It's been a tradition on my blog to post one of this year's Putnam problems. Usually I post it on the Tuesday after the exam, and this year will be no exception, so expect a discussion about a Putnam problem in tomorrow's post.
That's right -- I just spent a full week writing about State Meet Saturday, the big XC race, and McFarland USA, and you're probably tired of hearing about that by now. (No, I didn't sub for that Spanish class today -- hopefully they finished the movie with either another sub or their teacher.) So instead, expect a full week of writing about Putnam Saturday and the big math test. (And this is more logical for me to post, because a, this is a math blog and b, I actually participated in the Putnam, unlike the State XC Meet.)
Each year, I claim that I'll post Problem A1, since that's theoretically the easiest problem -- one which bright high school students might be able to understand. Yet each year, I find a different problem to post instead -- indeed, sometimes the A1 question is about Calculus. (That shows how difficult the Putnam is, if the "easiest" question on the test is Calculus!)
This year, I'll post problem A6, because its main subject is Geometry -- and this is, after all, a Geometry blog. I might mention some of the other problems anyway -- this year's A1 isn't Calculus, but just number theory (with a little Algebra), so students might understand some of this problem.
Normally, each year after the Putnam, I'd link to the Art of Problem Solving website. Dr. Kent Merryfield, a local math professor here in Southern California, would post dozen threads there, one for each Putnam problem, and others can discuss the solution. But unfortunately, Dr. Merryfield won't be posting any Putnam problems this year.
That's because Dr. Merryfield has passed away.
While the nation mourns the passing of former President George HW Bush, I'm just as saddened by the passing of Professor Merryfield. It just happened six days ago, on Tuesday. I only visit the Art of Problem Solving website once a year, right around the time of the Putnam. If he had passed six weeks ago, or six months ago, I probably wouldn't have known about until now. But it's ironic that it would happen just days before the Putnam.
Here's a link to the Art of Problem Solving website:
https://artofproblemsolving.com/community/c7h1745283_in_honor_of_kent_merryfield
His son, who's also a prolific AoPS poster, wrote the following in this thread:
J. Merry:
This seems like the place to talk about what he was to me in terms of math. I'll remember long rambling discussions about a range of topics - at first, almost lectures, but later I kept up and could bring in my own points. We bounced ideas off each other so much that we couldn't always remember who showed it to the other first. I definitely learned more math from him than anyone else - but of course, he had an unfair advantage in that category.
The article linked in scrabbler94's post includes pictures of his office door, an office he used for decades. That's where my love of contest math began; I would sometimes come in on afternoons as an elementary/middle school student, and at least one of his colleagues liked to throw neat little problems at me. When I got into the bigger contests and started doing really well (CA team for national Mathcounts, 1996), that got him involved in the contest world - eventually leading to the Math Day at the Beach regional high school contest (2000-present) and the Southern California ARML team (2002-present).
We came to the AoPS community at the same time, in June 2004. That year, the ARML team included a San Diego contingent (not yet enough to form their own team), and Richard Rusczyk was one of the adults with the San Diego group. He mentioned his site to us on the trip, we joined, and we were off. I've faded in and out sometimes, but he's been a consistent contributor the whole time.
And later, he writes:
J. Merry:
The timeline, as I understand it:
He taught his midday class (the Putnam preparation seminar, which meets once a week). After that, he made an utterly routine post here (on a private forum, so you won't see it on a search unless you have access), saying that there would be no new homework, as next week's meeting would be the Putnam postmortem* session.
Then he went off to lunch, and died.
*No, this wasn't the original wording. I just couldn't resist a bit of humor.
Another poster even quotes Dr. Merryfield's last post, which he made mere hours before his death:
haha0201:
Kent Merryfield wrote:
which with just one or two simple row elimination steps can be seen to be singular. (The rest doesn't print well on this blog, so I omit it.)
Just as the others wrote in the thread, the Putnam just won't be the same without Dr. Merryfield.
Last week during tutorial, I saw one student working on her Geometry assignment. She was working in the lesson on indirect proofs. More often than not, an indirect proof is required to solve one of the problems on the Putnam exam. For example, A5 can be solved with such a proof. I mentioned the big college-level briefly to this girl. If I can find myself in an actual math class this week, I'll discuss the Putnam with them in more detail.
