- "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.
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