I'm doing a unit on matrices in my Algebra 2 class and am looking for some application type projects to help them understand the uses for matrices. If anybody has anything or knows of any sites where I can find materials, it would really be appreciated.
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.
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.