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1731 clicks; posted to STEM » on 30 Jul 2021 at 3:25 AM (8 weeks ago)   |   Favorite    |   share:

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Ask me how I know this method was taught in high school Algebra 30+ years ago. Factoring ain't exactly new.

I was thought this method 23 years ago. It only works if the numbers are nice integers. As soon as decimals crop up it becomes hell to find the two numbers for which it works.

Yes, easier if the numbers are easy to find. Just going by the longer way is easier on the long run.

snowjack: Ask me how I know this method was taught in high school Algebra 30+ years ago. Factoring ain't exactly new.

Clarification: apparently the "new" part isn't really about solving quadratics, it's just a more deterministic way to find the factors. But that's not new either -- I had a math teacher go over this method.

DerAppie: I was thought this method 23 years ago. It only works if the numbers are nice integers. As soon as decimals crop up it becomes hell to find the two numbers for which it works.

Yes, easier if the numbers are easy to find. Just going by the longer way is easier on the long run.

Well, let's try it and see if decimals throw it off.

I'll use x^2 - 8.572x + 12

(4.286 + u)(4.286 - u)     4.286 is 8.572/2
18.369796 - u^2 = 12
u^2 = 6.369796
u = sqrt(6.369796)
4.286 +- sqrt(6.369796) = 6.8098454786... and 1.7621452123...

A quick trip to Wolfram Alpha tells me the roots of x^2 - 8.572x + 12 are 6.80985 and 1.76215, but it rounds more.

The point is that all you need is -B/2 and C, you don't need to hunt for any numbers, or more that the process finds them for you.

LrdPhoenix: 1.7621452123

Whoops, that should be 1.7621545213, not sure how that happened.

LrdPhoenix: DerAppie: I was thought this method 23 years ago. It only works if the numbers are nice integers. As soon as decimals crop up it becomes hell to find the two numbers for which it works.

Yes, easier if the numbers are easy to find. Just going by the longer way is easier on the long run.

Well, let's try it and see if decimals throw it off.

I'll use x^2 - 8.572x + 12

(4.286 + u)(4.286 - u)     4.286 is 8.572/2
18.369796 - u^2 = 12
u^2 = 6.369796
u = sqrt(6.369796)
4.286 +- sqrt(6.369796) = 6.8098454786... and 1.7621452123...

A quick trip to Wolfram Alpha tells me the roots of x^2 - 8.572x + 12 are 6.80985 and 1.76215, but it rounds more.

The point is that all you need is -B/2 and C, you don't need to hunt for any numbers, or more that the process finds them for you.

I never said it couldn't be done, just that it stops being easier.

As soon as I need to grab a calculator to see what the roots are, it stops mattering what formula I use.

It wouldn't matter subby. If you had math teachers like I did they would make you learn the long painful way first then tell you the newer, easier way.

I also had a pre-cal teacher who would be five minutes into a proof before she realized she'd farked it up.

In public school, there is only one way to teach math, and all other ways will be severely penalized.  This new method will only be tolerated if it is more confusing than the old way.

Subby... Seriously?

developed by Dr. Po-Shen Loh at Carnegie Mellon University

Being published in an MIT Review has no bearing on who came up with this, which btw was a single mathematician at Carnegie Mellon.

Maybe we should send back an English teacher to teach you to read instead.

Boudyro: It wouldn't matter subby. If you had math teachers like I did they would make you learn the long painful way first then tell you the newer, easier way.

I also had a pre-cal teacher who would be five minutes into a proof before she realized she'd farked it up.

Well, it sucks your pre-cal teacher wasn't competent enough to check the proofs in her own lectures before delivering them.

But there's a damn good reason why new ideas are developed and then proven to work in math courses: rote mechanical knowledge without understanding is, at best, useless, and at worst, dangerous.

Unless of course your argument is that the concepts behind high school level math were too difficult to grasp, in which case there's not much discussion left to be had.

olrasputin: Unless of course your argument is that the concepts behind high school level math were too difficult to grasp, in which case there's not much discussion left to be had.

I don't think quaternions are ever taught in public school, mostly because no one has ever taught them in public school before, and it's only the most important mathematical tool for the new millenium.  Show them one of the YouTube videos, for Gods sake.

The problem is that stupid kids will just think +u, -u are roots of the original quadratic.

duckpoopy: The problem is that stupid kids will just think +u, -u are roots of the original quadratic.

They're called 'number sentences' now

/wishes he was making this up
//if there's a reason to hate common core, this is it

Boudyro: It wouldn't matter subby. If you had math teachers like I did they would make you learn the long painful way first then tell you the newer, easier way.

Best math teacher I ever had was the man- Jones- who taught me business calculus. At one point,, he was explaining a concept, stopped, looked up and said "a mathematician will tell you this isn't how it works. I'm not a mathematician, I'm an engineer. This is exactly how it works."

Has anyone checked to see how it works with complex roots?

PartTimeBuddha: Has anyone checked to see how it works with complex roots?

Examples: A Different Way to Solve Quadratic Equations

That's the method I was taught back in the early 80's.  If factoring wasn't obvious or easy to find you went back to quadratic formula.   This is the method I also taught my students when I was a math teacher.

