High School Mathematics-Physics SMILE Meeting
Mathematics: Miscellaneous

11 September 2001: Monica Seelman (Williams and St James Schools) Venn Diagrams
Discussed an example of a Venn Diagram.  In Particular, she considered the following sets:

• A:  set of all integers from 1 to 1000
• B:  set of all multiples of 11
• C:  set of all numbers below 500 with identical digits; eg, 11, 66, 111, 444

She pointed out that all two-digit numbers in set C are also in set B, but that the three-digit numbers in C are not in B, because they are not divisible by 11.  But the four digit numbers with identical digits are divisible by 11 [5555 = 101 ´ 55]. As an extension, all numbers with an even number of identical digits are divisible by 11, whereas those with an odd number of digits are not.  Interesting  results in "eleven-ology"!

11 September 2001: Earnest Garrison (Robeson HS, Physics)
handed out a write-up of a Paper Clip Lab, in which fatigue and fracture of solids was studied using paper clips.  The idea is to determine the distribution in the number of times one must bend a paper clip back and forth in a controlled fashion to get it to break at the "little loop", the "big loop", and on the "straight section".

Earnest also showed us an exercise in estimating the area of an irregularly shaped lake on a map, by overlaying a square lattice of dimension 1 cm.  The idea is to estimate the area as follows:

• J:   Number of squares completely covered by the lake
• K:  Number of squares more than 50 % covered by the lake.
• L:   Number of squares 50 % covered by the lake.
• M:  Number of squares less than 50% covered by the lake.
• N:   Number of squares not covered at all by the lake.

The area of the lake is estimated to be J + K + L/2.  This estimate is fairly accurate, in practice! And, students ware surprised at how closely their results agree with one another.

11 September 2001: Fred Schaal (Lane Tech HS, Mathematics)
suggested that "conditional logic" should be considered as an alternative to "deductive logic", and "inductive logic".  In conditional logic we have the statements A ® B and B ® C, from which we conclude that A ® C. As an example, he considered the syllogism

All animals named Flicka are horses.
All horses have four legs:
\ All animals named Flicka have four legs.

Note that A ® B and C ® B does not permit the conclusion A ® C, so that the following syllogism is incorrect.

All horses have four legs.
All animals named Flicka have four legs:
\ All horses are named Flicka.

20 November 2001: Estellvenia Sanders (Chicago Vocational HS) Digital Numerics
Estellvenia uses these activities with her high school students.

She gave us a sheet with the numbers 1-20 located randomly on it, and we were told to touch as many of the numbers as possible over a given time period (10-20 seconds), timed by a partner using a stopwatch.  We were to touch the numbers in increasing order (1 ... 2 ... 3 ... ) with an index finger. We recorded the total number touched  by each of us over three trials.  Then, we analyzed and compared the data.  By this exercise, some of the students will be able to remember and identify the numbers more quickly.

We then saw how sign language digits (numerically) can be combined with standard American Sign Language [http://www.lessontutor.com/ASLgenhome.html] symbols to speed up sign language, in that some letters have both "letter" and "number" signs in the 1867 version.  In the modern version of sign language, all letters have letter symbols http://www.masterstech-home.com/The_Library/ASL_Dictionary_Project/ASL_Tables/Alphabet.html, and numbers have separate number symbols, http://www.masterstech-home.com/The_Library/ASL_Dictionary_Project/ASL_Tables/Numbers.html, so that no mixing of  numbers and letters occurs.  Very interesting, Estellvenia.

04 December 2001: Bill Colson (Morgan Park HS, Mathematics) New Toys
Bill
used his \$100 equipment allotment from CPS to obtain blackboard drawing materials from the K-12 Mathematics and Science Catalog for Fall 2001 of the EAI http://www.eaieducation.com/.

Eric Armin Inc (EAI) Education
567 Commerce Street
PO Box 644
Franklin Lakes NJ 07417-0644
1 - 800 - 770-8010

In particular, he purchased these items:

• the Overhead Safe-T Compass [part number 530081 - semi-clear] \$1.40 each
• Clever Catch Balls [Algebra and Geometry: http://www.24hours7days.com/Games/All_Clever.html] with questions printed on it. When a student catches the ball with both hands, s/he should answer the question closest to the thumb on the left hand.

