Lesson 0-6: Knot Designs (Day 6)

This is what I wrote last year about today's lesson:

Lesson 0.6 of Michael Serra's Discovering Geometry is called "Knot Designs." Knot theory is a very recent field of topology.  Two figures are topologically equivalent if one can be bent, stretched, tied, or untied to form the other.

Of course, knot theory isn't a suitable topic for middle school science. But Lord Kelvin -- of temperature fame -- once believed that atoms were knots in the ether:

"Although his theory was not true, the mathematical study of knots is a very current topic today."

...and so it may ultimately have a link to physics after all.  Meanwhile, let's see what Serra has to say about knots. This is Lesson 0.6 in my old Second Edition, while it's Lesson 0.5 in the modern editions, as those editions omit my 0.5.

Serra begins:

"Knots have played very important roles in cultures all over the world. Before the Chinese use ideograms, they recorded events by using a system of knots."

The book depicts a Celtic knot. As Serra explains, the ancient Celts carved various knot designs in stone. He writes:

"Knot designs are geometric designs that appear to weave in or interlace like a knot."

By Serra's definition, the Olympic rings form a knot. Today's worksheet is based on Serra's definition, so there is much emphasis on rings. Of the three questions I selected, one of them has the students draw a knot using a compass (so the shapes will end up being rings). The other two are puzzles involving interlocking rings.

One of the puzzle questions asks students to sketch five rings linked together such that all five can be separated by cutting open one ring. This is easy -- just link four rings to a center ring. The other question is a classic -- the Borromean rings are three rings such that all three are linked, and yet no two of them are linked.

Here is a link to a solution to the Borromean ring puzzle:

http://im-possible.info/english/articles/borromeo/index.html

According to the link above, the Borromean rings are physically impossible -- unless the rings deviate slightly from perfect circularity. Hence the link labels this as an "impossible figure" not unlike the op art from Monday's lesson. Of course, we can draw op art, including perfectly circular Borromean rings, on paper with a compass.

Actually, I found a few interesting links involving Borromean rings. Here is Evelyn Lamb, who writes a math column for Scientific American once or twice per month ("Roots of Unity"):

https://blogs.scientificamerican.com/roots-of-unity/a-few-of-my-favorite-spaces-borromean-rings/

Recall that knots and science are related. Here is a Borromean ring consisting of three atoms:

https://www.livescience.com/9776-strange-physical-theory-proved-40-years.html

And here's a link to Borromean onion rings, courtesy one of my favorite mathematicians, Vi Hart:

https://www.khanacademy.org/math/math-for-fun-and-glory/vi-hart/thanksgiving-math/v/borromean-onion-rings

Actually, Serra mentions the Gordian knot as well, even though I didn't include this question on my worksheet.

The recreational math website Cut the Knot is actually named after the Gordian knot. Here is a link to Alexander (Bogomolny) the Great, who explains why he chose that name:

https://www.cut-the-knot.org/logo.shtml

Here is the Blaugust prompt for today:

Shoutouts!  Give a shout-out to a former teacher, a colleague, or someone in your school or community who is a difference maker.

Well, I don't have any colleagues, unless you count my fellow subs. But as for a former teacher -- actually, for Blaugust two years ago, I wrote about my favorite teachers. So let me cut-and-paste in from that post:

• My favorite elementary teacher was my second grade teacher -- who later became my fifth grade teacher as well. She was one of the first to notice that I was good at math, and so she came up with the idea of having a Pre-Algebra teacher from the high school (which went from Grades 7-12 in my district) send me a textbook. As a second-grader I would work on the assignments independently, then my teacher would send my work to the high school before I worked on the next assignment. By the time I reached the fifth grade and was in her class again, she had convinced the high school teacher to send me the textbook for "APA," or Advanced Pre-Algebra.
• Incidentally, my favorite math teacher was that teacher who sent me the advanced work. I finally met her when I was placed in her Algebra I class in the seventh grade. I was the only seventh grader in a class full of eighth graders, but she made me feel welcome in her class.
• Just like Fawn Nguyen, I had my favorite history teacher when I was an eighth grader. He was also in charge of the Thespian Club at our school, and so he decided to teach history in a unique way -- he would dress up as a historical figure and lecture as if he were that character. Therefore his lectures were more memorable to the students. A few years ago, he retired from teaching, and many of my classmates held a big party for him.
• My favorite science teacher was my junior-year teacher. I was an up-and-down student when it came to science -- the first two years of Integrated Science were more biology-leaning and I struggled a little, but the third year had more emphasis on physical science, which is more closely related to my strongest subject, math (as we spent over a month discussing with Kline's book). And so I did very well in this teacher's class -- indeed, she told me that I would finish the whole test in a few minutes and spend the rest of the time making my writing neat, and of course my answers were correct. She wondered why I wasn't enrolled in the magnet program, and I replied that I had moved to my new district as a freshman, while magnet students are recruited in the eighth grade. And so my science teacher convinced the school to admit me to the magnet program as a junior. Even though I was no longer in her class, she was still my most memorable science teacher for this reason.
• My favorite English teacher was my senior-year teacher -- or to be precise, one of two English teachers I had that year. You see, the magnet program I'd entered a year earlier was a year ahead in English -- that is, junior-level English for neighborhood students was equivalent to sophomore English within the magnet. This meant that I would have to double up on English my senior year in order to graduate from the magnet -- and I didn't look forward to this, since my strongest subject was math, not English. So even though I was the only senior in a class full of juniors, I enjoyed this English teacher's class the most. This teacher allowed us to be creative in our writing -- I remember that for extra-credit, I wrote parodies of the literature we were reading, except with my friends and me as the characters. There was also an essay contest for seniors in which we were to write about a journey we had taken -- I wasn't going to participate, except that the junior English teacher whose class I had to take decided to assign the same topic for an in-class grade! I was in the unique position of writing an essay for class and submitting the same essay to the contest.  So I wrote about my journey through my education (much of which I just wrote about in this post) -- and won $200.

When I reflect upon my favorite teachers, I notice that they have some traits in common. Two of my teachers taught subjects I didn't enjoy, English and history -- and made them enjoyable by presenting them in a unique way. The other teachers taught my stronger subjects, math and science -- and they recognized that I was talented enough in those subjects to move me up to the next level.

Some traditionalists lament the fact that the Common Core accountability movement encourages teachers to focus on the weaker students at the expense of the stronger students. They say that some strong students want to move ahead in their classes, but the teachers, who claim their hands are tied by Common Core, won't let them.

I'm torn whether I should focus on my stronger or weaker students as I get ready to teach in the middle school classroom this year. On one hand, neglecting the weaker students is why many people spurn tracking, so I want to help my weaker students get ahead. But on the other hand, I myself am the beneficiary of certain teachers noticing my special talents and allowing me to succeed in more challenging classes. Therefore I owe it to my stronger students to support them and celebrate their talents just as my own teachers celebrated my own talents.

This is so important that it bears repeating. I owe it to my stronger students to support them and celebrate their talents just as my own teachers celebrated my own talents.

Recall back on Square Root Day the story I told about teaching my second grade friend the square roots of 0, 1, and 144. I admit that this incident, along with my admiration of my second grade teacher, formed the foundation of my desire to become a teacher. At first I didn't know that Grades 7 and higher even existed -- I knew that my elementary school was K-6, and I'd always believed that students went directly from sixth grade to college. I remember that as a kindergartner, to me the sixth graders looked like grown-ups, and so I expected that they were nearly college students.

Naturally, it was the arrival of my Pre-Algebra text that alerted me to existence of 7th grade. I wasn't sure whether I wanted to be a teacher because I wasn't sure I'd be good enough at any subject other than math, but the benefactor who gave me the Pre-Algebra text was a single-subject teacher who taught math and nothing else. And so I knew at that moment that I wanted to become a single-subject math teacher -- which meant that I'd most likely teach in a high school.

