Trig Curve Sketching

Previously, my Pre Cal students were messing around with the graphs of sine and cosine. Some common errors rose to the surface. The assessment that came a couple days later revealed that students were having trouble calculating periods and getting the subtle details of cosine correct. To top it off, now I'm throwing tangent graphs at them and they just found it weird.

Luckily, Frank says it best.

Some days before we had started any of this trig graph stuff, Kate Nowak shared an excellent curve sketching game for Algebra II. Instantly I knew I needed to try it. It was too good not to. In fact, it spawned activities for Pre-Cal AND Calculus. More on the Calculus one later.

Setup

To summarize Kate's game: distribute decks of cards with different features on them, require students to draw something that has those features, and judge one another for accuracy.

I made a few modifications for my purposes.

• Deck of 22 half index cards
• One six sided die
• Whiteboards
• Markers

On the cards were sets of features that could apply to a sine, cosine, or tangent graph. Students pick a judge, the judge flips a card and rolls the die. A 1/2 means you draw a sine with those features, 3/4 cosine, and 5/6 a tangent. Student groaning when it comes up tangent is optional.

I printed this off on Avery 5160 labels (just like Log War and Inverse Trig War):

I threw a couple goofy ones in there because that's how I roll.

Let's Play

I told someone to keep score and to make sure they rotated the judges. For two classes we awarded a point for the "best" interpretation of the graph. I noticed the same kids were getting the point every time so I changed the rules for class three. Anyone who does the graph correctly gets a point.

I thought they'd get through about 15 cards in the 40-45 minutes. In reality it might have been closer to ten. A few cards caused some confusion. In a few cases the judge wasn't sure if anyone had done the graph correctly. I suspect if we played this towards the end of our time with trig graphs I would've had fewer debates to settle. There were still lots of good things going on here. If anything it hammered out the differences in sine and cosine for lots of them. And we solved the period calculation problem. The goofy cards produced some excellent results.

It was a lot of good graphing practice and discussion without feeling like graphing practice and discussion. It kept their attention for quite a while. I think we could easily do this again for some other stuff. The Calculus version went equally well.

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Trig Design

More in my desire to help the students less this year.

I've had a good schtick going for a few years. I take something that ordinarily would be classwork and pass out colored paper and markers and have the students do a display version that goes in the hallway. In my life as an academic teacher, these assignments were pretty scripted. In this instance we're discussing trig graphs.

Rather than the standard "here are twelve functions, everyone pick two" I came up with these specifications:

• create a set of 3 sine functions and 3 cosine functions
• "negative" amplitude must be demonstrated
• a "faster" function must be demonstrated
• a "slower" function must be demonstrated
• a vertical translation must be demonstrated
• create two graphs in desmos, one for each type
• print and identify your creations

Negative amplitude means a reflected function. Faster and slower reference the length of the period. If you've ever struggled with the whole why do I calculate the period of sin(2x) by determining 2pi / 2, try asking the student how they would complete a 10s task twice as fast. It nixes a trick and builds better understanding all at once. This is a handy if you ever want to discuss models of electrical signals.

This activity is quick thanks to the infrastructure I've been building for a few years: tons of iPads and a color printer in the room. No computer lab sign up sheets necessary.

I liked the vast variety of functions. It also brought some misconceptions to the surface. A lot of students could create a faster or slower function, but weren't sure how to tell me the actual period. This created an excellent opportunity for follow up the day after this activity. I also fielded a lot of questions about whether vertical translations affect things like amplitude and period. Again, great follow up material for the next day.

Something you may notice. If you study the work closely, there are errors. It happens. While on display, it's practice. I expect there to be errors in practice. Students self-correcting their errors between the time we practice and we assess is what I care about. If I were to nitpick these as they went up, I counteract my goal of minimal help.

Start to finish this was one and a half class periods. I'm also getting better at anticipating activities that finish early and having something ready for the ones that finish early.

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All My Triangles

Every year in Pre Cal I approach right triangles differently. This year is no different. The number of times you could say I "deliver a lesson" is starting to dwindle. I design tasks. The students spend class time doing those tasks, and we find the interesting questions within. Here's a short primer on how we progressed through right triangles.

I passed out protractors and some paper and told them to make a bunch of right triangles. Specifically, make a bunch of 30/45/60 right triangles. Side measurements are up to you. I had them make nine, it should probably be half that because the process took FOREVER. Useful as a reminder on how protractors work.

After the long process of making triangles, we labeled them with opposite, adjacent, etc. For each triangle they computed the decimal value of opposite/hypotenuse and adjacent/hypotenuse and recorded the results. No explanation as to why, just entertain me for a minute here.

Wait, what?

I found this website helpful during my discussion of the results. In one class not every kid got this far, again because TOO MANY TRIANGLES, MAKE FEWER NEXT YEAR. But we worked through it. Lots of interesting stuff here. The ratios seem to be close no matter the triangle, no matter the kid that made them. And what's that? Repetition? Is there some kind of pattern here? But what are these goofy decimals about?

