## Archive for November, 2013

### Common Problems

This is pretty cool. What is the value of the following expression?

(π^{4} + π^{5})^{1/6}

This reminds me of a series of problems that I call Maximum Mileage:

- What is the maximum possible product of two positive integers whose sum is 100?
- What is the maximum possible product of two prime numbers whose sum is 100?
- A set of positive integers has a sum of 100. What is the maximum possible product of the numbers in this set?
- A set of positive numbers has a sum of 100. What is the maximum possible product of the numbers in this set?

Bonus Question: Why does the expression at the beginning of this post remind me of that series of problems?

The following sentence might be a hint:

It is fortuitous that both conundrums incur a commonality of solution.

### The Math of Thanksgivukkah

Oish. Enough already.

I know it’s rare that Hanukkah and Thanksgiving coincide. But if one more person tells me that it’ll be another 70,000 years before this happens again, I’m gonna scream.

This may be the single dumbest statistic I’ve ever heard. Here’s why.

Consider some of the reasons that cause Hanukkah and Thanksgiving to coincide this year.

- Jews rely on the Shmuelian calendar for religious holidays, which is why Hanukkah seems to vary so greatly from year to year. It wouldn’t appear to vary quite so much if you followed the Shmuelian calendar, but if you’re like most of the world, you rely on the Gregorian calendar. (On the Shmuelian calendar, by contrast, it would seem that Thanksgiving varies a lot from year to year. For instance, Thanksgiving this year occurs on 25 Kislev, next year on 5 Kislev, in 2015 on 14 Kislev, and in 2016 on 23 Cheshvan.)
- The Shmuelian calendar has a 19-year cycle, while the Gregorian calendar has a (roughly) 7-year cycle. So you might expect that the calendars would coincide about every 133 years. And they sort of do. However, the last time that the first day of Hanukkah fell on November 28 was in 1861, two years before Abraham Lincoln declared Thanksgiving an official U.S. holiday in 1863.
- One year on Earth is approximately 365.25 days — but not exactly. In fact, it’s closer to 365.2422 days. That slight difference is about 11 minutes. Not a big deal, really, but over 400 years, the calendar would incur a discrepancy of about three days. That’s why Pope Gregory, in 1582, decreed that years divisible by 100 but not divisible by 400 would
**not**be leap years. But Rav Shmuel, who organized his calendar in the first century, didn’t have access to such specific solar measurements, so the Shmuelian calendar does not make similar accommodations.

Put all that together and — voila! — an amazing coincidence.

Because the Shmuelian calendar gains one day on the Gregorian calendar every 165 years or so — see the third bullet point above — it’ll be tens of thousands of years before they coincide again.

But here’s the thing. **It’ll never happen.** Not a chance.

There are lots of reasons why not.

**First,** Thanksgiving has been around for 150 years, but there’s no reason to think it’ll last another 70,000 years any more than the Romans should have thought we’d still be celebrating Saturnalia today. Countries and empires come and go, and so do their traditions.

**Second,** smart money says that when the Shmuelian calendar gets far enough out of whack that Passover no longer occurs in spring, there will be an adjustment. Or maybe there’ll be an adjustment to the Gregorian calendar first, for as yet unknown reasons. Or perhaps an entirely new calendar will appear on the scene. Who knows?

**Third,** zombies. Just sayin’.

Dr. Joel Hoffman gives a more detailed and eloquent description of Why Hanukkah and Thanksgiving Will Never Again Coincide over at *Huffington Post*.

**co·in·cide
**

*verb*

**1.**what you should do when it starts to rain

Speaking of things that coincide…

Parallel lines meet at infinity — which must make infinity a very noisy place!

An unfortunate coincidence…

The grad student stood up in his cubicle and shouted, “Why do things that happen to dumb people keep happening to me?”

And a funny coincidence…

After a long day of teaching, grading papers, and doing research for a paper, a mathematician headed to the pub where he was supposed to meet his wife. Seeing her across the bar, he walked up behind her, spun her stool around, and kissed her on the lips. She pushed him away violently, at which point he realized the woman wasn’t his wife.

“I’m very sorry,” he said. “I thought you were my wife. You look exactly like her.”

“You rotten, good-for-nothing son-of-a-bitch,” she said, and slapped him across the face.

“Funny,” he said. “You talk like her, too.”

### Annotated Amazon Review of *MJ4MF*

The paperback version of *Math Jokes 4 Mathy Folks* was released on August 9, 2010. During its first three years on Amazon, it received 17 reviews, with an average rating of 4.76. Recently, however, an unimpressed reviewer gave it just 2 stars:

This reminds me. If you’ve read *MJ4MF* and liked it, please post a review on Amazon. (If you disliked it, please post your review on MySpace.)

