I think an important element here to ask with these kind of problems is “does this headache exist any more or is it a historic holdover we even teach this?”

I think that you are right to frame this question from the perspective of why mathematicians care. I believe that it is easier to generate interest/engagement in this problem by connecting it to solving equations, ie x^2 + 7x + 10 = 0.

I would have the students test in groups that a “guess and check” method is a waste of time. Naturally for this question they’d probably only pick positive answers thus missing the mark. Rather than telling them guess and check is frustrating and ineffective, have them experience it.

I might then go on to explain that after a while, mathematicians started recording solutions along side their problems (I can’t validate for sure but it seems reasonable someone did at some point) and realized sometimes the answers were LITERALLY embedded in the equations. I would take this opportunity to show then two tables. One where the pattern is more visible and one where the pattern is less visible.

At this point, you’ve (I’ve) set the stage for some group discussion. You’ve cause a head ache. What is that pattern? How are the numbers in the solution mixed into the equation?

You might have a worksheet to record conjectures and examples to test them out, but I believe the key is how you construct the polynomials in each set of tables.

Feedback? Let me know if my lesson needs further elaboration. Also let me know if using “equal to zero” and “solving” is too drastic of a change. While one can make an area-model argument for factoring a polynomial, I have not found much success with this actually providing deeper meaning. I prefer to think of factoring a polynomial as a solving technique to break down a higher power into many linear cases with the zero product property as almost a hack, exposing the underlying structure of the polynomial to suit our needs.

This is a great question! I have a few perspectives to offer.

# My own
I rarely have a reason to factor polynomials unless a teacher has told me to (and I love math and work as a programmer/statistician!). The only practical reason I have found is when trying to do statistics we often want to find the minimum of some function. So then we find the derivative, set it equal to zero, and solve. Even then, I only factor polynomials in these cases to understand an example problem. In practice computers always solve these problems for me.

# History
According to ehow [1] and its reasonably reputable sources, it seems like the historical pursuit of factoring polynomials was done mostly for sport and the love of math.

# The web
A few people mentioning finding out things about the path of a falling projectile [3].

# My summary
The difficult to admit truth is that factoring polynomials is not very useful. Most people will never need the ability outside of contrived situations. Then again, the same thing could be said about running.

We factor polynomials for the same reasons we run, to become fit. Mathematical problems make us into fit thinkers. It doesn’t matter much if we do riddles, board games, or problems that fit into common core, as long as they exercise our brain.

But learning polynomials needn’t be a chore any more than running needs to be. I enjoyed learning them in high-school. I enjoy running now (but not then). People often like trying artificial challenges. We call that play.

Playing isn’t always fun, though. I didn’t like running in high school. It was hard. I was bad at it. I didn’t get any social rewards for getting better. Later (near puberty of course) I got worried about my appearance. I started running, lost weight, got attention, and learned to like this form of play as well.

Ultimately, we should learn to factor polynomials for the same reasons that mathematicians did, because it is fun to play. The question is, how do we present mathematical play so that more people find the joy in it?

[1]: http://www.ehow.com/info_8651462_history-polynomial-factoring.html
[2]: http://betterexplained.com/articles/understanding-algebra-why-do-we-factor-equations/
[3]: http://math.stackexchange.com/questions/83837/what-is-a-real-world-application-of-polynomial-factoring

I’ve had little use of factorable quadratics in the real world as an engineer. Using the quadratic formula to find a solution is much more common. When is the dimensions of a concrete beam with it’s area of steel which is round rebar (hello pi and decimal equivalents) going to give me an equation that looks like x^2+2x-3=0?

I think approaching with puzzle questions like: what’s a number that when you square it you get the same number as if you just add 6?

x^2 = x+6 Well 3 seems to work, are there any other numbers?

(-2) takes a few hints to get to. With these sort of puzzles you can build a structure that makes sense out of the type of problem as a puzzle solving strategy which can be interesting unto itself.

Here is a potential approach:

Speaking broadly, it is often easier to understand a new object when we can use analogies/metaphors to compare it to a known object.

Switching domains, suppose you are in a US history class learning about the role of the UK Prime Minister. Probably a good way to get an initial foot-hold on this is to say that it is sort of like the US President. Then you can begin to articulate some of the differences. As you learn new details, you can make new comparisons (e.g., US Congress vs. UK Parliament) and match up the systems (in this example: US government vs. UK government) in a way that draws on what one already knows. These matchings are only approximations, but they are a way to set the stage for more nuanced understanding.

Back to mathematics.

