Odd that this is being treated as a new discovery. I am a retired math teacher and at Piedmont High School in Northern California, we emphasized the geometry and symmetry of parabolas to understand quadratic equations and the quadratic formula beyond memorization. It helps students retain what they learn and uses multiple modes of thinking and communicating (verbal, geometric, numeric, & algebraic). We knew there were ancient antecedents but never thought it was news worthy, just pat of our job. I would guess this is likely true elsewhere as well. Nice to see an article about mathematics in print, but does not seem like the big deal that is portrayed.
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I'm kind of baffled... my son is in 8th grade math in California, and this approach (find two numbers that have a particular sum and a particular product) is embodied by the "diamond diagrams" that my son has been drilled on for literally *years*. Isn't this technique already a standard part of the California math curriculum?
Is it really difficult to remember the Quadratic equation for the student as highlighted in the article?. The power to remember the formula,chemical reactions and equations depends on one's interest and involvement in the subject. Quadratic equation isn't an equation that is difficult to recall
from our memory cell or its work with a parabola. Why don't the author think of an easy way to find a surface area of an asteroid?.
Appreciate the author's exploration of Babylonians method to solve the quadratic equation. The real challenge I look forward in the future is the equations for calculations in the field of anti gravity used by the architectural experts while construing 'hanging or floating pillars in temples of ancient and medieval India.
As every industrial application of the chemical reaction can't give the cent percent of its theoretical yield, application of every equation that gives perfect result on the paper has an infinitesimal 'left -over'. This "left-over" accelerates one's inquisitiveness for research and development in every field.
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I just watched Dr. Loh's video that was linked in the article (this one: https://www.poshenloh.com/quadraticdetail/). I actually watched it before finishing the article. Since so many criticisms I see here in the comments are about the parabola graphs, I would recommend watching Dr. Loh explain it all *without* the graphs.
1
BTW here is John Savage's article:
FACTORING QUADRATICS
John Savage
The Mathematics Teacher
Vol. 82, No. 1 (JANUARY 1989), pp. 35-36 (2 pages)
Published by: National Council of Teachers of Mathematics
1
I thought I was going to learn something. Perhaps I should not admit this, but I have no idea where you got the r and the s. Based on other people's comments, I would say this could have been much clearer and far more informative. Instead of Eureka there has been a big duh.
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Forget the plot. Sum of roots is b/a, so average of roots is b/2a. Roots are b/2a +u and b/2a - u. Product of roots is c/a, solve for u. Voila - the quadratic formula.
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People keep posting the claim that the cables of a suspension bridge follow a catenary curve.
So I did a Google Books search for "catenary bridge", and the first book* returned has a section titled "8.3.2. Catenary bridges", and that section is part of a section titled "8.3. Suspension Bridges".
So part of the confusion seems to be terminological.
The curve of the cable is extensively discussed in section "8.3.7. Theoretical understanding of suspension bridges".
Unfortunately, the discussion, which includes some history, is too complicated to summarize, so I will only note that both the catenary and the parabola are mentioned.
* "Bridge Engineering: A Global Perspective" by Leonardo Fernández Troyano (2003).
It seems to me that Dr Loh's method is a mnemonic to enable rapid "solution" of quadratic equations. It's not a proof. You need to "remember" 2 things, that the solutions are equidistant around x0=-b/2a and that the solutions (r and s in the article) obey rxs=c/a. These results are not proved in the article but are made plausible by the graphs (graphing is always a good idea). A proof similar to the logic of the article is to shift the origin of the x coordinate, rewrite the equation and choose the shift to eliminate the linear term. This would prove the quoted results. I must say I think it is great that an article on quadratic equations garnered so many comments. Let's have more!
This is SO much more complicated than the method we teach in community colleges, and not needed when lead coefficient is a=1.
It's called the ac method. (Students are advised to try guess and check first, but to switch to ac if they don't find a result quickly.)
Example: 15x^2 - x -2
ac = 15 x (-2 )= -30.
Factor of -30 that add to b =-1 are -6 and +5.
Splitting linear term gives (15x^2 - 6x) + (5x -2) = 3x(5x-2)+ 1(5x-2) = (5x - 2)(3x + 1).
Advantages: builds number sense (sadly lacking these days); reinforces factoring by grouping, need when expression has 4 terms; writing out all 4 terms when checking result demonstrates validity -i.e. matches the post-split 4 terms; and, of course, no guessing.
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Is it really so difficult to remember the quadratic formula? I think it's ingrained in my bones.
So much for the popular parabola, but let's not overdo it. The curve of suspension bridge cables is not described by a parabola, but rather by a catenary curve. If in doubt, look it up (https://www.google.com/search?q=catenary+curve&rlz=1C1CHBD_enUS803US804&oq=catenary&aqs=chrome.1.69i57j0l7.9718j1j7&sourceid=chrome&ie=UTF-8).
1
whatever happened to FOIL, first outside inside last? .. worked for me, a boomer, in about four seconds
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@Sparky FOIL works, but it can't always be used! For example, what about (4x^2 -2x + 1)( 3x + 5)? So, at some point, teachers need to explain how FOIL is a special case of distribution and practice with examples like the one above. And BTW, FOIL works only in English!
1
If I'm not mistaken, this is the method presented on some slide rules for solving quadratics. Remember slide rules?
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No clue.
1
Much ado about nothing. Use the quadratic formula and be done with it
Absolutely *nothing* new here.
Completing the square is a good idea, part of a good bag of tricks; it reduces the solving of a degree 2 polynomial to the solving of one where there is no term of degree 1, so X^2 = c, which is easy, real or complex, who cares. Going down one degree, the idea of completing a square, adapted of course, reduces the solving of a degree 1 polynomial to solving one with no term of degree 0, so X = 0, easy. Going up one degree, completing the cube reduces the solving of a degree 3 polynomial to the solving of one without a degree 2 term, so of the type X^3 + aX + b = 0, and Cardan kicks in. (Without this completing of the cube, it is absolutely hopeless to try to write down and/or remember a formula or algorithm for solving a cubic.)
Symmetry works for degree 2 ONLY. It is a cute trick, but a one trick pony. Completing the square is a 3 or 4 trick pony. It is also a very good example for young people of the idea of reducing a problem to a less complex one, then another ...
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I remember going to a back-t0-school night in another life and listening to my son's Math teacher saying that Math is really just a language and when you know the rules you can master it. That always stuck with me.
I think the method succeeds pedagogically, if it actually does, because it is different from the methods the students are already working with.
But it assumes a number of things without proof. They can be proved, of course, and are proved in analytic geometry, but not in Algebra I.
1. That the graph of a quadratic is symmetric about a line perpendicular to the x-axis. (It says “parabola” but you can use this approach without using any other properties of the parabola than the symmetry.) Many students come to believe that any such curve, like, say, y = x^4, is a parabola. Not true. That the trajectory of a projectile is a parabola, a result discovered by Galileo, requires knowing what a parabola actually is.
2. That the graph of the quadratic you’re solving intersects the x-axis at the two solutions. This is “obvious” to the mathematically informed, but news to algebra students who haven’t studied analytic geometry and may not even be aware that EVERY equation relating x and y has a graph.
3. That the sign of the coefficient of x in the parabola, that’s b, with the sign changed, so it’s -b, is the sum of the roots. Without that assumption, you can’t find the axis of symmetry by dividing -b by 2. (Also works only if the coefficient of x^2 is 1.)
4. That there are at most 2 real solutions.
1, 2, 3, and 4, can be derived from the quadratic formula, or from solving the equation by completing the square. They aren’t obvious.
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Maybe this approach could be another alternative for teaching math. Girls and women may better comprehend this approach. Teachers giving their students a choice of which approach they prefer, may be the answer. It would be interesting to know if females showed a preference of one way vs the other.
Interesting. I was taught this method in Austria many decades ago. I have before me a "Mathematik Repetitorium", dated 1951, which I had to buy in 1952 -r '52. Very useful, which is why I;ve kept it.
On page 39, it shows the formulas for r,s, and refers to Viete.
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It’s been a long time since I even thought about quadratic equations. I was good in math and learned to solve them without difficulties. And we also graphed paranoias. But I do not recall ever having been shown how a, b, and c in the equation relate to the placement of the parabola on the number line i.e. the width of the shape and the location of the center on the x axis. Graphical representations like this are wonderful. Thanks to both the professor and the ancients.
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In your quick review of a quadratic you allude to a suspension bridge as a parabola, but actually that is a catenary.
I could barely read through the examples. I know I'm not alone.
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So, how does the teacher explain to the student how to find the line of symmetry? Because without that, you cannot start this process. The sketch is always useful, but if the line of symmetry sits on an irrational value for x, it's not usually obvious from the student's sketch what value that is.
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"... how does the teacher explain to the student how to find the line of symmetry?"
Say that the axis of symmetry is the vertical line: x = -b / 2a.
And then give a few examples.
A proof would require calculus.
See, also, the vertex form of the parabola:
y = a(x-h)^2 + k
In that form, (h,k) is the location of the vertex of the parabola, and the vertical line, x = h, is the axis of symmetry.
@Robert It's the midpoint of the two solutions, or (r+s)/2. And we know r+s = –b. So it's –b/2 (or –b/2a for a not equal to 1). The axis symmetry is a visual reminder, not an essential part of the method.
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@Robert If you set up the two necessary equations for the two solutions of a quadratic equation, as this article lays out, and use simple algebra to solve them simultaneously, all you'd doing is getting and then calculating the quadratic formula. It's just much easier to memorize the formula and be done with it than to use this method or to complete the square. That is true whether the two solutions are real (different or identical) or are imaginary. I want advantage of sketching a parabola is that you can see in a graphic way why all this is correct, but if you going to that much trouble, you might actually see on the graph where the parabola crosses the x-axis and read the roots off the graph.
Its amazing to think how there's still so much to learn even with "basic" concepts that are taught early on. Its also scary to think how smart the people of the ancient world actually were. However, I think this method isn't necessarily the easiest but it does help give a sense of the actual math behind it if that makes sense. This would've been great to learn when I was in middle school but I think I'll stick to the quadratic formula or factoring.
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Correctness has frustrated editors for millennia.
Thank you for writing about this.
A clarification, though. The phrase "through the air" sent me in an unexpected direction.
If you had wanted to find the trajectory of a ball thrown by Patrick Mahomes to a certain point downfield with a certain peak height, you would have been solving a quadratic equation. The same would be true if you want to find the trajectory of a ball thrown with a given speed at a given angle or the distance it would travel along the ground to a receiver.
If I am not mistaken, if you wanted to estimate how far a ball travels along its path through the air, you would have to integrate, as described here: https://www.math.drexel.edu/~tolya/arc_length_x%5E2.pdf
You describe "completing the square" as though it's some deeply complicated process, but it's really basic - they teach it to 13 year olds here. And this process seems no easier, in fact it seems pretty convoluted. Completing the square or the formula are both much easier and simpler for quadratics that can't be easily factorised, and the description of factorising as "guessing" seems kind of ridiculous, it's not like you just randomly guess, the numbers you try are usually pretty obvious.
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@Zach It's not superhard math. Then again, lots of students stumble over this. Can you tell at a glance if 7x² – 35x + 54 is factorable? Unless you're taking an algebra test where the problems have been contrived to be factorable with integer solutions (in which case the factorization is contrived to be reasonably obvious), it's not obvious, and yes, you're guessing through the combinations of factors.
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Kenneth: "Can you tell at a glance if 7x² – 35x + 54 is factorable?"
That depends on what you mean by "at a glance".
The rigorous method is to look at the sign of the discriminant, D = b^2 - 4ac.
Thus, D = 35*35 - 4*7*54 = -287. That's negative, so the quadratic is not factorable.
Kenneth: "... it's not obvious, ..."
Admittedly, I had to use a calculator to figure that out, but some people could do that in their head.
Kenneth: "Unless you're taking an algebra test where the problems have been contrived to be factorable with integer solutions ..."
Some teachers allow calculators to be used during tests. However, contrived problems are very misleading, because they give the impression that all problems are that easy to solve.
This one definitely requires a calculator, or a trip to wolframalpha.com:
1.2 x^2 - pi x + sqrt(2)
And that's why calculators should be allowed during tests -- the student still has to understand how to apply the discriminant.
However, mathematically deriving the discriminant cannot be done with any calculator ...
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@Zach This method doesn't have to be taught in place of completing the square. I teach completing the square as well (but only with the area model - otherwise it becomes very abstract and too confusing for many of my students). I think this method and completing the square complement each other well. This method ties together the axis of symmetry and a visual understanding of the graph. Completing the square more directly relates the solution to factoring and multiplying. When you solve using both methods side-by-side, there are many similarities - as well there should be!
Next year, I plan to teach BOTH methods and ask students to choose which one works better for them. This can empower students and make the math more meaningful to them. Both methods, I believe, have advantages over the quadratic formula (which I won't ban students from using but won't emphasize either).
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Or... you can have WolframAlpha solve it for you.
I love brilliant people. They make life easy for the rest of us. I just wish instead of algebra and physics they offered tax preparation in high school.
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Maybe this is easier for a 15 year old taking a math quiz. (Personally, I'm skeptical.)
Anyone putting this to practical use needs to know the arithmetic equation. No one wants their apps slowed down by those steps.
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totally lost....and Ihad so much math in college and grad school
The happiest day of my life was when I passed Freshman Math and knew I would never have to take it again. This article confirmed the feeling 40 years later.
Look, Mom, no STEM— and I still had a rewarding career!
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We do not need to teach 'amazing tricks.' We need to teach concepts and foster understanding. Teaching multiple approaches to look at problems and how they relate to one another arm students with the tools they need to attack more difficult problems.
The best teachers know and practice this.
The rest teach amazing tricks.
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"Figuring out the factors that work is essentially trial and error. “The fact that you suddenly have to switch into a guessing mode makes you feel like maybe math is confusing or not systematic,” Dr. Loh said."
Having completed two and half years of college level math, up through advanced engineering math, I can attest that math is indeed both confusing and not systematic. There are many problems that cannot be solved by following a rote algorithm, but require a flash of insight, false starts, dead ends, and trial and error stabs in the dark that may or may not lead to a solution.
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The kids at my school preferred guess and try over formal techniques. It was quicker and required no further writing.
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This method is more or less the same as the quadratic formula.
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Why do high school students have to learn this kind of math anyway? It was torture for me in high school, I never understood it no matter how hard you explained it, and I'm delighted that it has no real use in real life, unless you're a mathematician. This kind of math should be an elective in high school, not obligatory. I've happily gone through life without knowing how to solve a quadratic equation!
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The zeros of quadratic equations give us the solutions of differential equations that describe the motion of a spring and the current of an electric circuit. Those are real life applications; they are a all around us, and they are beautiful.
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@Andrew So happy you enjoy it, there you go it's an elective YOU can choose! Should not be forced on every student.
The real title of this article should be "The Death of Critical Thinking Skills in American Education".
So a math professor figured out how to solve quadratics by constructing a highly stylized example, basically simplifying the Quadratic Formula applied to that example, and demonstrating how it works by graphing a parabola.
What an astonishing accomplishment. As if any decent math teacher demonstrating how completing the square works, or doing a good job of explaining the proof of the Quadratic Formula, hasn't been doing essentially the same thing for who knows how far back.
Maybe if we weren't allowing our politicians to sabotage education and monopolize educators' time to focus on rote obedience in students instead of critical thinking skills, something like this wouldn't be hailed as so rare it's worth publishing in a peer-reviewed academic journal, and then the NY Times.
This is utterly shameful.
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I don't think it's any simpler or clearer, when you are writing it more formally.
Usual solution by completing the square:
$$ax^2+bx+c=a\left[(x+\frac{b}{2a})^2+\frac{4ac-b^2}{4a^2}\left]$$
"new trick":
$$r=-\frac{b}{2a}-u$$
$$s=-\frac{b}{2a}+u$$
$$rs=\frac{c}{a}=(-\frac{b}{2a}-u)(-\frac{b}{2a}+u)=\frac{b^2}{4a^2}-u^2$$
Then
$$u^2=\frac{b^2-4ac}{4a^2}$$
Which is similar to the formula above.
Doesn't look simpler to me, and you have also to prove that the axis of symmetry of the parabola is at $x=-\frac{b}{2a}$, or even that there is an axis of symmetry.
Actually this method requires much more work.
On the other with the usual method the symmetry is an immediate consequence, and you get the coordinates of the vertex as well.
There may be a good reason this "amazing trick" is not used in teaching, after all.
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"Doesn't look simpler to me, ..."
The notation you posted is not simple in any case. Please suggest an online interpreter or post a notation that is readable in a Times comment.
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The quadratic equation was the beginning of the end for me in learning math. I could see the practical uses of the solving of algebraic equations that came before it, and some homework and exam problems were stated in real-life terms--you know, the old two-trains-hurtling-towards-one another and such.
But I was taught the quadratic equation in a very rote, just-memorize-it-and-do-it fashion as well as much of what followed. I knew we were working with parabolas but remember hearing nothing about suspension bridges, thrown footballs, or other real-life applications. That would have helped me a lot. Maybe all math teachers work that way now, but not back when I was in high school.
On the plus side, though I couldn't have reproduced this myself from memory, I do recall going through this material before. It just didn't engage me then.
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"Maybe all math teachers work that way now [by giving examples] ..."
I don't know about math teachers, but introductory math books are filled with examples. Check your library for books on pre-calculus, such as:
"Pre-calculus for dummies" by Mary Jane Sterling.
"Precalculus demystified" by Rhonda Huettenmueller.
I somehow passed both algebra and calculus in school (by the skin of my teeth, may I say), yet I have NEVER once needed any of it in my real life or career, not even for a millisecond.
OTOH, I have *very* much needed knowledge of human anatomy and physiology, which are utterly ignored by school systems far and wide - even though THAT information is germane to every single person alive!
Other than people who are going into STEM-related careers, schools would better serve their students by making human anatomy & physiology a requirement, and putting algebra and calculus off to the side as optional extras for those who are interested in those topics.
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@L: Human anatomy and physiology was taught in my high school as an alternative class for those like me who were not taking physics, and I was surprised to love it, do well in it, and still remember and use some of it. I also still use basic algebra, but not trig and not much if any algebra that came after the quadratic equation. Didn't try calculus. Parts of chemistry are very important in understanding human anatomy.
1
When I was a junior in high school a long time ago, my math teacher taught us this trick. She focussed on the axis of symmetry, helping us to understand that perception of the problem is as important as thinking on the problem. Her goal was not just mathematical, but creative: helping us to keep fresh eyes and open minds, to reframe the question, and develop our own novel solutions. This were important as lessons to take with us in life. And. we learned this "trick" as a bonus.
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Watching Professor Loh explaining his method put a big smile on my face. I will let others argue the benefits and deficiencies (whatever those might be) of his method. I, for one, am just overjoyed to see his passion for the process of discovery and communication. Thank you, Dr. Loh.
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What is described here is in fact a simple linear transfer of the variable X. Blew are the steps leading to the proposed approach, using the terminology familiar to mathematicians.
1) Set X=mU+n;
2) Choose n to vanish the linear term in the new equation;
3) Select m such that U=+1 and -1;
4) Calculate X from X=n+m and X=n-m.
"Choose n ..."
"Select m ..."
That sounds like guessing is required, but guessing is exactly what Dr. Loh wants to avoid:
'Figuring out the factors that work is essentially trial and error. “The fact that you suddenly have to switch into a guessing mode makes you feel like maybe math is confusing or not systematic,” Dr. Loh said.'
My only D in school was pre-calc my senior year. I was never so confused in my whole life, and I distinctly remember this equation coming up in the middle of nowere numerous times.
I have no idea how, or why, anyone would use this, but I'm truly grateful to those that do.
interesting but not sure it is easier than plugging into the standard quadratic formula.
You know, I cared little when I was mastering algebra and calculus as a student in public school and college, and now at the age of 74 I care not a whit!
I still, however, delight in the way parabolas and other equation graphs look. Here mathematics is invested with elegance, simplicity, and purity.
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Sometimes the solution to a quadratic equation is a complex number. The methodolgy seems preclude that possibility.
3
@Jp It doesn't. The sketching of parabola helps less in that case, but that's just meant as a visual aid to remind you how to recast the variable from x to u.
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The step u^2=9 gives u=3 and u=-3.
If it were u^2=-9, then we would get u=3i and u=-3i.
@Jp Agreed. If the parabola doesn't intersect the x-axis, there is no geometric solution on the RxR cartesian plane. Complex roots (solutions) cannot be found that way.
Both methods appear in Elements of Algebra, by Euler, paragraphs 638-645. The moral is to read books "by the masters, rather than by the pupils".
I myself learned the "new method" from writings of Lagrange. Here is his explanation:
in a quadratic equation x^2 - bx + c = 0, with roots r and s, the basic fact is that b is the sum of the roots, and c is their product, i.e. b = r+s and c = rs. Now one observes that if only one knew also the difference of the roots, i.e. if one knew r-s, then one could add that to b and get 2r, or subtract it from b and get 2s, which would solve for r and s.
There is in fact a nice relation between the sum, difference, and product of two numbers, involving their squares,
i.e. one always has (r+s)^2 - 4rs = (r-s)^2, or in our case, b^2 - 4c = (r-s)^2. So taking the square root of (b^2-4c), does give us r-s, hence:
2r = (r+s) + (r-s) = b + sqrt(b^2-4c), and 2s = (r+s) - (r-s) = b - sqrt(b^2-4c).
In the present article, the focus is on the difference between the roots and their average, which is (r-s)/2, (this is Euler's y, where x = y +b/2).
In both cases the key is essentially to search for r-s, since finding it reduces to taking a square root, or as Euler puts it, reducing a "mixed" quadratic equation such as x^2 - bx+c = 0, to a "pure" one, such as y^2 = d.
The boost from the version here I think is the geometric picture, helping the student see geometrically why finding r-s is useful. Thanks for the insight!
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Could you cite a specific source for the Lagrange method?
Gail paraphrasing Lagrange: "the basic fact is that b is the sum of the roots, and c is their product, i.e. b = r+s and c = rs."
Because:
(x-r)(x-s) =
x^2 -rx -sx + rs =
x^2 -(r+s)x +rs
Note that Lagrange uses "-b" for the linear coefficient.
@GAK Lectures on elementary mathematics, Lagrange, p. 47-48:
https://www.gutenberg.org/files/36640/36640-pdf.pdf
(which he attributes to Diophantus):
Notice Lagrange changes the meaning of x, using it also for(r-s) in my notation.
“ ….To find two numbers the sum and the product of which are given.
If we call the sum a and the product b we have at once, by the theory of equations, the equation x^2 −ax+b=0.
Diophantus resolves this problem in the following manner. The sum of the two numbers being given, he seeks their difference, and takes the latter as the unknown quantity. ……Calling the given sum a, the unknown difference x, one of the numbers will be (a+x)/2 and the other will be (a−x)/2. Multiplying these together we have (a^2-x^2)/4 . The term containing x is here eliminated, and equating the quantity last obtained to the given product, we have the simple equation (a^2 − x^2)/4 = b, from which we obtain x^2 = a^2 − 4b, and from the latter x = sqrt(a^2 − 4b)."
Thus for Lagrange, and Diophantus, the unknown is r-s, whereas for Dr. Loh it is (r-s)/2.
1
@GAK Lectures on elementary mathematics, Lagrange, p. 47-48:
https://www.gutenberg.org/files/36640/36640-pdf.pdf
(which he attributes to Diophantus):
Notice Lagrange changes the meaning of x, using it also for(r-s) in my notation.
