From Manipulatives to Geometry, Algebra, and Technology:
Solving an Ancient Optimization Problem
Betty M. Gasque
May, 2008
Euler said: “Nothing takes place in the world whose meaning is not that of
some maximum or minimum.” (Tikhomirov, 1990).
The study of optimization problems began approximately twenty-five centuries
ago and they are important in many areas of life today. These problems
provide excellent examples for the application of mathematical concepts but
relatively few students have the opportunity to solve “real world” optimization
problems. These problems have traditionally been relegated to the calculus
curriculum due to the level of analytic skills required for solution.
Strategies that could be used for solving optimization problem are sometimes
neglected in favor of analytic methods. Many of these strategies can be
considered using the TI-NspireTM. The TI-NspireTM allows students to
investigate a problem through linked geometric, algebraic, and numeric
explorations and facilitates multiple representations.
Heron of Alexandria was interested in all forms of measurement as well as
optics and mechanics. He considered this problem in the 1st century A.D.:
A and B are two given points on the same side of line L. Find a point D on L
such that the sum of the distances from A to D and from D to B is a minimum
Tikhomirov, 1990)
Problems of this type are still found in calculus texts. This is a modification of
one example: Town A is 14 kilometers from a river and town B is nine
kilometers from that same river. Town B is on the same side of the river as
town A. The river follows a straight path between locations O and C as
shown below. The distance from town A to town B is 13 kilometers. A
pumping station is to be built along the river to supply water to both towns.
Where should the pumping station be built so that the sum of the distances
from the pumping station to the two towns is a minimum?
Betty Martin Gasque
bgasque@aol.com
1
Many calculus teachers find that even those students who are able to
successfully solve optimization problems seldom enjoy them – many are put
off by the tedious calculations needed to solve these problems. Simpler
geometric methods of solution may be neglected and Lagrange’s approach is
the only one found in many classrooms. “The methods I set forth require
neither constructions nor geometric nor mechanical considerations. They
require only algebraic operations subject to a systematic and uniform course.”
(Lagrange quoted in Tikhomirov, 1990)
The power of the TI-NspireTM provides other interesting solution options.
Problem 1: An Investigative Method
As shown in Figure 1, information on the problem is provided on page 1.1 of
the document. On page 1.2, a Graphs & Geometry page, a model of the
problem is constructed (Figure 2). Students should be encouraged to drag D
and simply estimate the best location for the pumping station (D).
Figure 1
Figure 2
On page 1.3, students are instructed to find the distance from point O to point
C and to then move to page 1.4 (Figure 3). When they drag point D on page
1.5, the distances from A to D, from B to D, and the sum of the two distances
are displayed (Figure 4).
Figure 3
Betty Martin Gasque
bgasque@aol.com
Figure 4
2
Students are asked to estimate the best location for the pumping station in
terms of the distance from point O to point C.
Teachers may collect students’ estimates and find a class average at this
point in the investigation.
Problem 2: Heron’s Method
The problem posed by Heron concerned the physical behavior of light rays.
According to historians, the law of reflection of light was not new - it was
known to Euclid, Aristotle, and probably also Plato. Boyer and Merzbach
(1989) note that Heron gave a simple geometric argument in his work on
reflection (Catoptrics) to prove that the angle of incidence equaled the angle
of reflections. Kline (1959) stated that Heron used the “shortest possible
path” characteristic of light waves and congruent triangles in his proof. Kline
also noted that the shortest path requirement is found in many other
applications.
In the TI-NspireTM document (problem 2), students are encouraged to use a
geometric method to solve the problem. Page 2.3 (Figure 5) provides
directions. Page 2.4 provides a figure with a different scale and a hint (Figure
6). Point B’ is the reflection of point B about the line segment OC. Students
can use TI-NspireTM tools to determine the best location for the pumping
station as shown in Figures 7 and 8. They are asked to justify their solutions.
Figure 5
Betty Martin Gasque
bgasque@aol.com
Figure 6
3
A line segment from point A to point B’ is drawn and the intersection point of
segments AB’ and OC is found and labeled (Figure 7). The distance from
point O to point D is measured (Figure 8).
Figure 7
Figure 8
Problem 3: A Geometric Method
In problem 3, students are encouraged to explore another geometric solution
(Figures 9 and 10). Similar triangles may be used to solve the problem. A
calculator application and a notes page are provided for student work.
Figure 9
Betty Martin Gasque
bgasque@aol.com
Figure 10
4
Problem 4: An Algebraic Method
In problem 4, students are encouraged to explore an algebraic solution
(Figures 11 and 12).
Problem 5 – A
Figure 11
Figure 12
Different strategies may be employed to solve the problem algebraically.
Students may find the slopes of segment AB and segment AD (or segment
DB’), set the two slopes equal to each other, and solve for x.
Alternatively, students may find the equation of a line through points A and B’
and find the zero of this function (x, 0).
Problem 5: A Minimum Function Value Method
In problem 5, students find a function that expresses the sum of the two
distances in terms of x (the distance from O to D). They graph the function
and find the minimum function value (Figures 13 and 14).
