Elementary Mathematics-Science SMILE Meeting
09 May 2000
Notes Prepared by Earl Zwicker


Henry Heald Lecture
Professor Leon Lederman
Nobel Laureate in Physics and
Chair, Teachers Academy for Mathematics and Science

Monday, 22 May 2000
Perlstein Auditorium [33rd & State, NW Corner]

for a discussion about

The Essential Transition from 19th to 21st Century
High School Science Education

buffet supper 6 pm
discussion 7 pm
RSVP to (312) 567-8820 or parson@iit.edu

          OUR NEXT MEETING...
                             ...will be the start of the Fall
          semester on September 5, Tuesday. You will receive a
          reminder in the mail. You may have registered already...

                              SEE YOU THERE!!

Section A:

Mikhail Siddiq (Raymond School)
asked, "Would you want $20,000 in cash, or 1 cent doubled each day for 365 days?" Several people opted for the cash until Mikhail showed us that on day 22 the "income" would be $20,971.52, and the cumulative total by that day would be $41,943.03!! Using a calculator, Ken Schug (Chemistry, IIT) went to 46 days and got about $1,470,000,000,000 ($1.47 trillion)! The power of doubling! Amazing!

Mamie Hill (Woods School)
passed out large styrofoam cups, a piece of Saran WrapTM and a section of comic strip. We cut off the top rim of the cup, taped the Saran Wrap to the rim, then put a few drops of water on the Saran Wrap which formed a lens-like shape. Using this "lens," we could look through it and see the magnified comic strip lying below. (Handout: Liquid Lens) What can be learned from this? It may be used to view bugs and other small things, and it's something students can improvise at home and experiment further with it. A great idea!

Iona Greenfield (Carnegie School)
gave out milk cartons (8oz), popsicle sticks, sticky tape and construction paper. From these materials we made sailboats, using the bottom part of the milk carton for a hull. The clay, stuck to the bottom of the "hull," held the popsicle stick vertically to serve as a mast for a sail made of the construction paper. With the sailboat floating in a small tank of water, we could propel the boat by blowing on its sail, and explore the answers to various questions posed in the handout (Concept: Moving Air Can Do Work). Hands-on science at work!

Chandra Price (Burnham School)
(handout: Simple Science Experiments with Everyday Materials by Muriel Mandell) passed out soda straws, each enclosed in a sanitary paper wrapper. We tore off one end of the wrapper, pushed down on the remainder to compress it to about 1 inch in length, and removed it from the straw to lay on the table. When we dripped a few drops of water onto it, it GREW and writhed around almost like it was alive! (Why?!) Chandra had several potatoes on the table, and we found if we took a straw, and pushed it hard and fast, it would go straight into a potato! Why didn't the straw buckle or bend (as it does if we move the straw slowly)? The entire straw has "inertia," the tendency of a body in motion to continue in motion. Next, how can a straw in an empty glass be bent without touching it? Add water to the glass. It will appear to be bent where it enters the water (refraction)! Try it! Then we experimented with raw and hard-boiled eggs. How to tell them apart? Just spin them. The raw ones slow down and stop quickly, because the liquid inside absorbs the energy of spin. Not so with the hard-boiled ones, since their centers are spinning with the egg when set into rotation. How to clean a penny? Soap and water doesn't work well, but when we used a few drops of lemon juice, the penny got nice and bright.

Ken Schug thought it was due to complexing Cu++ with citrate, which makes the copper oxide on the surface soluble and it washes away. How to make an egg float? Place it in a glass of water. It sinks to the bottom. Add salt and stir. The salt water is more dense than the egg, so the egg then floats. Finally, to some milk in a cup, we added a little vinegar and stirred. The acid vinegar denatures the milk protein to form curds, and the clear liquid left is the whey. Lemon juice, being acid, would also work. Great stuff, Chandra!

Joyce McCoy (Spencer School)
placed on the table: construction paper, markers, a stapler, paper streamers (10 inch strips), and ribbon (10 inch). She gave us handouts (Art Activities - Wind Sock), and showed us an example, and we all got busy making our own take-home wind socks. With a wind sock hanging outside a window, one can judge how windy it is by watching how it behaves. The more days you observe it, the better idea you have of how windy it is! Neat!

Barbara Baker (Doolittle West School)
passed out a worksheet titled All About Condiments. And then we each got several packets: (as from fast food restaurants) ketchup, Dijon mustard, mayonnaise, duck sauce, honey, soy sauce, sweet sour sauce. By reading the packets, we had to write down their weights (units!), ingredients, and then place the packet in a cup of water to see if it sinks or floats. If it floats, it is less dense than water. And, which condiment do you like best?