“ ….To find two numbers the sum and the product of which are given.
If we call the sum a and the product b we have at once, by the theory of equations, the equation x^2 −ax+b=0.
Diophantus resolves this problem in the following manner. The sum of the two numbers being given, he seeks their difference, and takes the latter as the unknown quantity. ……Calling the given sum a, the unknown difference x, one of the numbers will be (a+x)/2 and the other will be (a−x)/2. Multiplying these together we have (a^2-x^2)/4 . The term containing x is here eliminated, and equating the quantity last obtained to the given product, we have the simple equation (a^2 − x^2)/4 = b, from which we obtain x^2 = a^2 − 4b, and from the latter x = sqrt(a^2 − 4b)."
Thus for Lagrange, and Diophantus, the unknown is r-s, whereas for Dr. Loh it is (r-s)/2.
Note also the essential formula, expressing a product as a difference of two squares, occurs geometrically as Prop. 5, Book II of Euclid, which is cited also by Girolamo Cardano in Ars Magna.
1
Answering the question of estimating how far Mahomes football traveled thought the air requires a few more teachable steps. Patrick cooperated by using his immense talents to throw a football that follows the quadratic of this example. The vertex lies on the center of symmetry at x = 2. The height of the football at that location is determined by inserting 2 for x in y = x^2 -4x -5, yielding y = -9. (Making sense of this requires negative values to be height above ground). The distance traveled through the air requires calculating the length of 2 sides of an isosceles triangle. The length from (-1,0) to (2,-9) is sqrt(3^2 +(-9)^2) which is sqrt(90). The total length through the air is twice this which is approximately 19. We can assume the units are yards. The ground separation is a forward pass of 6 yards but it traveled 19 yards through the air to get there.
A more accurate determination of how far that football travels through the air is determined by calculating the arc length of a quadratic. This requires an integral and can be evaluated to a closed form expression involving natural logarithms. An AP calculus course might touch on this subject.
I looked up most of the literature to which I have access, looking for ancient mathematicians working on quadratic equations. Surprisingly, I could not find any reference to Babylonian mathematicians.
The oldest references I could find to this subject were the works of Persian (that is, Iranian) mathematicians such as Omar Khayyam - mostly known for his poetry masterpiece (Rubáiyát of Khyyam) - and Muhammad ibn Musa al-Khwarizmi - known for his book al-Jabr (Algebra) and the word Algorithm.
Curiously, some references identify both these mathematicians as Iraqis. That cannot be correct for Iraq did not exist as a country until the 20th century (when the British created it by partitioning the Ottoman Empire). In contrast, both these mathematicians were living in or before the 12th century. Also, Omar Khayyam was born in Neyshabur and Muhammad ibn Musa al-Khwarizmi was undoubtedly born Khwarazm; cities that were once (and they may still be) part of Persia.
What seems to be the cause of this "confusion" is the current animosity between the US government and the regime in Tehran. This is not the first time when the US, after designating a country as "enemy", has tried to wipe out from the literature contributions of that country to humanity and world civilization. During the cold war, much of the scientific work done in Russia (Soviet Union) were re-introduced in the US as American findings, years after Russian (Soviet) mathematicians and physicists discovered them.
1
"... I could not find any reference to Babylonian mathematicians."
There is a link in the article that leads to Dr. Loh's paper at arxiv.org. In the first paragraph, he cites:
Katz, Victor J. (2008). "A History of Mathematics", 3rd ed.
Robson, Eleanor. (2008). "Mathematics in Ancient Iraq: A Social History".
Other sources are cited throughout the paper. Do a text search for "babyl" in the PDF version of the paper to find them.
It's not the destination, it's the journey that makes the trip worthwhile. I have taught mathematics (grades 5 through 16) for over 40 years. I have had wonderful conversations with my algebra and calculus students while looking at the quadratic formula. There would be a lot lost had I not spent the time to prepare them for the conversation. The conversation always ends with, "See, there really is no reason to have to memorize a formula."
2
Hello. Your example, though accurate is simplistic. The Quadratic Formula generally gives students a lot of trouble because there is A LOT of algebra involved which means a lot of chances for students to make mistakes. On the NY state exam, on the reference sheet, the Quadratic Formula is one of the formulas given so there is now no need for students to memorize this though most learn in middle school via a song. I learnt this relationship/method as the Theorem of Vieta which I show my students Alg1. Students in Alg2 & higher math courses can begin to have an appreciation of this method which is better when the leading coefficient, a, is no longer positive 1. So "Amazing" yes, will I change how I teach solving quadratic equations and adapt this as my go to method, not any time soon.
1
If a student is willing to learn, she can certainly learn how to derive and use the quadratic equation formula. On the other hand, if a student is unwilling to learn, he cannot learn anything no matter how simple it is.
1
As a high school math teacher, this "solving method" and the authors' description of it as "amazing" bothers me for several reasons. First, it appears the trick trades memorizing one thing (the quadratic formula) for memorizing another thing (a formula for finding the x coordinate of the vertex). And things get messy very quickly with this trick if the coefficient of x^2 isn't one or if the coefficient of x is an odd number. The quadratic formula was a great achievement in elementary algebra, and memorizing it isn't hard. (If you think it is, I can teach you a few funny songs or poems to help you.) I also feel this article makes "guess and check" sound evil, which it most assuredly isn't at the middle and high school level. These types of puzzle problems (two numbers that add to something and multiply to some thing else) can be a great way to help kids develop their numeracy skills, which are tremendously important in life. In this case, kids can use factoring to come up with the two numbers that add to the coefficient of x and multiply to the constant term. Effective teaching at the middle and high school level involves helping kids understand the value of productive struggle. Or as the standards for mathematical practice say, we are there to help students "make sense of problems and persevere in solving them."
21
@Mike M As a math teacher, you'll appreciate Bobson Wong's experience of teaching quadratics using this approach: http://bobsonwong.com/blog/27-on-a-different-method-for-solving-quadratic-equations
3
@Mike M
Having forgotten most of the details and variations on the use of the quadratic equation, I wonder the following: Is there an analytic geometry equivalent of what appears to be a 2D graph solution for the case of 3D? Would that be used to model the path of rain falling on an umbrella? Would the velocity of flow of rain introduce multiple solutions up to the point of producing a true parabolic conic? And would it require that the shape of the umbrella be a true parabolic conic also? What about more than 3 dimensions? Sigh ... so much to learn, so little time.
@Mike M Many of us found Math and instructors of your ilk ("Effective teaching at the middle and high school level involves helping kids understand the value of productive struggle."), frustrating ciphers. If you aim to have students become mathematically literate, embracing innovations that simplify the intricacies of opaque equations should be your priority.
2
As far as technique it is neither just completing the square or factoring, as some people seem to be saying, but rather borrows elementary relationships used by both. It really is the rewriting of the problem in U that is the difference here, at least in relation to U.S. textbooks, which does seem to intuitively connect the nature of the roots to what you are doing in moving those symbols around. Just another elementary relationship? Sure. But also not the point.
So much of studying elementary Algebra and Calculus is just that, moving symbols around. There aren't too many opportunities to really understand what one is really doing or showing. It happens here and there, but you just have to spend so much time practicing moving the symbols around (and you really do) that there isn't all that much time for navel gazing. It's great when techniques themselves actually lend intuitive understanding, as when learning about the properties of e.
Also, rewriting expressions in terms of new relationships not explicitly accounted for in an original equation is a technique students must eventually become comfortable with in future coursework, so this is also in some little way a nice introduction to thinking algebraically along those lines.
Finding innovative solutions it’s always interesting...but to solve a quadratic equations the solution here proposed is: to solve a second quadratic equation. Why not to solve the first one? Even because there’s not guarantee that the second one is more simple..
1
I do not believe that this was something all that new. Students in Russia learn and solve Vieta’s formula. It is a faster and easier way of solving the quadratic formula. For example x^2 2x 3
x1 x2=-b and x1 times x2=c
so, x=-3, 1
2
The real news here is that the quadratic equation has a practical purpose beyond bedeviling Algebra I students. Had somebody told me that I could use this to predict where Bart Starr's pass would fall I might have paid more attention.
21
This really does not deserve the relevance it is getting. I was recently going through quadratic equations and parabolas with my son and I came up with this "new" method during our chat. One just need to be sufficiently literate in math. To use this technique, one needs to know out to find the location of the minimum/maximum of the parabola which is not discussed at all here (the result, the "axis of symmetry", is simply stated). "Completing the square" is not any more convoluted than this. In fact, it is technically simpler. This derivation is pedagogically great but it was "discovered" many times before and it will be rediscovered many more times in the future.
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@Simone Maybe (probably?) it has been discovered many times before, but it seems that almost no one wrote it down or added it to a textbook, which seems odd if it is a cleaner, better explanation.
18
@Simone Good for you that you came up with this on your own. But not all students have you to help them with their math homework or are “sufficiently literate in math” like you. If this article shines a light on this technique and encourages more teachers to present this as an alternative that may be easier for students to grasp, then this article is relevant. Moreover, that you rather than your son, came up with this technique exemplifies why this should be taught in schools. If we are to provide equal education to everyone (not just children with educated parents who can help them with their school work), then SCHOOLS need to provide this education AT SCHOOL.
7
@Simone I agree. I have been a secondary math teacher (including in the International Baccalaureate) and have kids discover this on their own. Also, factoring when y = o is NOT guess and check - it involves knowing factoring. The graphing method is great but graphs tend to turn kids off. Quadratics is a great topic but needs a discovery approach - and definitely not the CONIC SECTION way which I was taught 40 years ago.
5
does the method also work if the roots are complex numbers?
@Ralph M Yes. The graphical bit about sketching a parabola isn't as useful since the parabola would not intersect the x-axis. But that is meant as a reminder about how to recast the variables and is not an essential part of the method.
1
My friend who teaches actuarial science at University of Connecticut tells me that the quadratic formula can be sung to the tune of either the Notre Dame Fight Song or Pop Goes the Weasel. And he demonstrates both each term.
2
I did not enjoy the way this story was written, or the way the reporter responded to comments. I do not need the cultural references or the breathless hype, or the argumentative tone of the reporter's responses to comments. The reader is always right, even when he is wrong.
I would rather see a clear, simple, well-written explanation of the technique. Maybe that is unrealistic (I am not being sarcastic here), but that is how I feel about it.
I will spend time trying to work this technique out, but I doubt that the exercise will be useful to me in any way.
Yes, I am angry that The New York Times will not publish a straight, low-key story about high-school mathematics that might be useful to me in some way.
"Maybe that is unrealistic ..."
It's a newspaper article, not a textbook, so maybe you are being "unrealistic".
"... a straight, low-key story about high-school mathematics that might be useful to me in some way."
The "story" is that an enthusiastic math teacher is excited about a new way to explain an elementary topic in math.
If you want less "story" and more "straightness", check your library for some books on pre-calculus. See, for example:
"Pre-calculus for dummies" by Mary Jane Sterling.
"Precalculus demystified" by Rhonda Huettenmueller.
Interesting article, which I almost skipped over because the title makes it sound like clickbait.
1
Nice! I like it. I wrote a blog post about it. Comments welcome. http://blog.passionatelycurious.com/2020/02/06/that-new-quadratic-equation-solving-technique/
We are delusional if we assume history has been defined solely by progress. We often underestimate our ancient ancestors. This is a fabulous example! As is the Antikythera Mechanism, circa 100 C.E.
I found this article amusing. I once quipped that after Sputnik if a student could solve a quadratic equation without looking up the formula in a book, he or she was scholarship material. If you want to get excited about elementary math, show your students how Charles Babbage evaluated any polynomial with no operations other that additions. The machine he designed in 1744 could calculate the roots of a seventh order polynomial with 31 digit accuracy.
1
"... Charles Babbage evaluated any polynomial with no operations other that additions ..."
You are referring to Babbage's "difference engine", which evaluated polynomials *numerically*.
"The machine he designed in 1744 could calculate the roots of a seventh order polynomial with 31 digit accuracy."
That is not the same as finding an algebraic expression that solves polynomial equations.
A math teacher should explain that distinction and why numerical methods are essential for *approximating* the roots of polynomials of degree greater than 4.
Specifically, there are no algebraic formulas for polynomials of degree greater than 4. See the "Abel–Ruffini theorem".
@Anon Don't know about your claim, but Babbage's "difference engine" work was circa 1840 ; he was born in 1791, per Wikipedia.
@Kevin H. I think he meant 1844 and had a typo.
Great example of thinking out of the box. I used to tell my math students (grades 1-12) there is more than one way to get to the right answer. As long as it is a correct method, use what works for you.
Many schools, especially elementary schools, teach only one method and if the student deviates to another method that is correct he/she is penalized for not using the singular method. What an awful way to turn kids off from math and to prevent parents from helping their children with math.
3
Why do we teach completing the square? It is rote and prescriptive, and it doesn't increase understanding of mathematics, honestly. Teach the geometry of the parabola and introduce the vertex form, which can be related visually to a parabola. Desmos is a great tool for this and can keep students engaged while learning. I have taught math at the high school level, and I don't see anything novel in the method described in this article. It feels like attention-seeking.
Of course, this method only works for parabolas with a vertical axis of symmetry. What do you say when some curious student asks about a "leaning" parabola? It happens.
'What do you say when some curious student asks about a "leaning" parabola? It happens.'
1. A "leaning" parabola is not a function of one variable.
2. A smooth, planar curve can be described analytically by a function of the form f(x,y) = 0 or as a parametric curve.
3. A rotation of the curve or the axes can be used to transform the curve so that its analytical form is simpler and so that an axis of symmetry is parallel to the y-axis.
Most of that could be introduced in one lecture with a few diagrams.
A second lecture could introduce Bézier curves and splines, which were originally used to describe the shapes of auto bodies. That would nicely answer the "What is all this stuff good for?" question.
Correcting myself:
"2. ... _an equation_ of the form f(x,y) = 0 ..."
I’m all for awesome new ways of teaching, but I’m honestly puzzled why we teach the quadratic equation at all. Even when I was an engineer I never used it, and the vast majority of students don’t need to know it. I say we focus on communication or writing skills instead.
1
"Even when I was an engineer I never used it, ..."
What math or software DID you use?
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@Peter "why we teach the quadratic equation at all": Actually, I think that that is a good question, and I should say that I studied electrical engineering and physics, and I've used the traditional quadratic solution equation on many occasions. One might ask, why did my HS English teachers assign me readings in Shakespeare, and Greek/Roman mythology? Perhaps to expose me to something I wouldn't have given a 2nd glance to, when I was 14 or 15, and who knew then, I might've gone into law (or writing for Monty Python?). Some of use actually enjoyed toying with quadratics, as some others enjoyed medieval English literature....
I am from Jakarta, and I also experienced the same hurdle to solve this basic algebra problem. It was always guessing game of the r and s. I was wondering "Isn't there any more exact method in Math to find these r and s?
Thank you Mr. Loh for making highschooler's works everywhere a little bit much easier!
I discovered this method on my own (one year ago) when I was preparing for university exam in Turkey and I think many of students either discovered or were taught.
If guessing 2 factors is easy, this method doesn't help.
But where guessing is difficult, does it work?
Solve 6x^2 +19x−7=0
I almost fainted trying above method :(
2
Seventeen years ago, an eighth-grade student of mine named Juaton Boutte discovered the germ of this idea. I helped him flesh out the details. I then used it continuously in my teaching and presented it to teachers at conferences and teacher in-service events. As example, I had 50 teachers attend a workshop at Asilomar, California in 2016 and prior to that I taught the method at a weeklong course for teachers at the Phillips Exeter Academy. In our version, we compare a rectangle and square with the same perimeter. This method led to connections to other ancient topics like the harmonic mean, Pythagorean triangles, and hand-calculating square roots. Of course, I learned early on about the Babylonian use of this method.
1
This kept me thinking about how the school program needs to change. I spent a good deal of time mastering quadratic equations in high school, got an A, and... never used them ever since. In my opinion, it was a waste of time for most students (unless you plan to become a statistician or something).
This method has been taught in other countries for ages and as the dude said was used by Kharazmi (the Persian mathematician who we got algorithm and algebra from). I however never considered it easier than guessing the sum and multiplication although I was taught to use it if guessing was impossible. I've taught math for years and the problem is not having easy formulas so people don't need to guess as he claims. The real problem, especially among US students, is understanding why the method works.
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@Panda Al-Khwarizmi came up with "completing the square" — that is, the standard method of solving quadratics and deriving the quadratic formula. That's an amazing achievement, but it's not exactly what Dr. Loh is describing here. https://www.maa.org/press/periodicals/convergence/completing-the-square-quadratics-using-addition
Dr. Loh's comparison between "completing the square" and his method: https://www.poshenloh.com/quadraticcompsqr/
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Omar Khayam used these methods to solve 2nd degree equations.
His method to solve 3rd degree equations with similar methods is fascinating. I think it would have been interesting for readers if you’d delved some more into the history.
Edit: Omar Khayam used these methods to solve 2nd degree equations. His method to solve 3rd degree equations with similar methods (comic sections) is fascinating. Would have loved to see more of the history in the story.
I am a visual learner in many circumstances. I did horribly in Algebra because I couldn’t see/get what was going on. The opposite was true in Geometry where I did just fine. I’ve always felt if there was a “visual” way for me to work through a problem I would have been much more successful. Perhaps this is a method. As it was my Algebra teacher told me she’d pass me if I promised I would never take match again. I gratefully promised.
Nice story, but for me using the standard quadratic formula is much easier (I lost it halfway through when he started talking about "u").
Excellent article - thank you for bringing this to widespread attention!
1
It may have been used by the babylonians but it's all greek to me.
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I must confess I didn’t get a lot of the reasoning at first read, (I last did quadratic equations about 50 years ago in an English grammar school, and I struggled then to fully understand.)
But I think I’m getting it now, thanks to extra reading, the popular “YouTube” program and a fantastic set of replies by the one of the authors of this article.
Many thanks!
1
I appreciate this article very much, and Mr. Chang's answers to all the questions posted by readers.
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Very shaggy math story.
First, some appreciation for the NYT for having readers who'll read about quadratic equations and a writer who makes them engaging.
Even more appreciation to the math teachers reading this and participating in the discussion: you guys (all-gender) have a really important job and our society doesn't pay enough attention to it.
It's hard to think of anything more important than teaching kids math and science. After college, LBJ found himself teaching all subjects in one of the poorest schools along the Mexican border. His most potent argument to parents about keeping their seven- and eight- year-olds in school: a man who can't check his boss's math on the paychecks isn't much further than a slave.
3
Gee, I learned a different way.
Multiply “a” by “c’
Completely factor that number to find all the numbers that when multiplied give you a*c
Look for a combination of those factors that when added yield “b”
The next step, although easy is hard to write in prose. So I give an example:
2
X. 4x - 21
a = 1
c = -21
b = 4
So a*c = -21
Factors of -21 are 3 and 7 with one of these numbers being negative. Find which combination gives you 4:
-3 and 7 when added together = 4 which is “b”
Rewrite the equation as:
2
x - 3x 7x -21
Rewrite this as:
x(x - 3) 7(x - 3)
2
Therefore (x 7)(x - 3) = x 4x - 21
2
So glad there are people out there who can argue about this.
2
I think this article would have benefitted from a real life example showing how the quadratic equation can be used to solve problems rather than just mentioning it in the beginning and then using numbers and figures to illustrate how it works.
As someone who did my share of math including 1.5 years of calculus in college, I have to say I use very little of it these days other than simple arithmetic. What would be more useful to most people is learning statistics (which is more conceptual than mathematical for basic stats).
In a way, I'm not surprised kids and so many people are put off by math. My stats teacher was superb and would invited weekly someone from the community who used stats in their work ranging from economists to historians to scientists to sociologists to talk to our introductory college class.
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@ms Amen.
Two things that should be required in High School are elementary probability/statistics and personal finance.
An informed electorate must understand probability and statistics to make informed voting decisions. People also need to understand probability and statistics to make rational decisions in their day-to-day lives.
The lack of understanding of personal finance in the general populace is astounding. Even simple compound interest seems to be above most people's heads.
If High School graduates understood statistics and personal finance better, perhaps they would be less likely to take out crippling student loans to get degrees that have almost no value in the market (beyond "being a college degree" - i.e., "the new High School diploma").
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@Bob Richards I am totally with you up until your last paragraph. First, I don’t think a poor understanding of math or personal finance is why some individuals choose degrees in fields that don’t pay well. More likely, they were just uninformed (or misinformed, sometimes intentionally by unscrupulous for-profit schools) about their real job prospects. These are smart people who just don’t know their way in the world yet.
Second, and more importantly, while I agree students need to go into their choice with their eyes open and knowing what they’re getting into, I don’t think the right solution ultimately is for people to stop pursuing degrees in fields that have poor market value, but for our society to perhaps better recognize the intrinsic value of knowledge and expertise to the strength and richness of civilization; we need to find a way to pay them better. We’d all be much poorer for a society that no longer had domain diversity. It’s easy to mock someone with a big student loan for an art history degree, but where would we be without educators, social workers, researchers, historians, linguists, sociologists, philosphers, the arts, and a hundred other occupations I’m sure I don’t even know exist, but which make up the workings of advanced civilization?
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@Bob Richards I am totally with you up until your last paragraph. First, I don’t think a poor understanding of math or personal finance is why some individuals choose degrees in fields that don’t pay well. More likely, they were just uninformed (or misinformed, sometimes intentionally by unscrupulous for-profit schools) about their real job prospects. These are smart people who just don’t know their way in the world yet.
Second, and more importantly, while I agree students need to go into their choice with their eyes open and knowing what they’re getting into, I don’t think the right solution ultimately is for people to stop pursuing degrees in fields that have poor market value, but for our society to perhaps better recognize the intrinsic value of knowledge and expertise to the strength and richness of civilization; we need to find a way to pay them better or make them more affordable. We’d all be much poorer for a society that no longer had domain diversity. It’s easy to mock someone with a big student loan for an art history degree, but where would we be without educators, social workers, researchers, historians, linguists, sociologists, philosphers, the arts, and a hundred other occupations I’m sure I don’t even know exist, but which make up the workings of advanced civilization?
As a mathematicians, I would love to see new teaching methods that make these basic topics accessible to more students. However, the article here has a bizarre insistence on crediting a professor with "discovering" this method, which is not a mathematical "discovery" in any sense. Similarly, the article talks about "nuances of logic", but the reasoning required is so elementary that mathematicians would not even view it as research. Indeed, Loh himself describes the method as a "trick", not as some kind of special new result. The reporter seems too eager to sell the result as some kind of progress in mathematics, rather than as a possible innovation in teaching material that is completely understood.
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@Carl M If I write, "Dr. Loh has not discovered something entirely new," how am I claiming that Dr. Loh has discovered something new?
The article is about how teachers might be able to present a basic algebra topic more clearly to students, not cutting edge research, which of course this is not.
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@Kenneth Chang When the article says, "although Dr. Loh filled in some nuances of logic in explaining why it works.", it seems to suggest that the mathematical community had any question about why the method works, or that the logic needed to be explained. I do appreciate the way the article makes the topic accessible and promotes the goal of finding better ways to teach mathematics.
9
'... the article talks about "nuances of logic", but the reasoning required is so elementary that mathematicians would not even view it as research.'
