The values of the minimum and maximum radii involve cube roots; they can be found by extremizing

*xy*(8 −

*x*−

*y*) subject to the constraint (

*x*− 4)(

*y*− 4)(

*x*+

*y*− 4) = ¼.

Here again are all the triangles with area 1 and perimeter 8, this time shown with their circumcircles.

The values of the minimum and maximum radii involve cube roots; they can be found by extremizing*xy*(8 − *x* − *y*) subject to the constraint (*x* − 4)(*y* − 4)(*x* + *y* − 4) = ¼.

The values of the minimum and maximum radii involve cube roots; they can be found by extremizing

I started following

Oral arguments happened on Tuesday. I've read contradictory tea-leaves analysis (1, 2, 3). I do wonder, though, if Roberts on Tuesday viewed Kennedy as a lost cause and sought to use the questioning time for purposes of damage control. Roberts emphasized the policy risks of a decision for the plaintiffs…might this convince Kennedy of the need to write as narrowly as possible in their favor? Or perhaps the difficulty of setting a manageable standard will deter Kennedy one last time from striking down a legislative map on the basis of partisanship—even a map like Wisconsin's that all nine justices would probably stipulate is corrupt. The Supreme Court might well end up adopting the view of the dissenting judge in the district court, who wrote, "I am … unable to conclude that Act 43's passage was anything other than the kind of “politics as usual” that courts have routinely either tolerated or acquiesced in."

- News outlets usually describe the Wisconsin case as a dispute between Republicans and Democrats. However, the parties in
*Gill v. Whitford*aren't the Republicans and the Democrats; the parties are the State and the citizens who are seeking relief. When Wisconsin legislators drew the challenged congressional districts, they clearly had party in mind; yet they weren't on party business. They were enacting law with the force of state government. Likewise, though the plaintiffs in the case belong to the party out of power, they bring their case as citizens alleging that the State has infringed their individual Constitutional rights. The more you can ignore the media's horse-race coverage, the better you will understand the arguments in the case.

- Kennedy is interested in gerrymandering as a First Amendment issue. To a layperson like me who tends to think of the speech part of the First Amendment as a "free-to-be-you-and-me" guarantee, this has been confusing. However, by putting together bits and pieces of Kennedy's previous decisions (as quoted in the district opinion), it seems that the First Amendment is supposed to function not only as a liberty guarantee, but also as an essential mechanism for democracy. It's as if the important speech from the Constitutional perspective is political speech. The give-and-take of public debate on the candidates and issues of the day is how we come to agree on what ought to be done or not done by government and who should represent us in that activity. And similarly, the kind of "free association" that matters for our system of government is particularly the getting together to have such discussions and debates. Perhaps this explains why Kennedy believes that corrupt districting is a First Amendment issue: it threatens to render coming together pointless, and threatens to remove from the occupiers of the legislature the burden of accounting for their behavior to the people at large in the public forum.

Amendment I

Congress shall make no law respecting an establishment of religion, or prohibiting the free exercise thereof; or abridging the freedom of speech, or of the press; or the right of the people peaceably to assemble, and to petition the government for a redress of grievances.

A key passage in the district court opinion (p. 56) summarizes a long process of sifting through relevant Supreme Court decisions, in which there was often little consensus. The takeaway is a proposed standard for deciding political gerrymandering cases:

"the First Amendment and the Equal Protection clause prohibit a redistricting scheme which (1) is intended to place a severe impediment on the effectiveness of the votes of individual citizens on the basis of their political affiliation, (2) has that effect, and (3) cannot be justified on other, legitimate legislative grounds."

Amplifying the notion of intent in (1), a footnote says it is "an intent to make the political system systematically unresponsive to a particular segment of the voters based on their political preference", for example by creating safe seats and/or by making for an entrenched minority party.

For (2), the efficiency gap (EG) is used as evidence, not as proof or a standard; and not as the only evidence: it is also important that two elections have happened with the map in question. (The "entrenching" of power highlighted by Kennedy in previous decisions has something to do with duration. Also important to avoid hypotheticals.)

For (3), it is noted that that the maps went through many drafts, each one consistent with legitimate principles, yet each one better for the party in power than the one before—making legitimate principles no justification for the final map. Also, while the defense of the state's "natural Republican geography" is qualitatively valid, it is not quantitatively enough to justify the map.

(A footnote…In Zimblog-related news, Judge Ripple's written decision makes correct use of the abbreviations

This year I watched all four of the major King Kong movies:

On with the show:

**Oscar-Kong!**

The award for **Best Visual Effects** goes to 2017's *Kong: Skull Island*. The effects in this movie were uniformly excellent. In fact, the final fight in *Skull Island* might be the best monster battle in the history of monster movies.

(Last place goes to the 1976*King Kong*, basically a movie about a guy in a gorilla suit.)

**Best Actor in a Leading Role**. Duh—it's Kong! But which one? There are good arguments for the 2005 Kong (utterly convincing as a primate) and for the 2017 Kong (utterly convincing as a monster). But I give the award to 1933 Kong, because even with primitive special effects he manages to evoke our sympathy.

**Best Actor in a Supporting Role**. Samuel L. Jackson gave a committed performance in *Kong: Skull Island*, as did John C. Reilly. But the Oscar goes to Charles Grodin in the 1976 *King Kong*, because he was the only actor in the film who seemed to be in on the joke. Always great to watch Charles Grodin camping it up.

**Best Actress**. Fay Wray (1907–2004) was a Hollywood icon (an obituary is here). She starred in many films, but the role of Ann Darrow made her a legend. In his review of the 1933 *King Kong*, Roger Ebert recounts an amusing anecdote:

"At a Hollywood party in 1972, I saw Hugh Hefner introduced to Fay Wray. 'I loved your movie,' he told her. 'Which one?' she asked."

There's only one Fay Wray. But the award for Best Actress in a King Kong movie goes to…

Naomi Watts. Watts is the 2005 film's center of gravity. She rightly has top billing in the film. She honors the historic role while making it hers. Watts has skills that few actors in any era possess. Her physicality and facial acting deliver the film's beauty-and-beast narrative, which is primary in the 2005 retelling. (Watts also has a good partner in Kong himself, played naturalistically by Andy Serkis via technology.)

**Best Supporting Actress**. There is no supporting actress in a King Kong movie.

**Best Cinematography**. The 2005 *King Kong*.

**Best Director**. The directors in 2017 and 1976 didn't seem sure of what they were trying to do. The award goes to Peter Jackson, whose vision is fully realized in the 2005 King Kong.

**Best Original Song**. The award goes to Max Steiner (1888–1971) for "Aboriginal Sacrifice Dance," the music that accompanies the Skull Island marriage ritual in the 1933 *King Kong*. Steiner was a Hollywood musical legend who scored countless films including *Casablanca*, *Gone with the Wind*, and *The Searchers*. Steiner's Kong-song is both completely inauthentic and completely groovy, a complicated verdict that artist Rico Gatson might agree with: his 2001 video installation "Jungle Jungle" draws upon footage from the 1933 ritual sequence. (I watched the loop many times when I came across it playing in an art museum a dozen years ago.) Peter Jackson added a nice touch to his 2005 film when he used Steiner's music during the "Eighth Wonder of the World" sequence in New York.

**Best Original Story**. Naturally this has to be the 1933 *King Kong*. Most movies that get made and re-made over the decades tell stories from literature: *Dracula*, *Frankenstein*, *Hamlet*, *Pride and Prejudice*, *Romeo and Juliet*. King Kong is the only perennial movie I can think of that tells a story that was originally developed specifically for the screen.

**Best Picture**. This isn't close. Peter Jackson's 2005 *King Kong* is not only the best Kong movie, it's a wonderful movie in any genre. Jackson's *King Kong* is a vintage pleasure that can only be described with the language of a 1930s movie poster. It's got romance! Adventure! Thrill and chills! I watched it last of the four movies, because freeing up three-plus hours to watch it wasn't easy. But once I settled in, I appreciated its luxurious storytelling. (Admittedly, some of the set-pieces went on a bit long.)

