Exasperating Problem

So a while ago, I mentioned to some of you on twitter that I was getting really frustrated with a particular problem we were presented with. I have a conjecture that I’m almost certain is true, but I can’t prove it.

Consider the unit circle x^2+y^2=1. Plot n equally spaced points on the circle starting from (1,0). Now draw the n-1 chords from (1,0) to the others. What is the product of the lengths of all of these chords?

(There is an extension problem, which is changing the unit circle to an ellipse 5x^2+y^2=5, for those who already have seen or find the original problem too easy.)

So feel free to write your own blog post with your solution, or throw your solution in the comments (just write SPOILER at the top so we know…).

What I’m interested in is if we could get a precalculus class to get the solution to this problem. Where they actually understand it. So if you had, say, 15 non-honors precalculus students and one week to work on this problem, how would you design the lesson?

I guess you have to have solved it or have seen a solution to know how to design the lesson. But even if you didn’t solve it (a la me!)… if there’s a solution you’ve read that someone posted in the comments… what would you do?

UPDATE: Mr. Ho has a great GeoGebra applet at his site; Mimi has some nice colorful diagrams and some explanation up at her site. Also, for those who want to wording for the ellipse problem… This extension I haven’t seen before, so I am citing Bowen Kerins (see comments below!) or Darryl Yong: “Take the diagram you drew in [the unit circle problem] and stretch it vertically so that the circle becomes the ellipse 5x^2+y^2=5. All the points for the chords scale too. What is the product of the lengths of all of these chords?”

Looking past teachers to teaching

Today I attended a session where three university profs — ed researchers — formed an informal panel. There was one important point that came up at the beginning, and became a riff for a few minutes. It was, as you prolly suspected from the really innovative title of this post, about the power of looking past teachers to teaching.

It’s a slight distinction, but crucial to the reorientation that I’m having about teaching.

Some points that came up in the conversation:

  • Replacing teachers won’t change things; replacing teaching methods will.
  • Focusing on teaching and not on teachers is the basis of lesson study (and the Seattle video club I talked about in the last post). It focuses the conversation on teaching/teacher moves.
  • Changing the conversation from teachers to teaching more readily implies that teaching is learnable. So we have to look past individual teachers to the methods of teaching. That being a good teacher can be taught. Another way to think about it: teaching is a complicated activity, rather that something owned by a particular person.
  • There are universal tasks to teaching that we can investigate (e.g. which ideas to privilege in a classroom).
  • It gets us away from the “I taught it but they didn’t learn it” phenomenon. That phrase doesn’t really make any sense when focusing on teaching and not the teacher.
  • The greatest untapped resource we can use in the classroom are our students and their insights. And by focusing less on the teacher and more on teaching moves, we can tap into that.
  • This outlook shifts the conversation away from teacher bashing (but one should also be cautious of going in the other direction of student bashing).

Yes, I know. There are some inconsistencies, and worse, this is all very abstract. And I HATE THAT. But this all tapped into the idea I wrote about recently, about how teaching moves are something that one can pay attention to. One can learn. One can revise. And through this process, hone the craft of teaching.

In other words, the focus on teaching instead of teachers is that it puts the emphasis on the ways teachers can do their jobs by focusing on students and learning.

So that was one part of the talk. In another part of the talk, there was a question about the constant tension between the jam-packed curricula with a zillion micro-pico-standards and getting students to really grapple with big ideas.

One speaker said that we “need more effort and courage” from teachers. I drew a sad face in my notebook next to that.

The second speaker actually spoke articulately, in defense of having common standards in theory [1]. He also said that he doesn’t see the problem as having a zillion pico standards. It’s that we go through all these little ideas that never get added up to any big ideas. His suggestion for dealing with this is to outline learning trajectories, with big ideas as the landmarks on the way. I don’t know what precisely he had in mind, but I figured that it probably involves student drawing connections by working on unfamiliar problems that force relationships among mathematical ideas (e.g. systems of equations with matrices; asymptotes for the tangent graphs and asymptotes of rational functions; absolute value equations and absolute value inequalities; etc.).

The third person then finished up speaking about the Common Core Standards — and eloquently continued the second speaker’s defense of standards.

That’s about it for the maths stuff I want to write about. (It’s late and I have lots to do tomorrow.)

