Monday, April 16, 2018

Lesson 14-5: Vectors (Day 145)

This is going to be yet another jam-packed post, because I have so many things to discuss today, including the biannual library book sale and the books I purchased there.

But let's begin with Pappas. Today on her Mathematics Calendar 2018, Theoni Pappas writes:

The circle's diameter is sqrt(32). What's the [inscribed] square's perimeter?

Notice that this question involves a special right triangle -- and as we've seen, special right triangle problems are fairly common on the Pappas calendar. It took forever for us to reach Chapter 14 -- but now that we're there, we can finally use special right triangles to solve today's problem.

We notice that the diameter of the circle equals the diagonal of the square -- and such a diagonal divides the square into two 45-45-90 triangles. The hypotenuse of a triangle also equals the diameter of the circle, which is sqrt(32). So we use the Isosceles Right Triangle Theorem of Lesson 14-1:

x sqrt(2) = sqrt(32)
x = sqrt(32)/sqrt(2)
x = sqrt(16)
x = 4

So a side of the square is 4. The question asks for the perimeter of the square though -- and even though we didn't cover Lesson 8-1 on perimeter formally, we know that to find the perimeter of a polygon, we just add up all the sides:

p = ns
p = 4x
p = 4(4)
p = 16

Therefore the perimeter of the square is 16 units -- and of course, today's date is the sixteenth.

Ordinarily, I don't write about Sunday's Pappas question in a Monday post. But yesterday's question happens to be related to today's lesson, and so I might as well mention the problem.

Lesson 14-5 of the U of Chicago text is called "Vectors." In the modern Third Edition of the text, vectors appear only in connection with translations in Lesson 4-6. In other words, Lessons 14-6 and 14-7 of the Second Edition have no counterpart in the Third Edition.

Yesterday on her Mathematics Calendar 2018, Theoni Pappas wrote:

A boat heads out of the harbor at 57 degrees. After 12 miles it changes its heading to 222 degrees for 9 miles. What degree angle is formed between the two vectors of lengths 12 & 9 miles?

Notice that the question does not ask for the sum of the two vectors -- indeed, we'd have to use trig to find the sum (which we will in Lesson 14-7). Instead, the question only asks for the angle formed between the two vectors. Such as problem doesn't appear in the U of Chicago text -- but perhaps students might be able to draw a picture and figure it out after today's Lesson 14-5.

When sailors give the heading of a boat as 57 degrees, it means 57 degrees east of north. Whereas in Precalculus we usually define the positive x-axis as 0 degrees and measure angles counterclockwise, sailors define the positive y-axis as 0 degrees and measure angles clockwise. Our text would give the heading as "57 degrees east of north" rather than simply "57 degrees."

The key to solving this problem is to notice where 237 degrees (57 + 180) is -- this turns out to be 57 degrees west of south. If our boat were to head 57 degrees and then 237 degrees, it would have to make a complete half-turn, so that the angle between the two vectors is 0 degrees. This is tricky unless we draw the picture. The angle would not be 180 degrees -- indeed, 180 degrees between vectors would mean that the heading didn't change from 57 degrees. (Again, draw it to see why!)

So a zero angle would imply that the new heading is 237 degrees -- but it's actually 222 degrees. So this difference must be the angle we seek -- 237 - 222 = 15. Hence the desired angle is 15 degrees -- and of course, yesterday's date was the fifteenth.

This is what I wrote last year about today's lesson:

Lesson 14-5 of the U of Chicago text is on vectors. Much of physics deals with vectors. Force is a vector quantity, as Einstein knew all too well.

I remember back when I was a high school senior taking AP Physics C, and our teacher wanted us to remember one thing about vectors:

Vectors operating at right angles are independent.

This means that if two vectors are perpendicular to each other -- most notably if one is parallel to the x-axis and the other to the y-axis -- then they are linearly independent. Motion along one vector has nothing to do with motion along the other. We notice this most clearly when one is throwing a ball, and we resolve the velocity into its horizontal and vertical components. The horizontal component of velocity is slowed down slightly by the force of air resistance, while the vertical component is slowed down -- and its direction is ultimately reversed -- by the force of gravity.

