Circles

9 Components
Initial Task
Square Peg
Big Ideas
Big Idea 1 Big Idea 2 Big Idea 3
Formative Assessment Lesson
Inscribing and Circumscribing Right Triangles
Re-engagement
Re-engagement for Unit 7
End of Unit Assessments
End of Unit Assessment for Unit 7 End of Unit Assessment Teacher Resources for GEO Unit 7 Find aligned Regents questions with Quiz Banker
Geo U6: Circles
Unit Outline
Initial Task
Square Peg
Big Ideas
Big Idea 1 Big Idea 2 Big Idea 3
Formative Assessment Lesson
Inscribing and Circumscribing Right Triangles
Re-engagement
Re-engagement for Unit 7
End of Unit Assessments
End of Unit Assessment for Unit 7 End of Unit Assessment Teacher Resources for GEO Unit 7 Find aligned Regents questions with Quiz Banker
None
Geo U6:

Circles

How do circles define the relationships among segments, angles, and arcs?
4 weeks
See Standards
Common Core: Major Standards:
G-SRT.B.5

Prove theorems involving similarity. Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Clarification: ASA, SAS, SSS, AAS, and Hypotenuse-Leg theorem are valid criteria for triangle congruence. AA, SAS, and SSS are valid criteria for triangle similarity.

G-SRT.C.8

Define trigonometric ratios and solve problems involving right triangles. Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems.★

G-GPE.B.4

Use coordinates to prove simple geometric theorems algebraically. 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).

G-MG.A.1

Apply geometric concepts in modeling situations. Use geometric shapes, their measures, and their properties to describe objects (e.g., modeling a tree trunk or a human torso as a cylinder).

G-MG.A.3

Apply geometric concepts in modeling situations. Apply geometric methods to solve design problems (e.g., designing an object or structure to satisfy physical constraints or minimize cost; working with typographic grid systems based on ratios).★

Common Core: Supporting Standards:
G-CO.D.12

Make geometric constructions. Make formal geometric constructions with a variety of tools and methods (compass and straightedge, string, reflective devices, paper folding, dynamic geometric software, etc.). Copying a segment; copying an angle; bisecting a segment; bisecting an angle; constructing perpendicular lines, including the perpendicular bisector of a line segment; and constructing a line parallel to a given line through a point not on the line. Clarification: Constructions include but are not limited to the listed constructions. Example: constructing the median of a triangle or constructing an isosceles triangle with given lengths.

G-CO.D.13

Make geometric constructions. Construct an equilateral triangle, a square, and a regular hexagon inscribed in a circle.

Common Core: Additional Standards:
G-C.A.1

Understand and apply theorems about circles. Prove that all circles are similar.

G-C.A.2

Understand and apply theorems about circles. 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.
Clarification: Relationships include but are not limited to the listed relationships. Example: angles involving tangents and secants.

G-C.A.3

Understand and apply theorems about circles. Construct the inscribed and circumscribed circles of a triangle, and prove properties of angles for a quadrilateral inscribed in a circle.

G-C.B.5

Find arc lengths and areas of sectors of circles. 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.

G-GPE.A.1

Translate between the geometric description and the equation for a conic section. 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.

Unit seven is about circles. Students extend their thinking about the Pythagorean theorem and coordinate distance to write the equation of a circle given the center and radius, and graph a circle given its equation. Students examine how central, inscribed, and circumscribed angles relate to chords, radii, and arcs of the circles. The idea of radian measure is introduced through the idea that all circles are similar. Throughout the unit, students use constructions to make visual sense of the parts of a circle (e.g., radius, chords, etc.) and use relationships to solve problems. This unit is a nice culmination of the year and can be used to review concepts from units 1 through 6 as they are related to circle relationships.

Essential Questions:

  • How can we model real life situations?
  • How can we use multiple representations to make sense of the world?
  • How do we measure change?
Printable Unit Plan
  • Geometry Unit Plan for Unit 7
    Resource:
    Geometry Unit Plan for Unit 7

    This is a printable version of the entire unit plan. You can make a copy of it and edit it or download and print it as needed.

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Unit Outline

9 Components

Every unit begins with an Initial Task and ends with a Balanced Assessment, both focusing on core mathematics of the unit. The core mathematics is developed through a series of resources around Big Ideas; as you move through the unit, keep students focused on how these ideas are connected and how they address mathematical problem solving. Before attempting the Balanced Assessment, students have an opportunity to synthesize their knowledge through a multi-day Formative Assessment Lesson and teacher-designed re-engagement lessons.

Initial Task See 1 itemHide 1 item

Big Ideas See 3 itemsHide 3 items

Formative Assessment Lesson See 1 itemHide 1 item

Re-engagement See 1 itemHide 1 item

End of Unit Assessments See 3 itemsHide 3 items