If Factoring Trinomials Is Aspirin, Then How Do You Create The Headache?

This Week’s Skill

Factoring quadratic trinomials.

eg. We can express quadratic trinomials like x2 – 7x – 18 as the product of the two binomials: (x – 9)(x + 2).

If you find that language disorienting, if it makes you wonder why anyone would even bother on a sunny day like today, you’re in good company with lots and lots of math students. At the secondary level, there are few skills that seem less necessary to students and few skills that seem harder to motivate for math teachers than factoring quadratic trinomials. (Sample their stress.)

What You Recommended

Mercifully, very few of the 70-ish comments on my last post suggested an instructional strategy for this skill without also describing the theory that gave rise to the strategy. We need more of that kind of conversation, not less.

Here are three theories I found particularly interesting:

The first two solutions seem to me very clearly defined and very easy to implement but also very far-fetched.

Why am I interested in the history of a topic that has terrorized me for years? I’d like to know less about its history than more. (Even Andy, who suggested the idea, admits its vulnerability.)

And if learning this tough skill now helps me learn some tougher skill later (what Josh G. described as “passing the buck upwards“) I can save myself the trouble twice over and learn neither.

The last design theory has a lot of promise. People love puzzles after all, even the kind that are far from thereal world.” But it seems very difficult to implement.

I consulted two textbooks – one Algebra 1 text from McGraw-Hill and Pearson each. One attempted to tie the skill to the real world. The other said, “Look we’re just going to teach you this stuff okay.” The latter approach is honest, if uninspired. The former approach seems overly self-satisfied. After checking off the “real world” box, they proceed to teach the skill abstractly through a series of worked examples.

What a Theory of Need Recommends

First, ask yourself, if the skill of factoring quadratics is aspirin, what is the headache? How does the skill relieve a mathematician’s pain? The strongest answer, I think, is that the skill helps us locate zeros. The number that evaluates the expression x2 – 7x – 18 to zero isn’t obvious. In its factored form however – (x – 9)(x + 2) – the zero product property tells us the answer quickly: x = 9 and x = -2.

This is Guershon Harel’s “need for computation,” particularly the need for efficiency in computation.

Once that need is clear, the activity becomes much easier to design. Students should experience inefficient computation before we help them develop efficient computation.

So with nothing on the board, ask your students to: “Pick a number between 1 and 10. Write it down.”

Not a problem. Now put the expression x2 – 7x – 18 on the board and ask students to evaluate it for their number.

This is an unreal and irrelevant task, admittedly, but no one asks “When will I ever use this?” because students tend to ask that question when they feel disoriented and stupid. This prompt is relatively clear and accessible.

Now you ask: “Who got zero? Anybody? Anybody? Raise a hand. Nobody? Okay. Not a problem. Try a different number. Try a different number. Try a different number. Don’t stop until you get a zero. Call me over when you do.”

If someone did get a zero, ask them to get another one. (Later question: how do they know there are only two solutions to this equation.) Record the solution next to the quadratic on the board. Put up three more. Ask them to find more zeros.

Tease the possibility that a more efficient method than guess-and-check exists.

After 5-10 minutes of guess-and-check, help them learn that method.

What This Is and Isn’t

I’m not saying this activity will be your students’ favorite day in your class all year. Factoring quadratics was never going to be that.

But I’ll make a mild claim that this activity will be motivating for students. We’ve created a task with a clear goal state and a low entry and a high exit, a task that is iterative with timely feedback. These features are all common to the most intriguing puzzles. Of course a student could ask, “Why do I care about finding zero?” But they could ask similar questions about Sudoko, Tetris, and other puzzle games. They don’t because puzzles are, by definition, puzzling.

I’ll make a strong claim that this activity will endow factoring quadratics with a sense of purpose that it often lacks. Not purpose in the world of work or surfboards or trains leaving Philadelphia traveling west, but a purpose in the world of math. By tying the skill of factoring quadratics into a network of older skills (especially “guess and check”) we strengthen all of them.

I’ll make a strong claim that this is an example of taking a theory of instruction and enacting it. Finding a workable theory of instructional design is hard enough. Enacting it is even tougher. I love that work.

Doug Mackenzie asks an important question which I’m about to ignore:

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

I’m not making any recommendations here about how students should learn that more efficient method for finding zeros. Tell them that method directly. Let them discover it. I know what I would do. We can draw from research. But that isn’t what this series about. This series is about creating the need for new learning, not satisfying it.

Next Week’s Skill

Exponent rules.

It’s like foldables were invented for exponent rules. Students can memorize a bunch of rules and write them down in something organized and pretty.

But why do we need them? If exponent rules are aspirin, then how do we create the headache?

Other Great Comments About Factoring Quadratics

It turns out I was on a similar frequency as Eric Fleming, Joshua Greene, Chuck Collins, and others.

Tim Hatman does some really impressive work exploiting Harel’s “need for certainty”:

So here’s my headache. Graph y=(x2 + 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?

