High School Mathematics-Physics SMILE Meeting
22 April 2003
Notes Prepared by Porter Johnson

Gary Guzdziol [Carol Rosenwald School -- Science Teacher]        Implosion of Steel Drum, Continued
again put a little water into the drum, heated it vigorously for about 15 minutes until steam was pouring out, and sealed the drum.. We waited for the drum to implode ... and we waited ... and we waited .. and we waited.  Nothing happened during the entire class!   Why? We concluded that either the drum had a  pinhole leak somewhere --- or else he had gotten a super-drumGary  promised to show us his home-made video of an imploding drum at the next meeting. 

We look forward to the video  --- thanks, Gary!

Leticia Rodriguez [Peck Elementary School]        Mass Concepts  + Fraction Game
first made a presentation on the concepts of mass and weight aimed at primary level.  She showed us these four objects:

W:  Wooden  sphere    : G:  Glass sphere       S:  Steel ball     P:  Plastic cube
She asked us to rank-order these objects in decreasing mass, based our visual examination. A typical answer might be
S    >  G     >     W     >     P
Then, she compared the objects on a small, equal-arm balance, and showed us that the actual ordering was
G    >  S     >     W     >     P
In fact, we could see that the following relations were approximately valid:
G = 20 P        G + S + 31 P          P + W < S          P+ W +S < G
She pointed out that preconceived notions are not necessarily correct, even when the answers seem obvious.  After some discussion, we guessed that the plastic cubes and steel balls were probably hollow.  How would you show such a thing?  Larry Alofs suggested using buoyancy, and determining the average density of the material. Also, he noticed that the plastic cubes were open at one end, and they floated.   The air remained trapped inside, no doubt due to surface tension.  After vigorous shaking of a cube under water,  Larry got most of the water out, and the cube then sank.  He also tried to use a straw to draw the air out of the cube, with limited success.  

Remark by PJ:  In the immortal classic, The Leatherstocking Tales  [http://www.mohicanpress.com/mo06058.html] by James Fenimore Cooper, Nathaniel  Bumppo [hawkeye, la longue carbine, etc], Chingachgook [The Last of the Mohicans], and his son Uncas [a Delaware --- American Indian cultures are invariably matriarchal!] hid from their pursuers by lying underwater among the reeds on the edge of a lake, while breathing through reed straws.  Does this actually work, and if so, how and why?

Leticia then showed us how to play The Fraction Game.  She handed out a template with six rows, containing the following items

  1. Row 1: One circle, marked 1/1
  2. Row 2: Four circles, each split into two regions of equal halves, with each region marked 1/2
  3. Row 3: Four circles, each split into three regions of equal thirds, with each region marked 1/3.
  4. Row 4: Four circles, each split into four regions of equal fourths, with each region marked 1/4.
  5. Row 5: Four circles, each split into five regions of equal fifths, with each region marked 1/5.
  6. Row 6: Four circles, each split into six regions of equal sixths, with each region marked 1/6.
Then she took out a pair of dice, and rolled them.  They came up with 5 and 3 -- and dividing the smaller over the larger --  she called out, "Three Fifths".  We were instructed to shade complete regions -- such as 3 regions marked 1/5  -- amounting to 3/5, the number called.  We were instructed to shade in complete regions only, but could reduce fractions.  For example, if 3 and 6 were rolled, she would call out "Three Sixths", and we could distribute it appropriately into sixths, thirds, and / or halves. The game -- played either "bingo style" or as a contest between individuals or teams that take turns in rolling the dice -- lasts for a specified time, and participant with the largest number of completely filled circles on the sheet [out of a possible 25] is the winner. A happy way to learn fractions!

Good lessons and a good game!  Thanks, Leticia!

Bill Blunk [Joliet Central HS, Physics]       100 Year Old Spinthariscope
showed us a Spinthariscope that was marked with the date 1903. But, just what is a Spinthariscope?  The following description is an adaptation of that taken from the Kenyon College (Gambier OH) Physics Department website; URL http://www2.kenyon.edu/depts/physics/EarlyApparatus/Miscellaneous/Spinthariscope/Spinthariscope.html :

Alpha particles impinging on a screen coated with zinc sulfide will produce tiny flashes or scintillations of light. William Crookes [his biography: http://chemistry.about.com/od/famouschemists/p/williamcrookesbio.htm] was one of the discoverers of the effect in 1903, along with Julius Elster and Hans Geitel.

The spinthariscope [dictionary definition at  http://www.bartleby.com/61/44/S0644400.html] is a brass tube with a magnifying eyepiece at one end and a screen of zinc sulfide [scintillator] at the other. A small thumb-wheel allows the alpha particle stream from a uranium compound to be directed toward the scintillator. When used in a dark room, bright flashes may be seen with a dark-adapted eye. 

