Friday, February 15, 2019

Lesson 11-2: The Distance Formula (Day 112)

Today on her Mathematics Calendar 2019, Theoni Pappas writes:

A = (-3, -2)
B = (6, -14)
AB = ?

To find the length of segment AB, we can use a certain formula:

AB = sqrt((x_2 - x_1)^2 + (y_2 - y_1)^2)
AB = sqrt((6 + 3)^2 + (-14 + 2)^2)
AB = sqrt(9^2 + 12^2)
AB = sqrt(81 + 144)
AB = sqrt(225)
AB = 15

Therefore the desired length is 15 -- and of course, today's date is the fifteenth.

To solve this problem, we used a certain formula known as the Distance Formula. We haven't covered that lesson in the U of Chicago text. Let me check my calendar to see when I'm scheduled to do it...

...and hey, it's today! Today is the day that I'm scheduled to do the Distance Formula. I believe that this is the first time since I started blogging Pappas that a problem related to a certain lesson in the text lands on the exact same day as that actual lesson. (There were some very close calls where the Pappas problem and the relevant lesson were a day or two apart.)

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

Lesson 11-2 of the U of Chicago text is called "The Distance Formula." In the modern Third Edition of the text, the Distance Formula appears in Lesson 11-5.

Let's get to today's lesson. Many students have trouble with graphing throughout Chapter 11, and furthermore, today we learn the Distance Formula, which of course will be difficult for some students.

In the past, I combined Lesson 11-2 with Lesson 8-7, on the Pythagorean Theorem (and indeed, this lesson in the Third Edition is titled "The Pythagorean Distance Formula").

David Joyce has more to say about the Distance Formula:

Also in chapter 1 there is an introduction to plane coordinate geometry. Unfortunately, there is no connection made with plane synthetic geometry. Here in chapter 1, a distance formula is asserted with neither logical nor intuitive justification. Of course, the justification is the Pythagorean theorem, and that's not discussed until chapter 5. In that chapter there is an exercise to prove the distance formula from the Pythagorean theorem. The Pythagorean theorem itself gets proved in yet a later chapter.

Fortunately, the U of Chicago text avoids this problem. Our text makes it clear that the Distance Formula is derived from the Pythagorean Theorem.

Today I post an old worksheet from a few years ago. It introduces the Distance Formula -- but of course, it teaches (or reviews) the Pythagorean Theorem as well  -- including its similarity proof, which is mentioned in the Common Core Standards.

Today is also an activity day, so let's add an activity to this old worksheet. The one Exploration Question given in this lesson is:

22. The distance from point X to (2, 8) is 17.
a. Show that X could be (10, 23).
b. Name five other possible locations of point X. (Hint: Draw a picture.)

Of course, there are infinitely many possible locations for X. The assumption is that students will look for lattice points (points whose coordinates are both integers). These are based on the 8-15-17 Pythagorean triple. Even though we haven't covered vectors yet, it might be helpful for teachers to think in terms of vectors when correcting the student work. Basically, the solutions are to add all versions of the vector <8, 15> to the original point (2, 8). This includes changing one of both signs of <8, 15> as well as switching the x- and y-components. There are seven possible lattice points in addition to the given (10, 23), and students only need to find five of them.

I decided to add today's Pappas problem to the worksheet. It might be years until the next time a Pappas problem lands on the same day as the relevant lesson. So I might as well take full advantage of today's coincidence!

President's Day is on Monday, and so my next post will be Tuesday.


Thursday, February 14, 2019

Lesson 11-1: Proofs with Coordinates (Day 111)

Today I subbed in the same special ed middle school English class as Tuesday. I promised that I would treat this as a multi-day class and do "A Day in the Life" even though there's an aide, since management during multi-day subbing assignments is important even with the aide.

I choose to do "A Day in the Life" today rather than Tuesday because I didn't want to keep discussing the English Performance Task. Actually, a few students in each class are still working on these, though most have finished. Recall that the teachers are out today in order to grade those tasks, so I hope they submitted them to Google Classroom in time for the grading.

8:15 -- The day at this school begins with homeroom and announcements. Here HR is the same as third period, not first period as some other middle schools. And all the periods rotate, so HR/third period doesn't always start the day. Indeed, today starts with sixth period.

There is an eighth grade ASB rep in the class. She keeps track of everyone participating in this week's Spirit Week dress-up. For Valentine's Day, the students are to wear red today. She actually includes yours truly in the count, since I'm wearing red too.

8:25 -- Sixth period arrives. This is the first eighth grade class of the day. It was the best eighth grade class of the day on Tuesday, and so I give these students a reward of V-Day pencils and candy.

The main assignment for all classes is an article on cell phones in the classroom. The article contains both the point (phones can enhance learning) and counterpoint (they are distracting). The students must answer nine questions about the article and then do a current event on the same article.

9:20 -- Sixth period leaves and first period arrives. This is the second eighth grade class of the day.

10:15 -- First period leaves for snack.

10:25 -- Second period arrives. This is the first seventh grade class of the day. It was the best seventh grade class of the day on Tuesday, and so I give these students a reward of V-Day pencils and candy.

I'd purchased a bag of 25 candies (Kit-Kats, Hershey's, etc.) for the reward along with an equal number of pencils. There are eleven 8th graders and only seven 7th graders present today in the two classes that had earned the rewards. And so there's just enough for me to give the seventh graders a second chocolate and second pencil.

11:20 -- Second period leaves. But as it turns out, it's raining today. I've explained what rainy days entail here in California -- rainy-day schedules. Instead of one lunch for all, students attend one of two possible lunches. My class is assigned the first lunch. The rule is that all students go to third period for attendance and to drop off their belongings, and then two minutes after the tardy bell is when the dismissal bell rings for lunch.