As it happens, I was checking the website of my old elementary school and -- believe it or not -- my second/fifth grade teacher still teaches there, 30 years after I was a student in her classes! (At least, she taught there last year, when the site was last updated.) And according to the website, she is now a special ed & intervention teacher.

Today we'll look at the blog of another Blaugust participant, Danielle Reycer:

http://www.daniellereycer.com/2018/08/my-11th-first-day.html

Danielle Reycer is a Colorado high school teacher. It's hard to tell whether which level of math she teaches -- it appears to be either Geometry or Algebra II.

If we were to ask Reycer to follow today's prompt and give someone a shout out, it would be either to her family or the MTBoS. That's because in her district today is the first day of school -- and her entire first day lesson plan comes from two famous members of the MTBoS.

The first is the Twos Through Nines Challenge. The MTBoS member who made this challenge famous is probably Sarah Carter of Oklahoma, whose blog I've linked to many times before:

https://mathequalslove.blogspot.com/2018/08/twos-to-nines-challenges-aka-best-first.html

I first read about this challenge back when I was a young elementary school student (probably back when I was in my favorite teacher's class). The manual to my old TI-34 calculator (the same one that mentioned the "handal" base five system) also had a page titled "for four 4's."  The challenge is to write as many integers as possible combining four 4's with the four operations:

0 = (4 - 4) * (4 - 4)
1 = 4/4 * 4/4
2 = 4/4 + 4/4
3 = (4 + 4 + 4)/4
4 = (4 - 4) * 4 + 4

and so on. But I never imagined this as a school activity until Carter used it in her class. Originally, she used it as a New Year's Day activity, using the digits of the New Year. (Theoni Pappas sometimes would something similar for her January 1st problem. This year she used (2018!)^0 = 1 which is invalid, but 2^0 * 1^8  = 1 might work if exponentiation is allowed. But in Carter's class the students would have to do 2 = (2 + 0) * 1^8 and so on.)

But this year, Carter changed it to an opening activity. Instead of the digits of the date, the students must use four identical digits (more like my TI-34 problem). And instead of four 4's, each group is assigned a different digit from 2 to 9 (hence the name of the activity).

After the Twos Through Nines Challenge, Reycer writes that the students' first homework assignment is to write a personal quiz. This also goes back to Sarah Carter:

https://mathequalslove.blogspot.com/2016/08/day-1-done.html

Now Reycer writes about one more activity she does on the first day of school, Name Tents. Many teachers have blogged about this activity, but the one who made it famous is most likely another Sara(h) -- Sara VanDerWerf of Minnesota:

https://saravanderwerf.com/2016/08/07/week-1-day-1-name-tents-with-feedback/

After I left my classroom a year and a half ago, I wrote the following here on the blog:

In delivering the pizza, I entered the co-located LAUSD elementary school, but no one from my charter school actually saw me. I laid the pizza in front of my old classroom door and knocked on it, then hurried down the steps. One of the sixth graders opened the door and saw the pizza, but by then I was gone -- I didn't want to stay and become a distraction. After all, the reason I'd left was that the sixth graders were too loud when I was there.

Along with the pizza, I'd left a personalized "Good Luck" card for each student. I'd originally planned on giving each student such a card on the last day of school, but due to the circumstances I decided to hand out the cards right then on Pi Day.

A few days later, I received an email from the dean. He told me that the students were touched by my pizza and "Good Luck" notes, and in fact, the sixth graders had cried for twenty minutes! I agree that this year's Pi Day was bittersweet -- how dearly I wished that I had been in the classroom celebrating Pi Day with the students.

The idea of writing personal "Good Luck" notes goes back to my days as a student teacher -- on my final day, I wrote a personalized "Good Luck" note to each student. But during the year I was at the charter middle school, I asked myself, why wait until the end of the year to connect to each student on a personal level? My plan had been to write personal notes at the end of each trimester. But the end of the first trimester was hectic -- between endless Benchmark Tests and my car breaking down, I wasn't in the mood to write personalized notes! The notes I mentioned above were my second trimester notes -- I never made it to the third trimester.

