Download Circles, Part 2 - Providence Public Schools

Geometry, Quarter 4, Unit 4.3
Circles, Part 2
Overview
Number of instruction days:
8–10
(1 day = 53 minutes)
Content to Be Learned
Mathematical Practices to Be Integrated

Use a variety of methods of proof, both formal
and informal, to justify conjectures about
properties and attributes of circles and their
component parts.
2 Reason abstractly and quantitatively.

Use a diagram of a circle as an aid to solve
problems with other circles.

Construct inscribed and circumscribed
triangles and tangents to a circle.

Create representations of angles and arcs and
their measures.

Use similarity to prove that the length of the
arc intercepted by an angle is proportional to
the radius, and define the radian measure of
the angle as the constant of proportionality.

Attend to the meaning of the radian as a unit
of measure.

Derive the formula for the area of a sector and
apply it to real-world problems.
3 Construct viable arguments and critique the
reasoning of others.

Use relationships among lines and angles in
circles to justify conclusions, communicate to
others, and respond to the arguments of
others.

Critique the arguments of others by comparing
the effectiveness of two plausible arguments,
distinguishing correct logic or reasoning from
that which is flawed, and, if there is a flaw in
an argument, explain what it is.
5 Use appropriate tools strategically.

Providence Public Schools
Use technology, patty paper, and compasses
for constructions and to solve real-world
problems.
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Essential Questions

How can you justify conjectures about
properties and attributes of circles and their
component parts?

What kind of real-world problems can you
solve using the formula for the area of a
tangent?

How does our knowledge of circles help us in
determining measurement of angles, arcs, and
lines relating to circles?

What are the relationships among the angles
and lines or line segments that can be drawn
in or about a circle, and what are their
implications?

What are the relationships between inscribed
and circumscribed polygons and circles?

Why is the concept of proof important in
geometry involving relationships in circles?

What is the difference between the length of
an arc and the measure of an arc?
Standards
Common Core State Standards for Mathematical Content
Geometry
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.
Find arc lengths and areas of sectors of circles [Radian introduced only as unit of measure]
G-C.5
Derive using similarity the fact that the length of the arc intercepted by an angle is
proportional to the radius, and define the radian measure of the angle as the constant of
proportionality; derive the formula for the area of a sector.
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Common Core State Standards for Mathematical Practice
2
Reason abstractly and quantitatively.
Mathematically proficient students make sense of quantities and their relationships in problem
situations. They bring two complementary abilities to bear on problems involving quantitative
relationships: the ability to decontextualize—to abstract a given situation and represent it symbolically
and manipulate the representing symbols as if they have a life of their own, without necessarily
attending to their referents—and the ability to contextualize, to pause as needed during the
manipulation process in order to probe into the referents for the symbols involved. Quantitative
reasoning entails habits of creating a coherent representation of the problem at hand; considering the
units involved; attending to the meaning of quantities, not just how to compute them; and knowing and
flexibly using different properties of operations and objects.
3
Construct viable arguments and critique the reasoning of others.
Mathematically proficient students understand and use stated assumptions, definitions, and previously
established results in constructing arguments. They make conjectures and build a logical progression of
statements to explore the truth of their conjectures. They are able to analyze situations by breaking
them into cases, and can recognize and use counterexamples. They justify their conclusions,
communicate them to others, and respond to the arguments of others. They reason inductively about
data, making plausible arguments that take into account the context from which the data arose.
Mathematically proficient students are also able to compare the effectiveness of two plausible
arguments, distinguish correct logic or reasoning from that which is flawed, and—if there is a flaw in an
argument—explain what it is. Elementary students can construct arguments using concrete referents
such as objects, drawings, diagrams, and actions. Such arguments can make sense and be correct, even
though they are not generalized or made formal until later grades. Later, students learn to determine
domains to which an argument applies. Students at all grades can listen or read the arguments of
others, decide whether they make sense, and ask useful questions to clarify or improve the arguments.
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
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website, and use them to pose or solve problems. They are able to use technological tools to explore
and deepen their understanding of concepts.
Clarifying the Standards
Prior Learning
Students begin in kindergarten with the identification and description of shapes, including circles.
Students develop the concept of fractional degree measurement in a circle in fourth grade. In fifth
grade, students extend this knowledge to the coordinate plane. In Grade 8, students experimented with
rigid motion through transformations that preserved distance and angle measures, and they described
the effect of dilations on two-dimensional figures using coordinates. Conic circles are studied in Algebra
I.
Current Learning
In this unit, students justify conjectures about properties and attributes of circles and their component
parts. They construct figures related to circles, such as inscribed and circumscribed triangles as well as
tangents to circles. Use of construction tools is a fluency in Geometry. Work with circles is a critical area
in the CCSS at this level. This is the first time students are introduced to radians as a unit of measure.
Students prove relationships among arcs and angles. They solve real-world problems using the formula
for area of a sector. All standards addressed in this unit are classified as additional content by the PARCC
Model Frameworks for Mathematics.
Future Learning
In Algebra II, basic circle concepts will be extended to the unit circle. The concept of circles will be
extended to hyperbolas and parabolas in Precalculus. Careers such as graphic design, science,
automotive technology, plumbing, HVAC technology, and engineering use these concepts.
Additional Findings
Understanding the process of proofs is challenging for students. Research shows “Alternative
approaches, helping students expand their understanding of the nature of proof, may be more
successful (Driscoll, 1983). Such approaches may include cooperative investigations in which students
make conjectures and resolve conflicts by presenting arguments and evidence, prove nonobvious
statements, and formulate hypotheses to prove (Fawcett, 1938; Human & Nel, 1989). (A Research
Companion to Principles and Standards for School Mathematics, p. 168)
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Assessment
When constructing an end-of-unit assessment, be aware that the assessment should measure your
students’ understanding of the big ideas indicated within the standards. The CCSS for Mathematical
Content and the CCSS for Mathematical Practice should be considered when designing assessments.
Standards-based mathematics assessment items should vary in difficulty, content, and type. The
assessment should comprise a mix of items, which could include multiple choice items, short and
extended response items, and performance-based tasks. When creating your assessment, you should be
mindful when an item could be differentiated to address the needs of students in your class.
The mathematical concepts below are not a prioritized list of assessment items, and your assessment is
not limited to these concepts. However, care should be given to assess the skills the students have
developed within this unit. The assessment should provide you with credible evidence as to your
students’ attainment of the mathematics within the unit.

