Monday, April 11, 2016

Lesson 15-3: The Inscribed Angle Theorem (Day 137)

Today I subbed in a special ed class. This class is self-contained, so I ended up covering all subjects, including math.

The students in this class are at different levels. One of the students is working on subtraction of decimals -- specifically money. Subtraction of decimals is considered a Grade 5-6 skill under the Common Core -- that is, fifth grade for "place-value strategies" and sixth grade for learning the standard algorithm. General multi-digit subtraction is a Grade 3-4 skill under the Core.

But the problems on this worksheet are unusual. I notice that "" is written at the bottom of the page, and so I use that to locate the actual worksheet on the web. Here is the link:

The student is working on "Version C" from the above link, but all of the worksheets there have these same unusual subtraction problems. He has already completed the first 25 problems on the worksheet and has now arrived at the following problem:

- $9.99
- $8.53

And now we see what makes this problem so strange -- there are three numbers, not just two!

I remember back when I was very young -- it could have been in second grade when I was learning (and teaching!) square roots, but probably even earlier, in first grade or kindergarten. I had seen several problems where we must add three or more numbers, but never in subtraction. So for example, I might have seen something like:


I knew how to add these values using the standard algorithm. We simply add the three numbers one column at a time. So 3 + 9 + 3 = 15, which means that we write 5 and carry 1. Then we have the next column and 1 + 6 + 9 + 5 = 21, so we write 1 and carry 2. This is a big difference between having just two addends and having three -- sometimes we have to carry digits larger than 1. Indeed, we may end up carrying digits as high as one less than the number of addends.

But I had never seen a subtraction problem with more than three numbers. I remember being excited to receive one of those workbooks stores sell to help kids learn over the summer -- I believe it was called "Advanced Addition and Subtraction." I hoped upon hope to see a subtraction problem with three numbers and wanted to learn the algorithm for doing them -- but unfortunately, there weren't any such problems. Of course there were addition problems with three or more numbers, but not any such subtraction problems. I completed the workbook, of course, but I was so sad that I had to answer the same-old subtraction problems with two numbers.

And I never did see any workbook or worksheet with subtracting three numbers -- until today. So now, I am finally able to work on the problems that I've been waiting almost 30 years to solve!

The truth is, there is no standard algorithm for subtracting three numbers. Let's look at the problem the student and I have to solve again:

- $9.99
- $8.53

Let's attempt to use the standard algorithm for subtraction on this problem away. We begin with the rightmost column and see that we can't subtract 3 - 9. So we borrow from the next column -- 6 becomes 5, and 3 becomes 13. Then 13 - 9 = 4 -- and then we can subtract that 3 that's in the bottom of the rightmost column, 4 - 3 = 1, so we write 1. In the next column, we must borrow again -- 8 becomes 7, and 5 becomes 15. Then 15 - 9 = 6 and 6 - 5 = 1, so we write 1. To the left of the decimal, we have 17 - 9 = 8 and 8 - 8 = 0, so there are no dollars. The final answer is 11 cents.

But if we try to subtract the next problem this way, we stumble:

- $3.96
- $7.58

We can't subtract 0 - 6, so we must borrow -- 3 becomes 2, and 0 becomes 10. Then 10 - 6 = 4 -- but now we must subtract 4 - 8, and we can't. So now what do we do? Should we borrow again? But now it's not obvious whether we should borrow from the 2 again to make 1, or from the 9 in the next column to make 8, or whatever! Perhaps we should have borrowed twice at the same time in the first step -- 3 becomes 1, and 0 becomes 20. (Compare to the addition problem from earlier -- when adding three numbers we often must carry the 2, so it figures that in subtracting three numbers we sometimes must borrow 2.) Now we can see why most texts avoid this type of problem!

In subtraction problems a - b, a is often called the minuend and b -- the value that is actually being subtracted from a -- is called the subtrahend. Now we must work on a problem like a - b - c. We know from Algebra I that a - b - c is equivalent to a - (b + c). Both b and c are being subtracted from a in this problem. So it makes sense to call both b and c "subtrahends," while a remains the only minuend in this problem.

So this gives us another way to solve the problem -- we first add the subtrahends together. So we find the sum of $3.96 and $7.58 as $11.54. Then we subtract this from the minuend -- $14.30 - $11.54 turns out to be $2.76.

In the end, I just break each problem into two subtraction problems. So for the first problem, I have the student subtract $18.63 - $9.99 first to obtain $8.64, and subtract $8.53 from the difference. I show him the standard algorithm, but notice that for this particular problem a nonstandard algorithm can be used -- $18.63 - $10 is $8.63, and since we subtracted one penny too much, we add it back to obtain the first difference.

