## If Math Is The Aspirin, Then How Do You Create The Headache?

June 17th, 2015 by Dan Meyer

Several months ago, I asked you, “You’re about to plan a lesson on concept [x] and you’d like students to find it interesting. What questions do you ask yourself as you plan?”

There were nearly 100 responses and they said a great deal about the theories of learning and motivation that hum beneath everything we do, whether or not we’d call them “theories,” or call them anything at all.

- “How can [x] help them to see math in the world around them?”
- “How can I connect [x] to something they already know?”
- “How can I explain [x] clearly?”
- “What has led up to [x] and where does [x] lead?”

You can throw a rock in the math edublogosphere and hit ten lessons teaching [x]. They might all be great but I’d bet against even one of them describing some larger theory about learning or mathematics or describing how the lesson enacts that theory.

Without that theory, you’re left with one (maybe) great lesson you found online. Add theory, though, and you start to notice other lessons that fit and don’t fit that theory. When great lessons don’t fit your theory about what makes lessons great, you modify your theory or construct another one. The wide world of lesson plans starts to shrink. It becomes easier to find great lessons and avoid not great ones. It becomes easier to *create* great ones. Your flywheel starts spinning and you miss your highway exit because you’re mentally constructing a great lesson.

Here is the most satisfying question I’ve asked about great lessons in the last year. It has led to some bonkers experiences with students and I want more.

- “If [x] is aspirin, then how do I create the headache?”

I’d like you to think of yourself for a moment not as a teacher or as an explainer or a caregiver though you are doubtlessly all of those things. Think of yourself as someone who sells aspirin. And realize that the best customer for your aspirin is someone who is in pain. Not a lot of pain. Not a migraine. Just a little.

Piaget called that pain “disequilibrium.” Neo-Piagetians call it “cognitive conflict.” Guershon Harel calls it “intellectual need.” I’m calling it a headache. I’m obviously not originating this idea but I’d like to advance it some more.

One of the worst things you can do is force people who don’t feel pain to take your aspirin. They may oblige you if you have some particular kind of authority in their lives but that aspirin will feel pointless. It’ll undermine their respect for medicine in general.

Math shouldn’t feel pointless. Math isn’t pointless. It may not have a point in job [y] or [z] but math has a point in math. We invented new math to resolve the limitations of old math. 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.

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. As with everything I have ever done in math education, your comments, questions, and criticism will push this project farther than I could push it on my own.

The first skill I’ll look at it is **factoring trinomials with integer roots**, ie. turning x^{2} + 7x + 10 into (x + 5)(x + 2). All real world applications of this skill are a lie. So if your theory is “math is interesting iff it’s real world,” your theory will struggle for relevance here.

Instead, ask yourself, “Why did mathematicians think this skill was worth even a little bit of our time? If the ability to factor that trinomial is aspirin for a mathematician, then how do we create the headache?”

on 17 Jun 2015 at 12:15 pm1 Mark HorleyExciting project. On your first task, I sometimes start with “which is bigger 2x or x^2?” This then leads on to the need to plot these things as x varies. Eventually we end up at Trinomials with integer roots (or Quadratics as we called them in the UK). I’m looking forward to seeing other ideas, because I’m not convinced this is the best way…

on 17 Jun 2015 at 12:18 pm2 Jason DyerI 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?”

on 17 Jun 2015 at 12:23 pm3 Dan Meyer@

Jason, I’m going to refrain, during this series, from editorializing about the value of a given skill to modern mathematics. It’s a fair question but I’m going to leave the answers to other people.on 17 Jun 2015 at 12:27 pm4 Eric FlemingI 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.

on 17 Jun 2015 at 12:43 pm5 Roy WrightUnfortunately, a lot of the deeper aspects of factoring polynomials originated in Descartes’ attempt to develop a general method for finding tangent lines (before Barrow/Newton/Leibniz came up with much better ideas).

For me, some of the best “headaches” related to polynomial factoring are certain basic questions in number theory — e.g., “Why is it that 4 seems to be the only square number I can find that comes right after a prime number?”

on 17 Jun 2015 at 12:55 pm6 Ben HaleyThis 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

on 17 Jun 2015 at 1:01 pm7 HunterAs a kid, I liked factoring strictly for the puzzle-y aspect of it. The noticing of patterns. Like if b is negative and c is negative then it must break into a (x+b)(x-c) where the larger is the neg. I try to instill this same puzzli-ness in my students. For example, give 24 alg tile units, make a rectangle. Given 2x+4 alg tiles make a rectangle. Given x^2 +7x+10 alg tiles, make a rectangle. (If possible). I increase difficulty as I go. Students buy in to the puzzle. The headache is knowing your pieces, but not how to arrange them.

on 17 Jun 2015 at 1:30 pm8 SteveI’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.

on 17 Jun 2015 at 1:47 pm9 l hodgeOne could start with algebra tiles. Using your example, ask if it is possible to turn x^2 & 7x & 10 (in tiles) into a large rectangle.

After a few of these, ask a similar question involving a large number of tiles. The tiles have become unwieldy and we need a better way to express what we are doing (factoring).

on 17 Jun 2015 at 1:51 pm10 MQHere 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?!)

on 17 Jun 2015 at 2:08 pm11 Doug McKenzieI like this question. I think it helps explain some of those lessons that seemed so great last year, but fall flat this year…I rush to hand out the aspirin without setting the stage well enough, or creating a headache.

