Iowa Results, Live

3:52 pm: You know what happens when you assume; I thought the Caucus was tomorrow, but as I was working on a companion piece, Prelude to Iowa I discovered that results were already coming in. So I’ll jump back and forth between the two posts. Prelude to Iowa will be published when there’s something complete enough to share.

Early Lead for Sanders: We already have some results as the good folks in Ottumwa caucused earlier today. The final tally was 9 for Bernie Sanders, 6 for Elizabeth Warren and 3 for Pete Buttigeig. Klobuchar and Yang had some support in the first alignment but neither was viable. M*A*S*H fans will remember Ottumwa as the home town of Radar O’Reilly. I’m sure Walter would be proud.

6:08 pm: It sounds like Amy Klobuchar has won a satellite caucus somewhere in Florida. If there’s a surprise tonight, it will be her beating expectations, but I don’t think she’ll break into the top tier, NY Times endorsement or no.

7:00 pm: Prelude to Iowa is now live. More to come, only “How do the Caucuses Work?” is done for now.

7:53 pm: “Beware of Paradoxical Results” has been added to Prelude to Iowa. The caucuses are set to start any minute, I’m going to start paying attention to the news coverage.

8:00 pm: Turnout sounds high, possibly 15% over last year? That would be good news for Sanders.

8:03 pm: Biden group looks tiny in Iowa City. Entrance polls indicate a 4 way race, Biden, Sanders, Warren and Buttigieg.

8:16 pm: Nicole Wallace just claimed that traditionally Democrats are married to substance over elect-ability. That seems wrong. Lots of candidates look strong in Des Moines. MSNBC is promising full first and final alignment results.

8:33 pm: In Debuque, Warren and Klobuchar missed viability at least on the first alignment. Buttigeig is ahead of everybody there.

8:50 pm: Klobuchar will probably beat expectations. Based on the buzz the surprise might be bigger than I thought.

There’s a woman on the tv now making a strong case for Warren.

Record turnout on a number of different places. Oldest group is looking smaller while the youngest group is getting bigger. All good news for Sanders.


The difference between the under 30 and over 65 groups is stunning.

9:18 pm: There seems to be a lot of Amy/Pete synergy.

9:23 pm: Pete looks like he’s a lot of people’s #2. Could that be enough to push him to the top?

9:31 pm: Pete and Amy kill it in Clive, IA. Biden and Warren still viable. Bernie didn’t make viability there.

9:50 pm: There’s very little data being released. In Cedar Falls only Sanders, Warren and Buttgeig are viable. The formula for assigning delegates appears to be really complicated.

This lack of data is getting kind of tedious. I wonder if they’re worried about the different narratives the three different sets of numbers will tell.

11:20 pm: Finally something is happening. Klobuchar took the stage. This is smart. If no one else does, it will get tons of airplay.

11:30 pm: The secretary from Iowa Precinct 1-1 could not get his smart phone app to work and has been on hold to the “hotline” for over two hours.

12:19 pm: It’s a shame that Elizabeth Warren’s speech was tape delayed; it’s probably the strongest one of the night.

It doesn’t look like we’ll be getting any results tonight. That’s not a good look for the Iowa Caucus; there’s already talk about whether on not there will even be a Caucus in four years. If the nomination comes down to delegates from Iowa, we’ll all be up to our asses in conspiracy theories.

It could be doubly delayed, Biden’s lawyer sent the Iowa Democratic Party a letter wanting to see the results before they are released. That’s not a good look for the Biden campaign, although without any data what-so-ever it may tell us all we need to know about how he did tonight.

Pete: “Iowa you have shocked the nation!” That made me laugh really hard. He’s the last major candidate to speak I think, and they all did okay, but Warren still wins.

Good night!

Image Credits:

  • Featured Image: Citizensharp [Public domain]

Star Trek: Picard – “Remembrance”

The first episode of Star Trek: Picard just premiered three days ago as I write this. There will be some spoilers, I won’t know how big they’ll be until I write them. You are warned.

I’ll prattle on a bit first to give some spoiler-free space before I get into Remembrance.

