Chapter 13 Test (Day 140)

This is my 24th post this month -- the most times I've ever posted in a month. I usually post only on school days, and a month can have at most 23 weekdays (31 days - 4 Saturdays - 4 Sundays). It's the Big March, and spring break is completely in April this year. So I posted all 23 weekdays this month -- plus a special post on Pi Day Sunday, to make 24 posts.

My previous monthly record was 22 posts, achieved in both October 2018 and 2019. October is often a month with just one student-free day (around either Columbus Day or the end of the first quarter.) The most times I've ever posted in March was 20, in both 2015 and 2016. The calendar in those years was based on my LA County district, where spring break is always in March, rather than tied to Easter.

Today I subbed in a sophomore World History class. It's in my first OC district. Since it's a high school class that's not math, I won't do "A Day in the Life" today.

Indeed, it's another one of those classes where the regular teacher is available on Zoom, so all I have to do is watch the in-person students. I've seen (and covered) several of these classes in my new OC district, but this is the first time I've done so here in my first OC district. That's all the more reason not to do "A Day in the Life" today.

The regular teacher ends his Zoom link with about 15-20 minutes left in each class, which gives me time to continue handing out Easter pencils and candy to all in-person students. And since this group hasn't heard it yet, I do sing "The Big March" as well today.

Today is Nineday on the Eleven Calendar:

Resolution #9: We pay attention to math as long as possible.

Naturally, I don't have much to say to the students about this resolution today. It's not a math class, and I don't do any teaching.

Today is the Chapter 13 Test. This is what I wrote two years ago about today's test:

Here's an answer key for the test:

1. a. 90 degrees. I could have made this one more difficult by choosing a heptagon, or even a triskaidecagon, but I just stuck with the easy square.

b. Here is the Logo program:
TO SQUARE
REPEAT 4 [FORWARD 13 RIGHT 90]
END

Notice that the side length is 13. I'll still find a way to sneak 13, if possible, into each problem.

2. a. If a person is not a Rhode Islander, then that person doesn't live in the U.S.
b. If a person doesn't live in the U.S., then that person isn't a Rhode Islander.
c. The inverse is false, while the contrapositive is true.

Notice that Rhode Island is the thirteenth state.

3. y = 10.

4. There is a line MN. (M is the thirteenth letter of the alphabet.)

5. Every name in this list is melodious.

6. The equation has no solution. (This question references 13, as 13x appears in the expansion.)

7. a. 13, 11, 9 (descending odds).
b. 13, 17, 19 (increasing primes -- of course, Euclid proved that this sequence is infinite).

8. a = 2, b = 1, c = 3.

9. kite.

10. I discussed this problem earlier this week. It is the same as the problem from the Glencoe text, except that I only drew half of the figure -- the part where a contradiction appears.

Assume that the figure is possible. Then ABC is isosceles, therefore angles A and C are each 40 degrees (as the third angle of the triangle is 100). Then ABO is isosceles (as it has two 40 degree angles), so AO = BO = 3. Then by the Triangle Inequality, 3 + 3 > 8, a contradiction.

11. Through any two points, there is exactly one line. (This is part of the Point-Line-Plane Postulate.)

12. a. KML measures 13 degrees.
b. K measures less than 167 degrees.
c. L measures less than 167 degrees. (This is the TEAI, Exterior Angle Inequality.)

13. a. Law of Ruling Out Possibilities.
b. You forgot to rule out another possibility -- that nothing bad will happen to you today. Hopefully, this will be true for you.

Since it's test day, this is a traditionalists post. None of the major traditionalists have posted this week, except for Darren Miller.

Actually, Miller hasn't really posted this week either. His school is one of those that takes Holy Week, the week before Easter, off for spring break -- and he announced that he won't post this week either.

Instead, let's go back to his posts from last week -- the first week of hybrid at his school. It may seem odd that schools would reopen for hybrid just before spring break. Here's the thing -- our governor gave schools a huge financial incentive to reopen by April 1st. The problem is that April 1st has to be so close to Easter this year -- so the deadline to reopen  is right when schools want to give teachers and students a week off for the holiday. This means that schools had to open for just a few days before break just so they can say that students have seen the inside of a classroom no later than April 1st -- and this must occur at both Holy Week and Bright Week (the week after Easter) schools.

