MATH 110 - Fall 2019
Discovering in Mathematics

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

Office Hours: M: 2:30-3:45pm, T: 3:45-4:45, W: 1:00-2:30pm, F: 1:30-2:30pm, and by appointment
Class Schedule: held TTh 2:00-3:30pm in Napier 201

Course Syllabus
Course Grade Scale
Journal Entry Form

READING/EXAM WEEK: December 11 - December 17

Review Session: Friday, December 13th 1:00pm-2:00pm in Eaton 111. Bring questions!

Office Hours:

• Wednesday, December 11: 11:00am--1:00pm
• Thursday, December 12: 3:30pm--5:00pm
• Friday, December 13: 3:30pm--5:00pm
• Monday, December 16: Noon--1:30pm
• By appointment

Final Exam: Monday, December 16th 7:00PM until 10:00PM in Napier 201.

Remember your journal is due at the final exam!

WEEK 15: December 9 - December 10

Journal Homework for class Tuesday, December 10:

• Bring a laptop or other devise, if you have one, on which you can complete the course evaluations.
• Review your class notes from Thursday's class on complements, planar graphs and more! If you did not finish working through number six on last Tuesday's group worksheet or any part of Thursday's group worksheet, try to complete them before class.
• Read Section 6.2 in our text (this covers pages 401-408). This discusses the results that you were finding in class group work on Thursday. It also shows how we can use the results to show that there are only five regular solids (those things that were on the front table in class)!
• Work Mindscape 6 and 13 from Section 6.4 (pages 450-452) on the Journal Entry Forms and place it in your journal.
• Complete THE COMPLEMENT DEGREES Mindscape: (a) 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. (b) 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. Complete this Mindscape on the Journal Entry Form and place it in your journal.
• Work Mindscape 15, 17 and 26 from Section 6.2 (pages 409-411) on the Journal Entry Forms and place it in your journal. Be sure that you explain your answers carefully and show your work!
• Email me by noon on Tuesday if you would like to discuss a journal question in class. In your email, include the question with its section number (if applicable) you have questions about. I will choose the question that has the most requests.

WEEK 14: December 2 - December 6

Have a great Thanksgiving Break!!!

Great work making conjectures about graph theory and proving results! Don't let anyone tell you that you cannot do mathematics!

Journal Homework for class Tuesday, December 3:

• Review your class notes from class Thursday, November 21st. If you missed class that day, be sure to get the notes from someone before class on December 3rd so that you know what is going on. The material that we covered 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.
• In particular, review Prim's and Kruskal's Algorithms. Look at the graph for which we applied Prim's Algorithm. Try applying Kruskal's Algorithm and see what you obtain. List the edges in the order you would choose them. Is it the same spanning tree? Does it have the same weight? How was the process different? This work can go with your class notes and should not be in your journal.
• Complete this TREE OR NOT TREE Mindscape: 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.) Complete it on the Journal Entry Form and place it in your journal AFTER the blue evaluation page for Exam 2 that I added to your journal.
• Complete this COMIC Mindscape: 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. Complete this Mindscape on the Journal Entry Form and place it in your journal AFTER the blue evaluation page for Exam 2 that I added to your journal.
• Complete THE FRIENDS Mindscape: 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. Complete this Mindscape on the Journal Entry Form and place it in your journal AFTER the blue evaluation page for Exam 2 that I added to your journal.

Journal Homework for class Thursday, December 5:

• Note that the homework for Friday must be done on the hand out. There are copies with the journal entry forms in the box outside my office.
• Review your class notes from Tuesday's class on Kruskal's algorithm, spanning trees, complements and more!
• 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!
• 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 type a report (roughly one page) that describes an application of interest to you and put it in your journal. I have a three hole punch if you need one or print it on a Journal Entry Form!
• Complete this GRAPH EXPLORATION Mindscape: 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. Complete this Mindscape on the Journal Entry Form and place it in your journal.
• Work Mindscape 10 from Section 6.4 (page 451) on the Journal Entry Forms and place it in your journal.
• Read Section 6.4 in the textbook (pages 434-449). This talks about spanning trees and Hamiltonian cycles!
• Email me by noon on Thursday if you would like to discuss a journal question in class. In your email, include the question with its section number (if applicable) you have questions about. I will choose the question that has the most requests.

