A grade goeth before a fall

And it's all my fault

While I doubt it registers with my students, I am at pains every semester to explain to them that they earn grades. I do not merely give them. Unfortunately, the students who most need to hear this message seem to be the least likely to retain it.

I recently taught an algebra class in an accelerated format. Students were warned at the outset of the course's brisk pace and the need to work diligently to stay abreast. The faint-hearted quickly folded their tents and stole away. The braver students stuck it out to the end—a bitter end for a few of them. Overall, though, the success rate was over 80 percent. I was happy that so many of my students passed the class.

One of the students was less than enamored with her “success.” Yes, she passed the class, but she passed it with only a C after having spent most of the semester at the B level. She had spectacularly flunked the comprehensive final (earning fewer than half the possible points on it) and her average plummeted. I declined to award a B to a student who couldn't even earn a D on the final exam. She called me up to complain at the injustice of the result.

Her particular complaint focused on what she perceived as the inequity of students getting a C grade with composite semester scores of 68.5 while she was being denied a B despite a composite score of 78.5. Why did I “round up” the scores near the C-D boundary but not hers at the B-C boundary?

Several factors influenced my decision. First of all, the C-D boundary is basically academic life versus death. A grade of D forces you to repeat the course for credit. I give very close scrutiny to the scores of all students teetering on the precipice of the C-D divide. Furthermore, the three students in question had all beaten my complainant by several points on the final (and the weakest of the three was in the enviable “hammock” position). Unlike my former B student, they had not used the final exam to demonstrate utter confusion and lack of subject-matter retention (a consideration of some significance in a prerequisite course like algebra).

Then, of course, there's the other tiny factor: Among the students with passing grades, the student in question had one of the lowest participation rates in the quizzes that I used throughout the semester to gauge my students' progress. To be fair, it was not chronic absences that caused her to miss so many quizzes (although her attendance did suffer near the end of the semester). No, it was her refusal to submit her paper to me when I collected them, even when I made a point of asking her directly. “No,” she'd say. “It's no good.” Brimming over with sweet reason, I would explain, “Five points on a ten-point quiz may be a little embarrassing, but five points in the grade book is significantly better than zero points!” She'd shove the crumpled quiz into her binder and resolutely refuse: “No, I don't want you to look at it. It's no good.”

In the end, she withheld or missed over twenty percent of her quizzes. A series of truly bad decisions. Those points were not there to reinforce her against a bad result on the final exam, which turned out to be a significant matter in the end. I guess her problems weren't just in algebra.

Even lotteries have winners

Lucky Larry hits a grand slam

It has been noted that state lotteries are basically a tax on innumeracy. You're better off dealing with the house percentage at a casino in Nevada. Nevertheless, even lotteries have winners. Just don't expect it to be you. Lightning is not going to strike you.

Of course, sometimes it strikes near by.

One of my students won the calculus lottery during the exam on integration, beating very long odds indeed. The result was the most bizarre “Lucky Larry” of my years as a math teacher. My colleagues were as flabbergasted as I was when I shared the student's “solutions” with them. Her work was nonsense, yet her answers were correct. Three times in a row. Of course, when that happens one suspects a hidden underlying pattern that produces valid results, contrary to all expectations. In this case, though—no. It was a giant fluke.

Or, rather, three flukes in a row. My flabber, she is as gasted as possible.

The problem on the integration exam was one of my “conceptual” exercises. One of my tasks as a calculus teacher is to clarify the meaning of the definite integral, ensuring that my students grasp its significance. Of course, one of the most common (and visual) interpretations of the definite integral is as the area under a curve. Surely any first-year calculus student must understand at least that much.

Accordingly, I presented my students with the graph of a simple function and asked them to evaluate three definite integrals of that function by inspection of the graph. I did not forbid them to use antidifferentiation and the fundamental theorem of calculus, but I emphasized that the simplest of calculations would suffice.

What, pray tell, is the value of the definite integral of f(x) from x = 1 to x = 2? A cursory examination of the trapezoidal region spanning the space between the x axis and the graph of the function reveals the area (and thus the definite integral) to equal 1.5. Easy! Not satisfied, however, with such a trivial computation, one of my students rolled out the big guns:

Damn! What a coincidence! The calculus is bogus, but the result is accidentally correct. No one expects lightning to strike twice, of course.

Brace yourself.

