Chapter 9 Test (Day 100)

Today is Day 100. As I've explained in previous years, Day 100 is significant in many kindergarten and first grade classrooms. Indeed, last year our K-1 teacher (who eventually succeeded me as middle school math teacher) celebrated Day 100. But the district whose calendar is observed on the blog is a high school only district. Thus there will be no celebration at any school following this calendar,

Meanwhile, today was also the "super blue blood moon" I mentioned in my New Year's Eve post. So here in California, the lunar eclipse occurred at around 5:30 this morning. Even though my bedroom window faces north, I was actually able to see the eclipse through the window.

How was this possible? Yes, the sun and moon are usually in the southern sky. But sunrise and sunset are only due east and west at the equinoxes. In spring and summer, sunrise and sunset are slightly north of due east/west, while in fall and winter, they're slightly south of due. Back in my August 21st post (the day of the solar eclipse), I mentioned an annular eclipse from a few years earlier. That eclipse occurred in May just before sunset. Thus the sun was north of due west, and so it was actually visible from my bedroom. (But I didn't have eclipse glasses that day, and so the visibility of the eclipse from my bedroom was irrelevant.)

Now that's the sun, but what about the moon? Well, the full moon and sun are approximately in opposition -- and they're exactly in opposition during an eclipse. (After all, that's why it's an eclipse in the first place!) Thus the full moon and sun follow the opposite rules -- moonrise is at sunset, and moonset is at sunrise. In winter, sunrise is south of due east -- thus moonset must be slightly north of due west. And so once again, the eclipse was visible from my bedroom. (And since it's a lunar eclipse, I didn't have to worry about eclipse glasses!)

When I woke up, most of the moon was reddish. A few minutes later it changed to its usual color, but only a small crescent was visible. The moon sank below the garage of house next door before more than a crescent could appear -- and besides, by then it was approaching sunrise anyway.

Of course, we could have figured this out if we had finished our Spherical Trigonometry book. Recall that we began Glen Van Brummelen's book this summer (but then I abruptly stopped as uncertainty regarding my employment situation grew). Actually, Van Brummelen writes only a little about lunar eclipses (and nothing about solar eclipses) in his book. First, he writes:

"Centuries before that, Aristotle had given several arguments for the sphericity of the Earth, including the observation that the shadow cast by the Earth on the Moon during a lunar eclipse is always a circle."

This, by the way, is related to the plane sections of Chapter 9. Since any plane section of a sphere is a circle, the shadow cast by a sphere is likewise always round. On the other hand, were we living on a disk, the shadow could be rectangular, since cylinders also both rectangular and round plane sections.

Here is Van Brummelen's other reference to eclipses:

"One way around this is to observe during a lunar eclipse, which takes place simultaneously for all Earthly observers."

The fact that lunar eclipses take place everywhere at the same time means that it is visible in any time zone where it is nighttime. The eclipse was visible from the west coast of North America as well as Asia and Australia. But in Europe, Africa, and the east coast of the Americas, it was daytime, hence the eclipse was not visible there.

If I were still teaching in the classroom today, it's possible that I might have taught a lesson on lunar eclipses as a follow-up to the solar eclipse activity of August 21st (our first day of school). Now Wednesdays would be Learning Center days, so I'd have to make it fit. Again, die cuts could be used to cut out moon shapes. It's also possible for me to award extra credit to any student who takes a photo of the eclipse. This would be the rare day when cell phones are allowed, as they would be part of the lesson.

In addition to eclipse day, today is also test day. Here are the answers to the Chapter 9 Test:

1. Draw two triangles, one not directly above the other, with corresponding vertices joined.
2. Draw a picture identical to #3.
3. Draw and identify a circle and an ellipse.
4. Draw and identify two circles.
5. circle (Yes, I had to make an eclipse reference!)
6. a. Draw a circle. b. Draw a parallelogram (not a rectangle). c. Draw a rectangle.
7. c or d.
8. Draw #7c or d again. (A cube is a rectangular parallelepiped!) The faces don't need to be squares.
9. a. 144 square units b. 8 units
10. a. Draw #3 again, with both heights labeled. b. 25pi square units
11. a. 2 stories b. 3 sections c. back middle
12. a. tetrahedron, regular triangular pyramid b. 6 edges c. (AB, CD) or (AC, BD) or (AD, BC)
13. sphere
14. solid sphere (This is yet another eclipse reference!)
15. rectangular solid
16 a. yes b. 3 planes
17. lw(h + 2)
18. xy(x + 2 + y)
19. pi r^2 (4 + r)
20. planes

There's no Euclid on the test, but let's look at the next proposition in Euclid anyway:

Proposition 13.

From the same point two straight lines cannot be set up at right angles to the same plane on the same side.

Euclid's proof of this Uniqueness of Perpendiculars proposition is indirect.

Indirect Proof:
Assume that point A lies in plane P, and both AB and AC are perpendicular to plane P. Since three noncollinear points determine a plane, A, B, and C lie in some plane Q. Since two planes intersect in a line, planes P and Q intersect in some line containing A -- call it line DE. By definition of line perpendicular to a plane, CA perp. DE. And for the same reason, BA perp. DE. Then in plane Q, there are two lines through A perpendicular to DE, which contradicts planar Uniqueness of Perpendiculars (implied by Angle Measure/Protractor Postulate). Therefore through a point on a plane, there can be only one line perpendicular to the plane. QED

Today is a traditionalists post. But today, I want to get as far away from traditionalism as possible. I was recently reading about perhaps the most progressive school possible:

http://www.sudval.org/

This is the link for Sudbury Valley School, a Massachusetts private school. Sudbury Valley School is indeed completely different from what the traditionalists want. It's better for me to link directly to the Sudbury Valley website than to explain what it is:

http://www.sudburyvalley.org/01_abou_01.html

Sudbury Valley School, since its founding in 1968, has been a place where children can enjoy life, liberty and the pursuit of happiness as they grow up in the newly emerging world. From the beginning of their enrollment, no matter what their age, students are given the freedom to use their time as they wish, and the responsibility for designing their path to adulthood.

In our environment,students are able to develop traits that are key to achieving success: They are comfortable learning new things; confident enough to rely on their own judgment; and capable of pursuing their passions to a high level of competence. Children at Sudbury Valley are adaptable to rapid change, open to innovation and creative in solving new problems. Beyond that, they grow to be trustworthy and responsible individuals, and function as contributing members of a free society.

At Sudbury Valley, students from pre-school through high school age explore the world freely at their own pace and in their own unique ways. They develop the ability to direct their own lives, be accountable for their actions, set priorities, allocate resources, deal with complex ethical issues, and work with others in a vibrant community.

So this is the Golden 50th Anniversary of Sudbury Valley. Now let's see in more detail how Sudbury Valley differs from traditionalism.

