My List of Published Articles and Letters

PUBLISHED ARTICLES

"More About P = N2 + M3".  Journal of Recreational Mathematics, 4(1),
January 1971, pp. 45-49.
"Number Patterns from Digit Sums".  The ARITHMETIC TEACHER, 18(2)
February 1971, pp. 100-103.
"Polynomial Functions and Vanishing Triangles".  Bulletin of the KATM,
February 1971, p. 20.
"Magic Triangles of Order n".  JRM, 5(1), January 1972, pp 28-32.
"Five Non-trivial Number Games".  The ARITHMETIC TEACHER, 19(7),
November 1972, pp. 558- 560.  (Reprinted in Games and Puzzles for
Elementary and Middle School Mathematics: Readings from the
ARITHMETIC TEACHER, NCTM, 1975, pp.148-150)
"Some Identities for the Triangular Numbers".  JRM, 6(2), Spring 1973,
pp. 127-135.
"Perimeter-Magic Polygons".  JRM, 7(1), Winter, 1974, pp. 14-20.
(Abstracted in ZENTRALBLAT FUR MATHEMATIC, Berlin, Germany)
"Kaprekar".  Math Lab Matrix, Fall 1976. p. 8.
"Back-to-Front Multiplication". The Oregon Mathematics Teacher,
February 1978, pp. 22-27.
"A Reader Reacts".  TOMT, March 1978.  p. 28.
"Phone Number Equations".  The Illinois Mathematics Teacher,
September 1978, pp.27-28.  (Also TOMT, May 1979, p.19.)
"Distinct-Digit Squares: A Calculator Activity".  TOMT, September
1978, pp. 16-19.
"Almost Magic Squares".  TOMT, October 1978, pp.
"Celebrating 1980 with Pythagoras".  TOMT, January 1980,
p. 12- 13, 19.
"Number Bracelets: A Study in Patterns".  The ARITHMETIC TEACHER,
27(9) May 1980, pp. 14-17.
"Product Dates". [in the FROM THE FILE feature]  The ARITHMETIC
TEACHER, October 1983, p. 53. [referenced in Mathematical Challenges
for the Middle Grades, William D. Jamski. NCTM, 1990, p. 35.]
"Calculator Poker".  MATHEMATICS TEACHING in the MIDDLE SCHOOL,
3(5) February  1998, pp.  366-368.
"Yppy's Year of Life: A Problem for the New Year".  Ohio Journal
of School Mathematics.  Spring 1998; #38 pp. 2-3.
"Keith Numbers".  OJSM, Spring 1998, #38. p. 23.
"The M.O.M. Game ("Matching Our Multiples"): LCM's via the Calculator".
OJSM, Winter 1999, #39. pp. 14-16.
"The Thinking of Students: The 100th Letter".  MATHEMATICS TEACHING
IN THE MIDDLE SCHOOL.  5(4); Dec. 1999.  pp. 258-262.
"Let's Take Another Look At Pi Day".  MATHEMATICS TEACHING
IN THE MIDDLE SCHOOL.  7(7); March 2002.  pp. 374-375.



[letters in NCTM journals...
1.   [topic: Palindromes] AT,  Nov.  1979, p.  52.

2.   [topic: Magic Triangles]  AT, (ca. 1975-78)

3.   "Beautiful 1992".  MT, Nov. 1992, p. 682.

4.   "14 April 1993" [MT Calendar Problem response].  MT, Nov. 1993, p. 712.

5.  "An In-school Field Trip".  MT, Dec. 1993, p. 779.

6.  "Personalize the Investigation!".  MTMS, April/May 1995, p. 375.

7.  "More from the ‘Menu'".  MTMS, Nov/Dec 1996, pp. 124-125.

8.  "3n + 1 Revisited" MTMS, Oct 1998, pp.84.

Here is the text of letter #3:

I also enjoyed reading Xu Zhaoyu's letter about how beautiful 1991 was
(September 1991), so naturally I enjoyed Monte Zerger's reply (March 1992).
Such number play is useful in livening a class discussion about 
the "personalities" of numbers.  Bravo, friends!
    But you editors asked, "How beautiful is 1992?"  To start the ball 
rolling, I humbly offer this observation.  Notice the prime factorization 
of 1992: 2^3 x 3 x 83.  Then note that 2^3 = 8.  Therefore,

                         1992 = 8 x 3 x 83.

It is of interest to note that if we require that the two-digit number be
a prime, 1992 is the only year in this century having a factorization that 
behaves in this way.  (Only two other years in the period 1000-1999 A.D.
manifest this same structure: 1316 = 4 x 7 x 47 and 1533 = 7 x 3 x 73.  
However, dropping the two-digit-prime requirement does give us two years 
in this century: 1950 = 6 x 5 x 65 and 1995 = 5 x 7 x 57.)
    The 1992 relationship was easily understood by the middle school students
that I teach.

