Continuous Everywhere but Differentiable Nowhere

A few days ago, JD2718 wrote a post about “Birthday Triangles” — having students create three coordinates out of their birthdate and then analyzing the triangle that these coordinates make.

Even though it’s just a bit of fun, and you could have students work with any sets of points you give them, there is something great to be said for students creating their own problems that they feel ownership of. As JD2718 writes:

Best evidence, (and mind this, please) almost every class, when they first plot their own birthday triangle, there is one or two sad looking kiddies (it’s not come to tears, but I’ve seen the quivering lip) who thinks their own triangle is ugly. “Nooo” I say “Yours is obtuuuse. Does anyone else have an obtuse triangle that looks as nice as Anna’s?” (it’s usually a girl)

I thought that this idea could work in calculus too, creating “Birthday Polynomials.” My first thought was exactly JD2718’s: take the three birthday coordinates and find a quadratic that would fit them. But that would be a precalc assignment. (With bonus question: With what birthdays could you not create a quadratic?)

But I wanted more. I wanted to come up with something awesome. Something calculus. Something that would knock my students’ socks off. I initially thought something like this… if I was born on April 21, 1978, the birthday polynomial could look like: [1].And questions could be: where are the local maxima and minima? where is it concave up and concave down? where is it increasing and decreasing? And of course you could do things with integration too…

But there’s something unsatisfying about that type of question. It’s nice, but I want to wow! my students. I want to knock their socks off. Show them something elegant and unexpected. So I thought…

I want them to create a polynomial using their birthdate which would have an inflection point that was their age.

I was planning on using this amazing property [if there’s a cubic equation that hits the x-axis three times, then there’s a point of inflection, and it will be the average of these three x-intercepts] they would have to discover.

So if my birthday were January 25, 1980 (it is not), and we evaluated this polynomial on March 30, 1980 (after I celebrated my birthday), a birthday polynomial might look like this:

.

[Note that the month and date play no role when finding the point of inflection... they are red herrings.]

But there are many annoying problems with this… First of all, that 3 is annoying. Second of all, that 2008 gives some of the fun away. I guess multiplying it by 3 and writing it like 6024 would help disguise it, but not much. Third of all, if I worked the problem on January 5th (or anytime in the year before January 25th), it would get my age wrong by a year. Fourth of all, it’s not elegant.

I’ve spent a little time tweeking it, and thinking of ways to rework it… but I haven’t anything elegant or clever yet. For now, it’s going to have to go on the backburner. Spring break is over and school is starting tomorrow and I have too much on my plate.

I’ll post an actual, good, interesting way to come up with a birthday polynomial with some amazing property (that somehow magically spews out your age, perhaps) when I have time…

[1] Of course I had to do a google search on “birthday polynomial” to make sure I wasn’t reinventing the wheel. One calculus teacher in Texas did something similar.

I created an online mixtape, on one of the most graphically sleek sites I’ve seen in a long while [muxtape].

You can hear it by clicking below. For bonus points, try to figure out the common thread which binds these songs together.

Update: Many people also swear by mixwit, which has a nice interface too. And you can make multiple mix tapes.

Update: Answer to the question in the subject line in the comments.

I just spent 3 hours watching Leonard Susskind [Wikipedia page] deliver two lectures on Classical Mechanics via iTunesU. I started this over Winter Break (I watched 4 lectures), but I stopped because I didn’t have the proper time to devote. Once you stop, and enough time passes, you have to start all over [1]. Plus I hadn’t bought a pretty $15 notebook from Paris to take notes in.

But I started over, notebook in hand.

The end result of this is: I’m inspired. I love love love learning about trajectories, generalized coordinates, lagrangians, and principles of least action. Plus, there are some pretty neat digressions or mini-lectures I could use for my multivariable calc class next year.

I highly recommend it. You just need calculus (and some basic multivariable can’t hurt). It’s a bit hard to find on iTunes, but this link might take you to the first lecture. It’s his Stanford PHY 25 course taught in 2007 (October - December).

[1] It reminds me of borrowing the 1st season of LOST from my friend, who had the last seventh disc out on loan to a different friend. Once I finished the first six discs, I was hungry for the last. But still, the other friend refused to return it. And to this day, I have not seen the entire first season of LOST. Now, since so much time has lapsed, I will have to start the season all over again. (Double sigh.)