A few highlights that show various elements of film-making working together at a high level:

**King Kong Scorecard!**

*King Kong*(1933)*King Kong*(1976)*King Kong*(2005)*Kong: Skull Island*(2017)

- First are my Oscar picks: this is the way I would hand out selected Oscars if the competition were only open to King Kong movies.
- After revealing the Oscar picks, I present my "King Kong Scorecard." Think of this as the way the Oscars might have been organized had they
*only ever been open*to King Kong movies.

On with the show:

POW! |

(Last place goes to the 1976

Suave, right?? |

"At a Hollywood party in 1972, I saw Hugh Hefner introduced to Fay Wray. 'I loved your movie,' he told her. 'Which one?' she asked."

There's only one Fay Wray. But the award for Best Actress in a King Kong movie goes to…

Naomi Watts. Watts is the 2005 film's center of gravity. She rightly has top billing in the film. She honors the historic role while making it hers. Watts has skills that few actors in any era possess. Her physicality and facial acting deliver the film's beauty-and-beast narrative, which is primary in the 2005 retelling. (Watts also has a good partner in Kong himself, played naturalistically by Andy Serkis via technology.)

Naomi Watts in King Kong (2005) |

From "Jungle Jungle" (2001), by Rico Gatson |

- The transporting Art Deco aesthetic in the opening titles, New York sets, costumes, and of course the Empire State Building itself.
- The steady current of homage to the original 1933 movie, including a conversation in a taxi in which director Carl Denham asks if Fay Wray is available to star in his movie; a restored "insect pit" sequence (the sequence along these lines was cut from the 1933 movie when it horrified audiences); and the Max Steiner music noted earlier.
- The nightmarish encounter with the Skull Islanders. And here let's pause to note that the basic Kong myth, um, raises issues. In 2017, the makers of
*Kong: Skull Island*moved to table such questions, by their handling of the Skull Islanders and by heavily de-emphasizing the beauty-and-the-beast arc. But the 2005*King Kong*handles the touchiest issues by doubling down on them. For example, King Kong's "thing for blondes" is handled with tongue in cheek while Kong searches for Ann in the streets of New York. And the Skull Islanders in the 2005 film are the furthest thing you could imagine from a politically correct ideal of aboriginal peoples. There was no clichÃ©, however; I was shocked, and gripped, by Jackson's nightmare vision of an ecstatically murderous society clinging to a landscape of barren crags. (Do they know that the other side of the island looks like Malibu?) - The lovely Central Park "ice-skating" sequence, an inspired addition to the traditional story.
- The Depression-era framing, which created atmosphere and added depth to Ann's character.
- The winsome sequence when Ann Darrow charms Kong on Skull Island.

1933 | 1976 | 2005 | 2017 | |
---|---|---|---|---|

Pre-island buildup / sense of going on a voyage | A | B | A | A |

Mysterious island / pulp-era delights & horrors | A | C | A | A |

"Beauty and the Beast" theme/pathos | B | B | A | C |

New York mayhem | A | A | A | F |

Special effects quality | C | C | A | A |

—From

In politics, I think people are best served by plain writing that is dense with testable claims and publicly verifiable facts. Writers do try to "energize the base," and in campaigns there is a place for that. But few stylists are capable of elevated prose about live political issues. The more energetic a piece of political writing, the worse that piece of writing probably is.

And the less convincing it probably is. ClichÃ©s can make a writer sound like somebody on a rant—like a loud TV you just want to turn off. Extra words subtract gravity from what you're saying. Stock phrases cover up the fact that no argument is being made.

Enough preamble—let's see if the following examples speak for themselves. In every case, I think that striking words would have made for a more effective statement. Sources for the original statements are listed at the end.

Weaker: I have no swastika or Third Reich related tattoos. PERIOD.

Stronger: I have no swastika or Third Reich related tattoos.

Weaker: But there wasn’t a shred of evidence that any insurer had “abused” the boy or his mom.

Stronger: But there was no evidence that any insurer had abused the boy or his mom.

Weaker: Jessie Ford, who’s like this great sociologist at NYU, …

Stronger: Jessie Ford, a great sociologist at NYU, …

Weaker: If we recognize men and women who identify with the genders they were assigned at birth (cisgender) and we recognize men and women who do not identify with their assigned gender (transgender), then surely we agree this difference is worth recording.

Stronger: If we recognize men and women who identify with the genders they were assigned at birth (cisgender) and we recognize men and women who do not identify with their assigned gender (transgender), then this difference is worth recording.

I may gradually add more examples to the list over time. Here are the sources of the original statements, in order:

From an earlier post:

*Amy multiplied an eight-syllable number by an eight-syllable number and obtained a four-syllable number. What could her numbers have been?*

*(These are whole numbers we're talking about.)*

To solve this, I first reflected that an eight-syllable number is necessarily pretty large…therefore the four-syllable number must be pretty large…but what kinds of large numbers only have four syllables? Probably something like "twenty trillion." A number like that ends with a lot of zeros, which means that it contains a lot of factors of 10, or in other words a lot of 2s and 5s.

So, I tried putting powers of 2 against powers of 5, and pretty soon I hit upon the following fact:

This is (6-syllable) × (6-syllable) = (4-syllable). Not far from Amy's problem! To patch it up, I added "thousand" to both factors; doing so increases the syllable count in each factor by two, but doesn't change the syllable count of the answer (because "sixteen thousand" becomes "sixteen trillion").

Now we have (8-syllable) × (8-syllable) = (4-syllable), as desired. Amy's numbers could have been 128,000 and 125,000.

Using a computer, I also found some interesting cases:

1) The smallest instance of Amy's numbers that I could find was 1,120 × 6,250 = 7,000,000.

2) Some extreme versions of the puzzle, with answers.

3) The four-syllable number 9 septillion can be written as a product of two 13-syllable numbers in three different ways:

4) The four-syllable number 1,000,000,002 can be written as a product of two 11-syllable numbers in two different ways:

That's it for Amy's puzzle! In a later post, I'll share my code for generating number names. Once you have the number name, counting syllables is easy; but converting a digit string into a string of words is harder. UPDATE 9/21/2017: Instead of sharing my code, here is a webpage with number-naming code in many programming languages.

To solve this, I first reflected that an eight-syllable number is necessarily pretty large…therefore the four-syllable number must be pretty large…but what kinds of large numbers only have four syllables? Probably something like "twenty trillion." A number like that ends with a lot of zeros, which means that it contains a lot of factors of 10, or in other words a lot of 2s and 5s.

So, I tried putting powers of 2 against powers of 5, and pretty soon I hit upon the following fact:

128 × 125 = 16,000

This is (6-syllable) × (6-syllable) = (4-syllable). Not far from Amy's problem! To patch it up, I added "thousand" to both factors; doing so increases the syllable count in each factor by two, but doesn't change the syllable count of the answer (because "sixteen thousand" becomes "sixteen trillion").

128,000 × 125,000 = 16,000,000,000

Now we have (8-syllable) × (8-syllable) = (4-syllable), as desired. Amy's numbers could have been 128,000 and 125,000.

***

Using a computer, I also found some interesting cases:

1) The smallest instance of Amy's numbers that I could find was 1,120 × 6,250 = 7,000,000.

2) Some extreme versions of the puzzle, with answers.

Amy multiplied a 22-syllable number by a 22-syllable number and obtained a 4-syllable number. What could her numbers have been?

6,103,515,625 × 1,474,560,000,000,000,000,000 = 9 nonillion

Amy multiplied a 26-syllable number by a 69-syllable number and obtained a 4-syllable number. What could her numbers have been?

2,147,483,648 × 1,396,983,861,923,217,773,437,500 = 3 decillion

3) The four-syllable number 9 septillion can be written as a product of two 13-syllable numbers in three different ways:

9 septillion = 156,250 × 57 quintillion, 600 quadrillion

9 septillion = 7,812,500 × 1 quintillion, 152 quadrillion

9 septillion = 39,062,500 × 230 quadrillion, 400 trillion

4) The four-syllable number 1,000,000,002 can be written as a product of two 11-syllable numbers in two different ways:

1,000,000,002 = 11,829 × 84,538

1,000,000,002 = 23,658 × 42,269

That's it for Amy's puzzle! In a later post, I'll share my code for generating number names. Once you have the number name, counting syllables is easy; but converting a digit string into a string of words is harder. UPDATE 9/21/2017: Instead of sharing my code, here is a webpage with number-naming code in many programming languages.

