Peterik Chapter 7: Using Rhymes in Your Songs
Table of Contents
Introduction
We're getting back to Peterik's book quickly this week. Most people who know anything about songs know that most songs rhyme, and so today's chapter is all about rhyming.
Rapoport Question of the Day
Today on her Daily Epsilon of Math, Rebecca Rapoport writes:
f (y) = tan(2y)
Find f '(pi/6).
First of all, my blogging policy is that between Tau Day and Pi Approximation Day, all math problems should be written in terms of tau, not pi. So let's change this problem to:
Find f '(tau/12).
This is a Calculus problem. Let's begin by taking the derivative of f:
f '(y) = 2sec^2(2y)
f '(tau/12) = 2sec^2(tau/6)
Now we can use the unit circle to simplify this. We recall that tau/6 is one-sixth of the way around the unit circle, or 60 degrees. Its cosine is 1/2, and so its secant is the reciprocal of this, or 2:
f '(tau/12) = 2(2)^2
f '(tau/12) = 8
Therefore the desired derivative is eight -- and of course, today's date is the eighth. Unfortunately, my Calculus students would have struggled with this problem -- and this is part of the reason why my seniors scored 1's on the AP exam.
Like many AP problems, this one contains two parts. The first part is Calculus, and the second part is Algebra, or more precisely here, Trigonometry. The Calc part is easy if the students remember that the derivative of tan x is sec^2 x, and that via the Chain Rule, we must multiply by 2, the derivative of 2y.
But I suspect that my kids would have had trouble with the second part of the problem -- simplification using Trig (and I assume that this would have been in the non-calculator section). The problem is that higher-scoring students would have learned Trig in their Precalculus class last year -- but I have absolutely no clue what Precalc was like for my students last year.
Again, recall that my entire district had distance learning the entire 2020-21 school year, and thus all of their Precalc learning would have been at home. Once, I did ask my students about what their Precalc class was like last year, and their reply was that it was shaky. Apparently, they switched teachers in the middle of the year. I know that this year, the other math teacher at my school (my partner teacher) handled the Precalc classes, but once again, this year is not last year.
And of course, there's the whole problem with distance learning in math. It's hard to tell how much learning is being accomplished during distance learning. Students try to figure out how to get a high grade in the class while doing the least amount of work. This begins with keeping the cameras turned off during Zoom classes and continues with spending long stretches of class away from the computer and visiting entertainment sites on phones during class time.
Then again, I don't want to accuse my specific Calculus students of having done any of these things during their Precalc classes last year. (But at least the likelihood that they at least kept the cameras off during Precalc is high -- no student wanted to be seen by the other kids as the only student with a camera turned on.)
And of course, there's also the possibility that many students were cheating in math last year.
The Photomath Problem
One website that students might use to cheat in math is called Wolfram Alpha. This is one of the oldest sites on the net -- indeed, its predecessor, integrals.com, existed all the way back when I was a young Calculus BC student. (If we try to visit integrals.com today, the browser then redirects you to the current Wolfram Alpha site.)
But no, I didn't use the old Wolfram integrals site to cheat. Indeed, at the time I first found out about it, we were currently on the derivatives section of the text -- and then by the time we reached the integrals section, I'd forgotten all about Wolfram.
As a teacher, from time to time I privately use Wolfram Alpha, mainly to double-check the answers to questions I've written (Exit Passes and written exam questions). This is because as a teacher, I must not make a mistake, period -- otherwise, the students get confused and end up not learning. Indeed, it's embarrassing to write an Exit Pass, only to have the answer not be the date because I made an error, and a mistake on a test question is a big mess. So I check my answers on Wolfram Alpha. Still, I said to myself that I must never tell my students about the Wolfram Alpha website.
Now of course, students cheat all the time online, so they already know where to go. But the site of choice isn't Wolfram Alpha, but something called "Photomath." And in many ways, Photomath is even worse than Wolfram Alpha, since all that's needed is a phone to take a photo (hence the site name) of the problem. At least Wolfram requires you to make the effort of typing in the problem.
