1. Introduction
2. A Dilation Postulate?
3. EngageNY and the Side-Splitting Theorem
4. Key Common Core Standards
5. Teaching the Key Standards
6. The Spring Semester of Geometry
7. Traditionalists: Singapore Math 7
8. Common Core: Math 8 and Integrated Math
9. Hawking, pages 1-50
10. Conclusion
Introduction
This is the sixth post of my summer "Natural Geometry" series. Each post of this series is named after a transformation, and today's transformation is "dilations."
Of course, we already know that dilations and similar triangles only exist in Euclidean geometry. So it would appear that "dilations" don't even belong in a "natural" geometry course (where "natural" is taken to include Euclidean and spherical geometry).
But one purpose of this post is to observe how dilations and similarity fit into a course that we try to keep natural as long as possible before introducing a single postulate (Point-Line-Plane) that forces the geometry to become Euclidean. Clearly, by the time we reach Unit 6 on dilations, we're firmly in Euclidean territory.
Another reason is that dilations and similarity are more important in Common Core Geometry than in pre-Core versions of the course. Dilations are used to prove the properties of similar triangles, and then similar triangles are used to prove the Pythagorean Theorem and slope formulas, among another common theorems. Thus it's important to think about how dilations and similarity are taught in a Common Core course.
A Dilation Postulate?
Let's start with the U of Chicago text, where dilations and similarity appear in Lesson 12-1. There we see the following as our first theorem of Chapter 12:
Theorem:
Let S_k be the transformation mapping (x, y) onto (kx, ky).
Let P' = S_k(P) and Q' = S_k(Q). Then:
(1) Line P'Q' | | PQ,
(2) P'Q' = k * PQ.
A coordinate proof using the slope and distance formulas then appears. This theorem is then used to prove the general properties of dilations in Lessons 12-2 and 12-3.
But this is unacceptable in Common Core Geometry. This is because dilations and similarity are supposed to be used to prove the slope and distance formulas! Thus a proof where slope and distance are used to prove the properties of dilations suddenly becomes circular.
An easy way out here is simply to declare the above theorem to be a postulate:
Dilation Postulate?
Let S_k be the transformation mapping (x, y) onto (kx, ky).
Let P' = S_k(P) and Q' = S_k(Q). Then:
(1) Line P'Q' | | PQ,
(2) P'Q' = k * PQ.
In fact, this is what I once suggested here on the blog. After all, we have a Reflection Postulate which asserts that reflections preserve the ABCD properties. And since all isometries are the composite of reflections, all isometries also inherit the ABCD properties from reflections. But since dilations aren't the composite of reflections, they don't inherit anything from the Reflection Postulate -- so it would appear that having a Dilation Postulate to state the properties of dilations would be nice.
But here's the problem -- earlier, I wrote that only a single postulate is supposed to distinguish Euclidean from spherical geometry. That postulate is the Point-Line-Plane Postulate. Thus we can't have a second postulate, a Dilation Postulate, that's valid in Euclidean but not spherical geometry.
Indeed, there's a big difference between a Dilation Postulate and, say, an Area Postulate that asserts that the area of a rectangle is lw. On the sphere, rectangles don't exist. Therefore, the Area Postulate, which can be written as:
Area Postulate:
If a quadrilateral is a rectangle, then its area is lw.
holds vacuously on the sphere as its antecedent is always false.
On the other hand, the antecedent of a Dilation Postulate is not always false on the sphere. There is nothing stopping us from defining a coordinate plane and S_k on the sphere. We've seen that a coordinate plane is valid for up to one hemisphere, and provided k isn't too large, (kx, ky) exists on the sphere as well. Yet the consequent is false -- clearly (1) fails as there are no parallel lines on the sphere, and as it turns out, (2) fails as well. This explains why we can have an Area Postulate without any trouble, but not a Dilation Postulate.
EngageNY and the Side-Splitting Theorem
There are two ways out of this jam. One, suggested by Dr. Hung-Hsi Wu, involves generalizing the Midpoint Connector Theorem (which is proved using parallelograms, not similarity) via something he calls the "Fundamental Assumption of School Mathematics." But this might be tricky for high school students to understand.
The other method is used by EngageNY, a curriculum from the state of New York. This method uses the areas of triangles to prove the Side-Splitting Theorem -- which is actually the last theorem proved in U of Chicago Chapter 12. Then Side-Splitter is used to prove the properties of dilations.
