High School SMILE Meeting
13 March 2001
Notes Prepared by Porter Johnson
Bill Colson (Morgan Park HS, Math)
3 . 1 4 1 5 9 2 6 5 3 5 ...
noted that tomorrow is p Day [3.14
--- get it?].
As an multicultural mnemonic to remembering the digits of p,
he presented the following phrases, in which the number of letters in
each word form the sequence
PJ Comment: Note that these messages constitute a code that loses
meaning in translation.
The French statement describes remembering the value of p,
whereas the meaning of the Spanish phrase is the following: Sun
and Moon and World acclaim the eternal
author of Cosmos. The code is, of course, broken in virtually
Many theologians and clerics argue that religious texts such as the Bible
Koran / Qur'an [http://www.usc.edu/dept/MSA/quran/],
understood properly only in their original languages.
||Que j'aime a faire connaître ce nombre utile aux
Archimède illustre inventeur quie de ton jugement peut priser la
Pour moi ton problème est de pariels avantages ...
||Sol y Luna y Mundo proclaman al Eterno Autor del Cosmo
||See, I have a rhyme assisting my feeble brain, its tasks
Bill also handed out copies of an article Test Yourself [SAT
I questions from recent tests designed to
measure verbal and math skills] from the 12 March 2001 issue
of Time Magazine:[
Fred Schaal (Lane Tech Park HS, Math)
- a6 = (z2)3 - (a2)3
= (z2 - a2)(z4 + a2z2
+ a4) = (z - a)(z + a)(z4 + a2z2
considered the algebra problem of factoring the following sixth
z6 - a6. He first pointed out that, if you
this as the difference of two cubes, and then consider z2
as the difference of two squares, you get the result
On the other hand, if you consider the polynomial
z6 - a6 as the difference of two squares,
use the appropriate cube formula on each of the factors, you get
z6 - a6= (z3)2
- (a3)2 =
(z3 - a3)(z3 + a3) = (z -
a)(z2 + a z
+ a2)(z + a)(z2 - a z + a2)
Since both of the answers must be correct, it must be true that
(z2 + a z + a2)(z2
- a z + a2) =
z4 + a2z2 + a4
Brandon verified this assertion by straightforward and tedious
algebra. Note: Since the LHS is symmetric
transformation z ® - z, all
odd powers of z must cancel on
the RHS as well. Porter Johnson commented on the
related problem of
finding all solutions of the sixth order polynomial equation z6
= 0. The solutions z0, z1, z2,
z4, z5, consist of the factor a multiplied by
each of the
6 sixth roots of unity; these complex numbers may be expressed in polar
zk= a e2ikp/6
k = 0. 1. 2. 3, 4, 5.
Here are the roots:
The factors zk form the following regular
hexagonal pattern in the complex plane
||a/2 [1 + i Ö3
||a/2 [1 - i Ö3
||a/2[-1 - i Ö3
||a/2 [-1 + i Ö3
The polynomial in question may thus be factored in the form
z6 - a6 = (z - z0)·(z
- z2)·(z - z3)·(z - z4)·(z
- z5 )
roots z0 and z3 are real,
whereas (z1, z5)
and (z2, z4) are complex conjugate
pairs. Thus the
products (z - z1) · (z - z5) and
(z - z2) · (z - z4) are real.
(z - z1)·(z - z5)=
z2 - a z + a2 ;
z2 * | * z1
z3 | z0
z4 * | * z5
(z - z2)·(z - z4) = z2 + a z + a2
You can also use these relations to get other identities easily:
(z - z1)·(z - z2)=
z2 + a2 -
a z ;
(z - z4)·(z - z5) = z2 + a2+
a z .
(z - z1)·(z - z2)·(z
- z5 ) =
z4 + 2 a2z2 + a4 - a2z2
z4 + a2z2 + a4
For the fascinating history of the Mathematician Evariste
Galois, who studied roots of polynomial equations, see the website
Here is a comment, taken from that website, by one of his teachers:
"It is the passion for mathematics which dominates him, I think it
be best for him if his parents would allow him to study nothing but
this, he is
wasting his time here and does nothing but torment his teachers and
himself with punishments."
