# A “Mean” Product

My mother once taught in a small country three-room schoolhouse, ampoule where students of different age levels often had to study the same course together. For the course in American History, salve the kids in the 6th, 7th, and 8th grades were combined into one class. This means that their ages were between 10 and 14.

If the product of their ages was 41,760,576, what was the average, or arithmetic mean, of their ages?

# The Rub-a-dub-dub Restaurant

In Mother Goose City, the most elegant restaurant where the elite meet to eat is the Rub-a-dub-dub Restaurant. It is owned and operated by those three wild and crazy guys: the butcher, the baker, and the candlestick maker. (Perhaps you will recall, they often traveled by tub!)

When they decided to become partners in this expensive project, each promised to contribute as much money as he could, according to the funds he had in his bank account. The results can be described as follows:

1. the ratio of the funds contributed by the butcher and baker was 3 to 5, respectively.
2. the amount of money contributed by the candlestick maker was equal to twice the amount of the butcher less one-half that of the baker.

3. the total amount of money raised by the three investors was greater than 100 mogolas and less than 125 mogolas. (A mogola is an informal unit of money in Mother Goose Land, similar to our use of “grand” to mean \$1000.)

Assuming that each man’s contribution was an integral number of mogolas, how much money did they have to start this culinary adventure?

We’re buying carrots, and we’ve found two sellers who have vegetables we like. The first will sell us four pounds and six ounces for a price of two pounds seven shillings and three pence.

The second grower had a bigger crop. He asks for four pounds thirteen shillings and ten pence, and in return will give us nine pounds fourteen ounces of his carrots.

Which batch of carrots should we buy, to pay the least amount per pound?

Solving hints you should know:

(A pound of weight is sixteen ounces. A pound of money in long ago England is exactly 20 shillings, and a shilling is twelve pence. So don’t confuse the pounds of weight that measure fatness with the pounds that mark expense.)

This problem came from The Gnarly Gnews, a humorous, bi-monthly math newsletter, published by the SMP Company, PO Box 1563, Santa Fe, NM 87504, Copyright 2002 by Montgomery Phister, Jr. It appeared in the Jan-Feb issue for 2003.

# Inching Along Down the Football Field

One fine fall day, Fred, the famous football player, decided to do a little math activity with his favorite sport. So he put his ball down on one of the goal lines of the 100-yard field. He then moved it forward half the distance — that is, 50 yards — toward the other goal line. Next he moved it half the remaining distance, 25 more yards.

He planned to continue in this manner as long as he could, always advancing the ball half the remaining distance toward the other end. Of course, it should be obvious that sooner or later his moves will be quite small, so small in fact that we could say he was truly “inching along”.

The question now is: on which move of the ball did the distance for the first time become less than one inch?

Bonus: what was the total distance that Fred had moved the ball after that move? Give your answer rounded to the nearest 100th of an inch.

When you get this problem’s two answers, please e-mail me. Be sure to explain “how” you arrived at your conclusions.

# Wordsworth

“What’s it worth?” is a common question in the business world to be sure. But what about a math classroom? The question of “What is the value of CAT or DOG?” certainly sounds intriguing to me and to elementary students I have taught. The idea becomes obvious if we give each letter its own value and we just add up the values of each letter to obtain the value of the word.

For starters, let’s give the letters the value of their positions in the alphabet:

A = 1, B = 2, C = 3, … , Y = 25, Z = 26

Now, here is how things turn out for some common 3-letter words:

```   C =  3        D =  4        F =  6        F =  6
A =  1        O = 15        A =  1        L = 12
T = 20        G =  7        N = 14        Y = 25
24            26            21            43

Where's the Challenge?```

Finding the value sums for three-letter words is, admittedly, not a great and difficult thing to do. So, why am I presenting this with such fanfare? It’s because if we reverse things — á la JEOPARDY — and ask, “Can you find a word that is worth 50 points? Or 25 points? What word has the greatest/smallest value?” etc., then things become an experience in true problem solving. It’s not so easy now, is it?

One sort of activity I have used is to ask the class, working as a team, to find words for each value number from some lower limit to some upper limit. For example, let’s start out with 3-letter words. I have found word sums as low as 6 (CAB) and as high as 66 (WRY) — and of course, every number inbetween.

As one begins such a project, it is convenient to just start putting down any 3-letter words that come to mind, compute their values, and compile an ordered list, leaving blank those numbers for which no value has been found so far. As spaces are filling up, soon you will start directing your attention towards the missing values. Then is when the fun — and the challenge — begins.

One suggestion would be to make a large poster on the bulletin board with a chart in this form:

 No. Word Expression Name Date 6 CAB 3+1+2 John S. Sept 10 7 BAD 2+1+4 Mary J. Sept. 11 8 CAD 3+1+4 Ann P. Sept. 12 9 DAD 4+1+4 Sue W. Sept. 10 10 BAG 2+1+7 Bill M. Sept. 11 … … … … …

[If you are using cooperative grouping in your class, this would make a good activity for students working in this way. And for the independent individual, he or she can do this “all by oneself”. In such structures, the chart is still a good recording strategy.]

Trivia Fun
Sometimes as a large body of word values are compiled, you can find interesting equal-value-pairs. In my not very extensive collection to date, I have noted this nice pair: FOX and FUR, both words having a total of 45, and the words themselves have an obvious connection in the real world. You and your students can surely find additional cases like this one. This is definitely a case of “two heads are better than one”; when large word lists are compiled, more interesting gems can be discovered by cooperating together.

By way of introducing the concept and formal notation of inequality, we can make light-hearted statements such as

## CAT < DOG

[Lest I start receiving angry email from the cat lovers of the world, I offer the following inequality to put things in perspective:

Okay?]

## SIX < TWO

Go for FOUR
Of course, there’s no rule that says you must limit yourself to 3-letter words; there are many more 4-letter words out there just waiting to be “valued”! My limit numbers so far go from 10 (BABE) to 79 (FUZZ).

And you say you want some trivia in this category? Here is one of my favorites: “MORE is less than LESS, MUCH is less than MORE and much less than LESS, whereas LOTS is lots more than all three of those.” Show this to be true by finding their values and writing out the appropriate inequality statement.

And this one has a unique flavor all its own:

## SHOE + 1 = SOCK

Returning for a moment to our numerical case above, what sort of inequality should we write here for FOUR and FIVE?

And on it goes…

I’m sure you’re beginning to see the possibilities for extending this activity as long as interest holds up. Contests could be on-going for extended periods of time for such ideas as

* the smallest value for 10-letter words;
* the largest value for 10-letter words;
* finding related words with the same or consecutive values;
* a short sentence (3 words or so) with equal-valued words.

The possibilities are limited only by one’s own creativity.

Sample Word Lists:

