Unit 7
Circles
Unit Narrative
In this unit, students investigate the geometry of circles more closely. In grade 7, students used formulas for the area and circumference of a circle to solve problems. In a previous course, students made formal geometric constructions. Earlier in this course, students studied similarity and proportional reasoning, and proved theorems about lines and angles. This unit builds on these skills and concepts. In later courses, the concepts learned in this unit will be helpful as students connect the unit circle to trigonometric functions.
First, students make connections between prior work with distance in the coordinate plane and the definition of a circle to create a generalized equation for a circle in the coordinate plane:
Then, students define the terms “chord,” “arc,” and “central angle” before observing that inscribed angles are half the measure of their associated central angles, and writing related proofs about congruent chords and similar triangles. Throughout this unit, students also construct lines tangent to circles and use their proofs that a tangent line is perpendicular to the radius drawn to the point of tangency.
Next, students prove properties of cyclic quadrilaterals, and they use their understanding of perpendicular bisectors from a previous unit to construct triangles with circumscribed circles and define “circumcenter.” Students then use angle bisectors to construct incenters of triangles and circles inscribed in triangles.
In the next section, students develop methods for calculating sector areas and arc lengths, and then students define “radian measure of a central angle” as the quotient of the length of the arc defined by the angle and the radius of the circle. They develop fluency with radian measures by shading portions of circles and working with a double number line.
In the final lesson, students apply what they have learned about circles to solve problems in context.
In this unit, students will do several constructions. A particular choice of construction tools is not necessary. Paper folding and straightedge and compass moves are both acceptable methods.
Students will continue to use and add to their reference charts. The completed reference chart for this unit is provided for teacher reference.