12:05 -- Third period arrives after lunch. This is the second seventh grade class of the day.

1:00 -- Third period leaves. It is now conference period.

2:00 -- Like many special ed teachers, today's regular teacher has four classes of her own and then co-teaches one class, which in her case is fifth period. This is a seventh grade English class. Like all English teachers, this resident teacher is out grading the Performance Tasks, and so the class ends up with two subs.

Normally, at this school, there is always silent reading built in to the schedule after lunch. On rainy days this is dropped (since "after lunch" is confusing when there are two lunches). But the resident teacher decides to specify silent reading for all periods in her lesson plans.

After silent reading, the students complete a crossword puzzle on academic vocabulary.

2:55 -- Fifth period leaves. But once again, the teacher I'm subbing for has been assigned after school duty this week. Thus I must watch the area between two buildings -- in the rain.

3:05 -- My duty is over, and so the four-day President's Day weekend in this district begins.

For multi-day subbing assignments, I automatically default to the first focus resolution:

1. Implement a classroom management system based on how students actually think.

An aide takes care of most of the management today. The same two students as on Tuesday are the most disruptive. Unlike Tuesday, today the seventh grade boy was slightly more talkative than the eighth grade boy.

My main management issue occurs during a certain two-minute stretch -- you guessed it! It's the two minutes when third period must return before going to the rainy lunch. The aide doesn't stay for these two minutes -- she just leaves right after second period.

The seventh graders try to go directly to lunch instead of stay for the two minutes. Actually, all of them have the same history class next door, and so they arrive at my room early. So they must wait out all of the four-minute passing period and then the two minutes begin.

This leads to arguments, of course. With the aide out (and for some reason, the history teacher must not have told them about the two minutes), I am the one who must inform them of the schedule. So they blame me -- they see me as the mean teacher who's stopping them from going to lunch even though I'm not the one who wrote the rainy day schedule.

(This isn't the first time when I was blamed for a schedule I didn't write. Two years ago at the old charter school, students complained when I didn't let them go to nutrition on a minimum day. The schedule specified that students go to break after the third, not second, class on minimum days.)

Meanwhile, my left literally falls apart, likely because of all the rain. (The aide is the wise one, as she's wearing boots.) Since the students are already upset with me because of the two minutes, they start making fun of me. One guy claims that I'm "breaking the dress code" by having only one shoe, while a girl starts hoping that my other shoe would fall apart too! (I end up purchasing another pair of shoes during the lunch break.)

When we return after lunch, the same guy starts complaining about the assignment -- that's there's not enough time to do it because "there's two lunches" and he's "not smart." The time claim is false -- since there's no silent reading time, this period is actually one minute longer than on a dry day! And besides, many students in previous classes had enough time to finish.

But I know by now that arguing this point doesn't work. Instead, I reassure him that yes, I believe that he really can finish the assignment. With the aide's help, he's the first in the class to finish. Afterward I tell him that I knew all along that he could do it -- after all, he's the only eighth grader in the class, so he should be able to show the seventh graders how it's done! (Presumably he's in this period only because it somehow fits better in his schedule.) He replies, "I'm smart!" That's more like it!

One thing that's difficult for me as a teacher is to avoid taking what the students say personally. The students get upset with me not because they hate me, but because they think that they should be allowed to go directly to lunch -- and I am the one preventing them. They make fun of my shoes falling apart because of this. But when they return after lunch, they stop making fun of me -- and start making fun of each other.

Two boys start complaining that their legs are hurting. It's likely that the injuries are caused by slipping in the rain during lunch. The other students respond by laughing at them.

I believe that laughing at the boys is worse than laughing at me. After all, I'm able to buy new shoes, but the guys can't buy replacement legs that don't hurt. Indeed, I'd prefer that the students make fun of me instead of them.

Once I learn how not to take student criticism personally, I can take it to my advantage. When kids start laughing at each other, I can encourage them to make fun of me instead. I remember one day at the old charter school when the special scholar was struggling with her Dren Quiz. Some other students wanted to make fun of her. So I got them to laugh at me instead. Back then, there was some phrase that I incessantly repeated. (I think it was, "Now here's the problem!") And so I started saying that phrase intentionally so that they'd make fun of me and not the special scholar.

This is the sort of thing that I can do more often. Some students naturally enjoy laughter, and unfortunately they often direct that laughter at an injured or struggling classmate. I should then do something silly so that they'll laugh at me instead. This is all part of the first resolution -- managing a classroom based on how students actually think.

Lesson 11-1 of the U of Chicago text is called "Proofs with Coordinates." In the modern Third Edition of the text, proofs with coordinates appear in Lesson 11-4.

Coordinate proofs are mentioned in the Common Core Standards:

CCSS.MATH.CONTENT.HSG.GPE.B.4
Use coordinates to prove simple geometric theorems algebraically. For example, prove or disprove that a figure defined by four given points in the coordinate plane is a rectangle; prove or disprove that the point (1, √3) lies on the circle centered at the origin and containing the point (0, 2).

In Lesson 11-1, we are given the coordinates of the vertices of a polygon, and we are asked to prove that the polygon is a parallelogram, right triangle, or rectangle. The key to these coordinate proofs is to find and compare the slopes of the sides.

But here's another Common Core Standard:

CCSS.MATH.CONTENT.8.EE.B.6
Use similar triangles to explain why the slope m is the same between any two distinct points on a non-vertical line in the coordinate plane; derive the equation y = mx for a line through the origin and the equation y = mx + b for a line intercepting the vertical axis at b.

