# MATH 110 - Fall 2016 Discovering in Mathematics

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

Office Hours: M: 1:30-3:30pm, W: 12:30-2:00pm, Th: 4:30-5:30pm, F: 11:00am-Noon, and by appointment
Class Schedule: held TTh 1:30-2:55pm in Eaton 110

Course Syllabus

One Last Journal Homework:

• Read Section 6.2 in the textbook (pages 401-408).
• Work Mindscapes 6, 9, 10, 12, 13, 15 and 26 from Section 6.2 (pages 409-411) in your journal.

Review Session: Monday, December 12th 1:00pm-2:00pm in Eaton 110.

Office Hours:

• Monday, December 12: 2:15pm--3:30pm
• By appointment

Final Exam: Tuesday, December 13th 1:30PM until 4:30PM in Eaton 110.

Remember your journal (including the last assignment above) is due at the final exam! (Check out this list for a condensed version of what should be in your journal.)

### WEEK 15: December 5 - December 9

Put in your calendar: Review Session Monday, December 12 from 1:00pm until 2:00pm in Eaton 110. Attendance is optional and you are welcome to come for any portion of the review session. Bring lots of questions.

Journal Homework for class Tuesday, December 6:

• MINIMAL SPANNING TREES: 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; or type a page and staple it into your journal) 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. (We will learn about this in class on Tuesday, so save this part for after class.)

Journal Homework for class Thursday, December 8:

• Review the worksheets from class. Make sure you understand all the concepts and bring any questions you have to class on Thursday. In particular, make sure you go back to number 5 on November 29th's worksheet and answer that question. Also complete questions 4 and 5 from Tuesday's worksheet if you did not finish those with your group (note that we discussed 5 indirectly!).
• Read Section 6.4 in our text (this covers pages 434-449). This will review a lot of the graph theory concepts we have been discussing recently.
• Work Mindscapes 6, 10, 11 and 13 from Section 6.4 (pages 450-452) in your journal.
• 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 $G$, $\overline{G}$? 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 $\overline{G}$. 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 $\overline{G}$? Explain.

Collected Homework (Due Friday, December 12, 3:00PM):

• Please be sure to be neat and staple your work; remember it can gain you an extra point!
• Remember if you discuss the homework with anyone else, it should be noted.
• Complete this worksheet. Note that the last page will make more sense after Tuesday's class.

### WEEK 14: November 28 - December 2

Journal Homework for class Tuesday, November 29:

• Review your class notes from last week's class. If you missed class last Tuesday, be sure to get the notes from someone before class on November 29th so that you know what is going on. The material that we covered Tuesday and much that we will cover for the rest of the semester is not in our textbook and so it is vitally important that you are taking good notes in class.
• Work Mindscape 26 from Section 6.1 (page 399) in your journal. This is what we were talking about in class on Tuesday!
• 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. Be sure you explain why!

Journal Homework for class Thursday, December 1:

• Review your Exam 2. Read ALL my comments and rework any questions on which you missed points. Please let me know if you have any questions no matter how big or small!
• Continue to work on the Weighted Graphs 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?
• TREE OR NOT TREE: 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.)
• In your journal, so the following COMIC QUESTION: 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 each vertex represent and what would each edge represent in the resulting graph. You need not actually draw a graph. Then prove that at least one person is mistaken.
• THE FRIENDS QUESTION: In your journal, 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.

Collected Homework (Due Friday, December 2, 2:00PM):

• Please be sure to be neat and staple your work; remember it can gain you an extra point!
• Remember if you discuss the Mindscapes with anyone else, they should be noted.
• Draw all distinct trees on seven vertices. (Hint: There are more than seven and less than fifteen.)
• Draw all distinct forests on five vertices. (Recall: A forest may have only one tree.)
• ***This was added on Tuesday!*** Write up solutions to Mindscapes 28 and 30 at the end of Section 6.1 (page 399). Note that there are five questions within Mindscape 28 so be sure you have answered all of them. Hint for Mindscape 30: create a graph model explicitly describing what the vertices and edges represent. Then use your graph to answer the question.
• 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.

### WEEK 13: November 21-25

Due an appointment, I need to shift my Monday office hours to start earlier. My hours will be 12:30-2:30.

There will be no collected homework due this week.

