Lesson 12-4: Proportions (Day 124)

Today I subbed in a high school English class. It is in my first OC district. Since it's a high school class that's not math, there's no need for "A Day in the Life" today.

Odd classes meet today, including first period, which really means zero period. This is a senior class that is reading Jon Krakauer's Into the Wild, a book about a young man who died in the Alaska wilderness back in the 1990's. This is followed by third period, a junior class that is also reading The Great Gatsby. Fifth period is the teacher's conference, and teachers with a first period don't usually have a seventh. Thus my teaching day is over by 10:00 (but there is still academic support in the afternoon).

Both the juniors and seniors have a writing assignment where they must take an object from the book that they're reading (or a creature in the case of Into the Wild) and create an "instruction manual" -- they should describe how to use it and keep it in working order (or alive).

Today is Nineday on the Eleven Calendar:

Resolution #9: We pay attention to math as long as possible.

This doesn't really come up today. One thing I do want to say about timing is that I was confused as to whether this teacher had a first (zero) period today, which resulted in my setting up the Zoom late, nearly halfway into the period. Some students quickly notice the message I put in Google Classroom for today's Zoom link, while others don't see it until class is almost over.

One girl doesn't figure out the link until after first period is over -- later on, she claims in a Google Classroom message that since I have the same first name as one of her other teachers, she thought that the link was for that teacher and not first period. Since so many people (including myself) have had trouble with Zoom, I'm inclined to believe her.

Since this is a high school class, I want to perform the Darren Miller wager. I mentioned this in last Wednesday's post -- some traditionalists like Miller want high schools to reopen for in-person learning, yet I've seen that when schools reopen, high school students opt out. So for this wager, I lose $2 for each junior and senior who opts in to in-person learning and gain $1 for each student who opts out.

Here are the stats for today's classes -- in first period, ten seniors opt in and two dozen opt out. This gives me a profit of $4. (Of those ten, four are in Cohort A, and only two of those four are present.) In second period, a dozen juniors opt in and 22 opt out, for a loss of $2. But if I were wagering with the actual traditionalist Miller, he'd point out that the junior class has one TA (most likely a senior) who attends in person, and so he should get an extra $2 for her. So I win $4 for first period and he wins $4 for third, making it a break-even day.

Notice that one of the first period seniors is listed as an in-person student but attends online today. But (as Miller) would point out, the wager is based on the roster, which lists him as in-person. Thus I lose money for that student.

Of course, we break even only because I raised the stakes to $2 for in-person juniors and seniors. If I'd kept it at 1:1 then I win easily. The point I was making was that only about a third of the upperclassmen attends in-person, so it's not as critical to reopen high schools as quickly as Miller insists.

The song for today is "Packet Rap." This is the second time I've performed it this month, despite there being no printed packets during the pandemic. I just like the song.

Lesson 12-4 of the U of Chicago text is on proportions. I wrote a lot about comparing the U of Chicago approach to those of other theorists, and I retain some of that discussion in today's post.

Let's start with Hung-Hsi Wu. His "Fundamental Theorem of Similarity" actually consists of some of the properties of dilations that appeared in yesterday's Lesson 12-3. If the image of PQ under a dilation with scale factor k is P'Q', then P'Q' | | PQ and P'Q' = k PQ.

Wu proves this in cases based on what sort of number the scale factor k is. For natural number k, Wu proves it using induction on k. This initial case, k = 2, is based on a special version of the Midpoint Connector Theorem of Lesson 11-5. The inductive case from k to k + 1 involves repeating the Midpoint Connector Theorem argument over and over.

For rational number k, if k = 1/q (q natural number), then Wu notes that a dilation of scale factor 1/q is the inverse of a dilation of scale factor q. And if k = p/q (p, q natural numbers), then Wu notes that a dilation of scale factor p/q is the composite of two dilations, with scale factors p and 1/q.

All that's left is to extend the argument to irrational k. Wu hand-waves over this by using what he calls the "Fundamental Assumption of School Mathematics" -- many theorems of pre-college math that apply to all rational numbers also apply to all real numbers. (This assumption also appears in Algebra II when defining what it means to raise a number to an irrational power.)

Even though Wu gives this proof, it's not the sort of proof we expect high school students to figure out easily. In the years since I posted it, I've regretted it. But every year since then, when I return to Lesson 12-6, I keep the Wu proof and change other parts of the worksheet!

As I wrote in the comment above, the EngageNY curriculum is based on the Wu proofs. But there's no way in the world EngageNY would use Wu's "Fundamental Theorem of Similarity" proof.

Instead, they replace it with a simpler proof based on the area of a triangle. I've mentioned the idea before how in proofs, area and similarity are often interchangeable -- this is why the Pythagorean Theorem has both area and similarity proofs. EngageNY's area proof removes the need for Wu's induction on k and subsequent extension to rational and real values of k.

But there's one problem here -- similarity is a "Module 2" topic, but area is a "Module 3" topic. This is the naive order suggested by the Common Core Standards -- since they mention similarity before area, all similarity lessons must appear before any area lessons. Once again, EngageNY justifies this by having students recall the triangle area formula from eighth grade (or earlier).

Notice that in the U of Chicago text, area (Chapter 8) appears before similarity (Chapter 12). Thus the U of Chicago could validly follow the EngageNY sequence of proofs -- except that it doesn't.