I’m not a fan of Star Trek: Discovery. I don’t hate it and they’ve done some good stuff, but as I said after the first episode, “This is not the Trek I’m looking for.” It’s a bit of a slog for me to get through the episodes. There was too much of a focus on war, but ultimately, it was the ethics that bothered me the most. I’m very much an old school Sci-Fi kind of guy and a fundamental part of that is that humanity is supposed to progress, to get better. The Federation and Starfleet have always represented that better version of humanity in Star Trek and even when they’ve lost their way, like in Insurrection, the plot revolved around the main characters setting it right. A first officer committing mutiny because she thinks she understands a situation better than her captain isn’t Trek. Even worse was the treatment of the tardigrade, which you have to compare to the Horta from the original series. In “Devil in the Dark,” Kirk, Spock, McCoy et. al. investigate a creature that’s killing miners on Janus VI. They take pains to understand the creature, discovering why it was attacking the miners and ultimately helping it. When the crew of Discovery encounters the tardigrade, they exploit the creature, even to the point of risking its life to basically make their ship go faster.

There are “reasons” and “context” for these actions and the Discovery crew eventually stop being horrible, but I think previous crews would have just dismissed the idea out of hand. Discovery got better by the second season, but it still leaves a bad taste. Like the first two Abramsverse movies, I’m left wondering how well the writers understand what makes “Star Trek” be Star Trek.

And here’s why I’m hopeful about this new series: it’s getting the ethics right. The universe is darker and there is some violence at the start of the episode that I found jarring in the context of a Star Trek episode, but the essential core is there. This was driven home in one particular moment for which I have to set the stage. Now-Admiral Picard has retired and is tending to the family vineyard in France. We’ve gotten some hints that his separation from Starfleet was contentious and we see that when he consents to an interview about the Romulan Supernova.

This event lead Jean Luc to step down from commanding the Enterprise in order to lead the rescue armada. Then, a group of synthetics attacked the colony on Mars, destroying the planet and the Utopia Planitia Shipyards where most Federation starships were constructed. Two subsequent decisions clearly bothered Jean Luc. The Federation banned synthetic humanoids and they canceled the rescue mission to Romulus. He resigned from Starfleet in protest, refusing to be complicit in the Federation turning its back on people in need. “The Federation understood that there were millions of lives at stake,” he told the interviewer. “Romulan lives,” she tried to clarify. “No, lives!” he replied. There it is. That is the core of Star Trek. We’re back to the classic situation, The Federation has lost its way and, in this case, it’s up to our titular character to put things right.

It is impressive how well constructed this episode is. The theme of memory is skillfully interwoven through the episode as one would expect for the pilot of a show built around a beloved character from a generation ago (not to mention an episode called “Remembrance.”).

There are also many, many so-called “Easter eggs.” You can check those out here.

And here. Thanks to Todd Egan and Forrest Meekins for the info and the links.

Really, calling these “Easter eggs” is an understatement. These are skillful callbacks to previous episodes and movies and they all point to what appears to be the central themes of the series; the rights of synthetic beings and the social evolution of the Romulans. Both were significant themes in TNG and they have philosophical and ethical heft. The callbacks to The Measure of a Man, one of the best and most significant episodes in all of Trek, were particularly acute. There’s an intriguing mystery developing involving both of these things and Jean Luc’s sense of right and wrong is right at the center.

One of the more intriguing allusions is the title of the stage-setting Short Trek “Children of Mars.” At first glance, the title seems straight forward; the story revolves around two school children who have parents on Mars. In rapid succession, we get to see their connection to those parents, a bit of their lives and then their devastated and devastating reactions as they watch as the synthetics’ attack on Mars unfolds in news reports. If you combine this with the fact that Romulus and Remus are not only the homeworlds of the Romulan Empire, but also the sons of Mars in Roman mythology, this short trek may be key to how the two primary threads of the series will weave together.

https://i.pinimg.com/736x/f0/4e/f3/f04ef3456d333bee3907dace0c0d4839.jpg

I think that a lot of the credit for the quality of this episode is due to Michael Chabon, Star Trek: Picard’s showrunner. I first encountered Chabon’s work 20 years ago when I read The Amazing Adventures of Kavalier and Clay. This novel centers around two young men who created a superhero back in the Golden Age of Comics, paralleling the story of Superman’s creators, Seigel and Schuster. The book is excellent; I’d recommend it to anyone with an interest in the genre. But, more than that, it is clearly a labor of love, very much like Star Trek: Picard. Chabon has been a fan of Star Trek since he was 10 years old and I expect that this new series will be a testament to how great new installments to venerable old properties can be when talented people who truly understand them are allowed to take the helm.

