Letters and Postcards

Linear programming is an engaging area to apply many algebraic skills and understandings, especially linear inequalities and systems of equations. It is also a topic where it is not too difficult to find worthwhile problems in standard textbooks.

"Letters and Postcards" is intended as an introductory lesson, prior to any explanation of the general approach one follows to solve such problems. Before handing out the worksheet, you might ask these questions:

- What would cost more altogether, sending a postcard or a letter?

- What would take more time to write?

- What information would you need to answer these questions?

Explain that they will explore this in the case of one person who was going to send both postcards and letters, many years ago.

Because the variables can only take whole number values, it is possible for students to find all the points in the feasible region on a Cartesian graph. By doing that first, they develop a feel for the nature of the problem. (This is done in #1-3.)

The next step (#4-6) is to put the problem's constraints in algebraic notation, and to see how those are represented on the graph. If you move on to this before everyone has completed #3, you may hand out and discuss a pre-made graph.

Finally (#7-8) we address the optimizing questions that are the standard topic of these sorts of problems. (In subsequent problems, the optimizing question will need to be posed up front, alongside the constraints, since that is what the problem is about. In this case, because we were setting the stage for a new sort of problem, we broke up the lesson into stages.)