"Yet the shapes and forms of architecture remain the three-dimensional objects of mathematics. Many are from Euclidean geometry, such as rectangular or square solids, pyramids, cones, spheres, cylinders."
This is the second page of the section on 21st century architecture. After summarizing all the solids used in architecture, Pappas finally speculates on what the future may bring -- since after all, to Pappas writing in 1994, the 21st century is still "the future":
"Any shape that tessellates a plane, such as the triangle, the square, the hexagon, and other polygons can be adapted for spatial living units."
And there we go with tessellations again. Of course, the triangle, square, and hexagon are the only tessellating polygons that are regular -- as Pappas explains later, "other polygons" refers to polygons that are not regular.
There is one photo with the following caption:
"This model of Paulo Soleri's visionary project is displayed at Arcosanti in Arizona."
I already linked to Soleri and Arcosanti last week. As much as I'd like to link to a photo here, I can't find online the exact photo from Pappas.
Today is my first day as a substitute teacher this year. It marks my grand return to the classroom after a tough year last year -- especially the second half. Today and tomorrow I'm filling in for a science teacher at a continuation high school. In the past, I usually didn't write about subbing for classes other than math. But this year, I will write about subbing for non-math classes because:
-- Being in a science class reminds me of the science I failed to teach last year.
-- Being in any class reminds me of the classroom management I failed to have last year.
In the past, when I subbed for non-math I often didn't even mention it on the blog. On days when I don't cover math, I should at least be focusing on management. I believe that if I had written about management issues on my subbing days (whether math or otherwise) and learned from them, I might have been a better manager and teacher last year.
Therefore I'm adding the "subbing" label to this post. You can expect me to write about classroom management in all subbing-labeled posts this year. This isn't the same as "crying over last spilled milk," since I'm not merely writing about the mistakes I made last year, but also what I do today, in an actual classroom today, to fix those mistakes. In the future, I'll refer to these posts to see how much improvement I'm making.
In fact, let me describe my day using the "Day in the Life" format of Tina Cardone:
8:00 -- I arrive at the continuation school. As it turns out, the science teachers are out having a meeting at the district office -- which is right next to the continuation school. Therefore I actually meet the science teacher, who tells me the lesson plan for the day.
Actually, the first two periods of the day aren't science, but "Orientation to High School" -- a strange name for a class at a continuation school (where there are no freshmen). I assume it means that this is the class where students learn valuable study skills.
The assignment is to go to Google Classroom on Chromebooks and open a document. It contains instructions on writing a letter to oneself. Students are supposed to write about what they like to do and why, as well as specific goals. The letter needs to be at least 450 words long, which the students check using the word count function.
The use of laptops reminds me of all the problems I had with computers and IXL last year. As it turns out, this time the slots are numbered, but the laptops aren't numbered (consistently that is -- some of them do have numbers). On the other hand, there are enough chargers for all the laptops. And so instead of enforcing placing the laptops in the right slot, I need to have the students charge them up at the end of class.
As is often true at many schools -- especially continuation schools -- the biggest problem in first period is tardiness. And so there's one student who arrives more than half an hour late, then at the end of class, claims that she "just barely arrived" and so she shouldn't have to do any work at all.
I've seen this happen before many times, both as a sub and as a regular teacher. A student arrives late, either due to tardiness or being summoned to the office. There are about 15-20 minutes left in class, yet at the end of class the student claims to have just barely arrived. In other words, they are telling me to pretend that 15-20 minutes are 15-20 seconds. Instead of challenging themselves to work twice as hard to finish as much of the assignment as they can, students who arrive 15-20 minutes before the end of class believe that they are entitled to 15-20 minutes of non-academic free time, and that anything else is unreasonable and unfair.
I am not sure what to do at this point -- but my mind is racing to determine how to avoid this excuse in the next class.
8:55 -- This is another Orientation class. This time, in order to encourage the students to do the work, I tell them that anyone who doesn't finish at least 100 words is in big trouble. While some students might have trouble coming up with 450 words, the only reasons not to have 100 words are talking, playing with cell phones, sleeping, and treating the entire class like a free period.
