276_homework_2017

Homework

05/09
  1. Prepare for the individual portion of the final exam. (To be given in class this Thursday, May 11.) If you study the first two midterms carefully, you should have a good idea of the topics that will appear on the exam, but there will also be a question about transformations and rigid motions.
  2. As part of your review for the final exam, you may make corrections to any of the individual questions (questions 1 through 4) on the second midterm, and hand them in at the start of class on Thursday. You can earn back up to half of the points you missed on any question by handing in a correct solution.
  3. The group final exam will have two questions: a question on RSA encryption, and a repeat of the groups question (Question 5) from the second midterm. If your group’s score on that question on the final exam is better than your score on the midterm, I will replace the midterm score with the one from the final. You may ask me questions about either final exam question at any time before the date of the final. (I have posted links to online modular arithmetic calculators on the “Useful Links” page; you may want to practice with those before the final.)
05/04
  1. May the Fourth be with you!
  2. Read the Introduction to this Wikipedia page on public key cryptography, and the “RSA Encryption” handout on the “Additional Notes and Files” page of the class website. Don’t worry if it doesn’t all make sense on a first reading: we will be discussing these ideas on Tuesday. Send me a question on the reading by Monday evening. Due 05/08.
  3. There will be a review session this Monday, May 8, 6:00-7:00 in KEN 1180.
04/27
  1. Complete any remaining activities from Section 4.3, making especially sure that you understand the subsections on Functions and Complex Numbers. (You will have a chance to ask questions on this material again on Tuesday.)
  2. Prepare for next Thursday’s exam, which will cover material through to the end of Section 4.3. The exam will be cumulative. In particular, it will use the material from Chapter 7 which we are now relating to modular arithmetic.
04/25
  1. Complete any Activities from the Modular Arithmetic portion of Section 4.3 that we did not cover in class today. (Make sure also that you can complete and understand all of Section 4.2.)
  2. Read the Functions and Complex Numbers portions of Section 4.3 for discussion in class on Thursday.
  3. Send me a question on Section 4.3 (either on the new reading, or on the “big picture” of this long section) by Wednesday evening. Due 04/26.
04/20
  1. Complete any Activities from the Matrices portion of Section 4.3 that were not covered in class today.
  2. Re-read the Modular Arithmetic portion of Section 4.3 for discussion in class on Tuesday. As always, feel free to send me a question on the reading if anything is not clear. (Since we did not cover anything on this section this week, don’t worry about completing all the Modular Arithmetic activities, as you were asked to do in the last homework: they will be re-assigned after Tuesday’s class.)
  3. Hand in Activities 4.42, 4.43, 4.47, 4.48. Due 04/27.
04/18
  1. Complete any Activities from the Modular Arithmetic portion of Section 4.3 that we did not cover in class today. (Make sure also that you can complete and understand all of Section 4.2.)
  2. (Re-)read the Matrices portion of Section 4.3 for discussion in class on Thursday. (As always, feel free to e-mail me questions on the reading, but it is not a requirement this time.)
04/13
  1. Complete any Activities from Section 4.2 that we did not cover in class today. In particular, complete a table of properties for the 6 operations on page 78 of the textbook. (I suggest you have one row for each operation, and list the 4 properties (commutative, associative, identity, inverses) across the top of your table.)
  2. Read Section 4.3 for discussion in class on Tuesday, and send me a question by Monday evening. Pay particular attention to the section on modular arithmetic. Feel free to skim the section on matrices, especially if you have not worked with matrices before. Due 04/17.
  3. Hand in Your table of properties for the 6 operations; Activity 4.20. Due 04/20.
04/11
  1. Complete any Activities from Section 4.1 that we did not cover in class today.
  2. Read Section 4.2 for discussion in class on Thursday. (As always, feel free to e-mail me questions on the reading, but it is not a requirement this time.)
04/06
  1. We studied compositions of reflections in class today. Experiment with compositions of translations and rotations, look for patterns, and make some conjectures as to what happens when you compose two translations, or two rotations. (Make sure you consider the case of two rotations about different centers.) Write up your conjectures, and any justifications you can find as to why they might be true, and hand them in next Thursday.
  2. Try to find a rigid motion (in the plane) which is neither a translation, a rotation, nor a reflection. “Rigid motion” here means a transformation which preserves distances between points; roughly speaking, something you can do with patty paper (without distorting or tearing the paper in any way).
