# Tag Archives: trigonometry

## Great Moments in Teaching: The Charge

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!”

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 tan2(x) + 1 = sec2(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.

## Teaching with Indirection

Technology is a great illustrator and indispensable for presentation. But as a student tool? Eh, not so much. Certainly not laptops.   I found laptops very useful in my history class, but primarily as a delivery and retrieval mechanism, or for their own presentations.  I haven’t found that a compelling reason to submit to the logistics of handing out and collecting laptops. But then, I’m a Luddite on this.  Recently, some colleagues were jazzed with several thousand dollars of cool science tools which I oohed and ahhed over politely. But….? Basically data collection. Fast data collection, which the students can analyze.  I guess. I don’t really do science.

A couple months ago, I used laptops and Desmos to teach transformations, and after two blocks that went….well, I suppose, I used whiteboards to do the same lesson in the last block. Far superior. I wouldn’t have even considered the hassle, but last year the school decided all algebra 2 teachers warranted a laptop cart and I want to occasionally acknowledge a gift intended to be useful. I would never–I mean no excuses never–book a laptop cart from the library to teach a lesson. But if it’s sitting around my classroom, I’m bound to try and find a way to use it. Still, even if I had a lesson that would be guaranteed superior to the same lesson on paper, I’d be tough to convince. Taking them out and putting them away takes up close to 15 minutes of classtime. Wasted. If all of my students had their laptops with them at every minute, waiting to be used….maybe. I’ve certainly found uses for phones on an occasional basis. But it’s not a huge gap I’m longing to fill.

Teaching is performance art. Sometimes the art lies in holding students’ attention directly, taking them point by point through a new topic. Other times, it lies in making them do the work. In both cases, the art lies in the method of revealing, of making them come along for the ride of understanding–even if it’s just in that moment.

It’s hard to do that if you put technology in the students’ hands. First, they’re too easily distracted. Second, it’s too easy to do without understanding.  A colleague of mine simply worships Dan Meyer, and loves all the Desmos activities.  They are neat. Without question or caveat. But I have limited time, and I’d rather have my students doing math directly, by hand even, than have them work on laptops or phones. Some Desmos activities do, absolutely, require the kids to work or show their math directly. Others are an interesting form of guess and check, designed (hopefully) to help kids understand patterns. The first, I like, but am unconvinced that the time and distraction suck are an improvement over handwritten work. The second, no. Not generally interested unless I have time for games, and I don’t.

This piece is only partially about technology, though. I wanted to talk about designing experiences, and for me, technology doesn’t give me the freedom to do that. Not with my kids, ability levels, and existing technology, anyway.

But how can I claim that technology is a distraction if I’m busy performing for the students?

Well, recall I said it was great for illustration and presentation. I love my smartboard, although I move pretty effortlessly between smartboards and whiteboard walls.

I have learned it’s very simple to screw up a lesson by speeding it up, but far more difficult to do slowing it down. I like introducing a topic, sometimes in a roundabout way, and having the students do the work alongside. Consider the example displayed here. These aren’t power points of my lecture. I start with a blank screen. I give the instructions, give the kids time to follow along, then use their input to make my own diagram. That way I can circle around, see that everyone’s on track, understanding the math, seeing connections.

I spend a great deal of time looking for ways to build instruction step by step, so that the vast majority of my students have no reason to refuse the effort. Draw a square. How hard is that? Besides, most of them enjoy drawing and sketching, and this beats posters.

Ideally, I don’t want them to see where we’re going. But then, remember I’m teaching advanced high school math. At various times, I want students to understand that math discoveries don’t always go where they were expected. The best way to do that, in my experience, is give them a situation and point out obvious things that connect in not so obvious ways.

Thus, a trigonometry class is a great place to start an activity that begins as a weird way of breaking up a square into similar triangles. The sketches in the first steps are just a way to get them started, suspend their disbelief.  The real application of knowledge begins at this step, as they identify the equivalent ratios for the different triangles. A geometry-level skill, one from two years ago, and one we try to beat into their heads. Proportionality, setting up cross products, is also something students have been taught consistently.  A trig class is going to have a pretty high percentage of functional students who remember a lot of what they’ve been taught a lot.

Which is important, because this sort of activity has to be paced properly. You have to have a number of pauses while students work independently. The pauses can’t be too short–you have to have time to wander around and explain–but not explain everything to everyone, which would take too long and kill the mood. Can’t be too fast, either, or why bother?

Ideally, students should be mildly mystified, but willing to play along. As I wrote several years ago, start slow, build student trust in your wild notions. If you keep them successful and interested, they’ll follow along working “blind”, applying their existing knowledge without complaint. Don’t deliver and they won’t follow. Which is why it’s important to start slow.

So in this particular activity, the students drew a square, some triangles, and found ratios without knowing when, or if, this was going to relate to trigonometry. Now, finally, they are using class-related knowledge, although SOHCAHTOA is technically covered in geometry and only reviewed in the early months of the year. But at least it does have something to do with Trig.

I’ve only done this once, but I was surprised and fascinated to note that some students were annoyed that I reminded them about the 1 unit substitution after they’d built the proportion statements.  I liked the structured approach of two distinct moves. They didn’t. “Why you make us do this twice?” griped Jamal, who is better at math than you might expect from his pants, defying gravity far south of his pelvis, much less his perpetually red-eyed stupor and speech patterns. (“He’s a c**n,” he informed me about a friend a month ago. I stared at him. “It’s okay. I’m half c**n, so I can say  that.  Like, my family, we all light-skinned but we c**ns.” I stared at him. “OK, I ain’t no c**n in your class.” I mentioned the discussion to an admin later, suggesting perhaps Jamal needed to be told that c**n isn’t n****r , and is an insult in any vernacular. “C**n?” she said, puzzled.  “Like….raccoon?” It took me a few minutes to realize that she was a Hispanic, so it was indeed possible she had no idea what the word meant. I should have gone to our African American admin.)

It’s not obvious to all students that the ratio labeling each triangle side is the length of that side. That is, if the base is one, then the length of the secant line will be the exact value of the secant ratio, and so on. Breaking the diagram into three distinct triangles helps, but I do recommend spending some time on this point.

So, for example, say if the angle is 30 degrees, what length would the side labeled sine be? What about cotangent? They already know about sine and cosine lengths, since I introduce this after we’ve covered the basics of the unit circle. But it helps to prod them into realizing that the cosecant length would be 2 units, and so on.

My students are familiar with my term “mother ship”. I use it in a number of contexts, but none so commonly as the Pythagorean Theorem. I ask them if they’ve seen Independence Day,  or one of the other zillions of alien invasion flicks in which the little independent saucers  all go back to the big behemoth. Because aliens will centralize, else how could humans emerge victorious? Just as all these little buzzing pods lead back to the big one, so too do so many ideas lead back to Pythagorean. Even its gaps. The Pythagorean Theorem doesn’t do angles, I point out. That’s why we started using trigonometry to solve for sides of right triangles. Originally, trigonometry was developed thousands of years ago to explain planetary motion, and was defined entirely in terms of spheres and chords. Not until Copernicus, a few hundred years ago, did we start to define trigonometry primarily in terms of right triangles.

Until this activity, I’d always taught the Pythagorean identities algebraically. I start, as many do, by reminding or introducing them to the equation for a circle, then talk about a radius of one, and so on. Then I derive the secant/tangent and cosecant/cotangent versions, which is pretty simple.

But I really like the geometric representation. The three triangles are spatial, physical artifacts of what is otherwise a very abstract concept. Ultimately, of course, these identities are used for very abstract purposes, but whenever possible, links to the concrete are welcome.

Besides, isn’t it cool that the three triangles reflect what the algebra shows? I suppose the fact that the triangles are all similar plays into it, but I’m not enough of a mathie to grasp that intuitively. The students, of course, don’t yet know the algebra. The Pythagorean identities are the one new fact set this lesson delivers.