He showed us how well the compass worked on the blackboard; then we tossed the ball around for a while and answered the questions. Useful stuff. Thanks, Bill!

05 March 2002: Roy Coleman (Morgan Park HS Physics) -- Probabilities

• He had us pick a number from the list 1 - 2 - 3 - 4:
He claimed that, in a free choice,  about 75% of people pick #3, "reasoning" that #4 is "last", #1 is "first", and #2 is "too close to first".  Believe it or not!
• Suppose that, in a given combat arena, the following probabilities are correct:
1. 60% Probability of a shell hitting its target
2. 60% Probability of a shell exploding on impact
3. 50% Probability of being killed when hit by an exploding shell
Which of these probabilities is obviously wrong***?
1. 36% probability of being hit by an exploding shell
2. 30% probability of being killed by an exploding shell aimed at you
3. 18% probability of dying
4. 6 shots required per fatality are required, in general
*** #3, because the probability of dying is obviously 100%.

Roy, we hope you get help soon!

11 March 2003: Bill Colson [Morgan Park HS, Mathematics]      T. G. I. P. --- Thank God It's Pi Day!
Bill
called our attention to the following websites from a recent National Council of Teachers of Mathematics [NCTM]  News Bulletin, which are appropriate for the up-coming Pi Day [3.14]:

Thanks for the timely reminder, Bill!

22 April 2003: Leticia Rodriguez [Peck Elementary School]        Fraction Game
Leticia
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!

21 October 2003: Bill Colson [Morgan Park HS, mathematics]        Molecular Expressions/ Florida State U Website
Bill
passed around information from their website,  http://micro.magnet.fsu.edu/, on the following topics:

These were absolutely fascinating!  Thanks for telling us about these sites, Bill!

Bill also passed around some geometrical questions concerning Tumbling, Spinning, and Plummeting, which appeared in the October 2003 issue of Discover Magazine in the feature article "bogglers" by scott kim"http://discovermagazine.com/.

14 September 2004: Bill Colson [Morgan Park HS, Mathematics]           Philately
Bill first showed us the new postage stamps issued in honor of R. Buckminster Fuller, the Man and Mind behind the Geodesic Domehttp://www.usps.com/communications/news/stamps/2004/sr04_043.htm. The top of his head is shown as a geodesic dome on the stamp.

14 September 2004: Fred Farnell [Lane Tech HS, Physics]           SPECIAL BELL SCHEDULE
Fred passed around a copy of the bell schedule at Lane Tech, which he had written on the board as  shown here:

 Division 830 to 900 1st 904 to 939 2nd 943 to 1018 3rd 1022 to1057 4th 1101 to 1136 5th 1140 to 1215 6th 1219 to 1254 7th 1258 to 133 8th 137 to 212 9th 216 to 241

Fred Schaal, his ever-alert colleague, saw the notice on the blackboard, and immediately identified a connection with mathematics. We could view these numbers as exponentials, such as 904  = 9 raised to the power 4 = 6561. Fred and Fred then developed the following list of interesting questions:

• Which would represent the largest number? The next largest number? ...
• Which would represent the smallest number? The next smallest number? ...
• Which line contains numbers that have the least differences?
• Can you develop other meaningful questions that could be asked about these numbers, if the times represent numbers written in exponential notation?

09 November 2004: Leticia Rodriguez [Peck Elementary School]           Interactive Fractions
Leticia passed around sheets showing wheels divided into halves, thirds, fourths, sixths, ninths, and twelfths, as well as a spinner for generating the numbers 1, 2, 3, 4, 6, 9, and 12.  We were to spin twice (say 3, 12), form a fraction by putting the smaller number over the larger (3/12), and color that fraction in one of the wheels.  The person who first colors all the wheels is the winner.  What a neat way for students to learn fractions!

Very interesting game, Leticia!

Bill also reported on Nextfest 2005 [http://www.technovelgy.com/ct/Science-Fiction-News.asp?NewsNum=409], sponsored by Wired Magazine -- a neat expo held at Navy Pier, June 24-26, 2005.   It was a festival of new innovations of various types, including visits by the Cloned Cats! Bill commented that teachers are able to see IMAX films at Navy Pier at no charge a few weeks after their opening.  Great, Bill!

