Natural Geometry Unit 6: Dilations

Table of Contents

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.

Natural Geometry Unit 5: Glide Reflections

Table of Contents

1. Introduction
2. Definitions in Natural Geometry
3. Placement of U of Chicago Lessons in Natural Geometry
4. Coordinate Plane in Natural Geometry?
5. Three Perpendiculars Revisited
6. Actual High School Proofs
7. Number Bases
8. Feynman, pages 161-250
9. Conclusion

Introduction

This is the fifth post of my summer "Natural Geometry" series. The fourth post of this series was my Pi Approximation Day post. There I discussed translations as well as a special "Three Perpendiculars Theorem" that's supposed to hold in both Euclidean and spherical (hence natural) geometry.

I decided to give glide reflections their own post, especially since the translations have been absorbed into the Pi Approximation Day post. Of course, there really isn't much more to say about glide reflections except that they are the composite of a reflection and a translation (in Euclid) or a rotation (on the sphere).

Instead, this post will tie up some loose ends regarding the first five units of natural geometry and what they would look like in an actual high school course.

Definitions in Natural Geometry

To review, our first five units are the following:

1. Point-Line-Plane
2. Reflections
3. Rotations
4. Translations
5. Glide Reflections

These five units are supposed to correspond to the first semester of a high school Geometry course -- that is, the first seven chapters of the U of Chicago text.

The most important idea is to be certain of our definitions. We've seen that just as there are inclusive and exclusive definitions of words such as "trapezoid" within Euclidean geometry, there are inclusive and exclusive definitions of words across geometries. If an object exists in Euclidean geometry but not spherical geometry, such as a "rectangle," then we must decide whether we want an inclusive definition of a "rectangle" (such as "equiangular quadrilateral") or an exclusive definition (a rectangle must have four right angles). Then under the inclusive definition, rectangles exist on the sphere, but under the exclusive definition, spherical rectangles don't exist.

I've been thinking about this in the last few days since my most recent posts. And anyway, here is what I decided:

First of all, the word "parallel" needs to keep its definition -- coplanar lines that don't intersect. After all, it's the existence or nonexistence of parallel lines that allows us to distinguish between Euclidean and non-Euclidean geometry.

Of course, I was playing around with "ultraparallel" in recent posts. The sole purpose of this was to define "translation" as "composite of reflections in ultraparallel mirrors" and then consider all rotations on the sphere to be translations (as some authors do). We're not going to use "ultraparallel" moving forward, so this means that we won't consider rotations to be translations anymore.

This means that the translations of Unit 4 exist only in Euclid. Thus this unit signifies the end of the theorems that are valid in both Euclidean and spherical geometry -- and the start of the theorems that are valid in Euclid only.

Since the word "parallelogram" contains the word "parallel" and parallel lines don't exist on the sphere, neither do parallelograms. As for "rectangles," even though this doesn't contain the word "parallel," the root "rect" means "right," so rectangles should have something to do with right angles.

Actually, there's a good reason to say that rectangles don't exist on the sphere, rather than try to consider equiangular, Saccheri, or Lambert quads to be rectangles. Here's the reason -- I've written that besides Point-Line-Plane, I don't want to change any other postulate in the text. This includes the Area Postulate of Lesson 8-3. Included in this postulate is the rectangle area formula A = lw. There is no spherical quad for which this area applies. Therefore, we conclude that there must be no rectangle on the sphere, for otherwise we would contradict this postulate.

Of course, since there is no spherical rectangle, there is no spherical square either. But there is one quad on the hierarchy which we might wish to keep on the sphere -- a "rhombus." As defined in the U of Chicago text:

• A quadrilateral is a rhombus if and only if its four sides are equal in length.

Nowhere in this definition are parallel lines mentioned. Indeed, such equilateral quads definitely exist on the sphere. Of course, in Euclid we prove that all rhombi are parallelograms -- and of course, this proof, given in Lesson 5-6, is spherically invalid. Notice that most of the quad hierarchy appears in Lesson 5-2 and follows almost immediately from the definition, but only "every rhombus is a pgram" requires additional thought -- and Euclidean geometry.

Thus we can allow rhombi to exist on the sphere. Actually, kites exist on the sphere as well -- and it follows trivially from the definition that every rhombus is a kite.

So far, we seem to be keeping all of the U of Chicago definitions, which force the definitions of "parallel" and of the special quads to be exclusive except for "rhombus" and "kite." Actually, let's look at the U of Chicago definition of parallel once more:

• Two coplanar lines m and n are parallel lines, written m | | n, if and only if they have no points in common, or they are identical.

Those last four words, "or they are identical," are the key here. According to this definition, every line is parallel to itself. Extending this to spherical geometry, we conclude that two lines are parallel iff they are identical. Thus the only translation that exists is the composite of reflections in identical mirrors -- in other words, the identity transformation.

So on the sphere, "parallel" means "identical" and "translation" means "identity." This degeneracy reminds us of similarity and dilations. On the sphere, "similar" means "congruent," and so "dilation" also means "identity" as well.

There might eventually be a situation where this definition is useful. We might be able to prove in natural geometry that two lines m and n are parallel. On the sphere, we then conclude that m and n are the same line, whereas in Euclid they need not be identical.

Placement of U of Chicago Lessons in Natural Geometry

Once again, the five units of natural geometry are supposed to correspond to the first seven chapters of the U of Chicago text. So let's start placing U of Chicago lessons into the five units.

Unit 1, "Point-Line-Plane," needs to include not just points, lines, and planes, but angles as well. So it needs to incorporate Lessons 1-1 through 3-3 of the text.

Unit 2, "Reflections," starts with Chapter 4, of course. It's definitely a good place to include Lesson 5-1 on isosceles triangles, since these have reflection symmetry. In the past, I also suggested that the triangle congruence theorems can be placed here as well, Lessons 7-1 to 7-4.

But Unit 3, "Rotations," is a problem. If we want to include figures with rotational symmetry in the rotations chapter, then parallelograms would be placed here. Yet we wish Unit 3 to be the last natural (spherically valid) unit, and so we don't want there to be "parallelograms" in this unit.

Let's think about this from another perspective -- the school calendar. Again, we want these five units to span the first semester of the year -- August through December. It's easy to think of the five units as corresponding to months -- Unit 1 in August, Unit 2 in September, up to Unit 5 in December.

If you think about it, we probably want Unit 1 in August to be light. Depending on the school, the first day of school might be midway through the month, plus it might take several days for the students to settle. So perhaps a good Unit 1 might start with something fun (like Lessons 1-4 or 1-5, as I did the first three years of this blog) and then go through the rest of Chapter 1, plus Lessons 3-1 and 3-2 to introduce angles.

Similarly, Unit 5 in December can be light as well. Once again -- depending on the calendar -- prep for finals, the exams themselves, and winter break can easily take up half the month. Thus "Glide Reflections" is an excellent December unit, as Lesson 6-6 is the only lesson that must be here.

Then Units 2-4 are the "meaty" units. It actually might make more sense to include all of Chapter 2 in the "Reflections" unit before moving on to Chapter 4.

The congruence theorems of Chapter 7 are placed in Unit 3, "Rotations" -- after Lesson 6-3. We do this because besides the congruence theorems, not much left in the U of Chicago text is natural.

This leaves Unit 4, "Translations," to contain Lesson 6-2 and all of the lessons that depend on the existence of parallel lines. This includes Lesson 3-4 (parallel line consequences) and parts of Chapter 5 (quadrilaterals) and 7 (parallelograms). Actually, notice that by doing this, all of our triangle theorems land in Unit 3 and quad theorems in Unit 4 anyway.

There are a few lessons that remain difficult to place. There might be a way to sneak parallelogram symmetry into Unit 3 (since there exist quads on the sphere with 180-degree rotational symmetry) without actually using the words "parallel" or "parallelogram." Once again, I recently pointed out that the following is easy to prove:

• If the opposite sides of a quadrilateral are congruent, then so are the opposite angles.

without any mention of pgrams. To do so, we simply divide the quad into two triangles and then prove those triangles congruent by SSS. We thus include this proof as an exercise shortly after teaching the students SSS and proofs with auxiliary lines. Later on, we then state the properties of parallel lines and pgrams, and then redo the proof.

