Constructing Parallelograms

This is a reflection of a talk I gave today. Hopefully these reflections (a) help me give better talks, and (b) help out some others.

Today I was worked from 6PM-8PM with the Intermediate group at the Berkeley Math Circle, middle school students maybe one or two standard deviations above the average honors student. My talk today was “All you have to do is construct a parallelogram!”.  Here is a link to the handout problems and their solutions. (Obviously I only went over a very proper subset of the problems during the lecture.)


Some background information: I had actually given an abridged version of the lecture to the honors geometry class at my Horner Junior High (discussing only 1,2,4,10). It had gone, as far as I could tell, very well. The HJH students audibly reacted as I completed the (short) solutions to their problems, meaning they not only understood the solutions but also could see them clearly enough to appreciate their elegance. A lot of students also thanked me after the lecture, and one of them asked for copies of my notes and told me I gave the best lectures. (Yay!)

The success of this lecture led me to adopt it for the Circle. The changes I made were:

  1. Add more problems (duh).
  2. Give more depth to the explanations, e.g. trying to explain motivations behind steps.
  3. Give students substantial time to try the problems themselves. Because IMO it spoils the fun for me to just keep giving solutions.

I plotted out in my head what direction the lecture would lead to. The most memorable part, I decided, would be problem #10 (by far my favorite on the set — try it!), and so I made a note to myself to end with that one. This turned out to be a good decision.


I started out by asking the students to name some properties of parallelograms. The important ones came out — parallel sides, congruent sides, and bisection of diagonals, along with some others I didn’t think of. I got a lot of people trying to answer this one. So far so good.

I then asked what conditions listed above were sufficient to prove a parallelogram. Again I got the two I wanted: two sides both parallel and congruent, and the bisection of diagonals. I then explained that we were going to be using the bisection criteria a lot.  I proceeded to present problem 1, asking the students for ideas, and was pleasantly surprised to end up with a solution that I hadn’t thought of. After congratulating the person who found the solution, I proceeded to present my solution, and tried to explain the idea of reflecting over midpoints to construct parallelograms.

I then let the students work for about 10 minutes on #2, #3, #4. Unfortunately, this is where the lecture started turning downhill. I walked around the room to see if there were any questions and to see what the students were coming up with. To my surprise, I found that a large portion of the students (more than half) were not doing the problems I asked them to do. I usually don’t mind if a few students aren’t interested, but the number of them really surprised me.

The second unpleasant surprise came when I began presenting the solution to #2. No one solved it in the 10 minutes, despite me having just told them to reflect points over midpoints (and there was only one possible reflection) — or if someone did, he/she didn’t saying anything. In retrospect I definitely should have written this on the board. Fortunately, I do coax the answer out of them by asking the right questions, but I now started to worry that the difficulty was too hard — ironic since I was initially worried about it being too easy.

The lack of solves on #2 means no one attempted #3 either (I also asked), so I am starting on a blank slate. Fortunately, this one turned out to be not as hard. The students quickly identify the good point and the steps towards the solution. #4 did not work as well, but this is mostly my fault because I made a mistake while presenting the solution and confused the students (and also kept mislabelling points). I did not catch it until the end.  After the solution to #4 I am a bit nervous, as the students have been in general rather quiet and were not reacting at all to the solutions (unlike the Horner students). I briefly ask the students whether I am making sense, if I am going too fast/slow, etc. (Of course, I know from experience that this is about as useful as asking someone what he/she wants for Christmas, but I cannot help doing so anyways.)

At this point it is 6:55 or so, and we have our 10-minute break. Herein, one student asks me for help with word problems — I tell him to chat with me after break (he never did). The other asks me if I am v_Enhance on AoPS. I guess my reputation precedes me.

After the break we present monthly contest awards (guess who’s the grader?). I then give the students a substantial amount of time (20 mins or so) to work on #6, #7, #10, #11. The ones that did seem to be working seemed mostly interested in #10 (good!). However, as before, less than half the students were actually doing the problems. The others are chatting quietly or drawing in their notebooks. Again I say nothing, but at this point I am slightly discouraged.

At 7:30PM I regroup and begin discussing #6. This one went very badly. The students seemed confused in general, and mostly fail to answer my questions. It is at this point I inadvertently learn that a few students do not actually know the similarity theorem for triangles (SAS~, SSS~, AA). By the time I finish, 20 minutes have passed (leaving 10 left) and the students seem tired.

At this point I start with #10, which is met with some enthusiasm as quite a few students are eager to see a solution. I start by asking the students to randomly guess where to place the fourth vertex until they get it right. (There are only \binom 52 \cdot 3 = 30 possible reflections, after all… :P). Nicely, the correct construction comes out on the fourth guess. :) I then proceed to explain the solution, which seems to be making much more sense. The timing works out great, and I deliver the punch line of the solution just as the lecture ends, to quite some applause. (You’ll have to see the solution to understand why!) Among the murmuring afterwards I hear lots of students saying that it was really good choice for ending. Quite a few students also thank me for the lecture afterwards (something that is actually quite rare), so it looks like these last ten minutes were more successful than anything else.


The thing that surprised me the most was the drawing in notebooks. I had assumed that the BMC students would be more motivated/interested than the kids at HJH. It turned out this was not the case. In retrospect, here are perhaps a few explanations why.

  • At a public school, the type of lecture I gave is really rare and stands out. In contrast, the BMC students regularly get good lectures on good mathematics, so it was harder for me to stand out.
  • I didn’t have the position of authority a classroom teacher has at BMC.
  • The HJH honors geometry students are actually very strong, as these are precisely the students that skipped a year of math. Many, if not most, are also my students in math club.
  • In that vein, many of the HJH students already know me well, from math club or otherwise.
  • The students who would have been the top students in Intermediate were long promoted to the Advanced group.

Here are some other thoughts.

  • I saw a lot of students staring at the diagrams I had provided on the page and not doing anything else. I wonder if next time I should omit diagrams from my geometry lectures, forcing the students to draw the diagrams themselves.
  • USAMO problems seem too hard under any circumstances. Even when the phrase “this is not a very hard geometry problem” appears in the rubric, and with significant hints.
  • I should have asked about the similarity. I kind of assumed everyone knew SSS~, SAS~, and AA.  This was false.

A couple more positive ones.

  • A couple times in the lecture I would ask the students for a Greek letter to name an angle. This actually drew a lot of attention and seemed to help with keeping the students awake; I should try this more in the future.
  • The #10 at the end really saved my lecture. So I will be thinking some more about keeping an “ace in the hole” so my talks finish on a memorable note.

Eh, teaching is hard.

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