The Kenyon College web page also contains a picture of the original Crookes device.

Bill turned out the lights, and during several minutes that our eyes were dark-adapting, he described how Ernest Rutherford, [ http://www.nobel.se/chemistry/laureates/1908/rutherford-bio.html] and his associates Geiger and Marsden, established the existence of the atomic nucleus by using such scintillations.  For more details on Lord Rutherford, see the biography Rutherford: Simple Genius by David Wilson [MIT Press 1983] ISBN 0-262-23115-8.  Bill handed out two spinthariscopes, which we passed around in the dark room to see the scintillations for ourselves. Great!

You showed us the light! Thanks, Bill!

Ann Brandon [Joliet West HS, Physics]      Waves and Resonance
Ann led us through three exercises to illustrate wave properties:

 Ann, you struck a resonance with our thoughts! Very nice!

Lovesea Jose [Du Sable HS, Physics]     Water Tube
showed us a plastic tube of outside diameter 8-10 cm, about 1 meter long. The tube was completely filled with water (dyed blue) and securely plugged at both ends.  Furthermore, we could see a white (Styrofoam®) ball  inside the tube.  When she held the tube vertically, we could see the ball gradually rise in the water, until it went to the top of the vessel.  There was a murmuring consensus that the ball rose in the water because the buoyant force on the ball acted upward, and was greater than the weight of the ballLovesea quickly turned the tube upside down so that the ball was initially at the bottom, and  it again rose to the top.  So far, so good!

Lovesea again turned the tube over, but then she tossed it up into the air.  We saw the ball initially rise a little, but it did not continue to rise when the tube was put into free flight.  Amazingly, the ball stopped in its tracks [relative to the tube!] just as she released it.  How come?  After some discussion, we developed the consensus that buoyancy occurs as a consequence of gravity, and that in  free fall, the tube, water, and ball move together in the same way.

Earl Zwicker showed us how this tube can be used as an accelerometer.

Great ideas, Lovesea!

Bill Colson [Morgan Park HS, Mathematics]        Geometry Puzzle
[passed around the drawing of an 8 unit ´ 8 unit square that had been divided into four pieces -- A, B, C, D --, as shown:

Bill cut out the pieces from the drawing and rearranged them into a 5 unit ´ 13 unit rectangle, as shown:

Note that we have been able to create a rectangle of area 65 units from a square of area 64 unitsHow come? The pieces fit together remarkably well,  -- at least as well as in a jigsaw puzzle -- and there were no evident gaps.  And yet, we decided that there was something seriously wrong with the second picture.  The total area of pieces A + B+ C+ D is 64 units, as before, but the area of the rectangle is 65 unitsBill said that there were several ways to explain the difficulty. Probably the easiest method is to note that the triangles formed by piece D and pieces D + A should be similar, so that 3 / 8 = 5 / 13! From this relation one could cross-multiply so that 39 = 40 --- which is ridiculous to everybody, except perhaps a Jack Benny aficionadoPorter said that the diagonal line of the rectangle could not be straight, since its total length must be Ö [132 + 52]  = 13.9284... , whereas the diagonal's two segments have lengths of Ö[52+22]  = 5.3852... and Ö[32+82]  = 8.5440... , respectively.  Their total, 13. 9292... , is slightly greater than the diagonal's length!

Bill had seen this problem, as well as a number of other interesting mathematical puzzles and quandaries, in the book One Equals Zero and Other Mathematical Surprises by Nitsa Movshovitz-Hadar and John Webb [Key Curriculum Press 1998] ISBN 1-559530309-0.The following excerpt appears on their website: http://www.keypress.com/x6049.xml:

"One equals zero! Every number is greater than itself! All triangles are isosceles! Surprised? Welcome to the world of One Equals Zero and Other Mathematical Surprises. In this book of blackline activity masters, all men are bald, mistakes are lucky, and teachers can never spring surprise tests on their students!

The paradoxes and problems in each One Equals Zero activity will perplex your students, arouse their curiosity, and challenge their intellect. Each counterintuitive result, false analogy, and answer that defies expectation will encourage students to look at familiar mathematical situations in a new light. By solving the paradoxes, your students will come to better understand both the possibilities and the limitations of mathematics."

Finally, Bill mentioned that the following trivia questions were answered in the book:
  1. For what physical characteristic is almost everybody "above average"?  [number of fingers]
  2. For what physical characteristic is everybody "average"? [number of heads]

So that's where the missing square went to, eh Podner! Very slick, Bill!

Notes taken by Porter Johnson