We also need to place lessons required by the Common Core Standards but not included anywhere in the U of Chicago text. This includes the concurrency proofs.

Coordinate Plane in Natural Geometry?

We shouldn't do slopes of parallel and perpendicular lines until after similarity in the second semester, but I still like the idea of seeing the coordinate plane in October, just before the PSAT, even if the Geometry students aren't actually reviewing linear equations.

It's possible, even in Unit 3 in October (where geometry should still be natural), to define some sort of coordinate plane. To do this, we let any point be the origin, and draw two perpendicular lines through this point to be the x- and y-axes. Provided that some sort of distance has been defined, then the points on these axes have coordinates. Then x = h is defined to be the line perpendicular to x-axis through point with coordinate h, and y = k is likewise defined to be perpendicular to the y-axis.

It's interesting to see what happens with this coordinate plane on the sphere. First of all, notice that the coordinates can't be made to correspond to latitude (and longitude), since the curves of constant latitude (the "parallels") aren't great circles, and hence aren't lines. If the origin is placed on the Equator, then the lines x = h do become meridians (of constant latitude). But if the origin is placed at one of the Poles, then both axes are meridians.

There is a problem with antipodal points -- if the two axes intersect at the origin, then they must also intersect at the point antipodal to the origin. Also, if the origin is on the Equator, then lines x = h (all meridians), intersect at the Poles. But the coordinate plane works provided that we stick to a single hemisphere (excluding the boundary).

Notice that if we consider the quad whose sides are the two axes, x = h, and y = k, then we notice that by definition, the angle between x -axis and x = h is right, as is that between y-axis and y = k. Since the angle between the axes is always right, we now have a Lambert quadrilateral. We can't prove that the angle between x = h and y = k is right (and on the sphere it can't possibly be right, since otherwise it would be a rectangle).

In fact, there might be enough here to prove some of the coordinate properties of reflections here, such as the image of (h, k) reflected over the x-axis is (h, -k). Notice that the quad whose sides are the y-axis, x = h, y = k, and y = -k is a Saccheri quadrilateral. In fact, we might be able to sneak in some properties of Lambert, Saccheri, and equiangular quads here just by trying to prove the coordinate properties of reflections and rotations. (At no point do we actually use the words "Lambert," "Saccheri," or "equiangular" here.)

In the special case that h = k, the Lambert quad becomes a kite. (On the sphere, this kite is not a rhombus, since a Lambert quad can't be a rhombus unless it's a square.) It might be possible to use the symmetry properties of a kite to prove that the image of (h, k) reflected over y = x is (k, h). (Note that the kite is used to prove that y = x is even a line, since we don't have any slope properties.) Of course, this requires that we teach kite symmetry in Unit 3 rather than wait for Unit 4.

Once again, I like the idea of placing these lessons in October just before the PSAT, even if only to remind the students of how the coordinate plane works. Of course, there's no harm in teaching the students linear equations, but only as a review of Algebra I rather than something that needs to be proved rigorously. I remind the readers that we need to wait until the second semester before we can fully prove the existence and properties of slope.

Three Perpendiculars Revisited

In my Pi Approximation Day post, I alluded to a natural "Three Perpendiculars Theorem":

Three Perpendiculars Theorem:
If three lines are perpendicular to the same line, then the three lines are either parallel or concurrent.

When I first posted this theorem last year, I wrote that, while it's valid in both Euclidean and spherical geometry, it's more of a curiosity than something we might want to teach students. Unless there's a compelling reason why we'd want to teach Three Perpendiculars, we probably shouldn't.

But let's think about it -- there's another theorem that students need to know, and it definitely has something to do with three perpendiculars and concurrency. You guessed it -- it's the concurrency of the three perpendicular bisectors of a triangle.

Let's think again about that incomplete proof of the concurrency of perpendicular bisectors as given in Lesson 4-5 of the U of Chicago text:

Construct the circle through the three noncollinear points A, B, and C.

Solution:
Step 1. Subroutine: m, the perpendicular bisector of AB
Step 2. Subroutine: n, the perpendicular bisector of BC
Step 3. m and n intersect at O. (Point Rule)
Step 4. Circle O containing A (Compass Rule)

If m and n intersect, it can be proven that this construction works. Because of the Perpendicular Bisector Theorem with line m, OA = OB. With line n, this theorem also justifies the conclusion OB = OC. By the Transitive Property of Equality, the three distances OA, OB, and OC are all equal. Thus Circle O with radius OA contains points B and C also. QEF

Earlier, we discovered that this proof is incomplete, because it takes it for granted that the lines m and n actually intersect. Also, we notice that the noncollinearity of A, B, and C isn't used in the proof. We found out that these are related depending on which geometry we're in:

Euclidean: If A, B, C are collinear then m | | n. If A, B, C are noncollinear then m, n intersect.
Spherical: All lines intersect, therefore m, n intersect.
Hyperbolic: If A, B, C are collinear than m | | n. If A, B, C are noncollinear then there's no conclusion.

Because of this, we can write a single theorem that incorporates both cases:

Theorem:
Let A, B, C be any three distinct points. Then the three perpendicular bisectors of AB, BC, AC are either concurrent or parallel.

Proof:
It's essentially the same as the U of Chicago proof. All we really need to change is drop Circle O and instead, use Converse Perpendicular Bisector Theorem to show that since O is equidistant from A, C, we conclude that O lies on the perpendicular bisector of AC. QED

Notice that without loss of generality, we can start by knowing that any two of the three perpendicular bisectors must intersect (AB & BC, AB & AC, or BC & AC) and conclude that the third bisector must pass through O as well. Thus the only way to avoid the existence of O is for all three perpendicular bisectors to be parallel. This is why we have validly shown that the three bisectors must be either concurrent or parallel.

This proof is valid in all three geometries -- Euclidean, spherical, and hyperbolic.

Actual High School Proofs

But even with this version of the Three Perpendiculars Theorem, we still haven't actually reached the proofs that we really want to see. Once again, Three Perpendiculars should ultimately lead to "a line and its translation image are parallel."

Indeed, as written, the theorem doesn't yet distinguish between the parallel and concurrent cases. We know that the lines are always concurrent in spherical geometry, and if A, B, C are noncollinear (as in the vertices of a triangle), then they are concurrent in Euclid as well. Thus, we should somehow be able to prove in neutral geometry that if A, B, C are noncollinear then the lines are concurrent. Such a proof can be placed near the end of Unit 3 -- but I can't figure out yet what the proof would look like.

There might also be a path to some interesting proofs if we were to begin with the concurrence of angle bisectors, rather than perpendicular bisectors. In this case, instead of points A, B, C, we begin with three lines a, b, c, and show that if these lines intersect then so do the bisectors of the angles that they form. Even though this proof would be mainly about intersecting lines, an indirect proof could be set up so that it refers to the properties of parallel lines instead.

Don't forget that we do have Playfair's Parallel Postulate available. Officially, Playfair is neutral (due to the "at most" one line clause). We know that Playfair must somehow be used in the concurrence of perpendicular bisectors proof, since the proof fails in hyperbolic geometry. This means that Playfair would be introduced before Unit 4 as well, even as other Euclidean parallel line properties aren't taught until Unit 4.

So there are still several proofs that I still need to figure out here. Still, it would be nice if we could could adapt the concurrency proofs so that they could help us out with the parallel line proofs.

Once we reach Unit 4, the geometry becomes fully Euclidean. So the original Euclidean version of our Point-Line-Plane postulate is used in proofs. Once again, we don't tell the students this -- we just quietly start using the "through any two points, there is exactly one line" clause.

Number Bases

In my Pi Approximation Day post, I briefly mentioned number bases -- mainly because I decided to include the Eleven Calendar (and hence its implied undecimal base) in my annual FAQ post.

Number bases is still one of my favorite topics, though I rarely have an opportunity to write about this topic on the blog, since it has nothing to do with Geometry. But since I'm thinking about it right now, let me squeeze some number bases into today's post.

