Matrices

Replies to This Discussion

Permalink Reply by Geoff St. Pierre on February 19, 2009 at 7:25am

A project I have used in the past is in encrypting messages. You can relate this to WWII, https (secure hypertext protocol), the NSA, and overall data security.

Before I do this lesson the students must know how to multiply matrices, compute inverses (a TI 8x can do it). I model an example of encryption to them as a lecture before I have them do the group activity. So this takes about 2 classes one to explain encryption and to model an encryption scenario and one to do the group activity.

Make a map a simple one could be:
" " --> 0 ; this is a space character.
A --> 1
B --> 2
.
.
.
Z --> 26

then you make an encoding matrix, like E = ( 1 5)
( -2 2)
then the key to decode the message will be E^{-1};

* Make 3 or 4 messages or one long one and chunk it into 3 or 4 parts (or however many groups you want).

* Give each group a part of the encoded message (you have to encode it using E) and have them decode it.
- They have to collect the message four numbers at a time and put them into a 2x2 Matrix row by row
(or whatever the size of the encoding Matrix E is that you used.)

* Then you can either give the students E^{-1} and have them decode, or give them E and have them "Break the code".

Then have the groups come together as a class to see the full message. You can give group grades as well as an overall grade.

Hope this helps.
Geoff

Permalink Reply by Geoff St. Pierre on February 19, 2009 at 8:45am

I forgot to mention that one of the benefits in the real world to encrypting with matrices is prevention of breaking the code with frequency analysis, since the letter "t" in a given message will not always be mapped to the same number in the encrypted message.

You can have your students compare this to a linear encryption like run the alphabet through the function f(x) = 2x - 1, or g(x) = x^2 - 30 in these "t" will always come out the same.

I have also had students come up with there own encryption methods before giving the matrix encryption lesson and project and then compare the security strengths and weaknesses in their methods and the matrix versions versus computational complexity. You can then reduce computational complexity using computers.

It goes on and on an on ... it depends on how much time you have to devote to this topic.


Geoff
http://jasperstreet.homep.net/wiki/index.php/Math

Permalink Reply by Jonathan Fjelstrom on February 19, 2009 at 9:27pm

This sounds like a great activity. I just started teaching Matrices with my precalculus class. I might have to look into this.

Permalink Reply by Geoff St. Pierre on February 20, 2009 at 7:45am

If you do this lesson, let me know how it goes.

thanks,

 87   85   18   20 

key:

       
1/14 [ 2   -5 ]

       [ 2    2 ]

Permalink Reply by greg on April 8, 2009 at 4:37am

Hi Melissa,
I made this webpage to help my student understand matrices: http://schools.u-46.org/index.pl?id=4479