Lesson 0.7 of Michael Serra's Discovering Geometry is called "Islamic Art." This is in the Second Edition -- in the modern editions, "Islamic Art" is Lesson 0.6. Serra begins:
"Islamic art is rich in geometric forms. Islamic artists were familiar with geometry through the works of Euclid, Pythagoras, and other mathematicians of antiquity, and they used geometric patterns extensively in their art and architecture.
"Many of [Muhammad's] followers interpreted his words to mean that the representation of humans or animals in art was forbidden. Therefore, instead of using human or animal forms for decorations, Islamic artists used intricate geometric patterns."
As usual, the questions I derive from Serra's text instruct the student to create Islamic-style art. This art is based on tessellations.
There are a few interesting things in this lesson. First, Serra includes a sidebar called "Improving Reasoning Skills -- Bagels I." As it turns out, Bagels is an old 1980's computer game. I never played it on my old computer, but as a young child, I actually had an old toy (Speak & Math) which included a version of Bagels (called "Number Stumper"). Here's a link to a modern version of Bagels:
http://www.dst-corp.com/james/Bagels.html
During the Responsive Classroom training at my old school, the presenter actually suggested Bagels as an opening week activity. In her version of the game, the word "Bagels" was replaced with "Nada," but the words "Pico" and "Fermi" were retained (so she called the game "Pico, Fermi, Nada"). Again, I don't post any version of "Pico, Fermi, Bagels/Nada," but if you want, you can use it in your own classroom instead of the "Islamic Art" lesson.
I do however include Serra's project for this lesson, "Geometry in Sculpture." This isn't directly related to Islamic art, though. Instead, he writes about Umbilic Torus, a sculpture. It was created by Helaman Ferguson and used as a trophy for the Jaime Escalante award -- named, of course, for the world's most famous math teacher.
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 three 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, 31 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.
For Blaugust today, we revisit the blog of Jenna Laib:
https://jennalaib.wordpress.com/
https://jennalaib.wordpress.com/2019/08/22/when-our-beliefs-become-compromised-part-1/
Recall that Laib isn't a math teacher -- she's an elementary math coach. And in this post, she quotes Cornelius Minor, who isn't a math teacher either:
Cornelius isn’t a math teacher. His background is middle school ELA, and and he authored the recent We Got This: Equity, Access, and the Quest to Be Who Our Students Need Us to Be (Heinemann, 2019). After hearing him speak at the annual Heinemann Teacher Tour in July, my head was swirling with thoughts about the math classroom.
In this post, Laib and Minor discuss the idea that students should feel that they belong in the ELA or math classroom. This is of course a noble goal -- and again, the reason that I selected the five teachers (including both math and ELA teachers) is that each made me feel that I belonged in their classes:
…and while I think my math-specific knowledge (content, pedagogical, PCK/content knowledge for teaching, etc.) is critical to this work, some of the work is the collective work of all educators. How do we make students feel like they belong? Like they are knowers and doers of our content area?
But then Laib's post quickly delves into politics:
As an educator — a white educator — how do I take what I have learned about society and our cruel systems and turn this into a productive force?
And here we go with a mention of race -- once again, the idea is to ensure that students of all races (and genders) feel a sense of belonging.
Once again, my goal is to keep these school year posts race-neutral and gender-neutral. Except to mention the races of the authors of this post (Laib states that she's white, and Minor -- assuming that's a picture of him on his book -- appears to be black), that's all I'm saying about race here. (Of course, if you don't mind reading a race-based post, nothing's stopping you from clicking the links above.)
But this means that when Laib writes:
There were times when I punished 10-year-old Nicole, even though she was doing the same annoying behaviors as her friend Haylee, because Nicole had annoyed me by talking through our entire literacy block — four hours earlier. There were times when I gave Bryant an easy worksheet on addition instead of the engaging, open ended task Richelle got to work on, because Bryant had “too many gaps” in his math understanding.
her underlying fear is, is she treating Nicole and Haylee differently because of their race? And the comment about Bryant and Richelle echoes Chakravarty's post from yesterday.
