Lesson 2-6: Unions and Intersections of Figures (Day 26)

Today I subbed in a seventh grade science. It is the second time I've subbed for this teacher -- the first time was back on May 21st. So you can read a little more about this class by referring back to my May 21st post.

That's weird. Back in May, I subbed for a special ed class in my old district, and three days later I subbed for Science 7 in my new district. This week, I sub for the same special ed teacher in my old district, and three days later I sub for the same Science 7 teacher in my old class. Hmm -- back in May, four days after Science 7 was U.S. History 8 at the same middle school. If I'm subbing for the same U.S. History 8 class four days from now (Monday), then someone's just playing with me!

(And it would have been even more eerie had the dates in each month lined up. In May I subbed on the 18th and the 21st, and this month I sub on the 17th and the 20th.)

Back on May 21st, I did "A Day in the Life" for this day of subbing. And so I am doing "A Day in the Life" for today -- the first DITL of the new school year.

Today I subbed in a seventh grade science class. I wish to focus more on classroom management in middle school classrooms, and so I'll do a "Day in the Life" today, with management as the focus resolution (first resolution).

8:15 --- The school day begins with homeroom and morning announcements.


8:25 -- The middle school rotation starts with fourth period today. (That's right -- even the period rotation today is the same as on May 21st!) The students are working on a group project -- they are to create a Mystery Materials poster where they must choose a common material (such as wood, metal, etc.) and, without naming the material, list only eight of its physical or chemical properties.

Physical and chemical properties of matter is yet another science topic that used to be an eighth grade topic under the old California Standards but a seventh grade topic under the preferred Integrated Pathway of NGSS. Every time I sub in a science class, I lament how I failed to -- oops, I promised that I wouldn't mention what happened in a certain school year while reading Eugenia Cheng's book. I intend to keep that promise.

9:15 -- Fourth period ends and fifth period begins. This class is still good for the most part -- but then again, both fourth and fifth period are designated as honors classes. I know that the real test will be the non-honors classes.

10:10 -- Fifth period ends and nutrition begins. Last year (if you recall from my May post), this school didn't have an actual snack break, so this is brand new for this year.
10:20 -- Sixth period is the first non-honors class -- the class that worries me. Before school, I actually reread my May 21st post and decide to prepare for the tough classes. In addition to the teacher's worksheet, I place my own note under the document camera. Like last year's, it reminds students about the good list of names, bad list of names for detention, and the naming of the best class of the day. I tell them that any group that can't even tell me which material they've selected for the poster will be placed on the detention list -- just like four months ago, when I gave detention to groups that couldn't tell me their topic for the project that day.

Some students talk loudly during my explanation of the project. Because of this, many of them inform me that they don't understand what exactly they're supposed to do for the project.

At the end of class, I ask the students to name their chosen material. One group doesn't name a material -- instead, they kept naming some of the physical properties like color, texture, and volume. And so true to my promise, I place the group on the detention list. One girl in the group becomes very upset -- but fortunately for her, a boy in the group convinces me that they genuinely believed that they were working hard on the project.

Instead, I place a different group on the detention list. One boy in this group claims that he can't start the project because he's "looking for a pencil" (which apparently takes the entire period). This is a lousy excuse -- yet it's one I've heard before, so I keep the three members of this group on the detention list.

11:10 -- First period is the teacher's conference period, and this leads into lunch. It gives me the perfect time to adjust my management so that the noisy sixth period experience isn't repeated during second and third periods -- especially second period, the other non-honors class.

I prepare another note to place under the document camera. This time it's some "Checking for Understanding" questions on whether the students actually understand the directions for their poster:

1. Name the nine materials each group can choose from.
2. T/F: You should write the name of the material on the poster.
3. How many clues should you include on your poster?
4. Name the three criteria your poster will be graded on.
5. T/F: You may color in the illustrations.
6. When are the posters due?

12:50 -- Second period arrives. At this middle school, the class right after lunch (no matter which period rotates into this spot) has silent reading.