On the non-math side of things, I had a wonderful night BBQing with friends and watching the sky change hughes, from orange, to light blue, to dark blue, to black. As the air got colder and the light retreated, the stars starting coming out, first slowly then quickly. As people left, conversations got less frenetic and more personal. And I left, after being regaled with a shooting star, at peace with Utah.

[1] Having these standards gets us focused on teaching. It also promotes the sharing of ideas; if someone gets it/does it right, then those lessons and approaches will be in demand.

Why I’m not Blogging

Part of me thought I would get to the Park City Math Institute and be blogging up a storm. I would sit down each night, twitching with excitement, ready to blog about all the ideas and problems and conversations I would be having.

And I am aflutter with excitement about this program.

But the problem is that I’m spending all day talking about math and teaching, and thinking and thinking and thinking, and I don’t have the motivation at the end of the day to organize my thoughts.

This is problematic. Because my memory:elephant’s memory::pebble:mountain.

So tonight I’m just going to jot down a few ideas/observations/thoughts that don’t fit into the larger posts I feel compelled to write on (a) lesson study, (b) math talk, and (c) PCMI-as-a-learning-community [1].

0. There’s a really personable, funny person at PCMI who writes a blog that I’ve been to a few times — but for some reason wasn’t in my reader. It’s awesome. From his about page: “I’m a recently tenured college professor teaching mathematics at a high school during my sabbatical leave. I’m blogging about my experiences mainly to record my successes, frustrations, thoughts and feelings.” The best part: it’s concrete and on the ground and honest. And being new to high school, he makes observations of things we don’t always notice — or that we’ve forgotten (example here). So go back through the archives and drink up! Adventures in Teaching

1. When kids are working in groups, and you want to start having them wind down without the time pressure, you ask them to hold up a 1, 2, or 3 fingers to represent how many more minutes they need. And to make sure everyone is participating, if they don’t need any more time, they hold up a fist.

2. If you’re having a problem with student attending class on time, on certain days you can give raffle tickets to students if they are in their seat before class starts. When class starts, you select a raffle ticket to win a prize. Sometimes the prizes can be lame, sometimes they should be good. (They do this with US — adults — to get to the morning session on time. It works for me.)

3. Watching videos of teachers is powerful. I would love to have DVDs of good teachers teaching. Nothing else – no text, no explanations, just the videos for me to watch and mull over.

4. A group of teachers from Seattle came for a week and presented the work they’re doing with “Complex Instruction.” Part of their work was building a supportive and hard-working community. No easy task. One of the things they do is video tape their lessons and have discussions about the tapes. When talking with one of ’em, he said that the teacher on the tapes of him was not anything like the teacher he thought he was. Powerful, and scary. I asked more about how they set up a safe space for teachers to look at the tapes, and feel supported, and not defensive, it was clear they had to do a lot of work beforehand to make sure that happened. There were stages. But what struck me the most was the norms they had when viewing the tapes. The teachers teaching weren’t individual teachers. They were any and all teachers. They said more than a few times that “it could be any of them.” To emphasize this, they only referred to the teacher as “Teacher” — not by the name of the specific teacher teaching. Everyone saw themselves as working almost as one collective, one Teacher, working towards improvement of practice through these videos and discussions. Just like we say how blogs and twitter have changed our lives when it comes to teaching, they say videos have done that. Scary, but I want to do something like this with a few teachers I feel safe with.

5. Teacher moves. This is a term I’ve picked up here. I generally hate jargon. I am a philistine. But I have come to really embrace it. Because it gives a name to something that we do all the time. We are confronted with a situation (whether it be academic, behavioral, social, blah blah blah) and we have about 2 seconds to decide how we’re going to deal with it. What I’m learning is that although we all have our own set of teacher moves — most that come naturally to us — we can work on expanding our repertoire and honing these teacher moves. How? First, by talking to (or reading blogs of!) other teachers — to see other teacher moves. Second, by thinking through hypothetical situations that might arise in the classroom and anticipating how you’re going to respond to them. For me, just giving these things a name — “teacher moves” — let’s us start having conversations about improvement.

It is letting me see the whole class as a set of discrete teacher moves. Moves I make consciously and unconsciously. But by starting to conceptualize my time in the classroom as a string of discrete teacher moves, I can start thinking about things in a somewhat more concrete way. It let’s me focus more on my actions.