Of course, I want to make sure that I'm treating all my own students equally. (My biggest fear is that I've mistreated students due to gender rather than race, but the same applies.) But once again, during these school-year posts, my focus should be on math-specific knowledge.
(Some people might find it ironic that I'm struggling to avoid mentioning gender and race in a post whose title mentions a religion -- Islam. Of course, we must be sure not to mistreat students due to their religion as well.)
Let's keep the rest of this post gender- and race-neutral by leaving Laib's blog. (Hmm, "Part 1" implies that she'll be posting a "Part 2" soon.) Instead, we return to the blog of Benjamin Leis:
http://mymathclub.blogspot.com/
http://mymathclub.blogspot.com/2019/08/seattle-mathjam-paper-folding-in-pub.html
Its been about a year since I last wrote about MathsJam: https://mymathclub.blogspot.com/2018/08/seattle-mathjam.html. So it feels like time to post an update. I've been hosting a meeting once a month now since last August. For the last several months we've been hovering at five attendees. They're never quite the same set of people and I've been growing the mailing list so it seems like we're on the cusp of getting a little bigger. I have yet to figure out the way to aggressively publicize something as quirky as this so the growth remains a slow burn. However, my long term vision is about two tables of folks per night and there are enough people already to make the get together fun so I'm not in a hurry.
Recall that Leis isn't a math teacher either -- he's a middle school club leader. Apparently, the other members of this "MathJam" are other teachers/club leaders.
The main topic of this meeting is paper folding something called a "hexahedroflex" -- which sounds and looks similar to the "hexaflexagons" mentioned previously on the blog. Since my blog is a Geometry blog, I couldn't help but notice his mention of our subject:
Finally: I added on my favorite small geometry piece from this month:
https://mymathclub.blogspot.com/p/collected-problems-4.html#p22
So what are we waiting for? Since Leis posted this problem, let me try to solve it:
ABCD is a square. P is on side
At first I noticed that not enough information is given to find any side length. In particular, we see that Angle BAQ can be anything between 0 and 45 degrees. So we let BAQ = theta, which then implies that DAP = 45 - theta. Now we can use the tangent function:
BQ = tan t (I'll write t so I won't have to keep rewriting "theta.")
DP = tan (45 - t)
There exists a subtraction formula for tangent, so let's use it:
DP = tan (45 - t)
= (tan 45 - tan t)/(1 + tan 45 tan t)
= (1 - tan t)/(1 + tan t)
Since ABCD is a square, all four sides are 1, including BC and CD:
CP = CD - DP
= 1 - (1 - tan t)/(1 + tan t)
= (1 + tan t - 1 + tan t)/(1 + tan t)
= 2 tan t/(1 + tan t)
CQ = BC - CQ
= 1 - tan t
Since we're eventually going to add the sides (to find the perimeter), let's rewrite CQ so that it already shares a common denominator with CP:
CQ = 1 - tan t
= (1 - tan^2 t)/(1 + tan t)
Angle C is obviously a right angle since it's an angle of square ABCD. Thus we can use the Pythagorean Theorem to find the length of hypotenuse PQ:
PQ^2 = CP^2 + CQ^2
= (2 tan t)^2/(1 + tan t)^2 + (1 - tan t^2)^2/(1 + tan t)^2
= (4 tan^2 t + 1 - 2 tan^2 t + tan^4 t)/(1 + tan t)^2
= (1 + 2 tan^2 t + tan^4 t)/(1 + tan t)^2
PQ^2 = (1 + tan^2 t)^2/(1 + tan t)^2
PQ = (1 + tan^2 t)/(1 + tan t)
And so the desired perimeter is:
Perimeter = CP + CQ + PQ
= (2 tan t)/(1 + tan t) + (1 - tan^2 t)/(1 + tan t) + (1 + tan^2 t)/(1 + tan t)
= (2 tan t + 1 - tan^2 t + 1 + tan^2 t)/(1 + tan t)
= (2 + 2 tan t)/(1 + tan t)
= 2
Sure enough, all references to tan t cancel out, so the perimeter is independent of theta. The desired perimeter is 2 cm.
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