Remember, I don’t use these images you see here in the lesson; rather, they represent a combination of what I say and draw during the lesson, pausing as the students work things out themselves.  Could I do this with technology? Sure. Could they? In my opinion, no. But it’s debatable, certainly. BUT–I also couldn’t do this with a book.
Is it just me, or do students take an absurdly long time, over many lessons and with many reminders,  to memorize the unit circle? I mean, my god, there are five values for each ratio. They go in order–big to small, small to big. How hard could it be? But after a couple years of students looking at me blankly at the end of the term when asked what the sine of pi over 6 is, I’ve learned to beat it into their heads. Some teachers never use the unit circle to teach ratios. I do not understand this. Steve teaches it all with co-functions and trig tables; I have taught any number of his students who know vaguely what it is, but have no conceptual understanding of it. They know the values, their operational ability is no different, but where’s the fun? The unit circle is an amazing entity.

I am a big fan of Desmos. At algebra 2 and higher, I ask my students to download the Desmos app. My students learn how to graph, how to create functions, how to explore functions. I want them to know Demos as a tool when it makes sense. Really.

So eventually–although I haven’t done it yet–I’m going to show my students this puny effort to automate the concepts we explored manually in this lesson.  Hey, I can use the laptops! It will be a great example of inverse calls.

But not right away. Look, my classes do a lot of repetition.  Plenty of worked problems. It’s not all discovery or exploration–in fact, relatively little time is spent on these. My students need to know how, building capacity. Why is the glue. The better a student is at the basics of math, the more important it is to smack them around with why, occasionally.

But I’m a performer.  English teachers talk about grabbing up front with the hook. But in math, ending big, revealing the path they’ve been wandering, is my goal. So when I draw in the circle, put in the coordinates, and hear “Holy sh**!” and various stunned gasps, following by a smattering of applause, I know my planning paid off.

“The f***? Damn. This been the unit circle all along. Shee-it.” That would be Jamal.

## Great Moments in Teaching: Or, Browbeating Psychoanalysis

One of my strengths as a test prep instructor was spotting weird mental glitches that was interfering with a student’s success. I miss this part of the job, but every so often I get the chance in classroom teaching.  In this case, summer school trigonometry. I taught first semester in block 1, second semester in block 2, but I taught the same material in both classes.

I had about eighteen kids in the two classes, but eight of them took both classes, meaning they’d failed both semesters. All eight students repeating both semesters were stronger than the three weakest students repeating the second semester, and the weakest student just repeating first semester. Remember what I said about GPAs? Shining example, right here.

So this is a conversation I had with Warren on the next-to-the last day of class. Before you decide I’m a rotten bully, understand that I had raised this issue several times with Warren, but the message had, like everything else, rolled right off his back like whatever water does with a duck.

He was taking the final test, and had asked me to check it over before he turned it in. (This is a normal part of my class routine).

“OK, you’ve got quite a few cases where I’m asking for onions and you’re giving me a Jeep.”

“I know.”

“No, you don’t. Like question 5. You’re using the Pythagorean theorem on a question asking you to understand and evaluate a trigonometric model.”

“I know.”

“No, you don’t.”

“I get it.”

“No, you don’t.”

“Yes. I see now.”

“NO YOU DON’T!” The class was now snorfling quietly, not out of mockery of Warren, but amusement at me. I was playing my aggravation very big.

“OK.”

“YOU DON’T UNDERSTAND HOW TO DO THIS PROBLEM.”

“I know.”

” Then why are you done with the test? This one is not just mildly wrong. It’s Jupiter and we’re Earth.”

“I know.”

“STOP SAYING THAT.”

“I kno…OK.”

“What’s OK?”

“I understand what you’re saying.”

“No, you don’t.”

“OK.”

“No, it’s not OK!”

“OK..I mean, I know…Oh, sh**.” Warren is no longer a duck, but a deer, frozen.

“Listen to me.”

“OK.”

“No, see, already you’re not listening. Don’t try to make me happy. Don’t try to give me what I want. You’re trying to figure it how to make me happy and that one task is consuming all your brain cycles. JUST LISTEN.”

“I know…no. OK. I get it.”

Half the class was howling by this point, and I shushed them.

“This problem is incorrect. Not mildly incorrect. Way off. DON’T SAY A WORD. Continue to listen. Cover your mouth if you must. Say nothing until I ask you a direct question.”

Warren stood. Affect way off, smiling nervously.

“You came up here, telling me you were done, asked me to just look through for minor errors. But as I look through the test, I see that you have no idea how to do at least three of the eight problems. SAY NOTHING!” Warren closed his mouth. “In two cases, you came up here earlier and asked me for help. I gave  you guidance, you said ‘I know’, I told you no, you didn’t, tried again, got nowhere. And now you’re up here saying you’re done. We have an hour left of class. You are MANIFESTLY not done. When I point out an error, you say ‘I know’ but you clearly don’t mean it because you are up here saying that you are finished! No–I haven’t asked a question. Stay put.”

Warren stood. But I could see the panic fade a bit. He was starting to actually listen.

“This is a trig modeling question. It’s about temperature in a room. Max and min temp. 24 hours in a day. Yet you are using the Pythagorean theorem. Why, Warren, are you using the Pythagorean theorem? That is a question. You can answer.”

But Warren stood mute. I waited. The class snickered and I ferociously signaled them to stop.

“I….I don’t know what you want me to say.”

“EXACTLY! That’s it. Exactly. Perfect. Now, continue to listen. The reason you are confused, Warren, is because you are uninterested in math at this particular second, and entirely interested in making me happy because you think it will help your grade. But I want you to learn. And that involves asking questions. It involves thinking. It involves furrowing your brow and asking for clarification. Normally, you ask your friends for help, or copy what they’ve done and think you understand the math. Sometimes you do. Mostly, you don’t.You just know how to go through the motions.”

Warren’s eyes widened. Apparently, he thought we teachers weren’t onto his scheme.

“But Warren, talking to me–talking to any teacher–is a conversation. A process. It is your job to communicate your confusion. It’s my job to try and give you clarity and undertanding. Our conversations are not mere rituals mandated by the Chinese American education canon. So let me ask the question a different way: When you were modeling trigonometric equations all this week, what pieces of information were relevant?”

“If I gave you the maximum and minimum points, how would you find amplitude and period?”

“I would sketch them and look for the middle.”

“Which is also the…..”

“Vertical shift.”

“OK. Now. Look at this problem. Do you see how this problem fits into that format? It describes the temperature in a corporate office. So what I want you to do now is go back and think about this problem. Think about how you could describe temperature in terms of max and min. Think about relating it to the time of day, hours past midnight. And then see if you can figure out how to work the problem.”

Warren obediently took the test and started to return to his seat, but stopped. “OK, but here’s what I don’t get. You’re asking us to solve an equation. But modeling is just building the equation. How come you’re asking us to solve the equation?”

I looked at the class. “Whoa. Did you hear what I heard?”

“A QUESTION!!!” and we all clapped loudly and genuinely for Warren, who smiled nervously again.

“Warren, I mentioned this over the past couple days: Trig equations don’t just occur in a vacuum. We build the equations to model the world. Then we look to the model to predict outcomes, which we do by solving for outputs given inputs, or vice versa. The problem covers both. It asks you to evaluate and explain the given model,  then it asks you to use the model as a trigonometric equation. In this case, I actually used function notation because I want to see if you understand it, but at other points, I’m using verbal descriptions.”

“OK.”

“Really?”

“Um. No. I don’t know how to start.”

I waited. The class waited.

“Could..could you give me a suggestion on how to start?”

“Is there something you could do to the given equation that might give you some insight?”

Pause.

“I could…graph it, maybe?”

“There’s a thought. Then look at it, look at the multiple answers, and see how it goes.”

As Warren walked back to his desk, I mimed collapsing in fatigue. “And now, everyone, entertainment’s over. Get back to work.”

