Monday, March 27, 2017

Lesson 12-7: Can There Be Giants? (Day 127)

Last weekend was the biannual book sale at our local library. Ordinarily, the spring book sale is held the first Saturday in April -- the only reason it would be the last Saturday in March instead would be to avoid Easter weekend. Yet the book sale was last Saturday, even though Easter is still several weeks away. Who knows what the librarians were thinking with the strange date here?

As usual, I purchased several math texts for one dollar each. For the first time, I actually found two Integrated Math I texts. The first was published by McDougal Littell in 1998 -- that's right, well before the Common Core. We know that some traditionalists oppose both Common Core and Integrated Math, but not only is it possible to have Common Core without Integrated Math, but vice versa as well. I see that the text comes from a school in Olympia (the capital of Washington state) -- and so I wonder what the book is doing here in California! But anyway, other states apparently had Integrated Math years before there was ever a Common Core.

The McDougal Littell text is divided into ten units:

McDougal Littell Integrated Mathematics 1
1. Exploring and Communicating Mathematics
2. Using Measures and Equations
3. Representing Data
4. Coordinates and Functions
5. Equations for Problem Solving
6. Ratios, Probability, and Similarity
7. Direct Variation
8. Linear Equations as Models
9. Reasoning and Measurement
10. Quadratic Equations as Models

The other text is published by Pearson -- so it's obviously a Common Core text. Two years ago, I wrote about the Pearson texts I found near the back of a classroom where I was subbing, but only the Integrated Math II and III texts were in the classroom that day. Well, since that day I remember actually seeing the Math I text in another classroom where I was subbing -- and yet I didn't mention Pearson Integrated Math I anywhere on the blog. I think what happened was that I saw the texts unused near the back of the room, but the students were actually working in another text. So I blogged only about the books the kids were actually using, not the Pearson texts.

The first difference between McDougal Littell and Pearson is obvious -- the latter is divided into two volumes, yet the former is only a single volume. I actually found only Volume 2 at the book sale.

Here is the table of contents for Pearson Math I Volume 2, which covers Chapters 8 to 14:

Pearson Mathematics I
Volume 2
8. Transformations
9. Connecting Algebra and Geometry
10. Reasoning and Proof
11. Proving Theorems About Lines and Angles
12. Congruent Triangles
13. Proving Theorems About Triangles
14. Proving Theorems About Quadrilaterals

Let's compare this to the contents of Pearson Math II Volume 1, which I wrote about two years ago:

Pearson Mathematics II
Volume 1
Chapter 1: Reasoning and Proof
Chapter 2: Proving Theorems About Lines and Angles
Chapter 3: Congruent Triangles
Chapter 4: Proving Theorems About Triangles
Chapter 5: Proving Theorems About Quadrilaterals
Chapter 6: Similarity
Chapter 7: Right Triangles and Trigonometry
Chapter 8: Circles

That's right -- the last five chapters of Math I and the first five of Math II are identical! I remember noticing that oddity two years ago -- and yet I never said a single word about it on the blog! Actually, the day I wrote about Math II and III, I noticed that these texts had one chapter in common, but of course I wrote nothing about Math I.

Let me repeat some of what I wrote two years ago about Pearson Math II Volume 1 text, since it applies to Pearson Math I Volume 2 as well. Note that we must add nine to every chapter number in Math II to obtain the corresponding Math I chapter.

Pearson's Chapter 2 is on Lines and Angles. Section 2-2 introduces a postulate on parallels -- and it happens to be the Same-Side Interior Angles Postulate. This is highly unusual, as most texts choose either Corresponding Angles (U of Chicago) or Alternate Interior Angles as a postulate. Section 2-3 contains the Parallel Tests -- and all of the are listed as theorems. The first test is the Converse of the Corresponding Angles Theorem, and its proof is said to be given in Section 13-5 -- but that chapter is on Probability. I once saw another text that does give the proof -- it's an indirect proof, and so Pearson probably intended to wait until Chapter 4 to give the proof, but it mistakenly omits it.