Malcolm Roberts names a central dilemma to all theories of instruction, not just this one:

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.

Simon names one key misunderstanding of factoring quadatics:

I 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}.

Chuck Collins names a second:

you’d be surprised how many college students don’t realize that the quadratic formula gives the same solutions that you get from factoring

Scott Hills recommends “diamond puzzles” in the weeks running up to this instruction:

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

About 

I’m Dan and this is my blog. I’m a former high school math teacher and current head of teaching at Desmos. More here.

39 Comments

  1. Ok, quick comment cos previously I thought ‘someone will mention this’ but as far as I can see this is not the case.

    First, I miss a relationship with multiple representations. For the sake of argument I will choose a slightly simpler format say
    x2 – 8x+15, this of course, if equated to zero and factoring yields (x-3)(x-5)=0, roots etc. but it can also be written (completing the square) as (x-4)^2-1=0 so as transformations of the basic graph x^2 with translation 4 to the right and 1 down, the max/min seen as well. Also can emphasise symmetry axes, and maybe some Desmos ;-)

    Second, I think it should be emphasized that factoring only makes sense if it’s related to equating to zero i.e., indeed as you do later on. Finding the roots. So again the visual aspect plays a role. But also explicitly state the equation, and also that it won’t work when not equal to 0.

    All in all I think I would like to emphasize that these representations and different algebraic appearances all boil down to ‘the same thing’ and that maths is a coherent structure. Much along the lines Chuck mentions the Quadratic Formula.

  2. My pre-calculus class in high school was totally centered around one problem: sketch graphs. I don’t remember how, but our teacher convinced us that graphing things was an important and satisfying activity — it somehow made you feel like you had conquered an equation, understood everything about it that was important. The more skills we learned, the faster and more accurately we were able to sketch graphs.

    Factoring was a HUGE part of that. Other things I remember doing are putting the y-intercept on the plot and deciding if it was concave up or down. We did those three things and felt like we had FIGURED OUT the equation in a deep way.

  3. When I taught IMP, there was a trinomial amongst a group where the area model was used to factor but the trinomial didn’t factor with integers. A number of students would naturally treat this problem as a puzzle and use decimals to get close to an answer. The first time I saw someone do this I was glad I didn’t give a math teacher response that the problem couldn’t be done. The kids just saw it as a better puzzle. They liked that headache!

  4. Seeing @Ben T’s comment reminded me of something I wanted to ask in the previous post:
    What’s the desired time-scale for the motivation?
    If you want to get through the lesson-of-the-day, then what Dan and others have suggested would work, but you’d be back here asking for an inspiring lesson for the next topic. Also, this short-term motivation might not help the students remember the important or valuable aspects of the material. However, if you take a broader approach, e.g. @Ben T’s precalculus course, then you have some level of motivation for each topic, and a hook for the students to remember the relevance of the topic long past the day’s lesson.

  5. I teach factoring polynomials by having my students do “factoring puzzles”. I teach them multiplication of polynomials by the lattice method, and then have designed a series of puzzles that are bits and pieces of the multiplications, which forces them to think about factoring and multiplying all mixed together, and then ultimately I give them the same puzzles with just the answers and they need to work the tables backwards to factor. It’s ultimately factoring by grouping, but it’s all the same tool (lattices), and I definitely spin factoring as “puzzles”.

    Here’s a link to a draft of a blog post that I hope to flush out with more details some day “when i have time”: http://rockyroer.blogspot.com/2015/06/factoring-puzzles.html

    There’s a few example worksheets downloadable, as well as links to my youtube videos that illustrate the process.

  6. RE: A Theory of Need (and “describing a theory”)
    “But that isn’t what this series about. This series is about creating the need for new learning, not satisfying it.”

    I have been writing about the concept of need in the context of [mathematics] education, which was also, incidentally, broached in the context of factoring quadratics:
    https://mathwater.wordpress.com/2015/05/17/needs-and-buying-in-versus-selling-out/

    You can find my underlying theory (Need-Can-Must-Work) sketched out in clearer language here:
    https://mathwater.wordpress.com/2015/06/03/clarify-dont-verify-interlude-as-re-introduction/

    And also the theory applied to a different context (professional development) here:
    https://mathwater.wordpress.com/2015/06/14/twlpd-the-work-lenses-professional-development/

    I have positioned need as one of three lenses, but not delved more deeply into its specifics. Perhaps what I have written will be of some help to others; certainly what others are writing in their [math] education blogs is of great help to me!

  7. I should mention my old post A Puzzle Equivalent to Factoring a Quadratic. What I found most interesting where the number of people who came across the post treating it as a real puzzle, like something of the NY Times blog or such.

    I don’t see your motivation working out when the leading coefficient isn’t 1. I almost universally find it faster to whip out the quadratic formula than any of the other procedures (and honestly, in general I’d rather a student write a factor like (x – 1/2) — as would come from using q. formula — rather than (2x – 1) — the typical thing in a lattice solve or grouping or whatnot).

