This is my third visit to this classroom. I described my most recent visit here in my May 8th post. My three visits to this classroom occurred in three different school years -- and each year, it seems as if this teacher has fewer math and more science classes. On my first visit, she taught mostly math, and last year, she had almost an even split of math and science classes. This year, the only two classes she has in her own room are both science -- her only math class is as a co-teacher.
Let me start by describing the science classes first, since the math class meets last period today. I end up covering five periods of science -- in five different classrooms! First, I had to cover a class for another sub who's running late and doesn't arrive until after the tardy bell, almost midway through the first period. Once he arrives, I return to the science class I'm supposed to be co-teaching first. Then I return to my own classroom for two periods.
I must cover the fourth science class during the conference period -- because the regular teacher there must cover for my teacher at a previously scheduled parent meeting! (The scheduling would have made perfect sense for my teacher -- have the meeting during her conference period. The fact that my teacher can't come to school today ruins the best laid plans there.) The fifth and final science class is the other co-teaching class.
Of these five science classes, two are seventh grade and three are eighth grade. Since I'm in so many science classrooms today, there's plenty of science material for me to compare to what I should have taught at the old charter school.
Let's look at the seventh grade classes first. The co-teaching class is starting a lab on arenology -- in other words, sand. The students analyze six sand samples under a microscope and record their observations in a chart. The special ed seventh graders in my own class are slightly behind -- they are just learning how to identify the parts of a microscope. This is one of the few lessons to use a printed science text -- the microscope instructions are in the appendix of the life sciences text. Then they return to Chromebooks to work on a vocab assignment.
We already know why science lessons are mostly online these days -- the NGSS and the Preferred Integrated Model. Indeed, the regular teacher mentions this in her lesson plan (to warn me that the students aren't used to looking at the printed science text). This is something that I should have figured out at the old charter school -- almost all of our work that year should have come from either the Illinois State or Study Island websites.
Recall that there were plenty of microscopes in my old classroom, so I could have easily assigned a lab that used them. I'm not sure whether there was a sand lab in the Illinois State text, since not every teacher throughout the country has easy access to sand. (Illinois itself is near Lake Michigan, but there are plenty of completely landlocked states without sand within a short reach.)
But suppose that I'd gone on my own and decided, as a Southern California teacher, to assign a sand lab to my charter students. Well, today's teacher had apparently gathered six samples of sand from beaches all the way from Ventura to Dana Point, plus Anza Borrego near the Salton Sea. Notice that Ventura and Anza Borrego are more than 200 miles apart, so that would have been at least a 400-mile round trip for the teacher to gather the sand. (That's the problem with science labs -- students see them and enjoy them, but no one thinks about how difficult it is for the science teacher to set them up until becoming a science teacher!) Unless I was going to send a tweet to Fawn Nguyen and ask her to ship me some sand samples from her nearby Ventura, there was no way I'd obtain all the sand.
And I can always tell the students to put the sand under a microscope, but unless I'm a trained science teacher, I don't know what to tell the students to look for. Today's students try describing the shape of the sand grains, place the sand under a magnet, and add a drop of vinegar. But unless this lab is mentioned in one of my texts (print or online), I wouldn't have been able to tell them all of this.
And to which middle school grade would I give this lab anyway? Once again, the fact that I was at the charter school in 2016, the very year of the transition to NGSS, would have thrown it off. (When the special ed students look up the California NGSS Standards online, they are even labeled on the website as the "2016 Standards.") Under the old standards, sand/arenology might have been thought of as a sixth grade Earth Science standard, but now it's clearly a seventh grade standard. Thus assuming that Grades 7-8 were grandfathered under the old standards and only sixth graders got the NGSS, there might have been no reason to assign a sand lab to any grade that year.
The closest I came that year to giving a lab similar to today's was in February, when there was an Illinois State lab involving measuring and identifying different substances. None of the substances was sand, and there were no microscopes used. After seeing today's lab, I see how I could have improved that lesson -- keep the various substances in different containers so that the groups can easily take samples for them to work on.