I've never visited the Photomath website, but I can figure out how it works. Symbolic equations are a cinch for Photomath (and for Wolfram as well). The following website is all about how to code (in Logo) an artificial intelligence program that can solve math word problems:
https://people.eecs.berkeley.edu/~bh/v3ch6/ai.html
So I can easily see how Photomath could have achieved the ideals mentioned at this link, and thus can solve more complex problems that it gleans from the photo.
Even so, the simple memorization problems are the easiest to do on Photomath -- and this includes memorizing the unit circle. So if a Precalc teacher posts a worksheet online that requires the students to find the exact values of trig ratios using the unit circle, it would be easy for a student to take a photo of the worksheet and send it to Photomath to get all the answers.
Again, I'm not accusing any of my specific students of having cheated in Precalc with Photomath. All I'm saying is that the temptation to do so was definitely there.
A student who cheats (or at least tunes out) during the trig units in Precalc will be a disadvantage when it comes to the AP exam and have to simplify sec^2 of a key unit circle angle. And indeed, they might not understand the difference between degrees and radians. The irony is that these students enter Calc and easily learn that the derivative of sin x is cos x (and that the derivative of tan x is sec^2 x), but they don't understand radians -- the only angle units for which those derivatives only work. And they get confused if they're asked to evaluate the trig functions at some fraction of pi.
(Of course, this is where tauists may jump in -- it's tough to explain why pi/6 is 30 degrees and pi/3 is 60 degrees, but tau/6 = 60 degrees is easier to remember. And it makes more sense, as one-sixth of the unit circle is indeed 60 degrees. Unfortunately, teaching tau in an AP Calculus class is definitely not recommended, as the exam writes angles in terms of pi, not tau.)
Oh, in case you're curious, what exactly is the derivative of sin x degrees? To find this, we must convert it to radians and use the Chain Rule:
f (x) = sin(x degrees)
f (x) = sin(x tau/360)
f '(x) = (tau/360)cos(x tau/360)
f '(x) = (tau/360)cos(x degrees)
This extra factor of tau/360 (or pi/180) occurs in all the trig derivatives. The only way to avoid it is to measure x in radians. But unfortunately, our Photomath students don't understand radian measure.
How to Deal with It
I don't know whether my own students cheated or not -- regardless of whether they did or not, there's nothing I could have done about it at the time, since I wasn't there. All I know is that my students entered my Calculus class a little weaker at Precalc, including Trig, than non-pandemic students.
And so I ask, is there anything I could have done about this? How could I have made up for any deficiencies in Calc and Trig so that my students could score higher on this year's AP exam?
Notice that Chapter 1 of our text was basically a review of Precalc (as limits, the first true Calculus topic, don't appear until Chapter 2). In non-pandemic times, Chapter 1 might have been sufficient as a brief refresher for students who had learned the material, but forgot it over the summer. But it's not good enough for students who never learned the material due to distance learning distractions.
Perhaps what I needed was a Chapter 0. A hypothetical Chapter 0 would be just like Chapter 1, except for the perspective of students who never fully learned the Precalc material, as opposed to those who learned and forgot it. Unfortunately, our Calculus text didn't have a Chapter 0. Instead, I'd be forced to come up with Chapter 0 material myself.
By including a Chapter 0, I can follow the same four-week pattern that I established for second semester, as well as the Math 8 units that I've mentioned in these summer posts:
(The difference between Calculus and Math 8 here is that we're using zero-based numbering for Calc and one-based numbering for Math 8. This is why Unit 2 is September in Math 8 while Chapter 2 is October in Calculus. The discrepancy extends into second semester -- in today's post, February is Unit 6 in Math, but February was Chapter 5 in Calculus.)
The opening-week activities are considered part of Chapter 0 -- recall that these took up all of Week 1 and part of Week 2. For the rest of Chapter 0, I'd have to gather worksheets and other materials to teach students the Precalc lessons that are critical to success in Calculus -- and this means that I'd needed to make the tough decision of which lessons are the most critical.
