Notice that this off-day has nothing to do with Columbus Day. In previous years, I used the term "Columbus Day" to refer to certain days in October when schools were closed, but those were in fact different districts. This is despite Monday, October 14th being Columbus Day observed.
We also notice that the first day of the semester was of course Day 1, and I already gave last day of the semester as Day 83. And so Day 42 is exactly halfway between 1 and 83. This shows us a more likely reason as to why Monday is a no-school day -- it represents the start of the second quarter.
Meanwhile, today I subbed in a middle school special ed class. Of course, this is in my new district, where today is Day 33. Thus it's not the end of the first quarter, but more like the end of the first hexter (in other words, the midpoint of the first trimester).
I came very close to subbing in an actual math class today. Originally, there was supposed to be a meeting today for all seventh grade math teachers. But then so many other teachers are calling in sick today that there simply aren't enough subs to cover them all. And so the math meeting is cancelled, and all subs have to cover for the sick teachers instead.
The administration should have known not to schedule the math meeting on a Monday or a Friday, since these are the two most common days for teachers to take off. And Friday is often worse than Monday, since teachers who get sick on Thursday don't want to come back to work on Friday.
Today's special ed teacher has two seventh grade classes and three history classes (two seventh and one eighth grade). The history classes all have aides, and so there's no "A Day in the Life" again. (So I sub at middle schools three days this week, and all three are some kind of special ed.)
For the first time in a long while, there is no lesson plan today. As I recently mentioned on the blog, that's what I originally intended my Conjectures/"Who Am I?" game for. And so today is my first opportunity to play this game in months. (No, there's no sign of a Thursday lesson plan, so the teacher likely just wakes up sick today. It's not another case of a teacher who gets sick on Thursday and then takes Friday off too.)
Of course, I only play the game in science class, since the aides help me figure out a lesson plan for the history classes. As usual, I begin with "What's my age?" and "What's my weight?" Then I look around the classroom for clues regarding what the students are learning now. There is an objective for science written on the board that mentions photosynthesis, and there is also a stack of collected notes also on photosynthesis. Thus my questions for the main part of the game are all about photosynthesis.
I came up with the idea of this game well before the idea of singing songs in class. But a few students who've had me before started asking for a song anyway. And so I go back to what has become my default song for seventh grade science, "Meet Me in Pomona," since it mentions life science (the animals seen at the fair). Of course, photosynthesis is all about plants, not animals (except that animals eat plants which get energy from the sun) -- and besides, seventh grade was the year for life science under the old California standards, not NGSS. Yet "Meet Me in Pomona" is my seventh grade science song anyway.
Lately I've been comparing all science classes to the "science" I taught at the old charter school. But there's not much to say about today's class since I see no lesson plan today (so I can't truly say what this science class is like). Like most special ed classes, they are behind the gen ed class, but it appears that these students have already finished scientific inquiries. But this class apparently hasn't reached metric measurement yet. (That's a shame, since I could have changed "What's my weight?" to "What's my mass in kilograms?")
Then again, when I described the seventh grade curriculum back in June, there was no mention of photosynthesis at all. I suspect, though, that this lesson is squeezed into an early unit. (I've also seen seventh grade science teach a lesson on bees near the end of the year, yet none of the late units on that pacing plan mention bees.)
The students in this class have notebooks, but again, they don't appear to be the interactive notebooks that I've seen in other science classes.
When the aides arrive for history, we come up with a more coherent lesson plan. The seventh graders are studying feudalism in medieval Europe. The aide has the students write a "Warm-Up" paragraph in notebooks, then finish a previous assignment on Chromebooks. The aide for eighth grade doesn't bother with notebooks or Chromebooks -- instead, she plays them a video on colonial America:
Notice that this video starts in 1619, the year that the slave trade began. Lately, this year has been mentioned in the news due to the quadricentennial -- the "1619 Project" of the New York Times.
