I usually think of a "multi-day subbing assignment" as occurring on consecutive days. Thus it's strange to consider a multi-day assignment as one where the two days are separated by a weekend -- much less a three-day weekend (since, as you recall, that district was closed on November 1st) -- let alone a three-day weekend where my other district was open and I actually subbed in said district.
Yet that's exactly what this is. The students have seen the neither the inside of a classroom nor their regular English teacher since Halloween. I covered this teacher for Days 47 and 48 according to her district's calendar. Therefore it is a multi-day assignment (because what I did in between those days doesn't matter). On Halloween she was out for meetings, but no one knows why she's out today, not even her aide.
My rule for "A Day in the Life" is that I do it for multi-day assignments, even if it's special ed with aides and co-teachers. (I once did "A Day in the Life" for two assignments separated by a day, though I didn't sub in the other district on the in-between day.) I figure that classroom management on my part becomes more significant on multi-day assignments when the students haven't seen their teacher for several days, even if there's an aide.
Today is my fourth visit to this class this year, my second visit in the past five days, and hence my first multi-day assignment of the year. So here is today's "A Day in the Life":
8:15 -- This school always starts with homeroom and first period. This is a study skills class.
The aide tells me that while the class is usually treated as a study hall, it's often awkward on Mondays and after long weekends, since the students might not have any assignments to do.
I don't always sing songs on special ed days, but with so much time and so littlework needed to be done, I choose a song. Back when I was at the old charter school, I associated "One Billion Is Big" (Fat Boys, Square One TV) with the first week of November, and so this is what I rap today.
Four students (two guys, two girls) end up on the good list. The girls actually get on the list for helping the teacher set up the wall calendar for November. (It's one of those dry-erase wall calendars used in schools.)
One guy, meanwhile, is acting up. Halloween was actually his first day in this class, since he's transferred from elsewhere in the district. Today is his second day, but he still hasn't met the regular teacher yet.
It happens too often in special ed classes -- after being on the bottom of the class for so many years, a student starts to become disillusioned about school in general. On Halloween, he blurted out that he's "not smart," and so he feels that there's no need for him to do any work. That day, I reminded him that the Washington Nationals had won the World Series the previous night.
Imagine if the Nats, after having failed to win the World Series for half a century (starting as the Expos), decided not even to try to win in their 51st year? Or from an individual player perspective, if Stephen Strasburg, just because he didn't win the World Series in his first nine years, decided not even to try to win in his tenth year? Or if Ryan Zimmerman, just because he didn't win the World Series in his first fourteen years, decided not even to try to win in his fifteenth year? Anyway, that's what it looks like when the new guy, just because he was at the bottom of his class for seven years, decides not even to try in the eighth grade. (Indeed, I tried to make this argument three years ago at the old charter school -- the year that the Chicago Cubs won their championship in over a century!)
Today, he isn't focused on schoolwork, but he does ask me why some words like "pterodactyl" start with a silent "p." I reply that it's because the word comes from Greek. He learns something today -- and I let him know this. He's now on the road to learning more things and turning around his educational career, just as the Nats turned things around after starting the season 19-31.
I sincerely hope that this student will at least consider choosing the other path.
9:20 -- First period ends. All classes after first period rotate at this school -- but on Mondays, the classes happen follow the natural numerical order.
Second period is the teacher's conference period. But I'm given a class to cover -- the regular teacher for that class arrives at school, teaches first period, gets sick, and goes home. So she must have period coverage the rest of the day.
This class turns out to be seventh grade science. As usual, it's time for me to compare this class to the science that I failed to teach at the old charter school.
Well, today's lesson is all about chemical changes. This marks another key difference between the old California and new NGSS Standards -- all of physical science used to fall in eighth grade, but now, Pre-Chemistry falls in seventh grade while Pre-Physics remains in eighth.
The students are just beginning to balance chemical equations. Notice that this is indirectly related to moles (and thus Mole Day), since an equation such as:
2 H_2 + O_2 -> 2 H_2O
means "two molecules of hydrogen gas combine with one molecule of gas to form two molecules of water" or "two moles of hydrogen gas combine with one mole of gas to form two moles of water."