Collected Homework (Due Friday, December 6, 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.
• Complete this worksheet. Please do all your work on the hand out. Copies of the handout are available in the box with the Journal Entry Forms outside my office. Note that the last page will make more sense after Tuesday's class.

WEEK 13: November 18 - November 22

Journal Homework for class Tuesday, November 19:

• Review what we discussed about trees on Thursday! Write out the definition of a tree and the three equivalent statements given what each group presented on Thursday. Practice drawing different trees.
• Come up with a conjecture for question 7 from the November 12th 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 (each with a PART TWO -- see below!) from Section 6.1 (page 396) on the Journal Entry Forms and place them in your journal AFTER the blue evaluation page for Exam 2 that I added to your journal. If you have not yet received your journal back from me, save the forms to put in your journal afterwards. There is currently a stack of these forms in a box outside my office. Note that these are already hole-punched! Don't wait to do your journal problems! Do each Mindscape as asked (they are looking for Eulerian circuits), and then ***complete a PART TWO for each replacing "Euler circuit" with "Hamiltonian cycle".
• Complete this Eulerian/Hamiltonian Mindscape: Prove or disprove the conjecture: If a graph is Eulerian, then it is Hamiltonian. Complete it on the Journal Entry Form and place it in your journal AFTER the blue evaluation page for Exam 2 that I added to your journal.
• Complete this Hamiltonian Parity Mindscape: 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. Complete it on the Journal Entry Form and place it in your journal AFTER the blue evaluation page for Exam 2 that I added to your journal.
• Be prepared to share your results in class on Tuesday!

Journal Homework for class Thursday, November 21:

• Review your class notes from Tuesday's class. If you missed class last Tuesday, be sure to get the notes from someone as soon as possible 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 Mindscapes 26 and 28 from Section 6.1 (page 399). This is what we were talking about in class on Tuesday! Complete this Mindscape on the Journal Entry Form and place it in your journal AFTER the blue evaluation page for Exam 2 that I added to your journal.
• Complete this Edges and Vertices of Trees Mindscape: 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 with at least one full sentence each! Complete this Mindscape on the Journal Entry Form and place it in your journal AFTER the blue evaluation page for Exam 2 that I added to your journal.
• Complete this Trees and Forests Mindscape: 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. Complete this Mindscape on the Journal Entry Form and place it in your journal AFTER the blue evaluation page for Exam 2 that I added to your journal.
• Email me by noon on Thursday if you would like to discuss a journal question in class. In your email, include the question with its section number (if applicable) you have questions about. I will choose the question that has the most requests.

Collected Homework (Due Friday, November 22, 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.
• These Mindscapes are not in your textbook; they are just listed below.
• Mindscape G1: Draw all distinct trees on seven vertices. (Hint: There are more than seven and less than fifteen.)
• Mindscape G2: Draw all distinct forests on five vertices. (Recall: A forest may have only one tree.)
• Mindscape G3: I claim that I have a tree on 2019 vertices with 2020 edges. Do you believe me? Why or why not? Write complete sentences justifying your answer.
• ***This was added on Tuesday!*** Mindscape G4: Determine if the following are degree sequences of a simple graph. If the sequence is a degree sequence of a graph, draw a graph with that sequence. If the sequence is not a degree sequence of a graph, explain why not. (a) 5, 3, 2, 2, 1, 1 (b) 8, 6, 5, 5, 4, 3, 3 (c) 5, 5, 4, 4, 4, 3, 3, 2, 1 (d) 3, 3, 2, 2, 1, 1, 1, 1

WEEK 12: November 11 - November 15

Journal Homework for class Tuesday, November 12:

