Friday, March 23, 2018

Lesson 13-5: Tangents to Circles and Spheres (Day 135)

Lesson 13-5 of the U of Chicago text is called "Tangents to Circles and Spheres." In the modern Third Edition of the text, tangents to circles and spheres appear in Lesson 14-4 -- the only lesson that's in Chapter 13 of the old text and Chapter 14 of the new text.

Spring break is definitely on many of our minds right now. As so often happens when I work in two districts, they end up taking different weeks off for spring break. In my new district, it's easy -- spring break is always the week after Easter (that is, Easter Week or Bright Week). Thus there is one week left until spring break in that district.

But the blog calendar follows my old district, not my new district. And in that district, spring break works a little differently. First of all, in this district, spring break has nothing to do with Easter. So
instead, spring break occurs during the same fixed week each year -- and that's next week. In other words, today is the last day before the vacation on the blog calendar.

We notice that today is Day 135 on the blog calendar, and 135 is 3/4 of 180. Despite this, spring break is not considered to divide the third and fourth quarters. This is because winter break, which really does divide the first and second semesters, occurred after Day 83, not Day 90. So the start of the fourth quarter is closer to the mathematical midpoint of the second semester, a few days ago.

I'm not sure why the district doesn't simply have spring break right after third quarter. Perhaps it's so that today can be the deadline for submitting the third quarter grades, or maybe today's the day the students actually take home the third quarter progress reports. I'm not sure, since of course I didn't sub in this district today.

But this district doesn't ignore Easter completely. The rule is that Good Friday and Easter Monday are always holidays in this district. Since the spring break week already contains Good Friday, an extra day is taken off for Easter Monday. And so the blog calendar will resume on Tuesday, April 3rd. In both 2017 and 2019, Easter falls on the third Sunday in April. So

Meanwhile, the LAUSD (and hence my charter school from last year) always takes off the week before Easter (that is, Holy Week), so they are closed next week as well. And just like the blog calendar, LAUSD is also closed an extra Monday -- not for Easter, but for Cesar Chavez Day. The labor leader's March 31st birthday ends up being part of spring break in years when it is close to Easter, as it obviously is this year.

Chavez Day is used to define spring break in the California State University system. Since the actual date falls on a Saturday, campuses are closed on Friday, March 30th instead. Spring break is thus considered to be all of next week -- there are no classes, but the campuses are open every day except Friday, Chavez Day Observed. In the University of California system (including my alma mater UCLA), Chavez Day is considered to be the last Friday in March, which is also March 30th. So once again, the UC spring break is next week as well.

Finally, I want to point out when spring break is in New York City -- the largest district in the nation and home to many MTBoS bloggers. In the Big Apple, spring break occurs during Passover, which is usually near Easter anyway. This year Passover starts on Saturday, March 31st. But New York, just like the blog calendar, has a rule that Good Friday is always a day off (though Easter Monday isn't part of this rule). And so New York takes off Good Friday and the week after Easter. (In some years, Passover falls a month after Easter -- in such years, Good Friday is a mere three-day weekend.) I am aware that there were closures in the city and other areas this week due to snow.

As usual, my plans for the blog are to post once or twice during "spring break" as observed by the official blog calendar. But I'll probably be subbing in my new district next week -- and I promised a "Day in the Life" if it's a math class. It'll look strange to make a "Day in the Life" post during "spring break," but it's possible.

Today, meanwhile, wasn't math, but the last day in the music class. Students all played the same songs as yesterday, except for seventh grade band. The student teacher explains that they're trying out different songs to play at a recital coming up in May. Today they played "John Williams: Movie Adventures," a medley of four Hollywood songs -- "Star Wars (Main Theme)," "Duel of the Fates," "Theme from Jurassic Park," and "Theme from E.T. (Extra Terrestrial)."

Being surrounded by all this music puts me in a mood for another music post -- and lately, I've been writing about music during vacations anyway. So expect a music post coming up for spring break. I am already thinking about ideas to include in this post:

  • I'll have more songs based on digits of mathematical constants.
  • For microtonality, I already have a 7-limit scale. Since I've been writing about the number 11 a lot lately (Eleven Calendar, Eleven Clock), I can't help but consider an 11-limit scale.
  • And I want to compose a new song for Easter.
OK, let's get to the lesson. I posted Lesson 13-5 last year, but I didn't have much to say about it. But remember that this is still an important lesson -- tangents to circles definitely appear on the PARCC and SBAC exams!