6. In my classroom, as I suspect in most classrooms across the country, the teacher is the sole authority of knowledge. What’s right and what’s wrong. It makes sense, of course. But I’ve been learning that this setup might not be the only approach. One of the most important things to instill in our kids is confidence in their abilities, and the ability to take intellectual risks. The way most math classes are structured, well that’s not so conducive for risk taking. And the teacher is the arbiter of knowledge — that’s right, that’s wrong. Students look to what the teacher writes on the board and starts to believe that only the teacher can be a math authority. I have no problem with the teacher being a math authority. But I’m starting to realize that in my classes, I am the math authority. I want kids to be math authorities too, and to view each other as valuable sources of information. It might be something to think about. Because I suspect that getting at this is also getting at independent learners. And getting at math confidence.

7. Me at the 4th of July parade with a bunch of math nerds! “We love math, how about you?” was the final two lines of our chant.

There are more things, but I’m tired. Sorry it isn’t so coherent.

[1] If I don’t write about these in a reasonable amount of time, feel free to harass me.

Blogotwitterversphere

I’m currently at PCMI in Utah (that’s the Park City Math Institute) for three weeks. There are about 50 middle and high school math teachers, all geeking out about teaching. Tomorrow I’m supposed to give a 5 minute talk about blogs and twitter. Little do they know it will be 7.5 minutes. Mwa hahaha. Talking about this stuff is not a big deal, and given a microphone and an internet enabled laptop, I could probably talk for a good hour. But to whittle away at my thoughts until I hit some core ideas that I can collapse into 5 minutes — that sounded like a fun challenge.

I decided to create a pecha kucha (20 powerpoint slides, 20 seconds per slide; see my favorite one here). Making one was new to me. And dang, it was hard. I failed. It turns out I didn’t get to 20 slides, and most have a little over 20 seconds of talking. So below you can watch the presentation that resulted out of the failed pecha kucha. Or, alternatively, the new style of presenting I like to call pechaka kuchaka.

Before diving right in and watching, I need you to watch one 18 second video. We couldn’t play it in the session for technological reasons.  So watch it. More enticement: there’s a BABY in it. The baby may or may not fly using magic.

Now for the presentation. Sorry about my voice. You’re not the only one who hates it.

Note this isn’t a post about how to blog or twitter. Or how to separate the wheat from the chaff when reading tweets and blog posts. Or how to not get overwhelmed with all the info out there in the blogotwitterversphere. This are just some of my current thoughts on some reasons why I do it.

I just want to share again Dan’s contribution to my presentation one last time, since it captures so much.

For those of you who are interested in these ideas and want to learn more about blogs and twitter, I’ve compiled a few links for you to explore more:

(0) My twitter page

(1) How to start your own blog (my thoughts, Kate’s thoughts, Elissa’s thoughts, Riley’s thoughts)

(2) My “Why Twitter” post.

(3) I made special note of The Moment when I started thinking of my tweeps (twitter buddies) as friends. Even though I don’t know them IRL [in real life].

(4) I save my favorite bits of twitter conversation, and aperiodically post them. I save more of the witty banter than the math substance stuff (which tends to get codified on peoples’s blogs). But you can see that we honestly do like each other a lot. Even though, again, I don’t know them IRL. To see these conversations, just look at the “FAVORITE TWEETS” page at the top of this blog. Or click here. If you’re looking to find some good peeps to follow, read these and pick the funniest ones. They’ll keep you going for days.

(5) If you want to see the blogs I read, just look on the right, at my blogroll. Some of them are defunct now, but I can’t quite delete them yet. The two most famous blogs are by far are Dan Meyer’s dy/dan and Kate Nowak’s f(t). (Apparently having mathematical notation in the title of your blog makes you an instant winner.) Our very own PCMIer Jesse Johnson has a blog (Math Be Brave) and Cal Armstrong does too (Things I Do).

(6) You can see all the blog posts that I find amazing here. It updates as I find more and more awesomeness.

(7) Some lists that people have made of math teachers on twitter are here and here.

Using a Cannon to Kill a Fly

Dan Meyer, in one of his recent WCYDWT, posted a picture of a roll of tickets (among other annuli).

So obvious is the question: how many tickets?

Of course more questions come tumbling out immediately, questions we need to ask to figure it out. What’s the inner and outer radius? What’s the length of one ticket? What’s the “thickness” of a ticket?

Dan sent me the information:

Inner Diameter: 27.77 mm
Outer Diameter: 168.65 mm
Length of a Ticket: 51.21 mm
Thickness of a Ticket*: 0.22 mm

The easy way to solve this is to find the area of the Annulus (the green area) and divide it by the area of the side of one ticket.