• Reread Section 3.4 in the textbook (pages 174-185), and review our discussions on the Continuum Hypothesis and Russell's Paradox.
• 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!
• Use your experience reading Section 3.5 to work Mindscapes 2, 6, 8, 9 and 10 from Section 3.5 (pages 200-201) on the Journal Entry Forms and place them in your journal AFTER the blue evaluation page for Exam 2 that I added to your journal. If you have not yet received your journal back from me, save the forms to put in your journal afterwards. There is currently a stack of these forms in a box outside my office. Note that these are already hole-punched! Don't wait to do your journal problems!
• Get ready to jump into the awesome world of Graph Theory on Tuesday!

Journal Homework for class Thursday, November 14:

• 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) on the Journal Entry Forms and place them in your journal AFTER the blue evaluation page for Exam 2 that I added to your journal. If you have not yet received your journal back from me, save the forms to put in your journal afterwards. There is currently a stack of these forms in a box outside my office. Note that these are already hole-punched! Don't wait to do your journal problems!

As announced, there will be a departmental colloquium this Thursday, November 14 at 4:30pm in Napier 201 with snacks starting at 4:15. William Smith computer science major Morgan Hamre will be giving a talk entitled A Brief History of Women in Technology. Be sure to check in with me after the talk to make sure I recorded your presence to count for bonus points!

Collected Homework (Due Friday, November 15, 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 13 at the end of Section 3.4 (page 187). Be sure to briefly explain your work.
• Write up a solution to Mindscape 7 at the end of Section 3.5 (pages 200-201). Be sure to do your journal work first and 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, and 20 (this is about Eulerian paths) at the end of Section 6.1 (pages 397-398). Note that identifying an Eulerian circuit or path by 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 do so. Use theorems and definitions to justify your answers.

WEEK 11: November 4 - November 8

Exam 2 will be in class on Thursday, November 7th covering Sections 2.5-2.7, 10.4 and 3.1-3.4 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 5:

• 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. I will not accept journals that are in binders that are too large.
• 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 its proof.
• Read Section 3.4 in the textbook (pages 174-185).
• Work Mindscapes 6, 8, 10, 14 and 19 from Section 3.4 (pages 186-188) in your journal.
• Prepare for your exam! Read the review sheet here. 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 should not 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. Make sure it follows all requirements!!!
• Have confidence in your abilities! You have been doing great work!

NO Collected Homework Due Friday!!!

WEEK 10: October 28 - November 1

Journal Homework for class Tuesday, October 29:

• Review your notes from Thursday's class and read Section 3.2 in the textbook (pages 147-156). Amaze your friends with the wild and wonderful world of infinity!
• Work on the Dodge Ball vs. Infinity handout from Thursday's class. Come to class with ideas about at least the first two questions to discuss with your group.
• Work Mindscapes 16, 30, 31, 32 and 36 from Section 3.2 (pages 158-160) on the Journal Entry Forms and place them in your journal. There is currently a stack of these forms in a box outside my office.
• Email me by noon on Tuesday if you would like to discuss a journal question in class. In your email, include the question and section number you have questions about. I will choose the question that has the most requests.

Journal Homework for class Thursday, October 31 (Happy Halloween!):

• Review your notes from Tuesday's class. Let me know if you have any questions on the material.
• 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. The main difference is that when we were making our x different in the ith decimal place from the real number corresponding to the natural number i, we chose our new digit randomly; in the text they specifically choose the number 2 or the number 4 so that their x will only have 2's and 4's as digits. Be sure you understand the text's argument. 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) on the Journal Entry Forms and place them in your journal. There is currently a stack of these forms in a box outside my office. Make sure you are caught up on your journal as you will be turning it in next week!
• Email me by noon on Thursday if you would like to discuss a journal question in class. In your email, include the question and section number you have questions about. I will choose the question that has the most requests.