You are making a false assumption about what Dr. Loh is trying to accomplish. As the article explains:
"Dr. Loh mentors some of the top high school math students in the country as coach of the United States Mathematical Olympiad team. But he also wants to improve the teaching of all math students."
And math teachers are always interested in better ways to teach their subject. Indeed, there is professional society that focuses on mathematics education called the "Mathematical Association of America".
As for being "elementary", please explain how to rigorously prove that a parabola has an axis of symmetry.
6
Easier?
I didn't know that using the quadratic equation formula was difficult.
113
@Bill The quadratic formula is not hard to use. Most algebra students, however, would not able to explain where that funny looking thing came from. Or how to solve the equation if perchance they didn't quite remember it. Is the 4 in the numerator and the 2 in the denominator or vice versa? This way, you can get the answer without remembering or re-deriving the formula.
18
@Kenneth Chang There is some value in making things simpler and teaching parabolas also helps use visual learning to complement math learning. However, I take some issue with the idea that math is "hard" because you have to memorize formulas and because you can't explain where that funny thing comes from.
Think of using your cell phone. You have to remember your PIN. And you don't have to understand how your phone interacts with cell towers to appreciate making a call.
The idea that all math has to be super easy and always trivial to understand feeds into the false narrative that math is only for a select few who magically get it. Math is actually applying formulas and ideas that really brilliant folks have derived through years and decades of work. It's a short cut for us commoners, even if we are engaged in highly technical mathematically intense fields to actually put that math knowledge to use to solve problems.
4
@Kenneth Chang
"This way, you can get the answer without remembering or re-deriving the formula."
But the same is true with completing the square.
Isn't it?
3
The method is almost exactly the same as “completing the square,” which BTW leads to the quadratic formula when applied to the general case. In the example given, the “square” is completed by adding the square of b to both sides of the equation. First rewrite the equation with the constant term on the right side, x^2 – 4x = 5. Next take half of b, square it, add it to both sides. x^2 – 4x 4 = 5 4. You do not need to factor the quadratic on the left side because you forced it be the square of x – 2, when you took half of b. The right side is 9 or the square of 3. If the squares of two numbers are equal, the numbers are either equal or opposite, so x – 2 is either 3 or –3. Therefore, x must be either –1 or 5.
The method described is correct, but no more transparent than “completing the square.” Both methods will challenge some students when the coefficient of x^2 is not unity and they must work with fractions. Completing the square also makes it obvious why irrational and complex solutions appear in pairs because the right hand side must be a perfect square (rational solutions), a non-square positive number (irrational solutions), or a negative number (complex solutions).
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@G-Man
Bingo, and any math teacher worth her or his salt can show why completing the square works.
This is the vastly superior way.
Bonus points for pointing out that complex solutions are basically 90 degree rotations of solutions in CxR three-space, even if only a few kids can follow the logic.
2
I'm not sure what was really accomplished here but simply deriving the quadratic formula in the most intuitive way possible using symmetry.
Dr. Loh is right that memorizing the quadratic formula isn't really helping students learn much, but I'm incredulous that this is somehow a novel teaching strategy that's not not been used since the babylonians. I'm pretty sure lots of (good) math teachers have done this for centuries with their students.
It's a trope to say that students shouldn't be memorizing everything to get the right answer all the time. Math itself is both a way of knowing how things really work, but it's also filled with tools that allow you to shortcut your way to an answer without having to understand the depth of it all the time. The act of "shortcutting" via formula memorization is actually filled with important strategic decisions that students learn to make and that is a useful thing to teach. The beauty and utility of math is that it allows both.
165
@Will Dr. Loh did extensive research, and he posted his paper back in October to solicit feedback from other people. As the article mentions, there was a math teacher in Canada who came up with almost the same approach in 1989, but that's about it. It's not what is in algebra textbooks. I'm sure tons of math teachers would have pointed that out.
13
@Kenneth Chang Fair enough. It's great that we're having a discussion about pedagogy, even if the underlying discovery is so subtle and implicit in what many students and teachers understand already. I do think this is a better approach to teaching that what's currently being done in texts. It does beg the question however of why we know that parabola graphs are symmetric.
The other quibble I have is with the hegemony of the visual and graphical. Many teachers say that visualizations are what we need more of, but I'm not sure that's always justified. Most graphical depictions are viewed quite differently by those who already understand a math concept compared to those who are learning one. Those who teach blithely with too many pictures often overlook a lot of implicit knowledge that helped generate that picture, and the kids just get more lost. It's also true that not every student (or mathematician for that mater) works well through the visual modality. And yet it's a trope that we always need more pictures....
22
@Kenneth Chang
I believe Will's point is that the approach suggested by Dr. Loh is mathematically equivalent to plugging the coefficients of the quadratic equation into the quadratic formula (which is found in every algebra textbook). The original contribution of Dr. Loh is that he provides an elegant way of deriving the quadratic formula. A student could indeed find it easier (and more insightful) to remember these simple steps (in effect deriving the formula every time) rather than memorize the formula itself. However, it could be argued that the student would be even better off learning it both ways.
5
I've taught this material to high school students. I do not see how this involves any less memorization than the quadratic equation. If you don't give the student the plot, the don't know that the axis of symmetry is at x=2, so....they have to remember to use -b/2. Then, they have to remember to set the two solutions to be r=2-u and s=2+u. If they can't remember the quadratic equation, they probably won't remember these three equations. Yes, this provides a more visual and intuitive approach, which will be great for the better students who are going onto calculus. For the rest, I can't see any benefit...it will still just be memorization for them.
5
@John Please read about Bobson Wong's experiences when he tried using this approach in an Algebra I class of high school freshmen and sophomores. What's noteworthy is these weren't top math students; indeed, he says many of them were repeating the class. It also helps put this in the context of an actual lesson plan.
http://bobsonwong.com/blog/27-on-a-different-method-for-solving-quadratic-equations
5
@John maybe this two equations are simpler. Not involves squares roots etc. And the 'a' seems to be indifferent.
@Kenneth Chang
Thanks for the extra reading. I really do like this approach, and I think it has real merit. I did note that B. Wong found that the students really had to be able to plot a quadratic to do this. The very simple form he used in which he just showed a generic curve, with the line of symmetry, x1, x2 and u labeled remains the same for EVERY problem. This is useful insight to him implementation in class. He did also note that some students struggled with the multi-step process still. He also made some rather inconsistent arguments about it being easier to solve, while giving examples involving fractions and the square root of 61...in fairness, he did say he let the students just use a calculator at that point. Hey, when teaching high school math, any different approach is worth a go. Thankyou.
1
The real insight here is: draw the parabola. That way you can see what you're doing and it makes sense.
If you know a little calculus, it's easy to see that the minimum is at dy/dx = 0, which is -b/2a. But even if you don't, you can look at a few cases and convince yourself that it makes sense.
Then you can immediately check what happens when you plug in this number for x and calculate y. Is y greater than zero? If yes, then there are no real roots (and you can see why!)
The key insight of the picture is to show you that you have a symmetric curve with a minimum and you want to find the two intercepts.
I will bet that if you put that picture on the board, most people (of any age) would fail to see how it related to the quadratic equation. But once they see it, they can't unsee it.
Math is perceived as difficult not necessarily because it is difficult, but because it is written in a language that many people never connect to real life - it's just formulas to memorise. It's like reading a text book in a language you don't really understand.
As soon as you make a connection to common sense, something you can see, something where a, b and c actually have a physical, visible meaning, suddenly math makes sense.
PS Was I the only one nerdy enough to try to work out the quadratic for the Mahomes pass?
Am I the only nerd who tried to work out the quadratic describing the Mahomes pass?
If he throws it with a velocity v at an angle A, then
- the height h(t) = v.sinA.t - (1/2)g.t^2
- the distance d(t) = v.cosA.t.
- total distance = (2v^2).cosAsinA/g
It works out relatively simple I suppose because at time = 0, the height and the distance both equal zero (assuming zero = the height of release, and the height at which the ball is caught, for simplicity).
1
I came up with this method myself in the eight grade as a way to solve quadratics before we were taught completing the square or the quadratic formula, just by playing around with an online graphing calculator and noticing the relationship between a, b, and c. Essentially they're all just different variations of the same formula—I wrote a proof myself after learning all three—but I found this to be the most intuitive way of understanding it. Completing the square and and especially the quadratic formula certainly mystify the logic that's at work. As a college junior six years later, this is the only method I can remember by heart.
A few international readers have said they'd actually been taught this method in school. Maybe that's why the US lags so far behind much of the developed world in math and science: intuitive ways of problem solving like this are treated as novelties while rote memorization is the norm.
1
"A parabola is a symmetrical curve that can describe the path of a projectile, like a thrown football, or the curve of a suspension bridge."
To which curve of a suspension bridge is this in reference? The curve of the cables holding the bridge up is a catenary, which is not a parabola. Don't know about the curve of the roadbed, but that would seem to be highly dependent on choices made by the designers/engineers, not a physical law which describes the curve of the chain hung between two points.
3
@Yaj With the weight of the road bed, the curve of a suspension cable changes from a catenary to a parabola. The Wikipedia article on catenaries explains this, with equations: https://en.wikipedia.org/wiki/Catenary#Suspension_bridge_curve
8
"The curve of the cables holding the bridge up is a catenary, ..."
Several commenters have made that claim, but a catenary is formed when a cable is *free-hanging*.
In contrast, a suspension bridge also has a road deck that loads the cables.
"... that would seem to be highly dependent on choices made by the designers/engineers, ..."
Obviously the cables on a practical bridge could follow a variety of possible curves, but the article is referring to an idealized suspension bridge.
See the Wikipedia articles, "Suspension bridge" and "Catenary".
@Yaj Sorry, but if you wish to view a catenary, look at any electric wires hanging between poles.
Or a clothesline with NO clothes hanging on it.
In the case of a suspension bridge, the catenary is distorted into a parabola by the weight of the bridge hanging under it.
2
I understand the method, but it's no simpler than completing the square. For the given example,
y = x^2 - 4x - 5
5 + 4 = (x - 2)^2
+-3 = x - 2
2 +-3 = x; x = -1, x = 5
Note that there's no memorization of the quadratic formula (or of anything else) needed for this method.
8
@Daniel Kirchheimer The example is probably too simple to effectively illustrate why this can be simpler and easier to understand to many people. Completing the square isn't particularly hard to do, but it's not immediately clear what you're doing with these algebraic manipulations other than following a recipe. In Dr. Loh's method, you're basically redefining the variable to eliminate the linear term. But there's nothing wrong with completing the square and if you know that method, this doesn't give you new knowledge.
2
@Kenneth Chang
The example in the article and that of @Daniel Kirchheimer is very simple to how quadratic equiations are explained in my country.
Firstly, by simple multiplication
(x+a)^2 = xx + xa + ax + aa = x^2 + 2 a x + a^2
(this is taught before general quadratic equations are even introduced).
If we have a general equation
x^2 + b x + c = 0
Trick 1. Let u = b / 2
Trick 2. Add and subtract u^2 in the left hand side
We get:
(x^2 + 2u x + u^2) - u^2 +c = 0
which is equivalent to
(x + u)^2 = u^2 - c
From here it is very easy to find roots.
If the first coefficient ("a") of the quadratic equation is not equal to 1 as in the example - simply divide the equation by "a" first.
3
Students today have a problem solving quadratic equations because they do not know how to multiply, divide or calculate a square root. They have been using calculators since grade school and never memorized the multiplication tables nor memorized formulas.
Modern math instruction doesn't recognize the value of math drilling, so has to step back and introduce concepts like stepping back and figuring out whether the answer makes sense, which good math students did unconsciously back in the day. Using a slide rule instead of a calculator reinforced the notion of getting a ball park number that made sense.
If you tell me how this equation can be of solid benefit in my daily life, I'll apply myself to learning it. Otherwise it's just a beautiful abstraction.
The great mistake in teaching math is in the failure to make it relevant to students.
1
I clicked on the link to read this story hoping that for the first time in my 68 years of life I'd be able to find a SIMPLE way to understand equations. Nope, one who is mathematically challenged will always find them, especially quadratic, as alphabet soup. I suffered through high school and college math classes being forced to try to learn this stuff. Strange how in all my years on this planet I've never needed to use one! Do I care about calculating the trajectory of a football? Nope just if it goes over the goal post. As to finances or econmics, I used to program portfolio management computer systems for banks and never needed a quadratic equation. Oh well, to the Sheldons of this world may they enjoy a new way to play with equations.
1
Completing the square seems easier to me. I'll let people look it up on the internet but it seems simpler than this solution.
One doesn't have to guess if one uses the quadratic formula.
This method is cool, but is only simple if you have x^2 and not 5x^2 or something other than 1 x^2.
Yes, one can easily divide by whatever number is in front of x^2 to "normalize" the equation to get it into the right format, but this is an extra step.
Nevertheless, a useful method, once one understands what is going on.
2
@TM Yes, you can just use the quadratic formula. I would bet that a large majority of students taking algebra do not know how to derive it or why it gives the correct solutions. Also, factoring is typically presented before the formula.
3
@TM In that case you cam just start by dividing everything through by five. Doesn't change the roots or the location of the vertex. You can actually work this through to equation form, but when you do that -surprise- you end up with the quadratic formula.
@Kenneth Chang Completing the square avoids the guesswork of factoring and does not require the memorization of the quadratic formula.
1
There are a lot of comments here, and as I prepare to teach class, I'm not sure if it's already been answered:
Does Dr Loh have a method for finding the axis of symmetry that is equally intuitive? -b/2a doesn't feel intuitively satisfying.
I wanna see this taught in action. I'd have to rethink the way I teach quadratics, which could be pretty cool...
@Mike Bell Dr. Loh actually doesn't use parabolas or axis of symmetry at all. He's really just using the fact there's a midpoint between r and s (the two roots). Try watching the video.
Bobson Wong added the bit about the parabola, just as a visual aid to remind students to recast the equations in terms of the distance from the midpoint. See his writeup of his experience here: http://bobsonwong.com/blog/27-on-a-different-method-for-solving-quadratic-equations
3
"-b/2a doesn't feel intuitively satisfying."
Since it sounds like you are a math teacher, I won't do anything but offer some ideas:
1. The standard form of the quadratic is not ideal for showing some its properties, so rewrite it in another form. Another commenter* notes that the "vertex form" is better in some ways:
f(x) = a(x-h)^2+k
In that form, h is the location of the axis of symmetry.
2. Take a look at this book:
"Maxima and Minima Without Calculus" by Ivan Niven (1981).
* In an earlier reply from Michael (Montclair)
https://nyti.ms/2ue49tx#permid=105040494
@Kenneth Chang I think you either misunderstood the question/comment. In order to use the method, one needs first to locate the mean of the two roots, whether or not one uses the graph. The questioner is merely asking if there is some intuitive way to suss out the fact that this occurs at the value -b/2a. If not, the point of departure for the method is knowing this value as a 'formula' which seems somewhat to undermine its appeal as an heuristic method.
Very cool.
However, we over-emphasize algebra in high school and instead under-teach statistics. While its great to have a basic understanding of algebraic tools, the vast majority of us will not use them, or will need to learn them at a far higher level for our respective professions, which can be taught at the university level.
We need to teach statistics in high schools so that people can read a newspaper on a daily basis and actually understand what is being reported. Most of life's decisions involve weighing some sort of probabilities and understand risks and rewards and we woeful under-prepare people for doing that.
4
Yes! to statistics and consider the enormous amount of misinformation or misunderstanding when advertisers use “relative” percentages without showing the absolute percentages.
This the standard trick to eliminate the linear term in any polynomial (quadratic or higher degree). Nothing new or "rediscovered", it has been there since there is algebra.
2
I think this method has been known for a couple of recent decades and was published on Purplemath. It's called the method of common differences (or finite differences). Not sure if you're allowed to use URLs in comments. But I would also point you to NumberSpiral and QTest - Natural Numbers for programmatic implementations.
@Michael M. Ross If you have a URL where this was demonstrated previously, yes please tell us. Just paste it into the comment box.
@Kenneth Chang It looks at the problem mainly from the reverse direction of deriving quadratics from integer sequences: https://www.purplemath.com/modules/nextnumb.htm
However, it can easily be turned around to derive sequences from quadratics.
(I used it to find: https://oeis.org/A155557)
Obtaining X intersects is somewhat secondary.
While I admire the elegance and utility of the parabolic approach, the difficulty of this area of math for students does stem from the "guessing".
So why guess?
6x2 + 13x+ 6=0
the middle coefficient is an odd number
so it must be the sum of one odd number and one even which in this case are products.
Odd products demand odd factors.
factors of 6 are 6,3,2,1,
so what are the odds? 3 and 3
this leaves the factors of 2 and 2
3x3=9; 2x2=4. 9+4=13
this is first grade math applied to a more complex computational series.
@Tom B The thing about factoring is that it only works for problems contrived for it to work. For example, what are the roots of x² – 4x – 6 = 0? Trying to factor this is a waste of time, so why do we have algebra students think this is the approach they should try first? When the simple answer doesn't exist, they think they're missing something obvious and that they're stupid when really they've missed nothing and factoring is generally stupid.
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@Kenneth Chang I agree that Tom B's manner of factoring is a waste of effort, but I disagree about factoring in general.
I was shown the quadratic equation, but the method I always use is basically the method shown in the article. It is based on factoring, algebraic factoring rather than numerical factoring.
Solutions to all polynomial equations are fundamentally solved by algebraic factoring, not just equations of order 2 nor even just equations with natural number exponents.
Like memorizing the quadratic equation, focusing on the parabola is a waste of effort because diagramming is not generally extensible. But while *focusing* on the parabola is a waste of effort, *illustrating* factoring using a parabola for order-2 equations helps students engage the non-textual parts of their minds and consequently gives them more handles with which to grab on to the idea and remember it.
@Clayton I understand there are pedagogical concepts that are being conveyed and I'm not anti-factoring. But it's intrinsically confusing that they get a test where all of the quadratics are factorable when this is not the right approach in almost every other instance. It's only after writing the article and delving into all these comments that it struck me how strange the teaching of quadratics is.
2
As a humanities person who appreciates the intricacies and applications of mathematics (but never really understood it), I was hoping to be able to understand this... but, like all the algebra hiterto encountered, I was completely lost by the time the article "reminded" readers of the quadratic. I know I had learned this--sometime in the '90s--but none of it came back, and the explanations and examples in this article lost me completely. Particularly, the introductions of numbers and variables that have no corresponding terms in the visual examples and seem to appear out of nowhere made this an impossible read. If the Times is written for a general readership, the editors would do well to remember that not every reader is an advanced (or what seems to non-mathematicians to be advanced) mathematician, and not everyone curious about mathematics grew up in an era where calculus was forced upon students by fifth grade... I would have liked to have learned (or re-learned) something here. Sadly, I just got frustrated.
5
@Steven Try skipping over the quadratic formula. It's an ugly looking thing, and one of the points I was trying to make is that, out of context, it's just a jumble of confusing symbols.
In the next section, I'm trying to solve a specific example where b = –4 and c = –5.
It's not particularly complex, but with math, you have to lay the groundwork, which unavoidably adds a pileup of symbols even before you get to the problem you're trying to solve.
Happy to try to step through the method for you.
@Kenneth Chang For anyone who would like me to unpack more of what's going on with this method, drop me a line at [email protected].
1
"As a humanities person ..."
Then it's safe to assume that you have a library card. :-)
Since a newspaper article can't replace an introductory book on pre-calculus, I would suggest looking for these books:
"Pre-calculus for dummies" by Mary Jane Sterling.
"Precalculus demystified" by Rhonda Huettenmueller.
Both of those discuss quadratics. If those are too advanced, look for ones with "algebra" in the title.
Great to see an article like this (that is this article) in a newspaper. I read it because it was there. Didn’t expect to understand it. I didn’t. But sometimes somethings deserve to be read because they were written.
I passed it on to a whizz math teacher in Ireland. I’m waiting for a response.
This is a nice article and a clever technique. As a research physicist with, trust me, a perfectly fine of algebra and parabolas, I'm still within the population who can't solve quadratics by inspection, don't like completing the square, and just plug it into the quadratic formula. I'll use this technique the next time.
And I admire the intrepid science writers who go out on limb and write articles like this while knowing they'll get scores of complaints from "experts" judging every word, expecting a NYT article to have the rigor of a peer reviewed journal, and ready to pounce on most everything as a symptomof our decaying society. I especially appreciate Mr. Chang's efforts to respond to some of the heated complaints!
2
Does a similar trick exist for the general solution of degree three polynomials? Students nowadays never learn the quite complicated formula for solving general degree three and four polynomials.
1
"Does a similar trick exist for the general solution of degree three polynomials?"
What you are asking is whether the method could be _extended_ or _generalized_ to higher degree polynomial equations.
The short answer is "no", because the method relies on the symmetry of quadratics. Higher degree polynomials are not, in general, symmetric.
"Students nowadays never learn the quite complicated formula for solving general degree three and four polynomials."
No one should have to "learn" those formulas, although introducing them would be a good idea, because that would lead to the important result that there are no formulas for polynomial equations with degree higher than 4.
Although proving it is not elementary, a teacher could also describe the "Abel–Ruffini theorem" and briefly discuss the life and work of those mathematicians. In particular, a teacher could explain why the theorem has the names of two mathematicians in it.
A web search for "Abel–Ruffini theorem" will find more info.
1
My math teacher in high school taught us a simple sequence for solving quadratic equations without guessing or formulas. What I can't remember is when I ever had to solve such an equation in real life.
@Harpo
I could solve these with some facility in high school algebra, and again when brushing up for a college test. But yet, to your last point, I think Bob Dylan said it best, “You don’t need a weatherman to know which way the wind blows”.
One of the reasons, maybe THE reason, I hate math, and numbers in general, is that a very long time ago I easily and consistently arrived at what I considered obvious answers to math problems, but they were always incorrect because I could not explain my process. There was no logic, no reason, to what math teachers were pushing. I hope this professor's discovery makes way for multiple ways of thinking in math classes.
Kudos to Chang and Corum for a lucid and illuminating look at a rich topic that rarely makes into the newspapers. And special kudos to Chang to dealing so patiently with the torrent of ill-informed responses.
Here's a suggestion.The article is really about how to approach complex problems, not about solving certain equations per se, and could be re-oriented in future iterations around that theme. These days you can get the solution of any given quadratic equation, and a lot of other data about it, with a couple of clicks, faster than you can chant "plus or minus the square root of b-squared minus four a c."
So what's to learn?
- A single difficult problem can often be broken up into several simpler problems. Work on them!
- It's often instructive to think about a problem in radically different ways. You can often transport insights from one domain into the other. The geometry of a parabola has a simple structure --- an axis of symmetry --- that is not anywhere near as evident in the algebraic equation. Make use of it to find those simpler problems whose solutions will solve the single complex one.
Perhaps the most intriguing sentence in the article is the one noting how varying the coefficients a,b,c leads to systematic variations in the shape and location of the corresponding parabola. This marks the transition from staring at an assortment of symbols to grasping the structure behind them. Now we're getting somewhere!
5
I am astonished at the grouchiness of so many of the comments. As though people were attached to a cumbersome and inelegant way to solve quadratic equations. Or that more confusing and inexact somehow means better. I share Dr. Loh's enthusiasm - this is a lovely simplification!
2
I'm not convinced this is a rediscovery...humanity has never forgotten parabolas are symmetrical about axes. I'm pretty sure countless students have stared at parabolas in the sleepy hours of the afternoon with their chins nestled in their books and come to Dr. Loh's conclusion. It's a helpful rewording of the geometry of a parabola, but it's not a long-lost secret.
