"In modern times we have witnessed the formation of the hyperbolic paraboloid (St. Mary's Cathedral in San Francisco), the geodesic structures of Buckminster Fuller, the module designs of Paolo Soleri, the parabolic airplane hanger, solid synthetic structures mimicking the tents of the nomads, catenary curve cables supporting the Olympic Sports Hall in Tokyo, and even an octagonal home with an elliptical dome ceiling."
This is the final page of introduction to the architecture chapter. Now Pappas lists several examples of geometric buildings from the past century or so. Many of these we have to see to appreciate, so let's try a Google image search.
St. Mary's Cathedral:
http://www.aviewoncities.com/sf/stmaryscathedral.htm
Paolo Soleri's planned town of Arcosanti, Arizona:
http://www.archdaily.com/tag/paolo-soleri/
Airplane Hangars (some, but not al,l are parabolic):
https://www.pinterest.com/fabricbuildings/custom-aircraft-hangars/
Yoyogi National Stadium in Tokyo (but a new stadium should be built in time for the 2020 Games):
http://www.alamy.com/stock-photo-yoyogi-national-stadium-in-tokyo-designed-by-architect-kenzo-tange-53753182.html
Octagonal Homes (not necessarily with elliptical ceilings):
https://www.pinterest.com/cindyg7811/octagon-houses/
Meanwhile, Pappas gives an example of "structures mimicking tents" in a photo on this page. Here is the caption:
"This tent-like structure illustrates the use of new materials and methods of constructions. Fashion Island, Foster City, California."
According to the following link, this mall no longer exists -- and apparently it was in decline even before Pappas wrote her book in 1994:
http://bigmallrat.blogspot.com/2006/11/mall-memories-san-mateo-fashion-island.html
Notice that so many examples in Pappas come from my home state of California. Well, Pappas did earn her degrees from Berkeley and Stanford, so at least Northern California is familiar to her. Oh, and that last example will have to wait. Buckminster Fuller actually merits his own section later in this chapter.
She concludes:
"In the final analysis, an architect is free to imagine any design so long as the mathematics and materials exist to support the structure."
Lesson 1-5 of the U of Chicago text is called "Drawing in Perspective." In the modern edition of the text, perspective doesn't appear until Lesson 9-4. This is more logical, as Chapter 9 in both editions is the chapter on three-dimensional figures.
Perspective appeared as Lesson 0.8 in Michael Serra's Discovering Geometry, which we already covered nearly two weeks ago on Day 8. This time, I'll reblog the old Lesson 1-5 post from last year.
Indeed, Lesson 1-5 is the other worksheet I taught in middle school last year as part of my opening week activities. This is what I wrote about it:
Speaking of class, today I gave the last of the opening week activities previously posted on the blog -- Designing Buildings. This is what I wrote earlier about this activity:
And as it turns out, Nguyen covered something similar to this in her class as well:
http://fawnnguyen.com/designing-buildings/
Nguyen's lesson takes a different approach to drawing three-dimensional figures. For one, the focus on this lesson is on buildings. Her lesson begins by having some buildings already drawn and the students counting the "rooms" and "windows." (As it turns out, one "room" is one cubic unit of volume, and one "window" is one square unit of lateral area.)
I like the way that Nguyen's lesson begins. Unlike the bridge problem, where I wanted to avoid beginning the school year with a problem that's impossible to solve, here we begin with a very solvable problem. The only issue I have is with the second question, because it requires materials. I work from the assumption that most classrooms don't have the blocks and isometric dot paper that Nguyen's classroom has.
(As an aside, notice that cubes drawn on isometric dot paper are definitely not in perspective. This is because, while edges perpendicular on the cube intersect at 120 degrees on the iso dot paper, edges parallel on the cube remain parallel on the paper. Therefore there are no vanishing points.)
Then again, my worksheet is very similar to Nguyen's. On the front side, I gave the same example as she did and the three buildings for the students also come from the Ventura County teacher. I used two of her easier buildings -- A and B -- and the more challenging Building F.
The back side of my worksheet differs slightly from Nguyen's, though. Her worksheet specified the number of rooms and windows and asked the students to draw the buildings. Mine, on the other hand, simply has the students draw four different buildings with eight rooms and then asks them to count the number of windows in each one.
Now that I'm giving this activity in an actual classroom, I don't have any interlocking cubes (which I can only assume means "Lego bricks"), but I did find some small manipulative cubes. There weren't enough for me to give every group eight cubes (as specified in the assignment) -- instead I gave five to each group of sixth graders and seven to each group of seventh graders. (Half the seventh graders were absent because they hadn't satisfied California's 7th grade vaccination requirement.) The eighth grade groups did receive the full set of eight cubes. I believe that having actual blocks certainly helped the students visualize the three-dimensional buildings.
By the way, here are the rules the middle school classes came up with as part of the Rules Posters. At last I'm done discussing the rules here on the blog:
1. Raise your hand
2. Be silent and listen when it's someone else's turn to speak
3. Stay in your seat
4. Keep your hands to yourself
5. Keep the desks free of drawing
6. Treat the books, papers, and any other resources like you would treat your own items
7. Keep your voice at a conversational level
8. Allow the speaker to finish before you raise your hand
9. Speak in a respectful manner
10. Stay on task, work hard, and do your best!
Returning to 2017, we notice that by this point, the lesson is no longer recognizable as an activity on perspective, since the buildings are not drawn in perspective. It was influenced by a Fawn Nguyen post and then modified yet again in my middle school classroom. Well, at least the lesson ties to architecture and hence to today's Pappas page. And between Lessons 0.8 and 1-5, the students should learn something about perspective in the end.
Of course, I mentioned the rules in that old post, which means that this is going to be yet another post on classroom management (and spilled milk). In fact, we can see why I had so many problems -- yes, these ten rules sound reasonable. But look at what happened in seventh grade early that day:
8:25 -- My first class, a seventh grade class, arrives. Today there is a confrontation with one of the seventh graders. She refuses to do her work, then argues with my student support aide, who asks her to leave the room. I am the teacher, so I should have tried to intervene sooner, though it still might not have made much difference. It is only Day 3, but I already know there's one girl I'll need to watch out for this year.
Let's see how many rules that girl broke that day. She broke Rule #1 (not raising her hand), #2 (not being silent), #9 (being disrespectful to my aide), and #10 (not staying on task). Yet, as we plainly see, I didn't intervene or attempt to punish her.
I've mentioned before where I went wrong -- the participation points system. The girl had earned a few cheap points (for turning in emergency papers), and my system stated that punishments begin after students lose all of their points. Since she hadn't lost all her points, I gave no punishment.
Obviously, I should have had a different system. Students can only gain participation points rather than lose them -- instead, punishments occur when students don't do what I tell them. In this case, I should have backed up my aide by threatening either a detention or a phone call unless the girl agreed to follow my aide out of the room. Instead, I ended up demonstrating to this girl -- as well as the other students in the class -- that the four rules she broke don't mean squat.
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