Of course the area of the red ticket is the “thickness” of the ticket multiplied by the length of the ticket.

I put the ticket on the side, lying flat on a table. So the height of that rectangle is the thickness of the ticket (exaggerated for dramatic effect) and the width of that rectangle is the length of the ticket. And you can see how that rectangle becomes curved when it is part of the roll.

Clearly if we want to find how many tickets can be curved to form the entire green roll, we’ll simply have to find the area of the green roll (the annulus) and divide it by the area of a single ticket.

Using Dan Meyer’s numbers, we get the number of tickets to be:

N=\frac{\pi(168.65/2)^2-\pi(27.77/2)^2}{51.21*0.22}

This comes out to be about 1929.07 tickets.

(The true answer is 2000 tickets).

But sometimes you want to kill a fly with something more powerful than a fly swatter. Something that will be slightly more challenging. Chopsticks? Been done. A cannon? Hard to find them lying around. Calculus? OF COURSE!

(Also, Dan asked if there was a way to solve this with Calculus.)

I had a couple ideas, but the most interesting for me was to try to come up with a function to model the tickets being rolled around a tube.** If I could find an equation which twists around like a roll of tickets, I could then use calculus’ arc length formula. If you don’t know what that is, it is pretty darn powerful. Given any normal function — curvy, straight, you name it — you can find how long the function is!

The idea behind it is the idea behind most calculus ideas. Break up the function into a bunch of pieces, and approximate the pieces with lines segments. Then add the lengths of those line segments together. You have an approximation for the true length of the function. If you break the function into more and more pieces, your approximation gets better. And if you start breaking the function into an almost infinite number of pieces and adding those line segments together, you’re going to get an almost perfect length. That’s calculus, and I’m not going to go into how you add an infinite number of line segments together here, but don’t be daunted. It sounds much more difficult than it is in reality.

Regardless, the whole point here is if we can find a function to model the tickets being wound around and around and around, we can just apply the calculus arc length formula and find the length of all the tickets if they were rolled out!

So the hard part about this problem is coming up with the equation to model the rolled up tickets. But it’s just a spiral. In particular, an Archimedean spiral.

In polar coordinates, the equation to get this spiral is the SUPER simple: r(\theta)=k\theta. You can’t get much simpler than that. It obviously takes 2\pi radians to make one full circle (to start and end at the same angle). And in that 2\pi radians, the spiral moves out from it’s previous location 2\pi k. That’s all we need. Seriously.

So first we’re going to model the ticket roll as if it does not have that hole in the center. It’ll first just be a spiral that will go on and on forever. We’ll then find where to start and stop this spiral to create the roll of tickets which starts at some inner radius and ends at some outer radius.

Look at that Archimedean spiral one more time. For our purposes, I want the distance between the red lines above to be the thickness of a single ticket. Does that make sense? That way the ticket’s are getting wound around and around and around, laying on top of one another.***

Remember that for each total revolution (2\pi), the spiral moves outwards 2\pi k  from it’s previous location. We want that distance it moves outwards to be the thickness of the ticket. So we simply set the thickness (0.22 mm) equal to 2\pi k to find that k \approx 0.035014.

HOLLA! We now have the function we’re using to model the tickets:

r(\theta)=0.035014\theta.

Dude, graph that on your TI-whatever and you’ll see a nice tight spiral.

Now to start and stop the spiral so it matches our roll of tickets. To do this, we need to graph it for only certain \theta values — so that we can get the spiral to start at the inner radius and stop at the outer radius. That’s just simple algebra, boys and girls! We’ll call the starting angle value \theta_{start} and the ending angle value \theta_{end}.

Since r(\theta)=0.035014\theta, we want to know what \theta is that will bring the spiral out to the inner circle. That will be our starting angle! So 27.77/2=0.035014\theta_{start}. Similarly we can set 168.65/2=0.035014\theta_{end}. The starting and ending angles are easily solved for, to get \theta_{start}=396.55566 and \theta_{end}=2408.32239.

Graphing the spiral with these endpoints looks like:

We’ve done all the hard parts! Now we simply use the arc length formula in calculus (for polar equations) to find the length of the spiral! Again, I’m not going to explain the derivation, but the equation for arc length is:

L=\int_{a}^{b} \sqrt{r^2+(\frac{dr}{d\theta})^2} d\theta

This gives us the length of a function, from one endpoint to another. Applying our equation for r(\theta) into this, we get:

L=\int_{396.55566}^{2408.32239} \sqrt{(0.035014\theta)^2+0.035014^2} d\theta.

