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.
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.
This assessment accompanies Unit 7 and should be given on the suggested assessment day or after completing the unit.
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Internalization of Standards via the Unit Assessment
Internalization of Trajectory of Unit
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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 |
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G.GPE.A.1
G.GPE.B.4
Derive the equation of a circle using the Pythagorean Theorem where the center of the circle is at the origin.
G.CO.A.5
G.C.A.1
G.GPE.A.1
Given a circle with a center translated from the origin, write the equation of the circle and describe its features.
G.CO.A.5
G.GPE.A.1
Write an equation for a circle in standard form by completing the square. Describe the transformations of a circle.
G.C.A.2
Define a chord to derive the Chord Central Angles Conjecture and Thales’ Theorem.
G.C.A.2
Describe the relationship between inscribed and central angles in terms of their intercepted arc.
G.C.A.2
Determine the angle and length relationships between intersecting chords.
G.C.A.3
Prove properties of angles in a quadrilateral inscribed in a circle.
G.C.A.2
G.C.A.3
Define and determine properties of tangents and secants of circles to solve problems with inscribed and circumscribed triangles.
G.C.A.2
G.C.A.4
Construct tangent lines to a circle to define and describe the circumscribed angle.
G.C.A.2
Use angle and side length relationships with chords, tangents, inscribed angles, and circumscribed angles to solve problems.
G.C.B.5
Define, describe, and calculate arc length.
G.C.B.5
Describe the proportional relationship between arc length and the radius of a circle. Convert between degrees and radians to write the arc measure in radians.
G.C.B.5
Calculate the sector area of a circle. Identify relationships between sector area, arc angle, and radius.
N.Q.A.2
N.Q.A.3
G.C.B.5
Use sector area of circles to calculate the composite area of figures.
Key: Major Cluster Supporting Cluster Additional Cluster
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