TRAPPING THE WILD WORD!

This mathtivity introduces kids to unimodal search methods. For example , you are assigned to determine the boiling temperature of hydroquinone, a chemical useful in photography. Given the typical sensitivity of most instruments, particularly those available in schools, this is only measurable as a RANGE, so that the process of REACHING BOILING TEMPERATURE AND GOING BEYOND resembles the familiar "bell curve" used (alas!) for grading. The graph of such a process is called UNIMODAL. I'll explain.

The word MODE is a general term for "average". Suppose you have a 10-data sample: 5, 7, 4, 7, 3 8, 5, 1, 9, 6. This sample has two modes -- or "two kinds of mostest" -- namely, 5 and 7, both of which occur twice in the sample, in contrast to once for the others. That is, this same is BIMODAL. Consider, on the other hand, the 10-data sample: 2, 1, 6, 8, 4, 6, 7, 4, 5, 6. Most of these are 1-timers; 4 occurs twice; 6 occurs thrice, as none other does. Hence, this sample has just ONE MODE (one "mostest kind"): it is UNIMODAL.

Many measuring processes and other types of experimental procedures are UNIMODAL. Since RESEARCH COSTS MONEY AND TAKES TIME AND SOMETIMES SEVERAL RESEARCHERS, you wish to MINIMIZE THE PROCESS.

There are three favored UNIMODAL SEARCH PROCEDURES:

  1. BINARY SEARCH;
  2. FIBBONACCI SEARCH;
  3. GOLDEN SECTION SEARCH.

Of these, BINARY SEARCH is the simplest and, usually, the cheapest to carry out. This mathtivity teaches kids the procedure of BINARY SEARCH with !

EQUIPMENT: TEACHER; STUDENTS; TWO COLUMN DICTIONARY, of 1000+ pages. (Is that cheap enough for you?)

  • One kid picks out a DEFINED WORD ("The Wild Word") ON A PAGE OF THE DICTIOARY, showing WORD and PAGE NUMBER to teacher. Also noted is the ordinal number of the word's position on the page: the first defined word; or the second; or the umpteenth.
  • Another kid, having been given the closed Dictionary, guesses the page number, then the position of the "wild word" on the page.
      BINARY SEARCH ENABLES THE KID TO FIND THIS WORD EXACTLY IN 15 GUESSES! How?

      Let's say that the Dictionary has 1017 pages. The critical thing to notice is that 1017 < 1024 = 210. Then BINARY SEARCH ALLOWS FINDING THE PAGE IN 10 GUESSES.

      Having the Page, the guesser counts the DEFINED WORDS on the page, and GUESSES ITS ORDINALITY. A typical 2-column Dictionary of 1017 pages has, at most, 32 DEFINED WORDS ON A PAGE. Now, 32 = 25. So it can be guessed in 15 or fewer guesses -- totaling, 10 + 5 = 15 guesses, for the entire procedure.

      As an example, suppose the word is on page 849, the 23rd DEFINED WORD on that page.

      Taking that 1017 page figure (the page number wouldbe know to all the students), and approximating it by the number 1024 = 210, the guesser asks, "Is the page number greater than 512?" (This is half of the number 1024.)

