The MTBoS, Week 4 (Day 98)

Before we get to the final lecture, let's get to the last week of the 2016 MTBoS Blogger Initiative. But I'm having a problem with this week's topic. This comes from Tina C., a Massachusetts teacher:

https://exploremtbos.wordpress.com/2016/01/31/week-4-of-the-2016-blogging-initiative/

[T]his week you’ll be sharing about a lesson. Perhaps you’ve already chosen it and compiled all your resources to share. If not, think to the week ahead and consider what you’re teaching (or observing) so that you can be sure to take a photo, gather samples of student work or record part of the conversation (#phonespockets anyone?) to be able to include direct quotes. None of these are required but all require planning ahead so think now about what you might want in your post!

In describing this topic, Tina C. quotes a more well-known MTBoS blogger, Kate Nowak, a New York curriculum developer.. Dan Meyer, the King of the MTBoS, once gave Nowak her own title -- "high priestess of the high council." Here's a link to the High Priestess's blog:

http://function-of-time.blogspot.ca/2016/01/in-defense-of-unsexy.html

Nowak doesn't want a spectacular lesson here -- she urges the MTBoS to "please be more boring." To fulfill this week's topic, we can teach an ordinary lesson -- the important part is to emphasize the interaction between the teacher and the students.

I am only a substitute teacher, which means that what I teach is based on the luck of the draw. Last week, the MTBoS Week 3 topic fell right into my lap -- I subbed for a math teacher who strongly believed in having the students question each other, when the MTBoS topic was about questions.

But this week, I'm covering a biology teacher who's out on paternity leave -- meaning that I taught very little math this week. Last week, many of the other participants struggled with their Week 3 posts while I cruised on by, and this week I'm the one who's having problems. This is why I won't consider myself a full member of MTBoS until I become a real math teacher, not just a sub. The real teachers with real classrooms and real lessons, of course, are having no trouble with Week 4.

Still, I started this Blogging Initiative, and I want to finish what I started. And so without further ado:

Date: About a year ago
The Lesson: Today I am posting a Pythagorean Theorem Activity from last year.
My Teaching: I've never taught this lesson in a classroom, but I had back when I was once a math tutor myself. I enjoy letting the students visualize exactly why the Pythagorean Theorem holds.
Student Responses: Whenever I have students try this out, it's always easier for them to complete the puzzle with the a^2 and b^2 squares than with the c^2, since it must be placed at an angle.

The worksheets that I'm posting refer to this Pythagorean Theorem activity. I'm hoping that I can teach this in the classroom soon.

Here are some math lessons that I was able to teach during the past week, despite spending most of it in a biology lab:

Date: Friday, January 29th
The Lesson: Just before being assigned to the biology class, I was placed in a middle school special ed class, but covered sixth grade math during the conference period. The students had a worksheet where they reflected points over either of the axes given their coordinates.
My Teaching: The class was already divided into six groups of six students each, so I played the Conjectures/"Who Am I?" game (MTBoS Week 2) with the students.
Student Responses: This game didn't go well at all. Group 6 was working way ahead of the game and answering most of the questions before I asked them. But Group 4 struggled with almost every question I gave them. I tried to help Group 4 out by letting them know which coordinates they had to keep and which ones they had to change the sign to. But the other groups grew restless, and I had to abide by the rules of the game by declaring Group 4's answers wrong and not awarding a point. This led to Group 4 getting upset, and one girl even started to cry.