Unit Circle

At this point I did a rare moment of lesson delivery, aka talking. Recall of special right triangles from geometry and the ratios present. While making the triangles, many kids asked why we had to bother with the 60 degree ones when duh there's a 60 degree angle in my 30 degree one. This part. This part right here.

We spend roughly a week with the unit circle and the six trig functions and the ratios and all of that. Important difference here is that I made an effort to talk about the unit circle as a collection of triangles. We always defined the trig functions in terms of triangle sides (not x and y), making note that in this very special collection of triangles, the hypotenuse is 1 and organizing the adjacent and opposite sides as a series of coordinates is helpful. An hey, in all cases the adjacent side runs horizontally (x) and the opposite runs vertically (y). Weird.

So often the unit circle is treated as a stand alone thing that we must do because the book told us to. But the unit circle is everything! Vectors, polar coordinates, and trig graphs make so much more conceptual sense if you can convince students to get the unit circle.

Now to branch out a bit. Right triangles that are not necessarily special. We wandered out into the school and found some objects that were tall. They measured a hypotenuse length and an angle.

That covers finding unknown sides when one side an angle are known. But what about unknown angles?

Armed with new knowledge of trig functions and that oh hey, the calculators has those buttons, I gave them a table to complete. Values of sin, cos, and tan from 0 to 90 in increments of 5.

I hand out a series of triangles where the corner (non-90) angles are unknown. Two sides are given. Students determine which trig ratio works the best based on given information and approximate the value of their angles. A few remember the inverse feature of the calculator and I was totally find with them using it. However, the idea here is to reinforce the idea that trig is big patterns. An observant student noted that the sin and cos tables seem to be going in opposite directions. Weird.

In conclusion, I had them create some artwork for their notebook. Display the four quadrants, orient a triangle in each one, make the sides whatever you want, show the values of the six trig functions for the reference angle, and finally, approximate the value of the reference angle.

In the end we covered right triangle trig faster than ever. And other than the unit circle, the kids spent more time listening to each other than listening to me.

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Floor Is Lava

I enjoy the goofy side of teaching. Entertainment value can go a long way in getting children to buy what you're selling. This one though, I mean, I've done dumb stuff before, but this is in the top five.

Ok, so kids lining up at the door. Not ideal. Step 1, make a restriction zone near the door.

Works well. Start jokingly referring to it as lava for a few days. Step 2, actually spell it out as such.

Immediately challenge them to jump the lava to gain entry. Step 3, line of teenagers at the door waiting to jump/avoid the lava.

I'll pass the mic to Kate here.

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Sharing is Caring

Some updates on two initiatives: providing resources and shutting up.

First, partly for accountability purposes, I'm doing all the Calculus homework and providing solutions via shared Dropbox folder. I passed out a bitly.com link which lets me track usage.

Homework is assigned Monday and due Friday. The giant spikes are Thursday nights. I'm fine with that. Due Friday means due Friday. Other AP classes are a tad more demanding because of reading, so this gives them a chance to find time when necessary. I have 50 students, and I'd say 2/3 of them said they use this resource regularly. The assignments aren't super long.

Second, helping less. They were struggling with related rates, I think that's like the law in Calculus or something. Here are some easy things, overthink them to death. Sigh. Anyway. They needed some more practice, I had a scheduled absence coming up. I pass out the problem set and say you have to do 5 out of 8, solutions are posted. I'm not going to much use to you (because I'll be absent but I'm not going to tell you). The gigantic spike on the right there is them accessing the solutions while I was away. Perfect.

PreCal is also getting the helping less treatment, which I'll save for a later date.

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Check for Understanding

I know teachers who don't like writing tests. I love it. For various reasons.

Kids are the best, bruh.

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Passion

Some weeks ago I was in a meeting. For the opening talking point we were asked to discuss our passion. Other than meaningful assessment, my passion revolves around the war on boredom. My students should be doing something as much as possible. As time goes on, that's become my students should be doing something genuinely interesting as much as possible (tricking them into learning as they like to say). I want to be the math teacher I (more or less) never had.

I had some nice teachers. But were they great? A brief recollection of my math classes (I attended 6 schools from K-12):

Elementary School (K-5)

I remember stressing about counting money in kindergarten. In first grade we did jumping jacks prior to our state testing. In third grade we learned times tables and long division. In third grade we played a game where a student would stand up, they would proceed to stand behind a class mate. If you were standing and got the fact right, you continued to the next class mate. If you were beaten, you traded. One kid (my academic "rival") blasted through most of the class, spitting out multiplication facts in an instant. I was excited when our teacher let us talk about big numbers, like millions. In fifth grade I stressed about fractions. We took a quiz on mixed numbers. I got a significant number of them wrong. This upset me. I (seriously) requested we review them all. The teacher snapped "I will NOT do all of them" and went over none of them. I still remember the stare. She retired only recently.