Since Mole Day was just twelve days ago, I feel justified in mentioning this to the students. I'm not quite sure whether the seventh grade curriculum actually mentions "moles" -- and indeed, at this level, the students might not yet be comfortable with scientific notation, which is why we might not want to introduce Avogadro's number as 6.02 * 10^23 yet.
Well, today I only tell them that Avogadro's number is more than 10^23 -- after all, we need only 10^23 to establish Mole Day as 10/23, October 23rd. Thus when the students learn about the Law of Conservation of Mass, I can say that we know that mass is conserved not by measuring molecules (which are too small to measure), but by measuring moles. Since molecules are so tiny, we need Avogadro's number to be huge in order for the mole to be a macroscopic, measurable quantity.
With no aide in this gen ed class, I definitely sing in this period. I've already prepared "One Billion Is Big" today, but notice that it's easy to modify the refrain so that it fits my science lesson:
Science Refrain:
One billion is big,
Avogadro's number is bigger.
Ten to the 23rd power,
That's one mole!
One boy, clearly fascinated with large numbers, informs me that one decillion is 10^33. I reply that 10^23 works out to be 100 sextillion. (Avogadro's actual number is more like 602 sextillion. In fact, the official definition of this constant is 602 sextillion, 214 quintillion, 76 quadrillion.)
The students today read a copy of the science text. I suspect that this is from a copy of a pre-NGSS eighth grade physical science text -- that is, this school's solution to the lack of Preferred Integrated texts is simply to copy pre-NGSS texts. The students take notes on a worksheet -- this might be part of an interactive notebook, but I don't see any such notebooks today. The worksheets are two-sided, meaning that gluing them into a notebook would be awkward.
OK, so let's think about science at the old charter school. The year I was there would be considered the first year of the California-to-NGSS transition, with only sixth grade having full NGSS. Thus this lesson, like all of physical science, would have been thought of as part of eighth grade my first year.
Of course, I only barely taught the main Pre-Chem and Pre-Physics lessons that year -- and for neither subject did I bother to create any songs that year. Otherwise I would have had an excellent Pre-Chem song to perform in class today.
As I explained back on Mole Day, I'm now torn as to whether I should have timed a Pre-Chem lesson to Mole Day that year. Since it's in late October, it's slightly later than when I should have introduced science lessons. Metric Week might have served as the ultimate deadline -- but then again, I could have followed a mid-October metric unit with a late October Pre-Chem unit. Then again, it didn't help that October 23rd fell on a Sunday that year.
I mentioned that elsewhere in this district, another seventh grade science teacher timed the metric unit to Metric Week, but the Pre-Chem unit was nowhere near October 23rd (even or November 23rd for that matter). I have no idea whether today's teacher started the current Pre-Chem unit by Mole Day -- if I were in that situation, I might have tried speeding up or skipping a few lessons (to return to later) just to make Pre-Chem land on Mole Day. (And of course, I don't know whether today's teacher had a metric unit near October 10th or not.)
I place five or six names on the good list in this class for reading out loud. One boy is disturbing others, and I end up kicking him out and sending him to the room next door.
10:15 -- Second period leaves for snack.
10:30 -- Third period arrives. This is the first of two eighth grade English classes.
These students have read a short story -- Roald Dahl's The Landlady. Their assignment is to write a narrative based on the reading. (In other classes, seventh graders have done a similar writing assignment based on Rudyard Kipling's Rikki-Tikki-Tavi.)
I place three or four names on the good list for writing the narrative and making suggested changes to their writing (such as avoiding the overuse of "said"). One student is on the bad list -- once again, it's for disturbing others. There is no time for songs in this class.
11:25 -- Third period leaves. Fourth period is the first for co-teaching a seventh grade English class.
These students (having already written narratives) are now working on writing for research. The regular teacher introduces a mnemonic: RACES-ACES. Here the letters stand for "Restate, Answer, Cite, Explain, and Significance" -- and here the repetition of ACES means that the students should continue the paragraph by answering again, and so on. The class is divided into eight groups, and each group is to create a poster on one part of RACES-ACES to present to the other students.
There is no time for songs in this class.
12:20 -- Fourth period leaves for lunch.