Recall that the hybrid model at Miller's school is different from the ones where I sub. He teaches live students in the morning and online students in the afternoon. Thus Cohort A goes to school Monday morning in-person (odd periods), Tuesday morning in-person (even periods), Thursday afternoon online (odd periods), and Friday afternoon online (even periods), with Cohort B opposite. Those who opt out of hybrid attend online all four afternoons. I'm not sure about Wednesday -- the only Wednesday his school had before spring break was reserved for the SAT.

Here are some excerpts from his March 23rd post -- I'll skip directly to the important lines:

https://rightontheleftcoast.blogspot.com/2021/03/stranger-than-you-can-even-imagine.html

Not one of them had watched the video for today's class.

They were just staring at me.

The student just stared at me.

They have forgotten how to be students.

I'm not the only one seeing this.

That's what they do.  They stare.  Children of the Corn, or something.

In most traditionalists' posts, I point out how many students, when given traditionalist p-sets, respond simply by leaving it blank. Well, this is the equivalent of leaving stuff blank when there's no paper -- a blank stare. Also, some traditionalists have complained about the "flipped classroom" even before the pandemic -- they assign a video for students to watch before class, but then hardly anyone watches it. In fact, one of his commenters says exactly this:

Ellen K:

It's not you-it's them. They have forgotten how to be students. And it's not just the remedial students struggling-it is ALL students. The hybrid approach doesn't work. Five years ago we were encouraged to do just the type of teaching you're using-present the material in advance for the students to consume on their own time and then put the knowledge in action during class time. The same thing happened.

(Note: At this point, Ellen's post becomes political. Recall that "Right" on the Left Coast means right wing, and so Miller and his readers like Ellen blame the opposing political party.)

Of course, the main idea of Miller's post is that his students would have been engaged with him had it not been for the pandemic. And it's not just the hybrid schedule that's the problem -- he's saying that they would have been engaged with him had the schools opened on March 16th, 2020 -- that is, if the 2019-20 school year had contained 180 days, 7 hours per day in person.

Going back to my long-term assignment, yes, there were a few students who weren't engaged, but it's nothing like what Miller describes here. Then again, my students returned in early October -- so their in-person layoff was only about half as long as Miller's (with much of that time being the usual spring and summer breaks anyway).

By the way, Miller states that he would have accepted a short closure of a few weeks (to "flatten the curve") -- we know his school takes Holy Week off, so he could have returned on Easter Monday, which was April 13th, 2020. Anything after that is unacceptable to him.

Of course, this says nothing about the students who would have opted out of in-person learning. I've been keeping track of the Miller wager -- for each student, I wager $1 that he or she would opt out of hybrid, while Miller's wager is that the student will opt in. (For today's classes, I would have broken even in second period and made a profit of $1 in fourth period. Sixth period is conference.)

But Miller would say that this wager doesn't matter. To him, the reason that any student would opt out is that they think that hybrid is a crappy system. To him, the perfect is the enemy of the good -- he's stated several times that he'd rather have full distance learning than have in-person for anything less than five days per week. And he believes that many students and parents agree. The same students who are opting out of his classes now would have attended school in-person on March 16th or April 13th, 2020.

In fact, Miller has also made a big issue in his posts about wearing masks. It appears that he'd rather teach fully online than in-person with a mask, even if it's five days per week. (And he implies that the students are the same -- if schools opened five days per week with a mask requirement, some students would opt out, but they'd suddenly opt in if the masks weren't needed.)

Miller is a zero-percenter -- he believes that for a person of his age or his students' age, the probability that anyone would die of the coronavirus (or the "'rona." as he likes to call it) is 0% to the nearest percent, even if everyone touched each other without masks. In short, they're just as likely to be struck by lightning as die of the virus -- and just as a rational person lives life as if the likelihood of being struck by lightning is 0, so should a young, healthy person live life with regards to the virus.

I've written about my LA County district that, like Miller's, has been stuck in the purple tier and is just getting ready to reopen. Recall that this district has four levels of reopening:

1. Full distance learning
2. Office hours (one in-person day per week)
3. Classic hybrid (two in-person days per week)
4. Near normal learning (four in-person days per week)

The district has been in Level 1, and will reopen at Level 2 this upcoming Tuesday. Meanwhile, my Orange County districts are currently in the equivalent of Level 3.

