Fowl Formulae for Thanksgiving

Okay, I know you’re going to find this hard to believe, but there is disagreement on the internet. And I don’t mean about some insignificant topic like gun control or taxes or health care or the value of 6 ÷ 2(1 + 2). This is big. This is important.

We’re talking turkey. Literally.

According to the British Turkey Information Service — yes, there really is such an agency — the amount of time you should cook your turkey at 375° F can be found with the following formula:

t =    \begin{cases}    20w + 70 & w < 4; \\    20w + 90 & w \geq 4,    \end{cases}

where t is the cooking time in minutes and w is the weight of the turkey in kilograms.

If you’d rather not do the math yourself, try the British Turkey Cooking Calculator, which will not only give you the cooking time but also the defrosting time and the size of turkey to buy for a given number of servings.

By comparison, the Meat Chart provided by FoodSafety.org says that turkey should be cooked at 325° F for 30 minutes per pound.

But the cooking website allrecipes.com says that only 20 minutes per pound is sufficient if you bake the bird at 350° F.

Whereas the good folks at delish offer the following guidelines:

Delish - Turkey Cooking Times

Cooking Times at 325° F from delish.com

which translates to the lovely formula

t =    \begin{cases}    5w + 125 & w \leq 10; \\    15w & w = 12; \\    7.5w + 120 & w \geq 14,    \end{cases}

but requires that you interpolate if your bird weighs an odd number of pounds. (Like 86 pounds, the world record for heaviest turkey ever raised. Even though the units digit is 6, you’d agree that 86 is an odd number of pounds for the weight of a turkey, no?)

As you might suspect, Wolfram Alpha has a more mathematically sophisticated formula:

\displaystyle t = T \times \left( \frac{w}{20} \right)^\frac{2}{3}

where t is the cooking time in hours, w is the weight in pounds, and T is a coefficient to account for cooking environment. For normal conditions, T = 4.5, and the equation reduces to

\displaystyle t = 36.64w^\frac{2}{3}

if you use minutes instead of hours for the unit of time.

But this feels a little like a math joke; below the formula, Wolfram offers the following:

using the heat equation for a spherical turkey in a 325° F oven

Falling into the wrong hands, that idea could lead to an horrendous modification of the spherical cow joke…

The turkeys at a farm were not gaining sufficient weight in the weeks leading up to Thanksgiving, so the farmer approached a local university to ask for help. A theoretical physicist was intrigued by the problem and offered his assistance. He spent several weeks at the farm, examining the turkeys and filling his notebook with equation after equation. Finally, he approached the farmer and said, “I have found a solution.”

“Oh, that’s excellent!” said the farmer.

“Yes,” said the physicist. “Unfortunately, it only works for spherical turkeys in a vacuum.”

The Wolfram formula is very similar to one suggested by physicist Pief Palofsky, who apparently dabbled in poultry when not winning the National Medal of Science.

\displaystyle t = \frac{2}{3} w^\frac{2}{3}

and when converted to minutes instead of hours, this becomes

\displaystyle t = 40w^\frac{2}{3}.

According to Turkey for the Holidays, the average weight of a turkey purchased at Thanksgiving is 15 pounds. The cooking times for a 15-pound bird, based on the formulae above, appear to have been chosen by a random number generator.

Recommender Time (min) Temp (° F)
British Turkey Information Service 226 minutes 375
Foodsafety.org 450 minutes 325
allrecipes.com 300 minutes 350
delish.com 233 minutes 325
Wolfram Alpha 223 minutes 325
Pief Palofsky 243 minutes 325

Even if you limit consideration to those who suggest a cooking temp of 325° F, the range of times still varies from just under 2¾ hours to a staggering 7½ hours. Wow.

With Thanksgiving just around the corner, where does all of this contradictory information leave us?

A number of sites on the internet claim that the only way to adequately check the doneness of a turkey is with a meat thermometer.

turkeyThe folks at recipetips.com claim that a turkey can be removed when the temperature is at least 170° F for the breast and 180° F for the thigh. Yet on the very same page, they claim, “Turkey must reach an internal temperature of 185° F.”

On the other hand, the folks at the Food Lab claim a turkey can be safely removed when the breast temperature reaches 150° F, because after resting 15‑20 minutes before carving, the amount of remaining bacteria will be minimal. They explain, “What the USDA is really looking for is a 7.0 log10 relative reduction in bacteria,” particularly Salmonella, which means that only 1 out of every 10,000,000 bacteria that were on the turkey to start with will survive the cooking process. And according to the USDA guidelines, a turkey that maintains a temperature above 150° F for 3.8 minutes or longer will reach that threshold for safety.

