General ed teachers typically have five classes. Special ed teachers in this district usually teach four classes and co-teach for one period, to add up to five classes. But this teacher is the opposite -- he teaches only one class (eighth grade US History) and co-teaches the other four! These classes are a mixture -- two English classes (one seventh, one eighth) and two math classes (both seventh).
The one class of his own is fifth period -- and fifth period, interestingly enough, just happens to be the start of today's middle school rotation. This means that I'm essentially done teaching after the first hour of the day. It goes without saying that it's not worth doing "A Day in the Life" today.
I will say a few things about today's subbing. An aide helps out in the one history class I teach -- together, the two of us agree to write two bad names (to get lunch detention). These students keep talking loudly about football and several topics other than the history assignment.
In the math classes, the seventh graders are learning about the distributive property. I notice several ideas made famous by Sarah Carter in these classes -- the students use interactive notebooks, and drawn on the board (presumably for this teacher's eighth grade classes) is Slope Dude. The regular teacher tells me that normally, her co-teacher (the man whom I'm subbing for) would divide the class into groups, sort of like Learning Centers from the Illinois State text at the old charter two years ago.
On days when I'm doing more observing than teaching, such as today, I like to notice ideas that might help me if I ever get my own classroom again in the future. The two English classes today both had subs -- a regular day-to-day sub in English 7 and a long-term sub in English 8.
I can't help but notice the long-term sub's struggles today and compare them to my own struggles especially from the old charter two years ago. It's obvious that today is one of her first days here. And indeed, one of the assignments today is for the students to divide into groups and discuss possible rules (within certain guidelines, of course) in preparation for writing a Behavior Contract. This is very similar to what I did on the second day of school two years ago at the old charter.
But throughout this assignment as well as the next (when she starts reading My Brother Sam Is Dead, a novel about the Revolutionary War), the students continually talk. This immediately reminds me of what happened in my class two years ago. First the students talk even when discussing rules. Then they talk through every lesson. Then they find other rules to break besides the "no talking" rule. In the end, the students take over the class.
For me, the final straw is when the phone rings, and many of the students continue to talk loudly while the long-term sub is answering. When she hangs up, I intervene. I tell the class that just because they have a new teacher, they can't keep talking and talking throughout the lesson.
After class, she tells me that for this period, her co-teacher (the man whom I subbing for) normally runs the class, and so today is the first day when she's in charge. She also asks me for some tips. and I tell her what I should have learned two years ago -- the "no talking" rule is the most important. If the students believe they can talk whenever they want, then they'll think they can break every rule.
In some of today's classes, I see the acronym "CHAMPS" written on the board. These letters stand for the classroom rules that are in effect at any given time. "C" stands for Conversation, and this can be set to S (silent), 1 (whisper), 2 (group talk), or 3 (normal voice). At the old charter, I tried to convince my students to be quiet by informing them "We're learning a new lesson now." But all that did was invite comments such as "Don't worry, we can learn this without being quiet!" -- and then, of course, they never learn the distributive property or whatever I'm teaching.
Instead, I should have done something similar to "CHAMPS" -- when the "C" indicator is set to silent, the students must be silent because I said so, and because they'll be punished if they aren't, even if they think they can learn without being silent. And I should have let them know this starting from the very day that we discussed the rules. (By the way, if I recall correctly, the "H" indicator stands for Help, and it might be set to R for raise your hand, N for no help during a test, and others such as ask your group for help or "three before me.")
In the end, I predict that this long-term sub will correct herself in a few days and become a very successful teacher. But for me, it might be too late. I never made the necessary corrections to my classroom management, and so I didn't finish the year at the old charter.
Meanwhile, this is my first post after Daylight Saving Time and the clocks falling back one hour. It is a tradition for my blog to discuss the DST debate right around each time change, and as you can see, this post is no exception.
Let me start with something I definitely wrote about last year -- the Kansen Chu bill that brings Year-Round DST to California. As it turns out, this bill has qualified for the ballot -- and indeed, it appears as Proposition 7.
Tomorrow, of course, is Election Day. This means that, unlike most of my previous DST posts, today's post is high stakes. Every time we change the clock, there's a debate about whether we should keep DST as it is or introduce Year-Round Standard Time or DST. But tomorrow, it's not merely talk anymore -- tomorrow we can do something about it.
