## Posts tagged ‘weight’

### 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.

### Heavy Cookies, Undervalued Coins, and Misconceptions

Simple question to get us started…

Which is worth more?

And **of course** the answer is, “The quarters, because 50¢ is more than 20¢,” right? But not to a kindergarten student or a pre-schooler who hasn’t yet learned how much coins are worth. A young student might argue, “Four is more than two.”

Why didn’t the quarter follow the nickel when he rolled himself down the hill?

Because the quarter had more cents.

Recently, I was asked to review an educational video for kindergarten math that had a similar question.

The video stated, “Can you tell the green, yellow, and orange cookies are heavier? That makes sense, doesn’t it? Because there are **more** of them!”

Uh, no.

This is the same logic that would lead one to claim that the value of four nickels is greater than the value two quarters because there are more nickels. It’s a huge misconception for students to focus on **number** rather than **value**. So it’s very frustrating to see this video reinforce that misconception.

For example, if each green, yellow, or orange cookie weighs 3 ounces, but each blue or purple cookie weighs 5 ounces, then the left pile would weigh 6 × 3 = 18 ounces, and the right pile would weigh 4 × 5 = 20 ounces, so the right side would be heavier. (Then again, are there really 6 cookies on the left and 4 on the right, or are some cookies hidden? Hard to tell.)

As far as I’m concerned, the only acceptable answer is that the pile of green, yellow, and orange cookies must be heavier — assuming, of course, that the balance scale isn’t malfunctioning — because the pans are tipped in that direction.

All of this reminds me of the poem “Smart” by Shel Silverstein.

SMARTMy dad gave me one dollar bill

‘Cause I’m his smartest son,

And I swapped it for two shiny quarters

‘Cause two is more than one!And then I took the quarters

And traded them to Lou

For three dimes — I guess he don’t know

That three is more than two!Just then, along came old blind Bates

And just ’cause he can’t see

He gave me four nickels for my three dimes,

And four is more than three!And I took the nickels to Hiram Coombs

Down at the seed-feed store,

And the fool gave me five pennies for them,

And five is more than four!And then I went and showed my dad,

And he got red in the cheeks

And closed his eyes and shook his head–

Too proud of me to speak!

### 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?

### The Math of Maker’s Mark

Last week, Maker’s Mark announced that they would change their recipe. According to COO Bob Samuels, the company was planning to reduce the alcohol content of its bourbon from 45 percent to 42 percent by replacing the removed alcohol with water. But outcry from thousands of bourbon drinkers convinced them to abandon their new 84-proof recipe and continue stocking shelves with 90-proof spirits.

This morning, sports reporters Tony Bruno and Harry Mayes suggested that bourbon drinkers were opposed to the change because they want to get drunk faster.

This got me to thinking about math.

As you know, mathematicians know a thing or two about alcohol.

Where there are four mathematicians, you’ll likely find a fifth.

And so do mathematical objects.

A definite integral walks into a bar. “Ten shots of whiskey, please.”

The bartender asks, “You sure you can handle that?”

“Don’t worry,” says the integral. “I know my limits.”

In the U.S., a “standard” drink is one that contains 0.6 fluid ounces of alcohol, but a standard drink does not necessarily correspond to a typical serving size. In practice, a typical drink of bourbon is a 1.5-ounce pour.

So what does this mean for Maker’s Mark? Sticking with 90-proof bourbon means that a 1.5-ounce drink will contain 0.675 fluid ounces of alcohol, whereas the revised 84-proof bourbon would have contained 0.63 fluid ounces of alcohol. Does that extra 0.045 fluid ounces really make a difference?

A little bit, but not much.

As shown in the table below, a 200-pound man would need to consume 4.76 drinks of 84-proof spirits to reach the legal blood alcohol content limit of 0.08, yet he would only need to down 4.44 drinks of 90-proof spirits. The difference is small. Another third of a drink isn’t much when you’ve already downed 4½.

Number of Drinks to Become Legally Drunk (0.08 BAC) |
|||

Weight (lbs) |
Typical Spirits(80 Proof) |
Proposed Maker’sMark (84 Proof) |
Original Maker’sMark (90 Proof) |

100 | 2.50 | 2.38 | 2.22 |

150 | 3.75 | 3.57 | 3.33 |

200 | 5.00 | 4.76 | 4.44 |

250 | 6.25 | 5.95 | 5.56 |

Note that this chart is for men; women of the same weight would require fewer drinks to reach the same level of intoxication. In addition, time is not reflected in this chart. Because of normal body processes, a person’s BAC is reduced by 0.01% every 40 minutes.

Still, the data seems clear. The revised Maker’s Mark recipe would cause intoxication almost as quickly as the original recipe, so if bourbon fans reacted simply because they want to get drunk faster, well, that seems misguided.

Then again, how many sh*tfaced bourbon drinkers have done this kind of analysis? Probably very few. But that does remind me of a joke.

How many bourbon drinkers does it take to change a light bulb?

Just one. Have him drink an entire bottle, then hold the bulb as the room spins.

Or this one that’s a little more mathy.

How many math department chairs does it take to change a light bulb?

Just one. He holds the bulb, and the world revolves around him.