So students are supposed to use similarity to prove the properties of slope. David Joyce, whom we mentioned throughout Chapter 9, also endorses the use of similarity to prove slope -- and indeed, he has harsh words to say about the treatment of coordinate geometry in most Geometry texts:

In a return to coordinate geometry it is implicitly assumed that a linear equation is the equation of a straight line. A proof would depend on the theory of similar triangles in chapter 10. At this point it is suggested that one can conclude that parallel lines have equal slope, and that the product the slopes of perpendicular lines is -1. The only justification given is by experiment. (A proof would require the theory of parallels.)

And in our text, similar triangles don't appear until Chapter 12. Thus to follow the Common Core and David Joyce, we should wait to teach Chapter 11 until after Chapter 12. Students need to have mastered similar triangles before they can begin learning about slope.

This has been the source of many headaches in my blog posts over the years. First of all, I'd start with Chapter 12, then go back to teach slope and some other Chapter 11 topics -- but I'd never actually reach Lesson 11-1.

The problem, of course, is that slope is an Algebra I topic. High school students are thus going to see slope well before they ever see similarity, because they take Algebra I before Geometry.

I also wrote extensively about Integrated Math courses. But even Integrated Math usually covers slope before similarity -- indeed, slope is a Math I topic, while similarity is a Math II topic. I once tried to devise my own Integrated Math courses that teach similarity before slope, but I failed. It's difficult to justify teaching similarity (from the second half of Geometry) before slope (which is from the first half of Algebra I).

In fact, we notice that the Common Core Standard requiring students to use similarity to prove the slope properties is an eighth grade standard, not a high school Geometry standard. This now makes sense -- students are introduced to slope in eighth grade in order to prepare to study it in more detail in Algebra I.

I think back to last year's eighth grade class. Of course, student behavior and classroom management were issues. But another problem was that I began teaching translations, reflections, and rotations -- and rotations, understandably, confused some students. The extra time spent on isometries meant less time on dilations -- and dilations are the bridge to similarity and slope.

I now sometimes wonder whether it's better to teach only one of the isometries -- perhaps reflections, since they generate all isometries (i.e., all isometries are the composite of one or more reflections) -- and then skip directly into dilations. But this contradicts the Common Core Standards that explicitly mention translations and rotations -- and these might appear in PARCC or SBAC questions.

At any rate, if the connection between similarity and slope is covered in eighth grade, then it doesn't need to be introduced in high school Geometry. And so we can write about slope in Chapter 3 without having to prove anything about similarity first. As I mentioned before, Chapter 3 is a great time to teach slope, since it's a review topic from Algebra I, and Chapter 3 is often taught right around the time of the PSAT (where slope questions will appear).

When David Joyce wrote about slope and similarity, he forgot that there's a class called "Algebra I" where students learn many things about slope and coordinates without proving everything. In the end, I did say that this year I'd adhere to, not Joyce's suggestions, but the order of the U of Chicago text.

And by the order of the text, I mean the order of the old Second Edition. Earlier, I wrote that Lesson 11-1 appears as Lesson 11-4 of the new Third Edition. So what exactly appears in the first three lessons of the modern version?

Well, Lessons 11-1 through 11-3 of the new text correspond to Lessons 13-1 through 13-4 of the old version of the text. Indeed, the new Chapter 11 is called "Indirect and Coordinate Proofs." You might recall that Chapter 13 of the old text has been destroyed, and its lessons are now included as parts of different chapters. And so the first half of the old Chapter 13 now forms the first part of 11. (There are now only three lessons instead of four because the old Lesson 13-2, "Negations," has now been incorporated into the other three lessons.)

Otherwise Chapter 11 remains intact in moving from Second to Third Edition. Chapter 11 of the old edition has six lessons, and these correspond roughly to Lessons 11-4 through 11-9 of the new text.

Let's finally take a look at the new Lesson 11-1 worksheet. We begin with the two examples from the text -- the first problem lists four ordered pairs and asks us to prove that they are the vertices of a parallelogram, while the second lists three pairs that may be the coordinates of a right triangle. In each case, students are to calculate the slopes of the sides formed by adjoining vertices, and show that these slopes are either equal or opposite reciprocals.

As usual, since today is Thursday, I must create a new worksheet. I decided to include Question #9, because students are asked to prove that EFGH is a rectangle -- and rectangles, unlike parallelograms or right triangles, are explicitly mentioned in the Common Core Standards:

CCSS.MATH.CONTENT.HSG.GPE.B.4
Use coordinates to prove simple geometric theorems algebraically. For example, prove or disprove that a figure defined by four given points in the coordinate plane is a rectangle; prove or disprove that the point (1, √3) lies on the circle centered at the origin and containing the point (0, 2).

The second part of that standard, on circles, will have to wait until later in the chapter. To complete the rectangle question, students must calculate the four slopes, and show that slopes of opposite sides are equal, while slopes of adjacent sides are opposite reciprocals.



Wednesday, February 13, 2019

Chapter 10 Test (Day 110)

Today I subbed in a high school English class. Thus there is no ¨Day in the Life" today.

Two of the classes are freshmen. No, there is no persuasive essay today -- they actually worked on the essays yesterday, and those who still need more time are finishing it at tutorial today. Instead, they are studying The Odyssey by Homer. One of the classes is special ed with an aide.

The other two classes are sophomore AVID. Believe it or not, this is my first time subbing AVID here. The last time I covered an AVID class was three years ago -- before I started working at the old charter school. (There were a few close calls -- I co-taught with AVID teachers during their non-AVID classes.)

