BONUS (5 points): Mindscape 15 on page 32.
• Remember that your solutions should be typed or neatly written, stapled if more than one page, and if you discuss the Mindscapes with anyone else, they should be noted.

WEEK 1

Welcome to Discovering in Mathematics!!!

Note that due to individual appointments, some of my open office hours are shortened this week and next week. The revised hours for the rest of this week are: Wednesday 2:30-3:30pm, Thursday 11:30am-Noon and Friday 1:30-2:00pm. If you cannot make these times and need to see me, please make an appointment.

Collected Homework (Due Thursday, September 3, 1:30PM):

• Write an autobiographical essay as assigned on the syllabus.

Journal Homework for class Thursday, September 3:

• Read the syllabus. We went through most of this in class, but you should make sure you have read all the details and don't have any questions about it. Also be sure to record the exam dates in your personal calendar/planner. Remember there are no make-ups.
• Read the Welcome!, Surfing the book, 1.1 and 1.4 sections of the book (pages xi-xxiv, 3-13, 27-28). You may also look at sections 1.2 and 1.3, but it would be best to work on 1.1 for a few days first!
• Work on stories 2, 3, 5, 6, 7 and 10 (this last one is from the handout in class), pages 3-13. Concentrate especially on the story your group was assigned, but take notes on all the problems and start trying to figure out each. Think about the techniques and ideas we discussed in class about approaching a new problem.

Collected Homework (Due Friday, September 4, 3:00PM):

• Write up solutions to Mindscapes 2, 6, and 7 at the end of section 1.4 (pages 28-29). Be sure to explain your answers carefully, including a short explanation of how you found them. There are hints after the Mindscapes. Try not to use them (it is ok if you do, but try not to), but do read the Problem-solving techniques in blue (for ALL the problems) after you finish working on the problems if not before.
• Remember that your solutions should be typed or neatly written, stapled if more than one page, and if you discuss the Mindscapes with anyone else, they should be noted.

Last modified: Friday 11 December 17:33:07 EST 2009