03 April 2001

Notes Prepared by Porter Johnson

**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.

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:

**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, Inc**: http://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?

\ |For various solid objects, we first hypothesized how they would behave inside the beaker, and then we put them in. Here are typical data:

| |

|------------|

| Veg Oil | Liquid

|------------| Separation

| Water | in

|------------| Beaker

| Honey |

|____________|

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

Next, he mentioned the ** golden rectangle** ratio

**( 1 + Ö5)/2 =
1.61803... **

This number arises out of the definition of a golden rectangle that the ratio of its heightGOLDEN RECTANGLE __________________ | | | b | h | | h | | | | | b | |__________________|

or

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

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

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

In particular, note that **89 / 55 = 1.6181818 ...** is
fairly close
to the limit. The sequence is generated from the first two
entries
**y _{1} = 1** and

**y _{n+1} = y_{n}+ y_{n-1} .**

Let us assume that the ratio **y _{n+1}**/

The iteration formula

**y _{n+1} = y_{n}+ y_{n-1} .**

is equivalent to

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

Thus the golden mean is the limit of the ** Fibonacci Sequence**,
independently of the starting seeds
**y _{1}** and

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,

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**,

**
**

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**