I'm able to give away the pencils and candy leftover from yesterday. Indeed, I bought 32 Kit-Kats and 16 pencils and gave away half of each yesterday to the eight special ed students. Since I usually the divide the class into groups of four for the Conjectures game, I give a pencil and candy to each member of the winning quartet in the two science classes. In seventh grade history, there's enough candy leftover for four members of my good list to get one. The eighth graders don't get squat -- but all they do is watch a video, so there's no need for any reward.
As for classroom management in the science classes, the teacher next door tells me that I can send bad students to his room, where they'll receive a double detention. This is another matter of crossing lines -- there are a few noisy students in one of the classes, but I'm unable to single any of them out for crossing the line. So I end up sending no one next door.
The aide in one of the history classes manages her class better. She makes sure that the students don't leave their seats without permission, so it's easier to determine who is causing trouble. One boy starts disturbing others for laughs. When she sends him to the double detention next door, he tries to blame another girl instead. This girl is able to convince the next-door teacher that she doesn't deserve the double detention.
OK, let's finally get to the U of Chicago text!
Lesson 4-2 of the U of Chicago text is called "Reflecting Figures." This is what I wrote two years ago about today's lesson:
Let me state the Reflection Postulate, because it is so important, directly from the U of Chicago text:
Reflection Postulate:
Under a reflection:
a. There is a 1-1 correspondence between points and their images.
This means that each preimage has exactly one image, and each image comes from exactly one preimage.
b. If three points are collinear, then their images are collinear.
Reflections preserve collinearity. The image of a line is a line.
c. If B is between A and C, then the image of B is between the images of A and C.
Reflections preserve betweenness. The image of a line segment is a line segment.
d. The distance between two preimages equals the distance between their images.
Reflections preserve distance.
e. The image of an angle is an angle of the same measure.
Reflections preserve angle measure.
This postulate corresponds to Dr. Franklin Mason's "Rigid Motion Postulate," in the old version of his Lesson 3.1 last year. Since then, Dr. M has completely changed his Chapter 3 -- this is almost certainly because isometries ("rigid motions") aren't emphasized on the Common Core texts nearly as much as either of us thought they would when we first read the standards. Nowadays, Dr. M uses the classical definition of congruent polygons (i.e., equality of corresponding measures). He assumes SAS as a postulate (just as the mathematician Hilbert did a century ago), and uses Euclid's ancient proof to derive ASA. But for SSS, Dr. M still uses rigid motions to move one of the triangles into place (similar to the start of the U of Chicago proof) before using SAS and Isosceles Triangle Theorem to prove SSS.
Part a is a very important part of the Reflection Postulate. Without it, a point A could have two reflection images -- there could be two distinct points B and C such that the reflecting line m is the perpendicular bisector of both
According to the text, reflections preserve:
Angle measure
Betweenness
Collinearity
Distance
a nice little mnemonic for the students.
The first theorem of this chapter is the Figure Reflection Theorem:
If a figure is determined by certain points, then its reflection image is the corresponding figure determined by the reflection images of those points.
This theorem is used to conclude, for example, that if A' is the image of A, and B' is the image of B, then
But that is the sort of proof that I don't want to confuse students with. It's best just to use the informal proof given in the text and save formal proofs for later.
The text states that when a figure intersects the reflecting line, the image must intersect the reflecting line in the same point or points. This follows immediately from the fact that the image of a point on the reflecting line is the point itself.
But what I find interesting is the related statement -- if a figure intersects its reflection image, then it must intersect the reflecting line in the same point or points. This statement is false in general, but it's true if the figure to be reflected is itself a line. This fact helps us greatly -- for example, consider Question 21 from the text:
The reflection image of Triangle ABC is Triangle XYZ. Now Lines AC and XZ intersect at a point -- which we now know must lie on the reflecting line. And Lines BC and YZ intersect at a point -- which we now know must lie on the reflecting line as well. And those two points determine exactly one line -- the reflecting line! So all the student has to do is draw the line through the two points of intersection.
