Participants will be mailed a copy of Number Theory Through Inquiry by Marshall, Odell, and Starbird (MAA, 2008) and asked to do some reading, including a few of the investigations from the book. This assignment will not be onerous and should be fun. This book was written as part of the development of the number theory course at UT. Participants will also be asked to email their questions and issues that arise from their own experience and from the readings so that the summer workshop can include addressing the real concerns of the participants.
In addition, participants should read a short narrative by Sam Wayne Young describing his experience in a Moore Method course. The narrative, Christmas in Big Lake, can be found here. Click here for the PDF version.
Updated Info
By now you should have received a copy of "Number Theory Through Inquiry", which is yours to keep, though we do ask that you bring it along with you to Austin for the workshop. We would like you to get a bit familiar with the style of the book. To this end we ask, as part of the pre-workshop assignment, that you try proving some of the book's lovely theorems (as well as answering some of its engaging questions and solving some of its fascinating exercises). Just do the best you can. Some are very straight forward and some are rather sophisticated (especially in the later chapters).
Being able to prove/answer/solve all of these successfully is by no means a prerequisite for the workshop. Just try to have some fun with them (but stick to the rules: no seeking help from any outside sources!). During the workshop we may have time to share some of the different solutions you will have discovered.
Here are some suggested sequences of theorems for you to try that may help to illustrate the book's strategy of guiding the students toward some concept or theorem. Of course, during the course, the students will have done basically all the preceding theorems, so you will be jumping into the middle.
Chapter 1: 1.32-1.35, 1.38
These theorems lead up to the development of the Euclidean Algorithm and then the first big
Linear Diophantine Theorem (a, b) = 1 iff there exist integers x and y such that ax + by = 1.
In addition, these also help lead to the idea of an inductive proof of theorem 1.38 (one of
the key ideas of chapter 1).
Chapter 2: 2.1, 2.7 - 2.9
These theorems lead through the FTA. I think 2.1 is useful in order to help set up a proof
of 2.7, and of course lemma 2.8 leads into 2.9. Again, induction is a very useful tool for these theorems.
Chapter 2 (2nd option): 2.32 - 2.35
This is the build up to the infinitude of primes. I don't think it is as interesting as
the 1st choice, but I wanted to give you guys something to pick from.
Chapter 3: 3.19, 3.20, 3.27, 3.28
These are meant to develop an understanding of linear congruences and then applying the connection
to Linear Diophantine equations to solve the Chinese Remainder Theorem.
Chapter 4: 4.5, 4.13 - 4.16
These are the theorems that lead up to Fermat's Little Theorem (both versions). If you are not familiar
with complete residue systems, see page 47.
Chapter 5: 5.1 - 5.4
These are all the theorems necessary to develop all the pieces of RSA Cryptography.
In practice, we usually just make it through Chapter 5 during a one semester course; however the later chapters contain some additional, wonderful number theory that you and your stronger students or independent study students will enjoy. Here are some sequences of activities that you might enjoy from the later chapters.
Chapter 6: 6.11-6.15
Chapter 7: 7.3-7.8
Chapter 8: 8.1-8.7
Chapter 9: 9.1-9.8
Chapter 10: 10.5-10.10