Tuesday, March 31, 2015

Section 10-7: Volumes of Pyramids and Cones (Day 138)

Today is an official state holiday here in California, but it's one of the least known holidays. Today is Cesar Chavez Day, to celebrate the 20th century farm worker and labor leader.

Some schools -- and most notably universities -- here in California observe Chavez Day. As I mentioned earlier, there's a trend to sever the link between spring break and Easter, in order to avoid having the school holidays be tied to a holiday that can vary by over a month like Easter. In the University of California and California State University systems, spring break is tied not to Easter, but to Chavez Day.

Notice that for schools whose the first day of school is after Labor Day, that end-of-summer holiday is the most important holiday on the calendar. The first day of school is either early or late depending on whether Labor Day is early or late, which means that the last day of every quarter and trimester, including the last day of school, is early or late depending on the previous Labor Day date. In 2014, Labor Day fell on its earliest possible date, September 1st, while this year, Labor Day will fall on its latest possible date, September 7th. This means that dates for the end of each semester will fall later during the 2015-6 school year than in the 2014-5 school year. Schools on an Early Start calendar may choose a different holiday to be the most important holiday on the calendar. For example, some Early Start schools are set up so that the last day of school is before Memorial Day, so that holiday is the most important. Others set up the year so that a fixed number of weeks occur before winter break, thereby making Christmas the most important holiday. The LAUSD Early Start calendar appears to have a fixed number of weeks before winter break -- but winter break itself is defined to begin three weeks after Thanksgiving. I won't know for sure until the holiday reaches its earliest possible date again on November 22nd, 2019, but it appears that Thanksgiving is the most important holiday on the LAUSD academic calendar.

Well, the most important holiday for the UC and Cal State college is Chavez Day. For example, at my alma mater, UCLA, the school year is divided into three quarters -- fall, winter, and spring. Spring break contains the observed Chavez Day and separates winter quarter from spring quarter. Since each quarter has a fixed length (ten weeks plus a finals week), and the length of the breaks is also fixed, all the dates in the UCLA academic calendar can be determined starting from Chavez Day.

There is a slight difference between the Cal State and UC calculations of the Chavez Day date. In the Cal State system, Chavez Day is on the labor leader's actual birthday, March 31st, and so spring break is the week containing the last day of March. Since most Cal States operate on a semester system, the spring break week is actually halfway during the spring semester, so that midterms can be given the week before the holiday. But at UC's -- well, at least at UCLA -- Chavez Day is defined to be the last Friday in March -- so that this year, March 27th was Chavez Day Observed. This means that at the Cal States, this week is spring break, while at the UC's, spring break was last week (just as it was on the blog) and spring quarter has already begun. Notice this means that there are often classes held at UC on the labor leader's actual birthday on the 31st, while there is never school this day at Cal State. I point out that in either case, it's more convenient, especially at UC, to tie spring break to Chavez Day rather than to Easter, in order to avoid quarters of differing lengths in years when the Christian holiday is exceptionally early or late.

LAUSD also observes a Chavez Day holiday. It is observed on the actual date, March 31st, in some years, while in others it's moved to the nearest Monday or Friday. In years in which March 31st is close to Easter -- recall that the district's spring break coincides with Holy Week -- Chavez Day is pushed to Easter Monday. So this year, LAUSD observes Chavez Day on Monday, April 6th -- six days after the labor leader's actual birthday and ten days after the UCLA holiday.

Two years ago, Easter actually fell on Chavez Day. There was a Google Doodle that day celebrating Cesar Chavez -- which angered some Christians expecting to see a Google Doodle for Easter. I point out that Chavez was a devout Catholic, and so one could argue that Chavez himself would have rather seen Google celebrate Easter than himself.

Section 10-7 of the U of Chicago text is on the volumes of pyramids and cones. And of course, the question on everyone's mind during this section is, where does the factor of 1/3 come from?

The U of Chicago text provides two ways to determine the factor of 1/3, and these appear in Exploration Questions 22 and 23. Notice that without the 1/3 factor, the volume formulas for pyramid and cone reduce to those of prism and cylinder, respectively -- so what we're actually saying is that the volume of a conic surface is one-third that of the corresponding cylindric surface. So Question 22 directs the students to create a cone and its corresponding cylinder and see how many conefuls of sand fill the cylinder. The hope, of course, is that the students obtain 3 as an answer. This is the technique used in Section 10.6 of the MacDougal Littell Grade 7 text that I purchased last weekend as well.

But of course, here in High School Geometry, we expect a more rigorous derivation. In Question 23, students actually create three triangular pyramids of the same base area and height and join them to form the corresponding prism, thereby showing that each pyramid has 1/3 the prism's volume. But this only proves the volume formula for a specific case. We then use Cavalieri's Principle to show that therefore, any pyramid or cone must have volume one-third the base area times height -- just as we used Cavalieri a few weeks ago to show that the volume of any prism, not just a box, must be the base area times height.

I decided not to include either of the activities from Questions 22 or 23. After all, there was just an activity yesterday and I wish to avoid posting activities on back-to-back days unless there is a specific reason to, such as during CAHSEE week.

But instead, I do wish to honor Cesar Chavez in one of the questions. Notice that Chavez dropped out of school early in order to work in the fields, so he most likely never made it to Geometry class. Still, Chavez was of Mexican descent, and there just so happens to be a question from this section of the text in which Mexico is mentioned. (A recent movie features a group of students working the fields in California at about the same age that Chavez once did, albeit some four decades later. But these students notably didn't drop out -- instead, they attended McFarland High School.)

In Question 11, we learn that the largest monument ever built is the Quetzalcoatl at Cholula de Rivadabia, a pyramid about 60 miles southeast of Mexico City. So in particular, the Quetzalcoatl is larger than any of the more famous Egyptian pyramids mentioned in yesterday's lesson. (But apparently, the Quetzalcoatl is well-known enough to avoid being flagged as a spelling error here on the Blogger editor.) Naturally, the students are asked to determine the volume of this large pyramid.

Question 15 invokes one of the simplest solids of revolution -- a BC Calculus staple. Of course, we only need to use the cone formula to find the volume and not anything from Calculus.

Finally, in the lesson I incorporated some of Section 10-6, "Remembering Formulas." I like the idea of having a section devoted solely to remembering these formulas -- since these are notoriously difficult for students to remember. The problem is that the way my lessons are set up, there is precious little time to devote an entire day's lesson just to remembering formulas. But I squeeze it in here by showing the same hierarchy of three-dimensional figures that appear in Section 10-6.

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