Today I subbed in a high school special ed class in my first OC district. Even though I've never subbed for this teacher before, I've subbed her class and recognize some of the students. It's a self-contained class with aides everywhere, and so there's no need to do "A Day in the Life" today.
And indeed, today's class is very non-representative on several levels. For starters, you might wonder what these students with very low skills are doing during the pandemic and the hybrid schedule. The gen ed students have academic support (mostly online) after lunch, but what about these students? The problem is that tonight is Open House -- or should I say, the "Showcase." As I explained on the blog last year:
I wrote that this new district has replaced its spring Open House with the "Showcase," used to help sixth graders choose a middle school for seventh grade, as well as help eighth graders choose a high school. As it turns out, this school has its "Showcase" tonight. For one thing, it means that today is a minimum day.
The fact that the "Showcase" was in January/February last year meant that these schools held their Open House before the coronavirus closure -- most schools that had Open House during the typical March/April timeframe probably didn't have it at all last year. This year the Showcase is being held online -- but the school still has a minimum day for it this year anyway.
For gen ed students, the schedule is basically the same as the usual hybrid schedule except that there is no academic support. It appears that some students with special needs, such as the ones I see today, do stay on campus after lunch -- but not on minimum days like today.
Like all Wednesdays, today is even periods. In second period, the students watch CNN 10 News and then answer questions about current events, in this case the East Coast winter storm and the second imp...ment trial that is about to start. (As usual, I try not to write out that politically charged word).
Fourth period is P.E. for these students. Recall that during the long-term assignment, students attended school for five periods because sixth period was independent P.E. class. But it's awkward for high schools to do something similar, since California only requires P.E. through sophomore year. Some juniors and seniors (especially those who are college-bound) may want six academic classes, and so high schools must still have six periods. (It appears that middle schools in my first OC district also have six periods.)
So I am curious to see what P.E. looks like in the pandemic era -- but today is again non-representative, because it's a special ed class. The students walk a few laps around the track and then participate in a quick 40-yard dash. I tell the students about my own days as a young high school Track athlete.
For sixth period on Wednesdays, a counselor comes in to provide Career Guidance -- but today is again non-representative, because it's a minimum day. It was decided that the special ed students should have lunch before they leave (whereas gen ed students who need lunch get it on their way home). So this ends up taking most of sixth period. The counselor ends up doing a short Careers lesson with only the online students, because the in-person students have already left for their bus.
As is my tradition, I do hand out pencils and candy to the special ed classes. It's close enough to Valentine's Day to start handing out heart pencils.
When I arrived at the school today, I found out that not only was there a new teacher for this class, but it was in a different classroom. And as it turns out, this was the very same room that I subbed the last three days before the coronavirus closure -- that's right, the room where I lost my old songbook! (So it didn't matter that I was switched last Friday to a science class -- the English teacher with the guitars has moved on to a different room.)
And so when the aide goes up to pick up one of the students from the bus, I immediately scan the room for my songbook. The former teacher and his guitars are gone -- all that remains is a poster of some ukelele chords.
And then -- near the back of the room, placed inside a manila folder, is my songbook. So, after almost eleven long months, I finally find my old songs.
I don't perform any songs today, as I almost never sing for this class (as it's run by the aides). But I will use post some of the songs that I thought were lost forever.
And let's start with the old Bizzie Lizzie songs. Elizabeth Landau's birthday was last week -- oh, how I wish I could have found the book in time to celebrate it. Well, at least I have it now. There's no way I want this to happen again, and so I want to preserve Lizzie's songs on the Internet forever.
Here is her original version of "American Pi," a parody of "American Pie." Landau has posted a newer version of this song on her current webpage, but I like the original version better.
AMERICAN PI
Copyright 1998 by Elizabeth Landau, aka Bizzie Lizzie
1st Verse:
A long long time ago, I still remember when,
My favorite number was pi.
And if I had some batteries, my efforts would have surely pleased,
That math fair judge, and that's no lie.