Now that we see the Math I text, this all makes sense. Chapter 4 of Math II is the same as Chapter 13 of Volume I, and this is indeed where the indirect proof of the Corresponding Angles Consequence is actually given. When renumbering the chapters, Pearson mistakenly kept "Lesson 13-5" instead of changing it to "Lesson 4-5" for Math II.

Looking at Lesson 13-5 in my Math I text, I see that in the exercise where the Corresponding Angles Consequence is to be proved, a hint is given: use the Triangle Angle-Sum Theorem. We've had several discussions regarding this proof and how other authors prefer geometry to be taught, such as David Joyce and Hung-Hsi Wu.

We know that Joyce wants there to be only one postulate for parallels. Pearson has two -- the Same-Side Interior Test and the Parallel Postulate of Playfair. Pearson uses Playfair as the first step of Triangle Angle-Sum, which is then used to prove Corresponding Angles Consequence. But as we've seen in the U of Chicago, we can use the Angle Measure (or Protractor) postulate in the first step of Triangle Angle-Sum instead of Playfair. Then Playfair becomes superfluous, and so there is only one parallel postulate, as Joyce suggests.

On the other hand, we ultimately use that one postulate in the proof of Corresponding Angles Consequence, but Wu shows us that the parallel consequences (as opposed to the tests) are provable in neutral geometry, without any parallel postulate at all. The proof requires replacing Triangle Sum with the weaker TEAI (Triangle Exterior Angle Inequality), which is provable in neutral geometry. I have tried something like this in the past on the blog, but it ended up being awkward. Trying to avoid a parallel postulate just because a result holds in neutral geometry turns out to be an unnatural restriction that only confuses high school students.

Let's get back to two years ago:

Pearson's Chapter 3 is on Congruent Triangles. We notice that in Sections 3-2 and 3-3, SSS, SAS, and ASA are given as postulates. But then in true Common Core fashion, congruence transformations (isometries) are used to verify these postulates in Section 3-8. I notice that as important as the isometries are to Common Core, translations, reflections, and rotations aren't defined anywhere in this section. I suspect that since these isometries form the foundation of Common Core Geometry, Pearson actually defines them in its Math I text. There's no way for me to know for sure since I don't have access to the Math I text.

Now I have the Math I text, so now we know -- transformations appear in Math I Chapter 8, while Math II starts with the same chapter as Math I Chapter 10.

Pearson's Chapter 5 is on Proving Theorems About Quadrilaterals. Most of this material appears in the same-numbered chapter of the U of Chicago, except some of the theorems on parallelograms don't appear until Chapter 7. Of course I had to check to see that Section 5-7 of Pearson gives the exclusive definition of trapezoid. As I said earlier, the inclusive definition was spotted in a PARCC question, but I have yet to see it. Then the rest of the chapter gets into coordinate geometry -- which I suppose makes sense as many coordinate proofs involve quadrilaterals. The harder concurrency theorems from Chapter 4 are given coordinate proofs here as well. Many of these coordinate proofs require the Distance or Midpoint Formulas, which are never introduced. Once again, I suspect that these formulas are taught in the Math I text and students are expected to remember them.

Apparently, looking at the Math I index, Distance and Midpoint appear in Chapter 7 of Math I -- the last chapter of Volume 1. I still don't have a copy of Volume 1, but considering what I wrote two years ago:

Notice that the four constructions in the Pearson text are exactly the same as the four in Prentice-Hall, even numbered in the same order! I checked the list of authors for the Pearson text and saw that there are several, since both algebra and geometry writers are needed. But I notice that one of the geometry authors is Laurie Bass, who is also, according to Joyce, the author for Prentice-Hall -- small wonder, then, that the same four constructions are here.