Collected Homework (Due Friday, November 1, 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.
• 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.
• 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 10 and 14 at the end of Section 3.3 (page 170). Remember that complete solutions include explanations with full sentences.
• ***This was added Tuesday!*** OPTIONAL BONUS (3 points): Write up a solution to Mindscape 18 at the end of Section 3.3 (page 171). Remember that complete solutions include explanations with full sentences.

WEEK 9: October 21 - October 25

The group Coding Project is due Friday, October 25th at 2:00pm. Thus there is no collected homework due this week.

If you were not in class on Thursday, come to office hours on Monday to pick up your journal and project assignments.

Journal Homework for class Tuesday, October 22:

• Meet with your Project Group. Finalize all calculations for Part I, work through calculations for Parts II-IV, 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.
• 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) on the Journal Entry Forms and place them in your journal AFTER the green evaluation page for Exam 1 that I added to your journal. There is currently a stack of these forms in a box outside my office. Note that these are already hole-punched! Don't wait to do your journal problems!
• Read Section 3.1 in the textbook (pages 140-142).
• Work Mindscapes 7, 9, and 12 from Section 3.1 (pages 143-144) on the Journal Entry Forms and place them in your journal AFTER the green evaluation page for Exam 1 that I added to your journal. There is currently a stack of these forms in a box outside my office. Note that these are already hole-punched! Don't wait to do your journal problems!
• Email me by noon on Tuesday if you were in class last Thursday and would like to discuss a journal question in class this Tuesday. In your email, include the question and section number you have questions about. I will choose the question that has the most requests.

Journal Homework for class Thursday, October 24:

• The class did great on the exam! Feel proud about your work! Also, 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!
• Review 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) on the Journal Entry Forms and place them in your journal AFTER the green evaluation page for Exam 1 that I added to your journal. There is currently a stack of these forms in a box outside my office. Note that these are already hole-punched! Don't wait to do your journal problems!
• Email me by noon on Thursday if you would like to discuss a journal question in class. In your email, include the question and section number you have questions about. I will choose the question that has the most requests.

Collected Homework (Due Friday, October 25, 2:00PM):

• Turn in your coding project, one for each group!

WEEK 8: October 16 - October 18

Have a Great Fall Break!!!

Journal Homework for class Thursday, October 17:

• 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 2.6 (pages 113-120) in the textbook.
• Work Mindscapes 4, 6, 7, 8, 10 and 11 from Section 2.6 (page 121) on the Journal Entry Forms and place them in your journal AFTER the green evaluation page for Exam 1 that I added to your journal. If you have not yet received your journal back from me, save the forms to put in your journal afterwards. There is currently a stack of these forms in a box outside my office. Note that these are already hole-punched! Don't wait to do your journal problems!
• 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.

Office Hours: Due to an appointment, I will need to move my Friday office hours to Thursday of this week. Thus I will have office hours Thursday, October 17: 3:45-4:45pm, and no office hours on Friday, October 18th. Please let me know if you have questions and cannot make my revised hours this week.

Collected Homework (Due Friday, October 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.
• 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.
• Write up solutions to Mindscapes 10 and 18 at the end of Section 2.6 (page 121). Be sure to explain your answers carefully! Hints: Use proof by contradiction for both, and, of course, the definition of rational! For Problem 18, be sure you have read the text first for hints!
• Write up solutions to Mindscapes 14 and 15 at the end of Section 10.4 (page 824). Note that the details that go with these question are written before question 10!

WEEK 7: October 7 - October 11

Important: The first part of your project assignment will be handed out this week. 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). Note that there will be eight groups of three, and one group of two. If you have not found a group of three, we may need to reshuffle to accommodate. Please let me know if you need assistance. Bring this with you to class on Tuesday or drop it off at my office on Wednesday.

Journal Homework for class Tuesday, October 8:

• Review your notes about the material we discussed from Section 2.5 last Tuesday.
• 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.
• Work Mindscapes 1, 2, 4, 5, 7, 16 and 17 from Section 2.5 (pages 109-111) on the Journal Entry Forms and place them in your journal AFTER the green evaluation page for Exam 1 that I added to your journal. If you have not yet received your journal back from me, save the forms to put in your journal afterwards. There is currently a stack of these forms in a box outside my office. Note that these are already hole-punched!