I believe that the EngageNY is probably the best possible sequence for teaching the properties of dilations without circularity or Wu's Fundamental Assumption. It's consistent with the U of Chicago order, since area (Chapter 8) appears well before similarity (Chapter 12). The main changes would be Chapter 12 itself (so that Lesson 12-10 and Side-Splitter come first) -- and of course, Lesson 8-7 on Pythagoras would be delayed until its similarity proof can be given (perhaps until Lesson 14-2, where said proof appears as an exercise).
A key difference between EngageNY and my curriculum is that in EngageNY, similarity appears to be a first semester topic (Unit 2 out of 6). For me, similarity is a second-semester topic -- indeed, my entire first semester corresponds mostly to EngageNY Unit 1 (on congruence).
Notice that for parallel lines, EngageNY uses the Wu proofs based on 180-degree rotations. Here I no longer support Wu's parallel line proofs -- it's based on rotations, and I want to keep the rotations unit (Unit 3) natural. This is why I want to use translations (Unit 4) to prove parallel line properties. But unfortunately, the translation proofs might contain gaps or are tricky, and so we might be forced to revert to Wu's proofs after all
Let's see what our second semester would look like so far (using U of Chicago chapters). We know that Chapter 8 must appear before 12:
8. Measurement Formulas (Area)
12. Similarity
Now it makes sense to have volume soon after area, so now we have:
8. Measurement Formulas (Area)
9. Three-Dimensional Figures
10. Surface Areas and Volumes
12. Similarity
Chapter 11 (Coordinate Geometry) can appear after Chapter 12, since the slope and distance formulas of Chapter 12 can now be proved. But still, this places Chapter 12 deep into the second semester.
The problem I have here is with Pi Day, March 14th. It's one of my favorite days of the year, and so if it's at all possible, I want the lesson on pi to appear near Pi Day. Of course, you might note that back on March 14th, I posted the Chapter 12 Test (along with Lesson 12-10, which is now suddenly the most important lesson of Chapter 12). But since I'm creating my own curriculum from scratch, I want to make every effort to place the pi lesson close to Pi Day.
We might be justified in delaying Lessons 8-8 and 8-9 (on circumference and area of a circle) until Chapter 15 (also on circles) and then teaching it near Pi Day:
8. Measurement Formulas (Area)
9. Three-Dimensional Figures
10. Surface Areas and Volumes
12. Similarity
15. Circles (including Lessons 8-8 and 8-9)
But we want cylinders, cones, and spheres to appear in the volume lessons of Chapter 10. The only way to avoid this is to place similarity at the start of the semester -- but then this places similarity after the area needed to prove Side-Splitter.
And all of this is just to get pi to land on March 14th. Technically, getting pi to land on Pi Day is secondary to covering all material in a way that students can learn and understand. But still, Pi Day is one of the most enjoyable days of the year. I think it's worth timing the lessons so that Pi Day can be celebrated to the fullest.
Key Common Core Standards
I've pointed out over and over that certain standards form the heart of Common Core Geometry as distinguished from pre-Core Geometry:
CCSS.MATH.CONTENT.8.EE.B.6
Use similar triangles to explain why the slope m is the same between any two distinct points on a non-vertical line in the coordinate plane; derive the equation y = mx for a line through the origin and the equation y = mx + b for a line intercepting the vertical axis at b.
And I've mentioned one big problem with this eighth grade standard -- in a typical eighth grade class, all of the EE standards are taught before any G standards are covered. Thus students are expected to use similar triangles to explain the slope formula in October or November -- at least 100 days before they see similar triangles in February or March.
We'll get back to eighth grade later, since for now we want to look at the high school standards. As we already know, one key standard is the fact that we must use similarity to prove Pythagoras:
CCSS.MATH.CONTENT.HSG.SRT.B.4
Prove theorems about triangles. Theorems include: a line parallel to one side of a triangle divides the other two proportionally, and conversely; the Pythagorean Theorem proved using triangle similarity.
Notice that the first theorem mentioned in this standard is actually Side-Splitter. The semicolon is important here -- only Pythagoras is required to be proved using similarity, not Side-Splitter. After all, if Side-Splitter needed to be proved using similarity, suddenly EngageNY would be invalid -- and so would my dilation unit if it's based on the EngageNY proof.
The Pythagorean Theorem is mentioned in another standard:
CCSS.MATH.CONTENT.HSG.SRT.C.8
Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems.
Even though I haven't mentioned it yet, Chapter 14 on trig should be part of the dilations unit. This is starting to make the dilations unit quite large, if Chapters 12, 11, and 14 are all included.
