Subliminal Text Messaging and Trig Identities

I'm sure most senior math teachers would agree that a lot of the difficulty kids have with trig identities has to do with the algebra involved, and not the trig. But it also comes from the fact that they often treat the identity as if it were an equation, and immediately start moving things from side to side or cross-multiplying, which is what their autopilot does as soon as it detects that = sign.

The subtlety that they're missing, and that I wanted to get across at the outset, is that when they solve an equation, they are already assuming it's true. It's the logical equivalent of saying "it's true because it's true." But identities are to be proven - and proving something is true is a lot trickier than assuming it's true - just ask a lawyer.

So while the rest of this week will be devoted to reinforcing their algebra skills, today I wanted to introduce some basic logical ideas, without actually saying them out loud. Instead I used my subliminal messaging powers, which will appear here in red italic text, which is why I have called this Subliminal Text Messaging!

The trap:
I did this today with the whole class at first, no notes, no recorded lesson. The part you see below took about 15 minutes, after which, they worked in groups of 2-3. I said pretty much these actual words, but their answers are of course composites.

Me: True or false?
(x + 3)(x - 3) = x² - 9   

All of them, immediately: True!  (Identities are about algebra that you already know)

Me: Convince me.        

Them: Well if you foil you get x² - 9. (a volunteer did this on the board): (Work on the LHS only)

Me: So what? What does that have to do with anything? (Wait a minute - what was the question again?)

Them: Well...it's the same as the other side.

Me: So what?  

Them: Well since it came to the same thing as up here (point to x² - 9 on RHS) then we were right, it was true. (Are you saying that if LHS = something, and RHS = that same something, then LHS = RHS?)

Me: Assuming of course that your "foiling" was correct. 

Them: Yes. Oh. Was it? (Just messing with their minds. :) And that we convince by using things we already know to be true.)

Me: It was, no worries. While you were doing this "foiling", did you need to look at the x² - 9?

Them: No.  (This is different from solving an equation - you're not doing something to both sides here, you're looking at one side only, then comparing it to the other.)

Me: What about this - true or false?
(x + 2)(x - 8) + 6x = (x + 4)(x - 4)

Them: ........

Me: What's the matter? Why isn't anyone answering me? 

Them: We're working on it.... (Students' likely subliminal message: Geez Miss, take a pill.)

Me: Oh this one isn't quite so obvious, eh? How come? (What's the difference between this one and the last one?)

Them: Because there's more steps. 

Me: Well how about this: Susie you simplify the LHS, Johnnie, you do the RHS, and we'll see what happens: 
(Two different people = the two sides are being done completely independently of each other - again, this ain't no equation being solved)

Susie: My side comes to x² - 16
Johnnie: My side comes to x² - 16
Them: It was true!

Me: How does that mean it was true? (Even when both sides got algebra-ed, if LHS = something, and RHS = that same something, then LHS = RHS? Sure about that?)

Them: Both sides came to the same thing, so they must have been equal. Like "this equals that". (Students' likely subliminal message: Isn't that just common sense?!?)

Me: Susie and Johnny, while you were doing your side, did you have to look at the other side in order to proceed?  

Susie and Johnny: Nope. But I did at the end. (The only reason to check the other side is to see if it's the same)

Me: Great! Now how about this:

Them: ........true?

Me:  Ah but I didn't ask you this time if it was true or false......in fact, simply by using the word "Prove", I'm already telling you that it's.....

Them: ...that it's true? ....(it's not about deciding true or false, it's about convincing by using other things we already know to be true, like algebra, trigonometry, and common sense....)

Me:  Right! But now explain to me why you thought it was true.....

We then did the above really simple example together, then off they went in their groups to do harder ones. I caught a few egregious algebra crimes and nipped them in the bud, and gave some groups harder ones to sink their teeth into, so it was a good opportunity for differentiation.

I plan to have them submit three proved identities on their blogs later this week. If I got my message across, I'll see proofs, rather than autopilot solving. More later!

If you have had success in helping your students with trig identities, or if you have your own subliminal messages to share, please do!

(This was also posted at The Flipped Learning Journal.)

The Big Beautiful Blur

I have done to my flip-o-graphic pretty much the same thing I've been doing to my lessons, which is chopping it up. There seems to be no end to the number of phases one goes through in the flip, and it just makes more sense to have fewer shorter graphics than one ginormous one. I'm also prettying it up a bit, because since I created the first one, I have discovered the pie tool in powerpoint....oh how much time that would have saved me....

