Thursday, April 5, 2018

Lesson 13-8: Exterior Angles of Polygons (Day 138)

Lesson 13-8 of the U of Chicago text is called "Exterior Angles of Polygons." In the modern Third Edition of the text, there again is no separate lesson for today's material. Part of it appears in Lesson 5-7 along with yesterday's lesson, and the rest of it (the computer part) has been dropped.

But there's a problem with the blog calendar with regards the Chapter 13 Test. The digit pattern has us giving tests every tenth school day -- Day 10, Day 20, Day 30, and so on. According to this pattern, the Chapter 13 Test should be on Day 140.

So far, the multiples of ten have landed on Tuesday, Wednesday, Thursday, and Friday. We've been lucky enough to avoid the tenth day falling on Monday -- until this Monday, that is.

Sometimes I wonder whether it's better to give a test on Monday -- an opportunity for the students to study over the weekend -- or on Friday -- an opportunity for the teachers to grade over the weekend. I make the assumption that teachers are more likely to grade tests over the weekend than the students are to study over the weekend. Hence it's my policy to avoid Monday tests whenever possible. The obvious exception is the final, since finals week in the district (observed by the blog calendar) is often Monday through Wednesday.

So tomorrow will be the Chapter 13 Test. Then Monday can be an activity day, in order for the digit pattern to resume with Lesson 14-1 on Tuesday (Day 141). This means that today, in addition to Lesson 13-8, must be the review for tomorrow's test.

In two weeks, the same problem will occur again, with Day 150 on Monday, April 23rd. Fortunately, Chapter 14 contains only seven lessons, so it works out more comfortably with Lesson 14-7 on Wednesday, Chapter Review on Thursday, Chapter Test on Friday, activity on Monday.

This is what I wrote last year about today's lesson -- which includes a discussion of Logo, the programming language mentioned in Lesson 13-8:

The simplest way to get to infinity is by a simple program. We write it in BASIC:

10 PRINT "HELLO"
20 GOTO 10

but we can also write it in Logo:

to hello
print "Hello
hello
end

Ironically, we must end the procedure with "END" even though it doesn't end. Last year, I mentioned an infinite polygon program written by Brian Harvey:

to poly :size :angle
forward :size
right :angle
poly :size :angle
end



I decided to keep the Logo problems in my worksheet. Unfortunately, unlike BASIC, Logo isn't easy to convert into TI-BASIC. Logo is mentioned in this section because the angles mentions in the RIGHT commands are in fact exterior angles. If the students don't have access to Logo (Harvey discusses how one can download Berkeley logo on his webpage), one can change it to simple angle questions -- for example, my Question #3 becomes, "What is the exterior angle of a regular octagon?" (rather than draw one in Logo).

So that is what I wrote last year. This year I decided to go back to the Berkeley Logo and show you some more interesting programs. One of my favorites is Tic-Tac-Toe:

https://www.cs.berkeley.edu/~bh/v1ch6/ttt.html

This set of procedures combines both list processing (for the strategy) and turtle graphics (to draw the X and O symbols). We can recognize most of the turtle graphic procedures:

to drawx
setheading 45
pendown
repeat 4 [forward 25.5 back 25.5 right 90]
end

This procedure obviously draws the X's. The only procedure with which we might be unfamiliar is the line setheading 45. Of course the 45 refers to 45 degrees. What this line does is tilt the turtle 45 degrees so that the symbol looks more like an X than like a + symbol.

Now that we've seen drawx, we may be curious to see what drawo looks like -- perhaps it's similar to the 180GON that we see in the U of Chicago text. Well, actually it isn't:

to drawo
pendown
arc 360 18
end

Of course, we can easily figure out what the line arc 360 18 does -- the 360 obviously means 360 degrees, and I think that 18 refers to the radius of the arc (in turtle units).