Traditionalists believe that teachers are "the sage on the stage," while progressives believe that teachers are "the guide on the side." Well, Sudbury Valley doesn't deign to call its adult employees "teachers" at all!. Instead, they are called "staff members." And these staff members not only avoid giving homework, they don't even give tests either.

The traditionalist wonders, how are students' grades determined without tests? The Sudbury answer is, students don't receive grades at all. So A-F, 1-4, and other grades are distinctions that Sudbury doesn't make.

The traditionalist wonders, by what criteria are students promoted to the next grade level without letter grades? The Sudbury answer is, students aren't divided into grade levels. Kindergarten, first grade, and other grade levels are distinctions that Sudbury doesn't make.

The traditionalist wonders, by what criteria are students promoted from elementary to middle school, or relatedly from middle school to high school, without grade levels? The Sudbury answer is, students of all ages, from four to the teens, attend the same school. Elementary, middle, and high school are distinctions that Sudbury doesn't make.

The only distinction that Sudbury makes is graduation. The following link is supposedly about how students can graduate, but it doesn't explain much at all:

http://www.sudburyvalley.org/01_abou_07.html

Apparently, prospective graduates are to write some sort of thesis statement, and the oral defense of this thesis is what determines who gets to graduate.

The following link is to a FAQ page:

http://www.sudburyvalley.org/01_abou_09.html

Let's skip to what matters the most to the traditionalists -- college admissions:

Daria: Let's go back to the learning strategy of the school. We presuppose that all children learn, they learn at their own pace, and they learn because they're passionate about learning. How does this prepare a child, for example, to go on to college when they must have SAT scores and must function in a very different setting in a college situation?

Mimsy: Kids who leave here are usually extremely well-prepared to go to college. First of all, they're quite knowledgeable, and they're very articulate. If you want to go to a college for which you need SAT scores (which certainly is not every college at all) then that's one of the things you're motivated to do, and you apply yourself to learning how to do well on the SAT's. This is not a strange idea. Recently, a guest came to the school, and he said, "Oh, I knew someone from this school once," – this person was a teacher – "I tutored him in math. He had graduated from your school and yet he knew very little math and he wanted to take the SAT's.

But within six weeks, he had learned everything I had to teach him." That speaks to motivation. This child was not interested in learning math until he needed it for the SAT's and when he needed it for the SAT's, he learned it quickly. Colleges are not as different from Sudbury Valley, I think, as high schools are, because you're expected to have a lot more autonomy and a lot more responsibility for doing what you need to do in college than you have in most high schools.

Really -- students can study for the SAT math section in six weeks? We know that traditionalist SteveH complains that three years (of "Pre-AP" math) is too little time to prepare for AP Calculus, and now Sudbury claims to get a student from "knowing very little math" to success in SAT math in 1/20 of that time!

It's interesting that Sudbury Valley is in Framingham, Massachusetts -- just 20 miles away from a little town called Cambridge. And that town is home to two universities that you might have heard of, namely Harvard and MIT. I'm sure that traditionalists are very interested whether Sudbury students are getting into colleges like Harvard and MIT and not just Framingham Community College. So far, I don't see much evidence of Sudbury grads entering the Ivy League.

The Sudbury website contains a blog. Notice that traditionalist schools don't require blogs to justify their existence the way that Sudbury schools do. And yes, I'm aware that I myself am a blogger criticizing another blog. The difference is that I'm just trying to explain Common Core Geometry, while the Sudbury bloggers are trying to explain how Sudbury Valley qualifies as a "school." And I know that I often criticize traditionalists. But compared to Sudbury, I and nearly everyone reading this blog are traditionalists. I can't see imagine that many Sudbury students are studying high school Geometry, not even the SAT crammer (as there's little Geometry on the new SAT).

Here is a link to the most recent blog post:

http://blog.sudburyvalley.org/2018/01/whole-language-phonics-and-two-boys-learning-to-read/

The author writes that one Sudbury student is learning to read via whole language while the other is reading via phonics:

One thing that I love about Sudbury schools is that children not only have the freedom to chose when and what to learn, but also how to learn. While the experts debate, our children are lucky to be at a school that respects children’s innate wisdom and intelligence and supports them on the unique path it takes them on.

Traditionalists, by the way, prefer phonics as the only valid method of teaching reading.

The second most recent blog post merely announces the 50th anniversary of Sudbury:

http://blog.sudburyvalley.org/2018/01/fifty-years-front-and-center/

Throughout the 50 years, we have expected excellence. The expectation has guided us unfailingly, and is, I believe, what has brought us to this point. It, more than anything, has fostered a continuously supportive community, inspired by our vision and our efforts to make that vision real.

Here's an article from October in which math is mentioned:

http://blog.sudburyvalley.org/2017/10/lost-over-the-summer/

Naturally, I find myself wondering what lessons learned by my SVS children have been “lost” over the summer?  Maybe they have forgotten the exact list of ingredients for gingerbread. Perhaps they forgot what time JC meets? Perhaps they have forgotten the price of their favorite snack at concession?  But these are all simple facts that can be easily re-accessed when the need for them arises. In the research on Math skill loss there is discussion about how the kids lose knowledge on how to carry out mathematical procedures.  This I can imagine. Once in my adult life, I was surprised to find myself needing to solve a quadratic equation. The only part I could remember was something about “4ac.” This was before the Internet, or at least before I knew how to use the Internet, so I found an old notebook that had common math equations on the inside front and back pages, and there it was! It took me much longer to solve my one problem as an adult than I could have done it in high school. But does that really represent an important “loss?”  If I forced myself to 15 or 20 of them, I bet I could rapidly regain my high school proficiency level, but for what purpose?

I experienced a similar problem more recently, when my daughter asked for help doing a long division problem. Much to my surprise, I had actually forgotten the mechanics of doing long division by hand. Quite embarrassed, I spent a few minutes on the Internet looking at examples, before I could help my daughter.

Hmm, so it appears the author is trying to justify not teaching the students traditional math because they'll easily forget it anyway, and any math needed could be easily looked up (in a notebook in 1968 and on the Internet now). I can't see traditionalists accepting that justification. In fact, let's go back to the article that traditionalist Barry Garelick posted two weeks ago:

We forget. All the time. Especially when we don’t understand why the algorithm works or if it has been a long time since we last used it. I can guarantee if I took a survey at Starbucks right now, and asked people how to perform long division, convert a mixed number to a fraction, recite the quadratic formula, factor a binomial, and complete the square — most would fail. Not because math is too hard or people are bad at it, but because memorization and algorithms are not the best ways to retain information. We remember through context, understanding and application.

Garelick:

We also remember through continued application. For example, if I haven’t worked with percent calculations for a while, I have to brush up on it. Same with finding derivatives of certain functions. The survey of people at Starbucks might be different if the majority of customers were practicing engineers. That people forget how how to do something if they haven’t worked with it for years is not evidence that the traditional method of math teaching is ineffective. And like many authors of similar rants and polemics, she also does not provide evidence that her methods are superior to those that she feels do not work.

