Unit 7: 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.
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)
This assessment accompanies Unit 7 and should be
given on the suggested assessment day or after completing the
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The central mathematical concepts that students will come to understand in this unit
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Topic A: Equations of Circles
Derive the equation of a circle using the Pythagorean Theorem where the center of the circle is at the origin.
Given a circle with a center translated from the origin, write the equation of the circle and describe its features.
Write an equation for a circle in standard form by completing the square. Describe the transformations of a circle.
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Topic B: Angle and Segment Relationships in Inscribed and Circumscribed Figures
Define a chord to derive the Chord Central Angles Conjecture and Thales’ Theorem.
Describe the relationship between inscribed and central angles in terms of their intercepted arc.
Determine the angle and length relationships between intersecting chords.
Prove properties of angles in a quadrilateral inscribed in a circle.
Define and determine properties of tangents and secants of circles to solve problems with inscribed and circumscribed triangles.
Construct tangent lines to a circle to define and describe the circumscribed angle.
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
Define, describe, and calculate arc length.
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.
Calculate the sector area of a circle. Identify relationships between sector area, arc angle, and radius.
Use sector area of circles to calculate the composite area of figures.
The content standards covered in this unit
— Prove that all circles are similar.
— 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.
— Construct the inscribed and circumscribed circles of a triangle, and prove properties of angles for a quadrilateral inscribed in a circle.
— Construct a tangent line from a point outside a given circle to the circle.
— 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.
— 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.
— 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).
— 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.
— Define appropriate quantities for the purpose of descriptive modeling.
— Choose a level of accuracy appropriate to limitations on measurement when reporting quantities.
Standards covered in previous units or grades that are important background for the current unit
— Explain how the criteria for triangle congruence (ASA, SAS, and SSS) follow from the definition of congruence in terms of rigid motions.
— Construct an equilateral triangle, a square, and a regular hexagon inscribed in a circle.
— Use coordinates to compute perimeters of polygons and areas of triangles and rectangles, e.g., using the distance formula.
Modeling is best interpreted not as a collection of isolated topics but in relation to other standards. Making mathematical models is a Standard for Mathematical Practice, and specific modeling standards appear throughout the high school standards indicated by a star symbol (★). The star symbol sometimes appears on the heading for a group of standards; in that case, it should be understood to apply to all standards in that group.
— Know the formulas for the area and circumference of a circle and use them to solve problems; give an informal derivation of the relationship between the circumference and area of a circle.
— Use facts about supplementary, complementary, vertical, and adjacent angles in a multi-step problem to write and solve simple equations for an unknown angle in a figure.
— Apply the Pythagorean Theorem to determine unknown side lengths in right triangles in real-world and mathematical problems in two and three dimensions.
— Apply the Pythagorean Theorem to find the distance between two points in a coordinate system.
— Recognize and represent proportional relationships between quantities.
— Use the method of completing the square to transform any quadratic equation in x into an equation of the form (x - p)² = q that has the same solutions. Derive the quadratic formula from this form.
— Complete the square in a quadratic expression to reveal the maximum or minimum value of the function it defines.
— Given two figures, use the definition of similarity in terms of similarity transformations to decide if they are similar; explain using similarity transformations the meaning of similarity for triangles as the equality of all corresponding pairs of angles and the proportionality of all corresponding pairs of sides.
Standards in future grades or units that connect to the content in this unit
— Understand radian measure of an angle as the length of the arc on the unit circle subtended by the angle.
— Explain how the unit circle in the coordinate plane enables the extension of trigonometric functions to all real numbers, interpreted as radian measures of angles traversed counterclockwise around the unit circle.
— Make sense of problems and persevere in solving them.
— Reason abstractly and quantitatively.
— Construct viable arguments and critique the reasoning of others.
— Model with mathematics.
— Use appropriate tools strategically.
— Attend to precision.
— Look for and make use of structure.
— Look for and express regularity in repeated reasoning.
Three-Dimensional Measurement and Application