Thursday, August 19, 2010

Vectors and Wait Time

I've been working with my (regular) Precalculus kids for a full week on vectors. Since I didn't want to teach them just rote algebra procedures, I made them do everything graphically. Adding and subtracting vectors graphically, scaling graphically, etc. As predicted, some kids figured out for themselves that <a, b> + <c, d> = <a + c, b + d>, and I told them that it's all good to use this algebraic method to double-check their graphing work. But, I'm pretty pleased that...
  1. Kids were able to answer this question using only graphical addition: List 5 vectors whose sum is <-3, 4>.

  2. When I wanted to introduce the graphical subtraction of vectors, I first gave the kids some diagrams of pairs of vectors to look at and I asked them to choose a pair that represented a vector and its "negative." Almost every kid got it right! They figured out for themselves that vectors of equal magnitudes and pointing in opposite directions represent "negatives" of one another. And we discussed how V + (-V) = <0, 0>, just like how 5 + (-5) = 0. That was pretty neat, because it was a tiny reference to set theory. (Or some such goodness in higher math.)


And I am truly amazed by how great these juniors have been in class. We spent a good chunk of class today working on using protractors and rulers to carefully duplicate vectors, and I was fully expecting them to throw frustration tantrums because of the precision I required. But, they were super! I am not holding my breath that this will remain the case, but I'm certainly doing my best to keep the class extremely smooth-running to avoid the 3-week-in rebellion. (I won't lie; it also helps to have a handful of kids from my honors class last year now in regular Precalc with me. I still think it's extremely silly of them to drop out of honors, but given that they're taking many other honors classes, I guess I can sympathize. MAYBE.)

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On the Geometry side, I had a really great experience with the shapes puzzle I had posted a link to. Kids were talking to each other using geometric vocabulary; they wanted more time on the assignment before discussing solutions as a class... I actually felt bad when I cut them off at 15-20 minutes of individual efforts struggling with the puzzle, because I could tell that the kids were really trying to figure it out on their own.

It was awesome. It made me think again about the issue of wait time. Giving kids just 5 more minutes on a thoughtful assignment sometimes makes such a huge difference.*

*Note: I think the issue of wait time is intrinsically tied with the timing and type of hints you give. For example, my first regular Geometry period felt significantly more frustrated on the assignment, because I gave them no hints while working. My second period (also regular Geometry), started to make visible breakthrough when I said after a good 15 minutes of individual efforts: "By the way, here is a hint: everytime you tag on a triangle to a shape you have already found, you end up changing the number of sides in that polygon."

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Geoff comes back this weekend. :) I'm so excited!!!

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