We've already seen the slope standard for eighth grade, but here it is for high school:
CCSS.MATH.CONTENT.HSG.GPE.B.5Prove the slope criteria for parallel and perpendicular lines and use them to solve geometric problems (e.g., find the equation of a line parallel or perpendicular to a given line that passes through a given point).
Notice that while dilations are used to prove that slope exists, we might be able to use translations to prove the slopes of parallel lines, and rotations (90 degrees, of course) for perpendicular lines.
All of these are used in coordinate proofs, which are required by the following standard:
CCSS.MATH.CONTENT.HSG.GPE.B.4
Use coordinates to prove simple geometric theorems algebraically. For example, prove or disprove that a figure defined by four given points in the coordinate plane is a rectangle; prove or disprove that the point (1, √3) lies on the circle centered at the origin and containing the point (0, 2).
Teaching the Key Standards
So we expect our dilations unit to incorporate all four of the key standards above. How will be able to accomplish this?
Well, let's look at Chapter 12 of our text in more detail -- except this time, we'll put Lesson 12-10 first, since we know that Side-Splitter is used to prove the others:
10. The Side-Splitting Theorem
1. Dilations on a Coordinate Plane
2. Dilations Without Coordinates
3. Properties of Dilations
4. Proportions
5. Similar Figures
6. The Fundamental Theorem of Similarity
7. Can There Be Giants?
8. The SSS Similarity Theorem
9. The AA and SAS Similarity Theorems
Then we follow this directly with Chapter 11 material:
1. Proofs with Coordinates
2. The Distance Formula
3. Equations for Circles
4. The Midpoint Formula
5. The Midpoint Connector Theorem
6. Three-Dimensional Coordinates
First of all, the parallel and perpendicular slope proofs, mentioned in the standard above, are used throughout Lesson 11-1, so they can be placed there. Here the Pythagoras proof of Lesson 14-2 should be used before the Distance Formula of Lesson 11-2. And of course Lesson 11-3, the equation of a circle, is also explicitly mentioned in the standards. So now we have for Chapter 11:
(Parallel and Perpendicular Slopes)
1. Proofs with Coordinates
(Lengths in Right Triangles, Pythagoras Proof)
2. The Distance Formula
3. Equations for Circles
(Prove or Disprove...Coordinate Proofs)
4. The Midpoint Formula
5. The Midpoint Connector Theorem
6. Three-Dimensional Coordinates
All that's missing here is that last standard, on trig. Except for Lesson 14-2, which is needed for the Pythagoras proof, none of Chapter 14 has been covered yet.
In the end, it's probably best to save the rest of Chapter 14 for a different unit. We don't want our units to be too long.
The Spring Semester of Geometry
This implies, of course, that the dilations unit will not span the entire second semester of Geometry. I want to include the entire second half of the text -- Chapters 8 through 15 -- in our second semester.
In various past posts, I've named the various second semester units after some of the more obscure transformations that exist. In particular, Chapter 8 on area would be called "transvections" (or shears), an affine transformation that preserves area, but not length or angle measure. Chapters 9 and 10 on volume would be called "screws," a 3D isometry that exists in no lower dimension.
Notice that in these units, we don't actually emphasize transvections or screws. For the most part, these are just cute names to go along with the other units named for important transformations. We might mention these in passing -- for example, say that a transvection maps a triangle (trapezoid) to another triangle (trapezoid) with the same base(s) and height. (Transvections also appear in 3D, which can be mentioned in connection with Cavalieri's Principle.) High school students will not be expected to answer test questions on transvections or screws.
Of course, Chapters 11 and 12 are in the "dilations" unit. As I've mentioned above, Chapter 12 needs to appear before Chapter 11. Chapter 13, meanwhile, can be placed anywhere. Recall that the modern Third Edition of the text eliminates Chapter 13, spreading its old material over several chapters. We might do the same with our Chapter 13, or perhaps combine it with Chapter 8 since this chapter sits alone in the "transvections" unit.
This leaves only Chapters 14 and 15. I'd like to include Chapter 14, on trig, in our unit on dilations and similarity, but that would make the unit too long. This is why it's better to combine Chapter 14 with Chapter 15 on circles. If we wish to name it after another transformation, then one that might fit is "circle inversions." Once again, we only mention circle inversions in passing -- perhaps including the interesting fact that the composite of two circle inversions with the same center is a dilation.
And of course, the final unit introduces spherical geometry. This unit occurs either during the SBAC (or other state test) or afterward -- perhaps depending on how many students are of testing age (juniors) in the Geometry class. If there are many juniors, then we'll want to give some sort of review, or else go easy on them during the week of the SBAC math test. During the week of the ELA or science test, we begin the spherical geometry introduction. If, on the other hand, most students are freshmen or sophomores, then more time can be spent on spherical geometry.
Of course, the introduction shouldn't span the entire unit. Here in California, we might wish to add some probability to this final unit. If we wish to emphasize probability more, we might move it up into the unit on area/"transvections," since some questions on geometric probability are closely related to area.
So let's try to place these units in order on a calendar. Here we assume that each unit spans a month (to make planning easier) and that the second semester starts right after winter break in January. If we more or less follow the U of Chicago order, then we obtain:
January: Unit 6, Area/Transvections (Chapters 8 & 13)
February: Unit 7, Volume/Screws (Chapters 9 & 10)
March: Unit 8, Dilations (Chapters 12 & 11)
April: Unit 9, Trig/Circle Inversions (Chapters 14 & 15)
May: Unit 10, Intro to Spherical Geometry
Depending on when the SBAC is given, Unit 9 should be the last before the test. If the test is given too early in April, at least be sure to cover Lesson 15-3 (Inscribed Angle Theorem) before the SBAC.
Once again, Pi Day is in March. But the only units in which an introduction to pi fits are the ones on area or circles, but neither of these fall in March. It's difficult to change up the order of these units, since area must appear before volume, area must appear before dilations (as explained above), and dilations must appear before trig.
The only reasonable way to get pi to land on Pi Day might be something like this:
January: Unit 6, Area/Transvections (Chapters 8 & 13)
February: Unit 7, Dilations (Chapters 12 & 11)
March: Unit 8, Volume/Screws (Chapters 9 & 10)
April: Unit 9, Trig/Circle Inversions (Chapters 14 & 15)
May: Unit 10, Intro to Spherical Geometry
but where only polygon area appears in January. Then March begins with Chapter 9 (intro to 3D figures) for the first week or so, so that Lessons 8-8 and 8-9 land on Pi Day. Then the rest of March is for surface area and volume, where we combine the 3D figures of the first week with the pi formulas of the second week to find formulas for cylinders, cones, and spheres.
Traditionalists: Singapore Math 7
There has been some recent traditionalist activity over the weekend:
https://traditionalmath.wordpress.com/2019/07/27/advice-on-the-teaching-of-standard-algorithms-before-common-core-says-it-is-safe-to-do-so-dept/
Our main traditionalist, Barry Garelick, is discussing the only algorithms that he believes young arithmetic students should be taught -- the standard algorithms:
EdReports.org is an organization that rates textbooks/curricula with respect to how well they align with the Common Core standards. There are no ratings on the effectiveness of a curriculum or textbook–just whether it adheres/aligns to the standards.
What captures my attention about this is the “no concepts assessed before appropriate grade level”. Sounds similar to “no wine before its time” but it has more sinister implications in my opinion.
Recall that the standard algorithms do appear in the Common Core Standards -- the problem Garelick and other traditionalists have is that they appear too late:
In my investigations and writing about Common Core standards, I have heard from both Jason Zimba and Bill McCallum, the two lead writers of the math standards. They have assured me that a standard that appears in a particular grade level may be taught in earlier grades. So for example, the standard algorithm for multidigit addition and subtraction appears in the fourth grade standards. This does not prohibit the teaching of the standard algorithm in, say, first or second grade. A logical take-away from this would be that students need not be saddled, therefore, with inefficient “strategies” for multidigit addition and subtraction that entail drawing pictures or extended methods that have been known to confuse rather than enlighten.
Hmm, first and second grade seem a bit early to me for the standard algorithms for multidigit addition and subtraction.
Here is my compromise between Garelick and the Common Core -- second grade for standard addition, third grade for standard subtraction. Based on what I've seen of actual students, it's actually the standard algorithm for multiplication that students complain about -- they want to use another algorithm instead, the lattice method. I say, teach the lattice method for multiplication in fourth grade.
Of course, Garelick's post is all about a website -- edreports.org -- which rates various texts based on their adherence to the Common Core. His concern is that based on their scoring system, it's impossible to get a high score for a text that teaches standards above grade level. Thus even a text based on my suggested compromise might get a low EdReports score.
In the comments, one poster describes another text rated low by EdReports -- Singapore Math 7. I'm excited by this comment only because until now, I hadn't seen a traditionalist discuss any Singapore text higher than sixth grade:
sugihs:
For example, in our public school district when we were considering “Math In Focus” (MiF), those opposed to Singapore Math cited EdReports’ low scores as reasons against MiF. For example, in the EdReports’ review, MiF Grade 7 received a score of just a 4 out of 12 (i.e. “Does Not Meet Expectations”) in “Gateway 1: Focus & Coherence” because:
“The instructional materials reviewed for Grade 7 do not meet expectations for assessing material at the Grade 7 level. There are too many concepts assessed that are beyond the Grade 7 CCSSM, and the alteration or omission of these items would significantly impact the structure of the materials. In chapters 1, 6, 7 and 8, there are assessment items that most closely align to standards above Grade 7 grade, and their inclusion is not mathematically reasonable for Grade 7. The alteration or omission of these items would significantly impact the underlying structure of the materials.”
The commenter "sugihs" had posted to Garelick's site a few times this year before this comment. This is the first time that I've highlighted sugihs here on my blog.What I wish I had was access to a Table of Contents for this text. (I performed a Google search but to no avail.) I'm curious as to what's in the four specified chapters (1, 6, 7, 8) that EdReports considers to be above grade level for Common Core Math 7.
sugihs:
Given the CCSS-M standards push Algebra 1 standards out to 9th grade, this implies that almost all math curricula that use pre CCSS-M standards will be punished for introducing concepts too early.
I disagree -- most pre-Core texts weren't based on eighth grade Algebra I. Yes, California texts were based on eighth grade Algebra I for a while, but most students weren't Californians. Thus I'd say that there are many pre-Core texts based on freshman Algebra I.
Another commenter in this thread is the usual SteveH:
SteveH:
Few talk about the fundamental change in education since I was in school in the 50’s and 60’s. That’s the idea of full inclusion. While this is a nice goal, it increases the range of abilities and willingness of students. In my son’s high school, it’s handled as a full inclusion environment, but with three academic levels for each course. However, in his K-8 schools, it’s handled in an age-tracking equal academic environment. They claim that differentiated instruction (differentiated learning) deals with the difficulty and claims on top of that that they achieve a higher level of understanding. It doesn’t happen. Some schools offer grade level academic grouping for part of the time, but that just groups the kids who get needed (mere) facts and (rote) skill mastery at home. This hides needed academic tracking at home and allows educators to believe that either their differentiated instruction works or that their ideas of natural learning work. I had many teachers tell me that “kids will learn when they are ready.”
First of all, here we go again with "full inclusion." As usual, the only thing I completely know about "full inclusion" is that traditionalists don't like it. Later on, he uses "age-tracking" instead. Again, he's criticizing those who say that they're opposed to "tracking" by claiming that the current system is also a form of tracking, "age-tracking" (that is, placing all five-year-olds in kindergarten, all six-year-olds in first grade, and so on). And once more, I point out that that in the end, there are two forms of tracking -- one by age, the other by race or income.
In this section, SteveH's uses of "(mere)" and "(rote)" are sarcastic. His opponents consider facts and mastery to be "mere" and "rote," while to SteveH they are the meat and potatoes of any math class.
He ends this section by criticizing the line "Kids will learn when they are ready." But what would SteveH do instead -- teach kids before they are ready? He assumes that if we just drill students with lots and lots of skill mastery, they'll learn it, even if they aren't ready. In fact, it's almost as if SteveH isn't really opposed to "full inclusion" at all -- he only prefers that all students are "fully included" in a traditionally-taught class.
SteveH:
High school AP/IB math teachers know that Common Core math in K-8 is not good enough. Parents in our town fought back to replace CMP with proper Glencoe math textbooks in 7th and 8th grades so that kids are ready for Geometry as a freshman in high school. However, the schools still use Everyday math in K-6 and that requires mastery help at home or with tutors to get into the Pre-Algebra class in 7th grade. If you don’t do this, then the probability of getting into any STEM field is greatly reduced. It’s all over by 7th grade.
I'm not quite sure what "CMP" is, but it's apparently a curriculum that traditionalists don't like, since SteveH contrasts it with "proper Glencoe" texts.
On the other hand, I definitely know what "Everyday Math" is -- it's the U of Chicago text for elementary school. Traditionalists criticize this text all the time. Technically, the U of Chicago elementary texts prepare the students for their seventh grade Pre-Algebra ("Transition Mathematics") and eighth grade Algebra I texts. But the senior text isn't Calculus, because "Functions, Stats, & Trig" is inserted between the Algebra II and Pre-Calc texts.
SteveH:
Low expectations. That is real social injustice.
Even though SteveH doesn't mention race or income here, race and income are usually implied when the terms "social justice" or "social injustice" are mentioned -- and once again, it all goes back to the relationship between tracking and race/income. I have much more to say on this topic, but since this post is already jam-packed full, I'll save it for later.
Once again, here's my compromise -- use the Singapore texts through middle and high school. Even though traditionalists like the Singapore texts for K-6 (or K-7), according to SteveH, Integrated Math "has already lost the battle." I guess by "losing," he must mean that it's the approach that Singapore and all countries other than the US and Vietnam have. If we're going to promote another nation's curriculum, then we should do so completely -- including Integrated Math.
Common Core: Math 8 and Integrated Math
Speaking of Integrated Math, the unit plan that I listed earlier is for traditional Geometry, not an Integrated Math class -- but of course, Integrated Math must contain some geometry. And, as we already know, transformational geometry first appears in the eighth grade standards. So how do my units based on transformations fit Common Core Math 8 and Integrated Math courses?
Well, the one thing about the G units in Common Core Math 8 is that formal proofs aren't needed. We notice that several eighth grade standards begin "use informal arguments:"
CCSS.MATH.CONTENT.8.G.A.5
Use informal arguments to establish facts about the angle sum and exterior angle of triangles, about the angles created when parallel lines are cut by a transversal, and the angle-angle criterion for similarity of triangles. For example, arrange three copies of the same triangle so that the sum of the three angles appears to form a line, and give an argument in terms of transversals why this is so.
Thus two-column proofs and the like shouldn't be needed here. Actually, I like the idea of suggesting the parallel line proofs based on translations and 180-degree rotations without actually giving the complete formal proofs. (This is how I tried to teach parallel lines to my eighth graders at the old charter school, and it might have worked, but behavior became an issue.)
One of these standards mentions Pythagoras:
CCSS.MATH.CONTENT.8.G.B.6
Explain a proof of the Pythagorean Theorem and its converse.
Here the area-based proof can be given, saving the similarity proof for high school. Notice that a standard proof of the converse mentioned here typically combines forward Pythagoras with some sort of uniqueness statement such as SSS. Again, "informal arguments" can be given here.
That only leaves Integrated Math to worry about. In the past, I've pointed out that Integrated Math I corresponds roughly to the first semesters of Algebra I and Geometry, while Integrated Math II corresponds to the second semesters of Algebra I and Geometry. Thus Math I focuses more on congruence, while Math II is more about similarity.
In this situation, I'd forget about trying to introduce spherical geometry or keep the geometry natural as long as possible. The emphasis in an Integrated Math class should be on making connections between the algebra and geometry taught in the courses.
It would be nice if Integrated Math could correct the error in the Math 8 standards, and teach similarity before slope. But this is difficult even in an integrated class -- similarity is a second semester Geometry topic, while slope is a first semester Algebra I topic. Not even Integrated Math teaches second semester Geometry before first semester Algebra I. (I once tried to post a curriculum that accomplishes this, but it was a mess.)
I've pointed out numerous times that the algebra content of Math 8 and Math I is nearly identical. It's only the geometry content that differs substantially. Math 8 contains similarity, while Math I doesn't (as it's saved for Math II). Conversely, Math I contains the proofs that Math 8 lacks.
But again, when it comes to the algebra content that needed to prepare for Calculus, Math 8 and Math I are nearly interchangeable. This is why I also sometimes propose Integrated Math as a way to accelerate students towards Calculus -- follow Math 8 with (freshman) Math II. Traditionalists never suggest this pathway because they hate Integrated Math, but I suspect that this is in fact closest to what Singapore teaches.
I admit that the geometry doesn't quite work out when jumping from Math 8 to Math II -- there's too much similarity and not enough proofs. But if the goal is to get to Calc, the geometry standards matter less than the algebra standards.
Hawking, pages 1-50
It's time for us to begin our other summer side-along graphic novel, Jim Ottaviani's Hawking. It's all about the famed physicist Stephen Hawking. I've written about him before on the blog, in connection with the movie A Theory of Everything (a biopic about him).
The book begins with Hawking as a twelve-year-old, talking with his two friends, Basil and John:
Basil: Is.
John: Isn't.
Basil: Is.
John: Is not. Sti'll isn't.
Basil: Wait a minute, Stephen. Is there life on other planets like in the Arthur C. Clarke books?
We flash backwards to Hawking's birth. His mother, while waiting for the doctor to arrive, reads An Atlas of Astronomy.
"I was born on January 8th, 1942," Hawking narrates.