Notice that these levels don't correspond to academic quarters, nor do they correspond to color-coded tiers (that is, Level 1 = purple, Level 2 = red, and so on). Indeed, today Orange County was promoted to the orange tier, and LA County may follow suit as early as Easter Monday. But the district still plans on reopening at Level 2.

Meanwhile, the social distancing requirement has been dropped from six to three feet. Because of this, my first OC district plans on moving to the equivalent of Level 4 on April 19th -- and the new OC district may follow suit, but it's still under discussion.

Check that -- it will follow suit. I received an email just minutes ago (about a half-hour before the time stamp of this blogpost) stating that it will move to Level 4 on April 26th. There will be an asynchronous day of transition (and in fact, many districts will have an asynchronous day, including LA County schools transitioning out of Level 1).

So we might have four in-person days for most students soon. In fact, Miller writes that the same may happen at his school in a comment:

Now that the feds and the state have reduced so-called social distancing to 3' instead of 6' in schools, there's talk--or is it just rumor?--of combining our Mon/Tues in-person students with our Thurs/Fri in-person students so that they can be at school 4 days a week instead of two, but still for only 3 hrs a day, and I'll still have to reteach my classes to the at-home kids in the afternoon.

Yes, that's the thing -- students will still opt out of hybrid. I wonder what Miller would rather do with those students than have to reteach them later -- but he never tells us. After all, he implies that if we reopened five days per week with no masks, all of those students would suddenly attend in-person

Let's look at the rest of the posts in the comment thread. There's even a Steve in this thread, but it doesn't appear to be SteveH:

Steve USMA '85:

You are scaring me Darren.

There's an anonymous comment about schools reopening:

Some parents think that going back to in person high school for only 6-8 weeks before the end of the school year is too disruptive (it is) or are concerned about Covid infections (will it ever be safe?) If you expect your child to go back to in-person school in August, then shouldn't you start getting them readjusted to school now? If they don't physically go back to school now, it will have been 17 months since they last stepped into a classroom.

Miller has expressed agreement with this sentiment. This is the one time when the perfect isn't the enemy of the good after all -- a hybrid schedule now makes it more likely that there will be a five-day reopening this fall.

The main commenter appears to be Ellen K, whom we've seen earlier. Her second comment is completely political, so let's look at her third comment:

Ellen K:

I retired in May 2019 on the heels of a meeting with parents who were INCENSED that in an AP Studio class I wouldn't accept anime doodles on notebook paper as entries for a year long portfolio. In the meeting I tried to explain that as a working artist having more skills and more diversity made their student more marketable. I was assailed with the comment "We raised our children that they could be anything they wanted to be." That was never the point, the point was they had to learn to do more than one thing. This is true of any profession.

Then Ellen throws the race card, so I'll cut off her comment here. (Yes, race frequently comes up in these traditionalists' posts.)

My focus here is on the statement "You can be anything you want to be." In some posts, it's the traditionalist who agrees with this statement and the student who disagrees. The student says something like "I can't do math" and responds by leaving the p-set blank, or "I can't do art" and responds by leaving the canvas blank. There are limits to what a student can do, but those limits are determined by factors beyond the student's control -- either one is a "math (or art" person" or isn't. The traditionalist tries to convince the student that he or she can do math or art, so don't leave the paper blank.

But in this post, it's the student who agrees with "You can do anything" and the traditionalist, Ellen, who disagrees. There are limits to what a student can do, but those limits are determined by what sacrifices he or she is willing to make. The student wants to be able to get a good grade in class without ever doing anything she doesn't want to do -- and ultimately, to be successful in life doing only things she wants to do.

At this point, this post (and most other posts in thread) retrace common traditionalist territory. Ellen's mention of parents comes up often -- in some past year (say 1955), if a student refused to make sacrifices in class, that student would be grounded for a month, but nowadays parents prefer to accommodate their child's unwillingness to make sacrifices. To them, a student who makes no sacrifices should still get a good letter grade.

There is no simple solution to this debate. Instead, let's worry about the letter grades that our students will receive on the Chapter 13 Test.