These phases are not measured in time, mind you. They are measured in what I call Flip-ages.

Flip-age 1: The most basic flip

I think I was in this phase for about a month or so, maybe even less:

Flip 101 - The Shift by amcsquared

I jumped in whole hog right away, mainly because I had all kinds of lessons in powerpoint format at the ready. I didn't change them much before transforming them into recorded lessons. I just uploaded those puppies to voicethread. The thing is, all those powerpoints were designed as in-class lessons, and had plenty of opportunities for students to be active, like warmup activities, guided note-taking, and examples. So my first flip-ees did all of that too, which meant that they really needed something like an hour to watch the voicethread and do all the writing that accompanied it. Those critics who said flippers were just off-loading drudgery onto the student's evening time? They were, to some extent, right. Mean, but right.

Flip-age 2: In which I stopped being the most important player in my class

This is also where the biggest transformation happened, the one that I still think was only possible, for me, by flipping, and rescuing the f2f time:

The New Centre by amcsquared

This phase, for me, was all about what to do during class. I spent a lot of time reading about Ramsey Musallam's Explore-Flip-Apply, Crystal Kirch's WSQ, Stacey Roshan's techie musings, Kate Nowak's everything, Andy Schwen's everything, John Golden's geogebra stuff, Dan Meyer's three-act-math tasks....so many blogs by so many greats. I've experimented with lots of their ideas and I've also written quite a lot about my experiments in f2f time, and of course, it's an ongoing process. This year I aimed for creating activities that are collaborative, engaging, and fun, as well as strategies for helping students - those that ask for help as well as those who don't. I can't say that this phase is over, but a new flip-age is dawning nevertheless....

Flip-age 3: The Big Beautiful Blur:

And now it's all about the emerging overlap and interactivity of the three sectors:

Flipographic 3.0 by amcsquared

Now the lesson and the activity are becoming one and the same thing, students' questions are shaping the lesson, or help arises as a natural consequence of an activity. I seem to spend almost as much time gathering, organizing, and responding to student feedback as I used to spend making those 100-slide powerpoints. Yes, that's right. One-hundred. Sorry, students of my past, especially my first flip-ees.

And the next flip-age? 

I hope it will have something to do with learning and assessment that is initiated by my students. Maybe, maybe, it won't fit into this flip-o-graphic, and I'll have to come up with something in three dimensions. Maybe I'll have to use Minecraft to create it, and I'll have to get my own son to teach me how!

Teaching vs. Telling - Do Tell!

As the lines between teacher and student have blurred in my class, so too have the lines between lessons and activities. I find myself trying to reduce, as much as possible, both the length of my recordings and the very necessity of sending my students off to watch them. I'm starting to feel that my recorded lessons are a kind of cop-out on my part, as if by delivering info that way, I'm saying, I can't think of a better way to teach this than by just telling it to you.

I'm just talking about myself here, so please don't take this as a criticism of anyone else. I am truly struggling with this, and maybe I will come to a realization that you already have, that maybe it's a bit unrealistic to think that some day I'll be able to "teach without telling" all year. And anyway, I really have no idea how anyone with a fixed curriculum and a limited amount of time, which is pretty much every high school teacher in the world, would realistically do that. So for the time being, I'll settle for minimizing the passive learning and maximizing the active learning as much as possible. On the other hand.....

Maybe, sometimes, telling is okay!

For example, after you've made your students think, struggle, discuss, compare, and wonder, then it's okay to tell them what's what, or at least better than if you just told them at the outset. I've always felt that it's okay after you've made them sort of suffer a bit, because then they appreciate the relief your facts offer them.

I think I did a pretty good job of that this year in my trig functions unit. I completely rearranged things, and eliminated a few "lessons" at the outset, by having them graph the trig functions without knowing they were trig functions, and without any preconceived idea of what the curve would turn out to be. Once they'd done their graphs, they compared with their peers, and I actually heard them wondering - Is this what it's supposed to look like? Why does it look like this? What kind of math operation would give this kind of curve?

They decided that the wave was the right one, and I told them, yes that's right, and then moved on to explain how trig gets into the act. Next year, I'll try to coax that out of them somehow. But it felt right to validate their intuitions at that point. They were not passively absorbing information at that point - they were primed and very ready to receive it.

I think it's also okay to "tell" once the right question has been asked. In my second year of teaching, I remember having a breakthrough during one of my classes. WARNING: THERE WILL BE MATH!