Most of the link above discusses Tic-Tac-Toe strategy and how to implement it. Harvey writes:

At the beginning of the discussion about strategy, I suggested that one possibility would be to make a complete list of all possible move sequences, with explicit next-move choices recorded for each. How many such sequences are there? If you write the program in a way that considers rotations of the board as equivalent, perhaps not very many. For example, if the computer moves first (in the center, of course) there are really only two responses the opponent can make: a corner or an edge. Any corner is equivalent to any other. From that point on, the entire sequence of the game can be forced by the computer, to a tie if the opponent played a corner, or to a win if the opponent played an edge. If the opponent moves first, there are three cases, center, corner, or edge. And so on.

Actually, the problem of Tic-Tac-Toe strategy has already been completely solved. The solution is given by the website xkcd, which describes itself as "A webcomic of romance, sarcasm, math [ha! -- dw], and language":

https://xkcd.com/832/

By the way, xkcd has created webcomics for several math topics. I was able to find two different webcomics involving Zeno's Paradoxes!

https://xkcd.com/1153/
https://xkcd.com/994/

Harvey goes on to write:

If you're tired of tic-tac-toe, another possibility would be to write a program that plays some other game according to a strategy. Don't start with checkers or chess! Many people have written programs in which the computer acts as dealer for a game of Blackjack; you could reverse the roles so that you deal the cards, and the computer tries to bet with a winning strategy. Another source of ideas is Martin Gardner, author of many books of mathematical games.
[emphasis mine]

What the -- here's yet another Martin Gardner reference! Well, that just goes to show how popular Gardner and his math games column was!

Actually, before taking up Harvey's suggestion the challenge would be to implement xkcd's game strategy in Logo. Notice that xkcd's Tic-Tac-Toe strategy is in fact foolproof, in that if the computer were programmed to follow it, the computer will always either win or tie. So in the procedure:

to ttt
local [me you position]
draw.board
init
if equalp :me "x [meplay 5]
forever [
  if already.wonp :me [print [I win!] stop]
  if tiedp [print [Tie game!] stop]
  youplay getmove                         ;; ask person for move
  if already.wonp :you [print [You win!] stop]
  if tiedp [print [Tie game!] stop]
  meplay pickmove make.triples            ;; compute program's move
]
end

we can delete the line that says You win! (where "you" refers to you, the computer user) because that line can never be reached.

And this is what I wrote last year about today's review worksheet -- which includes a story about indirect proofs:

Here is a link to a common indirect proof that sqrt(2) is irrational:

http://www.math.utah.edu/~pa/math/q1.html

Neither the U of Chicago nor Glencoe gives the proof outright. But both hint at it -- I just mentioned the U of Chicago's square root proofs. The Glencoe text asks the students to prove that if the square of a number is even, then it is divisible by four. As we can see at the above link, this fact is directly mentioned in the irrationality proof.

I remember once reading the proof of the irrationality of sqrt(2) in my textbook back when I was an Algebra I student. Until then, I had always heard that sqrt(2) was irrational, but I never realized that it was something that could be proved. So I was fascinated by the proof. Naturally, the text only included this as an extra page between the main sections, so it was something that the teacher skipped and most students probably ignored.

The irrationality of sqrt(2) has an interesting history. It goes back to Pythagoras -- he was one of the first mathematicians to use sqrt(2), since his famous Theorem could be used to show that the diagonal of a square has length sqrt(2). The website Cut the Knot, which has many proofs of the Pythagorean Theorem, also contains many proofs of the irrationality of sqrt(2):

http://www.cut-the-knot.org/proofs/sq_root.shtml

Now there is a famous story regarding sqrt(2) and Pythagoras. At the following link, we see that Pythagoras was the leader of a secret society, or Brotherhood:

http://nrich.maths.org/2671

Now Pythagoras and his followers believed that only natural numbers were truly numbers. Not even fractions were considered to be numbers, but simply the ratios of numbers -- numberhood itself was reserved only for the natural numbers. In some ways, this attitude resembles that of algebra students today -- when the solution of an equation is a fraction, they often don't consider it to be a real answer, even though modern mathematics considers fractions to be numbers. (The phrases real number and imaginary number reflect a similar attitude about 2000 years after Pythagoras -- that some numbers aren't really numbers.) So of course, the idea that there were "numbers" that weren't the ratio of natural numbers at all was just unthinkable.