Many traditionalists attack the Wu/EngageNY plan from not following Euclid's geometry. But ironically, Wu/EngageNY actually follow Euclid more closely than traditional texts do! His Book VI, which teaches similarity, begins with Proposition 1 (ratios between areas of triangles). This is followed by Proposition 2 (Side-Splitter Theorem), from which both Wu/EngageNY derive their first similarity theorem (Proposition 4, AA~).

Of course, there's also the idea of deriving the similarity theorems "classically" -- that is, by assuming AA~ as a postulate -- and then ultimately deriving the properties of dilations. This is the method used by the old PARCC test, which should be irrelevant since California isn't a PARCC state, and backlash against the Core has led to most PARCC states dropping that test. But I still end up posting old PARCC questions from time to time -- including the Lesson 12-3 worksheet from yesterday.

Here is today's worksheet.

Lesson 12-3: Properties of Size Changes (Day 123)

 Today on her Daily Epsilon on Math 2021, Rebecca Rapoport writes:

What is the minimum number of faces required to form an Archimedean solid?

Of course, until we know what an Archimedean solid is, we can't even start this problem. Many of us have heard of Platonic solids -- a Platonic solid is what the U of Chicago text calls a "regular polyhedron" in the Exploration section of Lesson 9-7:

"A regular polyhedron is a convex polyhedron in which all faces are congruent regular polygons and the same number of edges intersect at each of its vertices."

As it turns out, an Archimedean solid has almost the same definition, except for a few changes:

"An Archimedean solid is a convex polyhedron in which all faces are regular polygons that are not all congruent and the same number of edges intersect at each of its vertices..."

Notice that we can't simply drop the word "congruent" from the Platonic solid definition -- otherwise it would lead to an inclusive definition in which all Platonic solids are Archimedean. In particular, the tetrahedron would become Archimedean, and the answer to today's Rapoport question would be four. I will just say it now -- the answer isn't four. The definition of Archimedean solid is exclusive, so that no Platonic solid is Archimedean.

So how can the faces of a solid be all regular yet not all congruent? There are only two ways for regular polygons not to be congruent -- either they have different side lengths or different numbers of sides. But the faces of a solid share edges, thus forcing all side lengths to be equal. Thus in an Archimedean solid, the faces must have different numbers of sides -- for example, it might contain triangles and squares.

Oh, and there's another part of the definition of Archimedean solid that we must consider:

"...and is not a prism..."

It's easy to form a triangular prism with two equilateral triangles and three squares as faces. The definition of Archimedean solid excludes prisms, otherwise the answer to this problem is five. Here's a hint -- the correct answer isn't five.

One way to form an Archimedean solid is to take our simplest Platonic solid -- the tetrahedron -- and modify it so that it becomes Archimedean. For example, from each triangular face we might try cutting off a small triangle from each vertex.

If we try cutting halfway down each edge and removing the corners, what's left is a smaller triangle -- and so doing it to the whole tetrahedron, we're left with just a smaller tetrahedron.

Instead, we try cutting one-third of the way down each edge rather than halfway. Then voila -- what's left of each face is a regular hexagon. If we try doing to the entire tetrahedron, then the original four faces become regular hexagons, and the parts where we cut the four vertices become four new equilateral triangles.

Since we formed this new solid by cutting off or truncating the vertices of a tetrahedron, the resulting solid is called a truncated tetrahedron:

https://mathworld.wolfram.com/TruncatedTetrahedron.html

As it happens the truncated tetrahedron with its eight faces is the simplest Archimedean solid. Therefore the desired answer to today's problem is eight -- and of course, today's date is the eighth.

By the way, here's a link to the Wolfram page for all thirteen Archimedean solids. While eight is indeed the fewest number of faces one can have, the most is 92:

https://mathworld.wolfram.com/ArchimedeanSolid.html

According to this link, there's one part of the definition that we left out:

"...or an antiprism."

Antiprisms are tricky to define -- they're sort of like prisms, but not really. But as it turns out, the simplest antiprism also has eight faces, so we don't need to know what antiprisms are in order to get the Rapoport problem correct.

Also, Wolfram mentions another term, semiregular polyhedron. Since the U of Chicago text uses the term "regular polyhedron" for Platonic solid, we ought to use "semiregular polyhedron" instead of Archimedean solid. There's just one difference though -- a prism (or antiprism) counts as a semiregular polyhedron but not an Archimedean solid.

Archimedean solids are, of course, named for the Greek mathematician Archimedes, just as Platonic solids are named for Plato.

Lesson 12-3 of the U of Chicago text is called "Properties of Size Changes." In the modern Third Edition of the text, properties of size changes appear in Lesson 12-1.

This is what I wrote last year about today's lesson. Admittedly it isn't much.

Finally, here are the Geometry worksheets for today. They are based on Lesson 12-3, with an extra page for the proof of the Dilation Distance Theorem -- this proof comes directly from PARCC.

Meanwhile, the old first page makes an reference to FTS (an old Hung-Hsi Wu proof). Even though I no longer include Wu's FTS as part of the lesson, it's actually too much work to redo the entire page just to get rid of two little FTS mentions (especially when I'm already posting this late). So you'll just have to ignore FTS.

Lesson 12-2: Size Changes Without Coordinates (Day 122)

Today I subbed in a seventh grade PE class. It's in my first OC district. Of course, it's another class for which it's not worth doing "A Day in the Life."

