Edutopia - Comments for How to Teach Kids About Factoring a Polynomial
http://www.edutopia.org/crss/node/450306
enI think this a really good
http://www.edutopia.org/comment/138476#comment-138476
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<div class="field field-name-comment-body field-type-text-long field-label-hidden"><p>I think this a really good way to explain factoring. Geometrical representation helps the student understand the concept much better.</p>
</div>Tue, 02 Sep 2014 18:28:45 +0000Nikita-Hertz-824636comment 138476 at http://www.edutopia.orgA. Men. I was taught not to
http://www.edutopia.org/comment/136096#comment-136096
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<div class="field field-name-comment-body field-type-text-long field-label-hidden"><p>A. Men. I was taught not to use FOIL and instead use the area/volume model to teach multiplication and factoring, and I never looked back. I have worked with lower achieving students and I say that this method not only teaches the concept, but allows them to think visually (and kinesthetically, if you use algebra tiles), where their number sense trips them up. </p>
<p>In my mind, you want models to be too cumbersome to serve at some point, to provide impetus for abstraction. Also, doesn't understanding of what is and what is not possible open up whole new worlds of ideas? Teaching with models does not forbid abstraction, or even impede it. Rather, it deepens connections so students know, understand, and remember concepts without (sometimes silly) acronyms.</p>
</div>Thu, 21 Aug 2014 22:52:18 +0000Allison Weisel - 166082comment 136096 at http://www.edutopia.orgAmen, amen, amen. I stopped
http://www.edutopia.org/comment/134316#comment-134316
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<div class="field field-name-comment-body field-type-text-long field-label-hidden"><p>Amen, amen, amen. I stopped using FOIL my 6th-7th year.</p>
</div>Wed, 13 Aug 2014 18:44:34 +0000Peter-Ford-808821comment 134316 at http://www.edutopia.orgThank you, Bon, and you raise
http://www.edutopia.org/comment/133801#comment-133801
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<div class="field field-name-comment-body field-type-text-long field-label-hidden"><p>Thank you, Bon, and you raise legit questions here. Here's the thing, Bon. At some point, every "model" breaks down for special cases. That's why I'm playing with it right in front of you. At some point, you *must* transition into the abstract to deal with those pesky non-nice trinomials or any other polynomial. Really, it's a way to push our thinking just a bit to examine why we do what we do, or even what made us bring up polynomials at all.</p>
</div>Sun, 10 Aug 2014 00:48:50 +0000José Vilson - 183139comment 133801 at http://www.edutopia.orgI feel similarly about SOAP a
http://www.edutopia.org/comment/133796#comment-133796
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<div class="field field-name-comment-body field-type-text-long field-label-hidden"><p>I feel similarly about SOAP a) because I've never heard of it and b) because I'm not sure if that's going to help students progress upwards ...</p>
</div>Sun, 10 Aug 2014 00:43:56 +0000José Vilson - 183139comment 133796 at http://www.edutopia.orgThank you for your comment. I
http://www.edutopia.org/comment/133791#comment-133791
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<div class="field field-name-comment-body field-type-text-long field-label-hidden"><p>Thank you for your comment. I might suggest that I would *have* to make the connection from the geometric to the abstract because it's hard to define a fourth dimension, unless we're talking about time or something. In any case, I guess if you MUST ... but personally, I think FOIL is a gateway drug whereas if we focus on the idea of distributive property, there's a better fluidity so people don't *have* to rely on first-outside-inside-last.</p>
</div>Sun, 10 Aug 2014 00:42:26 +0000José Vilson - 183139comment 133791 at http://www.edutopia.orgI saw the cubic model as a
http://www.edutopia.org/comment/133726#comment-133726
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<div class="field field-name-comment-body field-type-text-long field-label-hidden"><p>I saw the cubic model as a Montessori device in my new classroom (leftover from previous teacher). Thought it was interesting. </p>
<p>What I'm wondering is how we decide the "size" of the x length. And how things would look if we varied that - as it is a variable after all.</p>
<p>Also, I have a problem teaching factoring of 3rd degree polynomials this way without saying out loud and multiple times that there are VERY FEW of these that have nice factorizations. There are also ridiculously few quadratics with factorizations, but we have the quadratic formula to (hopefully) show that the "non-nice" factorizations can also be irrational or even imaginary.</p>
