Today is the fifth day of my assignment in the high school math class. That's right, fifth day -- the regular teacher is unexpectedly out an extra day today. She has an appointment this morning, but she's already sent out emails declaring that she plans on returning tomorrow. So this really should be the final day of this assignment, with no sudden "sixth day of my assignment" tomorrow.
In fact, she leaves for her appointment just ten minutes after the start of third period Algebra I -- and this class has the Unit 1 Test coming up on Thursday. Once again, it would be a waste for someone with a math background just to sit there without trying to help the students get ready for the test.
And so I do help the students prepare today. After the regular teacher leaves for her appointment, the students pick up copies of the review worksheet, and I give them time to try it on their own. Then I sing the GCF Song -- last week the Tuesday cohort had only five students and today there are only four (all guys again), but since the other cohort got the song on Thursday, it's only right for me to sing it to this cohort as well. Again, I tell them that the song will help them when it's time for factoring polynomials.
But that's next week. This week's test is on adding, subtracting, and multiplying them. The worksheet again comes from Kuta, and I demonstrate about half the problems for them. I won't post the worksheet to the blog (since it's Algebra I, not Geometry), but I will post the ten problems that I show them today:
The last question is quite interesting -- and I tell the guys this. First, I point out that on this test, the students learn how to add, subtract, and multiply polynomials, suggesting that the answer is Division. I also mention the relevant Common Core Standard:
CCSS.MATH.CONTENT.HSA.APR.A.1Understand that polynomials form a system analogous to the integers, namely, they are closed under the operations of addition, subtraction, and multiplication; add, subtract, and multiply polynomials.
I've discussed this standard several times on the blog ("system analogous to the integers" = "ring").
Lecture 5 of Prof. Arthur Benjamin's The Mathematics of Games and Puzzles: From Cards to Sudoku is called "Practical Poker Probabilities." Here is a summary of the lecture:
- This lecture is devoted to the game of poker, one of the most popular card games ever invented.
- The most valuable hand is a straight flush, followed by four of a kind, full house, flush, straight, three of a kind, two pair, and one pair. A mnemonic to remember the hands from low to high is 1, 2, 3, straight, flush, 2-3, 4, straight flush.
- In Texas Hold 'Em, each player gets two cards. Three common cards (flop) are dealt, followed by a fourth (turn) and fifth (river) card. Bets are placed before each round of cards. Whoever has the best hand without folding is the winner.
- The player to the left of the dealer is the small blind and must bet 50 cents first. The next player is in the big blind and must bet $1.
- Suppose you were given 7:1 odds that all three cards in the flop are red. The probability that the first card is red is indeed 1/2. But the probability that the second card is red is only 25/51, and the probability that the third card is red is only 24/50. So the probability of three reds is 2/17.
- The probability of a rainbow flop of three different suits is 1 * 39/51 * 26/50 = 0.398. So the fair odds of a rainbow is approximately 3:2.
- If you are dealt AK (ace-king), what is the probability of getting an A or K on the flop? The probability of no ace or king is 44/50 * 43/49 * 42/48 = 0.68, so the probability of getting an A or K is about 32%.
- An "out" is a card that makes your hand a winner -- for example, if you have four cards to a flush, then you have nine outs.
- The Rule of 2 states that if you have x outs heading into the river, then the probability of winning your hand is approximately 2x%. So with ten outs, you win approximately 20% of the time.
- The Rule of 4 states that if you have x outs heading into the turn, then the probability of winning your hand is approximately 4x%. So with ten outs, you win approximately 40% of the time.
- What are the best two cards to be dealt? The obvious answer is two aces -- AA wins 85% of the time against a random hand. The next best hands are KK, QQ, JJ, TT (pair of tens), 99, 88, AK suited (big slick), 77, AQ (little slick). There are many online poker calculators.
- If you could choose AK off-suit, 22, or JT suited, which should you choose? Surprisingly, the deuces beat AKo 53% of the time. But JTs beats the deuces 53% of the time. And AKo beats JTs 59% of the time. These are non-transitive probabilities.
- Suppose you have 72o (WHIP, worst hand in poker) in the big blind and only three big blinds' worth of chips left. One player raises and everyone else folds to you. The pre-flop raiser shows you AKo (Anna Kournikova) and is forcing you all-in. Believe it or not, you should call the bet.
- These tips apply to other types of poker, including Video Poker. In Jacks or Better, you are dealt five cards. You can keep some or all of the cards. Then you are dealt five more cards and are paid according to the hand. You break even if you have a pair JJ, QQ, KK, or AA.
- Here is basic strategy: If you have two pair or higher, keep the winning hand. If you have one pair, draw three cards rather than keep any kicker. With no high cards, draw five cards. With one or more high cards, keep at most two high cards (suited if possible, otherwise two lowest).
- Exceptions: if you have 4 to straight flush, go for it (even with pair JJ+). Without JJ+, if you have a 4-flush or 3 to royal flush, go for it. Without any pair, if you have open-ended straight or 3 to straight flush, go for it, and if you have 10+suited high card, go for royal flush. EV = -0.54 cents.
Chapter 9 of the U of Chicago text is called "Three-Dimensional Figures." In past years, we skipped over this chapter and jumped directly into Chapter 10. After all, most questions relating to 3D figures on standardized tests are asking about their surface areas or volumes -- the purview of Chapter 10. As we are following the digit pattern this year, we will cover all of Chapter 9 starting today, Day 91.
I think back to David Joyce, who criticized a certain Geometry text. He writes:
Chapter 6 is on surface areas and volumes of solids. It is strange that surface areas and volumes are treated while the basics of solid geometry are ignored.
In summary, there is little mathematics in chapter 6. Most of the results require more than what's possible in a first course in geometry. Surface areas and volumes should only be treated after the basics of solid geometry are covered.
And so Chapter 6 of the Prentice-Hall text is just like Chapter 10 of the U of Chicago text. Joyce laments that students don't learn "the basics of solid geometry" before surface area and volume. But we can't fault Prentice-Hall for this. Even before the Common Core, most states' standards expected students to learn the 3D measurement formulas and hardly anything else about 3D solids.
We can't quite be sure what Joyce means by "the basics of solid geometry." But it's possible that some of what he wants to see actually appears in Chapter 9 of the U of Chicago text. Thus, by teaching Chapter 9, we are slightly closer to Joyce's ideal Geometry course.
And incidentally, there is one Common Core Standard in which 3D solids are mentioned, but not surface area of volume. We'll look at this standard in more detail later this week, in Lesson 9-4.
Lesson 9-1 of the U of Chicago text is called "Points, Lines, and Planes in Space." The first three sections of Chapter 9 are the same in both the old Second and modern Third Editions. (As it turns out, the new Third Edition squeezes in surface area in Chapter 9, saving only volume for Chapter 10.)
This is what I wrote last year about today's lesson:
The heart of this lesson is the Point-Line-Plane Postulate. We first see this postulate in Lesson 1-7, but now it includes parts e-g:
Point-Line-Plane Postulate:
a. Given a line in a plane, there exists a point in the plane not on the line. Given a plane in space, there exists a point in space not on the plane.
b. Every line is a set of points that can be put into a one-to-one correspondence with the real numbers, with any point on it corresponding to 0 and any other point corresponding to 1.
c. Through any two points, there is exactly one line.
d. On a number line, there is a unique distance between two points.
e. If two points lie in a plane, the line containing them lies in the plane.
f. Through three noncollinear points, there is exactly one plane.
g. If two different planes have a point in common, then their intersection is a line.
There are several terms defined in this lesson -- intersecting planes, parallel planes, perpendicular planes, and a line perpendicular to a plane.
Actually, I'm still thinking about Joyce's "basics of solid geometry." I know that his website also links to Euclid's Elements. So Book XI of Euclid is a reasonable guess as to what Joyce wants to see taught in class:
https://mathcs.clarku.edu/~djoyce/java/elements/bookXI/bookXI.html
Let's look at some of the definitions and propositions (theorems) here and compare them to the contents of Lesson 9-1. We'll start with Definition 3, since Definitions 1 and 2 will actually appear in tomorrow's Lesson 9-2.
Definition 3 appears in Lesson 9-1 as a line perpendicular to a plane. Definition 4 appears in this lesson as perpendicular planes. But Definition 5, the angle between a line and a plane, is only briefly mentioned in the U of Chicago text.
Again, Definitions 6 and 7 are about the angle between two planes, which is not discussed in our text at all. Definition 8, of course, appears in today's lesson -- but just as with lines, the U of Chicago uses an "inclusive" definition of parallel where a line or plane can be parallel to itself. Intersecting planes (our remaining term) are implied in Definition 8 as planes that are not parallel.