What if we ask for the definite integral from 1 to 3 instead? We get a little more area now. Take a look at the new graph, in which a second trapezoid now joins the first. We get an additional 2.5 square units which, added to the original 1.5, gives us 4. My student swung into action and unlimbered her surreal calculus calculation again:

I was now quite beside myself, shaking my head in astonishment as my red pen hovered over the page. Twice in a row! (What were the odds?)

Fortunately, I knew that I could count on part (c) to set the record straight and demonstrate to my student the error of her ways. It was, in fact, the simplest part of the problem. A kind of gift to the student possessing a clue. Can you find the area of a rectangle measuring 2 by 4? Of course! The answer must be 8.

My student presented her solution:

Nooooooooo!

Time to hit my head against the desk a few times.

In a million years, this will never happen again. (For one thing, this problem is going straight into the waste can, never to be recycled.)

I need to go lie down for a few minutes.

Dan Quayle speaks!

Stupid is as stupid does

My goodness, how we have missed Dan Quayle, the man elevated above his station in life as impeachment insurance by George Herbert Walker Bush. Quayle spoke up this week to endorse Mitt Romney for the Republican nomination for president of the United States. The man who spent four scary years one heartbeat away from the presidency stressed the conviction with which he was backing the former Massachusetts governor. Addressing Romney in front of television cameras and radio microphones, the illustrious former vice president said:

I am confident that you will be our nominee, and I am even more confident that you will be the next president of the United States of America.

Okay, you got that? Dan Quayle thinks Mitt Romney is all but certain to be the Republican standard-bearer ... and even more certain to become president. That's tricky, since the consequence cannot be more likely than its prerequisite.

Poor Dan is as good at probability as he is at spelling.

I think that I shall never see

Sic transit gloria arboris

The house was nearly ready. The front yard was nothing but dirt, but it was already cleared of most of the construction debris. Paulinho had planted a small evergreen tree that he intended to use as a local landmark when it grew larger. “The house with the fig tree” was his father’s. His would be “the house with the evergreen.”

—From an unpublished novel

The sentence of death was announced on Thanksgiving. The decision had been made earlier, but it was revealed only when I was present to hear it in person. One more eternal verity is about to hit the dust—quite literally in this case.

The pine tree in my parents' front yard was planted the month before I arrived on the scene. The family photo album is full of pictures of the first-born standing next to it. The tree's growth quickly outpaced mine and soon it towered over everyone and everything. For many years my siblings and I referred to it as simply “the Christmas tree,” in honor of its once-a-year decoration with lights and in recognition of its uniqueness. No other house on the dairy farm was so adorned. Deciduous trees and spindly palms dominated the landscape, while our evergreen stood out in singular splendor.

For all I know, the execution has already been carried out. My parents and their friendly neighborhood tree surgeon were simply waiting for a mutually convenient date to do the deed. My Christmas visit will tell the tale, and I will know the outcome while still several miles from the family farm. The tree's absence on the horizon will be more than obvious. The loss of the lifelong landmark will be acutely felt.

My parents did not make a casual and unfeeling decision to raze the tree. The decades had inflicted significant damage on the evergreen. A dangerous crack in the upper reaches of the trunk had already forced a hasty topping of the tree before it dropped its crown on the house. No other remedy was possible. The truncated tree was still taller than anything other than the oldest palm trees (it's framed by the two tallest in the above photo), but its glory days were now clearly over. The loss of its upper third caused the tree's remaining branches to spread out in renewed vigor, extending them to the point that they began to sag and threaten to break. The old tree required either a serious and continuing pruning regimen or ... removal.

My parents made an economical and prudent decision, so the tree's fate was sealed. I passed the information along to my manuscript editor, who was aware of the tree's supporting role in my novel. He quickly replied to my message with a painfully apt poem by Seamus Heaney:

Clearances VIII

I thought of walking round and round a space
Utterly empty, utterly a source
Where the decked chestnut tree had lost its place
In our front hedge above the wallflowers.
The white chips jumped and jumped and skited high.
I heard the hatchet's differentiated
Accurate cut, the crack, the sigh
And collapse of what luxuriated
Through the shocked tips and wreckage of it all.
Deep-planted and long gone, my coeval
Chestnut from a jam jar in a hole,
Its heft and hush became a bright nowhere,
A soul ramifying and forever
Silent, beyond silence listened for.