Of course, staff members at Sudbury Valley already know that their school is far from traditional. In the past 50 years, other schools inspired by Sudbury have opened around the world. I found out that there's even a Sudbury school right here in Southern California:

http://www.openschooloc.com/wp/

Like most Sudbury schools, the Open School of Orange County has a blog to justify its existence. I link to the most recent blog entry below:

http://www.openschooloc.com/wp/2018/01/21/why-do-we-have-academic-categories/

The author of this article, by the way, is an alumna of the original Sudbury Valley School, and now she's a teacher -- um, "staff member" -- at the Open School. She writes:

The artificial separation of categories — say, separating “mathematics” from the entire rest of the world — creates a series of illusions that educational institutions then spend an overwhelming amount of time and energy working against. Once you accept that “mathematics” is a separate category of activity from knitting, Pokemon, music, cooking, graphic design, physics, writing sonnets, hiking, sales, or getting a weekly allowance, you have undermined the mathematics intrinsic to all of those common life activities.

Then you’re stuck with dissociated word problems like “Sue has eleven apples. If she gives five to Peter, how many will she have?” In the meantime, all the activities I listed require exactly that kind of problem, while also yielding some additional product, such as a scarf, a sonnet, social time with your friends, exercise, a poster, or a song.

I can already imagine the traditionalist rebuttal to this -- students actually have to know math before they can apply it to cooking, physics, and so on -- and they won't ever know any actual math unless they are taught it traditionally. Otherwise students are being asked to "put the cart before the horse."

It goes without saying that Sudbury schools lack bell schedules. They do have school calendars, and students are required to attend school for a certain number of hourse (such as five) each day.

Meanwhile, Garelick has posted more recently than the MLK Day post I linked to above, but we might as well discuss the old post since I already linked to it. Let me link to the original article that Garelick is discussing here:

https://medium.com/i-math/common-core-math-is-not-the-enemy-c05b68f46b3e

As usual, SteveH comments on it:

“Why is Common Core Math Hated by Parents?

Because the Common Core Math standards are trying to teach number sense and mental math techniques through various forms of diagrams and step-wise procedures that are new and look confusing.”

SteveH:

Rote strawman 2 – Really. Stop reading! I guess I’m not paying attention.

Forget the fact that CCSS is non-STEM by definition, starts in Kindergarten, and that the highest level expectation is no remediation for community college math. 

Of course, Sudbury schools oppose Common Core as well, as they oppose all formal standards or requirements that students learn something by a certain time. Sudbury schools are private schools, since no public school could get away with Sudbury pedagogy due to state and federal requirements.

SteveH:
Forget the fact that the College Board knows that this is a problem and has defined Pre-AP in ninth grade as an attempt at crossing the chasm of content and skills between fuzzy K-6 CCSS math and AP Calculus.

Well, at least three years of Pre-AP math are better than six weeks of SAT Prep math.

“Respect the Teachers
They are experts on child education. They have the best intentions for the students in mind. This teacher made a decision [5X3 = 5+5+5 marked wrong] based off a lot more information about the student and class setting than we can tell from a photo. We don’t have to agree with it, but we can respect it. If you are confused, ask them why they did something before you discredit a teacher on the Internet.”

SteveH:

I had to put my hip boots on for that one. They are the experts of their own opinions. They are especially not experts in math. They might not be held to peer review, but mathemeticians are, and the mathemetician of this opinion piece is getting some bad peer reviews.

Well, at least they actually are teachers and not just "staff members." (I explained the controversy surrounding 5 * 3 = 5 + 5 + 5 back in October 2015, when it first arose. Here I'd actually agree with SteveH, since omega * 3 = omega + omega + omega in Cantor's ordinals.)

Any traditionalists reading this post might think I'm playing with them -- I'm defending mainstream progressive math by comparing it favorably to Sudbury. It's just like a thief's justification that at least he's not a killer.

Actually, it's possible to go in a different direction. Perhaps if we support traditional schools, then we should support them completely with a traditional pedagogy, and if we oppose traditionalism, then we should oppose it completely and support Sudbury. Maybe it's the middle ground -- mainstream progressive pedagogy -- that needs to be questioned.

Well, as of now I'm still not sold on Sudbury as a viable school model. I'll continue to post about Common Core Geometry, including worksheets based on it. And I'll continue to use the traditionalists label to compare Common Core to what they prefer.

Let's end today's post with a link to the second commenter from the Garelick post:

BL:

I never did memorize the quadratic formula. But my math teacher showed us how it’s derived, and I quickly did that before each test in the margin of the test paper.

Problem solved. (And, as a result, I never forgot about “completing the square”!)

That's interesting -- I'd imagine that more students have trouble remembering how to complete the square or derive the formula than the formula itself! The Sudbury author from earlier lists both the Quadratic Formula and completing the square as things that the average Starbucks customer can't remember from math class.

If I were teaching the Quadratic Formula, I'd forget about both BL and Sudbury and just teach it to the students using the tune of "Pop Goes the Weasel." As I've mentioned in previous posts, I like using music to help my students learn and remember math.

Here is today's test:

Chapter 9 Review (Day 99)

Today is the review for tomorrow's Chapter 9 Test. This is a brand new chapter for us on the blog, and so it's a brand new chapter for me to write.

In many ways this is a light chapter. Whereas the modern Third Edition includes surface area in Chapter 9 (Lessons 9-9 and 9-10), my old Second Edition stops after Lesson 9-8. Then again, students who have trouble visualizing three dimensions will struggle on tomorrow's test.

Well, at least our students won't have any Euclid on their test. Let's return to his next proposition:

Proposition 12.

To set up a straight line at right angles to a give plane from a given point in it.

Yesterday we construct the perpendicular from a point not on the plane, and now we construct the perpendicular from a point on the plane. This construction uses yesterday's as a subroutine.

This construction is even sillier than yesterday's to perform inside a classroom. This time, we have a point A on the floor and we wish to find a point directly above it. First, we label point B on the ceiling and use yesterday's construction to find a point C directly below it. Then we construct the line through A that is parallel to BC. This last construction is the usual plane construction -- but Euclid performs this construction in the plane containing A, B, and C. This plane is neither the floor nor the ceiling, but an invisible vertical plane that isn't even necessarily parallel to a wall. There is no reasonable way to perform this construction in the classroom.

And so there's no way that our students can physically perform this construction. There will be nothing like this on tomorrow's test, even though ironically, it would be easier to answer test questions about this construction than physically perform it.

Notice that the U of Chicago text doesn't actually provide the construction for drawing a parallel to a line through a point not on the line (which is a simple plane construction). The only way implied in the text to perform this construction is to make two perpendicular constructions (which we did in yesterday's post).

