Let's start with the math texts. One book I purchased is Math on Call: A Mathematics Handbook. So yes, it's a "handbook" (dated 1998), not a textbook. But for fifty cents, it was worth buying anyway.
Here is its Table of Contents:
- Numeration
- Number Theory
- Computation
- Algebra
- Graphs and Statistics
- Geometry
- Ratio, Proportion, and Percent
- Probability and Odds
And since this is a Geometry blog, let me give the contents of that chapter in more detail:
Geometry
- Elements of Geometric Figures
- Angles
- Plane Figures
- Solid Figures
Given the significance of pi in Geometry, it's only fitting that this chapters begins on "page" 314. (I point out that "page" is a misnomer -- it's more like "section" 314. Some sections are less than one page, while others are more than one page.)
Next, Barry Garelick will be delighted to learn that I purchased my second Dolciani text. Dated 1972, this is the teacher's edition to her Modern School Mathematics: Course 1.
Here is its Table of Contents:
- Numbers and Sets
- Operations of Arithmetic
- Properties of Arithmetic
- Number Names and Operations
- Other Numeral Systems
- Topics in Number Theory
- Sets and Geometry
- Rational Numbers and Fractions
- Computing with Fractions
- Computing with Decimals
- Percentage
- Measurement and Geometry
- Statistics and Probability
- Angles and Circles; Prisms
- Positive and Negative Numbers
Let's look at the contents of the three Geometry chapters (7, 12, 14) in more detail:
7. Sets and Geometry
- Points and Space
- Line Segments
- Congruent Segments
- Lines
- Half-Lines and Rays
- Planes
- Intersections of Lines
- Half-Planes
- Angles
- Intersection of Planes and Lines
12. Measurement and Geometry
- Units of Length
- Precision
- United States Units of Length
- Metric Units of Length
- Converting Measurements of Length
- Polygons
- Perimeters
- Transformations
- Perpendicular Lines: Parallelograms
- Triangles and Trapezoids
- Area of a Rectangular Region
- Areas of Other Polygons
14. Angles and Circles; Prisms
- Angle Measurement
- Special Angles
- Vertical Angles
- Simple Closed Curves; Circles
- Angles and Circles
- Circumference of a Circle
- Area of a Circular Region
- Right Angles in Space
- Rectangular Prisms: Surface Areas
- Volume of a Rectangular Prism
Even though this is the teacher's edition, nowhere is an exact grade level for this text mentioned. We recall the other Dolciani text that I bought last year was called "Course 2," so today's "Course 1" immediately precedes it.
It's logical to assume that "Course 1" is sixth grade while "Course 2" is seventh grade, so that "Course 1" means "first year" of middle school. The problem is that in Dolciani's day, sixth grade was still considered elementary school, not middle school. It's awkward to use "Course 1" to mean "highest elementary grade."
On the other hand, if "Course 1" means seventh grade (first year of the old junior high school), then "Course 2" would be eighth grade -- yet to Garelick, eighth grade should be Algebra I. It's likely that to Dolciani, "Course 1" is indeed seventh grade, but Garelick uses her "Course 2" with his seventh graders in order to prepare them for eighth grade Algebra I.
If we were following the digit pattern, then today, Day 142, would be Section 14-2, "Special Angles" (where the terms "acute," "right," "obtuse," "supplementary," and "complementary" are defined.) But using the digit pattern is awkward since, for example, Chapter 12 has a dozen sections.
In fact, here in the teacher's edition, Dolciani provides her own pacing guide. Today, Day 142, would instead be Section 13-2, "Constructing Bar Graphs."
Meanwhile, Section 12-8 is called "Transformations." Even though traditionalists act as if Common Core invented transformations, here they are in Dolciani's text! Here the traditionalists' queen defines "rigid transformations," "slide," "flip," and "rotation."
Of course, we're aware the difference between Dolciani and Common Core. Dolciani doesn't use transformations to define "congruence," while the Core does. Indeed, in this level text, Dolciani only defines "congruent" for segments (Section 7-3) and angles (Section 14-1).
The final text I bought is Calculus and Analytic Geometry (G.B. Thomas, 1966). I've been meaning to replace my old Calculus text that is falling apart.