A theme across mathematics is that there are multiple representations for a given object, and thinking about how one representation matches up with another (even when the matching is only an approximation) can similarly set the stage for more nuanced understanding.

f(x) = x^2 + 7x + 10 has multiple representations, too.

This particular representation makes it pretty easy to find a derivative (not a topic for those initially investigating how to factor trinomials, but worth noting here). Other representations can facilitate different actions. For example, what if we want to know when this function is 0? Better yet, what if we want to connect algebra and geometry — after all, they’re both areas of the “same” subject, mathematics — and try to illustrate the function graphically? Are there advantages to presenting it in its factored form with respect to figuring out what the graph looks like?

A note about Eric Fleming’s comment on guessing and checking the roots of f(x), in which he suggests that it’d be wasted time: “Naturally for this question [the students would] probably only pick positive answers thus missing the mark.”

If this is “naturally” what happens, then I might wonder a bit about how students are thinking about such a quadratic. After all, isn’t the easiest evaluation at f(0) = 10? As one puts in positive numbers, is it clear that the answer will only get bigger than 10 (since x > 0 means that x^2 and 7x are both greater than 0)? Would students really “only pick positive” values of x? This depends on the students, sure; but I bet someone would catch on. (Would struggling with only positive x values really be “a waste of time”?) And maybe at some point there is an opportunity to ask: Is there any x > 0 for which f(x) = 0? Why or why not?

Here are some headaches:

1. How do I graph this expression’s corresponding function?
2. Can every trinomial be factored?
3. Suppose two different students have different factorizations of the same trinomial. Can they both be right?

(Even with more than two students: Suppose we have (2x+2)(4x+8) and (4x+4)(2x+4) and (x+1)(8x+16). Can these all represent the same trinomial? How do we check? How do we decide on which representation is the “best”? Is rewriting the previous expressions as 8(x+1)(x+2) “allowed”?)

4. Can we connect factoring trinomials back to factoring integers? In the original example of x^2 + 7x + 10 = (x+5)(x+2), plug in x = 0 on both sides. Then we find that 10 = 5×2. We’ve found 10’s prime factorization! Can we always turn these trinomial problems into [prime] factorization problems?

Alternatively, in #4 plug in x = 1. Then we have 18 = 6×3. That’s not a prime factorization because 6 is not prime. Instead, we should have: 18 = 2x3x3. Does this prime factorization relate to [trinomial] factorization? (Never mind the aspirin for a moment — isn’t this causing a headache?!)

I’m going to take the summer and work out this theory, once per week, with ten skills in math that are a poor fit for other theories of interest and motivation.

What are these other theories? You call out “real world is interesting.” I remember you writing about Paul Silvia’s balance between the novel and the familiar (here). Are there others?

Math shouldn’t feel pointless. […] My challenge to all of us here is, before you offer students the new, more powerful math, put them in a place to experience the limitations of the older, less powerful math.

Is your theory of interest one about feeling a certain way or about thinking or believing or something else?

You talk about feeling in the post. Does that feeling need to be current when the learning is happening? (i.e. do you predict that interest occurs when there is a presently a feeling of need?) I would think that this wouldn’t be the case, because the feelings of need would be distracting from the feelings of interest.

Probably you mean that a student needs to experience feeling a need, and then remember that feeling later in order for interest to spike (“intellectual need”). That’s interesting, and is different than what happens with Asprin. (“Oh jeeze anyone got a pill? Yesterday my head killed.”)

If the analogy with Asprin breaks down in this way, then it seems to me that we haven’t hit theoretical rock bottom yet. Why would having a memory of a felt need lead to a spike in interest when that need has passed?

Possible answer: “Expectation of future use.” But then we have a new theory of interest, right? “People are interested in things that they expect to use in the future.” Maybe “intellectual need” is about planting the seeds for that expectation?

Possible answer: “Having a felt need ensures that a piece of learning is both comprehensible and novel.” This is Silvia’s, from your earlier post.

For me, the headache that trinomial factorization solves is related to my appreciation of mathematics as a system that builds upon itself: that x^2 + 7x + 10 consists of the (x + 5) and (x + 2) pieces. The thing is, I don’t know that I fully developed this appreciation until after I started teaching. It wasn’t until I started spending evenings with Geometer’s Sketchpad that I realized, “A parabola is just the product of two lines!” It was as though Nick Jackiw et al had pulled back a curtain to reveal that reality was constructed of Legos. And that felt awesome, like seeing patterns in the Matrix.