This now is the Trek I’ve been waiting for.

Bottom Line:

References:

Our Man Bones

The first time I visited Los Angeles, I went, like many visitors to Hollywood Boulevard. There were two things that I wanted to see. One was the signatures of the Star Trek cast at Grauman’s Chinese Theater. The other was DeForest Kelley’s star on the Walk of Fame. That second one took a while. Even after cries of “There’s Shatner’s star. Isn’t that good enough?” I persisted. This was the actor who played my favorite character in my all-time favorite teevee show. I wasn’t about to settle for second… or fourth… best.

If he were still around, DeForest Kelley would have turned 100 today. That warrants a tribute.

Here’s just a few things to mark the milestone.

A nice tribute from the people who worked with him.
Some interviews from around the time of Star Trek VI

And here’s “Remembering DeForest Kelley” from TrekMovie.com.

There were two other things I wanted to include in the tribute. The first was a website written by a fan who had gotten to know De in his final years. It really drove home just how kind and decent a person he had been. No luck tracking that down just yet, but if I ever come across it again, I’ll update this post.

The other was a tribute, also called ”Our Man Bones,” that I’d seen on UPN shortly after De passed away. It took a while to figure out where I’d actually seen it. I remembered the segment being part of an Anniversary Special, but the timing didn’t make any sense. As it turns out, it was from a special called “Ultimate Trek: Star Trek’s Greatest Moments.” Judging from the single clip that I can find on-line, it was pretty awful. Jason Alexander lampooned Kirk with other actors reprising Spock and McCoy. “Our Man Bones” is the only part I remember at all. It, I hope, was pretty good. I’m sure it ended strong, with the clip below from Balance of Terror. Powerful and moving given the circumstances, we’ll end with it ourselves.

“In this Galaxy…”

Quick Take: The Professor and the Madman

Joanne played this audiobook for me on our last long drive; something was vexing me and this true, compelling yarn proved to be the perfect tonic.

I wouldn’t normally warn about spoilers for a book first published in 1998, but I learned today that this book was made into a movie starring Mel Gibson and Brad Pitt which premiered this year (2019) in limited release. If you’re worried about it, there are some slight spoilers below; you can come back and read this after you’ve seen the movie or read the book.

The story starts dramatically with a murder, an unprecedentedly brutal homicide by the standards of London in the 1870s. Dr. William C. Minor, an American whose mental health had eroded since his service in the Civil War, shot George Merrett in the throat and then waited calmly for the police. It had been a terrible mistake he confessed; he was attempting to chase away a man who had broken into his room to torment him, a figment of his dementia. He was soon committed to the Broadmoor Criminal Lunatic Asylum, creating the circumstances where he would become one of the most prolific contributors to the creation of the Oxford English Dictionary.

The book proceeds, at least at first, as a bibiography, by which we understand a biography focused on two subjects. (Editors of the OED please note: this may be the first recorded use of this word as I have just now made it up. This is probably moot however as it doesn’t really seem to be a very good word after all; it sounds like a stutter and is too easily mistaken for ”bibliography.”) The second subject, the titular professor, is Dr. James Murray, who rose from humble beginnings to become the editor of the Oxford English Dictionary throughout the lion’s share of its creation.

Their stories and the interplay between the two men is enthralling, but ”The Professor and the Madman” is far more than a shared biography of two men. The author turns lovely phrases and paints vivid scenes. There is a fascinating account of the debate about whether the word ”protagonist” could ever be used in the plural. The story of the creation of the OED becomes a history of all English lexicography and is far more compelling than it has the right to be.

Like most biographies, the ending is bittersweet. Dr. Minor’s end is pitiful as his facilities continued to desert him. The end of Dr. Murray’s life is a study in unfinished business. He toiled on the OED until his passing just shy of his 80th birthday and did not live to see its completion. But at least there is the coda of the OED itself; completed and now well on its way to a third edition, it can stand as a monument to both of these men’s lives.

Thus, I strongly recommend the book. I can’t speak for the movie, but we plan to check it out at our earliest convenience.