Warren worked on the test for another hour. He forgot and said “I know” and “OK” reflexively a few times, but stopped himself before I could, to both of our smiles. He came up each time with a specific question. He listened to my response.  He went back and worked on the problem based on my response and his new understanding.

On the last day of class, after the final bell rang, Warren came up to chat with me.

“Thanks for yelling at me.”

“You know, I was working towards a good cause.”

“You were right. I was coming up to ask you questions because that’s what other kids did, so I figured that’s what you wanted. I never really thought about getting help from you. I just kind of…work through something using whatever I remember, until I’m done.”

“Don’t be a zombie.”

“Okay–wait. What’s a zombie?”

“Don’t just work problems without any sense of what’s going on. That’s why you flunked Trig the first time, I’ll bet.”

“Yeah. I didn’t always understand Algebra 2, but I could follow the procedures. But Trig, I just couldn’t do that.”

“Yeah. Zombie thinking. Don’t do that. I mean–zombie thinking is what you’re doing in math. You get the answers from friends, you don’t care about understanding the math. You just go through the motions. The driving me crazy saying ‘I know’ stuff, that’s different. Plenty of zombies do a better job of asking for help!”

“I understand math a lot more the way you teach it, but I also….I couldn’t always figure out your tests.”

“That’s why you ask for help. And not from your friends. Look–school is about more than getting an A. It’s about more than giving teachers what they want so you’ll get an A. It’s about learning how to learn. You have to start communicating with teachers–good, bad, indifferent–and learn how to figure out what they’re telling you. That starts with asking for what you need. If you can’t communicate with a teacher right away, don’t just ask a friend. Half the time, they’re just doing what you do! Find teachers you can work with. You’re a really bright guy. Don’t let school ruin you.”

Ten minutes later, as he walked out, he said: “Thanks again. I mean it.”

He knew a lot of math, and worked his way out of being a zombie. I gave him an A-.

## Realizing Radians: Teaching as Stagecraft

Teaching Objective: Introduce radian as a unit of angle measure that corresponds to the number of radians in the length of the arc that the angle “subtends” (cuts off? intersects?).  Put another way: One radian is the measure of an angle that subtends an arc the length of the circle’s radius.  Put still another way, with pictures:

How do you  engage understanding and interest, given this rather dry fact?  There’s no one answer. But in this particular case, I use stagecraft and misdirection.

I start by walking around a small circle.

“How far did I walk?”

“360 degrees.”

“Yeah, that won’t work.” I walk around a group of desks. “How far did I walk?”

“360 degrees.”

“Really? I walked the same distance both times?”

“No!” from the class.

“So what’s the difference?”

It takes a minute or so for someone to mention radius.

“Hey, there you go. Why does the radius matter?”

That’s always an interesting pause as the kids take into account something they’ve known forever, but never genuinely thought about before–the distance around a circle is determined by the radius.

“Yeah. Of course, we knew that, right? What’s that word for the distance around a circle?”

“Circumference!”

“Yes. And how do you find the circumference of a circle?” There’s always a pause, here. “OK, let me tell you for the fiftieth time: know the difference between area and circumference formulas!”

“2Πr” someone offers tentatively.  I put it up:

“So the circumference is the difference between this small circle” and I walk it again “and this biiiiigg circle around these desks here.” Nods. “And the difference in circumference comes down to radius.”

Pause.

“Look at the equation. 2 Π is 2 Π. So the only difference is radius. The difference in these two circles I walked is that one has a bigger radius.”

“So the real question is, how does the radius play into the circumference?”

“Well,” it’s always one of the better math students, here: “The bigger the radius is, the farther away from the center, right?”

“So then…you have to walk more around…more to walk around,” some other student will finish, or I’ll ask someone to explain what that means.

“Right. But how does that actually work? Can we know exactly how much bigger a circle is if it has a bigger radius?”

“A circle with a radius of 2 has a circumference of  4Π. A circle with a radius of 4 has a radius of 8 Π. So it’s bigger.” again, I can prompt if needed, but my class is such that the stronger students will speak their thoughts aloud. I allow it here, because they can never see where I’m going. See below for what happens if they start with spoiler alerts.

“Sure. But what’s that mean?”

Pause.

I pass out pairs of circles, cut from simple construction paper, of varying sizes, although each pair has the same radius.

“You’re going to find out exactly how many radius lengths are in a circle’s circumference using the two circles. Don’t mix and match. Don’t write annoyingly obscene things on the circles.”

“How about obscene things that aren’t annoying?”

“If you can think of charmingly obscene comments, imagine yourself repeating them to the principal or your parents, and refrain from writing them, too. Now. You will use one of these circles as a ruler. All you have to do is create a radius ruler. Then you’ll use that ruler to tell me how many times the radius goes around the circumference.”

“Use one of the circles as a ruler?”

“You figure it out.”

And they do. Most of them figure it out independently; a few covertly imitate a nearby group that got it. Folding up one of the circles into fourths (or 8ths) exposes the radius.

Folding up one circle exposes the radius.

It takes most of them a bit more time to figure out how to use the radius as a ruler, and sometimes I noodge them. It’s so low-tech!

Curl the folded circle around the edge of the measured circle.

But within ten to fifteen minutes everyone has painstakingly used the “radius ruler” to mark off the number of radius lengths around the circumference, and then I go back up front.

“Okay. So how many times did the radius fit into the circumference?”

Various choruses of “Over six” come back, but invariably, someone says something like “Six with and a little bit left over.”

“Hey, I like that. Six and a little bit. Everyone agreed?” Yesses come back. “So did everyone get something that looks like this?”

“Huh. And did it matter what size the circle was? Jody, you had the big two, right? Samir, the tiny ones? Same difference? Six and a little bit?”

“So no matter the circle size, it appears, the radius goes into the circumference six times, with a little bit left over.”

No one has any clue where I’m going, usually, but they’re interested.

“‘Goes into’ is a familiar term, isn’t it? I mean, if I say I wonder how many times 2 goes into 6, what am I actually asking?”

Pause, as the import registers, then “Six divided by two.”

“Yeah, it’s a division question! So when I ask how many times the radius goes into the circumference, I’m actually asking…..” The pause is a fun thing. Most beginning teachers dream of using it, but then get fearful when no one answers. No. Be fearless. Wait longer. And, if you need it:

“Oh, come on. You all just said it. How many times does 2 go into 6 is 6 divided by 2. So how many times the radius goes into the circumference is…”

and this time you’ll get it: “Circumference divided by the radius.”

“Yeah–and that’s interesting, isn’t it? It applies to the original formula, too.”

“Cancel  out the radius.” the class is still mystified, usually, but they see the math.

“Right. The radius is a factor in both the numerator and denominator, so they can be eliminated. This leaves an equation that looks like this.”

“The circumference divided by the radius is 2Π. Well. That’s good to know. Does everyone follow the math? Everyone get what we did? You all manually measured the circumference in terms of radius length–which is the same as division–and learned that the radius goes into the circumference a little bit over six times. Meanwhile, we’re looking at the algebra, where it appears that the circumference divided by the radius is 2Π.”

(Note: I have never had the experience where a bright kid figures it out at this point. If I did, I would kill him daid, visually speaking, with a look of daggers. YOU DO NOT SPOIL MY APPLAUSE LINE. It’s important. Then go to him or her later and say, “thanks for keeping it secret.” Or give kudos after the fact, “Aman figured it out early, just two seconds before figuring out I’d kill him if he spoke up.” Bright kids learn early, in my class, to speak to me personally about their great observations and not interrupt my stagecraft.)

And then, almost as an aside: “What is Π, again?” I always ask it that way, never “what’s the value of Π” because the stronger kids, again, will answer reflexively with the correct value and they aren’t the main audience yet. So the stronger kids will start talking yap about circles, and I will always call then on a weaker kid, up front.

“So, Alberto, you know those insane posters going around all the math teachers’ walls? With all the numbers?”

