Friday, two weeks from the end of school, and it’s rally schedule: chop off fifteen minutes from each block for a screaming session in the gym. It’s fourth block, my trig class, and although I try not to have favorites, this semester has been a bit low on students with energy and ability. But even the goof-offs in this class can remember the basics of trig, have put some effort into memorizing the unit circle, reciprocal values, the occasional Pythagorean identity, know the difference between sine and cosine graphs. And only two cheaters. The top kids are amazing, enthusiastic, and driven–and there are lots of them, many of whom I just taught Algebra 2. So a fun class, and really the only one with a genuine personality this semester.

I had given them some extra time to finish up a test from the day before, and it’s now just 35 minutes to rally.

“OK, I want to cover a couple things to set up Monday. Let’s….”

“NOOOOOOO!!!!” the blast of complaints hit me. I turned around and glared.

“Come on! It’s Friday! You can’t make us learn something new!” Tre, who last had a math teacher that wasn’t me in freshman algebra, put on his most ingratiating grin.

“It’s so hot, and my brain hurts. Please, no more math!” Patti slumped dramatically.

“QUIET!” I turned back from drawing a cosine graph to bellow them into submission.

“I just want to introduce a couple of interesting properties and get you thinking, once again, about…oh, for christ’s sake.”

“WHAT??? What happened?” the students crane their heads forward to see the object of my irritation. I was growling at a student whiteboard sitting on a desk.

“Oh, some student used a fricking sharpie to draw a self-portrait.” and I held up the board so the class could see the penis.

“HAHAHAHAHA!” TJ was cracking up and I whirled at him furiously.

“You know, we use these white boards every day, and if I can’t get the sharpie off, it’s ruined. You think it’s FUNNY that students destroy my stuff?”

TJ was genuinely puzzled. “No. You just called him a dick. Like, without saying so. That was cool.”

“Fine. Ruin the fun of yelling at you. Take one more ounce of joy from my day.” I grinned at him and sprayed cleaner on the board.

“Ain’t no cleaner taking off sharpie,” Ahmed sympathized.

“Dude, this is Kaboom,” Tre said. “Kaboom’s the bomb.”

“Best cleaner in the known universe.” I spray the board and let it sit. All my kids know I love Kaboom. I tell new teachers about Kaboom, an essential teaching tool. When the kids write F*** in Sharpie, it’s so incredibly satisfying to wipe the obnoxiousness out of existence with one spray. Lesser challenges–gang graffiti, pencil sketches, soda spills, even small patches of gum–all disappeared.

“I hate students, dammit.” I turned back to the board. “I mean, don’t get me wrong. I love you all. But I just hate students. Ruin my stuff, treat it like crap….” I stop, because students breaking my stuff can put me in a foul mood in a hurry.

“It wasn’t us!” Matteo protested.

“Dude, it was you.”

“Screw you, Furio, how do you know?”

“Cuz you’re a dick! That’s your picture!”

I laughed, feeling much better. “Look, back to work. So you know how there’s a line, and then we can square a line, or multiply it by another line, to get a…”

“Parabola,” a reasonable amount of the class chorused, but I could hear talking.

“Shush, whoever’s talking. What happens when we square the cosine function? Take a look at the function and let’s just square what we….BE QUIET BRIAN..see. Cosine starts at…QUIET.” I turn around, wait for quiet. “Cosine starts at what, Furio?”

“1.”

“So 1 squared is..?”

“1”.

I mark (0,1) in a different color, and move to the next hashmark. “Cosine is zero at pi over 2, zero squared is…QUIET.”

Most kids were paying attention, but there was this low level nattering that rose up every time I turned to the board. But we got through the first one quickly.

“So here’s the square of the cosine function. What do you notice?”

“It’s a cosine graph!” Vicky.

“Sure looks like it. Period? Amplitude?” and we identified all the parameters for a cosine function graph.

So the square of the cosine function can also be expressed as a regular cosine graph. Amplitude and vertical shift, one half, period one half the usual.”

Ahmed said with faux judiciousness, stroking his chin, “Ah, but how do we know this? It might just look like a cosine graph!”

“Good question. We can see the key points work, but maybe that’s just a coincidence. So pick a value and let’s plug it in. QUIET!”

“How about pi over six?”

Carla was impressed. “Wow, when you double the value, it becomes something entirely different.”

“Yes….QUIET!!! I’m always surprised at how the alignments happen. So now let’s go on to the sine function. What do you all think will happ….QUIET!”