Which has to make you wonder — if 3.8 minutes at 150° F is supposedly adequate, why then does the USDA Food Safety and Inspection Service recommend that the minimum internal temperature of the turkey in the thigh, wing, and breast should be at least 165° F? Who knows. I suspect it’s typical government over-engineering to remove all doubt.

So, how long should you cook your turkey? Hard to say. But if you put your turkey in the oven right now, it should be done by November 23.

When the turkey is finally ready, here are a few math jokes you can tell around the Thanksgiving table.

What do math teachers do on Thanksgiving?
Count their blessings!

What does a math teacher serve for dessert on Thanksgiving?
Pumpkin Pi.

How do you keep private messages secure on Thanksgiving?
Public turkey cryptography.

Thanksgiving dinners take 18 hours to prepare. They are consumed in 12 minutes. Halftimes take 12 minutes. This is not coincidence. 
~ Erma Bombeck

Gobble, gobble!

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November 14, 2017 at 12:32 pm Leave a comment

The Virginia Governor’s Race and MP.6

Hotly contested. Particularly nasty. Ugly. Uncivilized. Controversial. Disgusting. Dirty.

Those are just some of the words and phrases used to describe this year’s race for governor in my home state of Virginia. It’s part of the reason that we had the highest voter turnout for a gubernatorial election in 20 years. It’s also why every citizen in the Old Dominion was anxiously awaiting the results.

I was no different. At 8:30 p.m. EST, I strolled on over to POLITICO, where I was presented with more information that I could handle:

VA Election Results - 8:31 p.m.

VA Gubernatorial Results at 8:31 p.m. EST

It was surprising that Northam had a 5.8% lead, since some recent polls suggested that his lead had dwindled to as little as 3.3%. It was surprising that only 90 minutes after the polls had closed, many news organizations were already declaring Northam the winner. But it was outright astonishing that POLITICO was displaying the percent of precints reporting as:

72.68121590023384%

WTF?

On Feburary 13, 2011, in a post titled Statistically Speaking…, I presented the following joke:

69.8724% of all statistics reflect an unjustified level of precision.

Three years later, in a post titled Sound Smart with Math Words, I presented another version of that same joke, though this time the percentage was expressed to the millionths:

An unprecedented 69.846743% of all statistics reflect an unjustified level of precision.

Did I think the additional precision would make it more obvious that the sentence was actually a joke? Or did I just think it would make it funnier? I’m not sure.

But I do know that it would never have occurred to me to take the level of precision to 14 — count them, 14! — digits of accuracy beyond the decimal point.

But POLITICO thought it was necessary.

That’s right. They calculated and displayed the percent of precincts reporting to the hundred-trillionths place. Hundred. Trillionths.

That’s like stating the winning time for the Tour de France to the nearest millisecond.

Or estimating the weight of the Earth to the nearest gram.

In fairness to POLITICO, though, the percentages that they were reporting not only reflected an unjustified level of precision. They were also wrong.

According to the Virginia Department of Elections, there are 2,567 precincts in the commonwealth. If 1,865 precincts had submitted results, the percent of precincts reporting could have been displayed as:

72.6529%

If 1,866 precincts had turned in their results, the percent of precincts reporting could have been displayed as:

72.6918%

But there is no number of precincts for which the percent could have been reported as:

72.68121590023384%

So, either POLITICO was using an incorrect denominator, or their algorithm was incorrectly calculating percentages.

Oh, well. At least they got the election results correct. (I hope.)

In their defense, they did finally make a correction. When I checked the results at 9:24 p.m. EST, this is what was presented:

Virginia Election Results - 9:24 p.m.

VA Gubernatorial Results at 9:24 p.m. EST

The percentage of precincts was displayed as a more reasonable 97.74%. From this, I can surmise that 2,509 precincts had reported their results (since 2,509 / 2,567 = 0.9774) and that POLITICO had finally found someone who was nimble with a slide rule.

November 8, 2017 at 5:51 am Leave a comment

Interactions with Fractions

This math factoid, compliments of Learn Fun Facts, is just too good not to share…

\displaystyle \frac{18\,534}{9\,267} \times \frac{17\,469}{5\,823} = \frac{34\,182}{5\,697}

Ah, but maybe you missed it. Did you notice that each digit 1-9 appears exactly once in all three fractions? Pretty cool. But I have to say that this is still my favorite fraction equation:

\displaystyle \frac{1}{2} + \frac{1}{3} + \frac{1}{6} = 1

Simple. Beautiful.