Watch carefully and you'll see the straight lines both glowing green at the moment when the blue area leads the red area by the greatest amount. Were you in the ballpark?

Here is a video in MP4 format. And here is one-page walkthrough of the Calculus for the general case where ¼ is replaced by a general parameter Î».

At about which point in this movie would you say that the numerical difference between the blue and red areas is greatest?

A video file that you can play, pause, rewind, and download is here. One way to share an answer would be to pause the video at the moment of your best guess, take a screenshot, and email it to me at zimblogzimblog@gmail.com.

I sometimes wonder if studying Calculus or working in a Calculus-heavy field makes people any better at perception tasks like this. I do believe that teaching kinematics for so many years improved my direct perception of acceleration; being able to see a couple of derivatives down has helped me to avoid some erratic drivers over the years.

In case any Calculus-trained readers would like to analyze the problem symbolically, I'll give the formulas I used to make the animation:

- For any value of
*m*≥ 0, the blue region is defined by*x*≥ 0,*y*≥ 0,*y*≤*m**x*, and*y*≤*x*e^{−x}. - For any value of
*m*≥ 0, the red region is defined by*x*≤ 0,*y*≤ 0,*y*≥ ¼*m**x*, and*y*≥*x*e^{x}.

You should find a simple answer for

For some extra symbol pushing, replace

You may recall this puzzle from an earlier post on this blog (also in *Word Games 3*):

(Answer here.)

Today it occurred to me that we could make a version of this puzzle using multiplication in place of subtraction. Here's a puzzle along those lines:

(These are whole numbers we're talking about.) I found one solution by hand, which I'll share in my next post along with any other solutions I receive.

Amy's (8, 8, 4) scenario isn't the only interesting case to look at, so feel free to share any "near misses" too, such as

Amy subtracted a three-syllable number from a three-syllable number and obtained a thirty-seven syllable number. What could her numbers have been?

Today it occurred to me that we could make a version of this puzzle using multiplication in place of subtraction. Here's a puzzle along those lines:

Amy multiplied an eight-syllable number by an eight-syllable number and obtained a four-syllable number. What could her numbers have been?

(These are whole numbers we're talking about.) I found one solution by hand, which I'll share in my next post along with any other solutions I receive.

Amy's (8, 8, 4) scenario isn't the only interesting case to look at, so feel free to share any "near misses" too, such as

(7-syllable number) × (5-syllable number) = (3-syllable number),

(6-syllable number) × (6-syllable number) = (4-syllable number),

etc.

I just felt like looking at them.

There is a phase of the animation where the base and height both appear to remain constant over time, but that is only approximately the case. (I think if you had a rigorously constant base, then because the perimeter is constant, the third vertex would have to trace part of an ellipse, not a horizontal line. So the base and height can't both remain constant over time.)

Here's a problem that came home in my daughter's backpack:

Let's consider the first question first.*How many trays will be in each stack?*

I found the answer 24 by starting down the road of calculating 97 ÷ 4 far enough to see that result would be "24 and some stuff." (Too late, I remembered that I actually knew 4 × 24 = 96 offhand—it was still a good check on my answer.)

Thus my approach was to pursue, just far enough, a strategy of "divide and round down." If you work with computers, then you probably know that the operation "divide and round down" equals a computer command known as "DIV." For example, a computer will tell you that 16 DIV 3 = 5, and one way to check this answer would be to calculate 16 ÷ 3 = 5⅓ and round the result down.

I doubt this is how a computer actually calculates the value of 16 DIV 3, but "divide and round down" gives the right answer and shows the relationship between DIV and ÷, which is what I want to write about today.

A few more examples just to clarify how DIV works:

32 DIV 7 = 4

54 DIV 6 = 9

3 DIV 4 = 0.

A friendly term for the DIV operation might be "arraying." Ebenezer is trying to*array* 97 things in a 4-by-something array. It's to be understood that when you are arraying things, the equal groups are to be as large as possible, and there might be some articles left over. I think the word "arraying" has a discrete connotation that helps convey its meaning. We array pieces of silverware when we set the table, but we don't "array" the milk when we pour some into each child's cup. (Update 9/4/17: Maybe you could think of DIV as a "packaging up" operation. That phrase avoids the spatial connotations of arraying. For example, if an elevator fits 6 people and there are 20 people who need to get to the roof, then it is a packaging up operation more than an arraying operation. Anyway.)

One nice thing DIV enables us to do is to write down a formula for remainders. For example, if we know that 97 DIV 4 = 24, then we can multiply the 24 by 4 (result: 96) and then what's left over is the remainder; in symbols,

More generally, we have the following formula for remainders in a whole-number problem:

There is a second well known computer operation, called MOD, which finds the remainder; so we could write

*m* = *n *× (*m* DIV *n*) + *m* MOD *n*.

*r* = *s* × (*r* ÷ *s*).

Notice there's no remainder here; "division with remainder" is a contradiction in terms.

Of course, the relationship*r* = *s* × (*r* ÷ *s*) is simpler only if you understand rational numbers (or at least fractions). The curriculum in late elementary school often doesn't do enough in that respect, leaving students in middle school stuck with cumbersome (or even faulty) whole-number ideas about operations.

Naturally there are many word problems in which arraying (rather than division) is called for—because a cup of flour can be partitioned any which way, but lunch trays, marbles, wheelchairs, and teachers can't be partitioned and remain lunch trays, marbles, wheelchairs, or teachers. So in a word problem, the context determines the best answer to give—whether this be the exact quotient of two given numbers, or the whole-number part of the quotient, the quotient rounded to the next greatest whole number, or even the remainder. Getting the right answer to such problems is better thought of as a modeling skill than as a calculation skill.

In fact, I suspect that the existing curriculum spends too much time having kids practice bare whole-number division problems with remainders: that confuses the subject of arithmetic by giving division two incompatible meanings. Of course, I would give students plenty of distributed/repetitive practice with multi-digit long division; I'd just set things up so that in all those practice problems, either the divisor divides the dividend evenly, or else the answer is to be expressed correctly as a fraction or decimal. (So that, in both cases, the true relationship*r* = *s* × (*r* ÷ *s*) holds. You should always be able to check a quotient by simply multiplying.) Meanwhile, situations with remainders would be prevalent in the word problems students work on.

Perhaps that approach is simplistic, but certainly I'll say that I was appalled once to see my daughter working on a giant drill worksheet with dozens problems like this:

My daughter was terribly confused at first. "Isn't 5 divided by 4 equal to five-fourths?" she asked. I answered her by saying, "Yes it is. You are right. But for tonight, just write what they want." I didn't know what else to say. That worksheet penalizes students who have correctly learned basic fraction concepts and division concepts—concepts which, unlike whole-number concepts, are essential for the transition to algebra.

What a terrible and unnecessary conflict between ideas. There is a need for practicing calculations like 8,995 ÷ 7 = 1,285 so by all means, provide distributed/repetitive practice with those kinds of problems. But it can't be educational for students to rehearse statements like "5 ÷ 4 = 1 R 1," which lie about the division symbol in ways that grade-level students can detect.

I'll close with some better news, specifically in reference to this related earlier post about division. On a recent Saturday School I was elated to see, in a series of workbooks written in 2001 by my colleague Marsha Stanton, this page:

This is the best case I've seen so far of using a true equation to represent objects arrayed in equal groups with some objects left over. This would be worth doing as a way to represent word problems, as I wrote in the earlier post (one-page PDF at tinyurl.com/notdivision).

Ebenezer has 97 lunch trays. He will make four stacks of lunch trays. He will put the same number of trays in each stack. He will put as many trays as possible in each stack.If Ebenezer does this, how many trays will be in each stack? How many trays will be left over?

Let's consider the first question first.

