Monday, October 9, 2017

Chapter 3 Review, Continued (Day 38)

This is what Theoni Pappas writes on page 282 of her Magic of Mathematics:

"For example, a white circle #4 removes black circle #28 if 7 spaces lie between the two pieces."

This is the third page of the subsection called "The Old Mathematical Game of Rithmomachia." But unfortunately, the first two pages of this section were blocked by the weekend, thus both landing on non-posting days.

Well, there's no point in writing about Rithmomachia without giving the rules and other info that Pappas wrote on the missing pages 280 and 281. At least here's a link to the game and its rules, which should help you readers out:

As we can see from the following link, Pappas is right in the middle of explaining rule #3, which is the "Assault Capture" as listed at the link. She proceeds to describe the "Ambush Capture" next:

4) Ambuscade. If two pieces of a player can move on either side of an opponent's piece and the sum of these two pieces equals the opponent's piece number, then the player takes the opponent's piece. For example, if black triangle #12 can be straddled by white circles #4 and #8, then black triangle #12 is removed without actually moving the pieces.

As you can see, the assault involves multiplication while the ambush involves addition. You make think that this is a game that some math teacher made up in order to help the students review for an upcoming test. You might even guess that a member of the MTBoS invented the game, until you remember that it appears in the Pappas book, written while the Internet was still in its infancy (and so there clearly was no MTBoS yet).

But believe it or not, Rithmomachia is nearly a thousand years old -- math class or even schooling as we know it now didn't exist a millennium ago! In its heyday, it was even more popular than chess.

It sounds like interesting to play in math class, but I fear that the rules may be too confusing -- not the addition or multiplication part, but the application of the rules. When exactly do we add, and when do we multiply? Must we count the spaces or move the pieces? I can easily see frustrated students get stuck when they can't figure out the rules.

Oh, and we can't leave without Pappas telling us how to win the game. Let me write these out in full from Pappas, since the link above mentions victories only briefly. (Both Pappas and the link point out that only one of these can win any particular game, and this is chosen in advance.)

1) De corpore. [This is called the Victory of Body at the link above.] The players agree on a number of pieces to be captured, that will declare that player the winner.

2) De bonis. [This is called the Victory of Assets at the same link above.] Players agree on a number value target. If the sum of the number value of the pieces a player captures will total or exceed the target number, then that player is the winner.

3) De lite. [This is called the Victory of Proceeds at the link above.] Here a win depends both on the total number value of the pieces and the total number of digits of the pieces. By example, target value could be 160 with digits equaling only 6. So [oops -- this is the end of page 282, but we might as well go on to page 283 to finish this condition] a player with pieces 121, 9, 30 would win, while the player having 56, 64, 28, and 15 would not (these pieces have 8 digits in all).

I have more to say about the ways to win, but I'll save it for tomorrow, after we've seen the other four possible victories on page 283.

Meanwhile, two days ago was the first Saturday in October, and you know what that means -- yes, it's the biannual library book sale. And so, as usual, let me write about all the math books that I bought that day at the sale:

First, I purchased McDougal-Littell's Math Course 1. I already own the Math Course 2 book, and as we found out when I purchased the earlier book, these are middle school texts. Math Course 1 is intended for sixth graders, while Course 2 is for seventh graders. Notice that there is no textbook called Course 3 -- instead, the eighth grade text is called Algebra I. That's because these books are based on the old California standards, which encouraged eighth graders to take Algebra I. The book is dated 2008 -- notice that it was merely one year later, in 2009, when the Common Core Standards began to take hold, and the California standards went by the wayside.

Here is the table of contents:

1. Number Patterns and Fractions
2. Fraction and Decimal Operations
3. Integers
4. Expressions and Equations
5. Ratios and Proportions
6. Percents
7. Analyzing Data
8. Probability
9. Geometric Figures
10. Measurement and Area
11. Surface Area and Volume
12. Graphing: Review and Preview

Notice that some of topics covered in this sixth grade text are considered to be seventh grade topics under the Common Core, especially integers and probability. This is to be expected, since the old California standards prepared students for eighth grade Algebra I, while the Common Core Standards prepare students for ninth grade Algebra I.

Another Common Core 7 topic covered in this text is circle measurement. Indeed, circumference appears in Lesson 10.6, while area appears in Lesson 10.7. One of my favorite things to do whenever I get a new text is see whether it's possible for these lessons to land near Pi Day.

Well, this text has 12 chapters, which encourages us to teach six chapters per semester. If the school year is divided into quarters. then we're tempted to teach three per quarter (and four every trimester, at schools that use that quarter). On the Early Start Calendar, Pi Day usually falls near the end of the third quarter, so we'd be only on Chapter 9. The Labor Day calendar is even worse here, as Pi Day is near the end of the second trimester or Chapter 8. Neither takes us to Chapter 10 by Pi Day.

If I were teaching out of this book, I'd still aim for Chapter 6 by the end of the semester. This gives us about three weeks per chapter, which is ideal. Then I'd speed up a little during Chapters 7 and 8, since Stats and Prob are usually less important than Number System, Expressions, and Equations. Then we can slow down again for Chapters 9 and 10, which should get us to the circle lessons by Pi Day. We notice that Chapter 12, a "review and preview" chapter, makes sense to teach after state testing is over, so the goal is to finish Chapter 11 before testing.