I like some of what Eric F suggested. And his ideas raise a problem that I sometimes have: there is a nice pattern set up that suggests a method/”shortcut” (like using reciprocals), but not all students see it or remember it. Some are quite content doing it the long way. If we extend the “aspirin” theory just a little bit… is it bad/good theory to expect that they will “construct” their own aspirin? (Do we leave them in disequilibrium until they get themselves out?) Is it good/bad theory for teachers to deliver the aspirin, or should students only get aspirin from other students?

Here is another possible approach. It is much easier to multiply binomials to get trinomials than it is to factor them. Perhaps start with some linear equations, look at the graphs, discuss what the students are familiar with… slope, intercepts, etc. Then ask “We can multiply numbers… can you multiply functions? If I multiply two linear functions, that’s like multiplying two lines! What do you think that will look like?” The students can graph the products without actually finding the trinomial product and talk about those, then bring the distributive property in and find the equivalent trinomial. The exploration could then continue by looking at relationships between the binomial factors and the trinomial product, and asking: does any trinomial have factors (with integer coefficients) or only some? Are there “prime” trinomials? Are some trinomials factorable in more than one way?

You never know if it will work, but I know my best lessons are the ones where I don’t create a headache, I make them hungry (curious). And then instead of aspirin, I get to hand out snacks (new questions).

on 17 Jun 2015 at 2:27 pm12 Michael PershanWhat 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?

Is your theory of interest one about

feelinga certain way or aboutthinkingorbelievingor 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.

on 17 Jun 2015 at 2:40 pm13 Stephanie BowyerI’m excited to see what you come up with for factoring a trinomial. My first year of teaching (2 years ago) I had a REALLY hard time explaining to my students why we would ever want to factor a trinomial. I had never thought about it myself – I just knew it was a major part of what we do in Algebra. My second year (just ended), we used an exploration in one of the fall NCTM Mathematics Teacher journals to explore a quadratic function as the product of two linear functions. This allowed me to explain that factoring the quadratic was how we can figure out what those two linear functions are that were multiplied to get the quadratic. (Totally blew my mind when I first read about the idea.) This was helpful, but I wish I had spent more time letting students explore that relationship between the quadratic and the two linear equations. I’d love to see what other ways we can approach factoring to encourage greater comprehension. Factoring is something that the students at my school really struggle with, so we as teachers clearly need to find better ways to approach it.

on 17 Jun 2015 at 2:43 pm14 Karim AniFor 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.)

on 17 Jun 2015 at 3:02 pm15 Mark Chubbi think we need to start thinking like a student! Hard to imagine math class from other perspectives when most math teachers excelled at it as students themselves.

Mastering specific skills then solving story problems isn’t providing the “need” for the concept!

Many teachers aren’t sure how to start this way. Something we need to talk more about for sure!

on 17 Jun 2015 at 3:12 pm16 Chris HunterCan 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.

on 17 Jun 2015 at 3:57 pm17 Phil DateI think the headache here could be “What does the graph of y = x^2 + 7x + 10 look like? Sketch it (without using Desmos)”. Solving x^2 + 7x + 10 = 0 is now relevant because that will give us the roots and help us to sketch.

Of course, this begs the different question of why sketch a graph when we have Desmos available but I believe that this is part of the bigger picture and a very useful skill, particularly for those students who go on to study Maths at a higher level.

on 17 Jun 2015 at 4:25 pm18 Mike LawlerToo 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/

on 17 Jun 2015 at 4:41 pm19 Joey KellyThe headache could be completing the square.

When I taught Algebra 2, we started by exclusively working quadratics that were in vertex form. Once we got pretty comfortable with those, we started looking at quadratics in standard form and transformed* them into vertex form so that we could continue to apply our old procedures. It became clear pretty quickly that this was inefficient …

*Because we were only factoring perfect squares, it don’t think this really counted as factoring trinomials.

on 17 Jun 2015 at 5:12 pm20 John GoldenFor 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.

on 17 Jun 2015 at 5:19 pm21 Dan MeyerHere’s an example of a theory of task design from

Ben Haley:Again, it’s rare to find people stating their priors so explicitly so I want to call it out and commend it when I see it happen. (As Ben points out,

enactingthose priors is a harder task.)Others have been similarly explicit. Here’s

John Golden:Theories only explain so much and they have different liabilities.

John’stheory seems to be, students will be interested in math that makes mathclearer. This theory is vulnerable to students who might ask, “Why do I care about function behavior?” If the point of learning to factor is to explain function behavior, that student can make their life easier twice over by simply not learning either.I don’t want to say that “need” is the

onlyworthwhile design theory or that it’s without its own liabilities or that it’s mutually exclusive to all other theories. But more often than not, it’s led to revelatory mathematical moments with students, which is why I’m exploring it lately.Michael:The three I hear most often are these:

Too many lousy word problems exist related to trains, construction, and Playstations to put much stock in those theories.

