This is what Theoni Pappas writes on page 297 of her Magic of Mathematics:
"This wonderful magic square was created by Leonhard Euler in the 18th century. As in most magic squares its rows, columns, and diagonals total the same number, in this case 260."
This page is titled "Leonhard Euler & the Knight's Tour." It refers to yet another magic square, this time of order 8. As usual, here's the link:
The term "Knight's Tour" refers to a chess knight. Pappas says a knight "can land on every number of the entire square sequentially from 1 to 64, by just following the moves allowed of the knight."
If you read the comments at the above link, notice that the attribution to Euler is controversial. It's possible that the creator of this square is William Beverly, 65 years after Euler's death -- and in fact, the charge is that Euler never came up with a knight's tour at all, much less a magic square. Finally, it's pointed out that not only don't the diagonals add up to 260, it's provable that no knight's tour can be a magic square that includes the diagonals.
Well, we're done with magic squares in Pappas, but not with Euler. For the rest of this week she writes about something else attributed to Euler -- and hopefully this attribution is correct. One commenter, John V., says:
Regarding the attribution, people love appeals to authority and it is always easier to remember a famous name than one less famous. I think that is one reason for so many incorrect attributions.
Another related reason is hero-worship. People who admire a famous person are quite inclined to believe anything positive about them. I suspect many of the anecdotes about folks like Feyman, Erdos, [....]
Erdos -- well, let's not tell Hoffman that many of the Erdos stories in his book are false. Oh, and speaking of Erdos:
The final chapter -- more of an epilogue -- of Paul Hoffman's The Man Who Only Loved Numbers is called "We Mathematicians Are All a Bit Crazy." This chapter isn't numbered eight, but rather a sideways eight -- Chapter Infinity. Hoffman doesn't begin with a quote, but instead tells us a little something about himself:
"I am not a mathematician. I have never proved a theorem, let alone a surprising conjecture. My Erdos number is infinity."
Well, that explains why the chapter is numbered infinity. The title, meanwhile, comes from a quote Erdos heard from fellow mathematician Edmund Landau. As Hoffman says, many mathematicians indeed suffered from mental illness, most notably John Nash, the main character of A Beautiful Mind.
Hoffman ends the book by telling us how he meets Erdos in 1986, as he prepares to write this biography about him. The author had the famous mathematician critique his book, and Erdos told him to leave out the part about Benzedrine, because "I don't want kids who are thinking about going into mathematics to think that they have to take drugs to success." And here's Hoffman's response to close out this most enjoyable book:
"That was Erdos, always thinking about the epsilons."
Well, I interacted with the epsilons today. You see, today I subbed in a high school history class. As I've done before, I want to write about this day of subbing in "Day in the Life" format:
6:55 -- Believe it or not, this teacher has a zero period class. This is the first of three sophomore World History classes. I expect there to be many tardies in this class, and there are (about ten).
The students are learning about the Industrial Revolution and answering questions on three pages in the text. In the middle of this class, the dean enters and gives a lecture on classroom and school behavior rules.
Five students fail to answer even the first question, which asks for seven definitions from the text. I inform the regular teacher in my sub notes for him, but unfortunately there is no seating chart or any way to identify the student names. It's also difficult because many students say that they are skipping around instead of doing the definitions first -- and I know that some of the students making this claim are really trying to pass off yesterday's work as today's. Some students also give the excuse that the dean's ten-minute lecture prevents them from doing work the rest of the hour. I do have the foresight to tell a tardy student who arrives 20 minutes before class ends that 20 minutes is nonetheless enough time for him to do the definitions. Of course, he still doesn't do the work.
8:00 -- First period arrives. This is the only junior U.S. History class of the day. Two of the students are wearing Dodger shirts, and so I ask them whether they are looking forward to watching Game 1 of the World Series tonight. Notice that this is the first time a Southern California team has qualified for the finals in one of the four major sports since I created this blog -- the Kings having won the Stanley Cup a few months before I began the blog. (Yes, some Northern California teams have played in the Finals, most notably the Golden State Warriors, but this is Southern California.) It's the first class of the day for many of these students, and so there continue to be many tardies in this class too (about five).
In this class, students are researching the Ku Klux Klan and completing a written group report. They are allowed to use Chromebooks for this assignment. This time, I catch two students not doing the work, with one of them having a phone out. Once again, I don't have the names, but I've decided to write in the sub notes where they are sitting as well as a description of their backpacks. You see, I figure that the students will wear different clothes but the same backpack tomorrow, so describing the backpack gives the teacher a hint who the student is.
8:55 -- The second World History class arrives. This time, I'm more prepared for the students as I let them know up front that they are to complete at least the first question with all seven definitions. One student fails to finish the seven definitions -- but this time I manage to catch his name. A few students start working on their Chemistry assignment. I ask them whether they know what holiday it was yesterday. One girl knows that it was Mole Day. (And since these are tenth graders in Chemistry, they are "sopho-moles," according to yesterday's joke page.)
9:50 -- The students leave for nutrition.
10:10 -- The third and final World History class arrives. Two students fail to finish their definitions, and I must describe their backpacks again for the teacher.
11:05 -- Fourth period is the teacher's conference period. As it happens, I must cover another class during this time -- a special ed English class. Most of the students are sophomores, but there are a few freshmen and juniors in the class. Many of these students have just arrived in the country and speak mostly Spanish or Arabic, with very little English.
The students are supposed to be working on a speech about animals, but the regular teacher knows that they would struggle on this without her. So it becomes a study hall instead. I see that some of the students are working on math (Algebra I), and so I try to help two of them (a brother and sister, I believe). The lesson is on linear functions and mathematical modeling. Since this is the only mathematical part of my day, let me at least describe this in a little more detail.
CCSS.MATH.CONTENT.HSF.BF.A.1.A Determine an explicit expression, a recursive process, or steps for calculation from a context.
The example the students are given shows a pattern made out of tiles:
Figure 0: 3 tiles
Figure 1: 7 tiles
Figure 2: 11 tiles
Figure 3: 15 tiles
Figure 4: 19 tiles
I decide just to tell the students that the function is f(x) = 4x + 3, but I do show them why this function is correct by plugging in 0, 1, 2, and so on for x and pointing out that this gives the correct number of tiles. Students are then asked how many tiles Figure 20 will require. Of course, the answer is f(20) = 4(20) + 3 = 80 + 3 = 83 tiles. They are also asked to find which figure has 203 tiles. I tell them that they must set up an equation, 4x + 3 = 203, which they can solve easily -- 4x + 3 = 203, 4x = 200, x = 50, so it's Figure 50.
