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Geometry, Quarter 3, Unit 3.1
Circles
Overview
Number of instructional days:
10
(1 day = 45 minutes)
Content to be learned
Mathematical practices to be integrated

Use appropriate tools strategically.


Construct equilateral triangles, squares, and
regular hexagons in a given circle using a
compass and straightedge.
Construct the circumscribed and inscribed
circles of a triangle.
Construct two lines that are tangent to a circle
from a point outside the circle. Explore
relationships formed by this construction
(circumscribed angle, tangent radius theorems).

Prove properties of angles for a quadrilateral
inscribed in a circle.

Explore the relationships among the angles and
arcs in a circle.
Relationships to be assessed: 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.

Use basic construction tools for circles, lines,
and segments.

Use calculators for area and size computations.
Attend to precision.

Communicate clearly about different geometric
figures and their parts.

Use appropriate precision and accuracy with
calculations (especially radian measure) and
constructions.
Look for and make use of structure.

Recognize patterns when constructing different
inscribed and circumscribed figures.

Compare previous similarity concepts to
similarity in circles.
Time Permitting: Tangent-Chord, tangent secant,
two secant, two chord, and two tangent line and
angle relationships.
*Time permitting items should not be assessed on
mid-term and final exams.
Essential questions

What is the relationship between central angles
and inscribed angles in a circle?

What can you conclude about the angles of a
quadrilateral inscribed in a triangle?

How do you construct a circle that
circumscribes a triangle?

What are the key theorems about tangent lines
to a circle?

How do you inscribe a regular polygon in a
circle?

How do you inscribe a circle in a triangle?
Warwick Public Schools, in collaboration with
the Charles A. Dana Center at the University of Texas at Austin
C-25
Geometry, Quarter 3, Unit 3.1
Circles (10 days)
Written Curriculum
Common Core State Standards for Mathematical Content
Congruence
G-CO
Experiment with transformations in the plane
G-CO.1
Know precise definitions of angle, circle, perpendicular line, parallel line, and line segment,
based on the undefined notions of point, line, distance along a line, and distance around a
circular arc.
Make geometric constructions
G-CO.13 Construct an equilateral triangle, a square, and a regular hexagon inscribed in a circle.
Circles
G-C
Understand and apply theorems about circles
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.
Common Core State Standards for Mathematical Practice
5
Use appropriate tools strategically.
Mathematically proficient students consider the available tools when solving a mathematical problem.
These tools might include pencil and paper, concrete models, a ruler, a protractor, a calculator, a
spreadsheet, a computer algebra system, a statistical package, or dynamic geometry software. Proficient
students are sufficiently familiar with tools appropriate for their grade or course to make sound decisions
about when each of these tools might be helpful, recognizing both the insight to be gained and their
limitations. For example, mathematically proficient high school students analyze graphs of functions and
solutions generated using a graphing calculator. They detect possible errors by strategically using
estimation and other mathematical knowledge. When making mathematical models, they know that
technology can enable them to visualize the results of varying assumptions, explore consequences, and
compare predictions with data. Mathematically proficient students at various grade levels are able to
identify relevant external mathematical resources, such as digital content located on a website, and use
them to pose or solve problems. They are able to use technological tools to explore and deepen their
understanding of concepts.
Warwick Public Schools, in collaboration with
the Charles A. Dana Center at the University of Texas at Austin
C-26
Geometry, Quarter 3, Unit 3.1
6
Circles (10 days)
Attend to precision.
Mathematically proficient students try to communicate precisely to others. They try to use clear
definitions in discussion with others and in their own reasoning. They state the meaning of the symbols
they choose, including using the equal sign consistently and appropriately. They are careful about
specifying units of measure, and labeling axes to clarify the correspondence with quantities in a problem.
They calculate accurately and efficiently, express numerical answers with a degree of precision
appropriate for the problem context. In the elementary grades, students give carefully formulated
explanations to each other. By the time they reach high school they have learned to examine claims and
make explicit use of definitions.
7
Look for and make use of structure.
Mathematically proficient students look closely to discern a pattern or structure. Young students, for
example, might notice that three and seven more is the same amount as seven and three more, or they may
sort a collection of shapes according to how many sides the shapes have. Later, students will see 7 × 8
equals the well remembered 7 × 5 + 7 × 3, in preparation for learning about the distributive property. In
the expression x2 + 9x + 14, older students can see the 14 as 2 × 7 and the 9 as 2 + 7. They recognize the
significance of an existing line in a geometric figure and can use the strategy of drawing an auxiliary line
for solving problems. They also can step back for an overview and shift perspective. They can see
complicated things, such as some algebraic expressions, as single objects or as being composed of several
objects. For example, they can see 5 – 3(x – y)2 as 5 minus a positive number times a square and use that
to realize that its value cannot be more than 5 for any real numbers x and y.
Clarifying the Standards
Prior Learning
In grade 2, students partitioned circles into two, three, and four equal shares. In grade 7, students knew
and used formulas for area and circumference of a circle to solve problems.
Current Learning
Students construct an equilateral triangle, square, and regular hexagon inscribed in a circle. They identify
and describe relationships among inscribed angles, radii, and chords. (Relationships include the
following: the relationship between central, inscribed, and circumscribed angles; inscribed angles on a
diameter are right angles; and the radius of a circle is perpendicular to the tangent where the radius
intersects the circle.) Students construct the inscribed and circumscribed circles of a triangle and prove
properties of angles for a quadrilateral inscribed in a circle. They construct a tangent line from a point
outside a given circle to the circle.
Future Learning
Students will use this knowledge in Algebra 2 (with regards to the unit circle and trigonometric
functions), fourth-year courses, and college mathematics courses. Students will also use this knowledge in
artwork, engineering, education, construction, and computer science applications.
Warwick Public Schools, in collaboration with
the Charles A. Dana Center at the University of Texas at Austin
C-27
Geometry, Quarter 3, Unit 3.1
Circles (10 days)
Additional Findings
Atlas of Science Literacy, Volume One states, “A typical student belief about mathematical inquiry is that
formal proof is irrelevant to processes of discovery and invention. This type of belief limits a student’s
mathematical behavior.” (p. 26)
Warwick Public Schools, in collaboration with
the Charles A. Dana Center at the University of Texas at Austin
C-28