# Should Students Use Graphing Calculators in the High School Classroom?

As always, what should we accomplish? If our vision of the classroom is students who quickly and accurately manipulate symbols with pencil and paper, then our answer probably should be no.

But if we want a mathematics classroom that extends beyond pencil-and-paper skills to the understanding and realistic application of mathematics, then graphing calculators can be very helpful. Very briefly, here's how.

### An example for junior high school

Many pre-algebra or algebra classes assign students sentences like one of these:

4.95 + 0.05x = 0.08x
4.95 + 0.05x ≤ 0.08x
4.95 + 0.05x ≥ 0.08x

Traditionally, students learn to manipulate the symbols to solve for x. They might first multiply both sides by 100, move a term to the opposite side, divide, and solve. Then, at the end of the chapter, students might see a section of related word problems. But students would still ask, "When are we ever going to have to use this?"

Graphing calculators can change all of this. Instead of focusing on symbol manipulation, students often start with an application, such as costs charged by long-distance telephone companies. Suppose, for example, that Company Y1 charges \$4.95 monthly with a 5¢ per-minute charge and that Company Y2 charges 8¢ per minute with no extra monthly charge. Which company should you choose?

Students could first model these monthly costs with two linear functions: Y1 = 4.95 + 0.05x and Y2 = 0.08x.

Next, they could build a table of monthly costs. Here students can easily see relationships between costs for the two companies. Similarly, they could graph both together; the differences between the companies will be obvious.

But the real question is this: which company should you choose? For this question, a related problem is to find where the two linear functions intersect. This is easy with a table or a graph. (In the graph below, Y1 is the bold line and Y2 is the normal line.)

Students can quickly see that this is not a simple question to answer. Instead, the answer depends on how many long-distance minutes are used. From the table or graph, one can see that both plans cost the same for x = 165 minutes. That is, 165 minutes is the break-even point. If we talk for fewer long-distance minutes, then we should use Company Y2; if we use more minutes, we should use Company Y1.

### What is to be gained?

With graphing calculators, it is natural to teach mathematics from real applications (and not only in an optional section). Students will see numerical details, and a graph will help to put it all together. Mathematical richness is available for students since they focus on solving problems from at least four contexts:

Verbal context Statement of the problem
Symbolic context Model of the problem in equation or function form
Numeric context Various values for the models
Graphical context Picture of the models

These four contexts are not generally available to students in a traditional classroom. While these students learn much about symbolic expressions, they usually will not see a connection to real life, and they generally do not think about a graphical or numeric context to real problems.

So if your goal is the understanding and realistic application of mathematics, then graphing calculators can be very helpful.