  8. This builds a need for a better way for sure, though it does require a substantial amount of arithmetic computation. There is nothing wrong with this, but the success of the method Dan described will depend on the computation abilities of the class.

    It’s easy for a teacher, unfortunately, to reject this approach by saying ‘my students can’t handle the arithmetic for this’ and continue doing things in the traditional way. I think there’s a lot of potential here for building the need even with technology. A graphing calculator might make it too easy to generate a table of values and not generate sufficient exhaustion necessitating a better way. A scientific calculator might offer a good mix between the two. Building a spreadsheet is another approach with a higher bar for entry in terms of technology.

  9. I think you nailed it — finding the zeroes is what this could all be about. And I think you also nailed the difference between asking “what is factoring about?” and “what could factoring be about?”

    After 5-10 minutes of guess-and-check, help them learn that method.

    Does creating an experience of inefficient methods sufficient for creating an experience of intellectual need?

    I’d suggest, no. First, because the students might not realize that their techniques are inefficient. (Hell yes I guess and checked.) Second, because our feelings aren’t always transparent, and just because I experienced frustration doesn’t mean that I noticed my frustration, and it *certainly* doesn’t mean that I noticed that my frustration was due to inefficient methods.

    For these reason, I’m worried about just spending 5-10 minutes of guess-and-check followed by instruction.

    I’d prefer to spend a class period working on this, and I’d prefer if my students were in a position (curricularly speaking) to use something over and above guess-and-check. Maybe graphing? Maybe systematically making a table of values? The point is that we can more easily compare the features of methods (including their relative efficiency) when there’s more than one method on the table.

    If my goal for the lesson is for students to experience the insufficiency of their current zero-finding methods, I’ll want to make sure that they experience that by giving them problems that various techniques would naturally work on, and problems that they won’t. Since I want to make sure that my students notice the intellectual need for stronger methods, I’ll want to have a conversation where I ask them to reflect on the zero-finding experience.

    This is going to take longer than just a few minutes.

  10. Johanna Greenough

    June 25, 2015 - 5:06 am -

    I have been ruminating on this idea of “what’s the headache?” since your last post. I meant to post earlier but have been on vacation. I love the idea of finding the headache, and as a Montessori teacher, my first instinct was to say brain development. In my classroom we teach a squaring, square root, cubing and cube root curriculum that is materials-based and then transitions to abstraction. Sometimes parents ask, “why is my kid even learning this?” It’s a good question. After all, they can do that on a calculator. But it’s about the development of the brain. Being able to see the cube and its components in their mind and logically organize the parts of a cube root by hierarchy is an amazing feat of the brain. And it leads to mathematical habits that extend out into all areas of the curriculum.

    My other answer is perhaps more arguable. Why do people write poetry and never ask about its real life applicability? No one cares that poetry isn’t directly useful in everyone’s job. They appreciate it for its art and beauty. I think that factoring quadratics has a kind of beauty as well. I agree with the puzzle statements, and puzzles are often beautiful once solved, and therefore inherently motivating to solve.
    I told this to my brother and he laughed at the idea of math as an art. But I would say: will everyone write poetry? No, of course not. Not everyone’s drawn to that sort of beauty. But we still teach it to everyone, because then the 20% of students who are drawn to that will find their specific sort of beauty.
    So let’s teach quadratics, because then maybe the percentage of kids who do see beauty in that kind of puzzle and its unlocking will find joy there. I firmly feel that one of my most important jobs as a math teacher is presenting math in ways that allow students to see the joy.

    • Joanna, I love the way you put this. I love the puzzle aspect of factoring, but I also love the finding zeros (the way math interlocks in quadratics fascinates me), but it wasn’t until I started teaching it to others that I really began to feel the beauty of it – and want to help others unlock that beauty.

  11. I tried a completely different approach to factoring this year that turned out to really help the concept stick with my students. At a conference on teaching mathematics using a brain based model, I was introduced to the concept of “reverse thinking” and why it is so difficult for many students to think this way. The presenter gave many examples, but automatically factoring came to my mind. Given two binomials, students have a fairly easy time learning to multiply them. However, ask a student to factor and they don’t often even see the connection to what they did when they multiplied. This year, I tried a different approach to help them see the connection. As soon as they had mastered how to multiply 2 binomials, I started giving the students “puzzles” to work out. I started with giving them several trinomials–each had one binomial written next to it and then a blank space which they needed to fill in with another binomial. In the first set, I gave them a list of binomials to choose from and then for the second set I had them figuring out the second binomial without any help. I then moved on to just giving them a trinomial and asking them to find both binomials. Again in the first set they were given binomials to choose from. By the end the students were making up the problems–one student would multiply 2 binomials together and then pass their answer paper to their partner who would then have to factor the resulting trinomial (or binomial for the special case).

    The conversations that I heard from the students as they worked with their partners were wonderful. Because they saw each set as a puzzle to solve they were engaged the entire time. They started to recognize patterns almost immediately and some great class discussions grew from those patterns they noticed (some which I have never thought about in my 18 years of teaching).

