I. We begin with a brief excursus on chance and probability.
II. Here is an intriguing scenario about random events with surprising results. The experiment is referred to as Buffon's Needle.
III. We can use calculus to deduce what the exact probability should be.
IV. This experiment shows a method for estimating the value of pi.
V. One of the uses of calculus is in approximating values that we are interested in.
Starbird starts his lecture by discussing probability. In his first example, the Starbird family rolled a die 1,000 times, and the value 3 came up 164 times. Thus the experimental probability of rolling a 3 was 0.164, near the theoretical probability of 1/6.
The professor now refers to a French mathematician/scientist, known as Georges Louis Le Clerc Compte de Buffon (1707-1788). He was the one who devised the famous needle experiment that can be used to approximate a value of pi.
Notice that I actually linked to a video about Buffon's Needle back in my post on Third Pi Day back in November. (Actually it was just before Third Pi Day, as the exact day fell on the weekend.) So I don't need to repeat much of this video today, as you can refer back to that post and video. (In today's lecture, he uses breadsticks, which he states are more authentic to what Buffon actually used -- even though breadsticks tend to break.)
But I will say this: Starbird mentions that there are websites where users can perform Buffon's Needle over and over. I assume one such simulation appears in that video, but let me link to one anyway:
https://mste.illinois.edu/activity/buffon/
The professor performed such a simulation, and his needles landed on the lines 63,639 out of 100,000 total trials.
Starbird's proofs involve graphs, so I won't show them here. His proof is similar to the ones at either the link above or my November 8th video, although there's a slight difference. At the U of Illinois link above, the needle length and distance between lines are both 1. In my November 8th video the needle length is 1 and distance between lines is 2.
For the professor, the needle length and distance between lines are both 2. This allows him to write integral _0 ^pi sin theta dtheta, which he already found in the last lecture to be 2. This allows him to find the theoretical probability as 2/pi. He compares this to his experiment:
63,639/100,000 = 2/pi
pi = 2 * 100,000/63,639 = 3.1427269...
pi actually is = 3.1415926...
Note: When I refer to the November 8th video, of course I mean the "Method 1" part, which is based on trigonometry and Calculus. Method 2, Barbier's proof, doesn't directly mention Calculus (though limits are needed to change the needles from polygons into circles). This later proof might be worth mentioning in a Geometry class where we don't wish to invoke Calculus.
Indeed, it's interesting to wonder whether it's worth discussing Buffon's Needle in Geometry class, especially here in California where probablity is part of the Common Core Geometry course. Of course, I already posted my lessons on pi (U of Chicago Chapter 8) here about 2-3 weeks ago.
In my old district, today is Day 99. But in my new district, it's the day between the semesters. This PD day between the semesters is fairly common -- my old district had its PD day on January 6th, and for my new district it's today. The only difference is that in my old district, students think of the PD day as an extra day of winter break, but in my new district it's just a three-day weekend.
Thus the only way I'd sub today is in my old district. Calls from this district are rare -- and I'd just subbed there last Friday, so it was unlikely that I'd be called to sub today. And I wasn't.
Today is the review for tomorrow's Chapter 9 Test. In many ways this is a light chapter. While the modern Third Edition includes surface area in Chapter 9 (Lessons 9-9 and 9-10), my old Second Edition stops after Lesson 9-8. Then again, students who have trouble visualizing three dimensions will struggle on tomorrow's test.
Well, at least our students won't have any Euclid on their test. Let's return to his next proposition:
Yesterday we construct the perpendicular from a point not on the plane, and now we construct the perpendicular from a point on the plane. This construction uses yesterday's as a subroutine.
This construction is even sillier than last Friday's to perform inside a classroom. This time, we have a point A on the floor and we wish to find a point directly above it. First, we label point B on the ceiling and use yesterday's construction to find a point C directly below it. Then we construct the line through A that is parallel to
And so there's no way that our students can physically perform this construction. There will be nothing like this on tomorrow's test, even though ironically, it would be easier to answer test questions about this construction than physically perform it.
Notice that the U of Chicago text doesn't actually provide the construction for drawing a parallel to a line through a point not on the line (which is a simple plane construction). The only way implied in the text to perform this construction is to make two perpendicular constructions (which we did in yesterday's post).
Many texts that teach the construction of parallel lines use copying an angle (as in corresponding or alternate interior angles). Lesson 7-10 of the Third Edition is on constructions, and duplication of an angle is given, but still no parallel lines. (I also see some DGS "constructions" mentioned there -- I wonder whether this is similar to Euclid the Game, as alluded to in last Friday's post.)
If you must, here is a modernization of the proof of Proposition 12:
Given: the segments and angles in the above construction.
Prove:
Proof:
Statements Reasons
1. bla, bla, bla 1. Given
2.
The proof is trivial since both
Before I complete this post, let me point out that since I am a Southern Californian, there's no way I can go without mentioning the tragedy that occurred here yesterday. NBA star Kobe Bryant is no longer with us.
I've actually mentioned Bryant several times on the blog before. Just last week, I wrote about the P.E. class at the continuation school five years ago -- the one where I joined in all the fun by lifting weights and playing basketball. Well, even though I didn't say much about the class, I did post that I wore a Lakers jersey for the occasion -- and of course, it was a Kobe jersey. (It goes without saying that I wore the jersey again yesterday.)
Also five years ago, I used to tutor some math students, many of whom were Korean immigrants. I recall that many of the guys didn't know about American culture. But they knew about the Lakers -- and of course, they knew about Kobe.
One day when I subbed in a middle school English class, I needed an example of a simile, and so I quoted Bryant as he said that his team was "soft like Charmin."
The saddest part of all is the loss of the three girls -- including the star's daughter -- who had been on their way to participate in a basketball game. Based on their ages, the girls were all in middle school, likely the seventh or eighth grade. In other words, they were the same ages as some of the students I'll see this week.
Some of my New Decade's Resolutions are relevant here. Although today is the third day (on my calendar), the third resolution doesn't quite fit here as it mentions math: "We remember math like riding a bicycle." Then again, basketball came as easily to a natural talent like Bryant as bike riding comes to the rest of us.
The fifth resolution is "We treat the ones born in 1955 like heroes." Of course, Bryant was actually born in 1978, not 1955. Then again, NBA star Mychal Thompson, who was born in that year, spoke on sports radio today in tribute to Bryant.
And Kobe himself, like most sports stars, almost certainly lived by the tenth resolution: "We are not truly done until we have achieved excellence."
I end this post by quoting the legend's last words as a player: "Mamba out."
Anyway, here is the Review for Chapter 9 Test.
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