5
@Esther Maybe, but this particular formation — in particular, the recasting the problem in terms of u, the distance from the midpoint — is not what what is in algebra textbooks, and it's not how quadratics are taught.
6
@Kenneth Chang
Thank you for writing this article. Math instruction is happy for the shout out but dummy variables are a well established concept.
What you have discovered is that there is an unnecessarily high degree of alignment in what mathematical abstracts are presented.
1
@cleverclue I discovered bubble sort and the FOIL method on my own. I think that makes me a natural at mathematics but I can't claim credit for either.
I've discovered several concepts on my own that have names that I haven't trip over yet. I put a bunch of proofs in my dissertation that I couldn't track down.
1
I would not call the method a trick. I think a mathematician considers their options and chooses the best way to resolve the problem. Graphical methods solve problems as well as doing the algebra. With polynomials, you can test for the type of root, real, imaginary, positive or negative. Then select the method you wish to follow. It is not a trick, it uses what you know to find your answer.
@Ray '' a quick or artful way of getting a result''
This method is the most beautiful proof of the Quadratic formula that I have ever seen. I love it.
4
I went through school taking all sorts of advanced math courses, and worked for many years in machine design.Where was Dr. Loh when I needed him?
Wow, I am shocked by the hostility displayed by the commentators. Is it born of a fear of maths, or of the people who are fluent in maths? Is is part and parcel of the General American antipathy towards able thinkers? I am not good at maths, though I like numbers. Any tool that helps me better understand the language of mathematics is a blessing as far as I am concerned.
8
I, an unfortunately math liability, am extremely pleased US math deficits claim is disproved by the very capable replies and wish them continued astuteness. I am curious to know tho, if math is imaged one's mind or just is 'there,' My art, which seems be 'there,' may be in the place of math when I sustained a minor brain injury when 3.
I fell 12 feet straight into my head onto a concrete slap at about 10. Math got better, art not as good.
so, I don't know if anything was "rediscovered". for instance, you don't say where the formula for the axis of symmetry comes from. But anyone who has taken calc 1 knows the first derivative is the slope. And if you solve for the slope equal zero, you have the Apex of the parabola. so, ax^2 bx c produces a first derivative of 2x b. set equal to zero and solve for x = -b/2. This is all just restating an equation. it is probably good for explaining the quadratic formula to students and why some have imaginary answers as the don't intercept the y axis. But, not much more to it than the fact that there is a lot of math one can learn from parabolas.
2
Please check for 'caternary' curve. A simple suspension bridge doesn't form a parabola, per Wiki.
@Fred Prince Read farther down in the Wikipedia article on catenary. It explains why a suspension bridge (as opposed to a suspending cable) is a parabola.
11
Thanks NYT for bringing some math content to your platform! Much appreciated :)
2
Like the abacus and slide rule, interesting but not practical today. Students should have Excel skills for arithmetic and Maple or MATLAB skills for mathematics.
"If you had wanted to estimate how far a pass thrown by Patrick Mahomes would travel during the Super Bowl, you'd be solving a quadratic equation". Cool! Just by "having wanted" to know, I'd "be solving"? But, is this a Mahomes pass that continues to travel throughout the Super Bowl, or am I doing the estimating while the Super Bowl is underway? I couldn't read further, however much I have had wanted to.
Am algebra student and this looks way easier than ac method and quad formula
3
Amazing? Hardly. Old hat with hype. I’m shocked that Loh was “shocked” by such a “discovery.” Complete the square (which this method basically is), “tricky try,” use the formula. Whatever. This doesn’t deserve attention and wonderment. Move along. Nothing to see here
1
But you didn’t derive the key equation r s = -b.
1
@Kevin Cahill Yes, I do. The quadratic equation you're trying to solve is x² + bx + c = 0 (I'm doing the a=1 case, otherwise, just divide through by a.) You're looking for solutions of the form (x – r)(x – s) = 0. Multiplying this out, you get x – (r + s)x + rs = 0. The coefficients of the x terms must be equal, so r + s = –b. (This, and rs = c, is essentially the pedagogical reason for why algebra students go through the factoring rigamarole, I think.)
9
What more can we ever want? Another math trick. A trick which helps students take tests but does absolutely nothing to help them understand math.
More fast food! More sit coms with laugh tracks! More sugary cereals!
1
@Bostontrim It's not that kind of trick. For some students at least, this seems to be a way to learn the mechanics of solving a general quadratic equation as opposed to memorizing a formula (which by itself involves no understanding).
17
@Bostontrim I absolutely and completely disagree with your statement. In fact, the usual treatment of this subject, the quadratic formula "helps students take tests but does absolutely nothing to help them understand math." Generations of students have learned it by heart and regurgitated it on command. This is a method, not a formula, and it really takes you into the heart of the math in a way that is less painful than either the quadratic formula or completing the square. There is no added benefit from the added pain of those approaches. This is speaking as someone with a math PhD and lots of teaching experience, who is dead set against dumbing down material. It is important to oppose dumbing things down, but if you are going to do that, you need to know what you are talking about.
7
@Nat K I don't understand why completing the square is considered painful. It's simple and requires no memorization. For the example provided in the article,
y = x^2 - 4x - 5
5 + 4 = (x - 2)^2
+-3 = x - 2
2 +-3 = x; x = -1, x = 5
1
This is saving me so much time everyday!!
2
Math is not what you think it is. You think its the stuff taught in school.
Not really
its about solving real world problems. Now… school math can solve some problems, but not very many.
Why?
Real world problems are vastly more complex and variable.
So
Each and ever time you solve a real world problem, you invent a new form of math.
Example. I trimmed a poster by about 1/2 inch to fit a frame. How? You cant measure 1/2 inch very accurately and draw a line. Try it. Try drawing a straight line accurately. The solution: Put the frame’s backing sheet on the poster, then cut along the sheet. You dont have to measure or draw anything.
That is a new form of math! It is not “just” an application of existing math. Technically you are finding the difference between 2 coincident rectangles of different sizes.
You could argue that its regular geometry, and you might be right. But… when you program a computer, it has thousands of bits, and its all new math. A program cant be anything else, it must be only math, because it consists of numbers - bits. And its all new stuff, with its own axioms, rules, and theorems.
1
My colleague brought this article to me while I was on a break between teaching my college statistics classes today. Some students heard us discussing this and groaned. I can't wait to use it in class!!! Thank you again, New York Times.
Thank-you for stimulating my quadratic anxiety....thank-you...
While reading this article I had the same eureka moment I had sitting in my high school algebra class 50 years ago.....huh?
1
@Steve See.. A senior citizen who can still remember high school. Isn't algebra wonderful ....
Why is it necessary to have loud music drowning out what Dr. Loh is saying? To make it more exciting?
It is merely distracting and makes it more difficult to follow what he is saying.
2
for sure this guy contributes more to humanity than the present u.s. president
1
I’m crying into my corn flakes.
There is nothing really new about this method. It is just completing the square in another guise, as any professional mathematician would immediately recognize. Also, the claim that his method is "simpler" than the quadratic formula is disingeneous, because the quadratic formula is presented for the general equation ax^2 + bx + c = 0,
while his method works only for equations with leading term 1, that is the equation x^2 + bx + c = 0. The quadratic formula is also simpler when a = 1! Yes, you can always divide through by a to make the leading term 1, but this is not always desirable when a, b, and c are integers.
Articles about this "new" method have already appeared all over the internet. I am sorry that the NYT jumped on this bandwagon.
1
Re: "A parabola is a symmetrical curve that can describe the path of a projectile, like a thrown football, or the curve of a suspension bridge."
Path of projectile: yes.
Curve of suspension bridge is a catenary, different from a parabola.
1
@Robert Lipper It's a parabola. If it were just the suspension cable, it would be a catenary. With the weight of the road deck (much heavier than the suspension cable), it becomes a parabola. See, for example, the Wikipedia article on catenary under "Suspension bridge curve." https://en.wikipedia.org/wiki/Catenary#Suspension_bridge_curve
6
@Kenneth Chang "If the weight of the cable and supporting wires is not negligible then the analysis is more complex." So it is approximately a parabola in every case. Pretty good approximation maybe, but an approximation none the less. Mathematicians are forever trying to convince you that the problem they can solve is the one you wanted to solve. The really good ones solve your problem instead of their own. This is why applied math is so hard, and applied mathematicians and mathematical physicists deserve more admiration than they frequently get.
1
The solution works but as far as "easier" for students, clearly, Professor Loh has not taught high school students who struggle with math. What I have found is that the process is what stumps students. Students are very happy when you tell them "The quadratic formula will always give you the correct answer." Students who struggle would rather use the formula than factor or complete the square.
2
@MollyMu Po-Shen does work with students of all ages and abilities. But if you don't believe him, how about Bobson Wong, who is a high school teacher who has tried this? http://bobsonwong.com/blog/27-on-a-different-method-for-solving-quadratic-equations
3
Congratulations for introducing basic, crucial mathematics to the uncouth multitudes... Said multitudes absolutely need more intuitive grasp of mathematics to become cogent enough about the world to help sheperd our great leaders toward enough sanity to ensure survival of the species. Nice perspective on parabolas, and what the different coefficients thereof mean.
Not the easiest method to solve the quadratic equation, of course, as changing variables by taking X= (x+ b/2) as new variable is algebraically irresistible and solves the equation in 4 lines or so.
Next please try to do exponentials. Without a thorough grasp of exponentials, phenomena such as the CO2 catastrophe, or pandemics, can only escape the understanding of the commons or god-struck politicians. Exponentials grow at an instantaneous speed equal to their instantaneous value... exactly as a bacterial colony. Most catastrophes involve exponentials. Exponentials also illustrate all sorts of decays and, glued together, the most frequent probability distributions.
Mathematics is the language whose words are ready made thoughts (for example word-concepts such a “parabola”, or the “exponential” come with an arsenal of thoughts and inner logic).
Math is a metalanguage whose elements belong to, and depict, the world. Math is the skeleton of physics, which is how the world is made. To have more advanced thoughts on what the world is made of, mathematics not just the eyes, but the senses one can’t do without.
6
Oy! So much I have forgotten!
That short video would need to be extended to at least an hour or two to review for this senior what this all means.
Why the word: quadratic?
How does y = x squared make a parabola?
Talk to me more about parabolas...I need pictures.
"Function" means what again?
He jumped thru the part about
Need product = 12
Sum = 8
Why do you NEED product to = 12.
Product of what?
I was in Greece once, but darned if I remember any Greek...
2
Your example was fine and simple for the case that a = 1. Is it as just as simple to calculate when a != 1?
@Elli B Divide through a, and you get
x² + (b/a)x + c/a = 0
Then you can proceed as described. If you find fractions too messy, define B = b/a and C = c/a and you get
x² + Bx + C = 0
Yes, it's the same thing, slightly messier with fractions.
2
What this really means is that we haven't learned much at all over the last few centuries.
#(de)evolution?
2
Is the quadratic equation really that hard? Easily derived by completing the square.
2
"Einstein was a man who could ask profoundly simple questions about the workings of the universe. And what his work showed is that when the answers are simple too, it was then that you heard God thinking."
— Jacob Bronowski, "The Ascent of Man" (1973)
2
Thanks for reminding me why I hate math. The fact that some people get excited over this kind of thing bothers me even more.
The proof is a little deceitful. Most proofs would start out with a positive "b" instead of a negative one. The reason r+s = -b instead of just b is because you are really adding up the two solutions for x and in (x+r) * (x+s) =0, the two x's take on the opposite sign of r and s. 5 + (-1) = 4, not -4. I do see why this would be a little hard to explain.
2
Fifty years too late for me. Where were the Babylonians when I needed them?
6
Po-Shen Loh is certainly a respected professional mathematician, but I can't see the fuss about this result. It is just nothing new.
The quadratic formula gives the roots of x^2+bx+c=0 as
(-b+t)/2 and (-b-t)/2 where t is the square root of b^2-4c .
So one root is -b/2+u and the other is -b/2-u where u is half the square root b^2-4c. So u is half the square root of 4(b^2/4-c) . That is u = the square root of (b/2)^2-c.
So instead of remembering to use -b and the square root of b^2-4c the users remembers -b/2 and the square root of (b/2)^2-c.
It is the same formula.
3
@adm Yes, it is the same formula. It has to be the same formula. It's the same problem with the same solutions. If just memorize the formula, there is indeed no point to this. if you're happy with completing the square, that's fine, too. But for many students, this could be a better, clearer way of explaining the mechanics of solving a general quadratic.
6
@Kenneth Chang Thank you for your great article! But I disagree with " If just memorize the formula, there is indeed no point to this." If you understand this method you understand a lot about geometry and polynomials. It is possible to memorize the formula and apply it by rote and actually learn nothing more than "how to find the roots of a quadratic polynomial."
1
I learned this same method in high school. It useful to think of the full ax^2 + bx + c = 0 as unnecessarily complicated because of a poor choice of where x=0. If you do a shift... let x' = x - d, and then solve for the value of d that causes the coefficient of the term linear in x to vanish... the equation is way simpler:
a(x' + d)^2 + b(x' + d) + c = 0 ...
the coefficient of x' is... 2ad + b = 0, choose d = -b/2a
then the equation becomes
a x'^2 - b^2/4a + c = 0
only has a real solution if b^2/4a - c > 0 and then
x' = +- sqrt((b^2/4a -c)/a) = x + b/2a
so x = (-b +- sqrt(b^2 - 4ac))/2a
of course the graphic in the article is really helpful too.
Now the real fun is the cubic equation, first solved in the 1500's... some believe that solution was a watershed where then-modern Europeans realized they could surpass the knowledge of math passed down from antiguity.
1
I share many comments below as this being a self promotion.
I am no math wiz or a guru but a Caltech graduate - albeit almost 5 plus decades back.
I did not know this was such a big deal to solve this.
To me, it comes across as Math for dummies.
And I hate to say this - this presumes kids first know what a parabola is.
And the QB throw is not emphatically a parabola.
if anything - if Mahomes does it right with a spin - it's a gravity defying straight line - unless it's a hail Mary.
My opinion is math is not just solving a problem but the process of solving it.
Using the standard method - kids learn other things such as aquar root and what if they find a negative number under a square root.
Finally, as an aside - with pencil and paper fast disappearing - what do they draw this parabola on.?
2
This article just reminded me of why I didn't major in math--or minor, for that reason. I've heard the discipline described in almost-rhapsodic terms, and I get that. I just wish my right-dominated brain could figure it out. Having noted that, I am most-impressed with the quality of the responses from readers here. If only folks could think that much before posting their socio-political views. . . .
1
Hmm. Great to use the (visible) symmetry, but this method relies on the fact that x=b is the axis of symmetry. Is that obvious to those first learning or is it (practically) a given fact that needs memorization? Completing the square may not bring spatial reasoning into play but it does use pretty basic understanding of term distribution.
I agree that the quadratic formula has not served students well, and graphical interpretation of roots (zeros) is important. But completing the square should continue to be taught, even when there is an "amazing trick" that can also help. (For gosh sake, NYT, PLEASE stop using that term!)
2
To Kenneth Chang and Jonathan Corum:
Thank you for the stimulating article. Please continue to write more about the beautiful aspects of the Queen of Science.
1
There is so much important mathematics news that is overlooked by the news media because it doesn't understand that math is one of the sciences. It's a shame that they believe an article like this constitutes reporting the latest and greatest in math.
2
I had the benefit of attending a talk by Dr. Loh with my middle school kid at the Fields Institute in Toronto last year. It was eye-opening, I wish I had learned to solve quadratics this way instead of memorizing a formula :)
1
Most, or at least many, critics of this article clearly have a solid understanding of algebra, something I have never had. As someone in his 70’s, I still wonder why an otherwise A student struggled mightily with mathematics. So, I thank Kenneth for this fascinating article. Everything helps; maybe there’s hope for me yet.
Seems like an involved procedure, and more to remember. I like this line of thought though. It provides more information on the quadratic.
The coefficient c is the product of the roots. If the roots are integers, try the different factorizations of the number c. Otherwise, there is only 1 formula to remember.
In general, the coefficients of a polynomial are the alternating sum of the products of the roots taken 1 at a time, 2 at a time, 3 at a time, etc., for the increasing powers of the terms of the polynomial.
I’m sorry. The old way still works great, even to those of us with deteriorating mind at 75.
2
I'm glad he found this (it's always exciting when young people figure things out), but seriously we math teachers were using this method 30 years ago; it's just been overwhelmed by the testing mentality we have now leading the students to believe they must follow 1 and only 1 path to a solution.
This is a variation on the Babylonian method for solving quadratics that any person who's ever studied difference equations (not differential) knows.
5
you should examine matrices. you can solve algebra equations with a large amount of unknowns compared to just x and y.
1
The 'derivation' given here is correct but unnecessarily complicated. It can de done in a straight-forward way with
(x*x) + Bx +C = 0 with B = b/a, C= c/a.
Use (x - p)(x - q) = 0 and solve 2 similar linear equations for the square roots of p and q.
4
Very nice explanation and article. I think we need more of these examples esp in calculus. I am part of a group that is trying to make calculus a little more relevant to adults who are going into technical areas esp with regards to stats and everyday problems in analytics and AI. Every bit helps.
6
@St. Thomas Let me recommend a series of articles that Steven Strogatz, a Cornell mathematician, wrote a few years back in the NYT Opinion section, of all places: https://opinionator.blogs.nytimes.com/author/steven-strogatz/
2
The difference between differential calculus and integral calculus is:
Integral calculus is dental plaque which covers an entire tooth;
Differential calculus is dental plaque which is thicker at some parts of the tooth's surface than it is at other parts.
4
This a marvelous article. But I object to the headline characterizing this method as a "trick." So many people think math is a collection of "tricks," each problem having its own one-off shortcut to find the answer, and are hopelessly adrift when they can't match a new question with a "trick." The article shows this method of solving quadratic equations is a process of logical reasoning, as is all math. Send this headline writer back to the society page.
9
My parents are elderly retired math teacher and prof from Europe; they often shook their heads at methods taught here (including such simple things as subtraction), and taught me their more intuitive techniques. Tricks? Not really; “new” - nope, not either, just not known/taught here commonly, but often simpler and faster.
It’s been 25 years since I’ve had anything to do with functions, alas ... use it or lose it ...
... so regardless which way I learned, I enjoyed the read, and to re-wrap my mind about this thing I could once do as second nature!
7
Of course, what if the minimum of the quadratic is not a whole number and you're not entirely sure what it is? Then you need very basic calculus to take the derivative of the equation to find the minimum, which you can use in this methodology.
2
Most people are visual learners. I applaud anybody who can help people better understand mathematics. You gotta bring the subject to the student, instead of dragging the student to the subject, usually kicking and screaming.
But, I think that simple statistics and probability courses are the most applicable 'in the real world.' The ability to calculate the probability of events is of far more importance than the trajectory of a ballistic object. And it's pretty easy, really.
Also, chaos theory and fractal mathematics are far more interesting and help reveal some important aspects of nature.
I hope one day that we rearrange the typical school curriculum, not only to make mathematics more interesting, but to make them more widely applicable to our every day lives.
7
It’s also really just the quadratic formula, except the arithmetic is disfigured into a sequence of distinct ‘phases’ instead of the usual arithmetic prescribed by the formula. In any case, considering that we are apparently interested in measuring the learnability of the arithmetical instructions, surely it is wise to work to classify all permutations of the operations involved in solving the equation, the space of which is spanned by the kernel here — the quadratic formula, or equivalently completing the square, or equivalently this Babylonian method, etc.
We are left with the sneaking suspicion that Loh has indeed been working on a classification theory for this permutation space as it relates to the learnability metric.
3
I like this approach to teaching how to solve quads, both in solving specific quads and for deriving the quadratic formula. Some students have trouble grasping and remembering the completing the square approach usually taught.G
1
This is really just completing the square. But, perhaps thinking about it this way is easier for students to remember, in which case it should be taught this way (at least alongside completing the square).
9
@Vivek It's similar, but the methodology is simpler, because the linear terms cancel out, and you don't have think about "completing the square." Here Dr. Loh explains some of the subtler distinctions: https://www.poshenloh.com/quadraticcompsqr/
15
@Kenneth Chang But it's not simpler; the number of operations is similar, and I don''t understand the criticism, " you don't have to think about 'completing the square'." You have to "think about" whatever procedure you're employing, and completing the square involves no guesswork and no formula memorization.
1
Math teacher here. I don't see how this is a "trick," compared to simply "solving for x," but with another confusing step involved. I'm sure it works better for some kids, but I had to walk through it myself on some scratch paper to see what Loh's trying to say.
Instead, we should be teaching students how to derive the quadratic equation by completing the square. You can solve quadratics by plugging in numbers into a graphing calculator. However, a graphing calculator won't teach the deductive reasoning used to derive the equation in the first place. Pure mathematics is really the only place children are required to practice and demonstrate deductive reasoning skills apart from science and economics classes taken later in high school, so we shouldn't be attempting to find "tricks" to subvert the process.
22
if only we would listen to the babylonians on all matters dealing with accounting! a basic problem in their accounting schools was calculating compound interest, which, expanding exponentially, always outstrips the ability of the debtor to pay. Because of this basic mathematical fact, they instituted a jubilee to forgive farmers' and urban-dwellers' debts every several years.
6
I was surprised, but it does seem better than the traditional approach and it adds a little bit of understanding about what the equation is doing. In that respect, it could be used to teach a more informed understanding of quadratic equations.
Moreover, although "automated", it is less cumbersome than merely programming the formula
x = [-b +,- (b^2-4ac)^(1/2)] / 2ac
1
I think this exposition of the quadratic formula has an extra c in the denominator.
Interesting, and I'll give it a try - tomorrow. I've been using the traditional formula for about sixty years, have it locked in my memory - it's not difficult to remember - and can run off the solution by hand as fast as I can think it through. It still seems to work quite well.
1
While a different method is indeed interesting, it's not something that I would qualify as 'amazing' or really even easier. While this is not a formula to memorize, it is an algorithm, and from my decade of high school math teaching experience I can say that students struggle with both.
In addition, when saying that this provides a method for solving "all kinds of quadratics"... well, so does the quadratic formula. So that element alone doesn't make this any easier. And Dr. Loh's statement about math not being about memorizing formulas is accurate - but it's also not about memorizing algorithms.
When this moves into anything irrational or even when a=1, this becomes complex and would require a calculator for square roots. At that point, why not just graph it? Or why not use some other tool that provides an exact answer?
Math instruction should be about problem solving, not about becoming a human calculator. The example uses a graph because it's much easier - in a modern classroom, a graph of a quadratic can be produced in about 5 seconds. Why not just use the graph?
This is really just an interesting thought experiment, nothing more. I mean, I've developed an interesting way to help students find the squares of numbers, but I'm not publishing anything about it or getting mentioned in the NYT. Why? Because it doesn't really do anything to move mathematics forward in our classrooms. Learning in context & teaching problem solving does.
5
While I would have loved to learn with the Greeks and Babylonians, I had to settle for New York stat public schools in the 1960's. I'm pretty sure that is where I first learned this method. It is likely that one of my great Math teachers demonstrated it on the board for us.
1
I've been trying to understand why I'm so negative about this so called trick, and I think I have it. Algebra is not an end in itself. Rather, the ability to manipulate symbolic expressions and therefore prove identities or solve equations is its purpose. Algebra and trigonometry are pathways to calculus, the language of physics and engineering. If you are not able to solve a quadratic equation by completing the square, and need "tricks", then perhaps a career in science and engineering is not for you. If all you want to do is solve a quadratic equation, it is much easier to use the internet or a calculator.
