Five questions to get you geared up for Turkey Day.
Which weighs more?
- The weight of turkey that Americans will eat on Thanksgiving.
- The combined weight of the entire population of Chicago.
What percent of turkeys raised each year are eaten by Americans?
- About 90%.
- About 50%.
How close are humans to being pumpkins?
- About 75% genetically identical.
- About 90% genetically identical.
How often is Thanksgiving celebrated on the last Thursday in November?
- On average, 5/7 of the time.
At your Thanksgiving dinner feast, you’ve placed a name tent at each plate for yourself and nine guests. But your Uncle Huey, who’s too old to give a damn, has chosen his seat at random. Your other guests decide that they’ll come to the table and sit in the proper seat if no one is sitting there yet; if the seat with that person’s name tent is occupied, however, he’ll choose a different seat at random. As the host, you’ll be the last to sit. What’s the probability that you’ll get your assigned seat?
I’ll place a spoiler in the Comments on Thanksgiving Day.
She has plenty of company from Tina Cardone, author of Nix the Tricks, a free downloadable book of tricks that Cardone believes should be removed from the curriculum. (It also includes what and how things should be taught instead. Though Cardone is the author, she admits it was a collaborative effort of #MTBoS.)
But Gojak and Cardone are not talking about fun and mysterious math tricks, like this:
- In a bookstore or library, pick out two identical copies of the same book. Keep one, and give one to your friend.
- Ask your friend to pick a number, and both of you turn to that page in the book.
- Then tell your friend to pick a word in the top half of that page, and silently spell that word while moving forward one word for each letter. (For instance, if they picked “math,” they’d move forward four words as they spelled.) Have them do that again with the word they land on; then again; and again; till they reach the end of the page. They should stop at the last word that won’t take them to the next page.
- Do the same thing with your copy of the book — though make it obvious that you’re not copying them and using the same first word.
- Miraculously, you and your friend will land on the same word at the end of the page.
What Gojak and Cardone are talking about and, rightfully, railing against are the mnemonics and shortcuts that many teachers give to students in lieu of developing true understanding, such as:
- Ours is not to reason why; just invert, and multiply.
- Just add a 0 when multiplying by 10.
- The butterfly method of adding and subtracting fractions.
From my perspective, tricks are not inherently bad, as long as students have developed the conceptual understanding necessary to recognize why the trick works. In fact, the standard algorithms for multiplication and long division are, essentially, math tricks. But if students are able to reason their way to an answer — and, even better, if they can discern these tricks on their own from examples — then the ‘tricks’ can be useful in helping them get answers quickly.
The problem is that many teachers (or well-meaning parents) introduce tricks before students understand the underlying mathematics, and that can be to the students’ detriment. Instead of mathematics appearing to be a cohesive whole, students believe that it is a large bag of disconnected procedures to be memorized, remembered for the test, and then promptly forgotten.
My eighth-grade English teacher taught us the following:
You need to learn the rules of grammar, so you can feel comfortable breaking them.
I believe the corollary in mathematics applies here. If you understand the structure and algorithms associated with procedural fluency, then you can then feel comfortable using well-known shortcuts.
The Common Core identifies three pillars of rigor — conceptual understanding, procedural fluency, and application — and asserts that each pillar must be “pursued with equal intensity” (CCSSM Publishers’ Criteria for K-8 and High School). But I’ll take that one step further. I contend that conceptual understanding must happen prior to procedural fluency for learning to be effective. That contention is based on research as well as personal experience, and it’s consistent with the NCTM Procedural Fluency in Mathematics Position Statement, in which the Council states, “Procedural fluency builds on a foundation of conceptual understanding, strategic reasoning, and problem solving,” citing the Common Core State Standards for Mathematics, Principles and Standards for School Mathematics, and Principles to Actions: Ensuring Mathematical Success for All.
A mathematical trick, as far as I’m concerned, is merely the recognition and application of a pattern. A simple example is the 9× facts.