    I realize this doesn’t help the students to see the application of what they are doing, but seeing them so engaged in a topic that has been torture for many in the past was rewarding as a teacher. And the best part was that when it came time for the quiz and test on the subject, I’ve never seen such high grades.

  12. Some things I might be concerned about:

    “Making an expression zero” isn’t a strong enough stimulus to naturally motivate a lot of tedious computation.

    The focus becoming too much about computation and not enough about factoring and equivalent expressions.

    Factoring actually might not be more efficient than guess and check when one first learns to factor (your aspirin is no good).

    Michael’s point about successful guessers becoming entrenched in the guessing method.

    Evan’s point on computation issues, particularly if there is a leading coefficient different from 1, or students are substituting negative values.

    I think if you spent a day on guess and check, as Michael suggests, you very well might end up with good and interesting approaches to finding zeros. You might even have students developing some good strategies given ten minutes or so. This is actually another concern I have. Would I then abandon these very nice mathematical ideas that are being developed and start talking about parenthesis and numbers that multiply to whatever and so forth?

    Why not start out with binomials: “x^2 + 6x” or “3x^2 – 7x”? This removes some of the tediousness and computation issues, you throw them a bone with “0” working for all of them, and there is a relatively easy short cut that someone will likely notice if you use nice numbers. Toss in a third term to create the headache. Not convinced that making an expression zero is a better goal than simply factoring (making a rectangle) via the area model, though.

  13. +1 to Pershan. This mini-lesson idea fell flat for me. 5-10 minutes of computing doesn’t seem like that compelling a headache. Plus, I’m not seeing how it fits in a coherent mathematical story. It feels like just a one-off isolated skill.

    Also a suggestion: if you’re going to keep comparing how textbooks approach these, maybe add one into the mix like EDC’s CME series, which will have sound methodologies.

    Is this where we’re supposed to comment about exponent rules? This is one where I think too many teach it explicitly way too soon, and memorizing rules shortcuts understanding. I’d advocate for a set of problems distributed over at least a few days’ time where students need to reason about rewriting expressions using exponents when all they are sure of to begin with is the meaning of the exponent notation and that a/a=1 (for a =/= 0). They’ll start developing their own shortcuts and after a few days of that you can draw those out by asking them to explain how we know certain expressions have to be equivalent, for example 2^3000 is the same as 2^8000/2^5000 (even though handheld calculators choke on them), or 5^-1 has the same value as 0.2 (even though many students’ first instinct is to think that 5^-1 must be -5). And /then/ make them explicit (and make a foldable or add it to the word wall or however you codify it).

  14. mark schwartz

    June 25, 2015 - 8:43 am -

    i use a non-algebaic visual representation to introduce students to factoring trinomials and don’t call it factoring … it’s a game at first before presenting it algebraically.

  15. I checked the CCSSM document for “discriminant” and it’s not there. Nor is what to me is the fundamental thing about factors of a quadratic, namely, what do you get when you multiply out the factors (expand the brackets)(misuse the distributive law).

    (x + a)(x+b) = x^2 + (a+b)x + ab

    which contains all you need to know:
    coefficient of x is sum of the factor constants (a+b)
    constant coefficient is product of the factor constants (ab)

    So all you need to do is to find all possible pairs of factors of the constant term, and then add them and check.

    I take the example from Chris (1st comment)
    x^2 – 8x+15
    factors of 15 are 1,15 3,5 -1,-15, -3,-5
    sums are 16, 8, -16, -8 Whoopee! It’s the last one!!

    • Howard Phillips, Chris, you emphasize the factors and additions: my mentor teacher used this as a competition. The kids divided into two teams. A quadratic was placed on the board, and the first two from each team had to figure the answer in their head. Right answer, team got points. Most points won the game. The lesson came befor the game with variables only, (a+b)(a+b) first, asking the kids to notice the patterns created and begin predicting what each combination ( two plus, one plus, one minus, then two minus) would look like, then see the answer and reverse the pattern- he would then move to numbers and the game. Our students are it up. As a teacher, I like the connections/basic ideas it built for the students: Pascal’s Triangle, multiplying Binomials to nth powers, formula work, rational numbers… And on into pre-calculus. (I know you can think of even more connections!)
      This also connects with the next idea of exponents.
      The headache of exponents comes from all the student misconceptions about them. They see exponents as something to multiply “by” instead of instructions to follow for the base number – maybe because it gets taught as multiplication, and kids misunderstand WHAT is being multiplied.
      I again go to variables, or images like Suns, squares, triangles, to illustrate the properties of exponents. Child-like, but it separates the act, from the numbers, so every child has an entry point. Then when I give numbers, I get less of the 3^2 is 6!

  16. Why are we teaching factoring by hand in the first place? Is it some kind of required skill for life? When I was a kid knowing how to find the square root of a number using the algorithm was a needed skill. How many people know how to do that any more? If I want the zeros to a polynomial I go to WolframAlpha or a calculator. It is why we want those zeros I think is important. When do we retire certain math skills when they are replaced by technological methods?