2
@Michael But nearly all students have to take at least Algebra 2, which is where this is taught. So it doesn't matter if they don't want a science or engineering career - state and national standards, colleges, and SATs / ACTs all say they need to know this.
Also, why is completing the square the gold standard? It's pretty inefficient in most cases. I would say that better mathematicians use the most efficient method for the scenario, whether it be quadratic formula, factoring, completing the square, or just converting to vertex form and solving algebraically.
1
At first I thought this was a clickbait ad.
Doctors say don't eat this to clean your gut!
But actually, I have a more serious question: why do we teach the solution of the quadratic equation in high schools? I have a PhD in physics, and I was having this discussion the other day with a math professor. Neither of us could come up with a good reason. Aside from lack of practicality, there is very little didactic value to learning it. I always thought teaching exponentials or basic statistics at the high school level would be much more valuable.
13
"I have a PhD in physics, and I was having this discussion the other day with a math professor. Neither of us could come up with a good reason."
Well I don't have a PhD in anything, and I can "come up with a good reason", although it depends on having a good teacher.
As you both must know, functions can be used to model various physical phenomena, such as the trajectory of a projectile.
Teaching students how to work with such functions has to start with some elementary examples and quadratic functions are especially informative because they are the next step in complexity after linear functions.
And quadratics can be used to introduce several extremely important ideas about functions: roots, extrema, boundedness, and curvature.
What a teacher should point out is that you can't always find a formula for the roots of a function. And even in the simple case of a quadratic, the formula is unintuitively complicated.
3
I feel that both of those topics are covered in pretty good detail in most high school curriculums. Also, do you not believe that quadratics are a foundational element of physics? Every textbook we've ever seen includes them.
4
"I always thought teaching exponentials or basic statistics at the high school level would be much more valuable."
Several classes of functions, including exponentials, are taught in pre-calculus. See:
"Pre-calculus for dummies" by Mary Jane Sterling.
1
Chang and Corum write, " If you had wanted to estimate how far a pass thrown by Patrick Mahomes would travel during the Super Bowl, you’d be solving a quadratic equation."
Actually, they are indirectly noting how much greater is the ability of the human mind to process multiple variables and indeterminates than are computers. You are the centerfielder. With a runner on third and one out, the batter hits a fly toward the gap. The ball will describe a parabola modified in three dimensions by the (often gusting) wind and the rotation of the ball. Your eyes see part of the ever-changing arc, as you race toward the place your mind effectively instantaneously tells you the ball will come down. Perhaps you will also unconsciously factor in the sound of the bat hitting the ball, even the swing by the batter. If the ball is clearly in play, you not only will calculate the speed and direction to run to intercept it but, also, what it takes for you to circle to ball to optimize your throw to the plate to nail the runner tagging up from third. If you (unconsciously of course) calculate the ball may just barely leave the park, you will calculate the trajectory you must take to jump and possibly snag the ball before it goes into the stands. At no point do you stop to consciously think of any of this.
2
Having been through this nearly five decades ago, I will say that it would be somewhat difficult to look over shoulders and try to copy
cuneiform figures from stone or clay tablets.
1
As an 8th grade student currently learning quadratics, I think that this formula is redundant— r and s are the x intercepts, which are the solutions to the equation. To find r and s would be to solve the equation preemptively.
4
This method works, of course, in general with the minor modification that the line of symmetry is x=-b/2a when the lead coefficient isn't one. It works, in fact, even in the case of complex roots, although in that instance the geometrical motivation of the method disappears, almost literally (the roots are no longer 'in the picture'). There are, however, other very good reasons for students to learn how to factor quadratic trinomials and there are methods (so-called 'factoring diamonds', for example) which can make arriving at correct factors (when the polynomial isn't irreducible) vastly easier for students (and even professors). In short, the method should augment not replace other methods (including, gasp!, solution by the quadratic formula).
Finally, I sure hope he doesn't get an academic publication out if this. If he does it's just one more indication of the vast erosion of standards at all echelons of academia.
3
As a retired graduate engineer who didn't do all that well in math my eyes glazed over trying to read the article. I think I once wrote a computer program to solve this or a similar problem by substituting values for the variables until a solution was found.
3
I was told there’d be no math in today’s NYT.
3
Then fix the Times stylebook, Kenneth. If it told you to jump off a cliff would you do that too?
8
@Halsy Look in the dictionary. In English, both millennia and millenniums are acceptable as the plural form of millennium. For consistency, we use millenniums. https://www.merriam-webster.com/dictionary/millennium
14
This why some mathematicians shouldn't teach math to 8th graders. This "new method" is so ridiculously confusing and needs so much more justification that is beyond the attention span of the average 8th grader, that I would hesitate to mention it to even a freshman in College. The usual method of completing the square is a far more direct and understandable method that one should present if one doesn't want the kids to just learn to solve to solve quadratics using the formula. What's really a stake here is stopping computers from taking over this subject and have the kids be just a bunch of key punchers.
5
When I look back upon those care-laden months, their prominent features rise from the abyss of memory….We were arrived in an ‘Alice-in-Wonderland’ world, at the portals of which stood ‘A Quadratic Equation.’ -- Winston Churchill
1
This whole discussion would be a lot clearer and useful if you just said clearly what the quadratic equation is used to measure in the real world. I gather it is the area under the curve. So please give a few real world examples of using this to get a useful piece of information that someone may need to make a decision. Gee wiz!
9
@G. Harris The third paragraph of the article:
Quadratics, which are introduced in elementary algebra classes, pop up often in physics and engineering in the calculating of trajectories, even in sports. If you had wanted to estimate how far a pass thrown by Patrick Mahomes would travel during the Super Bowl, you’d be solving a quadratic equation. The equations also show up in calculations for maximizing profit, a key consideration for anyone who wants to succeed in business.
17
@Kenneth Chang If you wanted to estimate how far a pass was you would see where it was thrown and where it landed. The quadratic formula would be of no use.
@Kenneth Chang That's actually a kind of ridiculous assertion. Even snipers (who depend on understanding ballistic trajectories) don't whip out the quadratic equation before taking a shot.
Also, understanding the theory behind solving quadratic equations is important, but computers are far better at actually solving them. Humans are woefully inept at solving these equations. It's a waste of time, really.
I took 2 years of calculus in college, and I have never once needed to solve a quadratic since then.
Dear NYT ...
I am fairly annoyed ...
You cannot speak of rediscover. This is a close to trivial approach at solving quadratic equations that is taught in various countries around the globe.
1) It is not novel. And it is not rediscovered.
2) Secondary school students have a good chance of finding this approach themselves. No "professor" needed.
Apart from my criticism it is a fairly nice geometric way of computing the interesting points of parabolas.
13
@Patrick Steinmüller As yet, I haven't seen this in any algebra textbook, and I know Dr. Loh has been fishing for precedents for months.
This doesn't overturn anything that anyone knows about quadratic equations, and it's by and large equivalent to completing the square, but it's not exactly that.
For someone who knows how to complete the square, it doesn't teach you new facts. But I would posit that if you ask 100 random people to derive the quadratic formula, roughly 100 of them would not be able to.
15
@Kenneth Chang Are you saying this is new for everyone that has not used quadratic formula long enough to remember? That is a pretty poor way of justifying it as revolutionary. By your definition taking a typical method but factoring and expanding unnecessarily multiple times would also yield a new method. The reality is it is just another version of a pre-existing method whereby the perhaps more difficult part (knowing how it works) has been replaced with his clever use of "same distant from the average". This is not a new method, but an equally valuable new explanation.
1
@Kenneth Chang Once upon a time, I could solve a quadratic equation (HS and college physics.) Not only do I not remember how to do so anymore, I had a hard time following the explanation!
If it makes learning the concept easier in HS, it will make it simpler in college, and maybe we'll get more math and science students (Go Andrew Yang!)
Given my age, I'll stick to what I've done for 37 years. The human body is simpler than math. ;)
Looking at the comments, I'm surprised by all the negativity. Thank you for a very interesting and fun read! (And the NY Times really should change their style guide. That "millenniums" is painful to read!)
4
Just complete the square and solve for x Done.
3
Not impressed. What happens when the solutions are complex numbers? Not all parabolas intersect the x-axis!
@Nathan Hansard Parabolas are not needed for the method. It's a visual cue that Bobson Wong, the algebra teacher, added to help his students remember what they were doing as they were recasting the equation in terms of the distance from the midpoint.
Dr. Loh's method works perfectly well for complex numbers.
6
@Nathan Hansard There actually is a geometric meaning to the intercepts in this method if the roots are complex. I haven’t had time to dive into it yet, but in my blog post I included some links to some resources.
Where was b defined?
@altair "A more general equation for a parabola is a quadratic function y = ax² + bx + c"
1
Po-Shen Loh is currently the national coach of the United States' International Math Olympiad team. Under his coaching, the team won the competition in 2015, 2016, 2018, and 2019—their first victories since 1994.
6
“a” changes the height, not the width.
Nope! ‘A’ stretched the parabola, hanging the width. All parabolas, unless limiting a variable, extend forever.
How are students who don’t know calculus ( to figure out the x of the maximum or minimum) supposed to know that x=-b/(2a) is the axis of symmetry? Or is it just one more piece to memorize ?
1
@kagni You can show the axis of symmetry is x = -b / (2a) geometrically by considering the quadratic ax^2 + bx, and then noting that the vertical transformation c units leaves the axis of symmetry unchanged.
@Michael: It can certainly be shown, but this is yet another bit of complex conceptual knowledge being used to solve a simple problem.
To evaluate the pedagogical value of this approach, I would have to see a discussion of the course context (what is being covered the week before and after the quadratic equation is introduced).
I like symmetry properties of equations, but am not sure many (most) students do.
4
@Michael to show that x=-b/2a you have to solve a quadratic equation.
Maybe I'm missing something, but it seems to take a leap of faith that the axis of symmetry is -b. But I needed elementary calculus (zero derivative) to show that the parabola is symmetric around -b/2a.
I guess some more algebra and redefinitions of variables can be used to show this without calculus, but the starting assumption that a parabola has its peak/trough at x=-b/2a seems non-trivial.
7
@Milo You don't need calculus. For one, Dr. Loh originally did not invoke parabolas or geometry at all. He simply noted that r + s = –b (follows from pure algebra), and then recast the equation in terms of u, the distance from the midpoint. For two, by plotting out the points along of the parabola, you can see it's symmetric, and the minimum would be at the midpoint.
5
I like the idea of using the axis of symmetry as a way of seeing the solutions for a quadratic because it builds intuition about analytic geometry, but the concept of factoring to solve for the zeros is pretty crucial once you move out of quadratics. There all sorts of interesting and novel ways to solve equations, but the fundamental principle of the zero factor property is really powerful and can be generalized to any equation.
1
My father taught me a close variation on this method -- which took all the guesswork out of the factoring approach to solving quadratic equations -- back in 1970. (My Algebra I teacher had decided the class was holding me back, and so let me do the course on independent study in 3/4 of the year. But the concept of factoring initially stumped me, so I asked my father -- then the head of the Math Department -- for help.)
My wife says she learned a similar method a few years later -- in a different high school in neighboring Massachusetts. (I grew up in NH.)
But when the two of US were math teachers (for a few years in the late 1980's, before she went into math textbook publishing and I went into "high tech") neither of us could ever find this approach in a math textbook -- and I had quite a collection of them to peruse! And because it wasn't in the textbook, we were actively discouraged from teaching this method to our first year Algebra students.
(My wife's mother was also a math teacher -- which seems, in retrospect, to be a recessive gene.)
Unfortunately, my father never told me where HE learned this method, and he is now suffering from dementia in a nursing home, so it is too late for me to ask him. (The moral of that story being: Talk to your parents as much as you can WHILE you can.)
8
That technique is basically doing the quadratic formula (mentioned above) in steps. One can simply use that to find the answer much more quickly. Also, this technique has to be modified if x^2 term has a coefficient, and if you do that, you are simply solving the quadratic formula again.
There's definitely a good geometric intuition that this technique reveals, but once again, the quadratic formula would give the same intuition if only they were taught like that in schools, rather than making students memorize it.
2
This is really cool, and a perspective worth sharing in the classroom. But let's not throw out the baby with the bathwater. I would argue that completing the square is still easier to understand provided you include the geometric interpretation.
If the American student finds the quadratic formula too difficult to memorize and to use, then in the words of Adam Schiff, "we are lost."
9
@Michael If you're memorizing the formula, you don't understand it. If you know how to derive it, great. It's not super hard math, but it's not trivial, either.
21
@Kenneth Chang. To clarify my point, I am not in favor of rote memorization. That being said, I am of the opinion that all students should leave high school knowing the quadratic formula. Not for its utility, but because it is an important piece of mathematical culture.
As a veteran teacher, I can't help but roll my eyes every time I see some technique that promises to make math easier for students. As I said before, I think it is a really cool approach, and I am always in favor of providing a different perspective for my students.
Here is why I believe this won't help aid in understanding in a meaningful way. To understand this technique, you need to be comfortable expanding the expression (x - r)(x - s). But it is not a huge leap in skill to go from expanding this expression to completing the square on an arbitrary quadratic.
Again, really novel approach, and I will incorporate in my teaching. But let us please dispense with the fiction that this is going to demystify quadratic equations.
5
@Michael
Personally, I find this new method a bit less cumbersome than completing the square. However, I haven't tried any examples with a coefficient to x squared other than 1.
This was a welcome relief from reading about Trump thuggery. Also, I want to say how much I admire the patience of anyone trying to teach quadratic equations to students by whatever method.
63
@Nat
And I am surprised that I read it, math not having been either a favorite or forte in high school ( there might still be hope . . ) But I definitely know the formula! But I credit a wonderful teacher, who got it through to his students that it's the reasoning, the logic, that matters. And that math is an elegant language with which to explain relationships in the universe.
1
This method is simply solving by completing the square, with the steps broken up.
0 = x^2 - 4x - 5
+5 +5
5 = x^2 - 4x To complete the square, add (b/2)^2
+4 +4
9=x^2 - 4x + 4
9 = (x-2)^2
3 = x-2 and -3 = x-2
5 = x and -1 = x
4
Did you read the article? It mentions completing the square and explains why that is difficult for students:
"That’s when students turn to the quadratic formula. But they often misremember it — the usual derivation is a bit convoluted involving a technique called “completing the square” — and get the wrong answers.
Dr. Loh’s method allows people to calculate the answers without remembering the exact formula. (It also provides a more straightforward proof.)"
@Midd American No. The article explains why *deriving the quadratic formula* is difficult for students. It says nothing about the difficulty of using completing the square to find the solutions to a quadratic equation -- and it's easy and requires no memorization or guesswork.
1
As a child in school in India, I learnt the formula for solving quadratics without a problem. So did the rest of my class. But the explanation is nice.
3
Yes, the quadratic formula works.
However, I found the exposition of the two methods of Dr. Loh and Mr. Wong charming.
Also, I suspect these alternatives might grab a segment of students who aren’t enthralled by the traditional proof or the completing the square approach.
3
Woops. (x-r)(x-s) = x^2 -(r+s)*x + r*s. Instead, you incorrectly gave the last term with a negative sign, -r*s.
1
@Larry Petro Fixed now. Thanks.
8
I am a professional mathematician, and I have to say that this article is silly, and someone is being very self-promoting about nothing at your expense. The steps involved finding that axis of symmetry are EXACTLY what the supposedly mysterious `completing the square' is all about. And then one moves a distance to the left or right: that is what that +/- square root of (b^2-4ac) is all about. And hopefully there are a decent number of high school teachers who explain things like completing the square with pictures just like this!
42
*Millennia
8
@Anna G The Times stylebook specifies millenniums.
4
@Kenneth Chang C'est domage. Big time.
5
To see this method described as "amazing" reminds me of a comment Paul Kugman made re Paul Ryan, the former Republican Speaker of the House.
Krugman said that Ryan was an idiot's idea of a genius. Likewise, anyone who finds this method amazing is a dim bulb.
5
"millenniums"? The plural is millennia.
9
@Peninsula Pirate The Times stylebook specifies "millenniums," and so that's what we use.
3
@Kenneth Chang Time to stand up for what's correct, ya think?
4
Multiplying out (x – r)(x – s) produces x² – (r + s)x + rs, not - rs!
2
@Jon Fixed, thanks.
1
@Jon Fixed, thanks.
2
Learned how to complete the square in 3rd grade back in Soviet Union. Students need to be challenged, not coddled!
6
My thoughts, in the words of the great Billie Eilish, DUH
Any self-respecting 12 year old figures this out by inspection
1
@Duncan McTaggart Fine. How quickly can you find the roots of x² – 6x – 135 = 0? Or x² – 6x – 136 = 0?
9
@Kenneth Chang I have to bat for Mr. McTaggart here. With a graphing calculator and the visual representation it produces - very little time at all.
I was hoping at some point that the article would articulate on a novel new way of teaching or quickly calculating quadratic equations, but the article is essentially saying "look at the graph". Maybe I have been blessed to have many good math teachers, but visual graphs and looking at them were always a part of my math education in middle school and even well beyond.
3
@Kenneth Chang Wow. Are you really replying to all this stuff? Can't you let things lie unless there is some serious issue? And the implied snarkiness? Necessary?
4
I saw. I read. I gave up. (BA, English Writing, 1979)
62
Why does this comment not have more recommends? Come on liberal arts folks!
5
Some things get rediscovered periodically. The modern hype machines on the internet blow everything out of proportion, and we end up with a NYT article! Still better than another article on Trump or his "acquittal."
5
Teach me how to time order to solve a pathe integral
2
@Kenneth Chang: love the fact that the author is actually responding to comments! That is something that should happen more often...
4
The plural of millennium is millennia
4
"But a math professor at Carnegie Mellon University in Pittsburgh may have come up with a better way of solving it.
“When I stumbled on this, I was just completely shocked,” said the professor, Po-Shen Loh."
This has the tone of enormous import. It may be a useful pedantic device for students, but it's hard to see why this is newsworthy for the front page of the NY Times.
7
What ever happened to:
-b +- (b2 -4ac)1/2/(2a) = x
Totally lost me!
1
Lovely!
1
This is HORRID.
Math as a descriptive subject, with some dumb tricks to solve stuff. It reminded me of my time in school in a third world country where the math teacher (brimming with contempt and self-righteousness) stood in front of the class, back turned, and wrote differential equations, rewarding students for rote learning. If you dared to ask questions about the epistemology, such as: what is this, why are we doing this, how is this related to anything . . . you were either thrown out of class, or had your fingers rapped by a cane.
Because I routinely failed my math and statistics classes right up to the college level, I teach statistics in ug and master's programs now (after a doctorate). I try - with fervor of converts - to teach math/stats as epistemology. Why is quadratic equations important, how did they emerge, what do they mean, what information do they contain, why do they contain it, what does a solution mean, and why is the solution expressed in two separate terms important.
This professor illustrates why a country that fashionably hates math, has no problem putting a man on the moon and inventing the iPod - and leads in just about everything. Show me a country that yaps about math, and celebrates this kind of pedagogy, and I will show you the third world.
2
Applaud this. My second novel (following an historical novel) is SciFi. Consequently, at age 74, I find myself going back to school in math on my computer. It's a joy to learn that math can be more fun than threatening. What a relief to learn that this student may not be quite as stupid as thought, or taught, in high school. Thank you.
3
It is pretty bad when an article about math gets something very trivial completely wrong... the curve formed by a suspension bridge is not a parabola, but a catenary... even the ancient Greeks knew that they are different curves!
10
@Ben But, but, the NYT Book of Styles says......
4
sorry to be the math teacher, but there's an error in the sample calculation in the article:
Multiplying out (x – r)(x – s) produces x² – (r + s)x – rs.
-r * -s = +rs, not -rs (-1 * -1 = 1). It needs to be (x+r)(x-s) or (x-r)(x+s). Also this makes the middle term (r+s)x incorrect. It will be either (r-s)x or (s-r)x, depending on which one is positive.
Otherwise, pretty nifty.
Otherwise, pretty nifty.
6
@Irving W. Fixed, thanks.
3
This is in the Israeli curriculum...
And from what I heard in other countries' curriculums as well..
4
@Shay Could you find an example from an Israeli algebra textbook? This _isn't_ completing the square, although the underlying idea is similar. Thanks.
2
Are you serious? This? On The NY Times? I know not everyone is fluent with algebra, but to many of us who are, this stuff is obvious. And yes, there are many of us.
8
@Will
Don't forget that most Americans get a terrible education.
Also, I'm a NYT reader and although I could follow this article, the truth is I haven't seen a math problem since I was 16. I'm 54 now. Am I really supposed to remember? If you have a career in a field that required college math, this may seem elementary, but I have a PhD and a JD and believe me, I had to really reach into the recesses of my mind to recall most of it. It has just never come up in life since junior year in high school. Ever.
Isn’t there an app for this?
1
Yes, this new method is so much easier than the quadratic formula. Ha ha. (Rolls eyes.) Lets call this 'new math'.
2
The music on the video is incredibly annoying.
1
Well the problem is that you have to solve 2 unknown values using 2 equations: r+s=4 and r*s=4, guess what? you need a quadratic equation again: r^2-4r+5=0. I do not see any advantage by using this method, unless you want to guess a few number to find r and s
Hahaha! I thought the article was rough. Then I read the comments. God bless you math people. I'm reminded once again of why I ended up in law school
3
In the example, the author writes, “To find u, we want the product of r and s to be equal to c, which in this example is –5”
How do we know this? And where is “c” in the setup?
2
Product of roots = c/a
Charlie (New York): "... tear the speech in half ..."
Removing that comment was a dumb thing to do! I prepared an entire *mathematical* reply to it and you people have consequently trashed my reply!
Here is is AGAIN:
I will subvert your attempt to inject politics into a math article by finding the mathematical content ...
If you examine the edge of a torn sheet of paper with a magnifier, you will see many paper fibers protruding from it.
That leads to some questions: What is the *edge* of the torn sheet? And how *long* is the edge?
There is a similar problem with defining the length of a coastline with all its capes and bays.
And in the theory of fractals, the problem is even more challenging, because a fractal edge is infinitely ragged.
For more, see the Wikipedia article, "How Long Is the Coast of Britain? Statistical Self-Similarity and Fractional Dimension", which is about a paper by the mathematician Benoit_Mandelbrot, who was an expert on fractals.
2
Fractals are fascinating!
1
Professor Loh's method is just completing the square, standard fare in any good algebra class.
1
This is just completing the square. (Of course it is--what else is there?) Maybe ask an uninvolved mathematician before you print stuff like this.
5
I am rather shocked how this article misrepresents the Vieta's forumlas and its usefulness in instruction, and then depicts the vertex method as something independent of the Vieta's formulas.
Quote: "Figuring out the factors that work is essentially trial and error."
This is totally untrue. Solving algorithmically using Vieta's formulas still works via completing the square, which involves the vertex form. Hence, the vertex form is also not something new or independent of the Vieta's formulas.
The authors failed to realize that Vieta's formulas are incredibly useful to show the algebra in quadratic equations, and is usually the first time a student will ever come into contact with Galois theory and the permutation group. And if the math teacher ignores that, and goes on to guessing the solution by factorization, it is a fault of the teacher/curriculum that failed to distinguish algebra from arithmetics.
