## Posts tagged ‘million’

### A Ton of Money (or Maybe More)

One of my favorite resources from Illuminations is Too Big or Too Small, a collection of three classroom activities that develop number sense, one of which is the following problem:

Just as you decide to go to bed one night, the phone rings and a friend offers you a chance to be a millionaire. He tells you he won $2 million in a contest. The money was sent to him in two suitcases, each containing $1 million in one-dollar bills. He will give you one suitcase of money if your mom or dad will drive him to the airport to pick it up. Could your friend be telling you the truth? Can he make you a millionaire?

This problem is from the book *Developing Number Sense in the Middle Grades* by Barbara Reys and Rita Barger, published by NCTM in 1991. So it’s not new, but it’s still good.

My first attempt to use this problem with students was dreadful (details below), but I’ve used this problem successfully many times since. Yet something about it always bothered me. I’m not opposed to fictitious scenarios if they get students interested. But this scenario, in which a friend claims to win $2,000,000 and needs a ride to the airport, seems too contrived and not adventurous enough. Luckily, I recently had food poisoning and spent an entire Saturday on the couch watching bad movies. While watching *Rush Hour* (1998), I found a scenario that I liked a whole lot better…

In the clip, the kidnapper asks for the following:

- $20 million in $50’s
- $20 million in $20’s
- $10 million in $10’s

Now the questions of “How much would that weigh? How big a case would you need to carry all of it?” seem a little more meaningful.

I’ll channel my inner Andrew Stadel here. For both the weight and volume:

- Give an estimate that you know is too low.
- Give an estimate that you know is too high.

Now, do the calculations, and see how close your intuition was.

When I first used this task with students, I was anticipating a great discussion about how to estimate the weight and volume of the money. I suspected that some students might estimate that you could fit 5 or 6 bills on a sheet of paper, there are 500 sheets of paper in a ream, a ream weighs about 5 pounds, yada, yada, yada. Instead, one student raised his hand and said:

A dollar bill weighs exactly 1 gram.

I asked how he knew that. “Do you collect money? Are you a numismatist?”

No. That’s how drug dealers measure cocaine. They put a dollar bill on one side of a scale, and they put the cocaine on the other.

“Oh,” I said.

Some days, your students learn something from you. And some days, you learn something from them.

After you estimate the weight and volume, check your answer by clicking over to reference.com.

If you use this video clip and activity in a classroom with students, I’d love to hear how it goes. Please post about your experience in the comments.

### If I Had a Million Dollars…

The Daily Prompt at the Daily Post @ WordPress for December 28 gave the following hypothetical situation:

You’ve just won $1 billion dollars in the local lottery. You do not have to pay tax on your winnings. How will you spend the money?

I would have written about this sooner, but I’ve been too busy (a) fantasizing about how I’d spend that kind of money, (b) sending emails to the Daily Post telling them that they needn’t be so greedy; a million dollars would be plenty, and (c) sending more emails to the Daily Post telling them that the lottery is a tax on the mathematically challenged, that it’s insane to think that such a lottery would have no tax implications, that no lottery has ever had a billion-dollar prize, and that promulgating the possibility of winning such an unlikely sum only gives hope to those who should be putting their money in a savings account instead.

But I digress.

Part of me thinks I’d heed the advice of the Barenaked Ladies…

If I had a million dollars

I’d buy you some art

A Picasso or a Garfunkel

But I’m not sure that a million dollars would be enough to afford a painting by the world’s greatest cubist or to purchase the poet who rode Paul Simon’s coattails to musical fame, so instead I’d commission a mathematical sculpture by Zach Abel. Or maybe I’d just buy binder clips and construct some sculptures myself — at $3.68 per dozen, I could afford enough binder clips to make 746,268 copies of *Stressful*:

Part of me thinks I’d just withdraw the money in $1 bills from the bank. But how would I get it home? Do you have any idea how much that would weigh? Take some time to figure it out… the result will surprise you. The FAQ at the Bureau of Printing and Engraving might be helpful, as might this picture of a million dollars, although it’s $100 bills, not singles:

And here’s a great problem about $1,000,000, which I learned from Martin Gardner:

On January 1, I deposited $

xin a bank account. On January 2, I deposited $yin the same account. Every day thereafter, I deposited an amount equal to the sum of the previous two days’ deposits. On January 20, I deposited exactly $1,000,000. How much did I deposit on January 1?

