AWOKK, Day 6: KenKen Glossary

KENgratulations! You’ve made it to Day 6 of MJ4MF’s A Week of KenKen series. If you happened to miss any of the fun we’ve had previously…

• Day 1: Introduction
• Day 2: The KENtathlon
• Day 3: KenKen Times
• Day 4: My KenKen Puzzles
• Day 5: Harold Reiter’s Puzzles

Robert F. Fuhrer is a toy inventor with a knack for coming up with creative names, including Crocodile Dentist, Gator Golf, T.H.I.N.G.S. (Totally Hilarious, Incredibly Neat Games of Skill), Rumble Bugs, Missile Toe (literally, a rocket in the shape of a toe), and many others. As the president of Nextoy, LLC, which holds the registered trademark for KenKen^®, Bob now uses his creative naming abilities for the appellations of KenKen-related products.

The word that started this post, KENgratulations, is just one of his many linguistic creations. I rather like the term he coined to describe the computer application that randomly generates KenKen puzzles.

KEN·er·a·tor n. the “machine” (computer application) used to automatically generate KenKen puzzles

Quick! We need more 6 × 6 puzzles. Crank up the Kenerator!

The name conjures images of a machine from Willy Wonka.

Oompa, loompa, doom-pa-dee-do,
I’ve got a perfect puzzle for you…

Numbers and operations go in, puzzles come out.

Bob can also take credit for the following:

KEN·cil n. a pencil used to solve KenKen puzzles

I prefer kencils rather than pens when solving KenKen puzzles.

KEN·grat·u·la·tions n. an expression of praise for solving a KenKen puzzle

Kengratulations for solving that puzzle in less than 7 years!

KEN·thu·si·ast n. someone who likes to solve KenKen puzzles

Most Kenthusiasts solve more than one KenKen puzzle a day.

There’s no doubt, Bob is good. But as you saw in a previous post, I’ve got a knack for coining terms, too…

KEN·tath·lon n. competition involving multiple KenKen events

I complete a Kentathlon consisting of a 4 × 4, 5 × 5, and 6 × 6 puzzle every morning.

e·go·KEN·tric adj. a person who thinks that they are better than others at solving KenKen puzzles

He’s completely egokentric, even though he’s never won a KenKen tournament.

KEN·tral of·fice n. where KenKen puzzles are made

The Kenerator resides in the kentral office.

KEN·te·nar·i·an n. person who has solved 100+ KenKen puzzles

He became instantly addicted to KenKen puzzles; he became a kentenarian in less than 3 weeks.

su·per·KEN·te·nar·i·an n. person who has solved 100+ KenKen puzzles in one day

She became a superkentenarian by completing all the puzzles in Ferocious KenKen on Saturday.

KEN·ta·gon n. the arena in which KenKen tournaments take place (analogous to the Octagon, the eight-sided chain-link enclosure used for Ultimate Fighting Championship matches, though usually less violent)

Enter the kentagon, prepare to solve!

KEN·o·pause n. the period in a puzzle solver’s life when KenKen ceases to be fun

Kenthusiasts who have entered kenopause usually solve fewer than one puzzle a day, on average.

KEN·i·ten·tia·ry n. where KenKen solvers are locked up if they’re caught cheating (syn prism)

If you copy off your neighbor at a KenKen tournament, you’ll be sent to the kenitentiary.

KEN·al·ty n. a disadvantage imposed on a puzzle solver at a tournament for an infringement of the rules

He was given a 15-second kenalty for “using a kencil in an unsafe manner.”

KEN·ais·sance n. the period from roughly 2008 to 2010 when KenKen puzzles experienced tremendous growth in popularity, likely the result of publication in The Times (London), the NY Times, and other newspapers

Harold Reiter’s interest in KenKen started long before the Kenaissance.

KEN·der·foot, n. an amateur; someone who has solved only a few KenKen puzzles

He’s such a kenderfoot; he doesn’t even know the X-wing strategy!

KEN·den·cies n. the habits of a KenKen solver; analogous to a “tell” in poker

He has a kendency to complete all of the addition cages before attempting any subtraction cage.

hy·per·KEN·ti·late n. to breathe heavily while solving a puzzle (usu., a result of having difficulty)

At the 2013 KenKen International Championship, she started to hyperkentilate when she had trouble with a difficult 6 × 6 puzzle.