Now let's move on to the eighth grade classes. The first class (the one where I spent only half a period) is an honors class that is working on some sort of science fair project. It appears to be a contest that most students can volunteer to participate in, but honors students are automatically signed up to participate in it. The closest analog at my old charter school might have been the Green Team -- the problem was that my science class ended up becoming dependent on the Green Team project, which didn't really start until right around the time I left.
The other two classes are both learning about waves (electromagnetic, sound, etc.). One of the classes watches a Bill Nye the Science Guy episode on waves. The other is creating a foldable to show what they've already learned about electromagnetism -- this class is apparently ahead of the other, though, since one of the sources they can use to make the foldable are their Bill Nye notes.
Back at the old charter school, I actually considered showing my students Bill Nye. Notice that Bill Nye the Science Guy and Square One TV are near contemporaries on PBS -- the first season of the former was the year after the final season of the latter. But I've subbed for several science classes where Bill Nye is shown, yet not even one math class that aired Square One TV. Part of the reason for this might be that a single Nye episode covers a single topic (such as waves), while a Square One episode would cover a hodgepodge of topics. Thus Nye fits a school curriculum much better than Square One.
Each episode of Bill Nye ends with a song. Today's wave episode ends with "Baby, I Love Your Wave," a parody of Big Mountain's "Baby, I Love Your Way." (Actually, Peter Frampton is the originator of this song, but Big Mountain's version is closer to the date of this Bill Nye episode -- and the performer of this song looks like a reggae singer.)
If I had played Bill Nye episodes at the old charter school, I might have considered incorporating some of his songs into my class, just as I did with Square One TV. But of course, I did neither. (In my March 15th post, I wrote that the students at my old charter school -- two years after I was a teacher there -- just barely missed meeting Nye at the LA County Fair.)
Recall that there was a recent traditionalists' post about Bill Nye -- a teacher complained that Nye makes science appear to be all fun and games, when in reality it isn't. Then again, even if for this reason I chose not to play Nye episodes in class, I could still write down the song lyrics and add them to my notebook without playing any episodes in class, just to have the songs.
Speaking of songs, I decide to sing to my only solo class -- a special ed seventh grade class. Since the lesson is on earth science, I sing "Earth, Moon, and Sun" to them. In the two co-teaching science classes, some students ask for a song anyway, and so -- after obtaining the resident teacher's permission, of course -- I oblige them with "Earth, Moon, and Sun" as well. Oh, and regarding that solo class, behavior problems aren't an issue there, especially since there are only nine students.
OK, that's enough about the science that five teachers presented in their classes today and that I failed to teach at the old charter school. Let's look at the one co-teaching math class.
These seventh graders have recently completed the EE (Expressions and Equations) standards and have moved on to RP (Ratio and Proportion). The students are learning how to solve proportion equations "algebraically." Actually, for an proportion such as:
x = 5
6 4
the resident teacher tells them to form "butterflies" -- pairing x with 4 and 6 with 5. The first "wing" is incomplete since it contains x, while the second "wing" has two complete numbers. Then the students should multiply the pair in the complete wing first (6 * 5 = 30) then divide by the remaining number (so x = 30/4 = 7.5).
We know that traditionalists dislike the Common Core "line diagrams" and "tape diagrams" used to solve proportions, but I wonder what they'd say about this method -- yes, we're actually writing an equation, even though we're still not writing out all the steps.
Of course, no algebraic method for proportions was available for me at the old charter school. This is because in the Illinois State text, all RP standards are taught before even a single EE standard begins.
Today may be Tuesday, but it's Friday on the Eleven Calendar. This is the first day of my week, and so we focus on the first New Decade's Resolution:
Decade Resolution #1: We are good at math. We just need to improve at other things.