On the other hand, I’m not sure I have a favorite fraction joke. I mean, how do you pick just one? The number of fraction jokes is a lot like

\displaystyle \lim_{x\to 0} \frac{1}{x}.

That’s right. There’s no limit.

5 out of 4 people have trouble with fractions.

I will express polynomials as partial fractions. I will compute the value of continued fractions. I will even find a least common denominator. But I draw the line between the numerator and denominator.

What is one-fifth of a foot?
A toe.

How many tents can fit in a campground?
Ten, because ten tents (tenths) make a whole!

Before you go, here’s a fun little fraction problem for you:

What is 1/2 of 2/3 of 3/4 of 4/5 of 500?

November 3, 2017 at 6:58 am Leave a comment

Is Your Gödel Too Tight?

I don’t care what Stevie Nicks says, thunder does not only happen when it’s raining. And sorry, Kelly Clarkson, I’m not standing at your door because I’m sorry.

Logical fallacies are rampant in song lyrics. (Don’t even get me started.) I’m therefore hopeful that you won’t attempt to channel your inner songwriter while trying to solve the following logic puzzles, arranged roughly in order of difficulty.

Here’s Looking at You

Jack is looking at Anne, and Anne is looking at George. Jack is married, George is not. Is a married person looking at an unmarried person?

Beer is Proof that God Loves Us

Three people walk into a bar, and the bartender asks, “Would all of you like a beer?” The first says, “I don’t know.” The second says, “I don’t know.” The third emphatically replies, “Yes!”

Why was the third one able to respond in the affirmative?

Five to the Third

A five-digit number is equal to the sum of its digits raised to the third power. Alphametically,

CUBED = (C + U + B + E + D)3

What is the five-digit number?

Martin Gardner’s Children

I ran into an old friend, and I asked about her family. “How old are your three kids now?”

She said the product of their ages was 36. I replied, “Sorry, I still don’t know how old they are.”

She then said, “Well, the sum of their ages is the same as the house number across the street.”

“I’m sorry,” I said. “I still don’t know how old they are.”

Finally, she told me that the oldest one has red hair, and I finally realized their ages.

How old are my friend’s children?

If At First You Don’t Succeed…

If you take a positive integer, multiply its digits to obtain a second number, multiply all of the digits of the second number to obtain a third number, and so on, the persistence of a number is the number of steps required to reduce it to a single-digit number by repeating this process. For example, 77 has a persistence of four because it requires four steps to reduce it to a single digit: 77-49-36-18-8. The smallest number of persistence one is 10, the smallest of persistence two is 25, the smallest of persistence three is 39, and the smaller of persistence four is 77.

What is the smallest number of persistence five?

The Hardest Logic Puzzle Ever

Three gods A, B, and C are called, in no particular order, True, False, and Random. True always speaks truly, False always speaks falsely, but whether Random speaks truly or falsely is a completely random matter. Your task is to determine the identities of A, B, and C by asking three yes-no questions; each question must be put to exactly one god. The gods understand English, but will answer all questions in their own language, in which the words for yes and no are da and ja, in some order. You do not know which word means which.

(This puzzle is attributed to Raymond Smullyan, but the twist of not knowing which word means which was apparently added by computer scientist John McCarthy.)

October 26, 2017 at 6:32 am Leave a comment

Morelli, Coleman, and Statistical Outliers

Lego NFL Referee

You won’t see Pete Morelli and crew officiating tonight’s Monday Night Football game — and Philadelphia Eagles’ fans couldn’t be happier.

At kick-off, more than 74,000 fans had signed a petition to have Morelli banned from serving as the referee for any Eagles’ game. That’s because last Thursday night, Morelli and his crew called 10 penalties for 126 yards against the Eagles, whereas they only called 1 penalty for 1 yard against their opponents, the Carolina Panthers.

But Philadelphia sports reporter Dave Zangaro pointed out that Morelli has a history of lopsided officiating against the Eagles. In the last four Eagles’ games that Morelli has covered, his crew has called 40 penalties for 396 yards against the Eagles, but only eight penalties for 74 yards against the opponents.

No doubt, that’s quite a disparity.

But I’m curious if any of the petition signers have actually checked the numbers. Statistical anomalies happen, and I suspect that the imbalance they’ve identified is likely one of many. I didn’t run the numbers to determine if Morellli’s stats constitute an outlier; that would be too much work. But, I did take a quick peek at the other referees in the league to see what I can see.