Bottom Line:

References:

Winchester, S., The Professor and the Madman, Harper Collins, 1998.
https://itunes.apple.com/WebObjects/MZStore.woa/wa/viewBook?id=360606279

Quick Take: The Billion Dollar Boy

A few weeks ago, I was thinking back fondly about Heinlein’s juvenile novels. I used to read those all the time even once I aged out of the YA target audience. These hold up nicely. Not too long ago, Joanne and I enjoyed Starman Jones and Have Space Suit Will Travel as audiobooks on a long drive or three.

My nostalgia turned to the Jupiter Novels. In the mid to late 90’s, Tor books published this series as an homage to these works of Heinlein. I read a couple and they were pretty good. Maybe I’d like to reread a couple of those or track down the ones I hadn’t read, I thought.

By an odd coincidence or a creepy not-a-coincidence, if Facebook is eavesdropping on us, I soon received an e-mail from Arc Manor Publishers offering a free copy of the e-book of The Billion Dollar Boy by Charles Sheffield. This was one of the Jupiter Novels and it was one that, to the best of my recollection, I had not read.

And it was fine. After the first couple of pages, I knew in broad brushstrokes how the plot would unfold. The main character, a spoiled rich kid would, through some contrivance, end up in space. He’d be forced to earn his keep while he discovers amazing things and some dramatic stuff occurs. By the end, of course, he’s no longer an entitled jackass. Wikipedia tells me this is essentially the same plot as Captains Courageous.

But even with the predictability, this is a fun read. The story moves along quickly. The happenings are engaging, the conflict is exciting and the resolution is satisfying. The Billion Dollar Boy is nothing more or less than the science fiction equivalent of ordering comfort food in a restaurant. It’s not exactly what you grew up with and it’s fundamentally unchallenging but it’s reminiscent enough to be enjoyable.

Bottom Line:

A Flag for Elmira

We’ve settled into the new house enough that we’ve finally installed the flagpole. That was a non-trivial project; there’s nothing like a simple task to remind you of your total lack of aptitude for all things handy. This is the first time we’ve lived in the City of Elmira; we’ve lived in Horseheads and Southport and Newfield, but not within the city itself. To mark the occasion, the first flag that we’re flying from our new home is a proposed flag for the City that was designed by Alex Chichester. Last month, I unboxed the very flag that we’re flying and provided links to Alex’s story, about the flag, why he designed it and some of the community’s response to the flag. You can find all of that here.

The current city flag can be seen to the right and it’s nice enough. I asked about purchasing a copy of this flag from the city and I may yet buy one, but to be honest, it’s somewhat bland. The seal is a standard, city seal, but it’s the sort of seal that’s designed for the printed page. It works best on letterhead or in a book where you can look at it carefully and appreciate all of the detail.

Today, I thought I’d look at the proposed flag, which I’m inclined to call the Chichester Flag through the lens of the North American Vexillological Association’s five principles for good flag design as delineated in Good Flag, Bad Flag. Alex’s design is an objectively good flag. Here we go!

Principle 1. KEEP IT SIMPLE: The Flag should be so simple that a child can draw it from memory.

Here the Chichester Flag shines. The basic design a purple background with a blue horizontal band across the center. Many flags have shared this basic design, which could also be described as three horizontal stripes in two colors. Latvia, Austria, and Cambodia among others, share this basic design. The two joined gold rings as well as the choice of colors make the flag stand out as distinct from the other flags with similar designs. Simplicity is important and it’s no coincidence that NAVA lists this principle first as flags are meant to be seen from a distance and to either drape or to move in the wind. The Confederate States of America had three national flags in its four years of existence because the first two were easily confused with other flags.

A few weeks ago, while I was driving, I encountered a good illustration of this principle as the car in front of me had a small Canadian Flag sticker on its rear window and it was recognizable as a Canadian Flag even when it was many car lengths ahead of us. You can see this in the leftmost picture below. I pasted the official Elmira flag (middle) and the proposed flag (right) into the same photo and the difference is evident. The proposed flag is recognizable while the official flag could be many other things at this distance, such as the flag of Anchorage, Alaska or Honolulu, Hawaii.

Principle 2. USE MEANINGFUL SYMBOLISM: The flag’s images, colors, or patterns should relate to what it symbolizes.

