Clock Arithmetic and Cryptography

• use factors, multiples, prime factorization, and relatively prime numbers to solve problems;

Student Prerequisites

• Arithmetic: Students must be able to:
  • perform integer and rational arithmetic, including multiplicative inverses
• Algebraic: Students must be able to:
  • work with simple algebraic expressions
• Technological: Students must be able to:
  • perform basic mouse manipulations such as point, click and drag.
  • use a browser, such as Netscape, for experimenting with the activities.

Teacher Preparation

Students will need:

• Access to a browser
• pencil and paper
• Copies of supplemental materials for the activities:
  • Clock Arithmetic Exploration Questions
  • Caesar Cipher Exploration Questions
  • Caesar Cipher II Exploration Questions
  • Caesar Cipher III Exploration Questions

Key Terms

• modular arithmetic
• cipher
• affine cipher
• factor
• multiples
• remainders

Lesson Outline

1. Focus and Review

   Remind students what has been learned in previous lessons that will be pertinent to this lesson and/or have them begin to think about the words and ideas of this lesson:

   • Ask students what multiples are. If needed, use the discussion on multiples.

   • Next, ask students what remainders are. The discussion on remainders is available to help.

2. Objectives

   Let the students know what it is they will be doing and learning today. Say something like this:

   • Today's class is about clock arithmetic -- also called modular arithmetic -- and cryptography -- which is a method of creating secret messages. Your knowledge of multiples and remainders will be useful when coding and decoding messages.

   • We are going to use the computers to learn about modular arithmetic and cryptography, but please do not turn your computers on or go to this page until I ask you to. I want to show you a little about these ideas first.

3. Teacher Input

   You may choose to lead the students in a short discussion on the relationship between clocks and modular arithmetic.

   Explain to the students how to do the assignment. You should model or demonstrate it for the students, especially if they are not familiar with how to use our computer applets.

   • Open your browser to the Clock activity in order to demonstrate it to the students.

   • Show students how to change the numbers on the clock.

   • Pass out the Clock Arithmetic Exploration Questions worksheet.

   • Have the students complete the worksheet with you, as you demonstrate how modular arithmetic works.

   Next, introduce students to the notion of cryptography through a discussion of simple shift and affine ciphers.

   • Open your browser to the Caesar Cipher activity in order to demonstrate it to the students.

   • Try coding a phrase with the students, such as "Once more back into the fray," and then checking it by running it through the Caesar Cipher activity.

   • Pass out the Caesar Cipher Exploration Questions worksheet.

   • Give students another phrase to code. Some examples: "Nothing ventured, nothing gained," or "Go for the gold," or "Take me out to the ball game."

   • Have students trade their codes and their values for A and B with another student in the class to practice solving.

4. Guided Practice

   Give students additional practice, this time with the Caesar Cipher II activity. This is an excellent way to practice students' reasoning skills, since there are naive ways to play this (run phrases through) and systematic ways of playing this (run a few single letters through).

   • Pass out the Caesar Cipher II Exploration Questions worksheet, and work with students to find the code.

5. Independent Practice

   As a final activity, have students compete in teams using the Caesar Cipher III activity. Students should be told that the phrases all come from children's nursery rhymes. The first team that decodes its phrase, finding the multiplier and constant correctly, wins.

   • Pass out the Caesar Cipher III Exploration Questions worksheet, and divide the class into teams to work on the codes.

6. Closure

   You may wish to bring the class back together for a wrap-up discussion.

Alternate Outlines

This lesson can be rearranged if there is only one available computer:

• After introducing the information in the discussions, have the students take turns working in groups or individually to practice decoding ciphers.

Extensions

• Patterns in Pascal's Triangle.