And what I found leads me to wonder, Why hasn’t anyone started a petition to get Walt Coleman banned from officiating Atlanta Falcons games? Maybe it’s because Coleman officiates in favor of the Falcons.

Check it. In the last six Falcons’ games that Coleman has officiated, the Falcons have been penalized only 29 times for 216 yards. Their opponents, by comparison, have been penalized 53 times for 463 yards. That’s an average of four fewer penalties and half as many penalty yards per game.

And it’s even worse if you consider only home games. In those four games, the advantage is just 16 penalties for 111 yards against the Falcons to 37 penalties for 320 yards against their opponents.

Don’t believe me? Take a look…

Date Game Opponent’s Penalties Opponent’s Penalty Yds Falcons’ Penalties Falcons’ Penalty Yds
9/30/12 Panthers @ Falcons 9 64 2 15
1/13/13 Seahawks @ Falcons 6 35 3 11
9/29/13 Patriots @ Falcons 9 93 6 55
12/23/13 Falcons @ 49ers 7 45 5 37
12/4/16 Chiefs @ Falcons 13 128 5 30
9/24/17 Falcons @ Lions 9 98 8 68
Totals 53 463 29 216

 

Admittedly, those numbers aren’t quite as stark as Morelli’s, but they don’t exactly paint a picture of Coleman as an impartial ref, either.

In 2012, replacement official Brian Stropolo was banned from working a New Orleans Saints’ game when pictures of him donning Saints’ attire were found on his Facebook page. So there is precedence if the NFL wants to use my analysis to ban Coleman from Falcons’ games, or if they want to accept the petition and ban Morelli from Eagles’ games.

But let’s keep this in perspective and remember one thing: It is Philadelphia, after all. I mean, we’re talking about a sports town where fans threw snowballs at Santa Claus and threw batteries at Eagles quarterback Doug Pederson — the same Doug Pederson, in fact, who is now the Eagles coach. So if Morelli and his crew are deliberately blowing the whistle more against the Eagles than their opponents, who cares? This type of denigration couldn’t be offered to a more deserving team.

October 23, 2017 at 9:56 pm Leave a comment

Number Challenge from Will Shortz and NPR

Typically, the NPR Sunday Puzzle involves a word-based challenge, but this week’s challenge was a number puzzle.

[This challenge] comes from Zack Guido, who’s the author of the book Of Course! The Greatest Collection of Riddles & Brain Teasers for Expanding Your Mind. Write down the equation

65 – 43 = 21

You’ll notice that this is not correct. 65 minus 43 equals 22, not 21. The object is to move exactly two of the digits to create a correct equation. There is no trick in the puzzle’s wording. In the answer, the minus and equal signs do not move.

Seemed like an appropriate one to share with the MJ4MF audience. Enjoy!

October 18, 2017 at 8:07 am 2 comments

The Homework Inequality: 1 Great Problem > 50 Repetitive Exercises

Yesterday, my sons Alex and Eli were completing their homework on fraction operations, which included 39 problems in 4 sets, wherein each problem in a set was indistinguishable from its neighbors.

The worksheet contained 12 problems of fraction addition, 9 problems of fraction subtraction, 9 problems of fraction multiplication, and 9 problems of fraction-times-whole-number multiplication. That’s 39 problems of drudgery, when 10 problems would’ve been sufficient. Here’s a link to the worksheet they were given, if you’d like to torture your children or students in a similar way:

Frustrated by the monotony of the assignment, I told the boys they didn’t have to do all of the problems, and they could stop when they felt that they had done enough from each set.

“No,” said Alex. “We’re supposed to do them all.”

My sons are responsible students, but I’m frustrated by teachers who take advantage of their work ethic. Just because they’re willing to complete 50 exercises for homework doesn’t mean they should be assigned 50 exercises for homework.

My colleague at Discovery Education, Matt Cwalina, puts it this way:

Cwalina HW Quote

Some say that a picture is worth a thousand words. I say,

A great problem is worth a thousand exercises.

Personally, I would much rather have students think deeply about one challenging problem than mindlessly complete an entire worksheet. Luckily, my sons take after their daddy and love number puzzles, so I spontaneously created one.

Find three fractions, each with a single-digit numerator and denominator, that multiply to get as close to 1 as possible. Don’t repeat digits.

