## Factoring trinomials by first rewriting them

So back in December, I gave this problem to my students:

To my surprise, it got a lot of traction with the kiddos. We spent the entire period talking about it. The idea was for them to see how rewriting a trinomial with four terms helps us to factor it. I’ve used this approach, often called the “CAB method” and used with a large “X” to organize the product and sum of the A and C terms, to factor trinomials for the last several years and I really like it for two reasons:

• It doesn’t matter if a is greater than 1.
• It naturally integrates factoring by grouping. Traditionally, grouping is learned after factoring trinomials. But with this approach, I teach grouping before we even see trinomials. Yeah:

So, yeah, this is all great, but as I was explaining this approach to a colleague, she asked me why it works. It was in that moment that I realized that I had no idea.

Well, it turns out that later that day she went ahead and wrote up a proof of the method.

I read a quote somewhere or heard someone say that the real usefulness of algebra is the ability it affords us to rewrite things in order to help reveal their underlying structure. This method surely epitomizes that idea.

bp

## Why isn’t there more public-private collaboration?

Definitively, I can’t say much about the success of my problem-based learning this year. Mainly because I’m still trying to figure out what the heck I’m doing.

Because of my ineptitude with the PBL structure I’ve adopted, naturally, I am seeking out others that are doing a better job than me with it.

Enter: private schools.

Or are they called independent schools? Or prep schools? Boarding schools anyone? I’m  confused on what to call them — I think it depends on the individual school, though. Either way, in my brief experiences with problem-based learning, it seems like these schools tend to use problem-based learning (or at least how I understand it) more than their public school counterparts. It helps that they usually have much smaller classes.

For this reason, I’ve been exposed to private schools in ways that I have never been before. This exposure has inspired me to get out of my classroom.

In October, I made the hike to New Hampshire and visited Phillips Exeter Academy. Then last week I squeaked out a visit to Horace Mann. The takeaways from each were awesome and unique. But as a public school teacher planning and making these visits all on my own, I couldn’t help but think about the generally nonexistent collaboration that exists between public and private school teachers. Maybe I live in a bubble, but in all my years of teaching (besides my recent experiences), I’ve never actually collaborated locally with a private school math teacher.

I do have a tendency to isolate myself, so I’ll own it. At the same time, there have never been deliberate, organized attempts by my bosses (or my bosses’ bosses) to help me learn from the private sector (or have them learn from me). And that’s a shame because there’s so much to learn from each other. Besides, I don’t care what anyone says, the public and private worlds aren’t really all that different. Math is math. Kids are kids.

Through all my complaining, I don’t want to discount online communities like MTBoS that help teachers from every school type learn from one another. Through it, I’ve connected with folks spearheading great PBL work like Carmel Schettino, Joseph Mellor, and Johnothon Sauer — all of whom I’m indebted to for helping me create and iterate my PBL classroom. That said, all of our interactions have been online.

There’s an annual Twitter Math Camp (free) in the summer that I’ve attended twice. There’s also NCTM conferences (not free) that I’ve never attended. NCTM may be regional, but neither one of these is truly local. The MfA Summer Think that helped plan last year is only for MfA public school teachers.

The closest thing locally I can think of that allows public-private collaboration is Twitter Math Camp NYC, which is local, free, and open to all. I have been once. But even TMCNYC isn’t deliberate about connecting public-private teachers. If it happens at the conference, it’ll be coincidental.

I don’t think things will change much. Most teachers, like me, live in our worlds — public or private. With the day-to-day grind that teaching requires, it is really hard to physically get out of our own classrooms.

But I’m not going to give up. I can dream. It’s fun visiting private schools and classrooms, and enlightening! And to speak to the individual, there are folks like Sam Shah and Michael Pershan who both teach at private schools (I think) and who are two of most thoughtful teachers I know. Best yet, they both teach here in NYC. I wish I could visit their classrooms. Maybe I should ask?

bp

## Interleaving and problem writing

It’s official: my shift to problem-based learning has consumed me. It’s all I seem to be thinking and writing about for the last three months. Radical change can do that, I guess.

Like I’ve said before, it has been really fun to think about the algebra 2 curriculum in this new way. Instead of running through units like what is done traditionally, I’ve been nonlinear about my planning. One glaring example of this is how I’m teaching trigonometry and trigonometric functions. In all my previous years, I’ve had one or more isolated units in the spring. Now, instead of doing one big chunk of trig near the end of the year, my kids are learning it in bite-sized pieces over the course of the entire year. I’ve realized that there’s no reason why they can’t explore the unit circle while also learning how to factor or analyze exponential functions.

And this sort of interleaving happens through problems, which is the other reason why I’ve been so excited this year. Specifically, it has been sequencing and writing the problems that has me on the edge of my seat. This is been a challenge that I’m now addicted to.

Before this year, if someone would have asked me to write the problems that I give my students, I would have cried in fear. Why would I want to do that? My life consisted of going to jmap.org or the MTBoS search engine and that was it. The point is that I would find problems, not write them. Besides, even if I wanted to write them, it would have taken waaay too long.

That has changed.

First, I have found the time. Second, PBL requires that I write problems if I want to adequately meet the needs of my kids. That said, I’m not writing all of the problems the kids are doing, but definitely a majority of them. Thanks to Exeter and folks like Carmel Schettino, there are so many problem gems already out there to use and adapt.

As such, I’ve been thinking about problems in new ways. Specifically, I’ve never deliberately thought about the different types of problems that exist and when to throw any given one to the students based on the math that I want them to explore.

Recently, I’ve been especially interested in the student-work based problems, like algebra by example. In essence, they force the kids to analyze math work and then transfer that analysis to a new problem. The work can have mistakes. The problems usually spur some good discussion, too. (Mine haven’t, but I’ve heard they do.)

I don’t claim that they’re great problems and should be used by others, but I’ll close with a few that I’ve written and used with my students.

That’s all for now.

bp