MIddle School (6-8)

Sixth grade was the best. This guy asked the most interesting questions. I was fascinated by prime numbers, Roman numerals, and the very adult multiplication and division notation we learned. We looked at the Game of Life. Kids were divided into ability groups. There were different tests depending on what you could handle. I was so happy to be in the coveted C group. He'd pull us aside, give us Fawn Nguyen like things to tackle. We changed groups often, we had to come up with a name and a logo. There was an in-class economy, using tiny black and white bills. You earned a salary, he had a long list of things you could spend money on. He was super organized. He expected the same. I was not. It caught up to me a couple times during the year. We did a travel project at the end of the year. It was part of a national contest of some sort? You made up a road trip and had to calculate all the expenses and show the work.

Seventh grade was pre-Algebra. I was excited to play with equations and letters. The teacher would call a bunch of people up to the chalk board. You'd have an assigned problem and had to solve it for the class. I was that guy who would try to do it as fast as possible to impress ever--no one. She'd have grades ready, you could elect to have it shouted out loud or go up to the desk. I always went up to the desk.

Eighth grade was whatever. I'm told the teacher did a "many angry phone calls from many parents" kind of job. I don't know. A girl who sat near me raised her hand and said "I'm confused" a lot. She's a lawyer now. I would get bored and flip ahead in the book.

High School

This was kind of forgettable. Geometry was interesting, the teacher didn't enjoy jokes. Constructions were exciting. Proofs were whatever. I took Algebra 2 from a guy who said "what" like 1000 times per class. He'd answer way too many questions during tests. There'd be a line 15 people deep during a test. I moved to Texas, was put in a Function/Trig/Stats class. Felt really bad at math. Couldn't figure out transformations.

I took Pre Cal as a semester block. Teacher would go over two sections, then we start our homework. She'd then sit at her desk. Every day. I figured out mathematical induction, I was proud. Modular division and binary/hex numbers were neat.

Calculus was full of new and interesting notation. There was a related rates problem about a pool that was way easier than it looked. We spent a long time as a class working through it. The valedictorian was in there, he made me feel inadequate. I figured out f, f prime, and f double prime curve relations though. He didn't. The teacher was very friendly and cared and made an attempt to learn something about you. She let me draw things on the whiteboard whenever I wanted.

Ok, that was great...

...but what's the take away from all that?

I expect my experience is typical. One alleged disaster. A lot of mediocrity. One teacher who inadvertently convinced a 5th grader that it was unacceptable to not know something. Nice teachers who may not have done a ton of interesting teaching. But among that group, who had a passion for the job? When asked, what would those teachers say?

Then there's 6th grade. That was 20 years ago. I remember so much about it. It was fascinating. He was fascinating. Math was the best. I want that. My students deserve that. Math is more than problems 1-29 odd. Just because an experience is typical does not mean it's ok. I want to be better.

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Thanks Textbooks?

Stumbled on this the other day.

Holt Geometry

Two observations:

The abstract use of "geometry software" with no hint about what it might be, where I could obtain it, or how to set up what's being shown.

The dull task that could be replicated with paper in 30 seconds.

Do the people that write these things ever attempt to teach? Ever attempted these half-wit "labs" they come up with?

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Derivative Facts

I had a scheduled absence, and a need for Calculus students to occupy themselves for 90 minutes. Amid their normal Throwback Thursday routine, I had them create some art.

Take some colored paper, and demonstrate the four major derivative rules: power, product, quotient, and chain. A generic example, specific example, and written explanation were required.

Pretty pleased with the results. General consensus is that AP = lecture format. Boo to that. Sadly, something so simple presented at your average Calculus workshop might be a little too mind blowing.

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Native Document Cameras

A niche use case, but a neat feature. If you need to demonstrate something from an iPad, iPhone, or iPod, running the app AirServer on your teaching computer is the way to go. It acts like an AppleTV without the need for a physical AppleTV or switching projector inputs or any of that mess.

There's irony here somewhere

But, your school wireless network may not support it, or like me, AirServer misbehaves from time to time and you need a back up.

On the off chance you teach from a Mac, and on the off chance you recently updated to the latest OS X, you have a built-in option now.

Step 1

Plug your device into the computer using a USB cable.

Step 2

Open QuickTime Player and select "New Movie Recording"

Previously, this was a way to record using any built-in cameras or USB cameras attached to your computer. Now, it supports iOS devices as external "cameras," designed for recording the screen of the device, but also handy as a document camera if you open the camera app.

Handy for recording a workflow for your students (it will grab audio from your computer microphone). Handy for a wired document camera solution without the need to fiddle with your projector.

You could also connect a random student iPad and show off something if that's your thing. And yes, the window will change its aspect ratio if you rotate your device to landscape.

Requires the device to be running iOS 8, and the Mac to be running OS X 10.10 Yosemite.

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