Lesson 1-7: Postulates (Day 17)

This is what Theoni Pappas writes on page 251 of her Magic of Mathematics, right in the middle of a list of Buckminster Fuller's creations:

-- 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.

Lesson 1-6: The Need for Undefined Terms (Day 16)

This is what Theoni Pappas writes on page 250 of her Magic of Mathematics:

"Richard Buckminster Fuller was an inventor, a designer, an engineer, an author. He was an architect of ideas, a man whose visions were often ahead of their time."

This is the first page of the first section, "Buckminster Fuller, Geodesic Domes & Buckyballs." Here Pappas gives us an excellent description of Buckminster Fuller. Affectionately known as "Bucky," he was one of the most famous American architects.

Pappas includes a drawing of Bucky's most famous invention on this page, but she doesn't describe it until page 251, so I'll have to wait until tomorrow to blog about it.

Instead, I write about one of his other inventions from page 250, the Dymaxion House (since we are after all in the architecture chapter). Pappas tells us that Dymaxion Houses "were designed to be completely transportable living units." Of course, this description is just begging for a corresponding photo, so here's a link to pictures of both it and the Wichita House:

https://www.inverse.com/article/13715-forget-geodesic-domes-buckminster-fuller-s-dymaxion-house-was-his-masterpiece

By the way, the word "Dymaxion" can be defined -- it means "dynamic maximum tension." But now let's move on to some words which can't be defined.

Lesson 1-6 of the U of Chicago text is called "The Need for Undefined Terms." (It appears as Lesson 1-4 in the modern edition of the text.) Earlier this week we revisited Lessons 1-4 and 1-5 as they were my opening week activities last year. But those lessons are exceptions -- most of my Geometry lessons have nothing to do with middle school. For most lessons, we'll have to go back two years to find my previous notes on the lesson:

This is what I wrote two years ago about today's lesson. I have updated the post to reflect the number of results of a certain Google search.

Lesson 1-6 of the U of Chicago text is where the study of geometry formally begins. This section states that three important words in geometry -- point, line, and plane -- are undefined. This may seem strange, for mathematics is all about definitions, yet these three important concepts are undefined.

In college-level math, one learns that these undefined terms are called primitives, or primitive notions. Just over a hundred years ago, the German mathematician David Hilbert declared that there are in fact six primitive notions in geometry: point, line, plane, betweenness, lies on, and congruence. But most textbooks list only the first three as undefined terms. This is because texts actually define the last three using concepts from other branches of mathematics. "Lies on" or "containment" -- that is, what it means for, say, a line to contain a point -- is defined using set theory (which is why the very first sentence of this section states that a set is a collection of objects called elements). "Betweenness" of points -- that is, what it means for a point to be between two other points -- is defined later in this chapter in terms of betweenness for real numbers (their coordinates of course). And the definition of "congruence" is the cornerstone of Common Core Geometry -- we use reflections, rotations, and translations to define "congruence." So we're left with only three primitive notions -- points, lines, and planes.

Lesson 1-6 is a fairly light lesson. So point, line, and plane are undefined -- big deal! Of course, we can do things with points, lines, and planes, but that's not until 1-7. So instead, I use this as an opportunity to remind the students the reasons for taking a geometry course.

The students in a geometry course are around the age where thoughts such as "I hate math" become more and more common. This is the age where they wonder whether they'll ever have any use for the math that they're learning. They begin to wonder whether they'll ever use any math beyond what they learned in elementary school and wish that math classes were no longer required beyond elementary school, for can't they live very successful lives not knowing anything higher than fifth grade math?

As of today, a Google search for "I hate math" returns 602,000 results. And we can easily predict the most common reason for hating math -- of course it's because it's hard. We don't hate things that are easy -- we hate things that are hard. And the class that turns so many off from math is algebra. Indeed, if you choose some school and tell me only its standardized test scores in ELA and math, I can very reliably tell you whether it's an elementary or a secondary school. If the math score is higher, it's probably an elementary school -- if the ELA is higher, it's likely a secondary school. And so now we, as geometry teachers, have the students for the math course right after the one that caused them to hate math in the first place.

The number of search results for "I hate math" has increased almost 50% over two years ago. But I notice that much of the increase is generated by a number of tutors that have taken the name "I hate math" -- that is, they tutor for students who hate math, as opposed to hating math themselves.

But we do see some results that are obviously from genuine math haters. One girl has posted a YouTube video of about 6 1/2 minutes on why she hates math. The girl in the video is an eighth grader who is struggling with the Quadratic Formula in her Algebra I class. She says that she hates math because without the class, she'd have a 4.0 GPA, but with math she struggles just to get a D+. I don't link to the video, but anyone can find it via a Google/YouTube search.