But teachers' gazes made me shiver with every grade that they'd deliver,
Bad news in history -- just needed that time to study.
I can't remember if I cried when I realized that I hadn't tried,
Before something clicked off deep inside, then it, my calculator died.
Refrain:
And I was thinking, why, why can't I calculate pi?
I just want to see the numbers 3 point 1 4 1 5.
And if that's all, then let's keep it alive,
'Cause my calculator may have just died.
My calculator may have just died.
2nd Verse:
Did you write the Law of Sines and did you color outside the lines,
And did someone tell you to?
And did you ever make it past Trig, are tangents not something you dig,
And did you ever catch the number flu?
Now I had my TI-83 when a great misfortune came to me,
My screen began to blur -- I couldn't even graph a curve!
I was a lonely teenage computer freak -- my pencils were chewed but my glasses were sleek,
And the future was looking pretty bleak the day my calculator died. (to Refrain)
3rd Verse:
Now for three days I'd been on my own just using whatever my friends could loan,
But that's not how it used to be.
When I entered the math fair, I could have sat in the first chair,
'Cause I had a model 83.
But while I fumbled with the screws, a new formula began to fuse,
I'd have to use arctans -- those weren't in my plans.
I had a great idea that could really fly, but I couldn't even verify,
That this function was in terms of pi the day my calculator died. (to Refrain)
4th Verse:
I replaced all the double A's, but the new four didn't even faze,
My old beloved TI.
In the meantime I checked out a book to which I hadn't given a single look,
So much for originality. (Sigh!)
It said 2 arctan 3 over 79 plus 5 arctan 1/7 would be just fine,
If multiplied by four...I couldn't read anymore!
My calculations had been correct except for one small, minor defect,
It seemed like my entire life was wrecked the day my calculator died. (to Refrain)
5th Verse:
We know it's not the pie you eat -- this pi's a far more delicious treat,
And it can be so much fun.
Now pi's a movie, pi's a perfume, the digits of pi could fill a room,
But could you ever find the last one?
And as I told this to the school, I remembered one large, vital rule:
"Don't ever get up there if the audience doesn't care!"
As they all laughed, I felt disgraced, like I just gotten a pie in the face,
I knew I'd really lost the race the day my calculator died. (to Refrain)
6th Verse:
I met a guy in period three who seemed like he could help me,
At least he didn't turn away.
After a quick glance at my old friend, he finally had some advice to lend,
But he didn't put in a nice way:
"I can't believe you were so dumb -- all you needed was a new lithium!"
I asked, "What do you mean?" He wouldn't even come clean,
And tell me that there are two sets of batteries, not just the quartet.
So until I get some and it resets, I think I'll eat some pie. (to Refrain)
(Refrain is sung one last time, with the final line as below)
'Cause my calculator didn't really die!
Even though I've already posted Bizzie Lizzie's other song, I'll repost it here for completion:
THE DIGIT CONNECTION
Copyright 1998 by Elizabeth Landau, aka Bizzie Lizzie
1st Verse:
Why are there so many debates about pi?
And what's on the other side?
Pi is a ratio of random proportions.
Its digits have nothing to hide.
So we've been told and some choose to believe it,
But I know they're wrong, wait and see!
Someday we'll find it, the digit connection,
Mathematicians, logicians, and me.
2nd Verse:
Who said that everything has some sort of pattern,
Consisting of nothing but math.
Somebody thought of that, and someone believed it.
Now we're all caught in its wrath....
What's so hypnotic in something chaotic,
And what do we think we might see?
Someday we'll find it, the digit connection,
The optimists, the theorists, and me.
All of us under its spell,
We know it must be math-e-magic...
3rd Verse:
Have you been half asleep? And have you heard voices?
I've heard them calling my name.
Is this the sweet sound that calls the young sailors,
The voice might be one and the same....
I've heard it too many times to ignore it,
Irrational, random, and free.