Use definitions, properties, and theorems to identify and describe relationships among:
inscribed angles,
circumscribed angles,
radii,
chords,
central angles.

Use definitions, properties, and theorems to prove properties of angles for a quadrilateral inscribed
in a circle.

Make the following constructions:
inscribed and circumscribed circles of a triangle.

Find the arc length of a circle.

Derive and apply the formula for area of a sector.

Identify and apply the radian measure of the angle as the constant of proportionality.
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Instruction
Learning Objectives
Students will be able to:

Identify key parts of circles (including central angles, major arcs, minor arcs, and semicircles) and
determine their angular measures to solve problems.

Determine arc lengths and compare arc lengths to arc measures in problem situations.

Investigate how formulas for circumference and area of a circle relate to finding the area of a sector
and define radian measure.

Investigate and apply relationships between arcs and chords to solve problems.

Use a variety of methods of proof, both informal and formal, to verify conjectures about
relationships between arcs and chords.

Investigate and apply properties of inscribed angles in circles, including properties of inscribed
polygons.

Use a variety of methods of proof, both informal and formal, to verify conjectures about inscribed
angles and polygons.

Review and demonstrate knowledge of important concepts and procedures related to circles.
Resources
Geometry, Glencoe McGraw-Hill, 2010, Student/Teacher Editions

Sections 10-2 through 10-5(pp. 692 - 725)

Section 11-3 (pp. 782 – 788)

http://connected.mcgraw-hill.com/connected/login.do: Glencoe McGraw-Hill Online

Chapter 10 Resource Masters (pp. 11 – 35)

Teaching with Manipulatives
Using Overhead Manipulatives – Locating the Center of a Circle (p. 138)
Using Overhead Manipulatives – Investigating Inscribed Angles (p. 140)
Geometry Lab – Inscribed Angles (pp. 141 - 143)
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Using Overhead Manipulatives – Constructing a Circle to Inscribe a Triangle (p. 144)
Using Overhead Manipulatives – Inscribing a Circle in a Triangle (p. 147)

Interactive Classroom CD (PowerPoint Presentations)

Teacher Works CD-ROM
Exam View Assessment Suite
Algebra 2, Glencoe McGraw-Hill, 2010, Student/Teacher Editions (see the Supplementary Materials
Section of this binder)