“Oh, yeah. That’s Π, right? 3.14.”

“Right. So Π is 3.14 blah blah blah. And we multiply it by two.”

That’s when I start to get the gasps and “Oh, MAN!” “You’re kidding!”

“….so 3.14 blah blah times 2 is 6.28 or…..”

“SIX AND A LITTLE BIT!” the class always shouts with joy and comprehension. And on good days, I get applause, too, from the stronger kids who realized I misdirected them long enough to get a deeper appreciation of the math, not just “the answer”.

******************************************************

So a traditionalist would just explain it, maybe with power point. I don’t want to fault that, but I have a bunch of students who would simply not pay any attention. They’ll take the F. I either have to figure out a way to feed them the math in a way they’ll remember, or fail more kids than I’m comfortable failing.

A discovery-oriented teacher would probably turn it into a crafts project, complete with pipe cleaners and magic markers. I don’t want to fault that, but you always get the obsessive artists who focus on making a beautiful picture and don’t care about the math. Besides, it takes forever. This little activity has to be 15-20 minutes, tops. Remember, there’s still a lot to explain. Radians are the unit measure that allow us to talk about circles in terms akin to similarity in polygons–and that’s just the start, of course. We have to talk about conversion, about the power that radians gives us in terms of thinking of percentage of the entire circle–and then actual practice. I don’t have time for a damn pipe-cleaning activity.

As I’ve written before somewhere between open-ended, squishy discovery and straight discussion lecture lies a lot of ground for productive, memorable teaching. In my  opinion, good teachers don’t just transmit information, but create learning events, moments that all students remember and can use as hooks for further memories of learning. In this case, I want them to sneak around the back end to realize that  Π is a concrete reality, something that can actually be counted, if not exactly.

Teaching as stagecraft. All the best teachers use it–even pure lecture artists who do it with the power of their words (and an appropriate audience).  Many idealistic teachers begin with fond delusions of an enthralled class listening as they explain math in terms that their other soulless, uncaring teachers just listlessly put up on the board. When those fantasies are ruthlessly dashed, they often have no plan B. My god, it turns out that the kids really don’t find math interesting! Who do I blame, myself or them?

I never had the delusions. I always ask my kids one simple question: is your life better off if you pass math, or if you fail?  Stick with me, and you’ll pass. For many, that’s a soulless promise. To me, that’s where the fun starts. How do you get them interested? How do you create those moments? How do you engage kids who don’t care?

It’s not enough. It’s never enough.

But it’s a good way to start.

## Great Moments in Teaching: When Worlds Collide

I’m on vacation! I actually took a whole half day off to add to my spring break, spent a couple days with my grandkids (keep saying the phrase, it will get more real in a decade or three), then embarked on an epic road trip through the northwest. My goal to write more posts is much on my mind–despite my pledge, I’ve only written 10 posts this year. But I’ve gotten better at chunking–in years past, I would have written one “teaching oddness” post, rather than three.

So this new semester, new year, has already seen some teaching moments that are best thought of as crack cocaine, a hit of adrenaline that explodes in the psyche in that moment and every subsequent memory of it, the moments you know that all those feel-good movies about teaching aren’t a complete lie. Not all moments are big; this one would barely be noticed by an outsider.

I was explaining slope to one of my three huge algebra 2 classes, the most boisterous of them. Algebra 2 is tough when half your kids don’t remember or never learned Algebra 1, while the rest think they know all there is to know, which is y=mx+b and the quadratic formula (no understanding of what it means or how to factor). Meanwhile, my recent adventures in tutoring calculus (be sure to check out Ben Orlin’s comment) has increased my determination to improve conceptual understanding among my stronger students, even if my weaker ones get a tad bored.

“I want you to stop just thinking of slope as a number, something you can only get by looking at two points, subtracting y1 from y2, then x1 from x2. The simplest way to start this process is to consider the slope triangle, which I know a lot of you use to find the slope, but don’t really think about.”

“But think of slope as represented by an actual right triangle. The legs represent the relative change rates of the horizontal and vertical (the x and the y). The hypotenuse is the slope. You can see the rate of change. It’s not just a number. Evaluate slopes by their triangles and you can see the ratio in action.”

I’m skipping over some discussion, some give and take. As I drew pictures, I “activated prior knowledge“, elicited responses as to what slope was, what the slope-intercept form represented, etc. But this was pretty close to pure lecture. I can read the audience–they’re not hanging on every word, but they get it, I’m not preaching to snoozers.

“How many of you remember right triangle trigonometry last year, in geometry?” A few hands, mostly my top kids.

“Come on, SOHCATOA?”

“Oh, yeah, that stuff” and most hands go up.

“So when I teach right triangle trig, I do my best to beat into your heads that the trig identities are ratios. Trigonometry is, in fact, the study of the relationship between the ratios of triangle legs and the triangle’s angles.”

“And that means you can think of the slope of a line in terms of its trigonometric ratio. Take a look at the triangle again, but now use your geometry lens instead of algebra.”

“The slope of a line is rise over run in algebra. But in geometry, it’s opposite over adjacent. The slope of a line is identical to the slope triangle’s tangent ratio.”

“Holy SHIT.” Every head turned around to the back of the room (where the top kids sit), where Manuel, a big, rumpled, exceptionally bright sophomore was staring at my board work.

I smiled. Walked all the way to the back of the room, to Manuel’s desk, tapped it lightly. “Thanks. That means a lot.” Walked back all the way to the front.

Remy smiled knowingly. “That was like some sort of smart-people’s joke, right?”

“Naw,” I said. “His worlds just collided.”

I could do a bit more, explain how I followed up, but no. You either get why it’s great, or you don’t.

<mic drop>

## Jake’s Guest Lecture

Our well-regarded local junior college is the top destination for my high school’s graduates, a number of whom are more than bright enough to go to a four-year university but lack the money or the immediate desire to do so. Case in point: Jake, my best case for the hope that subsequent generations of Asian immigrants will adopt properly American values towards education, now at the local community college with a 4.0 GPA. He earned it entirely in math classes, having taken every course in the catalog–and nothing else. This from a kid who failed honors Algebra/Trig for not doing homework, and didn’t bother with any honors courses after that.

Jake visits four or five times a year, usually coming during class to see what’s up, working with other students as needed, then staying afterwards to chat. This last week he showed up to my first block trig class, with the surly kids who mouth off. We were in the process of proving the cosine addition formula.

The day before, I started with the question: “cos(a+b) = cos(a) + cos(b)?” and let them chew on this for a bit before I introduce remind them of proof by counterexample. A few test cases leads to the conclusion that no, they are not equal for all cases.

Then we went through this sketch that sets up the premise. I like the unit circle proof, because the right triangle proofs just hurt my head. So here we can see the original angle A, the original angle B, and the angle of the sum. Moreover, the unit circle proof includes a reminder of even and odd functions, a quick refresher as to why we know that cos(-B) = cos(B), but sin(-B) = -sin(b).

Math teachers often forget to point out and explain the seemingly random nature of some common proof steps. For example, proving that a triangle’s degrees sum up to 180 involves adding a parallel line to the top of the triangle and using transversal relationships and the straight angle.

Didn’t I make that sound obvious? You have this triangle, see, and you wonder geewhiz, how many degrees does it have? Hmm. Hey, I know! I’ll draw a parallel line through one vertex point! Who thinks like that? The illustration of a triangle’s 180 degrees is much more compelling than any proof.

So when introducing a proof, I try to make the transition from question to equation….observable. Answering the question requires that we define the question in known terms. What is the objective? How does the diagram and the lines drawn get us further to an answer?

Point 1 in the diagram defines the objective. Points 2 and 4 allow us to represent the same value in known terms–that is, cos(A) and cos(b). And thanks to some geometry that is intuitively obvious even if they’ve forgotten the theorem, we know that the distance between Point 1 and Point 3 [(1,0)] is equal to the distance between Point 2 and Point 4.