Date: Wednesday, February 3rd
The Lesson: On the first day in the biology class, the students actually had an assignment that involves mathematical modeling. This is the classic predator-prey problem, where students had to model the populations of sea lions (the predators) and sardines (the prey), and see how the population of one affected that of the other. Each group of four students had a square foot of blue construction paper to represent the bay, 50 small squares of paper to represent sardines, and several small squares of paper to represent sea lions. For the first generation, the students would toss two sea lions into the bay, and however many sardines each sea lion was touching it would eat. Any sea lion that didn't eat three sardines would die, and for every pair that survived, a new sea lion was born. Then this would start the next generation, and students would repeat the process for 25 generations.
My Teaching: The students actually began this activity on Tuesday, the day before I arrived. I tried to understand the rules of the activity but I was confused. Based on my reading of the instructions, the sardines that were eaten were removed from the bay. But then I couldn't see how there could ever be any sea lions after the 17th generation -- during the first generation, one sea lion eats its quota of three fish while the other sea lion fails to catch any fish and dies. Then for generations 2 through 16, the one remaining sea lion eats three sardines per generation, for a total of 48 fish eaten. This leaves only two sardines available for the sea lion at Generation 17, and so it dies.
Reflection: Simulating a predator-prey model is a great idea, provided that the students are given much clearer instructions. Of course, then students would have to keep track of both the sea lions and the sardines after each generation unless they're given a simple rule, such as "Replace all 50 sardines after each generation."

Today the bio students took a test on this unit. There were two multiple-choice questions directly related to this question. The first asked, what curve shows hows population increase when there are unlimited resources? Nearly every student correctly answered "exponential" -- only a few students answered "logistics curve." The other question presents a logistics curve and asks the students to identify which of five labeled points corresponds to the carrying capacity. The correct answer is E, the rightmost point, but some students chose C, where the curve is increasing the fastest. Notice that many Common Core Math questions in modeling are similar to this one.

Date: Thursday, February 4th
The Lesson: The biology teacher has an AVID class during third period. Twice a week, on Tuesdays and Thursdays, college tutors come in to help students with various lessons in any subject. I predicted that many of the students would have questions about math -- and I was absolutely right. The class consisted of all freshmen. Most were in Integrated Math I, but some were in the Honors version of the same class. Regular students were working on systems of equations, while Honors kids were working in a chapter on radicals. I found this surprising as Integrated Math I students typically don't even work with quadratics, much less simplifying radicals. Then again, I'm not completely sure what the Honors Integrated Math I curriculum is like.
My Teaching: I first helped one of the Honors students with simplifying sqrt(144) + cbrt(8) -- where "sqrt" and "cbrt" are ASCII for square root and cube root, respectively. Of course, the question wasn't written in ASCII, but in standard radical notation with an index of 3 to indicate cube root. The student thought that maybe the 3 was a coefficient, like sqrt(144) + 3sqrt(8), until I told her that it stood for cube root. Then I moved on to a regular student who had to solve this system: {x + 5y = 28, -x - 2y = -13}, and she decided to use substitution. I told her that it was much better to use elimination for this system, since the x-terms were x and -x.
Student Responses: The girl then started to multiply one of the equations by -1. When I told her not to do so, she tried to multiply the other by -1 instead. Then one of the college tutors intervened and told me that it was better for me to use an inquiry approach (shades of the MTBoS Week 3 topic), rather than just tell her not to multiply by -1. But then the tutor added, "Don't worry. Almost every sub has tried to just tell them the answers."
Reflection: Perhaps a good question for me to have asked the girl is, why would we need to multiply an equation by -1? Then perhaps she would have learned that it's the coefficients of the variables (including their sign) that determine which equation(s) need to be multiplied.

Notice what one student wrote on her Tutorial Request Form:

Prompt: I gained a new/greater understanding of my point of confusion by/when ...
Student: When our substitute [yours truly] explained to me that it's an exponent and not a multiple.

Oh, and that reminds me! In David Kung's Mind-Bending Math, Lecture 24, his final lecture, he discusses "The Paradox of Paradoxes" -- if paradoxes so difficult to figure out, why are they fun? He explains that people naturally enjoy the mental challenge of solving a puzzle. He mentions the famous 4 T's puzzle, which has a similar answer to the c^2 puzzle -- fit the 4 T's in the box at an angle! He also summarizes all the paradoxes described throughout the course. Most real numbers are not rational, most functions are continuous nowhere, and most functions that are continuous everywhere are differentiable nowhere. (Thanks, Sam Shah!) This idea that what's most familiar is least common permeates science -- most life forms are bacteria, most animals are insects, most matter is dark matter, most ordinary matter is plasma, and so on.