I wonder, though, how much of that was attributable to Sketchpad (or whatever problem I was trying to solve with it), and how much was due to where I was in my own intellectual development. I agree that a deep appreciation of math for its own sake is an important goal of math education. I just wonder how much it depends on external factors like good tasks (or curriculum), and how much depends on internal factors like a desire to understand the nature of reality. Which is to some extent to say, how much depends on age.

I’m a big believer in the notion that students are as smart as the opportunities they’re offered; some of the most thoughtful conversations I’ve ever had about income distributions (for instance) are ones I’ve had with sixth graders. At the same time, I remember having a conversation with my Algebra I class about whether zero was “true” – shouldn’t “no thing” and “thing” be mutually exclusive? Should we have to choose between zero and the rest of the number line? – and it falling utterly, absolutely, devastatingly flat. Maybe I taught it poorly, or maybe the topic itself fell too far beyond their zone of interest.

Are all ideas interesting at all ages, or are there some that we as humans grow into? I don’t know. Do teenagers dig on metaphysics? I have no idea, but I’d love to know. I’ll be very interested to see what kind of headache you come up with, though. I’d love to see what that looks like. (I’d also be very interested to know whether it can happen in a single task, or whether it must be part of a larger story, i.e. a longer-form math-as-fascinating-phenonemon curriculum.)

Can factoring trinomials be an aspirin for a computation headache? For example, evaluating x^2 — 17x + 72 for x = 13 is a pain whereas evaluating (x — 9)(x — 8) for x = 13 is easy mental math. I have full confidence that math teachers could create sets of exercises designed to cause this headache.

But factoring doesn’t always make this easier. For x = 0, I’m sticking with the trinomial. Maybe there’s a lesson about fluency (which I’ll define as the flexible, accurate use of procedures rather than the use of procedures w/o having to think) here?

(I’m not convinced that factoring to simplify computations is the way to go here, but @Jason beat me to it with his comment.)

Another thought… does factoring a trinomial provide any additional insight? Insight right here, right now, not down the road with zeros of a quadratic function. I’m thinking about how equivalent equations for a linear/quadratic relations can tell us more about the characteristics of a line/parabola (or the situation they might model).

Curious to hear your answer to @Michael’s Q re: these other theories.

Too long for a comment, so I wrote something up on my blog:

https://mikesmathpage.wordpress.com/2015/06/18/responding-to-dan-meyers-quadratic-question/

For me, this is a math for math’s sake situation. One of the big stories of algebra to me is that it is the generalization of arithmetic. Just as mathematicians have discovered that to understand numbers we need to understand their multiplicative structure, the same is true of factors. So I don’t teach that the skill to factor is relevant, I teach that knowing the factors is powerful. We can explain function behavior, make accurate predictions, solve equations and more. There are so many better ways to find factors than factoring. Graphing, technology, etc. We see the traditional way and use it to try to understand multiplication and division in the algebra setting, relying heavily upon the area model. We want to know factors because they explain why polynomials and rational functions work the way they do.

I’m with Steve. This year, I introduced as a way to solve puzzles – especially to find all of the solutions to a puzzle. Two consecutive odd numbers multiply to (some huge number). A number multiplied by 6 is equal to the number squared.
I had a bunch of these this year. Most were easy to stumble upon one solution. Writing an equation and solving allowed them to find all possible solutions.
(I had seniors who had not been successful at factoring previously, so we graphed on the calculator and found the zeroes instead. When they *had* to factor, they were encouraged to graph first to find the magic combo of numbers that work, then write up the factoring solution as if they had figured it out themselves.)

Thanks for the responses, Dan. I appreciate your making explicit some of the theories that you’re going after — real world, work world, kid’s world — and I agree that none of these hold up very long to inspection. I also agree that it’s important to engage with these beliefs, because they’re commonly held.

…but, come on, it takes two seconds to dismiss those theories, and you’ve done so expertly in the past. Hey, they played hard, kept it close, no regrets. Who’s left in the second round, besides for headache/aspirin?

Re feeling/believing/thinking, I see what you mean. I think what I was trying to get at was the concern about the timing of intellectual need. We don’t usually say that we are feeling a headache a day after we got one, and without that thump in your head there’s not currently a need to take aspirin. This person, though, would likely know that there’s a need for aspirin in general, i.e. aspirin has a valuable purpose. And their past feeling can help that justify that belief.

What I’m trying to disentangle is whether you are saying that things are interesting if kids currently have a headache (e.g. just-in-time instruction is the most interesting) or if the experience of having a headache makes aspirin more interesting (e.g. experience before formality sparks interest).