Circle art at the Pompidou in Paris (click for my photo):

Bernar Venet

I wonder what I could do with this and radians… A good quiz question is forming…

When I was away in Paris for Spring Break, people didn’t stop blogging. I spent a good number of hours catching up, while I’m sick and not in the mood to do anything really active. (When am I ever really in the mood to do active things, though?) So I logged into netvibes and buckled down… carnivals… posts… links…

There’s a lot of good stuff out there. Here’s some that I want to highlight:

1. A primer on the zeta function: you know this person knows how to break something down and present it in a clear way. It builds up, from smaller simple examples to build intuition, to a grand finish.
2. An analysis of whitespace — physical, and metaphorical: “I like to talk. I really wish I could just and listen to myself, because the information that I spew out is just awesome stuff. My students might disagree, though. If I reduce the amount of noise that I make, my students will be more likely to hear the important things I tell them. As a side note, I find that the misbehaviors in my class seem to happen when I am talking or the students are otherwise disengaged. So the less I talk and the more I work, the better! “
3. A funny (but insightful) take on parent teacher conferences, which I had to pass along to a number of my teacher friends. To whet your appetite: “I also feel I must apologize. I am sorry that I sent your child to the nurse the other day when he complained of a toothache. I don’t know where my head was. Thank you for the quick analysis of my motives via email that afternoon. Had you not pointed it out, I would have never picked up on my underlying desire to lessen the number of students in my class by sending them to the nurse for innocuous ailments. I got your message loud and clear though. Your use of 18 point font, bold print, all caps text really aids in the reading process. From now on, I will not send him to the nurse for toothaches.”
4. I struggle with homework, and it’s nice to know others do too. Good ideas for other forms of assessment are in the comments after the post. Huzzah!
5. A small, silly, cutsie way to get students engaged when dealing with coordinate points.
6. Carl Sagan on Flatland (from Science After Sunclipse)… Amazing expository. Good teaching. I was hooked and I know all this.

On Pi Day…
In my seventh grade class, I had to forge forward with the curriculum, but I came up with something great to punctuate the class work with. We have been working with areas of geometric shapes (specifically circles), radicals, and the pythagorean theorem. One day, a few days before Pi Day, I drew The Perfect Circle on the whiteboard. They were impressed. I was lucky — but I didn’t play it off as luck, but practice and skill. I rehearse, I told them.

That night, I sent them this video over email:

And again, they were impressed. So I decided that in honor of Pi Day, I would hold a freehand circle drawing contest. They came to class psyched. One said he had practiced drawing circles in the air so much that his arm hurt, and another used a whiteboard marker on a mirror. We forged forward, and as we worked, we punctuated the class with the competition (3 students at the board at a time; the winner of each round got to compete in the final championship round).

The winner of the contest got to create — with me — the bonus problem for the next test.

3 point

14159265358979323846264338327950288419716939937510

58209749445923078164062862089986280348253421170679

82148086513282306647093844609550582231725359408128

48111745028410270193852110555964462294895493038196

44288109756659334461284756482337867831652712019091

45648566923460348610454326648213393607260249141273

72458700660631558817488152092096282925409171536436

78925903600113305305488204665213841469519415116094

33057270365759591953092186117381932611793105118548

07446237996274956735188575272489122793818301194912

98336733624406566430860213949463952247371907021798

60943702770539217176293176752384674818467669405132

00056812714526356082778577134275778960917363717872

14684409012249534301465495853710507922796892589235

42019956112129021960864034418159813629774771309960

51870721134999999837297804995105973173281609631859

50244594553469083026425223082533446850352619311881

71010003137838752886587533208381420617177669147303

59825349042875546873115956286388235378759375195778

18577805321712268066130019278766111959092164201989

38095257201065485863278865936153381827968230301952

03530185296899577362259941389124972177528347913151

55748572424541506959508295331168617278558890750983

81754637464939319255060400927701671139009848824012

85836160356370766010471018194295559619894676783744

94482553797747268471040475346462080466842590694912

93313677028989152104752162056966024058038150193511

25338243003558764024749647326391419927260426992279

67823547816360093417216412199245863150302861829745

55706749838505494588586926995690927210797509302955

32116534498720275596023648066549911988183479775356

63698074265425278625518184175746728909777727938000

81647060016145249192173217214772350141441973568548

16136115735255213347574184946843852332390739414333  


and so on and so forth...

I’m going to be heading off to Paris on Sunday, taking a much-needed break from the daily grind. I might post something before I leave, but I’ll be gone until March 26th.There’s a good chance I’ll be bringing my lappy toppy with me, so you might get an update or two from the French capital, but in case you don’t hear from me, I’m (probably) a-okay 

Update: Bill Fitzgerald and Dan Meyer are now on a similar quest.

