It’s Been Too Long
I can’t help but channel my inner Foo Fighter as I start this post.
This is a call to all my past resignations;
It’s been too long…
Too long, indeed. My last post was August 8. I’ll use starting a new job and moving my family across the country as my excuse, but you deserve better. To get back into the swing of things, and to try to earn back your trust, I’ll start with a listicle of sorts. Let’s call it 12 Math Jokes You Should’ve Heard By Now. (Think that’s enough click-bait to get this post a thousand likes? We’ll see.)
—
Knock, knock.
Who’s there?
Pi.
Pi, who?
Don’t listen to me. I’m irrational.
I picked up a hitchhiker, and he seemed like a good guy. We had a pleasant conversation for a few minutes, and then he asked, “Thanks for picking me up. But weren’t you afraid I might be a serial killer?”
“Nah,” I said. “The odds of two serial killers in one car is extremely unlikely.”
I had a calculus test this morning. I thought about praying for a good grade. But I know God doesn’t work that way. So instead, I copied off my classmate who’s been accepted to Harvard, and I prayed for forgiveness.
I asked my wife, “What would you do if I won the lottery?” She said she’d take half and leave me. “Great!” I said. “I just won $10. Here’s $5. Don’t forget to write.”
Why did the math student ask a chemist for help?
He heard chemists have a lot of solutions.
Why was the fraction skeptical about marrying the decimal?
Because one of them would have to convert.
Atheists have difficulty with exponents because they don’t believe in higher powers.
The nurse apologized after realizing he’d put the splint on the patient’s incorrect finger. “You were really close,” said the patient. “You were only off by one digit.”
How is x^{2} + 2x + 4 = 0 like an artificial holiday tree?
Neither have real roots.
At a job interview, tell them you’re willing to give 110%. Unless you’re interviewing to be a statistician.
My girlfriend is like the square root of -100. She’s a perfect 10, but purely imaginary.
My wife calls me obtuse triangle, because I’m never right.
Ch-Ch-Ch-Changes — in Job and Location
A few days back, I mentioned that I had a new job and had moved across the country, and I said I’d write more about that later. Well, it’s later.
After six wonderful years of developing a highly-rated, award-winning, interactive math textbook at Discovery Education, I’ve taken a new position at the Math Learning Center, a non-profit organization in Portland, Oregon. The Math Learning Center (MLC) is the publisher of Bridges, an award-winning elementary math curriculum.
The reason for the change? Well, actually, there are several…
- MLC is not-for-profit, so any money raised from curriculum sales is used to improve the materials and professional development offerings.
- The mission of the Math Learning Center is “to inspire and enable individuals to discover and develop their mathematical confidence and ability.” It’s pretty easy to get behind a goal like that.
- Last but not least, the MLC staff might be the friendliest group of individuals I’ve ever met. To boot, they’re bright, hard-working, and dedicated to the organization’s mission.
With all that, the decision to join MLC was a rather easy one. If you can’t tell, I’m pretty excited about the change. I’ll be the new Chief Learning Officer, affectionately known as the CLO.
Time out for a puzzle.
Can you fill in the blanks to form a 16-letter math term that contains the letters CLO in order? Hint: think about transformational geometry or turning off the faucet.
_ _ _ _ _ _ _ C L O _ _ _ _ _ _
Relocating from Virginia to Oregon is a big deal. It’s nearly 2,800 miles — or 14 states, or 42 hours in a car — from our old house to our new one. Consequently, we hired a moving company to help with packing and shipping. When Lily from the moving company arrived, she asked if we had any “high-value items” to be transported, such as expensive jewelry or fur coats. (But not a real fur coat. That’s cruel.) I said that I didn’t think so, but then I asked what they consider a high-value item. Lily’s answer used a completely acceptable but surprising unit rate:
anything over $100 per pound
With that metric, it was suddenly obvious that we had several high-value items in our home. The first was a pair of diamond earrings that I had given my wife recently for our 15th anniversary. Since 5 carats = 1 gram, these small hunks of rock have a retail value of nearly $4,000,000 per pound, significantly above the moving company’s threshold.
The other high-value items were, well, us. The “value of statistical life,” or VSL, is a measure of the value of a human life. Its exact amount depends upon which federal agency you reference. The Environmental Protection Agency (EPA), for instance, pegs the VSL at $10 million. That means that I’m worth approximately $50,000 per pound, my petite wife is worth nearly $80,000 per pound, and our twin sons are worth well over $100,000 per pound each.
Granted, our value density isn’t as high as diamond, but we’re still pretty darn valuable.