The right tool for the right job.

The mathematician is from Carnegie-Mellon University. MIT Tech. Review just reported it. At least that's what the article says.

Gonz: Boudyro: It wouldn't matter subby. If you had math teachers like I did they would make you learn the long painful way first then tell you the newer, easier way.

Best math teacher I ever had was the man- Jones- who taught me business calculus. At one point,, he was explaining a concept, stopped, looked up and said "a mathematician will tell you this isn't how it works. I'm not a mathematician, I'm an engineer. This is exactly how it works."

My brother had similarly positive things to say about Business Calc. He was amazed at how straightforward it was. I had the more formal classes and, by 3/4 the way through calc 2, I was pretty much burned out on page-long derivations for fluid flow through a small and limited subset of possible reservoirs that will probably never exist while being annoyed because, in a real situation, I could calculate the answer in a few minutes or less using any computer and numerical integration (an approach that we spent almost zero time on). I appreciated and understood what we were doing, as it was genuinely impressive, but it seemed like a waste of time for something I was never going to use and was going to be completely out of practice with by the next semester. Though I could say the same thing about most math classes.. a huge focus on things I'll never do and neglect for the things I would actually do.

I just invented a device which is going to revolutionize travel. What you do is create an object with a circular cross section. Has to be a circle. And in the middle of the circle you put a bearing and a shaft, which you can attach to whatever you want. I call it the HyperRound. I'm going to make a fortune.

snowjack: Ask me how I know this method was taught in high school Algebra 30+ years ago. Factoring ain't exactly new.

They got foiled?

olrasputin: Boudyro: It wouldn't matter subby. If you had math teachers like I did they would make you learn the long painful way first then tell you the newer, easier way.

I also had a pre-cal teacher who would be five minutes into a proof before she realized she'd farked it up.

Well, it sucks your pre-cal teacher wasn't competent enough to check the proofs in her own lectures before delivering them.

But there's a damn good reason why new ideas are developed and then proven to work in math courses: rote mechanical knowledge without understanding is, at best, useless, and at worst, dangerous.

Unless of course your argument is that the concepts behind high school level math were too difficult to grasp, in which case there's not much discussion left to be had.

I admittedly have the higher math skills of a potato. Even algebra and geometry took a great deal of effort and my grades were very sensitive to the quality of the teacher. I made a B, with a fantastic teacher. Once. The pre-cal teacher would do her proofs freehand on an overhead projector, while lecturing, and often get confused, screw them up, and have to backtrack. I failed that class hard. There may also have been a pretty girl in that class that had something to do with it . . .

My problem was I was very smart at anything that wasn't math so I was lumped in the College prep crowd. I needed regular math classes and College-level everything else, but they didn't split things that way. At least not back then.

While I talk down my math skills, I took the SAT in 1991 and earned a 1300 with the math only being 20 points lower than my language score. So I wasn't a complete idiot, math just wasn't my strength.

falkone32: Gonz: Boudyro: It wouldn't matter subby. If you had math teachers like I did they would make you learn the long painful way first then tell you the newer, easier way.

Best math teacher I ever had was the man- Jones- who taught me business calculus. At one point,, he was explaining a concept, stopped, looked up and said "a mathematician will tell you this isn't how it works. I'm not a mathematician, I'm an engineer. This is exactly how it works."

My brother had similarly positive things to say about Business Calc. He was amazed at how straightforward it was. I had the more formal classes and, by 3/4 the way through calc 2, I was pretty much burned out on page-long derivations for fluid flow through a small and limited subset of possible reservoirs that will probably never exist while being annoyed because, in a real situation, I could calculate the answer in a few minutes or less using any computer and numerical integration (an approach that we spent almost zero time on). I appreciated and understood what we were doing, as it was genuinely impressive, but it seemed like a waste of time for something I was never going to use and was going to be completely out of practice with by the next semester. Though I could say the same thing about most math classes.. a huge focus on things I'll never do and neglect for the things I would actually do.

Well, I always prefaced calculus classes I taught with basically that: the class isn't about the problems themselves, or even finding solutions to said problems.

The purpose of the class is two-fold:

1) To understand the theoretical framework behind things like numerical integration, so that you have a handle on how close numerical estimates are to the "true" answer for a given numerical scheme. Also, more importantly, to understand that numerical problems generally arise because of underlying analytical problems--so when your program fails to converge, you can step back and note things like "Oh, this function oscillates at a very high frequency, so each successive grid refinement is returning drastically different function values. The root problem is that, with respect to finite grid spacing, this function is behaving a lot like it has a ton of discontinuities."

2) You practice math problems for the same reason the average person learns a foreign language. Not to use that specific skillset in the long run, but for the benefit gained from training yourself to think in a different way about things. In the case of math, the point is to learn greater care in following logically connected steps, and ideally, how to assess whether or not your answer is correct. The latter is super important, since people won't be paying you to solve problems in a situation where someone can just come along and say if your solution is right or wrong.