There are also images that say "I'm still waiting for the day that I will actually use xy + (420) > x - 5y[2 + 9 = 7] in real life." Well, of course we will probably never use a non-linear (because of the xy term) inequality such as that one in the real world. This image would have been much funnier if, instead of that inequality, the image contained the type of equation that traditionalists lament don't appear in Common Core texts, such as "I'm still waiting for the day that I will actually use (a quadratic-in-form equation with radicals) in real life."

So why do we require students to take so much of a class they hate in order to graduate high school? As it turns out, we can answer this question from one of the sections that we've skipped, Lesson 1-1:

"A point is a dot."

And this section gives many examples of dots -- the pixels on a computer screen. The shapes that appear on our screens consists of dots, which can be modeled in geometry by points. We look at images on our TV screens all the time. And one of the most geometry-intensive computer programs that we have are video games -- we must create images consisting of dots that move rapidly.

The point of all this is that we can surely have math without entertainment, but we can't have entertainment -- at least not most modern forms of entertainment -- without math. We can only imagine how much technology would disappear if math were to disappear.

Elementary school math -- at least early elementary arithmetic (before the dreaded fractions) -- is easy. And college majors majoring in STEM know the importance of learning math. The problem is those in-between years in middle and high school. If math were merely an elective in secondary school, many students would avoid it and choose easier classes. Then there wouldn't be enough STEM majors in college because they wouldn't have had the necessary algebra background. The only way to bridge the gap between "math is easy" (early elementary) and "math is important" (college STEM majors) is to require the subject during the intervening middle and high school years. Otherwise we'd have no modern technology or entertainment.

When I give notes in class, I prefer the use of guided notes. This is not just because I think the students always need the extra guidance, but that I, the teacher, need the guidance. In the middle of a lesson, I often forget what to teach, or forget how to explain it, unless I have guided notes in front of me.

And so today's images consist of guided notes. I begin with Lesson 1-6 and its definitions. Here I emphasize the fact that point, line, and plane are undefined by leaving spaces for the students to write in their definitions -- which they are to leave blank (or just write "undefined")! Notice that Lesson 1-6 distinguishes between plane geometry and solid geometry -- a crucial distinction in Common Core Geometry because the reflections, rotations, etc., that we discuss are transformations of the plane.

Then I move on to Lesson 1-1. This is based on an online discussion I had a few years ago on why students should learn math. I also include it as guided notes so that the students are listening when the teacher gives the reasons that they are taking this course. (The answers to the blanks beginning with the conversation are MBA, polynomial, investing, data, supermarket, and -- the object Americans use that has more computing power than the A-bomb -- cell phone!)

In the year since I first posted this lesson, I've been thinking about how to rewrite the lesson so that students are more responsive to it. In particular, I was thinking about last week's bridge puzzle, on which I wrote, "Back then, people spent their Sundays taking walks over bridges." Think about that statement for a moment -- entertainment back then was limited to Sundays. Back then, six days a week were workdays, on which no one expected to be entertained. Even on Sundays, the morning were devoted to church, so only the afternoons were amusing. And when we finally get to Sunday afternoon, all people did was cross bridges -- something that we wouldn't find entertaining today.

What has changed since the 18th century? The answer is technology -- that is, mathematics. Just as I mentioned in the worksheet, one especially widespread form of entertainment is the cell phone. We don't have to wait until Sunday afternoon for entertainment -- with our modern phones, we can be entertained at almost any time. Games and videos can be played anywhere, and if our friends live across the bridge, we don't need to cross it, since we can call or text them. All of this technology is available now because of mathematics.

Yet the greatest paradox is that, while math makes all of this technology possible, students use this technology to justify avoiding the study of mathematics. Traditionalists don't like the fact that students don't study as much now as they did in the past. Nowadays, the idea that one should study for two hours at once -- that is, go two hours without cell phones, TV, or other entertainment -- is unthinkable for many students, yet before modern technology, the idea of being entertained as often as once every two hours was equally unthinkable. The girl in the YouTube video says that she must study two hours per day just to pass her Algebra I class -- and that she's lucky if she can finish her weekend homework by Sunday afternoon. If the hypothetical "Math God" that she mentions in the video could make math disappear, she'd have a 4.0 GPA, and much less time needed to study -- but then the technology that makes YouTube possible would no longer exist, and she'd be spending Sundays crossing bridges to entertain herself.