<p>Consider x^2 - 10x +34. It "factors" into (x-5-3i)(x-5+3i) and you'd only get that mess by setting the polynomial to zero, using the quadratic formula to find the zeros/roots and apply the idea that (x-c) is a factor iff c is a zero/root.</p>
</div>Sat, 09 Aug 2014 11:25:27 +0000Bon Crowder - 287961comment 133726 at http://www.edutopia.orgCan't speak for anyone else,
http://www.edutopia.org/comment/133601#comment-133601
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<div class="field field-name-comment-body field-type-text-long field-label-hidden"><p>Can't speak for anyone else, but I prefer actually understanding how the results of the multiplication of the binomial and the trinomial show you precisely why the signs have to be that way. Examining the flow between factored and unfactored forms of the expressions are supposed to eventually sink in as to what's going on. If a mnemonic helps kids get through the exam, that's fine, if the goal is getting through an exam. But there's a downside. I like mnemonics best when they are used for arbitrary information, e.g., order of operations or the order of the cranial nerves or the names of the five Great Lakes. There's just no logical way to deduce the order or the names. So having aids to memory makes sense, if for nothing else than as a checklist to make sure you've got them all in the last case, and for the ordering in the second one. Order of operations does have a kind of logic to it, but it's still got an arbitrariness, too. </p>
<p>When there's something conceptual at play, however, I prefer to draw students' attention to that. Algebra is at least in part about structures, and I like to point back to how two-digit by two-digit multiplication can be broken down into (10a + b)(10c +d) followed by using the distributive property or the lattice method, or any reasonable approach that stresses the underlying structure, using specific numerical examples, and then returning to the algebraic setting. Doesn't work for every student (or elementary teacher, for that matter), but it's worth looking at. And I think we've got things like that going on with sums and differences of perfect cubes. I love that Jose is playing with this visually. Of course, as someone will surely want to say, that model breaks down when we go to one higher dimension, so obviously it's a complete waste of time to even consider. :^)</p>
</div>Fri, 08 Aug 2014 20:02:49 +0000Michael Paul Goldenberg - 43437comment 133601 at http://www.edutopia.orgI guess I'm getting just a
http://www.edutopia.org/comment/133591#comment-133591
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<div class="field field-name-comment-body field-type-text-long field-label-hidden"><p>I guess I'm getting just a tad obsessed here. I hadn't realized until just now that @CCSSI Mathematics is the same person(s) behind Five Triangles Math. Seeing a link on the latter site to the former was a rather large hint, given that there are no other links offered other than internal ones. And of course, Five Triangles is also anonymous. My, but there seems to be quite a lot to hide going on. </p>
<p>At any rate, all three problems linked to are, of course, NOT trinomials. And they're not in one variable. Coincidence? Hardly. They're hand picked to NOT be relevant to the issues Jose is exploring and that James Tanton discusses in his free course and that countless students have to struggle with every year, often to ill effect (if you consider it an ill effect when something in mathematics that could be intriguing and beautiful - again, see those Tanton videos - is made tedious, pointless, and frustrating to the point where it becomes yet another hurdle that eliminates a sizeable number of, ahem, unfit students. Because clearly, if a kid can't factor quadratic equations given the classic treatment they are given in most US mathematics classrooms, s/he must be intellectually defective, fit only for a job a McDonalds, as long as the electronic cash registers are working. Otherwise, it's flipping burgers forever. </p>
<p>So here are the gems we were asked to consider: </p>
<p>First, x^2 - yz + xy - xz</p>
<p>Well, that's a little wicked. Of course, don't put the terms in a convenient order. Too easy. But factoring by grouping does seem to rear its head here. So let's try grouping them this way:<br />
x^2 + xy | - yz - xz</p>
<p>The left group factors into x (x + y). The right factors into -z(x + y) and the distributive property gives us (x + y)(x - z). </p>
<p>Of course, if you have 3 variables rather than one, students are likely to freeze up, so this is a great problem to make a lot of them feel paralyzed and inferior. Always a goal for a certain breed of mathematics teacher. </p>
<p>2) a^2 - ac - b^2 + bc</p>
<p>Again, make sure not to put the terms in the most pliable order, because that would cut the frustration level a bit. </p>