This might have been an excellent resolution to show the special ed students, save for the fact that I only teach the special ed students science today, not math.
I do show one student in the co-teaching class this resolution. The resident teacher tells me that both she and the teacher I'm subbing for have had trouble reaching him. Anyway, today he at least copies the work onto the paper. The teacher's method is so fast that other students have answered the question before he has a time to think about it, so he just ends up copying from the board. I do get him to do two of the problems on a calculator before the answers are revealed. I hope that he'll take today's resolution to heart and know that he's indeed good at math.
Lecture 7 of Michael Starbird's Change and Motion is called "Derivatives the Easy Way." Here is an outline of this lecture:
I. Derivatives would be of no practical value if we had to an infinite process at each point of time. Fortunately, we don't.
II. The derivative reduces the number of bumps in the graph of a function.
III. Let's consider a function that is defined geometrically on a circle.
IV. If we have an expression for the position of a moving car, we can quickly know the expression for the velocity if it's one of these.
V. If we are trying to find the answer to a question that involves derivatives, we will be able to solve it in a practical way.
This lecture is all about finding derivatives using simple rules (such as for polynomials). Starbird's first example is p(t) = 2t, whose derivative is the slope p'(t) = 2. In general, if p(t) = ct, then p'(t) = c.
Then the professor moves on to p(t) = t^2. He reminds us of the definition of derivative and shows us to find ((t + delta-t)^2 - t^2)/delta-t = 2t + delta-t. (Ignore the algebra, he says!) As delta-t approaches zero, we conclude that d/dt(t^2) = 2t. Similarly, if f(x) = 5x^2, then f' (x) = 5(2x) = 10x, and then if f(x) = kx^2, then f' (x) = 2kx.
Starbird now moves on to a cube. He uses a geometrical argument to show that if f(x) = x^3, then the incremental volume is 3x^2 delta-x and the rate of increase is 3x^2 delta-x/delta-x = 3x^2. (Think of adding small slabs of base area x^2 and thickness delta-x to three faces of the cube.) Thus the derivative is f' (x) = 3x^2.
The professor generalizes. If f(x) = x^n, then f' (x) = nx^(n - 1), the famous power rule. This can be used to find derivatives of all polynomials: if f(x) = 5x^3 + 2x + 3, then f' (x) = 10x^2 + 2. It's because finding these derivatives is so easy and mechanical that Calculus is so useful.
Now Starbird shows us some graphs. The graph of x^3 - 3x^2 - x + 3 contains two "bumps," while its derivative 3x^2 - 6x - 1 contains only one "bump." Likewise, x^4 - 5x^2 + 4 contains three bumps, while its derivative 4x^3 - 10x has only two bumps.
The professor now explains what the sine function is using a unit circle -- if the angle from the positive x-axis is theta, then the vertical distance of the point on the circle is sin theta.
He then shows us how to find the derivative of the sine function. If we increase the angle by a small amount delta-theta, then how fast is the vertical distance changing? First, he defines the cosine as the horizontal distance of the point on the circle. To find sin(theta + delta-theta) - sin theta, he notices that this is the height of a small right triangle whose hypotenuse is very close to delta-theta (the length of the arc). This small triangle is similar to the big triangle with sides sin theta, cos theta, 1. Thus since corresponding sides are proportional, (sin(theta + delta-theta) - sin theta)/delta-theta = cos theta/1, and so d/dtheta(sin theta) = cos theta. (It's a clever argument that looks better when he does it all on the screen.) Likewise, we can show that d/dtheta(cos theta) = -sin theta.
Chapter 9 of the U of Chicago text is called "Three-Dimensional Figures." In past years, we skipped over this chapter and jumped directly into Chapter 10. After all, most questions relating to 3D figures on standardized tests are asking about their surface areas or volumes -- the purview of Chapter 10. As we are following the digit pattern this year, we will cover all of Chapter 9 starting today, Day 91.
