West Haven Public Schools Unit Planning Organizer Subject: Circles and Other Conic Sections Grade 10 Unit: Five Pacing: 4 weeks + 1 week Essential Question(s): 1. What is the relationship between angles within a circle and their arcs? 2. How are tangents and chords related to a circle? 3. How are the area of a sector and the arc length related to circle? 4. Why is the distance formula important to circles? Big Idea(s): 1. Within a circle there are angles and arcs. The measure of a central angles and their arcs are the same. The measure on an inscribed angle is one half of its arc. Inscribed angles on a diameter are right angles. The radius of a circle is perpendicular to the tangent where the radius intersects the circle. 2. A tangent is a line outside of a circle. It is perpendicular to the radius of the circle at its point of tangency and a chord is a segment whose endpoints are on the circle. 3. The area of a sector is a portion of the area of a circle. The arc length is a portion of the circumference of the circle. 4. The Distance Formula is used by completing the square to find the center and radius of a circle given by an equation. 1
Common Core State Standards (includes West Haven s Priority Common Core Standards in BOLD and Supporting Standards) CC.9 12.G.C.2 Identify and describe relationships among inscribed angles, radii, and chords. Include the relationship between central, inscribed, and circumscribed angles; inscribed angles on a diameter are right angles; the radius of a circle is perpendicular to the tangent where the radius intersects the circle. CC.9 12.G.C.3 Construct the inscribed and circumscribed circles of a triangle, and prove properties of angles for a quadrilateral inscribed in a circle. CC.9 12.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; derive the formula for the area of a sector. CC.9 12.G.C.1 Prove that all circles are similar. (Honors Geometry) CC.9 12.G.GPE.1 Derive the equation of a circle of given center and radius using the Distance Formula; complete the square to find the center and radius of a circle given by an equation. CC.9 12.G.GPE.2 Derive the equation of a parabola given a focus and directrix.(geometry Honors Enrichment) 2
Unwrapped Concepts and Skills, and Bloom Levels (BL) Concepts(Need to Know) Skills(Able to Do) BL relationships among IDENTIFY DESCRIBE 1 1 o central angles o inscribed angles o circumscribed angles o radii o chords o tangent length of the arc intercepted by an DERIVE (using similarity) 5 angle is proportional to the radius DEFINE 1 the radian measure of the angle DERIVE 5 formula for the area of a sector equation of circle DERIVE 5 center and radius of a circle FIND (complete the square) 3 Assessments Common Formative Pre Assessment (Followed by Data Team Analysis): See Test A on Circles and other Conic Sections Pre Requisite Skills Assessments A, B, C Dipsticks (Informal Progress Monitoring Checks): Classroom Teacher Made Data driven decisions Common Formative Post Assessment (Followed by Data Team Analysis): See Test B on Circles and other Conic Sections 3
Instructional Planning Suggested Resources/Materials: Prentice Hall Geometry Text and Recommendations from Pacing Guide Unit Five Pacing Guide Essential Question: Standard: CC.9 12.G.C.2 What is the relationship between angles within a circle and their arcs? 5 Days Section 7 6: Define: ALL Vocabulary; Theorems 7 13, 7 14; Examples 2, 3, 5 Section A #9 26, 34 39, Section B #42 47, 58 59, 63 65 Define Radian: The ratio between the length of an arc and it s radius. Apply the following conversion tool: How to convert Degrees to Radian Measure: X multiplied by π 180 4
Standard: CC.9 12.G.C.5 Essential Question: How are the area of a sector and the arc length related? 5 Days Section 7 7: Define: Sector of a Circle; Theorems 7 15, 7 16; Examples 2 only Section A #7 16, Section B #31, Section C #40 Standard: CC.9 12.G.GPE.1 Essential Question: How can the Distance Formula be used to derive the equation of a circle given center and radius? 5 Days Section 11 5: Define: Vocabulary; Theorems 11 13; Examples 1, 2, 3, Section A #1 21 Section B #27 39, 42 44 5
Standard: CC.9 12.G.C.2 Essential Question: How are tangents and chords related to a circle? 5 Days Section 11 1: Define: ALL Vocabulary; Theorems 11 1, 11 2; Examples 1 and 3 Section A #1 3, 10 12 Section B #20 22 Section 11 2: Define: chord; Theorems 11 4, 11 5, 11 6, 11 7, 11 8; Examples 1, 2, 3 Section A #1 16, Section B #17 18 Section 11 3: Define: ALL vocabulary; Theorems 11 9, 11 10; Examples 1, 2, 3 Section A #1 20 Section B #21 24 6
Unit 5 Performance Task Standards: ALL Essential Questions: ALL 4 Days Enrichment/Reteach PH page 582: Investigation: Exploring Properties of Tangents PH Enrichment 7 7: Still life with Circles and Polygons PH page 598: Investigation: Exploring Inscribed Angles PH Enrichment 11 2: N Chords and Triangular Numbers PH Enrichment 11 3: The Impossible Dream PH Enrichment 11 5: Re Teaching skill Re teaching: Prentice hall re teaching worksheets Re teaching: Data Driven Suggested Resources/Materials: Chapter 7 Sections 6, 7, Extra Practice p 696 # 13 20 Chapter 11 Section 1, 2, 3, 5, Extra Practice p 700 # 1 20 http://www.llustrativemathematics.org/ 7
Suggested Research based Effective Instructional Strategies: Vocabulary/Word Wall Enrichment/Extension Interdisciplinary Connections Circle Center Radius Diameter Central angle Semicircle Minor arc Major arc Adjacent arcs Circumference Concentric circles Arc length Congruent arcs Sector of a circle Segment of a circle Tangent to a circle PH page 582: Investigation: Exploring Properties of Tangents PH Enrichment 7 7: Still life with Circles and Polygons PH page 598: Investigation: Exploring Inscribed Angles PH Enrichment 11 2: N Chords and Triangular Numbers PH Enrichment 11 3: The Impossible Dream PH Enrichment 11 5: Circles PH: page 391: # 49 54, 60 PH page 39: # 28, 29, PH page 587: # 24 26 PH page 594: # 20 22, 39 PH page 602: # 27, 32, 37 PH page 617: Example 4 and #22 26 page 618 8
Point of tangency Inscribed in Circumscribed about Chord Inscribed angle Intercepted arc Radian Conversion Tool: x multiplied by π 180 9