Schoenfeld, and others, would argue it’s difficult to disentangle those three. Intellectual need invokes the

beliefthat what we’re learning has purpose, even if its purpose is only within in the world of math. Intellectual needcaninvoke a feeling of interest and self-efficacy. And when students are in that moment of need, their thinking may change also.Of course, that’s before we even factor the teacher’s efforts into the model.

on 17 Jun 2015 at 5:23 pm22 Stephanie ReillyI’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.)

on 17 Jun 2015 at 5:47 pm23 Karim AniPer the above, I’m not sure a “on the nature of math” approach would necessarily work with students. It may be too abstract. But what about an approach that asked students to look for patterns, eg:

1. Graph x^2 + 7x + 10

2. Discuss features, eg opens down, vertex at (-3.5, -2.25), y = 0 when x = -2 & -5

3. Play around. See what happens.

If you add an a coefficient to the x^2 term, students might quickly realize the relationship between its sign and how the parabola opens. (They’d also see that when a = 0, you’re left with a line.) That’s perhaps the easiest realization, and also the one that’s generally first to be taught.

If you remove the coefficient or fix a = 1, students might also realize that the x-coordinate of the vertex is always -1/2 of the b term. Make b = 10, the x-coordinate of the vertex shifts to -5. Make it -100, it’s now 50. (Once students discover this, you could go back to adjusting a, and some students might generalize that the vertex occurs at x = -b/2a…though perhaps that’s unnecessarily tangential for now.)

So what about the roots? When b = 7 and c = 10, the roots are x = -2 and x = -5. Students might reason that c is the product of these, while b is the sum of…something. Their opposites? Their absolute values? So you play around some more. You change the equation to be x^2 + 12x + 8 and see that the roots are x = 2 and x = 6. The conjectures re: c and b seem to hold, though the b part is still ill defined. Make it x^2 + 3x – 10. Now the roots are x = -5 and x = 2. Yet another nod towards the “c is the product of the roots” claim. And now, it definitely seems like b is the sum of the roots’ opposites, or the opposite of their sum.

But this isn’t what we’re after; we don’t want to have to graph the parabola to figure out the roots. We want to use the equation to reason what they must be. Here’s how I could see us getting there…

Students already know that the original parabola crosses the x-axis when x = -2. Put another way, y = 0 when x = -2. Similarly, y = 0 when x = -5. We can rewrite x = -2 as x + 2 = 0, and x = -5 as x + 5 = 0. Since y also equals 0 at these points, then x + 5 = y and x + 2 = y. (I’m not clear yet on how to motivate these last two moves.) If students graph these lines, they’ll see that they pass through their respective x-intercepts.

So maybe there’s something special about these lines? What happens if we add them? What if we divide them? Multiply them? (Aha!) The parabola must be the product of these two lines. Is this always true? Let’s play around. Yes, it appears to be true: the quadratic is a product of two linear functions whose lines pass through the parabola’s roots.

So given a quadratic equation, how can we deconstruct it to determine what the lines must be…and therefore the roots? Whatever the lines are, we know that their y-intercepts, when multiplied, will yield the c term of the quadratic equation. Meanwhile, the opposite of their y-intercepts, when added, will yield the b term of the quadratic equation. Even if students aren’t totally clear on how exactly to do that, I imagine they’d have more of a headache to figure it out than they do with, say, “reverse FOIL.” Give it a few days and Desmos, and I bet kids would figure out the patterns.

In the end, I’m not sure students would “care” about this from a pure math angle; I’m not sure how much they’d care about the realization that a quadratic is the product of two lines. Even if students didn’t view this activity as illustrating some deeper mathematical truth — even if didn’t enhance their appreciation of math for math’s sake — I’d be pretty psyched if it enhanced their relationship with their own intellect and their confidence in solving puzzles.

(Incidentally, there are of course great applications that require multiplying lines, eg every profit maximization problem. Since these start with the lines, though, it would be pretty inauthentic to reverse engineer the quadratic.)

on 17 Jun 2015 at 5:57 pm24 Michael PershanThanks 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

needto 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).

on 17 Jun 2015 at 6:46 pm25 Karim Ani@Michael, I disagree that making math relevant doesn’t “hold up very long to inspection.” I agree with the notion that “real world” isn’t a panacea to engaging students more deeply in math. We absolutely need to facilitate an appreciate of math even when it has no obvious practical application.

However, we should be careful not to confuse “not sufficient” with “not necessary.” Many teachers and content developers artificially impose so-called “real world” contexts on math, eg using Putt-Putt to motivate similar triangles, and with poor results. But I don’t think this is a referendum on real-world tasks. Rather, I think it’s a referendum on inauthentic, forced tasks, and an indication that we should do a better job articulating what real-world tasks actually are/entail/serve.

In this particular context, it’s possible — perhaps likely — that a real-world approach isn’t the best strategy; I can’t think of any applied problems that are solved by factoring trinomials, and trying to force it would be counterproductive. 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? (And Dan, I’d love to hear your thoughts on that, too.)

on 17 Jun 2015 at 9:26 pm26 josh g.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.)

on 17 Jun 2015 at 9:30 pm27 Dan MeyerKarim:This begs the definition of “authentic,” though. Not a small task.

josh g.:Like this reference and analysis.

on 17 Jun 2015 at 11:03 pm28 XavierSimply 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.

on 18 Jun 2015 at 3:43 am29 Joshua GreeneSomehow, factoring quadratics has a special place in the family of “how do you make [x] interesting?” Is that a combination of how little motivation is provided for it and the amount of time spent studying it or something else? I don’t know, but wouldn’t be surprised if this is a topic that frequently figures into personal math histories: “I was doing well at math until we go to [topic] and then I didn’t understand and was never good at math after that.”