I found the answer 24 by starting down the road of calculating 97 ÷ 4 far enough to see that result would be "24 and some stuff." (Too late, I remembered that I actually knew 4 × 24 = 96 offhand—it was still a good check on my answer.)

Thus my approach was to pursue, just far enough, a strategy of "divide and round down." If you work with computers, then you probably know that the operation "divide and round down" equals a computer command known as "DIV." For example, a computer will tell you that 16 DIV 3 = 5, and one way to check this answer would be to calculate 16 ÷ 3 = 5⅓ and round the result down.

I doubt this is how a computer actually calculates the value of 16 DIV 3, but "divide and round down" gives the right answer and shows the relationship between DIV and ÷, which is what I want to write about today.

A few more examples just to clarify how DIV works:

32 DIV 7 = 4

54 DIV 6 = 9

3 DIV 4 = 0.

A friendly term for the DIV operation might be "arraying." Ebenezer is trying to

One nice thing DIV enables us to do is to write down a formula for remainders. For example, if we know that 97 DIV 4 = 24, then we can multiply the 24 by 4 (result: 96) and then what's left over is the remainder; in symbols,

remainder = 97 − [4 × (97 DIV 4)].

More generally, we have the following formula for remainders in a whole-number problem:

remainder = *m* − [*n *× (*m* DIV *n*)].

remainder = *m* MOD *n*

*m* MOD *n* = *m* − [*n *× (*m* DIV *n*)].

or using the previous expression,

Rearranging this, we can say for any integer *m* and any nonzero integer *n*,

This is the whole-number analogue of the much simpler relationship for rational numbers *r* and *s* ≠ 0,

Notice there's no remainder here; "division with remainder" is a contradiction in terms.

Of course, the relationship

In fact, I suspect that the existing curriculum spends too much time having kids practice bare whole-number division problems with remainders: that confuses the subject of arithmetic by giving division two incompatible meanings. Of course, I would give students plenty of distributed/repetitive practice with multi-digit long division; I'd just set things up so that in all those practice problems, either the divisor divides the dividend evenly, or else the answer is to be expressed correctly as a fraction or decimal. (So that, in both cases, the true relationship

Perhaps that approach is simplistic, but certainly I'll say that I was appalled once to see my daughter working on a giant drill worksheet with dozens problems like this:

5 ÷ 4 = 1 R 1.

My daughter was terribly confused at first. "Isn't 5 divided by 4 equal to five-fourths?" she asked. I answered her by saying, "Yes it is. You are right. But for tonight, just write what they want." I didn't know what else to say. That worksheet penalizes students who have correctly learned basic fraction concepts and division concepts—concepts which, unlike whole-number concepts, are essential for the transition to algebra.

What a terrible and unnecessary conflict between ideas. There is a need for practicing calculations like 8,995 ÷ 7 = 1,285 so by all means, provide distributed/repetitive practice with those kinds of problems. But it can't be educational for students to rehearse statements like "5 ÷ 4 = 1 R 1," which lie about the division symbol in ways that grade-level students can detect.

I'll close with some better news, specifically in reference to this related earlier post about division. On a recent Saturday School I was elated to see, in a series of workbooks written in 2001 by my colleague Marsha Stanton, this page:

This is the best case I've seen so far of using a true equation to represent objects arrayed in equal groups with some objects left over. This would be worth doing as a way to represent word problems, as I wrote in the earlier post (one-page PDF at tinyurl.com/notdivision).

In the Riddle of the Mad King, we shared approximately 68% of the king's fortune with one son and shared the remainder with the other son, so as to simultaneously (1) preserve the king's bragging rights and (2) anger one of the sons.

What if there were three sons? Or many sons?

This raises the question of how to break 1 into parts*x* > *y* > *z* > … so that the least of the ratios 1/*x*, *x*/*y*, *y*/*z*, … is as large as possible.

For the case of two sons, we maximized the lesser of the two ratios by making both ratios equal. Let's assume that circumstance generalizes. If we give*n* sons the respective amounts *x*_{1}, *x*_{2}, …, *x*_{n−1}, 1 − *x*_{1} − *x*_{2} − … − *x*_{n−1}, then some algebra shows that we'll have equality of the ratios

if*x*_{1} is the unique root of the equation *r*^{n + 1} − 2*r* + 1 = 0 in the range 0 < *r* < 1. Once you know *x*_{1}, you can easily find the rest of the shares by repeatedly applying the same ratio 1/*x*_{1} that the king uses.

For example, if there are three sons, then solve*r*^{4} − 2*r* + 1 = 0 for *r* ≈ 0.544, which is the first son's share. The second son's share is the square of this number, and the third son's share is the cube. Numerically the shares come out to be, approximately,

*x*_{1}: 54%
*x*_{2}: 30%
*x*_{3}: 16%

Here is a graph for the case of three sons that generalizes the graph of the objective function *f*(*x*) from last time; here the shading is darker where Min{1/*x*, *x*/*y*, *y*/(1 − *x* − *y*)} is greater—except that the small white dot shows the solution *x*_{1} = 0.5436…, *x*_{2} = 0.2955….

What if there are infinitely many sons? You don't need any algebra to make all the ratios equal in this case, as long as you happen to know that

^{1}/_{2} + ^{1}/_{4} + ^{1}/_{8} + ^{1}/_{16} + … = 1.

What if there were three sons? Or many sons?

This raises the question of how to break 1 into parts

For the case of two sons, we maximized the lesser of the two ratios by making both ratios equal. Let's assume that circumstance generalizes. If we give

1/*x*_{1} = *x*_{1}/*x*_{2} = *x*_{2}/*x*_{3} = … = *x*_{n−2}/*x*_{n−1} = *x*_{n−1}/(1 − *x*_{1} − … − *x*_{n−1})

if

For example, if there are three sons, then solve

What if there are infinitely many sons? You don't need any algebra to make all the ratios equal in this case, as long as you happen to know that

We can take advantage of this fact to solve the problem in the case of infinitely many sons. Let's line everybody up: first the king, followed by an infinite lineup of his sons. The king starts by giving all of his money to the son on his left—who then gives half to the brother on *his *left, who then gives half of *that *to the brother on *his *left, and so on. In the limit, each son has twice as much as his neighbor, and the king's original fortune is twice what the richest son has.

My wife's initial recommendation was to pass two-thirds of the fortune to one son and pass one-third of the fortune to the other son. That way, you sow unrest because one son gets twice as much as the other son; and at the same time, you protect the king's legacy because the king's fortune is one and a half times what the richer son's fortune will be. Likewise, most readers' recommendations landed in the 60%–70% range. (I also received some Machiavellian political calculation and clever recommendations for trust arrangements.)

Looking in particular at my wife's proposal of two-thirds/one-third, a person might observe that if the king cares equally about the two considerations in the problem (the legacy aspect and the unrest aspect), then he might be unsatisfied with the two-thirds/one-third solution—because the legacy ratio (1.5) isn't as great as the unrest ratio (2). If we care equally about the legacy ratio and the unrest ratio, shouldn't we seek a solution in which the two ratios are the same?

So if 1 represents the entirety of the king's fortune, and *x* is the fraction given to the son who gets more, and 1 − *x* is the fraction given to the son who gets less, then let's arrange matters so that the legacy ratio 1/*x* and the unrest ratio *x*/(1 − *x*) are the same:

1/*x* = *x*/(1 − *x*).

There is one positive solution to this equation, and it's the reciprocal of the golden ratio Ï†:

(Making the comparison in the opposite sense, the ratio of the king's fortune to the larger share is the golden ratio: 1/*x* = 1/(1/Ï†) = Ï†.)

So by this analysis, we give one son about 62% of the fortune and give the other son about 38% of it.

So by this analysis, we give one son about 62% of the fortune and give the other son about 38% of it.

The golden ratio Ï† = ½(Sqrt[5] + 1) = 1.61803… is famous from architecture and the sciences. You can read about it on Wikipedia, or buy Mario Livio's book *The Golden Ratio: The Story of Phi, the World's Most Astonishing Number*. (I haven't read it.)

