## Posts tagged ‘word’

### Number Words and Learned Helplessness

How about some number word puzzles? Here’s a well-known puzzle that you’ve likely seen before:

What is the first positive integer that, when spelled out, contains the letter a?

And here’s a modification of that puzzle that you may find a little more difficult:

What is the first positive integer that, when spelled out, contains the letter c?

And taking it one step further:

What letters are never used in the spelling of any positive integer?

Who says that math isn’t useful in English class?

One more problem in a similar vein:

Pick any positive integer you like, and count the letters when that number is spelled out. Now count the letters when the resulting number is spelled out. Continue ad infinitum. What do you get?

Maybe those weren’t your cup of tea. Perhaps anagrams are more to your liking, so here are two (related) puzzles for you.

Try to make an anagram for each of the following three words.

• whirl
• slapstick
• cinerama

Too tough? Then try these three words instead.

• bat
• lemon
• cinerama

If you had trouble with the first set, you’re in good company. There are no anagrams for the words whirl or slapstick.

These two sets of words were used by Charisse Nixon, a pyschologist at Penn State–Erie, who gave the first set of words to half her class and the second set of words to the other half. She instructed them to find an anagram of the first word on their list; those students who had received the second set were successful. Nixon then instructed them to find an anagram of the second word on their list; again, those students who had received the second set were successful. When she then instructed them to find an anagram of the third word on their list — of which there is exactly one, American — those who hadn’t found anagrams for the first two words were less successful than their peers, even though the final challenge was identical.

Afterwards, students who received the first set of words admitted to feeling confused, rushed, frustrated, and stupid.

Nixon was studying learned helplessness, a condition in which a person suffers from a sense of powerlessness, often arising from persistent failure.

This has implications the math classroom. Students who perform at a fourth-grade level but are asked to participate in an eighth-grade class are surely as confused and frustrated as the subjects in Nixon’s experiment. Students need to occasionally feel success, or else they’ll shut down. If you’re a teacher, you don’t need me or a psychological research study to tell you that. So the question is, how can you get students to feel success? That is, what can you do to prevent learned helplessness?

My suggestion is to look for acceptable and accessible entry points.

Consider the following problem, which might be seen in a middle school classroom:

What is the maximum possible product of a set of positive integers whose sum is 20?

As written, that problem contains three words — maximum, product, and integers — that may confound some students. For middle school students who do understand the terminology, finding an appropriate strategy might be daunting.

In my opinion, the following is a better way to present this problem so that all students have an entry point:

Find some numbers with a sum of 20. Now, multiply those numbers together. Compare your result with a partner. Whose result was greater? Can the two of you work together to find a product that’s greater still?

Even a struggling middle school student could start this activity. Surely he could find some numbers with a sum of 20. Certainly, he could multiply them without a problem.

Why is this a better presentation? The wording is simplified. There is encouragement to work with a partner. It feels more like a collaborative game than a traditional math problem. It sounds — dare I say it? — like fun.

When a struggling student is able to get into a problem, and they’re able to make some strides in the right direction, and they’re rewarded by your positive encouragement, they attain some level of success. Maybe they won’t solve the problem entirely, but who cares? For many students, trying is progress.

And for students who are having trouble finding any success, perhaps the following words of encouragement will help.

If at first you don’t succeed, call it version 1.0.

If at first you don’t succeed, destroy all evidence that you ever tried.

If at first you don’t succeed, blame someone else and seek counseling.

If at first you don’t succeed, then skydiving is not for you.

If at first you don’t succeed, get new batteries.

If at first you don’t succeed, try two more times so your failure is statistically significant.

### Word + Letter = Math Term

On a recent Sunday Puzzle on NPR, Will Shortz gave a letter and a word, and the contestant was to guess the name of a popular TV show using an anagram of the letters (“Coming to TV This Fall: Anagrams,” Oct 12, 2014). For instance,

M + NAMED

This struck me as an interesting puzzle format. My only criticism is that it just wasn’t mathy enough.

But I’m not a problem maker, I’m a problem solver… so rather than cast aspersions at the puzzle, I’ll instead use the format to offer my own version.

