Two of the classes are eighth grade classes and two are seventh grade classes. This time, all classes have the same assignment -- a packet on bullying. Students must read the passage, answer four quiz questions and some vocabulary words, and type a paragraph on bullying. Like most special ed teachers, this teacher also co-teaches one period. It's another seventh grade English class, but these kids are studying A Christmas Carol.
As usual, an aide is in charge of most of the classes, so there's no "Day in the Life" today. As always, middle school periods rotate, and today the rotation actually starts with third period (which is the same as homeroom at this school).
Oh, and one more thing -- today it's raining, and you know what that means here in California. The school is on a rainy day schedule today. I described the only other time I've subbed in this district back in my May 31st post (but the minimum day schedule we had the Friday before Thanksgiving was similar to a rainy day schedule).
Adding to the confusion is that today is Wednesday -- the day of the weekly Common Planning meeting for teachers, and so students go home early. At this school, the snack break is after the second class on regular days and after the third class on dry Wednesdays -- but it reverts to after the second class on rainy Wednesdays. Meanwhile, lunch is before the fifth class on dry regular days and after the fifth class on dry Wednesdays -- and both before and after that class on rainy days.
Thus whether a student has first or second lunch depends on the location of the class that rotates into the fifth position -- today, that's first period. Our first period class has second lunch today.
Of course, the seventh graders are confused by today's schedule. In theory, the eighth graders could have remembered this schedule from last year, but recall that rainy days are relatively rare here in California, and rainy Wednesdays even more so. Thus there might not have been rainy Wednesdays at all last year. And besides, last year this school had no snack break, so the timing of snack on a rainy Wednesday still catches some eighth graders off guard. (Perhaps I should have done "A Day in the Life" today anyway just to illustrate the rainy Wednesday schedule!)
Back in September, I wrote that the worst-behaved class of the day was third period English 7. Today, third period is the best class of the day. Once again, this is why middle schools have period rotations in the first place -- in the morning students are sleepy, in the afternoon they are noisy.
So it makes second period, another English 7, the worst-behaved class of the day. One boy asks the aide to change seats, and she allows him to -- but as we know, this is a red flag. She seats him in a position where she can see his Chromebook screen -- nonetheless, the student visits an unauthorized website instead of typing his paragraph. (During fourth period conference, I'm assigned to cover a special ed World History 7 class, and likewise the aide there and I catch one girl playing games instead of doing her assignment.)
Meanwhile, today my Mathematics Calendar 2019 has arrived from Amazon. (I assume it was delivered by UPS -- there's no US Mail service today due to the National Day of Mourning.) So we'll be able to enjoy another year of Pappas problems.
Oh, and speaking of the National Day of Mourning, I'm still mourning a recent passing -- and no, I'm not talking about George HW Bush. Of course, I'm still thinking about Dr. Kent Merryfield and the Putnam exam. In particular, Question B3 of this year's Putnam reminds me of the late professor.
Did you know that I actually met Dr. Merryfield in person? It was most likely during one of the years that I took the Putnam as a student, either 2000 or 2001.
No, Merryfield never taught at my alma mater, UCLA. But Merryfield once invited me to give a talk about math at his own state university. I don't remember the full circumstances of why he invited me in particular, but I do remember what I spoke about. At the time, I was obsessed with the idea of tetration, or power towers. (I alluded to tetration most recently in my February 7th post, but I blogged about it in more detail back in 2015 and 2016.) So I spoke about tetration and number theory.
Here is Question B3 of this year's Putnam:
https://artofproblemsolving.com/community/c7h1747717_putnam_2018_b3
Find all positive integers
for which simultaneously 
divides 
, 
divides 
, and 
divides 
.And let's just skip to CantonMathGuy's answer:
There are four such
: 4, 16, 65536, and 
.In fact, we will prove that all the solutions are of the form
. Recall the following well-known fact:Lemma. If
and 
are nonnegative integers then 
iff 
.Also, note that
if and only if is a power of 2. Now we are in business.1. If
, then is of the form 
.2. If
, then 
, so 
, so 
is of the form 
.3. If
, then 
, so 
, so 
, so 
, so 
is of the form 
.Thus
. (These all work as the lemma and note are bidirectional.)Now simply note that
while 
to finish.And notice that 2^2^2^s is basically a power tower. In fact, if we had added the condition that n - 3 must divide 2^n - 3, then our solutions would be of the form 2^2^2^2^t, an even higher power tower.
Notice that if n divides 2^n, n - 1 divides 2^n - 1, and n - 2 divides 2^n - 2, then what we're saying is that n, n - 1, and n - 2 all divide 2^n - n. What I basically proved in my Merryfield speech is that if phi(a) divides 2^n - n, then a divides 2^2^n - 2^n. (Here phi(a) is Euler's totient function, and the proof is essentially Euler's generalization of Fermat's Little Theorem.) It means that the difference between two consecutive tetrations of 2, such as 2^2^2^2^2 - 2^2^2^2, has many factors.
I haven't thought much about that speech for Dr. Merryfield since that day. And so it's eerie that something similar would show up as Question B3 on the Putnam just days after Merryfield leaves us.
Lesson 7-3 of the U of Chicago text is called "Triangle Congruence Proofs." Let's forget about dilations and get back to the proofs we've been working on.
This is what I wrote last year about today's lesson:
Lesson 7-3 of the U of Chicago text discusses triangle congruence proofs. Finally, this is what most Geometry students and teachers think of when they hear about "proofs."