Journal Homework for class Tuesday, November 22:

• Continue working on your secret worksheet from class. Don't discuss it with people outside your group! Make sure you understand your new definitions and work on examples for exercises 1 and 2 on the worksheet. Remember that your group will be presenting on Tuesday, so the more comfortable you can become with the ideas before Tuesday the better! The graphs that you come up with in exercises 1 and 2 should give you ideas for conjectures in exercise 3.
• Come up with a conjecture for question 7 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?
• Work Mindscapes 6 and 7 from Section 6.1 (page 396) in your journal. Do each Mindscape as asked (they are looking for Eulerian circuits), and then complete them replacing "Euler circuit" with "Hamiltonian cycle".
• 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 the property that the degrees of all its vertices have the same parity (that is, all are even or all are odd), then the graph is Hamiltonian.
• Be prepared to share your results in class on Tuesday.

Have a great Thanksgiving Break!

### WEEK 12: November 14-18

Journal Homework for class Tuesday, November 15:

• Read Section 3.4 in the textbook (pages 174-185).
• Work Mindscapes 8, 11, 14 and 19 from Section 3.4 (pages 186-188) in your journal. Be sure to follow the directions especially for 11 and 14 where you are looking for a set using Cantor's method. Mindscape 19 is about Russell's Paradox which is discussed on pages 184-5 in our text - another mind-bending idea!
• Read Section 3.5 in the textbook (pages 190-199). This is looking at cardinality in terms of geometry. Check out the cardinality of the points in the square versus the cardinality of points on a line segment. Wow!

Journal Homework for class Thursday, November 17:

• 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 6 from the groupwork sheet.) Be sure to bring your handout with you to class on Thursday!
• Read Section 6.1 in the textbook (pages 385-395). Make sure the idea of why we need vertices of even degree to have an Eulerian graph really makes sense!
• Work Mindscapes 1, 2, 3, 8, 11 and 19 from Section 6.1 (pages 395-398) in your journal.

Collected Homework (Due Friday, November 18, 2:00PM):

• Please be sure to be neat and staple your work; remember it can gain you an extra point!
• Remember if you discuss the Mindscapes with anyone else, they should be noted.
• Write up solutions to Mindscapes 1, 4 and 13 at the end of Section 3.4 (pages 185-187). Be sure to briefly explain your work.
• ***This was added on Tuesday!*** Write up solutions to Mindscapes 10 (if you need to add edges, justify you have used the fewest possible), 12 (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) at the end of Section 6.1 (pages 397-398). Use theorems and definitions to justify your answers.
• BONUS (5 points): Mindscape 22 at the end of Section 3.4 (page 188). A thorough and clear explanation is required for full credit.

### WEEK 11: November 7-11

Exam 2 will be in class on Thursday, November 10th covering Sections 2.7, 10.4 and 3.1-3.3 of the text. Remember to bring your journal to be turned in!

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

Journal Homework for class Tuesday, November 8:

• 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.
• Review the ideas we were discussing in class, especially Cantor's Power Set Theorem and beginning to prove it.
• Start reading Section 3.4 in the textbook (especially pages 174-178).
• Work Mindscapes 3, 5, 6 and 10 from Section 3.4 (page 186) in your journal.
• Prepare for your exam! Read the review sheet found here. I will not bring copies of this to class, but I will project it with the document camera. Bring questions to class. And as mentioned above: catch up on journal work.

Preparing for the exam Thursday, November 10:

• 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. Don't forget to highlight properly!!! (Check out this list for a condensed version of what should be in your journal.)

### WEEK 10: October 31 - November 4

Happy Halloween!!!

Journal Homework for class Tuesday, November 1:

• Read Section 3.2 in the textbook (pages 147-156). Amaze your friends with the wild and wonderful world of infinity!
• Work Mindscapes 16, 30, 31, 32 and 36 from Section 3.2 (pages 158-160) in your journal.

The first departmental colloquium is this Thursday, November 3 at 4:45pm in Napier 201 with snacks starting at 4:30. Yu (Phoebe) Cai and Casey Coffey will be speaking about their summer research experience (with Prof. Rusinko!) in a talk entitled Mathematical Phylogenetics. Be sure to check in with me after the talk to make sure I recorded your presence to count for bonus points!