I suppose if we really want to force a traditionalist algorithm for, say, the second subtraction, we would first add the rightmost digits of both subtrahends together -- 6 + 8 = 14. Then since we can't subtract 0 - 14, we know we must borrow twice -- so 3 becomes 1 and 0 becomes 20, so that we can now subtract 20 - 14 = 6. But this is likely to confuse students -- especially the third graders to which traditionalists would like to teach a subtraction algorithm. And besides, we can't really call this a "standard algorithm" because it isn't standard at all.

It's interesting that my young self never considered multiplying three numbers at once. Perhaps it was because of my disappointment with three-number subtraction, or maybe it was because I knew that multiplication makes numbers grow so fast that it would be too hard to multiply three at once.

For the rest of today's post, I will compare some textbooks, and then I'm going to announce my newest side-along reading book for the blog.

And now you're wondering -- what? I just barely finished reading my last side-along book -- Rudy Rucker's The Fourth Dimension -- and now I'm already starting another book!

Well, last weekend, my local library held its biannual book sale. I purchased two books -- one of which is an old textbook, and the other will be my side-along reading book.

The text I purchased is HRW's Algebra One Interactions: Course 1, dated 1998. I already own the HRW Geometry text, and so now I own their Algebra I text as well. I spent much of my last post comparing various texts, and I'm doing the same in this post.

Let's remind ourselves how the HRW Geometry text is organized into chapters:

1. Exploring Geometry
2. Reasoning in Geometry
3. Parallels and Polygons
4. Congruent Triangles
5. Perimeter and Area
6. Shapes in Space
7. Surface Area and Volume
8. Similar Shapes
9. Circles
10. Trigonometry
11. Taxicabs, Fractals, and More
12. A Closer Look at Proof in Logic

I've already discussed how this text fits in the context of the school year. Even though there are twelve chapters, I feel that a better semester break is after Chapter 4, since Chapters 1-4 in HRW correspond roughly to Chapters 1-7 in the U of Chicago. Then Chapters 5-10 in HRW must be covered in the second semester before the state testing, as Chapter 11 is a recreational chapter that covers topics that don't appear on most standardized tests. (Chapter 12 in HRW corresponds approximately to Chapter 13 in the U of Chicago -- a chapter whose material can be split between the semesters or given all at once in the second semester.) Even though Chapters 5-10 in HRW in the second semester is a bit tough, they correspond to Chapters 8-15 in the U of Chicago -- and we're just about to wrap up these chapters this week on the blog. Chapter 6 in HRW is like Chapter 9 in the U of Chicago -- each introduces solid geometry, but most test questions are on surface area and volume, to be covered in the subsequent chapter in each respective text.

But now I'm looking at the HRW Algebra I text. Here is how this text is organized:

1. Data and Patterns in Algebra
2. Patterns With Integers
3. Rational Numbers and Probability
4. Geometry Connections
5. Addition and Subtraction in Algebra
6. Multiplication and Division in Algebra
7. Solving Equations and Inequalities
8. Linear Functions
9. Systems of Equations and Inequalities
10. A Preview of Functions
11. Applying Statistics
12. Applications in Geometry

At first, it appears that there are many topics from Algebra I missing from this text. But then I realized that this text is not intended to be a full Algebra I text at all!

You see, back in the pre-Common Core days, many schools had a two-year Algebra I course. So naturally, this course was intended for the weaker math students. So this text covered the first year of the two-year course.

Because what I bought yesterday was actually the teacher's edition, the table of contents for Course 2 is actually included:

1. Functions, Equations, and Inequalities
2. Linear Functions and Systems
3. Matrices
4. Probability and Statistics
5. Transformations
6. Exponents
7. Polynomials and Factoring
8. Quadratic Functions
9. Radicals and Coordinate Geometry
10. Rational Functions

Notice that there are several topics that rarely appear in any Algebra I course anymore. Matrices are now delayed to Algebra II, as are rational functions. I've even seen some old Algebra I texts that include radical equations, but considering that Chapter 9 is called "Radicals and Coordinate Geometry," I assume that the "radicals" occur in the context of the Distance Formula. (Remember what I wrote about the use of square roots last week!)

Now these two texts are not equivalent to the two volumes of most modern math texts. Each volume of a modern text is intended to be completed in a semester, while each of these two HRW texts was take a full year to complete. Some schools had two-year Algebra I courses without having two separate texts like HRW. Instead, the teachers simply spend six weeks on each chapter rather than three weeks in a one-year Algebra I course. (Of course, that class I subbed in last week was apparently treating Common Core Math 7 as a two-year course, as the class was still only in Chapter 2 in April!)

But now in the Common Core era, two-year Algebra I is rare. Indeed, notice that the HRW Course 1 text is almost equivalent to a Common Core Math 8 course -- especially considering that HRW added chapters on geometry and statistics, which also occur in Common Core 8. I've stated in the past that Integrated Math I is nearly equivalent to Common Core 8, so the HRW Course 1 text can almost double as an Integrated Math I text as well.

