So factoring is super useful, yes. But at the Exeter Conference, one of the keynote speakers was making an impassioned, clarion call for CAS in the classroom and threw up an image. It was of which quadratics are actually factorable, and which aren’t. I tried to make my own 15 minute version of that to show you below (where and are non-negative, just because I got lazy). Apologies if there are any mistakes.
This image struck me so hard I can’t even tell you. Because although we teach the quadratic formula, in reality, most of our assessments which come after the quadratics unit give factorable quadratics. But in one powerful image, we are reminded that most quadratics are not factorable (at least, over the rationals). And we all know why we give factorable quadratics all the time — and it’s nothing to be ashamed of. We don’t want to have students spend all their time using the quadratic formula (and possibly generating incorrect answers) when we’re trying to teach an unrelated skill.
Still, the implicit lesson we’ve taught our students, by always giving nice, factorable quadratics is that most things are factorable. I mean, how many times have you been asked “is there a mistake in this question?” when you’ve given students a non-factorable quadratic on a test not on the quadratic unit? I thought so.
So next year I vow to show my students this chart, and remind them that most things in this amazing universe are NOT factorable. Heck, most quadratics that come up in engineering won’t even have integer coefficients, I will say, while showing ’em a picture of a falling cow and the equation governing its vertical motion in metric units. And that I tend to give more factorable quadratics than unfactorable quadratics because I want to save them computing time, not for any other reason.
Great chart. I’ve always warned my kids how limited factoring was as a technique, but this image really hits it home. I’ll have to show it to my classes too.
Sadly, factoring is such a testable skill, I’ve tutored students who were taught nothing else for nearly an entire semester.
Well using another point of view the “number” of factorable quadratics and non-factorable quadratics are the same (infinitely countable).
@watchmath: True. I haven’t thought about this a lot – but I suspect that if you took b and c to be random integers (a=1), and you calculated the probability of having a factorable quadratic over all possible quadratics, you’d get 0. What do you think?
Using real numbers from 1-100 the probability is roughly .025.
The probability of getting imaginary numbers is about .125.
An inevitable question would be, “Is there a good reason for learning how to factor anymore?” I know of one independent school in the L.A. area that, at least as of a few years ago, removed factoring from their program in favor of using the calculator.
What about with completing the the square so you can rewrite a function in vertex form… needed for Conics. Can all of that be done on the calculator as well? Maybe I need to catch up on the calculator applications! ;)
Which leads to the question: “Why is factoring quadratics an essential standard?” Still haven’t got a good answer for this one.
You are so correct about the limitations of factoring as a method to solve quadratic equations. But how can we leave it out completely as one responder suggested? Teaching my unit on rational functions is difficult enough when students have been exposed to factoring. How would they perform the many operations required on rational expressions without factoring? Just wondering? Maybe someone has a new and more modern way than I do. I’m always open to new ideas.
@Jim: In fact, the context in which this chart was given was to say that to motivate lower-performing students, to let them explore patterns, you can let them use the CAS to do the factoring for them. The argument was that this sort of algebraic work was a barrier to the exploration of mathematics. But honestly, in my school, I would never let factoring go. I think it is really important. I feel like factoring with integers leads to factoring things like x^2-2 into (x+sqrt(2))(x-sqrt(2)), and that sort of thing does come up in higher math. Or x^2+2 can be factored into (x+i*sqrt(2))(x-i*sqrt(2)). Factoring polynomials is one of those beautiful things about math — that a n degree polynomials can always be factored into n factors (over the complex numbers).
@Cheryl: I think it depends on the calculator. Most can’t do it automatically, but you can certainly write programs to do it for you.
@David: Same response as @Jim.
@Mrs. H.: I generally agree.
The number of “Y”s you see can easily be found by finding the number of factors of your “c” term.
Since the number of “Y”s in each row is really how many pairs of factors multiply to be the “c” term, you can find out how many “Y”s you would find in any row.