3
Another technique is to use an app that takes a picture of the equation and displays the answer.
3
Your parabola basics are misleadingly simplistic. a does adjust the width of the parabola and c does slide it up and down, but b does more than simply translate it left to right (yes, technically you state that it shifts the axis of symmetry to the left or right, which is true - but folks will interpret that as simply translating the parabola in the x or -x direction which is not true).
I mean...sure. I just don't understand why he is portraying this as some sort of discovery. It's using basic algebra anybody already knows and is literally how calculators figure out what the roots are.
2
I find it a bit sad that something as trivial mathematically as the quadratic formula would be the focus of an article in a newspaper supposedly addressed to educated readers. The quadratic formula is trivial (and simple to derive). It would be more interesting were there to be some article relating to, say, Galois theory which shows why a number of classical problems in algebra and geometry cannot be solved (the most famous example being the insufficiency of radicals for solution of the general quintic equation and polynomials of higher degree). And then some uneducated troll shows ignorance in mistakenly claiming that the quadratic formula cannot be used to find complex roots. He is characteristic of Americans who believe that they understand things of which they have no understanding. It is a sign of very poor education. More useful than this formula would be why the quadratic formula is not a good algorithm for use on computers (there is a numerically stable algorithm far less subject to the vicissitudes of finite word length). In fact, if the quadratic formula is a "mathematically" complex to you, then you are not an educated person. If you do not understand basic mathematics through calculus (rigorous, not as formulas), basic genetics and evolution (which is about survival of species and not climbing a ladder) and physics (what is heat? why is Newtonian celestial mechanics falsified?), your ceremonial certification does not indicate any education.
2
Despite the headline, Dr. Loh is not saying he discovered a new solution. His paper and his talks make clear he is popularizing a solution that has existed for some time because he thinks it will aid students in understanding and solving quadratics, (something I failed at in high school so many years ago). That this “solution” has been found before is recognized, the change is that Dr. Loh has enough of a presence in the math teaching community that it is getting greater reach than before.
3
Sounds great, but it’s way over my head
Path of a projectile yes, but not the curve of a suspension bridge - see: catenary.
4
This is not rocket science. This is really just completing the square, which I was taught 40 years ago.......
1
It's just the method taught under the name of `completing the square'. Especially when a=1 it's really easy.
1
I am someone who has done a lot of math, and this method is something I'm very familiar with. It may not be the most common way to teach solving quadratic equations, but it definitely is not something people were unaware of before. Basically, this is Po-Shen Loh's preferred approach to teaching the subject and the media is hyping it like some great new discovery.
2
Americans need to improve their math skills.
Tricks are not helpful for this. Hard work is.
2
If understanding and/or solving quadratic equations is sso difficult, we are doomed from as a people.
All the senior Chinese leaders are scientists or engineers.
I have 2 questions/concerns:
* How is teaching the "u" different from teaching [sqrt (b^2 - 4ac)]/2a? If teachers were doing a good job (myself included when I was in the classroom) having students conceptualize the relationship between the zeros and the axis of symmetry, they would already get this.
* I'm afraid many will end up teaching "this is how you get the answer"...
1. Find half of "b"
2. Solve u = (b/2)^2 - c
3. x = b +/- u
1
The advantage of this method is enhanced by drawing a picture. Does anyone draw pictures anymore, and with what?
2
Ti-84...!
This is the same method explained on pages 12 and 13 of the Girls' Angle Bulletin, Volume 11, Number 1 from October/November 2017.
2
First, the headline reminds me of the many "try this weird trick" idiot ads on the internet. Oh, well.
Second, using a graphical method is always excellent; a picture is worth many words. However, one does not always have a piece of graph paper around every time one needs to know the roots of a quadratic equation. However, I can always write down the quadratic formula and then just "plug and chug". In my experience teaching chemistry to undergrads, students were always in search of a formula they could just plug numbers into. Or they wanted a list of steps to memorize. I wonder if remembering the steps in this method is any harder than memorizing the formula (and not forgetting that a does not always = 1). For example, when the answer is not a whole number or when it is difficult to read the value off the graph, the frustration comes back. I suppose if all students have graphing calculators (AND know how to use them), this could be avoided. But, sadly, few students are comfortable using those very powerful calculators.
Finally, as discouraging and off-topic as this is, I quote from Edward Gibbon: The power of instruction is seldom of much efficacy except in those happy dispositions where it is almost superfluous.
Looking back on my teaching experience, I have difficulty finding instances that contradict Gibbon's observation.
1
Weird. Nothing new here. This has been the way it was taught in South East Asia for decades.
Half a century ago, this would have all made sense.
2
I share your feelings. I graduated in 1967 from high school, and went on to take calculus in college. However, I was having trouble doing the simple algebra here. How discouraging. Oh well.
I had a high school math teacher that would say whenever you are trying to solve a math problem and you are stuck, remember that the number zero is your friend.
"Millenia."
Not "milleniums."
JFC, no stylebook, Times?
2
@hula hoop Their stylebook probably allows "criteria is" and "kudo".
1
That's hilarious. This decomposition is taught to first year high school students in Italy. He's just presenting it in a exceedingly verbose and poorly written way.
2
Great idea. Can someone help me please to solve the general fifth degree equation with a similarly insightful method?
Just the chart alone, explaining what a quadratic equation is and what the variables and parameters actually represent, was worth the price of this article.
Why not complete the square? You can even draw a picture of the square that needs completing. And if your goal is to solve
Ax^2 + Bx + C = 0 when A is not 1, and A is not 0, then solve
A^2x^2 + ABx + AC = 0, that is u^2 + Bu + AC = 0 where u = Ax? And what happens if the solution of your equation is not a real number, such as x^2 + 4x + 5 = 0? I agree that memorizing the so-called quadratic formula in no way demonstrates that the solver knows what is going on here, and that completing the square recapitulates the derivation of this formula. But, hey, as Rod Stewart sang, Every Picture Tells a Story.
1
It’s good to learn multiple solutions a problem, but this shouldn't be a substitute for the quadratic formula. The QF quickly solves any quadratic equation, real or complex, and the derivation isn’t any more “convoluted” than the approach described here. Yes, you generally memorize the result, but the above method involves memorizing a solution “recipe”, which isn’t any different. Mathematics at every level requires things you see proven once and then you use the result – that's how it works in real life. You can't go back to first principles all the time.
IMO the “new” method doesn't demonstrate the underlying principles any better than deriving the QF, and still involves computing a square that isn’t so obvious for non-integer coefficients. I’m an engineer, and if I had to go through this procedure for every quadratic equation it’d be impractical, even more so for symbolic equations. Math teachers now seem obsessed with “magic bullet” methods that work for overly simplified problems in the classroom, but don’t give students the skills required to go into mathematical fields later. The textbooks now don't seem to want to teach symbolic or real-valued math at all – just problems that work out to integers kids can guess at – and that's extremely dangerous.
The QF is the standard approach for a reason, and it’s not that difficult to prove. If you don’t remember it, look it up. There's nothing "amazing" about this new method, and it isn't very useful in practice.
1
I agree that factoring, which we spent way to much time on when I was in high school, is idiotic, but I disagree with this technique. Completing the square is worth doing once or twice, but then just CTS on the general equation and DERIVE the quadratic formula and move on. Live life, be happy. Once you derive it you have it memorized. Of course you have to take a break from Snapchat for 15 minutes.
Of course once you memorize it you should immediately forget it. It is a curious fact that finding roots of quadratics is not much used in later classes. It's been a long time, but don't recall it coming up in calculus or linear algebra. Used it a little in differential equations for finding solutions for second-order equations. Never encountered this topic in higher math like analysis or abstract algebra.
I guess parabolas come up in physics a lot, but then you're more interested in differentiation instead of roots.
1
I think this method would be quite useful where the sum is a larger number or one whose quotients are not obvious.
Gosh, this is one of the ways how I learned quadratic equations as a child. Didn't realize this was "new".
Ha, my dad (educated in India) taught this to me when I was a kid. I don't think it's new, maybe just not found in a recently published textbook perhaps.
1
There is no excuse for not remembering the quadratic equation. Just plug in a, b and c and you have your answer.
He finds that easier than just plugging coefficients into the quadratic equation? How?
When I tutored Algebra I gave my students a similar method but I also let on a secret that was being held back from them...I taught them a tiny bit of Calculus. The derivative y'(x)=2ax+b vanishes at the axis, or x=-b/(2a), which is the same result as the "average" proposed here. Then the -b/(2a)+u and -b/(2a)-u roots pop out and easy to solve, as in this method.
...
Later, when I tutored trigonometry I let on another secret that was being held back from my students, I taught them a tiny bit of complex analysis. In particular, if you know exp(i*x)=cos(x)+i*sin(x) then many manipulations in trigonometry become very simple.
...
So I often wonder why students are kept in the dark about concepts that will dramatically improve their understanding and ability to solve problems, particularly when it is very easy to learn. Now that I am a university professor I don't mess with my students this way, I give them all of the tools, help them to see the bigger picture, and then unleash them to tackle challenges they never dreamed possible.
1
"...pop up often in physics and engineering in the calculating of trajectories..."
Only a non-engineer would say something like that
Just plug it into Wolfram Alpha. Solving algebraic and calculus expressions has no use even in professional settings—chances are you have far more complex problems and math to worry about than getting a simplification right.
Suspension bridges are catenarys (cosh x), not parabolas.
3
"x² – 4x – 5 = (x – r)(x – s) = 0" seems misleading, as it implies complete equivalency between x² – 4x – 5 = 0 and (x – r)(x – s) = 0. If that were true then the equations are solved whenever r=x or s=x. For example, when x = 7 and r = 7 we'd have (7 - 7)(7 - s) = (0)(7 - s) = 0 for any value of s.
Perhaps the author meant, "Solve for r and s when *both* x² – 4x – 5 = 0 *and* (x – r)(x – s) = 0 are true.
1
My math ability and keen wit have allowed me to escape the vast array of drudgeries associated with modern day life. In high school I graduated in the top 5% of the bottom 10% of my class. I’m not good at math, but I know how many grams are in an ounce. My numerical perversions have served me well.
2
Negative numbers. (pfft) Did you know negative numbers weren't widely accepted until the 1800s? Descartes didn't accept them. His coordinate graphs showed only a positive x and positive y axis.
If you have one apple on the teacher's desk, how can you take away two? What on earth is a negative apple? It's nonsense. A number smaller than zero? What? Zero is nothing. al-Safr. The empty thing. Not even a number. A place holder. An open circle on the number line. Division by zero isn't undefined. It's no division at all. Don't even get me started on imaginary numbers. They're called that for good reason. 2^2 = 4. -2^2 = 4. So how can the square of anything be negative? They’re just making stuff up!
Numbers. (pfft) They can be awfully irrational.
1
@Tyrone Greene
The number i told pi to be rational, then pi told i to get real.
1
Just super. This guy gets on and is going to explain a concept that makes this calculating process EASY. So, math terrified guy that I am, I buy in and watch the video. Simple steps. Simple conclusion... ancient and absolutely faultless! Except for one small factor: I HAVE NO IDEA WHAT THE HECK HE IS TALKING ABOUT! Story my life with math. Assumption is that I get the process and the problem. I don't even know what those symbols mean. Anyway, too old to worry about it now. But if he ever wants a great Italian meal, I bet I can cook it better than he can. Sigh
1
@kermit
The goal here is to find a number or numbers (there is usually more than 1) that satisfies the equation, So if you x^2 -3x+2=0, you can plug 1 into the equation and see that it works: 1 squared minus three times one equals negative two. Add two to that and you get zero. The number two is also a solution to this equation.
Why would anyone in his right mind ever want to do this? I have no idea. It's no good if you're trying to impress girls. I tried it.
2
I never got math and at 55 it still looks like a jumble to me. I really really look at this graph and it makes no sense in my head. I wish languages get as glorified as math.
1
Nothing new. We engineers would call this "plug and chug", get a good initial point and plug in the numbers until you got close to the zero. We always cover this in a numerical methods course for zero finding. I guess the math guys have better marketing, engineers just want to solve the problem. Always funny to find out what people think "interesting". Oh well.
@how bad can it be
My son is a high school student who is going to major in math. He took multivariable calculus, linear algebra, and physics with the college engineering majors and beat them all out.
Intelligence level is another difference between math guys and engineers.
The quadratic equation has frustrated math students for millenniums.
I'm not a math brain, but I scored in the 99th percentile in the verbal portion of the GRE. The plural of millennium is millennia.
1
I was kind of let down reading the article. I find these methods too gimmicky and rarely usable except for a few simple cases.
1
Any way that helps people is good. Thank you.
Proof of quadratic formula
ax^2+bx+c=0
ax^2+bx=-c
divide by a
x^2+(b/a)x=-(c/a)
complete the square
x^2+(b/a)x+(b/2a)^2=-(c/a)+(b/2a)^2
[x+(b/2a)]^2=-(c/a)+(b^2/4a^2)=(b^2-4ac)/4a^2, putting over a common denominator of 4a^2
[x+(b/2a)]=+/- sqrt(b^2-4ac)/2a, taking the sqrt of both sides
x=-(b/2a)+/- sqrt (b^2-4ac)/2a
x=[-b+/- sqrt(b^2-4ac)]/2a
Reading this I cannot believe I ever solved a quadratic equation.
1
Aren't suspension bridges described by hyperbolic equations, not quadratic equations.
1
Neither the “guess method” nor this “alternative” is important. Even the quadratic formula is unimportant. The truly important idea/concept is the method of completion of the squares (the quadratic formula is a mere consequence of this). The reason for this is that you can work on any field this way. For example, try to solve x^2 – 4x + 2 = 0 in Z7. Easy: completing the square you get (x-2)^2 = 2, but (in Z7) u^2=2 implies u=3 or 4 (=-3). Then, since x = u+2, it follows that x=5 or 6.
More than once the authors mention how difficult it is to memorize the formula for the roots of quadratic equation. Students in many countries learn in the fifth or sixth grade how to derive the formula (not difficult at all) and memorize it so that they remember it for the rest of their lives. Nobody complains. This is not to discount the method described in the article.
2
This is a simple example that shows the mysterious connection between algebra and geometry. The method is clearly limited because the pictorial connection fails in some cases. So also, solutions to higher order equations cannot be easily interpreted in a similar manner. When they both work, is one better than the other? An engineer who is designing something will never calculate these results from first principles. Extremists in both fields have paid the ultimate price. The geometer who tried to convert integer numbers into "squares" stumbled into the world of "irrationals" and was put to death! So also the algebraicist (Galois) pushed the study of roots of equations discovered Group Theory - and was put to death. Not worth the risk. Use a cheat sheet. Most of us do, at some point or the other.
1
Great method, but nothing new to me. This is exactly how I learned to solve the quadratic equation in junior high (Sweden in the 80's). And later my high school math teacher and university calculus professors used the same method, so this method was well known and well used (at least in Sweden).
We never called the coefficients a, b, and c. We called them p and q, and if there was a coefficient before the squared term, we always divided with that number first:
x^2 + px + q = 0
Roots are centered symmetrically on -p/2 and offset from this center by the square root of center squared minus q. Or with algebra:
x = c +- sqrt(c^2 - q)
where c = -p/2
While the method is not new to me, I'm glad if American students can learn some math with more insight and less formulaic approaches, though.
1
While you rightly acknowledged the Babylonians and Greek inspirations, later in the article you said, it was only later people came up with the Zero and negative numbers. Why not acknowledge the Hindu’s contributions, particularly, the mathematician, Brhamagupta, with his definition of zero and contributions on negative numbers?
3
To solve a quadratic equation you first bring it into normal form, i.e., it should start with x^2, and then you use the method of quadratic substitution to apply the binomial formula. I did never see the suggested ansatz to write the equation in the form
(x-r)(x-s)=0. The well-known method of quadratic substitution is at least as simple as the "amazing discovery" described in the article.
1
Hm, hard to believe nobody in these comments pointed out that you get a much simpler formula if you always make a=1 by dividing the original formula by a, so you get x^2+px+q=0 (as in the simplified example). In this case the oh-so-complicated quadratic formula becomes -p/2+-sqrt(p^2/4-q) which is the only formula I ever memorized. You can make it even simpler if you define x^2+2px+q=0 so you get -p+-sqrt(p^2-q)
Everybody should be able to memorize that, so tricks and no geometric contortions needed, just plain and simple math.
1
I'm a math professor. Any good professor has already explained this, basically, when they taught the quadratic formula, since you should explain that it's clear in the quadratic formula that the roots are evenly spaced about the vertex. The vertex is -b/(2a), and in the quadratic formula you then have +- \sqrt{b^2-4ac}/(2a), which = u from the article. Just rewrite, (-b/(2a)- u)(-b/(2a) + u) = c and solve. You get b^2/(4a^2) - u^2 = c. Voila. Very common, and I've seen the u term discussed in a variety of ways.
3
I love the idea of introducing symmetry arguments. But the method seems to me to simply swap what it is you want the students to memorize (which is an unfortunate irony given Dr. Loh's quote about not memorizing formulas).
Instead of having them learn the quadratic formula, they are now asked to memorize two separate facts: that the sum of two roots "happens to be" equal to -b/2a and the product of two roots "happens to be" equal to c/a. I suppose there is the appeal of memorizing two less threatening formulas instead of one imposing one.
The way I learned it in school, and the way I teach it, is to have them properly learn the quadratic formula, which is, yes, painful for some students. But then you can use the formula to derive the sum and product formulas. The sum formula is particularly illuminating for the students, because they realize that the symmetry of the parabola is already encoded in the formula (via the plus and minus square roots)!
1
... and of course my sum formula is off by a factor of 2 (see?!)
I miss my dad daily. He was an engineer & I was a daughter who didn’t take to math. I certainly wish I could share this article with him. If I’d known this in the late ‘70s I wouldn’t have been sitting at the dining room table after dinner, at night, pulling my hair out. Dad: you’ve been gone over twenty years, but still I wish you were here to share stories I read in the Times!
6
The portion of the American population that still believes 2 + 2 = 4 is a hoax will still have a difficulty with solving quadratic equations.
Besides, in today's scorched earth capitalism, only unlimited exponential growth matters with no planetary limits. Though imaginary numbers not of the negative square root kind are suitable for politicians.
3
I'm unsure why this is better than the quadratic formula? I get that maybe middle school kids will find the memorization daunting but finding the midpoint of a parabola is done most easily by setting 0=2ax+b, which requires at least some elementary calculus to explain quickly. At least the quadratic formula is provable using only rudimentary algebraic procedures.
A more fun math fact in the same neighborhood (perhaps for people that don't see math as often) is why there's no general solution to the quintic polynomial (that is, y=ax^5+bx^4+cx^3+dx^2+ex^1+f).
2
@Brian
My son is in sixth grade and he has learned both factorization and the quadratic formula. He does not think the quadratic formula is "difficult", and most middle schooles can certainly master it. We keep on underestimating kids and their amazing brains.
Concerning your last point, in these times of computers and spreadseets, learning how to obtain numerical approximations to roots using the bisection or the secant method is more useful and illustrative than learning gimmicky tricks.
1
A former math teacher and someone who has seen students struggle with the quadratic equation....
The graphic approach and use of relationships are interesting. But I wonder, are they better than the general solution for the quadratic equation? Working with whole numbers helps illustrate the principles, but when dealing with decimals, real or unreal variables, might not the general equation be the better approach?
1
While I so appreciate your enthusiasm for it, this method is really just a special case and not a new method.
The quadratic equation accounts for shifts and scales of the parabola. We can derive it using dummy variables much like you have here. This is a special case where a=1.
Another way of thinking about the linear term -4x and the bias -5 follows: We add a line that tilts the original parabola, then we add a bias term that lowers the parabola.
Notice the linear term is what shifts the parabola off axis. We can learn more about that by looking up odd and even functions.
2
I’d never use this as a fast “trick” for solving the quadratic equation, but it does seem like an interesting pedagogical alternative for teaching students. I do feel like trial-and-error gets a bit of a bad rap here. “Guessing” the solution is a time-honored technique in math (and physics), and works better and better the more intuition that you build up. It’s not really a way to gain understanding initially, but it’s a useful (and fast!) approach once you understand the material well. So, I do think “guessing” is actually a skill worth practicing too.
2
Interesting article, but, unfortunately, starting with the first sentence - “The quadratic equation has frustrated math students for millenniums” - there is an underlying tone of how mysterious, baffling, and uniquely opaque math is. Math is no more challenging than other technical subject, and is an essential underpinning for mastering practically any science, as well making simple decisions daily. It is much more logical and easier to comprehend than many subjects. It just requires a desire to succeed and competent teaching. However, if we continue to treat mathematics as something arcane, we will continue to scare off students from this important discipline and ensure that our citizenry and voters will be innumerate and easily confused about basic knowledge needed to function effectively in our society.
13
@Gil H,
I think unfortunately, there are a lot of “mathophobes” who have the notion that math is difficult and not very important and only for strange people.
Rather than it is a domain of beauty and pleasure. And offers great power too.
It is a shame.
3
I looked at the sample equation provided y = x2 – 4x – 5 and determined in about 2 seconds by factoring that x = -1 and x = 5.
Maybe the new method works better for more complicated formulas - but how complicated can an quadratic equation get.
4
@J Colletti x² – 6x –135. How quickly can you solve this? (The example in the article is probably too trivial to illustrate why the method is more useful in general.)
55
@Kenneth Chang
What about the method of completing the square?
Write as x^2-6x+9-9-135=0, which is easy enough, and then as (x-3)^2-144=0, so that x-3=+/-12 and x = 15 or x=-9.
The method in the article is nice in the geometric perspective it gives, but there are good algebraic methods and not just guesswork as well.
5
@Kenneth Chang Well, consulting the coefficient on x, the expression is s obviously (x-3)squared equals... 9 + 135, or 12squared. I didn't time it but I would say less than 20 seconds. But I think this is basically the method in the article, which is basically the method I learned 50 years ago, which is also how you derive (never memorize) the formula. Not sure what the fuss is about.
3
So much space and so many comments on something trivial for those that have learned the 'old math' before the 70s.
People are unaware that the old math was not about calculating things, it was about the application of logic.
Logic is not only used in math, it is also used in law. According to the logic of the law, Trump should have been impeached.
But according to the different logic of the application of the principle 'might is right' and the installation of fear, he was not.
4
Really interesting and possibly helpful article for those of us mathematically challenged. One problem. The background music was TOO LOUD! I couldn't hear Professor Po-Shen Loh's explanation which might have cleared some of my life-long difficulties in understanding mathematics. Mix it down please - or change the selection to something less dynamic.
1
One more point. You write that factoring becomes a guessing game and that it is cumbersome with large number and only works well when the solutions are integers. All true (except the guessing is fairly informed). Which is exactly *why* people use the quadratic equations in those cases, and when "a" is not 1. This 'new' method isn't replacing factoring. Or rather, if a quadratic is simple enough to factor easily, then this method is not better than factoring, and if the eqn is hard to factor, well then you are using the quadratic eqn, and I've already commented on that (See DarthVader thread). All that said, as mentioned, I do show this geometry to my students already and am open to any new perspectives that help students. But this method is not easy unless a = 1, and so not exactly a cure-all for the quadratic eqn.
12
@Elizabeth x² – 6x –135. Can you factor? How about x² – 6x –136? Except for pretty trivial cases like the one in the article, it's a slog to determine whether a quadratic is factorable, and you could waste a good amount of time trying to factor something that isn't factorable. One way is to just plug into the quadratic formula, which is blind recipe following. Or you can use Dr. Loh's method, where you can see how the answer comes about.