These lessons aren't in the book (which starts at Chapter 1) -- and they certainly aren't on AP Classroom (which starts at limits, the equivalent of Chapter 2). DeltaMath would be an ideal source of material, except that my partner teacher didn't tell me about the existence of DeltaMath until Monday Week 4 -- just before I'd be ready to start Chapter 1. Still, I could use DeltaMath for the Chapter 0 assessment -- which I could count as a quiz.
Then the four quizzes of the semester are for Chapters 0, 1, 2, 3, while the four tests of the semester are Chapters 1, 2, 3, Final. And in case you're thinking about the minimum days (after the big deal I made about them this year), the quizzes for Chapters 1 and 2 are just before minimum day Mondays.
The Chapter 3 Quiz could be given just before Veteran's Day. (I didn't wish to give a long Chapter 3 Test before the holiday, but a shorter quiz would have fit.) Then when I visited my counterpart at the main high school, we would have been right around the same point in Chapter 3. (He told me that he covered Chapters 1-2 slowly -- he likely did so in order to fill in the gaps the students had due to distance learning along the way, as opposed to having an explicit Chapter 0.)
Once again, I'm not quite sure whether this Chapter 0 idea would have helped my students or not. But since we're thinking in terms of four-week chapters/units now anyway, it's nice to see how these chunks would fit the entire year and thus help keep me on pace in Calculus.
My Tweets
I made one tweet today. It connects today's Rapoport problem to my Calculus class:
This is a standard Calc AB problem that might appear in the non-calculator MC section. My own students might have had trouble on such a problem due to struggles during Precalc distance learning.
I made this post very late today, so no one has had time to read it or like it.
Popular Tweets
The label "Adopt A Teacher" is now trending. Of course, "Adopt A Teacher" isn't that much different from "clear the list" -- both labels implore Twitter users to donate Back to School supplies to relatively poor teachers.
By the way, you might wonder whether this is something I should have done during my year at the old charter school -- tweet "Adopt a Teacher" or "clear the list" and include items that are needed for my science projects. But this might have been tricky -- I didn't really have an opportunity to look at the science texts or what items I needed until after the first day of school, by which time most teachers have already been adopted, and most of their lists already cleared.
Peterik Chapter 7: Using Rhymes in Your Songs
Chapter 7 of Jim Peterik's Songwriting for Dummies is called "Using Rhymes in Your Songs." Here's how it begins:
"Welcome to the wonderful world of rhyming! Many writers consider this to be the fun part of the songwriting process, because the art of rhyming is similar to solving a brainteaser or putting together a jigsaw puzzle."
Once again, we expect songs to rhyme, and so this chapter definitely comes in handy. We begin by learning and identifying the two types of rhymes:
- Perfect rhyme: When the syllables of two or more words contain the same vowel and final consonant sounds but begin with different consonant sounds (such as bot and coat or bullet and pull it).
- Imperfect rhyme: Also called near, slant or false rhyme, imperfect rhyme is an approximation of rhyme. The two most common forms of imperfect rhyme are assonance, where two words share the same vowel sound (for example, prove and sooth, and love and hug) and consonance, where two or more words share the final consonant sound (as in young and song).
Next, we look at rhyming patterns:
"Internal rhyme, sometimes referred to as inner rhyme, occurs within the lines of the songs. External rhymes, or end rhymes, are the near or perfect rhymes that occur at the ends of the various lines of your lyric."
The most common rhymes are end rhymes. Here Peterik gives an example of one of his own songs, "Heavy Metal" (performed by Sammy Hagar). The rhyming pattern here is abab, where alternating lines rhyme. I won't write the entire lyrics here -- instead, I'll list only the words that rhyme:
leather (a), night (b), together (a), lights (b), power (a), overload (b), devour (a), hold (b)
There are several examples here, but to make it easy, I'll skip to the more of his own examples. Here is his "Vehicle" (performed by The Ides of March):
sedan (a), car (b), man (a), star (b)
"Check out other abab songs, if you want to learn by picking them apart. Here are a few to start with."