Sound Smart with Math Words

When law professor Richard D. Friedman appeared in front of the Supreme Court, he stated that an issue was “entirely orthogonal” to the discussion. Chief Justice John G. Roberts Jr. stopped him, saying, “I’m sorry. Entirely what?”

“Orthogonal,” Friedman replied, and then explained that it meant unrelated or irrelevant.

Justice Antonin Scalia was so taken by the word that he let out an ooh and suggested that the word be used in the opinion.

In math class, orthogonal means “at a right angle,” but in common English, it means that two things are unrelated. Many mathematical terms have taken a similar path; moreover, there are many terms that had extracurricular meanings long before we ever used them in a math classroom. Average is used to mean “typical.” Odd is used to mean “strange” or “abnormal.” And base is used to mean “foundation.” To name a few.

The stats teacher said that I was average, but he was just being mean.

You know what’s odd to me? Numbers that aren’t divisible by 2.

An exponent’s favorite song is, “All About the Base.”

Even words for quantities can have multiple meanings. Plato used number to mean any quantity more than 2. And forty used to refer to any large quantity, which is why Ali Baba had forty thieves, and why the Bible says that it rained for forty days and forty nights. Nowadays, we use thousands or millions or billions or gazillions to refer to a large, unknown quantity. (That’s just grammatical inflation, I suspect. In a future millennium, we’ll talk of sextillion tourists waiting in line at Disneyland or of googol icicles hanging from the gutters.)

Zevenbergen (2001) provided a list of 36 such terms that have both math and non-math meanings, including:

• angle
• improper
• point
• rational
• table
• volume

The alternate meanings can lead to a significant amount of confusion. Ask a mathematician, “What’s your point?” and she may respond, “(2, 4).” Likewise, if you ask a student to determine the volume of a soup can, he may answer, “Uh… quiet?”

It can all be quite perplexing. But don’t be overwhelmed. Sarah Cooper has some suggestions for working mathy terms into business meetings and everyday speech. Like this…

For more suggestions, check out her blog post How to Use Math Words to Sound Smart.

If you really want to sound smart, though, be sure to heed the advice of columnist Dave Barry:

Don’t say: “I think Peruvians are underpaid.”
Say instead: “The average Peruvian’s salary in 1981 dollars adjusted for the revised tax base is $1452.81 per annum, which is $836.07 below the mean gross poverty level.”
NOTE: Always make up exact figures. If an opponent asks you where you got your information, make that up, too.

This reminds me of several stats jokes:

• More than 83% of all statistics are made up on the spot.
• As many as one in four eggs contains salmonella, so you should only make three-egg omelettes, just to be safe.
• Even some failing students are in the top 90% of their class.
• An unprecedented 69.846743% of all statistics reflect an unjustified level of precision.

You can see the original version of “How to Win an Argument” at Dave Barry’s website, or you can check out a more readable version from the Cognitive Science Dept at Rensselaer.

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Zevenbergen, R. (2001). Mathematical literacy in the middle years. Literacy Learning: the Middle Years, 9(2), 21-28.

Making Progress, Arithmetically

Today is 11/12/13, a rather pleasant-sounding date because the numbers form an arithmetic sequence, albeit a trivial one. It’s not the only date in 2013 for which the month, date, and year form an arithmetic sequence. How many others are there?

Several nights ago, my sons asked if they could do bedtime math, but Eli asked if we could do problems other than those on the Bedtime Math website, because “they’re a little too easy.” So instead, I navigated to the MathCounts website and opened the 2013-14 MathCounts School Handbook. We scrolled to page 9 and attacked the problems in Warm-Up 1.

Things were going well until we reached Problem 8 in the set, which read:

The angles of a triangle form an arithmetic progression, and the smallest angle is 42°. What is the degree measure of the largest angle of the triangle?

Eli asked, “Daddy, what’s an arithmetic progression?” pronouncing arithmetic as “uh-rith-ma-tick” instead of “air-ith-met-ick.”

