Today is Day 107 -- at least in my old subbing district. It can be considered as the end of the fifth quaver, which we determine by noting that one-quarter of the way from Day 83 (the end of the first semester) to Day 180 is 107.25. Of course, in my new district today is only Day 99. Thus it's not yet the end of the fifth quaver in my new district, seeing as it's only the second week of the semester.
Oh, and this is a good time to mention another difference between the two district calendars. In my old district, Lincoln's Birthday and President's Day are on two separate Mondays. But my new district does something different -- this one and several other districts observe Lincoln's Birthday on the Friday before President's Day, so that there can be a four-day break in February.
(Last year, by the way, students observed a five-day weekend, from Thursday to Monday. But the first two days had nothing to do with Lincoln's Birthday. Instead they were PD days -- in other words, we teachers had only a three-day weekend. President's Day itself is the only school-closing February holiday in the LAUSD.)
What does this mean for the blog calendar? Well, I'm already committed to the calendar of the old district on the blog. This means that Lesson 10-8 won't be posted until Tuesday.
On the other hand, I'm likely to be called in to sub in my new district on Monday. This is an excuse for me to declare today as the last day to do a Day in the Life everyday that I sub, regardless of what subject I teach. Starting next week, I'll post a Day in the Life for math subbing only. Thus Monday is a non-posting day unless I sub a math class that day (in which case I make a "holiday" post).
OK, so today is the last Day in the Life for non-math subbing. Today I subbed for science, at the same high school that I subbed on Tuesday.
The class is officially an "Integrated Science" class. Most of the students are juniors, with a few sophomores and seniors in each class. I've discussed the idea of integrated high school science in previous posts, most recently a year ago. Before the class begins, the teacher next door warns me that many of the students are not college-bound, with many of them only caring about scraping by this class with a D grade.
Hmm, I can see traditionalists jumping on this right now -- of course integrated science isn't for college-bound students, who take classes called "Biology," "Chemistry," and "Physics" instead. Yet progressive pedagogists want to push integrated science down into middle school and integrated math up into high school.
Well, I have no response to this, since I don't know yet what science courses college-bound students take in this district. Nor do I know, for example, what science courses freshmen take, regardless of whether they're college bound or not. So we'll save the integrated science debate for another time, after I gather more information.
OK, let's start a Day in the Life now:
8:00 -- Second period begins. (Recall that "first period" is really zero period.) The Integrated Science lesson is actually about earth science. Students learn about earthquakes, volcanoes, and tectonics.
Students are working in packets. Originally, the teacher intends to collect them today, but instead they are to be collected on Monday. Many students come in expecting to turn in the packets or take a written quiz on them.
I help the students figure out how to graph latitude and longitude -- they're given the coordinates of earthquakes and volcanoes and must graph them on a map to discover the "Ring of Fire." They are allowed to research the Internet to write down the five most recent earthquakes. One of these would be Taiwan's magnitude 6.4 quake last Tuesday. This is the closest I get to math in this class today.
9:00 -- The second period students leave and the third period students arrive. One student finishes the packet early and begins working on a Geometry assignment instead -- so of course, I help him out on this worksheet. He is learning about similar triangles -- so apparently, his class is working on the equivalent of Chapter 12 in the U of Chicago text.
Most of the questions require the student to set up proportions and find lengths. But he struggles on one question, which actually requires a proof:
ABCDE is a regular pentagon.
I'm not sure what his teacher is looking for -- a two-column proof, or what? Anyway, I showed the student a picture and told him (orally) the following paragraph proof:
The sum of the angles is 540, so each angle is 108. Triangle ABC is an isosceles triangle with angle ABC equal to 108, so angles BAC and BCA are each 36. For the same reason angles FAD and FDA are also 36. This leaves 36 for angle CAD. Triangle ACD is isosceles (provable using SAS) with base angle 36, so angles ACD and ADC are each 72. It's also provable (using SAS) that triangles ABC and BCD are congruent, so angle CDB (same as CDF) is 36. Finally, angle DCF is found by subtracting angle BCA (36) from angle BCD (108), so angle DCF is 72. Thus triangles ACD and DCF have two pairs of congruent angles (CAD = CDF = 36, ACD = DCF = 72), and so by AA Similarity, triangles ACD and DCF are similar. QED
It's only fitting that I solve a pentagon problem on the first day of the Winter Olympics. The opening ceremony is being held today in a pentagonal stadium in South Korea. Apparently the number five stands for the five elements -- earth, fire, metal, water, wood.