I realize that the language is the biggest barrier for these siblings, not the math. But as I've written on the blog before, I had many English learners in one of my student teaching classes, and I was able to help them. I only know a few words in Spanish (mostly numbers and a few other key math terms), but fortunately they already know the basics (in both English and math), and so I think they can understand most of what I say.
As you may remember from last year, I like to hand out pencils around the holidays. Since there are only 13 students plus a TA, I give everyone a Halloween pencil.
12:05 -- In this district, students have a sort of "Interventions" or tutorial class. At all of the other schools in this district the time is embedded into the two-hour block schedule. But the school I subbed at today clearly does not have a block schedule. So instead, Interventions is considered part of lunchtime. All freshmen must attend Interventions, but older students only go if they need extra help or are failing a class. For some reason, Interventions is labeled as an "eighth period" class in attendance, even though there is no seventh period.
Since there are no freshmen in the history classes, only nine sophomores show up today for tutorial, plus one junior, a peer assistant. I see that she's wearing a Dodger "Fly the Pennant" blouse, and so I ask her about tonight's game. She tells that -- believe it or not -- her mom is taking her to the game tonight! I ask her about the cost of the tickets, and she informs me that they are $600 each -- but it would have been $1000 were the Yankees playing instead of the Astros.
Two sophomores from zero period complete their seven definitions during this time.
12:30 -- Lunchtime proper begins.
1:05 -- During fifth period, this teacher coaches the JV football team. I enjoy working out and actually lifting weights with the team. Of course, I can't bench press nearly as much as what these guys can lift!
1:55 -- Fifth period ends. Most students and teachers with a zero period don't need to attend sixth period, and so this ends my day. I go home to type up this blog entry.
Teachers are always working on improving, and often have specific goals for things to work on throughout a year. What have you been doing to work toward your goal? How do you feel you are doing?
I've been thinking about using backpacks to identify students ever since I left my old school and returned to subbing. I believe that I developed some bad classroom management habits back when I was a sub. The students would misbehave, I wouldn't know the kid's name, and then I end up not punishing the student at all. (Of course I don't even bother to ask for the student's name -- why in the world would students who know they're in trouble reveal their names?) The problem would be that I then developed the habit of not punishing students effectively, which persisted even after I became a regular teacher (and thus knew the students' names). Identifying students by backpack allows me to focus on effective responses to misbehavior, rather than attempting to figure out their names.
That being said, I still need to work on emphatically calling out a misbehaving student's name in cases when I do know the name. Only once today do I catch a student's name -- but that one time, I still fail to call out the name. It's as if I'm afraid to call it out, thinking that the student would then begin to argue -- even though I'm working on a teacher look to avoid such arguments.
Oh, and I do use my teacher look effectively during one of the classes. A group of girls start laughing during attendance, and I use teacher look to quiet them down.
It's time to review for the Chapter 4 Test. Two years ago, I didn't write much about the review but went directly to the worksheet, so here it is. (Notice that in the past, I referred to it as a Chapter 3 and 4 test and included questions from both chapters.)
This is what Theoni Pappas writes on page 296 of her Magic of Mathematics:
"Benjamin Franklin was a magic square enthusiast. In fact, while he was a clerk at the Pennsylvania Assembly, he often relieved the tedium of his work by making magic circles."
This page in Pappas is titled "Benjamin Franklin's Magic Circle." Here's a link to Ben Franklin's magic circle, but this link mentions other Franklin magic diagrams, including the order-16 square that we already saw last week.
According to the link, here are all the combinations that add up to 360 or 180:
The circle uses the integers from 12 to 75 plus another 12 in the center which is used for all summations.
The eight numbers in each radius plus the central 12 sum to 360.
The eight numbers in each circle plus the central 12 sum to 360.
The eight numbers in each eccentric circle plus the central 12 sum to 360. Franklin’s circle apparently shows 20 of these eccentric circles, but is very hard to read. I show only 8 such circles but obviously there are many more!
Any half circle in the top or bottom, plus half of the center number sum to 180.
Any half radius plus half of the center number sum to 180.
Any four adjacent numbers in an ‘almost square’ plus half of the center number sum to 180. For example, 73 + 14 + 15 + 72 + 6 = 180.
What other combinations are there?
Come to think of it, it's fitting that the magic constant of a circle would be 360 -- as in 360 degrees.
Chapter 7 of Paul Hoffman's The Man Who Loved Only Numbers is called "Survivors' Party." As usual, we begin with a quote -- this time it's a letter:
Dear Ron[ald Graham], Please send me Uncle Paul's birthdate if you have it. Then we will have something to look forward to and not be so depressed. Thanks, Ed P.S. I hope he is giving the SF all sorts of trouble.
And Ronald Graham replies that Erdos -- whose birthday is March 26th, 1913 -- is too busy reading the Book to give anyone much trouble. (By the way, Hoffman never explains who "Ed" is.)
As the title and letter imply, this chapter takes place after the mathematician's death. Every year around his birthday, there's a math conference in Memphis, Tennessee. In 1997, the first birthday after his passing, there is a "survivors' party" held in his honor.
Mathematicians at this reminisce about the days they've spent with Erdos. One of them, John Selfridge, talks about how Anyuka -- Paul's mother -- supports her son as he and Selfridge write a joint paper proving that the product of consecutive integers is never a power. At his funeral six months earlier, Selfridge helps deliver the mathematician's ashes to his mother's grave, and John speaks only the word "Anyuka" at the grave.
Two other mathematicians laugh as they remember how Erdos tells them "Good morning!" or even "Merry Christmas!" and then immediately starts talking about math.
At the University of Memphis where this party takes place, there are several cartoons posted on the walls of the math department. One of them is Doonesbury. The math professor is being rebuked by the chairman of the department because he dared to give a student a B+. The chairman tells him that this is a "new generation of students" who insist on a certain comfort level, and the professor can't fail the students even if they think that 1 + 1 = 3. Even though this comic is dated 1994, I can easily see many traditionalists, most notably Bill, agreeing that this definitely describes the current generation of college students.
That's right -- you may notice that today's post is labeled "traditionalists." That's because there's something I want to do on this special day.
For you see, today, October 23rd, is Mole Day. It's named after the mole as used in Chemistry. Also known as Avogadro's Number, one mole is 6.02 * 10^23. Thus Mole Day is the chemists' Pi Day -- the expression 10^23 is converted into the date 10/23, or October 23? (By the way, an upcoming redefinition of SI will define Avogadro's Number to be exactly 6.022140857 * 10^23.)
