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A new way to make quadratic equations easy

technologyreview.com

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Re: A new way to make quadratic equations easy

#71

Post by knzhou »

This kind of approach is familiar to me from competition math. Back in middle school math club we were taught this exact approach (under the name of "Vieta's formulas"), i.e. that thinking about the sum and product of the roots could be faster in some cases. Po-Shen Loh is the director of the US IMO team, so it makes sense he would like this approach.

However, I don't think it makes logical sense to teach it only this way. Here you start by assuming that a quadratic has two roots, which is not at all obvious the first time a kid sees a quadratic equation. (Especially because those roots can be complex numbers!) Completing the square tells you why there are two roots, and also naturally leads you to the necessity of complex numbers, i.e. when the "square" you end up making is negative. You can use the nice Vieta's formula tricks only after establishing that.

Re: A new way to make quadratic equations easy

#72

Post by knzhou »

thanhhaimai wrote: post #11

I have trouble understanding why this deserves a paper. This is what I learned in middle school back in Vietnam (although I did take advanced Math). For equation Ax^2 + Bx^2 + C = 0, the roots are: x1 = (-B + sqrt(B^2 - 4AC))/2A x2 = (-B - sqrt(B^2 - 4AC))/2A Now the author looks at a special case where A = 1. The equation becomes x^2 + Bx^2 + C = 0. Of course the roots simply become: x1 = -B/2 + sqrt(B^2 - 4C)/2 = -…

Yeah, half the elegance of the linked expressions just comes from setting A = 1.

This is definitely well-known. I learned this under the name of "Vieta's formulas" back in middle school. Vieta lived in the 1500s.

Re: A new way to make quadratic equations easy

#73

Post by jacobolus »

carapace wrote: post #68
sykick wrote: post #41

Earlier quoted context omitted.

I’ve taught elementary algebra for many years at a community college. The method described in the article is truly, in my opinion, very nice. I will be using this method from now on in elementary algebra. Students in elementary algebra don’t know what parabolas are. They haven’t been taught to complete the square and don’t know how to work with radicals. They have just been taught what square root is and basic binomi…

Apologies in advance for the harshness of my tone, but I nearly spit up my coffee... > Students in elementary algebra don’t know what parabolas are. The mind boggles. Have you never shown them a sliced cone? ( https://en.wikipedia.org/wiki/Conic_section ) I mean, they know what parabolas are: they live on a planet, with gravity. Every ball, every jump: parabolic motion, yeah? "Michael Jordan exerts his muscular power…

All of the world’s best scientists didn’t know that projectiles moved on parabolic paths until Galileo’s experiments on inclined planes in the 17th century. 21st century schoolchildren who haven’t been taught about it likely don’t either.

Here’s Tartaglia’s picture from the 16th century, drawn from his experimental research: https://www.maa.org/sites/default/files/images/upload_librar...

And here’s another similar diagram tossed up by image search: http://ej.iop.org/images/0143-0807/33/1/149/Full/ejp405251f1...

If you look for even earlier diagrams they show a projectile moving in a straight line for some distance and then suddenly dropping straight down.

Cf. https://en.wikipedia.org/wiki/Aristotelian_physics

Re: A new way to make quadratic equations easy

#74

Post by DavidSJ »

jacobolus wrote: post #73
carapace wrote: post #68

Earlier quoted context omitted.

Apologies in advance for the harshness of my tone, but I nearly spit up my coffee... > Students in elementary algebra don’t know what parabolas are. The mind boggles. Have you never shown them a sliced cone? ( https://en.wikipedia.org/wiki/Conic_section ) I mean, they know what parabolas are: they live on a planet, with gravity. Every ball, every jump: parabolic motion, yeah? "Michael Jordan exerts his muscular power…

All of the world’s best scientists didn’t know that projectiles moved on parabolic paths until Galileo’s experiments on inclined planes in the 17th century. 21st century schoolchildren who haven’t been taught about it likely don’t either. Here’s Tartaglia’s picture from the 16th century, drawn from his experimental research: https://www.maa.org/sites/default/files/images/upload_librar... And here’s another similar di…

Maybe Tartaglia just didn’t like quadratics. He did, after all, find the cubic formula. ;)

Re: A new way to make quadratic equations easy

#75

Post by sykick »

carapace wrote: post #68
sykick wrote: post #41

Earlier quoted context omitted.

I’ve taught elementary algebra for many years at a community college. The method described in the article is truly, in my opinion, very nice. I will be using this method from now on in elementary algebra. Students in elementary algebra don’t know what parabolas are. They haven’t been taught to complete the square and don’t know how to work with radicals. They have just been taught what square root is and basic binomi…

Apologies in advance for the harshness of my tone, but I nearly spit up my coffee... > Students in elementary algebra don’t know what parabolas are. The mind boggles. Have you never shown them a sliced cone? ( https://en.wikipedia.org/wiki/Conic_section ) I mean, they know what parabolas are: they live on a planet, with gravity. Every ball, every jump: parabolic motion, yeah? "Michael Jordan exerts his muscular power…

I deduce from your post that you have very little experience in teaching people at the level of beginning algebra. And while one might know geometrically what a parabola is there is a lot one must know before dealing with parabolas algebraically. I suggest that these things appear easy and obvious because you already know them and that you no longer remember what is hard for people learning this stuff for the first time.

I don’t show conic sections in elementary algebra. One typically really mentions the phrase “conic section” is pre-calculus which is 3 courses after elementary algebra. Over the past few centuries the order in which concepts are introduced has been developed. It’s not perfect but one should not be so quick to discount the way things are done without knowledge/experience in presenting these ideas to beginning students.

