Geometry Block - Unit 6

Unit 6a: The Unit Circle

Geometric & Spatial Reasoning (GSR)

G.GSR.7: Explore the concept of a radian measure and special right triangles. (GaDOE, 2022b, p. 13)

STANDARD & Expectations

LT & SC

Lesson Activities & Resources

G.GSR.7.1

Explore and interpret a radian as the ratio of the arc length to the radius of a circle.

LT:

• I am learning about radian measures of angles. 

• I am learning about the ratio of arc length to the radius of a circle.

SC:

• I understand the relationship between the radius of a circle, an arc length, and the associated radian measure. 

LA&R:

• Desmos and 3-Act Tasks; GeoGebra; Look for a great hands-on and technology visualization activity for this skill; Quizlet vocabulary & word wall; Flipped classroom activity for Pi

G.GSR.7.2

Explore and explain the relationship between radian measures and degree measures and convert fluently between degree and radian measures.

LT:

• I am learning about the relationship between radian and degree measures.

• I am learning to convert between radian and degree measures of angles.

SC:

• I can convert between degree and radian measures to solve real-life problems.

LA&R:

• Desmos and 3-Act Tasks; GeoGebra; IXL; Use the degrees off the Unit Circle to convert each degree to a radian to practice the skill, then check with a finished Unit Circle; Quizlet vocabulary & word wall.

G.GSR.7.3

Use special right triangles on the unit circle to determine the values of sine, cosine, and tangent for 30° 6, 45° 4, and 60° 3 angle measures. Use reflections of triangles to determine reference angles and identify coordinate values in all four quadrants of the coordinate plane.

LT:

• I am learning about special right triangles.

• I am learning how special right triangles determine the sine, cosine, and tangent values on the unit circle.

• I am learning about reference angles.

• I am learning to identify coordinate values in all four quadrants of the coordinate plane. 

SC:

• I can use Special Right Triangles to find the values of sine, cosine, and tangent for 30° 6, 45° 4, and 60° 3 angle measures.

• I can articulate the pattern associated with angle measures in all four quadrants of the Unit Circle. (Students will be using reference angles as degrees in this skill). 

• I can use reflections of triangles to determine reference angles.

• I can identify coordinate values in all four quadrants of the coordinate plane.

• I can measure the arc length of a circle using radians, converted from degrees.

• I know that the Unit Circle has a radius equal to 1.

LA&R:

• Desmos and 3-Act Tasks; GeoGebra; Teach & practice Special Right Triangles using the SAT formula sheet as a guide; Teach reference angles (as degrees); Build Unit Circle paper plates; Use Special Right Triangles to find the coordinates of the ordered pairs on the Unit Circle; Practice ways to memorize the degrees, radians and ordered pairs on a Unit Circle; Arc length; Quizlet vocabulary & word wall

Unit 6b: Circles

Geometric & Spatial Reasoning (GSR)

G.GSR.8: Examine and apply theorems involving circles; describe and derive arc length and area of a sector; and model and explain real-life frameworks involving circles. 

STANDARD & Expectations

LT & SC

Lesson Activities & Resources

G.GSR.8.1

Identify and apply angle relationships formed by chords, tangents, secants, and radii with circles.

LT:

• I am learning circle vocabulary. 

• I am learning about angle relationships in circles.

SC:

• I can identify and apply angle relationships formed by chords, tangents, secants, and radii with circles.

• I can identify angles based on the location of the vertex: central, inscribed, interior, and exterior.

• I can identify perpendicular segments when a radius and tangent segment meet at a point of tangency on a circle.

• I can solve problems involving triangles inscribed in and circumscribed about circles.

• I can solve problems involving opposite angles of a quadrilateral inscribed in a circle, which are supplementary.

LA&R:

• Desmos and 3-Act Tasks; GeoGebra; Discovery Learning using a circle where every 10 degrees is marked; IXL; DeltaMath; Quizlet Vocab (lots of words – may need two separate sets to manage student learning); Word wall; KutaSoftware

G.GSR.8.2

Using similarity, derive the fact that the length of the arc (arc length) intercepted by an angle is proportional to the radius; derive the formula for the area of a sector. Solve mathematically applicable problems involving applications of arc length and area of sector.

LT:

• I am learning about arc length in a circle.

• I am learning about the area of a sector in a circle.

• I am learning to use the arc length and area of a circle to solve circle application problems.

SC:

• I understand that arc length intercepted by an angle is proportional to the radius.

• I can solve application problems involving arc length and area of a sector.

LA&R:

• Desmos and 3-Act Tasks; GeoGebra; Show arc length is measuring a distance in a linear unit using a Unit Circle paper plate & string; Show the difference between the arc length & area of a sector formulas: Degrees (proportions) & Radians (derive then show how to use). Arc Length & Area of Sector practice; Quizlet vocabulary & word wall. 

G.GSR.8.3

Write and graph the equation of circles in standard form.

LT: 

• I am learning to write and graph circles in standard form.

Terminology:

General Form is x2+y2+Cx+Dy+E=0

Standard Form is x-h2+y-k2=r2

SC:

• I can identify the center and radius of a circle from an equation in standard form.

• I can identify the center and radius of a circle from the graph of a circle.

• I can write the equation of a circle in standard form given the graph of the circle.

• I can graph a circle from the standard form equation of a circle.

• I can use methods like completing the square to take the equation of a circle from general form to standard form. (Note: The leading coefficient of the quadratic terms should be limited to 1.) (What course will students learn completing the square?)

LA&R:

• Desmos and 3-Act Tasks; GeoGebra; TI-nSpire CAS calculator – students do the math then use the graphing calc to check that both forms graph the same circle. Desmos online graphing calculator; Quizlet vocabulary & word wall.