High School Mathematics-Physics SMILE Meeting
04 May 2004
Notes Prepared by Porter Johnson

Fall 2004 SMILE Schedule

14 September   First class / registration
28 September Second class / late registration
12 October Third class
26 October Hallowe'en week
09 November A week after national elections
23 November Thanksgiving week
07 December and  ... just 1 week later ...
14 December Last class

Benson Uwumarogie [Dunbar,  Mathematics]           Algeblocks 
Benson showed us how he uses the ETA/Cuisenaire Algeblocks system [http://www.etacuisenaire.com/algeblocks/algeblocks.jsp] and associated workmats [http://www.etacuisenaire.com/algeblocks/workmats.jsp] to illustrate the concepts and manipulations of abstract algebra by placing color-coded plastic, blocks on panels:  Basic Mat, Quadratic Mat, Sentences Mat, and a transparent Factor Track. Connections with algebra and geometry are more evident with this system.  For example, one may use the blocks representing x, y, and xy to build the geometric figure of area given by  algebraic equation (x+y)2Benson has found this system to work quite well in his classes.  Thanks for the info and ideas, Benson!

Karlene Joseph  [Lane Tech HS, Physics]           Birthday Candles and Spectroscopes
recently purchased from her local drugstore a set of Color Flame Birthday Candles [http://jdjiaxuan.en.alibaba.com/product/50003665/50029791/Color_Flame_Birthday_Candle/Color_Flame_Birthday__Candle.html]: 5 ´ 50 mm, which produce red, green, yellow, blue, and purple flames  and last for about 8 minutes Karlene passed out a supply of CENCO Quantitative Analysis Spectroscopes -- CP-30105-00, available from Sargent-Welch at  https://sargentwelch.com/store/catalog/product.jsp?catalog_number=WLS1809-96.  She lit the red candle, and we each looked at the spectral lines obtained for it, which included reddish hues, as well as other colors. After a few minutes, she lit the green candle, let us look at it for a few minutes -- and then the yellow -- and then the blue --- and then the purple candle.  It was fascinating to see the light from each candle, decomposed into its component wavelengths, and to associate them with different chemical elements.

Even though it wasn't our birthday, we saw the light! Thanks for sharing this with us, Karlene!

Fred Schaal   [Lane Tech HS, Mathematics]           Fraction Action 
Fred reminded us of a universal feature of programming algorithms on digital computers, in that they perform arithmetic exactly with integers, whereas with real numbers the calculations may be inexact because of round-off error.  Consequently, arithmetic with fractions can be done exactly -- provided the numbers that arise are not too large. On his trusty TI-81 programmable calculator, Fred went into fraction mode, and was able to verify the relation:

1 / 19 + 1 / 87 = 106 / 1653
On the other hand, the computer was not able to obtain the identity
4 / 123 + 5 / 841 = 3979 / 103443
giving some result like 0.0384656, since it encountered a denominator that was too large for storage, and had to retreat to real numbers.  Fred also pointed out that the TI-81 can be used for mixed fractions, in several different ways:
2 + 2 / 3 + 3 + 3 / 8 (frac)= 151/24
 ... or ...
2 + 2 * 3^(-1) + 3 + 5 * 8^(-1) (frac)= 151/24
Neato, Fred!

Richard Goberville [Joliet Central HS, physics]           Lightning Reaction
Richard recently purchased the Lightning Reaction toy [See this image]. Here is a description taken from The Stupid Store page:

"Here's how Lightning Reaction works -- Anywhere from two to four people can play at once. You remove a handle from the base and get ready. When you press the button in the center, a red light pulses and suspenseful music plays. As soon as the red light turns green, you press the red trigger button as quickly as possible. If you're the slowest player, you will get rewarded with a painful electric shock. If you were faster than your opponents, you can simply laugh as the loser screams in pain.
It was certainly a memorable experience when Bill Shanks, Don Kanner and others tested  the operation of this fine device, which contains three AA batteries, and presumably a step-up transformer as well!  Richard also showed us some cartoons with science-based components. Thanks, Richard!