And indeed, it helps that Pappas has a number base problem on her calendar today:

113 (base x) = 805 (base 9)
Determine x.

We might notice that just as 805 (base 9) means 8 + 9^2 + 0 * 9 + 5, we can expand 113 (base x) so that it becomes 1 * x^2 + 1 * x + 3:

113 (base x) = 805 (base 9)
x^2 + x + 3 = 805 (base 9)

This is just a quadratic equation in x, and so it can be solved like any other quadratic. We could do all the arithmetic in base 9 since the right side is already nonary. But to us native decimalists, it's easier just to convert the right side to decimal:

x^2 + x + 3 = 653
x^2 + x - 650 = 0
(x + 26)(x - 25) = 0
x = -26 or x = 25

There actually do exist such a thing as negative bases, but these are a bit awkward -- it's generally assumed in a number base problem that bases are natural numbers unless directly stated. Therefore the desired base is x = 25 -- and of course, today's date is the twenty-fifth.

The two bases in this problem are base 9 and base 25. Both of these are square bases. It turns out that square bases have a special property that non-square bases lack. Recall that a full repetend prime is a prime whose decimal expansion is of maximum length -- one less than the prime itself:

1/7 = 0.142857... (6 digits repeat)
1/17 = 0.0588235294117647... (16 digits)
1/19 = 0.052631578947368421... (18 digits)

Let's try to find the full repetend primes in bases 9 and 25. The calculations have already been completed on the Dozens Online website:

Base 9:

https://www.tapatalk.com/groups/dozensonline/everything-done-up-to-the-nines-t723.html

p	Len(M)/Maximum	M
2	1/1	4
3	–REGULAR–
5	2/4	17
7	3/6	125
12	5/11	07324
14	3/13	062
18	8/17	04678421
21	10/20	042327518
25	12/24	03462311507
32	15/31	02712148617674

Base 25:

https://www.tapatalk.com/groups/dozensonline/25-bingo-pentavigesimal-t787.html

n	dec.	1/n
2		0.ccc…	c
3		0.888…	8
5		0.5	5
7		0.3e73e7…	3e7
b	11	0.26kb926kb9…	26kb9
d	13	0.1n1n1n…	1n
h	17	0.1bj2nd5m1bj2nd5m…	1bj2nd5m
j	19	0.17m956ebl17m956ebl…	17m956ebl
n	23	0.1248h9je36d1248h9je36d…	1248h9je36d

And we see that unless we count 2, none of the primes are full repetend primes. It turns out that in odd square bases, 2 is the only full repetend prime -- and in even square bases (such as hexadecimal) there are no full repetend primes.

There's a simple reason for this -- each digit in a perfect square base corresponds to two digits in another base. For example, each digit of nonary (base 9) is two digits of ternary (base 3), and likewise with each digit of base 25 being two digits of base 5. Thus the maximum length of a repetend in a square base must be half of that same repetend in the square root base:

1/7 = 0.125... (base 9) = 0.010212... (base 3)
1/7 = 0.3e7... (base 25) = 0.032412... (base 5)

and so on. Thus 7 is a full repetend prime in bases 3 and 5, but not bases 9 and 25.

Before we leave number bases, here's a link to the Dozens Online for base 11:

https://www.tapatalk.com/groups/dozensonline/undecimal-the-unbelievable-t514.html

According to this link, undecimal has many full repetend primes: 2, 3, 12 (thirteen), 16 (seventeen), 21 (twenty-three), and 27 (twenty-nine).

It's also mentioned that one French Revolution mathematician, Joseph-Louis Lagrange, had (perhaps jokingly) considered an undecimal metric system. I'm not sure whether time (or angle measure) was included in Lagrange's proposal -- otherwise he might have stumbled upon the Eleven Calendar (or something similar).

Feynman, pages 161-250

It's time for us to continue our summer side-along graphic novel, Jim Ottaviani's Feynman. We left off with our hero offering $1,000 prizes for nanotechnology -- a miniature motor and a tiny book.

Unfortunately, our summary of this section of the book must be relatively brief. The book is due back at the library soon, plus there's a second book for us to read.

"I didn't expect it to take many years for someone to claim the money," Feynman narrates. "I also didn't expect all the kooks, and people who didn't get it."

Feynman (to various people holding large objects): No. Um, no. No, I said small.

"This went on for a few months, and I saw a lot of big boxes containing, well, big motors."

Feynman: Look, thanks for coming, Mr. McClellan. But anything that needs a crate of that size...
McClellan: No no, the motor's in here.
Feynman (seeing the small envelope): Uh-oh.

As it turns out, McClellan's envelope contained the proper motor, and so Feynman had to pay up -- if only he had set aside the prize money.

Unfortunately, in 1978, Feynman catches cancer. He loses his appetite, and so his doctor suggests that someone feed him his favorite foods so that he can eat again:

"So she [my friend Alix] came running from Brentwood to Pasadena with her vichyssoise, chocolate mousse, cakes, etc.," he narrates. "It worked very well. I loved to talk to Alix too. She was so interested in art and archaeology. She'd go to sites, and could stretch her imagination so fully so visualize how things had been."

Shortly thereafter, Feynman lectures on QED (quantum electrodynamics) in New Zealand. His goal is to explain QED so that an amateur like his friend Alix can understand it. And here is a small portion of this lecture:

Student: And naturally, the length of the arrow -- the amplitude -- is bigger for this one than it is for that one.
Feynman: No! They're almost exactly the same. It's the time that's different. Suppose you had to start at the light source and get to the eye, fast. You...running. And you had to touch the mirror along the way...
Picking A to touch isn't a good idea (too close to the light). Neither is touching point M (too close to the eye). The time it takes, if we plot it right underneath, looks like this (a parabola). The time for point A is much longer than for point B (between A and M near A). The stopwatch arrows turn with time, remember? So toward the center the different in direction of the arrows gets small, and then increases again as you go farther away...
But when we add them together, instead of two like I did before, it's millions of 'em, and we get a net result -- a total amplitude -- for the photon to arrive. It's this tremendous line. And it's length is mostly made up of arrows D through J.
Student: So, QED predicts that light reflects off the mirror.
Feynman: Very good! The contribution of these crazy bits near the edges is almost nothing -- all that staggering around cancels out.

Of course, this lecture would be easier to understand if you could see the pictures. The important thing here is that reflections here are much more complex than reflections in Geometry. This is because a mirror isn't perfectly flat -- Feynman draws various "arrows" (or vectors) to show which direction the photons are going. Only on average, he explains, does the angle of incidence actually equal the angle of reflection.

In fact, some Geometry texts provide the solution for a perfectly flat mirror. We consider the image of the eye in the mirror, and then draw a straight line from the light to the image of the eye. Then the light will bounce off the mirror in just the right place to reach the eye.

There are many other events in this chapter. Feynman ultimately receives his own prize -- the Nobel Prize -- for his work on QED. And the book ends shortly after the Challenger disaster, when the physicist -- after speaking to Houston NASA officials -- is invited to the Soviet Union to lecture on the causes of the disaster. He muses about the high school history teacher who died that day, namely Christa McAuliffe:

Feynman: She trained with astronauts. She had to know the true situation better than NASA officials.

"NASA has to deal in a world of reality to understand technological weaknesses well enough to actively try to eliminate them," he narrates. "They should propose only realistic flight schedules they have a reasonable chance of meeting. If they means we wouldn't support them, well...so be it. For a successful technology, reality must take precedence over public relations, for nature cannot be fooled. That was it. I was done. I only gave two interviews -- one to a local paper, one to the National Enquirer."

Feynman: ...Hey, if I said something dopey and they quote it, nobody will blame me! It's good to be home, but I'm going to leave this bag packed...we're going to Kyzy1!

Richard Feynman didn't make it to Tuva. The official, formal invitation from Moscow permitting the visit was dated February 19th, 1988 -- four days after he died. His last words, upon forcing himself awake from a coma: "I'd hate to die twice. It's so boring."

(I'm sorry, but I can't help but think about Niels Henrik Abel, the 19th century Norwegian who was invited to teach math in Berlin -- just days after he died. But at least Feynman lived a full life -- Abel lived only to 26.)