Someday we'll find it, the digit connection,
The lovers, the dreamers, and me.
3.1415926535 dot, dot, dot!
Of course, Pi Day is still over a month away. Then again, the seventh graders at my long-term school are still in the geometry unit -- and in fact, they reached the unit on pi last week. If my long-term had extended through January (and if I hadn't lost the songbook in this classroom), I could have performed these songs on Landau's birthday.
Then again, e Day is coming up this weekend (on a non-posting day). As I wrote before, Bizzie Lizzie also had a song about e (a parody of "Sugar Sugar" but I never wrote down all the lyrics before her website disappeared. I wrote a version on the blog, combining some of Lizzie's lyrics that I remember with a few words of my own:
Refrain:
e (2.718)
Ah, number number (281828)
You are my natural log,
And you got me calculating.
e (2.718)
Ah, number number (281828)
You are my derivative,
And you got me calculating.
1st Verse:
I just can't believe the loveliness of graphing you.
I can't believe you're more than two.
I just can't believe the loveliness of graphing you.
I can't believe you're more than two. (to Refrain)
2nd Verse:
I just can't believe your digits go forever now.
As long as a number can be.
I just can't believe your digits go forever now.
As long as you're the number e. (to Bridge)
Bridge:
Put a little cash in the bank, money.
Put a little cash in the bank, baby.
I'll make more next year, yeah, yeah, yeah!
Put a little cash in the bank.
100% interest on my money.
Compound it continuously, baby.
I'm gonna take the limit now, yeah, yeah, yeah!
My cash is multiplied by you, e. (to Refrain)
Since this is a song about the number e, it should be properly performed in the key of E major.
Meanwhile, there are some non-Lizzie songs that I find in my book as well. This includes some of the previously lost tunes of my songs "Another Ratio Song" and "Linear or Not." And so I must decide whether to keep the original tunes or the replacement tunes.
Yesterday, I decided on the fly to convert "Linear or Not" into a rap -- and I like the way it sounds. So I won't use the original tune I wrote years ago.
As for "Another Ratio Song," my book shows that it's in the key of G major, and the needed chords are ascending (G, Am7, Bm7, C, D7, and so on). And so I'm considering reverting this original tune, but perhaps updating it to fit 18EDL (since I first came up with these tunes before discovering EDL). With ascending chords, the song might serve as the "Do Re Mi" of 18EDL.
I hope to have another music post soon -- especially if I do return to the classroom of the teacher with the guitars and ukes. Well, at least I made it to the room with my missing songbook today.
Today is Tenday on the Eleven Calendar:
Resolution #10: We are not truly done until we have achieved excellence.
I tell the students that they aren't done with the 40-yard dash until they cross the finish line. I don't really pay "homage" to anyone today, though I suppose I indirectly pay homage to sprinters like Usain Bolt in inspiring the students to become better runners.
Lesson 10-2 of the U of Chicago text is called "Surface Areas of Pyramids and Cones." In the modern Third Edition of the text, surface areas of pyramids and cones appear in Lesson 9-10.This is what I wrote last year ago about today's lesson. Notice that I spent much of that post comparing the U of Chicago text to three other math texts (and I decided to preserve this discussion):One of the texts was published by Merrill, the others by McDougal Littell. I ended up purchasing the latter, which is dated 2001. I actually recognize this text from when I spent one month in an advanced seventh grade math classroom back in 2012.
[2021 update: This was actually for BTSA, which I explained on the blog during those long days when all the schools were closed. After that month, I was switched to a high school Algebra I class until I completed BTSA requirements.]
Geometry is covered in Chapters 8 through 10. Chapter 8 covers points, lines, polygons, transformations, and similarity. The transformation section covers reflections and translations (but not dilations in the similarity section), but of course, this is an old pre-Common Core text, so transformations aren't used to define congruence. Chapter 9 is officially called "Real Numbers and Solving Inequalities," but the real numbers portion of the chapter segues from square roots to the Pythagorean Theorem and to the Distance Formula.
