# 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 + 0.05

x= 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 *Y*_{1} charges $4.95 monthly
with a 5¢ per-minute charge and that Company *Y*_{2} 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: *Y*_{1} = 4.95 + 0.05*x* and *Y*_{2} = 0.08*x.*

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, *Y*_{1} is the
bold line and *Y*_{2} 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 *Y*_{2}; if we use more minutes, we should
use Company *Y*_{1}.

## 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 contextStatement of the problem

Symbolic contextModel of the problem in equation or function form

Numeric contextVarious values for the models

Graphical contextPicture 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.

Reprinted from Teacher to Teacher, Volume 5, Number 1.