That aside, I will admit that I actually do end up factoring polynomials fairly often, so here are some thoughts on headaches for which this is an aspirin:

(1) Algebraic geometry is, originally, all about zeroes of polynomials. If we have two curves of polynomial degrees n and m, they should intersect in n*m points. If we are clever and control most of those points (n*m-1), then we can factor and get something interesting out of the remaining point.

that’s a bit vague, so a concrete example: one way to generate pythagorean triples is to look at points of intersection between lines of rational slope through (-1,0) and the unit circle. In this case, we know there will be (at most) 2 solutions and we already have one, so we can factor nicely to find the other.

Another application is where we find too many points of intersection, then we learn that the intersecting curves are actually the same (or one is a component of the other).

(2) number theory/field theory

To create finite fields of a particular size, we actually want polynomials without roots. This is essentially the inverse problem, but naturally gives us questions about when polynomials factor and when they don’t.

In this area, integers are a kind of universal ring where we can always create a sensible (and unique) ring morphism from integers to any other ring we might be studying. That allows us to push polynomials with integer coefficients into any other ring and have a sensible interpretation. of course, if we have an integer factorization, that factorization also carries over to other rings.

(3) when was f(x) = c?

There are a lot of cases where it is interesting to know, in a functional relationship, when we hit a certain value. There are tons of methods, and basically all of them work and are interesting for quadratic equations that factor in the integers:

– naive guess and check

– factoring

– analytical formula for the roots

– calculus-based numerical algorithms

– approximating with simpler relationship

A thorough understanding of this special case, if used and presented correctly, can be a really useful reference for understanding the other techniques and also compare/contrast why some methods work or don’t in various other cases.

(4) Non-example? I was asked about factoring a while ago and thought there was an application for finding nash equilibrium mixed strategies for competitive games. Looking at it now, though, it seems we only have to solve linear equations.

on 18 Jun 2015 at 3:50 am30 Michael PershanWhen 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

reallyreal 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.)

on 18 Jun 2015 at 5:20 am31 Kyle PearceGreat read here. You had me thinking when you originally asked this question about factoring, so it is great to circle back to it and see how my thinking has evolved.

While I think we would all agree that factoring is useful for building number sense, algebraic reasoning skills, and many others, I’d like to take a stab at how factoring could be useful in making math ‘playful’ and ‘fun’ through curiosity.

Just recently I have tried to test the theory that playing with numbers without context could be fun, as long as the floor is low enough for everyone. Working with groups of grade 9 students who struggled through elementary mathematics, I attempted approaching solving equations as more of a puzzle than a procedure. Here’s what it (sort of) looked like:

Start with multiplication like: 9 x 12. Since students in the room typically are not completely comfortable with their times tables, 9 x 12 would have most students reach for their calculators. Instead, I encourage them to find a way to multiply it out WITHOUT a calculator and WITHOUT the standard algorithm. This was surprisingly difficult for most. Some students eventually figure out that they can break down the numbers into a sum of more friendly numbers to multiply.

We would commonly see something like:

9 x 10 + 9 x 2 which I would show as 9(10 + 2). We try to make it visual as well by doing something like this:

https://youtu.be/2Bq9Q3y0_aY

We try a few more and then I challenge them with something like: “If 5 times a number is equal to 45, what must the missing number be?”

We also write this algebraically as: “5x = 45”

Pretty easy stuff. Kids are feeling good. Eventually, I throw this at them:

5(x + 6) = 45

What must the missing number be?

Here’s how we make it visual: https://youtu.be/59JlCrmFMlc

Eventually, we can get kids chunking bigger numbers like:

16 x 13 to something like: (10 + 6)(10 + 3)

Soon enough, students will be chunking numbers down and multiplying binomials way before it is necessary (I mean, required by curriculum) to introduce.

Over time, students are willing (and eager) to take on problems like:

Find the ‘magic number’ that makes this true:

(x + 6)(x + 3) = 208

From here, we begin to remove some of the pieces that make these problems concrete and introduce more abstraction. Multiplying the binomials with variables and discussing how the result relates to the work done previously paves a nice pathway to then going backwards from:

208 = (10 + 6)(10 + 3)

If x = 10, then:

(x + 6)(x + 3) = x^2 + 9x + 18

From here, we can try to get kids thinking about other scenarios when we start from a trinomial and need to “go backwards” or factor.

I.e.:

What is the magic number?

x^2 + 7x + 12 = 132

Hopefully, the building up to this point will inspire students to WANT to find the answer because it is fun and challenging. If we’re lucky, we can encourage them to think about what matters in this problem? … in any problem like this? Are there any patterns? Could YOU find the missing number without doing a ton of work?