We don't need to go back to Euler's day, 300 years ago, to find generations of students who were willing to work hard and forego entertainment. But some traditionalists go back to 100-year-old texts because they feel that newer texts have too many pictures. Technology progressed so much that photography, even in the mid-20th century, was inexpensive (going back to "a point is dot,") but that photo technology made texts even as early as then too entertaining, and therefore, not educational enough for the traditionalists.

The phrase Millennial Generation refers of course to the millennium. Strictly speaking a millennial is one who was born in the old millennium and graduated from high school in the new millennium. By this definition, I am not a millennial, since I was born in December 1980 and graduated high school in June 1999 -- still the old millennium. But some authors, such as Mark Bauerlein, consider the Dumbest Generation to be anyone under 30 at the time of its publication (2008). By this definition, I am a member of the "dumbest generation."

Naturally, most traditionalists and members of older generations who criticize millennials blame the problems of our generation on technology. This is why, when I teach this lesson, I want to point out that using technology to justify being a "dren" who can't count change makes us -- including myself as a member of the generation -- look bad. Of course, in a few years, I can't credibly claim to be in the same generation as my students -- some incoming students starting high school this year are already born in the new millennium (and so are no longer "millennials"). The important thing is that all of us, my age and younger, need to avoid being the "dren" who can't solve simple math problems and instead work on becoming the hero whose knowledge of math saves the day. This is what I want my students -- including those like the girl from the video, if she ever scrapes by Algebra I and is placed in a Geometry class like mine -- to realize.

OK, let's return to 2017. Just as many students may hate math, I bet many of my readers hate it when I "cry over spilled milk" and write about my class from last year. But I'd doing so yet again in today's post, because that comment about Mark Bauerlein's book reminds me of something from last year.

Almost exactly one year ago today, I told my eighth graders about The Dumbest Generation, and I mentioned it here on the blog. But what I never blogged was what happened when I tried to mention the book to my seventh graders the following week.

Once again, this was just before the seventh graders took their first Dren Quiz. I wanted to explain to the students why they had to take a quiz on multiplying by ten, and so I tried to tell them about Bauerlein's book.

But the seventh graders were talking loudly. And this was right at the point when I realized that the students were quiet for my support staff member, but not for me. I knew that my experience as a teacher would be miserable unless I could get the students to be quiet. So I knew I had to do something at that point.

So I told the seventh graders to line up and stand outside until they were quiet. But they never did become quiet, and I knew that if I didn't proceed with the lesson, the students might even consider standing outside as a reward. Thus I continued with the lesson -- which, as you recall, was about Bauerlein's book and the Dren Quiz.

But because the class was still loud, not everyone heard all of what I was saying. One girl had missed everything I spoke except the word "dumbest." And when her mother came to pick her up, she told her what I'd said, and the mom angrily ran up to me and asked, "How dare you insult a 12-year-old girl like that!"

Eventually the mom and I met in the dean's office, where I calmly explained that I was referring to our generation, not to her daughter. The mom apologized, and she scolded her daughter for not telling her the full story. But I don't believe the girl ever forgave me for she thought I said. She was one of the better behaved students and I think she could have enjoyed my class, but instead she always spoke coldly to me and even refused to give me a high-five when I greeted the class in the mornings.

There are several issues here. First of all, the whole incident occurred because -- as was so often true in my classes -- the students were talking loudly and couldn't hear me. I knew that I had to do something about the fact that they obeyed my support aide and not me -- but clearly, making them stand outside changed nothing.

I've said before that I should have made it clear from the first day of school what my expectations were -- namely that they remain silent for most of the period. In particular, I should have ignored all complaints that making them be quiet is "unfair."

But suppose I'd made it to that point (almost a month into the year) and it became obvious that the students only listened to others and not to me. Then this is what I should have done at that point -- first, when my support aide is in the room, I say, "I like how quiet the class is now. This is how quiet I expect the class to be all the time, including when the aide isn't present."

Then I get ready to send my aide out of the room for any reason -- it could be as simple as making copies of anything (such as the Warm-Up sheets). At this point, I anticipate that students will start talking the instant she leaves the room. I make sure that all students have pencil and paper before she leaves, so that they can't claim "I was trying to borrow a pencil!" as a reason to talk.

At this point, my aide leaves the room. I expect that the first student will talk possibly as soon as she opens the door, or at the least before she closes it as she leaves. When this is happening, I scan the room like a hawk, looking for the first student to speak. Then I call out that student. If the student claims "That's unfair!" or "I wasn't talking!" then I give that student the teacher look, and proceed as I mentioned in previous classroom management posts. By following this, the students get the message that I mean it when I tell them not to talk.