The Chichester Flag has nice clear symbolism. The blue band represents the Chemung river which runs through the city. This is apt; the original settlement in the area was founded at the meeting of the Chemung River and the Newtown Creek. It was the completion of the Chemung Canal that connected the Chemung River to Seneca Lake and ultimately to the Erie Canal System. This allowed Elmira to become a regional center of manufacturing and shipping. There is little doubt that the Chemung River had a profound influence on the city and the community.

The two golden interlocking rings represent the uniting of the city’s north, south, east and west sides into a single community. This works both figuratively and concretely as the four sections of the rings that cross the blue band can be seen to represent the four driving bridges that cross the river within the city.

The colors of the flag are also significant. The Elmira College colors are purple and gold and those colors are ubiquitous on campus. The college, which sits in the heart of the city was founded in 1855, nine years before the city was incorporated from the village and part of the town of the same name. The new flag’s designer, Alex said, “I personally identify the city with the color purple,… It’s probably a lot to do with Elmira College. Also, there’s purple and golden wildflowers all over town.”

The wildflower rationale is frequently echoed in stories about why the college chose purple and gold as their school colors, although those stories usually invoke irises, the school flower. The other reason the college usually shares as an inspiration for their choice of colors is that purple and gold were among the colors of the Women’s Suffrage Movement (you can read a bit more about that here). This reason might be even more salient. Elmira had significant links to the women’s movement and this includes the college itself, which was the first college in the U. S. to offer degrees to women that were equivalent to those that were being offered to men.

Elmira also played an important role in the abolitionist movement and the Underground Railroad. When we consider this along with the ties to the women’s movement, the fact that the purple portion of the flag evokes an equals sign (=), the flag elicits all of this history.

Principle 3. USE 2 TO 3 BASIC COLORS: Limit the number of colors on the flag to three, which contrast well and come from the standard color set.

There’s very little that needs to be said here; the flag contains three colors that contrast nicely. A useful measure of this is whether the flag remains recognizable and attractive when rendered in grayscale. It does. The black and white version remains both striking and distinctive.

As an interesting side note, the NAVA manual defines the basic color set as “red, blue, green, black, yellow and white” and states that other colors “are seldom needed in a good design.” At least part of the reason is that “flag fabric comes in a relatively limited number of colors.” I wonder if this is still true; it seems to me that printed flags have become more common and easily obtained in the 14 years since the manual was written. Either way, the connections between Elmira and the color purple are significant enough to warrant its use.

Principle 4. NO LETTERING OR SEALS: Never use writing of any kind or an organization’s seal.

This is strongly related to the “keep it simple” principle. Seals are difficult to see at a distance and text on printed flags appears reversed on the back, making it difficult to read. To make the text readable on both sides dramatically increases the cost of the flag. The Chichester flag has neither text nor a seal.

Principle 5. BE DISTINCTIVE OR BE RELATED: Avoid duplicating other flags, but use similarities to show connections.

[Iroquois Confederacy flag]

The Chichester flag can claim both of these characteristics. The color choices and the interlocking rings set the flag apart from other flags with similar designs making it distinctive while it shares some similarities to related flags. The similarities to the Elmira College flag, shown above, are evident. The gold rings can also be seen as an homage to the current Elmira flag as they share its color and evoke its main motif, the circular seal. In a nice coincidence, the purple color refers back to the flag of the Iroquois Confederacy, who inhabited this area of North America before the arrival of European settlers.

Final Words:

This was an enjoyable project and a nice inaugural flag for our new home. I’d never ordered a custom flag before, but the process was easy and straightforward. It’s easy enough that I will probably order more custom flags in the future, when I’m interested in flying something that isn’t easily available.

It is also been exciting to be able to fly a flag that, as far as I know, has never been flown before. That’s been a great deal of fun and I want to thank Alex for allowing me to use his design.

And speaking of Alex, I hope that his campaign to have his flag become the official flag of the City of Elmira is successful. It’s a beautiful flag and his arguments about using the flag to invigorate the community and brand the city deserve careful consideration. If you’re interested in such things, please check out his videos. I think you’ll find them compelling.