So I’d done this all the day before in first block, setting up the equation and doing the proof algebra myself, and the kids were lost. In my second block class, I turned the problem over to the kids at this point.

The solution involves coordinate geometry, algebra, and one Pythagorean identity. No new process, nothing to “discover”. Familiar math, unfamiliar objective. Perfect.

I grouped the second block kids by 5 or 6 instead of the usual 3 or 4 (always roughly by ability), giving each team one distance to simplify (P1P3 or P2P4). Once they were done, they joined up with kids who’d found the other distance, set the two expressions equal and solve for cos(A+B). The group with the strongest kids were tasked with solving the entire equation, no double teaming.

Block Two kids worked enthusiastically and quickly. I decided to retrace steps and do the same activity with block 1 the next day. Which is when Jake—remember Jake? This is a story about Jake—showed up.

“Hey, Jake! You here for the duration? Good. I’m giving you a group.”

Jake got those who had either been absent or were too weak at the math to be comfortable doing the work. I kept a watchful eye on the rest, who tussled with the algebra. I tried not to yell at them for thinking (cos(A) + cos(B))2 = cos(A)2 + cos(B)2, even though they all passed algebra 2 (often in my class), even though I’ve stressed binomial multiplication constantly throughout the year but no, I’m not bitter. Meanwhile, Jake carefully broke down the concept and made sure the other six understood, while they paid much more attention to him than they ever did to me but no, I’m not bitter.

Result: much better understanding of how and why cos(A+B) = cos(A)cos(B) – sin(A)sin(B). One of my most hostile students even thanked me for “making us do the math ourselves” because now, to her great surprise, she grasped how we had proved and thus derived the formula.

And then she went on to ask “But we have calculators now. Do we need to know this?” She looked at me warily, as I’m prone to snarl at this. But I decided to use my helper elf.

“Jake?”

Jake, mind you, gave exactly the same answer I would have, but he’s just twenty years old, so they listened as he ran through the process for cosine 75 (degrees. 75 degrees. Jake’s a stickler for niceties.)

“But why is this better?” persisted my skeptic.

“It’s exact,” Jake explained. “Precise. When we use a calculator, it rounds numbers. Besides, who programs computers to make the calculations? You have to know the most accurate method to better understand the math.”

“Class, one thing I’d add to Jake’s answer is that depending on circumstances, you might want to factor the numerator, particularly if you are in the middle of a process.” and I added that in:

“Yeah, that’s right,” Jake confirmed. “like if you were multiplying this, I can think of all sorts of reasons a square root of two might be in the denominator. But other times you need to expand.”

I suddenly had another idea. “Hey. How about if we use right triangles?”

“Like how?”

I sketched out two triangles.

“Oh, good idea. Except you forgot the right triangle mark.”

I sighed. “Class, you see how Jake is insanely nitpicky? Like he’s always making me write in degrees? He’s right. I’m wrong. I’ve told you that before; I’m not a real mathematician and they have conniptions at my sloppiness. But…” I’m struck by an idea. “I don’t need to mark it here! These have to be right triangles. Neener.” (I nonetheless added them in, although I left them off here out of defiance.)

“This is good. So suppose you want to add the two angles here. These right triangles have integer sides, but their angle measures are approximations. Let’s find those values using the inverse.”

Ahmed has his calculator out already. “Angle A is…53 degrees, rounded down. Angle B is 67.38 degrees.”

Me: “Just checking–does everyone understand what Ahhmed did?” I wrote out cos-1(35). “He used the inverse function on the calculator; it’s just a reverse lookup.”

” Let’s keep them rounded to integers. So 53 + 67 is 120 degrees, which has a cosine of ….what?” Jake paused, waiting for a response. Born teacher, he is.

By golly, my efforts on memorization have paid off. Several kids chimed in with “negative one half.”

“Meanwhile, if we multiply all these values using the cosine addition formula…” he worked through the math with the students, “we get -3365“.

Dewayne punched some numbers and snorted. “-0.507692307692. That’s practically the same thing!” .

I had another idea. “You know how I said you should look at things graphically? Let’s graph this out on the unit circle.”

Jake was pleased. “This is excellent. So where would cosine(A+B) show up? We need to find the sine of each to plot it on the circle.” We worked through that and I entered the points.

Isaac: “Yeah, Dewayne is right. The two points are the same on the graph!”

“But this is a unit circle,” Jake said. “Just a single unit. As the values get bigger….I wish we could show it on this graph. Could we make a bigger circle? Or that probably wouldn’t scale.”

“How about if we just show all the values for every x? We could plot the line through that point? From the origin?”

“What would the slope be?” Gianna asked.

“Yeah, what would the slope be? Rise over run. And in the unit circle, the rise is sine, the run is cosine, so…”

“Tangent!” everyone chorused.

Jake was impressed. “See, this is why I should have taken trigonometry. I never thought about that.”

“OK, so I’m going to graph two lines. One’s slope is the tangent of 120, the other’s is the tan(cos-1(-3365))), which is just using the inverse to find the degree measure and taking the tangent in one step. Shazam.”

We then looked more closely at different points on the graph and agreed that yes, this piddling difference became visible over time.

“So the lines show how far apart the points would be for 120 and the addition formula number if you made the circle to that radius?” Katie asked.

“Yep. And that’s just what we can see,” Jake added. “The difference matters long before that point.”

When second block started, after brunch, Abdul rushed in, “Ahmed said we had a genius guest lecturer? Where is he?”

I faced a cranky crowd when I told them the genius had to go to class, so Jake will have to come back sometime soon.

*************************************************************************

Two months ago, Jake stopped by for a chat and I asked him about his transfer plans.

“Oh, I don’t know. Four year universities, I’ll have to take other classes, instead of what interests me.”

“You can’t be serious.”

“Well, maybe in a few years. But I have to wait a while for the computer programming classes I need to take, and the math classes are more fun.”

“Computer programming?”

“Yeah. That’s what I want to….what. Why are you laughing.”

“Do you know anything about computers?”

“No, but it’s a good field, right?”

“I think you’re one of the most gifted math students I’ve bumped into, and you’ve never shown the slightest interest in technology or programming.”

Jake sat up. “My professor told me that, too. He said I should think about applied math. Is that what you mean?”

“Eventually, probably, but let’s go back to why the hell you don’t have a transfer plan.”

“Well, should I go to [name of a local decent state university]?”

I brought up his school website, keyed in “transfer to [name of elite state university system]”.

Jake looked on. “Wait. There’s a procedure to apply to [schools much better than local decent state university]?”

“You will go to your counselor, tell her or him you want to put together a transfer plan. Report back to me with the results in no less than 2 weeks. Is that clear?”

“OK,” meekly.

Just five days later, Jake’s cousin, Joey, my best algebra 2 student, reported that Jake had a transfer plan started and was getting the paperwork ready.

“Oh, yeah. I’m scheduled to transfer to [extremely elite public university] in fall of 2017. I’ve been taking all math classes, so I have a bunch of GE to take. But it’s all in place.” He grinned wryly. “I didn’t think I’d be eligible for a school that good.”

“And that’s just the guarantee, right?”

“Yes, I want to look at [another very highly regarded public]. Do you think that’s a good idea?”

“I do. You should also apply to a few private universities, just for the experience. It’s worth learning if they give transfer students money.” I named a few possibilities. “And ask your professors, too.”

“Okay. And you don’t think I should major in computer programming?”

“Do you know anything about programming right now? If not, why commit?”

“I don’t know. I never knew about applied math possibilities. It sounds interesting.”

“Or pure math, even. So you’ve got some research to do, right? And keep your GPA excellent with all that GE.”

“Right.”

“And at some point, you’re going to think wow, I never would have done any of this without my teacher’s fabulous support and advice.”

“I already think that. Really. Thanks.”

Just in case you think his visits pay dividends in only one direction.