Wow, that was much longer than I thought. Hopefully I communicated this idea clearly enough as it is a big idea, but one I hope to explore further next year to identify what works and what doesn’t.

on 18 Jun 2015 at 5:44 am32 Paul JorgensThe 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.

on 18 Jun 2015 at 6:02 am33 TreyI’m not convinced that all or most concepts in the standards we are asked to convey are relevant to real world or can tie into students’ interests. I do think “word problems” are just a means of getting kids to identify with something. In small group settings, there is a common comfort to talk about the path golf ball travels in putt-putt rather than a line segment of a similar triangle. But I digress.

I show a video clip and then tell a story. The bottom line is learning to take something apart and retain it’s value is based on the premise you know how something is put together.

The video clip is one from an A&E show (can’t remember name) where a woman auctions off parts of a house that is going to be bulldozed (i.e. doors, windows, brick pathways, etc.).

The story is that taking something apart can be accomplished in several ways. One is you can bulldoze it. Just smash it apart, but then it has no value. in fact, it will probably cost you to have it hauled off as a pile of rubble. Useless.

The other is to disassemble it in the reverse order in which it is put together. Then, the value of it’s parts can be retained and even be made use of again. If we know how to build polynomials, then we can take them apart neatly and retain their value.

Factoring is just a tool we have to take something apart, find it’s constituent components, and then possibly use those parts to build something else anew.

on 18 Jun 2015 at 6:02 am34 Pawan KumarI 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.

on 18 Jun 2015 at 6:40 am35 Howard PhillipsFirstly, factoring quadratics comes in useful when doing partial fractions, which comes in useful when doing artificially simple integrations in calculus. So It might as well be left till a calculus class, and would take about 10 minutes to explain and 15 minutes to master, particularly if (x + a)(x + b) has been expanded to x^2 + (a + b)x + ab and discussed.

Secondly it is useful in dynamic systems behaviour in distinguishing between decaying oscillations and a slow approach to a steady state from one side, and then with simple examples used to get the ideas across.

Thirdly, for Karim , the expression “The parabola must be the product of these two lines.” is a somewhat confusing way of putting it.

Have fun, Dan, youv’e probably picked the hardest case to start with !

on 18 Jun 2015 at 6:46 am36 AndyThe 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.

on 18 Jun 2015 at 7:38 am37 AndyKyle, your videos and progressions are awesome.

on 18 Jun 2015 at 7:58 am38 Karim Ani@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

canonly 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 nonephenomenon — but it’s possible that the Story of Math may get lost. I hope not.on 18 Jun 2015 at 8:03 am39 Kyle Pearce@Andy

Thanks!

Your comment regarding area and factoring to find the dimensions reminds me of another animation I did. In this particular example, we are multiplying binomials where the unknown value is the length of the garden rake, but the opposite could also work:

https://youtu.be/5HNzMkGEvWw?list=PLDe-CvW870TFx5PQFUGrP9QWqjPJF_KCo

Here’s another one that could potentially be extended to factoring:

https://youtu.be/xpk2mvZX2Ms?list=PLDe-CvW870TFx5PQFUGrP9QWqjPJF_KCo

Enjoying the discussion, folks.

on 18 Jun 2015 at 8:06 am40 Amy BrossartPlease 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!

on 18 Jun 2015 at 8:11 am41 KeithIt seems to me, after many years of teaching quadratic factoring, that the best HEADACHE is a max/min application, such as using a projectile formula to find maximum height.

The ASPIRIN must consist of a combination of algebra-geometry, not just algebraic, binomial factors.

After creating and viewing the geometry of the parabola, students will eventually see the time-coordinate of the vertex corresponding to the midpoint of the two zeroes.

Then a further discussion could lead to the calculation of those two zeroes. Finally, ending in the theoretical relationship between the points (x,y) and the parameters: a, b and c of the quadratic.

on 18 Jun 2015 at 9:58 am42 ihor Charischak@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.

on 18 Jun 2015 at 10:28 am43 Joanne FitzpatrickYour aspirin? (x+5)(x+2)

Consider a class full of 14 yr old mathematicians in Algebra 1 have this headache:

“Well, students, you’ve been asked to graph this function but you’re not sure what you can expect it to look like. You wonder, ‘Where will it fall on the coordinate plane?’ Part of the answer is it will cross the x axis at x = -5 and at x = – 2. Remember how the y coordinates of all points on the x axis are zero? Well, if we set y to zero, voila! We can see that if either binomial equals zero, we can figure out the two possible values of x. BTW, students, that little tool is called the Zero Product Property and is a great one to have in your tool bag.”

“Physicians, finance people, etc graph data all the time. One area of interest is always where a graph comes close to or even crosses axes. It will always tell us something about the behavior of the graph. Just like you knowing where classmates hang out tells you something about their behavior and gives you insight if you want to become friends.”

“Oh, there’s the bell. Class over.”

I like math headaches for high school students, all the way up through AP’s. It’s how we teach ’em to crave the migraines in college and beyond.

on 18 Jun 2015 at 11:00 am44 Doug SmithI’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

on 18 Jun 2015 at 12:16 pm45 Chester DrawsI don’t struggle to teach how to factorise quadratics (trinomials). It’s not like it is hard to find the two numbers that multiply to the c term and add to the b term. Even chucking in negatives does not change the essential process, so inability with numerical methods (specifically to factorise and add negatives) is a bigger issue than ability to follow the simple method. Poor number knowledge is what holds the slower learners back, not the trickiness of the process.

