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Foreshadowing with compass and straightedge

March 22, 2009 am31 1:34 am

Teachers often teach, we are frequently told, when they are not certain what to do, in the way that they were taught.

But yours truly never was taught construction, and has nothing to fall back on. So I invent.

This term I have an off-track advanced class. They started geometry in February, and will take the state exam (Regents! yuch) next January.  Two weeks of logic (the kind I like), and onward…

I decided to teach construction from the first week. To introduce something new, or synthesize something they’d already done. Four weeks, we’ve killed full periods twice, half periods twice. The kiddies like the constructions, a lot.

Next week, do you think a kid will propose a good construction of a parallel line?

Each period I show them something new. Something small. And then assign a more involved construction that practices the skill (mini-project). Some weeks they get a choice. For example, first week, we learned to take a given radius and point, and construct circles. And they practiced and practiced. And after three weeks even the kids who could not get a decent circle are fairly good. Anyhow, the first week they did circles and had three choices: some kids just made pictures with the circles (pig’s faces), or concentric circles with equal distance between the circles, or an interlocking circle pattern (of my own invention, though it must be common – pick a point on the circle, that is the new center of a new circle, constant radius, new points of intersection become new centers, etc.)

Jump ahead, yesterday I teach congruent angles. And I want everyone working on the same mini-project. I’ll give you letters, though they didn’t have them. Start with \overline{ABCDE}, with AB = BC = CD.  Next, \overrightarrow{AZ} and then construct \overrightarrow{BY} so that \angle{ZAB} = \angle {YBC} and so that Y and Z are on opposite sides of \overline{ABCDE} . Next, \overrightarrow{CX} , on the same side as \overrightarrow{AZ} , so \angle{XCD} = \angle{ZAB} . Finally, one more on the opposite side, so that we have a twig with alternate branches.

In the diagram at right, A, is sort of close:

Anyhow, a question for you. Do you think, next week, when I ask for a parallel line construction (and we’ll have studied corresponding angles, alt int, etc, in the interim) do you think a kid will propose the standard construction? I will likely start with a helping device, perhaps “Given \angle{ABC} construct a line through C parallel to \overline{AB}

I think it is 50 – 50, but if it happens, that would be a bit more positive feedback for this… It would be really cool if, having been led to the vicinity, some students made the leap on their own. I already have them highly motivated by the hands on activity, which has value. I’d like some evidence that they are learning a) more, or b) better, or c) deeper.

5 Comments leave one →
  1. March 22, 2009 pm31 7:06 pm 7:06 pm

    Do they already know about angles made by parallel lines and a transversal? If so I’d think it’s more likely. But either way it will be interesting to see what they come up with.

    • March 22, 2009 pm31 7:12 pm 7:12 pm

      “Parallel” in general was Thursday, which is why I was a little surprised none of the kids mentioned the parallel lines in the construction on Friday. Then again, to this point, there really has been no link between our regular class work and the constructions.

      Monday we will deal with the parallel postulate and some corollaries. I like the history, since our book is so far away from Euclid. Tuesday we’ll be on to algebraic stuff with transversals, and we will deal with sums of angles in triangles (perhaps polygons as well) before we come back to the construction.

      So it is possible. But there is lots intervening as well. We’ll see. If a few kids figure it out, way cool. But even if they don’t, I’ll show them, and they are prepared to successfully complete this most annoying construction (annoying, or difficult, because it is indirect)

    • March 30, 2009 am31 12:53 am 12:53 am

      I had them start with 3 points, pass to their neighbors, label A, B, C, pass, construct angle BAC, and then call me over when their whole table (three kids) was ready. At that point I told them to construct a line through A parallel to BC.

      A few tables eventually got it, but my biggest surprise was the first group. One girl, without a pause, said “You make congruent angles here and here (motioning, correctly, with compass). What else should we do?”

      I asked her lots of questions afterwards… but she really had not looked anything up (I didn’t tell them what construction was coming next), and she really had never done constructions in a previous grade.


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