Some may question the wisdom of mentioning Bauerlein's book in justifying the Dren Quiz. On one hand, if the class is quiet, then I can tell them about Bauerlein without fear of anyone misconstruing it as a personal insult. On the other hand, my counterpart at our sister school had the students had the students fill in times tables almost everyday -- and I doubt she mentioned Bauerlein's book. So I could have just assigned the Dren Quiz and gave the reason as "Because I said so!"

Finally, I know that several students regularly told me that I was "unfair" -- and that includes the day that I made the students stand outside. But notice that the girl who thought I'd insulted her (that is, the girl who genuinely thought that I was being unfair) said nothing to me at all. Indeed, this is how students whom I genuinely treated unfair would act -- by being cold with me, not by calling out "Unfair!" over and over. Therefore, I should assume that anyone calling me "Unfair!" is just trying to get out of following the rules cheaply. Such students deserve only a teacher look -- not a full-blown argument over why my rules aren't "unfair."

I know that I keep using other blogs, such as Fawn Nguyen and Julie Reulbach, as excuses to keep going back to classroom management. Now it's Sarah Carter's turn:

http://mathequalslove.blogspot.com/2017/09/seating-preference-form.html

Carter writes about a "Seating Preference Form" where students can choose their own seats. I've written about some of the issues I had with seats in both the seventh and eighth grade classes. I wonder whether using this would have solved some of those issues.

For example, recall that there were four students per group, while the clique of five girls in eighth grade wanted to sit together. Maybe by knowing this, they might have decided for themselves which girl would have to sit elsewhere, rather than have me choose.

Meanwhile, in seventh grade, the four troublemaker boys needed to be separated. Most likely they would have requested to sit together -- but then I could tell them that in trying to seat everyone else according to their own preferences, the only remaining seats for them were separated!

Lesson 1-5: Drawing in Perspective (Day 15)

This is what Theoni Pappas writes on page 249 of her Magic of Mathematics:

"In modern times we have witnessed the formation of the hyperbolic paraboloid (St. Mary's Cathedral in San Francisco), the geodesic structures of Buckminster Fuller, the module designs of Paolo Soleri, the parabolic airplane hanger, solid synthetic structures mimicking the tents of the nomads, catenary curve cables supporting the Olympic Sports Hall in Tokyo, and even an octagonal home with an elliptical dome ceiling."

This is the final page of introduction to the architecture chapter. Now Pappas lists several examples of geometric buildings from the past century or so. Many of these we have to see to appreciate, so let's try a Google image search.

St. Mary's Cathedral:
http://www.aviewoncities.com/sf/stmaryscathedral.htm

Paolo Soleri's planned town of Arcosanti, Arizona:
http://www.archdaily.com/tag/paolo-soleri/

Airplane Hangars (some, but not al,l are parabolic):
https://www.pinterest.com/fabricbuildings/custom-aircraft-hangars/

Yoyogi National Stadium in Tokyo (but a new stadium should be built in time for the 2020 Games):
http://www.alamy.com/stock-photo-yoyogi-national-stadium-in-tokyo-designed-by-architect-kenzo-tange-53753182.html

Octagonal Homes (not necessarily with elliptical ceilings):
https://www.pinterest.com/cindyg7811/octagon-houses/

Meanwhile, Pappas gives an example of "structures mimicking tents" in a photo on this page. Here is the caption:

"This tent-like structure illustrates the use of new materials and methods of constructions. Fashion Island, Foster City, California."

According to the following link, this mall no longer exists -- and apparently it was in decline even before Pappas wrote her book in 1994:

http://bigmallrat.blogspot.com/2006/11/mall-memories-san-mateo-fashion-island.html

Notice that so many examples in Pappas come from my home state of California. Well, Pappas did earn her degrees from Berkeley and Stanford, so at least Northern California is familiar to her. Oh, and that last example will have to wait. Buckminster Fuller actually merits his own section later in this chapter.

She concludes:

"In the final analysis, an architect is free to imagine any design so long as the mathematics and materials exist to support the structure."

Lesson 1-5 of the U of Chicago text is called "Drawing in Perspective." In the modern edition of the text, perspective doesn't appear until Lesson 9-4. This is more logical, as Chapter 9 in both editions is the chapter on three-dimensional figures.

Perspective appeared as Lesson 0.8 in Michael Serra's Discovering Geometry, which we already covered nearly two weeks ago on Day 8. This time, I'll reblog the old Lesson 1-5 post from last year.