    Yes, I still teach factoring trinomials in my math classes. Why? Because I have been doing it for 30+ years and cannot get out of the rut. I spend maybe one day on it and then on to WolframAlpha and why we want those elusive zeros. Remember the only trinomials (or any polynomials for that mater) that are factorable are contrived by textbook authors. The number of real-world factorable polynomials is zero.

  17. the trouble with this topic is that knowing how to factor serves many, different purposes.

    yes, it helps find zeroes, but as Garth rightly points out, if this were its sole need, technology does that for us now.

    but it serves other purposes too, as noted by Tim Hatman:

    “Tim Hatman does some really impressive work exploiting Harel’s “need for certainty”:

    So here’s my headache. Graph y=(x2 + 7x + 10)/(x + 5)”

    even more useful is when it is critical to discovering more abstract relationships, such as in the proof for the indeterminate limit of sinx/x using the squeeze theorem.

    that is so pervasive in so many circumstances is what leads to the central problem: factoring is so useful in so many ways that it must be learned abstractly and yet many of its most immediately useful aspects are better handled by technology today.

    i don’t buy the zeroes lesson for this reason, and i think at a certain point we have to admit that some math skills are necessarily abstract because that is precisely what makes them so practical.

  18. I don’t buy factoring trinomials as being the aspirin for guess and check. Why? Because factoring often involves a bit of guess and check. Less so for someone with strong number sense, but that’s the case for your lesson as well. But when my students are learning to factor, they often try many different pairs of factors before they find the ones that sum to -b.

  19. From Dan: “Of course a student could ask, “Why do I care about finding zero?” But they could ask similar questions about Sudoko,” etc.

    I think I can answer that one! If you can find where a quadratic expression is zero, then you can *solve any quadratic equation,* period! i.e. f(x) will be equal to g(x) precisely when f(x) – g(x) = 0. So the skill of finding a zero of a polynomial is tantamount to the skill of solving a polynomial equation. Obvious next question from Dan’s hypothetical student: “Who cares about solving equations?” My response, “Because, um, that’s like what algebra is all about.”

    From Johanna: “Why do people write poetry and never ask about its real life applicability?” I think the difference is that people *know* what poetry is supposed to be, but perhaps there is some confusion in the math ed. community about what role factoring is supposed to play. If we were all just to *admit* that factoring is a mere puzzle, not some linchpin of the high school curriculum, and to *make it clear to students* that we’re simply learning this technique “for the heck of it,” then that would change the game. Right now I’d say many give the impression that learning to factor is somehow *a truly important math skill.* Is it?

    The truth, as I see it, is that factoring is a matter of *convenience.* i.e. we factor for the sake of quickly solving equations with *integer solutions,* and we pose problems with integer solutions so that we don’t get side-tracked by some weird-ass irrational values! This happens all the time thoughout the curriculum, when we want to analyze polynomials or rational functions — it’s just *nice* to deal with integers, because we don’t *learn much* by doing otherwise.

    A second view is that, what is actually really truly important, is to be able to factor things *symmetrically,* i.e. to complete the square. That is the skill a person needs to solve general quadratics, not just those with integer solutions, and it’s also the key that unlocks the structure of two-variable quadratics, whose graphs might be circles, parabolas, ellipses, etc — but which? Now there are some headaches for which “completing the square” is the aspirin!

  20. Last year I decided to throw in some factored equations to solve when solving linear equations. Just to give something different, emphasize that a solution is a number that makes an equation true, emphasize that equations can have more than one solution, and do some prep for later quadratic stuff. I might give one factored and equal to zero, and one factored and not equal to zero….

    (x – 2)(x – 3) = 0
    (x – 2)(x – 3) = 12

    Maybe something not factored… x^2 – 6x + 8 = 0?

    After everyone has had time to think about the first one, I’ll collect answers & discuss why two different numbers work, the zero product property, etc. They then try the next one and hopefully immediately see that it’s not bad to reason through, but that the zero product property doesn’t work quite so well.

    Another thing to do is graph quadratic functions that are in factored form and not in factored form. They can figure out that factored form is easier to graph, and that can provide motivation for factoring.

    By solving without factoring but given some factored problems, and graphing without factoring but given some factored problems, students become familiar with the uses of factoring without factoring. Then factoring has a little more of a purpose…

  21. Johanna Langill

    June 25, 2015 - 10:21 pm -

    I second the puzzle approach. My curriculum uses Algebra Tiles and the area model to “find the rectangle,” where students work backwards to figure out what the missing pieces of area or dimension would have to be. In fact, before we teach multiplying binomials we do some playing around with factoring with Algebra Tiles, we just say “make a rectangle with these tiles and find its dimensions” rather than “factor this trinomial.”

