-- in the field of cartography he created the World Energy Map of 1940, the Life magazine World Strategy Map of 1943, and the Dymaxion Air Ocean Map of 1954
-- the geodesic dome (triangulated space enclosing dome)
And we might as well cut off the list right at the invention we want to focus on. The geodesic dome is Bucky's most famous invention. According to his patent application, this dome is a framework for enclosing space, made out of plastic material, with a mass of only 4 kg per square meter rather than the typical 2500 kg per square meter.
I have much to say about the geodesic dome, as Geometry plays a key part in its construction. Let's see how Pappas describes the Geometry of this dome:
"Beginning with the icosahedron, he subdivided its faces into equilateral triangles. Now he circumscribed this with a sphere and projected its vertices onto the sphere."
Recall what an icosahedron is -- a polyhedron with 20 faces, each a equilateral triangle. Bucky could divide each face into four equilateral triangles by joining the midpoints of the edges.
Now we bring a sphere in, and suddenly Glen Van Brummelen's spherical geometry is relevant. In fact, Van Brummelen often projects points onto a sphere in his book -- we consider rays whose endpoints lie at the center of the sphere. Each ray that passes though a point (the preimage of the projection) also passes through the sphere at some other point (the image of the projection).
Pappas continues:
"The equilateral triangles do not remain congruent. Suppose he truncated the surface of this new solid."
OK, I can sort of see why the spherical triangles wouldn't be congruent -- some of the preimage triangles shared one vertex with a face of the icosahedron (with the other two being edge midpoints), while others share no vertex with a face (as all three are edge midpoints). Meanwhile, "truncation" usually means replacing a vertex with an extra face by cutting off a corner of the polyhedron. I admit that by this point it's difficult to visualize what Bucky is doing here.
"Now the geodesic structure's shape more closely approaches the shape of the sphere. A sphere has the least surface area for a given volume."
This should sound familiar. It is mentioned in Lesson 15-9 of the U of Chicago text as the space version of the Isoperimetric Theorem. Less surface area implies lower building costs!
Pappas ends the page by writing:
"In the geodesic dome, gravity's role is nearly irrelevant."
In fact, Pappas compares the equilibrium between the tension and air compression in a geodesic dome to the same equilibrium in a soap bubble. Bubbles are also mentioned in Lesson 15-9.
At this point, we can go on and on about the geodesic dome. But before I leave, let me say one thing about the name, "geodesic dome." A geodesic is the analog of a line -- the shortest distance between two points -- in non-Euclidean geometry. In spherical geometry, a geodesic is a great circle. We can see how spherical geometry and its geodesics appear in the construction of the dome.
The picture that appears on the previous page 250 is that of a geodesic dome. It's definitely worth linking to a photo of them:
http://www.vikingdome.com/geodesic-domes/
Geodesics in spherical geometry work differently from those in Euclidean geometry. We know so because the postulates in each geometry are different. And that takes us to the next section in the U of Chicago text.
Lesson 1-7 of the U of Chicago text is called "Postulates." (It appears as Lesson 1-5 in the modern edition of the text.) I've decided to restore the original inclusive definition of parallel (where a line can be parallel to itself). Therefore, this is what I wrote three years ago about this lesson:
Section 1-7 of the U of Chicago text introduces postulates. In the last section, the undefined terms -- the primitive notions -- point, line, and plane were introduced. Since these are undefined, we don't really know what they are unless we have postulates -- also known as axioms -- to describe them.
I reproduce the main postulate of this section, the Point-Line-Plane Postulate:
Point-Line-Plane Postulate:
(a) Unique line assumption: Through any two points, there is exactly one line.
(b) Dimension assumption: 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.
(c) Number line assumption: 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.
(d) Distance assumption: On a number line, there is a unique distance between two points. If the points have coordinates x and y, we define this distance to be |x - y|.
Let's look at each of these four assumptions -- since postulates really are assumptions, or statements that are obviously true -- in detail. The first assumption, that two points determine a line, goes all the way back to Euclid's First Postulate. In Hilbert's formulation of Euclidean geometry, this is Axioms I.1 and I.2.
The second assumption, about dimensions, are often different in other texts. Some texts, for example, emphasize that three noncollinear points determine a plane -- and give the example of a tripod standing on its three legs, the ends of which are the three points determining the plane of the floor. Hilbert's Axioms I.3 through I.8 roughly correspond to this assumption.
Assumptions (c) and (d) often appear in geometry textbooks as the "Ruler Postulate." The Ruler Postulate was first formulated by the American mathematician George David Birkhoff, about eighty years ago. The Ruler Postulate basically states that rulers work -- that is, we can measure line segments.
The section continues with its first theorem, the Line Intersection Theorem:
Line Intersection Theorem:
Two different lines intersect in at most one point.