Chapter 13 Review (Day 139)

Today I subbed in a self-contained special ed class. It's in my first OC district. There's no need for me to do "A Day in the Life" today, of course, with aides taking over the class.

The students do have a math lesson today, on equivalent ratios. They watch two Khan Academy videos and then answer a short quiz. Unfortunately, I don't have much opportunity to teach much math -- the two girls who are in-person finish their math quickly, while one of the aides runs the Zoom link.

And indeed, I usually don't have much opportunity to sing in classes like this. I was hoping to sing "Nine Nine Nine" from Square One TV today, since multiplication is involved in finding equivalent ratios. But instead, I quickly sing "U-N-I-T Rate! Rate Rate!" between the math and history lessons. This song make sense as I perform the first verse (on ratios). It also celebrates my anniversary as a UCLA student -- yes, today I am Twenty-Two Years a Bruin -- and it anticipates tonight's Elite Eight game vs. Michigan. (No, this class doesn't walk or take a "Big March" around the school either.)

Today is Eightday on the Eleven Calendar:

Resolution #8: We follow procedures in the classroom.

Of course special ed students in a class like this must follow procedures. But it's the aides who tell them what procedures to follow -- these students have issues, and they know how to handle them without getting into arguments.

Oh, and since the aides take care of attendance, I don't know how many students are in each cohort, and so there's no Miller wager.

You're right -- I don't have much to say about this class. I will say this, though -- despite my not having visited this school since before the pandemic, a few students do remember me as a sub. There's a freshman or two who recognize me from the corresponding middle school class. And one older girl remembers how I helped her with math two years ago, but there were some bad students in class that day who disrupted the lesson -- not her, of course.

(I wrote about that day on the blog -- it was about one week after Pi Day. It was an Algebra I class, and so I sang "Quadratic Weasel." It was one of the first times I sang in class since the old charter school -- and it reestablished my habit of singing in the classroom.)

It's time to review for the Chapter 13 Test. I keep changing how I would give this test, but I do have some review worksheets from last year and beyond, so I can post them today.

And this is what I wrote last year about today's review worksheet -- which includes a story about indirect proofs (including irrationality, since I just wrote about 1/sqrt(2) earlier):

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 -- yesterday's lesson, including both exterior angles and Logo.

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

For those readers who missed it at the bottom of my last post, the Orange County districts are both taking the week after Easter (Bright Week) off for spring break. Two years ago, Easter was later, and so the district closed the week before Easter (Holy Week) instead. Last year, all schools closed for coronavirus just before we made it to the spring holiday. But this year, I still have one week left of the Big March -- which explains why I sing seven verses of the "Big March" this year, one for each of the seven weeks between Presidents' Day and spring break.

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.

This is what I wrote two years ago about today's lesson -- which includes a discussion of Logo, the programming language mentioned in Lesson 13-8. 

[2021 update: I'm retaining that old discussion this year, even though my focus is now on other programming languages (Mocha BASIC and Java).]

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

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]

[2021 update: Actually, a very interesting game to attempt is the one that Alice and Bob play in Putnam problem 2020 B2. Since the goal for the Putnam participant is to find ways for Alice to win, we should let the user be Alice and the computer be Bob. Then our goal as the programmer is to find ways for Bob to win.]

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.

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

Here's another program I had fun with:

to namegame :name
  cleartext
  print sentence sentence :name :name sentence "bo word "b bf :name
  print sentence [Banana fana fo] word "f bf :name
  print sentence [Me mi mo] word "m bf :name
  print :name
end
namegame "David


Unfortunately in all the years I've posted, I've never full created a Lesson 13-8 worksheet. So instead I'll post what one of the eighth grade classes from last year used. It's actually a foldable that can also be included in an interactive notebook.

What's bad about this is that "interior angles" and "exterior angles" are apparently misspelled as "internal angles" and "external angles." Thus the only part that actually matches anything in our Chapter 13 isn't even spelled correctly. But it beats anything I've previously posted for this lesson -- and vocabulary, foldables, and interactive notebooks are all things that I need to incorporate into my own lessons more often.

[2021 update: That is, I needed to incorporate these before the coronavirus. I just did a virus-friendly lesson for Lesson 13-7, so we don't need one for 13-8.]