Leticia Rodriguez [Peck Elementary School] and Bud Schultz [Aurora West HS, physics]           Copernic Agent
Leticia and Bud touted the use of the Copernic Agent Basic search engine:  http://www.copernic.com/en/products/agent/basic.html. This software package, which requires only 8 kB of memory, can be down-loaded for free.  Its features include automatically combining the results from many search engines, producing a lists of results for each search, providing more convenient follow-up searches within these lists.  They tried, without success, to show us a video, The Lightning Story.  For a similar exhibition, see the National Geographic page Lightning:  The Shocking Story: http://environment.nationalgeographic.com/environment/natural-disasters/lightning-profile.html. Check out this new freeware! Thanks for telling us about it, Leticia and Bud!

Don Kanner [Lane Tech HS, physics]           Funnel Multiplication
recently learned of a novel method of multiplication, which he illustrated with an example such as 4567 ´ 9876 = 45103692.  He wrote the numbers directly  below one another, and performed the following manipulations:

numbers arranged in                     4 | 5 | 6 | 7
four columns of two ´ 9 | 8 | 7 | 6
product of numbers in same column |3 6|4 0|4 2|4 2|
sum of diag products; adjacent cols: 7 7|8 3|8 5|
sum of diag products; next col pairs |8 3|8 6|
sum of diag products; 1st and last cols: |8 7|
(now, add everything up) --------------------
(et, voilá!) |4 5|1 0|3 6|9 2|
We could write out the multinomial product of decimal numbers a b c d ´ e f g h in the following form: 
[103 a + 102 b + 101 c + 100 d ]
  ´ [103 e + 102 f + 101 g + 100 h ]
-------------- = -------------
106 (a ´ e) + 104 (b ´ f)  + 102 (c ´ g) + 100 (d ´ h) +
105 (a ´ f + b ´ e) + 103 (b ´ g + c ´ f) + 101 (c ´ h + d ´ g) +
104 (a ´ g + c ´ e) + 102 (b ´ h + d ´ f) +
103 (a ´ h+ d ´ e)
The rows in this algebraic expression correspond exactly to rows in the numerical expression above.  It works!

For discussion of an ancient description of this algorithm for multiplication, see the Iowa State University Department of Mathematics webpage Math Night Module:  Multiple Methods of Multiplication [ http://www.math.iastate.edu/mathnight/activities/modules/multiply/aboutmod.shtml] from which the following has been excerpted:

"The history of mathematics in India is ancient. The Hindu Vedic tradition is an oral tradition of knowledge passed down in short verses, dating to before the invention of paper. The Vedas encompass a broad spectrum of knowledge, including the sutras (verses) pertaining to mathematics. In the early 20th century Swami Shri Bharati Krishna Tirthaji Maharaja claimed to have rediscovered a collection of 16 ancient mathematical sutras from the Vedas and published it in a book called Vedic Mathematics. Historians do not agree on whether or not these were truly part of the Vedic tradition. If these sutras date back to the Vedic era they were certainly part of an oral rather than a written tradition. However, they are a novel and useful approach to computation: they are flexible in application and easy to remember. They can often be applied in algebraic contexts as well as in simple arithmetic. 'Vertically and Crosswise' [sic: URDHVATIRYAGBHYAM] bridges the gap between arithmetic and algebra: the algorithm is very similar to the standard algorithm used in the US and also is similar to the "FOIL" (first, outer, inner, last) rule used for multiplying binomials in algebra."
For additional discussion of this and other ancient multiplication methods see also http://www.math.iastate.edu/mathnight/activities/modules/multiply/ and  http://www.pballew.net/old_mult.htm.

Thanks, Don!

Walter McDonald  [CPS Substitute:  Radiology Technician at Veterans Administration]           Is a 2nd Earth out there?
Walter passed around a recent article by Pamela Simpson (AP) that recently appeared in the Chicago Sun Times, which involved the discovery of a solar system on star HU 70642, a star similar to our sun, which lies about 94 light years from earth, and around which a planet lies in an orbit similar in shape and distance to the planet Jupiter in our own solar system.  The article appears on the Red Nova News website at http://www.redorbit.com/news/scifi-gaming/11557/is_a_2nd_earth_out_there__jupiterlike_planet_found/index.html. See also the article Celestial Soulmate? Jupiter-like Plane Found in System Similar to Ours by Tariq  Malik:  http://www.space.com/scienceastronomy/jupiterlike_planet_030703.html.  According to Alan Penny of Rutherford Appleton Laboratory

"This is the first one that is really like our own solar system of the 110 that we've found.  We think it's a substantial step on the way to finding another earth.