A tale of two churches

Catholicism in transition

This is the weekend when Catholics in the United States begin to use the third edition of the English-language Roman missal, which makes several changes to the text of the mass. It is, overall, a more traditional translation, reinstating such things as the thrice-spoken “mea culpa” (rendered in English as, “through my fault, through my fault, through my most grievous fault”) and reverting to “And with your spirit” as the rendering of “Et cum spiritu tuo” (instead of the more mundane “And also with you”). Except for the hardcore ultramontanes who still pine for the old Tridentine mass in Latin, most conservative Catholics are gleeful, correctly seeing the new translation as further evidence of reactionary retrenchment in the Church—and a further diminution of the influence of Vatican II. Can veils for women be far behind?

In the past few months I have had occasion to step into two Catholic churches. (Before anyone asks, I will note that in neither case did anything shatter or burst into flames.) Both churches are modern constructions and had some notable features in common. In particular, they represented a big step back toward a more traditionally Catholic presentation, a far cry from the nearly featureless dark-paneled rectangular box that is St. Aloysius in Tulare.

I visited Our Lady of the Assumption on the occasion of a Portuguese festa in Turlock. The pastor's brother gave me a tour of the facilities. As someone old enough to have been an altar boy in the days of the Latin Mass, I have seen enough Church history to recognize a regression toward the mean. I told my guide that his brother's church represents a successful fusion of modern construction with traditional decor. My guide beamed, acknowledging that the Portuguese community in Turlock had aimed at that exact result when planning their church.

More recently I joined some family members at Holy Spirit Church in Fresno for the baptism of a nephew. The christening would follow the conclusion of the mass service, so I thought I was safe when I made a late arrival and loitered in the lobby. However, my eagle-eyed sister was too alert for me, noted my presence, and came out to collect me and take me inside. (As previously noted, no supernatural phenomena attended my entry into the sacred circle of mystical incantations and wafer transubstantiation.) The first thing I noticed was that Holy Spirit departs from the traditional parallel rows of pews in the same way as Our Lady of the Assumption. Unlike the Turlock church, however, the Fresno church has placed its crucifix so that it is invisible to those sitting in the side pews. From that perspective, where I was sitting with my sister's family, you might as well have been sitting in an Episcopal church. Holy Spirit's altar was a Protestant-compatible table and I'm sure the motley collection of art screens behind it provided ample peek-a-boo opportunities for the servers (both altar boys and altar girls at the service I attended).

The churches in Turlock and Fresno had another thing in common, and I regret not having any photographs to show you. Both of them have the Stations of the Cross (the “Via Dolorosa”) represented in mural form as a kind of frieze on the interior wall above the main entrance. In traditional churches, the fourteen Stations are usually wall plaques depicting the crucifixion of Jesus, seven of them equally spaced on the north wall and the other seven on the south wall (many old Catholic churches were preferentially oriented so that the altar was at the east end). The mural in Our Lady of the Assumption is dark and stark, graphically conveying the pain and anguish of the Savior's execution. I commented to my guide that it seemed more intense than some parishioners might prefer. He admitted that a few people in the community had lobbied to have the mural painted over after it had been unveiled, but that it was now generally accepted. The artist had had plans for other artwork in the interior of the church, but those had been shelved after the mural of the Stations of the Cross had been judged to sate the community's appetite for the artist's work.

By contrast, the Stations mural in Fresno's Holy Spirit is an exercise in kitsch, a truly unfortunate and distracting collection of excessively bright images in different sizes, cartoonish in conception and execution. The color palette appeared to be inspired by sidewalk chalk. If any venue cries out for disciplined and respectful depictions, I should think a church interior does. While the Our Lady of the Assumption mural pushed hard against the bounds of tradition in its display of angst (Jesus is amazingly serene in most of the crucifixion scenes in Stations of the Cross), the composition had a unity of purpose and conception. The Holy Spirit mural was a collage of disparate scenes united by garish colors and amateurish execution.

The results were occasionally unintentionally amusing (unless the artist was being deliberately subversive). The fifth Station depicts Simon of Cyrene, an innocent bystander, being impressed into service to help Jesus carry the cross lest the condemned prisoner die of exhaustion before the authorities get to nail him to it. The Holy Spirit mural makes it look as though Jesus is copping a feel of Simon's butt. In the tenth Station, Jesus is stripped of his garments. This scene in the Holy Spirit mural is so badly composed that it could be subtitled “Jesus flashes his Roman guards.” Both of the guards have stunned expressions on their faces, so they appear to be quite impressed. I made it through the service without chuckling aloud, but I suspect it looked like I was having a better time than the mass warranted.