Here are the three links for this week, two right above mine and then the next Blogspot. All of these appear to be Geometry lessons, so that's great:

https://alternativemath.wordpress.com/2016/02/05/geometry-constraints-and-trig/
https://edtechenthusiast.wordpress.com/2016/02/05/same-lesson-5-years-apart/
http://palmersponderings207.blogspot.com/2016/02/mtbos-blogging-initiative-week-4-teach.html

Geometry -- oh, that's right. Kung's course and the MTBoS have dominated my blog this past month, but after Lincoln's Birthday weekend, this will return to a Common Core Geometry blog. So below are images of the Pythagorean Theorem Activity and two Core-inspired activities.

For those who are reading this post as part of the MTBoS, thanks for participating in the 2016 Blogging Initiative. Hopefully there'll be a 2017 Blogging Initiative -- see you next year! Otherwise, my next Geometry post will be on Tuesday, February 9th.

Slope: Based in part on Lesson 3-4 (Day 97)

Lecture 23 of David Kung's Mind-Bending Math is called "Banach-Tarski's 1 + 1 = 1." As Dave Kung points out, this is the strangest paradox of all, one that seems to refute kindergarten-level math.

The main result of this lecture is to demonstrate that it's possible to divide a sphere into finitely many parts and put them together to create two spheres, each as large as the original. That is, for each piece of the sphere, we can use an isometry to map it to a piece of one of two new spheres.

Kung points out that the proof of Banach-Tarski is very long -- so long that he devotes this entire lecture to just this one proof. I will only give parts of his proof -- the parts that are the most relevant to Geometry.

Kung begins his proof by considering groups of "words" consisting of only a and b. There exist countably infinitely many such words. Now he wants these words to correspond to transformations, with a representing a 120-degree rotation and b a 180-degree rotation. Then ab represents the composite of a rotation of 120 degrees following one of 180 degrees -- that is, the 180-degree rotation is done first. Also, aaa, also written a^3, is the composite of three 120-degree rotations, and b^2 is the composite of two 180-degree rotations. Both of these are equivalent to a rotation of 360 degrees, which is the identity, also written as 1.

Here's an aside to discuss the notation. Why do we write ab to perform b first? Notice that the U of Chicago text does the same thing -- a o b means "a following b." After all, (a o b)(x) = a(b(x)) as opposed to b(a(x)) -- it's just how function notation works. So when the U of Chicago writes the notation r_m o r_l (in the Two Reflections Theorems for Translations and Rotations), the text really means to reflect over line l first.

But this leads to another question -- why write ab for a o b? As Kung points out, the set of all words form a group, and group theorists tend to write the group operation as multiplication -- this is why they also write the identity as 1, the multiplicative identity. But this is an endless source of confusion for trig students, where sin^2 implies multiplication as sin^2(x) = (sin x)(sin x), while sin^-1 implies composition as (sin o sin^-1)(x) = x.

Anyway, notice that ababa is not the identity 1, as when we apply these to the sphere, the centers of rotation are different. Notice that in 3D, the center of rotation is an axis. The axis for a might as well be the actual axis of the globe (from North Pole to South Pole), while the axis for b can be any angle that is not a rational multiple of 360 degrees from the axis of a. Notice that 1 radian works. (My house is approximately 1 radian from the North Pole.)

Kung proceeds by assigning every word a color (navy blue, orange, purple) as follows. Let W be a word consisting of a and b as follows:

The empty word (which is 1) is colored navy.
If W is navy, then aW is orange.
If W is orange, then aW is purple.
If W is purple, then aW is navy.
If W is either orange or purple, then bW is navy.
If W is navy, then bW is either orange or purple via a more complicated rule.

The rest of the proof is complicated, so I'll try to summarize it quickly. Not only do the words get colors, but so do the points on a sphere. Just as with the Vitali set formation from yesterday, we color two points the same color if there exists a composite of a and b mapping one to the other (where there are uncountably many colors). Then we choose (Axiom of Choice) one point of each color. This gives us a new unmeasurable set to which we can apply the words -- these map each point to another point of the same color. Finally, another trick allows us to make two copies of each point based on the color of the words -- for example, it's possible to use bW to map both all the orange words to all the navy words, and all the purple words to all the navy words, so that there are two copies of navy. If we do this enough times we generate two copies of the entire sphere.

