TRAFGRAFS FOR MAKING CHOICES

I AM THE CHOICES I CAN MAKE1 I AM THE ROLES I CAN PLAY!

This mathtivity shows how to use TRAFGRAFS to LEARN ABOUT CHOICE-MAKING. Another file will show how to use TRAFGRAFS to LEARN ABOUT ROLE-PLAYING.

THE FUNDAMENTAL PRINCIPLE OF CHOICE -- for KIDS gradually to LEARN is THIS:

  1. The NUMBER OF ITEMS is n;
  2. suppose them to be INDEPENDENT (choosing a particular item says nothing about choosing any other item};
  3. then each ITEM exists on TWO LEVELS:
    1. You have n ITEMS. Do you INCLUDE ITEM 1? YES, or NO. 2 possibilities for ITEM 1. Do yu INCLUDE ITEM 2? YES, or NO. 2 possibilities for ITEM 2. That totals 2 X 2 = 4 possibilities. And 2 (Y-N) possibilities for ITEM 3, making the current TOTAL: 2 X 2 X 2 = 8 possibiliit1es. Etc. With n ITEMS, you have 2 X 2 X X ... X 2 = 2n, THE PRODUCT OF n 2's. That's one level. And each one is a CHOICE-SET, CONSISTING of Y-N distinctions, say: Y N N Y N N ... Y. Imagine each one written on a line of tablet paper, for 2n ROWS.
    2. Now cut out each of the 2n ROWS and put in a shoebox to draw from. Do you draw a particular choice-set (strip)? Answer: YES or NO -- 2 possibilities. And 2 possibilities for the next choice-set (strip). And the next. And the next. Etc. What is the TOTAL? You have n choice-sets (strips), each representing 2n (Y-N) CHOICES. That's: 2n X 2n X 2n X ... X 2n. That is, 2n WRITTEN n TIMES AND MULTIPLIED. TOTAL: (2n)n CHOICES!

      Let's see what that means.

      CHOICE TABLE
      n Items2n(2n)n
      121 = 2(21)1 = 2
      222 = 4(22)2 = 42 = 4 X 4 = 16
      323 = 8(23)3 = 83 = 8 X 8 X 8 =512
      424 = 16(24)4 = 164 = 16 X 16 X 16 X 16 = 65536
      525 = 32(25)5 = 325 = 32 X 32 X 32 X 32 X 32 = 233554432
      626 = 64(26)6 = 646 = 64 X 64 X 64 X 64 X 64 X 64 = 68719476736
      See how FAST this GROWS? It soom becomes to LARGE to write easily in a TABLE. For n = 9 INDPENDENT ITEMS, THE TOTAL NUMBER OF CHOICES IS A SUPERGOOGOL! What's that?

      A 5-year-old boy thought up the NUMBER WRITTEN AS 1 FOLLOWED BY 100 ZEROS, and NAMED THIS NUMBER "A GOOGOL". But, GIVEN 9 INDENDENT ITEMS, THE NUMBER OF CHOICES IS MORE THAN 1 FOLLOWED BY 153 ZEROS!!! A SUPERGOOGOL. THERE ISN'T A GOOGOL OF ANYTHING IN THE ENTIRE UNIVERSE! But any Kid with 9 INDENDENT ITEMS (TOYS, BOOKS, CLOTHES, FOOD, etc.) HAS A SUPERGOOGOL OF CHOICES!

      Here's how you can show this to a PRESCHOOL KID with TINKERTOY items. Take the second row of the above TABLE.
      1. n = 2. Guide the Kid in putting 2 COUNTABLES (rocks or whatever) on a (physical) table.
      2. Each COUNTABLE can be CHOSEN, or NOT CHOSEN: YES-NO. What are the possibilities?
        1. Label a TINKERTOY RED spool "NO"; label a BLUE spool "NO"; connect the two spools by a rod.
        2. Connect by rod another RED spool, LABELED "NO", to a BLUE spool, LABELED "YES".
        3. Connect by rod another RED spool, LABELED "YES", to a BLUE spool, LABELED "NO".
        4. Connect by rod another RED spool, LABELED "YES", to a spool, LABELED "YES".
      This constructs THE CHOICE-SET (ALL POSSIBILITIES):
      	NO-----NO        NO-----YES    YES-----NO    YES-----YES
      Put a bowl of pennies (say, 20) on the table.
      • Ask, "Do we choose the NO-----NO rod?" YES or NO. 2 possbilities. So put 2 pennies out on the table.
      • "Do we choose the NO-----YES rod?" YES or NO. 2 possibilities. So put 2 more pennies on the table.
      • "Do we choose tne YES-----NO rod?" YES or NO. 2 possibilities. So put 2 more pennies on the table.
      • "Do we choose the YES-----NO rod?" YES or NO. 2 possibilities. So put 2 more pennies on the table.
      • We've asked the YES-NO QUESTION about EACH OF THE FOUR DIFFERENT RODS. That covers ALL THE CHOICE-SETS FOR 2 ITEMS.
      • Now, COUNT THE PENNIES: 1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16. 16 PENNIES. THEY REPRESENT A TOTAL OF 16 CHOICES FROM 2 INDEPENDENT ITEMS.
    CAN YOU GUIDE A CHILD TO USE TINKERTOY TOOLS TO IMPLEMENT ROW 3 OF CHOICE-TABLE? WHEN THE CHILD IS OLDER, ROW 4? And HIGHER?