  3. We have studied several different examples of operations in this class, but we have not yet defined carefully what an operation is! Read Section 4.1 (“What is an operation?”) for discussion in class on Tuesday. (This section makes reference to “functions of two variables”, which is the last section in Chapter 3. You should not need to read that section, but you will have to remember the definition of the Cartesian product of two sets.)
  4. Hand in Activities 3.34 and 3.35; your write-up of translation/rotation experiments and conjectures. Due 04/13.
04/04
  1. Complete any Activities from Section 3.4 that we did not cover in class today.
  2. Look up definitions of “transformation” (of the plane), translation, reflection, and rotation, and bring them to class for discussion on Thursday.
03/30
  1. Read Sections 3.3 and 3.4 of the textbook (“Compositions of Functions” and “Inverses of Functions”) for discussion in class on Tuesday. Send me a question on the reading by Monday evening. Due 04/03.
  2. Complete any Activities from Sections 3.2 and 3.3 that we did not cover in class today. (We will continue discussion of 3.3 on Tuesday, so you will have another opportunity to complete those Activities after that discussion.
  3. Hand in Activities 3.26 and 3.27. Due 04/06.
03/16
  1. In class today, we voted to attend the Marden Lecture on Tuesday, March 28, in lieu of our regular class. The lecture will be in the Lubar School of Business, room N140, from 4:00-5:00. The lecture hall has been full in previous years, so I suggest arriving early if possible. Our next regular class will therefore be on Thursday, March 30.
  2. If you were not in class today, complete Activities 3.6 and 3.7 (page 35 of the textbook) before March 30. Note that you are supposed to have a separate card for each function in these activities; you should have 21 cards in all when you are finished. Bring the cards to class, as we will be using them in additional activities in the next few sections.
  3. Re-read Section 3.2 of the textbook (“Some Properties a Function May (or May Not) Have”) for discussion in our next class (Thursday, 03/30). Send me a question on the reading by the evening before. Due 03/29.
  4. Have a good break!
03/14
  1. Complete any Activities from Section 3.1 that we did not cover in class today.
  2. Read Section 3.2 of the textbook (“Some Properties a Function May (or May Not) Have”) for discussion in class on Thursday.
03/09
  1. Complete any Activities from Section 2.3 that we did not cover in class today.
  2. Hand in Activities 2.20 and 2.22. Due 03/16.
  3. Read Section 3.1 of the textbook (“Definition and Representations of Functions”) for discussion in class on Tuesday. Concentrate on understanding the basic definition of a function (which is one of the harder things we will be studying this semester, so don’t worry if you find it confusing at a first reading) and examples (such as Example 3.3). Send me a question on the reading by Monday evening. Due 03/13.
03/07
  1. Re-read Section 2.3 of the textbook (“Power Sets and Cartesian Products”), for discussion in class on Thursday.
03/02
  1. Complete any Activities from Section 2.2 that we did not complete in class.
  2. Prepare for Tuesday’s midterm exam, which will cover material from Chapter 7, Chapter 1, and Sections 2.1 and 2.2. (There is a more detailed list of study topics on the “Additionl Notes and Files” page.)
02/28
  1. Complete any Activities from Section 2.1 that we did not complete in class.
  2. Read Sections 2.2 and 2.3 of the textbook (“Set Theoretic Operation …”, and “Power Sets and Cartesian Products”), for discussion in class on Thursday. Send me a question on the reading by Wednesday evening. Due 02/29.
02/23
  1. Complete any Activities from Section 1.4 that we did not complete in class.
  2. Hand in Activities 1.11, 1.12, 1.14, 1.18. Due 03/02.
  3. Read Section 2.1 of the textbook (“Basic Notions, …”), for discussion in class on Tuesday. Send me a question on the reading by Monday evening. Due 02/27.
02/21
  1. Activities 1.8, 1.9, 1.11, 1.12 through 1.15.
  2. Read Section 1.4 of the textbook (“Open Sentences and Quantifiers”) for discussion on Thursday. As always, do as many of the Activities as you can while reading. You do not have to send me a question, but please do so if anything in the reading is unclear.
02/16
  1. Hand in the “Exercises on prime factorization” exercise set. (You can find this exercise set under “Additional Notes and Files” on either class website.) Due 02/23.