The mantissa 6.02 is also used on this special day. We'd like to make it into a time, 6:02, just as the next three digits of pi, 159, are used to denote the time 1:59 on Pi Day. But unfortunately, 6:02 AM is before school while 6:02 PM is after school. Mole Day participants often declare that the holiday is to be celebrated for 12 hours, from 6:02 AM until 6:02 PM. Then this covers the whole school day. Oh, and notice that after I corrected my post count last week, today is my 602nd post!
Here's a link to the official Mole Day website. Apparently this year's theme is "The MOLEvengers."
Q: How would you describe a stinky chemist? A: Mole-odorous Q: What kind of fruit did Avogadro eat in the summer? A: Watermolens
Q: What kind of test do chemistry students like best? A: Mole-tiple choice.
Q: Why is Avogadro so rich? A: He's a multi-mole-ionare!
Today I wish to link to the aforementioned traditionalist Bill. This post is actually dated August 8th, but I actually linked to another article in my August 10th traditionalists post and forgot about Bill. You'll find out soon why I chose Mole Day to return to Bill.
This link is at the Joanne Jacobs website -- one of Bill's favorite places to comment:
As usual, Jacobs herself links to another site, EdSource. I've referenced EdSource directly in the past, but I want to remain at Jacobs since that's where Bill's comments are. The article refers to a remedial math class taken at Cal State Dominguez Hills. (Hey -- that's the second mention this month of the school where I earned my credential!)
For example, a traditional for-credit college algebra class usually meets three times a week for 50 minutes. In contrast, the co-requisite (Aida) Tseggai attended met for an hour and ten minutes three times a week for instructor Cassondra Lochard’s lectures; in addition, students had an extra group hour weekly with a teaching assistant plus one-on-one tutoring. In contrast to regular classroom protocol, the teaching assistant circulated among the desks during lectures, softly giving advice and reviewing students’ calculations and algebra formulations.
The article goes on to mention Raquel Herrera, a biochemistry student. (Chemistry -- oh, so that's why I'm posting this on Mole Day!) Herrera took -- and failed -- the remedial math class mentioned in the article. And here's Bill's response:
Perhaps Ms. Herrera has been lied to all of her life in school by being given grades rather than actually earning them. I would hope a major in biochemistry would require more math than simply algebra, how would she make it through two semesters of general chemistry, organic chemistry (which was a 2nd year course for chem majors), and Bio-Chemistry?
Another commenter, "Midwestern girl," confirmed that Calculus is indeed required. Bill continues:
That’s what I thought since doing chemistry usually requires math through at least Calc I/II, though my old 9th grade algebra teacher once said “You guys and gals have no trouble with algebra, you just can’t add, subtract, multiply, and divide)…LOLOL
If Ms. Herrera had failed Algebra twice (which she should have taken in high school), she would have never made it through calculus I/II…I can still do basic limits and derivatives, even though I haven’t taken calculus in 35 years (lord knows I have tried to forget it, but that math has warped my fragile little mind – E Cartman)
Let me try to figure out what is going on here. I want to guess Herrera's story here -- why she chose biochem as a major, as well as why she wants to change to "liberal studies" to be a teacher. But I don't wish to use the name Raquel Herrera in this story because my guesses may be false. So instead we will make up a hypothetical student -- let's call him Amadeo (as in Amadeo Avogadro, for whom Avogadro's Number was named). Oh, and the gender change is intentional, since I obviously don't want to give the impression that girls are the only ones who fail remedial math.
Little Amadeo always enjoyed science. When he grows up, Amadeo wants to become a scientist, or maybe even a doctor. So naturally he would choose biochem as his major.
But Amadeo doesn't like math. Ever since he was in the kindergarten or first grade, his teacher would ask him a math question, and more often than not she'd tell him that he was wrong. By the time he reaches the third grade, he's decided that he hates math. He is definitely what I'd call a "dren." When the teacher is giving a math lesson, Amadeo tunes out and counts down the minutes until it's time for another subject, such as his favorite -- science.
So we can see the problem here -- Amadeo is strong at science but weak at math. Bill tells us that if Amadeo wants to be a biochem major, he should have passes Algebra I/II in high school and be prepared to take Calculus I/II in college. As we already know, other traditionalists like SteveH would take it a step further -- since Amadeo desires a STEM degree like biochem, he should be taking Algebra I in eighth grade and AP Calculus as a high school senior. But surely these classes are for above-average math students, not below-average kids like Amadeo.
Amadeo manages to earn at least a C in his math classes since he's admitted to a four-year university, but Bill says that his teachers have lied to him (yes, just like the chairman in Doonesbury). Most likely, Amedeo passes his tests by learning just math enough the night before, then forgetting it the next day. But this is no excuse to Bill -- he states that he still remembers the Calculus he learned decades ago, so Amadeo ought to remember the Algebra in the few years between his high school classes and the university math placement test.
Instead, Amadeo fails the placement test, is placed into remedial math, and then fails the remedial class too. (Yes, this professor actually does give him a failing grade.) And so he chooses to drop Algebra in favor of a Statistics course, and changes his major to become a teacher.
Bill doesn't address this part of the story in his comments, but we can guess his reaction based on his previous posts. First of all, Bill doesn't like the idea of weak math students taking Stats. To him, a true Stats class is harder than Algebra II, and so the Stats class should be open only to those who earn A's and B's in College Algebra, not D's and F's.
Second, we see that Amadeo, who isn't good at math, wants to become a teacher. If he's trying to be an elementary school teacher, he must teach all subjects, including math. The fear is that Amadeo will pass his disdain for math to the next generation of Amadeos (and Raquels), who then likewise find their own paths to becoming scientists and doctors being blocked by weak math skills.
Perhaps Amadeo might want to become a secondary science teacher instead. To teach high school science classes he'd likely need to be stronger at math, but perhaps he could get away with teaching middle school science. As a middle school teacher, he'd be able to focus on the science -- and perhaps he could give a nod to the connections between science and math. Then maybe his weaker math students might be drawn to the path to a STEM major. They could see why it's important to learn math if they are interested in science.
It would be great if every middle school science teacher could demonstrate the connection between math and science in order to enlighten students. But I can tell you of one middle school science teacher who failed to make this connection last year -- and that teacher is yours truly.
Last year, I remember one seventh grade girl who told me exactly what she wanted to be when she grows up -- a veterinarian. I still remember one day when I thought I heard her having a lewd conversation about private parts with other students. It turned out that she was actually talking about the private parts of a dog. In other words, she was talking about her dreams, her future.