One student claims not to have a Google password. Five more students hardly write anything, saying that they don't know what to write while enjoying their non-academic free time. I wrote down all of their names for the teacher, informing him that I told these students multiple times to work and yet they refused.
9:45 -- The students go out to nutrition.
10:00 -- It is third period, and the first science class arrives. Officially, this is an Integrated Science class. Integrated Science is rare in high school, but I point out that I took Integrated Science as a young student.
The students have another online assignment -- this time to take notes in Google Classroom. These students were learning about taxonomy, where all organisms are divided into categories. The hierarchy that they are to fill in on a worksheet is Life, Domain, Kingdom, Class, Order, Family, Genus, Species.
In middle school, a simple version of the hierarchy might have appeared in the old seventh grade life science class. I'm not sure where it belongs under the new NGSS standards.
Meanwhile, in thinking about how I could have taught science to my middle school students, I suppose I could have set something up on Google Classroom, if I'd tried hard enough. The coding teacher did provide the students with Google accounts, and I believe that the history teacher painstakingly uploaded the entire history text online.
10:55 -- The second science class arrives. In each of these science classes, two students refuse to do any work. Again, I write down their names for the regular teacher, informing him that I told the students multiple times to work. In the end, this class ends up being the best behaved of the day.
11:45 -- At this continuation school, many students attend only for a half day. And so the AM students go home, while the PM students arrive. Some teachers only teach for a half day, but the teacher I'm covering has a class seventh period -- the last class of the day. And so I end up with a two-hour break before the final class.
The regular teacher comes in from his meeting lunch break and warns me about 7th period. He tells me that these students might be 10th and 11th graders, yet they act like middle school students. For this class only, he enforces a seating chart, and he tells me which students to watch out for.
With a long break before seventh period, this is an opportunity to be proactive. I pass out the taxonomy worksheets, and write names on them corresponding to the seating chart. I also write down instructions for the class on the board:
1. Be quiet while attendance is being taken.
2. Pick up a laptop from the cart.
3. Open up Google Classroom and take notes on Section 2.1.
4. Finish both sides of the worksheet -- at least one side to avoid big trouble.
5. Plug in the laptop to be charged.
2:10 -- I believe the preparation time works, as this class turns out not to be that bad at all. Two girls are absent, and two guys try to move into their seats. I remind them that their teacher wants me to enforce the seating chart.
With about ten minutes left in class, some students start lining up at the door. I tell them to return to their seats -- and they did so just in time, as the regular teacher walks in after his meeting.
The only real problem in this class is that someone takes out a phone to make lewd sound effects. It is impossible for me to determine who is making the sounds.
3:10 -- Class ends, and I go home to type up this blog entry.
For this version of "Day in the Life," there is only one reflection question:
Teachers are always working on improving, and often have specific goals for things to work on throughout a year. What have you been doing to work toward your goal? How do you feel you are doing?
My one goal, of course, is to become a better classroom manager. I think that overall, this day went fairly well. The only two things I don't like are first period students taking advantage of tardies to avoid work, and seventh period with the sound effects.
In the past, I often didn't write down the names of students who disobeyed me -- mainly because I didn't know what their names were. This led to bad management habits -- by not writing the names, I wasn't enforcing the rules, and that led to students breaking even more rules.
Of course, I don't ask students for their names, since who would tell me their names knowing that I'm trying to write them up for bad behavior? So instead, I pulled clever tricks to determine their names:
-- In second period, I take the laptops away from students who weren't working. These laptops are still logged in to their Google accounts, and so I get the names from there.
-- In third and fourth periods, three of the four students with blank papers at least have names on them, and so I used those names.
-- The remaining student had a completely blank paper -- but then instead of turning in the paper to the class folder, he places it in his personal folder! And so I see the name on the folder. The fourth period names are tricky, since each one shared his name with another student in the class.
This is a two-day assignment, but there is some confusion whether I will return to this classroom tomorrow or fill in for the other science teacher (the one with a half-day) instead. If I return to the same classroom, it will represent a true test of how much my management has improved.
Lesson 1-9 of the U of Chicago text is called "The Triangle Inequality." (It appears as Lesson 1-7 in the modern edition of the text.)
