A tentative schedule (PDF, TeX) was handed out on the first day of class. The calendar on this page will be updated to show what we actually did each day and what was assigned.
| Date | Summary |
|---|---|
| Tuesday, January 13 |
Introduction to the course. Everyone introduced themselves, said a few interesting things, and filled out the first-day questionnaire. We went through the course information sheet and talked about how to read mathematics. The first writing assignment was given; it is due on Tuesday, January 27. Handouts:
Read for next class: Section 1.1, pages 3–12
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| Thursday, January 15 |
We discussed Section 1.1 from the textbook, which was about identification numbers and check digits. In particular, we talked about the need for and advantages of identification numbers and the need for check digits to detect transmission errors. Some examples of identification numbers include Social Security numbers, UPCs, ISBNs, and credit cards (many of which use the Luhn algorithm for their check digit scheme); it is not important to know the details of these particular check digit schemes. We worked through a few examples of computation using check digits, given a check digit scheme; these examples showed how to determine whether an identification number is valid, how to find the check digit if it is not given, and how to find a missing digit in the middle of an identification number. Finally, we briefly talked about the Caesar cipher as an introduction to modular arithmetic. Read for next class: Part of Section 1.2, pages 18–22; Section 1.3, pages 37–50 Do for next class: Section 1.1, problems 3, 4, 9, 13, 14, 16, and 28; Section 1.3, problem 26 |
| Tuesday, January 20 |
We worked through Problem 28(a) from Section 1.1, and we discussed Section 1.3 from the textbook, which was about encoding data using binary codes. It is not important to know the particular details of the examples of binary codes given in the textbook, but you should be able to read and understand a description of a binary code well enough to encode and decode data. During the last few minutes of class we talked a little about modular arithmetic, which is discussed in Section 1.2; an interesting way to think about modular arithmetic is by using a clock face, as described on page 35 of the textbook. Handout: Postnet worksheet (PDF, WordPerfect) Read for next class: Section 2.1, pages 61–78; part of Section 2.2, pages 86–89 Do for next class: Section 1.3, problems 21, 22, 24, 34, and 35; Section 2.1, problem 9; Section 2.2, problem 2; and the Postnet worksheet. Optional: Section 1.2, problems 3, 4, and 46 (reading about clock arithmetic on page 35 will be helpful for problem 46). |
| Thursday, January 22 |
The Chapter 1 test took the first half hour of class. In the remaining time we talked about Section 2.1 from the textbook, including the reason that regular pentagons cannot tile the plane and the formula for finding the degree measure of each interior angle of a regular polygon with n sides. Important definitions from Section 2.1 include polygon, regular polygon, the names for polygons with various numbers of sides (most importantly, triangle, quadrilateral, pentagon, hexagon, and octagon), convex and concave, tiling (also called a tessellation), and regular tiling. It is not important to memorize these definitions word-for-word, but you should be familiar with what these terms mean so that when they appear in questions on quizzes and chapter tests you can understand what is being asked. The textbook also discusses how to form tilings from any triangle or any quadrilateral, which are nice things to know. In the last 15 minutes of class we played with polyominoes a bit, which are not covered in the textbook. Handout: Graph paper Read for next class: Part of Section 2.2, pages 86–89; Section 2.3, pages 112–125 Do for next class: Section 2.1, problems 4, 5, 10, and 26; Section 2.2, problem 14; Section 2.3, problem 17. Also, try the questions about polyominoes. |
| Tuesday, January 27 |
We spent the day working on a worksheet (PDF, TeX) which covered Chapter 2. See also the page about polyominoes. The Chapter 2 test will be Thursday; the worksheet would probably make a good test review. Read for next class: Section 6.1, pages 341–353 Do for next class: Work on the Chapter 2 worksheet. |
| Thursday, January 29 |
