Month: January 2013

Guest Post: Conics Project

This is a guest post from my friend Liz Wolf.

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“The conics section comes at a tough time in our curriculum.  It’s a few weeks after Spring Break, and kids are always antsy in class and have major spring fever.  I wanted a way to make conics less abstract and show the kids how often they come up in every day life.  I came up with a project that not only got them outside, but also got them looking at things in a different way.  The photo of the water droplet on the swing set was my favorite.  The students really embraced this and I was impressed with how well they embraced GeoGebra having never used it before.”

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Below are some examples of final products from her class, and the instruction sheet she used.

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Update: Liz sent me her Geogebra instruction sheet!

Swampped

This is one of those days where the one good thing was hard to find. I had to reprimand one class for not doing their work at home, I am having to deal with juggling because a ton of students have been out, I’m dealing with a lot of this and that, and I have a HUGE amount on my plate for this weekend. Like: impossibly large amount of stuff to do. So my anxiety is through the roof. And, yes, I have that tickle in the back of my throat which could mean something or it could mean nothing.

But that’s precisely why we need this blog. So I’m going to post some of the small good things that happened. None of them were GOOD (like, enough to undo my stress) but they were positive.

(1) Another multivariable calculus student turned in her aweeeeesome 3D function which is aweeesome possum…Image

 

(2) Some kids were really excited about showing me some of the stuff they did for their roller coasters in their calculus projects (which were due today).

(3) I took over a colleagues precalculus class while she took my multivariable calc kids (and one of her classes) to the Museum of Mathematics… and her kids were working on the same project mine are (the family of curves project). Her kids were soooo into it and were coming up with some stunning, beyond stunning in fact, pieces of artwork.

(4) A former kid who served for all my years on the SFJC came back to visit and we caught up after school. It was nice to hear what exciting things he has planned in the next five months!

That’s about it. When I’m overwhelmed and overextended, and when a lot of kids have their own things I’m dealing with, I can’t appreciate these small moments. So I am glad I took the time to force myself to think of these small moments, in a sea of mediocre ones.

Families of Curves #2

So today I started the Family of Curves project in Precalculus. Students are going to be given three in class days to work on this, and about a week or two of out-of-class to finish it on their own.

I started class showing around 4 or 5 minutes of this Vi Hart video with no introduction:

Then I showed a whole bunch of pictures… of tessellations, Escher prints, one of the things they were going to be creating on geogebra [but without telling them it was not a famous artist], and a few beautiful prints and the website for Geometry Daily.

Then I had them take out their laptops, and just get started working on Geogebra. The packet below takes them through the sequence command, and then shows them how the sequence command can create a family of curves…

Here’s the instructions getting kids started on Geogebra and what’s expected of them…

[.docx]

Note: My kids are getting more and more fluent with Geogebra… We have been using it on-and-off all year at various times.

They were silently working the entire class. I put on some music, and they started talking a bit. But since it’s an individual project, I suppose I can’t expect a lot of talking. Some kids have been asking me “how do you make circles?” and one student asked me how to fill in circles…

It took them pretty much the whole day today to do the geogebra introductory stuffs, so they didn’t all get to play around with their own functions. I expect tomorrow will be pretty awesome to watch them tinker and explore, and get cool things.

I don’t know if they are “into” this yet. I’ll see if I get any anecdotal evidence tomorrow.

One Good Thing

A short post:

For those of you who don’t know, Rachel Kernodle (@rdkpickle, http://sonatamathematique.wordpress.com/) has started a group blog called “one good thing.” She wrote about it here, and you can visit the blog here.

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The idea is that even in the most frustratingly upsetting days as teachers, there is at least one good thing that happens — as long as you keep your eyes open to it. We may feel we suck, we may get all arrrrgh at students, a lot of random stress can take over and fill us with anxiety… and we get our blinders on, and lose sight of the bigger picture. Looking for one good thing each day helps us see the bigger picture when our vision narrows. And it also helps us archive the little moments, which are oh so important!

Right now there are about 7 authors posting regularly. This is one of the many projects that math teachers have going on (others are here)! I know Rachel wants to invite others who want to contribute regularly or semi-regularly to join in (it’s not an exclusive club!) — so she said you can throw your email in the comments here in the next couple days and she’ll add you as an author to the blog. Or you can tweet her to get added or find out more information. That simple!

What’s nice is this blog will soon be populated with a million little stories from a bunch of (math) teachers all around the world. A beautiful pastiche of why we teach, with concrete, on-the-ground examples.

(My entries on the “one good thing” blog are archived here.)

Families of Curves

When I put out my call for help with Project Based Learning, I got a wonderful email from @gelada (a.k.a. Edmund Harriss of the blog Maxwell’s Demon) with a few things he’s done in his classes. And he — I am crossing my fingers tight — is going to put those online at some point for everyone. To just give you a taste of how awesome he is, I will just say that he was in NYC a few years ago and agreed to talk to my classes about what it’s like to be a real mathematician (“like, does a mathematician just like sit in a like room all day and like solve problems?”), and have kids think about and build aperiodic tilings of the plane.