Indeed, Lesson 1-5 is the other worksheet I taught in middle school last year as part of my opening week activities. This is what I wrote about it:

Speaking of class, today I gave the last of the opening week activities previously posted on the blog -- Designing Buildings. This is what I wrote earlier about this activity:

And as it turns out, Nguyen covered something similar to this in her class as well:

http://fawnnguyen.com/designing-buildings/

Nguyen's lesson takes a different approach to drawing three-dimensional figures. For one, the focus on this lesson is on buildings. Her lesson begins by having some buildings already drawn and the students counting the "rooms" and "windows." (As it turns out, one "room" is one cubic unit of volume, and one "window" is one square unit of lateral area.)

I like the way that Nguyen's lesson begins. Unlike the bridge problem, where I wanted to avoid beginning the school year with a problem that's impossible to solve, here we begin with a very solvable problem. The only issue I have is with the second question, because it requires materials. I work from the assumption that most classrooms don't have the blocks and isometric dot paper that Nguyen's classroom has.

(As an aside, notice that cubes drawn on isometric dot paper are definitely not in perspective. This is because, while edges perpendicular on the cube intersect at 120 degrees on the iso dot paper, edges parallel on the cube remain parallel on the paper. Therefore there are no vanishing points.)

Then again, my worksheet is very similar to Nguyen's. On the front side, I gave the same example as she did and the three buildings for the students also come from the Ventura County teacher. I used two of her easier buildings -- A and B -- and the more challenging Building F.

The back side of my worksheet differs slightly from Nguyen's, though. Her worksheet specified the number of rooms and windows and asked the students to draw the buildings. Mine, on the other hand, simply has the students draw four different buildings with eight rooms and then asks them to count the number of windows in each one.

Now that I'm giving this activity in an actual classroom, I don't have any interlocking cubes (which I can only assume means "Lego bricks"), but I did find some small manipulative cubes. There weren't enough for me to give every group eight cubes (as specified in the assignment) -- instead I gave five to each group of sixth graders and seven to each group of seventh graders. (Half the seventh graders were absent because they hadn't satisfied California's 7th grade vaccination requirement.) The eighth grade groups did receive the full set of eight cubes. I believe that having actual blocks certainly helped the students visualize the three-dimensional buildings.

By the way, here are the rules the middle school classes came up with as part of the Rules Posters. At last I'm done discussing the rules here on the blog:

1. Raise your hand

2. Be silent and listen when it's someone else's turn to speak

3. Stay in your seat

4. Keep your hands to yourself

5. Keep the desks free of drawing

6. Treat the books, papers, and any other resources like you would treat your own items

7. Keep your voice at a conversational level

8. Allow the speaker to finish before you raise your hand

9. Speak in a respectful manner

10. Stay on task, work hard, and do your best!

Returning to 2017, we notice that by this point, the lesson is no longer recognizable as an activity on perspective, since the buildings are not drawn in perspective. It was influenced by a Fawn Nguyen post and then modified yet again in my middle school classroom. Well, at least the lesson ties to architecture and hence to today's Pappas page. And between Lessons 0.8 and 1-5, the students should learn something about perspective in the end.

Of course, I mentioned the rules in that old post, which means that this is going to be yet another post on classroom management (and spilled milk). In fact, we can see why I had so many problems -- yes, these ten rules sound reasonable. But look at what happened in seventh grade early that day:

8:25 -- My first class, a seventh grade class, arrives. Today there is a confrontation with one of the seventh graders. She refuses to do her work, then argues with my student support aide, who asks her to leave the room. I am the teacher, so I should have tried to intervene sooner, though it still might not have made much difference. It is only Day 3, but I already know there's one girl I'll need to watch out for this year.

Let's see how many rules that girl broke that day. She broke Rule #1 (not raising her hand), #2 (not being silent), #9 (being disrespectful to my aide), and #10 (not staying on task). Yet, as we plainly see, I didn't intervene or attempt to punish her.

I've mentioned before where I went wrong -- the participation points system. The girl had earned a few cheap points (for turning in emergency papers), and my system stated that punishments begin after students lose all of their points. Since she hadn't lost all her points, I gave no punishment.

Obviously, I should have had a different system. Students can only gain participation points rather than lose them -- instead, punishments occur when students don't do what I tell them. In this case, I should have backed up my aide by threatening either a detention or a phone call unless the girl agreed to follow my aide out of the room. Instead, I ended up demonstrating to this girl -- as well as the other students in the class -- that the four rules she broke don't mean squat.