We started class with the Chapter 2 test, which took the first half hour of class. Afterward we talked about Section 6.1 for a little bit, which is the beginning of the chapter about graph theory, routes, and networks. Most of what we talked about was definitions: graph, vertex (plural vertices, also called nodes), edge, adjacent vertices, degree of a vertex, path, circuit (also called a cycle), Eulerian circuit, connected and disconnected graphs, and the components of a graph. We discussed the Königsberg bridge problem, which is also covered in the textbook. Chapter 2 test: PDF, TeX; solutions: 1, 2, 3, 4, 5(abc), 5(d) Quiz: PDF, TeX, MetaPost drawing, quiz-01-29.1 Handout: Writing assignment 2, due Tuesday, February 17 (PDF, TeX). For the first option, please see the Caesar cipher generator. Read for next class: Section 6.1, pages 341–353; Section 6.2, pages 360–370
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| Tuesday, February 3 |
In class we reviewed some of the concepts from Section 6.1, including the definitions (and introducing the term bridge, which I forgot on Thursday) and finding an Eulerian circuit if one exists. I gave a couple examples of situations that can be modeled with graphs in which loops make sense. Then we talked about Section 6.2. The main ideas of Section 6.2 are weighted graphs, subgraphs, trees (and spanning trees), and minimal spanning trees. We went through an example of finding a minimal spanning tree for a weighted graph; we used Kruskal’s algorithm, which is described step-by-step in the textbook on page 368. Lastly, I mentioned the idea of Eulerization, which is what you do if a graph doesn’t have an Eulerian circuit but you want it to. There is a handout about Eulerization with a few problems at the end for you to work for Thursday. Quiz: PDF, TeX, MetaPost drawing, quiz-01-29.1 Handout: Eulerization (PDF, WordPerfect) Read for next class: Section 6.3, pages 378–395 Do for next class: Eulerization handout; Section 6.2, problems 3, 10, 12, 16, 24, and 28; Section 6.3, problem 2. |
| Thursday, February 5 |
I mentioned a few things at the beginning of class: The Chapter 6 test is on Tuesday, February 10. There is a difference between the sentences “Every bird is not a robin” and “Not every bird is a robin”—it’s important to keep this difference in mind. Finally, when describing a subgraph, it isn’t enough to just list the vertices; you also need to describe where the edges are. The easiest way to describe a subgraph is to draw a picture of it (being sure to label the vertices). We then talked about Section 6.3 in the textbook, which talks about the Traveling-Salesperson Problem (TSP). [Note that there is a mistake in the book in Figure 6.51(a) on page 378: The edge AB should not be bold in the left-hand picture.] We talked about Hamiltonian paths and Hamiltonian cycles, and noted the differences between Hamiltonian paths and cycles and Eulerian paths and cycles. We defined complete graphs and derived the formula for the number of edges in a complete graph. Then we discussed the traveling-salesperson problem itself, using the idea of the cost of a path in a weighted graph. We discussed the inherent difficulty of solving the traveling-salesperson problem perfectly, and thus the need for approximation algorithms that are easy to do and usually give a good answer (though perhaps not the best answer). The particular algorithm that we discussed in class is the nearest-neighbor algorithm, described on page 386 of the textbook. The book also describes the cheapest-link algorithm, which is similar to Kruskal’s algorithm from Section 6.2, but we didn’t do an example of the cheapest-link algorithm in class. Quiz: PDF, TeX, MetaPost drawing, quiz-02-05.1 Read for next class: Section 7.1, pages 411–425 Do for next class: Section 6.3, problems 4–8, 14, 18, 20, 26, 29, 30, and 40 (and problem 42 if you want to try the cheapest-link algorithm); Section 7.1, problem 8 |
| Tuesday, February 10 |
We started class with the Chapter 6 test, and then talked a little about projects, tasks, digraphs, and precedence relations, which are covered in Section 7.1. Chapter 6 test: PDF, TeX; MetaPost drawings 1, 2 Read for next class: Section 7.1, pages 411–425; Section 7.2, pages 432–446 Do for next class: Section 7.1, problems 5, 6, 12, 16, 22, 28, and 33; Section 7.2, problem 5 |
| Thursday, February 12 |
We did some examples of scheduling projects—breaking them into tasks, estimating the time required for each task, and then scheduling the tasks in such a way that they get done as quickly as possible. We discussed how this relates to the order-requirement digraphs introduced in Section 7.1 and the Gantt charts introduced in Section 7.2. We talked about maximal paths and their relation to the critical time of a project (though see also the critical time worksheet below). We contrasted the critical time of a project with the finishing time; the critical time is calculated from the order-requirement digraph, while the finishing time is the actual time taken to finish the project. The finishing time is always at least as large as the critical time. We finished by running through an example of scheduling a project using the list-processing scheduling algorithm with the decreasing-time scheduling algorithm discussed in Section 7.2. Quiz: PDF, TeX, MetaPost drawing, quiz-02-12.1 Handouts: Critical time computation (PDF, WordPerfect); critical time exercises (PDF, WordPerfect) Read for next class: Section 7.2, pages 432–446; Section 6.3 problems 47–49, pages 401–403; critical time handouts Do for next class: Writing assignment 2 (PDF, TeX); Section 7.2, problems 4, 6, 13, and 30; critical time exercises; Section 6.3, problem 48 |
