Area of a regular hexagon with side 4throot(12)
In a regular hexagon there are many equilateral and 30-60-90 triangles. Indeed, by joining opposite vertices we obtain six equilateral triangles (with side length same as the original hexagon), and each of these can be split into two 30-60-90 triangles (with this side length as the hypotenuse). Thus the base of each 30-60-90 triangle is half of the original side length or 4throot(12)/2, and the height of each 30-60-90 triangle, same as the height of each equilateral triangle, is 4throot(12)sqrt(3)/2.
Then the area of each equilateral triangle is:
A = (1/2)bh
= (1/2)4throot(12)4throot(12)sqrt(3)/2
= (1/4)sqrt(12)sqrt(3)
= (1/4)sqrt(36)
= 6/4 = 3/2
and there are six such triangles in the hexagon. Therefore the total area is 6(3/2) = 9 square units -- and of course, today's date is the ninth.
Areas of regular hexagons aren't truly covered in the U of Chicago text. While area is taught in Chapter 8, the 30-60-90 triangles needed here aren't taught until Lesson 14-1. And while regular hexagons are mentioned in this lesson, only their heights -- not their areas -- appear there.
Notice that if the side of the hexagon is s, the same reasoning above gives s^2(3sqrt(3)/2) as the area of the hexagon. So if s is a whole number, then the area can't be a whole number because we can't get rid of the sqrt(3) factor. And if s is a whole number times sqrt(3), it appears as s^2 in the area so the sqrt(3) gets squared, so that we still can't get rid of the sqrt(3) factor in the area. Thus in order for the area to be a whole number, the side length must involve the fourth root of 3, not the square root. This explains why Rapoport chose a strange value of 4throot(12) in this problem.
By the way, this is the only Geometry problem on the Rapoport problem for all of September. So don't expect me to mention her calendar for the next few weeks.
This is what I wrote last year about today's lesson. I have updated the post to reflect the number of results of a certain Google search.
Lesson 1-6 of the U of Chicago text is where the study of geometry formally begins. This section states that three important words in geometry -- point, line, and plane -- are undefined. This may seem strange, for mathematics is all about definitions, yet these three important concepts are undefined.
In college-level math, one learns that these undefined terms are called primitives, or primitive notions. Just over a hundred years ago, the German mathematician David Hilbert declared that there are in fact six primitive notions in geometry: point, line, plane, betweenness, lies on, and congruence. But most textbooks list only the first three as undefined terms. This is because texts actually define the last three using concepts from other branches of mathematics. "Lies on" or "containment" -- that is, what it means for, say, a line to contain a point -- is defined using set theory (which is why the very first sentence of this section states that a set is a collection of objects called elements). "Betweenness" of points -- that is, what it means for a point to be between two other points -- is defined later in this chapter in terms of betweenness for real numbers (their coordinates of course). And the definition of "congruence" is the cornerstone of Common Core Geometry -- we use reflections, rotations, and translations to define "congruence." So we're left with only three primitive notions -- points, lines, and planes.
Lesson 1-6 is a fairly light lesson. So point, line, and plane are undefined -- big deal! Of course, we can do things with points, lines, and planes, but that's not until 1-7. So instead, I use this as an opportunity to remind the students the reasons for taking a geometry course.
The students in a geometry course are around the age where thoughts such as "I hate math" become more and more common. This is the age where they wonder whether they'll ever have any use for the math that they're learning. They begin to wonder whether they'll ever use any math beyond what they learned in elementary school and wish that math classes were no longer required beyond elementary school, for can't they live very successful lives not knowing anything higher than fifth grade math?
As of today, a Google search for "I hate math" returns 474,000 results. And we can easily predict the most common reason for hating math -- of course it's because it's hard. We don't hate things that are easy -- we hate things that are hard. And the class that turns so many off from math is algebra. Indeed, if you choose some school and tell me only its standardized test scores in ELA and math, I can very reliably tell you whether it's an elementary or a secondary school. If the math score is higher, it's probably an elementary school -- if the ELA is higher, it's likely a secondary school. And so now we, as geometry teachers, have the students for the math course right after the one that caused them to hate math in the first place.
The number of search results for "I hate math" has actually decreased slightly from last year. This recent result, which as of 2020 is no longer the top Google result, gives eight reasons why people hate math:
http://www.patrickbetdavid.com/8-reasons-people-hate-math/
[2020 update: Patrick Bet David also published a new book last month, Your Next Five Moves. The main subject of this book is business.]
The #1 reason given at this link is "it's like learning a new language" -- in other words, it's hard. Two other reasons mentioned here are related -- "it's cool to hate math" and "only nerds like it." This is, of course, why I first came up with the concept of a "dren" -- or reverse nerd. The idea is to counter the idea that being bad at math is "cool" and good at math is "nerdy" -- instead, if you're bad at least at basic math, then you're "drenny" (or "drennish").