Many texts that teach the construction of parallel lines use copying an angle (as in corresponding or alternate interior angles). Lesson 7-10 of the Third Edition is on constructions, and duplication of an angle is given, but still no parallel lines. (I also see some DGS "constructions" mentioned there -- I wonder whether this is similar to Euclid the Game, as alluded to in yesterday's post.)

If you must, here is a modernization of the proof of Proposition 11:

Given: the segments and angles in the above construction.
Prove: AD perp. Plane P

Proof:
Statements                              Reasons
1. bla, bla, bla                        1. Given
2. AD perp. Plane P               2. Perpendicular to Parallels (spatial, last week's Prop 8)

The proof is trivial since both BC perp. Plane P and AD | | BC are true by the way Euclid constructs these lines -- in other words, they are part of "Given." If (and that's a big if) we were to prove this in the classroom, it would be more instructive to show Euclid's proof of both yesterday's and especially today's propositions directly, than to attempt to convert the proofs to two columns.

Anyway, here is the Review for Chapter 9 Test.

Lesson 9-8: The Four-Color Problem (Day 98)

Lesson 9-8 of the U of Chicago text is called "The Four-Color Problem." This lesson doesn't appear anywhere in the modern Third Edition, because this is one of those "extra" lessons that we include mainly for fun.

In the past, I've mentioned several books and lectures which discuss the Four-Color Conjecture. One of these was David Kung's lectures. Two years ago, his lectures inspired me to create a worksheet for Lesson 9-8, even though we never formally covered Chapter 9. (Thus Lessons 9-5 and 9-8 were the only ones in this chapter for which I'd made a worksheet -- until this year.) I most recently mentioned the prover of Four Colors -- Wolfgang Haken -- two months ago, in my November 27th post.

And so today I post the worksheet from two years ago -- exactly. Yes, the actual date I posted the original worksheet was January 29th, 2016. That day, I also posted a worksheet on reflections on the coordinate plane (not three-dimensional space), and so I'll repost them both today. I admit that it may be awkward to include planar reflections during the same unit that spatial reflections are taught (part of Lesson 9-5). But again, I want to prepare students for the PARCC and SBAC, and these Common Core tests are likely to ask questions about planar reflections over an axis. Unfortunately, this topic isn't fully covered in the U of Chicago text (at least not my old Second Edition, anyway).

This is what I wrote exactly two years ago about today's lesson:

Then Kung moves on to a famous theorem -- the Four-Color Theorem. Just like Lesson 12-7 "Can There Be Giants" from yesterday, the Four-Color Theorem appears in the U of Chicago text -- in fact it's Lesson 9-8. We skipped over it only because we omitted Chapter 9 entirely (as the rest of the material, 3D figures, are covered more thoroughly along with volume in Chapter 10).

The Four-Color Theorem states that any map can be colored with at most four colors. The U of Chicago tells us that the theorem holds on either a plane or a sphere. Kung points out that on other surfaces, different numbers of colors are required. On a Mobius strip, six colors are needed. Both Kung and the U of Chicago text mention that for a special doughnut shape (or is it a coffee cup shape?) called a torus, seven colors are necessary.

CCSS.MATH.CONTENT.8.G.A.3
Describe the effect of dilations, translations, rotations, and reflections on two-dimensional figures using coordinates.

Notice that eighth grade is the first in which transformations appear.

Here are the two lessons that I'm posting instead. One is based on the reflections worksheet that the students worked on in class, and the other is based on the Four-Color Theorem -- another lesson inspired by Kung's lecture. Next week, Kung will continue with topology.

Before we return to Euclid, I want to make a few MTBoS-related links. First, the Queen of the MTBoS -- Fawn Nguyen, that is -- has spoken:

http://fawnnguyen.com/how-i-use-between-two-numbers/

Hmm, I could have sworn that Nguyen already explained what "Between 2 Numbers" is and then I repeated her explanation here on my own blog, but I guess neither of us ever did it. So anyway, Nguyen writes:

It has become one of our regular warm-up routines. (We do visual patterns and math talks, too. Duh.) Before I launched the site, I was just referring to this routine with my kiddos as a “tidbit.”

Take tidbit #5, for example. I ask three questions on Google Form, the first two are identical.

Question 1 is asking for a guess, an estimate, a gut check, a what-do-you-think.

I notice that all three of Nguyen's Warm-Ups are links. And in this example, students don't merely estimate some real-world value, but submit it online. I've mentioned the Warm-Ups that Illinois State required us to give. These are given online, and I project them onto the front board, but they aren't interactive the ways Nguyen's Warm-Ups are.

The other link I wish to give here is to a member of MTBoS who actually teaches the Four-Color Theorem in class:

http://eatplaymath.blogspot.com/2015/10/the-four-color-theorem-and-pumpkin-time.html

Lisa Winer is the author of this post that is over two years old. She doesn't specify in what state she lives, nor does she make it easy for me to figure out what grade or class this is.

Anyway, in Winer's class, she uses the term "chromatic number" to describe the fewest number of colors required to fill in a map. The Four-Color Theorem, therefore, states that the chromatic number of any planar map is four. On a Mobius strip the maximum chromatic number is six, and of course on a torus the maximum is seven.

It's time to return to Euclid. Of course, he writes nothing about Four Colors or reflections across an axis, and so we proceed with the next proposition instead:

Proposition 11.

To draw a straight line perpendicular to a given plane from a given elevated point.

Propositions 11 and 12 are both constructions. Many of Euclid's propositions are constructions -- indeed, "The First Theorem in Euclid's Elements" (that is, Proposition I.1) featured in Lesson 4-4 is actually a construction.

Classical constructions are performed with a straightedge and compass, and David Joyce writes about the importance of actually proving constructions as theorems. But it's awkward to ask our students to perform a construction in three dimensions.

In this construction, we have a point A and a plane P, and we wish to construct the line perpendicular to P through A. How can our students do this? Is P the flat plane of the paper and A a point floating up in space?

It might be interesting to attempt Euclid's construction in the classroom. Here's how: We choose A to be a point on the ceiling and P is the plane of the floor. Thus our goal is to draw a point on the floor directly below A.

The key to this construction is to hang a rope from point A -- a rope that should be longer than the room is high. We can pull the rope at any angle and double-mark the points where the rope is touching the floor. I say "double-mark" because the point on the floor (where the rope touches) is marked (say with chalk), and then the point on the rope (where the floor touches) is marked (say with a piece of tape). The rope now can serve as a compass -- the point of the compass is at A, and the opening of the compass is set to the distance between A and the tape. The locus of all points on the floor that are the same distance from A as the point marked on the floor is a circle, and the locus of all points on a given line on the floor that are the same distance from A is a pair of points. So if we have a point (say B) drawn on a line on the floor, then we could find the unique point C on that line such that AB and AC are congruent.

All the lines on the floor can be drawn in chalk. There will be some plane constructions drawn on the floor as well, so we could use a large compass where the pencil has been replaced with chalk.