...I wouldn't be surprised that Math I Chapter 7 is similar to Bass Chapter 1 -- the Prentice-Hall text that Joyce criticizes. Part of the criticism is that the Distance Formula appears well before the Pythagorean Theorem on which it depends.

Okay, that's enough from two years ago. What I want to do now is compare the Pearson text to the McDougal-Littell text that I purchased this weekend, so we can compare Math I to Math I.

First of all, we see that, for an "Integrated" Math text, algebra and geometry are separated! In fact, we could teach a complete Geometry course by starting with the last chapter of Math I Volume 1 and going up to the first chapter of Math II Volume 2, with the intervening volumes (Math I Volume 2 and Math II Volume 1) both being pure geometry texts.

On the other hand, McDougall Littell doesn't have separate algebra and geometry chapters. Many of the units include both topics. For example, Unit 6 is "Ratios, Probability, and Similarity." This chapter includes both the algebra of solving proportions and their obvious application to similarity, which is geometry. The following Unit 7, "Direct Variation," also includes connections, as its first lesson is "Direct Variation, Slope, and Tangent." This lesson connects similarity to slope via the trig ratio of tangent. Common Core tells us to tie similarity to slope, and here we see the connection made in this pre-Core text.

As it turns out, the transformations of translation, rotation, and so on also appear in this text -- again, these transformations existed long before the Common Core (as we've seen with the U of Chicago text, which is also pre-Core). Translations and rotations are in Unit 4, "Coordinates and Functions," while dilations obviously appear with similarity in Unit 6. Reflections don't appear until the final unit, "Quadratic Equations as Models" -- which in a way makes sense when we consider reflections along with the line of symmetry of a parabola. By contrast, the only truly integrated chapter of Pearson is Chapter 9, "Connecting Algebra and Geometry" (which covers perimeter, area, and slope).

McDougall Littell provides a complete listing of the topics that appear in each of the Integrated Math I, II, and III texts:

Course 1
Algebra: linear equations, linear inequalities, multiplying binomials, factoring expressions
Geometry: angles, polygons, circles, perimeter, circumference, area, surface area, volume, trig ratios
Stats/Prob: analyzing and displaying data, experimental, theoretical, and geometric probability
Logical Reasoning: conjectures, counterexamples, if-then statements
Discrete Math: discrete quantities, matrices to display data, lattices

Course 2
Algebra: quadratic equations, linear systems, rational equations, complex numbers
Geometry: similar and congruent figures, coordinate and transformational geometry, right triangles
Stats/Prob: sampling methods, simulation, binomial distributions
Logical Reasoning: inductive, deductive, valid, and invalid reasoning, postulates and proof
Discrete Math: matrix operations, transformation matrices, counting techniques

Course 3
Algebra: polynomial functions, exponential functions, logarithmic functions, parametric equations
Geometry: Inscribed figures, transforming graphs, vectors, triangle trig, circular trig
Stats/Prob: variability, standard deviation, z-scores
Logical Reasoning: identities, contrapositive and inverse, comparing proof methods
Discrete Math: sequences and series, recursion, limits

As we've seen above, some of the topics listed under "Course 2" are introduced in Course 1.

I've written before about the relationship between Common Core 8 and Integrated Math I. In many ways, Integrated Math I can double as a Math 8 course. But there are a few differences, depending on whether we use Pearson or McDougal-Littell as our Math I course.

I suspect that Pearson Math I Volume 1 (and I can only guess its contents since I don't have that text) matches the algebra content of Math 8. But the geometry content of Math I Volume 2 is a bit too advanced for eighth graders, especially with proof being introduced in Chapter 10.

With McDougal-Littell, it's the other way around. Most of its geometry content appears in Common Core Math 8 (or even Math 7), except for trig ratios. But the algebra content differs greatly from the Math 8 course. Multiplying binomials appears in McDougal-Littell Course 1 but not Math 8. Notice that some factoring actually appears in Math 8 (and even Math 7), but it's just GCF factoring, not full trinomial factoring.