It's my first time I've covered for this teacher since a two-day assignment in February 2020. This teacher likes to begin each period by having the students walk two laps. Last year when I covered for him, I sang "The Big March" while walking, since it was the first week of that dreadful period. And this year it's the Big March again, so naturally I sing this same song today. To reduce the monotony, I add another simple song to our long walk -- the "Row" parodies "Measures of Center" and "Same Sign Add and Keep."

The walk is followed by a weekly "Friday stretch" -- the students do ten different stretches with 10-15 seconds of rest between them, and then they measure their heart rate. This teacher did nothing like this last year (even though one of the days I subbed for him was a Friday). I want to keep in shape, so I do the Friday stretch along with them -- meaning that I do it three times today.

Today is Fiveday on the Eleven Calendar:

Resolution #5: We treat people who are great at math as heroes.

Obviously this is a bit tricky in a P.E. class. Indeed, I don't really mention heroes or pay homage to anyone at all today. After the stretch, the students take turns participating in a 50-yard dash, and so the default hero is sprinter Usain Bolt.

The fastest sprinters among today's seventh graders run sub-7 seconds for the 50. Some of the faster girls can run it in sub-8.

Today on her Daily Epsilon on Math 2021, Rebecca Rapoport writes:

What is ED/FG? ABCD is a square. E and F are midpoints [of BC and AD respectively].

[There's one more thing that's given in the diagram -- G is the foot of the perpendicular from A to BF.]

Since we're not given the side length of the square, let's just call it s -- most likely, it will turn out that the exact length doesn't matter. Actually, let's call it 2s -- since midpoints appear in the problem, we most likely will prefer that extra factor of 2 to avoid annoying halves later on. (In fact, if the exact length ultimately doesn't matter, we can get away with simply calling it 2 and avoiding s altogether, but let's keep the s here.)

This means that in right triangle CDE, CD = 2x (side of a square) and CE = x (half of a side). Since we know two legs of a right triangle, we use the Pythagorean Theorem to find the hypotenuse:

a^2 + b^2 = c^2, where a = s, b = 2s, c = ED

s^2 + (2s)^2 = ED^2

5s^2 = ED^2

ED = s sqrt(5)

Now we have one of the sides we need, ED -- all that remains is to find the other side, FG. We notice that FB = ED = s sqrt(5), but this doesn't give us FG right away.

The fact that we're asked to find a ratio ED/FG suggests that the answer has something to do with similar triangles. And indeed, we can use a theorem found in Lesson 14-2:

Right Triangle Altitude Theorem:

b. each leg is the geometric mean of the hypotenuse and the segment of the hypotenuse adjacent to the leg.

In the diagram in the U of Chicago text, this appears as a = sqrt(cx), where a is the leg, c is the hypotenuse, and x is the segment we're trying to find.

a = sqrt(cx), where a = s, c = s sqrt(5), x = GF

a^2 = cx

s^2 = s sqrt(5) x

s = x sqrt(5)

x = s/sqrt(5)

Thus we have ED = s sqrt(5) and GF = s/sqrt(5). It now remains to find the ratio:

ED/GF = s sqrt(5)/(s/sqrt(5)) = sqrt(5)^2 = 5

Therefore the desired ratio is 5 -- and of course, today's date is the fifth. We haven't quite reached this theorem since we're not in Chapter 14 yet. But the theorem ultimately goes back to similarity -- and that's what we're learning about now, similarity and dilations.

Lesson 12-2 of the U of Chicago text is called "Size Changes Without Coordinates." In the modern Third Edition of the text, size changes without coordinates don't appear on their own. The first lesson of the new text, Lesson 12-1, corresponds more closely to Lesson 12-3 of the old text. The opening dilation activity of the old Lesson 12-2 is nonetheless squeezed into the new 12-1.

OK, today is the day that I want to post a pandemic activity, which usually means Desmos. But it's tricky to find a dilation activity that doesn't contain coordinates. (While reflections and rotations are often considered without coordinates, translations and dilations tend to be taught on the number plane.)

I do finally find one to my liking:

https://teacher.desmos.com/activitybuilder/custom/5e13560bbefb180dcd92d427

This activity uses a "stretching machine" to perform the dilations -- and this stretching machine isn't dependent on a coordinate plane.

I'll post one of last year's worksheets to supplement this Desmos activity.

Lesson 12-1: Size Changes on a Coordinate Plane (Day 121)

Today on her Daily Epsilon on Math 2021, Rebecca Rapoport writes:

The surface area of a sphere of radius r divided by the area of a circle of radius r.

This is straightforward provided we know the formulas. The sphere surface area formula is 4pi r^2, and the circle area formula is pi r^2. Thus the desired ratio is 4 -- and of course, today's date is the fourth. In fact, Lesson 10-9 of the U of Chicago text points this out:

"Surprisingly, this formula indicates that the surface area of a sphere is equal to 4 times the area of a great circle of the sphere."

Lesson 12-1 of the U of Chicago text is called "Size Changes on a Coordinate Plane." In the modern Third Edition of the text, size changes on a coordinate plane appear in Lesson 3-7. Yes, Chapter 12 is officially the same in both editions, but for some reason, the new edition introduces transformations involving size ("dilations") very early in the text. Beginning with the old Lesson 12-3, most of the old Chapter 12 material does indeed appear in the new Chapter 12 as well.

