The lessons end up being about grilled cheese sandwiches (to practice reading as well as prepare for cooking tomorrow), relationships (taught by a special health teacher), and art. At lunch there's a Best Buddies meeting, and they're preparing for a talent show tomorrow. During this time I continue my holiday tradition and pass out Easter candy and pencils.
And with an electronic musical keyboard in the classroom, I can't resist playing music from my songbook yet again. It seems as if I've been finding excuses to sing almost everyday now. One gen ed student from Best Buddies requests "Mathematics of Love" from yesterday, followed by "Draw a Map," which I posted on the blog yet never actually sang in -- oops, ixnay on the arterchay athmay!
Many musical keyboards have certain songs already built in. This keyboard has 100 pre-programmed songs, and one of them is labeled as Bach's Etude. Oh, and speaking of Bach...
Dialogue 2 of Douglas Hofstadter's Godel, Escher, Bach is called "Two-Part Invention." Here's how it begins:
Actually, unlike yesterday's "prequel," today's dialogue actually is Lewis Carroll's short story. This time, I'll repeat the link from yesterday:
http://www.ditext.com/carroll/tortoise.html
and then only cut-and-paste some key lines.
...
...
Chapter 2 of Douglas Hofstadter's Godel, Escher, Bach, called "Meaning and Form in Mathematics," begins as follows:
"This Two-Part Invention was the inspiration for my two characters. Just as Lewis Carroll took liberties with Zeno's Tortoise and Achilles, so have I taken liberties with Lewis Carroll's Tortoise and Achilles."
In this chapter, Hofstadter introduces another formal system -- the pq-system. There are three distinct symbols of the pq-system:
p q -
-- the letters p, q, and the hyphen.
Once again, the author must now state axioms and rules for this system. He states that this system contains infinitely many axioms, all of which satisfy the following definition:
Definition: xp-qx- is an axiom, whenever x is composed of hyphens only.
The author refers to such an infinite set of axioms as an axiom schema. Then he states only one rule:
Rule: Suppose x, y, and z all stand for particular strings containing only hyphens. And suppose xpyqz is known to be a theorem. Then xpy-qz- is a theorem.
For example, take x to be '--', y to be '---', and z to be '-'. The rule tells us:
If --p---q- turns out to be a theorem, then so will --p----q--.
It turns out that all the valid theorems follow a certain pattern -- a decision procedure that determines whether a theorem is valid or not. All valid theorems consist of hyphens, a single p, more hyphens, a single q, and one last hyphen-group. The author reveals:
"Back to the decision procedure...The criterion for theoremhood is that the first two hyphen-groups should add up, in length, to the third hyphen-group. For instance, --p--q---- is a theorem, since 2 plus 2 equals 4, whereas --p--q- is not, since 2 plus 2 is not 1."
The author distinguishes between top-down and bottom-up. A top-down decision procedure starts with the theorem to be proved and works backwards until we arrive at an axiom. For bottom-up, we create a bucket full of theorems:
(1a) Throw the simplest possible axiom (-p-q--) into the bucket.
(1b) Apply the rule of inference to the item in the bucket, and put the result into the bucket.
(2a) Throw the second-simplest axiom into the bucket.
(2b) Apply the rule to each item in the bucket, and throw all results into the bucket.
(3a) Throw the third-simplest axiom into the bucket.
(3b) Apply the rule to each item in the bucket, and throw all results into the bucket.
etc., etc.
Eventually all possible theorems appear in the bucket. Notice that after (1a) the lone theorem with a sum of 2 is in the bucket, after (2a) all theorems with a sum of at most 3 are in the bucket, after (3a) all theorems with a sum of at most 4 are in the bucket, and so on.
Oh, I get it. Indeed, Hofstadter admits that why he chooses the symbols p and q, since p stands for "Plus" and q stands for "eQuals." He informs us that there is an isomorphism between pq-theorems and additions:
"This usage of the word 'isomorphism' is derived from a more precise notion in mathematics. It is a cause for joy when a mathematician discovers an isomorphism between two structures which he knows."
The author explains that there is a "lower level" of our isomorphism -- that is, a mapping between the parts of the two structures, an interpretation:
p <==> plus
q <==> equals
- <==> one
-- <==> two
--- <==> three
etc.
He writes:
"The idea of these people is to set up a formal system whose theorems reflect some portion of reality isomorphically."
But here's another possible interpretation that doesn't exactly reflect reality:
p <==> horse
q <==> happy
- <==> apple
Hofstadter warns us about trying to search for meanings in symbols:
"The difference between meaning in a formal system in a language is a very important one, however."