Each of the 26 letters of the alphabet has been paired with a common English word. An anagram of the pair will yield a common math word. How many can you find?

1. A + ERA
2. B + AGLARE
3. C + BITES
4. D + NOTICER
5. E + EDGERS
6. F + SAUCER
7. G + LEAN
8. H + OPERABLY
9. I + TANGLER
10. J + INDUCTIONS
11. K + SEW
12. L + POSE
13. M + RIPS
14. N + AIMED
15. O + PINT
17. Q + AURES
18. R + ENVIES
19. S + RECITED
20. T + HAM
21. U + RAIDS
22. V + EXERT
23. W + ROPE
24. X + SEA
25. Y + PENTHOUSE
26. Z + ORE

I don’t believe in providing an answer key, but you can find some help at Math Words, and you can click over to More Words if you run into real trouble. But give it the old college try before seeking assistance. Honestly, you’ll feel better about yourself if you solve these on your own.

### Science Word Game for Road Trips

For your next road trip, a fun, family-friendly game… especially if your family tree has a lot of geeky branches.

Couldn’t be simpler:
Combine abbreviations from the periodic table to create a word.

For instance, you could combine the abbreviations for barium and rutherfordium to make BaRf.

Or use chromium, iodine, titanium, carbon, and aluminum to make CrITiCAl.

Or my current favorite — combine titanium, platinum, oxygen, and einsteinium to form TiPtOEs. How fun!

Post your longest words in the comments. (If you choose to write a computer program to find the longest possible word, please don’t spoil everyone else’s fun.)

Uber-geeks will want to use a formula such as

Points = 3 × No. of Letters + 5 × No. of Elements Used

to score the game. And you can if you like; play ten rounds, and highest score wins. But I say just have fun, make words, and impress everyone riding in the car with you!

You might find the following lists helpful.

Single-Letter Abbreviations:
B, C, F, H, I, K, N, O, P, S, U, V, W, Y

Two-Letter Abbreviations, With A Vowel:
Ac, Ag, Al, Am, Ar, As, At, Au, Ba, Be, Bh, Bi, Bk, Br, Ca, Ce, Co, Cu, Er, Es, Eu, Fe, Ga, Ge, He, Ho, In, Ir, La, Li, Lu, Mo, Na, Ne, Ni, No, Os, Pa, Po, Pu, Ra, Re, Ru, Se, Si, Ta, Te, Ti, Xe

Two-Letter Abbreviations, No Vowel:
Cd, , Cf, Cl, Cm, , Cp, Cr, Cs, Db, Ds, Dy, Fm, Fr, Gd, Hf, Hg, Hs, Kr, Lr, Md, Mg, Mn, Mt, Nb, Nd, Np, Pb, Pd, Pm, Pr, Pt, Rb, Rf, Rg, Rh, Rn, Sb, Sc, Sg, Sm, Sn, Sr, Tb, Tc, Th, Tl, Tm, Yb, Zn, Zr

Three-Letter Abbreviations:
Uuh, Uuo, Uup, Uuq, Uus, Uut

Click Image for Larger Version

### A Puzzle of Few Words

Winston Churchill once said, “The short words are the best, and the old words best of all.”

Today’s post contains several puzzles, the answer to each is a short word, and at least one of the answers is a very old word, indeed.

But first, a joke about words and math:

Teacher: Use the word announce in a sentence.
Student: Yes, ma’am. Announce is one-sixteenth of a pound.

The first puzzle is about the game Hangman.

In the game of Hangman, the first player thinks of a word and reveals the number of letters. The second player then guesses letters. If a guessed letter is in the word, the first player reveals the position(s) of every occurrence of that letter within the word. If the guessed letter is not in the word, then the second player receives a body part for a man who is hanging from a gallows (hence the name, Hangman). If the entire man is completed before the word is guessed, the second player has been hanged and loses.

Various versions of the game use hangmen with different numbers of body parts. The number of parts typically ranges from 7 to 13. The Hangman game at www.playhangmangames.net contains 7 body parts, and my sons play a version with 11 body parts:

Jon McLoone at the Wolfram Blog ran a simulation to determine the best words for the game of Hangman. Of course, you could just click on that link, but it might be more fun to think about the following question before you do:

What is the best word to use when playing Hangman? And does the best word change, depending on how many body parts are in the version you play?