## Designing Multiple Answer Math Tests

I got the idea for Multiple Answer Tests originally because I wanted to prepare my kids for Common Core Tests. (I’d rather people not use that post as the primary link, as I have done a lot more work since then.)

About six months later (a little over a year ago), I gave an update, which goes extensively into the grading of these tests, if you’re curious. At that time, I was teaching Pre-Calc and Algebra 2/Trig. This past year, I’ve been teaching Trigonometry and Algebra II. I’d never taught trig before, so all my work was new. In contrast, I have a lot of Algebra 2 tests, so I often rework a multiple choice question into a multiple answer.

I thought I’d go into the work of designing a multiple answer test, as well as give some examples of my newer work.

I design my questions almost in an ad hoc basis. Some questions I really like and keep practically intact; others get tweaked each time. I build tests from a mental question database, pulling them in from tests. So when I start a new test, I take the previous unit test, evaluate it, see if I’ve covered the same information, create new questions as needed, pull in questions I didn’t use on an earlier test, whatever. I don’t know how teachers can use the same test time and again. I’d get bored.

I recently realized my questions have a typology. Realizing this has helped me construct questions more clearly, sometimes adding a free response activity just to get the students started down the right path.

The first type of question requires modeling and/or solving one equation completely. The answer choices all involve that one process.

Trigonometry:

I’m very proud of this question. My kids had learned how to graph the functions, but we hadn’t yet turned to modeling applications. So they got this cold, and did really well with it. (In the first class, anyway. We’ll see how the next group does in a month or so.) I had to design it in such a way to really telegraph the question’s simplicity, to convince the students to give it a shot.

Algebra II:

The rational expression question is incredibly flexible. I’m probably teaching pre-calc again next year and am really looking forward to beefing this question up with analysis.

Other questions are a situation or graph that can be addressed from multiple aspects. The student ends up working 2 or 3 actual calculations per question. I realized the questions look the same as the previous type, but they represent much more work and I need to start making that clear.

Trigonometry:

Algebra II:

I love the Pythagorean Ruler question, which could be used purely for plane geometry questions, or right triangle trig. Or both. The furniture question is an early draft; I needed an inverse question and wanted some linear modeling review, so I threw together something that gave me both.

I can also use this format to test fluency on basic functions very efficiently. Instead of wasting one whole question on a trig identity, I can test four or five identities at once.

Or this one, also trig, where I toss in some simplification (re-expression) coupled with an understanding of the actual ratios (cosine and secant), even though they haven’t yet done any graphing. So even if they have graphing calculators (most don’t), they wouldn’t know what to look for.

I’m not much for “math can be used in the real world” lectures, but trigonometry is the one class where I can be all, “in your FACE!” when kids complain that they’d never see this in real life.

I stole the above concept from a trig book and converted to multiple answer, but the one below I came up with all by myself, and there’s all sorts of ways to take it. (and yes, as Mark Roulo points out, it should be “the B29’s circumference is blah blah blah.” Fixed in the source.)

Some other questions for Algebra II, although they can easily be beefed up for pre-calc.

One of the last things I do in creating a test is consider the weight I give each question. Sometimes I realize that I’ve created a really tough question with only five answer choices (my minimum). So I’ll add some easier answer choices to give kids credit for knowledge, even if they aren’t up to the toughest concepts yet.

That’s something I’ve really liked about the format. I can push the kids at different levels with the same question, and create more answer choices to give more weight to important concepts.

The kids mostly hate the tests, but readily admit that the hatred is for all the right reasons. Many kids used to As in math are flummoxed by the format, which forces them to realize they don’t really know the math as well as they think they do. They’ve really trained their brains to spot the correct answer in a multiple choice format–or identify the wrong ones. (These are the same kids who have memorized certain freeform response questions, but are flattened by unusual situations that don’t fit directly into the algorithms.)

Other strong students do exceptionally well, often spotting question interpretations I didn’t think of, or asking excellent clarifications that I incorporate into later tests. This tells me that I’m on the right track, exposing and differentiating ability levels.

At the lower ability levels, students actually do pretty well, once I convince them not to randomly circle answers. So, for example, on a rational expression question, they might screw up the final answer, but they can identify factors in common. Or they might make a mistake in calculating linear velocity, but they correctly calculate the circumference, and can answer questions about it.

I’ve already written about the frustrations, as when the kids have correctly worked problems but didn’t identify the verbal description of their process. But that, too, is useful, as they can plainly see the evidence. It forces them to (ahem) attend to precision.

Of course, I’m less than precise myself, and one thing I really love about these tests is my ability to hide this personality flaw. But if you spot any ambiguities, please let me know.

## Teaching Math a Third Way

I was reading Harry Webb’s advice to a new secondary teacher, describing his usual classroom procedure for “senior maths”, as an addendum to his earlier post on classroom management. And I thought hey, I could use this to fully demonstrate the difference in math instruction philosophies.

Harry’s lesson is a starting activity, a classroom discussion/lecture, and classwork.

So here’s what I did on Friday for a trig class, which is certainly “senior maths”: brief classroom discussion, class activity (what Harry would call “group work”), brief classroom discussion. And I think it’s worth showing that difference.

The kids walked in, sat in assigned seats grouped in fours—strong kids in back, weakest in front. I often forget and start before the tardy bell, just laying out what we’ll do that day. I never check homework—the kids take pictures and send it to me, and I eventually get it into the gradebook. I don’t really care if kids do homework or not. They take pictures of it and text or email me. I eventually check. If kids have homework questions, they’re to let me know during the tardy pause and I’ll review them on an as-needed basis. But yesterday, the kids hadn’t had homework, so not an issue.

When the tardy bell rang, I had just finished sketching this:

(this next bit is what I think Harry would call classroom discussion):

“Can anyone tell me the relationship these triangles have?”

I got a good, solid chorus of “similar” from the room—not everyone, but more than a smattering. I picked on Patti, up front, and asked her to explain her answer.

“They have two congruent angles.”

“Good. Dennis, why do I only need to know about two of the angles?”

Dennis did the wait out game, but I’m better. After a while, he said, “I don’t know.”

“Do you know how many degrees are in a triangle?”

“180. Oh. OK. If they add up to 180, and two of them are equal, the third one has to be the same amount to get to 180.”

“See, you did know. Jeb, if two triangles are similar, what else do I know?”

Jeb, in the back corner, said “The sides have a constant ratio.”

“More completely, the corresponding sides of the triangle have a constant ratio. Good. How many people remember this from geometry?” All the hands are up. “If you had me for geometry, and about eight of you did, you may even remember me saying that in high school math, similarity is much more important than congruence, for high school math, anyway. Trigonometry will prove me right once again. So while I hand out the activity, everyone work the problem.”

When I got back up front, I confirmed everyone knew how to solve that, then I went on to this:

“I don’t want everyone to answer right away, okay? I’ll call on someone. Give people a chance to think. Which one of these variables can be solved without a proportion? Olin?”

Olin, very cautiously: “x?”

“Because…”

“I can just…see what I add to 8 to get 12?”

“Right. Now, that probably seems painfully obvious, but I want to emphasize—always look at the sketch to see what you know. Don’t assume all variables take some massive equation and brain work. Now, how can I find the length of the other side? Alex?”

“I’m just trying to figure that out.”

“You’re assuming the triangles are similar? Can she do that, Jamie?”

“Yes, because the lines are parallel.”

“Hey, great. Why does that help, Mickey?”

“I don’t know.”

“Cast your mind back to geometry. Which you took with me, Mickey, so don’t make me look bad. What did we know about parallel lines and transversals?”

“Oh. Oh, okay. Yeah. the left angles are congruent to each other, and the right ones, too.”

“Because….”

“Corresponding angles,” said Andy. I marked them in.

“Okay. So back to Alex. Got an equation yet?”

“I don’t know what I should match with what.”

“Okay. So this, guys, is the challenge of proportions. What will give me the common ratio that Jeb mentioned? I need a valid relationship. It can be two parts of the same shape, or corresponding parts from different shapes. Valicia?”

