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Mathematical LensAuthor(s): Dana Yancoskie, Ron Lancaster and Brigitte BenteleSource: The Mathematics Teacher, Vol. 101, No. 4, Mathematical Discourse (NOVEMBER 2007),pp. 265-267Published by: National Council of Teachers of MathematicsStable URL: http://www.jstor.org/stable/20876109 .
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MATHEMATICAL Dana Yancoskie
1. Approximately what percentage of this photograph is red? Explain your estimation strategy. Discuss the accu
racy of your technique.
2. What is the ratio of the size of the
upper larger cluster of bracts to the size of the lower smaller cluster of bracts? Explain your reasoning.
3. The bracts, arranged on a central
connecting stem, create various acute
angles with the stem. What are the measures of these angles, and how are
they related?
4. When this collection of Heliconia
plants was originally planted at the Fairchild Tropical Botanic Garden, the gardeners would have been greatly concerned about the spacing between these plants (as between any plants). Gardeners consult books to determine
the recommended spacing for a par ticular plant, or they use their experi ence and powers of observation about what seems to work the best. Once
they know the spacing, the next issue is to determine how many plants will be needed for a particular area. Table 1 shows the maximum number of
plants arranged on a rectangular or
triangular grid in a garden whose area
is 100 square feet. The table data were
obtained using the plant calculator available at www.math.umn.edu/
~white/personal/plantcalc.html, cre
ated by Dennis White, professor of
mathematics, University of Minnesota.
(a) Without using the regression analysis features of your calcula
tor, develop an equation for^ in terms of x.
(b) Find an equation that expresses a
simple relationship between z and
y. Use this equation to determine an equation for z in terms of x.
(c) The following claims appear at the Web site Installation and Mainte nance of Landscape Bedding Plants,
www.ces.ncsu.edu/depts/hort/hil/ liil-555.html. For each, explain how the result was obtained.
For square spacing, the distance in inches between plants within rows
(X) equals the distance in inches between rows (Y), and X = Y.
For triangular spacing, the dis tance in inches between plants
within and between rows each
equals X and the distance in inches
between rows (Y) equals 0.886 times X.
The Heliconia plant, commonly called lobster claw, is a spectacular plant related to gingers, bananas, prayer plants, and
birds-of-paradise plants. It is also some
times known as false bird-of-paradise because of its flowers' close similarity to true bird-of-paradise flowers. The focus of these questions is the geometric attributes of this plant's smooth, red, boatlike bracts, or leaves. For example, we will look at
their size and the angles formed by differ ent bracts. We will also look at different
geometric arrangements that could be used for planting Heliconia in gardens.
Table 1
Plant Arrangements
x Distance between Plants
10
11
12
y Maximum Number of Possible Plants for a
Rectangular Grid
_400_
_293_
_225_
_177_
_144_
_115_
100
Maximum Number of Possible Plants for an
Equilateral Triangle Grid
_461_
_339_
_259_
205_
_166_
_137_
115
"Mathematical Lens" uses photographs as
a springboard for mathematical Inquiry. The goal of this department Is to encourage readers to see patterns and relationships that they can think about and extend In a
mathematically playful way.
Edited by Ron Lancaster [email protected]
University of Toronto Toronto, Ontario M5S1A1 Canada
Brlqltte Bentele [email protected]
Trinity School New York, NY 10024
Vol. 101, No. 4 November 2007 | MATHEMATICS TEACHER 265
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f?i AT H E M AT IC A L LENS solutions
Some of these questions can be resolved
by pasting the digital image of the Heli conia into The Geometer's Sketchpad (GSP).
1. GSP was used to superimpose a rect
angular grid on top of the photograph (fig. 1). Each row was examined, and the squares that contained any red were marked with a red dot. Any squares that were at least 50 percent red were marked with a blue dot. The data were compiled in table 2.
With the rectangular grid, there are a
total of 20 x 15 = 300 squares. Let r be the amount of red in the photograph. Then
70 161 -< r <-,
300 300
so the value of r is roughly between 23 percent and 54 percent. The
accuracy of the estimate is inversely related to the size of the squares. That is, the accuracy increases as the size of the individual squares decreases; or, the more squares there
are, the better the estimate.