Surprisingly, McLoone found that there is a single best Hangman word, for any game with 8 to 13 body parts. A strong hint is included at the bottom of this post.

The second puzzle is a product value puzzle, in which the product value of a word is equal to the product of the value of the letters. Specifically,

Assign each letter of the alphabet a value as follows: A = 1, B = 2, C = 3, and so on. The product value of a word is the product of its letters. For instance, the word CAT has a product value of 60 because C = 3, A = 1, T = 20, and 3 × 1 × 20 = 60.

One fun puzzle based on this set-up:

Find an “acre” word, which is a word with a product value of 43,560, the number of square feet in an acre.

The (unique) answer happens to be one of my favorite English words.

Turning this idea around, another variation is as follows:

Find a four-letter English word with the largest possible product value.

According to Scrabble Australia, there are 16,739 four-letter words. However, many of those would not be considered common — such as euoi (an impassioned cry), nabk (berries you’ve never tasted, from a plant you’ve never seen), and zizz (a short sleep). The last of these examples has an impressive product value of 263 = 17,576, but it’s not the highest. Not even close, in fact; its product value is only 7.8% of the largest product value for a four-letter word.

You can explore similar puzzles with the Product Value Calculators on the MJ4MF website.

Hints for all three puzzle appear below the following joke about words and math:

A boy was told to write an essay about his favorite subject. He wrote, “I really love math,” and turned in his essay. Returning his paper, the teacher said, “Sorry, your essay needs to be at least 100 words.” So he wrote, “I really, really, really, really, really, really, really, really, really, really, really, really, really, really, really, really, really, really, really, really, really, really, really, really, really, really, really, really, really, really, really, really, really, really, really, really, really, really, really, really, really, really, really, really, really, really, really, really, really, really, really, really, really, really, really, really, really, really, really, really, really, really, really, really, really, really, really, really, really, really, really, really, really, really, really, really, really, really, really, really, really, really, really, really, really, really, really, really, really, really, really, really, really, really, really, really, really love math.”

Hints

The best Hangman word (according to McLoone’s analysis) has a product value of 6,760.

The defintion of the acre word is “alternating layers of silt or clay, usually of contrasting colors, that comprise an annual cycle of deposition in a body of still water.” That doesn’t help? Then just enter 43,560 into the second form on the Product Value Calculators page.

And the four-letter word with the largest possible product value? It’s an anagram of the mythological river that divides Earth and the Underworld, as well as an anagram of the 70’s rock band that sang Come Sail Away and Mr. Roboto.

### Word Squares Revisited

On December 22, I posed the following puzzle in the Square Deal post:

Create a 4 × 4 grid composed of common English words (you can use the list of official Scrabble four‑letter words as a reference) such that the sum of the point values of the 16 letters is as high as possible.

In that post, I mentioned that I had created a grid worth 62 points, and many readers have asked about it. Here it is:

 Z I Z Z I D E A Z E S T Z A T I

All of the words are acceptable when playing Scrabble. While IDEA and ZEST are very common, the other two are not:

• ZIZZ = a nap
• ZATI = a species of macaque

I am certain that a better solution exists, but I have neither the time nor the energy to search for it. Anyone out there want to write a computer program to solve it?

** UPDATE (1/14/11): Check out the comments section from the previous post. Veky Edgar wrote a computer program and found the maximum possible score (92 points), and Scandinavian manually created a grid worth 85 points.

The Math Jokes 4 Mathy Folks blog is an online extension to the book Math Jokes 4 Mathy Folks. The blog contains jokes submitted by readers, new jokes discovered by the author, details about speaking appearances and workshops, and other random bits of information that might be interesting to the strange folks who like math jokes.

## MJ4MF (offline version)

Math Jokes 4 Mathy Folks is available from Amazon, Borders, Barnes & Noble, NCTM, Robert D. Reed Publishers, and other purveyors of exceptional literature.