Here is its Table of Contents:
- The Rate of Change of a Function
- Derivatives of Algebraic Functions
- Applications
- Integration
- Applications of the Definite Integral
- Transcendental Functions
- Methods of Integration
- Determinants and Linear Functions
- Plane Analytic Geometry
- Hyperbolic Functions
- Polar Coordinates
- Vectors and Parametric Equations
Notice that this text is most likely for college, not high school AP Calculus. Actually, this is only Part I of the text -- Part II (Chapters 13-18) is on Multivariable Calculus (third semester of college).
Thus it's logical to assume that Chapters 1-6 is first semester Calculus and Chapters 7-12 is second semester Calc. Notice that AB Calc is usually worth one semester of credit and two for BC Calc.
But then again, the only chapter that's definitely part of BC but not AB is Chapter 7. Chapters 8-12 aren't usually part of any AP Calculus course. In fact, I've seen some material from four of those five chapters (excluding Chapter 10 on hyperbolic functions) included in the Honors Precalculus text from the class I subbed for last week.
Now it's time for me to announce our new side-along reading book:
Godel, Escher, Bach: An Eternal Golden Braid by Douglas R. Hofstadter
I was torn between this book and another one -- Sylvia Nasar's A Beautiful Mind. But this one had the fifty-cent label, and so my decision to get this one was easily made.
Once again, I admit that the books I buy for side-along reading aren't the same as those read by most math teacher bloggers. Other math teachers prefer to get books about teaching. In fact, I keep saying that I wouldn't mind getting that book "foreword by Fawn Nguyen" -- our reigning queen. (The book is Geoffrey Krall's Necessary Conditions.) But not only is that book much too new to appear at a library book sale, I've hardly ever seen it on the shelf at a real bookstore.
And so we're stuck with Hofstadter's book from 1979. (I actually have the 1980 paperback.) Besides, Google just did a Doodle for one of the three title geniuses, Bach. And so our reading of this book now can be considered perfect timing.
Once again, there will be no "spilled milk" posts during side-along reading. Expect "ixnay on the arterchay" whenever I'm about to mention my old school from two years ago. On the other hand, as for my other tiresome topics, traditionalists' posts won't go away, and of course I'll keep writing about music since Bach is one of our three subjects.
Indeed, I'm creating a new label "Hofstadter" for our reading, although I could have used the "music" label for Bach.
Indeed, I'm creating a new label "Hofstadter" for our reading, although I could have used the "music" label for Bach.
And so let our side-along reading for April (and probably May, since it's so long) begin. I'll devote today's post to the 26-page "Introducation: A Musico-Logical Offering." Hofstadter begins:
"Frederick the Great, King of Prussia, came to power in 1740. Although he is remembered in history books mostly for his military astuteness, he was also devoted to the life of the mind and the spirit."
And that included music. Indeed, the monarch wanted Bach's father to play in his court:
"For years the King had let it be known, through gentle hints to Philipp Emanuel, how pleased he would be to have the elder Bach come and pay him a visit, but the wish had never been realized."
Instead it was the younger and now more famous Bach, Johann Sebastian, who played for the King. I point out that here Hofstadter quotes Bach's first biographer. Johann Nikolaus Forkel:
"After his return to Leipzig, he composed the subject, which he had received from the King, in three and six parts, added several artificial passages in strict canon to it, and had it engraved, under the style of Musikalisches Opfer' [Musical Offering] and dedicated it to the inventor."
Notice that Bach did exactly what the Google Doodle Bach player did two weeks ago -- the monarch had given him a "Royal Theme" and the composer added to it. Here Hofstadter quotes a Baron named Gottfried van Swieten:
"The King is of this opinion, and to prove it to me he sang aloud a chromatic fugue subject which he had given this old Back, who on the spot had made of it a fugue in four parts, then in five parts, and finally in eight parts."
(The Google Doodle only had four parts.) The author continues:
"The trio sonata forms a delightful relief from the austerity of the fugues and canons, because it is very melodious and sweet, almost danceable."
Here Hofstadter defines a "canon" as "one single theme played against itself." The simplest example is a round -- recall that one of the guitar pieces from the music class I subbed for ten days ago was given as a round. Another example is an inversion, where one part is turned "upside-down" to create another part:
"For a simple example of inversion, try the tune "Good King Wenceslas." When the original and its inversion are sung together, starting an octave apart and staggered with a time-delay of two beats, a pleasing canon results."