There's actually not much that I have to say about this lesson. We already know much about how the proofs in this section go -- generally we are given two triangles with some of the corresponding sides and angles given as congruent. In the easiest examples, the given pairs are already enough to conclude that the triangles are congruent. The U of Chicago text points out that often we must work to get one of the needed pairs -- such as the Reflexive Property of Equality when the two triangles have a side in common, or perhaps the Vertical Angles Theorem to find a pair of congruent angles -- then we use SAS, SSS, or ASA to conclude that the two triangles are congruent. Finally, the students are usually asked to prove one more pair of parts to be congruent, which requires the use of CPCTC.
Now technically speaking, CPCTC was covered in Lesson 6-7, which we skipped over. The U of Chicago text uses the abbreviation CPCF, which stands for corresponding parts of congruent figures, for after all, the property applies to all polygons, not just triangles. But the abbreviation CPCTC is so well-known that I prefer to use CPCTC rather than CPCF. Even though we skipped Lesson 6-7, I believe that the students can figure out CPCTC quickly, so I incorporate it into today's lesson.
Last year I didn't create a worksheet, but instead just wrote down ten problems. This year, I will create a worksheet and include some of the problems from last year, but others I had to throw out. In particular, I had to drop the proof of the Converse of the Isosceles Triangle Theorem -- we certainly want to avoid the U of Chicago proof that uses AAS, since we haven't taught AAS yet. I want to cover it at the same time as the forward Isosceles Triangle Theorem -- I'm saving both for next week. I also had to throw out a few "review" questions that once again review lessons that we've skipped.
Notice that Question 5 mentions isosceles triangles, but it doesn't actually require either the Isosceles Triangle Theorem or its converse. We only need to use the definition of "isosceles" to get one pair of congruent sides -- the first S. The definition of "midpoint" gives us the second S, and the Reflexive Property of Congruence gives us the third S, so the triangles are congruent by SSS.
Indeed, this is a good time for me to bring up a point about two-column proofs. I can easily see a student being stuck on this question because none of the three S's are given -- they have to work to find all of them. One thing I like to point out is if the students see a long word like "isosceles" -- or even "midpoint" -- in a question, they will usually need the definition of that word. In particular, a big word in the "Given" usually leads to the students using the meaning half of the definition, and a big word in the "Prove" often needs the sufficient condition half. The meanings of the words "isosceles" and "midpoint" lead to two of the three S's, and then they'll probably see that the Reflexive Property gives the third S.
Of course, soon they'll learn about the Isosceles Triangle Theorem and its converse -- and once those appear, nearly every proof involving isosceles triangles will use one or the other. But before students learn about the theorems pertaining to a key term, it's the definition that appears in the proof. We've already seen a few proofs about parallel lines where it's the definition of "parallel" that matters -- but once we've proved the Parallel Tests and Consequences, we hardly ever use the definition of parallel.
Last thing -- in the U of Chicago text, the "Given" is never included as a separate step. I always include it as this is more common -- so our proofs will have one more step than those in the text.
There's actually not much that I have to say about this lesson. We already know much about how the proofs in this section go -- generally we are given two triangles with some of the corresponding sides and angles given as congruent. In the easiest examples, the given pairs are already enough to conclude that the triangles are congruent. The U of Chicago text points out that often we must work to get one of the needed pairs -- such as the Reflexive Property of Equality when the two triangles have a side in common, or perhaps the Vertical Angles Theorem to find a pair of congruent angles -- then we use SAS, SSS, or ASA to conclude that the two triangles are congruent. Finally, the students are usually asked to prove one more pair of parts to be congruent, which requires the use of CPCTC.
Now technically speaking, CPCTC was covered in Lesson 6-7, which we skipped over. The U of Chicago text uses the abbreviation CPCF, which stands for corresponding parts of congruent figures, for after all, the property applies to all polygons, not just triangles. But the abbreviation CPCTC is so well-known that I prefer to use CPCTC rather than CPCF. Even though we skipped Lesson 6-7, I believe that the students can figure out CPCTC quickly, so I incorporate it into today's lesson.
Last year I didn't create a worksheet, but instead just wrote down ten problems. This year, I will create a worksheet and include some of the problems from last year, but others I had to throw out. In particular, I had to drop the proof of the Converse of the Isosceles Triangle Theorem -- we certainly want to avoid the U of Chicago proof that uses AAS, since we haven't taught AAS yet. I want to cover it at the same time as the forward Isosceles Triangle Theorem -- I'm saving both for next week. I also had to throw out a few "review" questions that once again review lessons that we've skipped.
Notice that Question 5 mentions isosceles triangles, but it doesn't actually require either the Isosceles Triangle Theorem or its converse. We only need to use the definition of "isosceles" to get one pair of congruent sides -- the first S. The definition of "midpoint" gives us the second S, and the Reflexive Property of Congruence gives us the third S, so the triangles are congruent by SSS.
Indeed, this is a good time for me to bring up a point about two-column proofs. I can easily see a student being stuck on this question because none of the three S's are given -- they have to work to find all of them. One thing I like to point out is if the students see a long word like "isosceles" -- or even "midpoint" -- in a question, they will usually need the definition of that word. In particular, a big word in the "Given" usually leads to the students using the meaning half of the definition, and a big word in the "Prove" often needs the sufficient condition half. The meanings of the words "isosceles" and "midpoint" lead to two of the three S's, and then they'll probably see that the Reflexive Property gives the third S.
Of course, soon they'll learn about the Isosceles Triangle Theorem and its converse -- and once those appear, nearly every proof involving isosceles triangles will use one or the other. But before students learn about the theorems pertaining to a key term, it's the definition that appears in the proof. We've already seen a few proofs about parallel lines where it's the definition of "parallel" that matters -- but once we've proved the Parallel Tests and Consequences, we hardly ever use the definition of parallel.
Last thing -- in the U of Chicago text, the "Given" is never included as a separate step. I always include it as this is more common -- so our proofs will have one more step than those in the text.
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