Number Theory and Cryptology for Middle Level Teachers
Summer 2007
COURSE OVERVIEW
Graduate Credit Hours: 3
Usual Delivery Format: This course follows the on-site, one-week, summer course design (see
Instructional Approach in the Descriptions of all Math in the Middle Courses folder) with class
meeting 8:00 – 5:00 M – F. The course can easily be adapted to other schedules with 40 hours of
contact time.
Principal Developers/Instructors:
Kristie Pfabe, Department of Mathematics, Nebraska Wesleyan University
Michelle Reeb Homp, Center for Science, Mathematics and Computer Education, University
of Nebraska – Lincoln
Course Description: This course focuses on basic number theory results which are needed to
understand the number theoretic RSA cryptography algorithm (an encryption algorithm which is in
use today to secure information sent via the internet). As the number theory results are developed,
connections to middle level curricula are emphasized and proofs are carefully selected so that
those which are included in the course are particularly relevant and accessible to middle level
teachers. This portion of the course promotes a deep understanding of the integers and their
properties in connection with the operations of multiplication and division. Elementary ciphers
(methods for encoding and decoding) are included to introduce the nature of cryptology in
preparation for understanding the RSA method. The cryptology related activities are readily
adaptable as enrichment activities for middle level students. The connection of number theory to
the RSA encryption algorithm allows the participants to see and understand a very relevant, realworld application of mathematics.
Course Goals: The goals of the course are to introduce the teacher participants to: (1) the basic
results of elementary number theory, (2) the rigor of mathematical definitions, reasoning and
proof, (3) the application of number theory to cryptology, (4) and the connections between
Number Theory and the middle school curriculum.
Instructional Style: The course is designed in an interactive-lecture style (similar to a Socratic
method) with problem sessions, examples and activities designed for cooperative groups
distributed consistently throughout.
Course Schedule:
Day Topics
1
Course Introduction
Section 1: Integers and Divisibility
Copyright 2007. Number Theory and Cryptology for Middle Level Teachers. Developed by the Math in the Middle Institute
Partnership, University of Nebraska, Lincoln.
1
2
3
4
5
Video: “N is a Number”
Exploration: Even & Odd Numbers
Section 2: Primes and Factorization
Reading: “The Mathematical Universe”, Chapter A
Finish Section 2
Introduction to Cryptology (substitution and transposition ciphers)
Section 3: Linear Diophantine Equations
Reading: “The Mathematical Universe”, Chapter F
Section 4: Congruence
Application: Divisibility rules, ISBN’s
Video: “The Proof” (60 minutes)
Section 5: Linear Congruence Equations
Reading: “The Mathematical Universe”, Chapter P
Finish section 5
Section 6: Fermat’s and Wilson’s Theorems
Section 7: Euler Phi-Function
Video: “Decoding Nazi Secrets” (first 60 minutes)
Reading: RSA handout (included in course notebook materials, or see
section 2.5 of Heart of Mathematics, by Burger & Starbird, for
an elementary treatment of RSA cryptography)
Other Number Bases
RSA Public Key Cryptography
RSA Activity
Video: “Decoding Nazi Secrets” (second 60 minutes)
Course evaluations
Required Texts/Materials:
1. Course notebook containing handouts and problems which are included in Course Section
Folders.
2. Dunham, W. (1994). The mathematical universe. New York: Wiley & Sons.
Chapters A, F and P are assigned as readings.
3. NOVA movies N is a Number, The Proof, and Decoding Nazi Secrets
References:
1. Dudley, U. (1978). Elementary Number Theory (2nd ed). New York: W. H. Freeman and
Company.
2. Walker, Judy (University of Nebraska – Lincoln), The Joy of Numbers
Other Materials:
VCR/DVD player for movies “N is a Number”, “The Proof”, “Decoding Nazi Secrets”
Several white/chalk board markers and erasers
Scissors and paper strips for optional activity
Poster paper (for presentations and theorem displays)
Decks of cards
Computer lab availability (with internet access, password logins/names)
Copyright 2007. Number Theory and Cryptology for Middle Level Teachers. Developed by the Math in the Middle Institute
Partnership, University of Nebraska, Lincoln.
2
Cakes, napkins & utensils for final activity (see RSA Section Folder) – to be delivered the
last day of class
Acknowledgements:
This course has been significantly influenced by
 Professor Ron Rosier, who, using a grant from SONY, created a course entitled Joy of
Numbers at Georgetown University
and by
 Professor Judy Walker, who, using Rosier's materials, developed a
Joy of Numbers course at UNL, developed the cryptology content for All Girls/All Math at
UNL, and taught Number Theory for Elementary and Middle Level Teachers as part of an
NSF grant
Copyright 2007. Number Theory and Cryptology for Middle Level Teachers. Developed by the Math in the Middle Institute
Partnership, University of Nebraska, Lincoln.
3