Actually, my first day of school song "The Dren Song" is based on inversion. I've also seen musical inversion mentioned in some Common Core Geometry texts as an example of a rotation. In fact, the author goes on to describe a "crab canon" -- a musical reflection. Hey -- we could even count a round as a musical translation, then all three main Common Core transformations are playable:
"Such an information-preserving transformation is often called an isomorphism, and we will have much traffic with isomorphisms in this book."
And of course, "isometries" (or rigid transformations) are examples of isomorphisms. (Maybe Garelick and the other traditionalists need to read this book!)
The author explains that a fugue is like a canon, except that it starts with a single voice, then a second voice sings the same theme a perfect fifth higher while the first sings a "countersubject." Then he continues describing the Musical Offering:
"It is itself one large intellectual fugue, in which many ideas and forms have been woven together, and in which playful double meaning and subtle allusions are commonplace."
The author ends Bach by describing the composer's "Endlessly Rising Canon" -- a song that starts in the key of C minor, then D minor, then E minor, and continuing to rise by a whole tone until returning to C minor. (This effect only works in our standard 12EDO scale, so don't attempt this using the EDL Mocha scales.) And speaking of something that rises endlessly, Hofstadter uses this point to segue to our second genius -- the artist M.C. Escher:
"To my mind, the most beautiful and powerful visual realizations of this notion of Strange Loops exist in the work of the Dutch graphic artist M.C. Escher, who lived from 1902 to 1972. Escher was the creator of some of the most intellectually stimulating drawings of all time."
Of course, the author is describing Escher's Waterfall, which he compares to Bach's rising canon:
"The similarity of vision is remarkable. Bach and Escher are playing one single theme in two different 'keys': music and art."
The author now describes a similar Escher painting -- Ascending and Descending -- and asks how many steps, or even levels, there are on that staircase:
"It is true that there is an inherent haziness in level-counting, not only in Escher pictures, but in hierarchical, many-level systems."
And now Hofstadter moves on to the third genius -- the mathematician Kurt Godel:
"In the examples we have seen of Strange Loops by Bach and Escher, there is a conflict between the finite and the infinite, and hence a strong sense of paradox. And, just as the Bach and Escher loops appeal to very simple and ancient institutions -- a musical scale, a staircase -- so this discovery, by K. Godel, of a Strange Loop in mathematical systems has its origins in simple and ancient institutions."
We've discussed Godel on the blog before, and indeed in the context of paradoxes -- three years ago, when I wrote about David Kung's lectures on paradox. But here the "ancient institution" to which the author refers here is the Epimenides paradox -- the statement that according to Epimenides the Cretan, all Cretans are liars. Simply put, the paradox is "This statement is false." The author states Godel's Theorem simply as:
All consistent axiomatic formulations of number theory include undecidable propositions.
...that is, statements that are true yet unproveable. The author writes:
"The actual creation of the statement was the working out of this one beautiful spark of intuition."
The Godel sentence is a theorem of number theory -- which, at the time he wrote it, refers to the famous work Principia Mathematica by Russell and Whitehead:
This statement of number theory does not have any proof in the system of Prinicipia Mathematica.
As the author writes:
"The grand conclusion? That the system of Principia Mathematica is 'incomplete' -- there are true statements of number theory which its methods of proof are too weak to demonstrate."
At this point Hofstadter introduces a little mathematical logic to the reader. He starts with an example familiar to the readers of this blog -- non-Euclidean geometry:
"How could there be many different kinds of 'points' and 'lines' in one single reality? Today, the solution to the dilemma may be apparent, even to some nonmathematicians -- but at the time, the dilemma created havoc in mathematical circles."
The author moves on to Russell's paradox -- does the set of all sets that don't contain themselves contain itself? (He uses "self-swallowing" to refer to a set that does contain itself, while all other sets are "run-of-the-mill." So he asks, is R, the set of all run-of-the-mill sets, itself run-of-the-mill?"
