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La Cañada Math II Advanced Newsletter
Unit 9 – Circles
Unit 9 Overview
This unit will begin by defining basic terms related to circles (i.e. center,
chord, diameter, secant, …) that will be used in various proofs and
problems. Students will then be exposed to a variety of theorems, and
corollaries to those theorems, that will be used to solve problems. Each
theorem will be proven using previous definitions, postulates, and
theorems. Indirect proof, or proof by contradiction, will be introduced as
a tool for proving a theorem that does not have an obvious path.
Students will be introduced to the concept of indirect proof in a simpler
geometric problem and with numerical reasoning before using it in the
proof a new theorem. In particular, proof by contradiction will be used to
prove that the tangent of a circle is perpendicular to the radius at the
point of tangency. They will also construct a tangent line from a point
outside of a circle and explain the relevant geometry.
Theorems in this unit will be related to the tangents and radii of a circle,
the arcs of a circle and their relationship with central, inscribed, interior
and exterior angles, and the chords and secants of a circle. Once
students have analyzed, understood, and proved these theorems, they
will use them to solve problems. Some of these problems will involve
algebraic as well as geometric reasoning. Real-world meaning will be
constructed using the concept of latitude and longitude lines as the
measure of the central angle of a sphere. As the unit progresses, students
will develop a sense of when it is beneficial or useful to construct an
auxiliary line in order to create an effective proof. This skill, along with the
idea of indirect proof, will be beneficial throughout this unit and future
units.
In order to prove several of the theorems in this unit, students will also
need to consider the various cases which can occur within the
constraints of the context and use each of them as the starting point for
a proof. Students will work with several special angles in a circle, and the
segments that are related to them. For example, students will prove the
various cases where an inscribed angle intercepts an arc that is twice its
measure. Previous knowledge and problem solving skills will be used to
prove theorems about two tangents, two secants, or a secant and
tangent drawn in a circle and then use those theorems to draw
conclusions based on a given diagram
Important Dates
February 23
Unit 9 Test (Tentative)
Homework Policy
Daily Homework will be structured
more specifically to provide more
detailed information for students
and parents. Homework will
generally consist of two sections;
exercises and problems.
Exercises are typical of the type
of questions that have been
taught in class. Students should
have a model and/or framework
for this type of question and
should consult their resources to
maintain consistency from our
work as a class. Exercises have a
prescribed solution method.
Problems and proofs are
questions for which there is no
prescribed method. They will
require students to utilize their
problem solving skills, creativity,
and make connections among
mathematical ideas.
Weekly Homework will be a
component of the course, and
will emphasize algebra skills
learned in last year’s course in an
attempt to maintain important
ideas before entering LC Math III.
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Additional Unit 9 Information
Key Common Core Standards
G.C.2 - Identify and describe relationships among inscribed angles, radii, and
chords. Include the relationship between central, inscribed, and circumscribed
angles; inscribed angles on a diameter are right angles; the radius of a circle is
perpendicular to the tangent where the radius intersects the circle.
G.C.3 - Construct the inscribed and circumscribed circles of a triangle, and
prove properties of angles for a quadrilateral inscribed in a circle.
G.C.4 - (+) Construct a tangent line from a point outside a given circle to the
circle.
Essential Questions
-What relationships exist between segments in circles?
-What relationships exist between angles and segments in circles?
-Why do constructions related to circles make sense?
-What is indirect proof?
Helpful Resources
 Geometry Textbook – Chapter 9 (Sections 1-7)
Chapter 10 (Section 1)
 Construction Support:
www.mathsisfun.com/geometry/constructions.html
 Khan Academy:
https://www.khanacademy.org/math/geometry-home/geometry
Dr. Carruthers ([email protected])
Mr. McDermott ([email protected])
Spotlight on the
Standards for
Mathematical
Practice:
MP 3: Construct viable
arguments and critique the
reasoning of others.
Constructing arguments is an
important mathematical
practice. Mathematics, as a
field of study, is characterized
by a need to prove what is true.
As students advance in their
mathematics education, they
will be asked to justify and prove
mathematical relationships with
an increasing level of rigor and
formality.
This unit will extend students’
understanding of proof and
making a logical argument that
is supported by definitions and
postulated. In this unit, students
will be introduced to the
concept of proof by
contradiction, or indirect proof,
which provides an additional
proof strategy when traditional
methods become difficult or
increasingly complicated.
Students will use proof, and
deductive reasoning in problem
solving throughout this course
and in all future mathematics.
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