The Secret to the Good Split

I was doing the grade 9 factoring unit, and we had just finished factoring by grouping, wherein

this:                                                              2x² + 2x + 3x + 3

becomes this:                                               2x(x + 1) + 3(x + 1)

which then factors into this:                              (x + 1) (2x + 3)

From there we went to factoring trinomials like this:

                                                                   2x² + 5x + 3

which can be done by splitting the middle term like this:

                                                                  2x² + 2x + 3x + 3

so that the grouping can be done as above. But the thing is, you have to split that "5x" just the right way, otherwise, the grouping doesn't work. There are plenty of bad splits (eg 4x + 1x would be a bad split), but only one good split, and you could sit there and try them all until you hit the good one, or....you can use the secret!

I remember deciding to myself, right there in front of my class, that I wouldn't tell them the secret until someone asked me. So I just kept putting examples on the board, taking their suggestions for the split and working through the example. Sometimes they hit on the right one right away, and sometimes, happily, they did not. Finally, one student, Richard, sensed that I knew more than I was letting on, and asked The Question. "Miss, how do you know how to split it?"

Before The Secret - Trial and Error

Well, that was the right time to tell them, no? They, or at least Richard, had fallen into my teacher-trap. He had completed an intellectual journey, and deserved the pot of gold, which was the secret to the good split. I suppose that I could have justified it by then challenging someone to find out exactly why the secret works....maybe if I teach grade 9 again someday, I'll do that!

Reality check:

So as much as I'm trying to stop telling, I suspect there is a time for it. And I don't think we should just fall back on the old argument "Well, if we don't, someone else on the internet will, so what the heck?" I am also, like a lot of people, dazzled by the prospect of my students answering their own questions, and self-validating to boot, to become completely self-sufficient.

But is that an unrealistic goal to set everyday? And is there a place for telling?

What do you think? I would really love to know.

PLEASE TELL ME!

Am I the Peter Jackson in my classroom?

I love movies. I love watching them, talking about them, reading about how they are made, watching shows about them...I even love websites that list movie mistakes. I am one of those people that stays until the very last credit has rolled up and off the movie theatre screen, because I like to see how many people were involved, what their roles were, where the movie was filmed, etc. The sheer numbers of people involved always astounds me, heck, it astounds me that movies are ever completed, never mind that many are so good that they transport you, and sometimes even change your life.

Sometimes I see parallels in teaching.

I read somewhere that a teacher makes hundreds of decisions per day, so our job is probably proportionately as complex as the making of a movie. And I am always astounded at the end of the year to see how far my students have come, how much work we all did to get there, and the variety of things they produced - colourful blog posts, geogebras, videos, glogs, voicethreads, not to mention beautifully solved math problems.

Teachers can be control freak perfectionist artists, like many directors. I'm sure that every great director will always find things about their masterpieces that they think need fixing, even without websites like moviemistakes.com. And so it is for teachers. We're never done, are we? It's the double-edged sword that keeps you fresh, but never lets you rest. I suppose it is a small price to pay FOR BEING BRILLIANT. Back me up Steven Spielberg?

Movies also make me think about my my own career. I think I used to see my role as the actor, or at least the performer, to my students, and I probably still do, to some extent. I am always wrestling with my inner drama queen.

But those days are over, or at least, coming to an end. I can't stomach how teacher-centric I have been. So I've been working really hard to put my students at the centre of the stage, as it were, by flipping the class, taming my explainaholicism, trying to come up with activities that unleash their creativity, and handing over to others the privilege and joy of shining light in dark places.

But like usual, there's still something to fix.

I got a hint this morning about what it is.

I was just watching a show called "The Role that Changed My Life", in which Orlando Bloom was featured for his role as Legolas in The Lord of Rings movies. Of course various people were being interviewed as part of the story, and at one point, Viggo Mortensen said that all the actors and crew in the movie gave more than the director asked them to, because they were all genuinely interested in the story that they were telling.

Hmmmm. Dreaming really big here, but, how's this for a parallel:

If I am the Peter Jackson in my classroom...

.....and my students are the actors and crew, what would be the story that they are all taken up with, to the point that they would give more than I asked them to? A story that can only be told with the varied talents and inspiration of many people, who together create something  that could change someone's life? Is that too crazy, and is it even possible in today's schools?

The story sure as heck isn't going to be the curriculum as it is. Is it a project? And whose project? When a movie is made, the story is already written, but that doesn't take away from the devotion, hard work, and genius that the actors, costumers, crew, etc contribute to the final product.