In AVID classes, tutors typically work with students on Tuesdays and Thursdays. Since today is neither Tuesday nor Thursday, there are no tutors today. In the past, I mentioned the special boards -- VNPS -- that are used with the AVID tutors. I see the seven boards today, but the students do not use them on Wednesdays. Instead, they have two assignments -- one written, one on Chromebooks.


This teacher is out both today and tomorrow, but I am already locked in to cover the same middle school class tomorrow as yesterday. Since I will not return to this class tomorrow, I instead did my holiday incentive by giving out Valentine-themed pencils and candy to the top students in each class.

During tutorial, I help several students out in math, including Geometry. Many of these students are preparing for a quiz to be given before the four-day weekend. Chapter 7 of the Glencoe text (the first chapter fully in the second semester) is on similarity. I notice that one of the review questions is:

True or false. Corresponding angles in similar triangles are proportional.

One girl writes false as the answer to this question. Technically, the correct answer is true, since the angles are proportional -- it just so happens that the constant of proportionality must be 1. But I refrain from correcting her, as common sense implies that false is the intended answer. The question is meant to distinguish between angles and sides (the latter of which are proportional, while the constant may or may not be 1).

I help her and a few other Geometry students with their review. I always tell them about the Geometry student I once tutored years ago -- I asked him how he knew that a certain pair of triangles was similar, and his reply was that they were similar because we were in Chapter 7. This time, I make sure the students know why two triangles are similar before they try to set up a proportion. And of course I warn about the common trick (Triangles ABC and ADE are similar, but some might try to set up a proportion with BD or CE).


I continue to help students during AVID after they finish their assignment. This includes one Algebra II student who is learning about exponential functions, including e^x. Thatś right -- the Algebra II text used at this school is the one where exponential functions start the second semester. This is timed perfectly for e Day, which was last week on February 7th. (No, I avoid asking him whether his class celebrated e Day or not.)

Here is the Chapter 10 Test. Let me include the answers as well as the rationale for including some of the questions that I did.

1. 4.
2. 24 square units.
3. 9pi cubic units.
4. sqrt(82) * pi square units, 3pi cubic units.
5. 48,000 cubic units.
6. 98 square units.
7. 28,224pi square units, 790,272 cubic units.
8. 5.5. Section 10-3 of the U of Chicago text asks the students to estimate cube roots. If one prefers to make it a volume question, simply change it to: The volume of a cube is 165 cubic units. What is the length of its sides to the nearest tenth?
9. Its volume is multiplied by 343. This is a big PARCC question!
10. Its volume is multiplied by 25 -- not 125 because only two of the dimensions are being multiplied by 5, not the thickness.
11. The volume of Neptune is 64 times that of Earth.
12. A ring -- specifically the area between the the circular cross section of the cylinder and the circular cross section of the cone. This is Cavalieri's Principle -- recall the comments I made about Dr. Beals?

Instead, today is my "traditionalists" post. Actually, I already mentioned a traditionalist in this post -- Dr. Katharine Beals. Actually, Beals is no longer an active blogger -- instead, she's focusing on writing her own book about traditionalism and language.

But four years ago, Beals wrote a post attacking Cavalieri's Principle as fluff and a waste of time in high school Geometry classes. Since I reblog my old posts from past years every time I write about Cavalieri, I keep dragging up her name and the old debates.

Instead, let's look at today's active traditionalists. On February 6th, the tweeter CCSSIMath wrote about a sixth grade problem about the area of an irregular rectilinear polygon (to be found by dividing the polygon into rectangles). But according to CCSSIMath, this ought to be considered a fourth grade problem (as it would be in Japan) instead of a sixth grade problem.

https://twitter.com/CCSSIMath

We know that CCSSIMath likes to push Common Core recommendations down into lower grades. I point out that this requires elementary teachers with multiple subject credentials to be more comfortable with this sort of math. Two years ago at the old charter school, this problem appeared in the fifth grade text, and the fifth grade teacher had to ask me to help her solve the problem. Yet CCSSIMath tells me that I should have been helping the fourth grade teacher with this problem, not the fifth grade teacher!

On the other hand, in my current district sixth grade is still elementary school. And so even the sixth grade teachers might be uncomfortable with the problem if they lack math on their résumé. I like the idea of protecting teachers with multiple subject credentials from having to teach difficult math problems regardless of CCSSIMath and his/her preferences.

Ironically, five days later CCSSIMath gives a Common Core fraction problem that is "quite difficult for 5th grade arithmetic":

Leonard spent 1/4 of his money on a sandwich. He spent 2 times as much money on a gift for his brother as on some comic books. He has 3/8 of his money left. What fraction of his money did he spend on the comic books?

OK, I see what's going on here. The boy spent 1/4 on lunch and has 3/8 at the end, and so that leaves him with 1 - 1/4 - 3/8 = 3/8 for the gift and comics. Since he spent twice as much on the gift as the comics, it means that 2/8 was for the gift and 1/8 for the comics. The correct answer is 1/8.

Also, for this traditionalist post, let me write about something other than math. l address the subject I cannot help but think about lately -- English. Not only have I not subbed much math lately, but I have yet to see history or science since 2019 began. For some reason, I keep ending up in English classes -- even when it is special ed, it is English. And as we already know, English lately has been nothing but the district Performance Task.

During the LAUSD strike, I explained how district assessments were a point of contention. According to the settlement, district assessments would be reduced by 50%. My district is not LAUSD, and so that settlement has nothing to do with me. And this Performance Task is basically a district assessment, just like the ones the union wants to reduce.

