Surely this is a prank?
A new way to make quadratic equations easy
31–40 of 98 posts
Re: A new way to make quadratic equations easy
#32I think this is nice. It's a method of calculation derived from an important fact—that quadratic equations have two solutions and those solutions can be written in such a form.
I don't have the experience or interest to debate whether it's better or worse than any other method pedagogically. I can definitely imagine that I would have appreciated learning this method as well as completing the square as a child.
But honestly, the real interesting thing here is the (a) gathering information about what you think the solution ought to look like and (b) working backwards from there. That's a good general trick and worth having in your back pocket. It also emphasizes the importance of visualizing and examining your belief about what should occur.
People should suspect the fundamental theorem of algebra long before they prove it.
Re: A new way to make quadratic equations easy
#33This still seems way more complicated than it needs to be to teach it. I always used a much simpler way to avoid memorizing anything. Imagine you have a parabola y - c = k x^2 and want to solve for y = 0. Dead easy, right? To turn any other parabola into this form, you only need to scroll left or right on x until the minimum is at x'=0 (algebraically, this means eliminating any b*x' term). Teach students how to do ch…
I teach mathematics at a community college. This includes teaching a lot of elementary algebra courses. These are pre-college level math courses. In elementary algebra we introduce solving quadratic equations by factoring. In the next course intermediate algebra we teach the quadratic formula.
Students in elementary algebra are not equipped to understand change of coordinates. This is too hard of a concept at that stage in my opinion. The method described in the article is a very nice one and is appropriate for elementary algebra. It ties in nicely with factoring and makes solving quadratic equations quite easy. I will be using this method in the future.
Re: A new way to make quadratic equations easy
#34If you're going to rename the constants to make a=1, you might as well also replace b with b/2. The solution to x^2 + 2bx + c = 0 is
x = ± √(b^2 - c) - bRe: A new way to make quadratic equations easy
#35This is bizarre. It’s no less a mathematical trick than completing the square, and it doesn’t seem to be any simpler to use. The example given is to find the roots of x² - 2x + 4 = 0. Completing the square gives (x - 1)² + 3 = 0, from which you can immediately see that the roots are 1 ± √3 i. If anything this seems easier than the method of the article. Am I missing something? Added : The argument seems to be that yo…
Was going to reply precisely this. Completing the square is, I believe, already taught in standard curricula (at least, it was when I was in grade school), and is a trivial way to derive the quadratic formula.
Re: A new way to make quadratic equations easy
#36This seems more complicated and roundabout than completing the square. The average of the roots shows up that way too: We want to write x^2 + bx + c = 0 in the form (x+m)^2 + n = 0, so there's only one x left and we can rearrange for it. Expanding, (x+m)^2 = x^2 + 2mx + m^2, so we get the x^2 we want, and the coefficients of x tell us b = 2m, so m = b/2. We also get an m^2 (= b^2/4) we don't want, so let's take it aw…
Having taught elementary algebra for many years at a community college I think your perception of what is complicated and easy are not correct. The method in the article is far easier for elementary algebra students than what you describe.
You’ve proved the quadratic formula. This is a formula students in elementary algebra will struggle with. It’s a formula whose proof will be lost on them. What is shown in the article is an easy to apply mechanism for finding the roots. The method in the article is one that I can use in the classroom. There’s no way I’d attempt your explanation in an elementary algebra class.
Re: A new way to make quadratic equations easy
#37People are criticizing this because it is still the same quadratic formula. But of course it is! Math is consistent. But representation matters. A good chunk of mathematics is just about rewriting the same mathematical fact in a different way. For example the equation of a line could be written with coefficients or in slope/intercept form or in polar coordinates or in homogeneous coordinates or etc etc. Here the clai…
Perhaps people are skeptical about the claim of originality for a straightforward algebraic manipulation that can be carried out and understood by middle school students. (As far as I can see, that is what the article claims.) Bright people "reinvent the wheel" all the time. The rediscovery of the trapezoidal rule for numerical integration is a recent popular example.
Also, unrelated and probably unfair ... maybe a few cynical and overly skeptical people become a little cautious when they see a link to Tech Review.
Re: A new way to make quadratic equations easy
#38There is an error in the (pixelated) example that is given int the MIT tech review article (z² = 3 instead of -3). I think the fact that the author of the news missed this, that he probably took a screenshot of the formulas rewritten in Word, and that he was compelled to write such a long article on such a simple topic speaks for his level on the topic.
The fact that he only lists the formal article as a reference instead of the announcement, video, and accessible blog post by Po-Shen Loh really baffles me.
The original "disclosure" by Po-Shen Loh [0] is much less sensational and gives some context for his work (teaching middle school students). In the formal article, he is also stating that the method is very likely not __new__, but that he wants to popularize it in teaching.
I think, as many other commenters pointed out, that there is no great breakthrough here. However "his" method may have the advantage of training the intuition of young students, by helping them understand the concepts of average and "deviation" (I'm not really sure how to call it in that case), and maybe visualizing them.
Re: A new way to make quadratic equations easy
#39This is bizarre. It’s no less a mathematical trick than completing the square, and it doesn’t seem to be any simpler to use. The example given is to find the roots of x² - 2x + 4 = 0. Completing the square gives (x - 1)² + 3 = 0, from which you can immediately see that the roots are 1 ± √3 i. If anything this seems easier than the method of the article. Am I missing something? Added : The argument seems to be that yo…
Programming is full of these. Every day I see some new technique or style appear. Is it any good? We've got easier access to the biggest collection of free source code, the biggest and most diverse set of programmers, and the fastest automated testing of all time. Just do an experiment. You don't need to sell me on how cool it is. Show me how it's better.
- Does it lead to smaller programs? Don't show me a 10-line example. (I've heard medical researchers say: "Anyone can cure cancer in mice.") Convert a 10,000- or 100,000-line program.
- Does it lead to humans writing fewer bugs? Have a bunch of them try it, and measure their speed and defect rates. Everyone has designed a system that they themselves love, and nobody else can understand.
If you promote a new method which is logically equivalent to old ones, and not obviously much better, and without any data showing specific metrics that it improves, I'm forced to assume your metric is simply "I happen to like it better". Experimentation is cool, but it doesn't cause lasting long-term shifts.
Re: A new way to make quadratic equations easy
#40Um... what's the difference?
x = (-b ± √(b^2 - 4ac)) / 2a
x = (-b ± √(b^2 - 4c)) / 2
x = -b/2 ± √(b^2 / 4 - c)