And why am I assuming that I'm the director, anyway? Maybe I'm the writer. Maybe I'm the producer. Okay, enough with the movie terms. What would I be?

I try to blog about very practical, use-this-right-away stuff, but today I'm just dreaming. If you have any ideas of what kind of story, or project, or idea could do such a thing in a classroom, or if you are already doing that, I'd love to hear about it.

And thanks, movie people, especially you, Viggo.

It almost looks like I know what I'm doing

I made some huge changes in how I did trig functions this year, and I am pretty sure it was several orders of magnitude better. It's only now that it's over, and I'm summarizing it here, that I realize that a lot of what made it so much better is kind of flukey. I made some decisions early on, or rather I just did some random things, that ended up making it all fit together perfectly. If I had tried to make it this perfect, it probably never would have happened.

Here's the skinny:

1. The very first thing they saw, and did stuff with, was this ferris wheel.
2. I based EVERYTHING that followed on that concrete reference.
3. I gave almost no homework, or mindless practice examples, or whatever you want to call it.
4. I insisted on daily blogging. 
5. I rearranged the sequence of lessons by moving one lesson later.
6. The very last task brought them full circle back to the first activity. 

Details/Observations/Reflections about these changes:

1. The ferris wheel: I posted the details of this activity here.

Guess how much I like heights.

• So all they saw was a circle with a little car that they could move around themselves. I didn't say anything about waves, or angles, or sin or cos. But I did show them a picture of me the very last time I was on a ferris wheel >>>>
• They did not know what the graphs would look like ahead of time. That added a kind of suspense to the whole operation, and I heard feedback like "I really didn't think it would look like that." That is a good thing. It made them look FOR something, instead of look AT something I gave them.
• I didn't say anything about angles until I was asked that magical question - how can we be sure we're putting the car in the right place on the ferris wheel for 5 seconds, or 10 seconds? Then and only then did I tell them about the special secret tool hidden in the ggb file. And by the way, that tool was initially not great, it was more about sectors than angles, but I fixed it so that it's all about the angle. I did manage to bring their thoughts in to realm of angles nevertheless, but only once they had asked that question.
• Assumptions about the overall shape of the graphs were made based on very few points. It was funny to see how many people not only assumed that it was a series of jagged teeth, but also who followed other peoples' lead and stopped doing their own thinking. I revealed the answer to them in this series of slides, and when I added the last few points, right around slide 6, I got a big "OOOHHH!!!"
• Once they had the visual, concrete connections between height & time, and distance from wall & time, it was easy to go from there to height & angle, and distance & angle. The real key, for me,  was to connect time with angle, to get why we even bother with angles at all for circular motion.

2. I based EVERYTHING that followed on that activity.

• The unit circle became known as a much smaller, and weirder ferris wheel that was able to go underground (negative height) and behind walls (negative distance from wall). Once they saw the corresponding graphs of the weird ferris wheel's car, they recognized it from the first activity - same graphs, different sizes. THEY asked ME what kind of a rule would give a graph shaped like these crazy waves that they got. I mean, that was huge! And in answering that, I actually got to show the connection between sohcahtoa and unit circle NOW, when it makes a lot more sense. I used to put it in at the end of the whole unit, as a little bonus interesting tidbit for those few who were still with me and able to grasp it. Which last year was I think one kid.
• In each voicethread from then on, the graphs they drew on day one kept coming up, so that they slowly but surely chipped away at its parameters. All except for one. Parameter b. My nemesis. See below.

3. I gave almost no homework.

• Instead, I packed as much practice, and as many class activities as I could while still leaving them time to work and get help during class.
• This was important, not only because I wanted them to have time to blog (see below), but I wanted to see if I could actually do this - get enough done in class that homework wouldn't be necessary. It was a gauntlet I threw down for myself. It was time to teacher up.
• I say almost because they did have to do the blog posts, and they did have a few voicethreads to watch. Some practice came from a few minutes of warmup on the board, like converting degrees to radians, or finding amplitude from a graph. But most of it, I think, came from USING what they were learning right away, sharing it, and reflecting about it.
• I didn't even give them a checklist, which I got some complaints about. And I found that hard not to have, because some of them have really taken to letting me know important stuff privately that way.