Anyway, he sent me something about families of curves, and that got my brain thinking about how I could incorporate this in my precalculus class. Students studied function transformations last year in Algebra II, and we reviewed them and applied them to trig functions. But I kinda want to have kids have some fun and make some mathematical art.

First off, I should say what a family of curves is.

family

That’s from Wikipedia. A simple family of curves might be y=kx which generates all the lines that go through the origin except for the vertical line.

lines

I made this in Geogebra with one command:

Sequence[k*x, k, -10, 10, 0.5]

This tells geogebra to graph y=kx for all values of k from -10 to 10, increasing each time by 0.5.

Okay, pretty, but not stunning. Let’s mix things up a bit.

lines 2

Sequence[k*x+k^2, k, -10, 10, 0.1]

Much prettier! And it came about by a simple modification of the geogebra command. Now for lines with a steep slope, they are also shifted upwards by k^2. This picture is beautiful, and gives rise to the question: is that whitespace at the bottom a parabola?

Another one?

sin

Sequence[1/k*sin(k*x),k,-10,10,0.2]

And finally, just one more…

tan

Sequence[1/k*tan(x)+k,k,-10,10,0.1]

Just kidding! I can’t stop! One more!

sec

Sequence[k sec(x)+(1/k)*x,k,-10,10,0.25]

What I like about these pictures is…

THEY ARE PRETTY

THEY ARE FUN TO MAKE

THEY ARE SUPER EASY TO MAKE & TINKER WITH

THEY MAKE ME WANT TO MAKE MOAR AND MOAR AND MOAR

And then, if you’re me, they raise some questions… Why do they look like they do? What is common to all the curves (if anything)? Does something special happen when k switches from negative to positive? What if I expanded the range of k values? What if I plotted the family of curves but with an infinite number of k values? Do the edges form a curve I can find? Can I make a prettier one? Can I change the coloring so that I have more than one color? What would happen if I added a second parameter into the mix? What if I didn’t vary k by a fixed amount, but I created a sequence of values for k instead? Why do some of them look three-dimensional? On a scale of 1 to awesomesauce, how amazingly fun is this?

You know what else is cool? You can just plot individual curves instead of the family of curves, and vary the parameter using a slider. Geogebra is awesome. Look at this .gif I created which shows the curves for the graph of the tangent function above… It really makes plain what’s going on… (click the image to see the .gif animate!)

sec animation

Okay, so I’m not exactly sure what I’m going to do with this… but here’s what I’ve been mulling over. My kids know how to use geogebra. They are fairly independent. And I don’t want to “ruin” this by putting too much structure on it. So here’s where I’m at.

We’re going to make a mathematical art gallery involving families of curves.

1. Each student submits three pieces to the gallery.

2. Each piece must be a family of curves with a parameter being varied — but causing at least two transformations (so y=kx^2 won’t count because it just involves a vertical stretch, but y=k(x-k)^2 would be allowed because there is a vertical stretch and horizontal shift).

3. At least one of the three pieces must involve the trig function(s) we’ve learned this year.

4. The art pieces must be beautiful… colors, number of curves in the family of curves, range for the parameter, etc., must be carefully chosen.

Additionally, accompanying each piece must be a little artists statement, which:

0. Has the title of the piece

1. States what is going on with each curve which allows the whole family of curves to look the way they do, making specific reference to function transformations.

2. Has some plots of some of individual curves in the family of curves to illustrate the writing they’ll be doing.

3. Has a list of things they notice about the graph and things they wonder about the graph.

At the end, I’ll photocopy the pieces onto cardstock and make a gallery in the room — but without the artist’s names displayed. I’ll give each student 5 stickers and they’ll put their stickers next to the pieces they like the most (that are not their own). I’ll invite the math department, the head of the upper school, and other faculty to do the same. The family of curves with the most stickers will win something — like a small prize, and for me to blow their artwork into a real poster that we display at the school somewhere. And hopefully the creme de la creme of these pieces can be submitted to the math-science journal that I’m starting this year.

Right now, I have a really good feeling about this. It’s low key. I can introduce it to them in half a class, and give them the rest of that class to continue working on it. I can give them a couple weeks of their own time to work on it (not using class time). And by trying to suss out the family of curves and why it looks the way it does, it forces them to think about function transformations (along with a bunch of reflections!) in a slightly deeper way. It’s not intense, and I’ll make it simple to grade and to do well on, but I think that’s the way to do it.

What’s also nice is when we get to conic sections, I can wow them by sharing that all conics are generated by r(\theta)=\frac{k}{1+k\cos(\theta)}. In other words, conic sections all can be generated by a single equation, and just varying the parameter k. Nice, huh?

PS. Since I am not going to do this for a few weeks, let me know if you have any additional ideas/thoughts to improve things!