But this student, just like our fictional Amadeo, is a weak math student. I remember often when I gave her a "Monday Five" worksheet (see my February 13th post for more info). I remember how much she'd struggle with multi-digit addition problems, and she complained about being required to do them. Again, she believed that only smart "nerds" do multi-digit addition without a calculator, and ignored me when I told her the opposite -- that only "drens" need a calculator for multi-digit addition.
I can easily seeing this girl meeting the same fate as the fictional Amadeo and the real Raquel. She tries to major in biology or a STEM major, fails math, and is no longer able to take the classes needed to become a vet.
I could have told her that she needed to learn more math in order to become a vet. But this would have failed, because she could have retorted that surely she needed to learn more science in order to become a vet -- especially the life science taught in seventh grade under the California Standards. But I failed to live up to my end of the bargain and teach enough science last year, except a little to eighth grade (and hence almost none to the seventh grade).
Yes, this is going to turn into yet another post where I write about last year's science failure. As today is still Mole Day, I wonder whether I should have given a Mole Day lesson last year. I admit that I was considering it last year, but in the end I didn't. And so for the rest of this post, I wish to imagine what my Mole Day lesson would have looked like.
One reason I didn't give a Mole Day lesson last year was the fact that October 23rd, 2016 happened to fall on a Sunday, a non-school day. Then again, it's not as if it would have been any better this year with Mole Day on a Monday -- as in Coding Monday (provided, of course, that computer day is still scheduled for Monday this year).
Of course, in years when Pi Day falls on a Sunday, we observe it in school the previous Friday, so it's possible that I could have done the same with Mole Day last year. (It's just that I was already noncommittal about Mole Day last year, and the Sunday thing put skipping it over the top.)
But let's imagine that I really did observe Mole Day last year. I've written earlier that my math and science units should span the full week, not just Friday. So let's write out an entire Illinois State week for the days leading up to Mole Day:
Monday, October 17th: Coding
Tuesday, October 18th: Math Traditional Lesson
Wednesday, October 19th: Learning Centers
Thursday, October 20th: Science Project
Friday, October 21st: Math Weekly Assessment
First of all, notice that last year, the Mole Week project would have been for eighth grade. That's because last year I would have taught to the old California Standards, with physical science as the eighth grade focus. This year, I would have switched to the NGSS, and chemistry appears in seventh as well as eighth grade. (So my future vet would have celebrated Mole Day this year instead.) Of course, "moles" don't actually appear until high school, but it's a good idea to introduce them if we're going to have a Mole Week.
Now what should the math lesson for this week be? The week of October 17th is the eighth week after the Benchmark Tests, and so with one standard per week, this is the eighth standard. There are two standards in NS (Number System), so we should be in EE6 (Expressions and Equations):
CCSS.MATH.CONTENT.8.EE.B.6 Use similar triangles to explain why the slope m is the same between any two distinct points on a non-vertical line in the coordinate plane; derive the equation y = mx for a line through the origin and the equation y = mx + b for a line intercepting the vertical axis at b.
(By the way, in my last post I mentioned Kate Nowak and the textbook she was writing. On her blog, she mentions how she teaches similar triangles before slope in order to meet this standard. But we must follow the naive order where all EE standards appear before any G standards.) Anyway, this standard has nothing to do with moles.
Normally, the math standards and science standards don't line up. One big reason is that the NGSS doesn't divide the middle school standards into grades -- the states do that. So it's impossible to write a text in which the math and science standards correspond to each other. This is a shame, since we want to convince the Amadeos and Raquels and future vets of the world that it's helpful to learn math if they want to major in science. So it would help if the science lessons allowed the students to apply the math they're currently learning in the math classes.
Still, it would be great during a special week like Mole Week to make the math and science lessons actually correspond. So what math standard would fit during Mole Week? Hey, that's easy -- Mole Day is 10/23 because Avogadro's Number is 6.02 * 10^23. This number, 6.02 * 10^23, is of course written in scientific notation. And hey -- there's a scientific notation standard in eighth grade:
CCSS.MATH.CONTENT.8.EE.A.4 Perform operations with numbers expressed in scientific notation, including problems where both decimal and scientific notation are used. Use scientific notation and choose units of appropriate size for measurements of very large or very small quantities (e.g., use millimeters per year for seafloor spreading). Interpret scientific notation that has been generated by technology
This standard, EE4, is only two off of EE6, the prescribed standard for this week. So it would be easy to slow down the lessons so that EE4 is covered during Mole Week instead of EE6. (We'd just have to make it up later on so that G8, the last "major content" standard, is reached before the SBAC.)
I've stated before that the Friday assessment should based on the previous week's standard in order to give the students time to learn it. So the assessed standard would be EE3:
CCSS.MATH.CONTENT.8.EE.A.3 Use numbers expressed in the form of a single digit times an integer power of 10 to estimate very large or very small quantities, and to express how many times as much one is than the other. For example, estimate the population of the United States as 3 times 108 and the population of the world as 7 times 109, and determine that the world population is more than 20 times larger.
Because of this standard, it's better to write Avogadro's Number as 6 * 10^23 (that is, a single digit times an integer power of 10) and save the mantissa 6.02 for high school. Students can then solve problems using Avogadro's Number, such as "How many atoms are in two (or whatever) moles," and similar questions. Tuesday's math lesson can begin with such problems on a worksheet.
There were a few things happening at my school during the week of October 17th. Wednesday of this week was an Illinois State observation, but the observed class was seventh, not eighth, grade.
More importantly, there was the California Earthquake Drill on Thursday. This drill always occurs on the third Thursday in October, and in theory the time should match the date -- that is, last year it was on 10/20 at 10:20 (and this year it was 10/19 at 10:19). In reality, the drill occurred at our school at 9:00 in order to avoid elementary recesses. (California state leaders should have anticipated that elementary schools would want to avoid recess. The drill should have been the third Thursday in September instead of October, so that the corresponding time would be in the 9:00 hour, before most schools have recess.)
But I arrived at school that day believing that the drill would be at 10:20 -- in the middle of eighth grade math class -- instead of 9:00, in the middle of seventh grade math. So that week, I would have made a lesson plan based on a 10:20 drill and having only half a class that day. Instead of a science project, I might have given the EE3 assessment instead. It's also possible that, with the previous Wednesday (October 12th) being Yom Kippur, this would mean two fewer school days between the introduction of EE3 and its assessment. I might have used this as a reason to give a Dren Quiz instead of the EE3 assessment.