<p>So we'll write instead a^2 - b^2 | + bc - ac</p>
<p>Neatly, we get the difference of two perfect squares on the left, suggesting</p>
<p>(a + b)(a - b). With a little thought, we can factor out a negative c on the right to get -c(a - b)<br />
[It never hurts to think about what result MIGHT be desirable and then see if there's a way to get it]. So now we can use the distributive property to get (a-b)(a + b - c). Check to see that this multiplication of a binomial times a trinomial gives the original expression. Oh, so tricky-san is our Five Triangles host. Why, you'd almost think that these problems were created to "prove" that modeling something that is very different, namely quadratics in one variable, just won't yield to methods that of course we already KNOW they won't yield to. Hmm. What would the goal be in bringing them up in a conversation about something different? </p>
<p>Finally, we get this gem: x^2 - 2xy + y^2 - 1</p>
<p>This one is neatly designed to lead you down various primrose paths to failure. </p>
<p>For example, seeing that it is four terms, and given the previous two problems, we are likely to want to factor by grouping into pairs of binomials. And it would seem that a promising approach would be to group, say, x^2 - 1 | + y^2 - 2xy</p>
<p>But not only does this fail (you can factor both expressions, but not so you get a common factor), but so will other grouping into pairs (with grouping x^2 + y^2 obviously doomed before we start). Despair begins to set in, and our oh-so-clever problem-poser has once again made sure to list the terms in an unhelpful order. </p>
<p>However, we aren't going to be daunted. We try writing x^2 - 2xy + y^2| - 1<br />
Since the left side is a perfect square trinomial, namely (x - y)^2, we have the difference of two perfect squares after all, but not as we first thought. </p>
<p>(x - y)^2 - 1 then factors as (x - y + 1)(x - y - 1). Bingo. But of course this is the product of two trinomials, not two binomials. We've gotten awfully far from ax^2 + bx + c = 0, haven't we? But then, that's the point. So it's not that the model being used to explore quadratic equations of the form most often dealt with in school algebra fails, but that it doesn't address problems that it was never claimed to explain. In a now archaic phrase that somehow still resonates, No Duh!</p>
</div>Fri, 08 Aug 2014 19:53:07 +0000Michael Paul Goldenberg - 43437comment 133591 at http://www.edutopia.orgHaving looked at the problems
http://www.edutopia.org/comment/133581#comment-133581
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<div class="field field-name-comment-body field-type-text-long field-label-hidden"><p>Having looked at the problems you linked to, @ccssi, I conclude that you've managed to prove. . . nothing at all. Any basic algebra book, after dealing with factoring trinomials, looks at additional factoring methods, including factoring by grouping. And yes, there are some nifty special cases that require a "combination of ingredients" (well, methods/techniques) to do the job. You must have felt all sort of excitement at finding three of those cases in order to torpedo the very idea of a model for factoring trinomials. But of course, you didn't succeed in doing so. </p>
<p>Try looking at James Tanton's brilliant (free) online course, complete with both video lectures and accompanying text with additional practice problems. Best thing I've ever seen for what many students never learn or do so only mechanically at best. And not a second of snark from Tanton. You could take a lesson just from his demeanor. My students love those videos, which I use in class in a particular way to give them a chance to really think about what he's doing and why it makes sense. When we then go through the traditional completing the square methods and finally explore the quadratic formula, they have an arsenal from which to choose if and when they need to factor/solve equations of the form ax^2 + bx + c = 0 without using graphing or CAS-equipped calculators. Leave it to you to change the game by bringing in an entirely different family of expressions, then whinging about how a model doesn't work on them. Funny, but I'm pretty sure students are asked far more frequently to deal with typical trinomials than with the sorts of things you linked to. But you knew that, didn't you? </p>
<p><a href="http://gdaymath.com/courses/quadratics/">http://gdaymath.com/courses/quadratics/</a></p>
<p>p.s.: Tanton's courses on the same site, Exploding Dots and the one on Combinatorics, are both brilliant as well. But YOU shouldn't bother with them. They're way too student-friendly for someone like you, I'm sure. Nothing to learn there for a know-it-all.</p>
</div>Fri, 08 Aug 2014 18:48:04 +0000Michael Paul Goldenberg - 43437comment 133581 at http://www.edutopia.org