Why do districts give these sorts of assessments, anyway? I assume the main rationale is to prepare the students for the SBAC. The name ¨Performance Task" gives it away -- this is supposed to be similar to the Performance Task portion of the SBAC.

We can see what the districts are thinking -- they want to prepare the students for the SBAC so they will earn a high score on the big test. But this is what exactly the union wants to reduce -- not just standardizing testing itself, but all the time devoted to preparing for standardized testing. If the sole purpose of giving the district test is to prepare for the SBAC, then we are justified in counting all of the time devoted to it as test prep. Thus English classes just spent the last six weeks on nothing but test prep.

The ideal situation would be for there to be no test prep -- teachers just simply teach the curriculum and the students get high scores on the test. For math, this would entail eliminating tricky questions that require extra test prep to solve, but what sort of questions these are is debatable. All this week, I wrote about Katharine Beals and Cavalieri. Beals would count Cavalieri as such a ¨test prep¨ question, since many Geometry teachers might not cover it had it not been for its presence on the test. (I wonder whether she would count ¨derive the area of a sphere¨ as a test prep question.)

Hmm, even though the Beals math blog is no longer active, she does currently write a blog with another Katharine -- to be precise, another Catherine (Johnson). Their blog is less about math and more about writing:


But still, I am not sure what such traditionalists would say about writing Performance Tasks. My guess is that they would tell us to teach good writing early, starting in elementary school. I do see the two Kates telling us about examples vs. non-examples at the link above, so this could help. And I suppose that even making a claim, researching evidence, and citing evidence is also something students can learn early as well. In theory, no test prep time is needed -- the students can enter the classroom on test day, receive the prompt, write a persuasive essay, and get a good score without knowing anything about the prompt until the they arrive that day.

Earlier, I did say that there should be three levels, with only one test at each level -- district for elementary, state for middle school, and Common Core for high school. But how do we know that, if we have state or national writing tests for secondary students, the district will not throw its own writing tests on top in the name of ¨preparing¨ the students for the main test? And as we have seen firsthand, the test prep needed for writing can be extensive.

Once again, I am a math teacher, not an English teacher (despite all my Jan.-Feb. subbing), and so I recognize what is too much testing in math more than English. This is why I can say that the standardized math test should be reduced to no more than a standard class period. I cannot be sure how much time should be devoted to a proper writing test for ELA.

So let us just get back to math. Is that fifth grade question fraction above fair? I suppose that in the end, the best way is just to ask students and teachers. It is very possible, for example, that my fifth grade teacher at the old charter could solve the fraction problem, even though she was unable to solve the rectilineal area problem.

What the students and teachers can handle -- whether it is English or math -- is how we should ultimately determine what is grade-level appropriate. (When tests are inappropriate, we wind up with situations such as the Atlanta cheating scandal, which has returned to the news today.)

By the way, let me conclude this post by mentioning the death of Lyndon Larouche. He was a controversial presidential candidate, but I mentioned him on the blog only in order to discuss his music theory. He suggested that middle C on a piano should be tuned to 256 Hz (hertz).

Since 256 is a power of two, C256 is compatible with Kite and his color notation. We may refer to C256 as white C. Then the other otonal colors (yellow, blue, lo, tho) correspond to the other integral hertz values (since hertz/frequency is an otonal system). No, I definitely do not agree with Larouche and his reasons for C256 (the so-called natural frequency of the earth), but at least his system and the Kite colors are somewhat compatible. 

Tuesday, February 12, 2019

Lesson 10-9: The Volume of a Sphere (Day 109)

Today I subbed in a middle school special ed English class. This is the third time I've subbed in this class --  I last blogged about the class in my December 5th post.

Two of the classes are eighth grade classes and two are seventh grade classes. This time, all classes have the same assignment -- and by now, do I even need to say what it is? Of course, it's the same old persuasive essay.

As usual, an aide is in charge of most of the classes, so there's no "Day in the Life" today.  As always, middle school periods rotate, and today the rotation actually starts with fourth period (which is the same as my conference period).

It seems as if these days, almost every class I've subbed in is English, and in almost every English class (whether it's gen ed or special ed) it's the same persuasive essay. After writing about this essay so often in my January and February posts, now's a good time to explain how exactly this essay works in this district.

Apparently, this persuasive essay is considered to be a "Performance Task." The seventh grade topic is reality TV, and the eighth grade topic is violence in movies. In both grades, the students should make a claim as to whether their respective entertainment is harmless or harmful to youths their age.

The persuasive essay unit spans all the way from New Year's Day to President's Day. They spend the first few weeks reading articles to find evidence for their claim. Then this week, they type up and submit their essays in Google Classroom. While students could have discussed their evidence with each other and the teacher earlier, once the essay begins it's considered to be a test, and so all students must be silent. The deadline is tomorrow -- and then Thursday, the last day before the four-day President's Day weekend, is for grading the essay.

All English teachers need a sub that day since they'll all be at the district office grading essays. As it turns out, I'm already signed on as the sub for today's teacher. She's out today because she's a special ed teacher (for meetings, etc.), and she's out Thursday because she's an English teacher.

By the way, I'm not sure whether students not in middle school have this essay to write. I do sort of recall hearing about an essay a week ago Friday -- the last time I subbed in high school. It's possible that only the testing grades had an essay to write (high school juniors, and third grade and above in elementary school).

