Mathematics Physics SMILE High School Meeting
10 October 2000
Notes prepared by Earl Zwicker

Don Kanner (Lane Tech HS) showed us a "Test Tube Black Box." He held up a cardboard tube about 45 cm long and 7 cm in diameter. About 2 cm from the left end, a string passed through the tube through a pair of diametrically opposed holes.  (On each end of the string were small metal rings to prevent the string from coming free of the tube.) Another string passed through the tube at its right end, in an identical manner, except it was longer. Looking at us with a grin, Don pulled down on the left string, and the string on the right end shortened. When he pulled down on the right end string, the left end string shortened. But then he pulled UP on the right end string - and it moved straight up until it was stopped by its bottom ring. And the left end string did not become shorter or move at all! How was this possible!? After showing us again with some variations, Don challenged us to come up with an explanation or make our own version. He explained that a chemistry colleague at Lane Tech  uses this to catch the attention of his students and to make them put their minds to work. So ... how about us!? Any ideas? Maybe Don will show us more next time.

Ed Robinson (De LaCruz School)
gave each of us a sheet of blank paper, then challenged us to find a pattern in playing a game that he called "Nim Mod." The first step was to sketch a rectangle and divide it into 4 boxes; (N = 4). The game is played similarly to tic-tac-toe, with one player making Xs and the other making Os. The loser of the game is always the player who is forced to fill in the last empty box, because none other is left. But each player, on his turn, may fill in 1, 2, or 3 boxes with his X (or O). With N = 4, it is clear that the Starter player (S) can fill in three boxes, leaving only one box empty, and forcing the second player to fill in the last box to become the loser of that particular game:

Thus, we can designate that the Starter (S) would be the winner (unless he is makes a mistake!) for N = 4. Ed had us continue the game for N = 5, and then he worked it at the board, showing us that S may easily be forced to lose the game by his opponent, so for N = 5, S loses. He put us through these games:
N = 6, S wins
N = 7, S wins
N= 9, S ??
and so on. No matter how high N becomes, if the problem is divided into sets of N = 4, then we can find if S is a winner or loser by dividing N by 4, and finding the Remainder, R.
If R = 0, S wins, if R = 1, S loses, if R = 2, etc. 
(Try it and see for yourself!) His students enjoy the math and trying to figure out the pattern. Thanks, Ed! We learned another new idea today!

Marilynn Stone (Lane Tech HS)
gave each of us a resealable sandwich bag containing these items:

2 green rectangles
3 blue squares
4 red triangles.
She then challenged us to assemble the pieces together to form a rectangle. (Earl Zwicker was first to succeed - but he has had many years of practice doing Harald Jensen's Pythagorean puzzle to introduce the Phenomenological Approach to new SMILE teachers!) She then showed us how to prove the Pythagorean Theorem using the puzzle. She did this by projecting transparencies of the puzzle pieces so we could literally "see" the reasoning. Pretty! But then she showed us how to make a proof with just half the puzzle, using the large blue square and the red triangles at its sides to form an even larger square. Marilynn labeled each side of the blue square with a "c", and contiguous hypotenuse with a "c" also. Then the short sides of the 4 triangles were labeled "b", and the longer sides, "a".  From this it was clear that the areas obeyed the following relation:
(a + b)2 = c2 + 4 ´ (ab/2)
{Just draw the picture!) One needs then only to complete the algebra to show that a2 + b2 = c2. Very nice, Marilynn! She then told us she had generated the transparencies by printing from her computer printer onto a transparency sheet. Neato! She gave us a handout page for with directions for her students to actually measure a, b, c on four different right triangles and check out the Pythagorean Theorem numerically for each one.
Good reinforcement.

Fred Schaal (Lane Tech HS)
showed us "The Occurrence of Concurrence".  He explained that if 3 straight lines in a plane intersect in a single, common point, it is called "concurrence." With the aid of a meter stick, he constructed a large, nice looking triangle on the white board. With a large compass having a marker pen attached at its "chalk" end, Fred used the compass to construct the line which bisected one of the angles of the triangle. He did this in a contrasting color. Then he constructed the bisectors of the other two angles the triangle. If the board had not been so slippery and the compass marker had made legible marks, the bisectors of the three angles would have intersected at a single point within the triangle: concurrence! Unfortunately, the construction was not precise, and it didn't work out. But you made your mark, Fred! Thanks for an interesting lesson.

Roy Coleman (Morgan Park HS)
asked if anyone could tell him how their school deals with the scheduling of exams if every class is to be a 2 hour class. No explanations were forthcoming.

Betty Roombos (Lane Tech HS)
explained how she shows her students to do vector problems. We are given two displacement vectors:

6 km  at 30o E of S
5 km at 20o N of W
Find the resultant and its direction. Betty would have them draw the vectors using pencil, protractor and paper, and constructing and measuring the resultant. But then they would do it analytically, by resolution of the vectors into their compass direction components, addition of those components, and then using algebra and trig to determine the resultant vector. With the aid of us with calculators, Betty carried the numbers through at the board, and it worked out well!

Fred Farnell (Lane Tech HS)
explained why the fit of a straight line equation to experimental constant velocity points (and its subsequent v vs t graph) at the last meeting was so poor. It turned out that the constant (non zero) velocity occurred over an 8 second period, but the data taking ran for 15 seconds, and the velocity was zero for the last 7 seconds! When Fred ran new data and kept only the portion for non-zero velocity, the fit turned out great! He did this "live". Same story for motion of constant acceleration. Thanks for restoring our faith, Fred!

Notes taken by Earl Zwicker.