Listed here are "Love Is Here to Stay" (a standard written by the Gershwins), "Live to Tell It All" (Vince Gill and Sonya Isaacs), and "Girl" (The Beatles).
The author moves on to the aabb rhyming pattern. Here is his "Night of the World Stage":
frustrations (a), elevation (a), head (b), red (b)
"For more examples of the aabb pattern, check out the songs in the table."
Some of the songs listed here are "I Hope You Dance" (written by Mark Sanders and performed by Leann Womack), "Marrakesh Express" (Crosby, Stills, and Nash), and "Your Song" (Elton John).
Peterik now mentions the aaba rhyming pattern. Here is his "I Can't Hold Back" (for Survivor):
eyes (a), desire (a), dreams (b), fire (a)
An interesting example is mentioned here -- "Can't Help Falling in Love with You," which was originally written as "...in Love with Him" but changed to "You" for Elvis Presley (another example of a change in gender perspective):
"If you read through the lyric, it's apparent that all the rhymes were crafted to fit the original title. It would have been an abca rhyming pattern."
Likewise, we consider the author's own "Eye of the Tiger":
street (a), chances (b), feet (a), survive (c), fast (d), glory (e), past (d), alive (c)
The chorus rhyme scheme is a bit confusing, because the word "rival" in the third line rhymes with the original title "Survival." It can still be considered a near rhyme with the new title ending in "Tiger."
Other rhyme patterns and songs are listed here. An example of an aaaa pattern where the first four lines all rhyme is "American Pie" (Don McLean). Example of abcb patterns where only lines two and four rhyme are "My Girl" (written by Smokey Robinson), and "In My Life" (The Beatles).
Internal rhyme is less common than end rhyme, but examples of internal rhyme can be found -- for example, in "Long Day" (written by Rob Thomas):
"Shelf is an internal rhyme to the word myself in the following line. In John Mayer's song 'No Such Thing' (co-written with Douglas Cook) he adds internal rhyme in the very first line of the song. '...she said to me condescendingly.'"
At this point, we distinguish between perfect and imperfect rhymes. Some prefer to use perfect rhymes:
"Look up the lyrics to 'My Favorite Things' from The Sound of Music and check out the wonderful perfect rhyming schemes. The argument for this perfection is the absolute neatness of all the phrases, kind of like the way the military demands its beds be made or the way the woman down the road keeps a perfect lawn."
Other writers don't mind using near rhymes. Peterik provides one of his own examples in his "Talent for Loving You":
soft (a), off (a), rain (b), down (c), sound (c), name (b), on, move (d), talent, you (d)
And in his "Can't Say It Loud Enough," he rhymes window with wind blows. The first line of that song also contains an inner near rhyme with eyes and five.
Sometimes we must work backwards to find a rhyme. The author explains:
"Your next step may be to work back from that line to find appropriate rhymes for the rest of the verse."
And he quotes co-writer Johnny Van Zant:
"When I was writing the lyric to 'Can't Say It Loud Enough' for The Van Zant album with Robert White Johnson and Jim Peterik, all we had was a great second line, 'My daddy said that the truth is the truth and there just ain't no space in between it.'"
The writing trio then worked backwards to the first line, and ended up finding "believe it" as a near rhyme to "between it."
Sometimes the rhyming takes place across verses. Peterik gives the example "In the Chapel in the Moonlight" (Billy Hill), whose first verse ends in entwine, the second verse shine, and the third mine.
We now look at songs with little rhyme. The author's own example "Wild-Eyed Southern Boys" contains some rhyme in the verses:
Verse 1: joint (a), blues (b), tonight (c), news (b)
Verse 2: honor (a), act (b), holler (c), black (b)
But there's no other rhyming in this song. Indeed, the chorus just repeats the title, with no word to rhyme with boys. Other examples of songs with very little rhyme include "Let's Make Love" (written by Chris Lindsay, performed by Faith Hill and Tim McGraw) and "Shape of My Heart" (written by Max Martin, performed by The Backstreet Boys).
This takes us to the next "Practice Makes Perfect" section -- it's time for me to write a song now.