It's now time for me to put my plan into action. I know that one of the most difficult instructions for a sub to enforce is silent reading. And so I continually tell the students to be quiet. I decide to save time by taking attendance during silent reading (in earlier classes I took roll after giving directions -- tricky because some students start moving to get the materials). One student is absent -- but another boy decides to make an unauthorized seat change into the absent student's seat. This is, of course, a red flag that this boy might be a troublemaker. One thing that I'm working on this week is saying the names of students as a warning before I add them to the bad list, and so I say and learn this boy's name. Then this boy asks for a restroom pass -- and of course, he has to ask for the pass during the period right after lunch. Since I submit attendance online today, there is no attendance loophole, and so the student must wait until the midpoint of the period, 1:25.

Then I give the instructions using the note that I've prepared. The boy who tried to change seats is now calling out to a boy in another group. So I call out the second boy's name as a warning. The first boy also starts trying to mock my voice. This is a perfect time to reprimand him without using my voice -- the teacher look. I randomly choose students to answer the six questions. One of them happens to be the first boy, who correctly answers Question #6.

The students begin working on the assignment. It appears to go somewhat more smoothly than sixth period, though a few of them still have questions about the project.

1:25 -- The boy leaves for the restroom. While he is gone, another student pranks him by placing glue in his seat. I don't see what happens, and so I place the names of both boys on the detention list. The first boy protests that he's a victim (of the prank), so he shouldn't receive a detention. But I figure that his messing around earlier has provoked the prank, and so I keep his name on the list. I also tell the rest of the class that there would be no more restroom privileges for this class. One girl is currently in the restroom, and so when she returns, no one may leave.

But this girl then starts walking around the classroom and talking to members of other groups. I ultimately add her to the detention list as well.

At the end of class, another problem occurs. I collect posters from each group one at a time to make sure that they've written the names of all group members, the group number, and the period number. Even though the assignment isn't due until tomorrow, the regular teacher wants me to collect today's work. (After all, which of three group members gets to take the work home? What if that student is absent tomorrow?) But while I'm checking one group for the names and numbers, another group manages to escape when the bell rings with their poster.

2:00 -- Second period leaves and third period arrives. This is an honors class, so most of the behavior problems of the earlier classes disappear. But there is one problem here -- in an effort to avoid having group posters leave the room, this time I tell the students to give me their posters after including the names and numbers (rather than be forced to wait until I call their group on at a time). Yet I still have to tell some groups to add the missing info -- and while doing so, two groups take their posters home with them.

I was all set to name third period as the best class of the day. But it's wrong from me to declare them the best when neither fourth nor fifth period took any posters home. And so I name fifth period the best and fourth the next best.

2:55 -- Third period ends, and so does a long day of subbing.

In reflection, I wonder what I could have done to make this day go more smoothly -- to be more like the ideal classroom manager, one who easily gets hired for full-time teaching positions.

I believe that I'm more much fair today with the detention list in second period than in sixth. Instead of placing a group on the list, I write the names of individual troublemakers. The idea is that the students on the detention list are those whom I tell to do something and they don't. I see that for several reasons, requiring them to reach milestones (even simple ones like "name your chosen material") are problematic, since they might not reach it for reasons not their fault. A milestone such as "finish half of the poster" may sound logical -- since this is a two-day project, they ought to do half of it today. But "half of the project" is ill-defined -- some groups may indeed write and illustrate four of the eight properties, while others may write all eight properties before drawing any pictures. (I knew better than to say anything like "finish half of the poster" today.)

But the first boy in second period still wonders why I gave him detention -- after all, I warn him not to change seats and so he returns to his seat, and then he's the victim of a prank. I keep on the list because of the little things -- how he tries to mock my voice, and how he keeps laughing at the boy from the other group. I believe that his actions show that he doesn't "respect order" -- which, as I know by now, really means that he doesn't respect me.

I keep on repeating the instructions, yet so many students fail to follow them. One group tries to make their poster topic "a table" rather than one of the materials, while another group in another period correctly chooses a material, but then researches eight random facts about their material rather than eight physical or chemical properties. In general, the honors students follow directions better, but even they often fail to write the group names/numbers on their posters, or they take the posters home when they're not supposed to.