In fact, I remember learning about acceleration so many years ago, and I was confused as to why an object that is turning or slowing down is said to be "accelerating." After all, I thought that an object slowing down was decelerating. To find out the answer, we must look at vectors. If the velocity and acceleration vectors are perpendicular, then the object is turning. They must be perpendicular because of the theorem from Chapter 13 (Lesson 13-5) that the tangent and radius of a circle are perpendicular. So if the velocity and acceleration vectors are in opposite directions, then the object is slowing down. I was confused by the use of the word "acceleration" by non-technical English speakers, where it usually refers to an increase in speed without considering vectors at all.

As it turns out, we prove in Linear Algebra -- a college course beyond Calculus -- that any two vectors that aren't parallel are linearly independent. But the linear independence that we consider the most in physics involve right angles. We don't discuss this idea in Lesson 14-5, but we will look at both velocity and force vectors on the posted worksheet.

Vectors appear in the Common Core Standards, but not in the geometry section. Indeed, they appear in the "Vector & Matrix Quantities" section:

(+) Recognize vector quantities as having both magnitude and direction. Represent vector quantities by directed line segments, and use appropriate symbols for vectors and their magnitudes (e.g., v, |v|, ||v||, v).

(The U of Chicago text uses the first option, a single boldface roman letter. So v is a vector.) The point is that the various domains of the standards don't line up exactly with courses -- not even for the traditional (as opposed to the integrated) pathway. So there are standards that appear in the geometry section that don't appear in a geometry course (i.e., conic sections), as well as vice versa (vectors).

The various Common Core Standards for vectors are spread out among the last three lessons of the chapter, 14-5 through 14-7. One standard that appears in today's Lesson 14-5 is:

Add vectors end-to-end, component-wise, and by the parallelogram rule. Understand that the magnitude of a sum of two vectors is typically not the sum of the magnitudes.

This is only partly realized in Lesson 14-5 -- component-wise addition will appear in 14-6. But adding vectors end-to-end and by the parallelogram rule do appear in 14-5, and the U of Chicago does give an example where the magnitude of a sum isn't the sum of the magnitudes -- indeed, there's an example where the sum is shorter than one of the vectors being added! The text explains that this is to be expected since one vector is the velocity of the boat and the other is the velocity of the current (and the boat is moving nearly upstream).

So in a way, we are beginning this standard today as well:

(+) Solve problems involving velocity and other quantities that can be represented by vectors.

The text points out that vectors are closely related to translations. Many texts tend to define the translation in terms of its vector. The U of Chicago does the opposite. Instead, translations are defined as compositions of reflections. The Two Reflection Theorem of Lesson 6-2 tells us how to find the magnitude and direction of a translation in terms of the two mirrors. Of course, some would say that the magnitude and direction of the translation really means the magnitude and direction of the translation vector. But the word "vector" can't appear in Lesson 6-2, since it isn't even defined until Lesson 14-5. Instead, we see the following theorem:

Two vectors are equal if and only if their initial and terminal points are preimages and images under the same translation.

This theorem, therefore, shows how to connect the U of Chicago definition of translation (two reflections) with the definition found in other texts (vectors). Notice that in both cases, "equal vectors" are defined to be vectors with the same magnitude and direction.

[2018 update: This problem is resolved in the modern Third Edition of the text. Translations appear in Lesson 4-4 and vectors appear in Lesson 4-6, rather than eight chapters apart as in the old text.]
Finally, the text defines vector addition:

The sum or resultant of two vectors AB and BC, written AB + BC, is the vector AC.

David Joyce criticizes the use of the word "resultant" to refer to vector sum:

The section of angles of elevation and depression need not appear, and the section of vectors omitted. (By the way, who ever calls the sum of two vectors the "resultant" of the two vectors?)

But then again, U of Chicago rarely uses the word "resultant" and mainly uses the word "sum" -- the word that we normally use when referring to addition. Joyce would prefer to omit the section on vectors, but they are included in the Common Core Standards -- and as I mentioned, the U of Chicago devotes three sections to vectors, not just one!

[2018 update: Joyce would be glad that the modern Third Edition has only one section on vectors -- but then again, he wouldn't be thrilled by the emphasis on translations and other transformations in both the U of Chicago text and Common Core.]

OK, so let's get to the books that I bought at the biannual book sale last Saturday. (Yes, for some reason the book sale was later than usual -- most of the time it's the first Saturday in April.)

One book I purchased is How to Solve Problems: Elements of a Theory of Problems and Problem Solving, written in 1974 by Wayne Wickelgren. I've selected this book as my side-along reading book for the remainder of this month. Yes, I'm aware it's been a while since my last side-along book -- lately I've been distracted by Mocha music and traditionalists.

