A new way to make quadratic equations easy • Hacker News Board

Re: A new way to make quadratic equations easy

#91

Post by abnry » Sat, Dec 07, 2019, 1:41 AM UTC

edflsafoiewq wrote: post #90

jfengel wrote: post #82

Earlier quoted context omitted.

I have never understood "completing the square". I mean, I get why and how it works, but it's completely unintuitive to me, or why you'd do it that way. I think it is a historical hangover. It makes more sense viewed geometrically, and the technique is attributed to the same al-Khwarizmi for whom algorithms are named. But it's less intuitive as part of algebra, and we focus a lot more on algebra today than in the med…

Algebraically, my intuition is that the x^2 + bx part of x^2 + bx + c = 0 "looks" pretty close to the expansion of (x+b)^2. If you relabel b so this becomes x^2 + 2b, it's even clearer. So you can try to fiddle with the constant term to make the LHS exactly that square, and you get completing the square. This is similar to how you solve a first order linear DE, y' + f(x) y = g(x). The idea is that the LHS "looks" lik…

It's not intuitive for someone new to mathematics, but as someone with two math degrees, it is just the strategy of "do stuff to something until something about it looks similar to something you've seen before" that mathematicians use ALL the time. The additional part of this strategy is to dream optimistically about how you can make the thing you are dealing with "nice" after doing stuff to it. In the case of completing the square, you are hoping you can just straight up take a square root. That would be easy. Turns out if you try enough stuff, you can.

Re: A new way to make quadratic equations easy

#92

Post by jacobolus » Sat, Dec 07, 2019, 6:23 AM UTC

carapace wrote: post #79

sykick wrote: post #75

Earlier quoted context omitted.

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 alrigh…

I think a big part is that many (most?) children find school – i.e. lectures, textbooks, homework exercises, exams – unmotivating/boring at best, and often extremely stressful/frightening, which means that they aren’t fully focused on it and can easily miss important details. Beyond that, schools often fail to provide meaningful feedback or support when people suffer serious misconceptions or are missing fundamental prerequisite knowledge/skills, which makes it easy for students to fall behind and have great difficulty recovering. Someone who spends the exact same amount of time on academic work but for whatever reason (external help outside school, internal motivation, some insightful introspection, ...) manages to pay closer attention, stay more relaxed, think about things ahead of where the class is expected to be, connect new learning to material learned before and build a better-connected mental map, etc. can end up pulling far ahead.

A whole lot of this has to do with level of preparation before ever arriving at school. Some kids read with their parents for hours every day from age 1–5+, learn to play a variety of strategic games (and games involving basic arithmetic practice), build structures or mechanisms or electronics, practice making art, work through books of logic puzzles, etc. Other kids are left alone and bored without learning materials at a reasonable level, plonked down in front of developmentally inappropriate or just badly produced TV, or handed over to unthoughtful video games.

Then consider how many kids and parents have serious problems at home, with confrontational or even abusive relationships. Pile on work stress, financial stress, poor diet or even hunger, poor sleep, environmental toxins, illness, etc.

Re: A new way to make quadratic equations easy

#93

Post by carapace » Sat, Dec 07, 2019, 4:52 PM UTC

jacobolus wrote: post #92

carapace wrote: post #79

Earlier quoted context omitted.

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 alrigh…

I think a big part is that many (most?) children find school – i.e. lectures, textbooks, homework exercises, exams – unmotivating/boring at best, and often extremely stressful/frightening, which means that they aren’t fully focused on it and can easily miss important details. Beyond that, schools often fail to provide meaningful feedback or support when people suffer serious misconceptions or are missing fundamental…

I couldn't agree more.

Re: A new way to make quadratic equations easy

#94

Post by bigred100 » Sat, Dec 07, 2019, 5:26 PM UTC

jfengel wrote: post #82

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 have never understood "completing the square". I mean, I get why and how it works, but it's completely unintuitive to me, or why you'd do it that way. I think it is a historical hangover. It makes more sense viewed geometrically, and the technique is attributed to the same al-Khwarizmi for whom algorithms are named. But it's less intuitive as part of algebra, and we focus a lot more on algebra today than in the med…

Once you get to quadratic forms you can use it again.

