This is a really difficult problem. You deserve a lot of credit if you are trying to find a practical application of each topic in algebra. The strength and usefulness of algebra is all the mathematics, physics, computer science, economics, chemistry, actuarial science, business applications, etc that are built on an algebraic foundation.
On your computer there are very likely files with the extension .gif or .jpg or .zip. These are files that are compressed to reduce the size of the files. The particular extension identifies the algebraic algorithm that is used to compress the file and the algorithm that is needed to uncompress it. How can so much information be stored on your iPod? This is again compressed information, compressed using an algebraic expression.
The way that Google ranks the web pages on the internet is described with an algebraic expression. Once the information is collected a very large system of linear equations has to be solved in order to produce a database that can answer your query. This is a significant problem, how do you solve a system of linear equations with millions of equations and millions of unknowns? Brute force won't work, computers are just too slow. What is needed is a very sophisticated algebraic procedure.
Television is another place where mathematics, in particular algebra, plays a significant role. The information gathered by the TV camera has to be sent through a cable or satellite signal to your TV set, and then be displayed on your set as an image and sound. This is accomplished by a mixture of algebra and trigonometry called a Fast Fourier Transform.
In many situations algebra is the language that is used to convey mathematical information. Go to the front page of Math Central, and in the Quandaries and Queries section enter the keyword real life. What you will see is real mathematical problems sent to us by ordinary people with questions that have arisen in their lives. If you look through them you will see that in many situations the language, notation and techniques of algebra are needed to solve their problems.
I hope this helps. Good luck with your students,
One of the Quandaries and Queries consultants, Walter Whiteley from York University in Toronto suggested that you look at Algebra in Simplest Terms, http://www.learner.org/resources/series66.html The overview on the first page of this site says:
A video instructional series on algebra for college and high school classrooms and adult learners; 26 half-hour video programs and coordinated books
In this series, host Sol Garfunkel explains how algebra is used for solving real-world problems and clearly explains concepts that may baffle many students. Graphic illustrations and on-location examples help students connect mathematics to daily life. The series also has applications in geometry and calculus instruction. Algebra is also valuable for teachers seeking to review the subject matter.
Produced by the Consortium for Mathematics and Its Applications and Chedd-Angier. 1991.
I watched program 1, the introduction and it was quite good. You are required to sign up, but it is free and you need Windows Media Player and a fast connection to the Internet.