Boudyro:
My problem was I was very smart at anything that wasn't math so I was lumped in the College prep crowd. I needed regular math classes and College-level everything else, but they didn't split things that way. At least not back then.

I got that a ton with my students--it's really a common phenomenon. Math really does have its own internal structure and requires a pretty specific way of thinking about things. It's a skill that can be acquired, but it's certainly not a spectator sport; writing out proofs step by step on an overhead isn't gonna teach anyone in the room a damned thing.

I always made it a point to pick a relatable real-world problem where the required machinery appeared naturally in the solution. I'd ask people how to start, and how to proceed at each step. And sure enough, using only the knowledge they had so far, we eventually hit a wall. So I'd ask for suggestions on what went wrong and how to fix that, and so on. And then eventually after a few such examples, I'd start talking about more general cases and how that led to a proof.

And more broadly, studying math to the point where you learn how to prove theorems and potentially even publish some papers really does permanently change the way you think. But for myself, and most colleagues I've talked to about it, it's more of a switch you flip on when needed.

If you've ever seen Super Troopers, there's a scene where two cops are joking around as they pull up to a murder scene. As they walk towards it, the more senior guy tells the rookie to "put on his game face", and he immediately tilts his hat forward and goes into serious cop mode.

It's kind of like that.

But here's the thing:  this isn't different from the quadratic formula at all.

Ultimately the method is to compute sqrt(B^2/4 - C), and take -B/2 plus or minus that.  This is literally, exactly, the conventional quadratic formula.     sqrt(B^2/4 - C) is equal to sqrt(B^2-4AC)/2A  (the video arranges things so A=1).  So this guy is telling you to compute {-B +- sqrt{B^2-4AC}/2A.  It's the same freaking thing.

The only thing different here is what you do stepwise to compute it.   It's the difference between "y=x^4" and "okay, so you take an x, and then square it, and then square it again."  That's not a new way, that's the old way explained in a slightly more imperative fashion.

Consider 6x^2 -11x -10 = 0    Here's how I taught my students - easy repeatable works well for any ax^2 + bx + c = 0 where a, b,c are integers, a not= 0...... (use formula or completing the square for non-integer coefficients)

Find two integers that multply to -60 and add to minus 11 or differ by 11. (Write factor pairs-the correct pair will jump off the page at you.)  1,60. 2,30. 3,20. 4,15...
-15 and +4 are the numbers.  YES THIS IS "GUESSWORK" - SUE ME!!

Re-write the equation 'splitting the middle term' like this using the two numbers just found:

6x^2 +4x -15x -10 = 0  Now find the greatest common factor of the first two and the last two terms separately...

2x(3x +2) -5(3x +2) = 0  Note that (3x + 2) is a common factor so re-write as:

(3x+2)(2x-5) = 0   Set each factor equal to zero and solve.  Thus x = -2/3 or 5/2  QED
......................................​......................................​......................................​......................................​..............
This 'new' method:
6x^2 -11x -10 = 0   Divide both ides by 6 to make coefficient of the first term to be 1. (His way won't work otherwise)

x^2 -11x/6 -10/6 = 0    here -b/2 would be 11/12    sub into (-b/2 + u)(-b/2 - u) = -10/6  solve for u

(11/12 + u) (11/12 - u)= -10/6

121/144 - u^2  = -10/6

121/144 + 10/6 = u^2

361/144 = u^2

19/12 = u

so roots are -b/2 +/- u     --> 11/12 + 19/12  and 11/12 - 19/12 ---> 5/2 and -2/3
......................................​......................................​......................................​......................................​............
I dunno, mine seems simpler, avoiding some nasty fraction work....

Russ1642: I just invented a device which is going to revolutionize travel. What you do is create an object with a circular cross section. Has to be a circle. And in the middle of the circle you put a bearing and a shaft, which you can attach to whatever you want. I call it the HyperRound. I'm going to make a fortune.

Directions unclear: have been spinning in circles for an hour. I reccomend adding 2 circles instead of one, but I'm not sure.

AppleOptionEsc: Russ1642: I just invented a device which is going to revolutionize travel. What you do is create an object with a circular cross section. Has to be a circle. And in the middle of the circle you put a bearing and a shaft, which you can attach to whatever you want. I call it the HyperRound. I'm going to make a fortune.

Directions unclear: have been spinning in circles for an hour. I reccomend adding 2 circles instead of one, but I'm not sure.

I can't give further details until my patent is awarded.

Yeah, unfortunately, it took me taking Algebra 3 times before I found a decent instructor, and that was in college. It was a great class, the guy had set it up himself. It followed the same exact curriculum as Algebra 101, but we met twice as often. Anywhere there was a trick, he'd teach it to us. But most importantly, he explained what in the fark was going on. None of my 3 Algebra instructors had done so, they just told you to do them and went back to doing their own thing.

I don't remember any of the tricks anymore, but the guy was awesome. Unfortunately, for some reason, the credits don't transfer, so if I go back to school again, I have to take it all over.

Spectrum: PartTimeBuddha: Has anyone checked to see how it works with complex roots?

Will do, and thank you for that.

I need a tad more coffee first.

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