I sort of like the way topics are introduced in the McDougal Littell text. Algebra and geometry really are integrated throughout the text. I would recommend the text for both eighth grade and Integrated Math I classes. The last chapter, on quadratic equations, is a little advanced for Common Core 8, but the appearance of the Quadratic Formula in the final lesson (Lesson 10-8) reminds me of the old Dolciani text that I used as a young Algebra I student. That text also teaches the Quadratic Formula in the very last lesson.

If I were at a school that uses the Pearson text, I would note that with the last five chapters of Math I repeating as the first chapter of Math II, I'd only cover each chapter once. That is, I'd teach Chapters 1 to 12 of Math I the first year, and then Chapter 4 to 15 of Math II the second year. This means that exactly 12 chapters is given each year -- six chapters per semester, one chapter per three weeks. I've stated before that I like the idea of covering each chapter in three weeks.

This means that the switch from Volume I to Volume II each year doesn't happen exactly at the semester mark. But it does mean that the transition from algebra to geometry in Math I, as well as the transition from geometry back to algebra in Math II, does occur right at the semester, which is convenient for semester finals.

The Pearson text also provides a loophole for traditionalist teachers and principals who oppose both Integrated Math and the delaying of Algebra I until ninth grade, rather than their preferred eighth grade Algebra I and ninth grade Geometry. Traditionalists who are forced by their districts to use Pearson can start with Math I Volume 1 in eighth grade, which is all Algebra I. Then the freshmen can use Math I Volume 2 and Math II Volume 1 to make up their Geometry course. Sophomores take Math II Volume 2 and Math III Volume 1, which comprise a credible Algebra II class. Juniors wrap up Math III Volume 2 and then any missing topics they need to prepare for senior year Calculus.

The trickiest part of this idea is the eighth grade Algebra I course. Eighth graders go to middle school while ninth graders attend high school, so high school teachers would need the middle schools to cooperate, plus the Math I Volume 1 texts must be delivered to middle school campuses. The volume plan is more convenient with semesters, so it may be tricky at trimester middle schools. Finally, there's one problem that any eighth grade Algebra I class has in the Common Core era -- the students still have to take the PARCC or SBAC in May of the eighth grade year, and those tests include geometry and other topics not included in Algebra I. My preference is to include the other PARCC or SBAC topics in the first semester and then Math I Volume I algebra the second semester. Some topics like quadratic equations are missing from Algebra I, but of they will appear in Algebra II. (Here's an idea that fits into trimesters -- first trimester: Geometry and Stats/Prob, second trimester: Pearson Math I Chapters 1 to 4, third trimester before PARCC/SBAC: Chapters 5 to 7, after the test: end with the Quadratic Formula.)

The Pearson text is typical of many texts in the Common Core era. Both the Illinois State and Pearson texts are divided into two softcover volumes rather than one hardback volume. And it's telling that both Illinois State and Pearson appear to have connections to England -- I wonder why texts based on our national standards need to be developed in another nation.

There are a few other books that I purchased at the book sale. One is Guiding Children's Learning of Mathematics by Leonard Kennedy and Steve Tipps (2000). The book seems to focus mainly on elementary school math, but there are a few middle school topics mentioned there (such as decimals, Stats/Prob) and besides, the book only costs a quarter. Another book is Lee Canter's Succeeding With Difficult Students (1993). In my quest to become the ideal classroom manager, the instructional aide at my school recommends that I read Lee Canter. This book appears to be based on an "inservice video package" -- that is, Canter and associates would visit schools on their PD days and show the teachers some videos. Since I don't have access to these videos (but who knows, maybe they're on YouTube), this book won't be as effective -- again, but this one also sets me back only a quarter.

The last book I'll mention will also be my newest side-along reading book. And now you're asking -- what the heck is going on? I stopped writing about my middle school classes and even took two weeks off from posting due to lack of time during the Big March, and now suddenly I have time to go back to posting about side-along reading books on the blog! (I know that this post is long, but then again, today is coding Monday.)