- 1 × 9 = 9
- 2 × 9 = 18
- 3 × 9 = 27
- 4 × 9 = 36
By looking for patterns, students might notice that 9 times a number is equal to 10 times that number minus the number itself; that is, 9n = 10n – n. For instance, 8 × 9 = 80 – 8. This makes sense algebraically, of course, since 10n – n = (10 – 1)n = 9n, but I wouldn’t expect a second grader to be able to work this out symbolically. I might, however, expect a second grader to correctly reason that this should always work. From that, a student would have a way to remember the 9× facts that is based on experience and understanding, not just remembering a rule that was delivered from on high.
Taking this further, recognizing this pattern helps with multiplication problems beyond the 9× facts. Recognizing that 9n = 10n – n can also help a student with the following exercise:
- 9 × 26 = 260 – 26 = 234
As a more sophisticated example, students might be able to generalize a rule for multiplying two numbers in the teens. I’ve seen tricks like this presented as follows, with no explanation:
- Take two numbers from 11 to 19 whose product you’d like to know.
- For example, 14 × 17.
- Add the larger number to the units digit of the smaller number.
- 17 + 4 = 21
- Concatenate a 0 to the result.
- 21(0) = 210
- Multiply the units digits of the two numbers.
- 7 × 4 = 28
- Add the last two steps.
- 210 + 28 = 238
- And there you have it: 17 × 14 = 238.
Why this trick works is rather beautiful.
17 × 14
(10 + 7) × (10 + 4)
10 × 10 + 7 × 10 + 10 × 4 + 7 × 4
100 + 70 + 40 + 28
10 × (17 + 4) + 28
210 + 28
The steps requiring students to add the larger number to the units digit of the smaller number and then concatenate a 0 to the result are represented in bold above as 10 × (17 + 4). Those steps in the rule are just a shortening of the second, third, and fourth lines in the expansion above.
Presenting this trick with an explanation is better than presenting it without explanation, for sure, but better still would be for students to look for patterns from numerous examples and, if possible, generate their own rules. If you presented students with the following list:
- 15 × 16 =240
- 17 × 14 = 238
- 13 × 18 = 234
- 19 × 12 = 228
would they be able to generate the above rule on their own? Probably not. But that’s okay. The rule is not the point. The point is that there is an inherent structure and pattern within numbers, and students might notice some patterns in the factors and products in the four examples above — for instance, the smaller factor decreases from 15 to 14 to 13 to 12 as you progress from one example to the next; the larger factor increases from 16 to 17 to 18 to 19; and the product decreases by 2, then by 4, then by 6.
Something’s going on there, but what?
Ours is not to tell them why; just give them more examples to try!
I’ve always hated the I before E except after C rule. My hatred is simple: a rule is a “prescribed direction for conduct,” and, as far as I’m concerned, it should be accurate very close to 100% of the time.
The Triangle Inequality? That’s a rule that always works.
The sum of the angles of a triangle? It’s 180°, 100% of the time.
Ceva’s Theorem? Completely worthless, to be sure, but also completely correct.
But the I before E rule? I wasn’t sure how often it was inaccurate, but it only took a few seconds to come up with myriad counterexamples:
That’s the thing, right? Math rules always work. Else we wouldn’t call them rules. But grammarians, philosophers, artists — pretty much anyone with a liberal arts degree — will call anything a rule that works some of the time.
So with some help from MoreWords, I created the following Venn diagram:
There are 5,443 words that contain either EI or IE. Of those,
- 3,562 correctly contain IE not following C
- 62 correctly contain EI following C
That is, of the 5,443 words containing EI or IE, 1,591 words violate the rule by having EI without a C in front of it, and 162 words violate the rule by having IE with a C in front of it.
Which is to say, only 66.6% of the words that contain either EI or IE adhere to the rule I before E except after C.
Put another way, the rule is total bullshit.
These numbers are consistent with an analysis from Language Log, which looked at about 8.7 million words randomly pulled from a month of the NY Times. It was found that 174,716 words contained EI or IE, but only 114,070 words correctly followed the rule, which means the rule held about 65% of the time.
One of the readers of Language Log commented that the rule works with the following amendment:
When the sound is long E,
it’s I before E,
except after C.
I’ll call bullshit.
I didn’t even have to think to come up with a list of words for which that modified rule fails:
Speaking of rules…
Philosophy is a game with objectives and no rules.
Mathematics is a game with rules and no objectives.
Mathematics is a game played according to certain simple rules with meaningless marks on paper.
— David Hilbert
My boys have been asking to do Math Trivia before bedtime each night, and one of my favorite sites, GRiN: Good Riddles Now, has provided a treasure trove of fun puzzles that they can solve.
There are 100 coins on the floor in a dark room: 90 coins show heads, the other 10 show tails. If you’re not allowed to turn on any lights, how can you divide the coins into two piles so that each pile contains the same number of coins showing tails?
GRiN was started by Justin Zablocki, a math major cum computer scientist who enjoys logic and puzzles. He created GRiN as a way to practice his web development skills and to “improve upon an underdeveloped entertainment category.” (Hear, hear!)
His favorite math joke?
Why does no one talk to π?
He’s irrational and goes on forever.
His favorite riddle?
A murderer is condemned to death. He has to choose between three rooms. The first is ablaze with raging fires, the second is full of assassins with loaded guns, and the third contains lions who haven’t eaten in three years. Which room is safest for him?
Keeping with today’s theme, here’s a math riddle quiz for ya. Enjoy.
- How is the moon like a dollar?
- A plane with 56 passengers crashes on the border between Canada and the United States. Where do they bury the survivors?
- When spelled out, what is the first positive integer that contains the letter a?
- In a race of 548 runners, you overtake the last runner. What place are you now in?
- The first term of a sequence is a1 = 13. Every term thereafter satisfies a1 ∙ a2 ∙∙∙ ak = k! for k > 1. What is the 31st term of this sequence?
- There are four cookies in the cookie jar. You take three of them. How many do you have?
- If you remove the first letter, the last letter, and all the letters in between, what do you have left?
- What is the next number in the sequence 1, 4, 5, 6, 7, 9, 11, …?
- What is the product of all the digits on a telephone dialpad?
- If you have 6 apples in one hand and 7 oranges in the other, what do you have?
- What has a face and two hands but no arms or legs?
- What occurs once in a minute, twice in a moment, but never in a thousand years?
- A man has four daughters, and each daughter has a brother. How many children does the man have?
- Both have four quarters.
- You don’t bury survivors.
- One thousand.
- Trick question. It’s not possible to overtake the last runner, because you’d have to be behind him, in which case you’d be the last runner.
- The mailman.
- 100. The pattern of numbers are the positive integers that do not have a t in them when spelled out.
- Big hands.
- A clock.
- The letter m.
- 5. One son is a brother to each of the daughters.
With the election two days in the rearview mirror, three states remain undecided in their Senate election:
- Louisiana, which uses an archaic and easily manipulated run-off system;
- Alaska, where Mark Begich claims that there are many uncounted rural votes; and,
- Virginia, the Old Dominion — my home state — with Mark Warner and Ed Gillespie in an apparent dead heat.
As of this morning, Warner was ahead 1,071,283 to 1,054,556. That’s a lead of 16,727 votes with 99.9% of precincts reporting. Local newspapers have declared Warner the “apparent winner,” but no concession has been offered.
Why hasn’t Gillespie conceded yet? Perhaps it has to do with simple math.
While 0.1% sure doesn’t seem like a lot — and it’s not, if you’re talking about the amount of alcohol in a bottle of whiskey — it can represent a lot — like when you’re talking about the amount of alcohol in your blood.
It’s also a lot when you’re talking about millions of votes. If the 2,179,235 votes counted so far represent 99.9% of all votes, then the remaining 0.1% represents 2,181 votes. If Gillespie gets all of them, that would bring him within 14,546 votes of Warner. Were Gillespie to get all of the provisional votes that are yet to be counted — the number of which is unknown — well, he probably still won’t win, but I suppose you can’t blame a guy for trying.
And let’s not forget, we’re talking about politicians. For most of them, delusion is a normal state of existence.
A politician’s wife called him from the hospital. “Honey, I had triplets!” she exclaimed. The politician responded, “I demand a recount!”
To be fair, many politicians think realistically — they just don’t think very often.
A cannibal goes to the butcher shop and notices that mathematician brain is selling for $1 a pound, but politician brain is selling for $4 a pound. “Is the politician brain really that much better?” she asks the butcher.
“Not really,” he says. “But it takes a whole lot more politicians to make a pound.”
I make a point of not having heroes, but there are people I greatly admire. Tom Magliozzi, the co-host of Car Talk who passed away yesterday, was one of those people.
Not only was Tom able to make other people laugh, he was always laughing himself. He and his brother Ray hosted Car Talk from 1977-2012, making folks laugh — and think — for 35 years.
In case you haven’t noticed, laughing and thinking are two of my favorite activities.
Every week, Tom and Ray would try “frantically to come up with a mediocre new puzzler,” a logical or mathematical problem that wouldn’t have an immediately obvious solution. Sometimes I’d be able to solve them, sometimes I wouldn’t, but I’d always enjoy them… and I’d laugh out loud while Ray read the puzzler and Tom offered commentary.
Below are two of my favorites, but you can find the full list of puzzlers at the Car Talk web site.
This first one sounds so simplistic, but most folks get tangled up in the details. Share it at your next department meeting, and see how many colleagues can solve it. You’ll be disappointingly surprised!
A store paid $6.75 for a shirt, and they then sell the shirt for $12. A man visits the store, buys the shirt, and pays with a $20 bill. The clerk gives the customer $8 in change, as expected. But unbeknownst to the clerk, the bill was counterfeit — instead of Andrew Jackson’s picture on the bill, it’s got Michael Jackson’s! In total, how much did the store lose on the entire transaction?
That one reminds me of the Marilyn Burns horse problem: You buy a horse for $50, sell it for $60, buy it back for $70, then sell it again for $80. Did you make money, lose money, or break even?
This next one is a classic that’s taken many forms. Finding a solution isn’t too hard… finding the simplest solution may take a little effort, though.
You have a four-ounce glass and a nine-ounce glass. You have an endless supply of water. You can fill or dump either glass. You can measure six ounces of water using these two glasses. What’s the fewest number of steps in which you can measure six ounces?
RIP, Click. I’m sure you’re already making people laugh and think upstairs.
On a recent Sunday Puzzle on NPR, Will Shortz gave a letter and a word, and the contestant was to guess the name of a popular TV show using an anagram of the letters (“Coming to TV This Fall: Anagrams,” Oct 12, 2014). For instance,
M + NAMED
gave the answer
This struck me as an interesting puzzle format. My only criticism is that it just wasn’t mathy enough.
But I’m not a problem maker, I’m a problem solver… so rather than cast aspersions at the puzzle, I’ll instead use the format to offer my own version.
Each of the 26 letters of the alphabet has been paired with a common English word. An anagram of the pair will yield a common math word. How many can you find?
- A + ERA
- B + AGLARE
- C + BITES
- D + NOTICER
- E + EDGERS
- F + SAUCER
- G + LEAN
- H + OPERABLY
- I + TANGLER
- J + INDUCTIONS
- K + SEW
- L + POSE
- M + RIPS
- N + AIMED
- O + PINT
- P + MYRIAD
- Q + AURES
- R + ENVIES
- S + RECITED
- T + HAM
- U + RAIDS
- V + EXERT
- W + ROPE
- X + SEA
- Y + PENTHOUSE
- Z + ORE
I don’t believe in providing an answer key, but you can find some help at Math Words, and you can click over to More Words if you run into real trouble. But give it the old college try before seeking assistance. Honestly, you’ll feel better about yourself if you solve these on your own.