Thus concludes our reading of Ottaviani's Feynman. We'll move on to his latest book, Hawking -- as in Stephen Hawking -- in my next post.

Conclusion

At this point, this is what our natural geometry course looks like:

Unit 1: Point-Line-Plane
covers at least Lessons 1-4 through 1-9, 3-1, 3-2 of the U of Chicago text

Unit 2: Reflections
covers Chapters 2 and 4 of the U of Chicago text

Unit 3: Rotations
covers Lesson 6-3, 7-1 though 7-4, coordinate reflections/rotations, Playfair, concurrency proofs

Unit 4: Translations
covers Lesson 6-2, Chapter 5, Lessons 7-5 through 7-8

Unit 5: Glide Reflections
covers Lesson 6-6, review for first semester final

There are still several gaps to be filled in. These missing proofs won't be easy to find, but hopefully they'll be worth it. They'll shed some insight on parallel lines and what exactly distinguishes Euclidean from spherical geometry.

Pi Approximation Day 2019

Today is Pi Approximation Day, so-named because July 22nd -- written internationally as 22/7 -- is approximately equal to pi.

It also marks five full years since my very first post here on the blog. Yes, I've declared Pi Day of the Century -- March 14th, 2015, to be the ceremonial start date of this blog. But the chronological start date was July 22nd, 2014. I like celebrating both Pi Day and Pi Approximation Day -- including eating pie both days -- and so I celebrate both of them as launch dates for the blog.

Last year, I used the anniversary to create a FAQ about my blog and myself. I do the same this year -- and this year's FAQ will reflect the changes that have happened to my career since last year. As usual, let me include a table of contents for this FAQ:

1. Who am I? Am I a math teacher?
2. Who is Theoni Pappas?
3. How did I approximate pi in my classroom?
4. What is the U of Chicago text?
5. Who is Fawn Nguyen?
6. Who are the traditionalists?
7. What's with the "line and its translation image are parallel" proof?
8. What's "Mocha music"?
9. Mocha Music for Pi Approximation Day
10. How should have I stated my most important classroom rule?
11. Who is Wendy Krieger, and what is the Eleven Calendar?

1. Who am I? Am I a math teacher?

I am David Walker. I'm trying to start a career as a math teacher, but I must admit that my career isn't off to as good a start as I'd like.

Three years ago was my first as a teacher at a charter middle school, but I left that classroom. By now, I fear that I won't be hired to teach this year at all. If I'm not hired, then I'll remain a substitute teacher. But this will make the launch of my teaching career that much more difficult.

By the way, in my last few posts, I announced that my old school was in danger of closing because the charter renewal petition was denied. But I wasn't completely sure whether the school will really be closed, since they could have appealed to the county or state. And so I wrote that in a few days, I'd find out once and for all.

Well, last last week I drove past my old school -- and guess what? There is now a completely different charter school at that site. That settles it -- the old charter school is closed. It truly now is the old charter school, as in no longer existing.

I'll still blog occasionally about how I could have taught better there. Perhaps if I'd been a better teacher, I could have still been teaching there, and the school might still be open. Or maybe the school might have closed anyway even if I had remained, since the primary reason for the denial was financial mismanagement, rather than any academic reason. Thus if I'd stayed, then I'd still be in the same boat -- having to look for a new teaching job. The only difference is that I'd have three full years of teaching experience -- and that's crucial when trying to get a leg up in this competitive job market for teachers.

So the answer to this question is no, I'm not currently a math teacher, but I'm hoping that I'll be one again soon.

2. Who is Theoni Pappas?

Theoni Pappas is the author of The Mathematics Calendar. In most years, Pappas produces a calendar that provides a math problem each day. The answer to each question is the date. I will post the Pappas questions for each day that I blog, provided that it's a Geometry question.

The question for July 22nd this year is trig, but it's simple enough that it might be taught in a Geometry class:

10sin^-1(sin 2 1/5 degrees)

We notice that since sin^-1 denotes inverse sine, this means that for any angle theta, we know that sin^-1(sin theta) is just theta. Thus sin^-1(sin 2 1/5) = 2 1/5. So the problem becomes:

10(2 1/5 degrees)

The answer is 22 degrees -- and of course, today's date is the 22nd.

By the way, I don't necessarily like the notation "sin^-1" to mean inverse sine. This is tough on students, where "x^-1" means "1/x," that is, the multiplicative" inverse of x, while "f^-1" means the composite inverse of f.

Some authors use the subscript "o" (same as the composite "o") with f^-1 to emphasize that this is the composite inverse of f, but this looks terrible in ASCII: f_o^-1(x) or sin_o^-1(x). One good thing about the o is that it also looks like the degree symbol, so that sin_o^-1(x) means "find out how many degrees the angle has whose sine is x." Of course, this then breaks down as soon as the students learn about radians.

Of course, there's always the notation "arcsin(x)," which is what I prefer. The only problem is that most TI calculators use the notation "sin^-1(x)."

3. How did I approximate pi in my classroom?

Since today is Pi Approximation Day, I should write something about approximating pi.

The dominant approximation of pi in my classroom was, of course, 3.14. I didn't quite reach the unit on pi in my seventh grade class -- Grade 7 being the year that pi first appears in the current Common Core Standards. But I did reach a unit on the volume of cylinders, cones, and spheres in my eighth grade class.

The main driver of my use of 3.14 as the only approximation of pi was IXL. I used this software to review the volume problems with my eighth graders. The software requires students to use 3.14 as pi, even though I provided them with scientific calculators with a pi key. For example, the volume of a cylinder of radius and height both 2 is 8pi cubic units. IXL expects students to enter 8(3.14) = 25.12, even though 8pi rounded to the nearest hundredth is 25.13. IXL will charge the students with an incorrect answer if they enter 25.13 instead of 25.12. And even on written tests, I didn't want to confuse the students by telling to do something different for written problems, and so I only used 3.14 for pi throughout the class.

But there was another problem with our use of calculators to find volumes. Some of the calculators were in a mode to convert all decimals into fractions. Thus 25.12 appeared as 628/25 -- and I couldn't figure out how to put them in decimal mode. This actually allowed me to catch cheaters -- only one of the calculators was in mixed number mode. Here was one of the problems from that actual test, where students had to find the volume of a cylinder of radius and height both 3:

V = pi r^2 h
V = (3.14)(3)(3)(3)
V = 84 39/50

so it displayed 84 39/50 instead of 84.78 or 4239/50. So any student who wrote 84 39/50 on the test -- other than the one I knew had the mixed number calculator -- must have been cheating.

Here's an interesting question -- suppose instead of catching cheaters, my goal was to make it as easy as possible on my students as well as my support staff member, who was grading the tests. That is, let's say I want to choose radii and heights carefully so that, by using 3.14 for pi, the volume would work out to be a whole number (which appears the same on all calculators regardless of mode). How could have I gone about this?

The main approximation of pi, 3.14, converts to 314/100 = 157/50. It's fortunate that 314 is even, so that the denominator reduces to 50 rather than 100. So our goal now is to choose a radius and height so that we can cancel the 50 remaining in the denominator.

We see that 50 factors as 2 * 5^2. The volume of a cylinder is V = pi r^2 h -- and since r is squared in this equation, making it a multiple of five cancels 5^2. It only remains to make h even to cancel out the last factor of two in the denominator. Let's check to see whether this works for r = 5, h = 2:

V = pi r^2 h
V = (3.14)(5)^2(2)
V = 157 cubic units

which is a whole number, so it works.