Well, I'm done with Mocha music for now, but traditionalism, on the other hand, isn't going away. I purchased some textbooks over the weekend -- and these are directly related to traditionalism.

One of these is Statistics for Dummies, written by Deborah Rumsey in 2003. I chose this book because I never took AP Stats in high school (though I did take an intro stats class at UCLA), yet I want to be able to help students out when I'm subbing and seeing them work on stats assignments.

Here's how Rumsey begins her first chapter:

"Today's society is completely taken over by numbers. Numbers appear everywhere you look, from billboards telling of the latest abortion statistics, to sports shows discussing the Las Vegas odds for the upcoming football game to the evening news, with stories focusing on crime rates, the expected life span of someone who eats junk food, and the president's approval rating. On a normal day, you can run into five, ten, or even twenty different statistics (with even more on Election Night). Just by reading a Sunday newspaper all the way through, you come across literally hundreds of statistics in reports, advertisements, and articles covering everything from soup (how much does an average person consume per year?) to nuts (how many nuts do you have to eat to increase your IQ?)."

Rumsey's introduction here makes Statistics sound like an important class to take. Yet according to traditionalists, it's impossible to understand Stats fully without taking at least Algebra II, and possibly Calculus as well. So imagine if the only people who can take in all of those numbers on billboards, TV, newspapers, and the Internet are Algebra II and Calc students. But of course, Rumsey's book isn't intended for smart math students, but for "dummies."

Now let's move on to textbooks for classes that I actually took -- and indeed, I have a personal connection to both of the following texts.

The first book is yet another U of Chicago text -- Advanced Algebra (that is, Algebra II). Let's check out its Table of Contents:

1. The Language of Algebra
2. Variations and Graphs
3. Linear Relations
4. Matrices
5. Systems
6. Parabolas and Quadratic Equations
7. Functions
8. Powers and Roots
9. Exponents and Logarithms
10. Trigonometry
11. Polynomials
12. Quadratic Relations
13. Series, Combinations, and Statistics
14. Dimensions and Space

Notice that Lesson 9-7 of this text is called "The Number e." As it turns out, Day 97 in the new district where I'm subbing was February 7th -- that is, e Day (of the Century). So if I were teaching a class using this text, the digit pattern would place the e lesson on or close to e Day.

But then again, following the digit pattern might be tough for this text. Not only are there five chapters with ten lessons each (Chapters 2, 4, 6, 10, and 12), but two chapters have eleven lessons each (Chapters 8 and 13). With so many long chapters, the digit pattern is awkward. When, for example, would we teach Lesson 13-11 -- on Day 141, the same day as Lesson 14-1? And that's after having no time between Lessons 12-10 and 13-1.

As usual, I own the old Second Edition of the text. If we compare the modern Third Edition to the old version, we notice that the new version has only 13 chapters (just as the new edition of the Geometry text has one fewer chapter than my old edition). This time, it appears that Chapters 1-13 roughly correspond in both editions, and so Chapter 14 has been omitted.

That's a shame, since Chapter 14 contains so many interesting lessons. Indeed, let's look at this last chapter in more detail:

14-1. Three-Dimensional Coordinates
14-2. Equations of Planes
14-3. Solving Systems in 3-Space
14-4. Distance and Spheres
14-5. Solids and Surfaces of Revolution
14-6. Higher Dimensions
14-7. Solving Higher-Dimensional Systems
14-8. Fractals

So even though the focus of this chapter is definitely the third dimension, both higher (the fourth) and in-between (fractal) dimensions are introduced here. Meanwhile, if we were following the digit pattern, then today (in my old district) would be Lesson 14-5, solids and surfaces of revolution. Of course, finding the volumes of such solids is a Calculus problem, but at least the surfaces themselves are introduced here (including paraboloids).

As it turns out, the U of Chicago Algebra II text has a special place in my heart -- it is, in fact, the text I used myself as a young Algebra II student. I still recognize a few lessons from the text -- these include Lesson 11-8, where a polynomial is used to model how many slices it's possible to cut from a pizza, and Lesson 12-5, where I saw that a circle drawn in perspective is an ellipse.

If you recall, I took Algebra I in the seventh grade, and so I made it to Algebra II as a freshman. It was in the middle of this year when I moved to a different district -- and my old district didn't use the U of Chicago text. Some of my fellow freshmen used the U of Chicago Geometry text that the focus of this blog, but I was already in Algebra II.

I recall spending a lot of time of matrices when I first arrived, so it must have been Chapter 4. On the other hand, I don't recall studying Chapter 14 material at all -- after all, I would have remembered the fourth dimension and fractals.

In fact, I'm not sure whether the class reached Chapter 13 either. Perhaps I would know more stats today (and not need the Dummies book) if we'd reached the stats chapter back then. I'm not sure whether traditionalists would complain about stats here -- this is part of Algebra II after all (rather than in lieu of). And the Algebra II teacher apparently didn't have time to teach the stats anyway.

But as I wrote earlier, we definitely reached Chapter 12. This means that we most likely covered three chapters per quarter. When I arrived near the start of the second quarter, a student teacher instructed us on Chapters 4, 5, and 6, before the regular teacher took over for Chapter 7.

Of all the U of Chicago texts, perhaps traditionalists find Algebra II the least objectionable. The Geometry and earlier texts have too many transformations, and the elementary texts (Everyday Math) are the traditionalists' least favorite Chicago books. And after Algebra II, the series slows down, placing a Functions, Stats, and Trig text before Precalculus. This makes it more difficult for students to reach AP Calculus as a senior.

There's one more text to mention -- one that the traditionalists will like. The book is called Modern School Mathematics: Structure and Method, Book 2. And the lead author is -- Mary Dolciani.

Dolciani -- now that's a name the traditionalists hold in high regard. And the book is dated 1970, during the Golden Age of math texts, as far as they're concerned. I often find old books at the library book sale, so it was only a matter of time before I'd find a Dolciani text.

Here is the Table of Contents for the Dolciani text:

1. Rational Numbers
2. Addition and Subtractions of Rational Numbers
3. Multiplication and Division of Rational Numbers
4. Geometric Figures in the Plane
5. Congruence and Measurement of Plane Figures
6. Exponents and Scientific Notation
7. The Metric System
8. Precision and Accuracy
9. Decimal Numerals and Real Numbers
10. Open Number Sentences
11. Solving Open Number Sentences
12. Using Equations
13. Square Roots; Similar Figures
14. Pyramids and Prisms
15. Cones, Cylinders, and Spheres
16. Relations, Functions, and Graphing
17. Probability

This is definitely a middle school text. Many modern texts often use the name "Book 2" to refer to the second middle school grade -- that is, seventh grade. But recall that the concept of "middle school" didn't really exist in 1970 -- instead, there were junior high schools. Sixth grade was almost always the last elementary grade, and so "Book 2" must refer to eighth grade, not seventh grade. (As I've said before, some districts, including my new district, still consider sixth grade to be elementary school to this day.)

On the other hand, the push to teach Algebra I in eighth grade was also unknown in 1970, even though Dolciani's biggest defenders, the traditionalists, push eighth grade Algebra I. A traditionalist homeschooling parent, for example, might give this Book 2 to her seventh grader.

Here are a few things I notice about this book. Every lesson contains two sets of exercises -- Oral Exercises and Written Exercises. I've said before that I myself studied out of a Dolciani text (in my old district, before I moved) when I was a young seventh grader in Algebra I. She was one of the listed co-authors (along with a Brown), and the text was probably dated around 1990, but still, I definitely remember Dolciani's hallmark Oral and Written Exercises in every p-set.

Notice the name of Chapter 10 (and 11) -- "Open Number Sentences." We know that traditionalists hate the mention of "number sentences" in the Common Core. I wonder whether they know that their queen, Dolciani, uses the phrase herself. Of course, she uses the phrase "number sentences" for the same reason that the Core does -- to refer to equations and inequalities collectively.

We also see the phrase "rational numbers" in the first three chapters. Recently, some traditionalists complained about the distinction made between "fractions" and "rational numbers" in the Core. The difference is that rationals, unlike fractions, can be negative. Again, I wonder whether the traditionalists are aware that their queen, Dolciani, uses the detested phrase herself. After all, she writes on page 10:

"Now consider the set of numbers which can be represented by numerals in the form of fractions, a/b, in which a represents a whole number and b represents a counting number. In this book, this set of numbers will be denoted by the symbol A."

"The numbers a/b that make up the set A, together with their negatives, constitute the set R of rational numbers."

Let's now go back to Barry Garelick and his recent posts where he mentions the author Dolciani:

As for students taking algebra in 8th grade, I mentioned I use a 1962 textbook by Dolciani.  Here are two problems taken from the book.

