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Algebra II: Strand 7. Conic Sections; Topic 4. Applications of Conic Sections; Topic Notes
STRAND 7: CONIC SECTIONS
TOPIC 7.4: APPLICATIONS OF CONIC SECTIONS
Topic Notes
Mathematical focus
The mathematical focus of this topic is to use conic sections, their geometric properties,
and algebraic representations to model physical situations and solve application problems.
Topic focus
This topic contains six tasks:
Task 7.4.1: Radio Astronomy
Task 7.4.2: Earthquake Energy
Task 7.4.3: An Arching Dilemma
Task 7.4.4: LORAN
Task 7.4.5: Rocket Engine Design
Task 7.4.6:Communication Satellite Mission Design
The goal of this topic is for participants and students to solve application problems
involving physical representations of conics sections. They must be able to recognize the
conic that will model a situation, write its equation, and solve it either algebraically or
graphically for needed information. Finding and using the appropriate conic model is often
a matter of recognizing the locus definition in an applied context. In some situations, the
solution to the equation must them be interpreted in the context of the problem situation.
TExES standards focus
TExES Standard III.014 Geometry and Measurement. The teacher understands coordinate,
transformational, and vector geometry and their connections. The beginning teacher:
(F) Uses coordinate geometry to derive and explore the equations, properties,
and applications of conic sections (i.e., lines, circles, hyperbolas, ellipses,
parabolas).
TExES Standard IV.019 Mathematical Processes and Perspectives. The teacher understands
mathematical connections both within and outside of mathematics and how to communicate
mathematical ideas and concepts. The beginning teacher:
(B) Applies correct mathematical reasoning to derive valid conclusions from a set
of premises.
(C) Uses inductive reasoning to make conjectures and uses deductive methods to
evaluate the validity of conjectures.
TEKS/TAKS focus
TEKS 2A.3 Foundations for functions. The student formulates systems of equations
December 20, 2004. Ensuring Teacher Quality: Algebra II, produced by the Charles A. Dana Center at The University of Texas
at Austin for the Texas Higher Education Coordinating Board.
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Algebra II: Strand 7. Conic Sections; Topic 4. Applications of Conic Sections; Topic Notes
and inequalities from problem situations, uses a variety of methods to solve them, and
analyzes the solutions in terms of the situations. The student is expected to:
(A) analyze situations and formulate systems of equations in two or more unknowns or
inequalities in two unknowns to solve problems.
TEKS 2A.5 Algebra and geometry. The student knows the relationship between the
geometric and algebraic descriptions of conic sections. The student is expected to:
(B) sketch graphs of conic sections to relate simple parameter changes in the equation to
corresponding changes in the graph.
Materials
Materials needed
Graphing Calculator
Conics APP for TI-83 Plus
Chart Paper
Poster Markers
Blank Transparencies
Transparency Markers
Task
7.4.1
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Task
7.4.2
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Task
7.4.3
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Task
7.4.4
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Task
7.4.5
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Task
7.4.6
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Procedure
Facilitators may choose to do any of the Tasks.
Task 7.4.1 – Parabola
Task 7.4.2 – Circles
Task 7.4.3 – Ellipse
Task 7.4.4 – System of Hyperbolas
Task 7.4.5 – Parabola
Task 7.4.6 – System of 2 Circles and an Ellipse
Task 7.4.1 should probably be done together in class to illustrate how the locus definition
of a conic will be useful in determining the type of conic that can be used to model a
physical situation. Participants may need help to understand that some decisions in the
solution of the problem such as placement of the axes are left to their discretion and that
the answer to the equation or system of equations used must be interpreted in the context of
the problem situation.
Divide the participants into groups and assign one or two groups to do the remaining tasks.
They should record their work on a poster or transparency and be prepared to explain to the
group the decisions they made in the solution of their problem.
Summary
The ability to do these problems easily will be ample evidence that the participants or
students understand both the geometric and algebraic representations of the conic sections.
The connection between the locus definition of a conic, its equation, and an application
problem is very important if conic sections are going to be useful mathematical tools.
December 20, 2004. Ensuring Teacher Quality: Algebra II, produced by the Charles A. Dana Center at The University of Texas
at Austin for the Texas Higher Education Coordinating Board.
3
Algebra II: Strand 7. Conic Sections; Topic 4. Applications of Conic Sections; Topic Notes
Assessments
These tasks are assessments for the entire strand. Participants should have reviewed the
skills necessary to solve the problems by completing the tasks in Topics 7.2 and 7.3.
Teacher use only
Task 7.4.1
Task 7.4.2
Task 7.4.3
Task 7.4.4
Task 7.4.5
Task 7.4.6
Modify for students
Ready for students
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December 20, 2004. Ensuring Teacher Quality: Algebra II, produced by the Charles A. Dana Center at The University of Texas
at Austin for the Texas Higher Education Coordinating Board.