Clock Arithmetic and Cryptography

Abstract

The following discussions and activities are designed to lead the students to practice their basic arithmetic skills by learning about clock arithmetic (modular arithmetic) and cryptography. Although somewhat lengthy (approximately 2 hours), the lesson can easily be separated into two lessons.

Objectives

Upon completion of this lesson, students will:

  • be able to perform basic operations in modular (clock) arithmetic
  • be able to encode and decode messages using simple shift and affine ciphers
  • have practiced their multiplication, division, addition and subtraction skills

Standards Addressed

Grade 9

  • Numeration

    • The student demonstrates conceptual understanding of real numbers.

Grade 10

  • Numeration

    • The student demonstrates conceptual understanding of real numbers.

Grade 6

  • Number Sense

    • 2.0 Students calculate and solve problems involving addition, subtraction, multiplication, and division

Third Grade

  • Number and Operations in Base Ten

    • Use place value understanding and properties of operations to perform multi-digit arithmetic.

  • Operations and Algebraic Thinking

    • Represent and solve problems involving multiplication and division.
    • Understand properties of multiplication and the relationship between multiplication and division.
    • Multiply and divide within 100.

Algebra

  • Creating Equations

    • Create equations that describe numbers or relationships

Grades 6-8

  • Numbers and Operations

    • Compute fluently and make reasonable estimates

Grades 9-12

  • Numbers and Operations

    • Compute fluently and make reasonable estimates
    • Understand numbers, ways of representing numbers, relationships among numbers, and number systems

Grade 6

  • Number and Operations, Measurement, Geometry, Data Analysis and Probability, Algebra

    • COMPETENCY GOAL 1: The learner will understand and compute with rational numbers.

Grade 7

  • Number and Operations, Measurement, Geometry, Data Analysis and Probability, Algebra

    • COMPETENCY GOAL 1: The learner will understand and compute with rational numbers.

Grade 8

  • Number and Operations, Measurement, Geometry, Data Analysis and Probability, Algebra

    • COMPETENCY GOAL 1: The learner will understand and compute with real numbers.

Technical Mathematics I

  • Number and Operations

    • Competency Goal 1: The learner will apply various strategies to solve problems.

3rd Grade

  • Measurement

    • The student will demonstrate through the mathematical processes an understanding of length, time, weight, and liquid volume measurements; the relationships between systems of measure; accurate, efficient, and generalizable methods of determining the perim
    • The student will demonstrate through the mathematical processes an understanding of length, time, weight, and liquid volume measurements; the relationships between systems of measure; accurate, efficient, and generalizable methods of determining the perimeters of polygons; and the values and combinations of coins required to make change.

6th Grade

  • Numbers and Operations

    • The student will demonstrate through the mathematical processes an understanding of the concepts of whole-number percentages, integers, and ratio and rate; the addition and subtraction of fractions; accurate, efficient, and generalizable methods of multiplying and dividing fractions and decimals; and the use of exponential notation to represent whole numbers.

5th Grade

  • Measurement

    • The student will demonstrate through the mathematical processes an understanding of the units and systems of measurement and the application of tools and formulas to determine measurements.

4th Grade

  • Measurement

    • Standard 4-5: The student will demonstrate through the mathematical processes an understanding of elapsed time; conversions within the U.S. Customary System; and accurate, efficient, and generalizable methods of determining area.

Grade 6

  • Number, Operation, and Quantitative Reasoning

    • 2. The student adds, subtracts, multiplies, and divides to solve problems and justify solutions.

Grade 7

  • Number, Operation, and Quantitative Reasoning

    • 2. The student adds, subtracts, multiplies, or divides to solve problems and justify solutions.

Grade 8

  • Number, Operation, and Quantitative Reasoning

    • 2. The student selects and uses appropriate operations to solve problems and justify solutions.

3rd Grade

  • Measurement

    • 3.15 The student will tell time to the nearest five-minute interval and to the nearest minute, using analog and digital clocks.
    • 3.16 The student will identify equivalent periods of time, including relationships among days, months, and years, as well as minutes and hours.

7th Grade

  • Computation and Estimation

    • 7.5 The student will formulate rules for and solve practical problems involving basic operations (addition, subtraction, multiplication, and division) with integers.

5th Grade

  • Measurement

    • 5.12 The student will determine an amount of elapsed time in hours and minutes within a 24-hour period.

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 for experimenting with the activities

Teacher Preparation

Key Terms

affine cipher

Affine ciphers use linear functions to scramble the letters of secret messages

cipher

Ciphers are codes for writing secret messages. Two simple types are shift ciphers and affine ciphers

factor

Any of the numbers or symbols in mathematics that when multiplied together form a product. For example, 3 is a factor of 12, because 3 can be multiplied by 4 to give 12. Similarly, 5 is a factor of 20, because 5 times 4 is 20

modular arithmetic

A method for finding remainders where all the possible numbers (the numbers less than the divisor) are put in a circle, and then by counting around the circle the number of times of the number being divided, the remainder will be the final number landed on

multiples

The product of multiplying a number by a whole number. For example, multiples of 5 are 10, 15, 20, or any number that can be evenly divided by 5

remainders

After dividing one number by another, if any amount is left that does not divide evenly, that amount is called the remainder. For example, when 8 is divided by 3, three goes in to eight twice (making 6), and the remainder is 2. When dividing 9 by 3, there is no remainder, because 3 goes in to 9 exactly 3 times, with nothing left over

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).

  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.

  6. Closure

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

Alternate Outline

The 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.