If we had a cone rather than a cylinder, then there is a factor of 1/3 to deal with. We can resolve the extra three in the denominator by making h a multiple of six (since it's already even). Let's check that this works for r = 5, h = 6:

V = (1/3) pi r^2 h
V = (1/3)(3.14)(5)^2(6)
V = 157 cubic units

Spheres, though, are the trickiest to make come out to be a whole number. This is because spheres, with the volume formula V = (4/3) pi r^3, have only r with no h. Thus r must have all of the factors necessary to cancel the denominator. Fortunately, the sphere formula contains the factor 4/3, and the four in the numerator already takes care of the factor of two in the denominator 50. And so r only needs to carry a factor of five (only one factor is needed since r is cubed) and three (in order to take care of the denominator of 4/3). So r must be a mutliple of 15. Let's check r = 15:

V = (4/3) pi r^3
V = (4/3)(3.14)(15)^3
V = 14130 cubic units

Notice that the volume of a sphere of radius 15 is actually 14137 to the nearest cubic unit -- that's how large the error gets by using 3.14 for pi. Nonetheless, 15 is the smallest radius for which we can get a whole number as the volume by using pi = 3.14.

Now today, Pi Approximation Day, is all about the approximation 22/7 for pi. Notice that if we were to use 22/7 as pi instead of 157/50, obtaining a whole number for the volume would be easier.

For cylinders, the only factor to worry about in the denominator is seven. Either the radius or the height can carry this factor. So let's try r = 1, h = 7:

V = pi r^2 h
V = (22/7)(1^2)(7)
V = 22 cubic units

For cones, we also have the three in 1/3 to resolve. Let's be different and let the radius carry the factor of three this time. For r = 3, h = 7, we have:

V = (1/3) pi r^2 h
V = (1/3)(22/7)(3^2)(7)
V = 66 cubic units

For spheres, unfortunately our radius must carry both three (for 4/3) and seven, and so the smallest possible whole number radius is 21:

V = (4/3) pi r^3
V = (4/3)(22/7)(21)^3
V = 38808 cubic units

This is larger than the radius of 15 we used for pi = 3.14. But notice that there are many extra factors of two around in the numerator -- 4/3 has two factors and 22/7 has one. This means that we can cut our radius of 21 in half. Even though 21/2 is not a whole number, the volume is nonetheless whole:

V = (4/3) pi r^3
V = (4/3)(22/7)(21/2)^3
V = 4851 cubic units

To make it easier on the students, I could present the radius as 10.5 instead of 21/2. (Recall that the students can easily enter decimals on the calculator -- they just can't display them.) There is still some error associated with the approximation pi = 22/7, as the volume of a sphere of radius 10.5 is actually 4849 to the nearest cubic unit, not 4851. Still, 22/7 is a slightly better approximation than 157/50 is, with a much smaller denominator to boot.

With such possibilities for integer volumes, I could have made a test that is easy for my students to take and easy for the grader to grade. I avoided multiple choice on my original test since I didn't want to give decimal choices for students with calculators in fraction mode (or vice versa). With whole number answers, multiple choice becomes more feasible. Of course, the wrong choices would play to common errors (confusing radius with diameter, forgetting 1/3 for cones).

By the way,, I point out that the two summer circle constant days (Tau Day and Pi Approximation Day) can be used as alternatives to Pi Day parties for summer classes. My district divides the summer into A Session and B Session. Tau Day, on June 28th, was the second Friday of A Session, hence it wouldn't have worked for a party (since there is no summer school on Fridays).

Today, Pi Approximation Day is on the Monday of the last week of B Session. Thus teachers could throw a party today.

4. What is the U of Chicago text?

In 1991, the University of Chicago School Mathematics Project published a series of secondary mathematics texts. I was able to purchase both the Algebra I and Geometry texts for two dollars each at a local public library. One thing I discovered was that the U of Chicago Geometry text is based on the same transformations that appear in the Common Core Standards -- even though the U of Chicago published the text nearly two decades before the advent of the Core.

The common denominator is NCTM, the National Council of Teachers of Mathematics. This group was behind both the U of Chicago text and the Common Core Standards. Still, this means that one can satisfy most of the Common Core Standards by teaching out of the U of Chicago text.

There are newer versions of the Geometry text than the one that I own. I've read that while my old version introduces reflections in Chapter 4 and the other isometries in Chapter 6, in the new version all isometries appear in Chapter 4.

The U of Chicago text is not the only book that I mention here on the blog, not by a long shot. Many posts here refer to the Illinois State text. This is mainly because I was a math teacher, and I used the Illinois State text when teaching at my old school. 

To supplement the Illinois State text (mainly when creating homework packets), I used copies of a Saxon Algebra 1/2 text and a Saxon 65 text for fifth and sixth graders.

To learn more about some of these texts, here are some links to a webpage by Cathy Duffy. Duffy's goal is to help homeschooling parents choose a textbook for their children. Here is her review of the U of Chicago text:

http://cathyduffyreviews.com/math/geometry-ucsmp.htm

and the Saxon series:

http://cathyduffyreviews.com/math/saxon-math-54-through-calculus.htm

5. Who is Fawn Nguyen?

For years, Fawn Nguyen was the only blogger I knew who was a middle school math teacher. Back when I was a middle school teacher, I enjoyed reading Nguyen's blog, but now that I've left, it's not as important for me to focus on her blog over all others.

Nonetheless, let's take a look at Nguyen's blog today. She hasn't posted in six months, and so the following link is to her most recent post:

http://fawnnguyen.com/jelly-beans-or-no-jelly-beans/

Nguyen begins with a Warm-Up question called a "Would you rather...?"

Would you rather have 364 jellybeans and give 188 to friends or have 281 jellybeans and give 137 to friends? Whichever option you choose, justify your reasoning with mathematics.

But in her class, she decided to change this to a different type of "Would you rather...?"

As a student, would you rather be given the problem on the LEFT (jelly beans) or the one on the RIGHT?

And here's the question on the RIGHT:

Which problem below, A or B, yields a larger difference?
A. 364 - 188 =
B. 281 - 137 =

She writes that a majority of her sixth graders, 71%, chose A. Moreover, when she posted this as a survey on her blog -- with most respondents presumably being teachers -- an even larger majority, 90%, chose A. But Nguyen herself would choose question B:

Well, I prefer the one on the right [that I’d typed up]. How did I get it so wrong? I’m normally not this lame. But, truth be told, I don’t love the jelly beans question. At all. Maybe the one on the right is the wrong “fix” for the left one. If I could retype the problem on the right, I’d remove the equal signs since the question is just asking which one yields a larger difference, not caring exactly what each difference is. I want to believe that anyone who spends 5 minutes with me learns that I love mathematics.

I hope Fawn Nguyen posts again on her blog soon, especially since it's been six months! (By the way, if she goes a full year without posting, I'll assume that her blog is no longer active, and then I'll remove her from the FAQ.)

6. Who are the traditionalists?

I use the phrase "traditionalists" to refer to those who oppose Common Core for mathematical -- not political -- reasons. The traditionalists tend to believe that many of the topics in the Common Core Math Standards are taught too late.

My own relationship with the traditionalists is quite complex. I tend to agree with the traditionalists regarding elementary school math and disagree with them regarding high school math.

For example, they believe that the standards in Grades 4-6 that require the use of the standard algorithm for arithmetic should move down at least one grade level. I tend to agree with the traditionalists on this point here -- most of the reasons for opposing direct instruction, rote memorization, and drill don't apply to the youngest students. I remember once learning about two nonstandard algorithms for addition -- the Left-Right Method and the Plus-Minus Method -- on an old 15-minute math show for sixth graders, "Solve It," that used to air on our local PBS station back when I was that age. For mental math, these nonstandard algorithms are great, but for a young elementary student first learning to add two-digit numbers I prefer the standard algorithm.

On the other hand, the traditionalists also prefer that high school seniors take AP Calculus in order to open the door to college STEM majors. In particular, some STEM colleges here in Southern California, such as Cal Tech and Harvey Mudd, require calculus for admission. Even colleges that don't formally require calculus are likely to send students rejection letters if they apply without having calculus on the schedule, especially if they apply to STEM majors. Therefore they want the classes leading up to senior year to prepare students to take AP Calculus -- including Algebra I in the eighth grade. They oppose Common Core because its eighth grade math standards do not comprise a full Algebra I class. I only partially agree because I don't want seniors who can't handle AP Calculus or eighth graders who can't handle Algebra I to be labeled failures. Just as AP Calculus can open the door to STEM, it can close the door to a non-STEM career if students are forced to spend too much effort trying to pass their high school math classes.

