This is my milestone 1200th post. At the time of my 1100th post, I was wondering whether I'd even make it to my 1200th post -- the schools were closed due to the coronavirus pandemic, and no reopening was in sight. I was afraid that there would be no more subbing assignments, and so I was strongly considering searching for a career other than teaching. And if I were leave the field of education, there would no longer be a need for this education blog.
Fortunately, the schools reopened in Orange County, CA, in October, and I landed a long-term subbing assignment at a middle school. That job reawakened my commitment to teaching. And so this blog remains alive -- I make it to my 1200th post, and I plan on making it to 1300 as well.
Today is the fourth and final day of my assignment in the high school math class. Fridays are even periods, and Cohort B meets in person. As we saw on Wednesday, second, fourth, and sixth periods are all Geometry classes.
The students move on to Lesson 6.5 in APEX. This is mostly on the Inscribed Angle Theorem, which corresponds to Lesson 15-3 of the U of Chicago text. But the Kuta worksheet also mentions angles formed by chords -- the Angle-Chord Theorem appears in Lesson 15-5 of the U of Chicago text.
By the way, no, Lessons 15-2 or 15-4 of the U of Chicago text won't appear in APEX. These lessons appear in the U of Chicago text just for "fun" -- Regular Polygons and Schedules and Locating the Center of a Circle.
Today I had the largest Geometry classes of the week -- fourth and sixth period were tied for the most with eight students each. During tutorial, I do show some students the answers to Lesson 6.5, but not many. A few students work on APEX, while others prefer to print out the worksheet from home -- unlike 6.4, there are no copies of the 6.5 worksheet in the room.
But I will do some Shapelore today, for Lessons 15-2 through 15-5:
- In many games like Roulette, Craps, and Blackjack, your chance of winning can be calculated using the techniques we've learned in the previous lectures. But what happens when you're faced with an intelligent adversary whose goals are diametrically opposed to yours?
- When one player's gain is another player's loss, it's called a zero-sum game. It's not a winning strategy to be predictable in such a game.
- In Rock-Paper-Scissors, rock beats scissors, scissors beats paper, and paper beats rock. It is a zero-sum game because there are two players (Rose, Colin), the players choose strategy and reveal simultaneously, and a Rose win equals a Colin loss and vice versa.
- The payoff matrix for Rose looks like this:
- In the Penny Matching Game, each player chooses one side of a coin. If both players show heads, Colin wins $3 and if both players choose tails, Colin wins $1. If the coins don't match, then Rose wins $2 instead. Here is her payoff matrix:
- If Rose tells Colin that she'll choose at random by flipping her coin, then Colin can exploit this by always choosing heads. Rose's expected value = (1/2)(-3) + (1/2)(2) = -1/2 -- she'll lose 50 cents.
- So instead, she tells Colin that she'll play heads 3/8 of the time and tails 5/8 of the time. It turns out that her EV is actually positive -- it's +1/8 no matter what Colin chooses. Therefore 1/8 is the value of the game, V = 1/8. It's her optimal or equilibrium strategy.
- To implement her equilibrium strategy, Rose flips her coin thrice. If she gets exactly one head, then she shows heads -- otherwise she shows tails.
- To prove this, suppose Rose chose heads with probability x and tails 1-x of the time. Then her EV is -3x + 2(1-x) = 2 - 5x for Colin's head and 2x - (1-x) = 3x - 1for Colin's tail. This produces a system of equations, y = 2-5x and y = 3x - 1. The solution is (3/8, 1/8), which means that Rose should choose heads 3/8 of the time, and her EV will be 1/8.
- John von Neumann proved that every two-player zero-sum game has a value, and John Nash extended this to other types of games.
- The Nash equilibrium for Rock Paper Scissors is to choose each throw with probability 1/3. The EV in this case is zero.
- A variant of this game is called Rock Paper Scissors Lizard Spock. There is a poem to recall who wins this version -- in addition to the three classical symbols, Lizard beats Paper and Spock and loses to the others, while Spock beats Rock and Scissors. The Nash equilibrium is choose each throw with probability 1/5.