Well, my side-along reading book is The Magic of Mathematics by Theoni Pappas. I already planned on making up some "Pappas questions" that I would cover in my classes and on the blog. That fell apart when the administration started cracking down on questions from sources other than Illinois State and forced me to change my Warm-Up from Pappas to the Illinois State Daily Assessment. My so-called "Pappas questions" didn't come from Pappas herself, but were just modified versions of her questions with the same idea that the answer was the date.

So now I'm replacing Pappas questions with an actual Pappas book, which she wrote in 1994. Here is the table of contents:

0. Preface
1. Mathematics in Everday Things
2. Magical Mathematical Worlds
3. Mathematics & Art
4. The Magic of Numbers
5. Mathematical Magic in Nature
6. Mathematical Magic from the Past
7. Mathematics Plays its Music
8. The Revolution of Computers
9. Mathematics & The Mysteries of Life
10. Mathematics and Architecture
11. The Spell of Logic, Recreation & Games

I'll still going to follow the Pappas tradition and tie in the date somehow. And here's how I'll do it -- today is March 27th, the 86th day of the year. So I'll read page 86 today. Surely I should have time to read one page per day. This will take us to page 311 (the solution page) sometime in November. Of course, since I just bought the book I could just label this Day 1 and start from there. But it's not as if I won't be skipping pages anyway, since I don't post here everyday.

Page 86 is in the middle of Chapter 3, "Mathematics & Art." The current section is called "Projective Geometry & Art," which begins on Page 85. Pappas writes:

"Projective geometry is a field of mathematics that deals with properties and spatial relations of figures as they are projected -- and therefore with problems of perspective [emphasis hers]. Just as topology studies the properties of objects that remain unchanged after they have undergone a transformation, projective geometry studies properties of plane figures that do not change when they undergo projections."

She adds that a circle, when viewed in perspective, becomes an ellipse, and a square is projected into a different quadrilateral -- but she doesn't specific which one. Perspective is mentioned in Lesson 1-5 of the U of Chicago text, so maybe we can read it to figure out what happens to our square.

In the U of Chicago text, we read:

"In perspective drawings, horizontal (or vertical) lines remain horizontal (or vertical) and parallel. But oblique parallel lines will intersect if extended. The box below has two vanishing points, P and Q. Vertical lines on the box remain vertical and parallel, but the box has been tilted to have no horizontal edges."

So assuming that this box is a cube, each face is a square. The vertical sides of the square are still vertical, so we have at least one pair of parallel sides -- a trapezoid. But this trapezoid is definitely not a parallelogram, since the other sides ultimately intersect at the vanishing point. So the answer is that a square is projected into a trapezoid.

The U of Chicago also gives a nonperspective drawing of a cube. In this drawing, the sides of the square do not meet at a vanishing point, so the image of the square is a parallelogram. But this is definitely not a projection.

Now Lesson 1-5 isn't today's lesson -- today is actually Lesson 12-7, so let's hurry up and begin.

Lesson 12-7 of the U of Chicago text is called "Can There Be Giants?" Last year we skipped over this lesson, so I don't have any old worksheets for it. This is one of those "fun lessons" that we can cover if there's time, but in the past we bypassed it to get to 12-8 and the all-important SSS Similarity.

As I wrote above, Lesson 12-7 naturally lends itself to an activity. The whole idea behind it is that while dilations preserve shape, they don't preserve stability. This is because of the Fundamental Theorem of Similarity -- a dilation of scale factor k changes lengths by a factor of k, areas by a factor of k^2, and volumes by a factor of k^3. Weight varies as the volume, or k^3, while strength varies only as the area (as in surface or cross-sectional area), or k^2. Therefore, the answer to the question in the title of the lesson is no, there can't be giants because their k^2 strength, couldn't be strong enough to carry their own k^3 weight.

Here are the worksheets that I've created for this lesson:

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