“Jesus Christ, Eduardo and Brian, will the two of you shut.up.? NO! Stop the innocent ‘who me?’ crap. Three times in the past three minutes. I tell you to be quiet, turn to the promethean, turn around and there you are yapping again. Do I need to move you?”

Eduardo (Manuel‘s younger brother) and Benny look abashed, hearing the edge in my voice. I was mad at myself more than anything these two had done. Note to new teachers: don’t push through without attention. Constantly shushing is a sign you don’t own the room.. Don’t push through, stop when you need to. And it wasn’t an accident I’d picked two of the top kids in the class to shut down; it showed everyone else I was serious, if the unusual edge in my voice wasn’t enough.

By now I was furious with myself, and boy, do I get global in a hurry. My rotten students ruin my whiteboards and never shut up. I’m an idiot who decided to teach something complex 30 minutes before the weekend. And there are times when I’ve decided it’s not worth it and call it quits–call a pop quiz, put a problem on the board as an exit ticket, something. But deep breath, act like nothing happened, and push on, vowing to give it one more shot before I bail on an exit ticket activity.

“Wait.” Joanie, probably my top math student this year, sat up and scowled at the graph dots. “How can that be a cosine, too? That’s weird.”

“What kind of cosine function? What’s different?”

“It’s reflected. So cosine squared is cosine, and sine squared is negative cosine?”

“Looks like it.”

“But what’s the point of this?” Vicky asked. “Since squaring a sine or cosine function just takes you back to cosine, why do it?”

“Well, math applications will quite often require you to square functions, so it’s good to know how they behave. However, I really just want you to think about exploring functions. Up to now, you’ve been working primarily with transformations or known formats with parameters you can just plug in. But now we’re investigating functions that aren’t familiar with. Notice, too, that we did this all graphically with a minimum of evaluation.”

“So just for fun, what if we add the two functions we just created?”

“Here they are together. So let’s add the five primary points.”

TJ puzzled. “They’re all one? Really? That’s weird.”

“Yeah, but you can see it in the graphs,” Juan observed. “They’re equal at one-half, at opposite ends at one.”

I join all the points.

“So the graph y= cosine squared plus sine squared is always….”

“One!” the class chorused.

And then I threw out casually, oh so casually, “And cosine squared plus sine squared is…”

“One!…”

The pause was the best part. I looked down, and waited as the recognition grew, until by god, the entire room was shouting in approval, clapping and stomping.

It’s one of those things that maybe you had to be there. But in half an hour, at the end of a day, in hot weather, right before a rally and a weekend, I’d not only gotten those kids to apply their knowledge of trig graphs in a new approach, but draw a connection from graphic to algebraic. They hadn’t recognized the familiar equation because their minds were in “graph” mode, and only when I asked about a Pythagorean identity, using almost exactly the same words, did they realize that they already knew what the graph would show. But not until then.

And they thought it was really cool that I’d pulled them around to this recognition.

Literally, a minute of stomping until I waved it down. “All right! Thank you. Remember during the first week, when I told you I’m a stickler for understanding the connection between algebraic and visual representations? Here you go.”

And then, “But what about tangent? What happens when you square that?”

Ten minutes left and I’ve got them asking questions. I realized I haven’t had to shush them once.

And just as the bell rings, we established that tan^{2}(x) + 1 = sec^{2}(x).

The kids rushed out to the rally. Rallies are my one Bad Teacher thing: I don’t go. I checked the whiteboard, Kaboom had wiped out most of the damage. Then I walked to Starbucks just completely charged, reliving the math and the applause. All the yelling, all the grouchiness, wiped away. I’d killed.

I keep telling you: Teaching is a performance art.

June 3rd, 2018 at 7:41 pm

Very uplifting. You are a dedicated teacher.

June 3rd, 2018 at 11:27 pm

[…] Source: Education Realist […]

June 7th, 2018 at 8:02 pm

That takes me back! I spent an entire afternoon squaring and/or combining all the trig functions in different ways. I don’t think anything reinforces the various important identities in a better manner.

June 17th, 2018 at 12:47 pm

This is the best story I’ve read about why teaching can be so rewarding (I teach math and physics).

Loving this ‘Great Moments in Teaching’ series.

June 20th, 2018 at 3:55 am

Thanks!

October 26th, 2019 at 8:39 am

[…] Seeing the link between algebraic equations and graphs just before the weekend. Figuring out the Third Dimension The electoral college and Trump Reading aloud to my ELL class. […]

October 28th, 2019 at 2:52 pm

I can’t figure out what the black dots are on the two diagrams showing sin squared? I see the red dots are the sin squared function. But what are the black dots?