Over the years I've had several problems with Lesson 1-9, and here's why. Three years ago -- by which I mean the 2014-15 school year -- I noticed that Lesson 1-9 presents the Triangle Inequality as a postulate, when it's in fact provable using the theorems of Lesson 13-7. And so I decided to delay Lesson 1-9 until after 13-7, so we could prove the Triangle Inequality Theorem.
But then two years ago -- the 2015-2016 school year -- I juggled Chapter 13 around again. I ended up covering other lessons in Chapter 13 at various times, but never 13-7. And because I never posted Lesson 13-7, I'd never post 1-9 either. (Recall that the new Third Edition of the text no longer has our version of Chapter 13.)
Last year, I covered some chapters after I left my middle school classroom. That includes all of Chapter 13 (on Days 131-140), but never Lesson 1-9 at all. Now you can see what I'm restoring the U of Chicago order, so that Lesson 1-9 is no longer dependent on Chapter 13 or any future lesson.
And so we must go back three years to find a true Lesson 1-9 discussion. That day, I wrote about how theorems from Lesson 1-9 can be used to prove the theorem:
And that takes us to the topic of today's worksheet -- the Triangle Inequality. In the U of Chicago, the Triangle Inequality was given as a postulate, yet it can be proved as a theorem. Many texts, including the Glencoe text, do prove the Triangle Inequality as a theorem, and this is what we will do.
The proof of the Triangle Inequality begins in Glencoe's Section 5-2, where we must prove two theorems, which the U of Chicago calls the Unequal Sides and Unequal Angles Theorems. My student told me that he had no problem understanding these two theorems -- he wanted just a quick review of Indirect Proof in Section 5-3 before moving on to the Triangle Inequality in 5-4. (This is why I'm squeezing in the Triangle Inequality now, rather than prove only the Unequal Sides and Unequal Angles Theorems today and save the Triangle Inequality for next week.)
Dr. Franklin Mason also proves these theorems. In many ways, Dr. M's Chapter 5 is similar to the same numbered chapter in Glencoe, except that Dr. M saves the concurrency results for a separate chapter, Chapter 10. Both Dr. M and Glencoe follow the same sequence of theorems, in which each theorem is built from the previous theorem in the list:
Exterior Angle Theorem (abbreviated TEAE in Dr. M)
Exterior Angle Inequality (TEAI)
Unequal Sides Theorem (TSAI)
Unequal Angles Theorem (TASI)
Triangle Inequality (too important to be abbreviated!)
SAS Inequality (Hinge)
The U of Chicago follows the same pattern, except that the Unequal Angles Theorem is not used to prove the Triangle Inequality. Instead, the Triangle Inequality is merely a postulate. And since Unequal Angles isn't used to prove the Triangle Inequality, the U of Chicago didn't have to wait until
Chapter 13 to state the Triangle Inequality. Instead, the Triangle Inequality Postulate is given in Chapter 1, and the SAS Inequality, which depends on that postulate in its proof, is given in Chapter 7, still well before Chapter 13.
My blog attempted to restore the Dr. M-Glencoe order by delaying the Triangle Inequality. But I screwed up by not delaying the SAS Inequality as well. This is why next year, I plan on delaying SAS Inequality, so that the full logical sequence is given. [2017 update: Of course all I did that year was make things worse!]
But the first four theorems in the list are proved in U of Chicago's Section 13-7. Since I briefly mentioned the Exterior Angle Theorem (TEAE) at the end of the first semester, and the TEAI follows almost trivially from TEAE, my worksheet skips directly to the Unequal Sides Theorem. Its proof is given in the two-column format. Here I reproduce that proof, starting with a Given step:
Unequal Sides Theorem (Triangle Side-Angle Inequality, TSAI):
If two sides of a triangle are not congruent, then the angles opposite them are not congruent, and the larger angle is opposite the longer side.
Given: Triangle ABC with BA > BC
Prove: angle C > angle A
Proof:
Statements Reasons
1. Triangle ABC with BA > BC 1. Given
2. Identify point C' on ray BA 2. On a ray, there is exactly one point at a given distance from
with BC' = BC an endpoint.