Given that we have the departmental colloquium during my usual office hours, my Thursday office hours will be moved earlier this week. They will begin at 3:45 and end at 4:35 in time for us to go down to the talk and get snacks!

Journal Homework for class Thursday, November 3:

• Read Section 3.3 in the textbook (pages 163-168). REALLY read it! This reviews what we have been discussing in the last week! Note that the author's version of Cantor's diagonalization argument is a bit different from what we did in class. Be sure you understand it. It will help you especially with Mindscape 17 if you do!
• Work Mindscapes 5, 11, 16, 17 and 20 from Section 3.3 (pages 169-171) in your journal.

Collected Homework (Due Friday, November 4, 2:00PM):

• Please be sure to be neat and staple your work; remember it can gain you an extra point!
• Remember if you discuss the Mindscapes with anyone else, they should be noted.
• Write up a solution to Mindscapes 13 and 17 at the end of section 3.2 (pages 157-158). Remember that complete solutions include explanations with full sentences. Note that to do Mindscape 17 you will want to read and think about Mindscape 16 first, so do your journal work!
• ***This was added Tuesday!*** 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.

### WEEK 9: October 24-28

The group Coding Project is due Friday, October 28th at 2:00pm. Thus there is no collected homework this week.

Journal Homework for class Tuesday, October 25:

• Meet with your Project Group. Finalize all calculations and make a list of questions you can bring to office hours if there is anything about which you are confused. Start writing up introductions and explanations. Remember to think about creativity! These should be very high quality papers.
• Read Section 3.1 in the textbook (pages 140-142).
• Work Mindscapes 4, 7, 9, 10, and 12 from Section 3.1 (pages 143-144) in your journal.

Journal Homework for class Thursday, October 27:

• Rework any questions on the exam for which you missed points. After you have solutions or specific questions about where you are stuck, come to office hours to review them with me. Come early and often!
• Be sure to read your notes from Tuesday's class. Let me know if you have any questions on the material.
• Read pages 147-150 of Section 3.2 in the textbook. Welcome to the wild and wonderful world of infinity!
• Work Mindscapes 3, 4, 6, 9 and 14 from Section 3.2 (pages 157-158) in your journal.

Collected Homework (Due Friday, October 28, 2:00PM):

• Turn in your coding project, one for each group!
• Here is a rubric I will use to grade your projects. Take a look to make sure your group has thought about what you need.

### WEEK 8: October 17-21

Journal Homework for class Tuesday, October 18:

• Read the Project Handout Part I carefully. Then meet with your Project Group to make sure you understand how ISBN codes work, start the first few exercises, and set a schedule for completing the project. Note that everyone in the group should work on every problem to make sure everyone understands the concepts. Also note that if you have not told me who your group is, you will not get Part II of the Project!!! Email me yesterday!
• Read Section 2.7 in the textbook (pages 124-133).
• 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 134-135) in your journal.
• Without reading ahead, think of alternate voting procedures that might be better than Plurality Voting when more than two candidates are in the running. Discuss this with friends and be prepared to share ideas in class.

Journal Homework for class Thursday, October 20:

• Pick up the last parts of the Project on Wednesday. Read ALL the Project Handouts carefully. Then meet with your Project Group to make sure you understand how ISBN codes work, finish the exercises in Part I, start on PartII, and set a schedule for completing the project. Figure out who will start writing up which parts, but remember that everyone should work through all the parts.
• Read Section 10.4 in the textbook (pages 811-822). Discuss what you have learned with friends and family. If they don't know about the different voting methods, teach them about it and see how they react to the different outcomes.
• Work Mindscapes 7 (check out the details of the paradox on pages 817-818), 8, 10 and 11 from Section 10.4 (pages 823-824) in your journal.

Collected Homework (Due Friday, October 21, 2:00PM):

• Please be sure to be neat and staple your work; remember it can gain you an extra point!
• Remember if you discuss the Mindscapes with anyone else, they should be noted.
• Write up solutions to Mindscapes 10 (use the hint if you need to, but explain why it works), 18, 20 and 22 at the end of Section 2.7 (pages 133-134). Be sure to explain your answers carefully! Note that Mindscape 22 has an extra step, but one that you have done before!
• ***Added on Tuesday!*** Write up solutions to Mindscapes 14 and 15 at the end of Section 10.4 (page 824).