I could have just answered Eli’s question by stating the definition:

An arithmetic progression is a sequence of numbers for which there is a common difference between terms.

But such a definition isn’t very helpful, since I’m not sure that either Eli or Alex know what sequence, common difference, or term mean. It would have led to even more questions.

Plus, I’ve always believed that kids understand (and retain) more when they discover things on their own. Call it “discovery learning” or “inquiry-based instruction” or any of myriad other names from educational jargon, it just means that giving kids the answer is not the most effective way for them to learn.

So instead, I said, “Let me give you some examples.” And then I wrote:

1, 2, 3

3, 5, 7

Alex said, “Oh, I get it! An arithmetic progression is a nice pattern of numbers.”

So I said, “Well, let me give you some patterns that aren’t arithmetic progressions.” And then I wrote:

2, 4, 8

“That’s a nice pattern, isn’t it?” I asked. “But it’s not an arithmetic progression.”

“Oh,” said Alex. He thought for a second, then revised. “You have to add the same amount every time.”

And there you have it. Three examples, and my sons were able to define arithmetic progression. It’s not as sophisticated as “a common difference between terms,” but “add the same amount every time” is a sufficient definition for a six-year-old.

So they generated an arithmetic progression with 42 as the smallest term:

42, 45, 48

Eli said, “I don’t fink vat’s enough.” When asked to explain, he said he thought that the angles in a triangle add up to 180 degrees.

“Are you sure?” I asked. He wasn’t. Nor was Alex. So I asked if they could convince themselves that the sum of the angles is 180°.

Alex said, “Well, the angles in a square add up to 360°, and you could cut it in half.” So we did:

They then reasoned that each triangle would have a sum of 180°. “But maybe that only works for a square,” I said. “How do you know it’ll work for other shapes?”

Eli suggested that we could cut a rectangle in half, too:

And again they concluded that each triangle would have a sum of 180°.

Understand, this is NOT a proof of the triangle sum formula. When they get to high school and need to demonstrate the rigor that the Common Core State Standards are demanding, well, then we’ll worry about formal proof. But for now, I’m okay with six-year-olds who can demonstrate that kind of reasoning.

They then took another guess, but this time they chose three numbers that added to 180:

42, 59, 79

Realizing that the difference between the first and second terms was 17 and the difference between the second and third terms was 20, they revised:

42, 60, 78

They concluded that the largest angle had a measure of 78°. And all was right with the world.

So why am I telling you all this?

Partially, it’s because I’m a proud father.

But more importantly, it’s because this vignette demonstrates that teaching is an art, and successful teaching doesn’t happen by accident. It’s not easy, as many people believe. What’s easy is the perpetuation of bad teaching, a la Charlie Brown’s teacher, or textbooks that simply present information with the belief that students will absorb it by osmosis. Good teaching, however, requires content knowledge and pedagogical knowledge, and it demands teachers who can handle unexpected classroom twists and turns and have the ability to adjust on the fly.

A student is convinced that a right triangle isn’t a right triangle because the right angle isn’t in the lower left corner? You better find an effective way to clarify that misconception. (Hint: Don’t use a traditional textbook where every picture of a right triangle shows the right angle in the lower left corner.)

Students think that 16/64 = 1/4 because you can “cancel the 6’s”? Uh-oh. Better find some counterexamples pronto, and help them understand why 16/64 can be reduced to 1/4.

Your students don’t know the definition of arithmetic progression? Then you better figure out a way to help them define it, and just writing your definition on the chalkboard isn’t gonna cut it.

Want to see what good teaching looks like? See Dan Meyer, or Christopher Danielson, or Fawn Nguyen. Or many, many others who don’t blog about it but inspire students every day.

Someday soon, I hope to add my project at Discovery Education to the list of examples of good teaching. Until then, I’ll just keep blathering about my sons.

Math Glossary Quiz

Thanks to the folks at Quizlet, I’m able to offer you an opportunity to prove how smart you are.

Know what a polygon is?

Do you know the value of a trillion pins?

Think you know about polar bears?

Think again.

Try your hand at this 26-question quiz. For each definition, just enter the term to which it is referring.

Math Glossary Quiz

Post your results in the Comments, or share other math definitions.

Good luck!