Circles

Students expand their knowledge of circles to establish relationships between angle measures in and around circles, line segments and lines in and around circles, and portions of circles as related to area and circumference.

Unit Summary

In Unit 7, Circles, students expand their knowledge of circles from middle school to establish relationships between angle measures in and around circles, line segments and lines in and around circles, and portions of circles as related to area and circumference. 

This unit begins with Topic A, Equations of Circles, where students make an algebraic connection to geometry by writing equations for circles and understanding how to graph and derive features of a circle from an equation. This understanding can be used to review the criteria for perpendicular lines as well as algebraic quadratic concepts such as completing the square. In Topic B, Angle and Segment Relationships in Inscribed and Circumscribed Figures, students use inscribed, circumscribed, and central angles to develop an understanding of the relationship of angle measures in and around a circle. In addition, students will develop theorems related to chords that build a basis for relationships about line segments in and around a circle. Finally, in Topic C, Arc Length, Radians, and Sector Area, students build on their understanding of area and circumference of a circle to determine lengths and areas of sectors and discover proportional and congruent relationships related to area and lengths of arcs in circles. Students will also be exposed to the idea of radians and will derive a radian. 

In Algebra 2, students will use their understanding of radians and unit circle relationships to further explore trigonometric relationships. In addition, the understandings developed in this unit of circles will carry into conic sections and tangent relationships, which is studied in AP Calculus.

Assessment

This assessment accompanies Unit 7 and should be given on the suggested assessment day or after completing the unit.

Unit Prep

Intellectual Prep

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Internalization of Standards via the Unit Assessment

  • Take unit assessment. Annotate for: 
    • Standards that each question aligns to
    • Purpose of each question: spiral, foundational, mastery, developing
    • Strategies and representations used in daily lessons
    • Relationship to Essential Understandings of unit 
    • Lesson(s) that assessment points to

Internalization of Trajectory of Unit

  • Read and annotate the Unit Summary.
  • Notice the progression of concepts through the unit using the Lesson Map.
  • Do all target tasks. Annotate the target tasks for: 
    • Essential understandings
    • Connection to assessment questions 

Essential Understandings

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  • The equation of a circle, while not a function, is derived from the Pythagorean Theorem and an understanding of the relationship between the endpoints of the radii. 
  • The property of a circle being the result of a rotation of 360 degrees around a center point gives rise to properties of angles and line segments within and around a circle. These properties of chords, inscribed angles, central angles, and tangents can be used to determine other relationships within a circle relating to polygons. 
  • Arc length and sector area are derived from the understanding of a proportion of a circle that is traced or covered by a sector or arc. These proportional relationships give rise to the concept of a radian, which is a key measure in trigonometric functions.

Vocabulary

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circle radius
diameter equation of a circle in standard form
complete the square equation of a circle in general form
transformation translation
dilation chord
central angle tangent
arc Chord Central Angles Conjecture
Thales's Theorem semicircle
inscribed angle intercepted angle
major arc minor arc
arc measure Intersecting Chords Theorem
inscribed quadrilateral cyclic quadrilateral
secant point of tangency
circumscribed angle arc length
radians sector area

Unit Materials, Representations and Tools

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  • Students will need a compass and straightedge for Lesson 9.
  • Unit 7 Glossary: This glossary includes visuals and descriptions for certain topics mentioned in this unit. We've noted in specific lessons when we think this glossary will be particularly helpful.

Lesson Map

Topic A: Equations of Circles

Topic B: Angle and Segment Relationships in Inscribed and Circumscribed Figures

Topic C: Arc Length, Radians, and Sector Area

Common Core Standards

Key: Major Cluster Supporting Cluster Additional Cluster

Core Standards

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Circles
  • G.C.A.1 — Prove that all circles are similar.

  • G.C.A.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.A.3 — Construct the inscribed and circumscribed circles of a triangle, and prove properties of angles for a quadrilateral inscribed in a circle.

  • G.C.A.4 — Construct a tangent line from a point outside a given circle to the circle.

Congruence
  • G.CO.A.5 — Given a geometric figure and a rotation, reflection, or translation, draw the transformed figure using, e.g., graph paper, tracing paper, or geometry software. Specify a sequence of transformations that will carry a given figure onto another.

Expressing Geometric Properties with Equations
  • G.GPE.A.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.

  • G.GPE.B.4 — Use coordinates to prove simple geometric theorems algebraically. For example, prove or disprove that a figure defined by four given points in the coordinate plane is a rectangle; prove or disprove that the point (1, √3) lies on the circle centered at the origin and containing the point (0, 2).

High School — Geometry
  • G.C.B.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.

High School — Number and Quantity
  • N.Q.A.2 — Define appropriate quantities for the purpose of descriptive modeling.

  • N.Q.A.3 — Choose a level of accuracy appropriate to limitations on measurement when reporting quantities.

Foundational Standards

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Congruence
  • G.CO.B.8

  • G.CO.D.13

Expressing Geometric Properties with Equations
  • G.GPE.B.7

Geometry
  • 7.G.B.4

  • 7.G.B.5

  • 8.G.B.7

  • 8.G.B.8

Ratios and Proportional Relationships
  • 7.RP.A.2

Reasoning with Equations and Inequalities
  • A.REI.B.4.A

Seeing Structure in Expressions
  • A.SSE.B.3.B

Similarity, Right Triangles, and Trigonometry
  • G.SRT.A.2

Future Standards

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Trigonometric Functions
  • F.TF.A.1

  • F.TF.A.2

Standards for Mathematical Practice

  • CCSS.MATH.PRACTICE.MP1 — Make sense of problems and persevere in solving them.

  • CCSS.MATH.PRACTICE.MP2 — Reason abstractly and quantitatively.

  • CCSS.MATH.PRACTICE.MP3 — Construct viable arguments and critique the reasoning of others.

  • CCSS.MATH.PRACTICE.MP4 — Model with mathematics.

  • CCSS.MATH.PRACTICE.MP5 — Use appropriate tools strategically.

  • CCSS.MATH.PRACTICE.MP6 — Attend to precision.

  • CCSS.MATH.PRACTICE.MP7 — Look for and make use of structure.

  • CCSS.MATH.PRACTICE.MP8 — Look for and express regularity in repeated reasoning.