# Formulaic Teaching

All trig, all day. Trig in Algebra 2, trig in Geometry. I’m a pretty happy camper when I’m talking about the unit circle and all its marvels, the interwoven patterns that emerge with every bit of new information. I’m not sure all of my students feel the same (actually I’m fairly certain many of them don’t,sadly), but Saraphina gave herself a private Double Thumbs Up when the derivation of the sine of the sum of two angles became clear to her. Gotta love that kid.

In my slower track Geometry classes this week, we’ve been working on the Law of Sines. We derived the formula together, and have been using it first to find side lengths, and then, using the inverse, to find angle measures. I began with the entire class working together, but the addition of the use of the inverse function definitely split the class into several levels of comprehension. So today I developed a tiered practice set (link below) and got them working – relatively independently. (Truth be told, I could use a clone or two in those classes.) This will take another full day of practice, and we haven’t even touched problem solving. I’m beginning to wonder whether introducing the Law of Cosines after this topic will have as much value as I had hoped. We are moving into Volume and Surface Area next, for which I have more hands-on and accessible problem solving activities.

In my Algebra 2/Trig ‘gifted’ track classes (intentional air quotes added), we began the exciting unit of trig formulas. I absolutely hate giving students formulas without context, and without deriving them. But here’s the thing – or things: (a) in the unstoppable march towards the Regents exam, there is no time to do this and still complete the curriculum; it doubles the lesson time, and (b) many of my ‘gifted’ students are not quite ready for the algebraic fortitude that deriving the formulas requires. I know they will revisit this next year in Pre-Calculus, and will definitely cover this ground in the future. And I have learned – the hard way – how easy it is to lose half of the class doing this.

And yet –

I won’t lower my expectations. I respect all of my students too much to say they can’t do it. So I have been reflecting on this week of practice with formulas and the value therein; it’s not terribly inquiry-based, or discovery-oriented. But after listening to a lot of conversations, and answering their questions with more questions, I’ve developed a more nuanced view of what might appear to be mechanical work.

In my Geometry classes, for instance, my students really, really want to solve these problems correctly, and consistently. This practice is pushing them to extend their understanding of triangles beyond the all hallowed Right Triangle to EveryTriangle. Many of them still struggle with orienting the angles and their ‘opposite’ sides, regardless of the tips and strategies I’ve suggested. They are calling upon the basic triangle content from the fall term to aid their understanding, and the connections, for many, are visible (and rewarding to see). There are students who need more intervention than I realized – this activity surfaced some problems very quickly and clearly, like not being able to distinguish between an angle measure and a side length. [How a student has gotten to junior year in high school without this being addressed is for another post, or rather, another rant.] When the bell rang today, the kids asked for another day to complete the practice. Growth mindset in action. I love their perseverance, and I love to watch them help each other.

In my Algebra 2/Trig classes, we began with the Cosine Sum/Difference formulas yesterday, and moved on to the Sine formulas today. I gave them a full day of just using the Cosine formulas, introducing the idea of using them to prove relationships (such as cos (-*x*) = cos* *(*x*) at the end of the class. I began the sine formula lesson in my first class of the day with deriving the difference formula; the students worked with me, but we had little time for practice. In my other classes, I let them practice first, leaving an open question on the board. After a while, students – who were comfortable enough with the work to think about this – began to ask for assistance in working this through. I found myself talking to pockets of students, only to look up and see others joining the group. (Link to practice worksheet below.)

And these days of practice have had value beyond their primary objective of mastering the use of the formulas. The connections in trigonometry are prismatic, and the more one explores and works with it, the deeper they become. Working on these problems reinforce the concepts of reference and quadrantal angles, Pythagorean triples, and the big idea of proof. The students support one another well (thanks to Visibly Random Groupings almost every day!), and are less hesitant to ask for help from me.*

Side note: In my Geometry classes, as I mentioned, I need a clone. EVERYONE wants help, and NO ONE wants to ask a fellow student first. The students become angry when I won’t tell them whether an answer is correct, but rather suggest a strategy for checking it. In diametrical opposition are my Algebra 2 students, who are LOATHE to appear stupid, and hesitant (complete understatement for most of them) to ask questions in front of the class.

When I put my plans together for this week, I was concerned about these double duty practice classes. I never imagined, actually, that I would be blogging about it. And yet watching my students work on what I was afraid was too mechanical has actually been a lesson for me, one which – naturally – poses more questions. How do I insure that my students get the procedural fluency they need all the time, while keeping the classroom full of inquiry and math talk? And the eternal question, how do I get them to do some of this learning independently?

I guess the questions are what keeps me going.