Let me post a little of what I wrote last year about Year-Round DST:
But if I were to choose a single clock to use year-round, I'd personally prefer Year-Round DST. Part of this could be that I don't have young children. Parents of young children often prefer Year-Round Standard Time as sunrise and sunset on this clock are closer to the wake and bedtimes of children, while adults without young children prefer Year-Round DST, where sunrise and sunset are closer to the wake and bedtimes of adults.
Notice that legally, there is no such thing as Year-Round DST. Officially, Year-Round DST is actually the Year-Round Standard Time of the next time zone. So my favored Year-Round DST in California is formally Year-Round Mountain Standard Time. States have the power to adopt Year-Round Standard Time or have a biannual clock change, but they don't have the power to change time zones (which is what Year-Round DST actually is) -- instead it requires an act of Congress.
Because of this, Prop 7 indeed requires an act of Congress to implement fully. It's not as if we can just set the clocks forward to DST on Wednesday in the event that Prop 7 passes -- the biannual clock changes must continue until Congress approves the change.
Meanwhile, at the time of the spring time change I wrote about efforts in the state of Florida to implement Year-Round DST. Here's what I wrote:
The “Sunshine Protection Act” means that Florida would not set their clocks back in the fall, when the rest of the Eastern United States does. This change would give Florida residents more sunshine in the evening during the winter.
Many states have made similar proposals, including the Kansen Chu bill in California [in other words, Prop 7]. But this is the first time such a bill has actually passed in a state legislature.
The bill now awaits the governor's signature -- but that's not enough for the law to have effect. Since it's a Year-Round Daylight Saving Time bill, Congress must approve of the change. Recall that officially, there's no such thing as Year-Round DST. Instead, Year-Round DST is actually Year-Round Standard Time for the next time zone. The new time zone is often known as "Atlantic Time" in parts of Canada -- and closer to Florida, Puerto Rico would also be in this time zone.
Instead, Governor Rick Scott has indeed signed the bill, but Florida still has two time zones and clocks still changed over the weekend, because Congress has not acted. Notice that unlike Florida, the California Legislature can't do anything unless Prop 7 passes. This is because DST was first implemented in California by a similar ballot measure in the 1940's -- and anything implemented by a ballot measure can only be undone by another ballot measure.
A main concern of Year-Round DST is late winter sunrise and its effect on school start times. There was an earlier proposal to require middle and high schools to start later. Even though this isn't related to Prop 7, later school start times and Year-Round DST fit together. Later school start times allow school to start after sunrise even under Year-Round DST, and the extra hour of afternoon light provided by Year-Round DST allows time for after-school practice before sunset even if school starts and ends later.
But Governor Jerry Brown vetoed the later school start time bill. I wonder, if Prop 7 passes and Congress makes it effective, whether he (or the winner of tomorrow's gubernatorial election) would reconsider the later start time bill.
I've mentioned in past posts that late sunrises under Prop 7 would be especially problematic in Northern California. I can't help but notice that Northern California newspapers are more likely to reject Prop 7 while Southern California papers are more likely to support it. For example, let's look at newspapers from the six largest cities in California:
Sacramento (No on Prop 7)
https://www.sacbee.com/opinion/election-endorsements/article218732660.html
San Francisco (No on Prop 7):
https://www.sfchronicle.com/opinion/editorials/article/Chronicle-Recommends-No-on-Prop-7-13242628.php
San Jose (No on Prop 7):
https://www.mercurynews.com/2018/08/18/editorial-for-kids-safety-vote-no-on-year-around-daylight-saving-time/
Fresno (No on Prop 7):
https://www.fresnobee.com/opinion/editorials/article218937670.html
Los Angeles (Yes on Prop 7)
http://www.latimes.com/opinion/editorials/la-ed-proposition-7-endorsement-20180929-story.html
Of these six cities, the four north of the Grapevine oppose Prop 7 while LA endorses it. There's one more large city to consider -- San Diego. Actually, SDUT endorses Prop 7, but with a caveat -- SDUT actually prefers Year-Round Standard Time to Year-Round DST:
http://www.sandiegouniontribune.com/opinion/editorials/sd-proposition-7-california-time-switch-20181026-story.html
Notice that adolescents are mentioned in this article. Clearly for this age group, Year-Round DST with a late start is preferable to Year-Round DST with an early start, but Year-Round Standard Time with a late start, as SDUT suggests, is even more preferable (provided that the main concern is making teens alert in the morning as opposed to giving them daylight for after-school sports). In any case, a single Year-Round clock (whether Standard or DST) is more preferable in SoCal, where the swing between summer sunrise and winter sunrise is less extreme than in NorCal.