I think back to David Joyce, who criticized a certain Geometry text. He writes:
Chapter 6 is on surface areas and volumes of solids. It is strange that surface areas and volumes are treated while the basics of solid geometry are ignored.
In summary, there is little mathematics in chapter 6. Most of the results require more than what's possible in a first course in geometry. Surface areas and volumes should only be treated after the basics of solid geometry are covered.
And so Chapter 6 of the Prentice-Hall text is just like Chapter 10 of the U of Chicago text. Joyce laments that students don't learn "the basics of solid geometry" before surface area and volume. But we can't fault Prentice-Hall for this. Even before the Common Core, most states' standards expected students to learn the 3D measurement formulas and hardly anything else about 3D solids.
We can't quite be sure what Joyce means by "the basics of solid geometry." But it's possible that some of what he wants to see actually appears in Chapter 9 of the U of Chicago text. Thus, by teaching Chapter 9, we are slightly closer to Joyce's ideal Geometry course.
And incidentally, there is one Common Core Standard in which 3D solids are mentioned, but not surface area of volume. We'll look at this standard in more detail next week, in Lesson 9-4.
Lesson 9-1 of the U of Chicago text is called "Points, Lines, and Planes in Space." The first three sections of Chapter 9 are the same in both the old Second and modern Third Editions. (As it turns out, the new Third Edition squeezes in surface area in Chapter 9, saving only volume for Chapter 10.)
This is what I wrote last year about today's lesson:
The heart of this lesson is the Point-Line-Plane Postulate. We first see this postulate in Lesson 1-7, but now it includes parts e-g:
Point-Line-Plane Postulate:
a. Given a line in a plane, there exists a point in the plane not on the line. Given a plane in space, there exists a point in space not on the plane.
b. Every line is a set of points that can be put into a one-to-one correspondence with the real numbers, with any point on it corresponding to 0 and any other point corresponding to 1.
c. Through any two points, there is exactly one line.
d. On a number line, there is a unique distance between two points.
e. If two points lie in a plane, the line containing them lies in the plane.
f. Through three noncollinear points, there is exactly one plane.
g. If two different planes have a point in common, then their intersection is a line.
There are several terms defined in this lesson -- intersecting planes, parallel planes, perpendicular planes, and a line perpendicular to a plane.
Actually, I'm still thinking about Joyce's "basics of solid geometry." I know that his website also links to Euclid's Elements. So Book XI of Euclid is a reasonable guess as to what Joyce wants to see taught in class:
https://mathcs.clarku.edu/~djoyce/java/elements/bookXI/bookXI.html
Let's look at some of the definitions and propositions (theorems) here and compare them to the contents of Lesson 9-1. We'll start with Definition 3, since Definitions 1 and 2 will actually appear in tomorrow's Lesson 9-2.
Definition 3 appears in Lesson 9-1 as a line perpendicular to a plane. Definition 4 appears in this lesson as perpendicular planes. But Definition 5, the angle between a line and a plane, is only briefly mentioned in the U of Chicago text.
Again, Definitions 6 and 7 are about the angle between two planes, which is not discussed in our text at all. Definition 8, of course, appears in today's lesson -- but just as with lines, the U of Chicago uses an "inclusive" definition of parallel where a line or plane can be parallel to itself. Intersecting planes (our remaining term) are implied in Definition 8 as planes that are not parallel.
Let's look at the propositions (theorems) now:
This is essentially part e of our Point-Line-Plane Postulate. Euclid calls it a proposition (or theorem) and even provides a proof, but Joyce argues that the proof is unclear. Thus we might as well consider it to be a postulate.
This is essentially part f of our Point-Line-Plane Postulate. If A, B, and C are the three noncollinear points mentioned in part f, then we can take lines AB and AC to be the two intersecting lines that appear in Proposition 2, and triangle ABC to be the triangle mentioned in this proposition.
