When we recently bought honey roasted peanuts at the grocery store, Eli speculated that there were 215 peanuts in the jar.
“Okay, so now we have to count them,” Eli said.
“No,” I said, explaining that I didn’t want them touching food that others would be eating. I then showed them the back of the jar, which said that one serving contained about 39 pieces and the whole jar contained about 16 servings. They knew that 39 × 16 would approximate the number of pieces, and they estimated that the jar contained 40 × 15 = 600 pieces.
But then they wanted the actual value, and I wondered how we could use the estimate to find the exact product. More importantly, I wondered if it was possible to find an algorithm that would allow an easily calculated estimate to be converted to the exact value with some minor corrections.
My sons’ estimate used one more than the larger factor and one less than the smaller factor; that is, they found (m + 1) × (n – 1) to estimate the value of mn. A little algebra should help to help to provide some insight.
The product had a value of 600, so further refinement led to:
This led to an algorithm:
- Find an estimate with nice numbers.
- Add 1.
- Add the larger factor.
- Subtract the smaller factor.
This gives 600 + 1 + 39 – 16 = 624. And sure enough, 39 × 16 = 624.
This method works any time you want to find the exact value of a product when the larger factor is one more than a nice number and the smaller factor is one less than a nice number. Just estimate with the nice numbers, then follow the steps. The method can be modified if the larger factor is one less than a nice number and the smaller factor is one more than a nice number:
- Find the estimate.
- Add 1.
- Subtract the larger factor.
- Add the smaller factor.
So if you want to find the product 41 × 14, then the larger factor is one more than 40 and the smaller factor is one less than 15. The estimate is again 40 × 15 = 600.
Then 600 + 1 – 41 + 14 = 574. And sure enough, 41 × 14 = 574.
The same idea can be extended to numbers that aren’t the same distance from nice numbers. But that’s not the point. The intent was not to find general methods for every combination; instead, the hope was to use an easily calculated estimate as the basis for an exact calculation. I’m not sure this method completely succeeds, but it was fun for an afternoon of mental gymnastics.
Bert Tolkamp et al. were awarded an Ig Nobel Prize for answering a question that has long been on the minds of readers of this blog, and likely on the minds of the populace at large:
Are cows more likely to lie down the longer they stand?
I mean, seriously, how many nights have you lain awake pondering that question?
Their research revealed two startling facts:
- The longer a cow has been lying down, the more likely that the cow will soon stand up.
- Once a cow stands up, it’s impossible to predict how long until that cow lies down again.
I, for one, will rest easier knowing that these questions have finally been answered.
If you suffer from insomnia, the full article may be more valuable than Unisom, chamomile tea, or counting sheep.
You have to wonder if the researchers used a cow-culator to calculate the probabilities. Or perhaps that had to rely on techniques from advanced cow-culus.
Why do milking stools only have three legs?
Because the cow has the udder!
What do Greek cows say?
What do you call a male cow that swallows a hand grenade?
What do you call the same cow 5 seconds later?
Here are some other mathy cow jokes I’ve posted in the past.
According to CareerCast, three of the four best jobs in 2014 are in STEM fields: mathematician, statistician, and actuary. And the other — tenured university professor — might very well be a STEM career, too.
The worst job? Lumberjack, with a median annual salary of $24,000, a bad work environment, high stress, and a dismal hiring outlook.
Even though they’re on opposite ends of the best job spectrum, math folks and loggers have a lot in common. Both appreciate natural logs.
I learned this at http://www.lumberjack.com, which has a few interesting tidbits. But not enough to keep me interested, so I logged out.
And we all know that the grass is always greener, which is why some mathematicians opt for a life in the forest…
A math professor had enough of academic life, so he decided to become a lumberjack. He was hired by a logging firm, and he was told that he’d need to cut down 50 trees a day. On his first day, he was handed a chainsaw, and he went into the forest. When he returned to the office at the end of the first day, the foreman asked him, “So, how many trees did you cut down today?”
“Six,” replied the mathematician.
“That’s not enough,” said the foreman. “You’ll have to do better. Get up earlier tomorrow.” So he did, and again he went into the forest with a chainsaw. He returned at the end of the day, sweaty and exhausted. “How many’d you get today?” the foreman asked.
“Twelve,” replied the mathematician.
So the next day, the foreman went out to the forest with the mathematician. He started the chainsaw, started to cut, and explained to the mathematician what he was doing. When he finished, he said, “And that’s how you cut down a tree. Any questions?”
“Yeah,” said the mathematician. “What the hell was all that noise coming from the chainsaw?”
The French Quarter Festival and the NCTM Annual Meeting took place concurrently in New Orleans last week. So following five days of spectacular conversations and presentations at the conference, I headed to the festival for stage after stage of live music.
I sat on the lawn in Woldenberg Park, and the woman next to me was movin’ and groovin’ to the sounds of The Dixie Cups. I introduced myself, and she replied, “Hi, I’m Rhonda.” And the first thought that went through my head was…
Hard-on is an anagram of Rhonda.
What the hell’s the matter with me?
If you’re looking for a silver lining here — and believe me, I am — it’s that there are no other one-word anagrams of Rhonda. So at least I didn’t ignore a more socially appropriate anagram and jump straight into the blue.
But you have to wonder why that happened at all, instead of just accepting her name at face value and politely, automatically responding, “Nice to meet you.”
My mind has played games for as long as I can remember, often without my consent. The following are a list of some of them:
- Playing License Plate Algebra with the letters and digits on a license plate. For instance, if a Pennsylvania license plate has TFT to the left of the keystone and 567 to the right, and the keystone is then replaced by an equal sign, and some simplifying is done, this reduces to T2F = 567, and I search for order pairs (T, F) that make that equation true.
- Riding in a car, I’ll pick a speck of dirt on the window and pretend that it’s a laser/bomb/WMD. As I ride along, anything that the speck appears to touch while I look out the window is destroyed instantly.
- Sometimes, I’ll try to figure out what I’d do if a normal, daily event turned into a life-threatening situation (like this).
- Eating M&M’s two-by-two, one for each side of my mouth. (See my ruminations about a quest to find The Perfect Pack.)
- Having to step on an equal number of cracks with each foot, when walking on the sidewalk through our neighborhood.
- While playing basketball and other sports, getting fixated on a word — say, precise — and when I’m not dribbling or shooting, I’m finding anagrams of the word in my head, or I’ll start to combine pieces of letters — for instance, a c and an i without its dot could be used to form an a — so now I try to make anagrams of p, r, e, a, s, and e. And sure enough, I’ll stumble onto serape. But that’s not good enough. I’ll then return to precise, combine the r and i to make an n, and now I’ll look for anagrams of p, n, e, c, s, and e. There are none, so I’ll spend the rest of the game in a futile mental search. And two seconds after I convince myself that there are none to be found, the buzzer sounds, and I realize our basketball team has suffered its seventh straight double-digit loss. The defeat wasn’t entirely my fault, but my distractedness surely didn’t help matters, either.
What stupid games does your mind play?
I don’t know how else to say it, so I’m just gonna say it.
Fractions are full of sh*t.
Okay, not really. But if I have to hear one more time about how fractions are useful because of applications to cooking, I may commit hari-kari.
Before I jump into a diatribe, though, I absolutely have to share this improper fraction cartoon from Fat Rooster Studios (warning: rated PG-13).
It’s really hard to continue after that. But I’m gonna try.
There are three reasons that fractions are not really important in cooking.
- First, fractions only appear important because Americans are stupid. We insist on using the imperial system, and we measure dry ingredients in fractional parts of a cup. In other parts of the world, they don’t add 1 3/4 cups of flour to their recipe for croissants. Instead, they use 450 ml of flour. So making a half, a third, or a double recipe doesn’t involve operations with fractions.
- Second, ratios are important when cooking, not fractions. The exact amount of flour, sugar and baking powder in your chocolate chip cookies isn’t critical, so long as the ratio is 96 : 48 : 1. Approximately. Cooking is not an exact science. If your ratio of flour : sugar : baking powder = 98 : 45 : 2, you should still end up with a tasty dessert.
- Third — and, in my opinion, most importantly — great cooking derives from experience and approximation, not from exact measurements. My mother used to drive me crazy when she’d state, “But I followed the recipe,” if her normally fantastic lasagna came out less than fantastic.
Don’t believe me? Then watch the chef on a cattle drive as he makes chili over an open fire, and notice how he throws in a bucket of beans, two buckets of tomato sauce, and as much ground beef as he thinks is appropriate. You can bet your ass that Cookie ain’t got no measuring cups in the back of the chuck wagon. Or better yet, watch him make a cup of “six shooter coffee,” where his recipe is one handful of ground beans per cup of water. How much coffee is in a handful? Depends on the hand.
Maybe you think it’s just cowboys who estimate. Nope. Watch Emeril Lagasse as he adds a pinch of this, a dash of that, and — BAM! — the result is a grilled pork chop for which tourists pay $30 when they visit New Orleans.
If you need proof that ratios are the key mathematical element to successful cooking, listen to Dr. Mark Hadley. He claims that perfect ravioli is obtained when the ratio of pasta : filling : sauce = 45 : 45 : 10, which includes just enough olive oil “to give a thin layer of 200 microns over the surface of all the pasta – enough to make it glisten, resulting in the perfect mouthful.”
But, you know what? We shouldn’t let reality get in the way of a good story. Let’s please continue to perpetuate the myth that fractions are important — nay, critical — by including exorbitant numbers of cooking problems in the fraction units of textbooks. As far as I can surmise, the majority of fraction problem authors have never actually cooked. Here’s a typical problem:
The following recipe for Blueberry Bubble Loaf makes 12 servings.
- 2 cups cereal that contains blueberries
- 1 cup brown sugar
- 1/2 cup butter
- 2 packages of refrigerated buttermilk biscuits
Rewrite the recipe so that it makes 4 servings.
Let’s assume that this isn’t stupid. (Though it is, right? I mean, it might be reasonable to make 6 servings, since that would require just one package of refrigerated buttermilk biscuits. But to make just 4 servings? That means you’ll only need 2/3 package of refrigerated biscuits. What are you supposed to do with the other 1/3 of the package?)
But as presented, the solution requires that each ingredient be divided by 3. That gives 2/3 cup cereal, 1/3 cup brown sugar, 1/6 cup butter, and the aforementioned 2/3 package of refrigerated buttermilk biscuits. I decided to make this recipe.
- I have a 1/3-cup measure in my cooking drawer, so the first two ingredients were no problem.
- I don’t have a 1/6-cup measure. I could have measured 1/3 cup of butter and used an educated guess to divide the amount in half. Instead, I can just filled a 1/4-cup measure, and decided that that was close enough. Good enough for government work.
- I’ll only need 6 2/3 of the 10 biscuits that come in a 12-ounce container of refrigerated buttermilk biscuits. WTF? I decided that 7 biscuits is close enough, and I gave 3 uncooked biscuits to my dog. He’s happy at this development. I hope he doesn’t get worms.
I cooked the blueberry bubble loaf as directed, and it came out fine. Except that the total mixture only filled 1/3 of a bread pan, and it created a loaf that was only one inch tall. That’s not a loaf; that’s a tortilla.
But generally speaking, there was no material difference between the original loaf and my reduced-height loaf, despite the imprecision in my measurements. And do you know why there was no difference?
Because fractions are full of sh*t.
Now check this out. The following is a cake recipe from About.com.
- 2 cups cake flour
- 2 teaspoons baking powder
- 1/2 teaspoon salt
- 1/2 cup butter, softened
- 1 cup sugar
- 3 large eggs
- 2 teaspoons vanilla
- 3/4 cup milk
And here’s a vanilla cake recipe from Country Living.
- 1 1/2 cups cake flour
- 1 1/2 teaspoons baking powder
- 1/4 teaspoon salt
- 1/2 cup butter, softened
- 1 cup sugar
- 2 large eggs
- 1/2 teaspoon vanilla
- 1/2 cup milk
The second recipe requires 3/4 as much flour as the first recipe. If fractions really mattered, then every ingredient in the second recipe should have an amount that is 3/4 as much as the first recipe. But they don’t. There is 1/2 as much salt, the same amount of butter, the same amount of sugar, 2/3 as many eggs, 1/4 as much vanilla, and 2/3 as much milk.
So I’ll say it again.
Fractions are full of sh*t.
At least when it comes to cooking.
Fractions are, however, fodder for some great jokes.
Five out of four Americans have trouble with fractions.
Sex has a lot in common with fractions.
It’s improper for the larger one to be on top.
It’s hard to tell the difference between a numerator and a denominator. There is a fine line between them.
Two-thirds of Americans have trouble with fractions. The other half can handle them just fine.
Son: Can you help me find the lowest common denominator of 1/2 and 1/3?
Dad: You mean they still haven’t found it? They were looking for that when I was a kid!
Last Wednesday evening, Steven Strogatz delivered the opening session at the 2014 NCTM Annual Meeting in New Orleans.
His talk shared a title with his bestselling book, The Joy of x. During the talk, he described five keys in bringing math to the masses, including what worked — and what didn’t — when he wrote a 15-part series for the New York Times Opinionator blog. He identified the five elements as follows:
- Listen to Your Wife (Husband, Partner, etc.)
I was ecstatic to see humor at the top of his list. As an example of humorous mathematics, he played the now infamous Verizon .002 phone call.
As it turns out, the week was full of humor. (Who’da thunk, at a math conference?) Bill Amend, author of the comic strip Foxtrot, delivered the closing session at the conference. Earlier the same day, yours truly gave my soon-to-be-famous Punz and Puzzles talk to a standing-room-only crowd.
Following the conference, Jennifer Silverman tweeted the following:
The joke I actually told was:
Why is 6 afraid of 7?
Because 7 8 9.
Why don’t jokes work in base 8?
Because 7 10 11.
But who cares? If her son is laughing, I’m smiling!
After my session, I was accosted by an overly gregarious gentleman who had written a collection of math jokes on a yellow sheet of paper in red ink. While a queue of people who wanted me to sign their copies of Math Jokes 4 Mathy Folks formed behind him, he proceeded to tell me ALL of the jokes that he had written. He shared one joke that I found funny:
Though funny may not be the right word. Perhaps interesting is a better choice, because Pythagorean serum was the name we used for the concoction that was served at my book release party.
And while at the conference, I was told a joke that I think works better visually than verbally…
Last but not least, I was sent the following image of Newton’s Cradle by Zachary Kanin with the suggestion that maybe I use it the next time I present:
It was 7:02 a.m. on a Saturday morning. Alex ran into my bedroom and woke me from an incredible dream — I was speaking to Riemann, Newton, Pascal, and several other dead mathematicians, and they were just about to reveal an odd perfect number.
“Deedy!” he yelled — somehow daddy has been transformed to deedy in my house — and I sat bolt upright.
“What?” I asked, rubbing the sleep from my eyes.
“Do you know what 58 × 46 is?”
“I have no idea,” I told him. “What is it?”
“I don’t know, either,” he said. “But it was one of the questions Eli gave me on this morning’s math quiz.”
A few minutes later, he had the answer to that exercise and several others that appeared on the quiz that his brother had created for him.
This is what my twin six-year-olds do. They give each other math quizzes. With two-digit multiplication exercises and slightly more complex combinatorics problems (“How many two-digit numbers don’t have a 3 in them?”). For fun.
So when they recently brought home a math game from school called One Less — in which each player rolls a die and has to place a token on a number that is “one less” — my only thought was, “Really?”
Verbatim, here are the directions to the game:
Each player gets 10 counters. Players take turns rolling a die and placing a counter on a number that is one less than the number rolled. The game ends when one player has placed all 10 counters.
Upon reading the directions, I had one question: RUFKM?
- Kids who perform multi-digit multiplication for fun are asked to do single-digit subtraction for homework.
- The game ends when one player uses all his counters. Mind you, no one actually wins — the game just ends.
Well, this will never do.
I opted not to send a note to the teacher about how they need to increase the rigor of their mathematics curriculum. Doing so would just make me that guy.
Instead, I decided to turn a bad game into a good game. So we modified the rules as follows:
On a turn, a player rolls a die and places a coin on a space with a value one less than the number rolled. Players alternate turns. A player earns a point each time she gets three of her coins in a row. Game ends when one player has used all 10 coins. The winner is the player with the most points.
This allowed for all kinds of interesting questions:
- What’s the maximum possible score in a game?
- What’s the best arrangement of numbers on the game board?
- Will the first player always win?
- How does the game change if points are awarded for two-in-a-row or four-in-a-row?
- How does the game change if scoring gives 1 point for one-in-a-row, 3 points for two-in-a-row, 6 points for three-in-a-row, 10 points for four-in-a-row, and so on?
- How much wood could a woodchuck chuck if a woodchuck could chuck wood?
We determined the answer to the first question (8 points), and we agreed that we didn’t much care to know the answer to the last question. It seems like the first player shouldn’t always win; but he did in all of the games that we played.
As for the best arrangement of the game board, I have no idea. But if you’d like to explore, several game boards are included in the PDF link below.
What modifications have you made to games to improve them or to make them more mathematically robust?