      ANSWER: "Yes." (Then the Page Number is between 513 and 1024": 512 pages. Half of that is 256, and 512 + 256 = 768.) GUESSER (2nd Guess): "Is the Page Number greater than 768?" ANSWER: "Yes." (The RANGE is now 769 to 1024: 256 pages. Half of 256 is 128. 768 + 128 = 896.) GUESSER (3rd Guess): "Is the Page Number greater than 896?" ANSWER: "No." (The RANGE now is 769 to 896: 128 pages. Half of 128 is 64. 768 + 64 = 832.) GUESSER (4th GUESS): "Is the Page Number greater than 832?" ANSWER: "Yes." (The RANGE now is 833 to 896: 64 pages. Half of 64 is 32. 832 + 32 = 864.) GUESSER (5th GUESS): "Is the Page Number greater than 864?" ANSWER: "No." (The RANGE is now 833 to 864: 32 pages. Half of 32 is 16. 832 + 16 = 848.) GUESSER (6th GUESS): "Is the Page Number greater than 848?" ANSWER: "Yes." (The RANGE is now 849 to 864: 16 pages. Half of 16 is 8. 848 + 8 = 856.) GUESSER (7th GUESS): "Is the Page Number greater than 856?" ANSWER: "No." (The RANGE is now 849 to 856: 8 pages. Half of 8 is 4. 848 + 4 = 852.) GUESSER (8th GUESS): "Is the Page Number greater than 852?" ANSWER: "No." (The RANGE is now 849 to 853: 4 pages. Half of 4 is 2. 848 + 2 = 850.) GUESSER (9th GUESS): "Is the Page Number greater than 850?" ANSWER: "No." (The RANGE now is 849 to 850: 2 pages. Half of 2 is 1. 848 + 1 = 849.) GUESSER (10th TIME): "Is the Page Number 849?" ANSWER: "Yes." Having found the Page Number (in this case, Page 849), let's find the DEFINED (WILD) WORD BY THE ORDINALITY OF ITS POSITION ON THE PAGE. (Let's said that's 7th.)

      Assuming a MAXIMUM of 32 DEFINED WORDS to a 2-column page, the RANGE is 1-32. Half of 32 i1 16. GUESSER (1st GUESS): "Is the Word Number greater than 16? ÄNSWER: "No." (RANGE now; 1-8, 8 Positions. Half of 8, 4.) GUESSER (2nd GUESS): "Is the Word Number greater than 4?" ANSWER: "Yes." (RANGE: 5-8. Half is 2. 4 + 2 = 6.) GUESSER (3rd GUESS): "Is the Word Number greater than 6?" ANSWER: "Yes." (RANGE: 5-6.) GUESSER (4TH GUESS): "Is the Word Number greater than 5?" ANSWER: "No. GUESSER: "Then the Word Number is 5". And, looking on Page 849, the Guesser finds the 5th DEFINED (WILD) WORD. (Good Hunting! Tally-Ho!)
      One more interesting comment about this HALVING PROCESS. Let's go back to the HUNTING OF THE PAGE NUMBER, to write down the sequence of ANSWERS: Y, Y, N, Y, N, Y, N, N, N, Y.

      REPLACING "Y" by "1"; "N" by "0", we find: 1,1,0,1,0,1,0,0,0,1.

      Now, we INVERT THAT SEQUENCE, and omit commas: 1000101011. A BINARY NUMBER!

      . EVALUATING the INVERTED SEQUENCE AS POWERS OF 2, we obtain a sum in DECIMAL TERMS: 1 + 0 x 21 + 0 x 22 + 0 x 24 + 1 x 24 + 0x25/sup> + 1 x 26 + 0 x 27 + 1 x 28 + 1x 29 = 1 + 16 + 64 + 256 + 512 = 849. Hah!

      THE BINARY DECISION-SEQUENCE TO FIND THE ANSWER TRANSLATES, WHEN INVERTED, AS POWERS OF TWO TOTALING THE DECIMAL ANSWER!

      "Trapping The Wild Word" is a MATHTIVITY that trains students for USING BINARY SEARCH TO CARRY OUT USEFUL MEASURING OR OTHER RESEARCH PROJECTS!

      USEFUL APPLICATION: A committee must choose ONE among many PROPOSALS or RESUMÉS or whatever. They can SEPARATE OUT THE WORST HALF OF THE ORIGINAL SET AND REJECT. Given the SMALLER SET, SEPARATE OUT THE WORST HALF AND REJECT. Etc. CONTINUING IN THIS WAY, a SET OF REASONABLE SIZE CAN BE REACHED SO THAT EACH REMAINING ONE CAN BE CONSIDERED CAREFULLY.

      This may be a surprise to most folks who think BINARY NUMERATION & COMPUTAIION IS "MODERN". Some may think it's a cruel joke of "The New Math" to plague teachers and parents, as well as kids. Some may known that COMPUTERS USE BINARY NUMERATION & COMPUTATION, but may think it was invented for this purpose. Actually, BINARY COMPUTATION -- BUT NOT BINARY NUMERATION -- is ANCIENT!