This is very obviously part g of the Point-Line-Plane Postulate. Joyce points out that this is yet another postulate, and that it holds only in 3D, not 4D and above.
According to Joyce, this is the first true theorem in Book XI. It asserts that if a line intersects a plane and is perpendicular to two lines in the plane, then the line is perpendicular to the whole plane. Joyce points out that the proof is a bit long, but it works. Theoretically, our students can prove it using the new Point-Line-Postulate and theorems from the first semester of the U of Chicago text.
https://mathcs.clarku.edu/~djoyce/java/elements/bookXI/propXI4.html
Here is a modern rendering of this proof. The idea is that line l is perpendicular to each of two lines m, n, in plane P, with all lines concurrent at point E. Our goal is to prove that line l is perpendicular to the entire plane P by showing that, if o is any other line in plane P with E on o, then l must be perpendicular to o as well.
Given: l perp. m, l perp. n with l, m, n all intersecting at E
Prove: Line l is perpendicular to plane that contains m and n.
Proof:
Statements Reasons
1. bla, bla, bla 1. Given
2. Choose A, B on m and 2. Point-Line-Plane part b (Ruler Postulate)
C, D on n so that
AE = EB = CE = ED
3. Exists plane P containing m, n 3. Point-Line-Plane part f (3 noncollinear A, C, E)
4. Choose F on l, 4. Planes contain lines and lines contain points.
and o in plane P s.t. E on o
5. Lines AD, o intersect at G, 5. Line Intersection Theorem
Lines BC, o intersect at H
6. Angle AED = Angle CEB 6. Vertical Angle Theorem
7. Triangle AED = Triangle CEB 7. SAS Congruence Theorem [steps 2,6,2]
8. AD = CB, Angle DAE = EBC 8. CPCTC
9. Angle AEG = Angle BEH 9. Vertical Angle Theorem
10. Triangle AEG = Triangle BEH 10. ASA Congruence Theorem [steps 8,2,9]
11. GE = EH, AG = BH 11. CPCTC
12. FE = FE 12. Reflexive Property of Congruence
13. Triangle AEF = Triangle BEF 13. SAS Congruence Theorem [steps 2,1,12]
14. FA = FB 14. CPCTC
15. Triangle CEF = Triangle DEF 15. SAS Congruence Theorem [steps 2,1,12]
16. FC = FD 16. CPCTC
17. Triangle FAD = Triangle FBC 17. SSS Congruence Theorem [steps 8,14,16]
18. Angle FAD = Angle FBC 18. CPCTC
19. Triangle FAG = Triangle FBH 19. SAS Congruence Theorem [steps 11,18,14]
20. FG = FH 20. CPCTC
21. Triangle GEF = Triangle HEF 21. SSS Congruence Theorem [steps 11,12,20]
22. Angle GEF = Angle HEF 22. CPCTC
23. EF perp. GH (i.e., l perp. o) 23. GEF, HEF are congruent and a Linear Pair 24. Line l perpendicular to plane P 24. Definition of line perpendicular to plane
So this is probably what Joyce wants to see more of. Propositions 5 through 19 aren't very much different from this one. But as I wrote above, our students will find such proofs difficult -- we had to prove seven different pairs of triangles congruent above, in three dimensions to boot. No modern text teaches such theorems, since no state standards -- pre- or post-Core -- require them.
The proof works -- the definition in Step 24 is satisfied because o is arbitrary. But notice that in the drawing at the above link, Euclid assumes that G, the point where lines AD and o intersect, is between A and D. But this is irrelevant for the proof -- all the congruence theorems used in the proof still work even if G isn't between A and D.
What's worse, of course, is if o is parallel to AD. Notice that AD | | BC (since DAE and EBC, the angles proved congruent in Step 8, are alternate interior angles), so o could be parallel to both. But that's no problem -- just switch points C and D in that rare case, and the proof still works.
Here is the worksheet for today's Lesson 9-1:
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