This diagram from Wikipedia shows why Ï† enters into our problem:

In the context of our problem, the two sons' shares are *a* and *b*, the king's fortune is *a* + *b*, and the total amount will be divided in the golden ratio when (*a* + *b*)/*a* = *a*/*b*; that is, when the total fortune is to the larger share as the larger share is to the smaller.

Reader jeff came up with this same answer, using a spreadsheet rather than an equation:

Reader jeff came up with this same answer, using a spreadsheet rather than an equation:

A 50/50 split would maximize Wealthiest but it would minimize Discord. A 100/0 split would maximize Discord but it would minimize Wealthiest. The midpoint of 75/25, therefore, seems like the right choice BUT it doesn't feel right. We're trying to find out the best possible difference between the king's wealth and the younger son's wealth while also finding the best possible difference between the younger's wealth and the elder's wealth.

I created a spreadsheet (table below) to do the calculations for me and came up with 61.8% for the younger son and 38.2% for the elder. That way the mad king's wealth is 1.618 times greater than his younger son's inherited wealth and the younger son's wealth is 1.618 times greater than his older brother's.

Large Share Small Share WealthDiscord(King/ Large) (Large/ Small) 0.75 0.25 1.333 3 0.7 0.3 1.429 2.333 0.667 0.333 1.499 2.003 0.64 0.36 1.563 1.778 0.63 0.37 1.587 1.703 0.623 0.377 1.605 1.653 0.622 0.378 1.608 1.646 0.621 0.379 1.61 1.639 0.62 0.38 1.613 1.632 0.619 0.381 1.616 1.625 0.6180.3821.6181.6180.617 0.383 1.621 1.611 0.616 0.384 1.623 1.604 0.615 0.385 1.626 1.597 0.614 0.386 1.629 1.591 0.613 0.387 1.631 1.584

This leads to the golden ratio because column 1 of the spreadsheet contains values of

***

But maybe we don't care about the two ratios equally? When I gave this riddle to my daughter, she said that one son should get ^{501}/_{1000} of the fortune while the other son should get ^{499}/_{1000} of the fortune. To this suggestion I replied, "That's helpful for making the king the richest person ever, but it doesn't sound like very much unrest to me." Her answer: "If my sister got even a little more than me, I'd be furious."

Well OK then!

***

Another way to approach the problem is to define an objective function and maximize it. For example, take the objective function to be

Maximizing *f*(*x*) means setting things up so that the lesser of the two ratios is as large as it can possibly be. This is a kind of "minimize the worst case" approach, typical of min-max problems.

If you graph *f*(*x*) on the interval ½ < *x* < 1 (see the heavy blue curve in the image below), you can see that the maximum value occurs where 1/*x* = *x*/(1 − *x*), that is, where both ratios equal the golden ratio Ï†. Now we can tell the king not only that Ï† yields equal legacy and unrest ratios, but also that any other approach would have led to one of the ratios being smaller than it is.

Will the king go for it? Maybe, maybe not. Perhaps, like my daughter, the king has a different take on sibling rivalry.

A mad king, nearing death, wanted to pass his entire fortune to his two sons. The king wanted history to record him as the richest man who ever lived; therefore, he wanted neither son to receive too much of his fortune. The king also wanted to sow unrest in the kingdom; therefore, he wanted one son to receive substantially more than the other.

As the king's counsellor, what fraction of his fortune would you recommend giving to each heir in order to fulfill his wishes?

Can you give a rationale according to which yours is the best possible recommendation?

As the king's counsellor, what fraction of his fortune would you recommend giving to each heir in order to fulfill his wishes?

Can you give a rationale according to which yours is the best possible recommendation?

1.

2.

3.

5.

6.

7.

8.

Other answers are possible. These follow my usual conventions of avoiding proper nouns, foreign words, slang, contractions, hyphenated words, acronyms, and jargon—although

I hope you like this doodle! It is a sad face that becomes happy.

The motion in the image is projectile motion. There are 500 small particles, each following its own trajectory under the influence of gravity—as if you threw a great handful of pebbles in the air. I chose the particles' initial positions, initial speeds, and initial launch angles so that they would form a sad face at the beginning and a happy face at the end.

Here is a version without color, so you can see the physics a little better.

How it's made:

1) Import a raster image of a happy face and a raster image of a sad face.

2) Simplify the image so that every pixel is either black or white.

3) Select a random sample of 500 black pixels from the happy face and 500 black pixels from the sad face.

4) Use a linear mapping to convert pixel row/column locations into spatial

5) For each initial/final location pair \((x_0, y_0)\) and \((x_1, y_1)\), there is one projectile-motion trajectory that occupies point \((x_0, y_0)\) at time \(t = 0\) and occupies point \((x_1, y_1)\) at time \(t = T\). This trajectory has equations of motion

\[x = v_0\cos\theta\, t\]

\[y = v_0\sin\theta\, t - \frac{1}{2}g t^2\]

where

\[x = x_1 - x_0\]

\[y = y_1 - y_0\],

\[\theta = \cos^{-1}\left(1+\left(\frac{y+\frac{1}{2}gT^2}{x}\right)^2\right)^{-\frac{1}{2}}\]

and

\[v_0 = \frac{x}{T}\left(1+\left(\frac{y+\frac{1}{2}gT^2}{x}\right)^2\right)^{\frac{1}{2}}\]

Now all you have to do is animate the 500 pebbles as they follow the specified trajectories from their initial locations to their final locations. The reason the happy face assembles itself is that the value of \(T\) is the same for all of the pebbles. In the animation, the time starts a little before zero and ends a little after \(T\).

My family and I are headed West to see the eclipse. If you're near the path of totality, here's hoping you have clear weather!

Occasionally I like to see what's going on with HD 101584, a supergiant star several thousand light-years away. He's visible from Earth perhaps, on the darkest of nights, although I haven't looked for him. I know that he shines through an exhaled cloud, and that he wobbles in time to the movements of an invisible companion. We met a long time ago—he's like a friend who lives across the country, one I never visit yet think of with pleasure.

I was introduced to HD 101584 one winter almost thirty years ago, when I spent a month in Ohio apprenticing to Nancy D. Morrison, an astronomer at the University of Toledo. It wasn't known then that the star was part of a binary system, and in fact it counted as scientific progress when Dr. Morrison and I determined that the shell of gas around the star was expanding at a speed of approximately 48 kilometers per second. Our calculation was based on spectroscopic data that Dr. Morrison had collected in Cerro Tololo, Chile, the site of a large international observatory. Chile has many good places to build a telescope—elevated regions with clear nights and good seeing conditions. Some weather stations in the Atacama Desert, north of Cerro Tololo, have yet to record a single rainfall in their entire time of operation.

The University of Toledo campus is located in an attractive part of Toledo, far from the dystopic*StraÃŸenbauprojekte-Welt* that drivers experience while skirting the city on I-280. I arrived for work in January of 1989 and was surprised to remember that I'd visited the campus once before. In my junior or senior year of high school, the university had brought me down from Michigan as part of a program to recruit talented science students. The professors and the facilities had looked solid, and there would clearly be generous financial aid, but I didn't end up applying to the school, and it seems I forgot the entire episode until I set foot there again.

Dr. Morrison spent a lot of time in the beginning showing me how to use the software we'd be using, and she taught me the stellar physics I would need to understand in order to work on the project. I found her to be a kind, supportive, and somewhat formal person who treated scholarship and scientific accuracy with the gravity they deserve. Her example influenced my developing scientific and scholarly attitudes. Neither of us drew out the other very much, although I felt I could glimpse a little of Nancy the person in the car she drove (a vintage Volvo station wagon) and in the adventurous tastes in food that she revealed when we ate together. She was, I believe, the first woman ever to receive a Ph.D. from the eminent astronomy department of the University of Hawaii. Today Dr. Morrison holds many titles, including former Director of the Ritter Observatory, Fellow of the American Association for the Advancement of Science, and member of the Executive Committee of the American Astronomical Society.

