A former fellow high school student, Ted L., recently contacted me. He wrote concerning our senior year, in which we shared a math course that included parametric equations.
“I recall being in the advanced math class with Mr. Miller where I struggled quite a bit. In a discussion about solving some problem, you presented an alternative solution. Mr. Miller quickly dismissed your idea in a rather condescending fashion, shaking his head and stating, “No Summers, [you’re wrong],” with a tone that suggested your idea was rather silly, perhaps bordering on absurd. But you persisted, in a back and forth between you, that lasted for several minutes.
During that discussion, I was completely lost, having no idea at all what either of you was talking about. After many gives and takes, Mr. Miller stopped talking, quietly thought for a few moments, then declared, “You’re right.” Though not able to follow the conversation at all, I was very impressed that by your thorough understanding of the subject you were able to win the argument. Do you have any recollection of that event?”
Not to Be Too Harsh
The answer is… yes, I did. First, let me say, I really enjoyed each of my mathematics teachers. They made what might not have been so interesting a subject truly come alive for me. Mr. Miller was one of the best. And if I was in Mr. Miller‘s place, I’m not certain I wouldn’t have responded in similar fashion.
The Gist of the Problem
The thing was this. The book we were using included a chapter on parametric equations. Until we reached that chapter, I took no issue to the book’s contents. It was with one element only I took unyielding exception.
Repeatedly, in solving the equations, the authors squared the equations. Squaring an equation usually introduces a root (solution) that is not in the original equation.
Parametric Equations Example
Consider an example that proves my statement. Solve, by eliminating (the parameter) “t” from the two equations
2x = t
y = √t
The correct answer is
y = √(2x)
But, suppose instead, that you squared the y = √t
2x = t
y² = t.
And you’d wind up with the incorrect answer,
y² = 2x.
Why is this answer incorrect? Because the act of squaring the equation in y and t introduced an additional root. In the first instance, the so-called real number solution requires t be positive. This means y has to be positive.
But squaring allows y to be negative as well! For instance, if x = 1/2, then y can be either +1 or –1. As another example, if x = 1, then y can be either +√2 or –√2.
And the Result?
So what was the outcome of my discussion with Mr. Miller? Commendably, he wrote the authors of the textbook. Can you guess what their response was? You’ve got it! There was no response.