For instance, if “c” were 99, then the number of factors is
99 = 2²·11 –> 3·2 = 6
So there are six factors, thus three pairs that would multiply to be 99.
1 * 99 -> b = 100
3 * 33 -> b = 36
9 * 11 -> b = 20
So there will be three “Y”s and 97 “N”s on your table.
The rule is slightly different for perfect squares, like 100 because of the double factor:
100 has nine factors, but FIVE factor pairs
1 * 100 -> b = 101
2 * 50 -> b = 52
4 * 25 -> b = 29
5 * 20 -> b = 25
10 * 10 -> b = 20
So only five “Y”s.
Partly, of course, we are just recapitulating the development of our mathematics.
But I agree with the others – there is value in factoring later on, and algebra I is the natural place for its inclusion.
There is value in learning algebraic manipulation in general.
And there is beauty in learning both to multiply and unmultiply.
I would actually guess that many of those who hold up the calculator as an alternative are actually frustrated with how difficult it can be to teach factoring. But this is a reason to look at how we teach it, rather than if we should teach it.
I am also concerned that the continued downgrading of the level of abstraction in our curricula reinforce each other and make it harder to teach abstract concepts later on.
@David: I think that if Calculus is a goal then factoring is indispensable.
Off the top of my head, understanding the behavior of functions (zeroes and multiplicities) makes certain parts of calculus more accessible. Integration of rational functions using partial fractions is a more laborious process without it.
You might argue to skip factoring and teaching only completing the square for quadratics to teach them a process-based method for getting factors (which I think I’ll see alot more of since wolfram alpha). Or just do repeated long division or synthetic division using the rational zero test. Neither is efficient.
I agree. But I don’t think that just looking at teaching methods when the students are already in our class is enough.
I think not requiring mastery of multiplication table and fraction is what makes everything else difficult. For kids who can do a little mental math, factoring is fairly easy. For kids who can’t do mental math well, they have to work out every step on paper. It’s a frustratingly huge amount of work and they never see the forest through the trees.
The beginnings of abstraction is difficult when you can’t ground it in anything you know. It’s hard for them to confirm a new result using their experience and intuition because very few have it. It’s hard for them to see why abstraction is elegant. It just looks more complicated and difficult to them.
@Sam:”Over the rationals” is what gets many students.
Our standardized tests are designed to isolate skills in the questions. If you’re testing factoring and it includes fractions, how would you know if a kid missed the question because they didn’t know factoring or because they were weak with fractions. So then you gotta have 2 questions, one with fractions and one without. Give same test to students but change the integers to fractions and the number of mistakes go up, way up.
I would suggestion caution when talking about this topic. I would be concerned that they won’t hear the rationals part, only the non-factorable part. Then again I’m not familiar with your student population.
Last point. There’s a way to generate this chart.
Start at 0, go over to the right 1 and down 0 and mark square.
Start at 1, go over to the right 1 and down 1 and mark square.
Start at 2, go over to the right 1 and down 2 and mark square.
Start at 3, go over to the right 1 and down 3 and mark square.
Start at n, go over to the right 1 and down n and mark square.
Your chart looks good.
Great. Very few quadratics (let alone higher powers) factor. A few years ago I used Mathematica to calculate the percentage of quadratic factorable as a function of the largest absolute value of the coefficients. The result looked like a corner hyperbola with the percentage rapidly approaching the x-axis. With coefficients from -20 to 20 only 7% factored.
Lin, I am very interested your answer. Are you just talking about coefficients b and c in x^2 + bx + c and integral coefficients? In that case, 7% would be about 118 of the 1681 possible. Is there any way to extend your experiment to ax^2 + bx + c and use integral coefficients from -20 to 20? I’m not familiar with Mathematica, so I have no idea how much work that would be. I’m really interested in finding out how many of those 68 921 quadratics would be factorable.
