## Posts tagged ‘dollar’

### 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.

### Getting Rich the Hard Way

Ask a silly question, get a silly answer.

Teacher: If you have $4, and you ask your father for another dollar, how much would you have?

Johnny: Four dollars.

Teacher: Young man, you don’t know your addition facts!

Johnny: Ma’am, you don’t know my father!

Johnny’s father and my dad seem to have a lot in common. But my dad would have been proud of me yesterday. While walking home from the local coffee shop, I noticed a corner of a dollar bill on the ground. Not the whole bill, mind you, just a corner that had been ripped off. I thought not much of it, until two feet later I saw another scrap of the dollar bill… then another… and another…

I know and understand Calculus, and I realized that a lot of little things can add up to a lot, so I spent 15 minutes scouring the area for as many pieces of the dollar bill as I could find. I took them home and asked my sons, “Wanna do a puzzle?” We spent a half-hour reconstructing the bill and taping it together. The pictures below show the before and after:

The bill was not in good enough shape to be accepted by a vending machine (too much tape, I suspect, and the missing piece on the right side surely didn’t help, either), but it was in good enough shape for my bank to give me four shiny quarters in exchange for it.

I know that a penny saved is a penny earned. But what is a dollar found?

And the bigger question: What should I do with my new-found wealth?

I decided to buy a lottery ticket. The state gambling commission organized a raffle that boasted an infinite amout of money as the prize. To my great surprise, I won! When I showed up to claim the prize, they told me it would be disbursed as 1 dollar now, 1/2 dollar next week, 1/3 dollar the thrid week, 1/4 dollar the week after that, and so on.

But the joke’s on them. My winnings for the third week will include a one-third cent piece, and that’s gotta be worth something, right?

(Note: Almost everything above is true. I really did find the pieces of a dollar bill on the ground yesterday. As best I can tell, the bill had been on the lawn when it was cut by the blades of a power mower. And my bank really did give me four quarters in exchange for the taped-up, reconstructed version.)

### 5 Math Strategy Games to Practice Basic Skills

The summer is a great time for kids to hike, bike, swim… and forget everything that they learned during the school year.

The son returned to school after summer break. At the end of the first day, his mother received a call from the teacher about his poor behavior. “Now, just one minute,” said the mother. “He had poor behavior all summer, yet I never called you once!”

In *Outliers*, Malcolm Gladwell purports that poor kids lose ground to affluent kids during summer break. Their experiences and academic progress during the school year are similar, he contends, but their out-of-school experiences during the summer are very different. Though minor at first, the cumulative effect of those summer losses becomes noticeable as children get older.

The following are five games/puzzles that can be used with young kids to prevent summer losses and, possibly, even elicit some summer gains. Each has the characteristics that I love about a good game for young kids: It requires students to use and practice basic skills, but there is a higher purpose for doing so.

**1. KenKen**

This is a game that’s kind of like SuDoku, but a million times better. If you don’t know the game, check it out at www.kenken.com. My sons noticed me playing it one afternoon and asked what it was. I explained, and they asked if they could do it with me. We now solve three or four games every afternoon. I used to help them a lot, but now they pretty much know all of their math facts up through 7 × 7. How do you not love a game that helps four-years-olds learn the times table?

**2. Wormhole**

This is a puzzle, not a game, and you can learn all about it at Math Pickle. The general idea is that you start with a sequence of numbers in a flower-like pattern. You then multiply two adjacent numbers, subtract 1, and divide by the number below. The cool and surprising part is that every intermediate result is an integer, so there are no ugly decimals for kids to deal with. And by the twelfth ring of petals, every result is 0. Happens every time.

**3. Squares of Differences**

The good folks at Math For Love reminded me of this great problem, and Josh Zucker discussed it at length on the NYTimes Numberplay blog. Draw a square, and put a positive integer at each vertex. Then at the midpoint of each side, write the difference of the numbers at the two adjacent vertices. Now connect the midpoints to form a rotated square inside the original square, and repeat. It seems that if you continue this process long enough, you’ll eventually get all 0’s. But does that always happen?

By the time kids test this conjecture with three or four attempts, they’ve done a hundred subtraction problems without even realizing it.