Journal Homework for class Thursday, October 10:

• Read Section 2.7 in the textbook (pages 124-133). All together the rational and irrational numbers make what?
• 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) on the Journal Entry Forms and place them in your journal AFTER the green evaluation page for Exam 1 that I added to your journal. If you have not yet received your journal back from me, save the forms to put in your journal afterwards. There is currently a stack of these forms in a box outside my office. Note that these are already hole-punched! Don't wait to do your journal problems!
• Start reading Section 2.6 in the textbook (pages 113-120). A proof that the square root of two is irrational is there and you can read it to see what our next steps in class should be.
• 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.

Collected Homework (Due Friday, October 11, 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.
• 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.
• Write up a solution to Mindscape 1 at the end of Section 2.5 (page 109). Be sure to explain your answer with a couple of sentences.
• PART 2 of Mindscape 1 (Section 2.5): Use the scheme on page 99 to encode the message "I am now a mathematician." Write a couple of sentences explaining the process you used.
• Solve the following Fermat's Modular Madness Mindscape: Find the mod 23 equivalents to the following: (a) 7^{22} [that is, seven raised to the power of 22], (b) 7^{88} [that is, seven raised to the power of 88], (c) 7^{90} [that is, seven raised to the power of 90]. Hint: each part helps you with the next! Hint 2: Do your Mindscapes due Tuesday FIRST!!! Be sure to explain your answers carefully!
• Write up solutions to Mindscapes 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! Do your journal work first before you do this one!

WEEK 6: September 30 - October 4

Exam 1 will be in class on Thursday, October 3rd covering Chapters 1 and 2 (through Section 2.4) of the text. Remember to bring your journal to be turned in!

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

Journal Homework for class Tuesday, October 1 (Happy October!):

• 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. I will not accept journals that are in binders that are too large.
• Start reading Section 2.5 in the textbook (pages 98-109). We will discuss this during the first part of class on Tuesday.
• Work Mindscapes 17, 19 and 38 from Section 2.4 (pages 94-96) on the Journal Entry Forms and place them in your journal. There is currently a stack of these forms in a box outside my office. 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.
• Prepare for your exam! Read the review sheet found here. Bring questions to class. And as mentioned above: catch up on journal work.

Preparing for the exam Thursday, October 3:

• Work through the review sheet and practice problems from class (these do not need to go in your journal).
• Have confidence in your abilities! You have been doing great work!

NO Collected Homework Due Friday!!!

WEEK 5: September 23 - September 27

Journal Homework for class Tuesday, September 24:

• Bring something to class that has a UPC symbol (the barcode with twelve numbers) 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) on the Journal Entry Forms and place them in your journal. These are questions to explore. Share your work and explain what conclusions you make from it, realizing that there may not be a final answer, but you might be able to make conjectures.
• Work Mindscapes 2, 7 and 8 from Section 2.4 (pages 92-93) on the Journal Entry Forms and place them in your journal. There is currently a stack of these forms in a box outside my office. As usual, be sure to show your work and explain your results.

Journal Homework for class Thursday, September 26:

• Review our work on modular arithmetic from class on Tuesday. You were doing great! Make sure you understand the examples we did in groupwork. Practice showing all your work including how to write numbers as a multiple of the mod number plus a remainder. You need to show this step for full credit; this shows that you understand that the equivalence is coming from the remainder.
• 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 and 13 from Section 2.4 (pages 92-93) on the Journal Entry Forms and place them in your journal. There is currently a stack of these forms in a box outside my office. 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. For the last two Mindscapes, we just started talking about the UPCs, so this is very new. Experiment with these two questions and then go back and check your work again after Thursday's class.
• Email me by noon on Thursday if you would like to discuss a journal question in class on Thursday. In your email, include the question and section number you have questions about. I will choose the question that has the most requests.