I was considering counting this as a multi-day assignment (today and Thursday). Then I'd post "A Day in the Life" today since I like to provide you readers more details for multi-day assignments even when they're special ed or non-math classes. But I see no point in saying "the students wrote their essays in Google Classroom" over and over. I might even do "A Day in the Life" on Thursday, since that's grading day (so the students clearly won't be writing essays that day).

The day begins with conference period, but I must cover a few minutes in another classroom again for a late-arriving teacher. As it turns out, it's the same special ed room I visited last Friday.

Since I'm returning to today's class on Thursday -- Valentine's Day -- it's a great time to do my usual holiday pencil and candy incentive. The best eighth grade and seventh grade classes today (which mostly means remaining silent during the Performance Task) will get their reward on Thursday.

The aide and I choose sixth period to be the best eighth grade class and second period to be the best seventh grade class. The regular teacher and aide allow some guys in each grade to switch periods (due to a previous agreement with the history teacher), and -- you guessed it -- those are the most talkative students.

Indeed, one eighth grader who switches from sixth to first period is so talkative that it almost looks like sabotage -- he gets first period to respond to him, so that first period loses the Valentine's reward and his own sixth period class wins it! The aide assigns him a lunch detention and informs me that he shouldn't receive a reward if he attends sixth period on Thursday. Then this repeats itself with the seventh graders -- the talkative guy in third period is officially enrolled in the second period class!

There are 17 eighth graders and 11 seventh graders to whom I owe the reward, so this gives you an idea of the class sizes today. (This might be reduced to 16 and 10 if we leave out the "saboteurs.")

If you're tired of hearing about the persuasive essays, let me reassure you that once we get past President's Day, you won't have to hear about them again. They will have been completed and graded by then.

Lesson 10-9 of the U of Chicago text is called "The Surface Area of a Sphere." In the modern Third Edition of the text, the surface area of a sphere appears in Lesson 10-7.

But there are two problems here. The first is that in past years, I rearranged the lessons so that Lesson 10-9 was taught close to Easter. (This was close to -- but not exactly on -- April Fool's Day both years.) And so my old Lesson 10-9 worksheet made a reference to Easter and "Spring Spheres." So instead of the holiday worksheet, today I'm posting an alternate activity based on both Exploration Questions 22 and 23 from Lesson 10-9.

This is what little I wrote last year about today's activity:
  • From the U of Chicago text: calculate the surface area of the earth. Then compare the area of the United States and other countries to that of the entire earth.
The problem today is that this is a nine-lesson chapter. Just as we did with Chapter 8 last month, today we must begin our review for the Chapter 10 Test. Tomorrow, Day 110, will be the Chapter 10 Test itself, and Thursday, Day 111, will be Lesson 11-1.

Both two years ago and three years ago, I rearranged the lessons. Three years ago, I added two lessons from different chapters to the Chapter 10 Test. Then two years ago, I dropped not only the extra lessons, but Lessons 10-8 and 10-9 on spheres as well. On review day I posted the first worksheet from the previous year, which left out all of the non-Chapter 10 questions as well as some, but not all, of the sphere questions.

This is what I wrote in past years about today's worksheet. Again, I referred to the Easter holiday, which was a huge part of why I changed the chapter order from 2015 to 2016:

You may notice that today's blog entry is called "Review for Chapter 10 Test." At this point you're probably wondering -- how can there be a Chapter 10 Test already? After all, we haven't covered the surface area or volume of a sphere yet!

The problem, of course, is that next week is spring break. Chapter 10 is long, and the Easter holiday ends up splitting the chapter. This is actually a domino effect caused by Pi Day falling on Monday -- I wanted to cover pi -- part of Lessons 8-8 and 8-9 -- on Pi Day Monday, so we didn't start Chapter 10 until Tuesday. So that ended up pushing back Lessons 10-8 and 10-9 on the sphere.

We know that the formulas in Chapter 10 are hard for students to remember -- that is, after all, why the U of Chicago text devotes a full lesson, 10-6, just for remembering formulas! So imagine how much harder the formulas will be to remember when we have a week of spring break separating the start of Chapter 10 from its end!

And so I decided to declare this week to be the end of the unit and give a test this week. This is the same rationale for the Early Start Calendar -- we want to test the students before they have a chance to forget the material over the vacation weeks.

2019 update: Even though today shouldn't be an activity day, I'm keeping the mini-activity because it's something light to do when the emphasis should be on the review worksheet.  If you want, you could add some more review questions.



Friday, February 8, 2019

Lesson 10-8: The Volume of a Sphere (Day 108)

Lesson 10-8 of the U of Chicago text is called "The Volume of a Sphere." In the modern Third Edition of the text, the volume of a sphere appears in Lesson 10-6.

Today is Day 108 -- at least in my old subbing district. It can be considered as the end of the fifth quaver, which we determine by noting that one-quarter of the way from Day 85 (the end of the first semester) to Day 180 is 107.75. Of course, in my new district today is only Day 99. Thus it's not yet the end of the fifth quaver in my new district, seeing as it's only the second week of the semester.

Oh, and this is a good time to mention another difference between the two district calendars. In my old district, Lincoln's Birthday and President's Day are on two separate Mondays. But my new district does something different -- this one and several other districts observe Lincoln's Birthday on the Friday before President's Day, so that there can be a four-day break in February.

(Two years ago at the old charter, by the way, students observed a five-day weekend, from Thursday to Monday. But the first two days had nothing to do with Lincoln's Birthday. Instead they were PD days -- in other words, we teachers had only a three-day weekend. President's Day itself is the only school-closing February holiday in the LAUSD.)

What does this mean for the blog calendar? Well, I'm already committed to the calendar of the old district on the blog. This means that Lesson 10-9 won't be posted until Tuesday.