If you read far enough down the list of refereed publications in Dr. Morrison's*curriculum vitae*, you'll eventually come to "Unexpected Effects of a Trap in CCD Echelle Spectra of B-Type Stars," which appeared in *Publications of the Astronomical Society of the Pacific* in 1990. I helped with the research for that article during a return visit to Toledo over the summer that followed the original HD 101584 project. That summer, my commute from Michigan to Toledo was 62 miles each way; to make it possible, my dad gave me the use of his 1974 Corolla Deluxe station wagon, a workhorse with a quarter-million miles on the odometer. The wagon format made the car an excellent mobile locker-room. In those days I could play basketball, especially street ball, and if ever I drove past a playground and saw a good pickup-game happening, I'd pull over, change clothes in the car, and get on a team. In Toledo the car sat in a sunny parking lot all day, and its plasticky heat felt good to me after eight hours of reducing data in an air-conditioned room—we used mainframes back then!

In the years that followed my visits to Toledo, physics and mathematics rather than astronomy increasingly consumed my energy, and Nancy and I fell out of touch. She probably doesn't know how often I think about her and that time of my life. I'll let her know.

The basic data for HD 101584 can be seen here or in this database; some recent findings about the object are here: 1, 2.

I was introduced to HD 101584 one winter almost thirty years ago, when I spent a month in Ohio apprenticing to Nancy D. Morrison, an astronomer at the University of Toledo. It wasn't known then that the star was part of a binary system, and in fact it counted as scientific progress when Dr. Morrison and I determined that the shell of gas around the star was expanding at a speed of approximately 48 kilometers per second. Our calculation was based on spectroscopic data that Dr. Morrison had collected in Cerro Tololo, Chile, the site of a large international observatory. Chile has many good places to build a telescope—elevated regions with clear nights and good seeing conditions. Some weather stations in the Atacama Desert, north of Cerro Tololo, have yet to record a single rainfall in their entire time of operation.

The University of Toledo campus is located in an attractive part of Toledo, far from the dystopic

Dr. Morrison spent a lot of time in the beginning showing me how to use the software we'd be using, and she taught me the stellar physics I would need to understand in order to work on the project. I found her to be a kind, supportive, and somewhat formal person who treated scholarship and scientific accuracy with the gravity they deserve. Her example influenced my developing scientific and scholarly attitudes. Neither of us drew out the other very much, although I felt I could glimpse a little of Nancy the person in the car she drove (a vintage Volvo station wagon) and in the adventurous tastes in food that she revealed when we ate together. She was, I believe, the first woman ever to receive a Ph.D. from the eminent astronomy department of the University of Hawaii. Today Dr. Morrison holds many titles, including former Director of the Ritter Observatory, Fellow of the American Association for the Advancement of Science, and member of the Executive Committee of the American Astronomical Society.

If you read far enough down the list of refereed publications in Dr. Morrison's

In the years that followed my visits to Toledo, physics and mathematics rather than astronomy increasingly consumed my energy, and Nancy and I fell out of touch. She probably doesn't know how often I think about her and that time of my life. I'll let her know.

The basic data for HD 101584 can be seen here or in this database; some recent findings about the object are here: 1, 2.

I never expected you to vote for Hillary Clinton, and I appreciated when you didn't vote for Donald Trump either. I don't expect you to agree with me about taxes, health care, the Supreme Court, or foreign policy. After all, someday I could change my own mind about those things—I don't need my friends to reflect my own views back at me, and I learn from discussion and debate.

And I know that the political news offers you plenty of fat targets to criticize, like identity politics and political correctness. I don't think you're wrong to criticize those things.

But as your friend, I want you to know that I'm starting to blame *you* for Donald Trump.

I'd been feeling this way more and more, but the tragic weekend in Charlottesville has called the question.

The President is unfit to lead. What will you do about it? Keep your head down for the next three years?

I hope not, because there are things that only a Republican can do.

First, a small thing: build a movement among conservative legislators to require President Trump to release his tax returns. I'm not a conspiracy theorist and don't have particular ideas about what might be found in the returns, but Trump's refusal to release them was a thumb in the eye of our politics. It was a signal that Trump wasn't going to submit to oversight of any kind. Your party let him get away with it. Correct your mistake now.

Second, start laying groundwork for a presidential primary challenge in 2020. Encourage a great Republican candidate to run; donate early; join the team as an advisor. (Let me know who you're backing, and I'll donate too.) A primary challenger would likely lose, but a primary challenger would also give a voice to true Republicanism while Trump is destroying it. Yes, a primary increases the chances that a Democrat will win the general election; but it's a mistake to see a second Trump administration as a victory for Republicans. It's the opposite—a point of no return for the party.

Maybe there are other things you can do—maybe there are things you're already doing to put your house in order. I hope so, because the time of choosing is here. You can't sit this one out.

Nobody who venerates a Confederate flag is an American patriot. The Confederates of the 1860s were traitors against the United States of America. A patriot doesn't wave the flag of an enemy nation.

Making war in defense of slavery was a sin in the eyes of a just God. That sin still haunts the South.

All men are created equal. The descendants of the enslaved are Americans in the fullest sense. Statues to Confederate generals are an affront to the descendants of the enslaved. Tear all the statues down—or leave them be as a reminder; but if a town government has made a decision that the statues will go, then have the decency to stay home and reflect, rather than marching to defend the indefensible.

Neo-Nazis who stage a rally should be protested—vigorously, and non-violently. Punching a neo-Nazi is neither morally defensible nor politically wise. "All men are created equal" also applies to neo-Nazis. Mock them, outflank them, neutralize them, defeat them in this decade and the next. Fantasies about eradicating them forever are just that, fantasies—and disturbing ones at that.

Above all: vote.

***

Update 9/11/2017: Here is Andrew Sullivan on the Confederate statues—I link to this because it's consonant with the above and because it includes an interesting comparison to British history (Sullivan is an American citizen but British by birth). The first part of the post is about a different issue, so CTRL-F "***" to get to the break.

Update 9/15/2017: Here is Jay Nordlinger, a senior editor of*National Review*, with "Seeing the Confederacy Clear."

Making war in defense of slavery was a sin in the eyes of a just God. That sin still haunts the South.

All men are created equal. The descendants of the enslaved are Americans in the fullest sense. Statues to Confederate generals are an affront to the descendants of the enslaved. Tear all the statues down—or leave them be as a reminder; but if a town government has made a decision that the statues will go, then have the decency to stay home and reflect, rather than marching to defend the indefensible.

Neo-Nazis who stage a rally should be protested—vigorously, and non-violently. Punching a neo-Nazi is neither morally defensible nor politically wise. "All men are created equal" also applies to neo-Nazis. Mock them, outflank them, neutralize them, defeat them in this decade and the next. Fantasies about eradicating them forever are just that, fantasies—and disturbing ones at that.

Above all: vote.

***

Update 9/11/2017: Here is Andrew Sullivan on the Confederate statues—I link to this because it's consonant with the above and because it includes an interesting comparison to British history (Sullivan is an American citizen but British by birth). The first part of the post is about a different issue, so CTRL-F "***" to get to the break.

Update 9/15/2017: Here is Jay Nordlinger, a senior editor of

In each puzzle below, nine letters are provided. Use the nine letters to form three words of exactly three letters each. (Use all nine letters, and don't use any of the nine letters more than once.)

1.**H P D L M Y J A I**

2.**U O Z M P E J G I**

3.**N F I W X Y U H M**

4. **I Y O J T V X P U**

5.**A U N V K H T M E**

6.**B K P O T J C I U**

Extra challenge: in each puzzle below, use the twelve letters provided to form four words of exactly three letters each. (Use all twelve letters, and don't use any of the twelve letters more than once.)

7.**Y S I A M N F D G O J L**

8.**K W Z I Y O E B M F C N**

1.

2.

3.

5.

6.

Extra challenge: in each puzzle below, use the twelve letters provided to form four words of exactly three letters each. (Use all twelve letters, and don't use any of the twelve letters more than once.)

7.

8.

On Reddit there was an AMA request asking to hear from people who write word problems. I'm not on Reddit and couldn't participate in the discussion, but a friend forwarded me the link, so I thought I'd take up the questions here.