We were given another handout (Dinosaur Lessons), and a model for Pangea was constructed. Barbara had us tear a more-or-less round piece from construction paper (perhaps 7 inches diameter). Then we were invited up to the table where some pans (about 18 inches long) were partly filled with water. Barbara tore her round piece into 4 - 5 pieces, then carefully placed them on the water surface so they were back together again in their spherical shape. Then she dripped a few drops of a liquid detergent onto the paper, and the pieces rapidly moved apart from each other! A model of continental drift, and showing why dinosaurs could be found on different continents. Of course, the forces moving the paper "Pangea" apart in our model (surface tension forces) are not believed to be the same forces that produce continental drift on our planet Earth, but the analogy is otherwise a nice one, and it shows how the continents fit together like a puzzle to form one land mass, Pangea. See the website http://kids.earth.nasa.gov/archive/pangaea/.  Thanks, Barbara!

Kenneth Onumah (Kozminski Academy)
passed out copies of graph paper on which students had traced outlines of their hands and feet. One may count the squares to estimate areas, and may graphically compare the effects of age and gender by graphing points with a foot dimension on the vertical axis against the corresponding hand dimension on the horizontal axis. Ken showed us such a graph, and the points showed a correlation in size, meaning people with larger hands usually have larger feet. Or one of those dimensions vs age, or vs gender. A most interesting investigation, which motivates students since they themselves are involved in a personal way!

Marva Anyanwu (Green School)
(handout: Monocots & Dicots) handed out monocot and dicot seeds. We also received reproductive parts of plants with red flowers, and could make comparisons with the information pictured on the handouts. An active way to gain insight into relationships in the world of the living!

Section B:

Pearline Scott (Franklin School)
showed us some differences between a Math Fair and a Science Fair. Both use poster boards, and scientific method should apply in some form. Topics make take somewhat similar forms; there are Math puzzles, games, olympiads, and history. The Pythagorean Puzzle came up:

a2 + b2 = c2.
Lee Slick (Morgan Park HS)
showed us puzzle pieces on the overhead and invited us to put them together to form a solid rectangle. And Roy Coleman (Morgan Park HS) showed the connection with the relation

(A+B)2 = A2 + 2AB + B2.

Go to the website http://www.iit.edu/~smile/ph9711.html to find more on how the puzzle may be used, K-12, thanks to Harald Jensen.

Earnest Garrison (Jones Magnet HS)
told us how Friends of the Chicago River test for water quality. See the website http://www.chicagoriver.org/ . He gave us samples of various kinds of water to taste:

Samples of river water are tested for macro invertebrates (which are necessary for higher forms of life), pH, and dissolved oxygen (which affects taste among other things). Interesting, Earnie!

Estelvenia Sanders (Chicago Voc HS)
did Science in Sign XV for us: Geometry for the Deaf. She educated us about Tactile Interpreting (last time she did Cued Speech with us). She made a sketch of an open left hand on the board. With the thumb on the left, she wrote an A on the top bone, and B on the bottom bone. The first (index) finger was labeled C,D,E starting at the top bone and ending at the bottom, then F,G,H for the middle finger, etc. The letters of words can then be spelled out by touching at the correct points on the fingers. Another way, you can push your hand into theirs to form each letter. Thanks, Estelvenia!

Carl Martikean (Wallace HS, Gary)
told us how to get information on a rocket launcher for an Air Impulse Rocket or "Stomp Rocket." Go to the website http://www.physics.purdue.edu/OUTREACH/rocket.html Thanks, Carl!

Pokemon Cards, continued
Porter Johnson

In a recent SMILE class Patricia Phillips [Arai Middle School] posed the question of how many Pokemon Cards [given randomly and with equal probability] one would have to get to have "reasonable assurance" of having a complete set of 6 cards. We drew a few cards during the class, and we able to confirm the claim that the "average" number drawn would be about 14, and that you would generally have to get several of the same kind before acquiring a full set.

