Student Actions

Teacher Actions

• Develop accurate and appropriate procedural fluency when using efficient procedures and accuracy in operations to determine a measure from the relationship between arcs, chords, secants, tangents, and angles of a circle

• Develop Mathematical Reasoning by applying properties of the different parts of a circle to explore how these parts relate to one another (i.e. arc length and angle measure, chord length, sectors, etc.)

• Develop the Ability to Communicate Mathematically by correctly utilizing the appropriate mathematical notation(s) to demonstrate understanding, with both written and verbal descriptions, of the relationships between measurements of angles, arcs, segments, lines, and rays belonging to a circle. 

• Implement tasks that promote reasoning and problem-solving by engaging students to correctly model strategies for finding unknown measures of arcs, angles, segments, and other parts of a circle as well as develop properties for the relationship between these parts of a circle.

• Support productive struggle by providing tasks that allow students to explore and/or compare a variety of solving strategies of varying challenge levels as well as analyze situations involving common errors.

• Pose purposeful questions which assist students in distinguishing among relationships between angles, arcs, segments, lines, and rays and how they relate to the circle. 

Key Understandings

Misconceptions 

• Lines and segments in circles have similar and different properties:

• A chord is a segment with endpoints on the circle.

• A secant is a line that intersects the circle in two distinct points.

• A tangent is a line, ray, or segment that intersects the circle in one distinct point.

• Angles of a circle have similar and different properties:

• A central angle has its vertex as the center of the circle and is formed by two radii of the circle.

• An inscribed angle has its vertex on the circle and is formed by two chords that intersect the circle in two distinct points.

• The measure of an inscribed angle is half the measure of its central angle in any size circle.

• When two lines intersect in various positions in relation to the circle, there is a relationship between the angles and the arcs that are formed. The lines can intersect inside, outside, or on the circle.

• Arcs are a portion of the circumference of a circle and are intercepted by both central and inscribed angles.

• The measure of an arc is the same measure as its intercepted central angle.

• The measure of an arc is twice the measure of its intercepted inscribed angle.

• The measure of an inscribed angle is half the measure of its intercepted arc. 

• When secants and/or tangents intersect with each other or the circle itself, there are relationships formed between the lengths of the various segments.

• Correctly Identify minor arcs, semicircles, and major arcs and use the correct notation for each. 

• Students may fail to use correct notation to distinguish between angles, arcs, segments, lines, and rays.

• Students may interchange and/or use the equations incorrectly for finding angle measures or segment lengths in relation to secants, chords, or tangents intersecting in various points inside and outside of the circle.

• Students may not correctly identify the appropriate relationship between angles, arcs, lines, and segments inside and/or outside of a circle. 

  Knowledge Connections

Prior Knowledge

Leads to 

• Use substitution to simplify and evaluate algebraic expressions. (PA.A.3.1)

• Use knowledge of solving equations with rational values to represent, use and apply mathematical models. (A1.A.1.1) 

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