Here is the text of letter #6:

The article by Browning, Channell, and Meyer, "Professional Development:
Preparing Teachers to Present Techniques of Exploratory Data Analysis"
(September-October 1994, 166-72), is of special interest to me because
my seventh-grade students and I explored the same topic a year ago---
color distributions of M&M's candies---in much the same way as described
by the authors.  But we did two different things that might be of 
interest to the readers of Mathematics Teaching in the Middle School.
First, we also investigated the data that resulted using peanut M&M's
and the compared bar graphs of both types of candies.
	Second, I asked my students to record the prices they paid for
each plain and peanut bag.  With this information we (1) computed the
average cost per bag, which was meaningful because the students had
bought their bags at different stores at different prices; and (2)
with the price data, we computed the unit cost of each type of candy.
In our local Salvadoran society the results were rather interesting,
especially the cost for one peanut candy.  We found that one peanut
had a mean cost of 22 to 23 centavos, whereas one of our traditional
thick Salvadoran-style tortillas could be bought for only 20 centavos!
When viewing a peanut candy beside one of our tortillas, the difference
of food quality is quite surprising, not to mention the nutritional
factor.  So we had personalized our investigation in our culture.
	Next my students want to investigate Skittles!



Here is the text of letter #7:

More from the "Menu"
The monthly "menu" of problems makes a nice change of pace in my class
when we study problem solving.  I found the March-April 1996 menus
especially nice, so I typed several on a handout for my seventh-grade
students.  But I went one step further on several items.  I added
additional questions to "squeeze a little more educational juice" from
them.
	The "Jack and the Beanstalk" item (March, problem 9) is a good
example.

   Jack climbed up the beanstalk at a uniform rate.  At 2 p.m.
   he was one-sixth the way up and at 4 p.m. he was three-fourths
   the way up.  What fractional part of the entire beanstalk had
   he climbed by 3 p.m.?

I began wondering about the time aspect, so I asked, "(a) At what time
did he start climbing? (b) When will he get to the top? and (c) How
long was his trip?"  These questions should be considered as natural
follow-ups in such a situation.  Of course, they have specific answers
[(a) 1:25:43 p.m.; (b) 4:51:26 p.m.; and (c) 3 3/7 hours].  But I also
added an open-ended question: "(d) And how tall is that ol' beanstalk
anyway?"  This answer depends on Jack's climbing speed as given in
units like m/sec or ft./sec.  So students must discuss with a partner
what might be a reasonable speed for a boy like Jack or what might be
a reasonable height for a fairy-tale beanstalk.  This approach takes
the given problem out of the routine variety, increases communication,
and makes it a true problem that is more in keeping with the ideas
presented in the NCTM's Standards documents.  Other changes that can
be made are in the two fractions and the two hours (i.e., use 2 p.m.
with, say, 5 p.m.).  The latter change is the more important one, as
it requires deeper thought in the solving process.  That is, one cannot
merely use the "average of the fractions" shortcut in this variation.
	Another extendable activity is the one involving Steve and
Keith (April, problem 12).

   Steve gave Keith an amount of money equal to the amount in
   Keith's pocket.  Keith then gave Steve an amount equal to
   what Steve had left.  Keith then had $14 and Steve had $8.
   How much money did Keith originally have in his pocket?

Here one can simply ask, "What happens if the boys continue giving
money to each other until it becomes impossible for one to give to
the other under the given conditions?"  This problem can then be
turned into a true investigation by altering in a systematic manner
the original amounts of money that each boy has at the start.  Making
charts and collecting data would certainly be useful strategies for
this work.
	One of the dangers of problem solving is to "get the answer
and stop there".  In the classroom setting we should show our students
that there is often more to a problem than meets the eye.  Creative
problem solving demands that we go further and make our teaching more
exciting.

Here is the text of letter #8:

I read in the February 1998 issue the nice letter from Barry D. Cohen
(p. 356) about the "3n + 1 problem", as he called it.  Wherever I saw
this problem originally, the comment was made that all numbers less
than some value in the several millions---I forget the specific value---
do behave as Cohen said: they "bring you to 1."  Thus, it was conjectured,
but not proved, that all numbers behave that way.
   Cohen noted that this operation seems to be without pattern as to how
many steps are needed to arrive at 1, and I believe that he is right.
However in just the first 101 numbers (1-101), I noticed some interesting
results.  For example, the three consecutive numbers 28, 29, 30 all
require the same number of steps, although they arrive at 1 by slightly
different routes.  I drew a tree-type, or river-tributary-type, of
diagram to observe the behavior of those 101 numbers and found it to be
quite interesting.  I found five more trios of consecutive numbers, two
sets of three consecutive even numbers, and one consecutive foursome
that also take the same number of steps to arrive at 1.
    When I teach this recursive algorithm to fourth graders, I also let
them continue on from 1.  Since 1 is an odd number, 3 x 1 + 1 = 4.  Then
4/2 = 2 and 2/2 = 1, and we are back to 1 again.  We have a cycle of
4-2-1-4-2-1....  This result is a good opportunity to introduce the
concept of periodicity to elementary students.
    Finally, to follow up the activity that Cohen presented, why not
alter the rule concerning odd numbers and substitute the value 3n - 1
whenever n is odd?  Keep the even-number rule unchanged.  This process
produces something even nicer than the original, as three outcomes are
possible!  A trichotomy can be made in the set of whole numbers.  Some
numbers do still arrive at 1, but under my keep-on-going point of view,
they produce a flip-flop effect, namely 1 gives 2, then 2 gives 1 back
immediately.  So we end with the cycle 2-1-2-1, and so forth.  I leave
the other two patterns of results for readers to discover.  They are
cycles, but quite different from the first one.  They are not hard to
find, and the "aha!" moment of seeing them is exciting.