“Can I match up 8 and 6?”

“Can she?”

“Yes,” said Ali. “They are corresponding. But we don’t know what the short leg is.”

“We don’t need to,” says Patti. “6 over 8 is equal to y over 12.”

After finishing up on that problem, I turned to the handout.

“I stole this group of common similar triangle configurations, just as a way to remember when they might show up. But we’re going to focus on the sixth configuration. Can anyone tell me what’s distinctive about it?”

“It’s a right triangle with an altitude drawn,” offered Hank.

“True. Anything unusual?”

“No. All triangles have altitudes.” He looked momentarily doubtful. “Don’t they?”

“They do. So take a look at this” and I draw a right triangle in “upright” position. “Where do I draw an altitude?”

“You don’t need to….Oh!” I hear talking from all points in the room, and pick someone up front. “Oscar?”

“That’s the altitude,” he points. I wait. “The—not the hypotenuse.”

“Melissa? Can you give me a pattern?”

Melissa, in back, quite bright but never volunteers. “If the leg is a base, then a leg is the altitude.”

“True for all triangles?”

“No. Just for rights. Because the legs are perpendicular.”

“The hypotenuse is the base.”

“Right. So it turns out that the altitude to the hypotenuse of a right triangle is….interesting. Turn over the handout.”

The above conversation, which takes a while to write out, took about 15 minutes, give or take. I would expect Harry Webb has similar stories.

The next part of my lesson is the “group work” that Harry and other traditionalist think leads to “social loafing” and wasted time.

The kids are in ability groups of four; they go to whiteboards spaced all around the room: two 5X10s, 3 4x4s, and self-stick on bulletin boards that works great—I even have graphs attached.

And I just give them instructions and say, “Go.”

Is this discovery math? Hell, no. I give them all sorts of instructions. I don’t want open-ended exploration. What I want for them is to do for themselves and understand what I would have otherwise explained.

In the next 50 minutes, using my instructions, each group had identified the three triangles:

There’s always a surprise. In this case, more of the kids had trouble proving the similarity (that is, all angles were congruent) than with the geometric mean. I actually stopped the activity between steps 1 and 2 to ensure everyone understood that the altitude creates two acute angles congruent to the original two–which I frankly think is pretty awesome.

Even before they’d quite figured out the point of the angles, they’d gotten the ratios:

Each of the nine groups found the second step, proving the altitude (h) is the geometric mean of the segments (x & y) on their own; I confirmed with each group. Once they’d established that, I reminded them that the third step was to prove the Pythagorean theorem and to look for algebra that would get them there. Four of the groups had identified the essential ratios, identifying that a2 = xc and b2 = yc.

At that point, I brought it back “up front” and finished the proof, which requires three non-obvious steps.

a2 + b2 = xc + yc (reminding them about adding equations)

Then I waited a bit, because I wanted to see if the stronger kids pick up on the next step.

“Just think, a minute. Remember back in algebra II, when you were solving for inverses.”

“…Factor?” says Andy.

“Oh, I see it,” Melissa. “factor out the c.”

“Right. So then we have a2 + b2 = c(x+y)”

“Holy sh**.” from Mickey.

“Watch the language.”

“That is so cool.” says Ronnie, who is UP FRONT!

“if you don’t know what they’re saying, everyone, look at the diagram and tell me what x+y is equal to.”

And then there were a lot of “Holy sh*–crap” as the kids got it. Fun day.

I wrapped it up by reminding them that we were just doing some preliminary work getting warmed up to enter trig, but that they want to remember some key facts about the geometric mean, the altitude to the hypotenuse of a right triangle. Then I go into my spiel on the essential nature of triangles and we’re all done. Homework: Kuta Software worksheet on similar right triangles, just to give them some practice.

This lesson would rarely be included in a typical trig class, whether reform or traditional. I described the thinking that led to the sequence. But it’s a good example of what I do. (Also, as many bloggers have pointed out, my attention to detail is dismal, both in blogging about math and teaching it. Kids usually pick up on stuff I miss, and if it’s something big, I go back and cover it.)

I vary this up. Sometimes I go straight to an activity they do in groups (Negative 16s and Exponential Functions), other times I do a brief classroom discussion/lecture first (modeling linear equations and inequalities). Sometimes I have an all practice day or two—I’ve covered a lot of material, now it’s time to work problems and gain fluency (that’s when the tunes come out).

I originally had more but somehow the length got away from me, so I’ve chopped this down.

I have developed this method because I was never happy with traditional math, whether lecture or class discussion. The difference is not solely about the method of delivery; my method requires more time, and thus the pace is considerably slower.

The jury’s in on reform math: it doesn’t work well in the best of cases, and is devastatingly damaging to low ability kids. Paul Bruno refers to reform math as the pedagogy of privilege, and I agree. But it’s worth remembering that reform math evolved as a means of helping poor and black/Hispanic kids. Why? Because they weren’t interested in traditional math methods, and were failing in droves.

Ideally, we would stop forcing all kids into advanced math. But since that’s not an option, I think we need to do better than the carnage of high school math as we see it today: high failure rates, kids forced to repeat classes two or three times Given the ridiculous expectations, traditional math is due for some scrutiny, particularly in its ability to leave behind kids without the interest or high ability to carry them through. Let’s accept that most kids can’t really master advanced math. We can still do better. This is how I try for “better”.

I still have problems with students forgetting the material. I still teach kids who aren’t cognitively able to master higher level math. I’m not pretending the problems go away. But the students are willing to try. They don’t feel hopeless. They aren’t bored. I don’t often get the “what will we use this for” question—not because my math is more practical, but because the students aren’t looking for an argument. (And when they do give me the question, I tell them they won’t. Use it.) However, as I mentioned in the last post, I now have had students two or three years in a row. They were able to pass subsequent classes with different teachers, but they haven’t lost the ability to launch into an activity and work it, having faith that I’m not wasting their time. That tells me I’m not doing harm, anyway.

## Opening Day as Opening Night

I really like our late start; why the hell are so many school districts kicking off in early August? (They want higher test scores, Ed.)

Anyway, I’m teaching trigonometry for the first time. In every course, I assess my kids on algebra I, varying the difficulty of the approach based on the level of math. What to do with trig? My precalc assessment was too hard, my normal algebra assessment too easy—or was it? I didn’t want to discourage them on the first day, but I also didn’t want to give a test that gave them the wrong idea about the class’s difficulty level. After much internal debate, I created a simplified version of an early algebra 2/trig quiz. I dropped the quadratics (we only had 45 minutes). Then, just to be safe, I made backup copies of my algebra pre-assessment. If the kids squawked and gave too much of the “this is too hard” whine, I’d be ready.

And so in they came, 23 guys, many of them burly, a few of them black, none of them both, and 11 girls. Fully half the students I’d taught before, two of them I was teaching for the third time. (one poor junior has only had one high school math teacher.) Perhaps their familiarity with me helped, but for whatever reason they charged right in and demonstrated understanding of linear equations, systems, a shaky understanding of inequalities, and willingness to think through a simple word problem. Good enough. Great class—rambunctious, enthusiastic, way too talkative, but mostly getting the job done.

I’m still not much of a planner, which is why I gave no thought to my trig sequencing until I saw how they did with the assessments. If they’d tanked, I would have done a simple geometry activity to give me time to regroup, start after the weekend with some algebra. But they didn’t tank, so how did I want to start?

Special Rights. Definitely. I would use special rights to lead to right triangle trig. All clear. But how to get to special rights? Algebraic proof of the ratios. But why special rights? It seems random to start there. As long as I’m going to be random, and since trigonometry has something to do around the edges with right triangles, why not start with right triangles? At that moment, this image popped into my head:

Hey. One step back to geometric mean, and I’ve got a nice intro unit all set up.

So the next day, I started with this:

Note: I told them the questions were separate—that is, the square was equal only to the area in #1 and only to the perimeter in #2.