Then Friday could be the science activity instead. Indeed, I like the idea of having a Mole Day party on October 21st as well, and it may be better to have a party on the same day as a project rather than an assessment. It actually depends on what exactly the science project is. In theory we should be using Illinois State projects only -- and there are some chemistry projects near the front of the Illinois State physical science text.
Provided that I gave a Illinois State project the previous week (meaning that I'd have fulfilled the every two-week requirement), I could create my own Mole Day project. For example, I could have the students pour 1.2 * 10^25 molecules of water into a beaker. I'd give them the information that one mole of water is about 18 grams or 18 milliliters, and so they need to pour 20 moles or 360 ml. (A crude approximation, by the way, is that 10^24 molecules of water is one fluid ounce.) I could also ask them the reverse -- pour a certain amount of water into the beaker and then figure out how many molecules it is. Notice that the standard allows students to use calculators ("scientific notation that has been generated by technology"), but they should avoid calculators during the water project in order not to get the calculators wet.
There was enough time to give this project before the 10:20 drill -- and even more time once I found out that the drill was really at 9:00. So our week would look like this:
Monday, October 17th: Coding
Tuesday, October 18th: Math Traditional Lesson (EE4, scientific notation)
Wednesday, October 19th: Learning Centers (incorporates math and science)
Thursday, October 20th: Science Project (counting molecules of water)
Friday, October 21st: Math Weekly Assessment (on EE3), Mole Day Party
Again, this was the week leading up to Mole Day Sunday last year. Some sources define Mole Week (or Chemical Week) as the Sunday-to-Saturday week containing Mole Day, and so Mole Week was actually the week of October 24th last year. Giving the Mole Week project one week later means delaying EE4 another week (so one less week to reach G8 by the SBAC). There was no earthquake drill or observation that week -- instead there was a meeting with an Illinois State consultant. There might have been more pressure on me to give an Illinois State project that week. In reality, I gave the lesson H2O + ? from the math STEM text that week -- a project without any actual H2O involved. So I feel justified in giving that project and then adding my own water project to it. Of course, I'd send only photos of the original project to Illinois State.
Sarah Carter, one of my favorite teachers to link to, is now a Chemistry teacher in addition to being an Algebra I teacher. So Mole Day would be the perfect holiday for her to celebrate. Unfortunately, there's no mention of Mole Day on her blog today (or of Hexaflexagon Day for that matter). Instead, she posts her weekly Monday Must Reads.
One of these links, though is a teacher (Katherin O'Hara) who actually gives the Monty Hall problem in her class. After talking about Monty Hall all month, I feel I should point this out. Unfortunately, O'Hara provides only a Twitter link, and I don't post link to Twitter here -- especially when the Tweet is already seven weeks old.
OK, let's finally get to the U of Chicago text.
Lesson 4-7 of the U of Chicago text is called "Reflection-Symmetric Figures." (This corresponds to Lesson 6-1 in the modern Third Edition.) Two years ago I skipped Lesson 4-6, which affected the way I covered 4-7 as well. So this is what I wrote three years ago about today's lesson:
Section 4-7 of the U of Chicago text deals with reflection-symmetric figures. A definition is in order:
A plane figure F is a reflection-symmetric figure if and only if there is a line m such that r(F)=F. The line m is a symmetry line for the figure.
In other words, it's what one usually means when one uses the word "symmetry." Some geometry texts use the term "line-symmetric" instead of "reflection-symmetric." Some geometry and algebra texts use the term "axis of symmetry" instead of "symmetry line" -- especially Algebra I texts referring to the axis of symmetry of a parabola. Some biology texts use the term "bilateral symmetry" instead of "reflection (or line) symmetry" - in particular, when referring to symmetry in animals. As animals are three-dimensional, instead of a symmetry line there's a sagittal plane.
Indeed, it is this last topic that makes symmetry most relevant and interesting. Most animals -- including humans -- have bilateral symmetry. I once read of a teacher who came up with an activity where the students look for the most symmetrical human face. The teacher blogged about how students who are normally indifferent to geometry suddenly came fascinated and engaged to learn about the relationship between symmetry and human beauty. Unfortunately, this was more than a year ago, and I can't remember or find what teacher did this activity -- otherwise I'd be posting a link to that teacher's blog right here!
In the Common Core Standards, symmetry is first introduced as a fourth grade topic:
CCSS.MATH.CONTENT.4.G.A.3 Recognize a line of symmetry for a two-dimensional figure as a line across the figure such that the figure can be folded along the line into matching parts. Identify line-symmetric figures and draw lines of symmetry. Later on, symmetry appears in the high school geometry standards:
CCSS.MATH.CONTENT.HSG.CO.A.3 Given a rectangle, parallelogram, trapezoid, or regular polygon, describe the rotations and reflections that carry it onto itself. Notice that if a reflection over a line carries a polygon to itself, then that line is a symmetry line. But symmetry lines for polygons formally appears in Chapter 5 of the U of Chicago text. Right here in Chapter 4, we only cover symmetry lines for simpler shapes -- segments and angles. The text reads:
"In the next chapter, certain polygons are examined for symmetry. All of their symmetries can be traced back to symmetries of angles or segments."
For segments, the text presents the Segment Symmetry Theorem:
A segment has exactly two symmetry lines: 1. its perpendicular bisector, and 2. the line containing the segment.
The text gives an informal proof of this -- as the mirror image of an endpoint, there can only be two possible reflections mapping a segment AB to itself. One of them maps A to B and B to A -- and that mirror must be the perpendicular bisector of AB, by the definition of reflection. The other reflection maps A to A and B to B -- which means that both A and B must lie on the mirror, since the image of each is itself. No other symmetry is possible. QED
But we also want to work with angles. The first theorem given is the Side-Switching Theorem:
If one side of an angle is reflected over the line containing the angle bisector, its image is the other side of the angle.
An informal proof: the angle bisector divides an angle into two angles of equal measure. The picture in the U of Chicago text divides angle ABC into smaller angles 1 and 2. Now the reflection must map ray AB onto a ray that's on the other side of the angle bisector BD, but forms the same angle with BD that AB does with BD. And there's already such a ray in the correct place -- ray BC. Notice that part b of the Angle Measure Postulate from Chapter 3 already hints at this -- the "Two sides of line assumption" gives two angles of the same measure, one on each side of a given ray. QED
The other theorem, the Angle Symmetry Theorem, follows from the Side-Switching Theorem:
Angle Symmetry Theorem: The line containing the bisector of an angle is a symmetry line of the angle.