Section 13-2 (pp. 819 – 820)
Note: The district resources may contain content that goes beyond the standards addressed in this unit. See the
Planning for Effective Instructional Design and Delivery section below for specific recommendations.
Materials
Compasses, rulers with centimeters and inches, patty paper, dynamic software, TI-Nspire graphing
calculator, protractors, scissors, straightedge (or index cards) for construction, white paper, white
boards with coordinate grid (optional), dry erase markers (optional), string
Instructional Considerations
Key Vocabulary
intercepted arc
arc measure
arc length
radian
tangent
Planning for Effective Instructional Design and Delivery
Reinforced vocabulary taught in previous grades or units: inscribed, circumscribed, construction, circle,
radius, diameter, circumference, equilateral, quadrilateral, triangle, chord, and similarity.
In this unit, students investigate and apply many theorems that relate to circles and lines within and
about circles. Because it is a long unit, use a variety of strategies to maintain student interest and
involvement, such as the ones suggested here and in the resource materials.
A kinesthetic model of a circle can be used as a cue (another cues, questions, and advance organizers
tool) and to help students organize the information in this chapter. The Central Angles and Arcs student
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worksheet later in this unit can serve as such an activity to accompany Section 10-2. Ask students
guiding questions as they work. Consider developing similar activities with cardboard circles for other
concepts and theorems in this unit.
The Teaching with Manipulatives activities provided in the resource section also provide additional
kinesthetic activities for enriching student’s experiences with circle. Students work cooperatively in
small groups to investigate the geometric concepts.
Refer to Section 13-2 in Algebra 2 to define a radian. Have students concretely arrive at an
understanding of 1 radian by using the following nonlinguistic representation strategy, which is a
kinesthetic representation:
1.
Trace circles of varying sizes.
2.
Fold the circles into quadrants to locate the centers.
3.
Using string, determine the diameter of one of the circles and cut the string to that length.
4.
Fold the string in half and cut for the radius.
5.
Select a point on the circle at which to begin and lay the “radius” along the circle.
Sketching radii to the beginning point on the circle and to the end point of the “radius” identifies the
o
degree measure of 1 radian (approximately 57.3° or exactly 180 ). If students continue this “radius

string,” laying it completely around the circle and marking off as they go, they should be able to see six
complete radians with a bit left over (approximately 0.28 “radius string”). Hence, there are
approximately 6.28 radians—or exactly 2 radians—in a circle, no matter what the radius. This discovery
activity is essential to student understanding of the connection between degree measure and radian
measure. Extend this understanding by applying the concept of similarity and the proportionality of the
length of an intercepted arc and the radius.
The proofs of Theorem 10.2 in Section 10-3 and Example 3 on page 711 in Section 10-4 provide great
opportunities to explore simple, formal proofs.
Based on Chapter 10 Resource Masters, page 21 problem 3, students could be given a piece of a paper
plate to determine the plate’s original size, radius, and circumference. This provides students with an
opportunity to show what they know about the relationships between these measures.
Students have learned about x-intercepts and y-intercepts in previous courses. In Section 10-4, they are
introduced to the concept of intercepted arc. Be sure that students understand the general concept of
the term intercept so that they will be able to access its meaning in this context.
Many of the properties of circles can be expressed using algebraic expressions. Students can use
technology, such as dynamic geometry software, to collect data about the properties of segment lengths
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and angle measures in a circle and then use that data to generate an algebraic expression describing the
property. Alternatively, students can use compasses and straightedges to construct the parts of the
circle being investigated and then use rulers and protractors to measure segment lengths and angles.
From their data, students can inductively generate the relationships among the different measures.
The Lines and Segments That Interact with Circles handout later in this unit is an excellent tool for
students to use to summarize their learning prior to an end-of-unit assessment or activity.
Use a modified version of problem 49 on page 715 as an extended application assignment. Have
students create a logo (for the school, your class, a school team, advisory group, etc.) as indicated in the
problem, then have them investigate properties of their logo related to the content of this unit.
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Central Angles and Arcs Record Sheet (accompanies Section 10-2)
You will investigate the relationships among central angles, arc measure, and arc length. An arc
encompasses the same number of degrees as the central angle whose endpoints define the arc. The arc
length is part of the circumference.
For this activity, you will use a cardboard circle, Wikki Stix, a metal fastener, a ruler, and a protractor.
1.
Place a Wikki Stix across your circle so that it forms a diameter. Make sure it goes across the center
of the circle. Use a metal fastener to attach the stick to the circle at the center point.
2.
Measure the diameter of the circle. Write the formula for circumference below and calculate the
circumference of the circle.
Formula for circumference: C =
Diameter of the circle =
Circumference of the circle =
3.
Think of the diameter as two radii. Move one radius to form a central angle and draw a mark to
indicate the angle. Form each of the following angles: 45º, 120º, 60º.
4.
Refer to the Key Concept on page 695 of your textbook to find the measure of the arc inscribed by
each of your angles.
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Lines and Segments That Interact with Circles
How are chords, diameters, radii, secants, and tangents similar?
How are they different?
The first column in the table lists characteristics of lines and segments and their relationship to circles.
Think about how each characteristic relates to each type of line or segment and fill in each blank space
with Always, Sometimes, or Never.
tangent
chord
diameter
radius
secant
Includes the
center of a
circle
Intersects the
circle at exactly
one point
Has points
outside the
circle
Has one or
more endpoint
on the circle
Has exactly
one endpoint
on the circle
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Notes
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