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My class did well constructing parallel lines last Friday. (I asked them to perform the construction without telling them how) Several students figured out how to complete the construction without direction, and most did fine with some direction (although we need to keep working on accuracy). One student, on being asked to construct a parallel through a given point, replied without a moment’s hesitation “make these two corresponding angles” (motioning with her compass) “what else are we going to do?”

So this week, I changed strategy on introducing triangle congruence postulate/theorems. We opened by reviewing what “congruent” means in relation to polygons, and then I had them take out compass and striaghtedge, sketch triangles, pass them to their neighbors, and I asked the neighbor to construct a congruent triangle. Now, there was some fussing. I shrugged my shoulders when asked what to do, and indicated they could discuss with their neighbors.

Now, this did use some class time. We have some confident constructors who finished in a few moments. They talked with their neighbors, one got up and moved around. But we also have some timid constructors, a bit afraid of making marks on their papers unless they are certain they are correct. So I allowed a solid 5+ minutes, at which point three quarters or so were complete, but before complete frustration had set in for the others.

And I put a triangle on the board, and called on a student. First kid described SSS. I noted that he had constructed 3 congruent sides, and no angles. I asked students who had used that method to pick a pair of corresponding angles, and use the compass to measure them. Are they close to congruent? (positive mumbles)

Did someone do something else? Next kid described SAS (though I was surprised that she started with the angle.) Write them up as “reasons to be used in proof” “If SSS ≅ SSS, then the Δs are ≅” and “If SAS ≅ SAS, then the Δs are ≅” Brief discussion of postulate vs theorem. American books treat both as postulates. Some treat SSS alone as a postulate. Euclid picked another as his postulate. In a sense, it doesn’t matter, once one is postulated, the others can be proven (though the proofs may be challenging)

Something else? One kid in the class tried ASA. Oops, one kid tried SSA. But look, the compasses were out… I set up an easy triangle to copy, where the obtuse and acute cases were easy to find.

We ended up with less practice time than I would normally have with this lesson. But we got to everything we needed, there is good practice in the homework, and that hands-on constructing, I have a feeling that it is helping to make the learning much more solid.

10 Comments leave one →
1. Kandi permalink
April 2, 2009 pm30 7:45 pm 7:45 pm

What a great way to include interactive learning in your classroom. I believe students will use their strengths in different ways when addressed with a problem. I find it interesting that some students were even willing to assist others throughout the room. Did you find that some students received no help at all? I am also curious whether some students choose to not ask or refused help? This is an intriguing concept in math instruction.

2. April 3, 2009 am30 1:13 am 1:13 am

you lucky dog you. if i’ve ever taught constructions
to *anyone*, it’ll have been back in public school, to
some student who wasn’t getting it from the teacher.
this is some of the coolest math there is at beginner level
and its an awful shame so many kids never see it at all.

looks like you’ve got the “get out of the way
and let the students work” thing down real well.
(which is of course what i *really* envy.
i struggle with this every quarter with
results that vary pretty wildly…)

• April 3, 2009 am30 8:13 am 8:13 am

Thanks for the kind words. I pay extra attention here, because I am teaching away from my strength. I still talk a lot in class – but finding moments to just watch, to offer individual help – that’s good. And I have a senior who comes in and helps the kiddies. They like that. And she does too. And it gives me more opportunities to watch and listen. It’s actually helpful, diagnosing what’s going wrong, to listen as a third party and not be involved trying to explain at that moment.

3. April 3, 2009 am30 3:43 am 3:43 am

I love constructions. I’m teaching Geometry right now, and I think the students like it best when I give them a problem to start off class (e.g. plot three points and then construct a circle through those points).

Incidentally, Euclid didn’t use any of the triangle congruences as postulates, though he proved SAS early on. [The book I’m using assumes just 5 postulates, but then adds two more than he didn’t state.] We’ve been working our way through Book I of Euclid and it’s been really neat (though slow) to see how much you can prove starting from almost nothing!

• April 3, 2009 am30 8:10 am 8:10 am

Of course we don’t actually follow Euclid in American high schools. One of the major differences: we use quite a bit of algebra in developing theorems.

He didn’t postulate SAS? Hm. That was the tough one. But a remember having a bear of a time with SSS (took a good geometry course – undergrad level, but as a grad student).

I believe that “between” and “on” and perhaps “congruent” need to be added to our list of undefined terms. At least that’s what I do. And my kids like challenging the book – imprecise, mostly. But it is another hook to get them involved. A kid reworded a theorem last week and was rightfully proud (… no more than one right or obtuse angle – changed to — at least two acute angles). Good stuff, I think.

I’m having fun, and in the course I like least. That’s good.

4. April 4, 2009 pm30 9:58 pm 9:58 pm

The more constructions the better, though I think the real question is what kind of questions do you add and where do you step back? Sometimes I ruin the magic by showing all the steps. I think kids need some basic construction lingo like: compass settings can copy lengths, and angles, perpendiculars etc. They should be challenged to define this stuff on their own as well. I like the precision and the deeper sense you get from making a construction. I actually had a good application for constructions a few months ago. I have a “study hall” period where kids do HW but if they have none I encourage them to play chess – problem was right before winter break nobody had any homework and I only had 2 chess sets. A chess board drawn with a ruler/protractor is ok, but a chess board constructed is much better/cooler. Might be a fun activity for kids who like constructions.

• April 5, 2009 am30 1:58 am 1:58 am

My high school yearbook has a photo of me and my friend David, cafeteria table, pieces between us, playing on a pencil-drawn board. I am, it turns out, in a lost position.

Very cool, to use construction. And as they get fast, a sharp, constructed board really takes barely more time than a freehand mess.

5. April 5, 2009 am30 2:24 am 2:24 am

hilbert’s axioms famously include
some about “between-ness”.
evidently the greatest mathematician
of the last-but-one turn of the century
agrees with you in perceiving a gap
in euclid here.

6. April 5, 2009 am30 2:44 am 2:44 am

via the Mathematical Geneaology Project I learn that Hilbert’s advisor’s advisor (Klein) was Whitehead’s advisor (4 generations down) who was my professor’s advisor (4 more generations forward)

And that’s the professor who told me about “on” and “between” and “congruent” though he uses a fancier word for “on.” He also has partial or majority credit, depending on the source, for closing one of Hilbert’s problems. Woot woot.