This is my second visit to this classroom. I wrote about my first visit in my April 1st post. That day, I posted an April Fool's joke that this was my new permanent classroom -- that I was now teaching special ed English instead of math.
Officially, three of the classes are seniors and two of them are juniors. But all classes are doing the same thing -- beginning Chinua Achebe's Things Fall Apart. Their first task is to go to the library and check out an actual copy of the book.
My focus in this post is on the classes without aides -- the first senior class of the day as well as both junior classes. And my first problem starts with the early seniors -- of the nine students, six of them refuse to go to the library to get the book! It's one thing to pretend that you're reading the book, but it's another to avoid even getting the book in the first place.
I try to remind one student that four years of English are needed to graduate, and that it's impossible to pass English without even getting the book, but there was no way to do so without it turning into a big argument. I notice a copy of the detention slip he received yesterday -- for saying the F-word to his regular teacher. And so I figure that there's nothing else I can say to a guy who would cuss out his regular teacher like that. Of course, now I know where the detention slips are, and so I prepare to use them -- now that I'm aware that the two unaided junior classes will cause problems. (I also see a manila folder containing notes from previous subs. Sure enough, these three classes were the problem periods for them, too.)
During tutorial, I notice the guitars hanging in the classroom and decide to play them. (There's also a pair of ukeleles in the classroom as well. Back on April Fool's Day, I played these ukes while the students were taking the SBAC in other rooms. And that day the students watched a movie, so there was no opportunity to play any music that day. Thus I didn't mention it on the blog.) Of course I go back to my current go-to songs -- "Meet Me in Pomona," "U-N-I-T Rate!" and "One Billion Is Big."
And during the senior classes with aides, the students request me to sing these songs, and I do. (Note that I don't always play songs in classes with aides, but the guitars are irresistible.) But I'm not sure whether my song incentive will be enough to get through the unaided junior classes. So I suddenly decide to switch to Conjectures/"Who Am I?" instead.
But first, I don't even let the students into the classroom until after they go to the library. Then I divide the classes into four groups. The assignment is to write a five-sentence paragraph based on the first ten pages of Achebe's book. So instead, I write out five fill-in-the-blank questions (after the usual age and weight questions). The students fill in the blanks for points -- and once they're done, they've indeed written five sentences in their notebooks.
This is a tough one. On one hand, I have a resolution against simply giving students sentences to copy (which was a problem at the old charter school). But on the other, I'd rather that students copy my sentences than say "I don't know what to write!" -- and thus write nothing. Based on the early senior class plus the notes from previous subs, these don't look like the classes where I can trust them to write on their own.
Well, did this work? In the first junior class, the students are able to fill in the blanks, though there is a problem with some students throwing objects. In the second junior class, the students complain because I wasn't awarding them points when one students fails to write in a complete sentence.
Chapter 5 of Ian Stewart's The Story of Mathematics is called "Eternal Triangles: Trigonometry and Logarithms," and begins as follows:
"Euclidean geometry is based on triangles, mainly because every polygon can be built from triangles, and most other interesting shapes, such as circles and ellipses, can be approximated by polygons."
In this chapter, Stewart describes two types of functions that we learn in Algebra II or Pre-Calc -- namely trig and logs. Of the mathematicians who painstakingly created the old trig and log tables, he tells us:
"Humanity owes a great deal to those dedicated and dogged pioneers. The basic problem addressed by trigonometry is the calculation of properties of a triangle -- lengths of sides, sizes of angles -- from other such properties. It is much easier to describe the early history of trigonometry if we summarize the main features of modern trigonometry, which is mostly a reworking in 18th century notation of topics that go right back to the Greeks, if not earlier."
The author tells us that trig started in astronomy. The Greeks were trying to calculate a certain angle that would allow them to find the relative distances to the sun and moon:
"The method was right, but the observation was inaccurate; the correct angle is 89.8 degrees. The first trigonometric tables were derived by Hipparchus around 150 BC."
Stewart tells us that for astronomy, spherical trig might be more useful. Of course, I've already devoted two summers (2016 and 2017) to spherical trig -- which is just like plane trig, except:
"Only the formulas change. By far and above the most important trigonometry text of antiquity was the Mathematical Syntaxis of Ptolemy of Alexandria, which dates to about AD 150."
The author describes Ptolemy's Theorem -- now known as the sum formulas for sine and cosine. And Ptolemy used these to make a trig table:
"It was an extraordinary tour de force, and it put astronomers in business for well over a thousand years. A final noteworthy feature of the Almagest is how it handled the orbits of the planets."
Basically, Ptolemy used "epicycles," or circles built on other circles. At this point, Stewart now jumps to the subcontinent:
"Early trigonometric concepts appear in the writings of Hindu mathematicians and astronomers: Varahamihira's Pancha Siddhanta of 500, Brahmagupta's Brahma Sputa Siddhanta of 628 and the more detailed Siddhanta Siromani of Bhaskaracharya in 1150. Indian mathematicians generally used the half-chord, or jya-ardha, which is in effect the modern sine."
Now the author moves on to Renaissance Europe. He tells us all about Vieta -- a mathematician mentioned in yesterday's chapter as well:
"He invented new trigonometric identities, among them some interesting expressions for sines and cosines of integer multiples of theta in terms of the sine and cosine of theta."
Stewart is done with trig, but logs are the other theme of this chapter. The story of logs begins with Scottish mathematician John Napier:
"It seems likely that he started with geometric progressions, sequences of numbers in which each term is obtained from the previous one by multiplying by a fixed number -- such as the powers of 2 (1, 2, 4, 8, 16, 32, ...) or powers of 10 (1, 10, 100, 1000, 10,000, 100,000, ...)."
The original use of logs was to convert products into sums, since addition is easier. Stewart tells us that ironically, the first formula used to convert products to sums was trigonometric, not logarithmic:
"The main method in practical use was based on a formula discovered by Vieta:
sin((x + y)/2)cos((x - y)/2) = (sin x + sin y)/2
If you had tables of sines and cosines, you could use this formula to convert a product to a sum."
But it was up to Napier to replace trig with logs here:
"Napier seized on the idea, and found a major improvement. He formed a geometric series with a common ratio very close to 1."
Later on, English professor Henry Briggs replaced this with what we now call common logs base 10:
"In 1617 he published Logarithmorum Chilias Prima, the logarithms of the integers from 1 to 1000, stated to 14 decimal places."
Of course, we can't leave logs without mentioning natural logs and the number e:
"It arises if we try to form logarithms by starting from a geometric series whose common ratio is slightly larger than 1."
Stewart concludes the chapter by asking, where would we be without trig and logs?
"Science could never have advanced without some such method. The benefits of such a simple idea have been incalculable."
The sidebars in this chapter are all about trig, what trig did for ancient mathematicians, and what it does for us.
Here is today's review worksheet for Chapter 4.
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