References:

Image Credits:

Tom Foolery

Charlie Jacobson shared an article from Slate yesterday: Tom Lehrer at 90: a life of scientific satire, honoring Tom Lehrer on the occasion of his 90th birthday. It reminded me of just how much I enjoy Mr. Lehrer’s work. It also reminded me that the last time I subjected a class to one of his songs, (We discuss arithmetic in other bases in Cryptography, so we listen to “New Math“) I resolved to write my first ever fan letter. Thing is, the article above is from April 2018, so Mr. Lehrer is now a bit over 91 and a half. To quote the great man himself, “I believe that if any songs are going to come out of World War III, we had better start writing them now.”

Mr. Lehrer is a genius! I believe that I purchased a copy of “That Was The Year That Was” while I was in junior high school and promptly wore it out. I would make a point of listening to Dr. Demento every Sunday hoping to hear his stuff. It was a delightful surprise when I realized that Lehrer also write “Silent E” which I had loved from the Electric Company back in second or third grade. I still find myself humming that song nearly a half-century later.

I often say he’s the only mathematician I consider a role model and when I told a class last week that by becoming a professor, I’d figured out how to stay in college for the rest of my life, I was intentionally riffing on Lehrer’s “attempt to extend adolescence beyond all previous bounds”. He’s probably one of the biggest influences on my sense of humor, such as it is, which I suppose isn’t a very nice thing to say, but there you go.

The Slate article reminds us of Lehrer’s quote about Henry Kissinger. The first time I’d heard it, I thought it was that “All other forms of political satire became obsolete when Henry Kissinger was awarded the Nobel Peace Prize” and that’s stuck with me even though I’ve tried and I haven’t been able to verify that version of the quote. It’s always struck me as much more interesting than the version that I usually see, which is merely to say that political satire has become obsolete. But “politics has become too nutty to satirize” is an easy joke, while the other version is profound. To expand on it: “Henry Kissinger has won the Nobel a Peace Prize after being instrumental in the bombing of Cambodia and other horrible things. The only way that could possibly make sense is if the award itself was actually an act of satire. And it is such a perfect and succinct bit of satire that we’ve clearly reached the apex of the art form. No one else should attempt political satire because everything else will seem empty and futile in comparison. Political satire is obsolete.” It may just be headcanon, but that’s what I choose to believe Mr. Lehrer actually said.

The Slate article continues “…and in 2002 he remarked, still less optimistically: ‘Things I once thought were funny are scary now. I often feel like a resident of Pompeii who has been asked for some humorous comments on lava.’” Brilliant.

It’s worth the investment of time to track down all of Mr. Lehrer’s songs, but for those of you who are just getting started, here are a few of my favorites.

We Will All Go Together When We Go:

That’s Mathematics:

Silent “e.”

Wernher Von Braun:

I Got It From Agnes

Who’s Next:

Coming Flag-tractions

In a couple of weeks, we’ll be moving into the city of Elmira. We’ve lived in the area for some time, but never actually in Elmira itself. To mark the occasion, we wanted an appropriate flag to fly for the first month or so in our new home.

Meanwhile, a young man named Alex Chichester has designed a new flag for the city and he’s been promoting it. It’s a lovely design and it deserves a chance to become our new flag. We decided to fly his flag when we moved to the city. I contacted Alex and he graciously allowed me to have a copy made. After speaking with a number of nice people, I decided to work with a company called Asley who produced a lot of the flags we’ve flown at home for the last year or so. The new flag finally arrived yesterday. Here’s our first look at the finished product.

I’ll have more to say when I start flying the flag, but if you’re interested, here are two of Alex’s videos regarding his flag. He promotes this better than I can.

And if this has piqued your interest in flags and vexillology, here’s an excellent TED Talk on city flags and flag design.

More soon (ish).

Factoring Quadratics

A couple of weeks ago I covered factoring in my College Algebra class, which we’ll mainly use for working with quadratic equations. This always makes me think about a method for factoring quadratics that I’d never seen before moving back to New York state. Years ago, I looked it up on what I think was an NYS Department of Education website which called it the “AC method” for factoring quadratics. When I tried to find this again, I only found it on YouTube called “slide and divide” and the “Berry Method.” Both of those names seem unnecessarily arcane to me, but for this post, I’ll call this process “slide and divide.” The references to an “AC Method” that I say today were actually talking about the “grouping number method” which we’ll come back to.

I hope that the fact that it’s become difficult to find references to what I think of as the AC Method is evidence that it’s going away. It needs to.