The regular teacher could have made copies of the instructions and passed them out. But the problem was that she had gotten ill unexpectedly on the recess day (Yom Kippur). Her original intention was for the class to work on a density project today -- the students were to take small objects and find their volume (by submerging them in water) and their mass. (On Sunday, Theoni Pappas asks the reader to find the volume of an irregularly shaped stone using the same method.) And the teacher already has copies of the density project prepared. Most likely, the teacher doesn't want the students doing a lab involving water on a sub day, and so she hastily switches to the Mystery Materials posters without a chance to make copies of the worksheet.

Speaking of Pappas, today on her Mathematics Calendar 2018, Theoni Pappas writes:

These coins' radii are each 3 2/3 cm. To the nearest cm find the height of this arrangement.

[There are three interlocking rows of coins, with n coins in row n.]

At first, this question seems easy -- the radius is 3 2/3 cm and so the diameter is 22/3 cm, and the height appears to be thrice the diameter or 22 cm. The problem is that the rows are interlocking, and so the total height is ever so slightly less than three diameters.

The fact that the arrangement of the coins resembles a triangle gives the game away -- if we join the centers of the coins, we form a equilateral triangle with each side twice the diameter, 2d. Let's keep everything in terms of d and then plug in the diameter later -- this is a common trick I use when answering Pappas questions where the given lengths are awkward.

We want the height of an equilateral triangle with side 2d. So we cut the triangle into two 30-60-90 triangles with shorter leg d and hypotenuse 2d, and we want the longer leg, which is d sqrt(3). This is the length of the segment joining the center of the top coin to the center of the bottom coin. The actual height of the arrangement adds two more radii (one on top, one on bottom) to this, and so the height is d(1 + sqrt(3)).

And now we finally substitute in d, which we've already found to be 22/3 cm:

d(1 + sqrt(3)) = (22/3)(1 + sqrt(3)) = 20.035 cm.

Rounded to the nearest whole centimeter, the desired height is 20 cm -- and of course, today's date is the twentieth.

Chapter 4 of Eugenia Cheng's The Art of Logic in an Illogical World, "Opposites and Falsehoods," begins as follows:

"There are only two debates I remember from the debating club at school. One was 'This house believes that Margaret Thatcher should go," which was particularly memorable because she actually resigned the morning of the debate. The other one was 'The house believes that strawberries are better than raspberries.'"

Again, I remind you that Cheng writes about race and politics throughout her book. If you prefer not to read this, then I suggest that you avoid this blog for the next three weeks and skip all posts that have the "Eugenia Cheng" label.

Debating is all about proving an opponent to be wrong. Of this, Cheng writes:

"Logic, mathematics and science are all ways of finding out what is true. But they are also ways of finding out what is not true. Negation is how we argue against things."

Notice that negations don't appear in Chapter 2 of the U of Chicago text -- instead, we must wait all the way until Lesson 13-2. (Even the modern Third Edition waits until Chapter 11.) Yet negations can fit with the basic logic of Chapter 2 -- and we're discussing them now along with Cheng's book.

The author tells us that there are two ways to argue against the view that the Asian education system is better than the British (or American) one:

1. Measured and calm: I don't think the Asian education system is better.
2. Extreme and excited: No way! The British education system is better!

According to Cheng, we might think of the second statement as the "opposite" of the original claim (which some traditionalists have argued for in the past). But the first statement is more like the "negation" of the original. To form a negative, we simply declare that it is not true. Here are some more of her examples:

Original statement: I think the EU is fantastic.

Opposite: I think the EU is terrible.

Negation: I do not think the EU is fantastic. (It could be terrible, or it could be in between.)

Original statement: Margaret Thatcher was the greatest Prime Minister.

Opposite: Margaret Thatcher was the worst Prime Minister.

Negation: Margaret Thatcher was not the greatest Prime Minister. (She could be in between.)