    Besides making guess and check more efficient, factoring also helps us be more precise. What happens if you’re guessing and checking and you know that your function changed from positive to negative, but you don’t want to keep on narrowing it down by trying the (infinite!) fractions on the interval between the closest integers. You can use graphing technology or a table to see about where the quadratic is crossing the x-axis, but factoring tells us exactly where it is. However, this feels less compelling to me than puzzles. It’s a reason to apply/use this factoring tool that I know, but I don’t know if it would be enough to motivate the entire factoring process.

  22. I’m gonna focus on exponent rules, but I will say that my thoughts about factoring all come back to the puzzle like question, “How much can I know about a function without graphing it with a calculator?”

    Why do we have exponent rules? To answer this question, I would like to propose the following question: What would math look like without exponents or their rules?

    I think of my 6th grade students writing down the prime factorizations of whole numbers (Why learn that? Oh yeah, to improve number sense and seeing structure behind numbers, among other reasons). When you work with something simple like 24, writing out 2*2*2*3 is not so bad. When you up the stakes to something like 256, writing out 2*2*2*2*2*2*2*2 becomes annoying. It’s not so much a headache as a tedious process that any normal person would like to make quicker and easier. At this point, I discuss exponents as a means of communicating all of that multiplication without having to write it all.

    From this basic understanding of exponents (they make things easier), moving into the rules becomes a matter of discovering how to make expressions/functions easier to understand and avoid tedious redundancy. Similar to your approach for factoring quadratics, I imagine giving students in an 8th grade class or freshman algebra class a function like f(x)= x^5/x^3. I would ask students to evaluate the function for various values. After some time with this process, I would ask students if anyone stumbled upon an easier way to evaluate the function than blindly crunching something like 5^5/5^3 in a calculator. Hopefully in the midst of the activity, some student would have gotten annoyed enough to realize that he/she could just take the square of each number.

    My point is that exponent rules are really just blindly memorizing what students are capable of discovering on their own by means of the need to make complicated things simpler, which in turn makes these complicated things not so complicated and easier to understand/communicate.

  23. A Russian friend told me that in Russia, this skill is taught in the opposite way. Students graph the function to find the zeros and then use those to factor it. I’m still looking for an authentic purpose for factoring trinomials because in most real models the zeros won’t be integers and Wolfram Alpha is the best way to go.

  24. Thanks, as always, team.

    Ben T. describes the virtue of coherence:

    My pre-calculus class in high school was totally centered around one problem: sketch graphs. I don’t remember how, but our teacher convinced us that graphing things was an important and satisfying activity — it somehow made you feel like you had conquered an equation, understood everything about it that was important.

    Even when the goal is an abstract one, if it’s clear, coherent, if its sub-goals are iterative, with timely feedback, it can become puzzling.

    Coherence is great. Rather than each day being about a new factoring trick (grouping, FOIL, etc.) it’s about finding zeros. And the guess-and-check introduction to that coherence is even better. If a student forgets a trick, they have no fallback. In the model I’m proposing here, they can fall back to guessing and checking.

    Chuck Collins doesn’t recognize the same coherence in the “find the zeros” approach:

    If you want to get through the lesson-of-the-day, then what Dan and others have suggested would work, but you’d be back here asking for an inspiring lesson for the next topic.

    Kate neither:

    Plus, I’m not seeing how it fits in a coherent mathematical story. It feels like just a one-off isolated skill.

    I don’t think so. For a large stretch of Algebra, the entire question could be summarized: “What are the zeros of this function? How many are there? How do you know?”

    For instance, as students eventually become comfortable factoring quadratics with a = 1 and integer roots, you can drop one with non-integer roots. Or add a larger coefficient to the quadratic. There’s your next headache.

    In general, I’m trying to locate the criticisms of Michael, Kate, Luke, and James on a continuum.

    It’s possible to …

    a) … disagree with the underlying theory. (“This aspirin business isn’t the right way to look at math, which should be about beauty.”)

    b) … accept the theory (for now) but disagree with the need I’ve identified. (“Locating zeros isn’t the most pressing motivation for factoring quadratics.”)

    c) … accept the need but disagree with how I’ve tried to instantiate it with students. (“Five to ten minutes of guess and check ain’t gonna be enough.”)

    I’m not exactly sure where to locate these commenters.

    Johanna Greenough:

    My other answer is perhaps more arguable. Why do people write poetry and never ask about its real life applicability? No one cares that poetry isn’t directly useful in everyone’s job. They appreciate it for its art and beauty.

    The Montessori system sounds like the best (only?) place to try to enact a theory of instruction like “math is beautiful.” And I don’t disagree with you. But poetry isn’t a required subject K-12, while math is. What does that do to the metaphor?

    kmorrowleong:

    Can we agree that there is no longer a reason to factor when a does not equal 1?

    What math is there a reason for? And what is that reason?

    Loads of people in this thread and the last have suggested “make math a puzzle” as a guiding theory for instructional design. Man, is the devil ever in those details. My opinion is that Sandy has a nice approach to puzzling quadratic factors, though.