The main topic required in Common Core 8 and Integrated Math I that is missing from HRW Course 1 is transformations. Actually, Lesson 10.6 of the HRW text is on transformations. But this section merely introduces translations, reflections, and rotations -- this is hardly enough to satisfy the Common Core 8 or Integrated Math I standards.

Notice that Lesson 10.6 appears in an Algebra I text, as part of a chapter on functions. Thus much of the emphasis is on transforming functions -- especially the functions f (x) = x^2 and f (x) = |x|. There are major differences between studying transformations in Algebra vs. Geometry -- in particular, the Algebra text stresses translations and reflections, but not rotations.

To see why, note that a graph is a function if and only if it satisfies the Vertical Line Test. So we must consider only transformations such that the image of a vertical line is also vertical. Translations map all lines to either themselves or to parallel lines, so all translations preserve verticality. Reflections in horizontal or vertical mirrors also preserve verticality -- this follows from the Line Parallel to Mirror and Line Perpendicular to Mirror Theorems. Dilations also map lines to either themselves or to parallel lines. In Algebra classes, we also see "dilations" in which graphs are stretched or shrunk only in one dimension, not both. Both horizontal and vertical stretches preserve verticality.

On the other hand, a rotation doesn't preserve verticality unless its magnitude is 180 degrees. So only 180-degree rotations are guaranteed to map functions to functions. Both f (x) = x^2 and f (x) = |x| after 90-degree rotations violate the Vertical Line Test badly. Of course, some functions may still end up remaining functions after rotations in smaller angles -- for example, f (x) = |x| is still a function after rotation in any angle up to (but not including) 45 degrees. But believe it or not, f (x) = x^2 is not a function after any rotation other than 180. The graph after rotating a small (acute) angle may appear to be a function, but the graph, if extended long enough, will end up curving back to intersect most lines a second time. Notice that the only lines that intersect a parabola in exactly one point are the axis of symmetry and any line parallel to it, and any tangent line (as in the slope = derivative Calculus sense).

We can't help but notice that the HRW Course 2 text devotes an entire chapter to transformations. In fact, this is more coverage of transformations than either the Course 1 or the Geometry texts! In the Geometry text, transformations first appear in Lesson 1.6, with dilations in Lesson 8.1.

Now since I don't own the Course 2 text, I can't tell the extent to which transformations are covered in its Chapter 5. As this is still an Algebra I text, I can easily see the emphasis being on translations, reflections in horizontal/vertical mirrors, and horizontal/vertical stretches. Also, since Chapter 5 appears after Chapter 3 on matrices, perhaps matrices are used to perform transformations. We perform translations by adding matrices, and reflections, rotations, and stretches by multiplying. As it happens, reflections in either one of the axes or the lines y = x or y = -x have simple matrices, as do rotations centered at the origin of magnitude 90, 180, or 270 degrees.

Since the Course 1 text is equivalent to Integrated Math I, we may wonder whether the Course 2 text is equivalent to Integrated Math II. Some of the more advanced Algebra topics (like matrices, radicals, and radical functions) actually make more sense in Integrated Math II than Algebra I. We know that complex numbers appear in Integrated Math II, but I doubt they'd appear in Course 2.

And of course, we expect there to be much less geometry in Course 2 than in Integrated Math II. For after all, Course 2 is to be followed by a full Geometry course, while Integrated Math II isn't.

In fact, I can justify beginning with the HRW Course 1 text in eighth grade (since it's just Common Core 8), then proceeding with the HRW progression (Course 2 and Geometry) in Grades 9-10. Then college-bound juniors can move on to the HRW Advanced Algebra text (Algebra II), followed by either Pre-Calculus or the governor's proposed senior math course.

Now let's get on to the other book I purchased. It is Morris Kline's Mathematics and the Physical World, first published in 1959. Just like Rucker's The Fourth Dimension, it is a Dover edition -- and at weekend book sale it sold for 75 cents.

You may question the wisdom of starting another side-along book so soon, but notice what I spent last week discussing last week on the blog when I had no side-along book to read. Three of the posts were about traditionalists and the other two were spent playing with calendar dates and digits. So for those of you who want something else to read, here we go with the book I just bought.

Then again, we're not really escaping the traditionalist debate. This is because Morris Kline, a New York math professor, was one of the first teachers to promote a progressive pedagogy. In fact, on the back of his book, it reads:

Since the major branches of mathematics grew and expanded in conjunction with science, the most effective way to appreciate and understand mathematics is in terms of the study of nature. Unfortunately, the relationship of mathematics to the study of nature is neglected in dry, technique-oriented textbooks, and it has remained for Professor Morris Kline to describe the simultaneous growth of mathematics and the physical sciences in this remarkable book.