That takes us to Chapter 10. As it turns out, much of Chapter 10 of this seventh-grade text matches up with the same numbered chapter of the U of Chicago geometry text. Here are the sections:
Section 10.1: Circumference and Area of a Circle
Section 10.2: Three-Dimensional Figures
Section 10.3: Surface Areas of Prisms and Cylinders
Section 10.4: Volume of a Prism
Section 10.5: Volume of a Cylinder
Section 10.6: Volumes of Pyramids and Cones
Section 10.7: Volume of a Sphere
Section 10.8: Similar Solids
It's often interesting to see how much surface area and volume appears in pre-algebra texts. Wee see that this text gives all of the volume formulas, while only the cylindric solids have their surface areas included in the text. But let's keep in mind that this text was specifically written for the old California state standards that we had before the Common Core.
The final chapter, Chapter 12, of this text is on polynomials. This chapter actually goes a bit beyond the seventh grade standards -- most notably, Section 12.5 is "Multiplying Polynomials" and actually teaches the FOIL method of multiplying two binomials. I was only in the classroom that taught using this text for a month, but I was told that the honors class would cover Chapter 12 around the start of the second semester, with the rest of the chapters taught in numerical order. (Non-honors classes would not cover Chapter 12 at all.) The next section, Section 12.6, may also seem a bit advanced for a pre-algebra class -- "Graphing y = ax^2 and y = ax^3" -- but it appears in the 7th grade standards.
If we compare this to the Common Core Standards, we see that much of Chapter 10 of the McDougal Littell text corresponds to an eighth grade standard in Common Core:
CCSS.MATH.CONTENT.8.G.C.9Know the formulas for the volumes of cones, cylinders, and spheres and use them to solve real-world and mathematical problems.
This is to be expected. The Common Core Standards are based on Algebra I in ninth grade, while the California Standards were based on Algebra I in eighth grade. So many eighth grade Common Core Standards must have been seventh grade standards in California.
Before we leave the McDougal Littell text, let me note that Section 4.3 is on "Solving Equations Involving Negative Coefficients." For comparison purposes, let's look at the McDougal Littell Algebra Readiness text in more detail:
1. Expressions, Unit Analysis, and Problem Solving
2. Fractions
3. Decimals and Percents
4. Integers
5. Rational Numbers and Their Properties
6. Exponents
7. Square Roots and the Pythagorean Theorem
8. Equations in One Variable
9. Inequalities in One Variable
10. Linear Equations in Two Variables
The purpose of Algebra Readiness was to prepare students for Algebra I. Therefore, as we can see, there is very little geometry in this text compared to the McDougal Littell Math 7 text. The only geometry that appears is in Chapter 7, with even less geometry content than Chapter 9 of the Math 7 text (as the Distance Formula doesn't appear in the Algebra Readiness text). Area and volume are nowhere to be seen in the Algebra Readiness text (except the appendix, "Skills Review Handbook").
In many ways, Algebra Readiness was more like a Common Core 7 text than Common Core 8, as Common Core 8 contains more geometry (and even a little more algebra) than the Readiness text.
Recall that the traditionalists were recently complaining about how there is too much Geometry and too little Pre-Algebra in Common Core 8, even for students headed for freshman Algebra I. This text might be more acceptable, but it still lacks the rational equations that they also complained about.
Now leave McDougal Littell and continue with the Merrill text:
I didn't purchase the Merrill Pre-Algebra text, so I don't recall how old the text is. But I glanced at it and noticed that all of the equations that appear in Chapter 10 of the U of Chicago Geometry text also appear in this text, with the exception of the equations involving a sphere. That is, the surface area formulas of all cylindric and conic solids appear in this text. This is unusual since, as we've seen, neither the CAHSEE nor the Common Core Standard expect students to learn the more complex surface area formulas before high school Geometry. Since today's lesson is Lesson 10-2 of the U of Chicago text, which is on surface areas of pyramids and cones, I want to discuss what I remember about the Merrill lesson on these surface areas.
