What is Caesar Cipher III?
This activity allows the user to experiment with decoding messages using modular arithmetic. Modular arithmetic has been used for years as a means of encoding messages. The letters of the alphabet are coded as:
A B C D ... Z
0 1 2 3 ... 25
and then the numbers are changed via an affine (linear) transformation to new, coded numbers.
The coding function has the form:
Y = A * X + B
where X is the uncoded number, A is the multiplier, B is the constant (both A and B are known
to allies, but unknown to enemies) and Y is the calculated, coded number. The arithmetic is
done mod 26 to ensure that we get numbers back that can be translated back to letters before
sending the coded message.
This method of coding messages is ancient. In fact, Caesar is said to have liked to use the cipher with A = 1 and B = 3. This method of coding is fun to play with, but easy to crack, as can be seen from this activity.
Modern coding techniques still make use of modular arithmetic, but in a highly sophisticated way.
How Do I Use This Activity?
This activity allows the user to experiment with decoding messages using modular arithmetic.
Controls and Output
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At the top of the activity, a phrase is displayed that you need to decode:
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Below the phrase that you have to decode are three text fields for you to enter your
guesses for the multiplier, constant, and the decoded phrase. The
multiplier is the number that is multiplied by the number value of each letter in your text, and the
constant is the number that is added to this product of the number value of each letter and the
multiplier.
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After you have entered your "guesses", pressing the
Check your answers! button will let you know which are correct and which are incorrect.
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To view your score press the
Show Score button.
- This activity will automatically record how successful you are at answering the questions. To view the score, press the Show Score button at the bottom of the activity and a pop-up window will appear with the scoreboard. To close this pop-up window press the Close button or click back on the main window.
- To pause the scoring, press the Active button at the bottom of the screen and it will change to a Paused button. To resume scoring, press the Paused button.
- To reset the scoreboard, open the scoreboard using the Show Score button and then press the Reset button.
Description
This activity allows the user to experiment with decoding messages using modular arithmetic. This activity would work well in mixed ability groups of four or five for approximately forty-five to ninety minutes.
Place in Mathematics Curriculum
This activity can be used to:
- practice students' long division skills
- demonstrate the applications of modular arithmetic
- practice the students' logical reasoning skills
Standards Addressed
Grade 7
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Algebra and Functions
- 1.0 Students express quantitative relationships by using algebraic terminology, expressions, equations, inequalities, and graphs
Third Grade
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Number and Operations in Base Ten
- Use place value understanding and properties of operations to perform multi-digit arithmetic.
Algebra
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Creating Equations
- Create equations that describe numbers or relationships
Grades 6-8
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Algebra
- Understand patterns, relations, and functions
Grades 9-12
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Algebra
- Understand patterns, relations, and functions
4th Grade
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Algebra
- Standard 4-3: The student will demonstrate through the mathematical processes an understanding of numeric and nonnumeric patterns, the representation of simple mathematical relationships, and the application of procedures to find the value of an unknown.
Elementary Algebra
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Elementary Algebra
- Standard EA-3: The student will demonstrate through the mathematical processes an understanding of relationships and functions.
7th Grade
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Algebra
- The student will understand and generalize patterns as they represent and analyze quantitative relationships and change in a variety of contexts and problems using graphs, tables, and equations.
8th Grade
-
Algebra
- The student will understand and generalize patterns as they represent and analyze quantitative relationships and change in a variety of contexts and problems using graphs, tables, and equations.
Algebra I
-
Algebra
- Students will describe, extend, analyze, and create a wide variety of patterns and functions using appropriate materials and representations in real world problem solving.
Algebra II
-
Algebra
- Students will describe, extend, analyze, and create a wide variety of patterns and functions using appropriate materials and representations in real-world problem solving, and will demonstrate an understanding of the behavior of a variety of functions and their graphs.
Algebra I
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Foundation for Functions
- 1. The student understands that a function represents a dependence of one quantity on another and can be described in a variety of ways.
- 2. The student uses the properties and attributes of functions.
- 3. The student understands how algebra can be used to express generalizations and recognizes and uses the power of symbols to represent situations.
- 4. The student understands the importance of the skills required to manipulate symbols in order to solve problems and uses the necessary algebraic skills required to simplify algebraic expressions and solve equations and inequalities in problem situations.
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Linear Functions
- 7. The student formulates equations and inequalities based on linear functions, uses a variety of methods to solve them, and analyzes the solutions in terms of the situation.
- 8. The student formulates systems of linear equations from problem situations, uses a variety of methods to solve them, and analyzes the solutions in terms of the situation.
7th Grade
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Patterns, Functions, and Algebra
- 7.19 The student will represent, analyze, and generalize a variety of patterns, including arithmetic sequences and geometric sequences, with tables, graphs, rules, and words in order to investigate and describe functional relationships.
Secondary
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Algebra I
- A.02 The student will represent verbal quantitative situations algebraically and evaluate these expressions for given replacement values of the variables. Students will choose an appropriate computational technique, such as mental mathematics, calculator, or paper and pencil.
- A.05 The student will create and use tabular, symbolic, graphical, verbal, and physical representations to analyze a given set of data for the existence of a pattern, determine the domain and range of relations, and identify the relations that are functions.
Textbooks Aligned
Grade Seven
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Accentuate the Negative
- Investigation Two: Adding Integers
- Investigation Three: Subtracting Integers
Book 1
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Module 8 - Our Environment
- Section 1: Adding Integers
- Section 1: Subtracting Integers
Book 2
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Module 2 - Search and Rescue
- Section 3: Adding Integers
- Section 3: Subtracting Integers
- Section 4: Modeling Functions
Book 3
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Module 2 - At the Mall
- Section 4: Operations with Integers
Book 1
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From Zero to One and Beyond
- Lesson 12: Powering Up
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Number Powerhouse
- Lesson 3: Know How to Read the Signs
- Lesson 4: The Problems Are All Yours
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Patterns in Numbers and Shapes
- Lesson 3: Crossing the River
Book 2
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Making Mathematical Arguments
- Lesson 1: Statements About Signs
- Lesson 2: Counterexamples and Cube Combinations
- Lesson 3: More Cases to Consider
Book 3
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Exploring the Unknown
- Lesson 5: Extending the Lab Gear Model
Grade 8
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Reflections on Number
- Divisibility and Prime Factorization
- Multiplication and Division
- Operations with Inverses
Grade 5
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Dry and Wet Numbers
- Positive and Negative Numbers
- Scale Drawings
- Number Lines
Grade 6
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Operations
- Operations with Integers
- Plotting Points
- Coordinate Plane
Be Prepared to
- supply some structure for them to use to make reasonable guesses
- show how to solve two step equations
- be flexible on the time spent on this activity.
- encourage the students that they can do this if they give it a good try even though it is tricky and involved.