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

In the past, I skipped over Lesson 12-1. This is because I was mainly concerned with circularity -- dilations are used to prove some of the properties of coordinates, but right in this lesson, coordinates are used to prove the properties of dilations.

But last year, I was fed up with juggling the order of the U of Chicago text (and I got in trouble trying to juggle the Illinois State text as well). This year I want to stick to the order as intended by the authors of the U of Chicago text. And furthermore, we've seen that the actual dilation problems on the PARCC and SBAC involve performing dilations on a coordinate plane -- not using dilations to prove properties of coordinates! So Lesson 12-1 is more in line with PARCC and SBAC.

Here is the main theorem of Lesson 12-1 along with its coordinate proof:

Theorem:

Let S_k be the transformation mapping (x, y) onto (kx, ky).

Let P' = S_k(P) and Q' = S_k(Q). Then

(1) Line P'Q' | | line PQ, and

(2) P'Q' = k * PQ.

Proof:

Let P = (a, b) and Q = (c, d) be the preimages.

Then P' = (ka, kb) and Q' = (kc, kd).

(1) Line P'Q' is parallel to line PQ if the slopes are the same.

slope of line P'Q' = (kd - kb) / (kc - ka) = k(d - b) / k(c - a) = (d - b) / (c - a)

slope of line PQ = (d - b) / (c - a)

Thus line PQ | | line P'Q'.

(2) The goal is to show that P'Q' = k * PQ.

From the Distance Formula,

PQ = sqrt((c - a)^2 + (d - b)^2).

Also from the Distance Formula,

P'Q' = sqrt((kc - ka)^2 + (kd - kb)^2)

        = sqrt((k(c - a))^2 + (k(d - b))^2)      (Distributive Property)

        = sqrt(k^2(c - a)^2 + k^2(d - b)^2)    (Power of a Product)

        = sqrt(k^2((c - a)^2 + (d - b)^2))       (Distributive Property)

        = sqrt(k^2)sqrt((c - a)^2 + (d - b)^2) (Square Root of a Product)

        = ksqrt((c - a)^2 + (d - b)^2)              (Since k > 0, sqrt(k^2) = k)

        = k * PQ                                             (Substitution) QED

At the end of this post, it's back to posting worksheets based on the U of Chicago text. This time, I post an activity from last year where students dilate cartoon characters. This activity makes more sense this year than last year since it requires using coordinates.

Oh, and before you ask, my next pandemic-friendly activity won't be until tomorrow's post.

This lesson could've actually helped my eighth graders as well. We were supposed to cover dilations earlier but we ran out of time. For that matter, slope and the Distance Formula are also part of the eighth grade curriculum. I wouldn't make eighth graders perform the two preceding proofs with so many variables, but specific numerical examples are within the reach of eighth graders.

Chapter 11 Test (Day 120)

Today is many things. According to the blog calendar, today is Day 120. Therefore, it is the end of the second trimester -- at least it is in my first OC district. (And as we found out last year, this district has trimester finals for middle school students.) But as we saw in my new district, my new district has quarters in middle school, not trimesters.

Today I subbed in a high school instrumental music class. It is in my first OC district. (Of course, since it's a high school, there are no trimester finals today.)

I won't do "A Day in the Life" today -- but as usual, I will say a little about the classes. It's an even period day -- second period is Band, fourth period is Strings, and sixth period is conference. In each class, most of the students are freshmen.

In Band, the students work independently on their respective instruments. They have three assignments to work on -- two of which involve the Bb major scale. This is a fairly common key for many woodwinds and brass instruments. The students consider practicing outside -- but they don't, due to the weather. Yes, it's a California "snow day" (that is, a rainy day). Instead, they work in the practice room where their instruments are stored.

Unfortunately, things don't go as smoothly in Strings. For some reason, the Zoom link that I've been using suddenly doesn't work. It's almost as if someone flipped a switch -- before school, both the second and fourth period Zoom links work, but after snack break, neither does. It's happened before.

When I complain about it later on, I hear that it has something to do with Chromebooks -- for some reason, many Zoom features don't work on them. This is why it's better for us to create our own Zoom link and then post it in Google Classroom. But that doesn't explain why the links worked before snack on my Chromebook and not after.

Luckily, this class has a few aides (or music "coaches"). One coach is able to get into the Zoom meet, declare himself the host, and then admit me into the meet and declare me a cohost. Then he sets up breakout rooms depending on section (basses, cellos, and so on). This class has three assignments too -- two of which involve the D major scale. Unlike band instruments where flat keys dominate, string instruments are more likely to use sharp scales.

But then the coach host has to leave early for a meeting -- and since no one else is a host (the other coaches and I are merely cohosts), the Zoom meet ends completely. One of the other coaches is able to restart the meet -- but by this point many of the online students leave and don't return.

I always like to sing songs when I'm in music class. The song that I plan for today is "Mousetrap Car Song," since it's an original simple song. And I try singing it in the keys that the students are learning now -- Bb major for Band, D major for Strings.

I originally wrote "Mousetrap Car Song" in the key of E minor, but right at the end I jump to the relative G major, so I can claim the song as a G major song. Then I simply transpose this song from G to either Bb or D major depending on the class. Still, the song spans one full octave of the E minor scale, which in my vocal range takes me from E3 to E4.