In the Quick Conundrum Kung shows a trick for tying a knot, one that I've seen before -- start with the arms already twisted, grab each end of a rope, then untwist the arms.

Today will be my traditionalist topic for the week. Notice that the proof of Banach-Tarski is ultimately based on rotations -- which traditionalists dislike. Indeed, it may be nice for Geometry teachers and traditionalists alike to watch Kung's lectures, starting from Lecture 19 (the start of the fourth disc), or possibly even Lecture 17, to see the relationship between translations, rotations, reflections and higher-level mathematics.

I want to mention a local high school student who's been in the news lately -- Cedrick Argueta, who is currently a senior. Last year as a junior, not only did he earn a 5 on the AP Calculus exam, but he earned a perfect score:

http://www.latimes.com/local/california/la-me-0103-lopez-yom-teacher-20160201-column.html

I  also want to comment on the continued backlash to the Common Core tests -- in particular, the compromise of using the SAT as the end-of-year test for juniors:

https://www.washingtonpost.com/news/answer-sheet/wp/2016/02/03/dad-my-state-now-requires-11th-graders-to-take-the-sat-not-my-daughter/

Each time I read an article like this one, or a similar story about Colorado's transition from ACT to SAT as the end-of-year test, I'm starting to believe that ACT is better than SAT. But even ACT doesn't avoid all of the problems mentioned in the article -- for example, the ACT also has an optional writing section (Problem #4 from the article).

This is what I wrote last year about slope:

Section 3-4 of the U of Chicago text is where slope -- that very important concept -- is defined. The definition in the text is simple enough and is typical for most high school math texts. Here it is, rendered in ASCII:

Definition:
The slope of the line through (x_1, y_1) and (x_2, y_2), with x_1 != x_2, is (y_2 - y_1) / (x_2 - x_1).

But this isn't good enough for Common Core. Let's look at the relevant standard:

CCSS.MATH.CONTENT.HSG.GPE.B.5
Prove the slope criteria for parallel and perpendicular lines and use them to solve geometric problems (e.g., find the equation of a line parallel or perpendicular to a given line that passes through a given point).
[emphasis mine]

So we have to prove that slope works. It isn't good enough just to say that slope formula works just because we defined it to work. To see why, consider the following definition:

Definition:
The favorite food of the class containing the student S is the food that S likes to eat the most.

And now you instantly see the problem with this "definition." Naturally, not every student in the class likes the same food. The food that student S loves to eat may be a food to which student T is indifferent and a food that absolutely disgusts student U. The "favorite food" of a class depends on which student we ask to name a favorite food. Simply writing the word "Definition:" and setting the phrase "favorite food" in bold doesn't magically force everyone to have the same favorite food.

Likewise, writing the word "Definition:" and setting the word "slope" in bold doesn't magically force the slope to be the same no matter which two points we choose. It could be that if we choose points (x_1, y_1) and (x_2, y_2), we get a different slope from if we choose  (x_3, y_3) and (x_4, y_4), just as if we choose student S we get a different favorite food (or favorite song) from if we choose student T instead of S.

In mathematics, we would say that "favorite food of a class" is not well-defined. In order for slope to be well-defined, we must prove that the slope is independent of which two points we choose to plug into the formula. We can't prove that the favorite food (or song) is independent of which student we choose -- indeed, it's trivial to find a counterexample: simply declare one student's favorite to be the class's favorite, and watch all the counterexample students call out how much they don't like the declared favorite!

And so the main theorem for today is to prove that slope is well-defined. We want to prove that slope of a line is independent of which two points we choose to plug into the formula. The trick for doing this will remind us of the proof of the Distance Formula.

Let's end this post with a joke. What's the anagram of Banach-Tarski?