The idea behind VanDerWerf Name Tents and Carter Getting to Know You Quizzes is for the teacher to connect with the students at the beginning of the year -- when the teacher isn't too busy trying to give online Benchmarks or calculate final grades. Thus these are opportunities to establish a rapport with the students before the hard work begins.

When the year began, I did give my students an activity called Personality Coordinates, which comes from the former King of the MTBoS, Dan Meyer. But Personality Coordinates allow the students to connect with each other -- not for me, the teacher, to connect with the students. I hardly looked at the Personality Coordinates at all past the first day of school. On the other hand, VanDerWerf Name Tents and Carter Quizzes require the teacher to respond to the students' questions.

When Carter first posted her Getting to Know You Quizzes, it was on her first day of school -- a week after my first day. I thought it might have been awkward to do them a week after school started. And  I needed an activity to fill out Benchmark Testing Week after the tests were finished. So instead, I used her Survival in the Desert activity. In hindsight, I wonder whether it would have been better to give the Getting to Know You Quizzes even during my second week of school. (I could have joked that they were my "Benchmarks" after the students finished their own Benchmarks.)

On the other hand, VanDerWerf first posted her Name Tents on August 7th, 2016 -- nine days before my first day of school at the charter. Therefore, I theoretically could have read her blog in time for me to use the name tents on my first day of school. But I'd already planned on using Meyer Personality Coordinates on the first day and was probably not open to considering any alternative.

Many Blaugust participants mention VanDerWerf Name tents in their posts. Most of these teachers write about how they used and liked them last year (2017) and will use them again this year, or else they first heard about them this year (2018). This makes sense -- VanDerWerf wasn't as famous in 2016 as she is now. As more MTBoS teachers learn about the name tents, they begin to implement them in their classes.

But one teacher used VanDerWerf tents as soon as she posted them in 2016 -- Danielle Reycer. To prove this, let's link to one of Reycer's old 2016 posts:

http://www.daniellereycer.com/2016/08/my-first-week-backso-far.html

This post is dated August 25th, 2016, and it links back to VanDerWerf August 7th, 2016. So Reycer read VanDerWerf's blog -- and in the following two weeks, Reycer quickly made the decision to implement them in her class. If Reycer could learn about and use name tents in 2016, then so could I.

VanDerWerf name tents are a bit like elementary school name tags. But they are foldable, because on the inside, students write notes to the teacher -- one for each day of the first week of school. Now according to VanDerWerf, on some days she would let the students write whatever they wanted to, and on others she'd prompt them to ask her a question so that she could answer them.

I have a feeling that my students would have enjoyed the name tents. Instead of waiting until Pi Day to connect with my students, I'm able to learn more about them from the first week. And to this day, I wonder whether I might have gotten my students to listen to me and obey me if I'd been able to connect with them more from the start of the year.

The only students who might not have liked the name tents are the special ed students who have trouble writing. (I wonder what VanDerWerf does with the special ed students in her classes as far as name tents are concerned.)

Notice that Reycer uses both VanDerWerf Tents and Carter Quizzes in her class the first day. This sounds like a bit too much -- but at least she's erring on the side of connecting too much with her students. I, on the other hand, didn't connect with them enough -- and that was my big problem.

The MTBoS Sara(h)s -- VanDerWerf and Carter -- clearly understand how important it is for a teacher to connect with the students on the first day of school, and so does Reycer. Again, that's why VanDerWerf, Carter, and Reycer are successful teachers, but I am not.

If I'm ever given the opportunity to teach in my own class again, I'll definitely consider using either VanDerWerf Name Tents or Carter Quizzes. Indeed, I'm definitely leaning towards using the VanDerWerf Name Tents. (As for four 4's and similar activities, I haven't decided whether I'll ever use them in a class or not.)