Big Idea:

Big Idea 1

A circle is uniquely defined in the coordinate plane using its center and radius.

1 week

Evidence of Understanding

  • describe the characteristics of any circle
    • construct a circle using a compass and justify that all radii are equidistant and congruent
    • explain that all points on the circle are equidistant from the center of the circle
    • define radius as a relationship between a circle’s center and points on the circle
      • justify circles are similar using scale factor
      • reason proportionally and connect the measurements of a circle with similarity
    • analyze examples and non-examples to distinguish radii, chords, tangents, and secants
    • use constructions to explain the relationship between the radius and a tangent line
    • construct perpendicular bisectors of chords to show they always pass through the circle’s center
       
  • analyze characteristics of a circle in the coordinate plane
    • graph a circle in the coordinate plane given its center and radius
    • explore whether a point lies on a circle whose radius and center point are known
      • identify and justify possible methods including the use of a compass, straightedge, the Pythagorean Theorem, the distance formula, etc.
    • prove whether a point lies on a circle given the circle’s center and another point on the circle
    • describe characteristics about a circle given its center, radius, points on the circle, etc.
      • Example: find the area of a circle whose endpoints of its diameter lie at (2, 3) and (10, -3), or determine the diameter of a circle centered at the origin and passing through (-5, 10)
         
  • determine the standard equation of a circle in the coordinate plane
    • describe connections between a circle’s equation and and its graph in the coordinate plane
    • generate the equation of a circle from its graph
    • derive the equation of a circle from the Pythagorean Theorem
      • describe the relationship between x, h, y, k, and the Pythagorean Theorem
    • generate a circle’s equation given information about its center, radius, or points on the circle
    • apply completing the square to write the equation of a circle in standard form

Develop conceptual understanding:

circle, radius/radii, equidistant, center of the circle, points on the circle, equation of a circle

Supporting terms to communicate:

similar, scale factor, coordinates, Pythagorean Theorem, completing the square

Core Resource

A core resource supports multiple days of instruction.

  • Defining Circles
    This incomplete Core Resource supports students in making connections between the Pythagorean theorem and the equation of circles.
    Resource:
    Defining Circles

    This incomplete Core Resource supports students in making connections between the Pythagorean theorem and the equation of circles.

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Instructional Routine: Sharing Skepticism

The primary goal of this instructional routine is to support students in constructing and critiquing mathematical arguments.

  • Is It On the Circle?
    Students use what they know about finding distances in a coordinate plane and right triangles to prove that a point lies on a circle through another point.
    Resource:
    Is It On the Circle?

    Students use what they know about finding distances in a coordinate plane and right triangles to prove that a point lies on a circle through another point.

    All Resources From: