4) De honore. This win depends on the number of pieces and their number value. For example, it can be agreed that 5 pieces totaling exactly 160 will win. It is clear that the players must be familiar with the number values of their pieces and their sums.
OK, Pappas is still in the middle of telling us how to win the old game of Rithmomachia. This is the fourth of seven possible victories. Let's return to the link from yesterday's post:
http://www.gamecabinet.com/rules/Rithmomachia.html
What Pappas calls a de honore victory is what the link calls "The Victory of Body and Assets," since it combines the criteria for the Body (number of pieces) and Assets (sum of pieces) victories.
Now the fifth victory, which Pappas calls a de honore liteque victory, combines all three criteria from the previous post. The link above refers to this as "The Victory of Body, Assets, and Proceeds" ("Proceeds" = digits). So let's go directly to the sixth victory:
6) Victoria Magna. Arrange three captured pieces in either an arithmetic, geometric, or harmonic progression. e.g. arithmetic example -- 2, 5, 8; geometric example -- 36, 49, 64; harmonic example -- 6, 8, 12.
Notice that Pappas has erred here -- 36, 49, 64 is not a geometric progression. We should remember this from Lesson 14-2 of the U of Chicago text -- the geometric mean of 36 and 64 is sqrt(36 * 64), which works out to be 48, not 49. So 36, 48, 64 is a geometric progression with common ratio 4/3, but 36, 49, 64 isn't. But there is no 48 piece in Rithmomachia.
The link above explains the geometric progression better than Pappas does:
In a Geometric Progression, the ratios between successive numbers are given by a single value (called the ratio of the progression). For example: 3 - 12 - 48 is a Geometric Progression of ratio 4.
Meanwhile, a harmonic progression isn't commonly seen in high school classes. For example, why exactly is 6, 8, 12 considered a harmonic progression? In other words, why is 8 considered to be the harmonic mean of 6 and 12? Well, here's an interesting related question:
An athlete jogs one mile at 6 mph and then runs one mile at 12 mph. What is his average speed?
Well, the first mile takes 10 minutes to jog, and the second mile takes 5 minutes to run. So the total time for two miles is 15 minutes, or one quarter hour. So the average speed must be 8 mph. And what do you know -- the average speed is not the arithmetic mean of the two speeds, 9 mph. Instead, it's the harmonic mean of 8 mph.
If a and c are numbers and b their arithmetic mean, then b satisfies the equations:
a + c = 2b
b - a = c - b (the "common difference")
But if we change b to the harmonic mean, then the equation to solve is:
1 + 1 = 2
a c b
As explained at the link above:
In an Harmonic Progression, the ratio of two successive differences is equal to the ratio of the end numbers. If the Progression is a - b - c, we have c/a = (c-b)/(b-a). This number is the progression's ratio. For example: 4 - 6 - 12 is an Harmonic Progression of ratio 3, as 12/4 = 3 and (12-6)/(6-4) = 6/2 = 3.
We've seen the term "harmonic" used in mathematics before. In all cases, "harmonic" refers to the use of 1/n rather than n. So we replaced a, b, and c with their reciprocals in the equation above to find harmonic progressions and means. The series 1/n is called the "harmonic series" (which, as we found out earlier, diverges). And 1/n appears in musical harmony (the origin of the word "harmonic"). In fact, we've discussed before how 1/2 of a string's length raises the note an octave, 1/3 of a string's length raises the note a tritave (i.e., the 3rd harmonic), and so on.
It can be proved that the geometric mean of two positive numbers is always less than their arithmetic mean and greater than their harmonic mean.
Let's return to Pappas for the seventh and final victory:
7) Victoria Major. Display four pieces that can be combined to form two out of the three possible progressions. For example, the pieces with value 2, 3, 4, and 8 give the arithmetic progression 2, 3, 4 and the geometric progression 2, 4, 8. Note also that 2, 4, and 8 are white pieces, while 3 is a black piece.
According to the link above, the victoria magna is called a "mediocre victory," while the victoria major is called a "great victory."
This is the final victory that Pappas describes, but something seems missing here. We have victories involving one and two of the progressions. This leads us to wonder, is it possible to have all three of the progressions (arithmetic, geometric, harmonic) at once? According to the link above, this is in fact possible -- and this is called an "excellent victory":
This is an Excellent Victory: Arithmetic (6 - 9 - 12, ratio 3) Geometric (4 - 6 - 9, ratio 3/2) and Harmonic (4 - 6 - 12, ratio 3)
Now that we've seen all seven ways to win, we wonder whether it's possible to play an actual game of Rithmomachia in math class. Here's how I'd do it:
-- Instead of having the players agree on which victory will end the game, the teacher decides.
-- First, the teacher chooses one of the simple victories, such as de corpore (of Body). This allows the students to learn how to move and capture the pieces.
-- Afterward, the teacher chooses one of the progressions. Let's keep it simple and play only for mediocre (magna) victories. In Algebra I, we choose the arithmetic progression, in Geometry, we choose the geometric progression, and in Algebra II, we choose the harmonic progression. (The equation to be solved is certainly an Algebra II equation).
Of course, for this to work in a classroom, we need boards and pieces. The Rithmomachia board is exactly twice the size of a chessboard, so we need two chessboards. But the pieces are tricky. We could use some sort of manipulative (like DIDAX from last year), or if push comes to shove, we just use coins. We can use masking tape to label the pieces with points before hand.
The logistics of this would be tricky, though. This is a two-player game, so what would the other students in the class be doing? Depending on how many boards we're willing to set up, perhaps it's possible to have the students compete in teams. One player could move the pieces, and the other teammates would be calculating (math class, remember?) the possible victories.
Then again, maybe Rithmomachia is too awkward to play in our math classes. We should play simpler games like Exploding Dots. That's right -- remember what Fawn Nguyen wrote about today being the first day of "Global Math Week"?
And as I wrote earlier, it's also Metric Week. This reminds me of what I was writing yesterday about the U of Chicago Transition Math text. If we had been following that text instead of the Geometry text, then today (Day 39) we'd be in Lesson 3-9, "Measuring Volume." This is part of Chapter 3 of the text, "Measurement." The emphasis is not on the volumes of prisms or cylinders (which appear later in Lesson 12-7), but on cubic units. Metric units are taught in Lesson 3-4 and continue to appear throughout the chapter, including liters in Lesson 3-9.
This means that the text is set up perfectly to teach the metric lessons of Chapter 3 near Metric Week, and teach the circle lessons of Chapter 12 near Pi Day!
Oh, and this is a great time to segue into today's traditionalists topic. I promised you that I'd write about how traditionalists might react to the U of Chicago Transition Math text.
First of all, traditionalists might complain when they see the U of Chicago listed on the cover. We already know that the U of Chicago is the most criticized math publisher by traditionalists. Then again, most of their ire is directed towards the elementary texts, called Everyday Mathematics. They actually have precious little to say about the U of Chicago secondary texts.
The second thing traditionalists will notice is the title, Transition Mathematics. We already know that traditionalists distrust unfamiliar "labels" and names. Some of them still object to the use of the term "number sentences" instead of "equations," even though a number sentence is defined as an equation, or an inequality.
And so I expect that traditionalists would freak out at the title Transition Math, even though it's really just nothing other than Prealgebra. I suspect that part of the reason they loathe Integrated Math is that they don't like the name "Integrated Math." The class doesn't have the familiar word "Algebra" in the name, so a priori it must be less rigorous than Algebra I. Of course, J.K. Rowling said it best, "Fear of a name increases fear of the thing itself."
Of course, we also have the phrase "Never judge a book by its cover," first said by George Eliot and most recently by the teenage creator of a certain viral video. Yet traditionalists are likely to judge this math text by its cover, first by criticizing the publisher and then by criticizing the title. But once we get past the cover, traditionalists may notice that the second text in this series -- the intended eighth grade text -- is called Algebra.
On one hand, traditionalists may like the fact that the U of Chicago, believe it or not, encourages eighth grade Algebra I! But there's one problem -- the twelfth grade text is a Precalculus text -- not the Calculus text that they would prefer.
We've seen before that traditionalists actually don't care about eighth grade Algebra I -- they really care about senior-year Calculus. Eighth grade Algebra I matters only in that it leads to their desired class, senior-year Calc. We've seen the traditionalist SteveH refer to the "AP Calculus track," the set of all classes he wants students to take in Grades 8-12. The name of this sequence, "AP Calculus track," makes it plain which class he considers to be the most important. And so to traditionalists, eighth grade Algebra I is a waste of time unless it leads to senior-year Calc.
The U of Chicago texts for Grades 8-10 are Algebra, Geometry, and Advanced Algebra (i.e, Algebra II), and so these texts do follow the AP Calculus track. The problem is that the junior-year text is called Functions, Statistics, and Trigonometry. This text takes us off the AP Calculus track by delaying Precalc to senior year.
I don't own the U of Chicago upper-level texts, so I can't be quite sure of their content. But based off of their titles, I can easily see the traditionalists recommend the Functions, Stats, and Trig text for college bound non-STEM students. Prospective STEM majors, on the other hand, should skip this text and take Precalc as a junior, so they can reach the desired Calculus as a senior.
Traditionalists will tell you that the focus should be on content, not labels. Returning to the Transition Math text, I'd say that this text is solid as a Prealgebra text. In fact, the modern Third Edition of the text is even more rigorous than the old Second Edition that I just purchased. Recall that there's a link to the new version from the U of Chicago website:
https://s3.amazonaws.com/ucsmp/TM+to+CCSS+Corr+Chart+8Apr15.pdf
Most notably, the new edition has students solve inequalities, while the old edition doesn't. (This is interesting in light of the observation that the phrase "number sentences" includes inequalities, while the word "equations" does not.)
