Download Common Core Geometry Critical Area 4: Circles

Common Core Geometry
Critical Area 4: Circles
Standards for Mathematical Practice
1. Make sense of problems and persevere in solving them.
2. Reason abstractly and quantitatively.
3. Construct viable arguments and critique the reasoning of others.
4. Model with mathematics.
5. Use appropriate tools strategically.
6. Attend to precision.
7. Look for and make use of structure.
8. Look for and express regularity in repeated reasoning.
Pacing: Weeks 19-22
Circles
Circles
G-C
A. Understand and apply theorems about circles
1. Prove that all circles are similar.
2. Identify and describe relationships among inscribed angles, radii, and chords. Include the relationship between central, inscribed, and circumscribed angles;
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.
3. Construct the inscribed and circumscribed circles of a triangle, and prove properties of angles for a quadrilateral inscribed in a circle.
B. Find arc lengths and areas of sectors of circles
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.
Expressing Geometric Properties with Equations
G-GPE
A. Translate between the geometric description and the equation for a conic section
1. Derive the equation of a circle of given center and radius using the Pythagorean Theorem; complete the square to find the center and radius of a circle given by an
equation.
Key Student Understandings
 Students will calculate circle measurements using unique properties
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and relationships among the parts of a circle.
Students will develop the equation of a circle through the
Pythagorean Theorem.
Instructional Guide for Common Core Geometry
Assessment
Formative Assessment Strategies
Evidence for Standards-based Grading
**Common SBG Evidence Items
Rubric
Circle Up (G-C.A)
Circle Logo (G-C.A, G-C.B, G-GPE.A, SMP.7)
Setting Up Sprinklers (G-C.B, SMP.7)
Property of MPS
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Common Core Geometry
Critical Area 4: Circles
Disciplinary Literacy
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Disciplinary Literacy Framework
Common Misconceptions/Challenges
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Students often confuse vocabulary and formulas.
Students interchange central, inscribed, and circumscribed angles and their relationships.
Students have difficulty with the tools of construction.
Conceptualizing the definition of a radian can be a struggle.
The general equation for a circle can be difficult for students.
Students may struggle to recall how to complete the square in order to find the center and radius of a circle.
Pi is not 3.14
Instructional Practices
Suggested
Timeline
Circle: its parts
and properties
(Weeks 19- 21)
Suggested Learning Experiences
Precisely define, illustrate, and notate
circle vocabulary.
Resources
**Common SBG Evidence Items
Discovering Geometry: Lesson 1.7, Lesson 3.7 (Investigation 2 and 3, pg. 183 #14), Lesson 6.1 (Focus on
Investigation 1 only), Lesson 6.2 (Investigation 1 and 2), Lesson 6.3 (Develop Investigation 4 into a proof),
Lesson 6.4 (pg. 332 #2, 3 and choose one from 5-7), Lesson 6.5 (pg. 339 #15, addresses standard of
applying formulas to design applications), Lesson 6.6, Lesson 6.7 (Attention to Example A), Lesson 8.5,
Lesson 8.6
Engage NY: Circle Practice, Proving Thales’ Theorem
Illuminations: Graphs from the Unit Circle
Additional District Created Items: Circle Up**, Circle Measurements
Illustrative Mathematics: Right Triangles Inscribed in Circles 1 (TE), Inscribing a Circle in a Triangle 1 (TE),
Inscribing a Triangle in a Circle (TE)
Mathematics Assessment Resource Service: Calculating Arcs and Areas of Sectors of Circles
Instructional Guide for Common Core Geometry
Property of MPS
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Rev 7.18.16
Common Core Geometry
Critical Area 4: Circles
Problem Solving
(Weeks 21-22)
Apply circle properties in appropriate
situations
Illustrative Mathematics: Two Wheels and a Belt (TE), Orbiting Satellite (TE)
Additional District Created Items: Circle Logo**, Setting Up Sprinklers**
Mathematics Assessment Resource Service: Sorting Equations of Circles 1
Differentiation
Literacy Connections
Geometry Teacher’s Edition - Differentiated Instruction CK-12 Foundation
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Academic Vocabulary
Have the students work on creating a diagram of a circle. Their diagram
must have the following things labeled: radius, chord, diameter, secant,
tangent line.
Encourage students to make their diagrams colorful.
Allow time for sharing when students are finished.
When writing similarity ratios be sure to simplify.
Define semicircle with a diagram.
Define major arc with a diagram- named with three letters.
Define minor arc with a diagram- named with two letters (endpoints).
Present diagrams first and then engage the students in a discussion about
the diagram.
Begin by defining a chord as a line segment whose endpoints are both on
a circle.
Show students a diagram to define a chord.
Write out each theorem on the board. Request that students write the
notes in their notebooks.
Review the terms diameter, bisector, perpendicular, and congruent.
Vocabulary Strategies
Literacy Strategies
Additional Resources
Discovering Mathematics Online Textbook
Desmos Online Graphing Calculator
cK-12
learnzillion
GeoGebra
Engage NY Geometry Module 5
Georgia Geometry Unit 4
Instructional Guide for Common Core Geometry
Property of MPS
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Rev 7.18.16