Elementary Mathematics-Science SMILE Meeting
03 April 2001
Notes Prepared by Porter Johnson

Section A: [K-5]

Carolyn McGee and Claudette Rogers (Manierre School) Handout: Cereal Venn Diagrams
led an exercise in sorting and classifying cereal by attributes using Venn Diagrams, using 2 kinds of multi-colored, multi-shaped cereal pieces (e.g. Froot Loops™ and Trix™), paper cups, and 4-foot lengths of yarn or string. Lay out two overlapping circles [shown below] and place pieces of cereal with a common attribute [e.g., all round] inside one of the circles, and those with a different attribute [e.g. all red] inside the other circle.  The pieces that have both attributes [e.g. both round and red] should be placed in the overlapping section of the diagram. The cereal pieces were glued in place on the paper to make a colorful display.

A, B, C, and 1, 2, 3 Graph:
Next we took "alphabet-shaped" cereal, and predicted which letter of the alphabet would occur most frequently in the box. Then we counted the number of pieces for each letter, using tally marks on a sheet. Then we made a big display on the board, gluing the cereal letter pieces to strips to indicate how many there were for each letter, and put it on the board.  It was similar to the bar graph on the sheet shown below:

We were given an excellent list of written materials:
• The Alphabet Tree by Leo Lionni [Knopf 1968].
• "How Many Snails?" Counting book by Paul Giganti Jr [Greenwillow 1988].
• Inch by Inch by Leo Lionni [Astro-Honor 1962].
• How Big Is a Foot? by Rolf Miller [Dell Young Yearling 1962].
• How Many Bugs in a Box? by David A Carter [Simon and Schuster 1988].

Tanisha Kwaaning (Pullman School) Science Activities with Plants Handout: Bloom Basics [McDonald Publishing Co 1997]
passed out a picture of a flower with the various parts [sepal, pollen, pistil (stigma and style), ovary, ovules, stamen (anther and filament), petal] marked.

• Fascinating Flowers

We came to the desk and colored the various parts of the image to make our flowers.

• Tanisha taped onto the board a completed example of what she wanted us to do with materials she had placed on the table. There was a green sheet (about 30 ´ 40 cm in size) showing the assembled Parts of a Seed Plant. On the handout sheet were drawings with the names of the parts. We colored them (flower petals, stems, leaves, pistils, etc.) with the indicated colors, cut them out, and glued them onto our own green page to make a pretty and informative kind of 3-dimensional flower on the page. We cut out and glued little boxes onto the page also, and identified the names of the plant parts by drawing lines connecting the box and the part. What a pretty and creative way for students to learn!

Tanisha showed some images obtained from the website of The Education Center, Inchttp://www.themailbox.com/.  Note that you must register on that site for entry, to obtain access to a number of detailed pictures of plants on the internet.  Also, there is a publication, Plants:  Investigating Science Grades 4-6 [The Education Center 2000] ISBN 1-56234-401-3.

Additional Information on Parts of a Seed Plant [See the article: How to Identify Plants: Important Features of Flowering Plants at the website http://www.biologie.uni-hamburg.de/b-online/e02/02.htm: In particular, the article states that ... the principal parts of a seed plant are the leaves, stems, roots, flowers, fruits (images), and seeds. Here is a diagram for labeling the various parts of a plant: http://www.urbanext.uiuc.edu/gpe/case1/c1m1app.html.

Notes taken by Earl Zwicker

Section B: [4-8]

Emma Norise (Dunbar Vocational Career Academy) Density and the Scientific Method
She passed out a handout, titled ON THE LEVEL, which asked these questions

• Why does cream float?
• How do you keep the ingredients of salad dressing from separating?
• Why does salad dressing eventually separate?
• How do liquids separate?
• Which objects float and which ones sink in a given liquid?
She gave us large beakers (about 400 ml) and measured equal amounts of honey, vegetable oil, and water [with dye], which we poured into the beaker.  The liquids formed three distinct horizontal layers, with honey on the bottom, water in the middle, and vegetable oil on top.
`                \             |                 |            |                   |------------|                 | Veg Oil    |       Liquid                  |------------|     Separation                 |  Water     |         in                 |------------|       Beaker                 |  Honey     |                 |____________|`
For various solid objects, we first hypothesized how they would behave inside the beaker, and then we put them in.  Here are typical data:

 Material Hypothesis Observation Pasta float on water float between water and honey Magnetic Ball sink to bottom sink to bottom Grape float in honey float between water and honey Large Lego Blocks float in oil float just in water

Porter Johnson mentioned that the modern processed foods such as salad dressing, and many natural foods such as milk and fruit juice, are colloidal suspensions of materials that normally do not mix. For example, soft drinks are held in colloidal suspension by addition of binding materials. One of the original binders, gum arabic (from the plant Acacia Senegal), is more valuable in its pure form than gold  by weight.  See the website http://hans.presto.tripod.com/cat018.html.

Monica Seelman (ST James School)
passed around and discussed the new book on the following history of the number Zero from the cave men to Einstein:

Zero:  The Biography of a Dangerous Idea, Charles Seife [Penguin 2000] ISBN 0-14-028647-6

The Romans and Greeks did not use the number zero, but considered it as "the void".  The Arabs developed the modern concept of zero, and invented a symbol for it.  Actually, the modern Arabic symbol is not the symbol "0" used in the rest of the world, but simply a dot:  "." .

Of course, the controversy as to whether the millennium ended with the year 1999 or 2000 is related to the fact that there was no "year 0", since the counting of  years went directly from -1 to +1.

Porter Johnson (IIT Physics)
talked about several other special numbers upon which books have recently been written; namely

p= 3.14159...     e = 2.71828...      i = Ö(-1)      ¥ = "infinity"

Next, he mentioned the golden rectangle ratio

( 1 + Ö5)/2 = 1.61803...

```
GOLDEN RECTANGLE
__________________
|                  |
|        b         |
h  |                  |    h
|                  |
|                  |
|       b          |
|__________________|
```
This number arises out of the definition of a golden rectangle that the ratio of its height h {short side) to breadth b (long side) is the same as the ratio of its breadth b to the sum of its height and breadth (h + b):

h ¸ b = b ¸ (b + h)

or

h ´ (b + h) = b2

If we define the "golden ratio" x as the long side b divided by the short side h;  or  x = b / h, this equation becomes

x2 = x + 1 or x2 - x - 1 = 0

This quadratic equation has two solutions, one positive and one negative. The positive solution is

x = ( 1 + Ö5)/2

This golden ratio can also be understood as the limit of ratios in the Fibonacci Sequence:

1 ¼ 2 ¼ 3 ¼ 5 ¼ 8 ¼ 13 ¼ 21 ¼34 ¼ 55 ¼ 89 ¼

In particular, note that 89 / 55 = 1.6181818 ... is fairly close to the limit.  The sequence is generated from the first two entries y1 = 1 and y2 = 2 by taking the sum of the two previous elements:

yn+1 = yn+ yn-1 .

Let us assume that the ratio yn+1/ yn approaches a limiting value, x, at very large n.; i.e. yn+1/yn ® x and yn /yn-1 ® x.

The iteration formula

yn+1 = yn+ yn-1 .

is equivalent to

yn+1 / yn-1 = (yn+1 / yn) ( yn / yn-1 ) = (yn/yn-1) + 1

At very large n, the ratios may be replaced by their limiting values to obtain this equation for the limit:

x2 = x + 1

Thus the golden mean is the limit of the Fibonacci Sequence, independently of the starting seeds y1 and y2

One may express any real number uniquely through its continued fraction expansion

A  = a + 1 / (b + 1 / (c + 1 /( d + 1 / (e +  ¼) ) ) )

where the  coefficients a, b, c, d, e, ¼ are positive integers.  If the number A is rational, the continued fraction expansion will terminate; otherwise it will go on forever.  We may identify the number with its continued fraction:  A = (a, b, c, d, e, ¼ ).  For the golden mean the continued fraction has the simplest form, in that the coefficients a, b, c, d, e, ¼ are all equal to 1. That is,

x = ( 1 + Ö5) / 2 = 1 + 1 / (1 + 1 / (1 + 1 /(1 + 1 / (1 +  ¼ )  )  )  ) =  (1; 1, 1, 1, 1, ¼ )

The golden mean is related to Penrose Tilings; see the website http://mathworld.wolfram.com/PenroseTiles.html. By terminating this continued fraction after various steps we recover the ratios of Fibonacci numbers,

2 / 1 ¼ 3 / 2 ¼ 5 / 3 ¼ 8 / 5 ¼ 13 / 8 ¼ 21 / 13 ¼

The continued fraction for e, the base of the Natural Logarithms, is relatively simple [see http://www.mcs.surrey.ac.uk/Personal/R.Knott/Fibonacci/cfINTRO.html#othercfsE]

e = (2; 1, 2 ,1, 1, 4, 1, 1 ,6 ,1 ,1, 8, 1, 1, 10, 1, ...)

On the other hand, the continued fraction expansion of p is less elegant looking:
p =

(3; 7, 15, 1, 292, 1, 1, 1, 2, 1, 3, 1, 14, 2, 1, 1, 2, 2, 2, 2, 1, 84, 2, 1, 1, 15, 3, 13, 1, 4, 2, 6, 6, 99, 1, 2, 2, 6, 3, 5, 1, 1, 6, 8, 1, 7, 1, 2, 3, 7, 1, 2, 1, 1, 12, 1, 1, 1, 3, 1, 1, 8, 1, 1, 2, 1, 6, 1, 1, 5, 2, 2, 3, 1, 2, 4, 4, 16, 1, 161, 45, 1, 22, 1, 2, 2, 1, 4, 1, 2, ... )

The rational approximations are 3, 22/7, 333/106, 355/113 = 3.14159292, ... . The last approximation is rather accurate, because the next number in the continued fraction, 292, is rather large.

Of course, in the Bible and other religious writings frequent reference is made to numbers; for example the number 666 is called the Mark of the Beast in the Revelation of ST John.  Although the triple six structure of the number 666 seems evocative of special mystical significance, this number may have been written at the time in terms of Roman Numerals; DCLXVI. One possible interpretation, as described in the book The Kingdom of the Wicked by Anthony Burgess [Washington Square 1986] ISBN 0-671-62631-0, is the following Latin Anagram:

 D C L X V I Domitianus Caesar Legatus csti Violenter Interfacit Emperor Domitian is violently killing the representatives of Christ

Note that chi: c or X was widely used symbol for the word Christ in the ancient world, as it is today.  A modern interpretation of 666  is addressed in the article Is "www" in Hebrew equal to 666? at the website http://home.wanadoo.nl/mufooz/Nwo-mc/English/www-666.htm.

Notes taken by Porter Johnson