Earlier this week, I wrote that we'd be able to prove the Converse of the Perpendicular Bisector Theorem after this section. As it turns out, the Side-Switching Theorem is the theorem we need.
Converse of the Perpendicular Bisector Theorem: If a point is equidistant from the endpoints of a segment, then it is on the perpendicular bisector of the segment.
Given: PA = PB Prove: P is on the perpendicular bisector m of segment AB.
Now I was considering giving a two-column proof of this, but it ended up being a bit harder than I would like for the students. But as it turns out, even though the U of Chicago text doesn't prove this converse, in Section 5-1 it gives a paragraph proof of what it calls the "Isosceles Triangle Symmetry Theorem," and the proof of this one and the Converse of the Perpendicular Bisector Theorem are extremely similar. After all, we're given that PA = PB -- so PAB is in fact an isosceles triangle!
Proof: Let m be the line containing the angle bisector of angle APB. First, since m is an angle bisector, because of the Side-Switching Theorem, when ray PA is reflected over m, its image is PB. Thus A', the reflection image of A, is on ray PB. Second, P is on the reflecting line m, so P' = P. Hence, since reflections preserve distance, PA' = PA. Third, it is given that PA = PB. Now put all of these conclusions together. By the Transitive Property of Equality, PA' = PB. So A' and B are points on ray PB at the same distance from P, and so A' = B. That is, the reflection image of A over m is B.
But, by definition of reflection, that makes m the perpendicular bisector of AB -- and we already know that P is on it. Therefore P is on the perpendicular bisector m of segment AB. QED
Let's think about what we're trying to prove here. We want the Converse of the Perpendicular Bisector Theorem -- and consider what I wrote earlier about the proof of converses. The proof of the converse of a statement often involves the forward direction of the theorem and a uniqueness statement -- and even though we didn't use the forward direction of the theorem here, we did use a uniqueness statement here. As it turns out, given two distinct points A and B, there exists only one line m such that the mirror image of A over m is B -- and that line is the perpendicular bisector of the segment AB. And so if we can somehow find out another way that the mirror image of A over m is B, we'll have proved that m is the perpendicular bisector of AB. So that's exactly what we did above -- we proved that a certain line (the angle bisector of APB) is the perpendicular bisector of AB.
In this section, we found symmetry lines for simple figures such as segments and angles. But can we find symmetry lines for the simplest figures? As it turns out, a point has infinitely many lines of symmetry -- any line passing through the point is a symmetry line. But a ray has only one line of symmetry -- the line containing the ray.
Finally, does a line have a line of symmetry? This is exactly the answer to Question 25 of this section, in the Exploration/Bonus Section. A line -- considered as a straight angle -- contains more than one symmetry line. This is because any point on the line can be taken as the vertex of that straight angle. Since straight angles measure 180, their angle bisectors must divide them into pairs of 90-degree angles. Therefore, any line perpendicular to a line (straight angle) is a symmetry line of the given line. This is what I called the Line Perpendicular to Mirror Theorem. It implies that a line (straight angle) has infinitely many symmetry lines. (Of course, the line has one more symmetry line that I didn't mention -- namely the line itself.)
I included Question 24, even though it appears to mention corresponding and same-side interior angles formed by two lines and a transversal. But nowhere in the question does it mention anything about the two lines being parallel.
I left out Questions 16 and 17, which give the construction of an angle bisector. I finally plan on going to constructions sometime next week. But here's another video from Square One TV, where doctors have to perform a "bisectomy" on an angle. (Unfortunately, only the entire 30-minute show is available on YouTube -- the "bisectomy" doesn't begin until the 11-and-a-half-minute mark.)
This is what Theoni Pappas writes on page 293 of her Magic of Mathematics:
3) There are many ways to transform an existing magic square into a new magic square.
This is the third page of the magic square subsection. On this page Pappas gives Example (a), which is to transform each term linearly. We're already familiar with this, since Pappas uses such linear transformations to make "19 94" appear in the magic square of order 3 -- and then we did the same thing to make "20 17" appear. Examples (b), (c), and (d) don't appear until page 294 -- but since tomorrow's the weekend and a non-posting day, let me post these examples today:
(b) If two rows or two columns, equidistant from the center are interchanged, the resulting square is a magic square.
(c) Interchange quadrants in an even order magic square.
(d) Interchange partial quadrants in an odd order square.
Most of page 293 is taken up by Ben Franklin's huge 16 * 16 magic square. Of course I won't write out all 256 entries here, so let me just provide a link instead:
Some of the properties of this magic square are mentioned at the link above. Nonetheless, I'll rewrite the entire caption from Pappas anyway:
"This is Benjamin Franklin's super duper 16 * 16 magic square. It has all the properties of the regular magic square except its corner to corner diagonals do not total to its magic number, 2056. But at the diagram illustrates its magic number pops up in so many ways, such as -- broken 8-diagonals, broken 8-parallel rows, any 4 * 4 square; and perhaps you can find more. In the 1952 Journal of the Franklin Institute, Albert Chandler contends that this magic square is not Franklin's original, but one that was set incorrectly by a printer."
Chapter 6 of Paul Hoffman's The Man Who Loved Only Numbers is called "Getting the Goat." Since we're covering only one chapter today, we should be able to cover it completely. As usual, the chapter begins with an opening quote:
"My only advice is, if you can get me to offer you $5,000 not to open the door, take the money and go home."
-- Monty Hall
Wait, surely that's sounds familiar -- Monty Hall! Yes, the game show legend passed away three weeks ago, and I spent my first two posts afterward writing about the Monty Hall problem. It goes without saying that Hoffman is mentioning Hall in this post not because he's into game shows, but because he wants to write about the game shows, but about the Monty Hall problem (actually, he will use the word "dilemma" instead of problem. So yes, this means that this will my third post which I'm devoting to the Monty Hall dilemma.
In this chapter, Hoffman writes about a famous solver of the Monty Hall dilemma. Marilyn vos Savant writes a weekly column, described by Hoffman as "Hints from Heloise" for the mind. Notice that vos Savant is still alive, at age 71 this year. Indeed, she's still a regular contributor to Parade, and here is her latest post, dated today:
Anyway, vos Savant writes about Monty Hall dilemma in her colum in 1990. She explains that the car probability of staying is 1/3. And here is the response she receives:
"You blew it, and you blew it big! I'll explain: After the host reveals a goat, you now have a one-in-two chance of being correct. Whether you change your answer or not, the odds are the same. There is enough mathematical illiteracy in the country, and we don't need the world's highest IQ propagating more. Shame!"