Of course, the Transition Math and Geometry texts (as well as the Algebra addendum) also include transformations, which traditionalists tend to oppose.
Before we leave the U of Chicago text, notice that associated with the new Third Edition is also a new text. In the Second Edition, the much-maligned Everyday Math covered Grades K-6, while the secondary texts covered Grades 7-12. But recently, the secondary series expanded to cover sixth grade as well (even though Everyday Math 6 still exists). This text is Pre-Transition Math. (Imagine if the seventh grade text were called "Prealgebra." Would the sixth grade text be "Pre-Prealgebra"?)
https://s3.amazonaws.com/ucsmp/PTM+to+CCSS+Corr+Chart+8Apr15.pdf
We might compare the Pre-Transition Math text to the McDougal-Littell sixth grade text that I also bought over the weekend (a text that's most likely traditionalist-approved). Unit 1 in McDougal, on Number Sense (Chapters 1-3), is spread out in the first seven Pre-Transition Math chapters. Stats and Prob are well-represented in both texts (McDougal 7-8, Pre-Transition Math 5, 10, 13). Geometry appears in both texts (McDougal 9-11, Pre-Transition Math 9, 11, 12). The only topic present in McDougal that I couldn't find in Pre-Transition Math is the area of a trapezoid -- Pre-Transition Math in return includes constructions and other geometry topics not included in McDougal.
In the end, we see that the two texts are roughly equivalent. And so any traditionalist who accepts McDougal Littell as a strong sixth grade text ought to accept Pre-Transition Math as well, despite its "unacceptable" title.
It's interesting, though, that the U of Chicago needed to develop a Pre-Transition Math text for sixth grade when there's already Everyday Math 6. I wonder whether this is because the U of Chicago "knows" that the much-maligned Everyday Math series isn't as strong as its secondary math series, hence the demand to extend the secondary series down to Grade 6. This, of course, goes back to SteveH's oft-repeated claim that the K-6 curriculum is the problem. (Recall that I tend to agree with the traditionalists regarding the lowest elementary grades.)
I also notice that like the Algebra text, Pre-Transition Math also has an entire chapter as an addendum posted online only. This extra chapter is on decimals. Apparently, the U of Chicago considers decimals to be a fifth-grade topic, included in the sixth grade text only to satisfy Common Core. (We see that the old California standards also teach decimals in Grade 5, and so they appear in McDougal only as a review topic.) I wonder, therefore, how decimals are covered in Everyday Math 5.
Before we leave the traditionalists, let's go back to our most active traditionalist, Barry Garelick. In his most recent post, he announces that he's writing another book about traditional math:
https://traditionalmath.wordpress.com/2017/10/08/shameless-self-promotion-dept/
His previous post criticizes an NPR article. Let's link to both the article and Garelick's response:
http://www.npr.org/sections/ed/2017/10/04/554316261/want-change-in-education-look-beyond-the-usual-suspects-like-finland
https://traditionalmath.wordpress.com/2017/10/08/oh-please-dept/
The comment thread is enlightening. One traditionalist icon, Ze'ev Wurman, makes a comment here:
I hope you realize that in most states it is illegal to demand fees from parents to enable access to curricular activities. Sue, or at least threaten to sue.
This is in response to another teacher, whose school is charging parents for online programs. Yes, it's illegal in most states, but it turns out that this school is Canadian.
But as usual, the dominant commenter is SteveH. The gist of the four comments written by SteveH (and you wonder why I call him the blog co-author) is that the author of the NPR is a graduate of Yale -- and Ivy League schools like Yale expect their applicants to have taken Calculus, for which the Common Core K-8 curriculum fails to prepare students. He criticizes public schools for failing to offer choices, especially the choice to take rigorous traditional math (and for that matter English). So to him, the author of the NPR article is a hypocrite.
Here are the answers to the test. Notice that I added a few multiple choice questions in order to make the test more PSAT-like. I hope I didn't make the test too hard -- but once again, I wanted the test to be of PSAT-like difficulty as well.
Here are the answers:
1. 90 < x < 180 (graphed on a number line)
2. D
3. 22
4. 150, 30
5. B
6. 72
7. 72
8. y = 4x
9. 11, -1/11
10. -8
11. A
12. Any isosceles triangle (three sides, two congruent)
13. Any nonconvex nonagon (nine sides, not convex)
14. The midpoint quadrilateral should look like a rhombus.
15. Any five lines that don't form a pentagon (because they cross or don't connect)
16. Isosceles and scalene both branch out from triangle.
17. Any linear pair (the picture from #4 works)
18. Use a protractor. (#5 claims to be 52 degrees, but in #6, the same angle is 72 degrees!)
19. 160
20. 80
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