| Tuesday, February 17 |
After the daily quiz, most of the class was spent talking about conflict graphs and how the idea of graph coloring can be used to solve scheduling problems. We also went through a few examples in which we found forward critical times of tasks (see the critical time computation worksheet from Thursday: PDF, WordPerfect) and made a Gantt chart. Quiz: PDF, TeX, MetaPost drawing, quiz-02-17.1 Handouts: Word graphs, from Making the Alphabet Dance, by Ross Eckler, St. Martin’s Press, 1996 (1, 2, 3, 4, 5, 6); writing assignment 3 (PDF, TeX, MetaPost drawing, writing-3.1); conflict graphs Read for next class: Conflict graph handout Do for next class: The three problems from the conflict graph handout |
| Thursday, February 19 |
We began with the Chapter 7 test. Afterward, we started Chapter 8 by talking about dot plots, stem-and-leaf plots, histograms, bar graphs, line graphs, and pie charts and the differences between them. At the end of class we took a quiz. Chapter 7 test: PDF, TeX; MetaPost drawings 1, 2, 3, 4 Handout: Worksheet 8a (PDF, WordPerfect) Read for next class: Chapter 8, pages 477–544 Do for next class: Section 8.1, problems 2, 4, 8, 12, 22, 25, 30, and 38; Section 8.2, problem 8; Section 8.3, problem 17; Worksheet 8a |
| Tuesday, February 24 |
Most of the class today was spent talking about Sections 8.2 and 8.3. Section 8.2 discusses the use of graphs and charts to compare different data sets, and Section 8.3 is about “enhancement, distraction, and distortion” of graphs—essentially, ways in which graphs are often made to be misleading, intentionally or unintentionally. Remember that the Chapter 8 test is on Thursday. I would recommend doing the assigned homework problems from Sections 8.1 through 8.3 as a review for the test. There is also a worksheet I forgot to hand out in class today, which is available below. Handout: Worksheet 8b (PDF) Read for next class: Chapter 8, pages 477–544 Do for next class: Section 8.2, problems 2, 7, 11, 15, 19, 25, 31, and 35; Section 8.3, problems 1, 3, 4, 9, 11, 15, 18, 23, and 27; Worksheet 8b |
| Thursday, February 26 |
We took the Chapter 8 test at the beginning of class, and then talked about Section 9.1. One key point from this section is the difference between a population and a sample in the context of conducting a survey or an experiment. Another important point is the idea of a representative sample, and the possibility of bias in a sampling method. Chapter 8 test: PDF, TeX; Excel 2004 worksheet, zinc graph, manganese graph, Oscars graph Read for next class: Section 9.1, pages 565–573; Section 9.2, pages 578–586 Do for next class: Section 9.1, problems 1, 2, 12, 13, 14, 17, and 18; Section 9.2, problem 2 |
| Tuesday, March 3 |
We spent the day talking about sampling methods, beginning with simple random sampling, which is described in the last part of Section 9.1, and continuing with independent sampling, systematic sampling, quota sampling, stratified sampling, and cluster sampling, all of which are described in Section 9.2. Please note that my office hours on Wednesday, March 4 have been moved forward to 12:30–2:30. If you would like to talk to me but cannot make it to my office during those times, please send me an e-mail and we can set up a different time to meet. Handout: How to avoid plagiarism (PDF, TeX) Read for next class: Section 9.3, pages 594–613 Do for next class: Section 9.2, problems 3, 6, 8, 16, 20, 25, 28, and 32; Section 9.3, problem 2 |
| Thursday, March 5 |
We discussed statistical summaries of data, described in Section 9.3. We talked about the mean, the median, the mode, the range, the first and third quartiles, the variance, and the standard deviation of a list of data values. We also covered the difference between the population mean and a sample mean, what it means for a distribution of data to be skewed left or skewed right, the idea of a weighted mean, how to draw a box-and-whisker plot, and the difference between the population standard deviation and a sample standard deviation. It’s not important to memorize the formulas for variance and standard deviation; I will give you these formulas on the test if you need them. On Tuesday, since I will be out of town, Courtney Gibbons will give a review session about Chapter 9. The Chapter 9 test will be postponed until Thursday. Read for next class: Section 10.1 Do for next class: Section 9.3, problems 3, 5, 9, 15, 19, 29, 31, and 41 |
