So, once kids are pros at both GCF factoring and 4-term “grouping” factoring (which is just more GCF), it’s time to introduce what I previously would have started with… the Easy Trinomial.
These are those lovely expressions like x2 + 7x + 12 that are so, so easy to factor but be careful! for as with all teaching, there are pitfalls that you with your superior knowledge will glide over while your students plunge down to their depths, which are filled with spikes, and possibly snakes.
Now, start with a challenge. “Wow, I totally want to factor this, but I can’t find a greatest common factor. Who knows what to do?”
Hopefully no one, but if some bright young thing pipes up right away, clap a hand over their mouth and promise them 100 bonus points if they shut up for the rest of the period. Let the class flounder for a bit: if anyone throws out solutions after a respectable time, put them up on the board, and — I’m glossing over this but WOW it’s important enough for its own post — have students check proposed solutions by multiplying. Maybe the correct answer is up there, maybe not, who cares, because you’ve bribed all the kids who know a sure-fire procedure into silence (you remembered to do that, right? I wasn’t kidding) so anyone who has a correct answer usually can’t explain it in a satisfactory way.
“Wow, I know this class is awesome at factoring,” you say, bringing pride into it. “What’s making this problem harder than those problems you were doing yesterday?” If they solved it, put up another problem, more challenging factors, change the signs on them if they’re sharp, whatever. For illustrative purposes, I’m sticking with x2 + 7x + 12.
Once someone can nail it down –“There’s not enough terms!” is my favorite response — emphasize that they are doing some real mathematician work here. If you like to digress, tell the mathematician putting out the fire joke, or point to your corny SOLVE A SIMPLER PROBLEM poster, or pull up Polya’s wikipedia page and assign a bunch of research papers. Obviously not really (they’ll just mock his hair, which will make you mad and derail the whole lesson), but emphasize that this is how for-real mathematicians DO WORK. Being a mathematician isn’t awesome because you solve super-super-hard problems all the time; it’s awesome because you take super-super-hard problems and make them look easy.
Now take the reins. Split up the middle term and explain how you’re doing it. “Read the expression backwards. We need two numbers that multiply to give (12) and (add) to give (7).” Ostentatiously circle the + in front of the 12. If kids freak out about negative 12 vs. positive 12, tell them to chill. Get the kids to give you 3, and 4, then you rewrite the expression: x2 + 3x + 4x + 12. Then, and this is the most important part… STOP TALKING.
If you’ve done the previous two days right and your kids aren’t defective, they will already be working. x(x + 3) + 4(x + 3) turns into (x + 3)(x + 4). Just like before! Switch the order of the 3x and the 4x and send them off again — same thing!
New problem, with the middle term negative. Hope you remembered to give them plenty of examples with negative GCFs!
Keep going with the final term negative. Ostentatiously circle that negative and say: “instead of adding to get to the middle term, what should we do?” Cheap, I know, but working memory is a thing and let’s not overtax it.
Now, doing it this way is harder than the shortcuts. But it’s not much harder, and splitting up the middle term and writing the two pieces explicitly in the expression is keeping the math real (and rigorous!) instead of having those numbers just magically appear in the parentheses.
Next day (or that same day, even) when they are ready, bust out the terms with the leading coefficient >1. All you need to do is get them to split up the middle term smarter, but multiplying the first coefficient to the last term (yes, I still call this a “rainbow” and I still draw a big arc from first to last and write the product at the top — I can’t give up all my little tricks). And then, instead of some big new scary thing, it’s the same exact factoring they’ve been doing since DAY ONE. Yeah, it’s harder in degree, but not harder in kind. Instead of this being the day where I had to drag kids along and lose a good bunch anyway and get the “When are we EVER gonna use this?” barrage, it’s like I’m barely in the room. I wander around, spiking questions back at them (“Is this right?” -Did you check it? “Not yet.” -Then check it!… “Okay, I checked it now.” -Did it work? “Yeah.” -What does that tell you? “I should move to the next problem?”) and focusing on those who missed class or can’t multiply single-digit numbers or require their cell phones to be surgically removed from their thumbs in order to do math.
Final day is Differences of Squares. Same drill, throw up x2 – 9 and let them try. They will! Because now they have a mental construct for factoring. It’s not just a random collection of tricks, it’s a coherent method that they are beginning to master. They will give you (x – 3)(x – 3) and be hilariously convinced that it works, but if you’ve built your “check your work” culture right (needs its own post), they will get shouted down. Ask “what’s the middle term?” and if they need a little more, rewrite it as x2 + 0x – 9, because this “special” factoring too is also the EXACT SAME method: Distributive property.
Surprise bonus is that you will see persistence from students like you’ve never seen before. Previously, their mental model was “I need to know the right trick,” and they would just stare at a blank page until I came around to pull a new trick out of the bag. Now, their mental model tells them “It’s the distributive property, same as always” and they have a vague surety that somehow, this can work. And my rate of saying: “Did you check for a GCF?” dropped precipitously (still way too high — don’t kid yourself, this ain’t magic) because they are checking for common factors all the time, every time. SOLID.
As for Sum or Difference of Cubes? Ha! Leave SODOC for the Algebra 2 teacher. Sure, you can write up x3 + 0x2 + 0x + 8 and have yourself a field day (it does work), but it’s much, much nicer to attack that in the context of finding zeros of polynomials once your kids can do polynomial division. By Algebra 2, kids have an actual reason to want to solve cubics, and they will be much more tolerant of memorizing shortcuts — derive it once and tell them to do it like that every time or spend some time with an index card. In Algebra 1, I’ll push the kids who really like factoring (if they can do x4 – 1 without help, they can handle it), but spare the rest of the class.
HOW I KNOW THIS WORKS: Two years after adopting this system, I had a colleague drop by my room and lay this on me: “Every year, I quiz the kids on factoring in September and split them into ‘Got it’ and ‘Needs Practice’ groups afterwards. Now I split them up beforehand based on whether they had you or someone else. It’s 95% accurate and it saves me a pile of grading. What the hell do you do with them?”
I had to say it ain’t anything I did. It’s all D. Propz.