In a language, the meaning comes first -- it is active -- and then we form sentences based on the meanings of the words. But in a formal system, the meaning comes last -- it is passive -- since the true statements comes from the axioms and rules, not from their meaning. He reminds us that:
--p--p--p--q--------
is not a theorem, even though 2 + 2 + 2 + 2 = 8. Indeed, theorems only contain two hyphen-groups while this statement contains five.
The author provides us another possible valid interpretation of the symbols:
p <==> equals
q <==> taken from
- <==> one
-- <==> two
etc.
He reminds us:
"An interpretation will be meaningful to the extent that it accurately reflects some isomorphism to the real world."
Now Hofstadter asks, "Can all of reality can be turned into a formal system?" He tells us:
"One could suggest, for instance, that reality is itself nothing but one very complicated formal system."
In other words, is the universe deterministic? No one knows the answer.
But here's a question to which we know the answer -- how much is 12 * 12? Of course it's 144:
"But how many of the people who give that answer have actually any time in their lives drawn a 12 by 12 rectangle, and then counted the little squares in it?"
The author tells us that most people just rely on the standard algorithm instead. (That's everyone except the Common Core, according to traditionalists.)
In other words, we rely on the basic laws of arithmetic:
"The same assumption can also lead you to the commutativity and associativity of multiplication; just think of many cubes assembled together to form a large rectangular solid."
Hofstadter tells us about "exceptions" to the laws of arithmetic, such as 1 + 1 = 1 (for lovers), as well as 1 + 1 + 1 = 1 (the Trinity). But arithmetic is all about the properties of "ideal numbers," whatever those might be:
"Once you have decided to capsulize all of number theory in an ideal system, is it really possible to do the job completely? The picture Liberation, one of Escher's most beautiful, is a marvelous contrast between the formal and informal with a fascinating transition region."
Here the author is referring to the artist's painting where the triangles transition into birds.
Now Hofstadter provides a version of Euclid's Theorem -- the infinitude of the primes. For this version of the proof he uses factorials. For any natural number N, we see that N! + 1:
can't be a multiple of 2 (because it leaves 1 over, when you divide by 2);
can't be a multiple of 3 (because it leaves 1 over, when you divide by 3);
Thus N! + 1 must be divisible by a prime larger than N -- in other words, N isn't the largest prime. At this point the author tells us that despite proving the infinitude of the primes, the proof isn't infinite:
"It gets around it by using phrases like 'whatever N is' or 'no matter what number N is.' We could also phrase the proof over again, so that it uses the phrase 'all N.'"
Hofstadter concludes the chapter as follows:
"A very important question will be whether the rules for symbolic manipulation which we have then formulated are really of equal power (as far as number theory is concerned) to our usual mental reasoning abilities -- or, more generally, whether it is theoretically possible to attain the level of our thinking abilities, by using some formal system."
Today on her Mathematics Calendar 2019, Theoni Pappas writes:
M is the midpoint of segmentAB. If the slope of AB is 5/6, determine the value of a + b.
[Here is the given info from the diagram: A(-9, -3), B(x, 17), M(a, b).]
Unlike most Midpoint Formula problems on the Pappas Calendar, this one requires us to use the Slope Formula as well. We begin by using the Midpoint Formula (Lesson 11-4) to find b, since we're given enough info to find it:
b = (-3 + 17)/2
b = 14/2
b = 7
Now we use the Point-Slope Formula to find first the line passing through (-9, -3) with slope 5/6:
y + 3 = (5/6)(x + 9)
and then a point of this line with x-coordinate a and y-coordinate of b = 7 -- which we can find by plugging in a for x and 7 for y:
7 + 3 = (5/6)(a + 9)
10 = (5/6)(a + 9)
10(6/5) = a + 9
12 = a + 9
a = 3
Thus the midpoint (a, b) is (3, 7). The question asks us not for a or b, but for a + b. We can find that sum in the pq-system as ---p-------q----------, that is 3 + 7 = 10. Therefore the desired sum is ten -- and of course, today's date is the tenth.
There is a STEM-related Google Doodle today -- the first image of a black hole. I'm not sure whether the artist Escher could have imagined this.
Lesson 14-4 of the U of Chicago text is called "The Sine and Cosine Ratios." In the modern Third Edition, the sine and cosine ratios appear in Lesson 13-6.
"This Two-Part Invention was the inspiration for my two characters. Just as Lewis Carroll took liberties with Zeno's Tortoise and Achilles, so have I taken liberties with Lewis Carroll's Tortoise and Achilles."