The initial enthusiasm of getting the cards wore off after a few trials. I decided to "automate the process" by writing a computer program to do a significant number of "random trials". Here, for your potential amusement, is the FORTRAN program to make and record these trials:

c  pokemon cards
c  monte carlo simulation
c  number of "draws" to get a complete set.
      dimension icard(6),ntimes(100), ncum(0:100)
c  set up random number generator
      nrand = 11
      call seed(nrand)
c  initializing counter of games to zero
      do 10 n = 1,100
10    ntimes(n) = 0
c  input: number of games [complete sets] to get
      write(6,*)' how many games do you wish to play?'
      read(5,*) kgames
      write(8,88) kgames
88    format(' num games = ', i10)
c  playing the games
      do 100 k = 1,kgames
c  k is the game number
      do 40 i = 1,6
40    icard(i) = 0
c  playing a particular game
      do 60 n = 1,100
c  n is the number of cards in a given game
      call random(x)
      xx = 6.*x +1.
      j = int(xx)
c  j is the card number received on nth draw
      icard(j) = icard(j) +1
c  icard(j) is number of times jth card has appeared
      t = icard(1)*icard(2)*icard(3)*icard(4)*icard(5)*icard(6)
c   testing for a complete set:  t is not zero
      if (t.gt.0) then
c   complete set is obtained; go to next game
      ntimes(n) = ntimes(n)+1
      goto 100
      end if
c  end of a particular game
60    continue
c  if more than 100 games are required to get a match, we stop
100   continue
c  recording of 
      write(8,*) ' number of draws for a match'
      write(8,99) (ntimes(n), n = 1,100)
99    format(10i8)
c computing mean and standard deviation of number of draws
      sum1 = 0.
      sum2 = 0.
      do 200 n = 1,100
      fn = float(n)
      fnum = float(ntimes(n))
      sum1 = sum1 + fnum*fn
      sum2 = sum2 + fnum*fn**2
200   continue
      fgames = float(kgames)
      fmean = sum1/fgames
      fmeansq = sum2/fgames
      stddev = sqrt(fmeansq-fmean**2)
      write(8,999) fmean, fmeansq, stddev
999   format(' mean =', f10.6, /, ' meansq =', f10.6, /,
     > ' stddev =', f10.6)
c  ncum(n) is probability of getting a complete set after n games     
      ncum(0) = 0
      do 300 n = 1,100
300   ncum(n) = ncum(n-1)+ntimes(n)
c  print of ncum(j)
      write(8,*) ' cumulative hit number'
      write(8,99) (ncum(n), n = 1,100)
c  quit and go home

Here is output for a run with ten million "games". In each game I told the computer to keep drawing until it got a complete set. A record of the number N of draws needed for each set is given, as well as the cumulative hit numbers [number of times that N draws or fewer is required to obtain a complete set]:

 num games =   10000000
  number of draws for a match
       0       0       0       0       0  154776  385006  600803  750090  825776
  843029  816335  762255  690521  613102  538025  466376  400611  342242  290821
  245911  206988  174815  147081  122821  102982   85649   71363   60185   50199
   41792   35076   29128   24394   20305   16716   14039   11830    9921    8163
    6622    5673    4692    3953    3353    2869    2282    1943    1591    1315
    1073     924     750     634     548     488     368     298     230     184
     184     150     132      99      88      66      54      51      51      35
      20      24      26      11      16      12      14      12       9       2
       3       3       2       8       1       4       1       0       2       2
       0       0       0       0       1       0       0       0       0       1
 mean = 14.701120
 meansq =255.126600
 stddev =  6.245292
  cumulative hit number
       0       0       0       0       0  154776  539782 1140585 1890675 2716451
 3559480 4375815 5138070 5828591 6441693 6979718 7446094 7846705 8188947 8479768
 8725679 8932667 9107482 9254563 9377384 9480366 9566015 9637378 9697563 9747762
 9789554 9824630 9853758 9878152 9898457 9915173 9929212 9941042 9950963 9959126
 9965748 9971421 9976113 9980066 9983419 9986288 9988570 9990513 9992104 9993419
 9994492 9995416 9996166 9996800 9997348 9997836 9998204 9998502 9998732 9998916
 9999100 9999250 9999382 9999481 9999569 9999635 9999689 9999740 9999791 9999826
 9999846 9999870 9999896 9999907 9999923 9999935 9999949 9999961 9999970 9999972
 9999975 9999978 9999980 9999988 9999989 9999993 9999994 9999994 9999996 9999998
 9999998 9999998 9999998 9999998 9999999 9999999 9999999 9999999 999999910000000

We see that the mean number of draws is about 14.70, whereas the spread in that mean is about 6.25, so that between 7 and 21 draws are required to get a full set most of the time. In fact, it is about 5.4 % probable to have a full set after 7 draws, about 51.4 % probable to have a full set after 13 draws, and about 87.3 % probable to have the set after 21 draws. However, the distribution of draws is quite "skewed", and the most likely number of draws required is 11, which occurs about 8.4% of the time. Of the ten million computer-generated games, one of the games required 100 draws to get a full set of cards.