2. Since the photograph is two-dimen sional and the two sets of bracts are somewhat similar and therefore would take up equal proportions of a
circle circumscribing them, GSP was
used to construct circles on top of the photograph (fig. 2). Points were
placed on the tips of the bracts, four on the upper larger cluster and three on the lower smaller cluster. Chords were drawn, three on the upper
larger cluster and two on the lower smaller cluster. Perpendicular bisec
tors of the chords were constructed. The intersection of each of the two sets of perpendicular bisectors deter
mines the center of each of the two circles. The area of each circle was
calculated. The ratio of the areas is 3.00. The ratio of the lengths of different attributes is therefore the
square root of 3.
3. Figure 3 shows a central axis
Fig. 2 GSP is used to inscribe the Heliconia bracts in circles.
drawn down the main connecting stems at the center of each cluster of bracts. Rays were constructed by
Table 2
Squares Containing Color
Row Number
20
19
18
17
16
15
14
13
12
11
10
Total
Number of Squares with a Red or a Blue Dot
10
10
10
10
161
Number of Squares with a Blue Dot
0
70
?I?^?f 3 10 15
Fig. 1 Heliconia and superimposed grid in GSP
266 MATHEMATICS TEACHER | Vol. 101, No. 4 November 2007
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Fig. 3 GSP can be used to measure angles
formed by the bracts and the main stem.
A I /?x
Fig. 4 In a triangular grid, a represents the
distance between plants and b represents the
distance between rows.
Have you ever seen a building, a bridge, a sign, or a natural
phenomenon that stimulated mathematical thoughts? Why not take a photograph and send it to NCTM, along with the mathematical questions that the
photograph inspires? The ques tions can be playful, imaginative, curious, and inventive; they can also be mathematical extensions
sparked by the photograph. If the photograph includes
identifiable people, the pho tographer must obtain signed release forms. Photographers must also obtain release forms if trademarked items are shown. Original photographs must be either in hard copy or supplied digitally as 300
dpi images in .jpg format. For details on releases and digi tal standards, please see the NCTM Web site. Photographs will not be returned.
Send the photographs, dia grams, list of questions, solu tions, and completed release forms to the "Mathematical Lens" editors.
Members who wish to use this month's
photographs in a classroom setting can download the image from NCTM's
Web site, www.nctm.org. Follow links to Mathematics Teacher, and choose Current Issue. Then select Mathemati cal Lens from the Departments, and look for the link to the image.
2 = 1.15 14,400 16,560
(c) On a square grid, the distance between the plants and between the rows would be the same. On a
triangular grid as in figure 4, the distance between the plants would be the lengths of the sides of
equilateral triangles, whereas the distance between the rows would be the altitude of the triangle. Let a represent the distance between
plants within and between con secutive rows. Let b be the dis tance between rows. Using basic
trigonometry, or the Pythagorean theorem, it can be seen that b =
a sin(60?), so that b is approxi mately equal to
\aS, or 0.886a, as claimed on the Web site.
Editors' note: For readers interested in other applications of mathematics to
botany, see the September 2007 issue of Mathematics Teacher, "Where Have All
the Flowers Gone?" pp. 88-92. ?o
identifying points on the tips of the bracts and points where the bracts meet the main stem. Angles were measured. The measures of three
of the angles on the upper cluster of bracts are very close, within one
degree of one another.
4. (a) Observe that as the value of x increases, the value of y decreases, bringing some sort of inverse variation to mind, such as
1 1 1 #
= = ?,or#
= ?. x x~ x
When the value of a* doubles
(from 6 to 12), the value of y goes from 400 to 100. This points to a function of the form
a
x~
Substituting (6, 400) in this
equation to solve for a, we
obtain
400 = ?= 14,400.
62
Therefore,
14,400 #
=-?
A"
(b) For any given row in table 1, the ratio of z to y is constant and
roughly equal to 1.15. Thus, z =
l.l5y, and so
Vol. 101, No. 4 November 2007 | MATHEMATICS TEACHER 267
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