It will take a few Sundays for practicing Catholics to work the kinks out of the new Roman missal, but I expect the complaints to be few. Regular mass-goers will quickly pick up on the changes and infrequent attendees (Easter and Christmas, anyone?) won't care. For former Catholics who outgrew religion and “put away childish things,” it's mostly a matter of curiosity and perhaps just a bit of nostalgia. The third edition of the Roman missal is yet another signpost that conservatives are in the ascendant in the Church, but we already knew that, didn't we?

Addendum

In searching the web for photos of the Turlock and Fresno churches, I ran into the following dyspeptic reaction to Our Lady of the Assumption, posted by someone who thinks highly enough of himself to use “St. Christopher” as his handle:

What madness! A Catholic Church that has mostly Portuguese Mass. Oh yes — a TLM [traditional Latin mass] thrown in, at the Chapel at odd times on Sunday. Having cultural loyalty is a fine thing, and Portuguese is a wonderful language — but this focus on whatever is prevalent (Klingon Mass, anyone?) obliterates the meaning of what the Mass is supposed to represent. There is no question but that the Church must return to Latin, and a single, uniform Order of the Mass, as soon as is possible. Let those that wish to participate in something else, go to something else.

Is there any chance that “St. Christopher” might consider taking his own advice? No one is making him attend a Portuguese-language mass. For my own part, however, I think it might be fun to attend a Klingon mass. Once, anyway.

Post-Addendum

The diligent searching of my friend Gene O'Pedia has uncovered a pair of on-line images of the Holy Spirit mural. The colors are more muted in the photos than they appeared to me in real life, but I recognize the compositions and can confirm that these are the Stations of the Cross that I saw in Fresno. Their resolution is not high enough to zoom in too closely on the panels of particular interest, but they can still convey a sense of what I was talking about. The first image depicts, right to left, Stations 6 and 7 (“Veronica wipes the face of Jesus” and “Jesus falls for the second time”). The flat perspective of Station 7 (not Station 5, as I said above) makes it look like Jesus is patting Simon on the behind. The other photo shows Stations 10 and 11 (again, right to left: “Jesus is stripped of his garments” and “Jesus is nailed to the cross”). Again, the resolution is limited, but you can just tell that the two Roman soldiers are gaping at the undraped Jesus in Station 10. It's a fine example of religious kitsch.

A moment's reflection

Another conceptual understanding problem

I gave my algebra students a pretty little problem involving the graphs of functions and their inverses. The prompt was fairly simple:

The graph of y = f(x) is shown in the figure. Use the graph to find the following function values and then sketch the graph of the inverse function y = f ⁻¹(x) on the same coordinate grid.

The student was asked to find the values of f(−3), f(1), f ⁻¹(2), and f ⁻¹(10). As you can see from the graph, I conveniently provided my students with several points highlighted on the graph. If one examines the point on the function curve where x = −3, it is fairly easy to discern that y must be 2. Hence f(−3) = 2. Similarly, f(1) = 10. It's elementary graph reading.

After reading the initial two function values, I expected my students to discover the method in my madness, noting that I'm asking them to figure out the value of the inverse function for the input values 2 and 10, which were the initial output. Since the inverse function, by definition, maps in the direction opposite that of the original function, it immediately follows that f ⁻¹(2) = −3 and f ⁻¹(10) = 1. What could be simpler?

Apparently, lots of things. Some of my students were quite irked:

“You didn't give us the function.”

“On the contrary. I certainly did. Its graph is right there before you.”

“No, I mean, you didn't give us the formula. We can't figure out the inverse function without the formula.”

“Leave that for a moment. Can you do the first part of the problem? Can you find the value of the original function at x = −3 and x = 1?”

“No, I already told you: You didn't give us a formula to plug into.”

“I recommend you try looking at the graph a little longer.”

In a few variations on the above theme, the querulous student suddenly lit up and rushed back to his or her desk to fill in the answers. In other cases, the student instead sat down, head shaking, and appeared to be muttering sotto voce imprecations at the instructor's expense.