    Mike provides a very helpful articulation of why this topic is so challenging to teach:

    That [factoring] is so pervasive in so many circumstances is what leads to the central problem: factoring is so useful in so many ways that it must be learned abstractly and yet many of its most immediately useful aspects are better handled by technology today.

  25. When introducing exponents we used whiteboards. First working off what they already knew from the previous year (repeated multiplication) and having them write out that multiplication. Start with 2^3 and move on to higher exponents and eventually (2^6)(2^3). Do a few more examples of that kind and without you prompting students will find the rule in no time and start spreading it. Throw in different bases and have them explain whether or not their shortcut still works. Would love to name a theory but I think its our basic human need for efficiency and pattern making with an added dose of agency and ownership because its is their rule, they found it. Throw in an expression equivalent to 2^135 for kicks and giggles.

  26. I find the idea of creating a puzzle out of finding the zeros to be very compelling. This part of our algebra unit is always tedious with the students and in the past I have employed a number of strategies to get the students to learn how to do this. Language is a big barrier for my students. Throwing out terms like polynomial, factorisation and binomial always creates a road block I and often find myself starting here and explaining the language before the maths. Doing the exercise as a puzzle and also the “diamond puzzles” as mentioned in the comments would be a good way to get some maths in before having to deal with the language. I also agree that expanding binomial products is a lot easier for students to grasp and how you teach that will dictate how you go about teaching the other way around.

    Unfortunately at my school we tend to teach factorisation and finding the zero in distinct units as only the top level students have exposure to graphing quadratics so finding the zero loses a lot of its meaning to the rest of the cohort.

    As for exponent rules I treat them in a similar way to Tom in the comment above. By showing the students the reason why we have a multiplication sign again (I don’t know why but at some stage they do disassociate it with successive additions) you can have the discussion about what we should do if there are successive multiplications. In my experience, students tend to think of squared as it’s own distinct thing and it doesn’t register straight away that it could be ^3 or ^8 etc. So by giving them the problem of coming up with their own symbol can lead to an interesting and often fun discussion with the class about why they chose their sign.

    “Hidden” multiplication signs are the bane of my classroom. By writing everything out students can get a better understanding on the exponent rules. But maybe “exponent rules” are the wrong type of aspirin? Aren’t we trying to move away from rules that we can rote memorise and come up with methods of solving the “headache”? Better yet, how to avoid the headache in the first place? I look forward to seeing your idea on how to create this headache and the discussion that follows.

    Lastly, another language issue. Having power, exponent and indice all mean the same thing is another barrier to this topic. Can’t we all just choose one and banish the rest?

  27. I am not sure where to locate myself either.

    a) I am not against using the headache-aspirin structure, but I am against having this as a guiding principle. Why force yourself into that box when thinking about a lesson?

    b) The finding a zero is not a very good lesson. The need from the student’s perspective seems weak to me, but maybe there is a clever way to highlight that need a little better. The devil is in the details with any lesson, not just the puzzle approach.

    c) Short initiation is needed, I think, if the goal is factoring via find the zero. A stretched out initiation might make for a more interesting lesson unrelated to factoring.

    Another thought riffing off your original initiation might be to ask how many got an “even” value and how many got an “odd” value. Everyone should have “even” which gives you a heads up regarding arithmetic issues. Have them try another number, ask for opinions/reasons on whether the expression can be made odd, etc. Factor, (x-9)(x+2) and continue the discussion. Prime the pump, if you wish, with questions like: is 16 x 91 even or odd. To be clear, I am not thinking of “is it always even” as a headache in need of the factoring aspirin – just kind of a puzzling situation that might catch a little interest.

  28. Malcolm Roberts

    June 29, 2015 - 5:43 pm -

    Having read how Dan would approach factoring trinomials I tried to think of something similar that would lead to exponentials. The first thing that came to my mind was to get the students to try to guess the answer to a number of variations of the (well-known) chessboard problem:

    “If a chessboard were to have wheat placed upon each square such that one grain were placed on the first square, two on the second, four on the third, and so on (doubling the number of grains on each subsequent square), how many grains of wheat would be on the chessboard at the finish?”.

    In doing so could we create a task with a clear goal stated, with a low entry and a high exit, a task that is iterative with timely feedback?

    I am not sure if this strategy would work, maybe the need generated by this series of tasks is not strong enough to create interest in the students. However, at this stage my concern is not whether this task is a good one for exponentials but rather the far more general question of the applicability of the “theory of need” as guiding principle for our teaching. I have no doubt that Dan will have much success in using the factoring trinomial strategy he outlines. But is that due to the fact that Dan is Dan and he has characteristics that some others of us don’t? Could it be due to the fact that this lesson is so different to other mathematics lessons that the class has experienced? What would happen if every class used this strategy, would the novelty wear off and the students lose interest?

  29. Just to clarify, my issue wasn’t whether or not the approach had coherence, but really a question of do we want the approach to be coherent to some bigger story. I agree that one could make zero-finding a ‘big thing’ and then this approach would fit in with it nicely, but if there is a different or no ‘big thing’ then this approach could come across as a one-shot trick.