A Song for February
In our last post, we looked at what Math 8 Unit 5 would have looked like at the old charter school with these songs. Let's look at the ideal Unit 6, spanning Weeks 21-24 of school. The math standards to be covered are the G standards on geometry.
As I wrote earlier, this time of year corresponds to Unit 6 at the old charter school and Chapter 5 in my Calculus class. And in Math 8, this is one of my favorite units of the year -- geometry! My old blog was a Geometry blog, and so I've devoted many posts over the years to transformational geometry.
Ironically, even though I've written lots of prose about my favorite topic, I haven't written songs about this topic at all. On the original timeline, I covered some of these topics in January -- but all the songs I performed were for Grade 6-7 topics (or the Hidden Figures field trip). And the long-term school followed the same pacing and covered the G topics in February, but the regular teacher was back.
During this stretch on the original timeline, I played two Square One TV songs related to love (that most common song topic according to Peterik) in time for Valentine's Day -- "Count the Ways" and "Mathematics of Love," so we can place these on the same weeks, Week 21 and 23. During Week 22 on the original timeline, I played another Square One TV song -- Weird Al's "Patterns." But I can also play another song here -- "Angle Dance," since 8.G1b mentions angles.
In fact, there are lots of Square One TV songs related to geometry topics. Unfortunately, those songs aren't directly related to transformational geometry as much.
What should my lessons look like during this unit? Standard 8.G1 is about the three main isometries -- rotations, reflections, and translations. But it's divided into G1a, G1b, and G1c -- and these discuss what the isometries preserve, not the isometries themselves.
We could devote one week to each isometry. If we went from easiest to hardest, then that would place translations in Week 21, reflections in Week 22, and rotations in Week 23. But due to Presidents' Day (more precisely, the student-free days leading up to Prez Day), Week 23 is a short week -- and we might not want to give the hardest transformation, rotations, that week. Indeed, rotations might fit better in Week 22 with G1b, since rotations and angles are intimately related.
We could save translations for the short Week 23 -- students might understand these in one day. But related to translations are tessellations -- and there's a "Tessellations" Square One TV song. So we might wish to do translations and tessellations during the same week -- a longer week, like Week 21, so I can sing both "Count the Ways" (Tuesday) and "Tessellations" that week. Of course, I shouldn't be letting songs dictate the pacing plan, but if I think the song will helpful to the students, I'll do it. (For example, at the long-term school I once delayed a lesson on solving equations beyond a short day Monday, just so I'd have enough time to sing the "Solve It" song.)
For simplicity, let's assume that these five Square One TV songs fill Weeks 21-23 -- "Count the Ways," "Tessellations," "Patterns," "Angle Dance," and "Mathematics of Love" in that order. Since we just completed Peterik's chapter on rhyming, let's look at the rhyming schemes for these five songs.
In "Count the Ways," the verses follow the pattern abab. The first verse has a perfect rhyme (length and strength) for the a lines, and a near rhyme (time and mind) for the b lines. The second verse is trickier, since (the late) Naomi Judd was speaking to her daughter Wynonna during the verse. It appears that the a lines contain a near rhyme (guy, spoken by Naomi, and mine, sung by Wynonna) while the b lines contain a perfect rhyme (equation, sung by Wynonna, and persuasion, spoken by Naomi). But if I were to change the song to a masculine perspective, "his love equals mine" should become "my love equals yours," breaking the near rhyme. (In fact, I often sing it as "your love equals mine" in order to preserve the near rhyme.) In its refrain (which could be considered a pre-chorus and chorus, or perhaps a chorus and post-chorus instead), we have a perfect rhyme (heart and apart) and a near rhyme (hour and year).