Pythagoras and his followers must have spent years searching for the correct fraction whose square is 2, but to no avail. Finally, one of his followers, Hippasus, discovered the reason that they were having such bad luck finding the correct fraction -- because there is no such fraction! And, as the story goes, Pythagoras was so distraught, afraid that the secret that sqrt(2) was irrational would be revealed, that he ordered to have poor Hippasus drowned at sea!

But as I said, nowadays students simply complain when they have a fractional, or worse irrational, answer to a problem. No one has to drown any more just because of irrational numbers.

Question 10 on my test review, therefore, is actually the final step of that proof, since that's the step where the contradiction occurs. They are given a triangle with sides of length 3 and 8, and two angles each 40 degrees (one of which is opposite the side of length 3). The students are to use the Converse of the Isosceles Triangle Theorem to show that the missing side must also be of length 3, and then the Triangle Inequality to show that 3 + 3 must be greater than 8, a contradiction.

When I wrote this problem, I had trouble deciding how difficult I wanted my indirect proof to be. For example, I considered giving 100 as the measure of the angle opposite the side of length 8, and give only one 40-degree angle instead. Then the students would have to use the Triangle Angle-Sum Theorem to find the missing angle as 40 degrees before applying the Isosceles Converse.

Or, to go even further, we can derive a contradiction without making the angle isosceles at all. For example, we could make the angle opposite the 8 side to be, say, 90 degrees instead of 100. Then the missing angle would be 50 instead of 40. If the triangle is drawn so that 50 degrees is opposite the 3 side, then by the Unequal Angles Theorem, the missing side would be less than 3, so the sum of the two legs would still be less than the longest side.

But this might confuse the students even more -- especially if the 90-degree angle is marked with a box (to indicate right angle) rather than "90." A right triangle might lead a student to use the Pythagorean Theorem to find the missing leg. Although this still eventually leads to contradiction -- the missing side would be sqrt(55), which isn't less than 3 -- that irrational side length might still cause some students to drown.

And so I wrote my Question 10 on the review so that it will actually help the students prepare for the corresponding question on the test. I balance out this tough question with some easier questions about logic (converse, inverse, etc.). Hopefully the test won't be too hard for the students.

The first question on today's review worksheet is on Lesson 13-8 -- today's lesson, including both exterior angles and Logo.

By the way, I spend so much time writing about the BASIC emulator Mocha, so now it's natural to wonder whether there's an online Logo emulator. Well, I found the following link:

http://www.calormen.com/jslogo/

To use this emulator, we must click on the arrow near the "Run" box on the bottom of the screen -- this allows us to enter procedures on more than one line. Then we can type in REGGON as written in the text:

to reggon
  repeat 18[forward 7 right 20]
end
reggon

Then we click the "Run" box to run the program.

This program works, but some of the others written in the book don't work. For example, 180GON doesn't work since apparently, procedure names must start with a letter, not a digit. So we can place an "a" in front of the procedure name:

to a180gon
  repeat 180[forward 3 right 2]
end
clearscreen
a180gon

It also appears that infinite loops don't work -- apparently the computer must complete the entire program before the turtle moves a single step. So we can't run "hello" or "poly" as mentioned earlier in this post.

On the other hand, loops that eventually stop will work. The program "tree" as given at the Brian Harvey link works:

http://people.eecs.berkeley.edu/~bh/v1ch10/turtle.html

In fact, "tree" is already pre-programmed in, and we can just click on it on the right side of the screen.

I tried some of the other programs from the Harvey page. His "face" works -- in the text editor, enter his program "to square" first, then "to face." I'd add these three lines before clicking "Run":

clearscreen
face
hideturtle

The last line hides the turtle so that there isn't a turtle in the middle of the face. The command to make the turtle reappear is:

showturtle








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