Last year, I wasn't in any classroom around Putnam season at all. But two years ago, I was obviously at the old charter school, and I told my eighth graders about the Putnam. This is how that went:
I mentioned the Putnam exam to my eighth grade class. This is the youngest grade to which I've ever talked about the test -- usually, I prefer to mention it to high school students for whom I'm subbing or tutoring. The problem I chose happened to be more puzzle-like, so even eighth graders could at least understand the gist of the problem. Indeed, the Putnam puzzle is eerily similar to a puzzle posted on Sarah Carter's famous blog last year:
https://mathequalslove.blogspot.com/2017/11/color-square-puzzle-from-puzzle-box.html
And since Carter's freshmen enjoyed the puzzle, it's not a stretch to think that eighth graders could appreciate it as well. But of course, as usual, the problem I had with any discussion that year was that students talked too much -- and so some students never heard what I was saying. Still, I hope that at least a few of them were inspired by my Putnam speech.
By the way, while I was looking up info about this year's Putnam, I found this old page on the website of UCLA, my old school:
http://www.math.ucla.edu/~thiele/putnam/history.html
And yes -- my name is listed here. My Putnam score in 2001 was 22 out of 120. This sounds bad, but this placed me among the top 500 test takers in the country. The most common score on the test, after all, is usually zero.
By the way, last year, a UCLA student scored 61 points, good enough for 34th in the nation:
https://www.math.ucla.edu/news/ucla-putnam-team-places-5th
Two other Bruins scored 60, including Ni Yan -- the highest scoring female in the nation. She therefore received the Elizabeth Lowell Putnam prize of $1000. The top UCLA student (Emre Girgin) also earned a $1000 prize (Basil Gordon prize).
Again, don't try to map Putnam scores to letter grades. In middle and high school classes, 61 out of 120 would be 50.8%, an F grade, but on the Putnam it's the 13th best score in the whole country!
This is what I wrote last year about today's lesson. Actually, I didn't say much about the lesson per se, but I did write extensively about proofs, the major theme for today's post:
I introduced the concept of low-, medium-, and high-level proofs. These categories aren't rigid, but here's an approximate division:
Low-level: Prove SAS Congruence from first principles (i.e. transformations, if it's Common Core)
Mid-level: Prove the Isosceles Triangle Theorem from SAS Congruence
High-level: Prove the Equilateral Triangle Theorem (i.e. that an equilateral triangle is equiangular) from the Isosceles Triangle Theorem
So we can somewhat see the difference among these levels -- in particular, we may use the lower-level theorems in the proofs of the higher-level theorems.
But there's a more important distinction among these levels in the Geometry classroom. Teachers are more likely to ask students to prove higher-level than lower-level theorems. Many Geometry texts, especially pre-Core, don't expect students to prove our low-level theorems, such as SAS Congruence from first principles. Indeed, they absolve students from the responsibility of proving SAS completely by making it a postulate!
And now we see where the opponents of Common Core come in. They point out that geometry based on transformations is too experimental to appear in the classroom. Instead, they favor the pre-Core status quo -- just declare SAS a postulate and throw out transformations altogether,
Now here's the problem with this thinking -- low-level is to mid-level as mid-level is to high-level. I can now imagine a hypothetical class where not only do we avoid the low-level derivation of SAS from transformations, but we can avoid the mid-level derivation of the Isosceles Triangle Theorem from SAS as well. Instead, just declare the Isosceles Triangle Property to be a postulate and throw out SAS Congruence altogether! Students can still prove interesting theorems from this Isosceles Triangle "Postulate," including the Equilateral Triangle Theorem -- even the first problem from that Weeks and Adkins page from two weeks ago can be proved using only the Isosceles Triangle "Postulate" (and its converse, which could be declared yet another postulate).
One may argue that no Geometry text actually does this -- but au contraire, there really is a text that does something like the above. In Lesson 5-1 (old version) of Michael Serra's Discovering Geometry, Conjecture 27 is the Isosceles Triangle Conjecture (and Conjecture 28 is its converse), while Conjecture 29 is the Equilateral Triangle Conjecture (stated as a biconditional), with a paragraph proof provided to show how 29 follows from 28. So even though all three of these statements are labeled as "conjectures," the net effect is that 27 and 28 are postulates (as no proof is given), while 29 is actually a theorem proved using postulate 28.
