Download Resource ID#: 9442

Detemination of the Optimal Point
Resource ID#: 9442
Primary Type: Lesson Plan
This document was generated on CPALMS - www.cpalms.org
Students will use dynamic geometry software to determine the optimal location for a facility
under a variety of scenarios. The experiments will suggest a relation between the optimal point
and a common concept in geometry; in some cases, there will be a connection to a statistical
concept. Algebra can be used to verify some of the conjectures.
Subject(s): Mathematics
Grade Level(s): 10, 11, 12
Intended Audience: Educators
Suggested Technology: Computer for Presenter, Computers for Students, Internet Connection,
LCD Projector, Microsoft Office, GeoGebra Free Software (Download the Free GeoGebra
Software)
Instructional Time: 3 Hour(s) 30 Minute(s)
Resource supports reading in content area: Yes
Freely Available: Yes
Keywords: optimal location, point of concurrency, median, mean, centroid, circumcenter,
incenter
Instructional Component Type(s): Lesson Plan, Worksheet, Problem-Solving
Task, Educational Software / Tool
Resource Collection: iCPALMS
ATTACHMENTS
OPTIMAL LOCATION.ggb
ConstructingPointsofConcurrence.docx
STUDENTWORKSHEETFOROPTIMALLOCATIONPROBLEMS.docx
TEACHERWORKSHEETFOROPTIMALLOCATIONPROBLEMS1.docx
LESSON CONTENT

Lesson Plan Template:
Confirmatory or Structured Inquiry

Learning Objectives: What will students know and be able to do as a result of this
lesson?
Students will develop connections between important geometrical concepts and the context of determining
the best point. They will also learn that "best" needs to be carefully defined in context. In addition, they will
learn to be skeptical and careful when generalizing from experiments. They will see how scientific
processes are useful, but not definitive, in developing mathematical ideas.

Prior Knowledge: What prior knowledge should students have for this lesson?
1.
Students should be familiar with the dynamic geometry software. There are several possible
software packages that could be used. One is Sketchpad, a commercial product, available in many
school computer labs. Another is GeoGebra, an open source, no-cost software package, that can be
downloaded: http://www.geogebra.org/cms/. If students are not familiar with the software, then
one or two simple constructions could precede the activity--e.g., constructing quadrilaterals and
their diagonals. Worksheets begin with instructions that are detailed and explicit.
2.
Students should be familiar with the definitions and basic mathematical properties of points of
concurrence of a triangle: median, incenter, cirucumcenter, orthocenter. A fifth concurrence point
-- Fermat-Torricelli point -- will be introduced in the activity; its fundamental property is that, if
segments connect it to each of the vertices, the angles at the FT point are all 120 degrees. Such a
point exists when all interior angles of the triangle measure no more than 120 degrees. If students
are NOT familiar with the points of concurrence, they will explore results of their constructions by
observing that concurrence occurs even as the triangle is varied.

Guiding Questions: What are the guiding questions for this lesson?
1.
2.

Which, if any, of the points of concurrence corresponds to the optimal point for each context?
Are there sufficient experiments and cases investigated to justify your conclusion?
Introduction: How will the teacher introduce the lesson to the students?
The teacher should indicate that one of the major uses of mathematics is the determination of a best
strategy or optimal values of variables. Complete teaching of the tasks will involve attempting to determine
the best points for a variety of conditions and contexts. One must be careful to examine several distinct
configurations before articulating conclusions; even then there remains the possibility that a conclusion or
inference might be valid only in a limited set of cases. The primary tool that will be used in the
investigation is dynamic geometry software.
Students will be instructed to open GeoGebra and begin the construction of medians, angle bisectors,
altitudes, and perpendicular bisectors.

Investigate: What question(s) will students be investigating? What process will students
follow to collect information that can be used to answer the question(s)?
The investigative tasks and questions are contained in the STUDENT WORKSHEET below.
The TEACHER WORKSHEET contains the questions and some answers/comments.
The "Constructing Points of Concurrence" contains detailed instructions for constructing the centroid,
circumcenter, incenter, orthocenter, and Fermat-Torricelli point (when it exists) for a triangle.
There is strong alignment with most of the Mathematical Practices. The contexts presented in the questions
have the potential to provide a deep understanding of ideas through multi-representions -- algebra,
geometry, measurement, and context--and critical thinking. Moreover, the contexts are all in one of the
most common areas in which mathematics is applied -- optimization.
There is strong connections with the Nature of Science since students will be performing experiments,
observing result, drawing inferences, and testing hypotheses. Some of the questions that students will be
addressing are aligned with the Nature of Science.

Analyze: How will students organize and interpret the data collected during the
investigation?
Students working in pairs will sketch each scenario within the dynamic geometry software. Distances will
be measured and defined within this environment. The unknown location will be varied (dragged) until the
optimum is attained. A wide variety of given locations will be investigated with the students seeking to
describe the optimal location in terms of some previously learned concept. Then students will attempt to
explain the connection between the concept and the context. Then students will have discussions of their
observations and inferences.

Closure: What will the teacher do to bring the lesson to a close? How will the students
make sense of the investigation?
The complexity of the questions necessitates a whole class discussion focusing on each of the given
questions.
The discussion should include the role of experimentation, conjecture, rigorous verification, connections
with context, statistical concepts (mean and median), and algebra.
The teacher could point out that points of concurrence having meaning in context, one must be thorough
when using experimental methods to solve mathematics problems, and that mathematical ideas can supply
the explanation for the experimental results. The combination of experimental methods and mathematical
concepts provide a powerful combination.
Finally, students could be asked to suggest extensions of the questions. For example, what happens if there
are four points, rather than three. However, it is preferable to have students complete additional examples
to support, refute, or modify their conjectures.