| Tuesday, March 10 |
Today was a Chapter 9 review day, led by Courtney Gibbons. The random-number table on page 570 was reviewed and some review problems were given. The various sampling methods from Section 9.2 were illustrated with visual aids. After the quiz, a worksheet over Section 9.3 was handed out, and the rest of the class was a question-and-answer time. Handout: Section 9.3 worksheet (PDF, LaTeX) Read for next class: Section 10.1 Do for next class: Chapter 9 review problems (pages 627–629), problems 1, 3, 6, 7, 8, 9, 11, 14, 15, and 19 |
| Thursday, March 12 |
We began the day with the Chapter 9 test. Afterward we began our study of probability with an experiment based on the Monty Hall problem (sometimes called the three-doors problem). We found that the best strategy is to switch (specifically, in our experiment 8 out of 11 people who switched won, while only 5 of 15 who stayed with their original choice won). At the end of class we had a short time to briefly cover a few definitions from Section 10.1, including the definitions of experiment, outcome, sample space, and event, as these words relate to probability. Read for next class: Sections 10.1 and 10.2 Do for next class: Section 10.1, problems 1–8, 14, 16, 20, 23, 26, and 32; Section 10.2, problem 3 |
| Tuesday, March 24 |
We talked about the difference between experimental probability and theoretical probability, introduced the union (written ∪), and intersection (written ∩) of events, and talked about events that are mutually exclusive. We performed experiments in class to determine the experimental probability of certain events, specifying the experiment, the possible outcomes, the sample space, and the event. Read for next class: Sections 10.2 and 10.3 Do for next class: Section 10.1, problems 28, 32, and 37; Section 10.2, problems 2, 4, 5, 6, 8, 12, 16, 18, and 24 |
| Thursday, March 26 |
We briefly discussed some formulas for determining probabilities. In a nutshell:
We also discussed drawing tree diagrams, described in Section 10.2, and talked about how to use these diagrams to count the number of possible outcomes and to determine the probability of a complex outcome based on probabilities assigned to the various branches of the tree diagram. Handout: Worksheet 10a (PDF, WordPerfect) Read for next class: Section 10.3 Do for next class: Section 10.2, problems 2, 4, 5, 6, 8, 12, 16, 18, and 24; Section 10.3, problem 32; Worksheet 10a |
| Tuesday, March 31 |
We spent the first part of class talking a bit about the project. We then discussed expected value and odds from Section 10.3, and worked through a few examples. I spent a few minutes showing everyone a neat little program called Dasher for entering text using only a pointing device (such as a mouse). Read for next class: Section 11.1 Do for next class: Section 10.3, problems 31, 33, 35, 37, 39, and 41; Chapter 10 review problems (pages 695–697), problems 1(abcd), 3(abd), 5, 7, 9, 13, 19, 21, and 23 |
| Thursday, April 2 |
We took the Chapter 10 test. Afterward we voted to have three more chapter tests, with the last one on the Tuesday of Dead Week; the Chapter 11 test will be next Thursday, April 9. We then began discussing Chapter 11, starting with normal distributions and how the population mean μ and the population sample deviation σ affect the shape of a normal distribution curve. Read for next class: Sections 11.1 and 11.2 Do for next class: Section 11.1, problems 1–4, 6, 10, 11, 12, 15, 16, 17, 20, and 31; Section 11.2, problem 2 |
| Tuesday, April 7 |
Today we talked all about normal distributions. We began by reviewing the idea of a histogram and how it relates to a normal distribution curve, noting that it is the area under a portion of the curve that gives us information about frequency. We discussed the standard normal distribution (for which the mean is μ = 0 and the standard deviation is σ = 1), and used Tables 11.1 and 11.3 in the textbook to find various areas under this curve. From Section 11.2 we discussed the 68–95–99.7 rule, saw how to compute the z-score for a data value, and went through an example showing how to use these z-scores and the tables from the textbook to find the area under a portion of any normal distribution curve. The Chapter 11 test will be on Thursday. Handouts: Writing assignment 4 (PDF, TeX); photocopy of Tables 11.1 and 11.3 from the textbook Read for next class: Sections 11.1 and 11.2 Do for next class: Section 11.1, problems 7–14, 17, 20, 31, and 32; Section 11.2, problems 2, 3, 5, 7, 8, 11, 12, 13, 14, and 23 |