The answer is Banach-Tarski Banach-Tarski! Stay tuned for the last paradox ... of all paradoxes.

The Pythagorean Theorem and the Distance Formula (Day 96)

Lecture 22 of David Kung's Mind-Bending Math is called "When Measurement Is Impossible." In this lecture, Dave Kung explains that some sets simply aren't measurable.

Kung begins by describing two different types of integration. One is called Riemann integration, named for 19th century German mathematician Bernhard Riemann -- this is what students in Calculus AB and BC are asked to calculate. Students in these classes begin by learning about Riemann sums, and then the shortcuts (power rule, u-substitution, etc.) that allow them to calculate integrals without performing the Riemann sums.

Another type of integration is called Lebesgue integration, named for the French mathematician Henri Lebesgue. Kung thinks of Riemann integration as dividing a region into vertical bars, but of Lebesgue integration as dividing the region into horizontal bars instead. Closely related to the Lebesgue integral is the Lebesgue measure. The Lebesgue measure of a set S equals the integral of a certain function -- f (x) = 1 if x is an element of set S and 0 otherwise.

If a function is Riemann-integrable, then it's also Lebesgue-integrable and the two values of the integral are equal. Yet there exists functions that are Lebesgue-integrable but not Riemann-integrable, such as the function f (x) = 1 if x is rational, 0 otherwise. This function has no Riemann integral, but it does have a Lebesgue integral equal to 0. The set of rationals has Lebesgue measure 0.

The Quick Conundrum involves stacking a wheel above two other wheels, placing a ruler on top, and then sliding the ruler forward. This causes the wheels to spin in the same direction as the ruler, but much faster. Kung says that this is similar to the paradox involving the spool of thread, seen in an earlier lecture.

Now we get to the main part of the lecture -- how can there be a non-measurable set? Kung states that we desire a measure that satisfies four properties. Let's think back to the Area Postulate, mentioned in Lesson 8-5 of the U of Chicago text:

Area Postulate:
a. Uniqueness Property: Given a unit region, every polygonal region has a unique area.
b. Rectangle Formula: The area of a rectangle with dimensions l and w is lw.
c. Congruence Property: Congruent figures have the same area.
d. Additive Property: The area of the union of two nonoverlapping regions is the sum of the areas of the regions.

As it turns out, the four desirable properties of measure that Kung mentions correspond almost exactly to these four parts of the Area Postulate! We want every region to have a measure -- the only difference is that Kung mentions that the measure should be nonnegative. Second, Kung's example is with 1D measure (linear, not square units), so he gives the 1D analog of the Rectangle Formula -- the Interval Formula, which is part of the Ruler Postulate (Point-Line-Plane in the U of Chicago).

Third, we want congruent figures to have the same area, but what does it mean to be congruent? That's easy -- there should exist an isometry mapping one to the other. So we see that Lebesgue used the idea of rigid transformations about a century before there was a Common Core!

We put these all together and come up with four reasonable properties of measure. But, as it turns out, the assumption that such a measure exists leads to a contradiction. As Kung points out, we've seen this several times before -- we come up with a set of reasonable properties for a system to have, and we prove that only some of the properties can be satisfied, not all. We think back to Godel's Incompleteness Theorem, Arrow's Impossibility Theorem, and mapping the globe onto a plane.

The counterexample involves what we call a Vitali set (Italian mathematician Giuseppe Vitali). Kung considers the interval [0, 1), but to make it easier, he places these numbers not on a straight line, but on a circle of circumference 1. He then colors the points such that any two points that differ by a rational have the same color -- this requires uncountably many colors. The Vitali set is formed by choosing (Axiom of Choice) one point of each color.

The Vitali set is uncountable, but the circle consists of countably many such sets -- indeed, Kung shows that simply rotating the Vitali set by a rational number gives another Vitali set, and the circle is the disjoint union of countably many such sets, one for each rational. Each of these infinitely many sets must have the same length since an isometry (a rotation) maps one to the other, yet they must all add up to 1, the circumference of the circle, a contradiction -- the sum of infinitely many zeros is 0, and the sum of infinitely many equal nonzero numbers is infinity. So the Vitali set is nonmeasurable.