Later, of course, when the exams were returned, I demonstrated what I had expected them to do. Since most of them had memorized the procedure for computing an inverse function—switch x and y in the formula y = f(x) and solve for y—they should have realized that the presence of the point (1, 10) on the graph of the original function implies the presence of (10, 1) on the graph of the inverse. Previously perplexed students rolled their eyes: “Oh, is that all? Why didn't you say so?”

I thought I did.

Hasta la vista, pendejo!

Telling more than they know

During the 6 o'clock hour on Friday morning, November 18, the KSFO talk-show hosts had some fun with the news item on the White House shooter. Babbler Brian Sussman and his trusty sidekick, “Officer” Vic, magnanimously agreed that it was important to protect the country's public officials (in stark contrast to their predecessors), but nevertheless found some cause for amusement.

Sussman:The media, though, has to really be bummed out. Because, okay, you look at the story, okay, think of this. He owns guns! All right?

Officer Vic:Yeah.

BS: He's from Idaho!

OV: Ah! That's two. We're getting close!

BS: He's a Christian!

OV: Oh! That's the big golden one right there.

BS: Oh, no, no, no. You really need a fourth one to really make this work.

OV: Yes.

BS: He needs to be white.

OV: Ah!

BS: Damn! His name is Ramiro Ortega Hernandez!

OV: Ah, darn it!

BS: He's Latin!

OV: Arrgh.

BS: We thought we had the perfect whitey. The bad Christian whitey from Idaho, who owned guns.

OV: They could even make him a tea-party guy!

BS: Oh, yes! Oh, we thought we had Idaho Whitey. The gun-owning man who's a Christian, who called Obama the Anti-Christ. But what's his name? What? His name's Ramiro Ortega Hernandez?

OV: Oh, no!

BS: Uh! Okay, wait—

OV: Can we anglicize it like we used to in baseball?

On the surface, of course, Sussman and his sidekick are simply mocking what they perceive as bias in the mainstream media (to which they apparently do not belong, despite being broadcast by a radio station that blankets the greater Bay Area). Without realizing it, though, they are making something else exceedingly clear: People with Hispanic surnames are automatically part of the constituency of the “mainstream” media. KSFO has no truck with such. Sussman and Vic draw the line of demarcation without a moment's hesitation.

And the right wing wonders why the damned Mexicans keep voting for the other guys.

I threw them a curve

Rote versus reason

Most math teachers would agree that we want two things from our students: (1) correct solutions to math problems and (2) an understanding of those solutions. Of course, some students are perfectly happy with mere technical facility: Please teach us the algorithm so that we can turn the crank on it, generate correct answers, get our college credit, and get the hell out of here. They balk when we probe for conceptual understanding. Other students, naturally, claim a profound knowledge of the conceptual underpinnings of the subject matter but lament their difficulty with the merely technical and computational aspects. Will the twain ever meet?

Course grades in math classes tend to be based mostly on the demonstrated ability to compute accurate results. It's more difficult to probe for evidence of their conceptual grasp. Occasionally, however, I give it the good old college try. Here's a graph I presented to one of my calculus classes. I asked my students to look at each of the points indicated by the red dots and make some judgments about the function and its first two derivatives.

My students had a little table to fill in. The instructions said, “Fill in the table, using +, –, 0, or DNE (for positive, negative, zero, and “does not exist,” respectively) for f(x), f ʹ(x), and f ʺ(x) at the indicated values of x.”

A small panic ensued. “Where's the formula for the function, Dr. Z?” “How can I compute derivatives if I don't have the formula?” I counseled them to calm down and consider that I wasn't asking for numerical values—yes, quibblers, except for 0—and that actual computations were unnecessary.

Consider, for example, the point corresponding to x = −1. The value of f(−1) is pretty clearly 5, hence positive. The point is also a local maximum, so a tangent line at that point would be horizontal; the slope is therefore 0 and that's the value of f ʹ(−1). Finally, the curve is concave down in the vicinity of a maximum, so f ʺ(−1) is necessarily negative.

No need to panic.

The trickiest case (if “tricky” is even the right word) is probably x = 3.2 (or thereabouts). It's approximately midway between a local maximum and a local minimum, suggesting that it must be at or near a point of inflection, where the concavity changes and the second derivative must be zero (or nonexistent). That takes a little discernment. In most cases, however, the answers should be evident to any first-year calculus student with a genuine understanding of the significance of the first and second derivative.