Summative Assessment
The teacher can give the students a related problem, possibly one of several natural extensions of the
activity, and observe their problem-solving strategy and ability to draw valid conclusions. The following
are some extensions.
1.
A family has just purchased a large triangular plot of land. Where should a house be built so that
the sum of the distances to the three streets is as small as possible?
ANSWER: At the vertex opposite the longest side of the triangle. If the triangle is equilateral, all
points in the plot are equally good.
2.
Three communities with equal populations wish to build a shared facility. There are many roads in
the region, but all are oriented either north-south or east-west. Where should the facility be located
in order to minimize the total distance traveled by all residents to the facility?
ANSWER: The location is (x,y) where x is the median of the x-coordinates of the student home
locations, and y is the median of the y-coordinates.
3.
Determine the optimal bus stop location if there are six children living along the road.
ANSWER: Any place between homes of the two middle children.
4.
Suppose children live 100m, 200m, 300m, 600m, 1000m, 2000m along a road and a school bus is
to make two stops along the road.
a.
Where should the bus stop to minimize the total distance walked by the children?
ANSWER: There are two scenarios that should be compared -- (1) picking up the first
three children at the first stop (others at the second stop), (2) picking up the first four
children at the first stop.
b.

Formative Assessment
1.
2.

Where should the bus stop to minimize the longest distance walked by any of the
children?
Students will be constructing the points of concurrence of a triangle. The teacher will circulate in
the computer lab asking students questions relating to the meaning of the centroid, in center, and
circumcenter.
Students will then determine which of the points of concurrence are relevant to each of the context
questions. The teacher will ask each student for his/her conjecture and to describe the
experiments/cases that led to the conjecture.
Feedback to Students
The activity will consist of two major stages -- an experimental stage using dynamic geometry software and
a more conceptual stage in which the experimental observations are placed in the context of geometrical
concepts. After students have made some progress with the first stage, a class discussion will elicit ideas
from the students, which others can use to compare their results. Often some students will make a
generalized conjecture which other students may notice does not apply for all initial triangles.
ACCOMMODATIONS & RECOMMENDATIONS

Accommodations:
Differentiated instruction is relatively straightforward due to the existence of several investigations.
Working in teams should mitigate difficulties due to visual and motor problems.

Extensions:
There are many extensions.
o
o
o


What happens if there are more than three residences or communities?
What happens if two bus stops are permitted?
Suppose the three communities are connected via straight roads. Where should a shared facility be
built that minimizes the total cost of new road(s) that connect to the existing roads?
o Generate contexts that would extend one or more of the scenarios to three-dimensions.
Suggested Technology: Computer for Presenter, Computers for Students, Internet Connection, LCD
Projector, Microsoft Office, GeoGebra Free Software
Special Materials Needed:
Since the activity is likely to require more than one class period, students will need to have a means of
saving their work, either on the computer hard drive or on flash drives (individual or shared).

Further Recommendations:
The discussion is crucial. Students must justify their conclusions. It is common for students to try two or
three examples and draw conclusions that are valid only for acute triangles.
Additional Information/Instructions
By Author/Submitter
This resource can be used in a variety of ways, depending on the background of the students and
the goals of the instructor.
One purpose could be to introduce students to the various points of concurrency.
Another could be to present students with an opportunity to investigate situations that they can
control.
Students could experiment to notice that various construction result in concurrence, then study
why this happens.
SOURCE AND ACCESS INFORMATION
Contributed by: Steve Blumsack
Name of Author/Source: Steve Blumsack
District/Organization of Contributor(s): Leon
Is this Resource freely Available? Yes
License: CPALMS License - no distribution - non commercial
Related Standards
Name
MAFS.912.G-CO.3.9:
Description
Prove theorems about lines and angles; use theorems about lines
and angles to solve problems. Theorems include: vertical angles
are congruent; when a transversal crosses parallel lines,
alternate interior angles are congruent and corresponding
angles are congruent; points on a perpendicular bisector of a
line segment are exactly those equidistant from the segment’s
endpoints.
Remarks/Examples:
MAFS.912.G-CO.3.10:
MAFS.912.G-CO.4.12:
SC.8.N.1.3:
SC.8.N.1.6:
Prove theorems about triangles; use theorems about triangles to
solve problems. Theorems include: measures of interior angles
of a triangle sum to 180°; triangle inequality theorem; base
angles of isosceles triangles are congruent; the segment joining
midpoints of two sides of a triangle is parallel to the third side
and half the length; the medians of a triangle meet at a point.
Make formal geometric constructions with a variety of tools and
methods (compass and straightedge, string, reflective devices,
paper folding, dynamic geometric software, etc.). Copying a
segment; copying an angle; bisecting a segment; bisecting an
angle; constructing perpendicular lines, including the
perpendicular bisector of a line segment; and constructing a
line parallel to a given line through a point not on the line.
Use phrases such as "results support" or "fail to support" in
science, understanding that science does not offer conclusive
'proof' of a knowledge claim.
Understand that scientific investigations involve the collection
of relevant empirical evidence, the use of logical reasoning, and
the application of imagination in devising hypotheses,
predictions, explanations and models to make sense of the
collected evidence.