1:05 -- Fifth period arrives. This is the second for co-teaching a seventh grade English class.
For some reason, there's only enough time for six of the eight groups to present -- yet somehow, I have time to sing one verse of "One Billion Is Big" (the original, not the science version).
2:00 -- Fifth period leaves. Sixth period is the second of two eighth grade English classes.
2:30 -- The aide leaves, since this is the end of her contracted working hours. This means that I cover the latter half of this class unaided. So of course, I tell the students that I will sing "One Billion Is Big" if they earn it.
While two or three students are placed on the good list, I must also write two guys on the bad list for throwing objects (first a coin, then a rubber band).
During the last few minutes of class, I finally tell the students what they must do to earn the rap -- put away the Chromebooks that they've been writing the narratives on. It makes sense that if I use a song incentive for a later period but not an earlier period, then it should be for something that only the later period needs to do, which is clean up. The students put away the Chromebooks, placing them in the cart to charge (with color-coded slots) with enough time for me to sing one verse of the song.
2:55 -- Sixth period leaves, thus completing a long, hard-working day.
I reflect upon my day, especially the 1 1/2 periods that I teach unaided (second period science and sixth period English). I wonder, is there anything I could have done to prevent having to send the seventh grade science student out of the room, or prevent the eighth graders from throwing objects (especially the latter, since throwing objects is dangerous)?
Indeed, on Halloween, I name sixth period as the best class of the day, but today, they are nowhere near the best (which I consider to be third period). Of course, this is an artifact of the period rotation, as sixth period on Thursday was before the aide's departure, whereas today, half of the period is after she leaves.
Still, I wonder whether there's anything I could have done better during that half-period. It's not as if the two guys involved were working hard on the narrative during the aide's half-period -- at the time I catch them throwing objects, each has only one written sentence displayed on his Chromebook. (If they'd been working hard with the aide, the work from that time would still be on the screen.)
There is no seating chart for this class, but I keep thinking that the two guys involved in the throwing had been sitting near the front of the room for the aide, but then moved to the back of the room in time to throw objects. Perhaps if I had kept track of where they are sitting, I might have been able to prevent the throwing.
In fifth period, I think I know now why there's not enough time for the last two groups to present. The teacher sees on the clock that it's almost 2:55, so she has the students clean up. But in reality, it's almost 1:55, not 2:55, because the clock is still set to DST. So there's still an hour left in school, and a few minutes left in the period (since each period is less than an hour). She realizes too late that she's asked them to clean up too early, and so with nothing else for the students to do, I fill the last few minutes of class with part of the "One Billion Is Big" rap. (It doesn't help that this school has period rotations, so fifth period kids in the class aren't necessarily a tip-off that there's an hour left.)
And this is a great place to segue into my biannual DST debate post.
EDIT: I no longer think so. I've cut out the DST part of this post and delayed this part to Wednesday, November 6th.
Chapter 14 of Ian Stewart's The Story of Mathematics is called "Algebra Comes of Age: Numbers Give Way to Structures." Here's how it begins:
"By 1860 the theory of permutation groups was well developed. The theory of invariants -- algebraic expressions that do not change when certain changes of variable are performed -- had drawn attention to various infinite sets of transformations, such as the projective group of all projections of space."
As the title implies, this chapter is all about Abstract Algebra (taught at the college-level). But as the quote above tells us, this was actually first inspired by Geometry and a discovery by Felix Klein:
"Now lengths and angles are the basic concepts of Euclid's geometry. So Klein realized that these concepts are the invariants of the Euclidean group, the quantities that do not change when a transformation from the group is applied."
Once again, the "Euclidean group" is the set of all Common Core isometries. We can think of this are the symmetry group of the entire plane (since every reflection, translation, and so on, maps the plane to itself). Klein figured out the symmetry group for every time of geometry except one -- Riemann's geometry of surfaces, which is more complicated:
"It does not quite fit into Klein's programme. Lie's and Klein's joint research led Lie to introduce one of the most important ideas in modern mathematics, that of a continuous transformation group, now known as a Lie group."