A cannibal goes into a butcher shop, and he notices that the market specializes in brains. He sees that the butcher is selling engineer’s brain for $1.50 per pound, mathematician’s brain for $2.25 per pound, and politician’s brain for $375.00 a pound. Flabbergasted, he asks the owner why the huge difference in price. The butcher replies, “Do you have any idea how many politicians it takes to get a pound of brains?”
In the end, neither the diamond earrings nor any member of our family were loaded onto the moving truck. A week later, we’re adapting nicely to Portland culture, and I start my job at Math Learning Center in just a few days. Wish me luck!
It’s About Time
I know, I know. It’s been a really long time since my last post. Nearly six months ago — February 25, to be exact.
I’ve got a good excuse, though. I took a new job, and I moved across the country. (More about that later.)
For now, I’m going to ease back into this with a simple post full of jokes. And I know what you’re thinking: “It’s about time!” So in honor of you, here are a collection of math jokes about time.
Did you hear about the hungry clock?
It went back four seconds.I lost my job at the calendar factory. My boss was mad that I took a few days off!
Mondays are an horrendous way to spend 1/7 of your life.
Traditional calendars are for the week-minded.
Did you hear about the two thieves who broke into a house and stole a calendar?
They each got 6 months.A broken clock is still correct twice every day.
The problem with calendars? In one year, out the other.
What’s the difference between a mathematician and a calendar?
The calendar has dates.The scientist dropped a watch into a beaker. She was hoping for a timely solution.
What did the hour hand say to the minute hand?
“Don’t listen to that other guy. He’s got second-hand information.”A calendar doesn’t feel well and visits the doctor. The doctor tells him, “I’ve got some bad news for you. You’ve got 12 months.”
My calendar was printed upside down. It was an interesting turn of events.
Did you hear about the calendar who owed money to a mobster?
His days are numbered!What type of candy never arrives on time?
Choco-late.When I was young, we were so poor that I had to use old calendar pages to wipe after defecating.
The worst days are behind me.What is a calendar’s favorite fruit?
Dates.How many months have 28 days?
All of them.How many seconds are in a year?
12: January 2, February 2, March 2, …
Okay, for reals regarding that last one. In a 365-day, non-leap year, there are 31,536,000 seconds. That’s kind of a fun number, because its prime factorization is…
…and the only digits in the prime factorization are the four single-digit primes. Cool stuff.
When Math Falls into the Wrong Hands
My brother-in-law recently forwarded an email that contained a lot of images plucked from various degenerate corners of the internet, and he suggested that this one could go into my next book:
I suppose it’s funny enough, and I guess it’s technically a math joke, but there’s a problem.
It doesn’t work.
I know, I know. Most people just read the joke, get the humor that the note’s author has used some odious expression to represent the PIN code, and go on about their day. Plus, I’ve heard that less than 1% of the world’s population has taken calculus, so there aren’t too many people who could actually check the math. Not to mention, how many of them would care enough to do so?
Uh… I can think of at least one person who cares enough.
While it’s certainly egotistical to think that I’m the only one in the intersection, it’s likely offensive to include anyone in the intersection who really wouldn’t want to be. So apologies to Matt Parker, Des McHale, Colin Adams, Ed Burger, or any of the other funny math folks who think they should have been included.
Anyway, where was I? Oh, right. Bad math.
The definite integral in the joke sent by my brother-in-law doesn’t yield a four-digit positive integer.
In fact, it yields a very irrational number with a lot of digits:
-2.58208625277854512796640677001459519299166472798789689499…
So unless the PIN code for that bank card has an infinity of digits, well, this is going to be problematic.
I propose, instead, that the joke be rewritten to use the following:
Would it be less funny? Probably. But at least it’d be accurate.
Not to mention, it would be a significantly more fair to Darling. Honestly, no one should ever have to do integration by substitution.
The Other Golden Ratio
You’re a math geek. I know that to be true, because you’re reading this blog. And I also know that when you hear golden ratio, you think of this:
Or this:
But there’s another, different — and, dare I say, better? — golden ratio that may be even more important to learn. Especially if you’re one of the mathy folks to whom this adage applies:
Wherever you find four mathematicians, you’ll likely find a fifth.
As Homer Simpson says, “It’s funny ’cause it’s true.”
Like many classroom teachers, I’m often ready for a cocktail on Friday afternoons. And like those teachers, I don’t want to spend a lot of time thinking about it; I don’t want to rummage for a recipe; I just want to relax and have a drink. But, I also don’t want to have the same cocktail every Friday; variety is the spice of life.
So, what’s a boy to do?