Meanwhile, today I subbed in a middle school special ed class. This is my third visit here -- the first was back on November 9th. As usual, you can refer to my November 9th post to learn more about this class.

This is a self-contained class where the same six or seven students (all boys) stay in our classroom for most of the day. The middle school rotation today starts with second period, which happens to be math. That's right -- this is the school where all the periods rotate, and the rotation has nothing to do with the day of the week.

Once again, there's no "Day in the Life" because there are special aides in charge. Apparently, this is the time of year when all the special ed teachers have their meetings, since every class I've subbed for this week is special ed.

The fact that the rotation starts with second period benefits me, since I'm a math teacher. If you recall from my November 9th post, the students often have free time during the last period on Fridays. And on Friday, November 9th, the rotation started with third period and ended with math. (That's what happens when the rotation isn't tied to the day of the week!) Today, it's first period science that gets the short end of the stick.

And thus I do get to see some math today. The students have five problems to solve -- three multiplication problems, one long division, and one fraction addition. Here is a sample of some of the problems the regular teacher left on the board:

1. 16 * 90 =
2. 4/8 + 3/24 =
3. 11 * 60 =
4. 7929/67 = (long division)
5. 629 * 84 =

(All of these problems are made-up except the fractions, which is the exact problem she left.) Two students do fairly well on these problem. One boy struggles -- mainly because his writing is so messy that he can't keep track of the numbers. He also tries to simplify 4/8 = 1/2 and thinks he's done without realizing that he's supposed to add the fractions. (It's too bad that the teacher didn't originally pose the question as 3/24 + 4/8 instead. If the boy simplifies 3/24 = 1/8 first, then all he'd needs to do is add 1/8 + 4/8 = 5/8.)

I wish I could have helped this student more. I wonder whether an alternate method from the Number Talks book would have helped -- perhaps setting up the multiplication in boxes (near the end of the multiplication chapter).

But then a traditionalist would counter that what he needs isn't another method -- he needs to learn how to print more neatly overall so he can do the standard algorithm correctly. (Later on in English, his printing is still messy.) And besides, there's no guarantee that he would draw the boxes neatly. (I wouldn't recommend the lattice method that many of my old charter students favored, since that requires even more neatness than the standard algorithm.)

As it turns out, one of the aides is also out today -- and his sub is also strong in math. He tells me that unlike me, he hasn't earned his teaching credential yet.

By the way, try guessing what the students are writing in English. Yes, you guessed correctly -- a persuasive essay! And this one is also about violence in the media, just like yesterday's class. So obviously this is a district-wide assignment. (Checking back to my subbing posts from last year, yes, the students were writing persuasive essays in February as well.)

Actually, there is a girl in the class today -- but she's only scheduled here for homeroom. She also returns just before 11:30, when her parents check her out for the day. (There's another girl on the roster for first period only, but she's absent the whole day.)

It's approaching Valentine's Day, so you know what that means -- pencils and candy. I give two eighth graders a Valentine's pencil -- including the girl, just before she leaves (while the seventh graders are in another room). At the end of the day, I give four seventh graders a pencil plus some chocolate that I bought during my long lunch/conference period (while eighth grade is in another room). Because of the schedule, the eighth graders miss out on the holiday sweets.

Today on her Mathematics Calendar 2019, Theoni Pappas writes:

The exterior angle of a 16-sided regular polygon is (pi/?) radians.

At first, this seems to be a straightforward application of the Exterior Angles of a Polygon Sum Theorem, which doesn't appear in the U of Chicago text until Lesson 13-8. The angles add up to 360, so all we should have to do is divide 360 by 16.

That's until we realize that we need to give the angle in radians which don't appear in most Geometry texts at all. Provided we know that 360 degrees is 2pi radians, it's simple -- instead of 360/16, our answer is 2pi/16, which reduces to pi/8. The question already provides the pi, so all we need to do is fill in the ? with 8. Therefore our answer is eight -- and of course, today's date is the eighth.

We might have missed science today, but let's not miss the science Google Doodle today. Friedlieb Ferdinand Runge was the 19th-century chemist who researched the caffeine molecule. (Missing science -- hey, that reminds me of the problems I had at the old charter school.)

This is what I wrote last year about today's lesson. Here, I compared the treatment of the volume in two different texts, an old McDougal Littell California seventh grade text and the U of Chicago text:

The McDougal Littell text, in Lesson 10.7, demonstrates the sphere volume formula the same way that it does the cone volume formula. We take a cone whose height and radius are both equal to the radius of the sphere, and we find out how many conefuls of sand fill the sphere. The text states that two conefuls make up a hemisphere, and so four conefuls make up the entire sphere.

But of course, we want to derive the formula more rigorously. Recall that Dr. David Joyce states that a limiting argument is the best that can be done at this level -- but I disagree. Dr. Franklin Mason, meanwhile, enthusiastically gives another derivation of the sphere volume formula, and Dr. M's proof also appears in the U of Chicago text. Recall that Dr. M considers this day on which the sphere volume formula -- Lesson 12.6 of his text -- is derived to be one of the three best days of the year.

The U of Chicago text mentions that this proof uses Cavalieri's Principle. But it was hardly the mathematician Cavalieri who first proved the sphere volume formula. Indeed, according to Dr. M, this proof goes all the way back to Archimedes -- the ancient Greek mathematician who lived a few years after Euclid. (It's possible that their lives overlapped slightly.)