Original statement: Climate change is definitely real.

Opposite: Climate change is definitely fake.

Negation: Climate change is not definitely real. (But it's the mainstream scientific view.)

Original statement: Sugar is good for you.

Opposite: Sugar is bad for you.

Negation: Sugar is not good for you. (It could be OK in small amounts.)

Original statement: I am male.

Opposite: I am female.

Negation: I am not male. (I could be intersex.)

Original statement: Barack Obama is black.

Opposite: Barack Obama is white.

Negation: Barack Obama is not black. (He is mixed-race.)

Cheng tells us that considering opposites rather than negations ignores the gray areas in life:

"People are not very good at dealing with gray areas, and in fact nor is logic. We'll come back to this later, but for now it's important to note that the gray area should be included somewhere, otherwise we're just ignoring part of reality."

According to the author, ignoring the gray area is formally called the Law of the Excluded Middle:

"It doesn't mean we've excluded the middle in the sense of throwing it away or ignoring it, it just means we've included it with one side or another so that there is effectively no middle any more."

At this point, Cheng draws some diagrams. She shows us a continuum of colors from white to black and asks us where to draw the line. We could draw the line between "black" and "not black," or between "white" and "not white." There could be three sections "white," "black," and "lost middle," or even two equal halves "white(ish)" and "black(ish)." Which division we choose depends on the context -- especially if "white" and "black" aren't merely colors, but races.

Cheng tells us that if we divide the continuum into "white," "gray," and "black," then there are two dividing lines that we draw:

"This is to some degree what happens with the terminology of 'heterosexual,' 'homosexual,' and 'bisexual.'"

The author will return to this idea later, but she warns us:

"Absorbing the gray area into one side or the other is a simplification, but at least not incorrect or contradictory. Whereas denying its existence altogether is where black and white thinking usually goes wrong."

Now Cheng draws some Venn diagrams, which I must describe to you. In her first diagram, the universe of people contains only one circle, representing white people. Then the complement of this set consists of all non-white people. In general, the region where A is not true is the outside of the circle representing A.

The whole point here is that it's possible for show a gray area in these diagrams. The interior of circle A is white and its distant exterior is black, but there is a ring of gray surrounding the white, thereby separating it from the black.

The author proceeds to describe truth values -- here 0 means "false" and 1 means "true":

"You might think mathematicians just love turning things into numbers, but remember, math isn't just about numbers, it's about many other things too. However, numbers are so familiar and easy to reason with that if we can turn a situation into some numbers it can be very helpful."

Cheng now presents the truth table for "not" or negation, which is easy to render in ASCII:

A     not A

1     0

0     1

But, as she reminds us, there are some statements whose truth value is currently unknown:

1. The universe is finite.
2. One day we will be able to cure all cancer.
3. A meteor caused the extinction of the dinosaurs.

Cheng asks, how do we negate an implication? So far, we don't know any way to do so other than just to say "does not imply." In her example, the negation of "If you have privilege then you are white" is "Having privilege does not imply that you are white." In symbols we just sort of cross out the implication arrow, but this looks a bit awkward in ASCII, where we must write:

A =/> B (or maybe A =/=> B).

Cheng proceeds with faulty implications. Here is her first example:

"Some black people are better off than me, therefore I don't have white privilege."

And here is a proposed proof:

1. Some black people are better off than me even though I am white.
2. If some black people are better off than you then you don't have white privilege.
3. Therefore I do not have white privilege.

Cheng refers to this as modus ponens:

1. A is true.

2. A implies B.

3. Therefore B is true.

Actually, modus ponens appears in some Geometry texts, including Lesson 13-1 of the U of Chicago text -- except there it's known by a non-Latin name, "Law of Detachment." To refute a proof that uses modus ponens, we must refute either step 1 or step 2, since as soon as we have both, 3 follows. In this case, Cheng refutes step 2 as a straw man argument.

Cheng now discusses the contrapositive, which appears in Lesson 13-2. Here are her examples -- from the above straw man argument, we have:

"If some black people are better off than a white person then that person does not have white privilege."