Figure 13
Betty Martin Gasque
bgasque@aol.com
Figure 14
5
Problem Six: A Tabular Method
In problem 6, students copy their function from problem 5, generate a function
table, and change delta x to find an approximation of the minimum function
value numerically (Figure 15 and 16).
Figure 15
Figure 16
Problems 7, 8, and 9: Using the TI-NspireTM – Linking Multiple
Representations
The linked representations of the TI-NspireTM can enable students to make
geometric, numeric, and algebraic connections.
In problem 7, the values labeled in bold on page 7.2 have been linked to the
spreadsheet on page 7.3. On page 7.1 of the document, students were
instructed to drag point D to observe the changes in the values. List lod
captures the length of segment OD. Lists lbd and lad capture the lengths of
segments BD and AD, respectively. As students drag point D, the values OD,
BD, and AD are stored in the spreadsheet (Figures 17 and 18). The fourth
list, sumab, is defined as the sum of lists lbd and lad.
Figure 17
Betty Martin Gasque
bgasque@aol.com
Figure 18
6
A scatter plot of the sum list (sumab) versus the length of segment OD list
(lod) is graphed on page 7.4 (Figure 19). Students can trace on the graph
and estimate the value of x that produces the minimum function value (Figure
20).
Figure 19
Figure 20
Students may also inspect the spreadsheet to determine the minimum value
in the sum list. They may need to be reminded that this is only the minimum
value of those data that were captured.
More visual connections…
To more closely connect the changing values in the geometric construction
and the scatter plot of the data, a split screen may be used to display the
construction with the plot of the data. Page 8.1 displays a scatter plot set-up
for a graph of data that will be stored in the spreadsheet on the next page
(Figure 21). As students drag point D, the data are collected and graphed
(Figures 22).
Figure 21
Betty Martin Gasque
bgasque@aol.com
Figure 22
7
An alternate method for displaying the scatter plot with the geometric
construction is shown in problem 9. Rather than using a split screen, the
construction is pasted into an analytic window (Figure 23 and 24).
Figure 23
Figure 24
Problem 10: An Analytic Method
In problem 10, students are instructed to solve the problem using calculus, an
analytic method. They are given instructions through a comment on a
calculator page and provided a notes page for their results.
Students are then asked to consider this question: What if the distance from
the river to each town had been twice as far (OA = 28 km and CB = 18 km)?
Problem 11: A New Problem
In problem 11, students are presented with a variation of this problem and
asked questions related to this new scenario (Figures 25, 26 and 27).
Figure 25
Betty Martin Gasque
bgasque@aol.com
Figure 26
8
Figure 27
Possible solutions are shown in Figures 28 and 29.
Figure 28
Figure 29
Discussing optimization problems, R. Courant and M. Robbins (quoted in
Tikhomirov, 1990) noted: “It can hardly be denied that our elementary
methods are simpler and more direct than the methods of analysis. In
general, when studying some scientific problems it is better to begin with its
individual peculiarities than rely on general methods.”
Independent of the method of solution, the use of historical problems in the
mathematics classroom enables students to appreciate the utility of
mathematics that was developed in other time periods and cultures. As
Swetz (1989) stated, “students can be placed in the role of mathematical
archaeologists and be led to discoveries.”
In contrast to Lagrange’s uniform course, the ability to analyze a problem and
choose an appropriate method of analysis is a skill that we hope all students
will develop. The TI-NspireTM provides a powerful tool for students to explore
and investigate multiple representations of rich problems.
Betty Martin Gasque
bgasque@aol.com
9
Bibliography
Boyer, C.B. and Merzbach, U.C. A History of Mathematics, Second Edition,
New York, NY: John Wiley & Sons, Inc., 1989
Cajori, F. A History of Mathematics. New York, NY: The Macmillan Company,
1926
Grifalconi, A. The Village of Round and Square Houses. Boston, MA: Little,
Brown and Company, 1992.
Kline, M. Mathematics and the Physical World. New York, NY: Thomas Y.
Crowell Company, 1959.
Levenson, M. E. Maxima and Minima. New York, NY: The Macmillan
Company, 1967.
National Council of Teachers of Mathematics. Curriculum and Evaluation
Standards for School Mathematics. Reston, VA: The Council, 1989.
Niven, I. Maxima and Minima Without Calculus. Washington, D. C.: Dolciana
Mathematical Expositions No. 6, Mathematical Association of America, 1991.
Pleacher, D. “Activities to Introduce Maxima-Minima Problems.” Mathematics
Teacher 84 (May, 1991): 379-382.
Swetz, F. J. “Using Problems from the History of Mathematics in Classroom
Instructions.” Mathematics Teacher 82 (May, 1989): 370-377.
Tierney, J. A. “Elementary Techniques in Maxima and Minima.” Mathematics
Teacher 46 (November, 1953): 484-486.
Tikhomirov, V. M. Stories about Maxima and Minima. Providence, RI:
American Mathematical Society, 1990.
Betty Martin Gasque
bgasque@aol.com
10