And then we have the definition of parallel lines:
Definition
Two coplanar lines are parallel lines if and only if they have no points in common, or they are identical.
I've discussed these in some of my introductory posts in July. The last four words of this definition are controversial: "or they are identical." But as I pointed out, using this definition often simplifies later proofs -- in particular, it often allows one to replace an indirect proof with a direct proof. And technically speaking, the proof of the Line Intersection Theorem is actually an indirect proof -- but it's so simple that the text includes an informal argument here while delaying other indirect proofs until Chapter 13.
So far, in the introductory posts, we wanted to prove that two (coplanar) lines are parallel -- using our definition that they are either non-intersecting or identical. We were able to do this two ways -- we could prove that if they have at least one point in common, then they must have every point in common -- or we could prove that if at least one point on one line fails to lie on the second line, then every point on the first line fails to lie on the second line. These are hypotheses and conclusions that can easily fit into the Given and Prove sections of a proof.
But as teachers, our priority is to make geometry easy for the students to understand. So which will confuse the students less: a definition of parallel containing those four extra words "or they are identical," or many indirect proofs? We can't be sure until actually teaching this in a classroom.
So for now, I will stick to the U of Chicago definition of parallel, with those words "or they are identical," and delay indirect proofs as long as possible. But on the following images, I'll just leave a space for "parallel" in the vocabulary section and leave it up to individual teachers whether or not to include those four extra words in the definition. (Notice that in my exercises derived from the U of Chicago text, I preserved the true or false question "a line is parallel to itself." Of course, the answer will depend on which definition the teacher decides to use.)
The section concludes with some Postulates from Arithmetic and Algebra. As I mentioned yesterday, I want to avoid mentioning "algebra" -- the subject that causes many students to hate math -- yet these are important properties that show up in proofs (for example, the Reflexive Property of Equality). And so I'll just call them "Properties from Arithmetic" and just leave "algebra" out of it.
As an aside, let me point out that I was once a teenager studying geometry. Like those of many students at that age, my thoughts turned toward girlfriends and boyfriends -- except that I was the geometry nerd who wasn't a part of all of that. Like most nerds, I could make the following assumptions:
David Walker's First Postulate:
I am physically attracted to no other person.
David Walker's Second Postulate:
I am physically attractive to no other person.
These are statements that were obviously true -- in other words, they're postulates.
If you thought that there would be no "spilled milk" in this post, guess again. That's because today's the day my old school made its annual field trip to the LA County Fair. So I can't help but think about last year's trip to the fair.
This is what I wrote last year about the field trip:
1) Teachers make a lot of decisions throughout the day. Sometimes we make so many it feels overwhelming. When you think about today, what is a decision/teacher move you made that you are proud of? What is one you are worried wasn’t ideal?
I think that the best decision I made during the first 22 days of school was to include a music break as part of my daily lesson. As I wrote in my First Day of School (August 16th) and August monthly posts, I try to sing a math related song three times a week. This motivates the students to want to sing along -- and by learning the words, they are learning math without realizing it. One of my most popular songs is the one I mentioned in my August monthly post, Count on It. Music break is ten minutes out of an 80-minute block -- but as an incentive, I extend the break to 15 minutes if the students are singing along.
As for the worst decision I made -- well, the field trip to the LA County Fair was two days ago, and so it's still fresh on my mind. There were a number of poor decisions I made on that trip. I know that this isn't supposed to be a Day in the Life post, but here is a brief overview of my field trip:
10:00 -- We arrived at the fair. All groups -- including mine of half a dozen sixth graders, five boys, one girl -- walked through the Jurassic Planet exhibit. My students were hungry and wanted to eat their lunch, but I tell them that all groups would eat near Mojo's Wild and Crazy Island.
12:00 -- The students eventually spent all of their money on the Extreme Thrills tickets. Since all of the other rides were now open, we walked towards the Carnival section -- only to find out that all of the rides require purchasing tickets. The kids kept walking hoping to find a free ride, but we didn't.
2:00 -- As we get ready to board the bus to leave, I met my Support Staff aide, who had a small group of sixth graders of her own. She told me that her group had taken a tram to the farm area, rode a few extreme rides, and still had money left over for the carnival rides!
At that point, one of my group members proceeded to blame me for giving them such a miserable day at the fair -- even though I wasn't the one who wouldn't let them ride. (That would be the carnies who told them that they needed tickets to ride.) On the other hand, he had a point, as there actually were a few things that I could have done to improve my group's experience at the fair.
Until I arrived, I didn't even know that there was a tram. That was something I should have looked into ahead of time -- when I was doing research for my song "Meet Me in Pomona, Mona." Finally, I should have found out that all of the rides require tickets -- perhaps if I'd told my students this, they would have saved money for the Carnival section.
