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La Cañada Math II Advanced Newsletter Unit 9 – Circles Unit 9 Overview This unit will begin by defining basic terms related to circles (i.e. center, chord, diameter, secant, …) that will be used in various proofs and problems. Students will then be exposed to a variety of theorems, and corollaries to those theorems, that will be used to solve problems. Each theorem will be proven using previous definitions, postulates, and theorems. Indirect proof, or proof by contradiction, will be introduced as a tool for proving a theorem that does not have an obvious path. Students will be introduced to the concept of indirect proof in a simpler geometric problem and with numerical reasoning before using it in the proof a new theorem. In particular, proof by contradiction will be used to prove that the tangent of a circle is perpendicular to the radius at the point of tangency. They will also construct a tangent line from a point outside of a circle and explain the relevant geometry. Theorems in this unit will be related to the tangents and radii of a circle, the arcs of a circle and their relationship with central, inscribed, interior and exterior angles, and the chords and secants of a circle. Once students have analyzed, understood, and proved these theorems, they will use them to solve problems. Some of these problems will involve algebraic as well as geometric reasoning. Real-world meaning will be constructed using the concept of latitude and longitude lines as the measure of the central angle of a sphere. As the unit progresses, students will develop a sense of when it is beneficial or useful to construct an auxiliary line in order to create an effective proof. This skill, along with the idea of indirect proof, will be beneficial throughout this unit and future units. In order to prove several of the theorems in this unit, students will also need to consider the various cases which can occur within the constraints of the context and use each of them as the starting point for a proof. Students will work with several special angles in a circle, and the segments that are related to them. For example, students will prove the various cases where an inscribed angle intercepts an arc that is twice its measure. Previous knowledge and problem solving skills will be used to prove theorems about two tangents, two secants, or a secant and tangent drawn in a circle and then use those theorems to draw conclusions based on a given diagram Important Dates February 23 Unit 9 Test (Tentative) Homework Policy Daily Homework will be structured more specifically to provide more detailed information for students and parents. Homework will generally consist of two sections; exercises and problems. Exercises are typical of the type of questions that have been taught in class. Students should have a model and/or framework for this type of question and should consult their resources to maintain consistency from our work as a class. Exercises have a prescribed solution method. Problems and proofs are questions for which there is no prescribed method. They will require students to utilize their problem solving skills, creativity, and make connections among mathematical ideas. Weekly Homework will be a component of the course, and will emphasize algebra skills learned in last year’s course in an attempt to maintain important ideas before entering LC Math III. Page 1 of 2 Additional Unit 9 Information Key Common Core Standards 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. G.C.3 - Construct the inscribed and circumscribed circles of a triangle, and prove properties of angles for a quadrilateral inscribed in a circle. G.C.4 - (+) Construct a tangent line from a point outside a given circle to the circle. Essential Questions -What relationships exist between segments in circles? -What relationships exist between angles and segments in circles? -Why do constructions related to circles make sense? -What is indirect proof? Helpful Resources Geometry Textbook – Chapter 9 (Sections 1-7) Chapter 10 (Section 1) Construction Support: www.mathsisfun.com/geometry/constructions.html Khan Academy: https://www.khanacademy.org/math/geometry-home/geometry Dr. Carruthers ([email protected]) Mr. McDermott ([email protected]) Spotlight on the Standards for Mathematical Practice: MP 3: Construct viable arguments and critique the reasoning of others. Constructing arguments is an important mathematical practice. Mathematics, as a field of study, is characterized by a need to prove what is true. As students advance in their mathematics education, they will be asked to justify and prove mathematical relationships with an increasing level of rigor and formality. This unit will extend students’ understanding of proof and making a logical argument that is supported by definitions and postulated. In this unit, students will be introduced to the concept of proof by contradiction, or indirect proof, which provides an additional proof strategy when traditional methods become difficult or increasingly complicated. Students will use proof, and deductive reasoning in problem solving throughout this course and in all future mathematics. Page 2 of 2