Marilynn Stone (Lane Tech HS, Physics)
showed a circuit board system containing a battery, 3 light bulbs, and
single switches, and one double switch [S5] attached to a plywood sheet
2 feet by 2 feet], as shown.
She then asked the students to accomplish these
tasks with the battery:
The last exercise may be rather difficult.
|Connect B1 in series.
||Connect B2 and B3 in series.
||Connect B1, B2, and B3 in
|Connect B1 and B2 in parallel
B2, and B3 in parallel.
||Connect B2 in Series.
|Connect B3 in series.
||Connect B1 and B3 in series.
||Connect B1 and B3 in parallel
|Connect B1 and B2 in series.
Kanner (Lane Tech HS, Physics)
began by taking a plastic straw, flattening one end to make a double
blowing into it to make a sound. He cut pieces off the
other end and blew to demonstrate that, as the straw becomes
pitch of the sound goes up. He made a very interesting
presentation at the
Elementary SMILE Meeting on 06 March [Section B] , which is
detail on the website em030601.htm.
Don then announced
that he would give us a surprise test -- which turned out to be a
test. To that end, he put a chart on the board that looked like this
Average Sensitivity of Human Hearing at Various Frequencies
He started to give us the hearing test, but we were saved because there
was too much
hum in his audio oscillator [perhaps because
of a problem in the amplifier driven by it]. The hearing test was
and Don concluded by making the following general comments:
- The human ear is most sensitive in the region around 2 KHz.
- The threshold of hearing goes up by 20 dB [a factor of 100
sound pressure] at around 100 Hz. In other words, we are
sensitive to low-frequency [base] notes than to high frequency [treble]
notes. Consequently, it is routine to amplify the low frequencies
stereo systems. One inevitably amplifies low frequency background in
process--- Dolby™ and certain other systems are designed to
reduce the enhancement
of low frequency noise, while preserving the signal.
- Many teachers cannot hear well at frequencies much beyond 8
KHz, whereas most
students can detect frequencies up to 20 KHz. Comment by
Porter Johnson: "There is little difference in hearing thresholds
and female children. Between ages 10 and 20, males begin to show
auditory sensitivity relative to females. Women continue to demonstrate
hearing than men into advanced age. These gender differences are
probably due to
greater exposure of males to noise rather than to their inherent
to its effects." Source: [http://www.nidcd.nih.gov/health/hearing/noise.asp]
- At about 20 KHz the "hearing threshold" and the
of pain" are both about 120 dB---in effect we cannot hear
sounds. Smaller animals can hear well at much higher frequencies
[dogs and cats can hear
well at about 40 KHz, as demonstrated by mechanical or
whistles], whereas elephants are very sensitive to low frequency
- 20 Hz] not heard by smaller animals.
Porter Johnson commented that our region of maximum hearing
sensitivity, 2 KHz, corresponds to a wavelength of about 10
distance between our ears. This rough correspondence also works pretty
animals. He also commented on transmission of sound in the
enables marine mammals to communicate over great distances.
"Acoustic thermometry, however, capitalizes on the presence of sound
channels present in the deep sea capable of trapping and transmitting
very long distances. The channels are created by the variation of
temperature with depth. Located at a depth of about 3,000 feet, these
super-highways act almost like a lens in focusing the sound and guiding
thousands of miles." Source:
Porter Johnson (IIT BCPS Department) Handout: Day and Night
used a globe and a small bed light and showed how the length of day,
angle of the local noon, and the direction of the rising and setting
on the latitude, as well as the time of year. The handout, which
technical details for calculating these quantities [neglecting
refraction, the finite size of the solar disk, and the eccentricity of
earth's orbit around the sun] is located on his website at URL
He made the following qualitative observations:
- The axis of rotation of the earth is tilted at an angle of about
23o to the direction of its orbital angular momentum
around the sun. This axis points roughly in the direction of the star
Polaris in the Little Dipper / Ursa Minor
- At the Winter solstice [21-22 December] the earth's axis is
tilted away from the sun. The sun appears lower in the sky
and it is Winter in the Northern Hemisphere, whereas it is and
Summer in the Southern Hemisphere.