A similar paradox is "Grelling's paradox":
"Divide the adjectives in English into two categories: those which are self-descriptive, such as 'pentasyllabic,' 'awkwardnessful,' and 'recherche,' and those which are not, such as 'edible,' 'incomplete,' and 'bisyllabic.'"
The latter type of adjective is called "heterological." So we ask, is "heterological" heterological? And here is another paradox, this time with two sentences:
The following sentence is false.
The preceding sentence is true.
The author tells us that the theory of types was created to avoid paradoxes in set theory, but it didn't erase any other paradoxes:
"For people whose interest went no further than set theory, this was quite adequate -- but for people interested in the elimination of paradoxes generally, some similar 'hierarchization' seemed necessary, to forbid looping back inside language."
But such hierarchization -- saying that I can't talk or write about myself -- can be quite complex:
"Besides, the drive to eliminate paradoxes at any cost, especially when it requires the creation of highly artificial formalisms, puts too much stress on bland consistency, and too little on the quirky and bizarre, which make life and mathematics interesting."
Hofstadter moves on to one mathematician whose goal was to formalize all of mathematics -- that is, to prove that everything is consistent:
"This question particularly bothered the distinguished German mathematician (metamathematician) David Hilbert, who set before the world community of mathematicians (and metamathematicians) this challenge: to demonstrate rigorously -- perhaps following the very methods outlines in Russell and Whitehead -- that the system defined in Principia Mathematica was both consistent,
(contradiction-free) and complete (i.e., that every true statement of number theory could be derived within the framework drawn up in P.M.)"
And this is what Godel proved impossible, thus demolishing Hilbert's program. The author now proceeds to describe what effect this would have on computers -- an idea that actually goes back to the nineteenth century with Charles Babbage:
"A character who could almost have stepped out of the pages of the Pickwick Papers, Babbage was most famous during his lifetime for his vigorous campaign to rid London of 'street nuisances' -- organ grinders above all."
Now he and his friend Lady Ada Lovelace are known for coming up with the idea of the Analytical Engine -- a computer that was never built. Computing would have to wait until the 1930's and 1940's:
"Those same years saw the theory of computers develop by leaps and bounds. This theory was tightly linked to metamathematics."
And now the author writes about Alan Turing (the Imitation Game protagonist) and artificial intelligence -- a computer that passes the "Turing test" of appearing to be human. Such a computer must follow certain rules in order to pass:
"Certainly there must be rules on all sorts of different levels. There must be many 'just plain' rules. There must be 'metarules' to modify the 'just plain' rules; then 'metametarules' to modify the metarules, and so on."
Hofstadter says of his book:
"This book is structured in an unusual way: as a counterpoint between Dialogues and Chapters. The purpose of this structure is to allow me to present new concepts twice: almost every new concept is first presented as a Dialogue, yielding a set of concrete, visual images; then these serve, during the reading of the following Chapter, as an intuitive background for a more serious and abstract presentation of the same concept."
And I'll try to respect this pattern as I write about his book. Thus in tomorrow's post, I'll begin with his first Dialogue before reading his first Chapter.
Hofstadter concludes the introduction as follows:
"But finally I realized that to me, Godel and Escher and Bach were only shadows cast in different directions by some central solid essence. I tried to reconstruct the central object, and came up with this book."
Today on her Mathematics Calendar 2019, Theoni Pappas writes:
2" squares are cut from a square of unknown size. A lidless box is formed whose volume is 32 cubic inches. What is the length of the square's side?
Notice that after we cut 2" squares out of each the corner, we simply fold up the box. (That's much simpler than anything in an Escher painting!) We cut out two inches from both sides, so that if the original side length is x, the length and width become x - 4. The height of the box is of course 2, so we just use the formula for the volume of a box:
2(x - 4)^2 = 32
(x - 4)^2 = 16
x - 4 = 4
x = 8
Therefore the side of the square is eight inches -- and of course, today's date is the eighth.
Lesson 14-2 of the U of Chicago text is called "Lengths in Right Triangles." In the modern Third Edition of the text, lengths in right triangles appear in Lesson 13-3.
Recall that Chapter 13 of the new edition corresponds to Chapter 14 of the old edition (since the old Chapter 13 has been split up into different chapters). The new Lesson 13-1 is the last lesson of the old Chapter 12, while the new Lesson 13-2 is on the Angle Bisector Theorem -- a theorem that doesn't appear in the old text (but occasionally appears on the Pappas calendar).
