Big Idea:

Big Idea 2

A circle is symmetrical and its points are related by a center and radius.

1 week

Evidence of Understanding

  • describe the properties of a circle
    • explore the relationship between a center point and a fixed distance (radius) rotated around it
      • define a circle using the center, radius, and lines of symmetry
    • use tools to inductively show the circumference of a circle is 2πr
    • explain the relationship between the central angle of a circle and its arc measure
      • describe a circle using 360o degrees and 2π  radians or 180o and π radians
         
  • graph the equation of a circle in the coordinate plane
    • given any equation (x - h)2+ (y - k)2 = r2, describe how x, y, h, k, and r impact the graph
    • find the center and the radius of a circle by completing the square
    • given a coordinate point, use the graph or equation toprove if it lies on the circle
    • given a coordinate point, use symmetry to justify other points on the circle
    • use Pythagorean Theorem to identify the radius, center, or points that lie on the circle
      • explain the connection of the equation of a circle and Pythagorean Theorem
         
  • use triangles and trigonometry to describe the graph of a circle
    • given the center, radius and the terminal side of any angle between 0 and 360, find the coordinates of the corresponding point on the circle
      • use the properties of special right triangles (30-60-90 and 45-45-90)
      • describe the relationship between cosine and the x coordinate of a point on the circle
      • describe the relationship between sine and the y coordinate of a point on the circle
    • prove sinθ + cosθ = r2
    • use transformations to describe and justify the relationship between coordinate points that share a reference angle
      • Ex: the points at 30 and 150 are similar because the right triangle is reflected
      • Ex: the points 30 and 120 are similar because the right triangle is rotated   
    • given the center and any point on the circle, use inverse trig functions (sin-1, cos-1, and tan-1) to find the measure of the circle’s corresponding central angle

Develop conceptual understanding:

circle, circumference, arc measure, radian, terminal side, special right triangle (30-60-90 and 45-45-90), cosine, sine, reference angle

Supporting terms to communicate:

center, radius, central angle, degrees, symmetrical, reflect, completing the square, quadrant, Pythagorean Theorem, right triangle, reflection, rotation, inverse trig
Core Resource
A core resource supports multiple days of instruction. COMING SOON!
    Instructional Routine: Connecting Representations
    These tasks are embedded within the instructional routine called Connecting Representations.