On the other hand, today is an activity day -- and it's the last school day before Easter. Last year I omitted this, but this year I'm bringing it back -- Spring Spheres! And not only that, but it actually fits this final lesson on tangents to circles and spheres.

And so this is what I wrote two years ago about today's activity (including more comments about the Easter Date):

Here's my idea of an activity: we take the idea of dividing the surface of a sphere into figures that are nearly polygons and run with it. Now Dr. M divides his surface into triangles, but here we will use square Post-it notes instead. After all, we measure areas in square units, not "triangular units." The task directs students to estimate how many Post-it notes it takes to cover the surface of a sphere before they actually try it.

I have decided to name this activity "Spring Spheres." The name actually refers to an an incident a few years ago (it was 2011 -- the year when Easter was very late) where a volunteer in a classroom was not allowed to bring Easter eggs to school because they were religious. So she decided to bring the students "spring spheres" instead. Here I twist the use of that name around -- it's springtime and we're finding the surface area of a sphere -- hence the name "Spring Spheres."

This activity is long and requires that there are several balls in the classroom -- and even if we divide the class into groups and ask some students to bring balls to school, they may simply play with the balls rather than complete the activity. So here are some other activities that I am posting today:
  • From the U of Chicago text: I decided to change the activity from two years ago to one that actually fits Lesson 13-5. It's about three important spheres -- earth, moon, and sun. One question is about a lunar eclipse -- think back to the "super blue blood moon" two months ago.
  • Let's balance out "Spring Spheres" with a question about Easter -- specifically the Easter date. Even though Easter is determined by a table, the table can be calculated using a formula. The following link gives a link to what is known as the Conway Doomsday Algorithm -- and that's Conway as in John Horton Conway, the mathematician who also argued for the inclusive definition of trapezoid. In fact, Doomsday is used to determine the day of the week -- and that's part of calculating Easter, since we need to know when Sunday is. The link also describes how to calculate the Jewish holidays of Passover and Rosh Hashanah, but these are more complicated than calculating Easter. A neat trick is to verify that the Easter calculation works this year, then calculate when it falls next year. Notice that this is a math lesson, but if your school is similar to the Washington state school where Easter eggs have to be called "Spring Spheres," then just stick to the Spring Spheres lesson in the first place.

If you pay attention to the dates of Easter and Passover from year to year, you will notice that although they usually fall within a week or so of each other, on occasion Passover falls about a month after (Gregorian) Easter. At the present time, this happens in in the 3rd, 11th, and 14th years of the Metonoic Cycle (i.e., when the Golden Number equals 3, 11, or 14). The reason for this discrepancy is the fact that although the Metonic Cycle is very good, it is not perfect (as we've seen in this course). In particular, it is a little off if you use it to predict the length of the tropical year. So, over the centuries the date of the vernal equinox, as predicted by the Metonic Cycle, has been drifting to later and later dates. So, the rule for Passover, which was originally intended to track the vernal equinox, has gotten a few days off. In ancient times this was never a problem since Passover was set by actual observations of the Moon and of the vernal equinox. However, after Hillel II standardized the Hebrew calendar in the 4th century, actual observations of celestial events no longer played a part in the determination of the date of Passover. The Gregorian calendar reform of 1582 brought the Western Church back into conformity with astronomical events, hence the discrepancy. 

Similarly, you will notice that in many years Gregorian Easter (the one marked on all calendars) differs from Julian (Orthodox) Easter, sometimes by a week, sometimes by a month. Again, this is due to the different rules of calculation. A major difference is that Orthodox Easter uses the old Julian calendar for calculation, and the date of the Vernal Equinox is slipping later and later on the Julian calendar relative to the Gregorian calendar (and to astronomical fact). Also, the date of Paschal Full Moon for the Julian calculation is about 4 days later than that for the Gregorian calculation. At present, in 5 out of 19 years in the Metonic Cycle--the years when the Golden Number equals 3, 8, 11, 14 and 19--Orthodox Easter occurs a month after Gregorian Easter. In three of these years, Passover also falls a month after Gregorian Easter (see above).

Yes, every time I turn around, it's Christmas or Thanksgiving -- or Easter, that is. Yes, the Big March is over, and it's now spring break. My next regular school post will be on Tuesday, April 3rd.

And so this concludes my last post before spring break. Once again, I plan on making one or two posts next week, during spring break itself.

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