OK, so my Dolciani is dated 1970, eight years newer than Garelick's text. Also, he tells us that this is the Algebra I text, not the "Math 2" text that I purchased.

A commenter wonders how Garelick obtained the Dolciani texts, since she doubts that his district would provide them to him. No, he didn't buy them from the library book sale:

 I didn’t tell them that’s what I’m using. I managed to procure about 15 of her 1962 edition of Algebra 1 at a penny per book; paid only for shipping. Good move, since now the same book is selling for $60. I like the book for its sequencing and problems; the explanations are too formal and confusing for some topics. But the scaffolding of problems is superb. I also draw from the official textbook (Big Ideas) for topics not covered in Dolciani (like fractional exponents, arithmetic series and some other topics, but not too many). Big Ideas has a dearth of good word problems and when they do have a standard work, or distance problem it’s a one-off type of thing–as if they are supposed to get the working of such problems from that one problem.

In the previous post, he reveals what text he uses with his seventh graders:

Look, I use a 1962 Dolciani algebra textbook to teach my algebra class. The word problems are plenty challenging for my students, though I’m fairly certain that the authors of said study would find such problems lacking in “real world relevancy” (as if my students care) and low cognitive demand.  Yes, I hear you saying “But they’re not from poverty and they would do well anywhere.”  Really? Got proof of that?
For my 7th grade class, I use JUMP Math, which uses micro-scaffolded approaches, but doesn’t skimp on the conceptual understanding behind the procedures either.  It has been given bad reviews by those who hole math reform ideologies in high regard as being “too procedural”.
I've never heard of JUMP Math. Meanwhile, I wonder whether Garelick would use the Dolciani Math 2 text I bought in his seventh grade class if he'd had access to enough copies of it.

Finally, let's start Wicklegren's How to Solve Problems. Don't worry, traditionalists -- just be glad the text is not called "How to Problem Solve."

Chapter 1 of Wicklegren's How to Solve Problems is an Introduction. He begins:

"The purpose of this book is to help you improve your ability to solve mathematical, scientific, and engineering problems. With this in mind, I will describe certain elementary concepts and principles of the theory of problems and problem solving, something we have learned a great deal about since the 1950's, when the advent of computers made possible research on artificial intelligence and computer simulation of human problem solving."

Throughout this book, Wicklegren will describe many problems, including some classic puzzles and other riddles. We won't see many of them in this introductory chapter -- instead, the author wants us to look at the big picture. He continues:

"You should pay careful attention to these problems and should not be discouraged if you do not perfectly understand the theoretical discussions."

He compares games and puzzles to real-world problems, but warns us:

"A practical problem such as how to build a bridge across a river is a formal problem if, in solving the problem, one is limited to some specified set of materials (givens), operations, and, of course, the goal of getting the bridge built. In actuality, you might limit yourself in this way for a while and, if no solution emerged, decide to consider the use of some additional materials, if possible."

Wicklegren describes the age-old method of reducing a problem to one previously solved -- a technique so common that people have made jokes about it:

"However, in cases where it is obvious that a particular problem is a member of a class of problems you have solved before, you do not need to make explicit, conscious use of the method: simply go ahead and solve the problem, using methods that you have learned to apply to this class."

He explains why learning how to solve problems is so important:

"Mastering general problem-solving methods is important to you both so you can use problems as a learning device and so you can achieve the maximum range of applicability of the knowledge you have stored in mind -- on an examination, or a job, or whatever. The goal of this book is to teach as many of these general problem-solving methods as I know about."

Hey, traditionalists -- even you can appreciate what Wicklegren writes next:

"Practice [emphasis his] in using the methods is essential to achieving a high level of skill. Everyone who solves problems uses many or all of the methods described in this book."

And he concludes by further describing the important of practice:

"This is the way of all skill learning, whether driving a car, playing tennis, or solving mathematical problems."

I don't disagree with Wicklegren and the traditionalists that practice makes perfect. The problem is that if the practice isn't interesting enough, students won't even begin to practice. Fortunately, the problems in this book should be fun enough for them to want to practice.

By the way, I remember SteveH writing something about a "Polya" and problem solving. Here Wicklegren mentions Polya, not in the intro, but in the preface:

"My greatest intellectual debts are to Allen Newell, Herbert Simon, and George Polya."

I assume that the name Polya will still raise a red flag for the traditionalists.

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