Lesson 1-4: Points in Networks (Day 14)

This is what Theoni Pappas writes on page 248 of her Magic of Mathematics:

"The stone structures of the Renaissance showed a refinement of symmetry that relied on light and dark and solids and voids. With the discovery of new building materials, new mathematical ideas were adapted and used to maximize the potential of these materials."

We are still in the admittedly lengthy introduction to the math and architecture chapter. Pappas is in the middle of listing some historical examples of the use of math in architecture.

On this page, Pappas provides a photo of an unusually shaped house. This is to demonstrate how important our subject, Geometry, is to architecture. Here's the caption:

"Each of the floorplans of the three levels of this house is designed from two overlapping equilateral triangles. This triangle motif is carried out throughout the interior supports and windows."

And in case you're wondering how a house shaped like two equilateral triangles can even stand, it turns out that the house was carved into the side of a hill.

Lesson 1-4 of the U of Chicago text is called "Points in Networks." (It is combined with the old Lesson 1-1 to form the new Lesson 1-3 in the modern edition of the text.)

Unlike Lessons 1-1 to 1-3, which I've never covered before this year, we are definitely familiar with Lesson 1-4 and the Bridges of Konigsberg from previous years. It is, therefore, the first lesson this year for which I can just reblog the worksheets and comments from last year.

But there is a twist here. I actually taught the Lesson 1-4 activity last year as it was my first day of school activity for my middle school classes. And so I'm actually going to reblog the experience of my first day as a teacher last year. Yes, this means that this will be yet another "crying over spilled milk" post that has dominated my blog these past few weeks. In this post, I'll only repeat what I wrote about my eighth grade class, as well as the commentary I wrote at the end.

11:25 -- My eighth grade class arrives. This is my smallest class, with only 12 students -- but there are only eight students present at the start of class. I begin the class the same way I start all my classes, with a Warm-Up question:

What is 2 * 2 * 2 * 2? (That is, 2 times 2 times 2 times 2.)

Most students answer correctly, although a few tried to add. A student or two is upset that the very first thing we do on the first day of school is multiply. I point out that the answer is 16 -- and that today is the 16th. I always go around to stamp correct papers -- many teachers point out that students enjoy getting stamps, and my students are no exception.

11:35 -- My student support aide arrives -- the English teacher and I are each assigned one. Actually, she arrives with the four missing students, all girls.

We move on to an Opening Activity -- the Konigsberg Bridge Problem. I've written about this problem previously on the blog and even suggested it as a first day of school activity -- well, now I'm finally giving the activity on an actual first day of school. This is a little of what I said about this problem here on the blog:

The Königsberg Bridge Problem is a famous math problem from nearly 300 years ago. Fawn Nguyen, a well-known math blogger and fellow Southern Californian -- she lives in Ventura County -- used this as an activity in her geometry class:

http://fawnnguyen.com/famous-bridge-problem/

As we all know, the Königsberg Bridge Problem is impossible to solve -- it has no solution. But I don't want to start the class with a problem that the students can't solve -- they're already frustrated enough with problems that do have solutions when they just can't find them.

The whole point of this lesson is to point out that students should look for patterns, and that sometimes it's just as important to know why something is impossible as it is to know why something is possible.

Let me complete this with a note on pronunciation. The U of Chicago text points out that the name Euler ends up sounding like "Oiler." But how does one go about pronouncing the name Königsberg? I once read that the o-umlaut ends up sounding like "uh," almost like "ur." A Google search reveals a ten-second video in which this name is pronounced:

By the way. some students believe that they have a solution to the Konigsberg problem, but actually they are crossing one of the bridges twice (they start on island D, cross the bridge towards C, but then head back to D). I start to explain about Euler and why the problem is impossible -- and as I do so, the student who earlier complained about 2 * 2 * 2 * 2 figures out that the impossibility has to do with there being an odd number of bridges from each land! I'm impressed!

12:05 -- Because I know how tough the 80-minute block schedule can be on middle school students, I provide a music break. I get out my guitar and I play the following inspirational song:

The Dren Song -- by Mr. Walker

[OK, I'll skip the song since I already reblogged it in a recent post.]

Along the way, I explain that a "dren" is a reverse-nerd -- a nerd is someone who's good at math, and a "dren" is someone who doesn't understand the basics of arithmetic. As it turns out, the student who complained about 2 * 2 * 2 * 2 enjoys this song and looks forward to my next song.

I show my students the September 2015 Boys' Life article about the mathematicians and scientists who work for NASA and the possible future of people traveling to and living on the moon. But as it turns out, eight of the 12 students in my class are girls, so I don't expect Boys' Life to motivate them.