"If you have white privilege than you are better off than all black people."

If you travel abroad you must have a passport.

If you don't have a passport you can't travel abroad.

These are contrapositive pairs. As we know from Lesson 13-2, a statement and its contrapositive are logically equivalent:

A => B

B is false => A is false (contrapositive)

B => A (converse)

A is false => B is false (contrapositive of the converse -- called "inverse" by U of Chicago)

Cheng shows all the possibilities in a chart, which I don't reproduce here. (Again, Lesson 13-2 shows all the combinations involving converses, inverses, and contrapositives.)

Confusing a statement with its inverse is a common fallacy:

If you are a US citizen then you can legally live in the US.

If you are not a US citizen then you can't legally live in the US (not equivalent).

Also, a contrapositive is not to be confused with the negation:

A =/> B

Cheng writes about evidence. For example, some people might believe:

Being of Chinese origin implies being good at math.

and claim every Chinese-looking math person (like Cheng herself) as evidence. But this statement is equivalent to its contrapositive:

Not being good at math implies not being Chinese.

and every non-Chinese non-mathematician qualifies as evidence -- for example, a Canadian goose or an American singer.

Cheng's final example takes us back to science (perhaps a bit more complicated than the science I saw as a sub today). Back in her own school days, her class performed an experiment to test Hooke's law of springs. The hypothesis is: the extension of a spring is proportional to the load that is hanging from it. Given the right kind of data, we might conclude:

There is sufficient evidence to suggest that this hypothesis is true, to within 95 percent certainty (for example).

And if we don't get the correct date, we conclude its negation:

There is not sufficient evidence to suggest that this hypothesis is true, to within 95 percent certainty.

So we might refine the hypothesis as follows:

The extension of a spring is proportional to the load hanging from it, within a maximum load limit.

And this is Hooke's law of springs. Cheng points out that a "law" is something that has been determined to be probably true, to within levels of certainty accepted by science.

This leads to Cheng's conclusion of the chapter -- a glib view of scientists and statisticians:

"My excellent math teacher, Mr. Muddle, taught us that when you work as a professional statistician, if you do not have the right data to support your hypothesis the correct negation is 'There is insufficient evidence to support this hypothesis and therefore we need more funding in order to pursue the matter further.'"

Lesson 2-6 of the U of Chicago text is called "Unions and Intersections of Figures." (It appears as Lesson 2-5 in the modern edition of the text.)

This is what I wrote two years ago about today's lesson:

Lesson 2-6 of the U of Chicago text focuses on unions and intersections. This is, of course, the domain of set theory.

In many ways, set theory is the basis of modern mathematics, and so many textbooks -- including higher math such as Precalculus and beyond -- mention set theory early on. Of course, the focus in this text is on unions and intersections of geometric figures. In particular, unions are used to define both polygon and angle, while intersections are used to define parallel lines.

The first three examples in the text, where the underlying sets contain natural numbers, real numbers, and points -- are OK. But I didn't like the fourth example, on airlines. I've decided to throw this one out -- if we want a non-mathematical example, why not just use sets of letters, such as {a, e, i, o, u}, the set of vowels?

One of the most important sets in mathematics is the null set, or empty set. According to the text, this set can be written as either { } or an O with a line through it (often called O-slash by students). Once again, since I can't represent that symbol on Blogger, let's use the strikethrough instead: O.

Now the text mentions that the intersection of two sets might be the empty set. But it doesn't mention what happens when one finds the union, or intersection, of the empty set and another set. As it turns out, the union of the empty set and another set is that other set -- so the empty set acts as the identity element for union, just as 0 is the identity for addition and 1 is the multiplicative identity. But the intersection of the empty set and another set is the empty set -- so the empty set acts as the absorbing element for intersection, just as 0 is the absorbing element for multiplication.

One question students often ask is, if { } is the empty set and O is the empty set, what's {O}? When I was young, I once heard a teacher point out that this is not the empty set because it's no longer empty -- it contains an element.