Unit 5

Circles

 

Geometry

Suggested Unit Length and Description:

 

25 days

 

In this unit, students prove basic theorems about circles, with particular attention to perpendicularity and inscribed angles, in order to see symmetry in circles and as an application of triangle congruence criteria. They study relationships among segments on chords, secants, and tangents as an application of similarity. In the Cartesian coordinate system, students use the distance formula to write the equation of a circle when given the radius and the coordinates of its center. Given an equation of a circle, they draw the graph in the coordinate plane, and apply techniques for solving quadratic equations to determine intersections between lines and circles or parabolas and between two circles. (Mathematics Appendix A, p.34)

 

        Understand and apply theorems about circles.

        Find arc lengths and areas of sectors of circles; emphasize similarity of all circles.

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

        Use coordinates to prove simple geometric theorems algebraically.

        Apply geometric concepts in modeling situations; focus on situations which the analysis of circles is required.

 

Standards:

 

G-C.A.1 Prove that all circles are similar.

 

G-C.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.

 

G-C.A.3 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 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 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.

 

G-GPE.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).

 

G-C.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:

G-C.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

G-C.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

G-C.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

G-C.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 (r1 and r2) and the arc lengths (s1 and s2) determined by congruent central angles are proportional, such that r1/s1 = r2/s2

        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

G-GPE.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

G-GPE.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)

G-C.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

 

Enduring Understandings:

 

        The circle is used for wheels, coins, and many other common objects because it is the most symmetric of all two-dimensional figures.

        Representation of geometric ideas and relationships allow multiple approaches to geometric problems and connect geometric interpretations to other contexts.

        Geometric representations in the coordinate plane are a useful way to model various problem situations and physical phenomena.

 

Essential Questions:

 

       How does geometry explain or describe the structure of our world?

       How can you find the amount of trim needed to go around the edge of a circular tablecloth?

       How can the area covered by a windshield be found?

       What relationships exist between the angles formed by segments intersecting a circle?

       What is the relationship between the segments that intersect a circle?