- The earth rotates counterclockwise [as seen from the North
Pole]. A period of about 23 hours and 56 minutes [sidereal
day] is required for one rotation, [or about 366 1/4 rotations
occur in one year]. The tracking motor on a telescope must be
designed so that it will make a full rotation in a sidereal day, rather
than in the more familiar solar day.
- At the poles there are 6 months of continuous darkness, followed
by 6 months of continuous sunlight. The transitions occur at the
Vernal Equinox [21-22 March] and the Autumnal Equinox [21-22 September].
- At any point above the Arctic circle or below the Antarctic
circle [latitude about 67o] there are 24 hour
periods continuous sunlight [summer] and of continuous
[darkness]. The sun lies very close to the horizon and its
horizontal projection lies in every direction at some point of the
- Everywhere on the equator there are precisely 12 hours of
daylight, and the sun rises exactly in the East and sets exactly in the
West---at any time of the year.
- On the Vernal and the Autumnal Equinox there are precisely 12
hours of daylight and the sun rises precisely in the East and sets in
the West, at all latitudes North and South.
- Between the Autumnal and Vernal Equinox [fall and winter] the sun
rises South of East in the Northern Hemisphere, whereas for spring and
summer it rises North of East in the Northern Hemisphere. The
times are reversed in the Southern Hemisphere.
(Gordon Tech HS, Physics)
handed out copies of a worksheet activity entitled Exploring Life
Science: Measuring Liquid Volume with a Graduated Cylinder.
Arlyn VanEk (Illiana Christian HS, Physics)
illustrated the behavior of electric motors and generators, using the
with the large-screen TV. First, he reminded us that the
directions of Current and Magnetic Field are related by the right-hand
Using your right-hand:
He showed how to give a large-scale visualization of the convention by
putting long paper
cylinders on the fingers of his right hand. He then used visuals
illustrate how a current-carrying loop of wire is made to rotate in a
magnetic field, and noted that
if the current does not vary with time [Pure DC], the wire will not
continue to rotate, but
will come to rest at an equilibrium
position. The changing current necessary to make an electric
motor can be
obtained with a split ring [commutator], so that the current
direction every half-rotation. The current will be of fixed magnitude,
direction will change every half-cycle.
- Point your thumb in the direction of the conventional current, I
[labeled by v in the diagram!]
- Curl your fingers into a half-circle around the wire. They point
direction of the magnetic field, B.
Next he showed how to make an electric generator
to convert mechanical energy into electrical current. The
device is a rotating split-ring coil in an external magnetic field, so
direction of flow of current reverses at every half-turn. The
voltage of 3-5 volts was monitored on a standard digital oscilloscope,
in the Radio Shack Catalog [
http://www.radioshack.com/], such as this model:
100MHz Cursor Readout Dual-Channel Oscilloscope
As seen from the trace, the output voltage is about 5 Volts
with full wave rectified
AC structure; that is, I(t) = Io |sin w t|.
In addition, the oscilloscope showed
spiked voltage pulses that correspond to the openings in the split ring
commutator. These pulses are similar to high frequency ignition
interference that may be heard on AM channels on the car radio, which
correspond to opening and closing of distributor points.
$1,199.99 Reg. Price; Brand: Instek
Cat #: 910-5360 Model: GOS-6103
Brandon (Joliet West HS, Physics)
passed out 4 sheets with various circuit problems designed to emphasize
approach in analysis of circuits involving resistors in series and in
These sheets, which she prepared in the "draw" program in
/Office 95 or 97, had the following titles:
Here is an image of one of the sample circuit problems from the third
The full set can be obtained from Ann Brandon: firstname.lastname@example.org
- Find the Total Resistance in Each Circuit
- Voltages in a Circuit
- Currents in a Circuit
- Currents, Voltages, and Resistances
Notes taken by Porter Johnson and Earl Zwicker.