On the way home from work day, I heard the DJ on the radio endorse Prop 7. He points out that going home from work in the dark is depressing -- and that its effect on school students in the morning is harmless, since today the sun would rise around 7:30 (still before school) instead of 6:30. But his argument is flawed -- today, November 5th, is not December 21st, the winter solstice. The sun rises almost an hour later on December 21st than November 5th. Just because there's enough daylight the morning of November 5th for students, it doesn't mean that there's enough on December 21st. (While sunrise changes drastically from November 5th to December 21st, sunset doesn't. It's all because of the Equation of Time.)
Before we leave this issue, there's one more proposal, if SoCal and NorCal can never agree:
http://www.zocalopublicsquare.org/2018/08/06/california-needs-three-time-zones/ideas/connecting-california/
Notice that Joe Mathews, the author of this proposal, gives the opposite of what Florida wanted -- instead of a state in a single time zone, this proposal divides a state into three time zones. Here NorCal is on Year-Round Standard Time (avoiding late winter sunrises for students) while SoCal is on Year-Round DST (allowing us to sync up with Arizona).
Here the proposed dividing line is longitude 120W, which is an interesting choice because 120W is the longitude on which Pacific Time is based. The proposal places Central California counties split by this longitude (including Fresno) n a third time zone which keeps the biannual clock changes.
This idea is interesting, though I'd admit I'd like it more if it matched up with Prop 7 endorsements of the newspapers in the respective cities. To make the time zone match the endorsements, we should follow the suggestion of Mathews and place LA on Year-Round DST (Cactus Time), but then all counties on or west of 120W would keep the status quo (Almond Time). But then there's no way to make San Diego fit the SDUT endorsement of Year-Round Standard Time.
I do like his name "Cactus Time" though -- in past posts, I've mentioned the Sheila Danzig plan of dividing the country into two time zones two hours apart. "Arizona Time" is a good name for the western time zone (since the Grand Canyon State already observes this time) while "Florida Time" is a good name for the eastern time zone (Danzig's home state, as well as my own recommendation for the Sunshine State above). Then instead of "Arizona Time" and "Florida Time," we can honor spring training baseball and call the time zones "Cactus Time" and "Grapefruit Time."
Hmm, dividing California into three -- where have I heard this before? That's right -- the Three Californias initiative would have been Prop 9 but it was dropped. (Then again, Three Californias had nothing to do with DST or time zones.) Well, as long as my house and the districts where I work aren't split into different time zones....
Anyway, so how will I vote on Prop 7 tomorrow? In the past, I said that I support Year-Round DST for California provided that another state join in -- Nevada. (Otherwise, in winter Nevada would be surrounded by states on UTC-7 while the Silver State itself is on UTC-8.) Prop 7 doesn't put California on Year-Round DST since it requires Congressional action.
Therefore I'll vote "yes" on Prop 7 tomorrow, in the hopes that Nevada might agree to adopt the same Year-Round clock as well. Then Congress could allow California and Nevada to adopt Arizona Time (or Cactus Time) simultaneously. If Nevada doesn't agree, then California should keep the status quo (the biannual clock change).
Since this has become my Election Day post, let me comment on another race on the California ballot, namely State Superintendent. It's generally accepted that of the two candidates, Tony Thurmond is pro-public school while Marshall Tuck is pro-charter. I've worked at both public schools and charters, so I can't fully take either side. Both men are from the same party (Democrats), and so party affiliation makes no difference.
I currently sub in two public districts, which probably pushes me towards Thurmond. If I were still working at the old charter from two years ago, then this might have made me lean towards Tuck.
As for the top of the ticket, it's hard to me to find information separating the two candidates, Gavin Newsom (the current Lieutentant Governor) and John Cox. In the past few days, Newsom expressed support for universal preschool. He is the favorite to defeat Cox, which probably explains why newspapers have less to report on Cox's educational views:
http://www.latimes.com/politics/la-pol-ca-governor-race-gavin-newsom-education-20181103-story.html
Oops -- I spent too much time in this post discussing DST. Let's get back to the U of Chicago text.