```3-LETTER WORDS

6: CAB   19: EGG   32: RAM    45: FOX   58: TOW
7: BAD   20: AND   33: FIR    46: MIX   59: RUT
8: CAD   21: HID   34: OAR    47: NOR   60: TOY
9: DAD   22: AIL   35: RAP    48: WET   61: YOU
10: BAG   23: BAT   36: AWL    49: NOT   62: YUP
11: FAD   24: CAT   37: PAT    50: OWL   63: TRY
12: BEE   25: ALL   38: COT    51: POT   64: STY
13: HAD   26: DOG   39: FIX    52: SIX   65: TUX
14: BEG   27: SAG   40: TOE    53: ROT   66: WRY
15: FED   28: FOG   41: BOX    54: OUR
16: FEE   29: AWE   42: FUN    55: NUT
17: DID   30: DAY   43: BUT    56: ZOO
18: JAG   31: PAN   44: MOP    57: PUT

4-LETTER WORDS

10: BABE   28: BEAT   46: GIRL 	 64: SPIT
11:        29: LAKE   47: SHOE	 65: LOSS
12: BEAD   30: BEER   48: SOCK	 66: LOTS
13         31: BELL   49: SING	 67: TORN
14: DEAD   32: HIGH   50: FORK	 68: XRAY
15: FACE   33: SAID   51: MORE	 69: SOOT
16: CAGE   34: PALE   52: SHIP	 70: SPOT
17: CEDE   35: GAVE   53: MANY	 71: WAVY
18: HEAD   36: HAVE   54: LOVE	 72: ROTS
19: BAKE   37: LIKE   55: LESS	 73: MUST
20: FEED   38: NEAR   56: OVEN	 74: MUTT
21: DICE   39: COAT   57: SORE	 75:
22: BEAN   40: FIVE   58: TORE	 76: PUTS
23: MADE   41: SAIL   59: VIEW	 77: PUTT
24: BAIL   42: FISH   60: ROLL	 78:
25: JACK   43: BOOK   61: MIST	 79: FUZZ
26: BEAR   44: COOK   62: VOTE
27: HAND   45: MUCH   63: JAZZ```

Footnote: Can you help me with the remaining blank spaces? I’ve been adding too much for this and my brain is tired. Just send me an e-mail. Thanks.

Reference: Phyllis Zweig Chinn, Coding for fun and mathematics. The ARITHMETIC TEACHER, December 1976, pp. 597-600.Update: 7/31/01

We have just finished researching the matter of applying this activity to the last names of the 43 Presidents of the United States. We feel we have found some interesting data worth sharing. Most of our data involves prime numbers.

First, there are 9 Presidents who numerical values are primes, ranging from Ford (43) to the two Roosevelts (131 each). The other 6 remaining prime totals are:

47, 61, 73, 79, 83, and 97
.

Can you connect each number with its corresponding President?

Second, if we add up the various individual totals from Washington up to another President, we get several more primes, in fact, this happens 7 times. Here are the results:

1. 241, up to Monroe;
2. 811, up to Tyler;
3. 2087, up to Roosevelt (Teddy);
4. 2287, up to Harding;
5. 2357, up to Coolidge;
6. 2857, up to Kennedy; and finally,
7. 3371, up to BUSH! (the newest one!)

[Note: Just so there is no confusion. Since Cleveland served 2 non-consecutive presidencies (22nd and 24th), his name is used twice to compute the totals.]

Finally, we’d like to mention two other interesting numbers that showed up. The squares of 64 and 121 are the values for Buchanan and Eisenhower, respectively.

The \$1 Word Game