So is the Area Postulate of Lesson 8-5 wrong? Notice that the text is careful to write that every polygonal region has a unique area. Without the word "polygonal," the region could be constructed similar to a Vitali set and the claim would be false.

By the way, you may be wondering, where is the MTBoS Week 4 post? Actually, I'm having a problem with Week 4. You see, the Week 4 prompt is supposed to be all about an actual lesson that we're teaching this week. As a sub, I'm not guaranteed to teach any math lessons this week. In fact, let's look at my subbing load this week:

Monday: Subbed in a special ed English/history class for the second day -- first day was last Friday.
Tuesday: Rejected an elementary school assignment. I'm not comfortable teaching the youngest students -- especially kindergartners, and the assignment was at a K-3 school. (I mentioned last year how some districts, including my own, separate K-3 and 4-5 at certain schools.)
Wednesday: Subbed in a high school biology class. This is another teacher who will be on paternity leave -- his lesson plans extend until February 19th. I'm at least assigned to cover this class for the rest of the week -- perhaps a biology specialist may be found to cover the next two weeks.

This bio teacher has sixth period conference, and I was assigned to cover for an Algebra II teacher (grandfathered for older students -- she also teaches Integrated Math II) who had to leave early for a dentist appointment. I ended up covering only half of the period. This class was learning about radicals, but there was a review worksheet on factoring, and I was able to help one student on this factoring worksheet.

Not only that, but today's biology class involved some math. They were working on a predator-prey model where they had to simulate the interaction between two species (sea lions and sardines). I was confused with how the question was worded, and so I was only able to help the students with graphing the data.

So I covered half a period of actual math, plus mathematical modeling in biology where I didn't fully understand the model. The bio teacher also has a period of AVID, and twice a week (Tuesdays and Thursdays) college tutors come in to help the students with different subjects. Sometimes I join in when I see them working on math. Otherwise I'm locked into the bio job, so there's no chance that I'll cover a real math class this week.

Putting it all together, this doesn't sound appropriate for a MTBoS post. I'll have to decide soon what I'm going to do about this week's post.

Now let's get back to Geometry class. It's been some time since I mentioned Theoni Pappas and her Mathematical Calendar, but I just can't help mentioning today's question:

(Figure shows AE and BD intersecting at C with right angles ABC, EDC.)

Triangle ABC can be shown to be similar to Triangle EDC most easily by:

(1) SSS~
(2) SAS~
(3) AA~
(4) not possible

The answer is (3) -- and notice that today's date is the 3rd. Pappas likes to set up the questions so that the correct answer is the date. Today's question fits perfectly as a warm-up, since it follows directly from yesterday's lesson on SSS~ and AA~.

This is what I wrote last year about today's lesson:

Lesson 8-7 of the U of Chicago text is on the Pythagorean Theorem, and Lesson 11-2 of the same text is on the Distance Formula. I explained yesterday that I will cover these two related theorems in this lesson.

The Pythagorean Theorem is, of course, one of the most famous mathematical theorems. It is usually the first theorem that a student learns that is named for a person -- the famous Greek mathematician Pythagoras, who lived about 2500 years ago -- a few centuries before Euclid. I believe that the only other named theorem in the text is the Cavalieri Principle -- named after an Italian mathematician from 400 years ago. Perhaps the best known named theorem is Fermat's Last Theorem -- named after the same mathematician Fermat mentioned in yesterday's lecture. (We discuss some other mathematicians such as Euclid and Descartes, but not their theorems.)

It's known that Pythagoras was not the only person who knew of his named theorem. The ancient Babylonians and Chinese knew of the theorem, and it's possible that the Egyptians at least knew about the 3-4-5 case.

