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:
Tell us about a favorite activity/lesson that makes you jump for joy when you get to use it.
Well, I guess that today's knot design activity might make me jump for joy. It's an opening activity that the students can enjoy as it allows them to exhibit their creativity.
Of course, that's probably not what Shelli meant when she posed this question. She obviously wants us to mention an activity or lesson that we actually taught in the classroom. This requires me to think back three years to my charter middle school classroom.
One day in November that year, I wrote about Benchmark Testing Week -- a very stress-filled week, since there were both print and online Benchmarks to take in both math and English, and they took hours to complete. But I was able to complete some activities during that week. I suppose that these activities made the students and me jump for joy -- if only because anything that wasn't a Benchmark Test made us jump for joy that week. (This week at my old school is probably the early-year Benchmarks, but here I'm referring to the end-of-first-trimester Benchmarks in November.)
Of the sixth grade class, here's what I wrote about a lesson I gave that week:
Yesterday I had a Bruin Corps member present during the sixth grade block. So here's what I did -- I began the class with a Warm-Up division problem from Illinois State -- 2400 / 51. Then I divided the class into two groups -- those who got the right answer and those who didn't. I gave the higher group to my Bruin Corps member, so he could help them move on, just as he and his fellow Bruin have helped with the higher grades this week.
But this led to problems. First of all, some students decided not to answer the Warm-Up. I remember one boy who chose not to answer, so I seated him with the students who didn't know division. Then when I assigned a division question, he answered it quickly -- meaning that he already knew all the steps and was just too lazy to do the Warm-Up. He complained that he had to sit with the students who didn't know division when he already knew it.
There were probably also some students who cheated and started copying the answers when they saw who was getting them right. I reckon that some teachers get around this by simply handing out colored cards rather than telling them where to sit. Then the students can't tell as easily who's getting the right answer, making it harder to cheat. Many of these problems persisted into the homework, where some students either skipped completely or wrote in nonsensical answers, such as "55 divided 8 is 55 remainder 3." And remember -- this is all despite the history teacher giving them time to do the math assignment in his class!
I tried to give the inspiration example I've mentioned earlier -- the Cubs. The North Siders have failed to win the World Series for 108 years before they won it all this week. They didn't let their past failure hinder their present success. Yet the students who can't divide aren't thinking like Cubs -- that is, like champions. They fail to divide, and so they keep coming up ways to avoid division rather than think, I don't know to divide now, but if I work hard (like the Cubs), I will.
Today during the IXL time, the struggling sixth graders continued to struggle. Indeed, I can't say that I've taught a single student to divide -- the stronger students already knowing division and the weaker students coming up with excuses. It doesn't help that I'm trying to crack down on students who don't remember their IXL passwords by telling them they can't use the computers. Weaker students who wish to avoid division just claim that they don't know their passwords!
As I reflect on this day two years later, this was probably one of the few times that I was able to implement Learning Centers that year. For about half the class, this was a good day, but for others, it didn't work. There were too many opportunities for students to avoid the work -- from the one boy who knew how to divide but didn't attempt the Warm-Up to the kids who tried to cheat during the Warm-Up to the ones who claim their computers didn't work. The whole idea of Learning Centers would have worked much better if I'd had tighter classroom management to make sure that students didn't attempt those tricks.
The seventh graders probably has the most enjoyable lesson of the week:
In fact, I realize that the seventh grade lesson is so easy that today, I actually go back and have the students work on the Orienteering STEM project from the Illinois State text. This decision is easy to make, since my support staff member and Bruin Corps member are both present during the seventh grade math block. So I take one group outside, give them compasses, and have them create a map to hide the "treasure" (the textbook), and then another group takes the compasses and map and uses them to find the treasure.
While all of this is going on, my Bruin Corps member watches the rest of the class. She sees that they've already mastered the concept of opposites, so she has them do some general addition of integers that are not opposites. She has them play a game for points similar to the "Who Am I?" games that I played as a sub last year, and buys pizza for the winning group. The winners turn out to be the same group that hides the first treasure -- that is, they are outside for part of the game, yet they still come back to win. And this is more amazing because the smartest student in that group is in fact absent today, so it's not as if he's doing all the work.