4. Daily blogging:

• This was a real slow starter. On the second day, when I should have had 22 blog posts, I had 4. Third day was better, but I had to keep at it. I had to remind them that if they wanted the option of doing a final blog post instead of a test, they had to blog everyday. This may seem like a bribe, but to me it was more of a natural consequence. I knew that if they did this everyday, it would help them understand, and would therefore improve their results on any assessment.
• I had an aha moment when I realized that at least some of that reluctance to blog came from their likely misconception that they had to get everything right the first time.
• At the end of the week, some told me they found that doing this helped them. But I wanted more authenticity than that. And it came in the second week, when I said "You don't have to blog tonight." and I had 2 ask me if they could anyway, and 2 who just did it.
• At the end of the first week I put this together, posted it on the classblog, and had them read it. LOTS of positive feedback from them, that it was fun to see what others had said, fun to see their own names and words. Also I think it made them see the value of reading each others' posts.
• It took a long time to do that snippet post, but it greased the wheels for the next week, when I almost never had to remind people to blog.

5. Change in sequence:

• The lesson I moved was the one on the exact values of sin and cos for angles like 45°, or 210°. I decided to skip over this as a result of the momentum we had going at the end of the first week. I knew that I would lose a lot of them if I went from the concrete to the abstract this soon, even for a day, so I waited, and when I did circle back to it, again, instead of homework, I had them do this, in groups, during class, to help them memorize. Feedback - Miss this was so much fun! - actual words.
• I now realize, after many many years, that this is likely where I lost many of them in the past. It used to come just before the basic functions lesson, just in time to make them get all squirrel-ly about numbers like √3/2 and 1/√2. Anything after that must have been all static to most of them. Sorry, students of my past.

6. The very last task - this was today. It was a thing of beauty. It was the most beautiful, intuitive, logical, wonderful thing, and it was all a fluke.

• This time they had to use what they had learned about sin, cos, amplitude, etc to regenerate the hand-made graphs they started with, but this time, by typing a rule into our beloved geogebra and getting it to match these original ones, for which they still had their own paper versions, plus these snapshots:

Height vs time

Distance from wall vs time

• Did I know I was going to do this when I started? NO!
• Unfortunately, it being the last day before March break, many kids weren't there. But those that were zoomed right in on the one thing that they were missing, the only parameter that I hadn't brought up in any of the voicethreads, parameter b. And why hadn't I brought it up? BECAUSE I FORGOT TO!
• But it was perfect, because one by one, they came to me and asked how to get parameter b, whereupon I had the privilege of watching them get the final piece of the puzzle, which is that parameter b takes the variable "time", and changes it into an angle. How? By multiplying radians per second by seconds. thereby giving you radians, which is an angle.

Want to know when I got that? When the teacher figured it out? Today. At about 10:35 am. 

A week without homework. Sort of.

Don't know where I'm going with this, all I know is, I am dazzled by the idea of getting enough real learning to happen in class that there is little to no need for mindless practice questions from the textbook or worksheets. I want as much as possible to happen in the 50 minutes we spend together. All I ask them to do outside of class is a blog post every single day this week. It can be a summary of what we did in class, what they learned in class, or it can be a summary of what they watched in the voicethread. Oh yes, there are also voicethreads BUT they are there mainly as a backup. If I don't get through all the class activities, or if I do but there's someone out there who needs to take another look, or insert a question, it's there for them. Is that cheating? Maybe it is. That's for next time. Baby steps.

Why the daily blog posts?

Three reasons:

First so that they can digest the information, process, and organize it in a way that makes sense to them.

Second, to get them actively learning. They'll be active in class, but I still think they have to do something on their own to pin down the concepts. But this way it's not just busy work, it's actual construction of their own knowledge, and not in a vacuum, because they can all check out each other's blogs. By now, everyone knows whose work to check out first! I'm hoping that the creativity that I experience when I'm writing happens for them, the urge to embed, link, colour, or otherwise clarify an idea takes over.

The third reason is that I want to give everyone another assessment option besides tests. I first tried this a few weeks ago, right after we had finished a unit called Optimization, which is really short, really simple, and in which most kids usually do very well. Some kids were disappointed in their test mark, though, so I gave them the option of doing a blog post on the chapter. The deal was, I'll only record the better of the two marks. I gave them very specific guidelines for it, which I put up on the classblog. and which I used to create this grid:


Seven students did it, and out of those, 5 got a better mark as a result.  Here are five of the posts. The results were pretty encouraging for many reasons, not the least of which was the quality of the work, and the enthusiasm they had for doing this type of assessment over tests. I asked for some informal feedback afterwards, and all those who answered felt that it helped them not only to learn the concepts and get the big picture, but it was a calmer, more authentic way to demonstrate their understanding.