And of course, those "dry, technique-oriented textbooks" are the ones that traditionalists favor -- such as the texts in which, say, the standard algorithm for subtraction appears. So we can think of Kline as the father of the progressive math movement, a movement that has lasted all the way from the book's publishing in 1959 to the present day with Common Core.

Here's a link to a biography of Morris Kline:

I could start reading the book today, but I've decided to wait until halfway next week. This is because that's when I'll be starting the long PARCC Released Test Questions review on the blog. So Kline's book will give me something to write about during the long drawn-out review -- and besides, it's easier to keep track of the chapters and questions, as I'll read Chapter 1 of Kline's book on the same day I cover PARCC Released Test Question #1. It also gives us an extra week and a half between finishing Rucker's book and starting Kline's.

I've decided not to give this post the "traditionalists" label. All of the posts in which I discuss Kline's book will be labeled "Morris Kline," and I'll begin his book in earnest the middle of next week. But I will still reserve the "traditionalists" label for when I contrast Kline's ideas with those of traditionalists in more detail.

Meanwhile, this is what I wrote last year about today's lesson:

Today we finally reach the last chapter of the U of Chicago text. Chapter 15 of the U of Chicago text is on Further Work With Circles, and this chapter belongs here at the end because it appears on the PARCC End-of-Year (EOY). Chapter 9 in the HRW Geometry text corresponds to this chapter.

Yet of the three circle lessons that are tested on the EOY, Lesson 15-3 is the only one that actually appears in Chapter 15. The other two lessons are last week's Lesson 13-5 on tangents to circles and Lesson 11-3 on equations of circles.

Lesson 15-3 of the U of Chicago text is on the Inscribed Angle Theorem. I admit that I often have trouble remembering all of the circle theorems myself, but this one is the most important:

Inscribed Angle Theorem:
In a circle, the measure of an inscribed angle is one-half the measure of its intercepted arc.

The text divides the proof into three cases -- depending on whether the center of the circle is inside, outside, or on the inscribed angle. The easiest case occurs when the center is on the angle, and this case is used to prove the other two cases. The exact same thing occurred when we proved the Triangle Area Formula back in Chapter 8. Then, the three cases were whether the altitude was inside, outside, or aside of the triangle -- and the case when the altitude was aside of the triangle (i.e., when the triangle was a right triangle) was used to prove the other two.

I will reproduce the paragraph proof of the Inscribed Angle Theorem from the U of Chicago:

Given: Angle ABC inscribed in Circle O
Prove: Angle ABC = 1/2 * Arc AC

Case I: The auxiliary segment OA is required. Since Triangle AOB is isosceles [both OA and OB are radii of the circle -- dw], Angle B = Angle A. Call this measure x. By the Exterior Angle Theorem, Angle AOC = 2x. Because the measure of an arc equals the measure of its central angle, Arc AC = 2x = 2 * Angle B. Solving for Angle B, Angle B = 1/2 * Arc AC. QED Case I.

Notice that the trick here was that between the central angle (whose measure equals that of the arc) and the inscribed angle is an isosceles triangle. We saw the same thing happen in yesterday's proof of the Angle Bisector Theorem -- the angle bisector of a triangle is a side-splitter of a larger triangle, and cutting out the smaller triangle from the larger leaves an isosceles triangle behind.

Let's move onto Case II. Well, the U of Chicago almost gives us a two-column proof here, so why don't we complete it into a full two-column proof. For Case II, O is in the interior of Angle ABC.

Statements                                                     Reasons
1. O interior ABC                                           1. Given
2. Draw ray BO inside ABC                            2. Definition of interior of angle
3. Angle ABC = Angle ABD + Angle DBC       3. Angle Addition Postulate
4. Angle ABC = 1/2 * Arc AD + 1/2 * Arc DC 4. Case I and Substitution
5. Angle ABC = 1/2(Arc AD + Arc DC)           5. Distributive Property
6. Angle ABC = 1/2 * Arc AC                         6. Arc Addition Property and Substitution

The proof of Case III isn't fully given, but it's hinted that we use subtraction rather than addition as we did in Case II. Once again, I bring up the Triangle Area Proof -- the case of the obtuse triangle involved subtracting the areas of two right triangles, whereas in the case where that same angle were acute, we'd be adding the areas of two right triangles.

The text mentions a simple corollary of the Inscribed Angle Theorem:

An angle inscribed in a semicircle is a right angle.

The text motivates the study of inscribed angles by considering camera angles and lenses. According to the text, a normal camera lens has a picture angle of 46 degrees, a wide-camera lens has an angle of 118 degrees, and a telephoto lens has an angle of 18. I briefly mention this on my worksheet. But a full consideration of camera angles doesn't occur until the next section of the text, Lesson 15-4 -- but we're only really doing Lesson 15-3.

No comments:

Post a Comment