OK, so I support the traditional pedagogy with regards to elementary math and oppose it with regards to high school math. But what's my opinion regarding traditionalism in middle school math -- you know, the grades that I'll actually be teaching? In middle school, I actually prefer a blend between the traditional and progressive philosophies. And the part of traditionalism that I agree with in middle school is making sure that the students do have the basic skills in arithmetic -- since, after all, that is elementary school math, and with elementary math I prefer traditionalism.

I've referred to many specific traditionalists during my five years of posting on the blog. I'd consider anyone who chooses option B above to be a traditionalist -- which would make Nguyen herself a traditionalist! Actually, I'd say that Nguyen combines both traditionalist and progressive pedagogy in her class -- and if I were to return to the middle school classroom someday, I'll try to emulate her in this regard.

But the traditionalist who is currently the most active is Barry Garelick. Here is a link to his most recent post:

https://traditionalmath.wordpress.com/2019/07/18/out-on-good-behavior-dept/

Garelick also teaches middle school math right here in California. One of his frequent commenters is SteveH, a traditionalist in his own right. I consider SteveH to be a co-author of Garelick's blog since he posts there so often.

Let's look at Garelick's post in more detail:

I am currently writing a series called “Out on Good Behavior: Teaching Math While Looking Over Your Shoulder”  When the series is complete, it  will be published in book form by John Catt Educational, Ltd.

Before you ask, no, Garelick's book, when it's published won't be our next side-along book (unless for some reason I see it at a library book sale someday for $1).

I've actually discussed his Chapter 1 in an earlier post, so let's look at his Chapter 2 for this FAQ:

https://truthinamericaneducation.com/education-reform/an-espresso-based-job-interview-a-1962-algebra-book-and-procedures-vs-understanding/

(Note: Truth in American Education is an anti-Common Core website.)

In the remaining two weeks at my previous school, I applied for the few math teaching positions that were advertised. I had the typical non-responses except for one—a high school that specialized in problem-based learning. I had applied there out of desperation never expecting a response. I received an email saying they were interested in interviewing me. Despite my skills at making my teaching appear to be what people wanted to see, I knew that this one required too much suspension of disbelief on both sides of the aisle.

So as we can already tell, traditionalists such as Garelick don't like problem-based learning -- which I also used at the old charter school with the Illinois State text.

A few days later I was at the school for a 2 PM interview. I tend to get a bit logy in the afternoon so I thought I’d have an espresso prior to coming in. The principal, Marianne and assistant principal, Katherine, interviewed me and asked the usual questions: What does a typical lesson look like, what are my expectations and so on. My inner voice tried to keep me from extended caffeinated responses.  I emphasized how I leave time for students to start on homework in class, do the “I do, we do, you do” technique, and in my controlled ramblings managed to get across that I am, by and large, traditional.

Oh, and speaking of books that can be purchased cheaply, Garelick wrote that he sneaked copies of his favorite traditional textbook into his classroom -- and look at how little he paid for them:

“I bought about fifteen of them over the internet when they were selling for one cent a piece about four years ago. So I was basically paying for shipping. But now the prices increased because of Amazon’s supply and demand algorithm, so they’re selling for about $60 a copy last time I looked. Which tells me a lot of people are buying them.”

A more recent text that he was able to use in his classroom is JUMP Math:

“I used an alternative textbook, which my school let me use: JUMP Math. It was developed in Canada and broke concepts down into very small incremental steps. It scaffolds problems down to incremental procedures and builds on those.”

In the end, Garelick gets the job and begins teaching at St. Stevens, a Catholic school. I'm not quite sure whether he's still there now.

7. What's with the "line and its translation image are parallel" proof?

Both this year and in past summers, I devoted several posts out of the blue to the proof of a statement from Geometry, "a line and its translation image are parallel." The reason for this proof goes back to the goals of Common Core Math and the traditionalist debate.

The Common Core expects students to prove statements that traditionalists take for granted, such as the triangle congruence properties (that is, SSS, SAS, and ASA.) These statements are proved using transformations -- reflections, rotations, and translations.

Now it's possible to take this a step further. The parallel line properties (that is, Corresponding Angles and its converse) can also be proved using transformations. One of the first mathematicians to do so is UC Berkeley professor Hung-Hsi Wu. The two things to note about his proof are:

• Wu uses 180-degree rotations to map one parallel line to another. Therefore, his main proof is for the Alternate Interior Angles Test rather than Corresponding Angles.
• Wu delays using a parallel postulate as late as possible. In particular, he is able to prove the Alternate Interior Angles Test (that is, if ... then the lines are parallel) without any need for a parallel postulate at all. So his proof is valid in both Euclidean and hyperbolic geometry. On the other hand, the converse requires a parallel postulate (Playfair's).

Using Wu's proof, we can avoid the parallel postulate until a certain point when it's needed. Then later on, we can mention that there's another type of geometry called hyperbolic geometry -- and that all the proofs so far are valid in hyperbolic geometry up to the point where Wu first invokes Playfair.

Unfortunately, I find two problem with this approach:

• Wu's proof depends on 180-degree rotations, but for most students, translations are easier to understand than rotations. I once saw a website (now defunct) that demonstrated translating one parallel line to another to show that corresponding angles are congruent. It would be nice if I could convert that animation into a rigorous proof.
• Wu's proof is valid in hyperbolic geometry, but who cares, since we usually don't discuss hyperbolic geometry in high school math anyway. If we're going to mention non-Euclidean geometry at all, it would be spherical geometry -- after all, we live on a sphere. Some honors classes even introduce spherical geometry at the end of the year. It would be nice if I had a proof that, if valid in a second geometry at all, is valid in spherical, not hyperbolic, geometry.

Last year, I finally posted a valid proof of the Corresponding Angles Consequence. But this proof turns out to be very complicated. For example, I used the following theorem:

Three Perpendiculars Theorem:

If three lines are perpendicular to the same line, then the three lines are either parallel or concurrent.

This theorem is valid in both Euclidean and spherical geometry -- in particular, the "parallel" part is Euclidean, while the "concurrent" part is spherical. (In other words, the theorem is valid in "natural" geometry, which incorporates both Euclidean and spherical geometry.)

But Three Perpendiculars isn't the sort of theorem we want to teach in high school. And besides, we're trying to prove something about parallel lines, and there are no parallels on the sphere. Thus it's silly to prove these theorems in both Euclidean and spherical geometry, even if technically they're valid (for example, "if two parallel lines are cut by a transversal" is vacuously true on the sphere).

I completed the proof only because I wanted to finish what I started. But my final result is definitely inappropriate for a high school class. Perhaps instead of working on both goals, it might be better to focus on one or the other:

• We might replace Wu's 180-degree rotations with translations to prove statements that are valid and meaningful in both Euclidean and hyperbolic geometry. At the end of the year, we might say that there's a geometry called hyperbolic -- show the students the Numberphile video.
• We might mention spherical geometry at the end of the year, but only in connection to statements that are true in both Euclidean and spherical geometry (such as SSS, SAS, ASA), not parallel lines.

I won't pursue this any further in posts during the school year, but I will discuss it in posts during the summer. During the school year, my lessons are based on the U of Chicago text, where both Corresponding Angles and its converse are postulates.

If I ever teach in a classroom again, then it depends on the grade level. In eighth grade, where transformations are introduced but no formal proofs are given, we can informally show them that we can translate corresponding angles to each other -- as well as map alternating interior angles to each other via a 180-degree rotation. That way, the eighth graders can learn that these pairs of angles are congruent without having to teach them vocabulary -- "corresponding" or "alternate interior" angles.

In a Geometry class, it all depends on what text is being used. If parallel line postulates appear transformations, then I could show them how transformations can be used to prove the parallel line statements without delving too deeply with "Three Perpendiculars" or other nonsense. If parallel lines appear before transformations (as they do in the U of Chicago text) then just forget about it.

By the way, the statement I was trying to prove -- "a line and its translation image are parallel" -- doesn't even appear in the Common Core Standards. On the other hand, "a line and its dilation image are parallel" does explicitly appear in the standards. So this is something that we can keep in mind -- but then again, parallel lines usually appear well before dilations in most texts, so it's not as if we could use dilations to prove the parallel line statements.