When I was in high school, I remember that factoring quadratics that looked like ax^2 +bx + c was pretty straight forward when a = 1, but when a was different from 1 it got trickier. The only tool we were taught was trial and error, but you need to develop some intuition to use trial and error efficiently. Both the grouping number method and slide and divide attempt to give students a systematic process that builds on the experience of the a = 1 case to help them with the a \neq 1 case. If you’re a mathematician, and you’ve seen the slide and divide approach, you probably found it horrifying. I know I did. It goes something like this: say you want to factor the following polynomial.

6 x^2 - x - 12

You begin by replacing the leading coefficient with 1 while replacing the constant term with the product of itself and the leading coefficient.

x^2 - x - 72

Why do we do this? I’m not sure it’s ever made clear. But this is a lot easier to factor since the leading coefficient is one. We just need to find two factors of -72 that add up to the coefficient of the middle term, -1, namely -9 and 8 . These become the constant terms of the factors

(x-9)(x+8)

We then replace each x with 6x

(6x-9)(6x+8)

…and then remove the common factor from each binomial.

(2x-3)(3x+4)

With that, our quadratic polynomial is factored.

The single redeeming feature of this process is that it actually works, but think about these steps. Where did the 6 in front of the x^2 go? Why does it make sense to multiply it into the -12? For that matter, why do the 6xs magically reappear? And why is it okay to just cancel the common factors from the penultimate step? If I thought about it for a moment, I bet I’d actually have even more questions.

I’ve had a few students who wanted to use this method to factor, but none of them was ever able to explain why it works. That’s the ultimate problem with this method. It’s a list of meaningless steps that, once forgotten, will be gone forever because it doesn’t attach to any understanding. If you teach mathematics, it’s incumbent upon you to make your material meaningful to your students. Some people naively believe that this means you have to teach mathematics in a context that is directly relevant to the students’ interest, but in my opinion, that’s not true. You need to help them understand what’s happening and why it’s being done. Each step should have a clear and understandable rationale that helps to drive the process forward. Slide and divide does none of these things.

This process is not irredeemable, however. It is possible to work slide and divide in a way that gives it these characteristics. Let’s go through this in our original example.

6 x^2 - x - 12

The idea behind this method is that, if we can rewrite the polynomial with a leading coefficient of one, it will be easier to factor. To do this we need the leading term to be a perfect square. We can make that happen by multiplying the polynomial by 6. Because we don’t want to change the polynomial, we’ll divide by 6 at the same time.

= \frac{36 x^2 - 6x - 72}{6}

A simple substitution will give us a leading coefficient of 1. We define u = 6x.

= \frac{u^2 - u - 72}{6}

Just like we did above, to factor our numerator we need two factors of -72 that add up to -1, that’s still -9 and 8 .

= \frac{(u - 9)(u+8)}{6}

To get back to expressions that involve xs, we undo the substitution. That makes it clear why the 6xs return.

= \frac{(6x - 9)(6x+8)}{6}

And now, we’d like to get rid of the 6s since we don’t need them any longer. Where did the 6 in the numerator go? It’s inside the common factors of the two binomials. We can factor those out…

= \frac{3(2x - 3)2(3x+4)}{6}

…and they clearly cancel with the 6 in the denominator. That leaves us with our answer.

= (2x - 3)(3x+4)

That’s pretty cumbersome and so slide and divide leads you to a no-win-scenario. The quick way doesn’t engender understanding while the more rigorous approach isn’t quick.

So, what should we teach instead? The grouping number method that I mentioned earlier. To set it up, let’s think about what happens in the following polynomial multiplication.

(4x-1)(2x-3)

=2x(4x-1)-3(4x-1)

=8x^2 -2x -12x +3

=8x^2 -14x +3

Normally, you’d just skip the second step, but it’s important to realize that what you’re really doing is using the distributive property twice. Additionally, let’s think about the -12 and the -2. It’s clear that they add up to the final coefficient of x, but it’s also true that their product (-12)(-2) = 24 is the same as the product of the leading coefficient and the constant term, (8)(3). Notice this has to be true. In either case, it’s the product of 4, -1, 2 and -3, merely in a different order.

So, how do we go in the other direction? That is, how do we factor instead of multiply? We’ll demonstrate the grouping number method on the polynomial 8x^2 - 14x + 3.