Of course, this may seem silly -- Serra doesn't avoid SAS altogether, but instead gives it later on in the same chapter (Lesson 5-4, old version). And ultimately when we reach the end of the book when two-column proofs are taught, students are asked to prove the Isosceles Triangle Theorem using SAS, just as in most other texts.
But it does show that the pre-Core status quo is attacked on two fronts. If you argue that students should learn SAS so that the Isosceles Triangle Property can be a theorem rather than a postulate, then why not take it further in the direction of more rigor, and teach the students about reflections so that SAS can be a theorem rather than a postulate? Or going the other way, if you argue that students shouldn't have to learn how SAS follows from reflections, then why not take it further and say that students shouldn't have to learn how the Isosceles Triangle Property follows from SAS? It's not at all obvious why the exact status quo (SAS a postulate, Isosceles Triangle Property a theorem) is neither excessively nor insufficiently rigorous.
Now there is an argument that, if true, would vindicate the status quo defenders. It could be that the mid-level derivation of the Isosceles Triangle Theorem from SAS is easy for the students to understand, thereby preparing them well for the rest of Geometry and subsequent courses, but the low-level derivation of SAS from reflections is too hard for them and makes them cry after the test, thereby discouraging them from taking subsequent math courses. If this can be demonstrated, then the status quo is exactly right. Then again, until such a demonstration can be made, it's just as likely that the Common Core transformation approach, or even Serra's conjecture approach, could be correct.
There's one more thing that we must take into consideration -- the distinction between the Common Core Standards and the Common Core tests. Much of all my chapter juggling has occurred because I'd originally set up my lessons to match the standards, only to see something else on the tests. The standards state that students should learn how SAS and the other congruence and similarity theorems follow from the properties of transformations -- but such low-level proofs don't appear on the PARCC or any other Common Core test. It's actually easier to test for medium- and high-level proofs on a test, and the PARCC is no exception.
The PARCC question where students have to derive the Alternate Interior Angles Consequence from the Corresponding Angles Consequence is a mid-level proof. The PARCC question where students have to prove that the sum of the exterior angles of a triangle is 360 is definitely a high-level proof (after all, the triangle sum ultimately goes back to alternate interior angles). A low-level proof in this tree would be to show how the Corresponding Angles Consequence goes back to transformations.
Mid-level proofs on the PARCC are problematic -- and it's these questions that drive most of the changes in my curriculum. We saw last week how although the Common Core Standards ask students to derive SAS Similarity from the properties of dilations, a PARCC question asks them to derive a mid-level property of dilations from SAS~ instead.
But high-level proofs cause the fewest curriculum problems. I consider the classic two-column proofs of U of Chicago's Lesson 7-3 -- where students use SAS to prove two triangles congruent, but the "S" comes from the Reflexive Property or the definition of midpoint and the "A" comes from the Vertical Angles Theorem or some other result -- to be high-level proofs. This is because they appear at the top of the proof tree, rather than branch out to be used in other theorems. Such proofs don't require the students to derive SAS as a theorem.
On this blog, I will present low-level proofs in worksheets in order to meet the Common Core Standards, but I don't expect students to reproduce them in the exercises or on a quiz or test. On the other hand, students will have to know and understand the mid- and high-level proofs.
Let's get to today's worksheet. Now as it turns out, not only did I begin Chapter 7 last year near the Thanksgiving break, but it was also when I was purchasing a new computer and working hard to get it installed and connected to the Internet. These two facts combined mean that I don't necessary have a great Lesson 7-1 worksheet from last year.
Last year I gave some sort of an activity, where students were given parts of a triangle such as SS, AA, SSS, AAA, and so on, to determine whether they are sufficient to determine a triangle. Then I would follow this with a discussion of the results followed by some "review" problems. Once again, juggling the lessons around means that the students would be "reviewing" concepts that I haven't taught this year yet, such as Triangle Inequality and Triangle Sum. (Notice that this are related to the SSS and AAA conditions, respectively.)
So I decided to keep the activity-like part of the lesson and replace the Triangle Inequality and Triangle Sum questions with some more information about isometries and polygon congruence. This is what the resulting worksheet looks like:
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