Instead, I tell them about the movie trailer that was released just yesterday -- Hidden Figures, about the scientist Katherine Johnson who worked for NASA and the Apollo projects in the 1960's. For those of you who have read my blog before, it goes without saying that I plan on watching this movie, and I highly recommend that my students watch it in January as well.

12:15 -- I proceed with my next Opening Day activity -- Personality Coordinates. This activity comes from the King of the MTBoS, Dan Meyer:

http://blog.mrmeyer.com/2013/personality-coordinates-icebreaker/

Each person in a group picks a dot and writes her name next to it.

Now the group’s job is to label the axes. Physical attributes don’t require all that much thought and don’t reveal all that much, so don’t allow them.

That’s it. It requires a surprising amount of creativity and conversation. Happy first day of school, teachers.

12:30 -- My support aide leaves, and this is a good time to end the period with an Exit Pass:

If you don't know the answer, ..

The answer is "at least know where to find it," which is posted in a corner of the room. (I mentioned this in an earlier blog post.) Some wrong answers are "ask the teacher" and "you're a dren."

12:45 -- My eighth grade class goes out to lunch.

If there's anything I could change about the way I ran the class today, it would be to teach the entire class in reverse order. That way, the Exit Pass becomes a Warm-Up, a scavenger hunt to find the rest of the quote, Personality Coordinates occur earlier in the class, and the 2 * 2 * 2 * 2 question doesn't turn off students right at the start of the period.

I would also rewrite the Konigsberg worksheet. I'd already changed the worksheet to add more bridge problems, including some trivial ones. But now I'd number those trivial problems #1 and #2 (rather than #3 and #4, as they were numbered today).

Another problem I have has to do with explaining my directions clearly. I was hoping to create a seating chart directly from the Personality Coordinates worksheet (since the students are already seated in groups of four), but I couldn't because some groups randomly labeled the dots rather than place the student sitting northwest in the upper-left corner of the page. Also, some students wrote the Exit Pass on a separate sheet of paper rather than the back of the Warm-Up.

I remember explaining my directions to the students -- but I could be remembering my explanations to the 6th and 7th grade classes, not the 8th grade class. Anyway, I know that I don't always explain instructions clearly to my students from my days as a sub, so I must give the students the benefit of the doubt whenever I see them misinterpreting instructions.

OK, let's return to 2017. There are a few things that I want to say about my reflection -- from the perspective of it being a year later and I not having returned to that classroom.

First of all, last year I wrote that maybe I should have reversed the order of the first day. Then the opening Warm-Up question, 2 * 2 * 2 * 2, becomes an Exit Pass.

The answer to that question is 16, and the first day of school last year was August 16th. This was an idea that came from Pappas -- making the answer to the Warm-Up question be the date. But, as we already know, this fell apart because the Illinois State Daily Assessment took over the Warm-Up.

Thus it would have been better for me to establish the Pappas tradition by making the date be the answer to the Exit Pass, not the Warm-Up. In other words, I should have reversed my Exit Pass and Warm-Up not only that first day, but everyday. This also serves another purpose -- after Exit Passes, I often never gave the correct answer because some students are already walking out the door while others are still trying to correct their mistakes. By making the date be the answer to the Exit Pass, the students already know the answer, so never finding out the answer isn't an issue. Again, this then frees Warm-Ups for the Illinois State Daily Assessment, which we would then work out on the board.

Last year, I wrote that I often had trouble explaining my instructions clearly. But in hindsight, I think this was part of my overarching management issue -- the students never stopped talking. I now believe that I often spoke quickly because I knew I needed to talk before the students did. And this applied not just to giving clear directions -- my math explanations suffered as well. I know that I could have taught math much better if I could count on the students being quiet during the lesson.

Obviously, at some point on the first day of school, I need to tell the students of the importance of being quiet. At the time, I actually didn't mind the students talking during the Warm-Up, but the problem is that they don't stop talking when I'm giving instructions, lessons, or tests. Therefore it's worth it to keep the kids quiet and in their seats during the Warm-Up as well.

By reversing the Warm-Up and Exit Pass, the new first day Warm-Up is the scavenger hunt to complete the phrase, "If you don't know the answer...." Unlike most Warm-Ups -- especially the Illinois State Warm-Ups (which would begin after that is set up online) -- this Warm-Up actually involves students moving around and talking -- so it's a bit awkward to show the importance of being quiet and sitting down at this point.

