The two classes I have in the resident teacher's classroom are both eighth grade English. Like last Friday's gen ed English 8 class, these students are learning about arguments and claims. Most of the management problems occur in the sixth period class -- the larger of the two classes (about 24 students) and is right after lunch on today's period rotation. One of the two aides (yes, an extra aide is assigned to this class only) must write down seven names on our bad list for continuous talking.
In order to introduce arguments, we play a video for the students. I often embed on this blog the videos I see while subbing, except I can't because it's hosted on TeacherTube, not YouTube. I'll still provide the link to the TeacherTube website:
https://www.teachertube.com/video/all-about-my-claim-the-argument-song-a-parody--431554
The song is called "All About My Claim" -- a parody of Meghan Trainor's "All About That Bass."
This reminds me of something -- when I first heard Trainor's song a few years ago, I envisioned creating a math parody of this song and singing it during music break. The word "bass" sounds like "base," and we have bases in math, so the song fits.
A quick YouTube search reveals that some "All About That Base" math parodies have indeed been already created. Most of these videos use "base" to mean "base and exponent." But since this is a Geometry blog, I'd prefer my song to use "base" to mean "base and height."
I was able to find one "All About That Base...and height" song. It's all about finding the areas of triangles and rectangles. My vision was for my parody to be about volume, not area. Indeed, I was thinking about this around November-December 2016 when I was trying to teach my eighth graders the volume formulas (the test the special scholar cheated on in my Epiphany post). But I never wrote the song because I kept focusing on songs I could sing my Grade 6-7 students. (This was around the time that one of my sixth graders taught me "Mode, Mode, Mode the Most.") Thus I probably would have written "All About That Base" had I been a pure Math 8 teacher who didn't have to worry about the lower grades.
Well, now we're right between the area and volume chapters in the U of Chicago text. Thus it's a great time to post some song parodies. Let's start with the area version that I found on YouTube. (We'll get to my volume version at the bottom -- or should I say the "base" -- of this post.)
Let me transcribe the lyrics here:
ALL ABOUT THAT BASE (Area Version)
Chorus:
Because you know I'm all about that base,
'Bout that base and height.
I'm all about that base,
'Bout that base and height.
I'm all about that base,
'Bout that base and height.
I'm all about that base,
'Bout that base base base base.
1st Verse:
Yeah, it's pretty clear base times height divided by two,
That equals, equals the area of a triangle. Woo!
'Cause I got those smarts, smarts that all the colleges chase,
And all the right brains in all the right places.
I see the formulas working the area.
We know that stuff is right, come on let's give it a shot!
If you got smart smarts, just let them know,
The height is perpendicular from the bottom to the top.
Pre-Chorus:
Yeah my momma, she told me to worry about your grades,
She said, "Teachers like effort, so you better get all them A's."
You know I won't be no failure up against you all,
So if that's what you're in to, then go ahead and move along.
(to Chorus)
2nd Verse:
I'm bringing smartness back! Go ahead and tell them who is
Really right! Naw, I'm just playin'. I know you
Think it's hard! But I'm here to tell you
Base times height is the area of a rectangle and square.
(to Pre-Chorus)
The author of this song is listed as "Mrs.Nguyen_Math." I assume that this "Mrs. Nguyen" isn't the same as Fawn Nguyen, but who knows?
Oh, and speaking of our queen, Fawn Nguyen posted yesterday. I must have missed her post since I was so distracted with the LAUSD teacher strike. Oh, and SteveH leaves a comment on Barry Garelick's page early this afternoon. I didn't want to make two traditionalists' posts in a row, but I can't help it that everyone decided to post around the time of the teacher strike.
Let's start with Nguyen's latest post:
http://fawnnguyen.com/jelly-beans-or-no-jelly-beans/
Nguyen begins with a Warm-Up question called a "Would you rather...?"
Would you rather have 364 jellybeans and give 188 to friends or have 281 jellybeans and give 137 to friends? Whichever option you choose, justify your reasoning with mathematics.
But in her class, she decided to change this to a different type of "Would you rather...?"
As a student, would you rather be given the problem on the LEFT (jelly beans) or the one on the RIGHT?
And here's the question on the RIGHT:
Which problem below, A or B, yields a larger difference?
A. 364 - 188 =
B. 281 - 137 =
She writes that a majority of her sixth graders, 71%, chose A. Moreover, when she posted this as a survey on her blog -- with most respondents presumably being teachers -- an even larger majority, 90%, chose A.
In fact, we can likely figure out what the only sort of people to choose B are -- the traditionalists. We know that the traditionalists have nothing against word problems per se, but questions that start with "Would you rather..." are probably too open-ended for the traditionalists. Well, actually, there's one more person who choose B -- Nguyen herself:
Well, I prefer the one on the right [that I’d typed up]. How did I get it so wrong? I’m normally not this lame. But, truth be told, I don’t love the jelly beans question. At all. Maybe the one on the right is the wrong “fix” for the left one. If I could retype the problem on the right, I’d remove the equal signs since the question is just asking which one yields a larger difference, not caring exactly what each difference is. I want to believe that anyone who spends 5 minutes with me learns that I love mathematics.