And as a corollary, it follows that if a line is parallel to the reflecting line, it must be parallel to its reflection image, Last year, I called this the "Line Parallel to Mirror Theorem" (where "mirror" refers to the reflecting line) But I will wait a few days before introducing that theorem to students. (Notice that the Common Core Standards state that a line must be parallel to its dilation image, so why not give the conditions when a line is parallel to its reflection, rotation, or translation images?)
Today is an activity day. Two years ago, I posted several activities after Lesson 4-4, and so I might as well reblog those activities today, in addition to today's Reflection Postulate worksheet. This is what I wrote last year about the activities:
Meanwhile, I created worksheets for the proof of reflection over the axes. This is all in addition to the original worksheet for today, the Centauri challenge.
The following activity is another one from Michael Serra's text -- it appears in his Chapter 15, since this one often goes with two-column proofs (and proofs appear in his book late). In this activity, strings consisting of the letters P, Q, R, and S are converted into others using four rules (that end up being our postulates):
Rule 1. Any two adjacent letters in a string can change places with each other. (PQ=>QP)
Rule 2. If a string ends in the same two letters, then you may substitute a Q for those two letters. (RSS=>RQ)
Rule 3. If a string begins in the same two letters, then you may add an S in front of those two letters. (PPR=>SPPR)
Rule 4. If a string of letters starts and finishes with the same letter, then you may substitute an R for all the letters between the first and last letters. (PQRSP=>PRP).
Then the text gives the following theorem, PQQRSS=>QRQ. (Notice that I chose to write "=>" where the text writes ">>" since, after all, we've already used the former to denote the hypothesis and conclusion of a conditional.)
Proof:
Statements Reasons
1. PQQRSS 1. Given
2. PQQRQ 2. By Rule 2
3. QPQRQ 3. By Rule 1
4. QRQ 4. By Rule 4
So for the students, this is a puzzle which gets them thinking about the logical structure of proofs without having to think about geometry.
The text calls this the "Centauri challenge," which I assume refers to Alpha Centauri, the closest star system to the sun. Notice that many of the Cooperative Problem Solving challenges in Serra are said to take place in a futuristic lunar colony. For this one, the inhabitants of this colony are trying to communicate with aliens from (Alpha) Centauri, but apparently, the Centaurian alphabet consists of only four letters.
My worksheet contains all of Challenge 1, then adds Challenge 2 as a Bonus. In case you're curious, here are my answers to Challenge 2.
1. Can you produce a string of five or more letters that cannot be reduced to RQ?
My answer is that I can't -- but now I must prove it. Here is my proof, in paragraph form:
Proof:
Our string has five letters, but there are only four letters available. So one of those letters must appear at least twice! (This is called the Pigeonhole Principle.) Let's call the letter that appears twice X. (I know, it's actually P, Q, R, or S, not X, but here I'm using X as a variable to stand for one of the letters P, Q, R, or S, since I want this proof to be as general as possible.)
Using Rule 1, we take the first appearance of X and change it with the letter on its left. Now take that X and change it with the new letter on its left. Keep doing this until the X is the first letter. Now take the last appearance of X and change it with the letter on its left. Keep doing this until this other X is the last letter.
Now our string begins with X and ends with X. So by Rule 4, it becomes XRX. Using Rule 1, we can change the first X and the R to obtain RXX. Finally, this string ends in the same two letters (whatever letter the X stands for), so by Rule 2, it becomes RQ. QED
After doing this, the second question becomes obvious.
2. One of the rules of Centauri can be removed without losing any of the first five theorems proved in Challenge 1. Which of the rules can be removed?
Look at which rules have been used in the post so far, and notice which one's missing. That's the rule that can be removed. I like the similarity between determining which theorems are provable with or without a certain rule, and finding out which theorems in geometry require a Parallel Postulate. This is what Russell, Whitehead, Godel, Erdos etc. did in general -- try to figure out what proofs require which axioms, or postulates, or rules.
Here is the activity:
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