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 10

Numeration

The student demonstrates conceptual understanding of real numbers.

Grade 9

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

Algebra

Creating Equations

Create equations that describe numbers or relationships

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.

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 perimeters of polygons; and the values and combinations of coins required to make change.

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

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.

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.

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.

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.

5th Grade

Measurement

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

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.

Student Prerequisites

Arithmetic: Student 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

Access to a browser

Pencil and paper

Copies of supplemental materials for the activities:

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

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.

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.

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.

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.

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).

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.