But let's return to the scenes when Hawking is a young 12-year-old. In this "heroes" series, I like to look at the heroes back when they're the same age as the students I'm trying to teach:
Young Stephen: Didn't expect that! Chemistry is definitely more fun.
Mrs. Hawking: Stephen!
Young Stephen: Everything's fine, Mother! Just doing a bit of an experiment.
Mrs. Hawking: Well...all right. But it's dinnertime. Would Basil like to stay?
Young Stephen: Yes. You'd like to stay, wouldn't you?
Basil: Sure.
Young Stephen (at table): Nonono, Monopoly isn't adaptable enough. I've an idea for a much more interesting game. I'll show you the detailed outline after we eat, but here's the gist. You start with...
Basil (on the way to school the next day): ...barely a word from anyone else, even to pass the food!
John: Who can understand that lot, anyway? They all speak almost as fast as Stephen. So what did you talk about?
Basil: There's this game he created called "dynasty." I can't figure it out, and I don't think he has it figured out, either. But as far as I can tell, playing will take forever, and I can't see how there's any way to win.
John: No way to win. That's our bet, mate.
Basil: Hah. We'll see. Hallo, [friend Michael] Church.
Michael: King, McClenahan. What are you doing?
Basil: Walking to school. What are you doing?
Michael: Right. So clever.
Basil: Okay, yeah. I was telling John about dinner the other day. Over at Hawking's house.
Michael: Was everybody reading? It's madness. Nobody talking to anybody else.
After completing his days at St. Albans School, he takes the entrance exams and applies as a young 17-year-old student to Oxford:
"I didn't think I had done very well," Hawking narrates as he opens his Hogwarts -- um, Oxford letter.
But of course, he does get into Oxford. (By the way, that Hogwarts reference is intentional. As an American, my only knowledge of the British educational system is via Harry Potter. For example, Hawking is shown taking his A-level exams. These are equivalent to NEWT's in JK Rowling.)
"Only 'grey men' applied themselves, you see," Hawking narrates. "Hard workers. Dull."
Derek: Steve. Come with us to the pub?
Hawking: Sorry. Can't until next year.
"Not my sort. Derek, Gordon, and Richard were the only other physics students in my year. Hard workers, actually, but not grey men. We stuck together."
Hawking: ...more than that! Galileo did three important things, besides dropping objects off the leaning tower of Pisa. And dying exactly 300 years...
Richard: ...before you were born. We know, Steve.
Indeed, this was one of Hawking's favorite pastimes with his college roommates -- discussing his own hero, Galileo, whose death day coincides with his birthday.
Hawking: That, plus basically giving us the scientific method...
Gordon: Okay, okay. Fine. I'm convinced.
"We didn't actually discuss history all that much, I don't think," Hawking narrates. "But we got our grounding in classical physics a la Newton...In other words, light as particles, equal and opposite forces defined as mass times acceleration, and calculus."
Newton: Not "particles": corpuscles. And it's not calculus: it's the "method of fluxions."
"...And also electromagnetism per Maxwell and Faraday -- light as waves, electricity and magnetism as two sides of the same coin."
But Hawking didn't spend all of his time at Oxford studying physics -- not by a long shot:
"When my friends and I had arrived in Oxford," Hawking narrates, "anybody who was anybody rowed, and never wore jeans."
Hawking: Set ready!
"I rowed from my second year on. Well, others rowed. I was the cox. Nicer clothes, and an opportunity to call the shots on the smart set."
Hawking: Bow pair -- fall in. Stern pair -- fall out.
"I was excellent at it."
Hawking: Sorry. Should've said holdwater, I s'pose.
"I was loud, at least. We still won a few races. And when we did, my crew were traditionalists when it came to rewarding the man at the rear."
Yes, I know this post is all about "traditionalists," but Garelick and SteveH have nothing on these rowing traditionalists. The tradition is to throw the cox -- Hawking -- into the River Thames (Isis).
Let's look at the last few pages of today's reading section in more detail. It's actually about some of the giants who preceded Hawking. Some scientists considered the existence of black holes:
"Which, it may surprise you to find out, Einstein -- no wait...EINSTEIN! ---found just as distasteful as quantum mechanics," Hawking narrates. "Eddington had his back on this too..."
Eddington: I think there should be a law of nature to prevent a star from behaving in this absurd way!
"This is one case -- his objection to quantum theory is another -- where Einstein's sense of aesthetics led him down the wrong path. There was yet another, but on that one he later changed his mind.
Einstein: "Even Homer nods," ja?