Lesson 5-7 of the U of Chicago text is called "Sums of Angle Measures in Polygons." This is one of the few lessons in this part of the book that is the same in both the Second and Third Editions.
This is what I wrote last year about today's lesson:
Lesson 5-7 of the U of Chicago text discusses the sum of the angle measures in polygons, including triangles, quadrilaterals, and higher polygons. To me, this is the most arithmetic- and algebra-intensive lesson in all of the first semester.
The lesson begins with a discussion of Euclidean and non-Euclidean geometry. The 19th-century mathematician Karl Friedrich Gauss wanted to determine whether Euclidean geometry was true -- that is, that it accurately described the measure of the earth -- by experiment. The text shows a photo of three mountaintops that Gauss used as the vertices of a triangle, and the mathematician found that the sum of the angle measures of the triangle was, to within experimental error, 180 degrees.
Later on, the text states that if Gauss could have used a larger triangle -- say with one vertex at the North Pole and two vertices on the equator -- the angle-sum would have been greater than 180. The geometry of a sphere is not Euclidean, but is a special type of non-Euclidean geometry -- often called spherical geometry. As stated in the text:
"In a plane, two perpendiculars to the same line cannot intersect to form a triangle, but this can happen on a sphere. The surface of the earth can be approximated as a sphere. A triangle formed by two longitudes (north-south lines) an the equator is isosceles with two right base angles! Since there is a third angle at the North Pole, the measures add to more than 180 degrees. Thus neither the Two Perpendiculars Theorem nor the Triangle-Sum Theorem works on the surface of the earth."
But hold on a minute. It's obvious that the Triangle-Sum Theorem only holds in Euclidean geometry, as its proof uses the Alternate Interior Angles Consequence that depends on the Fifth Postulate. But we were able to prove the Two Perpendiculars Theorem on this blog, without using any sort of Parallel Postulate at all! So the Two Perpendiculars Theorem ought to hold for all types of geometry, both Euclidean and non-Euclidean -- yet it clearly doesn't hold for spherical geometry.
The truth is that spherical geometry differs from Euclidean geometry much more strongly than hyperbolic geometry differs from Euclidean. We can obtain hyperbolic geometry from Euclidean simply by dropping the Fifth Postulate and replacing it with an axiom stating that there are many parallels through a point not on the line. But we can't obtain spherical geometry in a similar way.
First of all, what exactly is a line in spherical geometry? (Recall that line is one of the undefined terms, so we can't rely on its definition.) Any figure that we think is a "line" on earth goes all the way around the world, and so is actually a circle. What we want is for a "line" to be the shortest distance between two points. Notice that smaller circles on the globe clearly look curved, but larger circles that go around the world look like straight lines to a traveler. Therefore the most "linear" circle is the largest possible circle -- one that shares a center with the earth. This is called a great circle -- and this is why the example in the text mentions two longitudes and the equator -- these are great circles. But the so-called "parallels of latitude" are not great circles and so are not "lines" (geodesics).
Now what postulates does this spherical geometry violate? Notice that there are no parallel lines on the sphere, because any two great circles intersect. (Once again, note that "parallels of latitude" are not great circles.) Any two longitudes meet at the poles, and so the Unique Line Assumption part of the Point-Line-Plane postulate is violated -- through the poles there are infinitely many lines rather than just one.
But any two great circles that intersect at the North Pole must intersect at the South Pole. And any two great circles that intersect at one point intersect at the point directly opposite that point -- often called the antipodes, or antipodal point. So one way to avoid this problem is to declare that two antipodal points are actually one point. The resulting geometry is called elliptic geometry.
Yet elliptic geometry still violates the postulates. Here I link to David Joyce's website for more discussion of elliptic geometry:
http://aleph0.clarku.edu/~djoyce/java/elements/bookI/propI16.html
Notice that this is a link to the first theorem of Euclid that fails in elliptic geometry. It is the Triangle Exterior Angle Inequality Theorem, or TEAI. Dr. Franklin Mason follows Euclid and uses the TEAI to derive the Parallel Tests. The Parallel Tests do not hold in elliptic geometry (of course not, since they prove lines parallel and there are no parallel lines).
In discussing which of Euclid's five postulates that fail in elliptic geometry, the link above writes:
Elliptic geometry satisfies some of the postulates of Euclidean geometry, but not all of them under all interpretations. Usually, I.Post.1, to draw a straight line from any point to any point, is interpreted to include the uniqueness of that line. But in elliptic geometry a completed “straight line” is topologically a circle so that any pair of points on it divide it into two arcs. Therefore, in elliptic geometry exactly two “straight lines” join any two given “points.”
