given by an equation. G.GPE.4 Use coordinates to prove simple geometric theorems algebraically.; for example prove or disprove that the point (1, √3) lies on the circle centered at the origin and containing the point (0, 2). Note: Include simple proofs involving circles. Ability to connect experience with coordinate proofs from Unit 4 to circles Students can write the equation of the circle being described and then use substitution to demonstrate the coordinate of the given point satisfy the equation of the given circle thus proving that the point lies on the circle. G.MG.1 Use geometric shapes, their measures, and their properties to describe objects. (e.g., modeling a tree trunk or a human torso as a cylinder). ★ Note: Focus on situations in which the analysis of circles is required. Ability to connect experiences from Unit 2 and Unit 3 with two- dimensional and three-dimensional shapes to circles Apply geometric properties of circles to solve modeling problems.
similarity transformations to decide if two figures were similar. In 4th grade, students: learn the concept of angle measure in degrees. G.C.2 Identify and describe relationships among inscribed angles, radii, and chords. In future mathematics courses, students: will use the relationships established by this standard in more complex modeling situations. In 5th grade, students: classify two-dimensional shapes based on their properties. In 7th grade, students: use facts about supplementary and complementary angles to solve problems. G.C.3 Construct the inscribed and circumscribed circles of a triangle, and prove properties of angles for a quadrilateral inscribed in a circle. In future mathematics courses, students: will use the relationships established by this standard in more complex modeling situations. In 4th grade, students: learn the concept of angle measure in degrees. In 7th grade, students: use formulas for the area and circumference of a circle to solve problems. In 8th grade, students: describe a sequence that exhibits the similarity between two similar two-dimensional figures. G.C.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; In Algebra II, students will: understand radian measure of an angle as the length of the arc on the unit circle subtended by the angle.
Common Misconceptions This list includes general misunderstandings and issues that frequently hinder student mastery of concepts regarding the content of this unit. Topic/Standard/Concept Misconception Strategies to Address Misconception Secant and Tangent Lines/G.C.2 Students have trouble distinguishing between a tangent to a circle and a secant line. A line intersecting a circle in two places is referred to as a secant. The portion of the secant contained within the circle is called a chord. If a line intersects a circle at only a single point, it is called a tangent. Central Angles and Inscribed Angles /G.C.2 Students have trouble understanding the difference between a central angle and an inscribed angle. Definitions 1 - A central angle of a circle is an angle whose vertex is located at the center of the circle. Angle BOC in the figure below. 2 - An inscribed angle is an angle whose vertex is on a circle and whose sides each intersect the circle at another point. Angle CAB in the figure below. Clue students in that central angles bend at the center. Try this as a memory device to help students remember the correct definition for central angles. Also help students remember that inscribed angles bend on the circle. Just like all other inscribed figures meet on the circle.
Sample Assessment Items The items included in this component will be aligned to the standards in the unit. Topic Standards Addressed Link Notes Modeling with Circles G.MG.1 This site contains a modeling task related to circles. Modeling with Circles G.MG.1 This site contains a modeling task related to circles. Modeling with Circles G.MG.1 This is a link to Smarter Balance sample task which addresses concepts connected to circles. Circles and coordinates G.GPE.4 This is a link to Smarter Balance sample task which addresses concepts connected to circles. Circles and coordinates G.GPE.4 This is a link to Smarter Balance sample task which addresses concepts connected to circles.