This is what I wrote last year about today's lesson:
Lesson 14-2 of the U of Chicago text is on lengths in right triangles -- specifically, those lengths that are related to the altitude and involve the geometric mean.
Geometric Mean Theorem:
The geometric mean of the positive numbers a and b is sqrt(ab).
(Note: This may sound like a definition, but actually the U of Chicago defines geometric mean to be the number x such that a/x = x/b, so we need a theorem to get the geometric mean as sqrt(ab).)
Right Triangle Altitude Theorem:
In a right triangle:
a. The altitude of the hypotenuse is the geometric mean of the segments dividing the hypotenuse.
b. Each leg is the geometric mean of the hypotenuse and the segment adjacent to the leg.
In this lesson, I give the proof of the Pythagorean Theorem based on similarity, but this time I gave the proof in the book, which mentions the geometric mean. Let's look at the proof -- as usual, with an extra step for the Given:
Given: Right triangle
Prove: a^2 + b^2 = c^2
Proof:
Statements Reasons
1. Right triangle 1. Given
2. a geometric mean of c & x, 2. Right Triangle Altitude Theorem
b geometric mean of c & y
3. a = sqrt(cx), b = sqrt(cy) 3. Geometric Mean Theorem
4. a^2 = cx, b^2 = cy 4. Multiplication Property of Equality
5. a^2 + b^2 = cx + cy 5. Addition Property of Equality
6. a^2 + b^2 = c(x + y) 6. Distributive Property
7. x + y = c 7. Betweenness Theorem (Segment Addition)
8. a^2 + b^2 = c^2 8. Substitution (step 6 into step 7)
It is uncertain whether this is the proof that Common Core intends the students to learn, or whether my earlier proof that avoids geometric means suffices.
Actually, since posting this last year, I've decided to check both the PARCC and SBAC released test questions for those related to the proof of the Pythagorean Theorem. There were a few questions that required use of Pythagoras, but none directly related to the proof. Of course, some people lament that there aren't very many proofs on the Common Core tests.
Speaking of the Pythagorean Theorem, I've been thinking last week about how I presented it to my eighth graders...
...oops, that's ixnay on the arterchay athmay already! All I did was cut-and-paste from last year's Lesson 14-2 post and I have to ixnay it! (Well, for many of my readers, that's one good thing about it being side-along reading time again!)
The author explains that a fugue is like a canon, except that it starts with a single voice, then a second voice sings the same theme a perfect fifth higher while the first sings a "countersubject." Then he continues describing the Musical Offering:
"It is itself one large intellectual fugue, in which many ideas and forms have been woven together, and in which playful double meaning and subtle allusions are commonplace."
The author ends Bach by describing the composer's "Endlessly Rising Canon" -- a song that starts in the key of C minor, then D minor, then E minor, and continuing to rise by a whole tone until returning to C minor. (This effect only works in our standard 12EDO scale, so don't attempt this using the EDL Mocha scales.) And speaking of something that rises endlessly, Hofstadter uses this point to segue to our second genius -- the artist M.C. Escher:
"To my mind, the most beautiful and powerful visual realizations of this notion of Strange Loops exist in the work of the Dutch graphic artist M.C. Escher, who lived from 1902 to 1972. Escher was the creator of some of the most intellectually stimulating drawings of all time."
Of course, the author is describing Escher's Waterfall, which he compares to Bach's rising canon:
"The similarity of vision is remarkable. Bach and Escher are playing one single theme in two different 'keys': music and art."
The author now describes a similar Escher painting -- Ascending and Descending -- and asks how many steps, or even levels, there are on that staircase:
"It is true that there is an inherent haziness in level-counting, not only in Escher pictures, but in hierarchical, many-level systems."
And now Hofstadter moves on to the third genius -- the mathematician Kurt Godel:
"In the examples we have seen of Strange Loops by Bach and Escher, there is a conflict between the finite and the infinite, and hence a strong sense of paradox. And, just as the Bach and Escher loops appeal to very simple and ancient institutions -- a musical scale, a staircase -- so this discovery, by K. Godel, of a Strange Loop in mathematical systems has its origins in simple and ancient institutions."