Re: A new way to make quadratic equations easy

#95

Post by hollandheese » Sat, Dec 07, 2019, 8:50 PM UTC

ivansavz wrote: post #88

That's very cool. To the best of my knowledge the only purpose of learning how to complete the square in high school math is to be able to understand the derivation of the quadratic formula, so if we can use this alternative proof then students don't need to know "completing the square" anymore. Good riddance! The only other application of "completing the square" I can think of is the trig substitution in calculus, s…

Nope, completing the square is primarily used to translate a standard quadratic function into a vertex form quadratic so it can be graphed using transformations instead of guessing.

It is an essential multi-use tool in precalculus. The thing we should dump is the quadratic formula.

Re: A new way to make quadratic equations easy

#96

Post by knolax » Sun, Dec 08, 2019, 1:14 PM UTC

pfedak wrote: post #87

knolax wrote: post #83

Earlier quoted context omitted.

Why is MIT technology review on the front page so often. It's literally a student newspaper with the expected quality of one.

Are you perhaps thinking of https://thetech.com/ ? I'm not sure in what sense the Technology Review is a student newspaper - to my knowledge (and checking a few bylines), the journalists are not students.

Oh wow I just read their about page and I stand corrected. I'm not sure if the fact that they're a (supposedly) hundred year old newspaper makes me feel any better about the quality of their articles though.

Re: A new way to make quadratic equations easy

#97

Post by Tiruneh » Wed, Dec 11, 2019, 11:23 PM UTC

I have read the paper and it gives a good educational context from which the quadratic equation may be looked at. I have also worked out simpler way to solve quadratic equations using function evaluation without much memorization of symbolic formula:

for the equation : f(x) = ax^2 +bx + c = 0

X = Z +- Sqrt [-f(Z)/a] ; Z = -b/2a

for the equation f(x) = x^2 +bx+c = 0 it is even a bit simpler:

X = Z +- Sqrt[ -f(z) ] where Z = -b/2.

Example 1: f(x) = 3x^2 -8x-35 = 0

Z = -b/2a = - (-8)/(2.3) = 4/3

F(Z) = -121/3

X = 4/3 + or - Sqrt ( -(1/3)*(-121/3) = 4/3 + or - 11/3 = {5, -7/3)

Example 2: (Simpler form): f(x) = x^2 - 4x+ 3 = 0

Z = -b/2 = -(-4)/2 = 2

f(Z) = 2^2 - 4.2 + 3 = -1

X = 2 + or - Sqrt( -(-1) = 2 +- 1 = { 3, 1}

For detail of this method, please see the following pre-print

https://www.researchgate.net/publication/337829551_A_simple_...

Re: A new way to make quadratic equations easy

#98

Post by pflats » Thu, Dec 12, 2019, 8:16 PM UTC

jerf wrote: post #51

pflats wrote: post #47

Earlier quoted context omitted.

> The problem with deriving the quadratic equation that way is that for most people, that is a lot of symbols to keep track of, and you have to not only do an unintuitive "completing the square" step but you have to unintuitively do it fully generically. "Unintuitive" depends entirely on your introduction to the topic. If you're already completing the square, using it to solve quadractic equations you cannot factor i…

I think you're speaking from the perspective of someone very casually comfortable with symbol manipulation. This does not describe the average middle school student. When I said "unintuitive", I was speaking from the perspective of an average middle school student, for whom this is all either at the edge of the ability, or, often, a bit past it, and for a non-trivial number of them, way past it. In high school, I was…

I teach mathematics and write math curriculum professionally, and have literally taught the lesson I describe above for the better part of a decade. I say "we" in that post because in more than half of those classes, I did not teach it alone, but in a co-taught inclusion class for special education and general education students together. None of the classes where I used this lesson was an "honors"/"accelerated"/"pre-AP"/etc. class.

Every single one of those students was able to derive the quadratic equation by completing the square. It was not easy for some of them, but every single one did it.

Your parenthetical also implies to me that you think that the "truly disadvantaged" have less "mathematical skill". I would encourage you to reflect on that.