The log blogs that didn't happen:

Well, the next unit was logarithms, which is a bloodbath every year, and as we reached the end, just before the test, I offered the option of blog posting again, but it was way too late. This unit was huge, about 4 or 5 times bigger than optimization, and it's way, way harder. No takers this time. No surprise. But I felt I had really dropped the ball for them. What if I had had them blogging all along?

Enter trigonometric functions:

So for trig, I'm getting them started right now. I'm thinking that if they start now, as the days go on, they will not only keep track of the daily learning, but they will refer to their previous posts, compare their's to other peoples', see connections, and deepen their understanding as they go. Hopefully, by the time I get to the test, those that want to can opt to do a final post that authentically demonstrates their understanding - although I'll have to think about the guidelines for that. It'll have to involve some sort of new problem to be solved, like the optimization one was (which by the way, everyone had a different one to solve). Just got an idea. Involving geogebra. More later.

Day one: Today

Today's activity was about converting between degrees and radians, as well as coterminal angles on the unit circle. I just had them drawing angles on the unit circle, and doing conversions all together. It's Monday, so nothing fancy. Hm, I wonder how many have blogged already....let me check my google reader feed....ooooh three so far - here, here, and here!

Note: I have also posted this at The Flipped Learning Journal!

Follow up on new trig activity

Yesterday I posted this about a new introductory activity for trig functions, so here's what happened:

Despite a few technology hiccups, things went well, evidenced by the few graphs on the gdoc as of now. Hopefully by tomorrow there will be more.

Observations: (Followed by deeper thoughts):

• Thank goodness I had them print up the blank graphs just in case, because that turned out to be the only reason we were able to proceed when there were tech problems. Always have a Plan B that doesn't involve tech.
• Not everyone was able to open the ggb file, (miss I don't have java) but every group had at least one person who could. Always group when you're using a fancy-schmancy tool.
• Needed to walk through it a bit at first, emphasizing that the graph they are building is of time vs height, not of the little green ordered pairs that follow the car around in the ggb file. Did I miss an opportunity to do another hint-troduction?
• In the last class, I started out by getting them to mark the various times out on the circle - where would 15 seconds be....etc. But I feel like that was sort of spoon-feeding. Although this group needs more pushing, plus NO ONE could open geogebra, so I had to screen share it and move the car around for them. I forgive myself.
• Didn't get to the second part (distance from wall) in most cases. That's fine, they can do that on their own tonight. GACK - Homework.
• In one class, no one used the checkbox that revealed the angles, probably because no one got that far, but also because this group very early on decided that it would be linear. And none of the points they graphed disproved that - they didn't have enough of them to see the curving part. But that's okay, part of learning what something is, is learning what it isn't.
• In the other class, everyone ended up clicking the checkbox to reveal the angle lines. Got the thrill of my life when I saw "OHHHH!! THAT HELPS A LOT!"
• Nevertheless, I don't think anyone saw the angle as being significant - to them it was more about cutting the circle up into sectors. But maybe once they sleep on it, that will be more obvious. Maybe I need to change that ggb so that instead of sector lines popping up, it's an arm that rotates, and they make it rotate via a slider. And it changes colour when it hits 30, 45, 60, etc. Hmm. I wonder how to do that. Great. How long is THAT going to take me to do?
• Most of them predicted that the curve would be linear, or piece-wise linear, or an absolute value function. I heard some talk of quadratic too. I neither confirmed nor denied. It's not about getting the right answer at this point, it's about seeing the circular motion, and about how they should check out their assumptions, and how a few points can make all the difference.

Questions that I hoped would happen and did:

• Do we have to find the height for all the times you have in the table? 
• Aren't the four compass points enough?                                    
• Will it be a line?                                                               
• How can we find 5 seconds on the wheel?               
• Why did you give us such a wide graph?                   
• Is there a rule for this type of graph?                          

Comments that I hoped would happen and did:

• It's just going to repeat
• It looks like hills
• It looks like teeth
• This was fun!

Tomorrow:

We're going to look at all the graphs together. At this moment there is really only one that is correct and complete for the first cycle. They are going to tell me which one(s) are right. Then and only then will I show them this! And I will bring up angle as the quantity that interlocks with seconds. Then we'll start talking about the periodicity, etc, all those things I thought we'd get to today.