Spies, Lies, and Codes

ABSTRACT

            What would the world look like if the Allies had not won World War II?  Breaking the Axis military and diplomatic codes was an essential step in winning the war.  How does an imbedded CIA agent in an enemy country convey information back to agency headquarters in Washington, D.C. without being revealed?  Once students get introduced to the important roles codes and encryptions have had in the past and are having today in the security of our nation and once they can understand the underlying math, they will be able to better appreciate the field of mathematics.  This appreciation will lead to better reasoning and problem solving abilities and a better understanding of how math is applied in actual situations.  Students will be able to reason and use logic at higher levels and greater conceptual understanding of mathematics.

 

RATIONALE

            Imagine students who are truly excited about math, students who love math, and students who are quite adept at reasoning and problem solving.  For too long have my students asked me about the usefulness and purpose of the mathematics they are learning in algebra.  For far too long have we seen students give up on difficult problems that require reasoning and problem solving.  We believe that a long-term unit on codes and code-breaking will both increase students’ interest in mathematics and improve their reasoning skills. 

            We are not out to change the world or completely overhaul the way math is taught across the country, but we do want to energize our students and instill in them the same passion for mathematics that we have, the same understanding of its wide applications.  This unit is a good place to start in order to achieve those goals.  This unit will begin to expose students to the world of spies, the Central Intelligence Agency, and the National Security Agency.  Students will get the opportunity to come up with their own codes and attempt to break codes ranging in difficulty.  They will see the math behind some very common forms of encryption – those used to encode secret messages and those used in online financial transactions everyday.  Students will also be able to see how encoding/encryption have affected us throughout history, like how the U.S. Navy’s Signals Intelligence Division (SIGINT) broke Japanese military codes during World War II to how the NSA and CIA work to keep the United States one step ahead of its enemies today.  Movies like the James Bond and Mission Impossible series and television shows like Alias and 24 have already sparked a great deal of interest in the secret world of spies and codes. 

            Once students understand the necessity of mathematics in such a crucial field, we believe that they will develop a greater appreciation for the subject.  They will be able to see that a “dry” and “boring” subject can actually be exciting when they are able to put theory into practice.  This is crucial for my students to better their reasoning abilities and problem solving skills. 

 

DESCRIPTION

            From having the students provide a flashback of past events that occurred in history involving cryptography scandals to equipping students with mathematical tools to putting it into reality, this unit will give the students a sense of the importance of mathematics and the role mathematics plays in cryptography.

            To spark students’ interest in cryptography, one example in history of a cryptography scandal will be provided to the students via the PBS Nova website (http://www.pbs.org/wgbh/nova/venona/).  The students will also research the history of cryptography.  The students will be divided into groups and each group will provide an example in history of a cryptography scandal.  Each group will give their presentation to the class.  After presentations, lessons on cryptography will begin.  To start the long-term unit on codes and code-breaking, the question “How do you code a message so it is easy to decode and difficult to crack” will be posed to the students.  The lessons will familiarize students with commonly used cryptography terms, guide students through encoding and decoding messages as well as introduce various encryption techniques such as Caesar cipher and public key encryption.  To assess students’ level of understanding, the students can test their ability with the PBS Nova online activity (http://www.pbs.org/wgbh/nova/decoding/faceoff.html). As students gain more confidence in coding and decoding, their final project would be to develop their own secret codes and have their fellow classmates try to crack the codes.

            This unit project will give students a look back on the importance of cryptography in history as well as the important role cryptography plays in today’s society.  The lessons in the unit project will walk students through how to encode and decode messages.  From those tools, students can use their creativity to come up with their own codes.  Because they are able to relate cryptography to real life situations in the past and the present, the students will gain more appreciation for mathematics.

 

POTENTIAL IMPACT

Our project involves two teachers and twenty-five students.  Students will gain a deeper understanding of reason and the mathematics behind coding.  They will be able to see that math has very practical applications and understand its usefulness outside the classroom. 

At the end of the project, students will be required to give a presentation on the ciphers they have come up with on their own.  This will gauge students’ understanding of the material and how it can be used for practical purposes.  The students’ presentation materials will be saved and shown to our colleagues to see how effective this project was.  We will be able to see how well the project has fulfilled expectations and standards like those of the National Council of Teachers of Mathematics (NCTM) and the Texas Essential Knowledge and Skills (TEKS).  In collaboration with our math department, we may be able to enlist other teachers to adopt this project as a means to teach seemingly complicated conceptual mathematics to ninth and tenth graders. 

 

 

EVALUATION

            To see if goals are being reached, a series of assessments such as presentations, tests, and classroom assessment techniques will be given.  Each type of assessment will provide written feedback, which can be kept for future reference.  The feedback will gauge students’ level of understanding as well as misconceptions.  Thus, the level of difficulty of the lessons can be changed to meet the needs of the students.

            First type of assessment, presentations, will allow students to incorporate history, writing, and public speaking with mathematics.  Presentations include researching, writing up a summary of the research, and giving a five minute presentation with visuals.  This will accommodate students whose main strengths do not lie in mathematics, but like doing research, writing, or doing public speaking.  Tests will monitor students’ level of understanding.  Questions on the test will vary from multiple choice questions, true/false questions to free-response questions.  The type of questions varies to accommodate different learning styles.  Feedback from the test will see whether students are about to encode and decode messages as well as differentiate between the various encryption methods.  Other types of assessments are classroom assessment techniques such as probing background knowledge, filling in productive study time logs, developing concept maps, and documenting step-by-step solutions to problems are assessments will gauge students’ level of understanding.  These assessments are just a few types of assessments that are included in our unit project.

            Students will be required to create a demonstration poster for their final presentations.  These posters should include what type of encryption the students have chosen, the math behind it, and where all this sort of encryption can be used.  The posters should have both text and graphics, be visually pleasing, and convey a great deal of information quickly and concisely. 

 

 

 

CALENDAR

	MONDAY	TUESDAY	WEDNESDAY	THURSDAY	FRIDAY
WEEK 1	Introduction to coding – anchor video (CAT Background knowledge)	Shift Ciphers	Tom’s Lesson 1 (Shift Ciphers)	Presentations on History of Ciphers	Field Trip to World War II Museum
WEEK 2	Functions, Graphs, and Tables	Quiz over shift ciphers (CAT – Productive  Study Time Logs)	Tom’s Lesson 2 (Reversing Shift Ciphers)	Connie’s Lesson 1 (Order of Operations and Distributive Property)	Inverse Functions (CAT – Documented Problem Solutions)
WEEK 3	Modular Arithmetic	Modular Arithmetic	Introduction to Cracking Codes	Quiz on shift ciphers, modular arithmetic, and functions	Frequency Analysis
WEEK 4	Cracking shift ciphers	Connie’s Lesson 2 (Stretching a Shift Cipher)	Equivalent Expressions	Two-step process vs. multi-step process	Quiz on stretched shift ciphers
WEEK 5	Keyword Coding	Matrix Operations	Challenge students to design more effective codes	Sum up key concepts (CAT – Concept Map)	Presentations over Codes and Ciphers

 

 

BUDGET

Needed Materials:

TI Viewscreen                                      $550

Fieldtrip to WW II Museum                  $600

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Total                                                    $1,150

 

Provided Materials:

TI-83 Calculators (25)

HP notebook computers (25)

Buses for fieldtrip