In this chapter, Hofstadter introduces another formal system -- the pq-system. There are three distinct symbols of the pq-system:
p q -
-- the letters p, q, and the hyphen.
Once again, the author must now state axioms and rules for this system. He states that this system contains infinitely many axioms, all of which satisfy the following definition:
Definition: xp-qx- is an axiom, whenever x is composed of hyphens only.
The author refers to such an infinite set of axioms as an axiom schema. Then he states only one rule:
Rule: Suppose x, y, and z all stand for particular strings containing only hyphens. And suppose xpyqz is known to be a theorem. Then xpy-qz- is a theorem.
For example, take x to be '--', y to be '---', and z to be '-'. The rule tells us:
If --p---q- turns out to be a theorem, then so will --p----q--.
It turns out that all the valid theorems follow a certain pattern -- a decision procedure that determines whether a theorem is valid or not. All valid theorems consist of hyphens, a single p, more hyphens, a single q, and one last hyphen-group. The author reveals:
"Back to the decision procedure...The criterion for theoremhood is that the first two hyphen-groups should add up, in length, to the third hyphen-group. For instance, --p--q---- is a theorem, since 2 plus 2 equals 4, whereas --p--q- is not, since 2 plus 2 is not 1."
The author distinguishes between top-down and bottom-up. A top-down decision procedure starts with the theorem to be proved and works backwards until we arrive at an axiom. For bottom-up, we create a bucket full of theorems:
(1a) Throw the simplest possible axiom (-p-q--) into the bucket.
(1b) Apply the rule of inference to the item in the bucket, and put the result into the bucket.
(2a) Throw the second-simplest axiom into the bucket.
(2b) Apply the rule to each item in the bucket, and throw all results into the bucket.
(3a) Throw the third-simplest axiom into the bucket.
(3b) Apply the rule to each item in the bucket, and throw all results into the bucket.
etc., etc.
Eventually all possible theorems appear in the bucket. Notice that after (1a) the lone theorem with a sum of 2 is in the bucket, after (2a) all theorems with a sum of at most 3 are in the bucket, after (3a) all theorems with a sum of at most 4 are in the bucket, and so on.
Oh, I get it. Indeed, Hofstadter admits that why he chooses the symbols p and q, since p stands for "Plus" and q stands for "eQuals." He informs us that there is an isomorphism between pq-theorems and additions:
"This usage of the word 'isomorphism' is derived from a more precise notion in mathematics. It is a cause for joy when a mathematician discovers an isomorphism between two structures which he knows."
The author explains that there is a "lower level" of our isomorphism -- that is, a mapping between the parts of the two structures, an interpretation:
p <==> plus
q <==> equals
- <==> one
-- <==> two
--- <==> three
etc.
He writes:
"The idea of these people is to set up a formal system whose theorems reflect some portion of reality isomorphically."
But here's another possible interpretation that doesn't exactly reflect reality:
p <==> horse
q <==> happy
- <==> apple
Hofstadter warns us about trying to search for meanings in symbols:
"The difference between meaning in a formal system in a language is a very important one, however."
In a language, the meaning comes first -- it is active -- and then we form sentences based on the meanings of the words. But in a formal system, the meaning comes last -- it is passive -- since the true statements comes from the axioms and rules, not from their meaning. He reminds us that:
--p--p--p--q--------
is not a theorem, even though 2 + 2 + 2 + 2 = 8. Indeed, theorems only contain two hyphen-groups while this statement contains five.
The author provides us another possible valid interpretation of the symbols:
p <==> equals
q <==> taken from
- <==> one
-- <==> two
etc.
He reminds us:
"An interpretation will be meaningful to the extent that it accurately reflects some isomorphism to the real world."
Now Hofstadter asks, "Can all of reality can be turned into a formal system?" He tells us:
"One could suggest, for instance, that reality is itself nothing but one very complicated formal system."
In other words, is the universe deterministic? No one knows the answer.
But here's a question to which we know the answer -- how much is 12 * 12? Of course it's 144:
"But how many of the people who give that answer have actually any time in their lives drawn a 12 by 12 rectangle, and then counted the little squares in it?"
The author tells us that most people just rely on the standard algorithm instead. (That's everyone except the Common Core, according to traditionalists.)
In other words, we rely on the basic laws of arithmetic:
"The same assumption can also lead you to the commutativity and associativity of multiplication; just think of many cubes assembled together to form a large rectangular solid."
Hofstadter tells us about "exceptions" to the laws of arithmetic, such as 1 + 1 = 1 (for lovers), as well as 1 + 1 + 1 = 1 (the Trinity). But arithmetic is all about the properties of "ideal numbers," whatever those might be:
"Once you have decided to capsulize all of number theory in an ideal system, is it really possible to do the job completely? The picture Liberation, one of Escher's most beautiful, is a marvelous contrast between the formal and informal with a fascinating transition region."