I wasn’t happy with the questions. They gave too much away. But every rewrite I tried was even more confusing, and in a couple cases I wasn’t sure it was an accurate question. Besides, on the second day of school, you want to release to something achievable. Better too straightforward than have the kids feel helpless this early.

And it went great. Top kids finished in under five minutes; I had them test out the process for cubes vs rectangular prisms. All the rest completed the work in 15 minutes or less, with some needing a bit of reassurance.

I had to prompt them to recognize that the perimeter to side relationship is the “average” algorithm (that is, the arithmetic mean). “If I add two numbers and divide by 2, what is the result?” I think I noodged for a few minutes before someone ventured a guess.

I followed with a brief description of geometric mean, reminded them of the various measures of central tendencies, pointed out that now they all knew why the SAT followed “average” with arithmetic mean. Finished up with practice problems.

I was stumped briefly when a student noticed that the arithmetic mean always seemed larger. Argghh, I’d mean to look that up. I told them I’d look up the answer and get back to them. Meanwhile, I wondered, could the two means ever be equal? I made the stronger kids do some algebra, and let the others just talk it through.

Great lesson, not so much from the content, but from the energy. Look, I was winging it. I do that when I have a good idea that isn’t fully fleshed out. I cut back goals, keep things very simple, and watch for opportunities. I always advise new teachers to avoid mapping things out—they are often wasting time, because things will go off the rails early in some cases. Keep it broad, tell the kids that you’ll adjust if needed, and go.

The rest of the opening “unit”: a brief review of similarity and then use of geometric means in right triangles, leading to my favorite of the Pythagorean proofs. Then onto special right triangles, deriving the ratios algebraically. This puts things nicely in position for introduction of right triangle trig and I can drop in a quiz. Well, I’ll probably put in a day of word problems first.

After school today I ran into a group of football players waiting for practice to start, many of them previous students and two of them currently in that trig class. After hearing what they were all up to, how their summers had panned out, what the team’s chances were, Ronnie, one of the two current students, said, “I’m glad I have you; I would hate to be dumped for low grades my senior year.”

“Ah, yes, that’s my claim to fame. I’m not a great teacher, but by golly, I give passing grades.”

Shoney, the other of the two, a big, burly, not black senior, was laying along a school bench calmly watching the conversation, and spoke for the first time.

“You know. Trig was….fun today. It really was.”

Ronnie nodded.

The point is not oh, gosh, Ed is a fabulous teacher who makes kids love math. That’s never my goal, and it’s not what Shoney meant.

Recently, Steve Sailer writes that “school teaching can be thought of as a very unglamorous form of show biz, which involves stand-up performers (teachers) trying to make powerful connections with their audiences (students)”. He’s right. Education and entertainment are both, ultimately, forms of information transmission.

His next paragraph is dead on, too:

We are not surprised that some entertainers are better than other entertainers, nor are we surprised that some entertainers connect best with certain audiences, nor that entertainers go in and out of fashion in terms of influencing audiences. Moreover, the performances are sensitive to all the supporting infrastructure that performers may or may not need, such as good scripts, good publicity, and general social attitudes about their kind of performance.

People tend to construe the “education as entertainment” paradigm as “show the kids movies all day” or “keep the kids laughing”, but just as all entertainment isn’t comedy and happy endings, so too is education more than just giving the kids what they want.

I’m a teacher. I create learning events. I convince my audience to suspend disbelief, to engage. Learning happens in that moment. Some of the knowledge sticks. Other times, only the memory of learning remains, and I’m starting to count that as a win.

And so the year begins.

## Assessing “Upper Level” Math Students on Algebra I

A2/Trig

I am teaching Algebra II/TRIG! Not Algebra II. First time ever. Last December, I gave the kids a packet with the following letter:

Hi! I’m looking forward to our course.

Attached is a packet of Algebra I review work to prepare you for our class. If you are comfortable with linear and quadratic equations, then you’re in good shape. If you’re not, it’s time to study up!
Our course will be challenging and fast-paced, and I will not be teaching linear equations and quadratics in their entirety—that is, I expect you to know and demonstrate mastery of Algebra I concepts. We will be modeling equations and working with applied knowledge (the dreaded word problems) almost constantly. I don’t just expect you to regurgitate solutions. You’ll need to know what they mean.
I’m not trying to scare you off—just put you on your toes! But you should put in some time on this, because we will be having a test when you come to class the first full day. That test will go in the gradebook, but more importantly, it will serve as notice. You’ll know if you’re prepared for the class.

Have a great holiday.

Reminder: My school is on a full-block schedule, which means we teach a year’s content in a semester, then repeat the whole cycle with another group of students. A usual teacher schedule is three daily 90-minute classes, with a fourth period prep. I taught algebra II, pre-calc, and a state-test prep course (kids killed) last semester, and this semester I have A2/Trig and two precalcs.

(Notice that I am getting more advanced math classes? Me, too. It’s not a seniority thing. It’s not at my request. It’s possible, and tempting, to think they noticed the kids are doing well. I know the first decision to put me in pre-calc last year was deliberate, a decision to give me more advanced classes because they wanted a higher pass rate. But I honestly don’t know why it’s happening. Maybe they cycle round at this school, moving teachers from high to low and back again.)

So I said the first full day, and today was a half day, but the kids had a whole packet to work on and I wanted to understand I wasn’t screwing around. If they’d done the work, they’d do fine on the test. If they were planning on cramming, too bad so sad.

I was originally going to do a formal test, but decided to just throw a progression of problems on the board. Then I typed it up for next time, if I teach the class again.

How’d they do? About a third of them did well, given the oddball nature of the test. A couple got everything right. Most of them stumbled with graphing the parabola, which is fine. Some of them knew the forms (standard, point slope), but weren’t sure how to convert them.

Another three passed–that is, answered questions, showed they’d worked some of the packet. The rest failed.

Of the ones who failed, easily half of them had just blown off the packet but have the chops. The other half of that third I’m not sure of.

If you are thinking that kids in Algebra II/Trig should know more, well, they were demonstrably a step ahead of my usual algebra 2 classes. And I think some of them just didn’t know I was serious. Wait until that F gets entered, puppies. Like I told them today: “There’s a lower level option here. Take it if you can’t keep up.” Whoo and hoo.

Pre-calc

I’ve now taught pre-calc twice. The first time, last spring, I was stunned at the low abilities of the bottom third, which I didn’t really understand fully for two or three weeks, leaving some of them hopelessly behind. I slowed it down and caught the bulk of the class, with only four to five students losing out on the slower pace (that is, they could have done more, but not all that much more). So when I taught it again in the fall, I gave them this assessment to see how many students could graph a line, identify a parabola from its graph, factor, and use function notation. If you’re thinking that’s pretty much the same thing I do with the A2/Trig classes, well, yeah. Generally, non-honors version of course is equivalent of honors version of previous year.

I don’t formally grade this; the assessment happens while they’re working. I can see who stumbles on lines, who stumbles on parabolas, who needs noodging, who works confidently, and so on. I was able to keep more kids moving forward in the semester/year just ended using this assessment and a slightly slower pace. One of the two classes is noticeably stronger; half the kids made it through to the function operations before asking for assistance.

This assessment also serves as a confidence booster for the weaker kids. Convinced they don’t understand a single bit of it, they slowly realize that by golly, they do know how to graph a line and multiply binomials. They can even figure out where the vertex should be, and they might have forgotten about the relationship between factors and zeros, but the memory wasn’t that far away.

While I just threw together the A2/Trig course, I put a huge amount of thought into this precalc assessment last fall. I think it’s elegant, and introduces them to a lot of the ideas I’ll be covering in class, while using familiar models.

Part II is just a way of seeing how many of them remember trig and right triangle basics:

If you’re interested in assessing kids entering Algebra (I or II) or Geometry, check out this one–multiple choice, easy to grade, and easy to evaluate progress.