### Which is Closest?

Not too long ago, I published a blog post about end-to-end comparisons, those silly feats of computational gymnastics that try to reduce an overwhelming statistic to something more tangible. Something like this:

If each piece of candy corn sold in a year by Brach’s — the top manufacturer of the waxy confection — were laid end to end, they would circle the Earth 4.25 times.

In writing that post, I inadvertently formulated a statistic that rather surprised me:

If all the players on an NFL team were laid end to end, they’d stretch from the back of one end zone to the opposite goal line.

That the players would almost line the entire field struck me as an amazing coincidence. And it got me to thinking — might this be true for other sports?

Not one to let sleeping dogs — or professional athletes — lie, I decided to investigate. Based on that research, here’s a simple, one-question quiz for you.

Which of the following comparisons is the most accurate?

1. If all of the players on an NHL (hockey) roster were laid end to end, they would reach from one end of the rink to the other.
2. If all of the players on an NBA (basketball) roster were laid end to end, they would reach from one end of the court to the other.
3. If all of the players on an NFL (football) roster were laid end to end, they would reach from one end line to the other.
4. If all of the players on an MLB (baseball) roster were laid end to end, they would reach from home plate to second base.
5. If all of the players on an MLS (soccer) roster were laid end to end, they would reach from one end to the other.

As you begin to think about that question, some notes:

• Every professional baseball stadium has different measurements. Fenway Park (Boston) is a mere 310′ from home plate to the right field wall, whereas Comerica Park (Chicago) extends 420′ from home plate to straightaway center. Consequently, the distance from second to home is used in the fourth answer choice, because it’s the same for every field.
• To my surprise, MLS stadiums are not uniform in length and width. Who knew? The length of the field must be at least 100 meters, at most 110 meters, and anywhere in between is fine. Assume an average length of 105 meters for the fifth answer choice.

Before you read much further, let me say how much fun I’ve had discussing this question around the dinner table and at the local pub. In spite of hard facts, there is resolute disagreement about player height, roster size, and field dimensions. And the shocking (or should I say predictable?) results raise an eyebrow every time. I only mention that to persuade you to think about the question, alone or with some friends, before continuing.

Okay, you’ve cogitated? Then let’s roll.

In researching the answer to the question, I was struck by how close the total length of all players on the roster is to the length of the field, court, or rink. Coincidence? Of course, a larger field requires more players, so perhaps this is the evolution of roster size that one would expect.

To answer the question, you need to know the height of an average player, the number of players on a roster, and the dimensions of professional venues. All of that data can be found in a matter of minutes with an online search, but I’ll save you the trouble.

 League Average Height (in.) Players on a Roster Combined Height, Laid End to End (ft.) Dimensions NHL 73 23 140 200 feet (from end to end) NFL 74 53 327 120 yards (360 feet, from end to end) NBA 79 14 92 94 feet (from end to end) MLB 73 23 140 127 feet (from home to second) MLS 71 28 166 105 meters (345 feet, from end to end)

As it turns out, the MLS comparison is the least accurate. The combined heights of soccer players is only 48% of the length of their field. The NHL comparison is a little better, with players’ heights extending 70% of the length of the field. But the NFL and MLB are both very close, with the players’ heights equalling 91% of the field length and 110% of the distance from home to second, respectively. Astoundingly, if the players on an NBA team were laid end to end, they’d come just 22 inches short of covering the entire court, accounting for a miraculous 98% of the length!

So there you have it. D, final answer.

One last thought about this. I play ultimate frisbee, a sport with a field that measures 120 yards (360 feet). For tournaments, our rosters are capped at 29 players, and I suspect my amateur teammates are, on average, shorter than most professional athletes. If we assume a height of 5’10” for a typical frisbee player, then the combined height is 172 feet. That puts us in the realm of soccer, with our combined length covering just 48% of the field.

If, like me, you play a sport that isn’t one of the Big 5 in the U.S., I’d love to hear about your sport’s field and roster size, and how it ranks with the comparisons above.