I've heard of Lie groups before, but I've never taken a college course where these groups are studied in more detail. Unfortunately, today's post is not the best time for me to learn and introduce this new concept, since it's already jam-packed with "A Day in the Life" and DST/Forward Time stuff. So let me just quote some of what Stewart writes about Sophus Lie and his new idea:
"He was led to such transformation groups while seeking a theory of the solubility or insolubility of differential equations, analogous to that of Evariste Galois for algebraic equations, but today they arise in an enormous variety of mathematical contexts, and Lie's original motivation is not the most important application."
Stewart tells us that the set of all rotations with a given center is a Lie group, because the composite of two such rotations is another such rotation, and each rotation corresponds to a point on a particular manifold -- a circle. He also gives the set of all 3D-rotations with a given center as a Lie group, but:
"Unlike the group of rotations of a circle, it is non-commutative -- the result of combining two transformations depends on the order in which they are performed. In 1873, after a detour into PDEs, Lie returned to transformation groups, investigating properties of infinitesimal transformations."
These are now called Lie algebras. Meanwhile, Wilhelm Killing expanded upon Lie's work:
"The upshot of Killing's work is remarkable. He proved that the simple Lie algebras fall into four infinite families, now called A_n, B_n, C_n, and D_n."
Well, there are actually a finite number of exceptions:
"Killing actually thought there were six exceptions, but two of them turned out to be the same algebra in two different guises."
Lie groups were eventually generalized to more abstract groups:
"Among the first to come close to taking this step was Arthur Cayley, in three papers of 1849 and 1854."
One such abstract group was discovered by Gauss:
"Gauss started the process when he introduced what we now call Gaussian integers. These are complex numbers a + bi, where a and b are integers."
And these can be generalized even further by replacing i with the solutions of any integer polynomial and a, b with a_1, a_2, etc., coefficients:
"Taking a_1, ..., a_n to be rational, we obtain a system of complex numbers that is closed under addition, subtraction, multiplication, and division -- meaning that when these operations are applied to such a number, the result is a number of the same kind."
These are called fields -- and if we drop the division requirement, we obtain rings instead. (Yes, these are the "system analogous to rational numbers" and "system analogous to the integers" mentioned in the Common Core Standards. The set of polynomials in a single variable is a ring, and the set of rational functions in a single variable is a field.)
Meanwhile, work continued on trying to prove Fermat's Last Thoerem:
"In 1847 Gabriel Lame claimed a proof for all powers, but Ernst Eduard Kummer pointed out a mistake."
But Lame's proof did work for all powers below 23. Stewart now returns to fields and other generalizations, such as fields where multiplication isn't commutative:
"If this law fails, we have a division ring. An algebra is like a ring, but its elements can also be multiplied by various constants, the real numbers, complex numbers or -- in the most general setting -- by a field."
Stewart discusses finite simple groups, which can't be broken into smaller groups:
"The eventual classification of all finite simple groups was achieved by too many mathematicians to mention individually, but the overall programme for solving this problem was due to Daniel Gorenstein."
One simple group doesn't seem so simple at all -- the monster:
"Its order is [larger than Avagadro's number, larger than a decillion, but less than Avagadro's number times a decillion] and is roughly 8 * 10^53."
(There's no need for me to write its exact value -- look it up yourself!) Stewart continues to discuss more conjectures in abstract algebra:
"The biggest of all is the Taniyama-Weil conjecture, named after Yutaka Taniyama and Andre Weil. This says that every elliptic curve can be represented in terms of modular functions -- generalizations of trigonometric functions studied in particular by Klein."
And this leads to the ultimate proof of Fermat's Last Theorem:
"Andrew Wiles, as a child, had dreamed of proving Fermat's Last Theorem, but when he became a professional he decided that it was just an isolated problem -- unsolved, but not really important."
Instead, Wiles proved that Taniyama-Weil implies FLT, and he ultimately proved both of them.
"It was not only algebra that became abstract. Analysis and geometry also focused on more general issues, for similar reasons."
On this note, Stewart concludes the chapter as follows:
"And what seemed little more than formal game-playing yesterday may turn out to be a vital scientific or commercial tool tomorrow."
The sidebars in this chapter are biographies of Felix Klein, Emmy Noether, and Andrew Wiles, what abstract algebra did for them, and what abstract algebra does for us.
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.
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.
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