So far this summer, I've written posts labeled "Natural Geometry Unit 1, 2, 3" that incorporate the ideas mentioned in this section. Today's post counts as "Natural Geometry Unit 4: Translations," and the two bullet points above illustrate the contents of such a unit. Translations are defined and used to prove properties of parallel lines in Euclidean geometry. Implicit is the fact that these theorems are valid only in Euclidean geometry, as opposed to results taught in the previous three units that are valid in both Euclidean and spherical geometry. Hyperbolic geometry will not be mentioned in the high school geometry class at all.

8. What's "Mocha music"?

In many recent posts, I refer to something called "Mocha music." This is a good time to explain what Mocha music actually means.

When I was a young child in the 1980's, I had a computer that I could program in BASIC. This old computer had a SOUND command that could play 255 different tones. But these 255 tones don't correspond to the 88 keys of a piano. For years, it was a mystery as to how SOUND could be used to make music. Another command, PLAY, is used to make music instead, since PLAY's notes actually do correspond to piano keys.

Last year, I found an emulator for my old BASIC computer, called Mocha:

http://www.haplessgenius.com/mocha/

When we click on the "Sound" box on the left side of the screen, Mocha can play sounds, including those generated by the SOUND command. So finally, I could solve the SOUND mystery and figure out how the Sounds correspond to computer notes.

I discovered that SOUND is based on something called EDL, equal divisions of length. We can imagine that we have strings of different lengths -- as in a string instrument or inside a piano. The ratio of the lengths determine their sound -- for example, if two strings are in a 2/1 ratio, then the longer string sounds an octave lower than the shorter string.

The key number for SOUND is 261, the "Bridge" (or end of the string). Mocha labels the Sounds from 1 (low) to 255, so we subtract these numbers from 261 to get a Degree ranging from 260 (long string) to 6 (short string). The ratios between the Degrees determine the intervals. I found out that the Degrees corresponding to powers of 2 (8, 16, 32, 64, 128, 256) sound as E's on a piano, with Degree 128 being the E just above middle C (that is, E4).

Let's say we were to play the following two notes on Mocha:

10 SOUND 51,8

20 SOUND 86,8

The second number 8 indicates a half note, since 8 is half of 16 (the whole note). But we want to focus on the first numbers here, which indicate the pitches (tones).

We first convert the Sounds to Degrees. Since 261 - 51 = 210, the first note is Degree 210. The Degree of the second note is 261 - 86 = 175. Now the ratio between these two Degrees is 210/175, which reduces to 6/5. This is the interval of a minor third, so the two notes are a minor third apart. As it turns out, the two notes sounds as G and Bb -- "rugu G" and "rugugu Bb."

Let's try another example:

30 SOUND 144,8

40 SOUND 196,8

Warning -- we don't attempt to find the ratio 196/144 (which is 49/36 by the way). We only find the ratios of Degrees, not Sounds. The Degrees are 261 - 144 = 117 and 261 - 196 = 65. Thus the interval between the notes is 117/65 = 9/5, a minor seventh. (Using Degrees instead of sounds makes a big difference, since 49/36 would be an acute fourth or tritone, not a minor seventh.) The names of the two notes played by Mocha are "thu F" and "thugu Eb."

Where do all these strange color names like "gu/green" and "thu" come from? Actually, they refer to Kite's color notation, and the colors tell us which primes appear in the Degree:

• white: primes 2 or 3 only
• green: prime 5
• red: prime 7
• lavender: prime 11
• thu: prime 13
• su: prime 17
• inu: prime 19

Kite's color notation also uses colors such as yellow, blue, and so on. But these are "otonal" colors, while EDL scales/lengths of string are based on "utonal" colors only.

The website where Kite explains his color notation is here:

https://en.xen.wiki/w/Color_notation

Actually, here's another link where Kite's color notation is explained:

http://www.tallkite.com/forum/index.php?topic=86.0

"Kite" (or "Tall Kite") formerly used different colors such as "amber" and "ocher," and so many of my old posts mention these colors.

9. Mocha Music for Pi Approximation Day

I keep saying that I should use these exotic Mocha scales for composing new music, not simply converting music in our usual scale (12EDO) to the new scales.

But on holidays, I'm in the mood for converting old music to the new scales. For Pi Approximation Day, I was hoping to convert songs about pi -- specifically songs that used to be posted on other websites that are now defunct. This includes "American Pi" and "Digit Connection" from the old Bizzie Lizzie Sailor Pi site, as well as Danica McKellar's old pi song based on "Dance of the Sugar Plum Fairy."

I would convert these songs if I had access to the sheet music, which I don't. Maybe I'll convert them some day, but until then, it's easier just to find YouTube videos of the songs on which these are parodies ("American Pie," etc.) and sing the pi lyrics loud enough to drown out the real words.

So instead, let's just code a pi song based on 16EDL, similar to the song we played for Tau Day:

NEW
10 N=16
20 FOR X=1 TO 32
30 READ A
40 SOUND 261-N*(17-A),4
50 NEXT X
60 DATA 3,1,4,1,5,9,2,6,5,3
70 DATA 5,8,9,7,9,3,2,3,8,4
80 DATA 6,2,6,4,3,3,8,3,2,7
90 DATA 9,5

As is traditional, I stop just before the first zero. Then digits 1-9 map to Degrees 16 down to 8, with the lowest note played on E (line 10, N=16). We can change the value of N to any value from 1 to 16 to change the key.

Here's an actual song converted to 12EDL, a simpler EDL scale, the Sailor Pi theme song:

NEW
10 N=13
20 FOR X=1 TO 26
30 READ D,T
40 SOUND 261-N*D,T
50 NEXT X
60 DATA 8,4,8,2,9,4,9,2,10,4,11,4,9,12
70 DATA 9,4,9,2,10,4,10,2,11,4,12,4,10,12
80 DATA 12,4,12,2,10,4,10,2,8,4,6,4,7,12
90 DATA 8,4,9,4,10,2,11,6,12,16

Only the main verse is coded here. The "bridge" part -- which is instrumental in both the original Sailor Moon and Lizzie's Sailor Pi song -- is too hard for me to code without sheet music.

Here are the lyrics for the first verse -- the part which we coded above:

Fighting fractions by moonlight
Perplexing people by daylight
Reading Shakespeare at midnight
She is the one named Sailor Pi.

10. How should have I stated my most important classroom rule?

This is what I wrote three years ago:

Rule #3: Respect yourself and others.

Students respect themselves and each other as well as me, the teacher, by following all rules and allowing others to learn the material.

This rule worked for Fawn Nguyen, the teacher from whom I got this rule. But it didn't work in my classroom at all. Of course, a student playing with a phone case isn't respecting the teacher, but I needed a rule that would require the student to put the case away immediately.

Here's a much better rule:

Rule #1: Follow all adult directions.

And so if I say, "Put the phone case away," the student couldn't counter that phone cases were against the rules, because my direction was to put the case away.

If I ever find myself in the classroom again, this will definitely be my first and most important rule.

By the way, Garelick also mentions that his main weakness is classroom management:

They asked about my classroom management techniques. In any interview or evaluation process, one has to have some weakness to talk about and I freely admitted that classroom management is not my strong suit. I mentioned that my seventh grade math class had behavior problems even though there were a total of 10 students in the class.

“How did you handle the problems?” Marianne asked.

“I had a warning system; two warnings and they got a detention.  I wasn’t too faithful in carrying that out though.”

“Why was that?”

“When I gave a detention, the two main troublemakers were really good at carrying on about it and crying.”

“They cried?”

“I hated giving detentions. I always got talk-back, like ‘But I wasn’t talking’. And then the crying. Which they did with all the teachers, I found out.”

But then Garelick quickly returns to discussing traditional pedagogy.

In my old class, many troublemakers claimed "But I wasn't talking," but very seldom did anyone cry to trick me into thinking that punishing them was unreasonable.