Here the author is referring to the artist's painting where the triangles transition into birds.
Now Hofstadter provides a version of Euclid's Theorem -- the infinitude of the primes. For this version of the proof he uses factorials. For any natural number N, we see that N! + 1:
can't be a multiple of 2 (because it leaves 1 over, when you divide by 2);
can't be a multiple of 3 (because it leaves 1 over, when you divide by 3);
can't be a multiple of 4 (because it leaves 1 over, when you divide by 4);
...
can't be a multiple of N (because it leaves 1 over, when you divide by N);
"It gets around it by using phrases like 'whatever N is' or 'no matter what number N is.' We could also phrase the proof over again, so that it uses the phrase 'all N.'"
Hofstadter concludes the chapter as follows:
"A very important question will be whether the rules for symbolic manipulation which we have then formulated are really of equal power (as far as number theory is concerned) to our usual mental reasoning abilities -- or, more generally, whether it is theoretically possible to attain the level of our thinking abilities, by using some formal system."
Today on her Mathematics Calendar 2019, Theoni Pappas writes:
M is the midpoint of segment
[Here is the given info from the diagram: A(-9, -3), B(x, 17), M(a, b).]
Unlike most Midpoint Formula problems on the Pappas Calendar, this one requires us to use the Slope Formula as well. We begin by using the Midpoint Formula (Lesson 11-4) to find b, since we're given enough info to find it:
b = (-3 + 17)/2
b = 14/2
b = 7
Now we use the Point-Slope Formula to find first the line passing through (-9, -3) with slope 5/6:
y + 3 = (5/6)(x + 9)
and then a point of this line with x-coordinate a and y-coordinate of b = 7 -- which we can find by plugging in a for x and 7 for y:
7 + 3 = (5/6)(a + 9)
10 = (5/6)(a + 9)
10(6/5) = a + 9
12 = a + 9
a = 3
Thus the midpoint (a, b) is (3, 7). The question asks us not for a or b, but for a + b. We can find that sum in the pq-system as ---p-------q----------, that is 3 + 7 = 10. Therefore the desired sum is ten -- and of course, today's date is the tenth.
There is a STEM-related Google Doodle today -- the first image of a black hole. I'm not sure whether the artist Escher could have imagined this.
Lesson 14-4 of the U of Chicago text is called "The Sine and Cosine Ratios." In the modern Third Edition, the sine and cosine ratios appear in Lesson 13-6.
This is what I wrote last year about today's lesson:
Lesson 14-3 of the U of Chicago text is on the tangent ratio, and Lesson 14-4 of the U of Chicago text is on the sine and cosine ratios. I have decided to combine all three trig ratios into one lesson.
By the way, many of our students may have trouble with trigonometry, but one youngster who had no trouble with trig was -- you guessed it -- Ramanujan. When he was in the equivalent of seventh grade, an older friend lent him a trig text book, and the young genius mastered it that year!
Here's one more connection between Ramanujan and trig. The U of Chicago text tells us how to find some trig values exactly, but not others. For example, cos(60) = 1/2, but cos(20), cos(40), and cos(80) aren't as easy to find. Well, the Indian genius found an interesting formula connecting the three cosines whose values we can't find. (All values are in degrees -- "cbrt" is cube root.)
cbrt(cos(40)) + cbrt(cos(80)) - cbrt(cos(20)) = cbrt(1.5(cbrt(9) - 2))
A 20-degree angle is not constructible and so its cosine can't be written exactly using square or cube roots -- of real numbers, that is. Complex numbers are a different issue:
cos(20) = (cbrt(a) + cbrt(b))/2
where a and b are the complex cube roots of 1 -- that is, a = (1 + i sqrt(3))/2, b = (1 - i sqrt(3))/2.
Now you can see why we have high school students memorize cos(60) and not cos(20).
Lesson 14-3 of the U of Chicago text is on the tangent ratio, and Lesson 14-4 of the U of Chicago text is on the sine and cosine ratios. I have decided to combine all three trig ratios into one lesson.
David Joyce was not too thrilled to have trig in the geometry course. He wrote:
Chapter 11 [of the Prentice-Hall text -- dw] covers right-triangle trigonometry. It's hard to see how there's any time left for trigonometry in a course on geometry, but at least it should be possible to prove the basic facts of trigonometry once the theory of similar triangles is done. The section of angles of elevation and depression need not appear, and the section of vectors omitted. (By the way, who ever calls the sum of two vectors the "resultant" of the two vectors?) The one theorem of the chapter (area of triangle = 1/2 bc sin A) is given for acute triangles.