OK, let's return to 2017. First of all, I wrote that when I was writing the music break song, "Meet Me in Pomona, Mona" (as in Pomona, the city where the fair is located), I should have looked up info about the tram and how to get to the farm area.
And so let me fix that error today. Today I'll post a new, better version of the song. Last year, I was so obsessed with trying to match the lyrics of the song I was parodying ("Meet Me in St. Louis, Louis") that I kept writing about a "janitor" instead of the field trip itself. In this new version, I keep the first verse but change the second verse to reflect what the kids would see at the fair. Some of that info was included in my original refrain, and so I must change the refrain as well:
MEET ME IN POMONA, MONA
First Verse:
When Mona came up to the school, as she sat,
She hung up her coat and her hat.
She gazed around, but no teacher she found,
So she said "Where can the class be at?"
She remembered the noted, she flipped,
She saw it was a permission slip.
It said, "Hear, hear, it's too slow to learn here,
So let's go on this crazy field trip."
Refrain:
Meet me in Pomona, Mona,
Meet me at the fair.
Don't tell me that I'll learn science,
Any place but there.
The bus will leave for the fair soon,
We can stay there all afternoon.
Meet me in Pomona, Mona,
Meet me at the fair!
Second Verse:
At the fair Mona said, "Here I am!
So first I will get on this tram."
She went to the farm and she saw at the barn,
Cows, pigs, chickens, and even a ram.
And then Mona wanted to go,
To see the monkey named Mojo,.
Peacocks and giraffes and a whole lot of laughs,
And when to leave there she didn't know.
(Repeat Refrain)
While I'm at it, I might as well fix the Fraction Fever song as well, to add the extra verses that I was discussing last week:
FRACTION FEVER
First Verse:
Hey, if you've never
Played Fraction Fever
Here's how to get the action
You gotta get the right fraction!
Choose the wrong one and down you fall
(Down you fall!)
Through the hole and that's not all!
(That's not all!)
If you find the right one later
(Right one later!)
You'll go up in the elevator!
(Elevator!)
When you get to Floor 20
(Floor 20!)
You'll win plenty!
(Win plenty!)
Fraction! Fever!
Fraction! Fever!
Second Verse:
Hey, if you've never
Played Fraction Fever
Here's how to do addition
And also subtraction!
To go up in the elevator
(Elevator!)
Find a common denominator
(Denominator!)
Add or subtract the numerators
(Numerators!)
That will lead you to the elevators!
(Elevators!)
When you get the answer, always try
(Always try!)
To simplify!
(Simplify!)
Fraction! Fever!
Fraction! Fever!
Third Verse:
Hey, if you've never
Played Fraction Fever
Here's how to multiply
And also to divide!
Multiply the numerators
(Numerators)
Multiply the denominators!
(Denominators)
Don't forget when you divide
(You divide)
Flip the second one down upside!
(Down upside)
When you get the answer, always try
(Always try!)
To simplify!
(Simplify!)
Fraction! Fever!
Fraction! Fever!
Moreover, now that I found out what day the field trip, this makes a difference in thinking about how I would have organized the class if I had still been at my old school. Recall that the first two weeks school are for Opening Week Activities and Benchmarks. (This year, Opening Week was the week of August 21st-25th, while Benchmarks were August 28th-31st.) The means that the first week of actual instruction would be this week, September 5th-8th.
Meanwhile, in past posts I pointed out that my students struggled to learn many standards during the school year. I linked to the blogs of other teachers, who suggested that to overcome this, have a lag of one week (or even two weeks) between the introduction of a standard and its assessment. This means that in the Illinois State suggested weekly plan:
Monday: Coding
Tuesday: STEM Project
Wednesday: Traditional Lesson
Thursday: Learning Centers
Friday: Weekly Assessment
the standard to be assessed each Friday is the one introduced in the traditional lesson nine days earlier, not two days earlier.
In earlier posts I was noncommittal regarding whether I wanted to do this. But now that I know the exact date of the field trip, this makes the decision easier for me. The fact is that the first Friday other than Opening Week or Benchmark Week just happens to be field trip day. Since I couldn't give the weekly assessment on field trip day, it means that I'd either have to give the assessment on Thursday (with only one day between traditional lesson and assessment), or not at all. With a one-week lag, it means that I don't have to give an assessment during field trip week. The first week of instruction is September 5th-7th, and the first weekly assessment would be on September 15th.
