High School Math: using technology in the classroom

“It was much different in our times…” is a cliché that we often hear from our elders. But now, it is my turn to say it. When I was in high school, which corresponds to the late 1980’s, we did not have the luxury of exploiting technology in our math classes. In order to sketch the graph of a function, we had to perform a series of tedious steps which would yield a graph that we were never a hundred percent sure of, and unfortunately, we would have to count on the sketch presented in the text book. I can also remember going over a number of books and tens of graphing exercises prior to a really scary precalculus exam.

Once when I was in high school, I came upon a question similar to this one:

How many real solutions does the equation given by e^x = x^3 + 4 have?

At first sight, it really did not ring a bell and I desperately looked up the solution: Sketch the graphs of y = e^x and y = x^3 + 4 on the same set of coordinate axes and observe the number of intersection points. This was a complete awakening for me (and would later constitute a major corner stone in my math teaching career: I wrote a book, based upon how to exploit the capabilities of the graphing calculator, which was recognized by many as well as the US Department of Education).

Now, if I make the following claim, what would you think?

It is possible to solve any type of equation or inequality with a graphing calculator, whether it is algebraic, trigonometric, exponential, logarithmic, polynomial, or transcendental, in a similar way without having to perform tedious steps.

With a graphing calculator, yes, it is possible. However, it would be such a waste not to exploit other useful functions of these brilliant handheld computers that are given the modest name of “graphing calculators”. With our mathematical knowledge and what is already there within those handheld giants, the sky is the limit to what can be performed, mathematically speaking!

For instance, let us consider the following question:

What is the equation of the parabola that passes through the points (2, 0), (4, 6), (-3, 20)?

Here is one approach to the solution: Let the parabola be y = ax^2+ bx + c. Plug in the points and you get the three equations given by 4a + 2b + c = 0, 16a + 4b + c = 6 and 9a – 3b + c = 20. Now you have a linear system of three unknowns and three equations that you can use to solve for a, b and c to find the correct solution: a = 1, b = – 3 and c = 2.

Now, how can you use technology to help you? Can you use it to solve the linear system for a, b and c? Yes, indeed you can. But let me propose a rather “radical” approach. Does your graphing calculator perform quadratic regression? I bet it does. I suggest that you use quadratic regression to find the equation of the parabola; since you have three points and no more, quadratic regression will exactly give you the parabola that passes through all three of these points and in one easy step. I believe this approach is much better than what I used to be doing when I was in high school since it saves lots of time avoiding the danger of making a mistake while solving the system of three equations for the three unknowns.

Consequently, in the level of civilization attained today, we do have the luxury of using technology. All it takes is to use this option smartly by combining the capabilities of technology with our knowledge of mathematics. Thus, we can not only increase the quality of our teaching, but also speed up the teaching and learning process. Here is a humble opinion of mine: There will be times when the difference between a fair math teacher and a good one will depend on how competently that teacher employs technology in the classroom. Believe me, my dear friends, those times have already arrived!