Simple. Follow the advice from the folks in the Food Hacks division at Wonder How To, who claim that the following ratio will yield a delicious cocktail every time:
2 : 1 : 1 :: alcohol : sour : sweet
Right now, you’re probably scratching your head and thinking, “Can it really be that simple?”
I’m here to tell you, friends — it is.
For instance, 2 parts tequila, 1 part lime juice, and 1 part triple sec? That’ll get you a tasty margarita.
And 2 parts bourbon, 1 part lemon juice, 1 part simple syrup? None other than a classic whiskey sour.
If you combine 2 parts tequila, 1 part rhubarb liqueur, and 1 part malic acid — it’s what gives green apples their tartness — then you’d be getting close to a drink they call The Scarlet Lantern at Bar Congress in Austin, TX. (They also mix in black cardamom-strawberry shrub. Keep Austin weird, eh?)
Now that you know about the other golden ratio, here’s what you need to do: Organize your liquor cabinet into two parts, hard alcohol and sweet mixers. Then, make sure you keep a couple of sour mixers in your fridge. When you get home on Friday, just grab a bottle from each side of the liquor cabinet and one more from the fridge, pour, and — voila — instant happiness.
Wanna get a little crazy? Find your favorite cube-shaped random number generator, give it three rolls, then choose the appropriate item from each column in the table below.
Die Roll |
Alcohol |
Sour |
Sweet |
1 |
Tequila |
Lemon Juice |
Simple Syrup |
2 |
Vodka |
Lime Juice |
Triple Sec |
3 |
Rum |
Grapefruit Juice |
Cointreau |
4 |
Rye Whiskey |
Strawberry Shrub |
Gran Marnier |
5 |
Bourbon |
Bloody Mary Mix |
Honey-Ginger Syrup |
6 |
Mezcal |
Dry Cider |
Grenadine |
Personally, I’m hoping for 6-3-5, which is kind of like a Mezcal Paloma, sort of like a Honey and Smoke, but not really similar enough to be either one. So, I guess I get to name it. And given what I’ve heard about the fat-burning properties of honey and grapefruit juice, I’m going to call this newly minted beverage the Weight Watcher.
Now we just need to come up with names for the other 215 combinations. I’ll get started on that right away… soon as I finish this drink.
Two the Hard Way (and an Easy Way)
During our trip to Arizona for winter break, two problems surfaced organically while we were on holiday. (Sorry. Two math problems. There were lots of non-math problems, too, but we don’t have time for all that.)
On the plane, the interactive in-flight map showed the outside temp, toggling between Fahrenheit and Celsius. That led to the following MJ4MF original problem, which I thought — and still think — is pretty good:
The conversion between Fahrenheit and Celsius temperatures follows the rule F = 9/5 C + 32. Sometimes, the temperature is positive for both Celsius and Fahrenheit; sometimes, the temperature is negative for both Celsius and Fahrenheit; and other times, Fahrenheit is positive while Celsius is negative. What is the least possible product of the Fahrenheit temperature and its corresponding Celsius temperature?
You can pause here if you’d like to solve this before I present a spoiler.
Before I reveal two different solutions, allow me to digress. It could be that the statement about the temps sometimes being positive and sometimes being negative is denying a teachable moment. The graph below shows the linear relationship between the two temperature scales. Perhaps a good classroom question is:
When will the product CF be positive and when will it be negative?
Or maybe a better question is:
When are C and F both positive, when are they both negative, and when do they have different signs?
So, to the problem that I posed. As I thought about it on the plane, I concluded that if F = 1.8C + 32, then the product CF = 1.8C^{2} + 32C. I then used calculus, found the derivative (CF)’ = 3.6C + 32, set that equal to 0, and concluded that C = ‑8.89, approximately. The corresponding Fahrenheit temperature is F = 16, so the minimum product is roughly ‑142.22.
Using calculus was like rolling a pie crust with a steamroller, though. I could have just as easily graphed the parabola and noted its vertex:
If I had used the other form of the rule, namely C = 5/9 (F ‑ 32), things might have been a little easier. Maybe. In that case, setting the derivative equal to 0 yields F = 16, which is arguably a nicer number. But then you still have to find the corresponding Celsius temperature, which is C = ‑8.89, and the product is still roughly ‑142.22. So, not much easier, if at all, and again graphing the parabola and noting its vertex would have done the trick:
The only real benefit to using this alternate version of the rule is that it provides a reasonable check. Since both methods — and both graphs — yield an answer of ‑142.22, we can feel confident in the result.
But there’s an easier way to solve this one.