The lesson that both Garelick and I needed to learn was that we need to know outside of how the students are reacting whether our punishments are reasonable or not. Indeed, I see now that students who genuinely believe that they're being treated unfairly don't say "But I wasn't talking" -- instead, they do the opposite and give the teacher the silent treatment. (Indeed, this is what some people do when their spouses forget their birthdays.)

Who is Wendy Krieger, and what is the Eleven Calendar?

Wendy Krieger is an Australian who is interested in different number bases and innovative systems of measurement. She is a former member of the Dozens Online forum:

https://www.tapatalk.com/groups/dozensonline/

One day on this forum, Krieger mused about creating a calendar based on the number eleven. I quickly became interested, because I knew of no existing or proposed calendar based on 11.

If you were to choose a number and invent a calendar based on that number, your calendar will probably not be original -- someone else would have already thought of your idea. Consider the neighbors of 11 -- 10 and 12. Well, there's already 12 months in our current Gregorian Calendar, while the ancient Egyptian calendar used 10 days per week ("decans"). And suppose you were to look at the prime neighbors of 11 -- 7 and 13. Of course, the Gregorian Calendar has seven days per week, while the International Fixed Calendar has 13 months per year.

Despite all of this, no existing calendar is based on 11. And so, inspired by Krieger's post, I decided to create my own Eleven Calendar.

Suppose we want there to be eleven months per year. Then since a year contains 365 days, we divide 365/11 to obtain 33.18.... If we round this to 33 days, we notice that 33 is itself a multiple of 11. And so each 33-day month contains three weeks of 11 days each.

A simple version of the Eleven Calendar begins on March 1st, so that the first month contains the first day of spring (or autumn in Krieger's native Australia). Since 11 * 33 is exactly 363, we just tack in the extra days at the end of the year as blank days:

Month 1: March 1st-April 2nd
Month 2: April 3rd-May 5th
Month 3: May 6th-June 7th
Month 4: June 8th-July 10th
Month 5: July 11th-August 12th
Month 6: August 13th-September 14th
Month 7: September 15th-October 17th
Month 8: October 18th-November 19th
Month 9: November 20th-December 22nd
Month 10: December 23rd-January 24th
Month 11: January 25th-February 26th
Blank Days: February 27th, 28th, (29th)

So far, I haven't really given names to the eleven months or eleven days of the week yet.

Unfortunately, Krieger has been banned from the Dozens Online forum. I don't quite know why, but I suspect it was a dispute between her and the other members regarding number bases. As the forum name implies, most members of the forum like base 12. But Krieger's preferred base is actually 120, which she often calls "twelfty." Thus her banning might have been the result of her attempts to use twelfty on a dozenal forum.

Still, Krieger occasionally communicates with me via this blog. First, she came up with a clock to add to the Eleven Calendar. This clock divides the day into 22 hours, with 66 minutes in each hour -- and she also told me of an ingenious way to read a regular analog clock, based on 24/60, and convert it on the fly to 22/66 time.

A few days ago, Krieger wrote to me about dividing the world into 22 time zones to match the 22 hours on the clock. My problem was that it's awkward to divide the 360 degrees of our sphere into 22 time zones.

And so Krieger's latest idea is to convert each old degree into 1.1 new degrees. Thus there are now 396 degrees in a circle. She explains that Brisbane (presumably her hometown), which is at longitude 150E under the old system, becomes 165E in the new system.

In the past, I've often given my coordinates as 34N, 118W. As it turns out, in the new system, my longtitude is almost exactly 130W -- indeed, it's eerie how close my house is to 130W. Indeed, if I were to supply my latitude, you'd be able to find, or get within a block of, my house.

Of course, I don't want randos on the Internet to stalk me, so of course I won't post my latitude. To the nearest whole degree, my latitude in the new system is 37N. But the exact confluence 37N, 130W is in the Pacific, and so no one will be able to find my house using the information I supply here.

This is what I wrote as my response to Krieger:

Hmm, this is interesting. So now there are 360 * 1.1 = 396 degrees in a full circle. This reminds me of 400 "gradians" in a circle, since 396 is so close to 400.

My own longitude is extremely close to 130W under this system, so my time would be seven hours behind Greenwich, while your time would be nine hours ahead of Greenwich.

Thanks for an interesting sundial idea as well!

(For the rest of this post, there are 396 gradians in a circle. We don't care about the 400-grad circle.)

Krieger also mentions that we can call our 11-day weeks "decans" to reflect the ancient Egyptian weeks, even though our weeks are one day longer. She also mentions the ancient Chinese "ce" or "ke," originally 1/100 day but changed to 1/96 day to fit the 24-hour day. For her sundial, she divides each of the 22 hours into 11 "uncia," each six minutes long. (The word "uncia" really means 1/12.)

The concept of using 396 gradians for a circle for the Eleven Calendar is often mentioned on the Dozenal Forum and is called an auxiliary base. Another possible auxiliary base for 11 is 990 -- chosen due to its proximity to 1000 (so a semicircle is about 500 degrees, a quadrant 250 degrees, and so on). Under this system, my longitude becomes 325W.

But users of the Eleven Calendar might be using a purely undecimal (or "levimal") base, where everything is in base 11. Then the proximity of 990 to the decimal thousand means absolutely nothing to an undecimal user, so we might as well keep using 396 (330 in undecimal).

One problem with using 396 (or 990, for that matter) is that neither is divisible by eight, so that 45 degrees becomes 49.5 gradians. It's awkward to discuss 49.5-49.5-99 triangles in Geometry -- and this looks even worse in undecimal, where 49.5 becomes 45.555... gradians.

Oh, and since today is Pi Approximation Day, we notice that in the approximation 22/7, numerator 22 is divisible by 7, so it fits the our gradian system. Let's look at what happens here:

pi radians = 180 degrees = 198 gradians
22/7 radians = 198 gradians
1/7 radian = 9 gradians
1 radian = 63 radians

So using the approximation pi = 22/7, we find that one radian is approximately 63 gradians.

OK, here are some Pi Approximation Day video links:

1. Draw Curiosity

Notice that this video, from two years ago, actually acknowledges Pi Approximation Day.

2. Math Babbler:

The Math Babbler tends to post a video for Pi Approximation Day every year. In this one, he discusses how accurate the approximation 22/7 actually is.

3. Converge to Diverge

In this brand-new video from today, the speaker attempts to approximate pi just as Archimedes did it -- using a regular 96-gon.

4. Numberphile

No Pi (Approximation) Day is complete without a Numberphile video. In this video, we learn that pi is not just irrational, but is also transcendental -- that is, there doesn't even exist a polynomial f, with integer or rational coefficients, such that f (pi) = 0 (unlike sqrt(2), which is algebraic). This is even stronger than Legendre's result that pi^2 is irrational.

5. Sharon Serano

Well, I already gave ten facts about pi, and so this video is twice as good.

6. Vi Hart

No, I can't avoid another one of my favorite math YouTube posters, Vi Hart. Of course, in this video she argues that there's nothing special about pi at all

7. TheOdd1sOut

This video is specifically listed as a "Vi Hart rebuttal" to videos such as the previous video. 
8. A Song Scout

This is another pi song based on its digits. Unlike Michael Blake's song (listed below), it is in the key of A minor rather than C major.

9. Michael Blake

I like the idea of using music to teach math, so here's a favorite, Michael Blake's "What Pi Sounds Like." I tried to find a way to incorporate songs such as this one into the classroom.

10. tiradorfranco2

I'm surprised that I don't post this Square One TV video on Pi Approximation Day. The song is about the mathematician who first discovered that pi is approximately 22/7 -- Archimedes. The singer even mentions how Archimedes was "busy calculating pi" at around the 2-minute mark.

11. Coding Challenge #140

Today is Pi Approximation Day, and so this video is all about approximating pi.

Bonus: Numberphile

Earlier I wrote that Numberphile created some hyperbolic geometry videos. Here's one of the more interesting videos, about sports in hyperbolic space. This is just in case we decide we'd rather show our students hyperbolic than spherical geometry.

And so I wish everyone a Happy Pi Approximation Day. Notice, this is being posted shortly after midnight Pacific Time, so that it is still 22 July in Australia.