11
@Elizabeth I think this is very similar to teaching elementary school students that multiplication is the same as repeated addition, eg. 2x3 is the same as 2+2+2. You can solve the problem by repeated addition or simply memorize the multiplication tables. At some point, it is more productive to memorize the tables (or use a calculator) so you can solve more complex problems quicker. So it is with this method which IMO simply derives the quadratic formula each time. At some point, you are better off memorizing the quadratic formula which will save you time if solving a more complex problem.
7
@Kenneth Chang
Sir, It's not "a slog" at all to determine whether or not a quadratic equation is factorable.
For ax^2 +bx + c = 0, with non-zero value of a,
the LHS is factorable over the real numbers if and only if the discriminant, b^2-4ac, is non-negative.
If non-zero a, b, and c are rational, the LHS is factorable over the rational numbers if and only if the discriminant is a perfect square.
The method conveys no info not found in the quadratic formula, but if the method helps students "see" what's going on, then great!
2
It's a little surprising to see an article about quadratic equations in the NYTimes. And this method actually seems easier to explain than completing the squares. I guess I'll teach this way from now on.
We already spend too much time talking about quadratic equations anyway, so if it's faster to explain I'm up for it.
But I have to say, quadratic equations should be easy enough to learn by any method or else there is no point in learning them at all. And most of the time the students get confused not because the method is too complicated, but because they don't have enough background. You have to spend some time manipulating numbers and fractions to actually be able to understand high school algebra.
1
I love it. It’s a very simple visualization of a quadratic equation. Also it puts the emphasis on the x intersects, which gives a deeper meaning to the equations. A good foundation in the basics helps tremendously when approaching the more advanced subjects in math.
1
I had trouble following this article because I haven't taken algebra in 30+ years. (I don't miss algebra.) However, I'm hopeful my high schoolers will find it enlightening and I'm supportive of teachers helping students find methods that work for them. People learn differently and this approach could be very useful indeed.
1
I had trouble following this because I know too much math, and it’s hard to see how you can trivially recast the solution as new, especially when the way you derive the quadratic equation follows this same “Babylonian” method. If drawing pictures helps someone understand what’s going on then I’m all for doing what it takes to teach, but invoking Babylonians for mundane algebraic manipulation is a stretch. Now if you come across a trivial Babylonian explanation for why no three positive integers a, b, c satisfy a^n b^n = c^n for integer n>2 then by all means publish.
Why wouldn’t you just say graph the equation and pick off the two x values that give you the Y. Parabolas are not simply the one shown with the axis of symmetry being parallel to Y axis. The axis can be 0-359 degrees. I would agree with commenters that we should give students a better view of why these equations are important. My best memory of Geometry was the first time I proved something that was complex and that I previously only memorized, having used it in my prior classes and tests. It was a feeling of empowerment that I wish more teachers could give their students.
2
@Michael Blazin 1) All parabolas of the form y=x² + bx + c are vertical in orientation, and since we're talking in the context of quadratic equations, that's all we need.
In general, the x values would be irrational, so you can't get an exact answer looking at a graph or using a calculator. In addition, the solutions can also be complex when the parabola does not intersect the x-axis. Thus, this method is more general.
6
"If I have 2 numbers whose sum is 8 then they need to be the same distance away from their average."
Huh? I don't understand what this means in plain English much less what implications it has for this "easier" way to deal with quadratics. And this, precisely, is the problem with so many so-called teachers. They have such a depth of knowledge they throw language out there thinking it's self-evident, forgetting that most of us do not share that knowledge base.
But this just triggers my continuing PTSD with American education's fetishization of algebra and its misguided conceit that every educated person needs to know this stuff.
I hold a doctorate in counselor education and, except for studying for the GRE and now, agonizingly, attempting to help my son through this morass, I've needed almost nothing I was exposed to in algebra. As many have pointed out, algebra is the single largest reason high school students fail and drop out. Do we gain anything that balances this cost? So far I've seen no reliable study that finds success in algebra is positively correlated with anything other than success in math and engineering and some sciences.
At the same time, we have massive amounts of credit card debt and an electorate that can't reliably analyze economic policy nonsense. Instead of wasting so much time on something most will never use, why aren't we educating people to manage their financial lives and to make informed decisions about economics?
2
How can you understand macro and micro economic concepts without algebra? How can you understand lease vs. buy for a car without algebra? What is the break even? Without it we have people ranting memes about inequality and capitalism without understanding the underlying processes.
3
@MMcKaibab Visually, the average of two numbers is the midpoint between them on a number line, i.e. the "middle." The average of 3 and 15 is 9 because 9 - 6 = 3 and 9 + 6 = 15. In my classroom, I draw a picture to illustrate this. (And after a few examples, most of my students are able to see this.)
You raise a larger question about the relevance of algebra, which I don't think I can adequately address here. I understand the frustration you have about it. I see it every day in my classes! Math anxiety is real. I address it by using language and examples that are more accessible, building on prior knowledge, and making mathematical connections so students see that what they learned before in math relates to what they're learning now and what they'll learn in the future.
3
I can't believe someone would a PhD would call math a waste of time. I mean, by your reckoning, people shouldn't learn the law because it's bad for their credit score and (the loans are) a huge drag on the economy. Imagine telling people not to read Shakespeare because, when will you ever need to perform a soliloquy?
I agree that math is not taught well (and if you haven't been around math in a while, I agree, this is a bit opaque), but it was my calculus teacher who made me think of math in logical terms, even after I succumbed to the "not good math" fallacy that so many people, especially women, fall for because it can be challenging.
I also think there's a value to learning something challenging, like calculus or Shakespeare, for the sake of it, even if you never use it. Plus, if you want to do any kind of science or programming as a profession, this math is fundamental, and it would be a shame to wait until college to get one's head around it. I don't think it's out of reach for most students, and it saddens me that someone who plainly values education would call basic math a "waste."
I am not good at memorization. On my final in Topological Spaces (a senior college math course), I had forgotten the quadratic formula. Fortunately, it is simple to derive. I may have been the last to turn in the final, but I still aced it.
2
Yeah for great teachers. I’m thankful for those who helped me become an engineer and broadened my understanding of the world.
1
My miserable math education included absolutely zero context for quadratic equations beyond that they were to be found on the Regent's exam. I learned more about what quadratic equations are for from paragraph 3 of this article than I did in months of dreary, test-driven algebra class.
I'm happy to learn that there are smart people out there who know how to improve math education; I'm happy for their students; and I wish I had teachers like them, who actually convey that they are interested in the subject itself.
8
Fantastic!!!
I would have enjoyed knowing this back when I took the GREs. The factoring always took me way too much time.
Interesting angle, but nothing new here. What's left out, but vastly more important in teaching the student how to make useful generalizations, are the equations for which there are no solutions in real numbers; graphically, those for which the parabola has no x-intercepts.
For example, there are no solutions to x^2 -5x + 11 = 0. The graph of that equation has no value for y; it curves upward with its minimum point above the x-axis.
Equations like these, though they are more difficult in instruction, help give students a visual sense of the relationships between quadratics with real solutions (roots) and those without.
4
@JS Dr. Loh's method works for these equations too, although the parabola sketching is not quite as helpful. It still helps remind students to rewrite the equation in terms of u, the distance from the midpoint, but it's not obvious what the solutions are since the parabola does not intersect the x-axis. But, like the quadratic formula, you still end up with the complex solutions.
4
@Kenneth Chang ; also with complex solutions you have a path to the conundrum of i, the square root of -1. That leads to all sorts of classroom energy.
I can't count the number of lightbulbs that go on when a student discovers the x-intercepts are the solutions in reals.
2
@Kenneth Chang
I am highly skeptical of this, as are obviously many other posters....but I am open to a proof of this hand-waving. Can you top-post an explanation of how this is true, please?
Oh, and major kudos for being a write who reads and responds to these comments. Dr. Krugman sometimes does too. Jah respect!
1
Sorry, for those of us who haven't taken calculus in the last 20 years (and who were taught completing the square), is the concept of the axis of symmetry so prevalent that this will seem simpler? I have to admit, to my math-foggy mind, this seemed like adding a step in which one completes the square for u (which provides an answer rather than a guess, true, but would seem to take a long time if the numbers were not so neat).
I don't remember ever discussing the axis of symmetry in math class, but I do remember the quadratic formula, some 20 years after my calculus days ended. Perhaps this is just something I've forgotten, so I am honestly asking! I would be happy to hear that the pedagogy has advanced since I was a teen, as I had a great calculus teacher who helped me a lot, but some concepts (apparently, this one among them) that eluded my grasp.
1
A better question would be "why do we still require this to be taught in high school?" This sort of problem very rarely needs to be solved in the real world and, on the off chance you do need to solve the problem, there's dozens of Web sites and apps that instantly give you the answer. It's good for kids to know what a parabola is, but knowing the math for finding intercepts and lines of symmetry and refactoring the problem has little value. A much better use of class time would be teaching people basic programming skills, so they can be creative and understand how a computer works.
@efbrazil I read your comment with a tinge of sadness. Mathematics is the language of science. Algebra and trigonometry are foundational topics without which higher physics and engineering cannot be understood. There is a lot more to be done in this world than programming computers.
5
As a software engineer and CS degree holder I can very much assure that I use the ideas I learned in algebra and all of discrete mathematics quite regularly for my job.
4
mathematics is logical thinking. the flaw in mathematics education is that there is too much focus on solving specific mathematical problems rather than on the logical thinking that leads to the solutions. This country is in fire need of logical thinking.
1
I have always believed (and taught) that there are many ways to solve most math problems: algebra, geometry, trigonometry, calculus, ...
Be that as it may, I'm not sure that for the average student this method is "easier." Nor do I think that for the underperforming inner city student this method is easy at all.
But what the heck, if it helps at least one student, it's useful.
13
@More than one way to skin a cat I agree, which is why I put stock in what a math teacher like Bobson Wong said when he tried this approach with students who were not great math students. He provides a detailed account here: http://bobsonwong.com/blog/27-on-a-different-method-for-solving-quadratic-equations
His conclusions, in part:
"In general, I find that the method of solving quadratic equations described by Loh has several advantages over other procedures that I’ve taught in high school math:
"As Loh points out, students can use one method to solve all quadratic equations. Of course, students could also use the quadratic formula to solve any quadratic equation, but in my experience this method is easier for students to manage. Many students have difficulty substituting correctly into the formula. The algebraic manipulations used in this method are generally simpler and easier to remember than those required to substitute into the quadratic formula. I found that many students who had great difficulty correctly substituting into the quadratic formula had much more success with this method."
3
@More than one way to skin a cat I freely admit that many students may find the quadratic formula easier. Many of my students simply like substituting into a formula. But I think this method has the advantage of providing a visual understanding of solving quadratics. And as I said in my blog post, manyy students have a hard time substituting into the quadratic formula - they don't handle parentheses, taking the negative of a negative number, and squaring a negative number well. In contrast, this method requires students to be good with simplifying fractions. While this can be troublesome for many students, I find that most modern graphing calculators can perform operations with fractions quite easily (calculators can return a fraction in simplest form).
2
As a software engineer and CS degree holder I can very much assure that I use the ideas I learned in algebra and all of discrete mathematics quite regularly for my job.
1
This “discovery” about quadratic equations is like the “new math” of the 1960’s: It takes a mildly abstruse subject and makes it unnecessarily confusing and inaccessible. It adds unnecessary wonkiness.
Nevertheless, the ability to formulate and solve quadratic equations is basic to numerical competence. This is not a difficult skill to master.
EVERY student of STEM should be able to DERIVE the quadratic formula without breaking a sweat. Doing so is not difficult and only involves a few fundamental concepts of mathematics that everyone should know. I have done this many times when I have forgotten some detail of the formula. So have many other people.
The quadratic formula is simple, straightforward, doesn’t require a computer, and will ALWAYS work when applied properly.
Prof. Loh’s “discovery” does mot provide any new insights into quadratic equations beyond what existing methods already supply. It is much ado about nothing.
4
@JTS I respectfully disagree. Deriving the quadratic formula can be a tedious algebraic exercise. I've tried doing it in class and no matter what level of students I had, the result was the same - glazed eyes, confused expressions, and the dreaded question "Will this be on the test?" I and most other teachers I know don't spend a lot of time deriving the quadratic formula in class and certainly don't require students to derive it on their own. It just doesn't seem like the most effective use of our class time and would increase their math anxiety, which is bad enough as is!
2
@Bobson Wong
There is a good reason why proofs and derivations constitute a prominent part of mathematics classes: They lay bare the logical structure of mathematics as well as any underlying assumptions. This is what allows useful conclusions to be drawn and, more importantly, it helps identify and rectify errors in logic. Such is the foundation of logical thinking. This is a critical skill for young people to learn.
If your students are having trouble with the quadratic formula, you need to help them understand it so that they could do it in their sleep. If they can’t handle straightforward things like the quadratic formula, how can you expect that would be able to handle other important things like Maxwell’s equations?
While some might find this “trick” interesting or amusing, it really doesn’t help us with understanding and solving mundane, everyday problems.
I don’t remember this old method at all, and I’m pretty sure we just solved quadratics with ye olde TI-83 graphing calculator, which now seems like a very appropriate pedagogical compromise if the alternative technique was basically trial and error.
The quadratic formula popped into my mind as I was hiking along a local river last week. Why? I have no idea, hadn't thought about in decades - just amazed at the detritus squirrled away in dark recesses of my brain. Always liked math though not very good at it. In any case found this article informative and fun.
4
45 years ago, my Stuyvesant calculus teacher had a student from another of his classes make a presentation. We all used a textbook that already had many editions which had as "motivation" for calculus a simple minimum problem ostensibly about finding the most profitable height for a commercial building. The student was offended at both the nature of the problem and the fact that calculus was not needed: If I remember correctly he solved it using ideas that preceded calculus by at least 1000 years: the triangle inequality and trigonometry. We solved many problems in Math Team using symmetry. Today, that student is a math professor on the West Coast.
This method requires that the students know:
* The parabola is symmetrical about x = -b/2a.
* r*s = -c (see the prose).
Neither will be obvious to the students. Just more stuff for them to memorize.
To my mind, this method is no simpler than completing the square, which completely self-contained.
14
@Darth Vader You don't need to know anything about parabolas. Dr. Loh doesn't use them at all, but Bobson Wang found it helpful for his students to sketch a parabola, so they would remember how to rewrite the equation in terms of u.
All Loh does is rewrite the quadratic in terms of the midpoint, (r+s)/2.
From what you learn in factoring quadratics, you already know r+s = –b, and therefore the midpoint is –b/2. (Insert a in the denominator for the general case where a is not 1.)
r*s = c is the same thing everyone learns when factoring quadratics. It comes from expanding (x-r)(x-s).
9
@Darth Vader Exactly. I don't (always) remember the quadratic formula, but it only takes a few minutes to derive it and it makes quick work of the problem.
2
@Darth Vader all he is doing is completing the square anyway, but just not calling it that. It is just like doing synthetic division instead of longhand polynomial division. If someone finds it helpful great!
3
Yes! Almost any idea is better when it can encompass a graphical representation, "show me the picture". As a scinece teacher, with a math credential, I would have jumped all over using this.
I'm going to study this tonight at home. Freshman algebra was the closest I came to failing a class. And that was 1969. I'm going to keep at this until I get it!
3
Getting tired of all these miraculous cures for math anxiety. The real cure is ability. Those without should be left to develop the skills needed in retail service.
1
Your comment makes me wonder whether you were my terrible high school calculus teacher, whose inability to explain anything beyond the one way he’d prepared to finally turned me off math forever.
2
@Daedalus my, my that was snooty.
1
While this is all very interesting, what practical problems does it solve? I never used any of my Calculus, Trigonometry, analytic geometry, although I have used some very basic Algebra and see value in statistical analysis.
2
For those of us who didn’t have great math teachers or mathematically-inclined parents, this story has real merit. I also disagree that graphing is a non-starter. It is absolutely necessary to understanding algebra. Here’s a reality check for mathematicians and math teachers who cannot fathom those of us who didn’t fall in love with algebra: This is the first time anyone has explained to me why anyone would use a quadratic equation in real life. My math teachers never had clear answers about what was actually done with algebra. It is also the first time anyone has explained to me that the quadratic equation describes the function of a parabola. That’s a pretty crazy couple of educational gaps, when you think about it. And yes, I actually got good grades in math classes and standardized tests. How is that even possible? Of course, I never understood one single practical point of algebra until I saw chemistry computations. The math classes that I experienced were pretty standard, and ultimately not very useful. What I experienced was a sort of “pure grind” of memorizing various procedures that I quickly forgot. My math teachers taught me the details of moving around variables and guessing factors, and it ultimately seemed like a kind of school busy-work. Here’s my analogy: This is like giving someone a string of directions without saying where one is headed, why, and what the map looks like. At least this method gives someone some answers to those conceptual questions.
4
I still don’t get it. But I like the projection of the martinis in the picture behind Dr Loh.
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@Trish Yeah, that was a real cool problem where he was showing how people's mathematical intuition is way off sometimes. It's a bit hard to see, but which one of the martini glasses is half-full?
7
@Kenneth Chang Lets see...is it the one that looks least like it’s half full? No?
Thanks for writing this article. I love math in the mainstream. Not just in journal articles for the esoteric scholarly, though they have a contribution to make, too. Personally, I’m not a mathematician, just a math aficionado. I’ll play with this equation on the weekend.
3
@Trish A quadratic equation is a parabola. It’s symmetric over a line down thecenter point of the curve. If you fold on tha line the lines match. If you want yo find where it crosses the x axis ( horizontal line) those crossing points are the same distance from that center line point where it crosses the x axis.
2
You may be interested in a book by Beatrice Lumpkin called "Algebra Activities from Many Cultures" published by J. Weston Walch c1997
It has something called "systems of three equations: the Chinese solution," where you add three quadratic equations together to solve.
I really don't understand it but I found this book when I was home schooling and it seems very good if you could understand it. Interesting to see the history of mathematics from Africa, Guatemala, China, Egypt, India etc.
1
"We want r + s = –b" needs to show that the minimum is at x=b/2 ; not trivial for algebra students
"To find u, we want the product of r and s to be equal to c.." is easily shown using the standard formula, but that seems to miss the whole point.
All in all, this is more mysterious than completing the square..
5
@jwdooley r + s = –b comes simply from multiplying out (x–r)(x–s) = x² – (r+s)x + rs = 0 and comparing the coefficients with x² + bx + c = 0. Similarly, rs=c. You don't need to know anything about the parabola, but it helps when rewriting the equation in terms of the midpoint u.
3
@Kenneth Chang
Nice - thanks for that
1
Completing the square is a perfectly natural operation. It is just, like much in elementary Math, taught very badly.
3
"You might recall your teacher asking you to factor the jumble of symbols."
This is how we dumb down mathematics. I would never make such a comment to impressionable students.
When dealing with quadratic polynomials which cannot easily be factored, instead of using the memorized quadratic formula to find the roots, work through the steps involved in its derivation:
ax^2 + bx + c = 0
a^2x^2 + abx + ac = 0
a^2x^2 + abx = -ac
a^2x^2 + ab x + (ab/2)^2 = -ac + (ab/2)^2
(ax + ab/2)^2 = -ac + (ab/2)^2
ax+ab/2 = either Sqrt[ac + (ab/2)^2] or -Sqrt[ac + (ab/2)^2]
now solve for x
The above is a sequence of equations (not a jumble of symbols) expressing a sequence of ideas in mathematical notation.
2
What about for cases when the quadratic equation has no roots or one root?
1
@C. The method still works. (The drawing of the parabola is less helpful, though). You end up with an expression that includes the square root of a negative number. This is the part that the Babylonians didn't get. The case with one root is equivalent to a parabola with its vertex on the x-axis.
2
@C. The method will still work. Consider y = x^2 + 1, so that the axis of symmetry is the y axis and there are no real roots. This method reduces the problem to solving the equation -(u^2) = 1, which is the same as u^2 = -1, which gives the correct solutions to the original equation. But the diagram does break down, because you can't sketch u in the same way as the diagram in the article.
I don't see what the big deal is. I'm sure lots of people have figured this out. I've been doing it this way for years and figured it out on my own. We just need more teachers to understand it.
You can actually optimize it slightly from how it was described, since the description was trying to both show why it worked and show the calculations at the same time.
I always divide by a to start, making sure the coefficient of x² is one. So I'll call "b" the "new" coefficient of "x" (after dividing everything by a).
Then the x-coordinate of the vertex is -b/2. To get the y-coordinate, take that x-coord, square it, and subtract c. That's actually the negative of the vertex's y, but it gives you the positive distance from vertex to x-axis. The "spread" to each side will be the square root of that, as shown in the image... what they call "u".
That's what you add or subtract to the x coord to get the roots. To sum up: (1) get rid of "a" (2) x-coord = -b/2 (3) "spread" is square root of "x-coord squared, minus c". Add & subtract spread from x-coord to get roots.
(Note: if "x-coord squared, minus c" is negative, you get complex roots)
4
While many of the people replying here are arguing the finer points of why this may or may not be novel, or that they don't understand how it's "simpler" when the established method is so easy to understand, let's not forget that most algebra students don't get it, and this may help.
8
While this is probably an easier way if one uses the trial and error method, everyone that goes thru high school should remember the formula to solve quadratic equations. If remembering THAT is hard, then nothing can save you :)
Approaches like this are useful sometimes to understand what you are actually doing; but kids that understand this shouldn't have trouble remembering the old formula (or derive it).
3
A suspension bridge is not a parabola. It is a different shape, with a different equation, called a CATENARY
6
@thoughtful in new york If it were just the suspension cable, it would be a catenary. But a suspension bridge is a parabola because of the weight of the road deck.
See http://john.maloney.org/Structural/parabolas_and_catenaries.htm under the section, "The hanging chain: Incorporating external loading."
10
Unlike the assertion in this article, the curve of a suspension bridge is unquestionably NOT a parabola. It is a "Catenary curve", formed anytime 2 ends of a rope or wire, for example, are held up at the same height and the line is dropped naturally. It will fall exactly into a catenary curve - whether it's the Brooklyn Bridge, a jumprope, or electric line parallel to, and high above, the railroad tracks. This curve is commonly mistaken for a parabola but it isn't one, and the two have different mathematical equations that describe them
3
@semari A hanging rope is in the shape of a catenary. But a suspension bridge, which is supporting the weight of the road deck, is a parabola. Even the Wikipedia article on catenary explains this. Look under the section, "Suspension bridge curve." https://en.wikipedia.org/wiki/Catenary
7
@semari A lone rope or cable makes a catenary, but the suspension bridge with a deck supported by cables at a regular interval is a parabola. Look at Kenneth Chang's proof of that in his other comment if you can't believe it.
2
@David Weintraub It seems to me that, assuming the bridge deck is straight, the shape of the main cable is determined by the lengths of the smaller cables that run from the bridge deck to the main cable.
As a thought experiment, if there was only one connection between the bridge deck and the main cable, at the very center of the bridge, then the cable would have the shape of two straight line segments going from the tower to the bridge deck and back to the other tower.
Similarly, by making several connecting ables have the same length, it would be possible to make the main cable run parallel to the bridge deck in a region near the center of the bridge. This would be horrible engineering, I'm sure.
