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Transcript
Asbury Park Name of Unit: Circles With and Without Coordinates Content Area: HS Math - Geometry Big Idea: All circles are similar School District Unit #/Duration: Unit 5 / 25 days Grade Level: 10th An angle drawn from a diameter of a circle to a point on the circle is sure to be a right angle (Thales’ Theorem) A central angles is equal to its intercepted arc whereas an inscribed angle equals half of its intercepted arc Pythagorean Theorem can be used to derive the equation of a circle A tangent to a circle and the radius of the circle are perpendicular at the point of intersection Essential Questions: ● If 2 chord are equidistant from the center of a circle, what do we know? Explain why. ● What is Thales’ Theorem? It’s converse? ● How can we construct a rectangle with vertices on a circle? Is there more than 1 unique way to do this? ● What is an inscribed polygon? Can you sketch a few examples? ● Is it possible to inscribe a parallelogram that is not a I Can Statements: ● Apply Thales’ Theorem ( if triangle ABC has vertices on the circumference of a circle and it is a right triangle, one side is the diameter of the circle} ● Prove Thales’ Theorem ● identify the relationships between the diameters of a ● ● ● ● ● ● ● rectangle in a circle? Explain why or why not. How do minor and major arcs compare? How are minor and major arcs denoted? Explain why one of them uses 3 points and the other 2 ? What is an auxiliary segment and why might you draw one? How would you describe the secant angle theorem? Can you sketch a diagram to show which arcs are needed to use this theorem? Why do we use different forms of equations for circles? Can you support your answer with an example? How can we convert a general formula circle into a center-radius form? Why would we do this? ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● circle and other chords of the circle Inscribe a rectangle in a circle Understand the symmetries of inscribed rectangles across a diameter describe relationships between inscribed angles and central angles and their intercepted arcs prove the inscribed angle theorem understand that inscribed angles that intersect the same arc are equal in measure and use that relationship to solve problems compare and contrast central angles and inscribed angles identify minor and major arcs utilize inscribed angle theorem to find the measures of unknown angles prove relationships between inscribed angles and central angles Understand and explain why all circles are similar Utilize the arc angle measurement formula to solve problems Demonstrate arcs between parallel chords are congruent Show that congruent chords have congruent arcs construct tangents to a circle through a given point prove that tangent segments from the same point are equal in length use the fact if a circle is tangent to both rays of an angle, then its center lies on the angle bisector utilize tangent segments and radii of circles to conjecture and prove geometric statements use the inscribed angle theorem to prove other ● ● ● ● ● ● ● ● ● ● ● ● geometric theorems utilize the inscribed angle theorem to solve for missing “unknown” angles understand that the measure of an angle whose vertex lies in the interior of a circle is equal to half the sum of the angle measures of the arcs intercepted by it and its vertical angle find the measures of angles, arcs, and chords in figures that include two secant lines meeting outside a circle ( inference is required ) find missing lengths in circle-secant or circle-secanttangent diagrams write the equation for a circle in center-radius form, ( − h)2 + ( – k)2 = r2 and use it to solve problems identify the center and radius of a circle given the equation determine the distance between the centers of 2 tangent circles complete the square in order to write the equation of a circle in center-radius form factor perfect square trinomials