Although the AMA was about word problems, there is at least as much to say about writing "bare calculation" practice problems—see for example the carefully designed worksheet in this article—and about writing good conceptual questions, such as Phil Daro's eye-opening grade 3 problem of using a number line to show why^{4}/_{5} is closer to 1 than ^{5}/_{4}. Maybe procedural problems and conceptual problems could be topics for later posts. For now, let's talk word problems! The person making the AMA request asked some good questions.

**1. How did you get into the profession?**

I guess I've been doing this for a long time. As part of tutoring you have to make up problems, and I started tutoring my friends when I was in grade school. My first job writing problems*per se* was when I was a physics Ph.D. student, and I was tasked with designing a curriculum for introductory physics; this was part of an effort by UC Berkeley to expand on the success of Uri Treisman's groundbreaking work in Calculus. I believed then, and believe still, that problem quality is essential to the success of a curriculum. In 1992 Treisman wrote of his original project,

After graduate school I became a professor, and when you're a professor you have to make up tons of problems for your students. I collected my best problems about force and motion and included them in my physics textbook. Some of the puzzles I post on this blog could potentially be word problems in school (this one, say); others, not so much. Some of my Saturday School posts have word problems in them too.

Sometimes my job involves writing math word problems at the K–12 level. Often the point of these problems isn't necessarily to be copied down and given to students, but rather to be discussed by educators; I want to stimulate conversation about the nature of the curriculum itself. One valuable resource for that conversation has been www.illustrativemathematics.org. Another has been the material on www.achievethecore.org, and yet another has been Table 2 in this document, which describes the fifteen different kinds of basic addition and subtraction word problems. (Fifteen!)

At higher grade levels, here is a word problem that I recently included in a presentation:

In algebra, you often have to rewrite expressions in different forms. The point of this problem is to suggest that algebra teachers might try to include some problems in which rewriting a given expression has a*rationale*, however slight, within the context of the problem. My CPU problem isn't the best possible example…what I'm trying to do is encourage the field at large to work on creating more and better problems that serve the purpose. The inspiration for the CPU problem was a time last year when I was trying to speed up a loop in one of my own programs, and factoring an expression did the trick. (Math: it works!)

Writing word problems is rewarding, but I also like to find great word problems from the past and keep them alive. See my previous posts about Newton's problem of the oxen and Wells's*Academic Algebra *textbook from the 19th Century.

Here is a fascinating problem that was a favorite of Leo Tolstoy's:

Here is an elegant short problem in mathematician Roger Howe's essay "From Arithmetic to Algebra":

A last brief example is this penetrating and original problem about algebraic structure, from Illustrative Mathematics:

It might be the case that in recent decades, the best minds in math education have spent too much time injecting lengthy problems into the grades K–8 curriculum, and not enough time transforming and elevating the teaching and learning of simple problems.

I wouldn't hazard quite the same generalization about high school, because the high school curriculum probably needs more high-quality application problems, which tend to be lengthy. Here is a high school application problem I wrote once:

The next high school application problem was adapted from one published by the Shell Centre.

The next was adapted from a problem developed by COMAP.

(This problem should really be formulated using a google doc or Excel spreadsheet, not a printed page. I think spreadsheets are used too seldom in high school.)

**2. It's assumed that you write questions below your skill level. Do you write questions at your skill level?**

Not so much anymore, but "writing questions at your skill level" is a pretty good thumbnail description of doing research—so when I was a researcher, I did this all the time. See any of my peer-reviewed publications for examples.

The saga of the points in a circle problem on this blog was another example of a problem I made up that was at my own skill level; solving it took me a considerable amount of effort, and I had to create arguments of a kind I'd never made or seen before. It might not be a research-level problem, but I think it did attract some interest from a few mathematicians out there.

Sometimes I'll write a problem that isn't research-level, yet still is "at my level" in a sense, because it solves a problem that I legitimately wanted to solve. Here was one such problem; I wrote it after stepping in a pothole full of slush.

**3. Have you ever written a question that you thought would be simpler than it was?**

In physics, you often try to give your students a nicely bounded task, but then you realize that you made some inappropriate assumptions, and the problem can't be made as simple as you were trying to make it. I think it happened more than once that I jotted down a problem on my way to class and then was unable to solve it in front of my students! Whoops. Bad lesson planning.

Sometimes a word problem is just what you meant it to be, but then the students have a hard time with it, and it surprises you. Moments like that are valuable opportunities for me to adjust my own understanding of my students' brain-state. These moments also expose weaknesses of my curriculum up to that point, which I can try to correct the next time around. I wrote about an experience like that, describing a time when I gave one of my kids a word problem she hadn't been well prepared for. You can't smooth every transition in learning, but most transitions could be smoother than they are today.

**4. How do you decide whether to write about trains, apples and oranges, candy bars, etc.?**

In the best word problems, the math is just right and the context fits beautifully. Writing those problems is harder than it looks. A while ago I wrote this task to illustrate a geometry standard, and it's OK—it stars a ladybug—but a colleague later told me that her word problem about that standard had starred pirates and treasure. That's way better! Not because treasure is more exciting, but because the mathematical issue at hand was about pinning down a location, which is an important goal for treasure-hunters not ladybugs. The situation matches the math.

If I'm writing a word problem about, say, speed, then sometimes I'll have the numbers in hand and go looking for a context to put them in. So for example if it's a speed of around 8 mph, then I'll spend some time thinking about what sorts of things actually move at around 8 miles per hour. I think about what sorts of things from everyday life plausibly move at a pretty constant speed for an extended period of time. Working that way takes a lot of effort, but it tends to yield better problems.

Editorially, there tend to be a lot of constraints on the contexts you can use. Often there are rules like "No candy bars"—I guess parents don't like junk food in the curriculum. Also, no military applications. Sometimes these rules seem a little silly to me, and I wonder if they contribute to students' feeling that math is useless. But the rules are usually there for a reason.

**5. What's the process for writing a really good problem?**

Sometimes I think we should all spend less time writing problems, and more time curating excellent problems from every century and nation. In any case, I doubt there's a reliable process, or else there would be more good problems out there. Different people also have different views about what a good problem is—sometimes incompatible views. I'm eclectic and appreciate a well made problem in any genre, just as I like many genres of art.

In one mode of writing problems, you know what math you're trying to get at, and you choose a context to match. In another mode, you start with an interesting scenario, and then the math in the problem is whatever it needs to be to shed light on the scenario. Any mode has its pitfalls. When you're hunting for a context to match some specified piece of mathematics, it's possible to end up with junk like "At the store, Mary bought 78,123 cantaloupes and 831 fewer raisins…." On the other hand, when you're starting with the context, it's possible to overshoot the expertise of the audience ("How can this airline schedule its flights so that…"), or else fail to do justice to the context ("Treating the horse as a perfect sphere…").

Matching math to context sometimes sends me to the Internet to do research on different topics. What is a realistic weight for a songbird? Could there be a county in the U.S. that's roughly triangular? In order to create the problem about oil spills shown above, I studied the body of law that establishes penalties for chemical spills in the Columbia River Estuary (see WAC 173-183-840, Calculation of damages for spills into the Columbia River estuary). Once, I called a company historian at a vehicle manufacturer to make sure that a vehicle I was writing about could go as fast as I was saying it could go. This marvelous ratio/proportional relationship problem by Richard Stanley probably has some research behind it:

The first draft of a word problem is usually too long. The language needs to be condensed. It's possible to pare down the language too much, sacrificing authenticity, but usually the reverse happens, with unnecessary language demands getting between the student and the math. This is especially important because many students in the U.S. are learning English while they're learning math. The first version of the chemical spill problem had many more words than the version shown here; experts in bilingual math education helped me reduce the language a great deal.

Although the AMA was about word problems, there is at least as much to say about writing "bare calculation" practice problems—see for example the carefully designed worksheet in this article—and about writing good conceptual questions, such as Phil Daro's eye-opening grade 3 problem of using a number line to show why

I guess I've been doing this for a long time. As part of tutoring you have to make up problems, and I started tutoring my friends when I was in grade school. My first job writing problems

Most visitors to the program thought that the heart of our project was group learning. They were impressed by the enthusiasm of the students and the intensity of their interactions as they collectively attacked challenging problems. But the real core was the problem sets which drove the group interaction. One of the greatest challenges that we faced and still face today was figuring out suitable mathematical tasks for the students that not only would help them to crystallize their emerging understanding of the calculus, but that would also show them the beauty of the subject.