Collected Homework (Due Friday, September 27, 2:00PM):

• Please submit your solutions in the order they were assigned!
• 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.
• 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 265 to our original number, what is the remainder when this new number is divided by 36? Be sure to explain your answers carefully!
• Solve the following Modular Madness Mindscape: Find the mod 8 equivalents to the following: (a) 53 X 27 [that is, the product of 53 with 27], (b) 3^{756} [that is, three raised to the power of 756], (c) 3^{759} [that is, three raised to the power of 756]. Hint: Use your work from (b) to solve (c)!
• Write up a solution to Mindscape 15 (note that doing journal homework first might help!) at the end of Section 2.4 (page 94). Be sure to explain your answer carefully with WORDS! You likely will want to wait until after class on Thursday to do this question.

WEEK 4: September 16 - September 20

Journal Homework for class Tuesday, September 17:

• Remember to bring in your picture if you forgot to bring it to your appointment! Last chance on Tuesday! (Remember that it is part of the grade for your first assignment!)
• We worked through some tough concepts in class on Thursday! You asked some great questions and were doing really well! Have confidence!
• Review our justification of why the strategy of Fibonacci Nim works. Think about prime numbers and our goal to show that there are an infinite number of them by showing that for any natural number we can show there must be a prime number bigger than that number. Be ready to prove our theorem on Tuesday!
• Thoroughly read pages 68-71 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 2, 3, 8 and 12 from Section 2.3 (pages 80-81) on the Journal Entry Forms and place them in your journal. There is currently a stack of these forms in a box outside my office. These Mindscapes will get you thinking about prime numbers and help prepare us for the proof that there are an infinite number of primes.

Journal Homework for class Thursday, September 19:

• 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, 22, 24 and 35 from Section 2.3 (pages 81-83) on the Journal Entry Forms and place them in your journal. There is currently a stack of these forms in a box outside my office. 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.
• Email me by noon on Thursday if you would like to discuss a journal question at the beginning of class on Thursday. In your email, include the question and section number you have questions about. I will choose the question that has the most requests.

Collected Homework (Due Friday, September 20, 2:00PM):

• Write up a solution to Mindscape 5 at the end of section 2.3 (page 81). 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. Your solution should describe your list and also justify why it is infinite.
• Write up a solution to Mindscape 7 at the end of section 2.3 (page 81). Try values of n starting with 2. Justify your answer. How often do you think these numbers are prime? Explain.
• ***Added Tuesday: Write up solutions to Mindscapes 23 and 25 (The Division Algorithm is helpful in your explanation for both!) at the end of Section 2.3 (page 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!
• ***Added Tuesday: Write up a solution to this extra Mindscape: Does there exist a number n bigger than 10 such that both n and n+1 are prime numbers? If so, find such an n; if not, show why not. 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.
• 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 3: September 9 - September 13

Remember to keep your appointments!!!

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 10:

• Remember to bring in your picture if you forgot to bring it to your appointment!
• 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 discussing on Thursday about finding the ratio of adjacent Fibonacci numbers.
• Check out this website about Fibonacci numbers in nature. It discusses it discusses a bit about why these numbers might be used by nature.
• 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? Isn't it amazing how this is done?
• 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 review what we did in class on Thursday), 6, 8 and 10 from Section 2.2 (pages 61-63) on the Journal Entry Forms and place them in your journal. There is currently a stack of these forms in a box outside my office.

Journal Homework for class Thursday, September 12:

• Don't be afraid to experiment! You all have been sharing great ideas! Keep them coming!
• If you haven't met with me yet, be sure to email your schedule for Wednesday and Friday ASAP! This is part of your first assignment and I will be finalizing grades for that soon!
• Remember to bring in your picture if you forgot to bring it to your appointment!
• 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.
• Email me by noon on Thursday if you would like to discuss a journal question at the beginning of class on Thursday. In your email, include the question and section number you have questions about. I will choose the question that has the most requests.