In "Tessellations," the verses follow the pattern abcb. The first verse has the rhyme lines and fine (a near rhyme due to the plural -s) as well as guide and ride (perfect). The second verse has plane and same (a near rhyme), and meet and repeats (near due to the -s). Both verses have internal rhymes on the penultimate lines (done and fun in the first verse, gaps and overlaps in the second). The bridge has the rhyme right and tight (and possibly here and there as a near rhyme). As for the chorus, we have an aaaa scheme with words rhyming the title line -- sensation, vibrations, and imaginations. So the ending -ations makes these perfect feminine (that is, two-syllable) rhymes, or near rhymes if no -s. (It's hard for me to hear whether -s is there in each line. Here I include -s in vibrations to honor The Beach Boys and in imaginations since it's "our imaginations," but not in sensation.)
In "Polka Patterns," the verses follow the pattern aabb. The first verse has perfect rhymes (math and bath, smart and art). The second verse has one of the more interesting perfect rhymes I've ever heard (be bored and keyboard) and a near rhyme (health and self). The third line has two rhymes, one near (party and body) and one perfect (answers and dancers). The chorus has an interesting rhyme scheme -- even though no word rhymes with the title "Patterns," there are many words rhyming with each other here -- everywhere, there, stare, care, declare, flair, hair, wear, and (savoir) faire. In each case, there are three rhyming words followed by the title "Patterns." We could call this aaab (with b = "Patterns") if we consider the lines to be one bar each -- otherwise, it's abab with internal rhymes if we consider the lines to be two bars each. All the rhymes in the chorus are perfect.
In "Angle Dance," the verses follow the pattern abcb. All the rhymes in this song are perfect. In the first verse we have fad and mad, song and along, and merge and converge. In the second verse we have breeze and degrees, care and square, please and degrees (again), and toe and zero. In the third verse we have excuse and obtuse, hip and relationship, hun and fun, and correct and connect. The c lines in each verse also include internal rhymes -- meet and beat, knees and degrees (yes, it's an angle song with lots of "degrees"), dried and side, tight and right, beaut and acute, and inclined and find. I'm not sure whether this song is considered to have a true chorus -- instead, we have the hook title "Angle Dance, Angle Dance" followed by words that change after each verse (with their rhymes included in the lists).
In "Mathematics of Love," the verses follow the pattern aabccb. All the rhymes in this song are perfect, starting with glowing and overflowing, and tightly and lightly. The b rhyme is above and love (from the title line). On the video itself, the lead singer refers to the ccb lines as "the second verse," so this would make it a cross-verse rhyme. Later on, we hear the rhymes ways and days, shining and intertwining, and finally (the memory) of and love (the title line at the end of the song).
OK, so that's a lot about rhymes in Square One TV songs, but what about my own rhymes? My most blatant rhymes are in "The Packet Rap," where I keep making (perfect) rhymes with packet throughout the song -- hack it, back it, racket, backpack it, jacket, track it, attack it, unpack it, and lack it. Of course, I intentionally chose rhyming words -- why hide the papers under your coat when you can hide them under your jacket instead? And rather than telling the students they should "always have it" (their packet), I tell them to make sure that they "don't lack it."
The song "Solve It" starts with the lines "When you see an equation, or problems that involve it." The intention, of course, is to rhyme "involve it" with the title "Solve It." But you might wonder why a single equation might have many problems that "involve it" -- but if I were to make it singular, I'd not only destroy that perfect rhyme (reducing it to the near rhyme "involves it") but force in an extra syllable "a" (problem). But in the Peterik chapter, he mentions a certain near rhyme used by John Greenbaum -- problem and solve 'em. This sounds like a no-brainer rhyme to include in a math song -- "When you see equations, in any word problem, all you have to do is solve 'em." Unfortunately, if I were to make that change, I'd be forced to change my title to "Solve 'Em."
There's one more song to consider -- my opening day song now called "Heroes and Zeroes." Most of the rhymes in that song are perfect -- math and wrath (and later path), strange and change, degree and see, STEM and them, hero and zero (duh!), seem and dream, bad and add. But in the original version of the song, I needed a word to rhyme with "Calculus." So I went for the near rhyme "cool college." I could have said "good college," but the "c" and "l" in cool add to the consonance with "Calculus," which makes it more likely for our ears to accept this as a near rhyme. The new version of the song that I wrote last month is an eighth grade version, so Calculus and college are no longer relevant. (And of course I had to change many lines because they rhymed with the disparaging word for a zero-scoring student that I no longer use.)