For D major, the relative minor is B minor, and I sing the song from B2 to B3. But for Bb major, the relative minor is G major, and both G2-G3 and G3-G4 are difficult for me. I believe that I end up singing G3-F4 (as the highest G isn't emphasized), just as I did for "Whenever You Multiply."

For some reason, there is also a banjo in the main room, and so I decide to try it out. After writing about ukeleles all week, now I find myself playing yet another string instrument.

I know nothing about the banjo, but I notice that it has six strings -- and when I play them, they sound just like the standard tuning of a guitar, EADGBE. This suggests that I can play the same familiar guitar chords on the banjo.

In Band, I play "Mousetrap Car Song" in its original key, G major. But in Strings, I try transposing the song to the key of the week, D major. The guitar and banjo -- themselves string instruments -- share with the violin a preference for sharp scales, and so it's easier to play D major than Bb major. The tricky part is that the relative minor, Bm, is a barre chord on the guitar, as is its dominant F#7. I don't know whether banjo players use barre chords or not. I'd think that since a banjo has the same six strings as the guitar, we'd play the same barre chords as well -- and so I play the barre Bm and F#7 chords today on the banjo. (On the other hand, we saw that barres are unnecessarily on the four-string uke. Also, it appears that there is no preference for sharps or flats on the uke -- the easiest major chord to play is C major, the scale with no sharps or flats.)

Since the students are working independently in Band. there's enough time left for another song. I ask one guy to choose a song, and he chooses "Show Me the Numbers." Since I've found my old songbook, I wrote down part of my original tune from the old charter school and play it. Based on what I wrote, it's ambiguous whether I intended the song to be in the key of C or G major. Today I play it in G major on the banjo (again, I avoid Bb major on the banjo).

But I sing the new lyrics that I posted over the summer for this song. The old lyrics that I wrote at the charter school are obsolete. I also wrote that day that I wanted a new tune for this song using EDL scales -- I suggested using 20EDL. (That was before I found the songbook where I wrote part of the original tune.)

I could continue to write about 20EDL, but I won't. There are other things for me to discuss in today's post, and so I'm done with music for now.

Today is Sunday, the third day of the week on the Eleven Calendar:

Resolution #3: We remember math like riding a bicycle.

I mention this and all the other resolutions as part of the "Show Me the Numbers" song. I also point out that the students should remember how to play their instruments like riding a bicycle.

And finally, today is the day of the Chapter 11 Test. All those other things might be fun, but this is a Geometry blog, so I must post the test -- sorry.

Today is a test day -- hence a "traditionalists" post. Our main traditionalist, Barry Garelick, is still on his victory lap following the publishing of his fourth book. The post in which he announced the publishing has drawn a few more comments, but I won't spend much time on them now.

Instead, I wish to discuss an issue that's been all over the news -- the school reopening debate. I work for two OC districts where schools in all grades K-12 are open under the hybrid model. Some schools are fully open again, while others are still in full distance learning.

I'm surprised that our main traditionalist Garelick hasn't written much on school reopening -- but then again, he was preoccupied with the completion and publication of his fourth book. So instead, the traditionalist who's had the most to say about this is Darren Miller, of Right-(Wing) on the Left Coast.

Miller's high school is currently on full distance learning. But he's made it clear that he would prefer for his school to reopen full-time.

Once again, I'm torn over this issue. On one hand, I'm grateful that at least some schools are open, since teachers who work from home have less need for us subs, even if they get sick (in general, not necessarily sick from the coronavirus). On the other hand, I wonder whether there's any real reason to rush a reopening, especially considering that so many students are opting out of in-person learning at the schools where I sub. (Of course, older students are much more likely to opt out than younger students due to the supervision issue.)

Miller has mentioned mental health to support his side -- he sometimes points out that mental stress has increased in teens lately, and he blames it on the pandemic and school closures. I can't quickly find the specific post in which he makes this claim, but he likely cited reports like the following:

https://www.cidrap.umn.edu/news-perspective/2021/03/teens-mental-health-claims-skyrocket-pandemic

(Before we continue, I point out that I do not wish to offend the family members of anyone who has lost a young loved one due to mental depression. In this post, I refer only to mental stress and not to suicide, which I never want to take lightly.)

The claim here is that teens are stressed out due to the school closures, distance learning, the inability to see friends, and loneliness. The implication here is that teens really, really, want to get back to school, so if we were to open all high schools tomorrow, they'll be very happy and depression will plummet.

But if this were the class, then why do many high school students opt out of in-person learning. Why do I cover so many classes with just one or two students in-person and the rest at home? If distance learning is the source of depression, why are so many students choosing to extend that depression by opting out when they have the chance?

A student who is genuinely depressed due to loneliness might opt in to in-person learning -- only to find out that he or she is the only in-person student because everyone else has opted out (and so that student is still lonely). And even with sports -- where you'd think that students would be enthusiastic to get back to competing -- we see that students are opting out. Recall that at my alma mater, there are only a third as many girls (as I don't have the numbers for the guys) running Cross Country this year as last year.

So let's consider the following wager. For any high school class where in-person learning is allowed:

• I give you $1 for every student who opts into in-person learning.
• You give me $1 for every student who opts out and chooses full distance learning.

Of course, older students are more likely to opt out. In light of this fact, I give you $2 for every junior or senior who opts in to in-person learning, while taking only $1 for each junior/senior who opts out.

