Monday, October 29, 2012

How Bad Is Your Line?

This is ripped right off from PCMI 2011 Day 11. But I'm liking the Geogebra I made for it to calculate the sum of squares. Enjoy!

Regression Instructions on Nspire and 84

I have kids with both models of calculators, and I find that just giving them print instructions to work through at their own pace is the most efficient way for them to learn button-press procedures. It's either that or a live demonstration, and then you have to wait for everyone to keep up with you, and they don't have something they can look at later. On top of that, half the kids are not fluent English speakers, so I try not to make verbal instructions the only mode of communication, ever. The point of this is not for them to learn any math. We're doing other stuff for that. (So lay off about the inauthenticity of this task, please.) It's purely to learn the button presses. I spent way too much time making these, so please, take them so you don't have to duplicate my efforts. Also, disclaimer, these assume your handhelds have the latest operating systems. Which are way better than previous versions, so update yo' shit, people.

Nspire Regression Instructions (docx)

Nspire Regression Instructions (pdf)

84 Regression Instructions (docx)

84 Regression Instructions (pdf)

Monday, October 15, 2012

Why Algebra

I've been taking notice more this year with all the Why Algebra back-and-forth of how indispensable it is for learning Geometry. I know I'm not going to win any converts by answering "why Algebra?" with "because Geometry" but those about to rage quit this post are probably not going to be persuaded by a blog post anyway.

I don't mean the tepid "Algebra!" problems with the snazzy xy logo the textbook offers. I assign some of these, don't get me wrong, but if this is the sum total of the algebra used in your geometry course, you're doing it wrong.



I mean in the process of exploring how measurements on a plane relate to each other, algebra is a weapon the kids should be deploying like on the daily, in the cycle/ladder of examples, conjecture, prove, extend. Kids' resistance to this leads me to believe they aren't often asked to do this.

Expressing your conjecture. Here are some examples.





The resistance at first will be formidable. They will truck along obligingly until the last part, which they will leave blank and wait for someone to tell them what to put. I have to restate just what I want them to do a bunch of different ways, and ignore that they are pleading for me to just do it for them with their pleady little faces, and have the patience to wait them out. "Okay you saw examples of what results when angle B is 36, 48, 55 degrees. What about any angle of x degrees?" "5 sides 3 triangles, 6 sides 4 triangles, 7 sides 5 triangles. n sides ??? Triangles?" "how did you turn a 25 into a 130, and a 41 into a 98?" They will look at you like you are a crazy person. Meet this with incredulity. "You did take and pass an algebra course last year. Yes?" They will look at you with a face all dark clouds that says, "you bitch." Meet this with unrestrained confidence. "You, you can do this. Take a breath and focus on it for a minute. I wouldn't ask you to do something I didn't know you can do." They will. After the first few it gets easier.

Using Algebra to Prove Things
It's a major way we can know something is always true. "Is that always true? How do you know?" are sentences I probably utter in my sleep by now.

Here are some examples





The lovely part about using Algebra to make sense of Geometry is it offers context to hang your algebra on. In that last example, Mat originally ended up with e = -360 because he forgot to distribute the negative. He knew something was wrong, so I asked him to write it on the board so we could help. One of his classmates spotted the error in short order. Sometimes the stuff from the previous course doesn't gel until you have to use it in the next course.

I guess my point is, maybe "Why Algebra?" is the wrong question. If we agree that Mathematics (actual Mathematics involving logical, abstract ways of thinking about how quantities and measurements fit together, generalization, observing and using patterns, making predictions) is a valuable thing for an educated person to learn about, the question seems kind of silly. Of course Algebra. But, maybe we don't all agree with that. Or maybe we should talk about whether the content of Algebra 2 and beyond is valuable for everyone to learn about. Which the CCSS seems to have decided "yes" without consulting anybody. I guess I don't think the question has been very well defined.

Friday, October 12, 2012

Hours of Entertainment (Pew pew!)

Hey did you know underclassmen are almost as easy to entertain with laser pointers as kittens? It's true.

This challenge has had them going on and off for hours.

Hold this:


And move your body from one side of this board to the other:



while keeping the lasers on the stars. (There is a green Expo-marker star drawn on each side of the board.)

Other rules:
  • no changing the angle
  • hold the vertex against your sternum
  • always face the board, and no one stands between you and the board (safety, you know.)
A few of them are getting pretty good at it, so we appointed another kid to trace his path with chalk on the floor.

The children. They have some questions.

I know there are boring ways to get this point across with paper and pencil, but LASERS. THAT'S WHY.

Update: David Petersen made a Geogebra file to illustrate what is happening.


Wednesday, October 10, 2012

You Can Have My Compass and Straight Edge

when you pry them out of my cold, dead hands.

I apologize in advance that I'm going to get a little critical of people I don't know who are trying to do a good thing, and are probably very nice. This landed in my inbox today from someone who offhandedly described it as "cute" with no further commentary.



"How do we know something is true?" is a big, maybe The Big Question in Geometry. At least, in my course. I hope in yours, too. It's a big, bad, fun, important idea.

Don't get me wrong, the makers of this video did a very slick job with it. It is very, very well done. But I don't get the point of cutesy-ing up exposition on the topic. When is a learner supposed to watch it? Before or after they have looked at a bunch of examples of something and made a conjecture and paused to wonder if that thing always has to be true, and just how they can go about knowing that? Before or after they encounter a surprising consequence of a ho-hum construction? I really, really hope this isn't any learner's introduction to what proof is for. They need to get their brains in the weeds of puzzles they can't leave alone. They need to get their hands dirty. Please, teachers of Geometry. I am begging you, here.

I suppose maybe I'd show it after. Like, way after. Months from now. It is pretty cute. Maybe it will help snap into place some ideas they will have knocking around in their heads. But my prediction is it will not hold their attention.