We begin with the proof of the Pythagorean Theorem -- but which one? One of my favorite math websites, Cut the Knot (previously mentioned on this blog), gives over a hundred proofs of Pythagoras:

http://www.cut-the-knot.com/pythagoras/

The only other theorem with many known proofs is Gauss's Law of Quadratic Reciprocity. Here is a discussion of some of the first few proofs:

Proof #1 is Euclid's own proof, his Proposition I.47. Proof #2 is simple enough, but rarely seen. Proofs #3 and #4 both appear in the U of Chicago, Lesson 8-7 -- one is given as the main proof and the other appears in the exercises. Proof #5 is the presidential proof -- it was first proposed by James Garfield, the twentieth President of the United States. I've once seen a text where the high school students were expected to reproduce Garfield's proof.

So far, the first five proofs all involve area. My favorite area-based proof is actually Proof #9. I've tutored students where I've shown them this version of the proof. Just as the Cut the Knot page points out, Proof #9 "makes the algebraic part of proof #4 completely redundant" -- and because it doesn't require the students to know any area formulas at all (save that of the square), I could give this proof right now. In fact, I was considering including Proof #9 on today's worksheet. Instead, I will wait until our next activity day on Friday to post it.

But it's the proof by similarity, Proof #6, that's endorsed by Common Core. This proof has its own page:

http://www.cut-the-knot.org/pythagoras/PythagorasBySimilarity.shtml

Here is Proof #6 below. The only difference between my proof and #6 from the Cut the Knot webpage is that I switched points A and C, so that the right angle is at C. This fits the usual notation that c, the side opposite C, is the hypotenuse.

Given: ACB and ADC are right angles.
Prove: BC * BC + AC * AC = AB * AB (that is, a^2 + b^2 = c^2)

Statements                                Reasons
1. ADC, ACB, CDB rt. angles   1. Given
2. Angle A = A, Angle B = B     2. Reflexive Property of Congruence
3. ADC, ACB, CDB sim. tri.     3. AA Similarity Theorem
4. AC/AB = AD/AC,                 4. Corresponding sides are in proportion.
    BC/AB = BD/BC
5. AC * AC = AB * AD,           5. Multiplication Property of Equality
    BC * BC = AB * BD
6. BC * BC + AC * AC =         6. Addition Property of Equality
    AB * BD + AB * AD
7. BC * BC + AC * AC =         7. Distributive Property
    AB * (BD + AD)
8. BC * BC + AC * AC =         8. Betweenness Theorem (Segment Addition)
    AB * AB

I mentioned before that, like many converses, the Converse of the Pythagorean Theorem is proved using the forward theorem plus a uniqueness theorem -- and the correct uniqueness theorem happens to be the SSS Congruence Theorem (i.e., up to isometry, there is at most one triangle given three side lengths). To prove this, given a triangle with lengths a^2 + b^2 = c^2 we take another triangle with legs a and b, and we're given a right angle between a and b. By the forward Pythagorean Theorem, if the hypotenuse of the new triangle is z, then a^2 + b^2 = z^2. (I chose z following the U of Chicago proof.) Thenz^2 = c^2 by transitivity -- that is, z = c. So all three pairs of both triangles are congruent -- SSS. Then by CPCTC, the original triangle has an angle congruent to the given right angle -- so it's a right triangle. QED

Interestingly enough, there's yet another link at Proof #6 at Cut the Knot, "Lipogrammatic Proof of the Pythagorean Theorem." At that link, not only is Proof #6 remodified so that it's also an area proof (just like Proofs #1-5), but, as its author points out, slope is well-defined without referring to similar triangles!

The Common Core Standards only require that the Pythagorean Theorem be proved using similarity -- not the concept of slope (which we'll cover here on the blog tomorrow). So now I'm wondering whether it might be easier for the students to understand this derivation of slope.

It was David Joyce who pointed out that slope requires similarity to prove. He also criticized the area-based proof of the Pythagorean Theorem given in the Prentice-Hall text -- but this is because he wanted the Parallel Tests and Consequences to be taught first. (According to Cut the Knot, the Pythagorean Theorem is equivalent to the Parallel Postulate.)

For now, I think I'll stick to my plan to use similarity to prove slope -- but my proof may still be based on the one given at the above link.

Thus ends this post. Stay tuned for when Kung solves the mystery of 1 + 1 = 1...