Now the other book I purchased is an interesting -- it's my third U of Chicago text. The first book is of course the Geometry text we all know and love, and the second book is the Algebra I text. Well, the title of my latest book is Transition Mathematics.

What exactly is "Transition Mathematics," anyway? Well, the U of Chicago text consists of six secondary texts. Here they are listed with their intended grade levels:

7. Transition Mathematics
8. Algebra
9. Geometry
10. Advanced Algebra
11. Functions, Statistics, and Trigonometry
12. Precalculus and Discrete Mathematics

Since Transition Mathematics is the year before Algebra I, perhaps "Prealgebra" is a recognizable name for this course. Nonetheless, I'll continue to call it Transition Mathematics (or Transition Math for short), since that's what the U of Chicago calls it.

Here is the table of contents for Transition Math:

1. Decimal Notation
2. Large and Small Numbers
3. Measurement
4. Uses of Variables
5. Patterns Leading to Addition
6. Problem-Solving Strategies
7. Patterns Leading to Subtraction
8. Displays
9. Patterns Leading to Multiplication
10. Multiplication and Other Operations
11. Patterns Leading to Division
12. Real Numbers, Area, and Volume
13. Coordinate Graphs and Equations

Circle measurement appears in Lessons 12-4 and 12-5 of this text. If I were to teach this text and use the digit pattern, we notice that Days 124 and 125 on the blog calendar fall very close to Pi Day, and so the pattern works. Then again, there are more chapters with a full ten sections in this text (namely 1, 4, 5, and 10) than in the Geometry text. This means that we'd have to move quickly and squeeze in the tests in order to reach Lessons 12-4 and 12-5 by Pi Day.

This is the Second Edition of the text, dated 1995. Just as with the Geometry text, there is a more recent Third Edition. As it turns out, there are many more changes made to the Transition Math text than to the Geometry text. Indeed, only one chapter has an identical title in both the Second and Third Editions of Transition Math ("Patterns Leading to Division"), while one is very similar ("Uses of Variables" vs. "Using Variables"). On the other hand, as many as five chapters have the same title in the Second and Third Editions of Geometry.

You may be wondering, how do I know so much about the Third Edition of the text when I just bought the Second Edition two days ago. Well, we look at the U of Chicago link below:

where the Third Edition chapter titles are easily found.

We also notice the correspondences between the Transition Math and Common Core. We see that this text is still considered a seventh grade text -- correspondences to sixth grade standards are called "reviews" while those to eighth grade standards are labeled "anticipates."

This is interesting, because the U of Chicago eighth grade text is called "Algebra" -- and there is also a correspondence chart between the Algebra text and Common Core 8:

Ironically, the U of Chicago considers Algebra I to be an eighth grade course, even though the Common Core calls it a ninth grade course! Of course, the U of Chicago Algebra text doesn't contain any geometry. So the geometry strand of the Common Core 8 standards is squeezed into a special online only Chapter 14, in order for the U of Chicago to claim Common Core compliance. We know that the U of Chicago text is one of the first to emphasize the translations, reflections, and rotations of Common Core -- the problem is that it does so in its Geometry text (and even a little in the Transition Math text) and not the text for eighth grade (Algebra), the grade when transformations first appear in the standards.

I have much more to say about the Transition Math text, but I'll save it for tomorrow, since that's our scheduled traditionalists post. In particular, I want to guess how the traditionalists would react to such a text.

I also purchased two recreational math book as well. One of these is Paul Hoffman's The Man Who Loves Only Numbers, a biography of the mathematician Paul Erdos. The other is George G. Szpiro's Poincare's Prize, a book about the first discovered solution of a Millennium Prize problem. (Yes, I mentioned the Millennium Prize problems last week!)

And so we have our next side-along reading book. I was originally considering writing Szpiro's book, but the math is more difficult in that book (otherwise it wouldn't be worth a million-dollar prize), and so I think I'll start the book about Erdos on Wednesday. Recall that this is PSAT day and there will be no U of Chicago lesson, so it's a great day to start Erdos.

Ah yes, it's time to continue our review for the Chapter 3 Test. Now I decided to take another old worksheet from three years ago, which contain part of a test (as it starts with #12). So I need to create a new worksheet numbered #1-11.

Well, recall that this is PSAT week. Some may find it awkward to make students study for a full math test on Tuesday, implying that Wednesday's PSAT isn't important.

But now suppose Questions #1-11 are in fact PSAT-like problems. We know that the SAT, and by extension the PSAT, emphasizes (the Heart of) Algebra more than Geometry. Well, Chapter 3 is an excellent chapter to focus on algebra problems. Many algebra equations, for example, can be converted into Geometry problems simply by writing the left and right sides of the equation as the measures of vertical angles and then ask for the value of the variable. This chapter teaches vertical angles and linear pairs, as well as slope (another major PSAT/SAT topic).

Earlier, I wrote that I don't want to force algebra on our Geometry students so soon -- and indeed, I didn't force algebra on the students in Chapters 1 and 2. But Chapter 3 is a great time to begin slowly reintroducing algebra, since it's timed perfectly with PSAT week. (Again, the "Postulates from Algebra" don't appear until Chapter 3 in the new Third Edition of the Geometry text, again marking Chapter 3 as the "algebra" chapter.)

So notice the new Chapter 3 Review worksheet, with algebra Problems #1-11.

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