Naive product rule extendedAuthor(s): Lionel GarrisonSource: The Mathematics Teacher, Vol. 94, No. 7 (October 2001), p. 550Published by: National Council of Teachers of MathematicsStable URL: http://www.jstor.org/stable/20870793 .

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three dimensions. It was followed by "A Comment on Fruitful Mathematics'" by J. M. Sachs on pages 701-3 of the December 1974 Mathematics Teacher.

The content of another related article, "A Matter of Disks," by

William A. Ewbank, on pages 96-97 and 146 of the February 1986 Mathematics Teacher, arose

when the writer considered waste in cutting out disks.

If readers would like to extend the ideas mentioned by Peggy House in "An Extension: Gather Your Students around the Gath ering Circles," which followed Nakagomi's article, they can consult "A Bundling Fool Beats the Wrap," by Ian Stewart, in the "Mathematical Recreations" section of the June 1993 issue of Scientific American, on pages 142-44.

Marion Walter walter@math. uoregon.edu

University of Oregon Eugene, OR 97403-1222

Naive product rule extended In "Sharing Teaching Ideas: Is the Derivative of a Product the Product of the Derivatives?" on pages 26-27 of the January 2001 issue of the Mathematics Teacher,

Marsha Hurwitz writes that she invites her students to find pairs of nontrivial, that is, nonconstant, functions /"andg that obey the "naive" product rule (f

? g)1 = f

? g\

She offers one class of solutions:

where k is a constant, and she invites the reader to find others. One other family of solutions consists of functions of the type fix) - e andgM = ebx, where b = a/(a -1) and a 1.

One hopes that after struggling to find such bizarre exceptional cases, students do not soon forget

g(x) = x' Jl

and

the correct version of the product rule.

Lionel Garrison garrison@horacemann. or g The Horace Mann School Riveriate, NY 10471-3430

Analytical connections with triangular numbers In the article "Beyond the Golden Ratio: A Calculator-Based Inves tigation," on pages 138-44 of the February 2001 issue of the Mathe matics Teacher, Peter L. Glidden discusses the value R(n) of the repeated radical

+ + + "?,

where is a positive integer, finding it to be given by the analytical expression

(1) fi(B) = l?^?f?.

Table 2 of the article shows that R(n) is an integer for certain values of n, and seeing how to characterize such values of is interesting.

From equation (1), we see that R(n) is rational only when 1 + 4n is the square of an integer x. Since 1 + in is manifestly odd, x2 must also be odd; hence, is odd. Therefore, = 2k + 1, with k an integer. Then 1 + An = (2k + l)2, so must be given by

n = k(k + 1), which is just twice the triangular number A?, the number of dots in a triangular array of dots arranged in k rows, and the corresponding

R(n) is just k + 1. Therefore, R{n) is just the difference between successive triangular numbers, and R(n) is either irrational or an

integer. A possible student investigation

would be to see how these analyt ical connections with triangular numbers arise from the geometry discussed in Glidden's article.

Mark Snyder msnyder@tiac.net

msnyder@fsc.edu Fitchburg State College Fitchburg, MA 01420

Appropriate use of technology "Not instructing students on the [appropriate] use of technology is doing them a disservice."

This statement in Lin

McMullin's "Soundoff! Algemetic" in the February 2001 issue of the

Mathematics Teacher is right on the mark. Note that I added "appropriate" to that quotation.

Instructing students in the use of technology is very important; however, technology should be introduced gradually after students have used paper and pencil and manipulatives. For example, in trigonometry I used spaghetti and a calculator to produce the basic sine and cosine curves on

six-foot-long strips of butcher paper. My students did this activ ity in the school's hallways in teams of four students. They enjoyed this activity and recalled it later when they graphed more advanced curves in trigonometry. Parts of calculator capability are introduced to students gradually for specific analysis needs. Exam ples include using tracing to reveal values, maximum and minimum for curve-direction changes, intersection for same value for two or more curves, and tables for curve analysis.

In geometry, using patty paper for paper-folding constructions is appropriate initially. Then such technology as the TI-92 should be introduced to duplicate the hands on activities and subsequent student investigations. To use the technology effectively and effi ciently, students must be familiar with the basic properties being investigated. If a student asks, "What will happen if...," the easy reply is, "Try it!" The student? not the teacher?is responsible for finding the answer. Does a stu dent remember the answer better if the teacher gives it or if he or she learns it as a result of investi gating and using the calculator? One can argue that the teacher becomes the facilitator and that the student is guided by using the technology, thereby reinforcing and enhancing the concepts.

I maintain that if technology is introduced appropriately into the curriculum, the students are more involved and more responsible for their education, have a deeper comprehension of the concepts, and have a better appreciation of the mathematics being taught. Even with technology, students should be able to perform such basic skills as multiplication tables for the numbers 1 to 12 without a calculator. For example, teachers should use mathematics drills

without a calculator as part of periodic skills development. The key is appropriate use of technol ogy?not technology as a crutch.

Alan R. Brown ab55@erols.com Virginia Beach, VA 23454 2941

McMullin responds: I am tempted to argue that no inappropriate uses of technology exist in mathe matics. I certainly do not agree that technology should necessarily follow paper, pencils, and manip ulatives. Technology should not be added on the end of the curri culum; it should be an integral part of it from the beginning.

Students can do investigations with a computer algebra system (CAS), as well as with manip ulatives. For example, on the first day of teaching students about collecting like terms, what is wrong with beginning by letting a CAS do a few problems, like +

+ + + , and then discussing why the CAS changes the sum to 5x? This exercise should lead to a discussion of how multiplication is repeated addition and why 5x is equivalent tox + x + x + x + x. After the students get the idea, they can do a few problems by hand. When the "What if... questions occur?for example, What if I have 5x + 7s?-the CAS will be there for them to "try it."

I do not mean to say that in many instances other methods are not as appropriate as, or even better than, technology in fostering understanding and learning. I am saying that many topics that are currently the mainstays of algebra courses should be elimi nated in favor of doing them with a CAS; to name one: students do not need to be facile at factoring trinomials by hand. They do need to know what factoring means and when factoring can help them. They also need to know that the most efficient way to factor is to use the CAS. And the technology approach has an added bonus: polynomials of degree greater than two can be investigated right from the start.

Finding an algorithm and then programming a CAS to do it are

certainly good pedagogy. However, all the manipulative techniques of mathematics, the algemetics, are already there, in the CAS. Students need to be taught how

(Continued on page 593)

550 MATHEMATICS TEACHER

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