**4. Decimal Maze**

The Decimal Maze (PDF) comes from the lesson Too Big or Too Small on Illuminations. Trying to obtain the maximum value while traversing a maze with decimal operations, students learn about the effects of multiplying and dividing by decimals that are greater or less than 1. The activity is good for upper elementary and middle school students, but I’ve used modified versions with very young kids. For instance, a modified maze for kids in first grade uses single-digit positive integers while limiting the operations to just addition and subtraction; for older kids, a maze could include fractions or powers instead of decimals.

**5. Dollar Nim**

As I mentioned in a previous post, my wife created a great game that I call Dollar Nim. The idea is simple. Imagine you have 100¢, and on your turn you can remove 1¢, 5¢, 10¢, or 25¢. Players alternate turns; the player to reduce the amount to 0¢ is the winner. The optimal strategy is not obvious, and kids practice a whole lot of subtraction, especially as it relates to making change.

More generally, any one-pile nim game is great for the purpose of having kids practice subtraction without realizing it.

I hope you find some free time this summer to enjoy these games. I’ll leave you with a joke/truth about summer school.

I never understood the concept of summer school. The teacher’s going to go up there and go, “OK, class. You know that subject you couldn’t grasp in nine months? Well, we’re going to whip it out in six weeks.” – Todd Barry

### Dollar Nim

*The following post was featured at the NYTimes Numberplay blog during the week of August 8‑15, 2011.*

One-Pile Nim (a.k.a., Static Nim) is a game in which there is a pile of *n* objects, and each player can take up to *k* objects on her turn. The player who removes the last object wins. For example, on the TV show *Survivor: Thailand* in October 2002, the contestants were given an “immunity challenge” in which there were 21 flags, and a team could remove 1, 2, or 3 flags on a turn. (Using the notation above, *n* = 21 and *k *= 3.) Avinash Dixit claims that “the actual players [on *Survivor: Thailand*] got almost all of their moves wrong,” but the strategy for winning this game is not terribly difficult to figure out. If you’re not familiar with the game, you might enjoy determining the strategy on your own, so I won’t spoil your fun.

While riding back from a camping trip yesterday, my wife was keeping my sons amused by playing mental math games with them. However, she was using mostly drill-and-kill exercises, where she would state an expression like 21 – 6, and one of them would shout, “15!” Before I was able to suggest that she play a game that involved more strategy and less rote mathematics, she offered the following.

- Start with $1, or with 100¢, if you prefer.
- On alternating turns, players can remove any coin they like. (Well, technically, players remove a number of cents equal to the value of one of the four common U.S. coins — quarter, dime, nickel, penny — but such an overly complicated statement of the rules would have confused my sons.)
- The player who reduces the value to 0¢ wins.

That is, *n* = 100, and a player must choose to remove a value from the set {1, 5, 10, 25}.

I was duly impressed by my wife’s creation. (By “my wife’s creation,” I mean to refer to the game she made up, not to my sons, though the moniker would be equally applicable to the latter, and I must admit that I am often duly impressed by my sons, too.) It was a version of Nim that I had never seen before, and the optimal strategy was not obvious to me. Moreover, it had the characteristics of activities that I love to use with young kids: it causes them to practice some useful basic skill (in this case, calculating change for a dollar) for the purpose of trying to win a strategy game with more sophisticated mathematics.

My two sons, my wife, and I played Dollar Nim several times. During our third game, my wife took a dime to leave me with 29¢. “Daddy’s going to win,” Alex declared. Sure enough, I took a quarter to leave 4¢, and the outcome was decided. Both Alex and Eli had realized that if one of us was able to reduce the amount to 4¢, that person would win — everyone would be forced to take a penny.

Analyzing this game for two players is not terribly difficult, though once I had done it, I was intrigued by the patterns that appear in the optimal strategy. Analyzing the game for four players is a bit more difficult.

In the post on the *NYTimes NumberPlay *blog, Pradeep Mutalik offered the following extension question:

Since Dollar Nim is played with real money, it makes sense for the participants to keep the change they remove. This confers a reward for removing larger denominations. To offset this, the winner must be given an extra monetary reward. What should be the

minimumprize money for the two-player game so that no matter what happens, the winner comes out ahead?

I’ll leave it as an exercise for the reader to determine the optimal strategy for the two- and four-player versions of this game, as well as to determine the answer to Pradeep’s question.

**[Update, 6/30/11]** The sequence of “unsafe” values for two-player Dollar Nim is now listed as A192333 in the Online Encyclopedia of Integer Sequences.