Big Idea:

Big Idea 2

There is a constant proportional relationship between an angle and its arc measures on a circle.

1 week

Evidence of Understanding

  • describe proportional relationships between angles, arcs, and sectors of a circle
    • create visuals to describe a central angle in terms of the fraction of the circle it represents
      • Example: show an angle measuring 72° is equivalent to ⅕ of the circle
    • investigate the constant ratio between an angle’s arc length and a circle’s circumference
      • explain the proportional relationship with the radius
      • use similarity to derive the length of the arc intercepted by an angle
      • use tools to verify the relationship between an angle’s degree and radian measure
    • apply the relationship between degrees and radians to find a missing measure
      • determine and justify the length of 1 radian
    • explain how sectors are related to area of a circle using similarity relationships
      • derive and use the formula for determining the area of a sector
         
  • analyze and apply the relationships between the angles and arcs used to describe a circle
    • use examples and nonexamples to distinguish central, inscribed, and circumscribed angles
      • define and classify angles using radii, chords, and tangents
    • investigate relationships between angle and arc measure for a central, inscribed, or circumscribed angle and use them to solve problems
      • prove inscribed angles open to the same arc are congruent (and vice versa)
      • prove parallel chordsintercept congruent arcs
    • prove that a triangle inscribed in a circle with a leg through the center of the circle is a right triangle (hypotenuse is the diameter)
    • describe patterns relating the angle and arc measures on a circle by two tangents, two secants, a secant and a tangent, or a chord and a tangent
    • describe patterns relating the angle and arc measures resulting from two intersecting chords
    • calculate the value of an unknown angle or arc measure

Develop conceptual understanding:

central angle, arc length, circumference, degrees, radians, sector,  inscribed angle, circumscribed angle

Supporting terms to communicate:

equivalent, ratio, proportion, similarity, scale factor, dilation, angle, radius, diameter, area, chord, tangent, secant, parallel, alternate interior angles, hypotenuse

Core Resource
A core resource supports multiple days of instruction. COMING SOON!
    Instructional Routine: Contemplate then Calculate
    These tasks are embedded within the instructional routine called Contemplate then Calculate.
    Instructional Routine: Connecting Representations
    These tasks are embedded within the instructional routine called Connecting Representations.
    • Arc Length or Sector Area?
      Connect what students know about area and circumference of circles to match to diagrams representing sector areas and arc lengths.
      Resource:
      Arc Length or Sector Area?

      Connect what students know about area and circumference of circles to match to diagrams representing sector areas and arc lengths.

      All Resources From: