A question from Brain Quest Grade 4:
Eli responded belligerently.
It’s not just mathematicians. Everyone who knows that would call them “parallel.”
How do you like that? Not only is my son mathematically literate, but he’s a sarcastic smart-ass, too. I couldn’t be more proud.
In honor of Eli…
What do you call a two-headed canary?
What do you call a geometer who spent all summer at the beach?
What do you call the motherboard on your spouse’s computer?
What do you call two fishermen who fish standing up?
What do you call a number that can’t keep still?
A roamin’ numeral.
What do you call a math teacher who loses control of his pupils?
Today is a great date, and I almost missed it!
Today’s date (in U.S. format) is the last time this century that the month, date, and year are consecutive numbers. If you choose not to celebrate this momentous occasion, you’ll have to wait almost 89 years for this to happen again.
Another great date with consecutive numbers happened 5 years ago.
That’s the date that Math Jokes 4 Mathy Folks was published.
And I rather like 12/11/14. That’s just two days ago, when Math Jokes 4 Mathy Folks reached a rank of 2,210 on Amazon. That’s the highest sales rank it’s ever received. Woo-hoo!
Every year around this time, there is a significant spike in sales of MJ4MF. Ostensibly, it’s a good gift to give your engineer husband, statistician wife, or geometry teacher. And I am ecstatic that so many people are enjoying the book. But I’m wondering if we can blow the roof off of the Amazon rank; with a concerted effort, can we get the ranking of MJ4MF to below 1,000?
Here’s my request:
If you’re thinking of buying Math Jokes 4 Mathy Folks from Amazon for someone as a gift this holiday season, please make your purchase of MJ4MF between noon and midnight ET on Tuesday, December 16. (Use this conversion chart if you’re in a different time zone.)
Ordering by Tuesday, December 16 will still allow the book to arrive in time for Christmas or the last night of Chanukah, especially if you have Amazon Prime.
Since Amazon sales rank is based on a 24-hour period, any purchase on Tuesday will help with the ranking, so we don’t need to be much more specific than that.
And if you’re not thinking of buying Math Jokes 4 Mathy Folks this holiday season, well, what the hell is wrong with you? All the cool people are doing it.
What about 3 three-digit numbers?
Or 4 four-digit numbers?
Okay, let’s get really crazy… how long would it take you to find the sum of 10 ten-digit numbers?
You can decide whether you’ll do the calculation in your head, on a calculator, or with paper and pencil. Your choice.
With a calculator, it took me 91 seconds to find the sum of 10 ten-digit numbers.
Without a calculator, the winner of the Mental Calculation World Cup 2014 needed only 242 seconds to complete 10 problems in which participants were asked to add 10 ten-digit numbers. On average, that’s just 24 seconds to do in his head what took me a minute-and-a-half with technology.
Competitors at the MCWC do a number of mental calculation tasks. The following exercises will give you an idea of the computations that they do.
Exercise 1. Sum of 10 ten-digit numbers.
Exercise 2. Multiplication of 2 eight-digit numbers.
Exercise 3. Square root of a six-digit number.
Exercise 4. Day of the week for a calendar date.
Exercise 5. Multiplication of 3 three-digit numbers.
These sample exercises are taken from the examples in the MCWC 2014 Official Rules.
If you were able to complete all five of those exercises in, say, less than 3 minutes, then you might be ready for MCWC 2016. (Note that for Exercise 3, you needed to be accurate to within 5 × 10-6.)
But if you’re like me, you’ll probably want to skip the competition and keep your calculator close at hand.
Mike Birbiglia said:
I didn’t realize I was good with computers till my parents bought one.
My wife’s cousin Natalie — now in her 60’s — has a more pragmatic explanation for why older folks are less tech-savvy than the average bear.
Perhaps Baby Boomers, having grown up during the Cold War, are afraid of what might happen if you push the wrong button.
Many of us have love-hate relationships with computers. Some of us have hate-hate relationships with them. To wit:
There are two types of computers in the world: those that waste your time, and those that waste your time faster.
A computer once beat me at chess, but it was no match for me at kickboxing.
Men are like computers. They’ll do what you want, but not until they’re turned on and stroked in the right sequence.
A computer lets you make mistakes faster than any invention in human history, with the possible exceptions of handguns and tequila.
Problems in the MathCounts School Handbook are presented “shotgun style,” that is, a geometry problem precedes a logic puzzle and follows a probability question. (I worked for MathCounts for seven years and then served as a writer and chair of their Question Writing Committee, so I’m not unbiased.)
By comparison, textbooks often present 50 exercises on the same topic, each one only minimally different from the previous one. That tips the hand to students, methinks, and makes them realize, “Oh, I just need to do the same thing.” I prefer the MathCounts approach, where students have to dig into their bag of tricks to find a viable solution strategy.
With that in mind, here are a few problems I’ve encountered recently, each one not like the others.
Problem 1. The simple polygon is made from 73 squares, connected at their sides. What is the perimeter of the figure?
Problem 2. What is the expected number of times that a six-sided die must be rolled to get each number 1–6?
Problem 3. A wall is to be constructed from 2 x 1 bricks (that is, bricks that are twice as long in one direction as the other). A strong wall must have no fault lines; that is, it should have no horizontal or vertical lines that cut entirely through a configuration, dividing it into two pieces. What is the minimum size of a wall with no fault lines? The figure below shows a 3 × 4 wall that has both horizontal and vertical fault lines.
Please share great problems you’ve recently encountered in the Comments.
No answers, but here are some hints.
Problem 1. Look for a pattern.
Problem 2. Check out this simulation for the Cereal Box problem.
Problem 3. The smallest arrangement without a fault line is larger than 3 × 4 and smaller than 10 × 10.
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!