OK, so let's begin the construction. We start by drawing any line on the floor, and then we label any point on that line B. We now find C on this line exactly as given above -- we double-mark B on both the rope and floor, and then swing the rope to find C such that AB = AC.

Now we use the chalk compass to find the perpendicular bisector of BC. The midpoint is D.

Then we double-mark D with a second piece of tape, and then find the point on the last line we drew (that is, the perp. bisector of BC), to be labeled E, such that AD = AE. The second piece of tape must be higher up than the first since AD < AB, and so there's no danger of confusing which piece of tape is which.

Finally, we find the perpendicular bisector of DE. The midpoint is F. Euclid's G and H are any points on this last line -- their location doesn't matter. Only F is relevant here. AF is the desired line through A that is perpendicular to the plane of the floor, and F is directly below A.

Of course, this whole construction seems silly because of gravity. We can just hang a rope freely from A, label the point where the rope touches the ground F, and then we're done! The difference, of course, is that Euclid's three-dimensional space isn't physical space, and so there's no direction that's "favored" because of gravity or any physical force.

And so I'm not quite sure how David Joyce has in mind when he says he wants "the basics of solid geometry" to be taught better. Does he include Euclid's spatial constructions -- does he really want students to perform them? Or maybe he merely desires that students visualize the proofs in their minds while looking at the proof.

(Do you remember Euclid the Game, which is played on computers? Maybe in higher levels, players can make three-dimensional constructions that are difficult to perform in the real world!)

By the way, we can still modernize Euclid's proof:

Given: the segments and angles in the above construction.
Prove: AF perp. plane P

Proof:
Statements                              Reasons
1. bla, bla, bla                         1. Given
2. BC perp. plane (ED, DA)   2. Proposition 4 from two weeks ago
(Call it plane Q. In the classroom, Q is an invisible plane parallel to a wall.)
3. GH | | BC                            3. Two Perpendiculars Theorem (planar version)
4. GH perp. plane Q               4. Perpendicular to Parallels (spatial, last week's Prop 8)
5. AF in plane Q                     5. Point-Line-Plane, part f (A, D, F all in plane Q)
6. GH perp. AF                       6. Definition of line perpendicular to plane
7. AF perp. plane (GH, DE)   7. Prop 4 (AF perp. DE is part of "Given")
8. AF perp. plane P                 8. From construction (both lines were drawn in plane P)

It might be tricky to reconcile this proof with the "rope" construction from above. In Euclid's construction, AD is designed to be perpendicular to BC, likewise AF is perp. to DE. Both of these perpendicular constructions technically occur in planes other than the floor -- yet earlier I direct you to perform perpendicular constructions on the floor -- which is the wrong plane.

But think about it -- given a point A and a line, how do we construct a line through A perpendicular to the given line? The answer is that, using the compass, we find points B and C on that line that are equidistant from A, and then find the perpendicular bisector (in that plane) of BC.

But technically, all we really need is D, the midpoint of BC. Then the line through points A and D is automatically the perpendicular bisector of BC in the correct plane. It doesn't matter how we obtain the midpoint D -- all that matters is that we find it. This includes finding the perpendicular bisector of BC in the wrong plane (that is P, the plane of the floor). This is why Euclid is able to assert and use statements like AD perp. BC in his proof, even though this isn't obvious from our ropes. (And as it happens, the perpendiculars in plane P appear later in the proof anyway, so we might as well construct these.)

In the end, let's just stick to the Four-Color Theorem and two-dimensional reflections. Here are the worksheets for today:

Lesson 9-7: Making Surfaces (Day 97)

Lesson 9-7 of the U of Chicago text is called "Making Surfaces." In the modern Third Edition of the text, making surfaces appears in Lesson 9-8.

This lesson is all about making nets that can be folded to form polyhedra and other surfaces. Some figures have much simpler nets than others.

Today is an activity day -- and fortunately, today's lesson naturally leads to an activity. As I wrote in earlier posts, our activities will be based on the questions in the Exploration section. The first such question is the following:

25. A regular polyhedron is a convex polyhedron in which all faces are congruent regular polygons and the same number of edges intersect at each of its vertices. There are only five regular polyhedra; they are pictured here.

a. Determine the number of vertices of each regular polyhedron.
b. Determine the number of edges of each regular polyhedron.

Ah -- we've seen these before. The five regular polyhedra are also called the Platonic solids. I've mentioned these in previous posts -- three summers ago we explained why there are only five of them, and two years ago we discovered that there are six regular polytopes in four dimensions. The Platonic solids are the tetrahedron, hexahedron (cube), octahedron, dodecahedron, and icosahedron.

The Exploration section continues thusly:

In 26-30, use cardboard and tape to construct a model of the regular polyhedron from the net provided. The patterns below should be enlarged. Cut on solid lines, fold on dotted lines.

Many teachers have given Platonic solid lessons in their classes. Since I don't want to try to create the nets myself, I link to previously made lessons. The first page, based on Question 25 (counting the vertices and edges) comes from the following link -- an elementary school in Washington State:

http://wilderptsa.ourschoolpages.com/Doc/Math_Adventures/Platonic_Solids.pdf

Question 26-30, the nets themselves, come from the following link:

https://www.math-drills.com/geometry/net_platonic_solids.pdf

The Math Drills link provides two nets for the dodecahedron. I chose the second one, since it more closely resembles the net in the U of Chicago text. On the other hand, their icosahedron net is very different from ours in the U of Chicago text.

Several members of the MTBoS have had Platonic solid activities in their own classes. Let's link to some of them:

Our first link is to Pamela Lawson, a Maine charter high school teacher. She taught her class about the Platonic solids exactly two years ago today:

https://rawsonmath.com/2016/01/26/how-do-we-know-that/
https://rawsonmath.com/2016/02/07/more-3d-geometry/

Notice that these posts were part of the 2016 MTBoS Blogging Initiative. (And no, there's still no sign of a 2018 Initiative.) Lawson begins:

I’m teaching this 12 week geometry class focusing on 3-dimensional figures. It’s a brand new class, like many at Baxter Academy, so I get to make it up as I go. Since our focus is on 3-dimensional figures, I thought I would begin with some Platonic solids. So I found some nets of the solids that my students could cut and fold. Once they had them constructed, there was a lot of recognition of the different shapes and, even though I was calling them tetrahedron, octahedron, and so on, many of my students began referring to them as if they were dice: D4, D8, D12, D20. Anyway, I must have made some statement about there only being 5 Platonic solids, and they now had the complete set. One student asked, “How do we know that? How do we know that there are only 5?” Great question, right?

(She's teaching a 12-week Geometry course? That's right -- hers is one of the rare high schools that uses trimesters!) Of course, I'd already give a full explanation here on the blog, just after Independence Day in 2015. Let me repeat parts of that post here:

Legendre's Proposition 357 states that the sum of the plane angles that make up a solid angle must be less than [360 degrees]. He proves this essentially by "flattening out" the solid angle -- he takes a plane that intersects all sides of the solid angle and uses the previous Proposition 356 (which we've already proved here on the blog) to show that each plane angle of the solid angle is less than the same angle projected onto the new plane. A good way to visualize this is to imagine that the solid angle is formed at the vertex S of a pyramid -- the points A, B, C, etc., mentioned Legendre can be the vertices of the base of the pyramid, and the point O can be any point in the plane of the base -- for example, the center of the polygonal base.