"Observations did the trick. Einstein was just fine with reality teaching him a thing or two, and as we've seen, astronomers weren't idle while he was rethinking space and time for us. They were observing things that shook up how the universe began."
At this point, Hawking provides with with an even briefer history of the big bang (1928-1931):
"It's traditional to start these discussions with 'In the beginning...' and a tour of other creation myths. But like many things in nature, the scientific truth is much more interesting than any story we could make up. So we'll skip mythology and stay in the 20th century. The success of general relativity in predicting the behavior of light, and the curvature of space, came at a cost...When you use it to look at the Really Big Picture, you must eventually wonder what's stopping gravity from forcing everything to eventually collapse. Like most scientists of his time, Einstein thought the universe was static and eternal. Thought it, preferred it...and then fudged his own equation to ensure it. Einstein added a cosmological constant (lambda) to guarantee the answer came out the way he wanted it to."
In the pictures, we see Einstein erasing part of his equation to make room to add a lambda term.
"A few scientists -- very few at first -- didn't share this preference for a static universe. One was the Russian mathematician and physicist Alexander Friedmann. He took his cues from Einstein..."
Einstein: What is the speed of light in a vacuum is a constant? What if clocks don't always run at the same speed? What if space and time aren't separate at all?
Friedmann: What if the universe changes? Evolves?
Einstein: No, nonono. That's going to far.
Friedmann: And fine-tuning a constant to make this come out the way you want them to isn't?
"Friedmann solved Einstein's field equations without picking a lambda that ensured a static, eternal universe. The result of his work? Different kinds of curvature for space..."
And the final picture shows what the three kinds of curvature look like. Positive curvature leads to spherical geometry, negative curvature to hyperbolic geometry, and zero to Euclidean geometry!
Conclusion
Physicists don't yet know the overall shape of the universe -- that is, whether space is Euclidean, spherical, or hyperbolic (or even a hybrid -- recall that there are more than three geometries in 3D as opposed to 2D).
But in my Pi Approximation Day post, I linked to a Numberphile video which explains some weird things that would happen in a hyperbolic world -- baseballs become impossible to catch, golf balls become impossible to putt, and so on. But hold on a minute, you might say -- since we can indeed catch baseballs and putt golf balls in our world, doesn't this prove that the universe isn't hyperbolic?
Actually, it doesn't -- and here's why? Notice that everything that appears in that video is based on an approximate formula for the circumference and area of a circle -- pi * e^r.
Suppose in Euclidean geometry, we wanted to know the area of a circle of radius 1 mile. Should we plug in r = 1 mile, or r = 5280 feet, into the formula A = pi * r^2? In Euclid it doesn't matter -- if we plug in r = 1 mile we get A = pi square miles, and if we plug in r = 5280 feet we get something like A = 87,582,577 square feet, but 87,582,577 ft.^2 = pi mi.^2, so it doesn't matter.
But in hyperbolic geometry, if we plug in r = 1 mile we get pi * e square miles, and if we instead plug in r = 5280 feet we get pi * e^5280 square feet, which is much more than pi * e square miles. What's left unsaid in the video is that the formula is unit-dependent -- the area is pi * e^r only if r is written in the correct units.
In spherical geometry there is a "special distance" -- the radius of the sphere. In hyperbolic geometry there is also a "special distance." And one thing that makes this distance special is that the circle area formula only works if the unit equals this special distance. In the video, it's implied that the special distance is one foot -- which is why baseball and golf are weird at such small scales.
For those who believe that the universe is hyperbolic, the assumption is that the "special distance" is much, much greater than one foot. For example, it might be a billion light years. Then for circles whose radii is much smaller than a billion light years, their circumference and area are nearly equal to their Euclidean values. Only on the scale of a billion light years does the formula pi * e^r take effect.
Our justification for introducing spherical rather than hyperbolic geometry to high school students is that our planet is (essentially) spherical. But if someday the universe is proved to be hyperbolic, should we then start teaching hyperbolic geometry in high school? I'd say no -- no one will ever travel far enough (i.e., a billion light years) for hyperbolic geometry to make a difference, whereas spherical geometry affects airplane flights right here on earth.
Of course, it's also possible for the universe to be spherical. But once again, the special distance -- that is, the radius of the sphere (or hypersphere) -- is likely to be billions of light years. Thus no one will ever travel a full lap around the universe. On scales that people will actually be able to travel, the universe is more or less Euclidean.
So as exciting as it might be to apply non-Euclidean geometry to the universe, in the end, we might as well assume that it is Euclidean.
No comments:
Post a Comment