G.C.1.2
G.C.1.2 Use the distance and midpoint formula, where appropriate, to recognize and write the radius r, center (h,k), and standard form of the equation of a circle (x − ℎ)^{2}+ (y − k)^{2} = r^{2} with and without graphs.
In a Nutshell
Students will state the value of the radius and the coordinates for the center of a circle given either an equation of a circle or a graph. Students will write the equation of a circle in standard form given any of the following combinations: the radius and the center, two ordered pairs representing a center and radius or endpoints of a diameter, or the graph of a circle in the coordinate plane.
Student Actions

Teacher Actions


Develop accurate and appropriate procedural fluency by accurately representing the equation of a circle when provided various situations involving properties of the circle (coordinates for a radius, center, and/or diameter) and using efficient procedures with the distance formula and midpoint formulas when necessary.

Develop the ability to communicate mathematically by appropriately interpreting situations involving equations of circles and using consistent mathematical terminology when describing reasoning of strategies, which could include the distance and midpoint formulas, in both verbal and written explanations with teachers and peers.

Develop strategies for problemsolving when analyzing tasks to determine the most appropriate strategy in modeling the equation of a circle and assessing the reasonableness of a solution in context.


Use and connect the mathematical representations between the graph of a circle in the coordinate plane and the appropriate algebraic equation using various mathematical situations

Pose purposeful questions to support students in creating connections between different representations of circles.

Implement tasks that promote reasoning and problemsolving by allowing students to analyze various representations of a circle, both graphical and numerical properties, as well as develop strategies for modeling equations in different contexts.

Key Understandings

Misconceptions


When presented with the equation of a circle written in standard form, the center is represented by the values of (h,k), and the radius is found by taking the square root of the value following the equal sign.

The midpoint formula is used to locate the center point of the circle using the two coordinate pairs for the endpoints of a diameter.

The length of the radius is calculated using the distance formula when presented with the coordinate pair representing the center of the circle and another coordinate pair for a point on the circle in order to write the equation of the circle in standard form.

The equation of a circle can be derived from the Pythagorean Theorem.

Any distances that connect the center of a circle to any point on the circle are equivalent to each other. Any one of those distances represents the length of a radius of the circle.


Students misinterpret the minus signs inside the parentheses in the equation of a circle and model the incorrect coordinate to represent the center of the circle. For example, (x + 4)^{2} +(y  1)^{2} = 9, and the center the student shows is (4,1).

Students incorrectly represent the center of a circle when writing the standard form of the equation of a circle. For example, the center of a circle is (4, 1) and the equation is written as (x + 4)^{2} + (y  1) = 9.

Students forget to take the square root or divide the squared value by 2 when finding the length of the radius of the circle from the equation.

Students forget to square the radius value or multiply the radius value by 2 when writing the standard form of the equation of a circle.

Students think circles in the 3rd quadrant have a negative radius because x and y are both negative.

Students may confuse diameter and radius and forget to take half of the diameter value when determining the length of the radius.

Students may not recognize when it is necessary to use the midpoint formula to locate the center or the distance formula to determine the length of the radius of a circle.

Knowledge Connections

Prior Knowledge

Leads to


Write square roots and cube roots of constants and monomial algebraic expressions in simplest radical form. (A1.N.1.1)

Write the equation of the line given its slope and yintercept, slope and one point, two points, x and yintercepts, or a set of data points. (A1.A.4.3)

Apply the distance formula to solve problems. (G.RT.1.1)


Model realworld situations which involve conic sections. (PC.CS.1.1)

Identify key features of conic sections (foci, directrix, radii, axes, asymptotes, center) graphically and algebraically. (PC.CS.1.2)

Sketch a graph of a conic section using its key features. (PC.CS.1.3)

Write the equation of a conic section given its key features. (PC.CS.1.4)

Given an equation determine if the equation represents a circle, ellipse, parabola, or hyperbola. (PC.CS.1.5)

OKMath Framework Introduction
Geometry Grade Introduction
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