We've discussed Godel on the blog before, and indeed in the context of paradoxes -- three years ago, when I wrote about David Kung's lectures on paradox. But here the "ancient institution" to which the author refers here is the Epimenides paradox -- the statement that according to Epimenides the Cretan, all Cretans are liars. Simply put, the paradox is "This statement is false." The author states Godel's Theorem simply as:
All consistent axiomatic formulations of number theory include undecidable propositions.
...that is, statements that are true yet unproveable. The author writes:
"The actual creation of the statement was the working out of this one beautiful spark of intuition."
The Godel sentence is a theorem of number theory -- which, at the time he wrote it, refers to the famous work Principia Mathematica by Russell and Whitehead:
This statement of number theory does not have any proof in the system of Prinicipia Mathematica.
As the author writes:
"The grand conclusion? That the system of Principia Mathematica is 'incomplete' -- there are true statements of number theory which its methods of proof are too weak to demonstrate."
At this point Hofstadter introduces a little mathematical logic to the reader. He starts with an example familiar to the readers of this blog -- non-Euclidean geometry:
"How could there be many different kinds of 'points' and 'lines' in one single reality? Today, the solution to the dilemma may be apparent, even to some nonmathematicians -- but at the time, the dilemma created havoc in mathematical circles."
The author moves on to Russell's paradox -- does the set of all sets that don't contain themselves contain itself? (He uses "self-swallowing" to refer to a set that does contain itself, while all other sets are "run-of-the-mill." So he asks, is R, the set of all run-of-the-mill sets, itself run-of-the-mill?"
A similar paradox is "Grelling's paradox":
"Divide the adjectives in English into two categories: those which are self-descriptive, such as 'pentasyllabic,' 'awkwardnessful,' and 'recherche,' and those which are not, such as 'edible,' 'incomplete,' and 'bisyllabic.'"
The latter type of adjective is called "heterological." So we ask, is "heterological" heterological? And here is another paradox, this time with two sentences:
The following sentence is false.
The preceding sentence is true.
The author tells us that the theory of types was created to avoid paradoxes in set theory, but it didn't erase any other paradoxes:
"For people whose interest went no further than set theory, this was quite adequate -- but for people interested in the elimination of paradoxes generally, some similar 'hierarchization' seemed necessary, to forbid looping back inside language."
But such hierarchization -- saying that I can't talk or write about myself -- can be quite complex:
"Besides, the drive to eliminate paradoxes at any cost, especially when it requires the creation of highly artificial formalisms, puts too much stress on bland consistency, and too little on the quirky and bizarre, which make life and mathematics interesting."
Hofstadter moves on to one mathematician whose goal was to formalize all of mathematics -- that is, to prove that everything is consistent:
"This question particularly bothered the distinguished German mathematician (metamathematician) David Hilbert, who set before the world community of mathematicians (and metamathematicians) this challenge: to demonstrate rigorously -- perhaps following the very methods outlines in Russell and Whitehead -- that the system defined in Principia Mathematica was both consistent,
(contradiction-free) and complete (i.e., that every true statement of number theory could be derived within the framework drawn up in P.M.)"
And this is what Godel proved impossible, thus demolishing Hilbert's program. The author now proceeds to describe what effect this would have on computers -- an idea that actually goes back to the nineteenth century with Charles Babbage:
"A character who could almost have stepped out of the pages of the Pickwick Papers, Babbage was most famous during his lifetime for his vigorous campaign to rid London of 'street nuisances' -- organ grinders above all."
Now he and his friend Lady Ada Lovelace are known for coming up with the idea of the Analytical Engine -- a computer that was never built. Computing would have to wait until the 1930's and 1940's:
"Those same years saw the theory of computers develop by leaps and bounds. This theory was tightly linked to metamathematics."
And now the author writes about Alan Turing (the Imitation Game protagonist) and artificial intelligence -- a computer that passes the "Turing test" of appearing to be human. Such a computer must follow certain rules in order to pass:
"Certainly there must be rules on all sorts of different levels. There must be many 'just plain' rules. There must be 'metarules' to modify the 'just plain' rules; then 'metametarules' to modify the metarules, and so on."