Once they get the hang of it most students enjoy factorising, because it is fun to learn a new skill. Once they have it, they even enjoy teaching the students beside them.

If anything I have problems with multiple factor linear factorisation, since that involves actual *thinking* about which factors are in common.

The issue, as always, is making them repeat the factorisation sufficient times to get it embedded. They need to do dozens, if not hundreds, and they can’t be bothered.

I could teach my class the greatest lesson ever on factorising, and have them doing it within minutes, but if they can’t be bothered with practice, homework and revision it is wasted effort.

Now if you can get me the aspirin to *that* problem, you will have my undying gratitude.

on 18 Jun 2015 at 12:31 pm46 Dan MeyerKarim:This seems to restate the question, though. “Something external to math” is what most people mean by “real world.”

For instance, I have little doubt that the author of this task believes both that it is “real world” and that it will help students “better understand something external to math.” And yet it seems very different from many good real world problems, and I have little confidence that it will interest most students.

So I’m still with Michael. There’s something to “good real world problems” yet uncaptured by terms like “authentic” or descriptors like “better understand something external to math.”

on 18 Jun 2015 at 12:37 pm47 Paul CarsonInteresting topic. One thing that immediately struck me was your use of jargon, something I avoid. I’d never say ‘we’re doing trinomial factors with integer roots’ to a student. Second, my approach is built up from multiplication of two numbers and generalising that over time using intuition. The main place it comes from in my method is how I multiply 2 numbers together and then use the same technique with brackets.

http://www.youtube.com/watch?v=gABW7ie3hHU

Brackets

http://www.youtube.com/watch?v=d6LEmhWEXDQ

This also comes out in degree order.

From there factoring is just a reversal of that process (a concept that is well embedded in my students) and they can see why from the multiplication technique (the addition part).

Finally, isn’t it interesting that we see this from so many points of view? Naturally I think my way is best, but is it? Is it just another perspective on it? When I teach, am I just trying to get students to see it from my perspective? Are there multiple ‘correct’ perspectives? I thought poor teaching came from rote textbook reading from teachers who didn’t really understand the subject. Perhaps it is a mix.

on 18 Jun 2015 at 12:54 pm48 SteveI 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.

on 18 Jun 2015 at 1:11 pm49 Fred ThomasI have no illusion that either a real world application or a contribution to the theoretical structure of math is sufficient to create the sort of headache we want. I do believe, however, that making students aware of BOTH applications AND the place of a specific skill within the overall structure of math can substantially increase the likelihood that a learning experience will succeed. Fortunately, factoring trinomials with integer roots does have applications AND it can contribute to a broader understanding of math. Unfortunately, far too students see how this skill is connected to the real world or to other aspects of mathematics.

Mastering algebra as a practical tool or as a theoretical discipline certainly requires the ability to recognize that expressions which appear quite different can actually describe precisely the same relationship. The many students who believe that x^2 + 7x + 10 and (x + 5)(x + 2) are completely different “problems” are very limited in both their ability to use algebra and their ability to understand math. Factoring exercises can help students to see what matters and what does not matter. These exercises can also help empower students to select the form of an expression which is most appropriate for a particular application.

Ideally, they should recognize that “x^2 + 7x + 10” is the form which (for many audiences) best communicates the nature of this relationship. They should also recognize that “(x + 5)(x + 2)” describes the same relationship and that it has advantages in some situations. As part of a computer program, for example, the second form requires just 3 operations while the first requires 4 operations and thus more CPU time. Our Math Machines “Color Functions” software lets students compare the two forms in terms of both results and efficiency. In my trial, the first took 0.354 ms and the second took 0.3380 ms—a 5% reduction that might be very important as part of some critical task, such as deploying an airbag or animating a super-fast video game.

If presented in broader context, factoring trinomials is well worth including in an algebra class.

on 18 Jun 2015 at 2:25 pm50 Howard Phillipsfactoring 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!).

on 18 Jun 2015 at 2:57 pm51 Kate NowakI think some things are in standards because they are broadly important for many people to understand for their lives, and there are some things in standards important to mathematicians worried about the small number of students who eventually will want to appreciate the ring structure of polynomial expressions. (Or something… those words might actually be nonsense. I’m not a mathematician.) And factoring is there because of the latter reason, but i think you’re asking how can we make it compelling anyway.

I’m onboard with everyone who has suggested making it a puzzle. I’d add that students can find it rewarding when we set them up to see connections across mathematics they’re learning. Factoring is rich territory for this because they can mess with it numerically, algebraically, geometrically (area model), and with roots of functions. The connections headache manifests when they wonder how this new weird problem fits in with everything else, you hear things like “Where is this GOING, Nowak?!” and then a little while later, “Ooohhhhhhh!” excitement when they see how it fits and try to figure out why. (Which presumes attention to an attitude among students toward math where everything should fit and make sense, which is no small feat, and not something i claim total success with yet.)

Another great thing is when they have to use this abstract skill to solve a problem about a tangible thing, like see Fawn’s lesson about the picture frames. Especially when they don’t have forewarning about what mathematical skills will pop up and they figure out that something they learned can be useful here.

on 18 Jun 2015 at 3:22 pm52 Malcolm RobertsMy 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.

on 18 Jun 2015 at 4:47 pm53 Chester Draws@ Malcolm

The important thing though is that all students have the opportunity to engage in mathematics at the level that matches their interest.And when we do this we are immediately in trouble with the people that decide everyone must do the same courses, because to stream — even on interest, rather than ability — is discriminatory.

