Standards:
GC.A.1 Prove that all circles are similar.
GC.A.2 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.
GC.A.3 Construct the inscribed and circumscribed
circles of a triangle, and prove properties of angles for a quadrilateral
inscribed in a circle.
GC.B.5 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.
GGPE.A.1 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.
GGPE.B.4 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).
GC.A.4 (+) Construct a tangent line from a point
outside a given circle to the
circle.
Focus Standards of
Mathematical Practice:
MP.1 Make sense of problems and persevere in
solving them.
MP.2 Reason abstractly and quantitatively.
MP.3 Construct viable arguments and critique the
reasoning of others.
MP.4 Model with mathematics.
MP.5 Use appropriate tools strategically.
MP.6 Attend to precision.
MP.7 Look for and make use of structure.
MP.8 Look for and express regularity in repeated
reasoning.
Instructional
Outcomes:
Full Development of the Major Clusters,
Supporting Clusters, Additional Clusters and Mathematical Practices for this
unit could include the following instructional outcomes:
GC.A.1
·
I
can recognize when figures are similar. (Two figures are similar if one is
the image of the other under a transformation from the plane into itself that
multiplies all distances by the same positive scale factor, k. That is to
say, one figure is a dilation of the other)
·
I
can compare the ratio of the circumference of a circle to the diameter of the
circle
·
I
can discuss, develop and justify this ratio for several circles
·
I
can determine this ratio is constant for all circles
GC.A.2
·
I
can identify inscribed angles, radii, chords, central angles, circumscribed
angles, diameter, tangent
·
I
can recognize that inscribed angles on a diameter are right angles
·
I
can recognize that radius of a circle is perpendicular to the radius at the
point of tangency
·
I
can examine the relationship between central, inscribed and circumscribed
angles by applying theorems about their measures
GC.A.3
·
I
can define inscribed and circumscribed circles of a triangle
·
I
can recall midpoint and bisector definitions
·
I
can define a point of concurrency
·
I
can prove properties of angles for a quadrilateral inscribed in a circle
·
I
can construct inscribed circles of a triangle
·
I
can construct circumscribed circles of a triangle
GC.B.5
·
I
can recall how to find the area and circumference of a circle
·
I
can explain that 1° = 180 radians
·
I
can recall from G.C.1, that all circles are similar
·
I
can determine the constant of proportionality (scale factor)
·
I
can justify the radii of any two circles (r_{1} and r_{2})
and the arc lengths (s_{1} and s_{2}) determined by congruent
central angles are proportional, such that
r_{1}/s_{1} = r_{2}/s_{2}
·
I
can verify that the constant of a proportion is the same as the radian
measure, Θ, of the given central angle. Conclude s=rΘ
GGPE.A.1
·
I
can define a circle
·
I
can use Pythagorean Theorem
·
I
can complete the square of a quadratic equation
·
I
can use the distance formula
·
I
can derive the equation of a circle using the Pythagorean Theorem – given
coordinates of the center and length of the radius
·
I
can determine the center and radius by completing the square
GGPE.B.4
·
I
can recall previous understandings of coordinate geometry (including, but not
limited to: distance, midpoint and slope formula, equations of a line,
definitions of parallel and perpendicular lines, etc.)
·
I
can use coordinate 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)
GC.A.4(+)
·
I
can recall vocabulary: tangent, radius, perpendicular bisector, midpoint
·
I
can identify the center of the circle
·
I
can synthesize theorems that apply to circles and tangents, such as: tangents
drawn from a common external point are congruent; a radius is perpendicular
to a tangent at the point of tangency
·
I
can construct the perpendicular bisector of the line segment between the
center C to the outside point P
·
I
can construct arcs on circle C from the midpoint Q , having length of CQ
·
I
can construct the tangent line