I did this a long time ago on Mathmatica 3. It took about 6 minutes to run so it was considering many quadratics. I believe I set a to run from 1 to 20 (since -20 to -1) would double the number but not the proportion. The other 2 coefficients ran from -20 to 20 including zero. Only those with relatively prime coefficients were considered (this eliminated any common monomial factors).
It was a triple nested loop. not that much code actually, including plotting the results.
With the results plotted as percent factorable vs. largest absolute value of coefficient (i.e. x = 1, 2, 3, …20) the results approximated a corner hyperbola (k/x) approaching the x-axis as x got larger. At coefficients of 20 and less, only about 7% factored; and dropping fast.
Unfortunately, I’m not a programmer and not that familiar with mathematica either, so I’m hesitant to try to reproduce the result with Mathmatica 6. If I find the time …
Hope this helps.
Have you seen the problem where you put dots/points on a circle, connect all of them, and count the regions? It’s fascinating, and hard to solve if you don’t already know what to do with it. My solution process wasn’t elegant, wasn’t the most efficient way to do it. But I did use factoring as part of my solution process. I’ll post about it on my blog soon.
Don’t give up on factoring!
Late to the party, but I wanted to chime in with a historical note.
This isn’t the first time people have wondered if we should teach factoring. There was a trend in the 80s to do away with it and supposedly there were understanding problems later so it got put back in.
Some countries don’t (now and historically) bother with it and just teach the quadratic formula.
i’m later than late reading this, but…
i just found out last week that the new integrated algebra exam doesn’t even present kids with unfactorable quadratics. they aren’t required to know the quadratic formula, or use it on the exam.
so that’s just crazy.
i thought this post was awesome, sam. thanks so much!
Does anyone teach factoring by finding x-intercepts (using a calculator), then writing the factor as (x – intercept)?
This is my first year teaching Algebra, and we’re just exploring factoring now. I have kids who can already factor in their sleep and kids who think “2x+2” ~= “4x” in the same class.
To keep them (and myself, frankly) interested at all skill levels, I’ve been focusing on graphical factoring. I don’t have a writeup ready, but check out the ggb file embedded at http://larkolicio.us/blog/?p=50 , and take a look at the NCTM illuminations worksheets at http://illuminations.nctm.org/LessonDetail.aspx?ID=L282 . I’ve been having kids explore the properties of factors – what happens when a factor is zero, and what happens if factors of a quadratic are different signs, etc. I like this approach. I flew over some methods of factoring like differences of squares, etc, and then set kids loose on either 1) tons of factoring practice, including practice distributing for students with especially low skills, or 2) studying the significance of factors graphically and numerically for the kids who can already factor cubic polynomials analytically.
The kids exploring factors graphically are not limited to pre-made problems. And they are automatically discovering things like the fact that parabolas with no roots must not have any factors (made my day).
Please do proofread the NCTM worksheets before using them (I got a few nasty surprises during class).
Stumbled upon your blog and this entry on factoring. As a college math prof who frequently teaches future high school math teachers, here are my thoughts:
1) The most powerful feature of factoring is not the process itself but the fact that a polynomial can be broken down into linear factors (over the complex numbers).
2) An inordinate amount of time is spent on teaching factoring techniques, but as you can see from the grid, very few are factorable over the rationals. And it’s worse once you go to higher degree polynomials.
Much better to spend some time factoring, then after covering quadratic functions, recap factoring in general with connection to graphs and zeros. Factor “backwards” from the zeros of the graph. GeoGebra can be a big help there. (I have a few activities of this ilk in my web site, under GeoGebra)
3) Even when the quadratic formula gives irrational or complex roots, the quadratic expression is still factorable :(x-c_1)(x-c_2). Here, c_1 and c_2 can be any real number. However, most algebra courses don’t even go there.
By the way, a small correction to one of the comments – if the parabola does not have any real zeros – it is not factorable over the real numbers. However, it is factorable over the complex numbers – just use the quadratic formula to find the roots, c_1 and c_2.
Assoc. Prof. of Math , Kean University, NJ