New Zealand from Year 11 (US grade 10) actually does split its programs. A large proportion of students only do a non-algebraic course, so never do factorising trinomials, for example. The rest do algebra, graphing, etc as a “normal” Maths program. I believe the Brits have their O-levels split into a more theoretical and a more practical option.

Yet most countries insist that all students must learn the same Maths.

on 18 Jun 2015 at 6:44 pm54 CathyMy 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.

on 18 Jun 2015 at 6:50 pm55 CathyForgot to mention that making connections within and outside mathematics is important. This would be a case for connections within mathematics – multiplication and division, and their various representations.

on 19 Jun 2015 at 1:07 am56 Andy HaanAwake 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?

on 19 Jun 2015 at 4:18 am57 ChrisReading this instantly put the thought of a question from one of this year’s exam in the UK. A great deal of interest and amusement was had by all. All good for newspapers on a slow news day. Its a probability question which ends up being a quadratic/trinomial. All you have to do is search for Hannah’s Sweets. There’s a couple of articles even asking what is the point of putting context and ‘real life’ into exams and also the point of putting context and ‘real life’ into maths.

If only ‘real life’ was real life.

on 19 Jun 2015 at 4:37 am58 Tim HatmanThere 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.

on 19 Jun 2015 at 5:19 am59 Kyle Pearce@Tim

I agree that “math for math’s sake” and taking the angle of “math as a puzzle” won’t necessarily cause a headache if you can’t hook them in. Your comment is making me think more deeply about my original approach and I hope to come across a way to toss in a question midway through that makes kids scratch their heads. Good summer thinking ahead!

Your suggested headache is a great one. Without factoring, it would be a total pain to graph with students likely being frustrated upon completion. However, I’d argue you may have the same problem drawing students in just as the “math for math’s sake” and “math as a puzzle” cases. I like how your idea would prompt a pain in the brain, but I’m not so sure that students who aren’t already interested in math would take to graphing that function via table of values.

It would appear that Dan’s strategic selection of factoring quadratics to start was a good one.

on 19 Jun 2015 at 8:09 am60 Chuck CollinsGreat 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).

on 19 Jun 2015 at 8:54 am61 Fred ThomasIf you teach factoring (whether you teach it because “It’s going to be on the test” or “It’s a fun puzzle” or “It will help you understand pure math,” you need at least one example to help motivate those numerous students who want a connection to something they find interesting or something they might use outside of math classrooms. I reframed my suggestion (comment 49) into a 3-minute YouTube video which might help motivate some algebra students.

on 20 Jun 2015 at 10:58 am62 JennyA 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)

on 21 Jun 2015 at 5:16 am63 SimonI think one of things that’s important is that our students understand that 1) factorising doesn’t change the value of the expression and 2) why it is more useful. Too often I find students thinking (x-3)(x+2) only ‘works’ for x = {-2,3}.

I think a graph might be a good approach, how about showing them a graph sketch and they have to work out an equation that matches it?

You could start with limited info, and slowly add info to it, e.g. Intercepts, maximum point.

Be interested to see which way you take this Dan.

on 21 Jun 2015 at 5:20 am64 John GoldenIn 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?

on 21 Jun 2015 at 5:30 am65 John GoldenOne story about these: I saw something I’ve never seen before. After those experiences, I had multiple students factor a binomial out of a trinomial using the area model instead of division. I never showed them that, they just applied it organically. I will pose it as a problem in the future, I think.

on 21 Jun 2015 at 12:45 pm66 TimteachesmathSequences, 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.

on 21 Jun 2015 at 1:07 pm67 EricWow what a wonderful question….. I do not think you could’ve brought up a better topic to start Dan, and I look forward to seeing the other 9 you bring up.

This is one of the topics that I always find drives me to have a headache as a teacher…. I always see factoring as one of the major skills that leads to inequality in the math classrooms, especially as an Algebra II teacher who has a lot of students who have no interest in future math pursuits.

The students who can be lured in to “Do math for math’s sake” or who like the challenge of a puzzle can learn this skill and often become better at factoring with practice. However the students who have for years been told that they are “Bad at math” (either through poor grades or from classrooms that are structured in fixed mindset math ability) are the ones who truly need some sort of reason why a) this skill can be masted regardless of previous math experience or b) why this is a skill worth learning. I find that I have never in my years teaching been able to set up this skill as separate to previous math concepts. Dan I hope you can you can also find an asprin for teachers who are required to teach this topic.

on 21 Jun 2015 at 3:46 pm68 TreyI’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.

