Keeping Secrets


Author:

Kevin Tucker

Unit Title:

Keeping Secrets

Grade:

11 and 12

Subject:

Pre-Calculus

Estimated Duration:

5 – 48 minute class periods

Unit Activities:

Activity 1:  Introduction to cryptography

Activity 2:  Simple Ciphers   

Activity 3:  Using matrices to encode and decode

Activity 4:  Creating your own message  

Background Knowledge: 

Students will need to know basic matrix operations (multiplication, determinant, inverse). Students will need to know that matrix multiplication is not commutative. An inverse matrix exists if and only if the determinant is not zero. 

We will also discuss modulo to reduce matrix entries to numbers that can be converted back into our original alphabet.

Date:

July 2013

The Big Idea (including global relevance)

This unit is about how to keep your secure information secure.  How can we use math to create a secure message that will be difficult for someone to decipher.

The Essential Question
  1. What is the importance of keeping your information secure?
  2. How can we use mathematics (matrices) to encode and decode messages to make them more secure?
Justification for Selection of Content

Traditionally we do not use matrices in many applications. We use them to solve systems of liner equations using inverse matrices or reduced row echelon form but that is it.  

The Challenge

To use matrices to encode and decode a message and send it to another group in the class.

The Hook

National Treasure: Ciphers, Code makers & Code breakers

Teacher's Guiding Questions
  1. What are some methods for encoding and decoding messages?
  2. How do we decode the encoded message?
  3. What are the advantages and disadvantages of the methods for encoding and decoding messages?
  4. How can we use matrices to encode messages better?
  5. What happens when we multiply or matrix by the encoded matrix?
  6. What do we have to do to make it usable?
  7. How do we undo matrix multiplication?
  8. What are some advantages and limitations of using matrices to encode and decode messages?
ACS (Real world applications; career connections; societal impact)

A – (Real World Application):  Keeping information secure for banks, military, and government agencies.

C – (Career Connections): Information Technology, National Security Agency, Military Cryptographers, Private Sector Cryptographers.

S – (Societal Impact):  In our society where we are sending personal information over the internet it is more important than ever to keep your information secure.  Hackers can steal your bank records and even your identity.

Engineering Design Process (EDP)

The problem is for the students to encode and essentially decode a message using matrix algebra. In the process students will have to develop an encoding matrix (3 X 3) at a minimum. In order to make sure that their matrix works they will test small phrases or words. Not all Matrices will work.  It must be an invertible matrix with a determinant of 1 (to make the math possible in high school if the determinant is not 1 the students would have to understand inverse mod 27). 

After they find a square matrix that works they will send an encoded message and the encoding matrix to another group in the class. The new group will use matrix algebra to decode the message. The groups will present the original message the encoding matrix and the decode message to the class. 

Unit Academic Standard

HSN-VM.C.6 - Use matrices to represent and manipulate data

HSN-VM.C.8 - Multiply matrices of appropriate dimension

HSN-VM.C.10 - The determinant of a square matrix is nonzero if and only if the matrix has a multiplicative inverse.

HSA-REI.C.9 – Find the inverse of a matrix if it exists and use it to solve problems.

Unit Activities

Activity 1 and 2: Early Ciphers

Activity 1 and 2 will focus on introducing students to cryptography.  Students will have the opportunity to solve a crypto quip similar to ones found in the newspaper (single substitution).  Students will then investigate some early Ciphers that do not do a simple letter replacement.

            Activity 1:  Introduction to cryptography

                                    1.1.1. Keeping Secrets_codemakers and breakers video_KTucker_july 1 2013

            Activity 2:  Simple Ciphers    

                                    1.1.2. Keeping Secrets_simpleciphers_KTucker_july 1 2013

Activity 3 and 4: Using matrices to encode and decode messages

Activity 3 and 4 will allow the students to use matrix algebra to encode and decode messages that are more secure than the previous forms.  Students will be introduced to modulo as a way to convert encoded numbers back into the alphabet.  The challenge is for the students to create an encoding matrix and a message to encode in their groups.  The groups will then encode the message and give the encoded message and the encoding matrix to another group.  The new group will decode the message and present the process of decoding to the class along with the original message.

