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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

#41

Post by sykick »

robinhouston wrote: post #6

This 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…

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 binomial and trinomial factoring.

To me it is obvious that the method in the article is far superior than teaching completing the square. I’m teaching a pre-calculus course this semester and many of my students still can’t complete the square. Pre-calculus is 3 math courses beyond elementary algebra.

All of this is just my opinion of course and I have no data or studies to back up my opinion. I will be using the method described in the article from now on in my elementary algebra courses.

Re: A new way to make quadratic equations easy

#42

Post by [deleted] »

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 = -…

[deleted]

Re: A new way to make quadratic equations easy

#43

Post by pflats »

banachtarski wrote: post #35
robinhouston wrote: post #6

This 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.

You are correct. It is generally expected to be part of an Algebra I curriculum (as per Common Core Appendix A's Traditional Pathway).

> Common Core High School: Algebra » Reasoning with Equations & Inequalities » Solve equations and inequalities in one variable. » 4 » a

> Use the method of completing the square to transform any quadratic equation in x into an equation of the form (x - p)^2 = q that has the same solutions. Derive the quadratic formula from this form.

http://www.corestandards.org/Math/Content/HSA/REI/B/4/a/

Re: A new way to make quadratic equations easy

#44

Post by jerf »

robinhouston wrote: post #6

This 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…

I believe the objection is that deriving the formula itself from completing the square is the challenge, not the direct use of completing the square as you show in your example. (Although I wonder where you learned that, because I was certainly never taught that so directly.) 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.

For a professional mathematician this is barely a warmup, of course, but for the average person in middle school or early high school, this is distinctly nontrivial work.

(I'm not taking a side here, just trying to describe what the issue is, since you asked. I'm ambivalent. Neither of my kids are quite this far yet, so I'm not quite here yet.)

Re: A new way to make quadratic equations easy

#45

Post by pflats »

sykick wrote: post #41
robinhouston wrote: post #6

This 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…

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…

> 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 teaching by the teacher. Completing the square has uses beyond proving the quadratic theorem; this does not.

I should say: I find this an incredibly cool and level-appropriate proof of the quadratic equation, but I think its merits as an improvement in pedagogy are dubious.

Re: A new way to make quadratic equations easy

#46

Post by leggomylibro »

banachtarski wrote: post #35
robinhouston wrote: post #6

This 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.

Depends. I took a mid-level calculus class recently because it had been awhile since I'd integrated anything, and the professor had to spend a day reviewing how to factor and divide polynomials when they got to integrals which used trig identities. "Completing the square" was a new phrase to most students.

Everyone knew the quadratic formula, but most people don't pay attention to how formulas are derived, much less remember that sort of thing a few years later. If you want to guess whether an average student knows something, ask yourself, "would it be on the test?"

Re: A new way to make quadratic equations easy

#47

Post by pflats »

jerf wrote: post #44
robinhouston wrote: post #6

This 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…

I believe the objection is that deriving the formula itself from completing the square is the challenge, not the direct use of completing the square as you show in your example. (Although I wonder where you learned that, because I was certainly never taught that so directly.) 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 n…

> 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 is not unintuitive.

As far as doing it fully generically, well, how else do you get a generic formula? When teaching this, we would do some completing the square to solve quadractics, and then tell our students:

"You know, this is annoying to have to complete the square EVERY TIME. What if we just decided to solve it the really hard way once, with A, B, and C in the equation instead of the numbers, and see what we get?"

Re: A new way to make quadratic equations easy

#48

Post by bikenaga »

The old (Baudhayana, Omar Khayyam) geometric approach to completing the square (and so solving [some] quadratics) is very nice because it's visual, and it's what I show students in many classes. It explains why it's "completing the square".

Suppose you want to complete the square for x^2 + 6 x. Represent this as an x-by-x square and a 6-by-x rectangle:

       x
     .....
  x  .....
     .....
     *****
     *****
     *****
  6  *****
     *****
     *****
Cut the 6 x rectangle into two 3-by-x rectangles:

       x
     .....
  x  .....
     .....

     *****
  3  *****
     *****
  
     *****
  3  *****
     *****
Move the lower 3-by-x rectangle up next to the square. The L-shaped figure still has area x^2 + 6 x.

       x      3
     .....  *****
  x  .....  *****
     .....  *****
  
     *****
  3  *****
     *****
What do you need to add (what is the size of the small square on the lower right) to complete the (large) square? The small square is 3-by-3, so it has area 9:

       x      3
     .....  *****
  x  .....  *****
     .....  *****
  
     *****  +---+
  3  *****  |   |
     *****  +---+
You get x^2 + 6 x + 9 = (x + 3)^2. If the original x^2 + 6 x was on one side of an equation, you add 9 to both sides.

Re: A new way to make quadratic equations easy

#49

Post by sykick »

pflats wrote: post #45
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…

> 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 numbers I took the average of I add and then subtract some number from the midpoint. The geometry makes this easier to explain over using completing the square.

I’ve seen a shocking number of calculus students struggle with completing the square. The merits of the approach in the article are entirely obvious to me but like everyone else I’ve had my share of obvious beliefs turn out to be false.

Re: A new way to make quadratic equations easy

#50

Post by 6gvONxR4sf7o »

sykick wrote: post #33
6gvONxR4sf7o wrote: post #30

This 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 st…

How aren't they equipped for it? Do you mean change of coordinates in its full generality? I'm talking about simple substitution of x' = x + k, connected with the intuition sketched below. They know how to perform simple substitution by then, and they can surely follow this logic:

    x       0---1---2---3-->
    x'  0---1---2---3---4-->

    x' = x + 1
    ==>
    x = x' - 1

    y = f(x)
    ==>
    y = f(x' - 1)
If you can talk about Napa being an hour north of San Francisco and San Francisco being an hour north of San Jose, this kind of student should be able to tell how far north of San Jose Napa is.

Can they not even do that? If not, I'd questioning why we teach them to solve quadratic equations before they can do substitution.

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