By the way, this page now has a comment section. Bobby Oliver's comment (from around New Year's Day this year) is illuminating here:
Bobby Oliver:
My high school experience (Algebra I, II, and Geometry) went like this: Teacher demonstrates a few problems on the board. Makes reference to how it is explained in the textbook. Assigns 30 problems for you to do as homework. You go home and have difficulty working the problems. Next day you come to class. Most of the problems you have done wrong. You grade your own paper. Teacher calls roll asking you for your grade. If you make below a 75 (which was a "D" back then, you could rework all the problems at 100 % and she would raise your lower grade up to a 75 for your 100% work. Now you're doing 60 problems per night.... getting the other problems wrong a lot, too. Teacher works all the homework 30 problems at an overhead or doc cam. Teacher assigns more problems. Over and over, this is repeated. You go in after school of extra help. Teacher explains the problems the same way as she did in class. You go in for more help. Then comes the test: 30 problems to solve in class. So goes the cycle over and over. In geometry, now you have to do 45 problems per night. It was very frustrating. That type of teaching did teach me something that I never wanted to do as a foreign language teacher with my students.
Thus Oliver isn't really criticizing math itself, but the dependence on traditionalist p-sets. (I must admit that 45 Geometry problems per night does seem a bit excessive.)
As of 2020, here is the top result for "I hate math" (even though the article is dated 2018):
https://towardsdatascience.com/i-hate-math-part-1-4e793f5a8f72
(Note: This article is titled "Part 1," but there is no "Part 2" posted anywhere.)
The author, Pablo Casas, writes that at first he didn't understand why he had to learn about derivatives in Calculus, until he was able to apply them to artificial intelligence and how robots can learn:
They learn thanks to an algorithm called backpropagation…. by using derivatives!!!
So why do we require students to take so much of a class they hate in order to graduate high school? As it turns out, we can answer this question from one of the sections that we've skipped, Lesson 1-1:
"A point is a dot."
And this section gives many examples of dots -- the pixels on a computer screen. The shapes that appear on our screens consists of dots, which can be modeled in geometry by points. We look at images on our TV screens all the time. And one of the most geometry-intensive computer programs that we have are video games -- we must create images consisting of dots that move rapidly.
The point of all this is that we can surely have math without entertainment, but we can't have entertainment -- at least not most modern forms of entertainment -- without math. We can only imagine how much technology would disappear if math were to disappear.
Since I posted this last year, a huge change occurred in the video game world. If you asked someone what the most popular game in the world was, the answer would have been some established franchise such as Assassin's Creed, Call of Duty, or Madden NFL. But this year, a newcomer has suddenly become the world's most popular game -- Fortnite.
Here is a link to the a list of open jobs at Epic Games -- the creator of Fortnite:
https://epicgames.wd5.myworkdayjobs.com/en-US/Epic_Games
We select "Engineering" on the left side. As of the date of this post, 56 jobs are available (nearly as many as last year). One of them is:
https://epicgames.wd5.myworkdayjobs.com/en-US/Epic_Games/job/Cary-NC/Lead-Support-Engineer_R2286-1
Support Engineering Manager/Lead
Minimum Qualifications:- Bachelor's degree (computers, math, science preferred) or equivalent work experience
[emphasis mine]
Last year, there was even a job opening in animation that specifically mentioned solving systems of equations. Hey Algebra teachers, try mentioning this in class when you're in the unit on solving systems and students start complaining about having to do the work.
(Note: For future reference, I notice that the coding language at Epic Games appears to be C++, so all the Java I learned this summer won't help me here. Then again, I'm already familiar with C++ from my college days.)
Yet according to bet David, cool people hate math and only uncool "nerds" like math. It then follows that the Fortnite and Unreal Engine 4 programmers are uncool -- or by extension, that Fortnite and Unreal Engine 4 are themselves uncool. Therefore only "nerds" ought to play these games, and everyone else should avoid them because they're uncool.
And once again, math, including Geometry, is needed for video games because objects are made of pixels or dots, and...
"A point is a dot."
Last year, I was attending some festival. There was a booth for the Electrical Training Institute, and they were handing out flyers for an apprentice opportunity for the Inside Electrical Wireman program. Let me quote some of that flyer here on the blog:
Required Documents:
3. Must provide proof of completing an Algebra course by submitting one of the following:
A. Official Sealed transcripts showing at least: Two semesters of High School Algebra with a passing grade of "C" or better OR
B. Official Sealed transcripts showing at least: One semester of College Algebra with a passing grade of "C" or better OR
C. Certificate showing successful completion of Electrical Training Alliance's On-Line Tech Math Course.
I remember how one of our traditionalists, Bill, once responded to those who propose replacing college-prep courses with vocational courses for those with no interest in going to college. His response was that even vocational courses -- such as those for becoming an electrician -- require knowledge of "basic algebra." He never explains how much algebra is "basic" -- I was thinking that perhaps it meant the first semester of Algebra I, or the SAT's "Heart of Algebra."