I won't take the time to show the full proof of Proposition 357, but I will mention an application of this theorem. Suppose we want to figure out how many Platonic solids there are. Recall that a Platonic solid is a completely regular polyhedron -- all of its faces are congruent regular polygons. As it turns out, we can use Proposition 357 to find all of the Platonic solids.

We start with the equilateral triangle, with each angle measuring 60 degrees. Now each vertex of our Platonic solid forms a solid angle. We need at least three plane angles to form a solid angle, but there is an upper limit to how many plane angles there can be. Proposition 357 tells us that the plane angles must add up to less than 360 degrees, and since each angle is 60 degrees, there must be fewer than six of them (since 6 times 60 is 360). So there can be three, four, or five 60-degree plane angles. The Platonic solid with three 60-degree plane angles is the tetrahedron, with four is the octahedron, and with five is the icosahedron.

If we move on to squares with their 90-degree angles, we can have three 90-degree plane angles, but not four (since 4 times 90 is 360). Three 90-degree plane angles gives us the cube. Regular pentagons have 108-degree angles. Again, we can't have four of them (since 4 times 108 is more than 360), and three 108-degree angles gives us the dodecahedron. Regular hexagons have 120-degree angles, but 3 times 120 is already 360. Since each solid angle must contain at least three plane angles, we are done, since increasing the number of sides in the polygon only increases the angle. Therefore, there are only five Platonic solids -- tetrahedron, cube, octahedron, dodecahedron, and icosahedron.

Returning to 2018, let's go back to Euclid, who gives definitions of the Platonic solids:

Definition 25.

A cube is a solid figure contained by six equal squares.

Definition 26.

An octahedron is a solid figure contained by eight equal and equilateral triangles.

Definition 27.

An icosahedron is a solid figure contained by twenty equal and equilateral triangles.

Definition 28.

A dodecahedron is a solid figure contained by twelve equal, equilateral and equiangular pentagons.

We notice that the tetrahedron is missing. According to David Joyce, Euclid refers to the tetrahedron merely as a triangular pyramid. In Book XIII, he also proves that these are the only five Platonic solids -- and there, he refers the tetrahedron simply as "pyramid."

Since I don't wish to jump to Book XIII of Euclid, let's look at the next proposition here in Book XI:

Proposition 10.

If two straight lines meeting one another are parallel to two straight lines meeting one another not in the same plane, then they contain equal angles.

As usual, let's modernize the proof:

Given: l, m intersect at B, n, o intersect at E, l | | n, m | | o (lines not all coplanar)
Prove: The angle between l and m is congruent to the angle between n and o.

Proof:
Statements                                        Reasons
1. bla, bla, bla                                   1. Given
2. Choose A, C, D, F on l, m, n, o    2. Point-Line-Plane, part b (Ruler Postulate)
so that AB = DE, BC = EF
3. ABED, BCFE are parallelograms 3. Parallelogram Tests, part d
                                                              (one pair of sides is parallel and congruent)
4. AD | | BE, BE | | CF                      4. Definition of parallelogram
5. AD = BE, BE = CF                       5. Parallelogram Consequences, part b
                                                              (opposite sides of a pgram are congruent)
6. AD | | CF                                      6. Transitivity of Parallels (Prop 9 from yesterday)
7. AD = CF                                       7. Transitivity of Congruence
8. ADFC is a parallelogram             8. Parallelogram Tests, part d
                                                              (one pair of sides is parallel and congruent)
9. AC = DF                                       9. Parallelogram Consequences, part b
                                                              (opposite sides of a pgram are congruent)
10. Triangle ABC = Triangle DEF   10. SSS Congruence Theorem [steps 2,2,9]
11. Angle ABC = Angle DEF           11. CPCTC

We can't help but notice that the six points A, B, C, D, E, F are the vertices of a triangular prism. And indeed, we see that the translation that appears in the U of Chicago definition of prism is the same translation that maps Triangle ABC to Triangle DEF.

Of course, this requires us to show that if two lines are parallel, then a translation must map one line to the other. I've alluded to the proof of this in posts from previous years, but I no longer include it as part of our curriculum.

Meanwhile, in this post I wish to think ahead to Pi Day. One of my favorite pi websites is the old Sailor Pi website, created by "Bizzie Lizzie" -- indeed I linked to it in previous Pi Day posts.

But sadly, the Sailor Pi website no longer exists. Last year, I had been looking forward to singing some of her pi songs in my class, but of course I left my classroom before Pi Day. On Pi Day itself, recall that I delivered pizza to my old class, but I didn't stay and sing any songs.

But I did write some of Bizzie Lizzie's old songs in a notebook that I purchased that day. One problem in my old class was that I didn't use music break to its full potential. I recall one day when I sang a song about Mean, Median, and Mode when my seventh graders needed to recall measures of center during a Monday lesson with the coding teacher. This was about a month after I'd written the song for my sixth graders when they first learned about measures of center.

This is something that I should have done more often -- when my students need to recall an old lesson from a few months earlier, sing an old song from a few months earlier to jog their memory. But I rarely did so, because I'd written the songs on poster paper and then packed them away when it was time for the next song. A notebook full of songs would allow me to recall songs more quickly. And so on Pi Day, I bought the notebook and wrote in all my old songs and some of Bizzie Lizzie's pi songs.

And now I'm glad I did it, since the Sailor Pi website is now defunct. (I made this sad discovery back on Third Pi Day in November, but I didn't blog about it until today.) But now I'm curious as to the current whereabouts of this mysterious Bizzie Lizzie -- maybe she posted her songs at a new website.

A Google search for Sailor Pi I made back in November returned the following Instagram page:

https://www.instagram.com/sailorpi/

The owner of this Instagram page calls herself Liz. Hmm, is this Liz, as in Bizzie Lizzie? It's logical to assume that this Liz, with a user name of @sailorpi, is indeed Bizzie Lizzie, the author of the old Sailor Pi website. Nonetheless, I sought more evidence to prove that these are the same person.

I looked back to some of Liz's old pictures. One photo, dated January 26th, 2014, is of a cake, and on it is written "Happy Big 3-0, Liz!"

We can't conclude that Liz's birthday is exactly on the 26th -- she could have celebrated a few days before her actual birthday, or she could have a few days after the party to post the photo. But it's safe to say that she turned 30 in late January 2014. For simplicity, assume that her birthday is the 26th -- which, as you might notice, is today's date. That's the real reason that I chose today to blog about the Liz mystery -- it's her 34th birthday.

