So it turns out the secret to teaching factoring is the distributive property. I mean, who knew? I’d thought I’d been emphasizing that all along, but turns out the kids didn’t. They focused on the tricks, memorized a bunch of crap, and then forgot it. But riding with D. Propz was far superior.
Teaching strat: Greatest common factor, same as before, including a notebook file that goes step by step through a problem like 8x3 – 6x2, including a key slide where the terms are displayed as 2x2*4x – 2x2*3. I never, never, never write work out on slides because then I miss the opportunity to completely botch my algebra in front of the class and let kids see that it happens to the best of us, but THIS one I put on slides, step by freaking step because I do it on smartboard and on the second-to-last slide make sure the 2x2 tiles are grouped so that I can drag my finger and factor them out (more on this later). Lame, but whatever. Tablet-people, please just project Algebra Touch for that part.
After some practice on that, I went straight into grouping four-term and five-term expressions with a common common factor. Stuff like 6a3 + 15a2 + 4a + 10. Before, these intimidate kids solely because they had more terms and we did them last. And always, the ones who were absent that day had tons of trouble catching up. But now, they are easy — just pull out the GCFs of each pair — 3a2(2a + 5) + 2(2a + 5). Next comes the hard part and before, I thought it was my job to teach a bunch of tricks so that kids wouldn’t have to deal with the hard part, but guess what? THEY NEED TO UNDERSTAND THE HARD PART.
So we spent a ton of time on this, Socratic questioning-stylof1:
Etc. etc. until you get to the key insight that 2a + 5 is, at heart, just a number.
Now, depending on your students’ tolerance for this type of thing (which I call “sustained thought”), you can go in a few directions. I like substitution (or “chunking” in EDC-lingo — buy their books, srsly).
First, round on the kid that said “A NUMBER” and is trying to grandstand about winning this latest round of “guess what teacher wants”
Me: “So if 2a + 5 is a just a number, what number is it?”
Prize student, suddenly floundering: “…. uh….”
Me: “Well, this is ALGEBRA, Jimmyf2, what do we call numbers we don’t know?
Every other student, in unison: “CALL IT X!f3”
And then it’s time to do some damn teaching. If I write x = 2a + 5, then let’s get fancy with the whiteboard markers here and….3a2(2a + 5) + 2(2a + 5) becomes 3a2 x + 2x, which factors into x(3a2 + 2)
(NOTE: do NOT gloss over this part. It’s boring to write about, but make sure every kid is on board. x and 2a + 5 should both be written in the same special color while the rest of the expression should be flat black, the kids should help with the factoring, let them talk it through… do NOT screw this up on the home stretch by whiz-banging your way through)
Then bring it home….
Me: But what’s “x”?
S, excited, in unison: A NUMBER!!
Me, winding them up: WHAT number?
S, super-confident, in unison: WE DON’T KNOW!!!
Me, once all the tears have subsided: We don’t know everything about x, but we do know something
S, timorous: it’s 2a + 5
Me: then we get (2a + 5)(3a2 + 2) OH MY GOD YOU JUST FACTORED IT
Hard for kids to handle that cognitive load. Hard for them to know when to use that “chunking” move. I should get up on my pedestal and talk about crafting a unique cooperative learning experience that guides them to develop it as their own solution, but I would be BS ing cause I do that by Direct Instruction ALL. DAMN. DAY.
Now pull up the smartboard file, where I have cleverly written everything out slide-by-slide, just as before, again with the 2a + 5 common factors in the same color and grouped just like before. And I factor it out via gesture. Just. Like. Before.
Why can I do this? Because it’s JUST A NUMBER.
Minds get blown, high fives all around, and we spend the rest of class on whiteboards factoring big scary things like champs. Something with a negative factor gets put up as a challenge, kids talk it through, I push them by putting up pairs like (c + 2) and (2 + c) vs. (c – 2) and (2 – c) and we hash that all out. They go home, crank out some practice problems, and we come back again, stronger, and ready for the real challenge. HOLLA
f1:We always do Socratic questioning early in the year, and I always introduce it by getting kids to talk about Socrates b/c a few always know. Then I emphasize the point that Socrates asked the Athenian muckity-mucks questions that were so hard to answer that they made him drink poison. Full stop. You WISH you could be that annoying, kids.
So when I have to push them on stuff later in the year, I put on my Socrates hat (actually a toga) and remind them: “If you get annoyed, Socrates wins.” Lampshading it like that really increases their tolerance for being jerked around (or at least decreases my guilt, which is almost as good)
f2Feel free to substitute your student’s actual name here
f3“Oh wow, I see why he used a instead of x as his variable! To reduce the cognitive load on students who are inevitably trained to call everything x! Such brilliance!”
I know, I know, ultra-pro teaching going on here