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:
Time Capsule - revisit an old post and reflect. If you are new to blogging - find a post on this day from the past on someone else's blog-read, share, reflect.
Well, technically I've already reblogged last year's post for Lesson 0.7. There were some other things I wrote about in that post that I could add to my reblogging that's not directly related to 0.7:
This is what Theoni Pappas writes on page 235 of her Magic of Mathematics:
"This famous drawing by Leonardo da Vinci appeared in the book De Divina Proportione, which Leonardo illustrated for mathematician Luca Paoli in 1509. Leonardo wrote an extensive section on the proportions of the human body in one of his notebooks."
This is the first and only page of the section "Secrets of the Renaissance Man." Of course you can't see what drawing Pappas is referring to here, so let me provide a link:
https://leonardodavinci.stanford.edu/submissions/clabaugh/history/leonardo.html
Pappas explains:
"In his book, he also made reference to the works of Vitruvius, the Roman architect (circa 30 B.C.) who also dealt with the proportions of the human body."
The title, De Divina Proportione, refers to the "divine proportion," which is also known as the golden ratio or Phi. Both Vitruvius and Leonardo believed that Phi = (1 + sqrt(5))/2 appeared in certain ratios of the human body. This is explained at the following link:
https://www.goldennumber.net/leonardo-da-vinci-golden-ratio-art/
- In the distance from the Da Vinci’s guide line drawn at the hairline to the guide line at the foot, the following are all at golden ratio points:
- the navel, which is most often associated with the golden ratio of the total height and not the height of the hairline
- the guidelines for the pectoral nipples
- the guidelines for the collar bone
- In the distance from the Da Vinci’s guide line drawn at the elbow to the guideline at the fingertips
- the base of the hand is at a golden ratio point.
"Leonardo adds, The length of a man's outspread arms is equal to his height."
This estimation appears in a Square One TV song, "Rule of Thumb" by Kid 'n Play. The rappers are trying to measure the length of the floor. One member knows that his height is about six feet, so he concludes that the length of his outspread arms from fingertip to fingertip is also six feet. There isn't a separate video on YouTube for this song, but it does appear at the start of this YouTube video on Math Talk, a spin off of Square One TV. This was recently posted a few months ago:
How can I connect this back to the science class I taught last year? Well, the actual ratios of the human body isn't part of the curriculum, although the human body itself is.
Under the old California standards, here's how a seventh grade life science class was organized. It began with a little chemistry with an emphasize on the elements required for life (hydrogen, carbon, oxygen, and so on). Then the lessons focus on cell structure, DNA, and genes. Next would be evolution and the history of life on earth. This is usually followed by biodiversity, with lessons first on microbes and fungi, plants, and then animals. Within the animal unit, typically invertebrates are covered first, then the various orders of vertebrates -- fish, amphibians, reptiles, birds, mammals. So humans appear last in this section -- but then this is followed immediately by the human body, also known as anatomy. Since I should have followed the California standards for seventh grade, this meant that the unit on the human body should have appeared at the end of the year.
And if you prefer me to reblog a 2016 post (my year in the classroom) rather than 2017, then let me do so right here. I don't want another August post, so instead I'll choose a September 2016 post, halfway through the first trimester:
Today the students take a test. This is for all three grade levels, now that I've changed my original assessment schedule. The eighth grade test is on rational approximations. So far, many students fare well on the test, since much of it involves approximating a square root on the calculator and rounding it off from zero to three decimal places.
Day 30 marks the midpoint of the trimester. In the past I've referred to half of a trimester by a special name -- the "hexter."
I've written about my plans to give four tests this trimester. Therefore, I ought to have two tests during each hexter. But as it turns out, I ended up giving only one test the first hexter, since I printed up the progress reports before grading the tests. Furthermore, the last major grade before printing the first hexter progress reports was a Dren Quiz, which was easy.
As it turns out, all of my eighth graders are earning a C or better. But there are a few students who were failing until the Dren Quiz raised their grades to a C. This might make the progress reports misleading, since the grades were artificially inflated by a Dren Quiz -- oops! As it turns out, most of my failing students are seventh graders. The first test was difficult, and no Dren Quiz can erase all the 10% and 20% scores received on the test.
Here is the song for today:
UNIT RATES
If you want to find unit rates,
There's one thing you must know.
To find a unit rate,
All you do is divide!
To see if it's proportional,
All you do is divide!
Write it as a fraction,
Reduce it then you're fine.
Graph it at (0, 0),
Then just draw a line.
If you want to find square roots,
There's one thing you must know.
To find an estimate,
4 and below, round down!
To find an estimate,
5 and above, round up!
1 place for tenths, 2 for hundredths,
3 for thousandths, you're fine.
Graph it between two values,
Right on the number line.
Tomorrow is also the beginning of a new module. Learning Module 3 of the Illinois State text is called "What's the Best Advantage?" In this module, students will finish the mousetrap cars that they started back in Module 1.