Yahoo has started a new web 2.0 site for helping teachers create, manage, and share “projects.” I’ve dreamed of a site where teachers collaborate (as well as beg, borrow, and steal) online in a massive community. Yeah, bloggers read and comment, but that’s not what I dream of. I want a huge archive and discussions on projects.

This could be that, if we’re speaking with all the idealism and naivety of a ten year old. But for a site like this to work, people have to use it. Without having played with it, my initial spidey sense is telling me that instead of the website being adaptable to us, instead we’re going to have to adapt to it. Constrained by what the website constitutes a “project,” teachers are likely to think this site isn’t as natural as it could be. Instead of technology adapting to our needs, we might reconfigure our needs to adapt to this technology. And I’d like to have some… not promise… but strong indicator that it’s worth it before heading off into the technological blue.

It’s unclear to me how useful this is going to be (if at all), but I’m going to keep an open mind. One thing I’ve often noted is how hard it is to find smartboard presentations online. I create mine from scratch, but I also imagine that others would find them useful, as I would find looking at (and stealing parts of) their presentation of the same material useful. I secretly have a hope that one day I will be the facilitator to this giant city wide project which will get teachers who actually make lessons (smartboard, handwritten, typed, group projects, etc.) to upload them to some site to share.

In any case, the website is still in beta form: http://beta.teachers.yahoo.com/

I signed up for an invite forever ago, and just got invited to join today, but maybe anyone can get one now that they’ve opened it up for testing.

Introducing trigonometry has become even more of a challenge than I thought. I think about each part of the lesson really hard; I want to give it a flow and focus on teaching the concepts. What I don’t want trigonometry to be is a huge mess of ad hoc rules.

Today was my introduction to radians. Looking back, my presentation was a bit more complicated than it needed to be to get the idea across. And what wasn’t clear (although that was one of my objectives) was why we use radians instead of degrees. So I’m going to start off class tomorrow with a little silent slideshow, replete with my own histrionics to make extraordinarily clear why we! love! radians!

Without further ado: why radians? (PDF file) [Unfortunately, SlideShare is only showing 18 of 23 pages for some reason. The PDF is complete.]

I’ve also come to realize that more is going on with kids than this whole forest for the trees crap that I wrote about before.

There’s a second reason things are getting mucked up, and that doesn’t have anything to do with my concept behind each lesson, or the flow, or anything like that. I realized today that a lot of the things that kids get tripped up on are (surprise surprise) basic facts about numbers. Is “1/2 times pi” the same thing as “pi over 2″? Yes. Do they know that? Possibly.

Or, for example, there’s the issue of manipulating visual and fractional information in their heads. We’re learning about radians, and we learn that there are “2 pi” radians in a circle. Then I ask them to draw an angle of “3 pi over 2″ radians. It was as if I asked them to dance around like a chicken while singing Ave Maria. And since the “pi” was there, they thought that using the calculator wasn’t really going to help them.

I think we’re slowly getting it, but I’m not sure. I’m going slowly, but I have now started identifying key skills and concepts that need to be honed before we move on with radians. For example, because of what I noticed in my lesson on radians, we’re going to be practicing working with fractions (the eternal scourge of math teachers) and pi. (See my thrown-together worksheet here.)

Imagine (for surely, gentle reader, this has never happened to you before) that you’re at a mini golf course and you’re putting at the infamous and dreaded windmill hole. By mandate from the PuttPutt gods, you are not allowed to leave until you get the ball into the windmill. There’s a mini golf coach there, trying to give you advice and show you how to hold the club and how to swing. However, after 20 tries you aren’t getting it. And then you try another 2o times. No luck. Now tell me how you think you’d feel at your mini golf coach who has been standing there trying to help you.

There will be whining, complaining, anger, and frustration — anxiety — all directed to this coach.

The analogy isn’t quite right, and I hope that my students don’t direct those feelings to me (this was an extended allegory, duh), but I can’t help but notice that the anxiety level has shot up in my room in the past two weeks, when I feel that one of my teaching talents is keeping a totally relaxed atmosphere.

UPDATE: My presentation (see above) on”Why Radians?” took 5 minutes and I think did the trick. I did it in both classes, and both seemed to get it. And the levity of it all made the classroom less tense. And with Spring Break descending upon us, we’re going to have a much needed break.