My approach for motivating factoring so far has been something like the following:

Factoring stuff is weird right now. But later on it comes in handy.

(Specifically, eg. simplifying rational expressions with a common factor in the num/denom. Which I try to handwavey-tell to students in less formal language.)

I don’t really try to pretend that this makes factoring inherently interesting, and I try to appeal to puzzle-solving thinking along the way as well. But I hope it at least distracts the question of “WHYYY DO I NEED THIS” long enough for the puzzle-solving mentality to click.

It’s also related to a general theory of how to make students want to learn something, that goes something like this:

“Explain to them that they’ll need it later for [X].”
where X is some later academic studies.

I don’t really like this theory as a basis for one’s lesson plans, but since it seems to show up a lot by default it is probably worth mentioning. (Really it’s just passing the buck upwards.)

Simply question: for simplyfing algebraic fractions. Otherwise it is artificial and it could be achieve without this. The quadric function questions could be solved with general equation y = ax^2 + bx + c and the formula of the equation of second degree.

In my classrooms, I don’t waste time doing it. Most of the contents of the curricula are wasteless. I know that it could be sound drastic, but it’s the truth. We have to prepare kids for two things:
– for real life
– for understanding maths in later studies.

I think that in the lowest studies we have to prioritize the first, and when we are aproaching to higher studies, we could concentrate on second. But ideally, the curricula should have just the first one.

Michael, when you say it takes “two seconds to dismiss those theories,” do you mean in general or strictly vis a vis this particular problem?

When I said “dismiss those theories” I meant “find many many counterexamples to those theories.” I did mean that generally. In general, there are many, many real-world problems that suck and many, many fake-world problems that rock. I know we all agree on this…

…unless we say, well, the sucky problems aren’t really real world. Because real world isn’t about something going on outside of the math classroom. “Real world” means “authentic tasks.”

I’m not sure if this is the move that you were making, Karim, so apologies if I’m missing the point. But if it is, then my first reaction is similar to Dan’s. What is an authentic task?

My main worry is that “authentic” is a GOODWORD, and there is no such thing as a bad authentic task, by definition. This would mean that there’s literally nothing to talk about, because task quality is baked into the meaning of authentic. (e.g. doesn’t help me to say that the key to baking a great cake is to make it delicious; the key to effective medicine is that it has to work, etc.)

The headache? I have been teaching an Algebra class almost every year since 1985. I haven’t figured this out yet. Yet.
Perhaps influenced by CPM, I prefer to think of factoring as rewriting as a product (of factors). Certainly the puzzles that can be offered using algebra tiles will result in student growth in furthering the practice standards.

Product form and sum form will result in class when doing a problem like Leaping Frogs. I saw it first in Malcolm Swan’s Problem with Patterns and Numbers. Also can be found on the nrich mathematics site http://nrich.maths.org/1246. As groups share their models, there is certain to be discussion about the different results and whether they are really the same.

I do wonder if rewriting in product form is a helpful tool in understanding properties of parabolas.This desmos exploration led more to the GCF
https://www.desmos.com/calculator/xtqikpaeh7
We generated a class list of representations in standard form and noticed and wondered why they had the same intercepts. A similar exploration might be done generating a list of equations leading to different roots.

CPM had a unit problem called the Rocket Problem. I adapted it to a TI-83 and geogebra type game. The problem can generate the need or at least the value in using product form to represent a parabola. http://staff.pausd.org/~pjorgens/geogebra/rocket1.html

I don’t know if students need to factor. I do think the context of factoring can be used to further the practice standards. I think problems exist where factored form is a model that is useful to students. I think factoring can be helpful while reasoning about parabolas. Probably not enough to create the headache yet.

I feel that factoring is helpful when dealing with rational functions in general. When the numerator and denominator polynomials have common factors, things simplify quite a bit as soon as you get rid of the common factors.

It’s particularly handy in finding limits (calculus) for rational functions when the above-mentioned common factors are present.

The other thought I have is that the general quadratic form is probably the simplest non-trivial example of factoring. We can use it to motivate the general factoring skills and factoring as an inverse of simplification via distributive property. If used well, it can improve/reinforce the understanding of the distributive property.

I see this understanding as a useful tool when we go on to investigate things such as (1 – r^n)/(1 – r) or (a^n – b^n)/(a-b) more generally. I don’t have words to do justice to the magical joy that my 12-year old experienced when she interpreted that (a^n – b^n) is always divisible by (a-b) when a and b are distinct – she actually tested the divisibility using many large numbers. That alone would be reason enough for me to teach factoring to my other kids.