            Activity 3:  Using matrices to encode and decode

                                    1.2.3. Keeping Secrets_Using matrix algebra_KTucker_july 1 2013

            Activity 4:  Creating your own message 

                                    1.2.4. Keeping Secrets_Your encoded message_KTucker_july 1 2013

Where the CBL and EDP appear in the Unit

The CBL and EDP will take place in lesson 2 activity 4.

Identify Problem:

            To create a 3X3 matrix to encode a message of at least 45 chacters.

Brainstorm Ideas:

Many different matrices work.  For most students a matrix with a determinant of 1 will be needed.  If the determinant is not 1 the group will have to work harder to find the inverse because we are converting the numbers mod 27.

Design:

Groups will create their matrix and try to encode a message.  Groups may need to change their matrix to get it to decode.  Groups may have to redesign their matrix.

Present solution:

After the groups encode their message I will collect their encoded message and their encoding matrix.  I will give each group a different message and encoding matrix for them to decode.  The groups will present the encoded message and matrix, and the decoded message and decoding matrix.  The presentations will include the process that the groups went through

Misconceptions

Because we are using inverse matrices we must make sure that the determinant is not =0. We will only use matrices with a determinant of 1 so the lesson does not become overly complicated.  As a class it is worth mentioning that the determinant does not have to be equal to one but there are limitations.

Additional Resources

National Treasure: Ciphers, Codemakers & Codebreakers - http://www.youtube.com/watch?v=-413tlhPDZo

Iluminations code crackers - http://illuminations.nctm.org/LessonDetail.aspx?ID=L618

Project coding - http://jwilson.coe.uga.edu/EMAT6680Fa09/KimS/EMAT6690/3rd%20Write%20up/Project-Code/ProjectCoding.html

NSA.gov - http://www.nsa.gov/academia/_files/collected_learning/high_school/algebra/matrices_secret_weapon.pdf

National treasures 2: Book of Secrets - https://www.youtube.com/watch?v=y0YvaorWqP8

Inverse of a matrix mod 26 if the determinant is not equal to 1 - http://ca.answers.yahoo.com/question/index?qid=20120202144428AAub1Mj

Pre-Unit Assessment Instrument

1.1.1a. Keeping Secrets_PreAssessment_KTucker_july 1 2013

Post-Unit Assessment Instrument

1.2.4o. Keeping Secrets_PostAssessment_KTucker_july 1 2013

How to Make This a Hierarchical Unit

In a Middle school setting you could use the first 2 activities that involve simple ciphers.

If the class is learning matrices you could limit the encoding matrix to a 2 X 2. This will make finding a matrix with a determinant of 1 much easier.  Also Student could create their own method for encoding messages. The decoding process will make them think in terms of inverse operations (being able to undo the encoding process. It will also lead to the concept of 1-to-1 functions.

Reflection

    The largest obstacle for this lesson was time in my opinion.  The lesson was planned for 5 days.  In practice it took 15 days to get through the lesson.  My expectation was that all of the students in my CP Pre-Calculus class would have a graphing calculator and a computer with PowerPoint and internet access.  In reality one student in a class of 16 had a graphing calculator, and less than 6 had a computer.  Because the students did not have access to the needed technology everything we did had to be done in class.  All of the calculations done with multiplying 3x3 and 3xN matrices and converting the multiplied matrix mod 27 had to be done during class time.  This greatly slowed down the progress. 

    This was my first attempt at an Engineering Design Project.  I found it difficult for me to manage 5 groups that are not accustomed to this type of project.  Many of the students in this class have not learned how to take ownership of their own learning.  It was their first attempt at a project in which they were not being told exactly how to make it work.  There was some trial and error and none of the methods for encoding and decoding worked the first time.  Taking myself out of the equation and allowing the students the opportunity to develop the concepts and struggle with gaining the knowledge for themselves while keeping them on task was difficult.  Fining a balance between giving answers and pushing in the right direction was a struggle.

    For me the success in the lesson was simple.  As the student groups experienced more success they typically worked harder.  At the beginning of each activity as the groups started to work the conversations were, “This is too hard”, “I can’t do this”… by the end of each activity the conversation was “This is easy”, or “That’s not so bad”.  All of the students agreed that it was a lot of work, and it took a lot of effort to be successful.

How to Make This a Hierarchical Unit


Reflection
How to Make This a Hierarchical Unit


Reflection
How to Make This a Hierarchical Unit


Reflection