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.
Pacing: 16 instructional days (14 lessons, 1 flex day, 1 assessment day)
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 |
Materials
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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
1
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.
2
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.
3
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.
Topic B: Angle and Segment Relationships in Inscribed and Circumscribed Figures
4
G.C.A.2
Define a chord to derive the Chord Central Angles Conjecture and Thales’ Theorem.
5
G.C.A.2
Describe the relationship between inscribed and central angles in terms of their intercepted arc.
6
G.C.A.2
Determine the angle and length relationships between intersecting chords.
7
G.C.A.3
Prove properties of angles in a quadrilateral inscribed in a circle.
8
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.
9
G.C.A.2
G.C.A.4
Construct tangent lines to a circle to define and describe the circumscribed angle.
10
G.C.A.2
Use angle and side length relationships with chords, tangents, inscribed angles, and circumscribed angles to solve problems.
Topic C: Arc Length, Radians, and Sector Area
11
G.C.B.5
Define, describe, and calculate arc length.
12
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.
13
G.C.B.5
Calculate the sector area of a circle. Identify relationships between sector area, arc angle, and radius.
14
N.Q.A.2
N.Q.A.3
G.C.B.5
Use sector area of circles to calculate the composite area of figures.
Common Core Standards
Key: Major Cluster Supporting Cluster Additional Cluster
Core Standards
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Circles
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G.C.A.1 — Prove that all circles are similar.
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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.
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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.
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G.C.A.4 — Construct a tangent line from a point outside a given circle to the circle.
Congruence
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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
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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.
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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
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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
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N.Q.A.2 — Define appropriate quantities for the purpose of descriptive modeling.
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N.Q.A.3 — Choose a level of accuracy appropriate to limitations on measurement when reporting quantities.
Foundational Standards
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Congruence
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G.CO.B.8
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G.CO.D.13
Expressing Geometric Properties with Equations
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G.GPE.B.7
Geometry
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7.G.B.4
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7.G.B.5
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8.G.B.7
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8.G.B.8
Ratios and Proportional Relationships
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7.RP.A.2
Reasoning with Equations and Inequalities
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A.REI.B.4.A
Seeing Structure in Expressions
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A.SSE.B.3.B
Similarity, Right Triangles, and Trigonometry
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G.SRT.A.2
Future Standards
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Trigonometric Functions
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F.TF.A.1
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F.TF.A.2
Standards for Mathematical Practice
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CCSS.MATH.PRACTICE.MP1 — Make sense of problems and persevere in solving them.
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CCSS.MATH.PRACTICE.MP2 — Reason abstractly and quantitatively.
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CCSS.MATH.PRACTICE.MP3 — Construct viable arguments and critique the reasoning of others.
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CCSS.MATH.PRACTICE.MP4 — Model with mathematics.
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CCSS.MATH.PRACTICE.MP5 — Use appropriate tools strategically.
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CCSS.MATH.PRACTICE.MP6 — Attend to precision.
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CCSS.MATH.PRACTICE.MP7 — Look for and make use of structure.
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CCSS.MATH.PRACTICE.MP8 — Look for and express regularity in repeated reasoning.