-- Scott Smith, Ph.D., University of Florida
Here Smith is referring to vos Savant's measured IQ of around 200. Well, if you recall the description of the problem from earlier this month, we can see what's going on here.
Dr. Smith writes about "mathematical illiteracy in the country." Recall that this was in 1990, a few years after the infamous "third pounder" burger incident:
A mathematician like Smith would've been upset that so many people were tricked into believing that 1/3 < 1/4, and now here comes vos Savant claiming that 1/3 = 1/2. Indeed, after she receives many more letters, another mathematician, E. Ray Bobo, asks:
"How many irate mathematicians are needed to get you to change your mind?"
In reality, we know that all these mathematicians are really claiming that 1 = 2, while vos Savant is telling them that 1 < 2. And so no matter how many hundreds, thousands, or even more are making the claim that 1 = 2, it doesn't make the claim any more true.
Some of the letters attack vos Savant for her gender -- almost 200 years after Sophie Germain was criticized for her gender. But I'd argue that gender in this case is a red herring. If vos Savant were instead a male making the 1/3 car probability claim, she (well, he) would still receive just as many letters questioning the claim. The letters would instead omit any reference to gender.
Indeed, this reminds me of NBA players who complained about fouls charged to them by Lauren Holtkamp, the league's third female referee. The players make sexist comments against Holtkamp after receiving the fouls, saying that she "doesn't belong" on the court. But does this mean that the players would have meekly accepted the foul calls had a male ref made them? I doubt it.
I suspect the same is true of race as well. If a driver of another race cuts you off on the road, you might angrily tweet later on, "This [racial expletive] cut me off today!" But if a driver of the same race cuts you off, you'd be just as angry, except you'd replace the expletive with "moron." (Think back to when race was mentioned by a student during my last subbing assignment.)
Nonetheless, here's a YouTube video discussing how so many "mansplainers" were stumped by the Monty Hall dilemma. (And yes, this is the second video that begins with the words "I hate math" that I post to the blog.)
Finally, let's go back to vos Savant's website, where she keeps a record of the complaints:
Vos Savant, according to Hoffman, doesn't make it any easier when she criticizes both the Wiles proof of FLT and Einstein's Theory of Relativity -- but then again, this is in 1993, three years after the Monty Hall column. (And besides, the Wiles proof of FLT really was wrong in 1993 -- the correct proof isn't written until 1994.)
By now, you may be wondering what do vos Savant and Monty Hall have to do with Paul Erdos? As it turns out, one of the mathematicians skeptical of the 1/3 car probability is the great Erdos. He is told this problem by his childhood friend Vazsonyi. In the end, Vazsonyi shows him a Monte Carlo situation in order to help him see the answer. A Monte Carlo simulation is similar to the activity that the students do at the end of the video, except it's randomized by a computer.
After the similation, Erdos grudgingly accepts the result, but not the proof. To him, this is similar to the proof of the Four Color Theorem (Lesson 9-8 in the U of Chicago text), which is also proved by trial and error on a computer. Erdos would prefer an elegant proof worthy of the SF's book to a proof by computer.
The chapter ends with the death of the great Paul Erdos. His passing is on September 20th, 1996 at the age of 83. But there are still two chapters remaining in this book, so his story isn't over yet. But let's get back to Geometry.
Lesson 4-6 of the U of Chicago text is called "Reflecting Polygons." This lesson doesn't appear in the new Third Edition -- instead, its material is incorporated into Lesson 4-2.
Two years ago, I didn't cover Lesson 4-6 as its scheduled day was blocked by a subbing day. So instead, we must go back three years to find the lesson:
Section 4-6 of the U of Chicago text considers what happens when we reflect an entire polygon -- not just individual points or even a segment or angle.
Still, the section begins with a theorem on what happens when we reflect a single point twice. Suppose we have two points, F and G and a reflecting line m. Now suppose I told you that the mirror image of F is G. So where do you think the mirror image of G is? If we drew this out and showed it to a student, chances are the student will say that the mirror image of G is F. The book gives a proof of this fact -- by the definition of reflection, G as the mirror image of F means that m is the perpendicular bisector of FG. But FG is the same segment as GF, so its perpendicular bisector is still m. And so, by the definition of reflection again, this would make F the mirror image of G. QED
The text calls this the Flip-Flop Theorem:
If F and F' are points or figures and r(F) = F', then r(F') = F.
Recall that the text often uses the function notation r(F) to denote the reflection image of F. So the theorem can be written as:
If F and F' are points or figures and the mirror image of F is F', then the mirror image of F' is F.
And one can use even more function notation than the text and write the theorem as:
If F is a point or figure, then r(r(F)) = F.
So here's a two-column proof of the Flip-Flop Theorem:
Given: r(F) = F' Prove: r(F') = F Proof: Statements Reasons 1. r(F) = F' 1. Given 2. m is the perp. bis. of FF' 2. Definition of reflection (meaning) 3. FF' = F'F 3. Reflexive Property of Equality 4. m is the perp. bis. of F'F 4. Substitution Property of Equality 5. r(F') = F 5. Definition of reflection (sufficient condition)
Notice that this proof uses both the meaning and the sufficient condition parts of the definition of reflection -- this occurs in other proofs as well. For example, a proof of the theorem "all right angles are congruent" (Euclid's Fourth Postulate) uses both the meaning and the sufficient condition parts of the definition of right angle.
But the above proof is a little strange. We explained earlier the significance of Statement 3 in the above proof -- but the problem is that we need a reason as to why FF' and F'F are the same segment. There is no actual definition, postulate, or theorem that states this directly. The reason I wrote "Reflexive Property" above is that this often occurs in other proofs -- especially triangle congruence proofs that are used to prove that certain quadrilaterals are parallelograms. For example, in Section 7-7, we wish to prove that quadrilaterals with opposite sides congruent are parallelograms. The proof at the beginning of that lesson divides quadrilateral ABCD into two triangles, ABD and CDB, which the text then proves are congruent by SSS. But Step 2 of that proof reads:
2. BD is congruent to DB 2. Reflexive Property of Congruence
And so I did the same in the above proof. Of course, it's awkward to follow a statement that uses the "Reflexive Property" (that some object equals itself) with one that uses the "Substitution Property." (So we're substituting an object for itself?)
Some people may point out that now we're being overly formalistic here. The Flip-Flop Theorem is obviously true -- the two-column proof only serves to confuse the students. Perhaps if even I, as a teacher, have trouble filling in all the steps in the "Reasons" column (like Step 3 above), it means that the proof is so simple that it's better written as a paragraph proof (as the U of Chicago text has done) and not as a two-column proof.