Hofstadter says of his book:
"This book is structured in an unusual way: as a counterpoint between Dialogues and Chapters. The purpose of this structure is to allow me to present new concepts twice: almost every new concept is first presented as a Dialogue, yielding a set of concrete, visual images; then these serve, during the reading of the following Chapter, as an intuitive background for a more serious and abstract presentation of the same concept."
And I'll try to respect this pattern as I write about his book. Thus in tomorrow's post, I'll begin with his first Dialogue before reading his first Chapter.
Hofstadter concludes the introduction as follows:
"But finally I realized that to me, Godel and Escher and Bach were only shadows cast in different directions by some central solid essence. I tried to reconstruct the central object, and came up with this book."
Today on her Mathematics Calendar 2019, Theoni Pappas writes:
2" squares are cut from a square of unknown size. A lidless box is formed whose volume is 32 cubic inches. What is the length of the square's side?
Notice that after we cut 2" squares out of each the corner, we simply fold up the box. (That's much simpler than anything in an Escher painting!) We cut out two inches from both sides, so that if the original side length is x, the length and width become x - 4. The height of the box is of course 2, so we just use the formula for the volume of a box:
2(x - 4)^2 = 32
(x - 4)^2 = 16
x - 4 = 4
x = 8
Therefore the side of the square is eight inches -- and of course, today's date is the eighth.
Lesson 14-2 of the U of Chicago text is called "Lengths in Right Triangles." In the modern Third Edition of the text, lengths in right triangles appear in Lesson 13-3.
Recall that Chapter 13 of the new edition corresponds to Chapter 14 of the old edition (since the old Chapter 13 has been split up into different chapters). The new Lesson 13-1 is the last lesson of the old Chapter 12, while the new Lesson 13-2 is on the Angle Bisector Theorem -- a theorem that doesn't appear in the old text (but occasionally appears on the Pappas calendar).
This is what I wrote last year about today's lesson:
Lesson 14-2 of the U of Chicago text is on lengths in right triangles -- specifically, those lengths that are related to the altitude and involve the geometric mean.
Geometric Mean Theorem:
The geometric mean of the positive numbers a and b is sqrt(ab).
(Note: This may sound like a definition, but actually the U of Chicago defines geometric mean to be the number x such that a/x = x/b, so we need a theorem to get the geometric mean as sqrt(ab).)
Right Triangle Altitude Theorem:
In a right triangle:
a. The altitude of the hypotenuse is the geometric mean of the segments dividing the hypotenuse.
b. Each leg is the geometric mean of the hypotenuse and the segment adjacent to the leg.
In this lesson, I give the proof of the Pythagorean Theorem based on similarity, but this time I gave the proof in the book, which mentions the geometric mean. Let's look at the proof -- as usual, with an extra step for the Given:
Given: Right triangle
Prove: a^2 + b^2 = c^2
Proof:
Statements Reasons
1. Right triangle 1. Given
2. a geometric mean of c & x, 2. Right Triangle Altitude Theorem
b geometric mean of c & y
3. a = sqrt(cx), b = sqrt(cy) 3. Geometric Mean Theorem
4. a^2 = cx, b^2 = cy 4. Multiplication Property of Equality
5. a^2 + b^2 = cx + cy 5. Addition Property of Equality
6. a^2 + b^2 = c(x + y) 6. Distributive Property
7. x + y = c 7. Betweenness Theorem (Segment Addition)
8. a^2 + b^2 = c^2 8. Substitution (step 6 into step 7)
It is uncertain whether this is the proof that Common Core intends the students to learn, or whether my earlier proof that avoids geometric means suffices.
Actually, since posting this last year, I've decided to check both the PARCC and SBAC released test questions for those related to the proof of the Pythagorean Theorem. There were a few questions that required use of Pythagoras, but none directly related to the proof. Of course, some people lament that there aren't very many proofs on the Common Core tests.
Speaking of the Pythagorean Theorem, I've been thinking last week about how I presented it to my eighth graders...
...oops, that's ixnay on the arterchay athmay already! All I did was cut-and-paste from last year's Lesson 14-2 post and I have to ixnay it! (Well, for many of my readers, that's one good thing about it being side-along reading time again!)
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