### How Many Share Your Birthday?

This afternoon, we celebrated Alex and Eli’s sixth birthday with a Disney-themed Cinco de Mayo party. The kids all wore Mickey Mouse ears, while the parents drank lots of margaritas. Tonight’s “bedtime math” question for my sons was the following:

You celebrated your birthday on May 2. How many other people in the world do you think celebrated their birthday on May 2?

It’s a simple estimation problem for most of us, but ratio is a tough concept for six-year-olds. I wasn’t sure they’d make much progress… especially since the good folks at about.com make this claim:

You currently share your birthday with about 859,178 people who reside in the United States.

This estimate appears to have used 313,600,000 as the U.S. population, which is reasonable, and then divided by 365. My frustration is that they then display the result to six significant figures. That’s problematic for two reasons — first, because their population estimate has only four significant figures, but also because it’s not the case that exactly 1/365 of the population celebrates their birthday on a given day.

But I digress. Sure, I’m frustrated with about.com’s negligence, but I started this post to tell you about our bedtime math problem, and it highlighted why I hate traditional textbook problems even more than I hate bad math in the media.

Alex first suggested that maybe the number of people who have the same birthday could be found by calculating 1/14 of 7 billion. When I asked why he wanted to divide by 14, his response was, “Because it’s a multiple of 7.” When I asked a few more questions to probe his thinking, he changed his mind. “No, wait, maybe it’s 1/35.” This time, he said he wanted to divide by 35 because it was a multiple of 7 *and* a multiple of 5, and he knew that 7 billion was also a multiple of both 7 and 5.

Then it hit me. He wasn’t trying to solve the problem. He was just trying to make sure the answer was a “nice number,” that is, an integer that preferably would end in a couple of zeroes.

A few more questions, and he finally admitted he knew an estimate could be found by dividing 7 billion by 365. “But that doesn’t work when you divide,” he told me.

Arrgh.

I believe this is what happens when kids see too many traditional textbook problems where the answers are neat and clean. They get conditioned to thinking that math is never messy.

**[Update: 5/8/13]** Just read this on the About page at the Let’s Play Math blog and thought it was worth including here: “Math is like ice cream, with more flavors than you can imagine — and if all your children ever do is textbook math, that’s like feeding them broccoli-flavored ice cream.”

And that couldn’t be further from the truth. Math is unbelievably messy. At least, real math is. Solving real-world problems often means getting a little dirty. You’ll have to roll around in fractions, dig through some decimals, and — Heaven help us! — occasionally tangle with some irrational numbers and extraneous results.

Eli then offered, “If you divide 7 billion by 365, you won’t get an integer.” (He smiled, proud of himself for using the term *integer*.) “That’s the answer, but I don’t know how to do that.” What he meant is that he couldn’t compute the result in his head; nor would I expect him to. We then found an estimate by building on Alex’s idea — instead of dividing by 35, we divided by 350 to approximate the number of people who celebrated a birthday on May 2, since 350 is close to 365 but gives a much nicer answer.

Wow. There are roughly 20 million people who will celebrate their birthday on the same date as you. Crazy, huh?

All of this reminds me of a few jokes.

Recent research shows that those who celebrate more birthdays live longer.

And all the time, I tell my wife:

Honey, you’re one in a million. Which means that there are 7,000 people on Earth exactly like you, so just remember that it wouldn’t be that hard to replace you.