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

Today I subbed in a middle school -- you guessed it! -- special ed class. Actually, today is a bit chaotic -- due to being short one sub, I kept jumping around between science and history.

As usual, there are aides and co-teachers galore -- and this time there's even a student teacher. My schedule is rearranged to take advantage of these extra adults in the classroom and place me where I'm needed the most. So here's my schedule for today -- two science (one co-teaching, both with a student teacher) and three history classes (two with a full aide, the other with a one-on-one aide.)

Of course, there's no "A Day in the Life" today. I'll only briefly mention what happens in one of the history classes. Since this seventh grade class is the only without a full aide, this is the only one where my classroom management truly matters.

Here in California, history in Grades 6-7 consists of a two-year World History course. Seventh grade typically begins at the fall of the Roman Empire. This is the only class where I have the opportunity to sing any songs. I start with "Mathematics of Love" since this contains Roman numerals (as in the fall of that empire), and then I can't resist singing "The Palindrome Song" on this final day of Palindrome Week, before it's too late.

The students have packets, with reading on the first page, five quiz questions on the second page, and then a crossword puzzle on the third page. At first I try making the puzzle the task to earn the song, but many of the special ed students struggle with it. Since no one is placed on the bad list (as opposed to two guys in earlier classes with their heads down and blank packets), I sing the songs anyway, after I tell them the answer to one of the words in the puzzle (namely, that Emperor Constantine renamed "Byzantium" to "Constantinople"). I wonder whether I should have made the five comprehension questions the task to earn the song -- and then I make sure that the students are answering these questions on their own.

So let's go straight to where I wish to put most of my emphasis today -- the science classes. As I typically do on the blog, I'll compare the science classes I see today to the science that I should have taught three years ago at the old charter school, but didn't.

The first science class I observe today is co-teaching -- and as it turns out, the student teacher is also present for this class. Here is the lesson plan that is written on the board:

Monday: Study for Quiz
Tuesday: Quiz
Wednesday: Bubble Lab Procedures
Thursday: Bubble Lab
Friday: Lab Review

But in the end, the plans have changed. The quiz was indeed on Tuesday, but the rest of the week isn't devoted to this "Bubble Lab." Indeed, there are two parts of today's lesson -- the first part on creating graphs, and the second part on the scientific method.

For the first part of the lesson, we take advantage of the co-teaching structure and divide the class into two groups. I assume that ordinarily, the regular teacher for whom I'm subbing would take half of the class -- but instead, today the student teacher takes half of the class. And she actually takes her half to our own classroom. The student teacher introduces graphs, while the resident co-teacher applies graphs to the previous lab (the "Penny Lab"). Yesterday, the class was similarly divided into halves -- this time, the students report to the opposite station from yesterday.

I follow the student teacher into our classroom. In discussing how to draw graphs, she mentions something called "DRY MIX." At first, I thought that there must have been some dry mix (of soap, say) as one of the materials needed for the "Penny Lab" (whatever that was). But no -- as it turns out, "DRY MIX" is actually a mnemonic similar to Sarah Carter's DIXI-ROYD. Even though Carter's DIXI-ROYD is for linear equations and today's DRY-MIX is actually for bar graphs, the two ideas are quite similar:

DRY = dependent, responding, y-axis
MIX = manipulated, independent, x-axis

The key difference here is that "responding" and "manipulated" make more sense in a scientific experiment than in an equation for Algebra I. (Indeed, I wonder how Carter herself -- who, if you recall, actually taught some science for a while -- introduced graphs in her science classes.)

Oh, and speaking of Sarah Carter again, I see interactive notebooks in the science class. I've pointed out before that in a way, interactive notebooks are more important in science here than in math. This is because there is no science textbook (except online), so the only physical evidence of learning any science is in the notebook.

In the student teacher's own science class (that is, the one with no co-teacher), she repeats the same bar graph lesson, but at a slightly slower pace for the special ed students. I help out whenever I can, especially when the students fail to start the graphs at zero on the y-axis or otherwise miscount the scale on that axis.

(By the way, Square One TV does discuss bar graphs -- most notably in a skit that shows rutabaga consumption going down as Square One TV ratings went up. But there was no bar graph song on that show at all.)

So now let's compare this to the old charter school. I've already explained why I didn't use interactive notebooks that year -- I was so afraid that students would either refuse to buy the notebooks, or refuse to bring them everyday. After observing so many classes using the notebooks, I now know that I should have used them anyway.