The sheepdog returned to the farmhouse and told the shepherd, “All 200 sheep have been returned to their pens.”

“200?” asked the shepherd. “But we only have 196 sheep.”

The dog replied, “Well, yeah, but you know I like to round up.”

Rounding up has been a topic of conversation in college basketball this week.

Marcus Keene, a guard for the Central Michigan Chippewas, scored 959 points in 32 games this season, giving him a points-per-game (PPG) average of 30.0.

Sort of.

Technically, his average is 29.96875, just shy of the highly coveted 30 points-per-game mark that’s only been attained by a few dozen players in NCAA history. Since 1981, only 8 players have reached 30 PPG, most recently Long Island’s Charles Jones in 1996‑97.

Photo: Carlos Osorio, AP

But the controversy swirled this week because Keene didn’t actually average more than 30 points per game. He was one point shy. His lofty accomplishment was nothing more than smoke-and-mirrors due to round-off error, or so the critics say.

Per-game statistics are used to compare players with one another, because totals can’t be compared for players who have played a different number of games. And let’s face it, no one wants to get into the habit of comparing per-game stats to seven decimal places. The NCAA reports all per-game statistics to the nearest tenth, and the truth is that Keene’s PPG average would be reported as 30.0, 30.00, 30.000, and 30.0000 if rounded to tenths, hundredths, thousandths, and ten-thousandths, respectively.

It’s been a good year for math and basketball. Anthony Davis can have an asterisk for his record-setting 52 points in the NBA All-Star Game because no one played defense; and now Marcus Keene can have an asterisk for his 30.0 points-per-game average.

In related news, it was reported that 53% of men say that they will watch the NCAA Division I Men’s Basketball Championship (aka, “March Madness”). And just to prove the men are the dumber sex, 61% of them admitted that they’ll watch while at work. Simple math says that 32.3% of men will watch the tourney at work. Which means that if you’re a man with two friends who don’t like basketball, then you’ll be the one killing office productivity next Thursday.

### NBA, Discovery, and the Math of Basketball

Last week, Discovery Education and the National Basketball Association (NBA) announced a partnership in which real-time data from stats.nba.com will flow into Math Techbook, and students will use that data to solve problems.

How cool is that?

Eighty students from John Hayden Johnson Middle School in Washington, DC, participated in the event, which was emceed by Hall-of-Famer “Big” Bob Lanier and made silly by Washington Wizards mascot G-Wiz.

G-Wiz and students at John Hayden Johnson Middle School in Washington, DC, at a joint event of the NBA and Discovery Education.

The event received a lot of press coverage, and as you may have heard, there’s no such thing as bad publicity. But one of the articles quoted me as saying:

It’s not like a beautiful, traditional math problem.

That is not what I said. I am absolutely certain that I have never used the words beautiful and traditional in the same sentence. Well, perhaps when referring to a wedding dress or an Irish cottage, maybe, but certainly not when referring to a math problem.

I was also quoted as saying:

It’s going to be messy, for sure.

That is, in fact, one of the things I did say. Because by definition, good math problems are messy. For this project, our writing team created problems that don’t have one right answer. For instance, one problem asks students to generate a formula to predict which players should be on the All‑NBA 1st Team. Should they use points and rebounds as part of their formula? If so, how much weight should they give to each? And should there be a deduction for the number of turnovers a player has? All of that is up to the student, and it’s certainly possible that more than one formula would give reasonable results. (If you don’t believe me, do a search for NBA Efficiency, TENDEX, Thibodeau, VORP, or New SPM to get a sense of some formulas currently used by professional statisticians.)

A microsite with a four sample problems is available at www.discoveryeducation.com/nbamath. To see all 16 problems and to experience the NBA Math Tool, you’ll need to login to Math Techbook; sign up for a 60‑day trial at www.discoveryeducation.com/mathtechbook.

I’m ecstatic about the problems that our writing team — which includes folks who love both math and basketball, like Brenan Bardige, Ellen Clay, Chris Shore, Shauna Hedgepeth, Katie Rhee, Jen Silverman, and Jason Slowbe — has created. One of the simpler problems they’ve written, meant for middle school students, is to determine which player should take a technical free throw. It’s not a hard problem, but students get to choose which team(s) to examine and how to use free-throw data to make their choice. With the NBA Math Tool that we’ve created, which includes FTM and FTA but not FT%, one possible formula is =ROUND(100*FTM/FTA,1), which will display the free-throw percentage to one decimal place of accuracy — though there are certainly less sophisticated formulas that will get the job done, too, and students could bypass formulas entirely by using equivalent fractions.

But a different article said that the “questions may look something like” this:

Andrew Wiggins is making 49.1% of his two-point shots and 52.3% of his threes. Which shot is he more likely to make?

Actually, we would never ask a question like this in Math Techbook, either as part of this NBA project or otherwise. By the time students start working with percentages in middle school (6.RPA.3.C), we expect that they already understand how to compare decimals (4.NF.C.7). Though basic exercises are included in the service, most problems — and especially those based on NBA data — exist at a greater depth of knowledge.

But what we might do is ask students to use proportions to make a prediction.

As you know, basketball announcers and sportswriters make predictions all the time. They talk about players being “on pace” to score some number of points or to grab a certain number of rebounds. In fact, the Washington Post recently prophesied that Steph Curry will hold the NBA’s all-time three-point record before the next presidential election.

During the first part of the event at Johnson Middle School, the students set out to make a prediction:

How many assists will John Wall finish the season with?

John Wall recently set the Wizards franchise record for assists, so the context was timely.

To solve this problem, students explored the NBA Math Tool, which now resides inside Math Techbook. This tool allows students to analyze both NBA and WNBA stats. Students considered data for the Washington Wizards:

Row 6 shows that John Wall had 98 assists through 11 games. Good information, to be sure, but it led to more questions from students than answers:

• Some players on the Wizards have played 13 games. How many games have the Wizards played so far this year?
• How many games will John Wall play this year?
• How many games are in an NBA season?

Looking at team data in the NBA Math Tool, students learned that the Wizards have played 13 games so far this year. And one student knew that every NBA team plays 82 games in a season. Good info… but now what?

One approach is to set up a proportion with the equation

$\frac{11}{13} = \frac{g}{82}$

which yields the number of games (g) that we can expect John to play this year (69), and then the equation

$\frac{11}{69} = \frac{98}{a}$

can be used to find the number of assists (a) that we can expect John to record (602).

But the eighty students in the gym were sixth- and seventh-graders, and they weren’t ready for algebraic equations. Instead, they attacked the problem by noting that Wall had 98 assists through the first 13 games, so they estimated:

• He should have about 200 assists through 25 games.
• He’d have about 400 assists through 50 games.
• He’d have about 600 assists through 75 games.
• That’s 7 games shy of a full 82‑game season, and Wall should have about 50 assists in 7 games.
• So, we can expect him to finish the season with about 650 assists.

My role at the event was to lead students through the solution as a group-problem solving activity; and then, to work with them in the media center on the free-throw problem described above. It was an incredible day! I got to co-teach with Ivory Latta, point guard for the Washington Mystics:

I got to meet some incredible people, including current players, former players, and NBA executives:

(L to R) Etan Thomas; Felipe Lopez; Patrick Vennebush, MJ4MF and Discovery Education; Bill Goodwyn, President and CEO of Discovery Education; Ivory Latta, Washington Mystics point guard; Elizabeth Lipscomb, Discovery Education; Todd Jacobson, Sr. VP of Social Responsibility, NBA; and “Big” Bob Lanier, Hall-of-Fame player for the Detroit Pistons and Milwaukee Bucks.

But most importantly, I was finally able to let the world know about this amazing project, which my team has been working on for a year.