This early exposure to factoring can develop intuition that can later help with understanding more advanced concepts such as the factor theorem and the fundamental theorem of algebra.

I like to connect multiple concepts which would be treated separately in general. This develops the instinct for using multiple avenues to approach a problem and to validate a solution. In my opinion, leaving out the factoring skill from curriculum would be taking a narrow view (i.e. solving quadratics which can be done with the closed-form formula) of its utility.

The thing I find interesting about factoring quadratics is the idea that the Babylonians started using it with taxes. However, this interest in taxes of a farming culture does not transfer over to students. Starting with an area as a quadratic and creating a need for the factored form does create the intellectual need for some students though. What can be done to make the desire to factor more real? Perhaps, we can recreate the need of the Babylonians by using skittles or some other concrete item as our area; how is this possible though without creating the obvious answer of just count the length of the sides? The idea that the Babylonians used factoring necessarily to find taxes may or may not have any truth to it, but this idea is interesting to me when trying to come up with creating the need to factor. It brings me back to the question how do we make finding the sides of a quadratic interesting to all students? The answer is probably visual, but I don’t know what it is or how to continue to truly pursue it. Thank you for a series that causes me to think about how to create that productive struggle.

@Michael:

I think the question of authenticity is related to the purpose of the task. Asking whether a task is “authentic” seems similar to asking whether it’s “real,” a topic we explored in some length in a blog post last year.

To summarize, though, I think a lot of it comes down to whether we’re talking about a conceptual understanding task or an application task. The way I think about it is, the purpose of an applications task is to use math to better understand something external to math, i.e. the real world. This is where Mathalicious, Yummy Math, and others spend most of their time.

Meanwhile, the purpose of a conceptual understanding task is to illustrate something about the math. In some cases this is facilitated by a context: a water tank, a taco cart, a parking lot, etc. In other cases, though, it’s not. It seems to me that this is the scenario that Dan’s trying to get at here; forget real-world applications…what happens when there’s not even a context that would work to set up the tension?

There are any number of things that are interesting about quadratics for their own sake. When I woke up this morning, I was still really fired up by the notion of a quadratic as the product of two lines, and I began to expand this out (pardon the pun): What happens when we multiple a quadratic by another line? Does every function family consist of lines? What about trig functions? To echo Hunter, Steve, and Stephanie, maybe the best approach with the trinomial task is to set it up as a puzzle, e.g. what’s hiding within this quadratic??

But here’s the thing: even if the task doesn’t include a context, I suspect that it would be most successful — maybe even, it can only be successful — if the task itself is part of a larger context, namely a coherent curriculum. If the task is intended to stand alone, i.e. if it’s not part of a larger storyline about revealing the nature of math, then I wonder how “puzzling” it’ll feel in the absence of any context whatsoever.* Kate Nowak does a really good job of making math interesting for its own sake. Lots of her blog posts are about lessons that she’s done with students. But even those…I imagine that she’s weaving them into a pretty great narrative throughout her school year, even if we only see bits and pieces of it. Perhaps more than anything else in this thread, I’m curious whether Dan envisions this task as existing on its own, or whether it’ll be connected to the other nine in the series.

* I think this is a really important question in math education in general right now. Lots of schools and districts are shying away from buying core curricula, instead opting to modularize their curriculum using disparate pieces. There are obvious advantages to this — perhaps it’s a way to circumvent the jack of all trades, king of none phenomenon — but it’s possible that the Story of Math may get lost. I hope not.

Please please please! Substance for factoring. I struggle every year to make it real world. Too many teachers feel it’s “essential” but essential for whom? I’d love to see an application for my lower achieving Algebra 2 students. Looking forward to your post!

@Andy I think the babylonian idea could lead towards an antidote to the factor blah-blah-blah headache.

Here’s a strong candidate to find a cure.

https://plus.maths.org/content/101-uses-quadratic-equation

But it’s still not interesting enough story for kids.. but on the right track.

I’m glad this topic was brought up. Factoring has been a thorn for me.

The idea of “headache” reminds me of discrepant events in learning science, and relates to Essential Questions.

I posed this same question to my PLN last year and no one seemed to come up with a good EQ. The closest we came to, I think, is to show how easy it is to multiply two binomials and ask why multiplication is more difficult than division. For sure this would be a type of headache but it’s unclear to me if it would strictly lead to learning how to factor a trinomial.