Using the calculator to solve this, we get L=98787.8359  mm. That’s the whole length of the roll of tickets, if it the tickets were laid out flat instead of rolled up. So by simply dividing this by the length of a ticket (51.21 mm), we get the number of tickets.

This comes out to be 1929.07 tickets. That’s what we got with calculus.

It seemed a bit crazy to me that the answer without calculus would be identical to the answer with calculus. It took a few minutes of thinking, but then it dawned on me that it would be crazy if they were not identical.

I wrote to Dan Meyer, explaining that I got the same answer using calculus that he did without, and that ex post facto, of course it made sense. I said:

Once I saw that we got the same answer, it dawned on me that they SHOULD be the same answer. (Isn’t that always the way in math?) Why wouldn’t they be the same? We’re both using the same initial data (inner radius, outer radius, thickness, and ticket length), and we’re not approximating anywhere in our calculations (we’re being exact).****

So there you go.

I just love that this simple algebra solution and this calculus solution turn out to be the same. Instead of showing me how useless calculus is (why do we need it if we can do it without?), I can’t help but cogitate on how amazing calculus is! These two methods are completely different! One is geometry. The other is based in functions. But both solutions are reducible to the same thing. They are really the same thing.

Now if you’re a calculete and you’re really interested in finding an exact — and non-calculator based solution — to the arc length integral, I suggest you look at the bottom of this page. It’s not hard, but also not the point of this post. I wanted to make the general process clear to someone who might not know calculus, but interested in the idea of how we can use it to solve the same problem. Of course, it’s a bit silly, way more firepower than you need. But it really gets you thinking about the ideas undergirding calculus, and there’s where I see the value in doing explorations like these.

PS. If you’re wondering why we’re not getting 2000 tickets, I think the answer is simple. In the initial data (the inner radius, the outer radius, the length of a ticket, and the thickness of a ticket), there was probably some error. I suspect it was with the thickness of the ticket — the hardest thing to actually measure. If the thickness of aticket was found to be 0.2124 mm (instead of 0.22 mm), then we would get precisely 2000 tickets! So if Dan were able to measure to the hundredths or thousandths accurately, I bet we would get something way closer to 2000 tickets.

—

*I seem to recall that Kate Nowak Jason Dyer suggested removing a bunch of tickets (50?) from the roll, stack them up, and measure the height. Use that to get the approximate thickness.

**Of course concentric circles would be possible, but then you’d get a summation, and blah blah blah. It all seems more precalculus than calculus.

***If that doesn’t make sense, try looking at this horrible picture of a shrimp. I mean ticket roll.

**** If you say I rounded to 5 or 6 decimal places a few times, guilty. However, I actually calculated it without any rounding, and it comes out to be the same value. So there. Pfft. The rounding didn’t affect the answer to the hundredths place.

I’ll be in and out this summer

The title says it all. Since school has let out, I’ve wanted to do very little thinking about teaching. I’m in my time-to-vegetate-and-watch-tv-and-read phase. I don’t really want to think about planning for next year … yet. That’ll come, though, once I get sick of vegetating, which happens.

I will, however, at the end of this week be traveling to Utah for a three week math workshop thingie. I’m sure it’ll be fun and I’ll have lots to talk about on here. One of the things I’m doing there is taking a workshop on Japanese Lesson Study, which I’ve been intrigued by since I first heard about them a couple years ago.

So I’m not dead yet, but I’m not going to posting super regularly this summer. I’ll do it when I’m inspired, have thought of something, or whatever.

Always,

Sam

Join in the 4 photos fun

So Kate Nowak and I created a site where we are going to post the 4 photos idea. You know, the idea that by seeing someone’s fridge, wallet, bookshelves, and computer desktop, you can decide whether you want to date ’em. Or hang out with ’em. Because, we aver, you probably can tell so much about a person, their interests, their personalities, their habits, they’re style, from these pics.

My embarrassing photos are here. (Click each photo.) Clearly you know I’m totes awesome. You know…  those take out rice containers in the fridge, the red plastic lobster on the bookshelf… WINNER!

So yeah, if you want to be included in this little adventure, just email your four (or more) photos to 4photos4photos@gmail.com. Then Kate or I will put them on a page. More likely Kate.