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

June 24th, 2015 by Dan Meyer

**This Week’s Skill**

Factoring quadratic trinomials.

eg. We can express quadratic trinomials like x^{2} – 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:

- Tie the skill to its historical roots. “The thing I find interesting about factoring quadratics is the idea that the Babylonians started using it with taxes.”
- Tie the skill to its later uses in math. “We want to know factors because they explain why polynomials and rational functions work the way they do.”
- Turn the skill into a puzzle. “… they knew that finding integer roots can be a fun and satisfying puzzle to solve.”

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 the “real 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 x^{2} – 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 x^{2} – 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**

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=(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?

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.