# Letters and Postcards

An introduction to linear programming

by Henri Picciotto

Middle school version (an introduction to linear inequalities)

## Teacher Notes

### Introduction

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.

### The Worksheet

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.)

### Wrap-Up

But the lesson is not over at the end of the page! Presumably, students will find that the minimum and maximum costs and times correspond to vertices of the feasible region. To understand why it turns out that way, it helps to look at an interactive graph. (Alternately, you can download the GeoGebra file that I used to create the graph.)

Questions to discuss:

• What is the formula for the total cost? How is it represented on the graph?
• For a given total cost, where are the points representing combinations that would cost that much?
• How can we use the interactive graph to find the minimum and maximum cost for points in the feasible region?
• Why do the minimum and maximum have to be at vertices?
• Knowing the equations for the lines that enclose the feasible region, how can we find the coordinates of the vertices?
(Same questions for the time.)

Related pages on this site:
Constant Sums
Cell Phone Plans