| Thursday, April 9 |
After the Chapter 11 test, we began discussing Section 3.1. There are four voting methods described in this section: the plurality method, the Borda count method, the plurality with elimination method, and the pairwise comparison method. We talked about the first three of these, and went through some examples, showing the differences between them. Chapter 11 test: PDF, TeX; MetaPost drawing, ch-11-test.1 Read for next class: Sections 3.1 and 3.2 Do for next class: Section 3.1, problems 2, 6, 14, 16, 18, 20, and 22; Section 3.2, problems 1 and 2 |
| Tuesday, April 14 |
We began today by quickly reviewing the plurality method, the Borda count method, and the plurality with elimination method. We then talked about preference tables and their application to the plurality with elimination method. Next we embarked upon a fairly lengthy example involving a vote to decide the location of the capital of Tennessee, working through it for the plurality method, the Borda count method, and the plurality with elimination method, and introducing the pairwise comparison method. This example was taken from Wikipedia, which has good articles about each of these voting systems:
After the Tennessee example, we talked very briefly about ways to break a tie, and then started on Section 3.2. This section is about flaws in voting systems. Today we talked about the majority criterion; we will continue with the other criteria on Thursday. The Chapter 3 test will be next Tuesday, April 21. Read for next class: Sections 3.1 and 3.2 Do for next class: Section 3.1, problems 2, 6, 14, 16, 18, 20, and 22; Section 3.2, problems 1, 2, 4, 6, 8, 12, 16, and 24 |
| Thursday, April 16 |
We discussed four desirable properties of voting systems: the majority criterion, the head-to-head criterion, the monotonicity criterion, and the irrelevant-alternatives criterion. We went through a few examples in which the various voting methods fail one or more of these criteria. Then we talked about Arrow’s impossibility theorem, which says that it is mathematically impossible to create a perfect voting system—any voting system must violate at least one of these criteria if there are three or more candidates. Finally, we briefly discussed approval voting. Read for next class: Sections 3.1 and 3.2 Do for next class: Section 3.1, problems 3, 5, 14, 17, and 21; Section 3.2, problems 1, 3, 7, 9, 13, 21, and 31 |
| Tuesday, April 21 |
After the Chapter 3 test, we began talking about Chapter 5, which discusses apportionment methods. We considered the motivation for apportionment, went through an example showing why simple rounding doesn’t always work, and then introduced the standard divisor and the standard quota and how they are used in Hamilton’s method. Read for next class: Sections 5.1 and 5.3 (feel free to read Section 5.2 if you’re interested, but I don’t think we’ll talk about it in class) Do for next class: Section 5.1, problems 1, 2, 8, 10, 12, and 24 |
| Thursday, April 23 |
We finished talking about Chapter 5 today. We began by reviewing an example of Hamilton’s method, and then showed how the same example would work using Lowndes’ method, which is similar except for the use of the relative fractional part of the standard quota. We then discussed some potential flaws of apportionment methods, as described in Section 5.3: an apportionment method may violate the quota rule, or it may allow the Alabama paradox, the population paradox, or the new-states paradox to occur. (We didn’t talk much about the population paradox in class, but you should understand what the other two are.) At the end, we saw that it is mathematically impossible for an apportionment method to avoid all of these flaws if there are at least three states. The Constitution and Paradoxes is an interesting page about flaws of apportionment methods, together with some Java applets that allow you to experiment with the populations of the states. Read for next class: Sections 5.1 and 5.3 (feel free to read Section 5.2 if you’re interested) Do for next class: Section 5.1, problems 1, 7, 9, 12, and 23; Section 5.3, problems 1, 3, 7, and 13 |
| Tuesday, April 28 |
The Chapter 5 test was given at the beginning of class. Afterward I offered $20 to whoever could write the largest finite number on one side of an index card; Crystal won by writing “Graham’s number.” We then spent a bit of time talking about large numbers, and finished with the last quiz. Chapter 5 test: PDF, TeX; Excel 2004 worksheet |
| Thursday, April 30 |
Three people showed up today. We spent the class playing SET, thinking about nontransitive dice, proving that the square root of 2 cannot be written as a fraction, and other fun stuff. Handout: Puzzles (PDF, TeX; MetaPost drawings 1, 2, 3, 4, 5) |