Yes, this project from two years ago came from the Illinois State text. But since only some of the students are working on the project, it ended up turning into Learning Centers again.
As for the eighth graders, I finally attempted to teach them some science that week:
Well, I tried to start online Benchmarks yesterday during the math intervention time usually devoted to the other online program IXL. But the problem was that there was a power outage! I'd already charged the laptops before the blackout, but it was impossible to access WiFi during the outage, so the students couldn't access the online Benchmarks. The blackout began at the start of lunch and ended right at -- you guessed it -- 2:25 (that is, P.E. time).
The English teacher suggested that I have the eighth graders finish the written Benchmarks for her own class, since the students couldn't finish them during English class. Well, the students refused to work on them, and when I told them that they were supposed to be working on the essay, one girl called out, "Well, you're supposed to teach us science and you're not doing that!"
"Okay, then," I replied, "let's start the science assignment now."
It's a good thing that I purchased that Common Core Science book last month, since yesterday was my first opportunity to use it -- after all, science was the last thing on my mind with all of this Benchmark stuff. (Notice that there is no science Benchmark, even though eighth graders are supposed to take the NGSS science test.) So I just jumped into Chapter 7 of that book, which as I wrote in my October 10th post, is on Matter and Its Interactions. I just had the students start writing the definitions of some vocabulary from that chapter, but they only got through the first four terms after all the arguing about the essay and writing. Still, at least I got some science in at a time when I thought I'd have very little time for science.
Yes, I admit three years later that this science lesson didn't work as well as I wanted. But then again, this was a spur-of-the-moment science lesson caused when a power outage prevented me from giving the online Benchmarks. The class didn't have a science text because copies of the Illinois State science texts were still a few weeks away from arriving. In the meantime, we had to use the online science text -- which we couldn't because of the blackout.
I could have made that lesson more enjoyable by changing it into a game, similar to the impromptu game played by the seventh graders the following day. But as I wrote earlier, even writing words and definitions was enjoyable compared to taking the Benchmarks.
If you prefer that I answer Shelli's prompt with a Geometry lesson based on the U of Chicago text (outside of my one year of teaching), then one of my favorites is on the area of a circle, which I first posted in 2018:
If G can be divided into A unit squares, then G' can be divided into A squares each of length k. And the area of a square of length k is clearly k^2, so the area of G' must be Ak^2. For the circle problem k is the radius r, and A is the area of the unit circle or pi, so the area of a circle is pi * r^2. We can do this right on the same worksheet -- there's already a circle drawn of radius 10 times the length of a square, so instead of the length of each square being 1/10, let it be 1 instead. Then the area of the circle of radius 10 is equal to the number of shaded boxes, or 314, since the old unit square has been divided into 100 unit squares.
I wrote about several activities related to the circle area in that post -- which includes activities created by other teachers. Unfortunately, I've never actually taught this lesson in the classroom -- I was planning on giving my seventh graders a pi lesson for Pi Day, but I didn't quite make it to March 14th that year.
Today's Blaugust poster is -- um, nobody. So far, no Blaugust participant has made any post dated August 21st yet. Of course, that's probably because most real Blaugust participants are too busy teaching today to post.
The most recent Blaugust post is from Sue Jones, who works at a community college in Illinois:
https://resourceroomblog.wordpress.com/
https://resourceroomblog.wordpress.com/2019/08/19/first-day/
Mirabile dictu, I have student workers and managed to get my “first half fours” video made and up. It’s about how to figure out 4 x a number, you double it twice. I decided I’d stick to pictures on this one and on the “bigger ones and how to double anything,” show how x 2 is “doubling” so … x 2 x 2 is … x 4….
Clearly, Jones works with students who are far below college-level. It's a point I've made before on the blog -- students don't learn all their times tables at once, but learn the easier ones first. Here we see that a student who has mastered the 2's times tables (doubling) can learn the 4's simply by doubling twice.