Both Merrill and the U of Chicago give the lateral area of the pyramid as the sum of the areas of its triangle lateral faces. But only the U of Chicago gives the formula for a regular pyramid, which it defines in Lesson 9-3 as a pyramid whose base is a regular polygon and the segment connecting the vertex to the center of this polygon is perpendicular to the plane of the base. The formula for the lateral area of a regular pyramid is LA = 1/2 * l * p.
But now we must consider the surface area of a cone. The Merrill text does something interesting here, as it considers the area of the net of the cone. We cut out the circular base and a slit in the lateral region, and then flatten this lateral region. What remains is a sector of a circle. Then the Merrill text simply gives the area of this sector as pi * r *s (where s, rather than l, is the slant height) without any further explanation.
The U of Chicago text, meanwhile, gives a limiting argument for the surface area of the cone, as its circular base is the limit of regular polygons as the number of sides approaches infinity. But there is Exploration Question 25, where the Merrill demonstration is done in reverse -- we begin with a sector of a disk and fold it into a cone.
But neither tells us why the area of the sector (and thus the lateral area of the cone) is pi *r * l. Let me give a demonstration of why the area of the sector is pi * r * l.
We begin with the area of a circle, pi * R^2. The reason why I used a capital R is to emphasize that the radius of the circle that appears in Question 25 is not the radius r of the base -- indeed, it's easy to see that the radius of the circle becomes the slant height l. So the area of the circle is pi * l^2 -- that is, before we cut out the sector. We want to fit the area after we cut it.
Let's recall another formula for the area of a circle given by Dr. Hung-Hsi Wu: A = 1/2 * C *R -- and once again, R = l, so we have A = 1/2 * C * l. But neither one of these gives us the circumference or area of a sector. If we let theta be the central angle of a sector, we obtain:
x = theta / 360 * C
L.A. = theta / 360 * A
= theta / 360 * 1/2 * C * l
For lack of a better variable, I just let x be the arclength of our sector. But here I let L.A. be the area of the sector, since these equals the lateral area of the cone we seek. The big problem, of course, is that we don't know what angle theta is for the cone to have a particular shape. But we notice that we can simply substitute the first equation into the second:
L.A. = 1/2 * theta / 360 * C * l
= 1/2 * x * l
And what exactly is the arclength x of our sector? Notice that once we fold the sector into a cone, the arclength of the sector becomes the circumference of the circular base of the cone! And this we know exactly what it is -- since the radius of the base is r, its circumference must be 2 * pi * r:
L.A. = 1/2 * (2 * pi * r) * l
= pi * r * l
as desired. QED
Let's continue with the next proposition in Euclid:Proposition 15.If two straight lines meeting one another are parallel to two straight lines meeting one another not in the same plane, then the planes through them are parallel.This proof should be easy to modernize:Given: AB, BC distinct lines in Plane P, DE, EF distinct lines in Plane Q, AB | | DE, BC | | EFProve: Plane P | | Plane Q
Proof:Statements Reasons1. bla, bla, bla 1. Given2. Point G in plane Q so that 2. Proposition 11 from Thursday (construction) BG perp. plane Q3. H in Q so that GH | | ED, 3. Existence of Parallels K in Q so that GK | | EF (sometimes called "Playfair," but uniqueness is not needed)4. BG perp. GH, BG perp. GK 4. Definition of line perpendicular to a plane5. BG perp. AB, BG perp. BC 5. Perpendicular to Parallels (planar)6. BG perp. plane P 6. Proposition 4 from two weeks ago7. Plane P | | Plane Q 7. Proposition 14 from yesterday (a form of Two Perpendiculars)Euclid's original proof should be simple enough for high school students to understand without the need to convert it to two columns. All that's need is to replace phrases such as "therefore the sum of the two angles...is two right angles" with "Perpendicular to Parallels," for example.
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