Thinking this was a good problem — and because I like when my sons make me feel stupid — I gave it to Eli and Alex. Within seconds, Eli said, “Well, F is positive and C is negative between 0°F and 32°F, so the minimum will occur halfway between them at F = 16. That means C = ‑80/9, so it’s whatever ‑1280/9 reduces to.” (Turns out, -1280/9 = ‑142 2/9 ≈ ‑142.22.)
Eli hasn’t taken calculus, so he doesn’t know — or, at least, he hasn’t learned — that the minimum product should occur halfway between the x– and y‑intercepts of the linear graph. Yet, he had an intuitive insight that just happens to be true. As a result, what took me about five minutes of deriving and manipulating took him about five seconds.
The second problem arose at the grocery store. Among our purchases was a box of sugar cubes, which contained, surprisingly, 126 cubes.This number is surprising in the sense that it’s not a number you’ll see very often, except for an occasional appearance in the ninth row of Pascal’s Triangle, or maybe if you’re a chemist searching for stable atoms.
A question that could have arisen from this situation involves surface area and volume:
A rectangular prism with integer dimensions has a volume of 126 cubic units. What is the least possible surface area?
That’s not the question that was shared with Alex and Eli, though. (The answer, if you care, is 162 square units, which results from a 3 × 6 × 7 arrangement — which, in fact, is the exact arrangement of cubes in the box above. I suspect this is not a coincidence.)
The problem that I shared with my sons involved probability:
Imagine that the arrangement of cubes is removed from the box intact, and all six faces of the prism are painted red. Then one of the sugar cubes is selected at random and rolled. What is the probability that the top face of the rolled cube will be red?
The boys made an organized list, as follows:
Painted Faces | Number of Cubes |
3 | 8 |
2 | 40 |
1 | 58 |
0 | 20 |
Further, the boys reasoned:
- P(cube with 3 red faces, red face lands on top) = 8/126 x 1/2 = 8/252
- P(cube with 2 red faces, red face lands on top) = 40/126 x 1/3 = 40/378
- P(cube with 1 red face, red face lands on top) = 58/126 x 1/6 = 58/756
Therefore,
- (P of getting a red face) = 8/252 + 40/378 + 58/576 = (24 + 80 + 58) / 576 = 162/576 = 9/42
Wow! That seems like a lot of work to get to the answer. Surely there’s an easier way, right?
Indeed, there is.
Notice that the penultimate step yielded the fraction 162/576. The numerator, 162, may look familiar. It’s the answer to the question that wasn’t asked above, the one about the least possible surface area of the prism. That’s no coincidence. In total, there will be 162 faces painted red. And there are 6 × 126 = 576 total faces on all of the sugar cubes (that is, six faces on each cube). This again suggests that the probability of rolling a red face is 162/576.
Did you happen to notice that the volume and surface area use the same digits in a different order? Cool.
So there you have it, two problems, each with two solutions, one easy and one hard. Or as mathematicians might say, one elegant and one common.
It’s typical for problems, especially problems worth solving, to have more than one solution strategy. What’s the trick to finding the elegant solution? Sadly, no such trick exists. Becoming a better problem solver is just like everything else in life; your skills improve with practice and experience. It’s akin to Peter Sagal’s advice in The Incomplete Book of Running, where he says, “You want to be a writer? […] Just sit down and write. The more you write, the better a writer you will become. You want to be a runner? Run when you can and where you can. Increase your mileage gradually, and your body will respond and you’ll find yourself running farther and faster than you ever thought possible.” You want to be a problem solver? Then spend your time solving problems. That’s the only way to increase the likelihood that you’ll occasionally stumble on an easy, elegant solution.
And every once in a while, you may even solve a problem faster than your kids.
Silent Letter Night
Several weeks ago, Will Shortz presented an NPR Sunday Puzzle in which he stated a word and a letter, and the resulting collection would be rearranged to form a new word in which the added letter is silent. For instance, if Will gave RODS + W, the correct answer would be SWORD, in which the W is silent. (Note that the collection of letters is also an anagram of WORDS, but the W isn’t silent.)
At a time of year known for silent nights, it seems like a puzzle involving silent letters is completely appropriate. I’ve borrowed Shortz’s idea and extended it a bit; some of the clues in the list below have more than one silent letter added. Many items in the list are related to today’s holiday; and, because this is a math blog, the others are related to mathematics. In full disclosure, two of the answers are proper nouns.
Enjoy, and happy holidays!
- TO + W =
- TON + K =
- TOGS + H =
- GEE + D =
- TIN + G + H =
- SIN + G =
- SIN + E =
- CORD + H =
- HOLE + W =
- HEART + W =
- TINNY + E =
- COINS + E =
- NOELS + M =
- PILES + E + L =
- FRAME + T =
- REDACT + E + S + S =
- RACISMS + H + T =