For my eighth graders, this will be an excellent opportunity to integrate science in the lesson. As I wrote earlier, the next NGSS science lesson on the computer is on motion and force. I've been delaying it until the students can learn about force and Newton's Laws. Well, as it just so happens, the students are supposed to measure the force used to launch the mousetrap cars -- in Newtons! So the idea is to have the eighth graders use the first hour to use the mousetrap cars and then the second hour to complete the online assignment.
But the problem is that our mixed-up Wednesday schedule might finally be changing. Here's how the old schedule worked: first period I would have sixth grade, then second period I'd have the eighth graders for "science" (the online assignment), and then third period I'd keep the eighth graders for STEM (which I'd use for either math or an Illinois State project). The problem with the old schedule is trying to fit music into the schedule. According to the music teacher's schedule, eighth grade music started near the end of first period and was intended to last into second period. So the eighth graders began the day in the history classroom and switched to music when the music teacher arrived -- only to have it end 15 minutes later when the sixth graders arrived to the history classroom.
The new, more logical schedule has eighth grade music line up with second period. This means, among other things, that the eighth graders won't be in my room for both an online science lesson and a STEM project, since they'll still be in music. I assume that I will begin the day with the sixth graders in my room -- but I can't send them to the history classroom after first period, since the eighth graders will still be in there for the music lesson. So I'd either keep the sixth graders an extra period or have them go to English and have the seventh graders come to my room.
Under the old schedule, the seventh graders came to my room for fourth period -- but then their music lesson took place in my room, and I didn't see them for math or science at all! Frankly, I wouldn't mind seeing the seventh graders tomorrow, and I bet they'll enjoy beginning the project. If they're in my room only for music as usual, then I'll just have them do the project on Thursday -- indeed, I suspect the project will bleed into Thursday for all the grades no matter what.
And if I do lose an hour with my eighth graders, then I'll just do the project today and have them do the online science assignment tomorrow after lunch -- that time on Thursday is usually for online math assignments, but I'll just have them do science instead.
But this is a two-day post, and I won't know what happens until tomorrow. That's right -- we received an email informing us of the new music schedule, but we were never told what to do with the students outside of music time! The English teacher has given up trying to figure out the schedule and says that she'll just give an online English assignment to whatever kids show up in her classroom! So this is what Wednesdays are like at our middle school, even one hexter into the year!
Returning to 2018, the Blaugust prompt is to reflect on these old posts. Note that in 2016 I mentioned a planned science lesson, then in 2017 I wrote about my failure to teach science. Yes, in 2016 I'd made so many plans to teach science properly, but none of them turned out well.
In that old 2016 post, I seemed to give the uncertainty of the Wednesday music lessons as an excuse not to teach science properly. Of course, the decision to make Wednesday the day for science was, in hindsight, a rather poor choice. I probably should have made Thursday the science day -- after all, I was guaranteed to see all three grades on Thursday. So I could have taught science to all three grades that day without any problems.
Today let's check out the blog of another Blaugust participant, Megan Dubee:
https://www.megandubee.com/single-post/2018/08/23/Open-Ended-Questions
Megan Dubee doesn't give her location, but she does mention a definite subject -- Algebra II. In this post, she describes how she gives her honors and regular ("on level") Algebra II classes the same assignment, with surprising results -- the on level students fared better than the honors students!
Dubee's question was open-ended. Instead of giving a function and asking for its domain and range, she provides the domain and range and asks for a function. Based on the drawings that she posts, the given domain is [-4, 10] and the given range is [-4, 6].
So why does her honors kids struggle? In another post, she writes about what "honors" means:
https://www.megandubee.com/single-post/2018/08/20/What-does-honors-mean-anyway
Notice that Algebra II is the one class where the difference between "honors" and regular is simple -- Honors Algebra II contains trig, while regular Algebra II doesn't.
The idea of honors classes comes up in the traditionalist debates. Here's my speculation as to why the honors kids struggle today -- the these students are good at solving traditional math problems (as opposed to open-ended problems). In a traditional math problem, we start with the function and then ask for its domain, range, and other characteristics.
But grade level kids don't fare as well with these questions (otherwise they'd be in honors). Such students are often frustrated with questions that have just one correct answer, since to them, finding that one answer is like searching for a needle in a haystack. They might enjoy classes like English, where there isn't a single correct answer. Open-ended questions, therefore, were designed to help guide these weaker students.
The traditionalists' biggest fear, of course, is that in trying to help the weaker math students with these open-ended questions, we're hurting the honors students. They'd prefer the traditional math classes with problems with a single answer. Then the honors kids would thrive -- and these are the students who are likely to grow up and build things that we all use and enjoy in the future.
It's difficult to say which type of problems my students from 2016 might have preferred. The idea behind the Illinois State projects is for there to be more open-ended questions just like Dubee's. But much of the problems with the projects go back to the students not following the directions -- and this is avoided only with tighter classroom management.
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