I read the article with a general interest in viewing a way that someone can be enthusiastic about something, as he shares rediscovered knowledge. It links humanity and moreover, people within the past, present and future. Also, while I'm also a math phobic and typically declare (even about calculating a tip) "Too much math!", Dr. Loh helped lessen the terror a bit.
8
One thing I recall that my college algebra professor said was "you'll forget more math than you can remember."
That was a spot on prediction, as I had to take trigonometry, college algebra, and general chemistry all twice just to feel comfortable that I could get better than a passing grade.
Chemistry's use of stoichiometry to balance equations of mixed units was by far the most useful tool I could remember and apply everyday.
It makes me wonder why, when the Current Occupant of the Oval Office, fabricates complete false facts out of thin air, there should be an equation that would graphically show the theory of it. However, there has gotta be an equation that uses facts to prove how far off his arguments are...!
HOMEWORK ASSIGNMENT:
Could some academic group please, please, pretty please, graphically plot those lies verse facts at hand and show the equations...?
1
"HOMEWORK ASSIGNMENT: ..."
I'm not sure if you are trying to be funny or really want an answer, but, assuming the latter ...
There are numerous books on critical thinking, fallacies, and rhetoric. Here is a good one to start with:
"With good reason : an introduction to informal fallacies" by S. Morris Engel.
A search at a library or an online bookseller will find more.
3
This is a Double Jeopardy question, and I don’t have the time nor inclination to remember where to run this to ground.
Category: tRump’s declarations and predictions vs. facts...!
I guess I don't get why this is "simpler". It isn't. It simply rederives the quadratic formula. It appears simple on paper because, for simplicity, a has been assumed to be 1. You'd see that it is exactly the quadratic formula if you worked out the general case of a being different from 1 - as it should. I really don't see how this makes it simpler. A good algebra teacher should show their students the basis for the quadratic formula anyway versus just asking them to memorize it.
9
@Michael If you want the version for a not equal to 1, divide through by a. Then it's x² + (b/a)x + c/a = 0. Proceed as before.
Of course, it ends up with the quadratic formula. It has to! I personally find it's an easier derivation, and plus, I don't have to remember the formula, which I don't.
13
@Michael Not having tried teaching it, but having taught a lot of math - the fact that the method has a clear visual interpretation will make it more accessible to a number of students. Whether that makes it "simpler" is a matter of opinion, but people generally seem to do better with processes that have a small number of steps in which there is a clear motivation and picture for each step. I think this method is very intriguing, actually, because of it has those things. One thing that is not known to the general public is that there has already been a shift in teaching how to factor quadratics, from the pure "guess and check" method I learned to the "A-C" method that a number of contemporary high school books teach. The reason for the shift is that the "guess and check" method turns out to be very opaque to many students.
6
@Kenneth Chang Or multiply by the coefficient of x^2. I think the idea here in this article is to move the coordinate system to a friendlier spot. Since solving Ay^2 + D = 0 is easier, why not do a lateral shift, as we do for cubic equations. In other words, in Ax^2 + Bx + C = 0 replace x with
y+v and collect like terms in y. One sees that if you choose v = -B/(2A) you get an equation of the form Ay^2+ D = 0. Mathematics is frequently the practice of reducing new problems to ones you have already solved!
2
Doesn't this method require knowing the axis of symmetry? the old -b/2a to find the axis of symmetry in standard form is just a generalization from what you would get if you completed the square. Doesn't seem like anything too new here. Not sure why it is getting so much attention.
3
@Jesse Not really. It's taking advantage of r + s = -b. The average, or midpoint, of r and s is (r + s)/2, by definition, and thus the midpoint equals -b/2. In a lot of ways, this is the same as completing the square, but by recasting the equation in terms of the distance from the midpoint, the expressions are simpler because the linear terms subtract out. It's equivalent — it _has_ to be equivalent — because in the end, it's producing the same answers to the same equation.
1
First you have to know the axis or have memorized that (r+s)=-b. Next you have to remember that r*s =c. Then you have to solve for r and s. In other words there is a lot to remember here as well.
Why on earth is this any better than just memorizing the quadratic equation formula?
Absurd.
It may be nice to have the visualization, but it certainly is no trick nor does it save time or effort.
@Kenneth Chang The point, I think that Jesse made is that this method IS simply completing the square explained a different way. In fact, the steps are essentially identical (including how to deal with the leading coefficient a not being 1). Only the justification (explanation) differs. In both case the equation is written (assuming the a=1 step has been done already) with the second coefficient b as a negative (so it may need to be a negative-negative) and then by rote the one divides b in half (which might give a fraction), square it, subtract the third coefficient c, and then take the square root of that. The square root is then added and subtracted to get the two solutions. It's the same steps in both methods.
Algebra books in the 1950 for sure taught completing the square as one of four ways to solve quadratics (factoring, using the formula, and graphic were the other three, but of these graphing is only approximate).
I have no trouble at all remembering the quadratic formula, which is based on completing the square. It is imprinted in my head; I'd tell it to you if you woke me up from deep slumber.
1
For anyone who wants a more formal and detailed discussion of this, take a look at Po-Shen Loh's site: https://www.poshenloh.com/quadratic/
and his preprint:
https://arxiv.org/abs/1910.06709
1
One Algebra teacher's response: This approach presupposes that our relatively weak algebra student who can't rapidly factor the quadratic equation in the usual manner can graph it. Many students struggle with graphing, so this approach will be a non-starter for them. And, while it may be easy when a=1, so is the usual approach. And, if a does not equal 1, (or if a=1 but b is odd), then this "new" approach will immediately drop the student into the land of fractions which is also not a comfortable thing for many students. The beauty of the traditional factoring approach is precisely that the solver only has to deal with integers until the very end when the quadratic has already been factored into linear expressions.
13
@Jonathan Bender Factoring of quadratic equations is weird. What a weird coincidence that these integers that happen to work! And that's because these are the few contrived cases where the equation is readily factorable. As soon as you go to the more general case, you have to shift to the formula, and this is IMO a much better explanation of where that comes from.
4
@Jonathan Bender To find the vertex one only has to factor the x^2 and x terms, as changing the constant just lifts or lowers the graph but leaves the vertex unchanged. Solve the wrong equation (ax^2+bx=0) and use that for the axis. Since one factor is x itself this is not hard even for students you might label weak in algebra to get ax+b as the other. The middle value is then in hand. Hope this helps.
1
@Jonathan Bender I also taught this method when a is not equal to 1 or when the averages are fractions. Unfortunately, most of my students struggle with fractions. However, most modern graphing calculators can perform operations with fractions and return a fraction in simplest form. When some of my HS students struggle with multiplying signed numbers, I do what I can to help them but also encourage them to use a calculator. It's not ideal, but in the limited time I have, it's a concession I've been forced to make. Plus, getting them past simplifying fractions enables them to solve complicated quadratic equations, which can help build their mathematical confidence. That has far-reaching implications that go beyond simply solving quadratic equations.
1
The article and the comments (e.g. the comments that say, this is just completing the square) nicely illustrate how a slight change in approach can be helpful to understanding a problem and its solution. I object mildly to the use of the word "trick", especially "amazing trick". Mathematical discovery and insight usually come from thinking about a problem from different points of view. Calling the insight that comes from a different approach a "trick" makes mathematics seem less accessible, more like magic, and more dependent on special cleverness than it really is.
8
@Matthew A As a mathematician, I use the word "trick" to mean something different than "magic". Loh is using the word somewhat in this sense. There are many mathematical techniques called "tricks" - see the post "What do named “tricks” share?" on the site MathOverflow.net . Indeed, this method for solving quadratics may come to be called "Loh's trick".
1
I don't think this method is new at all. I wrote something about it on Facebook years ago as a means of deriving the quadratic equation.
I didn't think it was particularly novel, but then again I am not a maths professor.
2
@David
That was a pretty dope humble brag.
Re: ax² + bx + c = 0
Seems to me I first heard of this while reading Shakespeare.
It was called "Much Ado About Nothing."
3
What a delightful article! It is always beautiful when geometry (graphing) is combined with algebra.
1
My father, a nuclear physicist, taught me a similar method to solve the two-body elastic collision problem, which arises in the first few weeks of any introductory physics course. Conventionally, it is attacked by means of a quadratic equation. My father's method, which he attributed to Einstein, imagines the problem as viewed by a moving observer, from whose perspective the two bodies are traveling in opposite directions, with opposite momentums. To this observer, the bodies simply reverse direction. Solving the problem in this manner is intuitive, requires less arithmetic (and no square roots), and even generalizes to multiple bodies in higher dimensions. I taught it to my son, who taught it to his classmates, who used it to get the correct answer on a high school exam. The teacher failed them all because, as she said, the point of the exercise is to apply the methods taught in class, not simply to get the right answer. (sigh)
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@Loren Platzman
Amazing!
2
@Loren Platzman,
Very sad...
Hopefully, the students learn something about poor teaching that is valuable to them.
Perhaps you could write it for for the N.Y. Times and or do a YouTube video?
3
@Loren Platzman,
Very sad...
Hopefully, the students learn something about poor teaching that is valuable to them.
2
As an EE, the quadratic formula is etched into my soul (as are many others). It didn't happen by memorization but, rather, continual use. Often, the abstract is best learned from the concrete.
6
Much easier to just graph the parabola on a graphing calculator and observe the zeros. Same basic method.
1
@Bruce McClure Until the answers are complex. This method works for all quadratic equations.
17
@Bruce McClure
Two additional points to Mr. Chang's comment:
1. Working through the logic without a calculator or computer allows our brains to weave this concept with additional concepts that may be useful in conjunction.
2. We as a society and species must not lose the general ability to understand and perform calculations like this one else we regress in our grasp of the world around us.
8
@Bruce McClure It also depends on what you're doing with the answer. Sometimes it is valuable to have an algebraic answer.
3
The first proof my College Algebra students ever see is often the Quadratic formula, obtained by completing the square. I have heard audible gasps on occasion. They had no idea where it came from or that there was another method that does exactly the same thing.
K-12 education has stopped teaching how and why math works, and has focused solely on the mechanics. My college students understand almost none of the core concepts of math (the multiplicative identity is pretty useful, but they don't know it), but know dozens of shortcuts. On the off chance that they remember when the shortcuts work (and when they don't work), students still have absolutely no idea why.
It's like knowing that a Quarter-Pounder with cheese is called a Royale with cheese in France, and never having heard of the metric system.
6
Not clear to me how the Professor knew that the axis of symmetry was 2. I know how to calculate the axis of symmetry but it involves "completing the square" in its general form
ax^2 +bx +c = 0, but it isn't mentioned in this article.
4
@DJR The average of r and s is (r+s)/2; that's the definition of average.
What Poh's method does is recast x in terms of u, the distance from the average. Visually, the average is the axis of symmetry of the parabola and u is the distance from the axis of symmetry, but you don't actually need to know that. However, Bobson Wang found that made it easier for his students to grasp the concept.
Completing the square is mentioned in the article, it would have taken a good chunk of space to explain it.
4
"... how the Professor knew that the axis of symmetry was 2."
The article glosses over some important background points, and that is one.
In introductory calculus, you learn that the maxima and minima of a curve occur where the slope of the curve is *zero*. And you learn that the *first derivative* of a function gives you the slope of the curve.
In the case of a quadratic function, f(x) = a x^2 + b x + c, the first derivative is:
f'(x) = 2 a x + b
Setting f'(x) = 0 and solving for x gives:
x = -b / (2 a).
That is where the parabola has slope zero, and where the axis of symmetry lies. In the example, a = 1, so that simplifies to:
x = -b / 2.
The second graphics box uses that fact, although it is very hard to see how. What the box should say is:
"We want (r + s) / 2 = –b / 2 ..."
Rigorously showing that a parabola has an axis of symmetry is a separate problem ... :-)
4
@GAK One can simply verify that for all real z, the function has the same value at ((-b/2a)+z) and ((-b/2a)-z). This argument is not explanatory, of course, but it is straightforward as a verification that there is a symmetry across the line x = -b/2a. Or, more briefly: a coordinate shift that moves to origin to -b/(2a) converts the equation into the form y = ax^2. Working mathematicians would view it as a perfectly rigorous (and routine) proof.
1
Just glad we have mathematicians who can understand it. It has been so long that I did quadratic equations that I found both the original method and the "improved" method still puzzling. So much for my math literacy.
3
What's old is new again.
Reading these comments, one is struck by how many people express a sense of competence on this subject.
That's because the majority of adults today have had some exposure to algebra by at least the 9th grade.
As some have mentioned in these comments, the technique for solving quadratic equations mentioned in the article, has been taught for decades and centuries outside of the United States.
3
More than offering another way to solve (and appreciate) the quadratic equation, this method provides an opportunity to teach both history (ancient Babylonians) and math at the same time. Too often, students learn some formula named after a famous scientist or theoretician but never learn about the context of its discovery, which might point to an alternate view of its derivation.
3
This approach is virtually the same as deriving the quadratic formula. While some students struggle with remembering and executing that formula, this approach is no godsend. Students will stay have to understand the roles of r,s and u and execute as well.
Perhaps time would be better spent understanding why people can't use the quadratic formula, rather than replacing it with something comparable.
3
The method is identical to the well known formula for the two roots: {-b+sqrt(b^2-4ac)}/2a and {-b-sqrt(b^2-4ac)}/2a.
If a is reduced to 1 (as was done by Dr. Loh in his explanation), the formula is the method described in the paper. The paper does give an intuitive feel for the formula, and in that respect, it is worth reading.
This is a great method for solving quadratics, but it is hardly new. As the comments on one of the Youtube videos announcing this discovery indicate, many people whose schooling took place outside the US learned this method decades ago, and it has been taught around the world for decades if not centuries. I learned essentially this exact method at middle school in the 1980s in New Zealand. This method is "completing the square" with a few cosmetic differences (which the article describes as "a bit convoluted" but it's no more difficult than the method shown). Completing the square is the method by which the quadratic formula (which American students are apparently very familiar with) is usually derived, so it's clear that it's not a new idea.
44
@JL It is not quite the same as Dr. Loh explains here. The distinctions in logic are rather subtle and perhaps insignificant in practice, basically, "How do you know there are only 2 solutions?" https://www.poshenloh.com/quadraticcompsqr/
4
I fully agree with your comment. I remember this “ancient” technique being taught in a math class for 14 year olds in Austria.
1
I teach calculus at the college level, and I dare say students would have an easier time with basic algebra if they did not use the TI-84 or similar calculator in high school.
Something that easy to factor in the given example should take just a few seconds by looking at the factors of the constant.
4
I learned this technique in the early 80s in Wisconsin in advanced algebra. So I'm not sure this guy really "rediscovered" anything unless I'm missing something.
11
@Julie Usually what people learn is to "complete the square" rather than recasting the equation in terms of the midpoint. It isn't something completely new — it has to produce the same answers to the same equation — but it makes it easier for many students who are confused by the abstract notation of algebra to see and understand what's going on.
8
@Kenneth Chang
Recasting the equation in terms of the axis of symmetry for the graph is *precisely* what ''completing the square" does. The crucial step in completing the square for the example presented in this article is to write:
x^2 - 4x - 5 = (x - 2)^2 - 4 - 5
= (x - 2)^2 - 9
which displays the quadratic as a polynomial symmetric about the origin of the "shifted" coordinate x' = x - 2. Note that x' = 0 precisely when x = 2. And contrary to the article there is no guesswork involved in this; it's all spelled out in the completing-the-square algorithm.
I think that Professor Loh's contribution here is pedagogical rather than mathematical, namely, emphasizing the graphical interpretation of "completing the square" to complement the algebraic steps the method entails. This helps beginners who haven't yet learned to conceptualize algebraic operations in geometric terms. Nothing wrong with good math pedagogy, but it hardly rates the overblown, sensationalistic headline the authors employed. ("Ancient Babylonians"??!)
4
This does not seem generalizable to roots of higher order polynomials. Should we teach special techniques like these and leave kids with the impression that this is right way to understand?
1
"This does not seem generalizable to roots of higher order polynomials."
Actually, the importance of the method is that the *symmetry* of the quadratic is used to factor it.
In practice, a calculator or a scientific computing package would be used to find the roots of general higher order polynomials.* See the Wikipedia article, "Root-finding algorithm".
"Should we teach special techniques like these and leave kids with the impression that this is right way to understand?"
Dr. Loh is definitely interesting teaching the "right way to understand":
"Math is not about memorizing formulas without meaning, but rather about learning how to reason logically through precise statements."
* Indeed, there are no general formulas for polynomials with degree greater than 4. See the "Abel–Ruffini theorem" article in Wikipedia.
"Quadratics, which are introduced in elementary algebra classes, pop up often in physics and engineering in the calculating of trajectories, even in sports."
More precisely, the trajectories are for a constant-mass projectile moving under a constant force without friction. Physics and engineering students get to solve the harder cases in which there are varying masses or forces, and there is friction, such as air resistance.
Parabolic trajectories are beautifully illustrated in "The Birth of a New Physics" by the science historian, I. Bernard Cohen.
The illustrations are stroboscopic photographs of a ball arcing along a parabola. The photographer, Berenice Abbott, is best known for her photographs of New York City, but Abbott also worked as a scientific photographer for time.
See "Berenice Abbott : a life in photography" by Julia Van Haaften.
4
Wow, a lot of comments here say that the method he is showing is so technical, but in reality he showed it’s use on a super simple problem that didn’t really benefit from its use. Try doing this with more complex problems and you will find his method is the absolute best way to it! Honestly, I wish I was taught this when I was in school.
7
Not sure why it’s taught so technically to make it difficult. I learned it differently (and simpler) in high school in Nigeria.
Adding r and s to the problem just increases the complexity. I quickly and accurately got the solution the old way.
10
Too often math is taught as discreet abstract formulas somehow cloistered from real world applications. While Dr. Loh's method may still appear as gobbledygook to many, it takes a step in the right direction.
1
Is there a generalization for all even polynomial functions? Can this method be iterated? Does this work for functions that are symmetric about the x-axis instead of the y? Can a phase shift be added to make a function symmetric about the y-axis, the roots found, and the phase shift removed?
1
@PictureBook No. No. Those are not functions. That in essence is what is happening.
I get -5 and +1? If you FOIL (x-5)(x+1), you get x^2 + (1x) + (-5x) + (-5) which simplifies to x^2 -4x -5. What am I doing wrong?
You’re not doing anything wrong, you factored the equation as (x 1) (x-5) which means the solutions are x = -1 and x = 5. Plug these back in and you’ll see this is ok.
1
"... uses the fact that parabolas are symmetrical."
Mathematically proving that is a bit of a challenge, but using symmetry to solve problems is one of dozens of heuristics described by George Polya in his classic book, "How to Solve It: A New Aspect of Mathematical Method".
"We want r + s = –b, which happens when the average of r and s is –b ÷ 2."
That also needs to be proven. The vertex of a parabola is where the parabola is a maximum or a minimum. The usual method* for finding a maximum or minimum is to use calculus to find points where the slope of the curve is zero.
"... where the parabola crosses the x-axis."
Obviously, that doesn't happen with some parabolas, in which case, the quadratic expression cannot be factored.
The point is that a lot of background knowledge is needed to fully understand the method.
* See, however, "Maxima and Minima Without Calculus" by Ivan Niven (1981).
23
@GAK,
My ignorant thought (apologies in advance) is that for a parabola that doesn’t cross the x-axis, create a new axis (x’) which the parabola meets or crosses.
I won’t be surprised if it doesn’t work, but I would like to understand why.
Thanks.
1
Stephen: "... I would like to understand why."
Translating the parabola would give a different quadratic equation. Some quadratics simply don't have any real roots. A simple example is:
x^2 + 1 = 0
That can be rewritten as:
x^2 = -1
Since there are no real numbers with a square of -1, there are no roots for the quadratic equation. You can also try graphing:
y = x^2 + 1
However, your general idea is a good one. Transforming a problem into another problem that can be solved more easily is commonly used in mathematics.
A classic example is using the Laplace transform to solve certain differential equations.
Another advantage of thinking in terms of the parabola is that it makes it obvious when there are no real-number answers (when it doesn't cross the x-axis) or only one answer (when it is tangent to the axis).
5
The cables of a suspension bridge could be forced to fit the shape of a parabola, but generally they are not. Suspension cables - and the St. Louis Arch - follow a catenary curve.
54
@Paul Berger-Durnbaugh The St. Louis arch is a catenary. Suspension bridges, however, have the shape of a parabola (because of the added weight of the deck). There's an explanation on this page (fairly far down): http://john.maloney.org/Structural/parabolas_and_catenaries.htm
13
Actually if the bridge cables suspend a decking or roadway then they do follow a parabola. If it's just a hanging cable alone then it's a catenary.
4
@Paul Berger-Durnbaugh It depends on what you mean by “suspension bridge”: https://en.wikipedia.org/wiki/Simple_suspension_bridge.
Mr Chang, some editing comments (I admire your patience in replying to so many comments!):
"A more general equation for a parabola is a quadratic function:” would work better as "A more general equation for a parabola is:” followed by the equation. The equation determines a function (or class of functions) but is not itself a function and there doesn’t seem to be any need to talk about functions in the article. But the article needs to indicate what a quadratic equation is. After the equation y = x^2, one might say, “Equations with a squared term like this but no term of higher degree such as x^3, x^4, . . . are called quadratic equations."
"the symmetrical points r and s”: r and s are lengths, not points. The points (r, 0) and (s, 0) are reflections of each other across the axis of symmetry.
"The two solutions to the quadratic equation will be the axis of symmetry plus or minus an unknown amount”: The axis of symmetry is a line (e.g., x = 2), not a number (e.g., 2).
It can get a bit clunky to write all these things. What might work better would be to put the points and axis of symmetry in different colors and refer to them by color name.
3
The axis of symmetry is not -b/2. It is -b/2a. It works in your example because a=1 but wouldn't work otherwise. Probably worth correcting in the description of how the new method works.
261
@Jack Johnson Which is why the article says, "For simplicity, we’ll consider an equation where a = 1."
17
Agree. I was confused by this explanation because it was not obvious where they got the number for axis of symmetry (-b/2 seemed to come from nowhere)
But then I took the derivative (slope at the parabola inflection point = 0) which revealed it to indeed be 2.
So one still needs to remember a formula:
axis of symmetry = -b/2a
or know a bit of calculus.
10
@Kenneth Chang
So all we need to do is to replace -b/2 with -b/2a for a general purpose solution?
1
All very interesting but I don't get what's wrong with the quadratic formula. Yes there are still two roots and you have to decide which one is the one you are after -- in practical problems.
9
@KBD Nothing is said about something being wrong with the Quadratic Formula. The clear advantage of the new method, in my opinion, is that (1) you have visuals to guide process (which also makes clear what you are finding) and (2) all quadratic equations can be solved this way. None of this is true for the Quadratic formula.
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@KBD
The quadratic equation is a black box to the vast majority of students. Sure, if you remember it and apply it successfully, you can get answers but perhaps without a serious understanding. This method offers a graphical aid as well as anchoring the roots of the quadratic to conical sections. The quadric formula offers answers, this method can increase understanding. If I weren't retired, I'd begin teaching both.
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@KBD Nothing is said about something being wrong with the Quadratic Formula. The clear advantage of the new method, in my opinion, is that (1) you have visuals to guide the process of finding the roots and (2) this "guide" clarifies to the student what it is they are doing/finding). Neither is true for the Quadratic formula or factoring methods we offer our students, who simply try to memorize (what appears to be nonsense to them) for points - not knowing why it works, what they are doing, or what the numbers they find mean once they have them!
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