If Miller is correct that students really want to come back to school, then you should win easily. So do you take the bet?

I decided to calculate my gain or loss based on the specific classes that I see today. It turns out that today I will break even. In the Strings class, I earn a profit of $2 as slightly more students are opting out than in. In Band, an equal number of freshmen opt in as out -- but I lose money because there happens to be one senior in the class who has opted in. Since he's a senior, I lose $2, so I break even for the day.

Of course, for many high school classes I stand to make a profit. I quickly glance as the roster for an Advanced Strings class that meets on odd period days -- even without counting, I see that I'll make money with this wager, even with the extra $2 premium to you for juniors and seniors.

Yesterday's special ed English class appears profitable for me, as only three students attend each period (but since it's hybrid, I need to know how many students are opting in on the other cohort -- in-person students on either cohort count as wins for you). Monday was special day class, so we expect more students to opt in than usual. There's a slightly majority (five out of nine) students who are opting out, but I don't win the bet since (if you recall) one of the in-person students is a senior. At best I break even, and I might even lose if some of the other in-person students are juniors.

In the case of Miller's specific classes, I'm especially curious about his Financial Math class. I subbed for a similar class yesterday, and there was only one in-person student, a junior. So I assume that I'll win that bet (but once again, there could be ten in-person students on the other cohort as far as I know).

Then again, I suspect that Miller wouldn't take this wager anyway. He knows that he might lose as too many of his students opt out -- not because it's too dangerous to return, but because people act irrationally due to "probability neglect." He mentions "probability neglect" in this post:

https://rightontheleftcoast.blogspot.com/2021/02/he-nailed-it-year-ago.html

Miller is what I once called on the blog a "zero-percenter" -- he believes that the probability of any particular individual (especially one who is sufficiently young without preexisting conditions) dying of the coronavirus, rounded to the nearest percent, is zero. Thus it's rational to do everything that you did before the pandemic (full-time opening, 30-40 students in one classroom, and so on).

Once again, I'm not quite as sure as Miller is. I'm not sure whether my probability of dying is truly that close to zero -- and even if it is, the probability of getting sick from the virus is higher. I don't necessarily want to get sick just because I won't die from it. Then again, I'm still grateful that my OC schools are open -- I wouldn't want there to be a complete shutdown with no schools open in-person anywhere in the state.

And I will continue to keep track of this opt-in/opt-our wager for the high school classes that I cover in the near future. Since this is a music post, I'll repeat SteveH's music comment from Garelick's publishing thread:

SteveH:

If you want to see what works for under-privileged groups – as equals, see El Sistema for music. They take kids from the barrios and get them to play at Carnegie Hall and the BBC Proms by the end of high school. The key to it are the private lessons that enforce mastery of skills and start in the earliest grades and NOT the mixed-ability orchestra classes in school. These musicians do not have just mere facts or rote skills. Mastery drives understanding and excitement and hard work – NOT they other way around.

Here are the answers to today's test.

1. Using the distance formula, two of the sides have the same length, namely sqrt(170). This is how we write the square root of 170 in ASCII. To the nearest hundredth, it is 13.04.

2. The slopes of the four sides are opposite reciprocals, 2 and -1/2. Yes, I included this question as it is specifically mentioned in the Common Core Standards!

3. Using the distance formula, all four sides have length sqrt(a^2 + b^2).

4. Using the distance formula, two of the medians have length sqrt(9a^2 + b^2).

5. 60.

6. From the Midpoint Connector Theorem, ZV | | YW. The result follows from the Corresponding Angles Parallel Consequence.

7. From the Midpoint Connector Theorem, BD | | EF. The result follows by definition of trapezoid.

8. 4.5.

9. (0.6, -0.6). Notice that four of the coordinates add up to zero, so only (3, -3) matters.

10. At its midpoint.

11. 49.5 cm. The new meter stick goes from 2 to 97 cm and we want the midpoint.

12. Using the distance formula, it is sqrt(4.5), or 2.12 km to the nearest hundredth.

13. sqrt(10), or 3.16 to the nearest hundredth.

14. 1 + sqrt(113) + sqrt (130), or 23.03 to the nearest hundredth.

15. sqrt(3925), or 62.65 to the nearest hundredth. (I said length, not slope!)

16. -1/2. (I said slope, not length!)

17. (2a, 2b), (-2a, 2b), (-2a, -2b), (2a, -2b). Hint: look at Question 5 from U of Chicago!

18. (0, 5).

If you want, you can add the following questions, as the equation of a circle is still missing:

In 19-20, determine a. the center, b. the radius, and c. one point on the circle with the given equation.

19. (x - 6)^2 + (y + 3)^2 = 169
20. x^2 + y^2 = 50

Here are the answers:

19. a. (6, -3) b. 13
Possible answers for c: (19, -3), (18, 2), (11, 9), (6, 10), and so on.

20. a. (0, 0) b. 5sqrt(2)
Possible answers for c: (5, 5), (-5, 5), and so on.

Chapter 11 Review (Day 119)

Today I subbed in a high school special ed class. It's in my first OC district -- and yes, it's in the room I covered in my February 11th post, and the teacher for whom I subbed several times last year, including the last three days before the pandemic.