Does this mean that Nguyen is secretly a traditionalist? I doubt it. After all, in her solution to her preferred question B, she uses a method besides the standard algorithm to subtract:364 minus 188… I’d need 12 more to go from 188 to 200, then 164 more to get to 364, so the difference is 176. Similarly, to do 281 minus 137, I’d need 63 more and 81 more, or 144. Problem A has a bigger difference of 176.
(Ruth Parker calls this method "Add instead" in her Number Talks book.) And besides, a traditionalist wouldn't even ask question A in the first place, even if only to compare it to B.
Meanwhile, notice that (assuming that most choosers of B are traditionalists after all) only 10% of teachers are "traditionalist" yet 29% of students are "traditionalist." This is what the traditionalists themselves like to point out -- there is more enthusiasm for nontraditional math problems among teachers than among students. Even though a majority of students prefer nontraditional math, a larger majority of teachers prefer it.
I like to call Nguyen the Queen of the MTBoS, since she was the first to defend the MTBoS after Dan Meyer rejected it. But maybe she's slightly more traditionalist than most MTBoS members. In some ways, Nguyen bridges the gap between the traditionalists and the MTBoS -- after all, she likes teaching both A. and B. in her classroom. (I wish to bridge this gap myself here on my own blog.)
Meanwhile, let's get back to the real traditionalists -- Barry Garelick and SteveH. I only briefly mentioned Garelick's post yesterday, so let me quote a little more of it today:
https://traditionalmath.wordpress.com/2019/01/13/wish-list-dept/
I have heard many people express the thought that “Calculation is the price we used to have to pay to do math. It’s no longer the case. What we need to learn is the mathematical understanding.
There is much information that we do not have from such statements.
Then he lists several questions whose answers he'd like to research and find out. I'll list some of these questions here:- Was the education they received really devoid of any kind of understanding and all rote?
- Are these complaints limited to those who were educated in the era of traditional or conventionally taught math?
- What percent of the student population has had math tutors, or been enrolled in learning centers?
Yes, we know that the question about math tutors is one of Garelick's favorites. Let's look at SteveH:
SteveH:
Make sure you point out that slower is not equal. Researchers have to analyze the effects of slow separately. And of course, they have to define what constitutes a test of understanding. The rubrics my son had in K-6 were completely worthless as with the state’s annual test results (NECAP and later CCSS), which, in any case, were a year late and many tutoring dollars short. State tests offer a terrible feedback loop. What are teachers in K-6 – potted plant facilitators?
This comment is interesting in light of the current LAUSD teacher strike. SteveH writes that "state tests offer a terrible feedback loop" -- so I doubt he'd be a strong advocate of using test scores to measure elementary teacher accountability.
He goes on to mention his preferred measures of accountability in high school (AP, SAT, etc.), but doesn't state how to address accountability (for either students or teachers) in elementary. Notice that he even criticizes "NECAP" -- a pre-Core state test -- so it isn't just the Core.
(By the way, today, January 15th, marks the 25th anniversary of the day I first took the SAT -- all the way back as a seventh grader. It was part of the Johns Hopkins annual talent search. I scored fairly well, 700, in math, but my verbal 410 disqualified me for the talent search.)
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.
Since we finally started 3D Geometry in this post, let me post my 3D version of the "All About That Base" song. The chorus is the same as Mrs. Nguyen's, but the verses are different:
Chorus:
Because you know I'm all about that base,
'Bout that base and height.
I'm all about that base,
'Bout that base and height.
I'm all about that base,
'Bout that base and height.
I'm all about that base,
'Bout that base base base base.
1st Verse:
Yeah, it's pretty clear, I really want to,
Calculate your volume, volume, like I'm suppose to do.
'Cause I got that formula that all the students chose.
Just plug in all the right values in all the right places.
See that base! That's the area of the top.
We know the height, come on now make it pop.
If you got your calculator, 'lator, just multiply 'em,
'Cause every cubic inch is perfect from the bottom to the top.
Pre-Chorus:
Hey prisms, cylinders, don't worry about your size,
'Cause students all know the formula to find it right.
You know for the whole volume, it's just V = bh,
And for lateral area, it's L.A. = Ph.
(to Chorus)
2nd Verse:
I'm bringing area back! Go ahead with lateral
Area! Naw I'm just playin'. I know you
Want surface area! Then I have to tell you
First find lateral area then add the bottom and the top.
(to Pre-Chorus)
Notice that Mrs. Nguyen's version isn't just about area -- it mentions grades and A's, while mine just sticks to volume and surface area. Back at the old charter school, students got upset when I kept talking about grades and A's over and over again. But there's nothing wrong with singing about grades -- indeed, my old "Dren Song" and "No Drens" are ultimately about the consequences of bad math grades. The problem was when I talked about grades as a form of classroom management -- when students asked "Why do I have to follow this rule?" I'd reply "So you can get an A" instead of "Because I said so."
I could have played the area version of this song during Chapter 8 and saved my own volume version of this song for Chapter 10. If we started the semester with the area version, then the grades verse fits with the general start-of-the-semester mood, with a fresh start to all grades.
Here is the worksheet for today's Lesson 9-1:
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