Lesson 3-4: Parallel Lines (Day 34)

Today I subbed in a seventh grade English class. Only one class is special ed with an aide, and so it's worth doing "A Day in the Life" today.

8:15 -- This is the middle school that always starts with homeroom and first period, while the rest of the classes rotate.

First period is actually an AVID class. I've subbed for AVID before, though not recently (and I'm not sure how often I wrote about it on the blog). I do recall a few times when I subbed for a teacher with an AVID class, but not the actual period. I even wrote about the AVID tutors who come in and use VNPS, vertical non-permanent surfaces, with the students.

But there are no tutors today. Instead, the class divides into six groups, and each group is supposed to imagine their own island, including what money, holidays, and books look like there. The groups come up with posters, and I even have two of the groups present their posters to the class. The other students evaluate each group on a scale of 1-5. I place the two groups who present on the good list (which also informs the teacher which groups haven't presented yet).

I recall having an island assignment back when I was a young seventh grader, but I don't remember much about the assignment. I also had a similar assignment in senior-year Government, where the emphasis of course is on my imaginary government.

Meanwhile, I choose a song to sing for the incentive today. Since today is the day after (to observant Jews, the second day of) Rosh Hashanah, I sing the song I wrote on Tuesday, the second of Tishri three years ago at the old charter school -- "Earth, Moon, and Sun" (tune: Hava Nagila).

9:20 -- First period leaves. Since today is Tuesday, the rotation starts with third period. This is the first of the seventh grade English classes.

These students have a short story to read, about a boy whose family finds a valuable "Space Rock" and contemplates whether or not to sell it and get rich. Afterwards, they must answer questions out of a four-page packet.

Last week, I wrote that sometimes I must sacrifice the first class of the day when I don't know what the lesson plan is. I also must do so when I don't know how much work to expect from the students, when the assignment is a lengthy packet. Well, in this class one guy goes as far as two pages, while two others just check off a box (indicating who the main character is) and leave the rest blank.

I place the top student on the good list. The two girls who only check off one box clearly don't do enough work, and so I put them on the bad list. Two other students appear on the bad list because they disturbed others. With so many on the bad list, this class doesn't earn the "Earth, Moon, and Sun" song (a shame, as the song is indirectly related to outer space, hence space rocks/meteorites).

10:15 -- Third period leaves for snack.

10:30 -- Fourth period has a co-teacher -- but due to a shortage of subs, I'm pulled from this class since the co-teacher can present the lesson on her own.

Instead, I'm placed in a special ed English class. Fourth period is the second hour of a two-hour class for English learners. Thus this is the one class with an aide, who speaks Spanish and is able to help the students on their assignment.

11:25 -- Fourth period leaves and fifth period arrives. This is also a two-hour class, but these are English learners at a somewhat higher level. Thus this class does not have an aide.

The class begins with silent reading, followed by a short reading passage about epidemics (from the Black Plague up to Ebola).

12:20 -- The class spans fifth and sixth periods, but as it turns out, today's rotation has these two classes separated by lunch.

1:05 -- Sixth period returns from lunch. The students must answer questions on a worksheet based on the reading passage on epidemics.

With two hours to complete a shorter assignment, these students are able to finish in time to hear the "Earth, Moon, and Sun" song.

2:00 -- For second period, I return to the original classroom that I'm subbing for. This is an honors English class.

This time, I require the students to complete one page (of the four-page "Space Rock" packet) to avoid the bad list, and two pages to get on the good list. Even though some kids are loud, the students all finish the first page in time to hear the "Earth, Moon, and Sun" song.

2:55 -- Second period leaves to go home -- but I don't. As so often happens at middle schools, I'm assigned supervision duty after school.

3:05 -- My long day finally ends.

The only real problem of the day is third period. Notice that it's the luck of the period rotation that third period regular English occurs before second period honors English. Thus I used third period to set the bar, which the honors class easily clears. A few weeks ago, I used an honors science class to determine where to set the bar -- and the regular students failed to reach it, resulting in my placing many names on the bad list.

Lesson 3-4 of the U of Chicago text is called "Parallel Lines." (It appears as Lesson 3-6 in the modern edition of the text.)

In the past, I was always unsure how I wanted to teach parallel lines. Over the years, I kept changing the way I presented this topic, for various reasons. The U of Chicago text teaches parallel lines the way it's done in most Geometry books, with two postulates:

Corresponding Angles Postulate:
If two coplanar lines are cut by a transversal so that two corresponding angles have the same measure, then the lines are parallel.

Parallel Lines Postulate:
If two lines are parallel and cut by a transversal, corresponding angles have the same measure.

But Dr. David Joyce, an education critic, wrote that there should only be one parallel postulate. In fact, we know that not only does Euclid have only one Parallel Postulate (his famous fifth postulate), but he doesn't even need the postulate it in his proof of the "Corresponding Angles Postulate." Instead he uses something called the Triangle Exterior Angle Inequality, or TEAI. On the other hand, his proof of the "Parallel Lines Postulate" is based on his fifth postulate.

Another education writer, Dr. Hung-Hsi Wu of Berkeley, also proves the Corresponding Angles Postulate without need of any parallel postulate. Instead, Wu uses Common Core transformations -- specifically 180-degree rotations -- to attain this result.

Four years ago, I used Wu's approach and based my parallel lessons on these half-turns. But after that, I felt that this was unsatisfactory for two reasons:

• The Corresponding Angles results are more apparently related to translations than rotations -- after all, if two parallel lines are cut by a transversal, then a translation clearly maps one corresponding angle to another. Indeed, I often wonder why we use the same word "corresponding" to refer to angles formed by a transversal and to refer to parts of congruent (or similar) triangles. In the case of congruent (or similar) triangles, there is by definition an isometry (or similarity transformation) mapping one to the other. Well, in the case of angles formed by a transversal, there is a translation mapping one to the other. This justifies use of the term "corresponding angles."
• The idea of delaying the Parallel Postulate until it is needed comes from the idea of "neutral geometry," based on theorems proved using only the other four postulates of Euclid. But we found out that there are two types of neutral geometry -- Euclidean geometry (where the fifth postulate is true) and hyperbolic geometry (where it is false). If we're going to mention non-Euclidean geometry at all, I'd prefer spherical geometry -- as in the spherical earth -- over hyperbolic geometry. But unfortunately, hyperbolic geometry isn't neutral. At least one of the first four postulates fails in spherical geometry -- whereas the fifth postulate, ironically, does hold.

And of course over the summer I wrote about spherical geometry.

Should I attempt to solve these two problems? In the end, these issues don't matter unless they actually affect teaching in the classroom. The second problem, with neutral geometry, matters only if we want, say, to introduce our students to non-Euclidean geometry at the end of the year -- and if were to do that, we'd want to show them spherical, not hyperbolic, geometry. (I've mentioned before how some high schools actually do this.)

Over the summer, I came up with the conclusion that the answer is "no," especially for the second question.

It would be nice if there was a clean break between "natural geometry" (that is, Euclidean+spherical geometry) and Euclidean geometry so we can say, "these Euclidean results still hold in spherical geometry while those don't" -- just as there's a clean break between "neutral geometry" (that is, Euclidean+hyperbolic geometry) and Euclidean geometry (simply by introducing the parallel postulate).

But does the first issue really matter? I wanted to use translations to demonstrate the Corresponding Angles Postulate in my class.