The other idea that I tested out was to ask students to write a program (what a headache!) that produces trinomials from (dx+e)(fx+g). This task led them to a new level of understanding of what components make up a, b and c in ax^2 + bx + c

I remember being a little pissed off when my algebra teacher taught the quadratic formula which I quickly programmed into my ti-85 for the test. I thought, why the heck to we spend all that time factoring? We can solve it much more quickly like this and there’s no problem if it doesn’t factor nicely.

Of course, that’s the engineer in me. The mathematician in me finds some beauty in the factoring method.

factoring quadratics with “nice” factors is one of the many parts of algebra which are designed to develop the student’s pattern recognition skills, particularly those who are intent on pursuing a math based career, and those who have become addicted to math. Whether or not pattern recognition skills in algebra transfer to other areas of life is a difficult question to answer.
I did a quick calculation to find the proportion of nice looking quadratics with coefficients in the range 2 to 12 (+ or -) and came up with 14%, or one in seven. So factorising by inspection is not going to get you anywhere very often. Many of the comments are not about factorising but about finding roots, looking at plots and so on. If I want roots I use the formula (or a computer program!).

My comment is more about the underlying theories than the specific problem of creating a headache for which factorising trinomials is the aspirin.

Given that all learners are different, and that the context of learning varies every time we teach, it seems to me to be a near impossible task to create a situation that will be headache inducing for all (maybe even the majority of) students all (maybe most of) the time.

Thus, in my ideal mathematics education the task of creating interest in factoring trinomials would sit inside the task of creating interest in mathematics. To this end I would like to see explicit classes devoted to thinking and working mathematically running in parallel with content courses. In the thinking and working mathematically classes students would learn about problem posing, specializing, conjecturing etc. and are expecting to make mistakes, to not necessarily complete the task as originally given, to collaborate with others, to use whatever technology is relevant and so on. Skills in mathematical modelling would also be taught in these classes. Now such courses won’t get far without a good array of mathematical tools to go with them and that is where the content courses come in.
With this approach the reasons why we might be interested factorizing trinomials with integer roots could be covered by referring to the things talked about in the thinking and working mathematically class such as:
1. The process of building up successively more general mathematical tools
2. A mathematical challenge
and so on. Also, if the question “Where am I actually going to use this?” comes up you can legitimately say “I don’t know” but you would be able to point to the thinking and working mathematically classes where with every problem you face you are not certain just which mathematical tools you will use but where having more tools is definitely an advantage. More than likely you will have an example of a problem where in one approach to the problem factorizing comes up.

Returning to my original point, I believe that that this approach still won’t engage all students even some of the time nor many students all of the time. The important thing though is that all students have the opportunity to engage in mathematics at the level that matches their interest.

My initial response would be very simple, but I’m not on the right track. Multiplication and division are inverse operations and you can use both operations in a variety of ways and to solve problems. We learn these operations through stories and context and begin working with natural numbers. Then we add one more number and move into whole numbers, then integers and rational numbers. Along the way we introduce unknowns and variables and they become part of our learning.

There are many visual representations for multiplication and division and the array is particularly powerful. We can use arrays with rational numbers as well as algebraic expressions. I see factoring polynomials in the same way as, ‘Here is the product, what factors multiply to make this product?’ It seems like number sense to me and flexibility to move numbers or expressions around. I use algebra tiles with students and they create the arrays I referred to. Overall, I think multiplying and factoring polynomials is a continuation of our journey learning about what multiplication and division mean and what these operations can represent. Overall, I think it helps deepen our understanding.

Awake in the middle of the night, I made the mistake of reading this post – that is, if I planned to do back to sleep at all…

An initial thought (that might help me sleep if I write it down somewhere) is that we “all” want to know where things come from, where do WE come from, why are we here, how do we know it came from that, etc.? Whether an item’s similarity in design to something in nature (i.e. Velcro to a burr or a jet engine to a falcon) or an out of place object or a piece of evidence in a crime scene, “where did it come from?” can be interesting and even become obsessive.

Arithmetic skills, students’ “older, less powerful math” (“Mr. Haan, I actually used to be good at math!” or vice-versa), help to relatively easily describe sums and products of integers, even some rational numbers (and irrational? and real?); we can multiply and divide, add and subtract, can even multiply binomials now, but where is THIS coming from? How did THIS get here?

Why does it matter, and can I figure out a way to articulate where all things like this come from?

There are a lot of people commenting on the theoretical side of this post but not a lot (at least in the first half of the comments that I actually read) who are actually taking on your challenge of finding the headache. No offense to those who wrote it, but “math for math’s sake” and “it’s a puzzle” are only going to work as headaches for those that already love math and completely defeat the purpose of the finding a headache – a perplexing problem that makes EVERY students want a shortcut.