Let's look at the propositions (theorems) now:
This is essentially part e of our Point-Line-Plane Postulate. Euclid calls it a proposition (or theorem) and even provides a proof, but Joyce argues that the proof is unclear. Thus we might as well consider it to be a postulate.
This is essentially part f of our Point-Line-Plane Postulate. If A, B, and C are the three noncollinear points mentioned in part f, then we can take lines AB and AC to be the two intersecting lines that appear in Proposition 2, and triangle ABC to be the triangle mentioned in this proposition.
This is very obviously part g of the Point-Line-Plane Postulate. Joyce points out that this is yet another postulate, and that it holds only in 3D, not 4D and above.
According to Joyce, this is the first true theorem in Book XI. It asserts that if a line intersects a plane and is perpendicular to two lines in the plane, then the line is perpendicular to the whole plane. Joyce points out that the proof is a bit long, but it works. Theoretically, our students can prove it using the new Point-Line-Postulate and theorems from the first semester of the U of Chicago text.
https://mathcs.clarku.edu/~djoyce/java/elements/bookXI/propXI4.html
Here is a modern rendering of this proof. The idea is that line l is perpendicular to each of two lines m, n, in plane P, with all lines concurrent at point E. Our goal is to prove that line l is perpendicular to the entire plane P by showing that, if o is any other line in plane P with E on o, then l must be perpendicular to o as well.
Given: l perp. m, l perp. n with l, m, n all intersecting at E
Prove: Line l is perpendicular to plane that contains m and n.
Proof:
Statements Reasons
1. bla, bla, bla 1. Given
2. Choose A, B on m and 2. Point-Line-Plane part b (Ruler Postulate)
C, D on n so that
AE = EB = CE = ED
3. Exists plane P containing m, n 3. Point-Line-Plane part f (3 noncollinear A, C, E)
4. Choose F on l, 4. Planes contain lines and lines contain points.
and o in plane P s.t. E on o
5. Lines AD, o intersect at G, 5. Line Intersection Theorem
Lines BC, o intersect at H
6. Angle AED = Angle CEB 6. Vertical Angle Theorem
7. Triangle AED = Triangle CEB 7. SAS Congruence Theorem [steps 2,6,2]
8. AD = CB, Angle DAE = EBC 8. CPCTC
9. Angle AEG = Angle BEH 9. Vertical Angle Theorem
10. Triangle AEG = Triangle BEH 10. ASA Congruence Theorem [steps 8,2,9]
11. GE = EH, AG = BH 11. CPCTC
12. FE = FE 12. Reflexive Property of Congruence
13. Triangle AEF = Triangle BEF 13. SAS Congruence Theorem [steps 2,1,12]
14. FA = FB 14. CPCTC
15. Triangle CEF = Triangle DEF 15. SAS Congruence Theorem [steps 2,1,12]
16. FC = FD 16. CPCTC
17. Triangle FAD = Triangle FBC 17. SSS Congruence Theorem [steps 8,14,16]
18. Angle FAD = Angle FBC 18. CPCTC
19. Triangle FAG = Triangle FBH 19. SAS Congruence Theorem [steps 11,18,14]
20. FG = FH 20. CPCTC
21. Triangle GEF = Triangle HEF 21. SSS Congruence Theorem [steps 11,12,20]
22. Angle GEF = Angle HEF 22. CPCTC
23. EF perp. GH (i.e., l perp. o) 23. GEF, HEF are congruent and a Linear Pair 24. Line l perpendicular to plane P 24. Definition of line perpendicular to plane
So this is probably what Joyce wants to see more of. Propositions 5 through 19 aren't very much different from this one. But as I wrote above, our students will find such proofs difficult -- we had to prove seven different pairs of triangles congruent above, in three dimensions to boot. No modern text teaches such theorems, since no state standards -- pre- or post-Core -- require them.
The proof works -- the definition in Step 24 is satisfied because o is arbitrary. But notice that in the drawing at the above link, Euclid assumes that G, the point where lines AD and o intersect, is between A and D. But this is irrelevant for the proof -- all the congruence theorems used in the proof still work even if G isn't between A and D.
What's worse, of course, is if o is parallel to AD. Notice that AD | | BC (since DAE and EBC, the angles proved congruent in Step 8, are alternate interior angles), so o could be parallel to both. But that's no problem -- just switch points C and D in that rare case, and the proof still works.
Here is the worksheet for today's Lesson 9-1:
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