Earnest Garrison (Jones Academic Magnet HS)
showed some Discovery Kits on electromagnetism that he had obtained
from the
source:
Science Kit & Boreal LaboratoriesHe passed around booklet #6530505, Electric Motor, written by Dr Lawrence Lowery, University of California, Berkeley, which described several experiments. Then Earnest showed us the transparent box with a tunnel passing through it, containing iron filings suspended in mineral oil, for showing the magnetic field in 3 dimensions when a magnet is inserted into the tunnel. Then he showed us a simple kit for making an electric motor, which simplified the process of winding the wire and putting the loop into the armature.
P.O. Box 5003
Tonawanda, NY 141515003
http://sciencekit.com/ [order a free CDROM catalogue]
Phone: 18008287777
Fax: 18008283299
Fred Schaal (Lane Tech HS, Math)
Walter McDonald (CPS Substitute and Veterans Administration
Diagnostic
Radiation Technologist)
showed us this graph of the trigonometric functions [sine, cosine,
tangent,
cotangent, secant and cosecant] which was obtained from the
Microsoft Encarta
Encyclopedia.
Walter made the following points:
Fred Farnell (Lane Tech HS, Physics) What happens when waves
meet?
put some paper cups on the floor and stretched a slinky™ across
the floor, which was held at its ends by two assistants. An
assistant, by
rapidly moving his end of the spring back and forth once {transverse to
the
direction of the stretched string), sends a transverse wave pulse
toward the
other end. When the other end was held
fixed, the wave was seen to reverse its orientation
and direction after reflection at the fixed end. The cups were
put parallel to
the stretched slinky on both sides of it, and the goal was to set up
waves that
would knock down all of the cups. This was seen to be difficult,
if not
impossible. Then, the assistants set up waves coming in
simultaneously
from each end, so that we could see the slinky before, during, and
after the
overlapping intersection of the wave pulses in the middle. Very
interesting, Fred!
Bill Colson (Morgan Park HS, Math)
showed a workbook, Stretching Your Math Students' Achievement,
Motivation, and Involvement:
Grades 7  12 Resource Handbook by Irv Lubliner, recently obtained
from the following source:
Bureau of Education and ResearchFirst he showed a clear exposition from that book of the rope trick (topological puzzle) that has been shown several times in SMILE: [ph101398.htm].
915 118th Avenue, SE
PO Box 96088
Bellevue WA 98009
http://www.ber.org/
Tel: 1  800  7353503
Then he showed us how to play a game called MAXIT, illustrating the point with a 4 ´ 4 square lattice. He put an ´ into one location, and had us to call out numbers between 10 and +10 for the other locations, with the result as shown:
5  8  6  2 
3  ´  7  6 
1  3  0  1 
4  9  9  9 
People on one side of the room were the UP'S AND DOWN'S, whereas those on the other side were the RIGHT'S AND LEFT'S. After flipping a coin to see who moves first, the winner was allowed to move the ´  either up or down, or right or left, respectively, to another location. The number in that location is replaced by an ´, and they get the number of points corresponding to that number, and you cannot move into the location of an ´. The last team to be able to move ends the gameand the team with the largest point total wins.
The next game involved the creation of a magic square, such as the following one:
14  20  3  12  7 
32  38  21  30  25 
19  25  8  17  17 
11  17  0  9  4 
17  23  6  15  16 
The numbers in this table may look unrelated, but they have not been randomly chosen, because if you pick five numbers, each from a different row and a different column, and take the sum, you will get the total 79. For instance, the five numbers shown in bold give 14 + 25 + 25 + 0 + 15 = 79. We have, in fact, generated 120 different combinations of numbers adding up to the total of 79. Isn't that remarkable? Surprisingly, there is nothing unique about the number 79, and you can see how the table was made by adding another row and another column to it:
**  11  17  0  9  4 
3  14  20  3  12  7 
21  32  38  21  30  25 
8  19  25  8  17  17 
0  11  17  0  9  4 
6  17  23  6  15  16 
The inner numbers are generated by taking the sum of the corresponding numbers in the first row and first column. e.g. 21 + 17 = 38. The "magic number" 79which is merely the sum of the ten numbers in the first row and the first columncan by changed by changing those 10 "seed numbers".
Don Kanner (Lane Tech HS, Physics)
announced that he was traveling to the Bay of Fundy this summer
to
witness the high tides for himself. There was a question of the meaning
of the
term tidal bore, a steepfronted wave caused by the meeting of
two tides
or by the constriction of a spring tide as it passes up an estuary. The
following image is from
http://octopus.gma.org/undersea_landscapes/Bay_of_Fundy/
"This is a natural phenomenon seen in very few parts of the world. The Bay of Fundy is particularly noted for its extremely high tides, the highest in the world, and for its tidal bores. In the funnelshaped Bay of Fundy which is 48 miles wide at its mouth and narrows down along its entire length, the tide water enters the bay at its widest point. As it passes along toward the head of the bay it is, in effect, squeezed by the evernarrowing sides and by the constant “shallowing” of the bottom....see a river change its direction of flow before your very eyes! At the TIDAL BORE RAFTING PARK this advancing tide becomes a wave, varying from just a ripple up to 10 feet in height. This wave is referred to as a “TIDAL BORE”. Nowhere else in the world can a tidal bore of this magnitude be seen. This is where the bore rolls in, in its fullest ferocity, followed by 3 to 10 foot rapids. Thus, we have the phenomenon of a river changing its flow before your very eyes, created by the tidal wave, or bore, flowing in OVER the outgoing water. The size or height of the tidal bore varies according to the phases of the moon. Highest tidal bore occur around the full and new moons."
Don also
mentioned that he had notified the publisher of a physics text of some
errors in
the drawings in the book. Specifically, he mentioned a "strobe
timed" drawing of a cannonball just shot from the cannon:
____________

BOOM !!! 00 0 0 0 0 0 0 0 0
____________
The diagram looks suspiciously like one for uniformly accelerated motion, which should not occur after the cannon ball leaves the cannon.
Porter Johnson (IIT) mentioned the golden rectangle ratio
( 1 + Ö5)/2 =
1.61803...
GOLDEN RECTANGLE
or
If we define the "golden ratio" x as the long side b divided by the short side h; i.e. x = b / h, this equation may be written as
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 of successive number pairs 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_{2} = 2 by taking the sum of the two previous elements:
y_{n+1} = y_{n}+ y_{n1} .
Let us assume that the ratio y_{n+1}/ y_{n} approaches a limiting value, x, at very large n; i.e. y_{n+1}/y_{n} ® x and y_{n} /y_{n1 }® x.
The iteration formula
y_{n+1} = y_{n}+ y_{n1} .
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 y_{2}.
One may express any real number uniquely through its continued fraction expansion [http://www.cuttheknot.org/do_you_know/fraction.shtml]:
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://britton.disted.camosun.bc.ca/goldengeom/goldenpenrose.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.
Notes taken by Porter Johnson.