on 21 Jun 2015 at 4:09 pm69 Greg Benedis-GrabI am coming late to the party as I just read this post today 6/21. I am also perhaps not qualified to contribute since I am really a science educator more than a math educator. I love the question involving aspirin and headaches. I think it is time for us to tackle the deeper questions of education rather than falling back on the comfortable buzz words such as authentic, project-based, problem-based, and real world. I agree that mathematics should be relevant and we need to think more deeply about student motivation both from a life experience perspective and a conceptual development perspective. I am not trying to critique the original post, but rather offer a question into the mix. Is the headache analogy the right analogy for our classrooms? I am thinking about a piece that was written in the 70’s that also was built upon the work of Piaget by Eleanor Duckworth titled “The having a wonderful ideas.” In this article Duckworth challenges us to listen to students ideas and value the ingenious and original ideas of students as perhaps the most valuable outcome of education. She provides striking examples of how students are able to solve deeper questions ingenously. Of course the standards movement and years of backlash against progressive education has tainted this view to some extent. However, I just wanted to at least think about to what extent students should be finding the inspirational connections that we value and what types of learning environments can make that possible.

on 22 Jun 2015 at 6:30 am70 James Key“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?

on 22 Jun 2015 at 5:26 pm71 Paula Beardell KriegWhenever I am trying to sort out a tricky concept my own go-to is to talk about it with a dyslexic young adult, since their need for clarity is essential and their perspectives are beyond my ability to imagine without their help. Both of my children fit the bill, and are at just the perfect age to ask about quadratics. I asked them, completely separately, about your question, and I am happy to share their answers with you.

First, both of them (independent of the other) immediately said to make factoring quadratics into a game. To me this was an interesting response as it indicated that they both know that ” factoring trinomials with integer roots, ie. turning x2 + 7x + 10 into (x + 5)(x + 2)” is not a real world skill. It’s interesting to note that they had no problem with it not being a real-world skill and it reflected that they knew that finding integer roots can be a fun and satisfying puzzle to solve. (Now don’t make the mistake of thinking these kids are math whizzes. Even though they have great number sense, the dyslexia has a negative impact on math performance, but I digress. )

After a certain point in his math classes my son had made a decision regarding quadratic equations: he decided that he would always just use the quadratic formula every time to solve every quadratic equation. That way he didn’t have to think about strategies, keeping different methods in mind, or start then fail and have to start over again. This worked for him. But, knowing this about him, I questioned his opinion that the factoring could be a game, after all, it doesn’t sound fun to me to solve equation after equation with the quadratic formula!

The point, then, would be to know in advance that there are integer solutions, making the quadratic formula unnecessary.

This seems perfectly legitimate to him. He said, “Mom, in real life you would only ever have one problem that would need the quadratic formula. You’d never have a list of them.” Oh. Yeah, uh, that’s right.

My daughter was adamant about not using graphs to solve quadratic equations. It took some time teasing out what the problem was that she had with graphs. Turns out that she was reacting to her own experience of having to create a graph over and over again for each homework problem, having to label the points, having to precisely position the curves, in other words graphing was cumbersome to the point of being obnoxious. And what was even more obnoxious to her was if shewas given a an expression like say, x^2 +3x and she factored into x(x+3) she would get points marked off for not writing it as (x+0)(x+3). Her teacher once remarked that it was surprising that my daughter didn’t like graphing since it included drawing, and she knew my daughter likes to draw. I’ll let you imagine my daughter’s response.

The fact was that my daughter was fine with factoring, but she hated how cumbersome her teachers made it. She was only taught the long way of doing things. Once when we were going over her homework together she saw me do a typical shortcut, which was to first lay out the parentheses and fill in with x , then put in the appropriate signs so my work looked like this (x + )(x – )then, after some thought I filled in the integers. She was aghast. And angry. She said that she had continually asked her teacher, isn’t there a simpler way of solving quadratics, but her teacher insisted that, no, always you first you had to make a diamond shape, fill in the spaces, then make the square, then fill that in (I am going to assume you know the techniques I am referring to), and there were no shortcuts. She was furious that she had not been introduced to any simpler methods of thinking about her factoring.

So that’s my last bit of insight: don’t muddy up factoring with too many auxiliary steps. Sure, teach all the methods, but then let quadratic factoring be the fun puzzle that they are.

on 23 Jun 2015 at 6:41 am72 Scott HillsI 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.

on 24 Jun 2015 at 6:54 am73 dy/dan » Blog Archive » If Factoring Trinomials is Aspirin Then How Do You Create The Headache?[…] and few skills that seem harder to motivate for math teachers than factoring quadratic trinomials. (Sample their […]

on 24 Jun 2015 at 6:55 am74 Dan MeyerThanks for the insightful commentary here, everybody. I pulled a number of your comments into the follow-up post.

on 29 Jun 2015 at 2:18 am75 Kenneth Tilton“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.

on 09 Jan 2016 at 10:35 am76 David JonesThere is a particular problem in maths today, with non specialists judging and observing maths teaching. They are desperate to see a real world application or even just something they understand themselves (if it’s an A level lesson). Meanwhile the bright students don’t need an immediate application and are losing out by a certain amount of time wasting/dumbing down[I am in no way an elitist!]. They understand / know why we are doing this (the fact that neither I nor they can articulate why is not important!). An example .. yesterday I made a yr11 student aware of your cuboid factorisation puzzle that ‘1000 teachers couldn’t solve’). Today the student has emailed a fantastic response involving computer science and P=NP type problems. He understands this far better than me. I will send you an edited copy of his ‘solution’ if you email me. It is too long and not appropriate for this forum. Thanks for your excellent site

on 09 Jan 2016 at 10:52 am77 David JonesPS

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