Here is a Square One TV video about Archimedes:


We mentioned earlier that Archimedes used polygons to determine the value of pi (also known as Archimedes' constant) -- hence the line in the song, "He was busy calculating pi." He was also famous for using the principle of buoyancy (also known as Archimedes' principle) to determine whether the king's gold crown was a fake -- and this is also mentioned in the song. Legend has it that the Greek mathematician was so excited when he discovered his principle -- he had been in a public bath at the time -- that he ran down the streets naked and shouted out "Eureka!" to announce his discovery. The Greek word eureka, meaning "I have found," is the motto of my home state of California.

But Archimedes himself actually considered the discovery of the sphere volume formula to be his crowning achievement -- to the extent that he requested it to be engraved on his tombstone. So let's finally derive that formula the way that Archimedes did over 2000 years ago. And no, he didn't simply drop a ball into water to determine the formula. Archimedes' sphere formula has nothing to do with Archimedes' principle of buoyancy.

We begin by considering three figures -- a cone, a cylinder, and a sphere. We will use the known volumes of the cone and cylinder to determine the unknown volume of the sphere -- thereby reducing the problem to a previously solved one.

Our cylinder will have the same radius as the sphere, while the height of the cylinder will equal the diameter (i.e., twice the radius) of the sphere. This way, the sphere will fit exactly in the cylinder.

Our cone, just like the cone mentioned in McDougal Littell, will have the its height and radius both equal to the radius of the sphere. Such a cone could fit exactly in a hemisphere. But we want there to be two cones, so that their combined height is the same as that of the cylinder. We set up the cones so that they have a common vertex (i.e., they are barely touching each other) and each base of a cone is also a base of the cylinder. The two touching cones are often referred to as a "double cone" -- Dr. M uses the term "bicone." (A bicone is also used to justify to Algebra II students why a hyperbola is a conic section with two branches. A hyperbola is the intersection of a bicone and a plane, such that the plane touches both cones.)

The focus is on the volume between the cylinder and the bicone. The surprising fact is that this volume is exactly equal to the volume of the sphere! Here is the proof as given by the U of Chicago:

"...the purple sections are the plane sections resulting from a plane slicing these figures in their middles. These purple sections are congruent circles with area pi * r^2. At h units above each purple section is a section shaded in pink. In the sphere, by the Pythagorean Theorem, the pink section is a small circle with radius sqrt(r^2 - h^2). The area of this section is found using the familiar formula for the area of a circle.

"Area(small circle) = pi * sqrt(r^2 - h^2) = pi(r^2 - h^2)

"For the region between the cylinder and the cones, the section is the pink ring between circles of radius r and h. (The radius of that circle is h because the acute angle measures 45 degrees, so an isosceles triangle is formed.)

"Area(ring) = pi * r^2 - pi * h^2 = pi(r^2 - h^2)

"Thus the pink circles have equal area. Since this works for any height h, Cavalieri's Principle can be applied. This means that the volume of the sphere is the difference in the volume of the cylinder (B * 2r) and the volume of the two cones (each with volume 1/3 *B * r).

"Volume of sphere = (B * 2r) - 2 * (1/3 * B * r)
                            = 2Br - 2/3 * Br
                            = 4/3 * Br

"But here the bases of the cones and cylinder are circles with radius r. So B = pi * r^2. Substituting,

"Volume of sphere = 4/3 * pi * r^2 * r
                            = 4/3 * pi * r^3." QED

The sphere volume is indeed the crowning achievement of Chapter 10. We began the chapter with the volume of a box and end up with the volume of the least box-like figure of all, the sphere. The start of Lesson 10-8 summarizes how we did this:

"It began with a postulate in Lesson 10-3 (volume of a box). Cavalieri's Principle was then applied and the following formula was deduced in Lesson 10-5 (volume of a prism or cylinder). A prism can be split into 3 pyramids with congruent heights and bases. Using Cavalieri's Principle again, a formula was derived in Lesson 10-7 (volume of a pyramid or cone). In this lesson, still another application of Cavalieri's Principle results in a formula for the volume of a sphere."

So take that, Dr. Katharine Beals! After all, she was the one who derided Cavalieri's Principle as progressive fluff that the Common Core tests on instead of actual math. But without Cavalieri's Principle, we'd be stuck finding the volumes of only boxes and their unions. Well, I suppose if we simply declared the volume formulas by fiat (i.e., as postulates) rather than actually deriving them, then Cavalieri's Principle is not needed. But if we want to prove them, then the Principle gives us an elegant proof of the sphere volume formula that was discovered over 2000 years before there ever was a Common Core -- a proof that, if mastered, should permit one to date a mathematician's daughter (as Beals mentioned on her website regarding the Quadratic Formula proof).

Sadly, we don't know whether Archimedes ever dated anyone's daughter, or whether he ever had daughters of his own. His life ended tragically, being captured by an enemy army. Legend has it that he was busy working on a geometry problem when the Roman army captured him. His last words before he was killed are said to be, "Noli turbare circulos meos" -- Latin for "Do not disturb my circles!"

Today is an activity day. It's based on this lesson's Exploration question:

25. Unlike a cone or cylinder, it is impossible to make an accurate 2-dimensional net for a sphere. For this reason, maps of the earth on a sheet of paper must be distorted. The Mercator projection is one way to show the earth. How is this projection made?


Notice that the correct answer to this question is quite complex. Here's a link that describes both a misconception and the correct answer:

https://www.math.ubc.ca/~israel/m103/mercator/mercator.html

There is some controversy regarding the Mercator projection. The following link describes the problems some people have with this map (including a clip from the TV show West Wing).

Remember that Monday is Lincoln's Birthday in my old district (and on the blog calendar), and so my next post will be Tuesday.