Also, I.Post.2, to produce a finite straight line continuously in a straight line, is sometimes interpreted to include the condition that its ends don’t meet when extended. Under that interpretation, elliptic geometry fails Postulate 2.
Both of these are essentially part of our Point-Line-Plane Postulate. So this is the postulate that we'd have to rewrite if we want elliptic or spherical geometry. Our other postulates still hold -- we can still measure angles, we can still perform reflections, and we still have plane separation (of course, we'd call these halves "hemispheres").
Notice that ironically, our Fifth Postulate still holds in spherical geometry. Of course, it's vacuously true -- there are no parallel lines, so any statement of the form "if lines are parallel, then..." or something about a line intersecting two parallel lines, is vacuously true. The Parallel Consequences are also vacuously true in spherical geometry. Playfair also holds, provided that we write it the way that Dr. M writes it on his site:
"Through a point not on a given line, there’s at most one line parallel to the given line."
(emphasis mine)
"At most one" allows for the possibility of zero parallel lines. Technically, this is the form of Playfair that we proved earlier this week -- we only showed that at most one parallel line exists. The proof that at least one parallel line exists uses rotations and is not valid in spherical geometry.
Some teachers believe that we should briefly introduce high school students to non-Euclidean geometry -- and usually spherical geometry is suggested as it describes the earth. This is opposite what a college non-Euclidean geometry class would do -- in college, the emphasis is usually on hyperbolic geometry because its theorems are more similar to those of Euclidean geometry.
[2018 update: Recall that on Friday, I also co-taught with a Geometry teacher. The class was learning Lesson 4-2 of the Glencoe text, which is on -- Triangle-Sum. And so that lesson and the U of Chicago lined up almost perfectly.]
But it's often interesting to discuss with students how spherical geometry affects the earth. A classic brainteaser often goes as follows:
http://www.murderousmaths.co.uk/books/bearpuz.htm
- A bear hunter sets out from camp and walks one mile south.
- He sees a bear and is about to shoot it.
- The bear grabs his gun and eats it.
- The hunter runs away one mile east.
- He then walks one mile north and gets back to his camp and changes his underwear.
- What colour was the bear?
The answer is that the "colour" (sorry -- this is obviously from a British website) of the bear is white, since the puzzle describes a polar bear at the North Pole. Technically, this is not a spherical triangle, since the "one mile east" is along a parallel of latitude, not a great circle. It's not even close to being a great circle -- if the hunter ran approximately six miles east he would have walked in a complete circle around the pole.
Here's another puzzle related to spherical geometry. I've tutored students who've taken a long transoceanic flight, from California to Seoul, South Korea. Along the way, the plane ends up flying very close to Alaska. The question is, why does it fly so close to Alaska, rather than take a more sensible route closer to, say, Hawaii? The answer is that the flight near Alaska is actually shorter -- the flight follows a great circle, and the great circle through California and Korea passes near Alaska.
One final related question -- any two great circles meet at two antipodal points. Where exactly is the point on the globe that is antipodal to where we are standing now? Despite all the talk about "digging a hole to China," that country is not antipodal to the United States. As it turns out, most of the Lower 48 United States are not antipodal to land at all. If one dug a straight hole through the center of the earth starting anywhere in California, we'd end up in the Indian Ocean. But Hawaii is antipodal to parts of Botswana and Namibia in Africa, and of course Alaska is antipodal to Antarctica.
Here is a link to a map that calculates antipodes:
http://www.findlatitudeandlongitude.com/antipode-map/
Returning to Euclidean geometry, here's the proof of the Triangle-Sum Theorem given in the U of Chicago text. Since the book gives a two-column proof, I'll convert it to a paragraph proof:
Triangle-Sum Theorem:
The sum of the measures of the angles of a triangle is 180 degrees.