By the way, the proof above also leads to an interesting length calculation. Assume that the length of the side of the regular pentagon is 1, and we wish to find the length of a diagonal, AC. As triangles ACD and DCF are similar, we write:
AC/CD = CD/CF
x/1 = 1/CF
The one thing blocking us from solving for x (which is AC) is that we don't know what CF is. As it turns out, we can use the same reasoning from above to show that ABF is also a 36-72-72 isosceles triangle, and so AF = AB (which is known to be 1). So CF must be AC - AF or 1 - x:
x/1 = 1/(1 - x)
x - x^2 = 1
x^2 - x - 1 = 0
The positive solution of this equation is (1 + sqrt(5))/2, also known as Phi, the Golden Ratio. This the ratio of the diagonal to the side of a regular pentagon is Phi. This might have been an interesting exercise for Phi Day, had it not fallen on a Saturday during winter break.
10:05 -- The students leave for snack. The next period is the teacher's conference period, and so I have an extended break.
11:30 -- Fifth period arrives. This class is much smaller (12 students, all juniors), and officially the class is called "Intro to Integrated Science." A new student -- a sophomore -- transfers in from the sixth period class. This class is working on only one page of the packet which contains questions about earthquakes. They don't need to graph coordinates on a map.
Meanwhile, another student finishes her assignment early and works on Geometry. Instead of a worksheet, she's going over the entire similarity chapter for a quiz. I give her some pointers, including not to assume that triangles are similar just because you're in the similarity chapter (which some students I tutored tried to do a few years ago), as well as avoiding the Side-Splitter trap (where if ABC and ADE are similar, students write AB/BD = BC/DE, instead of AB/AD).
12:20 -- The students leave for lunch.
1:10 -- Sixth period arrives. This is an Integrated Science class.
2:10 -- Sixth period leaves and seventh period arrives. This is another Integrated Science class.
3:00 -- Seventh period leaves. This week felt as if it dragged on forever, since I subbed for four days this week in my new district after receiving so few calls in my old district.
Let's look at my next New Year's Resolution:
4. Begin the lesson quickly instead of having lengthy warm-ups.
Even though this class doesn't have warm-ups, I am able to save time in one of my classes. I already know coming in that sixth period would be the worst-behaved class, since I read the note left by yesterday's sub. But for some reason, this class has two student TA's. And so I have the TA's take attendance, thus saving precious minutes at the start of class. I go straight into describing the lesson, informing the students that electronics are to be used only for completing the packet.
Meanwhile, I have opportunities to practice some of the other resolutions. Since this isn't a special ed class, I can work on my classroom management. One student in that fateful sixth period class breaks the rule. He takes out a cell phone without working on the packet, and then starts arguing with another student. When I try to talk to him, he storms out of the room and keeps on using the phone right outside the classroom. I do the only thing I can do -- leave his name down for the teacher. The TA's, of course, tell me what his name is.
Also, the third resolution comes up in seventh period. I tell the students about the electronics rule, then start to tell them what happened the previous period as a warning. Then I suddenly stop myself, since it would violate the third resolution (bringing up past incidents). I definitely need to work on this resolution some more. As it turns out, another student takes out his cell phone without doing any packet work, and begins joking around. I don't catch his name, and so I write down for the teacher that it's the redhead who is being disruptive. I don't really want to single him out, but maybe students with distinctive physical features really need to avoid breaking the rules.
Let's return to the U of Chicago text. This is what I wrote two years ago about today's lesson:
Lesson 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 mentioned in last week's post as well. (By the way, I just realized that I mentioned four different math texts in last week's post!)
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.
2018 update: Today is an activity day, while yesterday wasn't. Therefore I've chosen to replace the second worksheet with the aforementioned Question 23. (On the other hand, Question 22 requires that we make a cone and cylinder and pour three conefuls of dirt/sand into the cylinder.)
Actually, I obtain the worksheet from the following website:
http://www.cutoutfoldup.com/971-the-volume-of-a-pyramid-is-one-third-that-of-a-prism.php
http://www.cutoutfoldup.com/patterns/0971_us.pdf
Notice that the original version of this page is copyrighted. I try to avoid posting any page that contains a copyright symbol Note the following conversation at the first link:
OK, so according to the author, you may use this webpage for a class assignment. Let me give the full author, so that is is a proper reference -- the author is Laszlo C. Bardos. He writes:
Everything on this site is copyrighted. You may print copies for your personal use and teachers may photocopy materials for classroom use. Please ask me if you would like to do anything more substantial than this.
I fear that posting a copy of the page here would be considered "more substantial," and so I post only a link to Laszlo's page instead of a copy of the page.
My next post will be on Tuesday, after Lincoln's Birthday -- unless I sub in a math class, in which case I'll post on Lincoln's Birthday itself.
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