As the trig functions for obtuse angles aren't covered, and applications of trig to non-right triangles aren't mentioned, it would probably be better to remove this chapter entirely.
Yet most geometry books include trig because most state standards require it. And this most certainly includes the Common Core Standards:
Define trigonometric ratios and solve problems involving right triangles
CCSS.MATH.CONTENT.HSG.SRT.C.6
Understand that by similarity, side ratios in right triangles are properties of the angles in the triangle, leading to definitions of trigonometric ratios for acute angles.
Understand that by similarity, side ratios in right triangles are properties of the angles in the triangle, leading to definitions of trigonometric ratios for acute angles.
CCSS.MATH.CONTENT.HSG.SRT.C.7
Explain and use the relationship between the sine and cosine of complementary angles.
Explain and use the relationship between the sine and cosine of complementary angles.
CCSS.MATH.CONTENT.HSG.SRT.C.8
Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems.*
Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems.*
And all three of these standards appear in this lesson.
Some people may wonder, why do we use the same name "tangent" to refer to the "tangent" of a circle and the "tangent" in trigonometry? A Michigan math teacher, Mike Shelly, discusses the reasons at the following link:
On the other hand, the reason that "sine" and "cosine" have the same name is less of a mystery. In fact, the U of Chicago tells us that "cosine" actually means complement's sine -- since the cosine of an angle is the sine of its complement. This is Common Core Standard C.7 above.
Last Friday's post was a whirlwind of ideas, and today's post continues these ideas. In the last two days, I linked to a variety of sources in search of answers to questions such as:
- Should activities be taught during the trig unit?
- Should a trig unit be taught during the Geometry class?
- Should a Geometry class be taught during high school?
We searched high and low, from traditionalists to their opponents, seeking these answers. I fear that when I post links to all these competing sources, my own opinions are obscured. The blog readers know what David Joyce and the traditionalists believe, but not what I myself believe.
Well, here's my belief -- I answer all three of those questions in the affirmative. High school should have a Geometry class, Geometry class should have a trig unit, and a trig unit should have activities -- and I posted my activity for the trig unit of a high school Geometry course yesterday.
I also think back to the activity that sparked this debate -- proofs and the courtroom. We saw how the traditionalists objected to the courtroom activity on the grounds that it is too long.
I admit that I'm fascinated with the idea of using a courtroom to highlight Geometry proofs. I took Geometry back during the 1994-95 school year -- the year of the famous OJ Simpson trial. And so I often fantasized that my Geometry class was a courtroom -- the People's Court. Actually, that TV show was off the air during that year. But it made a comeback in 1996, the first full year after the Simpson trial, as TV stations were trying to capitalize on the Simpson trial's popularity. (This was the same year that another famous courtroom show debuted -- Judge Judy.)
So I might organize a People's Court during my Geometry classes. When I would teach the lesson depends on what textbook I was using. If I had Michael Serra's text, People's Court would occur at the end of the year, around Chapter 13. With the U of Chicago text, court may occur in Chapter 3 (when the class first learns about proofs), and in many other texts, it may occur in Chapter 4 (where triangle congruence proofs appear).
One way to prevent the unit from taking too long is to assign each group a different medium-level proof -- then they present those proofs when the class actually reaches that unit! So one group may be assigned the Isosceles Triangle Theorem to put on trial a week later, while another is assigned some of the Parallelogram Theorems to put on trial a few months later. As long as all groups present before the end of the first semester, it works out in the end.
Here's one more connection between Ramanujan and trig. The U of Chicago text tells us how to find some trig values exactly, but not others. For example, cos(60) = 1/2, but cos(20), cos(40), and cos(80) aren't as easy to find. Well, the Indian genius found an interesting formula connecting the three cosines whose values we can't find. (All values are in degrees -- "cbrt" is cube root.)
cbrt(cos(40)) + cbrt(cos(80)) - cbrt(cos(20)) = cbrt(1.5(cbrt(9) - 2))
A 20-degree angle is not constructible and so its cosine can't be written exactly using square or cube roots -- of real numbers, that is. Complex numbers are a different issue:
cos(20) = (cbrt(a) + cbrt(b))/2
where a and b are the complex cube roots of 1 -- that is, a = (1 + i sqrt(3))/2, b = (1 - i sqrt(3))/2.
Now you can see why we have high school students memorize cos(60) and not cos(20).
OK, let me post the worksheet:
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