3. angle 1 = angle 2 3. Isosceles Triangle Theorem
4. angle 2 > angle A 4. Exterior Angle Inequality (with triangle CC'A)
5. angle 1 > angle A 5. Substitution (step 3 into step 4)
6. angle 1 + angle 3 = angle BCA 6. Angle Addition Postulate
7. angle BCA > angle 1 7. Equation to Inequality Property
8. angle BCA > angle A 8. Transitive Property of Inequality (steps 5 and 7)
The next theorem is proved only informally in the U of Chicago. The informal discussion leads to an indirect proof.
Unequal Angles Theorem (Triangle Angle-Side Inequality, TASI):
If two angles of a triangle are not congruent, then the sides opposite them are not congruent, and the longer side is opposite the larger angle.
Indirect Proof:
The contrapositive of the Isosceles Triangle Theorem is: If two angles in a triangle are not congruent, then sides opposite them are not congruent. But which side is opposite the larger angle? Because of the Unequal Sides Theorem, the larger side cannot be opposite the smaller angle. All possibilities but one have been ruled out. The larger side must be opposite the larger angle. QED
My student told me that he wanted to see one more indirect proof before showing him the Triangle Inequality, so why not show him this one? The initial assumption is, assume that the longer side is not opposite the larger angle. Since the angle opposite the longer side is not greater than the angle opposite the shorter side, the former must be less than or equal to the latter. And these are the two cases that lead to contradictions of Isosceles Triangle Contrapositive and Unequal Sides as listed in the above paragraph proof.
Now finally we can prove the big one, the Triangle Inequality. This proof comes from Dr. M -- but Dr. M writes that his proof goes all the way back to Euclid. Here is the proof from Euclid, where he gives it as his Proposition I.20:
http://aleph0.clarku.edu/~djoyce/java/elements/bookI/propI20.html
Here is the two-column proof as given by Dr. M. His proof has eight steps, but I decided to add two more steps near the beginning. Step 1 is the Given, and Step 2 involves extending a line segment, so that it's similar to Step 2 of the Unequal Sides proof. Indeed, the proofs of Unequal Sides and the Triangle Inequality are similar in several aspects:
Triangle Inequality Theorem:
The sum of the lengths of two sides of any triangle is greater than the length of the third side.
Given: Triangle ABC
Prove: AC + BC > AB
Proof:
Statements Reasons
1. Triangle ABC 1. Given
2. Identify point D on ray BC 2. On a ray, there is exactly one point at a given distance from
with CD = AC an endpoint.
3. angle CAD = angle CDA 3. Isosceles Triangle Theorem
4. angle BAD = BAC + CAD 4. Angle Addition Postulate
5. angle BAD > angle CAD 5. Equation to Inequality Property
6. angle BAD > angle CDA 6. Substitution (step 3 into step 5)
7. BD > AB 7. Unequal Angles Theorem
8. BD = BC + CD 8. Betweenness Theorem (Segment Addition)
9. BD = BC + AC 9. Substitution (step 2 into step 8)
10. BC + AC > AB 10. Substitution (step 9 into step 7)
To help my student out, I also included another indirect proof in the exercises. We are given a triangle with two sides of lengths 9 cm and 20 cm, and we are asked whether the 9 cm side must be the shortest side. So we assume that it isn't the shortest side -- that is, that the third side must be even shorter than 9 cm. This would mean that the sum of the two shortest sides must be less than 9 + 9, or 18 cm, and so by the Triangle Inequality, the longest side must be shorter than 18 cm. But this contradicts the fact that it is 20 cm longer. Therefore the shortest side must be the 9 cm side. QED
Notice that the U of Chicago text probably expects an informal reason from the students. A full indirect proof can't be given because this question comes from Section 1-9, while indirect proofs aren't given until Chapter 13.
Returning to 2017, let's post the worksheets. First of all, since I'm now following the U of Chicago order, students are no longer responsible for a proof of the Triangle Inequality, so I only post the questions that don't depend on a proof.
On the other side, I post a review for the Chapter 1 Test. Recall that the Chapter 1 Test must be given on Day 20, or tomorrow, since Day 21 is Lesson 2-1. If there are eight or fewer lessons in a chapter, then there's a separate review day, but if there are nine lessons in a chapter, then the ninth lesson falls the day before the test.
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