There's no need for "A Day in the Life" today, but I will say more about this class. Like most schools in this district, this school has a block schedule. Despite the block schedule, I actually have four classes today -- one of them is during zero period. One class is a US History class (for juniors, according to the California State Standards) and the rest are US Government (for seniors). The juniors watch a video called America: The Story of Us. Whenever I play a YouTube video in class, I like to link to it here, even if it's for history or another non-math class:
The senior classes are all working in a packet. Their main task is to choose one of five documents that ultimately influenced the US Constitution (Magna Carta, Mayflower Compact, etc.) and draw a short comic about it.
Since these are older students, I decide not to fool around with a song incentive. (If I had sung one of my songs, it might have been Packet Rap since they are working in packets.) The teacher has a strict cell phone policy (must place in pockets in front of room), and so just getting the phones back is enough of an incentive.
It's Day 18 in this district, so we're well past opening week activities. But there are remnants of such activities scattered around the room, so I might as well explain what these are.
The first packet of the year for the seniors starts with a thought activity -- zombies have taken over campus right on the first day of school! In many ways, this is much like Sarah Carter's "Survival in the Desert" activity (the one I actually presented to my classes back at the old charter school), except from a Government perspective (rather than science). Instead of considering the supplies needed to survive in the desert, students discuss what roles their group members should play to make decisions in a post-apocalyptic zombie world -- and this leads to the need for a government.
There are also signs posted on the walls referring to an "Escape Room" activity. I've seen some math teachers refer to "Escape Room" on their blogs. I'm not sure who the originator is -- it's probably either Jay Chow or Cathy Yenca. Each teacher mentions the other in the respective blog post, so yet me just link to both of them:
http://www.mrchowmath.com/blog/june-08th-2018
http://www.mathycathy.com/blog/2018/04/desmos-breakout-escape-room-activities-amazing-review/
(It appears that Chow was the first to come up with "Escape Room" for math, but Yenca took his idea and ran away with it) Anyway, Chow and Yenca heavily rely on Desmos for their activity, but today's Government teacher instead based her version on codes:
Clue #1: Alphanumeric (with picture of Philadelphia state house)
Clue #2: Morse Code (with picture of James Madison)
Clue #3: Pigpen Code (with picture of the Constitution)
Clue #4: Polybius Code (with picture of the Constitutional Convention)
More on three of the ciphers can be found at the following links:
https://www.braingle.com/brainteasers/codes/morse.php
https://www.braingle.com/brainteasers/codes/pigpen.php
https://www.braingle.com/brainteasers/codes/polybius.php
The last cipher, alphanumeric, is the one used in Square One TV's "Neighborhood Super Spy." Both the alphanumeric and Polybius codes had random numbers corresponding to the letters, rather than simply A=1, B=2, as on Square One TV. (Actually, the Polybius code used here placed A-E in the first row, F-K in the second row, etc., but in random columns within each row.)
Notice that the Illinois State text (the project-based learning curriculum we used at the old charter school) had a few projects for secret codes. These projects fell near the end of the text, and so it was unlikely that I'd ever reach them even if I had completed that year of teaching. The first project contained versions of the Caesar cipher and its generalization, the affine cipher. Two more projects contained codes that had nothing to do with letters or alphabets. (If I had somehow reached those projects, "Neighborhood Super Spy" would have been a natural song for my music break.)
As this assignment is called "Escape Room," I'm wondering from whom the students are escaping. Is it the zombies from their packets? I have no idea whether the "Escape Room" activity is in any way related to the packets.
Once again, it's always interesting to me, as a fledgling teacher, to see what more experienced teachers are doing in their classrooms.
Lesson 1-8 of the U of Chicago text is called "One-Dimensional Figures." (It appears as Lesson 1-6 in the modern edition of the text.)
This is what I wrote last year about today's lesson:
Lesson 1-8 of the U of Chicago text deals with segments and rays. The text begins by introducing the simple idea of betweenness. In Common Core Geometry, betweenness is an important concept, because it's one of the four properties preserved by isometries (the "B" of "A-B-C-D").