04 October 2005: Paul Fracaro (Joliet Central HS, math/physics)           Paper Plate Fractions + Whiteboard Demonstration
Paul
used paper plates to explore fractions. The thin (inexpensive) paper plates can be folded into halves, fourths, and eighths to demonstrate fractions. Then he used the same technique to show how he helps students understand better how to do simple additions and subtractions involving whole numbers and fractions, followed by additions and subtractions when the denominators are not the same.

Paul then showed us some white boards the size of typical letter paper printed with a rectangular grid, as well as X and Y axes. With markers, it is a convenient way for the students to graph. They are available from ETA Cuisinairehttp://www.etacuisenaire.com/. Neato!  Thanks, Paul.

18 October 2005: Walter McDonald (VA and CPS substitute teacher)               Hidden Magic Coin
Walter
handed out a sheet which contained directions for the hidden coin trick (from Hidden Tricks: Playthink #613 from the book 1001 Playthinks by Ivan Moscovich.Walter then, with Fred's help, performed the trick -- which worked perfectly!-- and which illustrated the mathematical concept of parity checking; see the Webopedia web page What is parity checking?http://www.webopedia.com/TERM/P/parity_checking.html. Walter then discussed the role of parity in computer operation.  It is used to check the accuracy of data sent from one computer to another.

Five coins were shaken and then scattered onto the table.  Walter looked at them, and asked a volunteer to turn over any two coins, and then cover up any one coin from view. Walter then looked at the four coins in view, and told us that the hidden coin was "heads".  We looked.  He was right -- and we applauded Walter! Walter repeated this feat a second time, and he made us all curious to know how he did this.  Marilynn Stone figured it out.  You need at first to count the original number of heads and remember it.  Then count the number of heads in the final configuration with one coin covered.  When you then flip two coins, only the following three cases can occur:

 Initial Final Change in #Heads HH TT - 2 HT TH 0 TT HH + 2
Note that, with one or more pairs of coin flips, the number of heads must change by an even amount (0, ±2, ±4, ±6, etc). If the final visible number of  heads  counted differs by an odd number from the original number of heads, the covered coin  is "heads". If the final number of visible heads differs from the original number of heads by an even number, the covered coin is "tails". Great work, Walter and Marilynn!

07 February 2006: Fred Schaal (Lane Tech)          Prose and Poetry Day
Fred
talked about a class period between the first and second semesters, which he uses as a prose and poetry day. He read us a poem with a whimsical tribute to the number three -- Threes by John Atherton  http://holyjoe.org/poetry/atherton.htm -- with apologies to Joyce Kilmer!  He also read a small portion of a poem that is a parody of the famous nonsense poem Jabberwocky, which appeared  in Through the Looking-Glass and What Alice Found There by Lewis Carrollhttp://www.jabberwocky.com/carroll/jabber/jabberwocky.html. He also shared some limericks, all having to do with topics in mathematics.  His poems were obtained from the anthology Fantasia Mathematica, edited by Clifton Fadimanhttp://math.cofc.edu/kasman/MATHFICT/mfview.php?callnumber=mf21.

04 April 2006: Fred Schaal (Lane Tech)                      Looking for 6174
Years ago, when calculators were new, Fred came across a very interesting phenomenon. Write down any 4 digit number (such as 1234) and arrange the digits so that there is the largest number possible (4321) and the smallest possible number (1234) and subtract the smaller from the larger. In this case you get 3087. Repeat the process with the new number 3087 (i.e., 8730 - 0378 = 8342). Keep repeating the process.  We obtained

1234 ® 3087 ® 8352 ® 6174 ® 6174 ® ...
Eventually (in the case in question by 4 iterations) the process produces a result  which no longer changes (in the case in question (7641 - 1467 = 6174). Porter pointed out that starting with any 4 digit number, the result will be the same repeating  number 6174.  We asked ourselves why this works, but could not come up with an answer. We then tried it with a two digit number to see what happened.  Here is what we obtained:
12 ® 09 ® 81® 63 ® 27 ® 45 ® 09 ® ...
For this case, the process has reached a sort of  loop in which the same five steps (09 81 63 27 45) occur over and over.  Fred also wondered if it would work in a base other than  base 10? We have no idea, just as we don't know for base 10 numbers of various lengths.  For four digit numbers of base 8 we obtain the following sequence:
234 ®3065 ® 6143 ® 5063 ® 6143 ® ...
Thanks for simulating our thinking, Fred.