The Monte Carlo approach to solving was first employed in large-scale digital problems shortly after World War II by an applied mathematician active in the Manhattan Project Stanislaw Ulam, shortly after he recovered from a very serious illness. There is no better introduction to the subject than this paragraph, taken from his autobiography, Adventures of a Mathematician, [U California Press 1991; ISBN 0-520-07154-9] pp 196-197.

The idea for what was later called the Monte Carlo method occurred to me when I was playing Solitaire during my illness. I noticed that it may be much more practical to get an idea of the probability of the successful outcome of a Solitaire game (like Canfield or some other where the skill of the player is not important) by laying down the cards, or experimenting with the process, and merely noticing what proportion comes out successfully; rather than to try to compute all the combinatorial possibilities which are an exponentially increasing number so great that, except in very elementary cases, there is no way to estimate it. This is intellectually surprising, and if not exactly humiliating, it gives one a feeling of modesty about the limits of rational or traditional thinking. In a sufficiently complicated problem, actual sampling is better than examining all the chains of possibilities.

How do we understand these numbers, and how do we begin to compute the odds when the chances of drawing cards are not the same, as is surely the case for Pokemon cards given with Happy Meals at McDonalds?

Let us suppose that the pokemon cards #1, #2, #3, #4, #5, #6 are generated with probabilities p1, p2, p3, p4, p5, p6, respectively. Each probability pi is positive, and the total probabilities must add to 1:

p1 + p2 + p3 + p4 + p5 + p6 = 1.

It is also true that

[p1 + p2 + p3 + p4 + p5 + p6]N = 1 .

It is remarkable, but true, that if you write out the algebraic expression for

GN(pi) = [p1 + p2 + p3 + p4 + p5 + p6]N

as a sum of monomials in pi, the probability that, in N draws, you get card #1 n1 times; card #2 n2 times; card #3 n3 times; card #4 n4 times; card #5 n5 times; and card #6 n6 times, respectively, is given by the monomial term occuring in the above expression; ie

N! / [n1! n2! n3! n4! n5! n6!] * p1n1 p2n2 p3n3 p4n4 p5n5 p6n6
n1 + n2 + n3 + n4 + n5 + n6 = N

in that polynomial, GN(pi). We wish to determine the probability that, in N draws, we get a complete set of cards. That probability is obtained by picking out the terms from the expression for GN(pi), and dropping the terms that do not contain all six pi. The concept is simple in principle, but tedious in practice!

When the dust clears, we obtain the following expression for the probability of having a complete set after N trials:

P(N) = [p1 + p2 + p3 + p4 + p5 + p6]N
- [p1 + p2 + p3 + p4 + p5]N - ...
+ [p1 + p2 + p3 + p4]N + ...
- [p1 + p2 + p3]N - ...
+ [p1 + p2]N + ...
-[p1]N - ...

In this expression the first line is complete; in the second line there are 6 terms that involve any five of the pi; in the third line there are 10 terms that involve any four of the pi; in the fourth line there are 15 terms that involve any three of the pi; in the fifth line there are 10 terms that involve any two of the pi; and in the sixth line there are 6 terms involving each of the pi. That is the answer, no matter what N is and what the individual pi happen to be.

We can simplify the answer considerably for the case in which the 6 cards are equally probable, by setting pi = 1/6. We obtain that, for any N,

P(N) = 1 - 6 * [5/6]N + 10 * [4/6]N - 15 * [3/6]N + 10 * [2/6]N - 6 * [1/6]N

I have written an EXCEL program to determine these numbers and to draw a graph, which is shown below: The numbers agree with those obtained in the Monte Carlo Simulation, to about four significant figures. Here is a listing of the numbers for small values of N:

N P(N) N P(N) N P(N) N P(N)
1 .0000 11 .3562 21 .8726 31 .9790
2 .0000 12 .4378 22 .8933 32 .9825
3 .0000 13 .5139 23 .9108 33 .9854
4 .0000 14 .5828 24 .9254 34 .9878
5 .0000 11 .6442 25 .9388 35 .9899
6 .0154 16 .6980 26 .9480 36 .9915
7 .0540 17 .7446 27 .9566 37 .9930
8 .1140 18 .7847 28 .9638 38 .9941
9 .1890 19 .8189 29 .9698 39 .9950
10 .2718 20 .8480 30 .9748 40 .9959

[Pokemon Curve]