Collected Homework (Due Friday, September 13, 2:00PM -- Happy Friday the 13th!):

• Write up a solution to Mindscape 12 at the end of section 2.2 (page 63). Be sure to explain your solution carefully, showing all your work.
• Write up a solution to Mindscape 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!
• ***Added Tuesday: 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.
• ***Added Tuesday: Suppose you are Player 1 and you know your Fibonacci Nim strategy well. How many sticks might you begin with if your first move is to remove 55 sticks? (Note there are actually many right answers to this question, you just need to find one! And if you work in a group for this question find a different number for each member of your group!) Explain carefully why your answer works in words thinking about the rules of the game and the strategy.
• 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 2: September 2 - September 6

Remember to keep your appointments!!!

Journal Homework for class Tuesday, September 3:

• 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 6, 9 and 12 (pages 28-31) on the Journal Entry Forms and place them 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 on a Journal Entry form in your journal. You can label this problem as Section 1.4, BONUS 1.

Journal Homework for class Thursday, September 5:

• 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 from week 1. Work through the proof that "every natural number is interesting". Does it make sense?
• There was a 2015 movie about these famous mathematicians. Read about it here.
• Work Mindscapes 4, 8 and 10 at the end of Section 2.1 (page 50) on the Journal Entry Forms and place them in your journal. I will try to keep a stack of these journal forms outside my office. Be sure to let me know if it runs out!

Collected Homework (Due Friday, September 6, 2:00PM):

• Write up solutions to Mindscapes 14 and 19 at the end of section 1.4 (pages 31-34). 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: How many people must we gather to insure that at least two of them must have the same birthday? How many people must we gather to insure that at least three of them must have the same birthday? Be sure to answer both questions and explain your answers carefully, including a short explanation of how you found them. Your solution should include explanations of how these questions relate to the Pigeonhole Principle.
• Added on Tuesday: Write up solution Mindscape 14 at the end of section 2.1 (page 50). Be sure to answer all parts and explain your answers carefully, including a short explanation of how you found them. Your solution 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 26 - August 30

Welcome to Discovering in Mathematics!!!

Collected Homework (Due Thursday, August 29 at 2:00pm):

• Fill out this autobiographical questionnaire. Print it two sided if possible. If not, staple it before submitting. Be sure to leave the top portion on the first side (above where you place your name) blank.

Journal Homework for class Thursday, August 29 (i.e. due as preparation for class):

• There are some three-ring binders outside my office free for the taking if you need one for your journal.
• 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 yellow 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!
• Work on the story your group was assigned. Think about the techniques and ideas we discussed in class about approaching a new problem. Fill out a journal entry for your story (use one of the forms you picked up in class or print this out), and put it in your binder journal. Write a short explanation and/or make a diagram to accompany your answer. Prepare for your presentation. Try to meet with your group before Thursday's class. This is not required, but would be a plus.
• Work on journal entries for the other groups' stories as well. The stories are 2, 3, 5, 7, 9 and 11 (this last one is from the handout in class), pages 3-13 in the text. Again think about the techniques and ideas we discussed in class about approaching a new problem. Use a different Journal Entry for each story (front and back can be different!). Be sure to have your work on all six stories in your journal. You should either copy the question in your journal or make a clear outline of it so that when you look back at your journal to review for an exam you know what the original question was asking. (This is true in general throughout the semester. For this particular assignment, I definitely would do the OUTLINE version!) Write a short explanation and/or make a diagram to accompany each answer. Remember to follow the journal guidelines in the syllabus. It is not expected that your work for the other five stories will be as detailed as the one your group is presenting. You may revise your entry after our class on Thursday.

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

• Write up solutions to Mindscapes 2, and 5 at the end of section 1.4 (page 28). Be sure to explain your answers carefully, including a short explanation of how you found them. Hint for Mindscape 2: Try experimenting with a smaller example; what happens if you only three politicians? Hint for Mindscape 5: Don't be afraid to experiment!
• There are more 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. These are on pages 35-41.
• 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.