Given: Triangle ABC
Prove: angle A + angle B + angle C = 180
Proof:
Draw line BD with the measure of angle 1 (ABD) equal to angle A. By the Alternate Interior Angles Test, lines BD and AC are parallel. Then angle 3 (the angle on the other side of BC -- the text doesn't name it, but we can call it CBE if E is a point such that BE and BD are opposite rays) has the same measure as angle C, by the Alternate Interior Angles Consequence. By the Angle Addition Postulate, angles 1, 2 (ABC), and 3 add up to 180 degrees. Substituting, we get that angles A, ABC, and C add up to 180 degrees. QED
Right now, I am a substitute teacher, but last year I interviewed for a position as a regular teacher, and one of the things I was asked to prove was the Triangle-Sum Theorem. (I also had to derive the Quadratic Formula.) I gave a two-column proof similar to the one given in the text, and the principal told me that it was satisfactory, but that he might have preferred something like this:
Statements Reasons
1. Draw line BD parallel to line AC 1. Uniqueness of Parallels (Playfair)
2. angle 1 = angle A, angle 3 = angle C 2. Alternate Interior Angles Consequence
3. angle 2 = angle ABC 3. Reflexive Property of Equality
4. angle 1 + angle 2 + angle 3 = 180 4. Angle Addition Postulate
5. angle A + angle ABC + angle C = 180 5. Substitution (steps 2 and 3 into step 4)
So we include step 3, to show students that we are making three substitutions. Calling the same angle by two different names -- angle 2 and angle ABC -- emphasizes the need for a Reflexive Property to show that this angle equals itself. The U of Chicago just changes angle 2 to angle ABC without any explanation whatsoever. On the other hand, the U of Chicago distinguishes between the Angle Addition Postulate and the Linear Pair Theorem (which is the just the Angle Addition Postulate in the case that the angles add up to 180). The hope is that this form of the proof is the best for students to understand, which is our goal.
The Quadrilateral- and Polygon-Sum Theorems are just corollaries of the Triangle-Sum Theorem, as we expect. As I mentioned earlier, calculating (n - 2)180 (and dividing by n to find each angle of a regular polygon) is the most complicated algebra that I want students to have to do in first semester of the geometry class.
http://www.findlatitudeandlongitude.com/antipode-map/
Returning to Euclidean geometry, here's the proof of the Triangle-Sum Theorem given in the U of Chicago text. Since the book gives a two-column proof, I'll convert it to a paragraph proof:
Triangle-Sum Theorem:
The sum of the measures of the angles of a triangle is 180 degrees.
Given: Triangle ABC
Prove: angle A + angle B + angle C = 180
Proof:
Draw line BD with the measure of angle 1 (ABD) equal to angle A. By the Alternate Interior Angles Test, lines BD and AC are parallel. Then angle 3 (the angle on the other side of BC -- the text doesn't name it, but we can call it CBE if E is a point such that BE and BD are opposite rays) has the same measure as angle C, by the Alternate Interior Angles Consequence. By the Angle Addition Postulate, angles 1, 2 (ABC), and 3 add up to 180 degrees. Substituting, we get that angles A, ABC, and C add up to 180 degrees. QED
Right now, I am a substitute teacher, but last year I interviewed for a position as a regular teacher, and one of the things I was asked to prove was the Triangle-Sum Theorem. (I also had to derive the Quadratic Formula.) I gave a two-column proof similar to the one given in the text, and the principal told me that it was satisfactory, but that he might have preferred something like this:
Statements Reasons
1. Draw line BD parallel to line AC 1. Uniqueness of Parallels (Playfair)
2. angle 1 = angle A, angle 3 = angle C 2. Alternate Interior Angles Consequence
3. angle 2 = angle ABC 3. Reflexive Property of Equality
4. angle 1 + angle 2 + angle 3 = 180 4. Angle Addition Postulate
5. angle A + angle ABC + angle C = 180 5. Substitution (steps 2 and 3 into step 4)
So we include step 3, to show students that we are making three substitutions. Calling the same angle by two different names -- angle 2 and angle ABC -- emphasizes the need for a Reflexive Property to show that this angle equals itself. The U of Chicago just changes angle 2 to angle ABC without any explanation whatsoever. On the other hand, the U of Chicago distinguishes between the Angle Addition Postulate and the Linear Pair Theorem (which is the just the Angle Addition Postulate in the case that the angles add up to 180). The hope is that this form of the proof is the best for students to understand, which is our goal.
The Quadrilateral- and Polygon-Sum Theorems are just corollaries of the Triangle-Sum Theorem, as we expect. As I mentioned earlier, calculating (n - 2)180 (and dividing by n to find each angle of a regular polygon) is the most complicated algebra that I want students to have to do in first semester of the geometry class.
END
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