Re: A new way to make quadratic equations easy

#76

Post by Isamu »

sykick wrote: post #49
pflats wrote: post #45

Earlier quoted context omitted.

> To me it is obvious that the method in the article is far superior than teaching completing the square. I disagree. I would need some convincing that "two numbers that multiply to C and sum to B must have an average of B/2, so they must be B/2 + z and B/2 - z, so (B/2 + z)(B/2 - z) = C" is by any means obviously superior to completing the square. Neither is immediately intuitive; both will require prompting and tea…

I doubt I can convince you. I’m just going by my experience teaching the topic. At the time students first learn solving such equations they have just been taught factoring and what it means to factor a trinomial. They know the product of the constant terms in the binomials must be c. It’s also easy to explain that the average of two numbers is the midpoint. And thus if I start with the midpoint then to get to the nu…

>but like everyone else I’ve had my share of obvious beliefs turn out to be false.

Refreshing candor! Wish it held true that more people saw it through to discover their obvious truths didn't hold up.

Re: A new way to make quadratic equations easy

#77

Post by infocom6502 »

abnry wrote: post #63

Let's get the criticisms of the article out of the way: --It has terrible formatting and typos. --It puffs up something more important than it is. --It is more about pedagogy than a mathematical idea. --It suggests the idea is original, when it almost certainly is not. --It doesn't link to the original (and better source). Okay, here is the good things about the approach: --It is good to shift your thinking about mat…

I agree. This is nothing novel and a fairly standard way of solving the equation.

Re: A new way to make quadratic equations easy

#78

Post by [deleted] »

knzhou wrote: post #71

This kind of approach is familiar to me from competition math. Back in middle school math club we were taught this exact approach (under the name of "Vieta's formulas"), i.e. that thinking about the sum and product of the roots could be faster in some cases. Po-Shen Loh is the director of the US IMO team, so it makes sense he would like this approach. However, I don't think it makes logical sense to teach it only thi…

[deleted]

Re: A new way to make quadratic equations easy

#79

Post by carapace »

sykick wrote: post #75
carapace wrote: post #68

Earlier quoted context omitted.

Apologies in advance for the harshness of my tone, but I nearly spit up my coffee... > Students in elementary algebra don’t know what parabolas are. The mind boggles. Have you never shown them a sliced cone? ( https://en.wikipedia.org/wiki/Conic_section ) I mean, they know what parabolas are: they live on a planet, with gravity. Every ball, every jump: parabolic motion, yeah? "Michael Jordan exerts his muscular power…

I deduce from your post that you have very little experience in teaching people at the level of beginning algebra. And while one might know geometrically what a parabola is there is a lot one must know before dealing with parabolas algebraically. I suggest that these things appear easy and obvious because you already know them and that you no longer remember what is hard for people learning this stuff for the first t…

You've got me. I wish I could delete the comment, or at least edit it.

It's a bit worse than just having no experience teaching algebra, I didn't have the same experience as most kids trying to learn it. I was kind of a freak. I was the weird quiet kid in the back of the class who always knew the answer to every question. (Other kids tended to not like that, but I'm also very disarming (in person) and so I did alright.) One year, I was misplaced into a basic geometry class, and the teacher very kindly let me pick out some calculus textbooks and sit in the back of the class teaching myself calculus. (Some bureaucratic reason for why I couldn't transfer, or the calc classes were full, or we didn't have any, or something, I forget.) Learning math for me feels like remembering things I always knew.

So, yeah, maybe I should keep my mouth shut when it comes to teaching normal people how to do math.

Or maybe we should try to figure out what my brain is doing and how to teach people to do that too?

Maybe I have a normal brain and I'm just using it differently than most people?

I like that idea better, because then, instead of a freak, I'm a front-runner. And, in addition, there's hope for a great improvement in didactic technique and humanity's general "numeracy" level, eh? If we could teach people to math like i do we could compress basic math education (up to calculus) into just a year or so.

"Something self-referential and hopefully unsnarky about how this is a thread on a how-to-math-better article." ~me, failing at rhetoric.

And for kicks, here's Iconic Math: http://iconicmath.com/ (No affiliation with me. I'm just trying to end on an upbeat, constructive note.)

Re: A new way to make quadratic equations easy

#80

Post by fyp »

knzhou wrote: post #71

This kind of approach is familiar to me from competition math. Back in middle school math club we were taught this exact approach (under the name of "Vieta's formulas"), i.e. that thinking about the sum and product of the roots could be faster in some cases. Po-Shen Loh is the director of the US IMO team, so it makes sense he would like this approach. However, I don't think it makes logical sense to teach it only thi…

I am also familiar with Vieta from math competitions but I think this is more than that.

Given two numbers, r1 and r2, if you know their arithmetic mean:

  m = (r1 + r2) / 2
and geometric mean:

  g = sqrt(r1 * r2)
then following Loh's derivation you get a very cute formula:

  r1 = m - sqrt(m^2 - g^2)

  r2 = m + sqrt(m^2 - g^2)
Vieta gives an easy way of finding those means from a quadratic equation: r1+r2=-b and r1*r2=c. So you can plug in m=-b/2 and g=sqrt(c) in the equation above.

The fact that you can state roots in terms of their means is the more novel insight to me. (note: Loh doesn't talk about geometric means but I thought using just product of the roots isn't as "symmetric")

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