Speaking of eighth grade, let's get back to the new Math 8 song for today. After so many Square One TV songs during this unit, Week 24 definitely needs an original song. On Tuesday Week 24, I sing "The Big March" as I often do after Presidents' Day. (This is a parody of "The Ants Go Marchin' In" and inherits its numerical rhymes from that song -- the near rhyme one and thumb, then the perfect rhyme two and shoe, and so on.) On the original timeline, I round out the week with "Solve It," matching my actual lessons that week. But here on the new timeline, we move "Solve It" to earlier in the year during the EE strand, and now we need a new song for the G strand.
Before we look at the song, let's double-check the science for this unit. For Week 22, the PS2-4 project has the students gather objects including books, a spring scale, and graduated cylinder and start measuring forces. For Week 24, sandpaper is used as they are also measuring the force of friction.
Week 23 is the short week. It's the third week of the unit, and so there must be a science test. For the eighth graders, this must be given on Wednesday, February 15th, in addition to the SBAC Prep tests for math and science that they take every Wednesday. The tricky part is what to do about the seventh grade science test, as I don't see the Grade 7 students on Wednesdays. Monday is their coding day, and Tuesday is a math lesson.
OK, so let's get to the Week 24 song. Let's look at our standard and start thinking about rhyming words:
CCSS.MATH.CONTENT.8.G.A.2
Understand that a two-dimensional figure is congruent to another if the second can be obtained from the first by a sequence of rotations, reflections, and translations; given two congruent figures, describe a sequence that exhibits the congruence between them.
What are the key words in that standard? Well, I see "rotations," "reflections," and "translations." While we might be able to find some words that rhyme with these, recall that the three main isometries also have informal names -- "turn," "flip," and "slide." There are lots of words that rhyme with these three.
We also see the key word "congruent." There are a few perfect rhymes for "congruent," such as "truant" (but I don't want our kids thinking abut that word) and "fluent" (which, in math, means mastery of the basic operations, but that doesn't help us here). If we think about near rhymes instead, then we might consider "Bruin" (go UCLA -- hey, let's do another UCLA fight song parody). In fact, there are many more near rhymes if we count doin' and the like.
The last line of the standard mentions "congruence" and "sequence." These could be considered a perfect masculine (one-syllable) rhyme with "-ence" (but not "-uence," as "qu" makes a "kw" sound rather than anything with the "u" vowel).
Now that we have the rhymes, let's start the song:
20 FOR V=1 TO 3
30 FOR X=1 TO 43
40 READ A,T
50 SOUND 261-N*A,T*2
60 NEXT X
70 RESTORE
As usual, click on Sound before you RUN the program.
This song is written in 10EDL -- the simplest EDL in which I've ever composed a song. The scale consists of five notes -- C, D, E, F#, A. If the F# sounds out of place, we could change it to G and make it a traditional major pentatonic scale. Once again, just hold your horses and let us get to Chapter 9 on melodies before I make a full post on musical scales. I ran the TI generating program twice -- eight bars for the verse, eight bars for the chorus. I kept the verse the way it came out, but changed the order of some of the chorus bars to make it sound better.
As for the rhymes, I end up matching "slide," "turn," and "flip" with "side," "learn," and "tip." Of these, "tip" is the trickiest rhyme -- but the "tip" appears in the chorus to follow. (The tip to students is that reflections preserve congruence.) This is also what the students "learn" in the second verse (namely that rotations preserve congruence).
As for "congruent," I choose "doin'." Here I don't count "movin'" as an internal rhyme (while "movin'" and "doin'" might be a near rhyme, "movin'" and "congruent" are a bit too different to be considered even a near rhyme). Still, "movin'" sounds good in this line, as it's the moves (the transformations) that appear in sequence to define congruence.
Conclusion
Well, at least I've finally written an original geometry song (albeit for Math 8, not Geometry). Perhaps I should have written more such songs while still on the Geometry blog.
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