So here’s my headache. Graph y=(x^2+7x+10)/(x+5)
Without factoring, the only way to graph this is to just start plugging in x’s and making a table – that’s a headache! But when you start plotting the points…Whaaaaaaaat?!? It’s a straight line! How did that happen? What’s the equation of that line (why is one point missing) and how can I get there through a shortcut?

In my opinion, this works better than solving 0=x^2+7x+10 because students often have other methods of solving this before they get to factoring. Needing one more method to solve a quadratic is not really a headache.

Great discussion. Two observations:

As a mathematician, I think the two most general purposes of factorization are (1) the connection between the roots and the factorization of a polynomial (re: @Joshua algebraic-geometry) (you’d be surprised how many college students don’t realize that the quadratic formula gives the same solutions that you get from factoring) and (2) part of the generalization of arithmetic to operations on polynomials & rational functions (re: as mentioned by many). These both seem hard to formulate as a ‘headache’ as they fall under the you’ll-need/understand-this-later (un)motivation.

Echoing @Malcolm: there’s lots of different things going on in a classroom and it seems difficult to come up with a clear set of sources for generating a ‘headache’, and, from the comments, this is especially true for factoring. My only thought is to try to build a bigger story for the entire course and then somehow tie this lesson to that story. If I can’t tie it in, and the story is right, then I’d see what options I had to reduce or remove or transform the topic (possible in higher ed, near impossible in K12).

A friend in high school taught me that I could multiply large numbers in my head by using polynomials. I loved it because I could impress others with rapid mental math.

Ex: 78 × 31 = (70 + 8)(30 + 1)

In support of the puzzle idea, I had two good experiences this spring. This problem (building on area approach) https://twitter.com/mathhombre/status/598321083314511873 produced both high engagement and some connection formation. And this puzzle from Illuminations did all kinds of work. http://illuminations.nctm.org/Lesson.aspx?id=2938 Nice because it’s a variation on area, but that did cause a little bit of confusion because it’s similar format. Both got at one of the fundamental math ideas to me: if we know this, what else can we figure out?

Sequences, patterns, and structure are interesting at any age. Like Pawan Kumar wrote, being able to factor x-1 from x^n – 1 (after all, 1 must surely be a root) allows you to transform 1 + x + x^2 + . . . x^(n-1) and determine the sum, giving power over many and soon infinitely many things. Factors reveal structure behind polynomials which makes Des-man easier to play.

As many others wrote above, facility with forms allows you to see problems from a new perspective, and perspective is what we come to math to gain and how real math is done. Students with Algebra skills playing with http://oeis.org/A000127 can conquer then predict this unwieldy sequence and make the connections to Pascal’s Triangle or, as a 7th grader recently showed me, to the Rascal Triangle.

I’m thinking…still thinking.

I remember Dan’s TEDTalk. There’s a video on the screen and it’s playing. In the video is an 8-sided clear container with water pouring into it and a timer on the screen. And it’s just playing. Water’s pouring. Timer’s going.

Kids are thinking”…man…how long is this gonna take!?”

My initial take-away from Dan’s presentation (some years ago) was planting the idea of knowing what’s important. What are the right questions to ask. What do the students need to know.

Now, I’m thinking about it as a headache (how long is this gonna take) and an aspirin (the math and figuring out what we need to know).

For factoring, could the headache be something as fencing in a garden? Coils of fencing only come in 50 and 100′ lengths. Task is to build raised beds for a garden where the beds are spaced enough to allow a wheelbarrow to pass (known value), maximizing area of gardening space (each bed is max 4′ across and 8′ long…because boards come in 8 foot lengths…but could be cut to smaller length). What’s the biggest each bed can be, which will make the most growing space, and be confined in …say, 100 feet of fencing?

And we’re going to build models in the classroom with popsicle sticks and …other stuff…

Maybe I just throw a bunch of material onto each table with the instructions of only “Build a raised bed garden”. Kids are already primed to think about asking what’s important.

That’s all I’ve got.

…except a headache.

“Why should we learn to factor trinomials with integer roots, ie. turning x2 + 7x + 10 into (x + 5)(x + 2)?”

I’ll answer your question with some questions of my own: why should we learn to factor trinomials (or more general polynomials), period? Perhaps if we have a master plan for this set of skills, we can place the issue of “integer roots” into its proper context. For that matter, why should we learn to factor *numbers?* Surely the issue of factoring polynomials is related to the issue of factoring numbers.