Much of my fear was based on the students' initial reluctance. I assume that for Sarah Carter herself, she usually has no problem getting her students to buy and bring the notebooks because they recall from the previous year how useful they are. But last year, she started at a new school -- with students who had never used them before and thus were initially reluctant to purchase them. I'm sure, though, that Carter pressed how important the notebooks were, and harshly punished students who didn't buy them or bring them the first few weeks. Once she got past those first few weeks, her new students willingly bring them as readily as her old students did. In contrast, I let fear of initial reluctance dictate how I ran the entire class. ("More than half of my kids won't buy them or bring them the first few weeks, so I simply won't use them at all!")

As for the graphing lesson, I do remember giving my sixth graders a graphing lesson during the week leading up to the LA County Fair field trip. At the time, I thought of this lesson as a math lesson rather than a science lesson (and these were coordinate graphs, not bar graphs). Then again, the graphs may have been important for Illinois State STEM projects.

Notice that these teachers explicitly show the students how to make graphs. By contrast, the Illinois State text takes for granted that the students know how to make them. I get it what the Illinois State developers are thinking -- why teach it when more than half the kids probably won't listen! But in this case, perhaps a few early lessons on how to draw graphs (or even the scientific method, just as in today's class) might have made both STEM projects and science lessons go more smoothly.

Today's student teacher appears to be a strong classroom manager -- at least she's stronger than I was as a student teacher, or even years later at the old charter school. Several times, she tells students to be quiet or else she'll send them out of the room. I rarely did this, except for a short stint as a sub right after I left the old charter school (possibly mentioned in some of my 2017 blog posts). But my threats usually didn't work, and I felt that I kicked out too many students. Today's student teacher is much more effective because her threats are accompanied by teacher tone and teacher look. These cause the students to start behaving immediately, so that she doesn't need to send anyone out.

She tells me that she's enrolled in an online program to earn her elementary special ed credential. I reply that with her strong classroom management skills, plus a credential in the in-demand field of special ed, she'll have no trouble getting hired. She'll have a much smoother path to a career than I did, with my less-than-stellar management skills.

By the way, yesterday I posted the Palindrome Song. Since I sing it in class today, let me post the lyrics, which are not available on the Barry Carter website:

PALINDROME SONG

Refrain:
Go forward then go backward.
If the number reads the same,
Then it's a palindrome.
Go backward then go forward.
If the number's still the same,
Then it's a palindrome.
It's not a palomino,
On the western plain,
Where cowboys love to roam.
No! The wonder of all wonders,
A backward-forward number.
It's called a palindrome!

1st Verse:
Let's see! Take a 33!
Read it in reverse.
Hey it reads the same,
So it's a palindrome.
And 505,
Or 2002,
So they're both examples of that one I love.
It's a palindrome!
It's a palindrome!
It's a palindrome!
(repeat Refrain)

2nd Verse:
Let's see! Take a 63!
Read it in reverse.
Now we're in a fix,
'Cause it's a 36.
But don't be sad,
All we do is add.
Add the 63 to 36 and see.
That's a 99!
Hey we're doing fine!
That's a palindrome! Whoa! Whoa!
(repeat Refrain)

Actually, the refrain changes slightly each time it's sung. After the first verse, the palomino is replaced with a pachyderm. But I'm really having trouble understanding those lines completely:

Then it's a palindrome.
It's not a pachyderm,
Per coming in the circus,
At the hippodrome.
No! The wonder of all wonders...

Hmm, a "pachyderm" is an elephant, and elephants do appear in a circus, but a "hippodrome" is an ancient horse track, not any place for elephants. Then again, sometimes the words "circus" and "hippodrome" were used interchangeably, such as the famous Hippodrome of Byzantium. (Or should I say, Constantinople?)

After the second verse, the following lines appear (starting in the middle at "Go backward"). These lines are more important, since they explain the process to generate a palindrome:

Go backward and go forward.
It's the one that's glad you came,
'Cause it's a palindrome.
If reversing and then adding,
Doesn't work the first time,
Repeat it 'til you're home.
'Cause you'll finally reach that number,
That backward-forward number,
You've reached a palindrome!*

(*unless, of course, you started with a Lychrel number like 196)

Today on her Mathematics Calendar 2019, Theoni Pappas writes:

AB = 48, PB = 29, PS = ?

(Here is the given info from the diagram: AB and RS are common tangents to two circles, and they intersect at P. Points A and S lie on one circle, with B and R on the other.)

This is clearly a common type of Pappas problem -- a "Two Tangents" problem. Many texts contain a Two Tangents Theorem which states that two tangent segments with a common endpoint outside the circle are congruent. (The proof requires setting up two triangles that are congruent by HL.) But unfortunately, the U of Chicago text is one text that doesn't provide this theorem.

Anyway, by Segment Addition (or Betweenness Theorem), PA + PB = AB, and since AB = 48 and PB = 29, we conclude that PA = 19. Now by Two Tangents, PA = PS = 19.

Therefore the desired length is 19 -- and of course, today's date is the nineteenth.

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.