Comment by PJ:  Planets in nearby stars, while quite undetectable through direct observation, can be identified through detection of time-dependent Doppler shift of light coming from that star.  For the case in which there is only one major planet, the star and major planet rotate about their center of mass, with a period determined through Newtonian gravitation in terms of the distance between the star and the planet.  The magnitude of the Doppler shift of light from the star is proportional to the the orbital velocity of the star about that center of mass.  In practice, only planets the size of Jupiter or larger can be found, and they must be fairly close to the star, since otherwise the Doppler shift would be too small for detection with current techniques. For HU70642, the planet is twice the mass of Jupiter, and lies at a distance of 3.5 Astronomical Units.  The Jupiter-like planet  would sweep the solar system of unwanted debris and stabilize the orbits of the inner planets -- perhaps one like our own planet Earth.

Very interesting Walter!

Bill Blunk [Joliet Central, physics]           Series Circuits  --- or What?
Bill showed us a simple-looking circuit that consisted of two identical light bulbs in sockets, hooked together in series and attached to wires with a plug on the end.  When the circuit was plugged into the 115 VAC line, both lights went on with equal intensity, as expected.  However, when Bill unscrewed one of the bulbs, the other one continued to burn.  How come?  A similar thing happened when he screwed that bulb back in and unscrewed the other bulb.  Although Bill claimed to be a magician who would not reveal his secrets, we suspected that he had slipped diodes under the sockets.  Are we right, or are we right? Better luck next time! Thanks for the show, Bill!

Ann Brandon [Joliet West HS, physics] Sophie Germain; phone cord
passed around the article Sophie Germain:  Genius with a Pseudonym [http://sciencewomen.blogspot.com/2008/11/sophie-germain-mathematical-genius.html], which appeared in the Program for the Goodman Theater production of the play Proof by David Auburn [http://math.cofc.edu/kasman/MATHFICT/mfview.php?callnumber=mf139], which deals with a fragile young woman without formal mathematical training who makes an important discovery concerning prime numbers.  Sophie Germain (1776-1831) [http://www-groups.dcs.st-and.ac.uk/~history/Mathematicians/Germain.html] --alias Monsieur Le Blanc -- obtained important results on Fermat's Last Theorem [http://www-groups.dcs.st-and.ac.uk/~history/HistTopics/Fermat's_last_theorem.html#40] for Germain Primes --- prime numbers n for which 2n+1 is also a prime.  See also the column This Month in Physics History in the APS NEWS,  02 May 2004   [http://www.aps.org/apsnews/] Revolutionary Pursuits: Circa May 1816: Germain Forms Theory of Elastic SurfacesSophia Germain was instrumental in saving the life of the most famous mathematician in the world, Carl Frederich Gauss [ http://www-groups.dcs.st-and.ac.uk/~history/Mathematicians/Gauss.html], from the invading armies of Napoleon BonaparteGauss once said that she must have "the noblest courage, quite extraordinary talents, and superior genius".

Ann also showed us a novel use for an ordinary phone cord, a coiled wire that connects the handset to the main body of the telephone.  These cords are available separately at low cost -- for example, try the Dollar Store.  Simply hold the cord at both ends and stretch it -- the cord can be used to display transverse waves much more easily and reliably than with our usual choice, Mr SlinkyVery nice ideas, Ann.  Excellent!

Sally Hill [Clemente HS, Dean of Students]           Optical Illusions
Sally passed around a set of optical illusions which she has assembled,. She uses them in her new position to put parents and visitors at ease.  Here is a partial list of the titles of the various illusions:

Several of these illusions are available on the BAM Magic Club website:  http://www.mandrgames.com/illusions.htm.  

Thanks for sharing these with us, Sally!

Notes prepared by Porter Johnson