Unlike February 11th, which was an even period today, today is an odd period day. This teacher has two English periods each on odd and even days. Sixth period is his conference period, and so he has one more class on odd days -- and that class just so happens to be Business Math. (No, he didn't teach a Business Math class last year.)

Since there's a math class today, that means I will do "A Day in the Life" today, even though there are aides covering the classes.

9:00 -- This is the district where "first period" really means zero period, and so odd days begin for most students (including most special ed students) with third period. It is a senior English class.

The students are still reading the same novel from my February 11th post -- The Great Gatsby. They have now reached Chapter 6. We already know that this teacher has very small in-person classes, due to the combination of it being a senior class (higher grades tend to opt out of hybrid) and a special ed class (which was smaller even before the pandemic).

I haven't performed "Big March Song" for this class yet, and so I sing it today. Once again, some of the seniors do remember me from last year.

9:55 -- Third period leaves for snack break.

10:10 -- Fifth period arrives. This is the Business Math class.

The students learn about stocks and bonds today. There actually isn't much calculation today -- instead, they learn the difference between them and why anyone would want to invest in them. The assignment is some sort of Google Slides assignment where they answer questions. This class has only one student who attends in the classroom -- a junior guy.

I tell the students about how I, as a young sixth grader, participated in a stock market project where we "purchased" stocks and followed their progress in the newspaper. It appears that one of the slides requires these students to do something similar, but the aides tell them to wait for their regular teacher to explain this project in more detail.

10:20 -- Suddenly, the regular teacher appears in the Zoom meeting. He needs to meet with one of the students to prepare for a special ed meeting.

I ask him about his guitar and ukeleles, and he tells me that they're still in his old classroom -- the same room where I lost and found my old songbook. (I'll consider looking for them the next time I sub in the self-contained class that's there now.)

Then I try setting up a breakout room on Zoom for the regular teacher and student -- but I end up pushing all the other students into another room as well. The aide must go to the other room in order to send them back to the main room.

Last year, I once subbed in another Business Math class (not this one), and there was a special song that I sang that day -- "Compound Interest Rap." Thus I perform this rap again today.

11:05 -- Fifth period leaves and seventh period arrives. This is a junior English class.

As we saw on February 11th, the juniors are also reading The Great Gatsby. There are three in-person students, including the same guy from Business Math.

I sing "Big March Song" again, especially since one in-person guy (and some of the online students) already heard "Compound Interest Rap" earlier.

12:05 -- Seventh period leaves for lunch.

Normally, after lunch is academic support. The aides inform me that the regular teacher usually cancels academic support on sub days, and so I don't need to stay. (Notice that February 11th was a minimum day with no academic support for anyone, so I didn't know that this is his policy.)

Today is Saturday, the second day of the week on the Eleven Calendar:

Resolution #2: We make sacrifices in order to be successful at math.

I tell the students this in the Business Math class. I also tell them that they must make sacrifices and work hard in order to be successful in English -- especially during the Big March when many of us are always tired. (I say this just before I sing "Big March Song.")

Yes, I include the "music" label today on a day when I sub for the guitar/uke teacher. It's not for any guitar song though, but so I can post "Compound Interest Rap." As it turns out, I've never posted the lyrics to the blog (and thus had I not found my old songbook in this teacher's old room, I wouldn't have had the words for today's performance).

"Compound Interest Rap" by Scalar Learning:

COMPOUND INTEREST RAP

1st Verse:

Nowadays we got money on our minds,

Devisin' ways to pull cash on the time.

It's a common situation in the western world,

Building dreams with steam and then we make it work.

A lot of folks pullin' all the right strings,

Make a dime on a feelin', pull a dollar on the whim.

But once we get that bank roll flowin',

The interest rate is how we keep it growin'.

I know a lot of you wanna keep it simple,

Percentage and time is all you tryin' to get through.

Well that's simple interest and it's pivotal,

Rate times the number of years and the principal.

That's the way we calculate our funds,

While the money stays in, the beat goes in.

But next verse we take a real look into this,

A lesson on finding the simple interest.

Refrain:

A equals P times 1 plus r over n, to the nt, yeah, yeah, yeah, yeah!

2nd Verse:

Step back to the terms of the simple,

To comprehend this piece is critical.

Principal, it's not about a head of school,

Instead it represent the initial cash pool.

It's like if we deposit a quick grand,

With a rate of interest set to 10 percent.

We leave it be for 5 years and it grows,

But just how much? I'll tell you how we know.

We take 1000 times point 10,

Which is the decimal equivalent of 10 percent.

And multiply again 5 to unlid,

The interest earned, which is 500.

It's pretty simple to follow these steps,

But the real banks pay interest on interest.

To make true life calculation,

Compound interest is what you makin'. (To Refrain)

3rd Verse:

The time has come, and we can move another level,

To answer the question that we came to settle.

Compound interest, how does it work?

As the money grows we recognize the new net worth.

So let's push back to the hook,

And jot down what we just got to make it work.

A is the final amount of cash,

And once again big P is the principal stash.

We got t for years and r for interest rate,

Small n is the number of compounds that we take.

Let's plug it all in, same example as before,

We got a G in the bank, 10 percent on a roll.

We keep it in for 5 years and we compound twice,

We get 16 hundred 28 89.

With the simple rate we had 15 bills, right?