As I mentioned a few days ago, for Hilbert, betweenness is a primitive notion -- an undefined term, just as point, line, and plane are undefined. Yet the U of Chicago goes on to define it! It begins by defining betweenness for real numbers:
"A number is between two others if it is greater than one of them and less than the other."
Then the text can define betweenness for points:
"A point is between two other points on the same line if its coordinate is between their coordinates."
But Hilbert couldn't do this, because his points don't have coordinates. Recall that it was Birkhoff, not Hilbert, who came up with the Ruler Postulate assigning real numbers to points. Instead, Hilbert's axioms contain statements about order (Axioms II.1 through II.4), such as:
"II.2. If A and C are two points of a line, then there exists at least one point B lying between A and C."
Since we have a Ruler Postulate (part of the Point-Line-Plane Postulate), this statement is obvious, since points have coordinates and the same is true for real numbers -- between reals a and c is another real b.
I've seen some modern geometry texts mention a Ruler Postulate, but nonetheless leave the term betweenness undefined. Now as we mentioned earlier with point, line, and plane, if a term such as betweenness is undefined, then we need a postulate to describe what betweenness is. This postulate is often called the Segment Addition Postulate:
"If B is between A and C, then AB + BC = AC."
Notice that this statement does appear in the U of Chicago text. But the text doesn't call it the Segment Addition Postulate, but rather the Betweenness Theorem. As a theorem, we should be able to prove it -- and since after all, the text defines betweenness in terms of real numbers, we should be able to use real numbers to prove the theorem. Indeed, the text states that we can use algebra to prove the theorem, but the proof is omitted.
Following David Joyce's admonition that we avoid stating a theorem without giving its proof, let's attempt a proof of the Betweenness Theorem. We are given that B is between A and C. Now let us assign coordinates to these points. To make it easy to remember, we simply use lowercase letters, so point A has coordinate a, point B has coordinate b, and point C has coordinate c.
We are given that B is between A and C, so by definition of betweenness, we have either a < b < c, or the reverse of this, a > b > c. Without loss of generality, let us assume a < b < c (especially since the example in the book has a < b < c). Now by the Ruler Postulate (the Distance Assumption in the Point-Line-Plane Postulate), the distance between A and B (in other words, AB) is |a - b|. Since a < b, a - b must be negative, and so its absolute value is its opposite b - a. (To avoid confusing students, we emphasize that to find AB, we just subtract the right coordinate minus the left coordinate, so that AB isn't negative. This helps us to avoid mentioning absolute value.) Similarly BC = c - b and AC =c - a. And so we calculate:
AB + BC = (b - a) + (c - b) (Substitution Property of Equality)
= c - a (simplification -- cancelling terms b and -b)
= AC
The case where a > b > c is similar, except that AB is now a - b rather than b - a. All the signs are reversed and the same result AB + BC = AC appears. QED
Don't forget that I want to avoid torturing geometry students with algebra. And so I simply give the example with numerical values, with the variables off to the side for those who wish to see the proof.
The text proceeds to define segments, rays, and opposite rays in terms of betweenness. Notice that these definition are somewhat more formal than those given in other texts. A typical text, for example, might define a segment as "a portion of a line from one endpoint to another." But the U of Chicago text writes:
"The segment (or line segment) with endpoints A and B is the set consisting of the distinct points A and B and all points between A and B."
The definitions of ray and opposite ray are similarly defined in terms of betweenness.
The section concludes with the notation for line AB, ray AB, segment AB, and distance AB. But although every textbook distinguishes between segment AB and distance AB, many students -- and admittedly, many teachers as well -- do not. The former has an overline, but the latter doesn't. Unfortunately, Blogger allows me to underline AB and strikethrough
Now if
To avoid confusion, in the following images I threw out Question 8 from the text, a multiple choice question which states that
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