But I digress. Back to my point. The disparaging review that appeared on Amazon contained just 21 words:

jokes are not very funny – seems like they were stretching it to find enough jokes to fill a book to sell

To fully understand this review, I offer the following annotations.

jokes are not very funny

“I wouldn’t know humor if it bit me. I often travel to Branson, MO, to see Yakov Smirnoff perform live, and I think that Carrot Top’s performance on Star Search is the funniest moment ever.”

seems like they

“I’m unaware that the author is a single person,” or possibly, “I’m not familiar with rules of English grammar.”

were stretching it

“I don’t understand common English idioms. A friend pointed out that the correct phrase is just ‘were stretching’ without the ‘it.’ Oops.”

to find enough jokes

“I failed to realize that the book contains 400+ math jokes, yet a Google search for ‘math jokes’ returns 2,830,000 results. Simple percentages show how selective the author has been. I also hadn’t visited this blog before posting my review; I now see that a significant number of jokes not in the book have appeared on this blog, so clearly the author did not exhaust the supply.”

to fill a book to sell

“The author is a money-hungry swine who would sell his grandmother’s secret recipe for Hungarian pierogi for 50 bucks.”

Sadly, this last claim is mostly true. But my grandmother’s pierogi were divine, and the recipe is worth far more than $50. Kindly submit your bid in the Comments.

But I’m not bitter. I don’t care that this review reduces my average rating by 0.15 stars or that it single-handedly drops the book to #19 when someone searches for ‘math jokes’ on Amazon and sorts by “Avg. Customer Review.”

Instead, I prefer to remember the *MJ4MF* review written by Caregiver x 2, who said:

This morning I gave this book to my son, he didn’t put it down for a long time. He was laughing and flipping the pages as fast as he could. And he was on his summer break!

She is wise beyond her years, and I appreciate that she took the time to share her insightful comments with the world.

### One Direction Don’t Know OoOps

What teenager isn’t compelled to act when Niall, Zayn, Liam, Harry and Louis pose a mental math problem?

Admittedly, I’m late to the party on *Your Math Skills Are Terrible*, a 2011 parody of One Direction’s hit *You Don’t Know You’re Beautiful*. But as far as I can tell, I’m the first one to question One Direction’s understanding of the order of operations.

The expression generated by the lyrics is as follows:

4 ÷ 2 + 6 × 60 + 2 – 100 + 24 ÷ 2 + 7 ÷ 3 + 60

And if you perform the operations left-to-right, you get the answer that One Direction claims: **130**.

But if you follow the order of operations as promoted by textbooks and math teachers, then you get a different answer: **338 1/3**.

Ah, well, did we really expect some young pop stars to get this right? At least their song taught me a new acronym: OAP = old age pensioner (an official term used to refer to retirees in the United Kingdom, though informally it just means an old person).

With this song, One Direction is hanging out at the intersection of math and music. Turns out, a lot of jokes hang out at that intersection, too.

What’s yellow, weighs 1,000 pounds, and sings?

Two 500-pound canaries.Lumberjacks make good mathematicians because of their natural log rhythms.

What’s the world’s longest song?

Aleph-null bottles of beer on the wall, aleph-null bottles of beer, …Music is the pleasure the human mind experiences from counting without being aware that it is counting. (Gottfried Wilhelm Leibniz)

### Approximate Answer to Life, the Universe, and Everything

Douglas Adams wrote that 42 is the answer to life, the universe, and everything. Today is 11/18/13, a date whose numbers sum to 42, so it seems an appropriate time to talk about variations on this important number.

**40**

The approximate answer.

**41.99**

The retail price of the answer.

**51.19**

The price of the answer after tax and tip.

**39.06**

The price of the answer at Wal-Mart.

**XLII**

The Roman answer.

**101010**

The binary answer.

**41.9999999982**

The answer as computed by an Intel Pentium processor.

**42*** i
*The imaginary answer.

**44**

The address of the answer’s next door neighbor.

**00042**

The ZIP code of the answer.

**42.00000**

The high-precision answer.

### Math Silliness

Jiminy, looking back at my posts during the past month, I’ve been waaaaaaaay too serious. Here’s something a little lighter — but be forewarned, it’s PG-13.

Overheard at the math department holiday party:

- I’m like π — I’m really long, and I go on forever.
- I’m algebraically divorced. Will you replace my
*x*without asking*y*? - What do math and my genitalia have in common? Both are hard for you!
- I know the first 1,000 digits of π. But that don’t mean nothin’ if I can’t get the 10 digits in your phone number.
- On a scale of 1-10, you’re
*e*^{π}. - You must be an asymptote… I keep getting closer and closer, but you won’t let me touch.

I was at an Internet cafe yesterday, and my server went down on me.

Please enter your new password: **penis**

*Sorry, your password isn’t long enough.*

Sex is better than logic, but I can’t prove it.

Were your parents married before you were born?

*Half.*

Half?

*Yes, my father was married, my mother was not.*

### 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 progressionis 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.