Two days ago I wrote about an article describing a faster way to multiply than the standard multiplication algorithm. I closed that post with "The article originally appeared in Popular Mechanics. They should
probably stick to what they know best, because they're way out of their
league here." I could close this post in exactly the same way, as PM has stepped in it again.
First, the article. Go ahead and read it, it's brief. Supposedly, a Dr. Loh at Carnegie Mellon has developed a way to solve quadratics equations that is easier than the quadratic formula or completing the square. In the video posted on his own web site, Dr. Loh states that while pieces of this new method have been known for hundreds of years or more, "How can it be that I've never seen this before, and never seen this in any textbook?" Well, I'll tell you, Dr. Loh. But first, let me bring up two points:
1) He didn't "develop" this, it's well-known to those of us adept at algebra, and
2) How practical is this method when the answers don't come up integers?
Here's the video mentioned above, in which Dr. Loh explains his new and exotic procedure. Take 3 minutes and 48 seconds to watch it.
After you've watched it, ask yourself this: How did he get the sum and the product needed to solve the problem?
It took me a moment to figure it out, but then it came to me. Follow my reasoning here.
Take any quadratic equation. Manipulate it so that there is 0 on the right side of the equal sign. Complete the square. In a couple more steps you've derived the Quadratic Formula, which will allow you to solve any quadratic equation. Bottom line: you know the quadratic formula works because you can follow each step of the "completing the square" process (assuming you know that, which is something any Algebra 1 student should understand).
Given a quadratic equation of the form
you can use the Quadratic Formula to solve for the two solutions, which will always be
Add those two solutions together and you get -b/a, multiply them and you get c/a. That's the information that Dr. Loh used in his method, which you would not understand unless you knew these answers from the Quadratic Formula
. In the video, -b/a=8 and c/a=12.
Also, we know that a quadratic of the form above has a vertical axis of symmetry which, being vertical, always crosses the x-axis and goes through the vertex of the graphed parabola. Thus, if there are any real solutions to this problem, the x-intercepts, they must be equidistant from this line. This distance is what Dr. Loh calls "u". So, if you need a sum to be 8, (4+u) and (4-u) work as the two roots because they sum to 8 and are equidistant from the line of symmetry, and then you multiply them together and set the product equal to 12.
So far, so good. Well, not good, but not horrible--Dr. Loh's method is great for those of us who already understand algebra, but it's what we call "mathemagic" for people who are learning algebra. One of the Algebra 1 standards in California's 1997 Math Standards was that students would be familiar with the derivation of the Quadratic Formula by completing the square. They would know where the formula came from, and that it wasn't conjured from thin air by some mystic. No such luck doing that with Dr. Loh's method, where you have to just accept his this-will-work pronouncements. His method is like that of the ancient Babylonian "algebraists"--just follow these step-by-step instructions, don't worry about where they came from, and your answer will be correct. It's very utilitarian, it leads to being able to get an answer but it doesn't lead to any understanding. In this way, Dr. Loh's method is the very antithesis of what is supposed to be one of the big selling points of Common Core: understanding.
In the Popular Mechanics article and in Dr. Loh's own video, the quadratic equation to be solved had a=1 (students love those because they're much easier) and the answers came out to be integers. What happens if those two conditions aren't the case? Dr. Loh never explicitly mentioned "-b/a" or "c/a", I had to infer the former from his work and the determine the latter from of my own knowledge. Here's how to solve such a problem using the Quadratic Formula as well as Dr. Loh's method:
click to enlarge
Is it obvious to you that Dr. Loh's method is better than the Quadratic formula? It's certainly not to me, especially in this problem where the solutions aren't integers. Dr. Loh's method is one of those cool applications that those of us already adept at algebra delight in, but as a tool for teaching the foundations of quadratic equations, I find it lacking in many ways.
The bottom line is that we should not be getting math information from Popular Mechanics--not if we want anything meaningful, that is.