Here's one final way to state the Flip-Flop Theorem:
A reflection is an involution.
An involution is simply a function or translation such that performing it twice on a point or figure gives the original point or figure. Therefore composition of an involution with itself is the identity. In function notation, f(f(x)) = x.
Now the other concept introduced in this chapter is orientation. The important concept, added to the Reflection Postulate as part f, is that reflections switch orientation.
But what exactly is the "orientation" of a polygon? The text explains that, in naming the vertices of a polygon, we can move either clockwise or counterclockwise around the polygon. The important idea here is that if pentagon ABCDE is clockwise and we reflect it, then A'B'C'D'E' is counterclockwise.
Then the book proceeds to tell us that "orientation" is undefined -- just like point, line, and plane. As we mentioned earlier, we only discover what an undefined term is by using postulate. So we have the Point-Line-Plane Postulate to tell us what points, lines, and planes are, and we have part f of the Reflection Postulate to tell us what orientation is. We may not know what orientation actually is, but we do know that whatever it is, reflections switch it.
The idea that reflections switch orientation shows up later on. In particular, translations and rotations preserve orientation, because they are the compositions of two reflections -- so the first reflection switches it, and the second switches it back.
Also, a question that often comes up is, if translations and rotations are the compositions of two reflections, maybe reflections are the composition of two rotations, or two of something else. As it turns out, this is impossible. Reflections can't be the composition of two of the same type of transformation, because of orientation. Either the orientation is switched and switched back, or it isn't switched at all. (If you want a reflection to be some transformation composed with itself, you must do something complicated, such as cut the plane into strips, then translate some of the strips and reflect the others.)
Is it possible to define "orientation"? We think back to Chapter 1, where the term "point," although undefined, can be modeled with an ordered pair. If we know all of the x- and y-coordinates of the vertices of the polygon, then we can plug it into a complicated formula such that if the answer is positive, then the orientation must be counterclockwise, and if the answer is negative, then the orientation must be clockwise. (If it's zero, then the points are collinear, which means that they don't form a polygon at all.) What's cool about the formula is that the number -- not just the sign -- actually means something. In particular, if we divide the number by two, we get the area of the polygon! But I won't give the formula here.
There's also a simpler version of the formula, but it only works if the polygon is convex. Notice the picture of octagon FGHIJKLM in the text. The book points out that determining its orientation is more difficult because it's nonconvex.
A much more intuitive way of thinking about orientation is if the preimage and image aren't figures, but words. If we hold up words to a mirror, then unless we're lucky and choose a word like MOM, the image will be illegible, since reflections reverse orientation. But if we translated the words instead, then we can still read the words (unless by "translation" we mean translation into another language).
One final note about orientation: A well-known math teacher blogger named Kate Nowak -- she calls her blog "Function of Time" or f(t) in function notation -- recently gave an Opening Task to her geometry classes:
Now Nowak gave her classes pairs of figures, and the students had to identify whether the two figures are "the same" or "not the same." As it turned out, the students easily reached a consensus if the two figures have the same orientation, but they disagreed if the orientations were different:
One group: "We said set C is not the same because you have to flip it." Me [Nowak -- dw]: "Great." Other group: "Wait a minute, we said set C is the same because we thought flipping was okay." Me: "Also great." Yet another group: "So which is it? We said they are the same." Me: "... ... ... because... ?"
Okay, let's return to 2017. Kate Nowak's website still exists, but she hasn't posted since the first day of school, and apparently she's creating a new middle school curriculum.
Today's an activity day. Of course, we could give the activity that Nowak describes in her 2014 post above, or even the Monty/Monte (that is, Hall/Carlo) simulation from above. But instead, let's go back two years when I wrote about Euclid the Game:
A few weeks ago, I mentioned the math teacher Lisa Bejarano, who had posted something called "Euclid: The Game" in one of her recent posts. And when I saw that part of the game reminded me of ancient geometer's Proposition 1 from Lesson 4-4, I couldn't resist checking the game out.
Apparently, this is a one-player game. The goal is, on each level, to construct the figure in the diagram at the top of each page. The possible moves are the same as those allowed in classical Greek construction -- drawing an arbitrary point, drawing a point at an intersection, drawing a segment given two endpoints, drawing a ray given the endpoint and another point, and drawing a circle given the center and a point on the circle.
Now Level 1 is indeed Euclid's first proposition -- to draw an equilateral triangle given a side. This one, despite being Level 1, may be tough for students seeing this for the first time -- but of course, our students who remember yesterday's Lesson 4-4 should have no trouble with this one. Notice that according to Kasper Peulen, the creator, this game is powered by Geogebra -- and we were just talking about John Golden and his Geogebra lessons this week. Yes, I'm definitely going to keep going back to Bejarano, Golden, and other teachers when looking for good geometry activities.
Level 2 requires students to construct midpoints. The usual way to perform this construction is to construct the perpendicular bisector -- it intersects the original segment at its midpoint. For our students, this will be a preview of next week's Lesson 4-5 on perpendicular bisectors.
Level 3 requires students to construct angle bisectors. As we've already seen here on the blog, angle bisectors appear on the Common Core tests, yet are given short shrift in the U of Chicago text. The construction is buried in a Question in Lesson 4-7. Here's how to bisect Angle AOB:
Step 1. Circle O containing A Step 2. Circle O intersects Ray OB at C. Step 3. Subroutine: Line PQ, the perpendicular bisector of AC As it turns out, the Euclid game has an equivalent of a "subroutine" -- like many computer and video games, passing a level unlocks a new "tool." In Level 2, I had already unlocked the midpoint tool. So I decided to follow the U of Chicago suggestion -- I drew a circle A (to label points of intersection B and C), found the midpoint D of BC, and then drew Ray AD. I passed the level with a minimum number of moves, three.
Level 4 requires students to find the perpendicular to a line through a point on the line. In the U of Chicago text, this is Example 2 of Lesson 3-6. This time, following the U of Chicago construction doesn't give me the minimum number of moves -- I needed four, but the minimum is three.
Level 5 requires students to find the perpendicular to a line through a point not on the line. It is the line given in this week's Uniqueness of Perpendiculars Theorem.
Of course, not every classroom has access to a computer -- then again, Euclid obviously didn't have a computer in ancient Greece either. So I decided to create worksheets for the first six levels of Euclid: the Game, and students will have to solve them the way that Euclid would have.