Eli started randomly suggesting products. “What about 4/5 × 6/7 × 9/8?” He’d work out the result, say, “I think I can do better,” then try another. And another. And another. Finally, he found a product that equaled 1. (No spoiler here. Find it yourself.)

Alex eventually found an answer, too. At the bottom of his homework assignment, he added a section that he titled “Bonus” where he captured his attempts:

I don’t know exactly how many calculations Eli completed while working on this problem, but I know that Alex completed at least seven, thanks to his documentation. Wouldn’t you agree that completing several fraction computations while thinking about this more interesting problem is superior to doing a collection of random fraction computations with no purpose?

There is a preponderance of evidence (see Rohrer, Dedrick, and Stershic 2015; definitely check out Figure 4 at the top of page 905) that massed practice — that is, completing a large number of repetitions of the same activity over and over — is counterproductive. Unfortunately, massed practice feels good because it results in short-term memory gains, which trigger a perceived level of mastery; but, it doesn’t lead to long-term retention. Moreover, students who learn a skill by practicing it repeatedly get really good at performing that skill when they know it’s coming; but, two months down the road, when they need to use that skill in an unfamiliar context because it’s not on a worksheet titled “Lesson 0.1: Adding and Multiplying Fractions,” they’re less likely to remember than if they had used more effective practice methods. One of those more effective methods is interleaving, which involves spacing out practice over multiple sessions and varying the difficulty of the tasks. Whether you’re trying to learn how to integrate by parts or how to hit a curve ball, be sure to make your practice exercises a little more difficult than you’re used to. Know that interleaving your practice will not feel as good as massed practice while you’re doing it; but later, you’ll feel better due to improved memory, long-term learning, and mastery of skills.

Interleaving is one of the reasons I love the MathCounts School Handbook, which can be downloaded for free from the MathCounts website. The topics covered by the 250 problems in the School Handbook run the gamut from algebra, number sense, and probability, to geometry, statistics, combinatorics, sequences, and proportional reasoning — and any given page may contain problems of any type! Veteran MathCounts coach Nick Diaz refers to this mixture as “shotgun style,” meaning that students never know what’s coming next. Consequently, similar problems are not presented all at once; instead, students are exposed to them several days or perhaps weeks apart. Having to recall a skill that hasn’t been used for a while requires more effort than remembering what you did just five minutes ago, but the result is long-term retention. It’s doubtful that the writers of the first MathCounts School Handbook knew the research about interleaving and massed practice… but they clearly knew about effective learning.

The other reason I like the MathCounts School Handbook is the difficulty level of the problems. Sure, some of the items look like traditional textbook exercises, but you’ll also find a lot of atypical problems, like this one from the 2017-18 School Handbook:

If p, q, and r are prime numbers such that pq + r = 73, what is the least possible value of p + q + r?

That problem, as well as the fraction problem that I created for Alex and Eli, would both fall into the category of open-middle problems, which means…

  • the beginning is closed: every students starts with the same initial problem.
  • the end is closed: there is a small, finite number of unique answers (often, just one).
  • the middle is open: there are multiple ways to approach and ultimately solve the problem.

Open-middle problems often allow for implicit procedural practice while asking students to focus on a more challenging problem. This results in a higher level of engagement for students. Moreover, it reduces the need for massed practice, because students are performing calculations while doing something else. You can find a large collection of open-middle problems at www.openmiddle.com, and the following is one of my favorites:

Use the digits 1 to 9, at most one time each, to fill in the boxes to make a result that has the greatest value possible.

Maximum Exponent Boxes

It’s a great problem, because random guessing will lead students to combinations that work, but it may not be obvious how to determine the greatest possible value. Consequently, there’s an entry point for all students, the problem offers implicit procedural practice, and the challenge of finding the greatest value provides motivation for students to continue.

I have a dream that one day, in traditional classrooms where 50 problems are assigned for homework every night, where procedural fluency is valued over conceptual understanding; that one day, right there in those classrooms, students will no longer think that math is simply a series of disparate rules with no purpose, but instead will experience the joy of attempting and solving challenging problems that inspire purposeful play and, as a side benefit, encourage students to practice the skills they will need to be successful learners.

There are myriad resources available so that teachers and parents can encourage their students to engage in these kinds of problem-solving activities, so it is my hope that this dream is not too far away.


As a special bonus for reading to the end, check out this Interleaved Mathematics Practice Guide that Professor Doug Rohrer was kind enough to share with me (and now, with you).

October 5, 2017 at 6:06 am 2 comments

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About MJ4MF

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.

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

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