Of course, I could tell the students to be quiet before starting -- and then afterward, I tell the students to be quiet again. It's important not to accept any excuses for not being silent, including:

-- You're mean!
-- You're the only teacher who makes us sit down and be quiet.
-- You're unfair!
-- You're unreasonable!
-- You're weird!
-- Making us be quiet is juvenile.
-- I wasn't talking.
-- Why do we have to be quiet?

Oh yeah -- I haven't said much about the actual Bridges of Konigsberg worksheet yet. Well, last year I wrote that the two easiest questions should be numbered #1 and #2, not #3 and #4. So I decide to fix this for today's posting. The simplest way for me to do this is to make the old #1 and #2 into more examples, and so #3-8 are renumbered as #1-6. With the order of activities being reversed, the Bridges worksheet is given later in class, so it's good to shorten it to make sure that we get to #6 (the Konigsberg problem) and discuss its impossibility, and get to the Exit Pass. In this class, this will be the first time that they have to do any arithmetic in the math class.

Lesson 1-3: Ordered Pairs as Points (Day 13)

This is what Theoni Pappas writes on page 244 of her Magic of Mathematics:

"Mechanics is the paradise of mathematical science because here we come to the fruits of mathematics." -- Leonardo da Vinci

Yes, Pappas just wrote about Leonardo and the human body last week, and now she quotes him again in this, the introduction to her architecture chapter. Of course, the idea is the same -- science and technology are the fruits of mathematics, which is why students should study it.

According to Pappas, math has been a tool in construction for millennia. Architects eliminate trial and error in building by using math -- indeed, this is true in many fields. Those who don't learn math are forced to use trial and error techniques -- and in construction, this would be quite expensive.

Pappas writes:

"As comprehensive as it may seem, this is but a partial list of some of the mathematical concepts which have been used in architecture over the centuries."

The list actually extends to page 245, but I'll give some of the items on this list anyway:

-- pyramids
-- optical illusions
-- polyhedra
-- Pythagorean Theorem
-- spheres, hemispheres
-- angles
-- symmetry
-- parabolic curves

The chapter introduction actually extends several pages, so it will be well into next week before we actually reach the first proper section of this chapter. Part of this is because there are so many photos in the intro. On page 244 is the first item on the list, pyramids -- it's the Temple of Kululkan, Chichen Itza, Yucatan. And as for the second item, optical illusions, we wrote about these last week as we were reading Serra's text. Yes, op art appears in architecture as well!

Lesson 1-3 of the U of Chicago text is called "Ordered Pairs as Points." (It appears as Lesson 1-2 in the modern edition of the text.) The main focus of the lesson is graphing points on the plane. Indeed, we have another description of a point:

Third description of a point:
A point is an ordered pair of numbers.

The idea of graphing points on a coordinate plane is a familiar one. But sometimes I wonder whether we should make students graph points and lines so soon in their Geometry course.

Once again, here's how I think about it -- the students coming to us just finished Algebra I. Some of them struggled just to earn a grade of C- or D- (whatever the lowest allowable Algebra I grade is in your district is so that the students can advance to Geometry). The students who just barely passed Algebra I are tired of seeing algebra. They may look forward to Geometry where they won't have to see so much algebra -- and then one of the first things we show them is more algebra.

Then there's also the issue, first brought up by David Joyce, that students should use similarity to show why the graph of a linear equation is a line. This idea appears in the Common Core standards for eighth grade, but it's awkward in high school. Graphing linear equations is a first semester Algebra I topic while similarity is a second semester Geometry topic -- and it's difficult to justify delaying graphing linear equations by three semesters just to conform to Joyce's wishes.

In the past, I've tried -- and failed -- to teach linear graphs after similarity. (This includes last year, when I tried to follow the eighth grade standards, but I left the class before graphing equations.) This year, my plan is simple -- I will conform to the order of the U of Chicago text. The U of Chicago text introduces linear graphs in Lesson 1-3, and so that's when I'm teaching it.

The bonus question asks about longitude and latitude. I've already located my own coordinates as being near 34N, 118W.

This blog isn't following the LAUSD calendar, so today isn't the Admission Day holiday. On the other hand, Monday is definitely Labor Day, so my next post will be on Tuesday.

Oh, and I can't let the day pass without mentioning that today, September 1st, 2017, is the Harry Potter equivalent of "Back to the Future Day," since the epilogue is set today. Fans gathered at King's Cross Station in London -- at Platform 9 3/4, of course -- and met Warwick Davis, the actor who played Professor Flitwick in the films. Recall that Flitwick is the example I gave of Lee Canter's "nonassertive" classroom manager. Sorry, Warwick Davis, but I need to be less like your character in order to become a successful, assertive manager.