### WEEK 7: October 12-14

Have a great fall break!

Important: [You should have done this for LAST Thursday! If you have not yet, please do so ASAP. Emailing your group list is acceptable.] Your project assignment will be handed out soon. You will be working in groups of three on the project. Find a group of three, agree to work together and turn in a piece of paper with your team members listed (one paper per group). Bring this with you to class on Thursday October 6th to hand in.

Journal Homework for class Thursday, October 13:

• Review Section 2.6 in the textbook (pages 113-120).
• Work Mindscapes 16, 19, 22 and 29 from Section 2.6 (pages 121-122) 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 14, 2:00PM):

• Please be sure to be neat and staple your work; remember it can gain you an extra point!
• Remember if you discuss the Mindscapes with anyone else, they should be noted.
• Write up solutions to Mindscapes 14, 18, 20 and 26 at the end of Section 2.6 (pages 121-122). Be sure to explain your answers carefully! Hints: Use proof by contradiction for 14, 18 and 26, and, of course, the definition of rational! For Problem 14, you may assume that you have already proved that the square root of 14 is irrational.

### WEEK 6: October 3-7

Exam 1 will be in class on Thursday, October 6th covering Chapters 1 and 2 (through Section 2.6) of the text. Remember to bring your journal to be turned in!

Important: Your project assignment will be handed out soon. You will be working in groups of three on the project. Find a group of three or two, agree to work together and turn in a piece of paper with your team members listed (one paper per group). Note that there will be only two groups of two and seven groups of three, so do your best to find a group of three. Bring this with you to class on Thursday to hand in.

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

Journal Homework for class Tuesday, October 4:

• 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 98-109). Carefully work through the details of how the public key coding scheme works. Try to explain it to a friend.
• Read pages 113-118 of Section 2.6 in the textbook. Try proving that the square root of 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 109-111) in your journal.
• Work Mindscapes 2, 3, 4, 6, 7, 8, 11 and 12 from Section 2.6 (pages 120-121) in your journal.
• Prepare for your exam! Read the review sheet found here. Bring questions to class. And as mentioned above: catch up on journal work.

Check out this list for a condensed version of what should be in your journal. If you find something missing (i.e. it is on the website, but did not make it on the list), inform me immediately; your classmates will thank you.

Preparing for the exam Thursday, October 6:

• Work through the review sheet and practice problems (these do not need to go in your journal). When doing modular arithmetic, remember to look for opportunities when Fermat's Little Theorem might be helpful. In what situations does it apply?
• Reread Section 2.6 in the textbook, including the last two pages (pages 113-120).
• 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 (pages 121-122).

### WEEK 5: September 26-30

Journal Homework for class Tuesday, September 27:

• Bring something to class that has a UPC symbol on it. You probably do this every day already, but feel free to get creative! Try to bring something that is not food. Note that books have ISBN codes which are different than UPCs, so a book will not count.
• Check out this cool diagram illustrating how even numbers up to fifty can be written as the sum of two primes. The diagram is on the right hand side of this page. Read the caption there as it explains how to read the diagram, and then check out the bigger version of the diagram here.
• Read Section 2.3 in the textbook (pages 68-80), and start reading Section 2.4 (pages 86-89).
• Work Mindscapes 16 and 27 from Section 2.3 (pages 81-82) in your journal.
• Work Mindscapes 2, 7 and 8 from Section 2.4 (pages 92-93) in your journal.

Due to needing to visit another colleague's class, my office hours on Friday will be shifted. My hours will be 12:30pm-1:30pm. Please contact me if you cannot make these times and have questions.

Journal Homework for class Thursday, September 29:

• Again: bring something to class that has a UPC symbol on it. You probably do this every day already, but feel free to get creative! Try to bring something that is not food. Note that books have ISBN codes which are different than UPCs, so a book will not count. We will use these in group work at the beginning of class.
• Read Section 2.4 in the textbook (pages 86-92). Review what we discussed in class. Bring specific questions about any of the material, especially that which we already covered.
• Work Mindscapes 3, 5, 9, 13 and 16 from Section 2.4 (pages 92-94) in your journal.