use standard form of a circle x2 + y2 + Cx + Dy + E =0 to solve problems find the equations of two lines tangent to the circle with specified slopes, given a circle given a circle and a point outside the circle, students find the equation of the line tangent to the circle from that point ● Common Core State Standards: ● G.C.2 ; G.C.1 ; G.C.3 ; G.C.4 ; G.C.5 ● G.GPE.1 ; G.GPE.4 ● G.SRT.9 Pre-requisite standards: G.CO.10 ; G.CO.11 ; G.CO.12 ; G.CO.3 ;G.CO.5 ; G.CO.9 G.SRT.5 8.G.7 ; 8.G.8 Mathematical standards highlighted: ● MP 1; MP 3 ; MP 7 Interdisciplinary Connections: Symbol: **IC Technology Integration: (Standards included only if students will be demonstrating knowledge/understanding/skill.) Symbol: ***TI Texts ● Primary Text: Eureka Math – 10 [ Engage NY – Geometry mod 5] ● ● Secondary/Supplemental Texts: Pearson Geometry HMH Geometry Suggested Instructional Activities/Strategies Topic A – Central and inscribed angles: Lesson 1: Thales’ Theorem Lesson 2: Circles, Chords, Diameters, and Their Relationships Lesson 3: Rectangles Inscribed in Circles Lesson 4: Experiments with Inscribed Angles Lesson 5: Inscribed Angle Theorem and Its Applications Lesson 6: Unknown Angle Problems with Inscribed Angles in Circles Topic B – Arcs and sectors : Lesson 7: The Angle Measure of an Arc Lesson 8: Arcs and Chords Lesson 9: Arc Length and Areas of Sectors Lesson 10: Unknown Length and Area Problems Mid-module assessment Topic C – Secants and tangents: Lesson 11: Properties of Tangents Lesson 12: Tangent Segments Lesson 13: The Inscribed Angle Alternate—A Tangent Angle Lesson 14: Secant Lines; Secant Lines That Meet Inside a Circle Lesson 15: Secant Angle Theorem, Exterior Case Lesson 16: Similar Triangles in Circle-Secant (or Circle-Secant-Tangent) Diagrams Topic D – Equations for circles and their tangents: Lesson 17: Writing the Equation for a Circle Lesson 18: Recognizing Equations of Circles Lesson 19: Equations for Tangent Lines to Circles Topic E – Cyclic quadrilaterals: Lesson 20: Cyclic Quadrilaterals Lesson 21: Ptolemy’s Theorem End – of-module assessment Teacher Resources ● ● ● ● ● ● Document Camera Epson Brite-board ( interactive) Ti-84’s Communicators Geometer’s Sketchpad Geogebra Vocabulary ● ● ● ● ● ● ● ● ● ● ● ● ● Domain Specific Academic Vocabulary (Tier 3) Arc length Central angle Chord Inscribed angle Sector secant Tangent line Inscribed polygon Cyclic Quadrilateral Diameter Minor arc Major Arc Radius Thales Theorem ● ● ● ● General Academic Vocabulary (Tier 2) circle line adjacent center Assessments Summative Assessment: ● Mid-module assessment ● End-of-module assessment ● Teacher-prepared quizzes Formative Assessments: ● Do now’s ● Class work ● Homework ● Exit Tickets ● Reflections Type Differentiation/Scaffolding (for example ELL, students who are classified, struggling learners, etc.) Visual Auditory Kinesthetic ● ● ● Language Development ● Poster of perpendicular bisector construction; Reread all instructions aloud; listen to video clips Hands-on , cut out activity for Thales’ Theorem ; cut out trapezoids (then cut 2 consecutive angles and place in a line to show they will always be supplementary) Vocabulary development with pictures and graphs to assist ELL students Appendix 1 (graphic organizers, rubrics, websites, activities, manipulatives, sample assessments, etc.) ● ● ● ● ● ● ● ● ● ● Kutasoftware.com Purplemath.com Mathisfun.com Coolmath.com Mathworksheets4kids Khanacademy.com him: http://en.wikipedia.org/wiki/Thales%27 Theorem Graphic organizer { chords of a circle ) p. 27, 33 Eureka Math Mid-module rubric & assessment p. 128-144 Eureka Math Appendix 2 (Quad D Exemplar Lesson Plan) http://map.mathshell.org/lessons.php?unit=9335&collection=8 [ Solving problems with circles and triangles] http://map.mathshell.org/lessons.php?unit=9350&collection=8 [ sorting equations of circles 1] map.mathshell.org/lessons.php?unit=9355&collection=8 [ sorting equations of circles 2 ]