Sometimes my job involves writing math word problems at the K–12 level. Often the point of these problems isn't necessarily to be copied down and given to students, but rather to be discussed by educators; I want to stimulate conversation about the nature of the curriculum itself. One valuable resource for that conversation has been www.illustrativemathematics.org. Another has been the material on www.achievethecore.org, and yet another has been Table 2 in this document, which describes the fifteen different kinds of basic addition and subtraction word problems. (Fifteen!)

At higher grade levels, here is a word problem that I recently included in a presentation:

In algebra, you often have to rewrite expressions in different forms. The point of this problem is to suggest that algebra teachers might try to include some problems in which rewriting a given expression has a

Writing word problems is rewarding, but I also like to find great word problems from the past and keep them alive. See my previous posts about Newton's problem of the oxen and Wells's

Here is a fascinating problem that was a favorite of Leo Tolstoy's:

A team of haymakers was assigned the task of scything two meadows, one twice the size of the other. The team worked half a day on the larger meadow. Then it split into two equal groups. The first group remained in the larger meadow and finished it by evening. The second group scythed the smaller meadow, but by evening there still remained a portion to do. This portion was scythed the next day by one haymaker in a single day’s work. How many haymakers were there in the team?

The Tolstoy problem isn't easy. Yet *elegant *shouldn't have to mean *difficult*. Sometimes, a simple word problem can be almost poetic in its spareness.

A last brief example is this penetrating and original problem about algebraic structure, from Illustrative Mathematics:

It might be the case that in recent decades, the best minds in math education have spent too much time injecting lengthy problems into the grades K–8 curriculum, and not enough time transforming and elevating the teaching and learning of simple problems.

I wouldn't hazard quite the same generalization about high school, because the high school curriculum probably needs more high-quality application problems, which tend to be lengthy. Here is a high school application problem I wrote once:

The next high school application problem was adapted from one published by the Shell Centre.

The next was adapted from a problem developed by COMAP.

(This problem should really be formulated using a google doc or Excel spreadsheet, not a printed page. I think spreadsheets are used too seldom in high school.)

Not so much anymore, but "writing questions at your skill level" is a pretty good thumbnail description of doing research—so when I was a researcher, I did this all the time. See any of my peer-reviewed publications for examples.

The saga of the points in a circle problem on this blog was another example of a problem I made up that was at my own skill level; solving it took me a considerable amount of effort, and I had to create arguments of a kind I'd never made or seen before. It might not be a research-level problem, but I think it did attract some interest from a few mathematicians out there.

Sometimes I'll write a problem that isn't research-level, yet still is "at my level" in a sense, because it solves a problem that I legitimately wanted to solve. Here was one such problem; I wrote it after stepping in a pothole full of slush.

In physics, you often try to give your students a nicely bounded task, but then you realize that you made some inappropriate assumptions, and the problem can't be made as simple as you were trying to make it. I think it happened more than once that I jotted down a problem on my way to class and then was unable to solve it in front of my students! Whoops. Bad lesson planning.

Sometimes a word problem is just what you meant it to be, but then the students have a hard time with it, and it surprises you. Moments like that are valuable opportunities for me to adjust my own understanding of my students' brain-state. These moments also expose weaknesses of my curriculum up to that point, which I can try to correct the next time around. I wrote about an experience like that, describing a time when I gave one of my kids a word problem she hadn't been well prepared for. You can't smooth every transition in learning, but most transitions could be smoother than they are today.

In the best word problems, the math is just right and the context fits beautifully. Writing those problems is harder than it looks. A while ago I wrote this task to illustrate a geometry standard, and it's OK—it stars a ladybug—but a colleague later told me that her word problem about that standard had starred pirates and treasure. That's way better! Not because treasure is more exciting, but because the mathematical issue at hand was about pinning down a location, which is an important goal for treasure-hunters not ladybugs. The situation matches the math.

If I'm writing a word problem about, say, speed, then sometimes I'll have the numbers in hand and go looking for a context to put them in. So for example if it's a speed of around 8 mph, then I'll spend some time thinking about what sorts of things actually move at around 8 miles per hour. I think about what sorts of things from everyday life plausibly move at a pretty constant speed for an extended period of time. Working that way takes a lot of effort, but it tends to yield better problems.

Editorially, there tend to be a lot of constraints on the contexts you can use. Often there are rules like "No candy bars"—I guess parents don't like junk food in the curriculum. Also, no military applications. Sometimes these rules seem a little silly to me, and I wonder if they contribute to students' feeling that math is useless. But the rules are usually there for a reason.

Sometimes I think we should all spend less time writing problems, and more time curating excellent problems from every century and nation. In any case, I doubt there's a reliable process, or else there would be more good problems out there. Different people also have different views about what a good problem is—sometimes incompatible views. I'm eclectic and appreciate a well made problem in any genre, just as I like many genres of art.

In one mode of writing problems, you know what math you're trying to get at, and you choose a context to match. In another mode, you start with an interesting scenario, and then the math in the problem is whatever it needs to be to shed light on the scenario. Any mode has its pitfalls. When you're hunting for a context to match some specified piece of mathematics, it's possible to end up with junk like "At the store, Mary bought 78,123 cantaloupes and 831 fewer raisins…." On the other hand, when you're starting with the context, it's possible to overshoot the expertise of the audience ("How can this airline schedule its flights so that…"), or else fail to do justice to the context ("Treating the horse as a perfect sphere…").

Matching math to context sometimes sends me to the Internet to do research on different topics. What is a realistic weight for a songbird? Could there be a county in the U.S. that's roughly triangular? In order to create the problem about oil spills shown above, I studied the body of law that establishes penalties for chemical spills in the Columbia River Estuary (see WAC 173-183-840, Calculation of damages for spills into the Columbia River estuary). Once, I called a company historian at a vehicle manufacturer to make sure that a vehicle I was writing about could go as fast as I was saying it could go. This marvelous ratio/proportional relationship problem by Richard Stanley probably has some research behind it:

Tree-measuring tape.There is a special tape measure used in measuring trees; when wrappedarounda tree, it gives as its reading thediameterof the tree. Describe how the special tape measure differs from a standard tape measure. Make a diagram to scale showing the marks on the first part of the special tape measure.

When I'm writing a word problem, my brain is totally engaged in the math, so in that moment I can't afford to worry about whether the person shopping for cantaloupes is going to be named Jason or something else. So, when I write a problem, I just write A, B, and C for the names. As soon as the problem is finished, I open up a spreadsheet that I downloaded a couple years ago from the New York City Department of Health, which lists the most popular baby names given out in New York. In my copy of the spreadsheet, I've added a random generator that selects one of the names every time I open the file. Whatever name comes up, I use that name. Over time, the Law of Large Numbers pretty much guarantees that my word problems will avoid bias in the names of the characters. Just now, I opened the spreadsheet and it gave me "Mya." I never would have thought of that name myself.

Part of the process of developing a problem should be testing it with students, then revising it. Some of the problems I see in textbooks don't look to me as though they had been honed to perfection over time. More generally, I fear that something in the economics of the market fails to reward excellence in math problems. What too few can see, too few will pay for. There is also a tradition in math education of using problems without attribution. The rules of scholarship don't seem apply to math problems, and the status of copyright law is unclear. I should say that the problem Tolstoy loved, the problem about the haymakers, was written by a student named Petrov. He died young, of tuberculosis.

Part of the process of developing a problem should be testing it with students, then revising it. Some of the problems I see in textbooks don't look to me as though they had been honed to perfection over time. More generally, I fear that something in the economics of the market fails to reward excellence in math problems. What too few can see, too few will pay for. There is also a tradition in math education of using problems without attribution. The rules of scholarship don't seem apply to math problems, and the status of copyright law is unclear. I should say that the problem Tolstoy loved, the problem about the haymakers, was written by a student named Petrov. He died young, of tuberculosis.

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