Collected Homework (Due Friday, September 30, 2:00PM):

• Please be sure to be neat and staple your work; remember it can gain you an extra point!
• Remember if you discuss the Mindscapes with anyone else, they should be noted.
• Write up a solution to Mindscape 28 at the end of section 2.3 (page 82). Be sure to explain your answers carefully, accompanying your words with examples!
• Solve the following Extra Mindscape: Suppose a certain number when divided by 72 yields a remainder of 45. If we add 269 to our original number, what is the remainder when this new number is divided by 36? Be sure to explain your answers carefully with WORDS!
• Write up a solution to Mindscapes 4, 15 (note that doing journal homework first might help!) and 38 at the end of section 2.4 (pages 92-96). Be sure to explain your answers carefully with WORDS!

### WEEK 4: September 19-23

Journal Homework for class Tuesday, September 20:

• We worked through some tough concepts in class on Thursday! Thank you for staying focused and asking questions!
• Review what we discussed as the introduction to the proof of the fact that there are an infinite number of primes. Experiment with the $N$ I gave you at the end of class. Do you see why it is NOT divisible by 2, 3, 4, and 5? How does this help us find a prime bigger than $M=5$?
• Thoroughly read pages 68-73 of Section 2.3 in the textbook. You might want to take a peek at what is beyond as well. Pages 72-73 review the idea behind the proof of the infinitude of primes that we alluded to in class on Thursday. We will discuss this in detail!
• Work Mindscapes 1, 2, 3, 8 and 12 from Section 2.3 (pages 80-81) in your journal.

Journal Homework for class Thursday, September 22:

• Review our work from class on Tuesday. We proved that there are an infinite number of primes! Make sure the proof makes sense to you.
• Continue working on the twin-primes and writing-numbers-as-the-sum-of-two-primes questions from your group worksheet at the end of class. Go straight to your groups as you arrive for class to compare your findings.
• Read pages 68 up to the top of page 79 of Section 2.3 in the textbook. Pay special attention to the story of Fermat's Last Theorem on the last two pages. A lesson to be learned: Have patience!
• Work Mindscapes 11, 19, 22, 24 and 35 from Section 2.3 (pages 81-83) in your journal. Remember that although journal work need not be as refined as collected work, you should still explain your answers! If you haven't already noticed this, after you have worked your journal problems be sure to check those that have solutions or hints in the back of the text to see if your solution agrees. Remember that different approaches are fine as long as they are right.

Collected Homework (Due Friday, September 23, 2:00PM):

• Please be sure to be neat and staple your work; remember it can gain you an extra point!
• Write up solutions to Mindscapes 5 (there is more than one answer; you will probably use words to describe your list as you cannot actually write an infinite number of things down in a finite amount of time...unless you are a superhero), 7 (Try values of $n$ starting with 2. Justify your answer. How often do you think these numbers are prime? Explain.), and 9 (Experiment with examples to obtain your solution.) at the end of section 2.3 (page 81). Be sure to explain your answers carefully, including a short explanation of how you found them.
• Write up solutions to Mindscapes 23 (The Division Algorithm is helpful in your explanation here!) and 25 at the end of section 2.3 (pages 81-82). Be sure to explain your answers carefully, including a short explanation of how you found them. Doing your journal work first will be helpful!
• BONUS (5 points): Mindscape 36 on page 83. A thorough and clear explanation is required for full credit.
• Remember if you discuss the Mindscapes with anyone else, they should be noted.

### WEEK 3: September 12-16

Last call for pictures/appointments! Final autobiography grades will be made Wednesday. All students who wish to be in the class must have submitted an autobiography and had their appointment by that time.

Please use office hours whenever you have questions! It was great seeing so many of you last week. I hope you will continue to come and continue to use the hours in the early part of the week especially!

Journal Homework for class Tuesday, September 13:

• Enjoy your pineapples! Look for Fibonacci numbers everywhere!
• Read pages 53-59 of Section 2.2 in the textbook. This includes some of the ideas we were using and will use Tuesday about finding the ratio of adjacent Fibonacci numbers.
• 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!!!
• Here's a two minute youtube video that shows someone preparing a pineapple to eat. Count the number of spirals the person makes with his carving tool as he cuts off the little "nodes" on the pineapple. Is it a Fibonacci number?
• Here's a youtube video that shows someone discussing Fibonacci numbers. If you find it fascinating and want to learn more, watch the next one (or two); it is a three part series.
• Work Mindscapes 5 (this will help you for what we will do in class!), 6, 8 and 10 from Section 2.2 (pages 61-63) in your journal.