Jones admits that this post isn't necessarily the best Blaugust post:
Resolving tomorrow to do a Blaugust #mtbos worthy blog…. and/or … the geogebra sharing idea. Whoops, that needs a video, too. 16 steps…
And by now "tomorrow" actually refers to yesterday, so I assume that the Blaugust-worthy post isn't really coming.
Another recent poster who appears on Shelli's Blaugust list is Elissa Miller. She was a major participant in previous challenges:
http://misscalculate.blogspot.com/
http://misscalculate.blogspot.com/2019/08/guest-post-mistakes-by-leila-chakravarty.html
First of all, when we click on the link, we first see a link to another blog, Sam Shah. I notice that Shah has signed his name on Shelli's Blaugust list. But at that link, we see no actual posts written by Shah himself, but rather an invitation to yet another challenge (that is, separate from Blaugust).
And then we see that Miller didn't even bother to participate in Shah's challenge herself. Instead, the post is written by yet another teacher, Leila Chakravarty.
We begin with Miller/Shah's prompt:
Prompt: How do you express your identity as a doer of mathematics to, and share your “why” for doing mathematics with, kids?
And the following response is written by Chakravarty. Notice that Chakravarty describes herself as being both "fast" and "slow" at math:
It started in first grade when two boys who I consistently outperformed were placed in second grade math, but I remained in first, getting sent to the back of the room for finishing my work too fast and bothering people who weren’t done yet. I would like to point out that boys who bother people because they are bored are “being boys” and “need a challenge.” Girls who do so are non-compliant. It continued in seventh grade when I was enrolled in a lower math class despite a strong placement score because my elementary school was notorious for lackluster math preparation.
Math was not for freshman girls who worked too slow and had too many questions, who needed to study for hours and go to the tutoring center. Math was not for queer brown girls who were homesick, navigating identity, or too distracted by music classes. Math was not for someone with her high school math teacher’s voice echoing in her head.
Clearly both gender and race are key issues in Chakravarty's post. Since this is a school-year post, I wish to keep this post as gender-free and race-free as possible. (I will point out that Chakravarty is a South Asian name, so this explains her use of "brown." I avoid interpreting her use of "queer" here.)
But clearly speed is a key issue in this post. Chakravarty's problem here seems to be the fact that many teachers equate "good at math" with "fast at math." Her counterargument is that she is very good at math, but not very fast at math.
If we think back to the traditionalists, they mention speed at math mainly when it comes to basic math, such as the arithmetic of single-digit numbers. They suggest that students who are fast at basic arithmetic have more time to spend on doing higher math, such as Algebra I. (For example, it's easier to factor a quadratic polynomial if one can add and multiply quickly.) Students who haven't mastered single-digit arithmetic tend to get frustrated when trying to solve an Algebra I problem.
In Chakravarty's case, notice that she was fast as a young first grader (when, in her pre-Core class, she probably learned how to add single-digit numbers). She only describes herself as "slow" only when referring to post-elementary school math. Thus the reason that she's slow at higher math most likely has nothing to do with the failure to master arithmetic, as the traditionalists suggest.
Was I guilty at equating "good at math" with "fast at math" as a teacher at the old charter school? It's possible to argue that any sort of multiplication quiz, such as a Dren Quiz, rewards speed. But my Dren Quizzes didn't have a strict time limit (such as 50 problems in one minute). Yes, I admit that I did cover one day for another teacher who gave such a timed quiz -- but that was another teacher.
There was one day when I was guilty of equating "good" with "fast." It actually happened about a week after covering for that other teacher. (Did she corrupt me that day?) My seventh graders were working on a quiz on opposites (the same lesson that I described earlier in this post), but everyone was talking loudly. After I collected a few quizzes, I then declared that everyone who hadn't turned one in would get a zero for talking during the quiz. Thus I inadvertently rewarded speed -- only those who were fast enough to turn it in would get a grade.
I remember one girl who was a strong yet slow student -- she became upset. I must admit that I had treated this girl just as Chakravarty's teachers had treated her, because I rewarded speed for no reason.
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