    Is the goal to create intellectual need so this day’s topic goes well, or so that the student creates connections between this topic and others and the learning persists?

    I don’t think these objectives are exclusive, but it makes some difference in how we talk about different approaches.

  30. Mark Schwartz

    July 15, 2015 - 3:07 pm -

    Consider this visual “factoring” method. The students spend one class doing the game and the next class I actually use the word “factoring”. When I cut and pasted this, the actual grid didn’t copy, so you’ll have to make up the grid marks vertically between the 3rd and 4th column and between the 2nd and 3rd row. When the students are given the grid, the area into which they put each color is clearly designated.

    Some students who had previously had Algebra recognized this as factoring, but I told them not to announce it. Please note that this area concept helped them “see” factoring. Further, there is the relationship to the quadratic. We proceeded from this to the more traditional demonstrations of finding zeroes, etc.

    I came to this because most students self-report as visual learners and I wanted to take this into account.

    The Game

    • The “grid” is divided into 4 sections. You have blue, green, red and yellow chips.

    • You will take a specified number of chips of each color.

    • You can place the colored chips only in the section with that designated color. For the section marked “green or red” you can’t place red and green; only red or green.

    • The chips must be placed such that the edges of any configuration of chips in any section must make contact with both the horizontal and vertical axes (see example below).

    • When all the chips have been placed in their designated section, the configuration of all the chips combined must be a rectangle or a square. (see example below).

    For example, take 6 blue, 7 green and 2 yellow chips and build the rectangle & you get:

    B B B G G
    B B B G G
    G G G Y Y

    • After you have made your rectangle with the designated number of chips, please record the set used and a picture of your rectangle.

    Use the following “sets” of chips to make your rectangles.

    Set↓ blue green red yellow
    1 2 1 6 3
    2 6 7 ─ 1
    3 10 5 6 3
    4 6 7 ─ 2
    5 12 ─ 11 2
    6 4 6 6 9
    7 4 4 3 3
    8 16 8 6 3
    9 6 4 ─ 4

  31. I’m strangely interested in factorable polynomials by Garth’s claim that a factorable quadratic has zero real world occurrences. Not “nearly zero”, but like fully zero. Nada. Never. Is that maybe interesting to anyone? We say, here’s some math, it works, it is truth, and it has no purpose ever, whatsoever, at all! Let’s learn all about it!

    My main purpose for commenting is that I wanted to build off two points made above. First, @Dan’s point above asserting the importance of coherence over introducing a new factoring technique each day, and second, @Mark’s comment above on using a visual puzzle to teach factoring.

    This year, I am teaching factoring as a visual/kinesthetic skill in which we use algebra tiles to re-arrange a polynomial into a rectangle. I use language like “Find a rectangular arrangement of x^2 + 7x + 5” and “Can you find all the rectangular arrangements of x^2 + ___x + 6”. I used Algebra tiles, and asked students to find what percent of the polynomials x^2 + ___x + ____ would make rectangles. We agreed to only allow the digits 0 – 9 to occupy the ___. This was a different way for me to approach this topic this year and I was surprised the other day when I realized that the rectangle analogy can extend and motivate the method of solving a polynomial equation by collecting the terms of the polynomial and using the zero-product property of multiplication. I think the biggest payoffs here are in terms of coherence. Students can think about factoring as rectangles, and they can easily see when the rectangle is improperly constructed.

    When we factor x^2 + 5x + 4 = 0 to (x + 1)(x + 4) = 0, students can visualize a rectangle with sides of length x + 1 and x + 4. Since multiplying the sides gives the area, the equation (x + 1)(x + 4) = 0 means that our rectangle has zero area, and students can determine for themselves that means that one of the sides must have a length of zero, and that choosing x = -1 or x = -4 will make our rectangle have an area of zero to satisfy the equation.

    This approach is simply trying to connect factoring to the geometric understanding that students have of rectangles. This is useful for relating polynomials which can be factored to composite numbers which can be made into rectangles, and prime numbers and prime polynomials which have no “interesting” rectangular arrangements.

    I suppose I should not still be surprised that when I search for factoring polynomials on youtube I find only explanations that move directly into symbolic manipulation, factoring algorithms and provide almost no mention of a visual approach to factoring which might give students a chance to figure it out for themselves.

    Here is a short animation I made to try and communicate the meaning of factoring. Potentially relevant to the discussion above

    https://www.youtube.com/watch?v=nbUxSK6XTCc

    Update: searching for “factoring algebra tiles” is where you can find the attempts to explain factoring visually. Not recommended for students — most videos are 15+ minutes and even the 2 minute videos make 2 minutes feel like more than 2 minutes and put us back at the beginning of this conversation — how do you generate the curiosity in students that would lead them to be interested in watching one of these videos to learn factoring? Also, I’m allowed to criticize these videos because I made one myself. For shame!