The my office hours on Thursday will be shifted. My hours will be 3:15pm-4:15pm. Please contact me if you cannot make these times and have questions.

Journal Homework for class Thursday, September 15:

• Reread the first part of Section 2.2 in the textbook and then finish reading it to the end (pages 53-61).
• Work Mindscapes 17, 19, 22, and 24 from Section 2.2 (page 64) in your journal. Be sure to show your work and briefly explain your thought process.

Collected Homework (Due Friday, September 16, 2:00PM):

• Please be sure to be neat and staple your work; remember it can gain you an extra point!
• Write up a solution to Mindscape 4 at the end of section 2.2 (pages 57-62). ***THEN look for patterns! In class we found this expression with all one's (also see Mindscape 3), in Mindscape 4 you did the same expression but with two' and three's. Looking at those three, can you guess (without calculating) what it would be with all four's? What makes you think that? Now calculate and see if you are correct. Explain your pattern.
• Write up a solution to Mindscapes 36 at the end of section 2.2 (page 66). Be sure to explain your solution carefully. Hint: Think of where the Fibonacci numbers come from. Note that there are two parts to this question!
• Write up solutions to Mindscapes 20 and 23 at the end of section 2.2 (page 64). Be sure to explain your answers carefully, including a short explanation of how you found them.
• Remember if you discuss the Mindscapes with anyone else, they should be noted.

### WEEK 2: September 5-9

Remember to keep your appointments!!!

Note that due to introductory appointments with many of you, some of my office hours have scheduled appointments this week. The open hours for Monday are: 2:10pm-3:30pm. If you cannot make those times and need to see me, please make an appointment.

Journal Homework for class Tuesday, September 6:

• 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 9, 10 and 12 (pages 29-31) in your journal.
• Optional and Just For Fun: Did you enjoy thinking about Story 7? Read this Let's Make a Deal article from the New York Times. Then try this version of the Let's Make a Deal game. Try each strategy (Switch or Stay) at least 15 times in a row. How do they compare? Does it seem to match our results discussed in class? Include your thoughts/work in your Journal.

Keep your appointments! If you haven't met with me and do not yet have an appointment, email me ASAP with your schedule! If you have met with me and forgot to bring your picture, remember to bring one with you to class.

Journal Homework for class Thursday, September 8:

• Buy a pineapple and bring it to class on Thursday. 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.
• 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?
• I referred to a recent movie about these famous mathematicians. Read about it here. Keep your eyes and ears peeled, the Mathematics and Computer Science Department may be showing this movie for a Movie Night!
• Continue your work on the following question: If we have five closed tennis ball cans (each can hold up to three balls), how do we know that two cans must hold the same number? You may keep your work on this in your class notes.
• Work Mindscapes 8, 10 and 11 at the end of Section 2.1 (page 50) in your journal.

Collected Homework (Due Friday, September 9, 2:00PM):

• Write up solutions to Mindscapes 6 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.
• ***Added on Tuesday*** Write up solutions to Mindscapes 4 and 14 at the end of section 2.1 (pages 49-50). Be sure to explain your answers carefully, including a short explanation of how you found them. Your solutions should include explanations of how these questions relate to the Pigeonhole Principle. Feel free to use the hint for Mindscape 14 at the back of the book.
• 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.
• Also remember that while you may talk to others about the ideas in these Mindscapes, you should be alone when you write up your solution. Your solution must be your own work.

### WEEK 1: August 29 - September 2

Welcome to Discovering in Mathematics!!!

Collected Homework (Due Thursday, September 1 at the beginning of class):

• Write an autobiographical essay as assigned on the syllabus.

Journal Homework for class Thursday, September 1 (i.e. due as preparation for class):

• Read the syllabus TWICE. Although we spent some time on this in class, we did not discuss every detail. You should be sure you have read all of it and understand what is expected. Please ask if you have questions. Note the paper copy I gave you is blue so that you can easily find it. Refer to it often. 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). Especially take note of the "Travel Tips" on page xv, and the "Lessons for Life" on page 27. 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!