In the CCSSM, the issue of “turning one expression into another,” as you have done in your example, is motivated by the desire to “reveal the zeros of the function.” So perhaps we should be discussing this: why do we care about the zeros of a function? The issue of integer roots, in my opinion, is just a matter of “keeping things simple.” i.e. we do this for the same reason that, when teaching students to add fractions, we are more likely to give 2/5 + 3/7 than 385/486 + 999/1002.

One lens through which to view the skill of factoring is “puzzle-solving.” Consider this sequence of learning experiences.

1. Students learn to multiply multi-digit expressions, e.g. 25 x 31.

2. Students pose challenges to each other to perform “multiplication in reverse.” For instance: “I have multiplied two 2-digit numbers together, producing 3 hundreds, 34 tens, and 63 ones. Find the original two numbers.”

3. Students learn to multiply multi-term variable expressions, e.g. (2x+5)*(3x+1).

4. Students pose challenges to each other to perform “polynomial multiplication in reverse.” For instance: “I have multiplied two binomial expressions together, producing 3 groups of x-squared, 34 groups of x, and 63 groups of 1. Find the original two binomials.”

I’ll list pros and cons to this approach, as I see them. Pro: ties between “number arithmetic” and “algebra” are clear in this approach — one is merely an extension of the other. Pro: ties between “multiplying” and “factoring” are clear — one is merely the reverse of the other. Pro: working in reverse serves to strengthen math skills, and also involves elements of reasoning. Con: why would I want to work in reverse in the first place?

I’ll repeat what I think are the main questions to resolve here:

1. How does learning to factor numbers make me a more powerful problem-solver?

2. How does learning to factor polynomials make me a more powerful problem-solver?

3. What are the connections between the two questions above?

4. Why is it important to “reveal the zeros of a function?” What do we gain from doing this?

I start out, about 2 weeks before factoring becomes a part of the math lexicon, with “diamond puzzles” in which students must first identify what 2 numbers add to a particular sum while multiplying to a particular product. The puzzle being the point, no mention is made of factoring.

After students start to become pretty good at this puzzle, I remove different pieces and encourage fluency with sums and products and keep the game interesting.

Factoring with a=1 seems to flow easily when students see the underlying mathematics as a game they already understand and find engaging.

Further discussion is made and the necessity of deeper thought is needed when a =/= 1. By this point, hopefully, students have developed an intrinsic desire to see how the game continues when a isn’t 1.

“I wonder how puzzling it’ll feel in the absence of any context whatsoever.”

Doesn’t that have it backwards. Consider Sudoku, word scrambles, crosswords — all addictive, none with any context. Even Chess puzzles are ridiculously contrived.

Tetris, what is the context there?

Meanwhile, I remember like it was yesterday the delight I took in Algebra back in the ninth grade: every problem is a puzzle!

Finally, consider DragonBox Algebra. Pure puzzle-solving, very popular, and without even the added context of “this is why this manipulation is valid”. (DBA just shows you the allowed transformation (such as distributing multiplication over division) and away you go.)

The only problem is when students sit down to work these puzzles and have no way of knowing if/when they have in fact solved one. But the puzzles are there, we just have to find a way to let them know how they are doing *while* they are solving them.

Meanwhile, sorry folks, the real-world meaningfulness just is not there. It would be great if it were, but that road to student motivation is dead end.

Which is OK: math in and of itself is a delight when made accessible.

PS
I am presently reading this book: Godel, Escher, Bach: An Eternal Golden Braid
by Douglas R. Hofstadter
My present view on maths is that it is more of a language than anything else. Yes it is a science, an art, a tool for real life! but also a LANGUAGE. If we propose this aspect more, then perhaps the non mathematicians will leave us alone.
To deal with the quadratic and its roots.
The equation, the graph, the roots etc, are just different representations of the same thing. Just thinking of the graph representation, this of course is only one way of showing it. The axis don’t have to be perpendicular or even linearly scaled.
Whatever you believe about heaven and God, let’s imagine God sees it perfectly and we only see different glimpses. (I have not thought this through fully, this is only and informal ‘thinking aloud’). Who knows in heaven we too might see things in their perfect beauty. We’ll be saying things like ‘on earth we decided to use x but we could have used any of the other 25 letters’. And God will say you could have used any one of an infinite set of characters. You preferred to write it down because you wanted it visual. But you could have communicated it to each other via pitch of sound or frequency of colour,or physically by throwing stones out of windows, or even by smell! (the dogs have been solving quadratics many years before you). I am losing credibility here but I hope my point is made