At the class's post-exam discussion of the results, the reviews for this problem were decidedly mixed. When pressed slightly, there was a grudging consensus that, “Oh, yes, it's clear now,” but my more computation-driven students remained unmollified. They preferred to demonstrate their differentiation chops on actual formulas using the rules they'd memorized.

The experience triggered an odd recollection with me. I remembered my grandfather at the dinner table, finishing off a meal my grandmother had prepared with a recipe she had never used before. She was eager for his verdict:

“Was it good?” she asked. “Did you like it?”

My grandfather nodded his head.

“Yes, thank you. It was very good. But don't make it again.”

A few of my students may despair, but I'm keeping that calculus problem in my recipe box.

Get on your knees

Amen!

Jeanne Phillips fumbles another easy one with a flaccid answer in today's installment of Dear Abby:

Dear Abby: My husband and I are not religious. We believe that people are entitled to their own beliefs. My problem lies with my brother-in-law and his wife. They are two of the most judgmental, sanctimonious people I have ever known. They “hate” (their word) Mormons, Catholics, etc. How would you suggest I respond to their criticism of our “lack” of Christianity and their offers to pray for us? —Biting My Tongue in Great Falls, Mont.

Dear Biting Your Tongue: If your relatives are an example of people who practice Christianity, heaven help the rest of us. If you must interact with them, practice selective deafness, and when they spout hatred, excuse yourselves.

Oh, Jeanne, “selective deafness” isn't going to work with these god-botherers. Otherwise they would have gotten the hint long ago that their religious babble isn't appreciated by the tongue-biter and her husband. By offering to pray for them, the self-righteous duo is setting up a perfect rope-a-dope situation. Seize the opportunity! For example, thus:

Dear Biting Your Tongue: Subtlety would be lost on your brother-in-law and his wife and direct confrontation could cause family strife you might prefer to avoid (though do discuss with your husband the possible advantages of being estranged from his brother and sister-in-law). Your best option is grateful acceptance of their offer to pray for you: “Oh, thank you! That is so considerate of you! You know that my husband and I aren't particularly religious, but it's clear that your faith is strong and in your hearts you're prepared to move mountains. You are welcome to pray for us as much as you want, but let's not speak of it again. We can patiently wait for your prayers to demonstrate their power.” Try to avoid a sarcastic tone while you say this. Keep it neutral. If they try to bring it up later, quash it quickly: “Oh, don't worry about it. I'm sure you're doing your best.” Repeat as necessary.

Garfield does math

And so do we

I always look at the comic sections of the newspapers I read, but I don't necessarily look at all of the comics. “Pearls Before Swine” always gets my attention, as does “Bizarro,” but others need to do something special to draw me in—like sprinkle their panels with numbers. “Garfield” did exactly that yesterday. (Is it true, as Stephan Pastis says, that cartoonists prefer to bury their weakest efforts in their Saturday strips?)

Everyone realizes, of course, that a giant mutant 98-year-old lady would be physically impossible, despite such earlier documentary evidence as Attack of the 50 Foot Woman. Galileo's square-cube law should have put that notion to rest (but Hollywood prefers to honor that law in the breach). But let's allow Garfield the same leeway that movie producers get. Let's accept that a giant 98-year-old lady is driving her 32-story 1965 Bonneville into town, threatening the entire community.

The 1965 Bonneville was a gigantic (in its own way) vehicle over 18 feet in length. Its height was about 4.5 feet (with allowances for tire pressure and passenger load). In the comic strip, the giant old lady's Bonneville is said to be scaled up to 32 stories in height. While architects are allowed quite a bit of variation in what constitutes a “story,” we can use 10 feet as a reasonable mid-range measure. In other words, the giant old lady's car is 320 feet tall, or (divide by 4.5) over 71 times as tall as a regular Bonneville. That's big.

And if your 98-year-old great-grandmother is five foot two, she'd be nearly 370 feet tall if she were scaled up to be the little old lady in the car.

Scary!

Now, about that turn-signal thing. Garfield says it's 16 feet tall (and blinking, of course). A look at the back end of a '65 Bonneville shows us that the rear lights were not quite half as tall as your basic license plate. If we call it 4 inches (being just a little generous—I don't have a Bonneville handy to actually measure), scaling it up by a factor of 71 results in 284 inches—or nearly 24 feet.

But Garfield said 16 feet. Oh, oh! But you know, that's probably good enough for the funny papers. Let's give him this one.