The NBA slogan is, This is why we play. But today I say, this is why we work: to develop rich curriculum resources that are fun, relevant, and powerful in teaching kids math.

#mathslamdunk

### I’m Playing Baaas-Ket-Baaall

Lego NBA Player

I recently had a meeting at the National Basketball Association (NBA) offices in New York City. I had gotten very excited about this meeting, thinking maybe I’d bump into Lebron or Kobe or Shaq or Dr. J or Jerry West or David Stern. (It could happen, ya know. Not so long ago, I bumped into Brooke Shields while attending a meeting for MoMath. All things are possible in NYC.)

But irony of ironies… when I arrived, I met no one famous; rather, one of the NBA staffers wanted to meet me because Math Jokes 4 Mathy Folks is his mom’s coffee table book. She’s a retired chemical-cum-mechanical engineer, so geeky jokes are her ilk.

Three engineers are arguing about God’s profession.

The first says, “God has to be a mechanical engineer. Look at the design of the joints and muscles.”

“No, no,” said the second. “Look at the central nervous system. All that wiring? Surely, God is an electrical engineer.”

“I think you’re both wrong,” said the third. “He’s got to be a civil engineer. Who else would put a waste management facility in the middle of a recreation area?”

Now, I know that this story likely sounds like an elaborate set-up.

Yo momma is so dorky, she reads Math Jokes 4 Mathy Folks.

Well, it’s not. All of this is true.

The wonderful young man who wanted to meet me was Daniel Feinberg. I asked about his mother’s favorite joke from Math Jokes 4 Mathy Folks, and he told me it was this one (which is sometimes known as the Pizza Theorem):

originally from Jay Fallon at Posterous Spaces,

Via email, Daniel told me:

It’s funny, because she [Daniel’s mom] hadn’t taken a look at the book in some time, and when I asked her for her favorite joke, she got sucked into reading the entire thing — again.

Now that’s a nice compliment.

Daniel isn’t an engineer or even a math guy. He loves golf, though, and his favorite joke from Math Jokes 4 Mathy Folks is:

A pastor, a doctor, and a mathematician were stuck behind a slow foursome while playing golf. The greenskeeper noticed their frustration and explained to them, “The slow group ahead of you is a bunch of blind firemen. They lost their sight saving our clubhouse from a fire last year, so we always let them play for free.”

The pastor responded, “That’s terrible! I’ll say a prayer for them.”

The doctor said, “I’ll contact my ophthalmologist friends and see if there isn’t something that can be done.”

And the mathematician asked, “Why can’t these guys play at night?”

Incidentally, Joshua Ferris included this same joke in his book To Rise Again at a Decent Hour, though the main character tells it with a priest, a minister, and a rabbi. Go figure.

I’d like to thank Daniel and his mom for their continued support. Hearing that MJ4MF made even one person smile is enough to think that it was worth writing.

Before you go, here are some basketball-related math jokes. Or maybe they’re math-related basketball jokes. Whatever. Enjoy.

What do basketball players call the last occurrence of the function that gives the greatest integer less than or equal to x?
The Final Floor.

What do athletes playing basketball and students taking a math test have in common?
They both dribble.

What’s the difference between the Knicks and a dollar bill?
You can get four quarters from a dollar bill.

Okay, maybe that last one isn’t very mathy, so here’s a mathy quote from basketball commentator and former coach Doug Collins:

Any time Detroit scores more than 100 points and holds the other team below 100 points, they almost always win.

Almost?

The Math Jokes 4 Mathy Folks blog is an online extension to the book Math Jokes 4 Mathy Folks. The blog contains jokes submitted by readers, new jokes discovered by the author, details about speaking appearances and workshops, and other random bits of information that might be interesting to the strange folks who like math jokes.

## MJ4MF (offline version)

Math Jokes 4 Mathy Folks is available from Amazon, Borders, Barnes & Noble, NCTM, Robert D. Reed Publishers, and other purveyors of exceptional literature.