(Oh, and speaking of science-related birthdays, there is a Google Doodle today for Wilder Penfield, an American-Canadian neurosurgeon. According to the doodle, he used burnt toast to more learn about the brain.)

How does this help us tell whether Liz is Bizzie Lizzie? Well, on the old Sailor Pi site, Bizzie Lizzie wrote a "2003 Season of Sailor Pi," which details her days as a college student. If Liz's 30th birthday was indeed 1/26/14, then she would have celebrated her 19th birthday in 2003. This is consistent with Bizzie Lizzie attending college in 2003.

As we look at some of Liz's other Instagram photos, we see her wearing pi blouses and celebrating Pi Day on March 14th. This suggests that the "pi" in her username @sailorpi really does refer to the number pi and not, for example, "Sailor P.I." So far, the evidence that Liz and Bizzie Lizzie are the same person is compelling, but I wanted one more piece of solid evidence.

Liz often took photos of her workplace. In June, five months after her 30th birthday, she posted photos of her last day working at Cable News Network (CNN) and first day working at Jet Propulson Laboratories (JPL) right here in Southern California.

Hmm, CNN is a news source, and it's possible to search the CNN website for articles. If Liz indeed worked at CNN, perhaps some of her articles are still archived there. And since she's enjoys Pi Day, maybe she'd have been called upon to write a Pi Day article for CNN.

We now perform another Google search for pi day site:cnn.com. The first two results are for Pi Day 2017 and Pi Day 2016, which are no good since we know Liz left in 2014. But the third result is dated Pi Day 2014, so this is promising. Here's a link to the article:

http://www.cnn.com/2014/03/14/tech/innovation/pi-day-math-celebrations/index.html

The author of this article is Elizabeth Landau. Hmm, Elizabeth...Liz...Bizzie Lizzie -- could it be? We now perform one last Google search for the name Elizabeth Landau. The first result is LinkedIn:

https://www.linkedin.com/in/elizabethlandau/

And in this post, we confirm that Elizabeth attended Princeton from 2002-2006, which places her in college in 2003, just as Bizzie Lizzie wrote on the Sailor Pi website.

Oh, and there's one more thing to clinch it -- Liz's hair on both LinkedIn and Instagram is red. On the old Sailor Pi website, she drew herself as a redhead playing the guitar and dancing to her songs.

The evidence is now overwhelming. Elizabeth Landau is indeed Bizzie Lizzie, the author of the old Sailor Pi website.

By the way, all of this makes me seem like a stalker. After all, who else would search various social networking sites just to find one particular person?

The reason I searched for Liz is that I want to keep track of her songs. On her old website, she would claim her songs as falling under a "copyright." And so I respect this copyright by searching for the author and citing her name when writing about her songs. I'm surprised that I was able to find Landau so easily -- it helps that she ultimately became a journalist. Indeed, according to LinkedIn, one her roles at JPL is "writing long-form articles about space," so she is still technically a journalist.

By the way, Landau has written some earlier Pi Day articles for CNN:

http://lightyears.blogs.cnn.com/2012/03/13/pi-day-how-3-14-helps-find-other-planets-and-more/
http://www.cnn.com/2010/TECH/03/12/pi.day.math/index.html
http://scitech.blogs.cnn.com/2010/03/12/geek-out-my-life-with-pi/
http://scitech.blogs.cnn.com/2009/03/13/pi-day-and-american-pi/

In the last of these, Landau wrote her song "American Pi," a parody of Don McLean's song "American Pie." I'd copied the words of this song from the old Sailor Pi site into my notebook. This 2009 version is slightly different from the original, but it's still recognizably the same song.

I wrote the lyrics to one more songs from the Sailor Pi website. Since these words are, as far as I know, no longer posted on the Internet, I'll post them here. The song is "The Digit Connection," a parody of Kermit the Frog's song "The Rainbow Connection" from The Muppet Movie. Here is the song -- with proper attribution, of course.

THE DIGIT CONNECTION
Copyright 1998 by Elizabeth Landau, aka Bizzie Lizzie

1st Verse:
Why are there so many debates about pi?
And what's on the other side?
Pi is a ratio of random proportions.
Its digits have nothing to hide.
So we've been told and some choose to believe it,
But I know they're wrong, wait and see!
Someday we'll find it, the digit connection,
Mathematicians, logicians, and me.

2nd Verse:
Who said that everything has some sort of pattern,
Consisting of nothing but math.
Somebody thought of that, and someone believed it.
Now we're all caught in its wrath....
What's so hypnotic in something chaotic,
And what do we think we might see?
Someday we'll find it, the digit connection,
The optimists, the theorists, and me.

All of us under its spell,
We know it must be math-e-magic...

3rd Verse:
Have you been half asleep? And have you heard voices?
I've heard them calling my name.
Is this the sweet sound that calls the young sailors,
The voice might be one and the same....
I've heard it too many times to ignore it,
Irrational, random, and free.
Someday we'll find it, the digit connection,
The lovers, the dreamers, and me.

3.1415926535 dot, dot, dot!

People of my and Landau's generation -- late X'ers and early Millennials -- often created our own websites on the early Internet. I remember seeing the website of one of my former classmates. She, like Landau, was also a huge Sailor Moon fan, and both girls recast their friends as various characters from that particular anime. To me, it's sad that nowadays, AOL and Geocities are disappearing, because youngsters are simply creating profiles on social networking websites like Instagram and LinkedIn rather than making their own webpages. Our generation showed much more creativity.

Anyway, I wish Elizabeth Landau a happy 34th birthday, wherever she may be. (Sorry, it just seems so strange to keep writing out her real name -- to me, she'll always be Bizzie Lizzie.) And I'll spend the rest of her birthday reading all the wonderful articles she links to on her LinkedIn page.

Here are the worksheets for today's activity:

Lesson 9-6: Views of Solids and Surfaces (Day 96)

Theoni Pappas, on her Mathematics Calendar 2018, has a Geometry question today:

This square is circumscribed [sic]. Find the circle's area to the nearest whole number.

(A side of the square is given as length 4.)

Pappas makes an error here in stating the question. The square is inscribed, not circumscribed. At any rate, our students can solve this problem using the formulas of Chapter 8.

The side of the square is 4, and so its hypotenuse -- the diameter of the circle -- is 4sqrt(2). In the modern Third Edition, students learn about 45-45-90 right triangles in Chapter 8. Even though Second Edition students won't reach Special Right Triangles until Chapter 14, they can still use the Pythagorean Theorem to find the hypotenuse.

Then the radius of the circle is 2sqrt(2), and so its area is pi(2sqrt(2))^2 = 8pi square units. If we use an approximation such as 3.14 for pi, then the area becomes 25.12 square units. To the nearest whole number, this is 25 square units -- and of course, today's date is the 25th.