The grouping number of a quadratic polynomial is the leading coefficient multiplied by the constant term.

So, in this case, it would be 8 \times 3 = 24.

Next, you want to look for two factors of your grouping number that add up to the coefficient of x. In this case, (-12)(-2) = 24 and (-12)+(-2) = -14. We use these two numbers to rewrite the middle term to get the following.

8x^2 - 14x + 3 = 8x^2 - 2x - 12x + 3.

In other words, to rewrite the polynomial in this way, you find two factors of your grouping number that add up to the coefficient of x and use those to break down the middle term.

Why rewrite the polynomial in this way? Because it sets us up perfectly to factor by grouping.

Factoring by grouping is a technique for factoring a polynomial with four terms. In a nutshell, to factor by grouping, you remove the greatest common factor from the first two terms, then remove the greatest common factor from the last two terms. If the resulting binomial factors are the same you can factor this out to get the product of two binomials. Notice, that’s using the distributive property twice, just in the opposite direction as before.

Therefore, to finish factoring the polynomial we can factor by grouping.

8x^2 - 14x + 3

= 8x^2 - 2x - 12x + 3

= 2x(4x - 1) - 3(4x - 1)

= (4x - 1)(2x - 3)

Notice this exactly reverses the steps of the multiplication with which we started.

In short, as long as you have no common factors, the quadratic polynomials where the grouping number can be factored into two numbers that add up to the middle coefficient are exactly the ones that can be factored by grouping. All the others are prime.

The Serapis Flag

The Serapis Flag

There are many variations of the United States Flag that were used in the early days of the republic. One of the more interesting is the Serapis Flag which we are now flying. Its existence is intimately tied to the Battle of Flamborough Head fought 240 years ago 0n 23 September 1779.

John Paul Jones

The story of the Serapis flag begins with the story of John Paul, a ship captain. Paul fled his native Scotland after killing a crewman who mutinied over wages, presumably refusing to trust his fate to an Admiralty Court. Paul emigrated to the colonies, renamed himself “John Paul Jones” and befriended Benjamin Franklin. He joined the Continental Navy, rose through the ranks and became the United States’ first famous naval commander. He is sometimes called the Father of the U. S. Navy. By 1779, Jones was in command of the Bonhomme Richard, named in honor Franklin’s pseudonym from Poor Richard’s Almanac.

The Serapis vs. the Bonhomme Richard

Bonhomme Richard was a former merchant ship that had been armed and upgraded to a ship of war. It led, under Jones’ command, a small squadron of ships. Near Flamborough Head, Yorkshire, this squadron encountered a British convoy under the protection of the HMS Serapis. Conflict ensued and Bonhomme Richard took severe damage from the better armed Serapis. When the British captain asked if the Richard had struck her colors (meaning that they had lowered their flag as a sign of surrender) Jones is said to have replied with the famous “I have not yet begun to fight!” The flag, in fact, had been destroyed in the battle.

Eventually, the tide of the battle turned and Jones managed to lash Bonhomme Richard to the Serapis. The American crew was able to board and capture the British ship while their own ship, badly damaged and on fire, sank into the North Sea.

But what does this have to do with the Serapis Flag? Only this. Jones, now in command of the Serapis, put into a neutral Dutch port for repairs. British officials accused Jones of being a pirate and demanded his arrest. He was, after all, sailing a captured ship which was not flying the colors of any known nation. Bonhomme Richard’s flag, remember, had been blown into the sea. A flag was hastily created for Jones’ ship and entered into the Dutch records, allowing the Dutch to officially recognize the captured ship.

The Serapis Flag was created according to a description of United States flag that was provided by Benjamin Franklin, who was then Ambassador to France.

“It is with pleasure that we acquaint your excellency that the flag of the United States of America consists of thirteen stripes, alternately red, white, and blue; a small square in the upper angle, next the flagstaff, is a blue field, with thirteen white stars, denoting a new constellation.”

The Revolutionary Diplomatic Correspondence of the United States

And thus, this slightly garbled recounting of the Flag Resolution of 1777 is responsible for the distinctive look of the Serapis Flag. Because of this, the flag is also known as the Franklin Flag. It remains popular today and is frequently used in historical displays because of its uniqueness and its recognizably American character.

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