Students will learn about modular arithmetic in order to decipher encrypted messages.

Objectives

Upon completion of this lesson, students will:

learn about modular arithmetic

Standards Addressed:

Grade 6

Number Sense

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

Grade 7

Number Sense

1.0 Students know the properties of, and compute with, rational numbers expressed in a variety of forms

Algebra

Creating Equations

Create equations that describe numbers or relationships

Grades 3-5

Numbers and Operations

Compute fluently and make reasonable estimates

Grades 6-8

Numbers and Operations

Compute fluently and make reasonable estimates

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.

6th Grade

Number and Operation

Content Standard 1.0 The student will develop number and operation sense needed to represent numbers and number relationships verbally, symbolically, and graphically and to compute fluently and make reasonable estimates in problem solving.

7th Grade

Number and Operation

The student will develop number and operation sense needed to represent numbers and number relationships verbally, symbolically, and graphically and to compute fluently and make reasonable estimates in problem solving.

8th Grade

Number and Operation

The student will develop number and operation sense needed to represent numbers and number relationships verbally, symbolically, and graphically and to compute fluently and make reasonable estimates in problem solving.

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.

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

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

paper

pencil

scissors

Prior to the lesson the teacher should prepare encrypted codes for the students to decipher.

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

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

modulus

A unit of measure. For example, when measuring days, a modulus could be 24 for the number of hours in a day. 75 hours would be divided by 24 to give 3 remainder 3, or 3 days and 3 hours. See also modular arithmetic

Lesson Outline

Focus and Review

Have a clock drawn on the board with hands aligned as if it were 12:00.

Ask the students according to the clock on the board what time it would be in:

3 hours

8 hours

13 hours

28 hours

334 hours

Ask students to think about the answers for a moment. See if they can recognize a pattern that
might make it easier to calculate an answer for the "What time will it be in 334 hours
question?"

Objectives

Students will demonstrate their ability to use modular math by deciphering several encrypted
codes.

Teacher Input

Explain to the students that an easy way to calculate what time it will be in x number of
hours is to divide x by the number of hours on the face of the clock (the mod), take the
remainder, and count that far on the clock.

Introduce this method in writing. For example: 334 mod 12.

Explain how to decode a message if they are given the multiplier.

Guided Practice

Have the students convert the alphabet into numbers setting A=0, B=1, C=2, etc.

Tell the students to write a short sentence about themselves that they won't mind someone else
reading.

Have the students convert the letters in their sentence to numbers using the alphabet numbers
they just calculated.

Explain the term shift.

Have the students shift their numbers by 5 and convert their list back into letters.

Explain that their sentences should now look like a bunch of letters.

Have the students swap messages and try to decode them using their current knowledge.

Independent Practice

Have the students pair up and open the
Caesar's Cipher applet.

Have each student use the computer to encrypt a message by changing the constant value.

Tell the students to swap messages with someone else in the class.

Instruct the students to decode the message they received from their classmate.

Now, have the students encrypt a message using a multiplier.

Have the students swap messages and decode the message they receive.

Explain that the messages you are about to pass out have been intercepted from a company that
has discovered the fountain of youth and that the encrypted messages will lead them to the
fountain.

Hand out several messages for the students to decipher.

Have the pairs of students work to decipher the codes.

In order to ensure that each partner is doing an equivalent amount of work require that each
student must be responsible for deciphering at least 3 messages.

Closure

Once the first group has finished give the class 10 more minutes to work on completing
deciphering their messages

Review vocabulary

Ask several groups to share their ciphers and solutions with the class

Suggested Follow-Up

You may wish to follow-up with the
Caesar Cipher 2 activity.