- In fact, real male lizards play Rock Paper Scissors to compete for females. Orange beats blue, blue beats yellow, and yellow beats orange.
- In a game called Hide and Seek, a $2 bill, $5 bill, or $10 bill is hidden. If you guess correctly, then you win the money. How much money would be a fair price to pay, and what would be the optimal strategy? Here is your payoff matrix:
- Suppose you choose 2, 5, 10 with probability x, y, z respectively. Your EV is 2x if the hider chose $2, 5y if the hider chose $5, 10z if the hider chose $10. All of these must equal V. And since the probabilities x + y + z = 1, we get V/2 + V/5 + V/10 = 1. Solving gives us V = 1.25, and so we get x = 5/8, y = 2/8, and z = 1/8 -- the probabilities are in inverse proportion to their value.
- In One Card Poker, there are three kings and three queens. Each player bets $1. Rose gets one card -- if it's a king she wins, and if it's a queen she loses. After the deal, Rose can either bet another $1 or fold. If she bets, Colin can either call or fold.
- Clearly Rose should always bet her kings. But should she sometimes bluff her queens? Here is her payoff matrix:
- This is a diagonal matrix, so it fits the pattern of Hide and Seek. Rose should bluff her queens 1/3 of the time and fold her queens 2/3 of the time. Her EV in this case is 1/3. Since there are three queens, she should bluff on, say, the Queen of Hearts and fold the other two queens. Meanwhile, Colin's strategy is to fold 1/3 of the time.
- Le Her is an 18th century variant of Poker. There are two players, Dealer and Recipient, and 13 cards, from ace low to king high. Both players are dealt one card face down. The Recipient looks at her card and decides whether to keep it or switch it with the Dealer's card. The Dealer decides whether to keep his (new) card or switch it with the top card in the deck. Then both players reveal, and the player with the higher card wins.
- Believe it or not, the Recipient has an advantage. We won't show the full 13 * 13 matrix here, but it reveals that Rose, the Recipient, should always switch 7 or lower, keep 8 80% of the time, and always keep 9 or higher. Colin, the Dealer, should always switch 8 or lower, keep 9 56% of the time, and always keep 10 or higher. The value is 0.547, an advantage to Rose.
- Three-player games are trickier to analyze. For example, in Final Jeopardy, the three players are Alice with $8,000, Betty with $7,000, and Charlie with $4,000. How much should they bet?
- Believe it or not, Alice should actually go for the tie and bet exactly $6,000. Betty should bet $1,000 in order to tie Charlie if Alice is wrong. Charlie should bet $2,000 in order to tie Alice if both ladies are wrong.
- So far, we've seen only simplified version of Poker. Real Poker, and its variants like Texas Hold 'Em, are much more complicated.
Four years ago at the old charter school, the mathematical second semester was when SBAC Prep began. Indeed, the bell schedule was changed so that SBAC Prep would replace most of P.E. time. Since we'd lacked a real conference period anyway, things truly became tough for me once SBAC Prep began.
Last year, I posted a Chapter 8 "Quest" -- part quiz, part test. And like last year, we didn't start Chapter 8 until Lesson 8-6. And so just like last year, today I'm posting the "Chapter 8 Quest."
Here's how today's "quest" will work. It will contain ten questions. It consists of last year's Questions 11-20, mainly because last year, I explained that Questions 1-10 are no longer valid.
This is what I wrote last year about today's quest:
11. Choice (a). The triangles have the same base and height, therefore the same area.Let's worry about the Chapter 8 Quest that I'm posting today on the blog. Here are the answers:
12. 3s^2.
13. 13 feet.
14. 6 minutes.
15. 780,000 square feet.
16. 133 square feet.
17. 1/4 or .25. Probability is a tricky topic -- the U of Chicago assumes that the students already know something about probability. Then again, it should be obvious that the smaller square is 1/4 of the larger square.
18. 4,000 square units.
19. 13.5 square units.
20. a(b - c) or ab - ac square units.