  2. Read Sections 1.1, 1.2 and 1.3 of the textbook (“Statements”, “Key Words”, and “Logical Equivalence”), doing as many of the Activities as you can while reading. You may find Section 1.3 more challenging than the first two sections, but get through as much as you can. We will discuss all three sections in class on Tuesday. Send me a question on the reading by Monday evening. Due 02/20.
02/14
  1. Look over the homework that was returned today, and make sure you understand all of my comments. Pay particular attention to careful definitions (MP6: Attend to precision), and the argument that when the Division Algorithm is applied to write b = qa + r, then gcd(b, a) = gcd(a, r). You should be able to check this equality of greatest common divisors in any given case by finding the factors of b, a and r, but you should also be able to explain the general argument without first finding factors.
  2. Practice using the Euclidean Algorithm to find the greatest common divisor of two positive integers: make up your own examples and work them out until you are comfortable with the algorithm.
  3. For each of the Euclidean Algorithm examples you make up, write gcd(b, a) as an integral linear combination of b and a; that is, find integers m and n such that ma + nb = gcd(b, a). Again, practice until you are comfortable with the procedure, and are convinced that it will always work, no matter what the original positive integers b and a.
02/09
  1. Make sure your group is ready for their presentation on Tuesday. (This presentation will count for half of this week’s homework.) Due 02/14.
  2. Hand in Problem 6(c)(e) from the “Exercises on divisors and the Euclidean Algorithm” exercise set. Due 02/16.
02/07
  1. Continue working on the Egg-Timer Problem, with the goal of contributing to your group’s discussion and presentation preparation on Thursday. What eggs can you time with the original 7-minute and 11-minute timers? What happens if you change the timers? How does all of this relate to our work from Chapter 7?
  2. Work problem 6 from the “Exercises on divisors and the Euclidean Algorithm” exercise set. Make up a few more examples of your own, and solve them, until you feel comfortable with the procedure.
02/02
  1. Read Section 7.6 of the textbook. As always, make notes of anything that seems familiar, and anything you do not yet understand or want to know more about. (I will not repeat this instruction from now on, but you should continue to read the text actively.)
    The abstract/symbolic description of the Euclidean Algorithm is probably harder to understand than examples of it, so make sure you read Example 7.2 even if you don’t yet fully understand the statement of the algorithm. (Theorem 7.8, why the Euclidean Algorithm works, should be familiar to you after today’s class.) We will discuss this section on Tuesday, so send me a question by Monday evening. Due 02/06.
  2. Work through the first 5 exercises on the “Exercises on divisors and the Euclidean Algorithm” exercise set. (You can find this exercise set under “Additional Notes and Files” on either class website.)
  3. Hand in the first 4 exercises from “Exercises on Divisors and the Euclidean Algorithm”. Due 02/09.
01/31
  1. Read Section 7.5 of the textbook. Make notes of anything that seems familiar, and anything you do not yet understand or want to know more about. Spend some time trying to understand what Theorem 7.6 (The Division Algorithm) is saying: it isn’t as complicated as the notation makes it seem at first sight. One thing that might help you with the theorem is working Activity 7.18, especially if you represent the solutions to that activity on a number line.
01/26
  1. Read Section 7.4 of the textbook. Make notes of anything that seems familiar, and anything you do not yet understand or want to know more about. We will work several of the activities in Section 7.4 in class on Tuesday, but try to see how many of them you can do by yourself before our discussion. (You should consider this instruction to be understood in any future reading assignments: read actively, not passively.) Send me a question on the reading by Monday evening. Due 01/30.
  2. Make sure you can complete every activity in Sections 7.1, 7.3 and 7.4. Also, make up your own example of Activity 7.17; i.e., choose a pair of natural numbers (different from the pairs in Activities 7.15 and 7.16!), find their GCD and LCM, and verify that the relation in Theorem 7.5 is satisfied.
  3. Hand in Activity 7.13 (both the statement and a proof of your theorem); Activity 7.14; your example of Activity 7.17. Due 02/02.
01/24
  1. Read Sections 7.1 and 7.3 of the textbook. Make notes of anything that seems familiar, and anything you do not yet understand or want to know more about. Send me a question on the reading by Wednesday evening. (We will work several of the activities in Section 7.3 in class on Thursday, but try to see how many of them you can do by yourself before our discussion.) Due 01/25.