By the way, notice that 25/8 is the ancient Babylonian approximation to pi. Hmmm, perhaps August 25th should be declared Babylonian Pi Approximation Day? Hmm -- never mind! (By the way, I sneaked today's Pappas problem in on the review side of today's worksheet.)

Lesson 9-6 of the U of Chicago text is called "Views of Solids and Surfaces." In the modern Third Edition of the text, views of solids and surfaces appear in Lesson 9-5. (Recall that Lesson 9-4 of the new edition is "Drawing in Perspective," which is Lesson 1-5 of the old edition.)

This lesson is perfect for an architect. We've discussed architecture in many previous posts, most recently in my September 18th post as we read about buildings in Pappas.

Only two words are defined in this lesson -- views and elevations:

"Underneath the picture of the house are views of the front and right sides. These views are called elevations."

There's not much more for me to say about this lesson, except to add that the modern Third Edition includes some examples using isometric graph paper. This reminds me a little of the (mis)adventures my middle school students had with isometric graph paper last year.

And so we return to Euclid. Of course, none of his definitions or propositions have anything to do with views of solids and surfaces, so we just look at the next proposition in order:

Proposition 9

Straight lines which are parallel to the same straight line but do not lie in the same plane with it are also parallel to each other.

This is the three-dimensional analog of yet another theorem, previously studied in Lesson 3-4:

Transitivity of Parallelism Theorem:
In a plane, if l | | m and m | | n, then l | | n.

Some texts omit "in a plane," (after all, as Euclid is about to prove, it holds in three dimensions as well) but only prove it for a plane. The U of Chicago text is more honest and admits that it is only proving it for the plane case. Indeed, the text gives an informal argument in terms of slope -- and of course, we only define slope for lines in a (coordinate) plane.

Euclid's proof is interesting in that unlike those of the previous propositions, he does not use plane Transitivity of Parallelism to prove the spatial case.

Given: AB | | EF, CD | | EF (noncoplanar)
Prove: AB | | CD

Proof:
Statements                              Reasons
1. bla, bla, bla                         1. Given
2. Choose G on EF,                2. Point-Line-Plane, part b (Ruler/Protractor Postulates)
H on AB with GH perp. EF,
K on CD with GK perp. EF
3. EF perp. plane(GH, GK)    3. Proposition 4 from last week
(call it plane P)
4. AB perp. plane P,               4. Proposition 8 from yesterday
    CD perp. plane P
5. AB | | CD                            5. Proposition 6 from Monday

But in this case, Euclid's original proof is simple enough to show to high school students as it is:

Let each of the straight lines AB and CD be parallel to EF, but not in the same plane with it.

I say that AB is parallel to CD.

XI.4

Now, since EF is at right angles to each of the straight lines GH and GK, therefore EF is also at right angles to the plane through GH and GK.

XI.8

And EF is parallel to AB, therefore AB is also at right angles to the plane through HG and GK.

For the same reason CD is also at right angles to the plane through HG and GK. Therefore each of the straight lines AB and CD is at right angles to the plane through HG and GK.

XI.6

But if two straight lines are at right angles to the same plane, then the straight lines are parallel. Therefore AB is parallel to CD.

Therefore, straight lines which are parallel to the same straight line but do not lie in the same plane with it are also parallel to each other.

Q.E.D.

Perhaps the only change we might make is replace "plane through HG and GK" with "plane P" in order to avoid repeating that cumbersome phrase.

David Joyce mentions another proof at the above link -- midpoint quadrilaterals, which are also known as Varignon quadrilaterals. The U of Chicago text only mentions midpoint quadrilaterals in an Exploration exercise (likewise in the new Third Edition, the only difference being that the new text actually mentions Varignon's name). But it's easy to prove that they are parallelograms -- just divide the quadrilateral into two triangles and apply the Midsegment Theorem of Lesson 11-5 to each.

But now Joyce points out that the theorem is true even for "space quadrilaterals." Our definition of polygon (and hence quadrilateral) states that the vertices must be coplanar. But even if we relax this requirement and allow for space quadrilaterals, Varignon's Theorem still holds. After all, even space quadrilaterals can be divided into two triangles (and each triangle lies in a plane), and so we can still apply the Midsegment Theorem to each one. The difference is that today's Proposition 9 is used to prove that opposite midsegments are parallel -- each is parallel to the the diagonal of the quadrilateral but the three lines aren't coplanar.

Notice that the final parallelogram always lies in a single plane -- this follows from Tuesday's Proposition 7 that two parallel lines and a transversal are always coplanar.

Meanwhile, I've been continuing to think about a proof of Proposition 4 that is based on rotations, since this might be simpler than Euclid's proof that requires seven pairs of congruent triangles. Let's recall what we're supposed to prove -- given that line l is perpendicular to both lines m and n, prove that l must be perpendicular to the entire plane containing m and n.

Euclid begins by defining points A, B on m, C, D on n, and E, F on l. (Actually, E is where all three lines intersect.) These points have the additional property that AE = BE = CE = DE. My goal was to demonstrate that a rotation of 180 degrees about axis EF maps A and B to each other, as well as C and D to each other. Of course, E and F are fixed points of this rotation.

Let's first find A', the rotation image of A. We first notice that rotations, like all isometries, preserve angle measure, and so Angle A'EF = AEF, which is known to be 90. Also, since rotations preserve distance, AE = A'E. Finally, since the magnitude of the rotation is 180, Angle AEA' = 180. The only point satisfying all these requirements is B, so A' = B. Similarly, we have B' = A, C' = D, and D' = C.

Of course, this requires a more rigorous definition of rotation about an axis. For plane rotations about a center O, we expect A' to be a point such that AO = A'O and Angle AOA'  equals the magnitude. But for rotations about an axis, we can't use the same definition unless we know where O is along the entire axis. Otherwise we won't know where to measure AO, A'O, or Angle AOA'.

The correct answer is that O is chosen along the axis such that AO is perpendicular to the axis. Since AE is perpendicular to the axis EF, E is the correct point to choose. So that's why AE = A'E and Angle AEA' must be 180. If AE weren't perpendicular to the axis, choose O to be a different point instead.

Once this is complete, then we choose line o with G and H the points of intersection. It then follows that H = G' since rotations preserve collinearity, and the proof is complete.

Or is it? I'm wondering whether I slipped and accidentally assumed what we're trying to prove. For example, can we be sure that the rotation maps o to itself unless we already know that o is perpendicular to l? Also, if we define a rotation as the composite of two reflections in intersecting planes (similar to the definition for plane rotations), then we might not be able to prove that rotations work the way I said they do unless we've already assumed that Proposition 4 is true! (Why, for example, do we choose O such that AO is perpendicular to the axis?)

Making the proof work requires much more thought than I'm willing to take now, considering that we're not actually trying to teach Proposition 4 in a classroom.

Here is the worksheet for today: Students can fill out the front of the worksheet with examples of both a 3D figure (such as a pyramid) and a block-building (similar to #2 on the back) with the respective front, right, and top views of each.