And since I'm posting a test today, that means it's time for a traditionalist' topic! It's been a long time since we looked at the traditionalists, since I usually don't post traditionalists during actual teaching assignments, including long-term subbing.
And as it turns out, our main traditionalist, Barry Garelick, hasn't posted much anyway. In fact, since the start of my long-term, Garelick only posted twice -- both were during Thanksgiving break. If I really wanted to, I could have made a traditionalists' post over Thanksgiving and then just ignored traditionalists the rest of my long-term. (I almost did anyway -- my Turkey Day post was on Eugenia Cheng's book, and her word "ingressive" is somewhat similar to "traditionalist.")
Of Garelick's two posts, I'll discuss only his November 22nd post, since he specifically mentions something that he taught in his own middle school class. Thus I can compare his method to the way I taught during the recent long-term assignment:
https://traditionalmath.wordpress.com/2020/11/22/solving-problems-in-multiple-ways-dept/
The current interpretation of the seventh grade Common Core Math Standards as it applies to ratios and proportion provides a case in point. One of the authors of the standards, Phil Daro, was apparently guided by an unmoving and unshakeable conviction that traditionally taught math was nothing more than “getting the answer”. He has spoken about proportional reasoning and how it has been taught with no regard to process or conceptual understanding. I suspect that he is the main reason why proportional reasoning is now taught with multiple methods.
Notice that at my middle school, RP was taught in Math 7 at the start of the school year -- Unit 1. This unit was completed just prior to my arrival, so I don't know how it was taught to the students.
But ratios did appear during my last week of the long-term as part of the unit on scale drawing. And I will say this -- the material was pulled from different sources, and each source seemed to teach it in a different way, so yes, the students did see multiple methods.
We began with a PowerPoint that taught the traditionalists' preferred method -- writing a proportion and then cross multiplying. The method taught on APEX was to write the equation:
original length * scale factor = new length
and then solve it -- since it's not a proportion, there's no cross-multiplying. There was also a worksheet that had the students multiply or divide without writing an equation -- but that worksheet never loaded properly on our online platform, and so it was ultimately dropped once the regular teacher returned.
The main assessment for this lesson was on Quizizz. I hear that most students succeeded on the quiz, but it was set up so that only the regular teacher could see the scores upon his impending return. I did glance at a few students' computer screens and I thought I saw a proportion there -- but I never saw whether the students cross-multiplied to solve the proportion or not. So unfortunately, I don't have much data as to the best way to teach ratio and scale.
The regular commenter SteveH posts in the thread:
SteveH:
Having a toolbox doesn’t help if you aren’t skilled with each tool. Sure, talk about different approaches, but only individual practice, practice, practice will make you better. That’s what they’re missing – individual practice with homework. Homework is not just about speed, but understanding and becoming more flexible.
I distinctly remember that word problems were a big turning point in my traditional math education. Students want to follow rote techniques, but that can’t happen with word problems. I don’t know how modern edu-math pushers claim that traditional math is just rote. They can’t just call it #FakeMath when traditional AP/IB math is the only proper path to a STEM degree in college.
Although he does briefly discuss Garelick's word problem, SteveH ultimately reverts to his favorite ideas -- more p-sets and the AP/IB path, that is, Calculus as a senior. (By the way, SteveH, while most of the Geometry students are sophomores, there are some freshmen in the class. The Geometry freshmen are on SteveH's path towards AP Calculus and STEM.)
By the way, four years ago at the old charter school, I did teach RP to Math 7 at the start of the year, as well as ratios/scale just after Thanksgiving. Unfortunately, I don't remember much about how I taught this, since I wrote only about only Math 6 and Math 8 on the blog at the time. I do remember teaching something very untraditional to the Math 6 kids -- tape diagrams and double number lines -- and it's likely that I showed these to Math 7 as well. I briefly mentioned that my Math 7 kids that year did well on the ratio/scale test -- but once again, I don't remember how I taught it.
OK, here is the Chapter 8 Quest. Let's hope that our students know enough Geometry not to leave it all blank.
By the way, MLK Day is on Monday, and so my next post will be on Tuesday.
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