GLaD againAuthor(s): Joel NeelySource: The Mathematics Teacher, Vol. 91, No. 3 (MARCH 1998), p. 238Published by: National Council of Teachers of MathematicsStable URL: http://www.jstor.org/stable/27970506 .

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when ABDC is a rectangle. The

general case, in which ABDC is

any parallelogram, then follows by viewing the parallelogram as the

image of a rectangle under a shear transformation parallel to AB.

The second construction, which we call interior-partition addition, also creates an {n + m)-partition from a given -partition and

m-partition of the opposite sides of a parallelogram. This construc tion is illustrated in figure 2 in the case m = 3 and = 4, although the construction is valid for 1 < m < . We see that the (m + ̂-partitioned segment EF is between segments A? and CD, where DF/DB = m/(m + n) follows

by considering similar triangles.

C_E_Al

\D = Ca F = E7 B=A3

Fig. 2

Interior-partition addition

The steps in the second construc tion are obvious from figure 2. In

particular, E is the intersection of the segments ACi and CAh and

Em+n_i is the intersection of the

segments BCn_i and DAm_i. Usu

ally, E and Em+n_i are distinct

points, the exception being when m = = 1. In this case, E and E2 are defined tobe the respective midpoints of AC and BD.

It is then a simple matter to use a sequence of partition-addi tion constructions to construct the lMh point along any given segment DB. First, as shown in

figure 3, erect any parallelogram ABDC. The midpoint E2 is then constructed with an = m -

1_

interior-partition addition on AB and CD. The remaining points F3, G4, .. use exterior

partition addition, in which the

one-partitioned segment A? is "added" to the most recently constructed -partitioned seg

ment to give an ( + l)-partitioned segment.

C HGF E_A

D H5\F3 E2 G4

Fig. 3 _

Constructing the 1/fith points of DB

Dave and Dan's Fibonacci par tition is achieved by "adding" the two most recently partitioned segments, as shown in figure 4. The points E2, G3, H5, /8, J13,... are at the Fibonacci points 1/2, 1/3,1/5,1/8,1/13,... from D toward B.

i c jJh g E_A

D I \Hs G3 E2

Fig. 4

Constructing the MFn points of DB, where Fn

- nth Fibonacci number

The alternative constructions shown above shed light on Dan and Dave's methods and have some additional applications. The reader may enjoy exploring why the three medians of a triangle are concurrent at their one-third

points (hint: extend any given tri

angle to a parallelogram) and the related result why the diagonal BD of a parallelogram ABCD is trisected by the segments drawn from A to the midpoints of sides BCmdDC.

Duane W. DeTemple Washington State University Pullman, WA 99164-3113

GLaD again As a former college teacher of mathematics and computing sci ence, and as a teacher, parent, and computing professional who

firmly believes in the value of

teaching problem-solving skills rather than cookbook answers to well-known problems, I would

like to offer my sincere apprecia tion to Goldenheim and Litchfield for having the ingenuity to use the tools available to them in dis

covering for themselves an ele

gant piece of mathematics. However, I must point out that

their teacher, their mathematics

department, and the NCTM have let them down most sadly. The construction that they present in the January 1997 issue not only is already known but appears to be quite commonplace in book

design and typography. As a former academic, I occa

sionally endure good-natured (I hope) wisecracks from colleagues who assert that anything that has become too useful no longer holds interest for the academic mind. This elegant little con struction certainly seems to have suffered that fate. Two examples will suffice to confirm my point.

Page 53 of Design of the Roman Letters by L'Harl Copeland (New York: Philosophical Library, 1966) includes a figure that is

equivalent to the one in the

presentation of Goldenheim and Litchfield, with the facing page containing the explanation in

passing: There is an interesting fact in

geometry that there is a progres sive sequence in the diagonals of the rectangle. It can be seen in the

drawing of the formula that the intersection of the two diagonals of the whole, or 1/1, bisects the rec

tangle at one-half both ways. The intersection of the diagonal of the one-half with the one of the whole trisects the rectangle both ways. The diagonal of the one-third with the one of the whole gives the fourth, and the sequence continues. It can be seen that the intersection of the one-third diagonal with that of the half gives the fifth. The for mula is: add the denominators of the fractions in all of which the numerators are one. For example, to find the septsection [sic]... the intersection of the diagonals of the 1/6 with 1/1,1/5 with 1/2, or 1/3 with 1/4 will all give the one-seventh.

I should point out that

Copeland is no mathematician; he merely offers this figure and

description in the context of lay ing out a page of text in pleasant proportions.

Stephen Moye's book Fontogra pher, Type by Design (New York: MIS Press, a subsidiary of Henry Holt and Company, 1995) is

about the design of electronic

typefaces for use in desktop pub lishing. The title page contains a

large graphic in which the uniform-division-by-diagonals construction is shown as the basis for establishing the proportions of page margins. A smaller ver sion of the same figure appears in the colophon, and a simplified version illuminates the initial letter of every chapter. Again, let me emphasize that I

have no criticism of the work of Goldenheim and Litchfield. Dis covery is valuable and exciting, and learning how to discover is even more so. However, I feel that a dangerous trend in our educational process, from ele mentary through university levels, prizes creativity above solid

grounding in what has come

before, perhaps in reaction to the excesses of the opposite nature in earlier years. We should be pro viding balanced guidance to the students of today. If we who should know better cannot distin

guish between new and known, how can we expect our students to do so?

Joel Neely jneely@ubdl. vdospk.com Memphis, TN 38116-8905

The use of mnemonics In "Reader Reflections: The new

grouping rule" (January 1997), Huang Sheng-zhu refers to the mistake students made in simpli fying the expression 1 -r 0.125 8.

Incidentally, a college senior

majoring in mathematics made the same mistake. The author

suggests a solution to this problem by advocating the use of a rule.

Students' inability to simplify such expressions correctly may be attributed to the prevalent? and sometimes indiscriminate? use of such mnemonics as PEMDAS (parentheses exponential-multiplication division-addition-subtraction) or BOMDAS (brackets-of multiplication-division addition-subtraction). Students

perform the operations of multi

plication and division or addition and subtraction in the exact order defined by the mnemonic; so divi sion follows the completion of

multiplication, and subtraction follows the completion of addition.

If we can help students under stand that division is a form of

(Continued on page 246)

238 THE MATHEMATICS TEACHER

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