Here's a summary of what the first three weeks of school would look like, from August 21st all the way to today:
First Week of School:
Monday, August 21st: Eclipse Activities (mentioned in that day's blog post)
Tuesday, August 22nd: Opening Week Activity (Personality Coordinates & Bridges)
Wednesday, August 23rd: Opening Week Activity (Classroom Rules)
Thursday, August 24th: Opening Week Activity (Buildings)
Friday, August 25th: Opening Week Activity (Sequences)
In 2016, the first day of school was on a Tuesday. This year, I transfer the entire first week of school to Tuesday-Friday of this year. This leaves Monday, the first day of school, open. I mentioned before that this was eclipse day, and on that day I listed the activities for that day.
Notice that even during Opening Week, it's possible to enforce the Illinois State weekly pattern. So on Tuesday, the students work in groups as if it were a STEM Project. It would still be a reverse of the first day of school last year, starting with Personality Coordinates -- obviously a group project. On Wednesday, the students write down the rules required for effective traditional lessons. On Thursday, the kids use manipulative blocks to form the buildings -- and manipulative blocks are one part of Learning Centers. On Friday, students complete the sequences as if it were a quiz -- but of course I don't grade the quiz.
Notice that we don't discuss the rules until the third day of school. Of course, I must still enforce the rules starting the first day of school -- especially rules relating to sitting down and being quiet. I enforce these rules exactly as I implied in many recent posts.
During music break, I sing both "The Dren Song" and "Earth, Moon, and Sun" (eclipse version) on Monday and Tuesday, and "Count on It" from Square One TV on Thursday and Friday.
Second Week of School:
Monday, August 28th: Benchmark Tests
Tuesday, August 29th: Benchmark Tests
Wednesday: August 30th: Benchmark Tests
Thursday: August 31st: Fraction Fever
Friday, September 1st: Admissions Day Holiday???
This week is a repeat of the second week of school in 2016. But again we follow the Illinois State weekly plan -- Fraction Fever should be completed using DIDAX manipulative blocks as required during Learning Centers.
I'm actually not sure whether my charter school followed the LAUSD calendar and observed the Admissions Day holiday, since the charter is no longer co-located with LAUSD. If there was school on the 1st, we follow the weekly plan again and give a Dren Quiz. Since nothing this week is graded, I might as well let it be a 1's Dren Quiz. Alternatively, I could count it as the first graded assignment of the year, in which case it should be a 10's Dren Quiz -- just as I began with 10's last year.
During music break, I sing "The Benchmark Test Song" on Monday and Tuesday, and "Fraction Fever" on Thursday (and Friday, if it's a school day).
Third Week of School:
Monday, September 4th: Labor Day Holiday
Tuesday, September 5th: First STEM Project
Wednesday, September 6th: First Traditional Lesson
Thursday, September 7th: First Learning Centers
Friday, September 8th: LA County Fair Field Trip
This week is not a repeat of the third week of school in 2016. Instead, I strictly enforce the pattern from Illinois State. Also, as per Illinois State, we follow the naive Common Core order of standards, so the standards to be taught are 6.RP1, 7.RP1, and 8.NS1.
If there was no school on September 1st, then I could squeeze in the Dren Quiz on the 7th. Since I definitely want all activities this week to be graded, it would be the 10's Dren Quiz.
During music break, I sing "Meet Me in Pomona, Mona" on Tuesday and Thursday, since this is the week leading up to the fair. In particular, I do not sing "Ratio Song" or "Another Ratio Song," even though I sang them last year during the 6.RP1 and 7.RP1 lessons on ratios. This is because the Pomona song takes priority. I can always save "Ratio Song" for next week -- it's not as if 6.RP2 and 7.RP2 aren't on ratios as well.
Also, notice that I do not sing "Need for Speed" (the mousetrap car song) from last year, even though I sang it last year during the third week of school. "Need for Speed" was the first project in the STEM book, but we're now following the pacing guide, which has the projects in a different order (as I mentioned last year). In fact, mousetrap cars only appear in the sixth grade pacing guide -- and it's not the first project of the year (it might have been the third or fourth project).
The quiz on 6.RP1, 7.RP1, and 8.NS1 will be next week, on September 15th. The first Illinois State online assignment will be on the 11th -- and it will be on the standards to the assessed that week (not the standards to be taught that week). So the "lag" applies to both homework and assessment.
Setting up the Learning Centers this week would be tricky. Most likely, I'll try to find a few usual DIDAX assignments to return to regularly -- usually algebra tiles or fraction blocks -- and I can even sing "Fraction Fever" on days that we use the fraction blocks. Notice that for the fourth week of school, there is a die cut assignment that's perfect for eighth grade (the square root project) as it lines up with 8.NS2. Otherwise, I'd just have to wing it to make sure that the students are receiving sufficiently many die cut assignments.
OK, that's enough crying over spilled milk for now. I know that I kept writing about my old class everyday this week, since Lessons 1-4 and 1-5 were my Opening Week activities from last year. So there is no reason for me to write about my old class next week.
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