But this way it's another 1 28 89. (To Refrain)

Today is the review for the Chapter 11 Test. This is what I wrote last year about today's lesson:

In particular, this test is based on the SPUR objectives for Chapter 11. As usual, I will discuss which items that I have decided to include and exclude, and the rationale for each:

One major topic that I had to include is coordinate proof, as this appears in Common Core. I did squeeze in some coordinate proofs involving the Distance or Midpoint Formulas, but not slope. So therefore, the coordinate proofs included on this review worksheet all involve either distance or midpoint, not slope. The only proofs involving parallel lines had these lines either both vertical or both horizontal. Once again, a good coordinate proof would often set it up so that the parallel lines that matter are either horizontal or vertical.

What good are coordinate proofs, anyway? Well, a coordinate proof transforms a geometry problem into an algebra problem. Sometimes I can't see how to begin a synthetic geometry proof, so instead I just start labeling the points with coordinates and see what develops.

So coordinate geometry reduces an unknown problem (in geometry) to one whose answer is solved (in algebra, in this case). Mathematicians reduce problems to previously-solved ones all the time -- enough that some people make jokes about it:

http://jokes.siliconindia.com/recent-jokes/Reducing-the-problem-nid-62964158.html

I ended up including six straight problems -- Questions 8 through 13 from U of Chicago. Most of these questions are from Objective C -- the Midpoint Connector Theorem. The text covers this here in Chapter 11, but we actually covered it early, in our Similarity Unit, because we actually used the Midpoint Connector Theorem to start the proof of the basic properties of similarity. Still, this was recent enough to justify including it on the test.

Next are a few center of gravity problems. This is straightforward, since all we have to do is average the coordinates. Afterwards are a few midpoint problems, including two-step questions where one must calculate the distance or slope from one point to the midpoint of another segment.

Then there are a few more coordinate proofs where one has to set up the vertices -- notice that some hints are given in earlier questions.

Desmos Geo. Coordinate Day 2 by Jenny White (Day 118)

Today is the second and final day of subbing in the self-contained special ed class in my new district.

I won't do "A Day in the Life" this time, since it's not too different from yesterday. The main difference in the schedule is that like most high schools in both districts, Mondays are Late Start days. For most classes, all this means is that the students log into Zoom a little later, since Mondays are all online. It's only in special ed classes like this one that has in-person students on Late Start Mondays.

One of the aides has brought her ukelele again. I forget which song she plays today -- I think she said it's a Neil Young song.

As for the song I perform today -- no, it's not "Another Ratio Song," since as I wrote in my last post, I'm still stuck on one chord. Instead, I switch to a song where chords don't matter -- "Packet Rap." This is the first time I've perform this song since the start of the pandemic -- after all, we don't really see too many packets these days for obvious reasons. Even this class doesn't actually have packets -- yes, the students do work on paper, but these papers aren't stapled into a packet. But I perform the rap anyway.

Also, I walk with a group of special ed students around the entire school. This is a different group from the one that heard "The Big March" last week -- and since I like to sing that song during long walks anyway, of course I sing it again.

Today is Friday, the first day of the week and first day of the year on the Eleven Calendar. I haven't really numbered the years on this calendar so far -- but since I left the old charter school in March, I've de facto counted the number of years since I left the old charter. It's been four years, so this would be the start of Year 5 in the calendar. It's the last time I use the phrase "four years ago" when referring to the charter -- soon I must start saying "five years ago." (Happy New Year -- also, this implies that the first full year after I created the calendar -- the same as the year I worked at the charter -- is Year 0.)

Resolution #1: We are good at math. We just need to improve at other things.

A few students do work on math today. One guy uses a calculator to count money, including $1 bills and common coins. I must tell him that he can't add 1 + 25 to add a dollar to a quarter -- instead, he must use a decimal point for the quarter. Also, I remind him to avoid the common error of forgetting the zero after the decimal point in .01 and .05 for the penny and nickel. Fortunately, he seems to learn fast.

There's also a girl who is practicing three-digit subtraction without a calculator. I use my own calculator as a random number generator. She makes the common errors that you expect in the standard algorithm.

The old fifth resolution about 1955 and avoiding the overuse of phones comes into play today. One boy is celebrating his birthday today -- technically, I should say man, since he's a senior celebrating his eighteenth birthday. As I did last Friday, I bring in some candy, including special Hershey's and Kit Kat bars just for the birthday man. He must earn the candy though by keeping his phone in his backpack -- which he has trouble doing because he received a special phone carrier for his birthday, which he feels obviates the need to keep the phone in the backpack.

I invoke the fifth resolution by reminding him that now that he's 18, he needs to think in terms of the real world -- soon he'll get a job to earn real money (and buy all the candy he wants), but if the boss tells him to put his phone anyway, he''ll have to, or else he'll lose his job and someone else will get the money instead. In other words, I threaten to give the candy to someone else unless he keeps the phone in his backpack. He does finally keep it away from lunchtime (when I buy and show him the candy) until it's time to give out the reward.

As for "homage," today is the first day of Women's History Month, so expect a lot of homage to be paid to women for the next few weeks. Today the students read an article on the late Ruth Bader Ginsburg, and so we definitely pay homage to her today.

Here's a link to Day 2 of the Desmos activity created by teacher Jenny White. I already linked to Day 1 of her activity on Friday.

https://teacher.desmos.com/activitybuilder/custom/5c92d05c53eee550160bda71