But this advertisement makes it clear that two semesters of Algebra I are required. So this includes such second semester topics as factoring and the Quadratic Formula. Once again, this math is required to work with electricity. So once again, let's ask our math haters whether they'd like to live in a world without electricity, since that's what a world without math would really look like.
Elementary school math -- at least early elementary arithmetic (before the dreaded fractions) -- is easy. And college majors majoring in STEM know the importance of learning math. The problem is those in-between years in middle and high school. If math were merely an elective in secondary school, many students would avoid it and choose easier classes. Then there wouldn't be enough STEM majors in college because they wouldn't have had the necessary algebra background. The only way to bridge the gap between "math is easy" (early elementary) and "math is important" (college STEM majors) is to require the subject during the intervening middle and high school years. Otherwise we'd have no modern technology or entertainment (like Fortnite).
When I give notes in class, I prefer the use of guided notes. This is not just because I think the students always need the extra guidance, but that I, the teacher, need the guidance. In the middle of a lesson, I often forget what to teach, or forget how to explain it, unless I have guided notes in front of me.
And so today's images consist of guided notes. I begin with Lesson 1-6 and its definitions. Here I emphasize the fact that point, line, and plane are undefined by leaving spaces for the students to write in their definitions -- which they are to leave blank (or just write "undefined")! Notice that Lesson 1-6 distinguishes between plane geometry and solid geometry -- a crucial distinction in Common Core Geometry because the reflections, rotations, etc., that we discuss are transformations of the plane.
Then I move on to Lesson 1-1. This is based on an online discussion I had a few years ago on why students should learn math. I also include it as guided notes so that the students are listening when the teacher gives the reasons that they are taking this course. (The answers to the blanks beginning with the conversation are MBA, polynomial, investing, data, supermarket, and -- the object Americans use that has more computing power than the A-bomb -- cell phone!)
In the years since I first posted this lesson, I've been thinking about how to rewrite the lesson so that students are more responsive to it. In particular, I was thinking about last week's bridge puzzle, on which I wrote, "Back then, people spent their Sundays taking walks over bridges." Think about that statement for a moment -- entertainment back then was limited to Sundays. Back then, six days a week were workdays, on which no one expected to be entertained. Even on Sundays, the morning were devoted to church, so only the afternoons were amusing. And when we finally get to Sunday afternoon, all people did was cross bridges -- something that we wouldn't find entertaining today.
What has changed since the 18th century? The answer is technology -- that is, mathematics. Just as I mentioned in the worksheet, one especially widespread form of entertainment is the cell phone. We don't have to wait until Sunday afternoon for entertainment -- with our modern phones, we can be entertained at almost any time. Games and videos can be played anywhere, and if our friends live across the bridge, we don't need to cross it, since we can call or text them. All of this technology is available now because of mathematics.
Yet the greatest paradox is that, while math makes all of this technology possible, students use this technology to justify avoiding the study of mathematics. Traditionalists don't like the fact that students don't study as much now as they did in the past. Nowadays, the idea that one should study for two hours at once -- that is, go two hours without cell phones, TV, or other entertainment -- is unthinkable for many students, yet before modern technology, the idea of being entertained as often as once every two hours was equally unthinkable.
We don't need to go back to Euler's day, 300 years ago, to find generations of students who were willing to work hard and forego entertainment. But some traditionalists go back to 100-year-old texts because they feel that newer texts have too many pictures. Technology progressed so much that photography, even in the mid-20th century, was inexpensive (going back to "a point is dot,") but that photo technology made texts even as early as then too entertaining, and therefore, not educational enough for the traditionalists.
The phrase Millennial Generation refers of course to the millennium. Strictly speaking a millennial is one who was born in the old millennium and graduated from high school in the new millennium. By this definition, I am not a millennial, since I was born in December 1980 and graduated high school in June 1999 -- still the old millennium. But some authors, such as Mark Bauerlein, consider the Dumbest Generation to be anyone under 30 at the time of its publication (2008). By this definition, I am a member of the "dumbest generation."
Naturally, most traditionalists and members of older generations who criticize millennials blame the problems of our generation on technology. This is why, when I teach this lesson, I want to point out that using technology to justify being a "dren" who can't count change makes us -- including myself as a member of the generation -- look bad. Of course, in a few years, I can't credibly claim to be in the same generation as my students -- some incoming students starting high school this year are already born in the new millennium (and so are no longer "millennials"). The important thing is that all of us, my age and younger, need to avoid being the "dren" who can't solve simple math problems and instead work on becoming the hero whose knowledge of math saves the day. This is what I want my students to realize.
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