The Geometry of a Circle - By: Dennis Kapatos

More documents

Recommendations

Info

$By: Melissa Hanel Math B/Grade 11 5 Day Unit Plan Technologies ...$

Unit Overview Unit Objectives: Students will learn a broad range of skills and content knowledge. In addition to all the theorems in each section, students will be able to make observations and conjectures and to test these conjectures using the technology and manipulatives at their disposal. Students will also be able to work cooperatively with other group members to investigate the properties of geometric figures (circles more specifically) and prove theorems. In addition to these, students will be able to recognize applications of circles and their related parts in the world around them. NCTM Standards: Number and Operation Students judge the reasonableness of numerical computations and their results. Algebra Students draw reasonable conclusions about a situation being modeled. Geometry Students explore relationships (including congruence and similarity) among classes of two- and three-dimensional geometric objects, make and test conjectures about them, and solve problems involving them. Students establish the validity of geometric conjectures using deduction, prove theorems, and critique arguments made by others. Students use Cartesian coordinates and other coordinate systems, such as navigational, polar, or spherical systems, to analyze geometric situations. Students use geometric ideas to solve problems in, and gain insights into, other disciplines and other areas of interest. Measurement Students make decisions about units and scales that are appropriate for problem situations involving measurement. Problem Solving Students build new mathematical knowledge through problem solving. Students solve problems that arise in mathematics and in other contexts. Reasoning and Proof Students make and investigate mathematical conjectures. Students develop and evaluate mathematical arguments and proofs. Students select and use various types of reasoning and methods of proof. Communication Students organize and consolidate their mathematical thinking through communication. Connections Students recognize and apply mathematics in contexts outside of mathematics. New York State Standards: G.PS.6 Use a variety of strategies to extend solution methods to other problems. G.PS.8 Determine information required to solve a problem, choose methods for obtaining the information, and define parameters for acceptable solutions. G.CM.5 Communicate logical arguments clearly, showing why a result makes sense and why the reasoning is valid.
G.CN.7 Recognize and apply mathematical ideas to problem situations that develop outside of mathematics. G.R.1 Use physical objects, diagrams, charts, tables, graphs, symbols, equations, or objects created using technology as representations of mathematical concepts. G.R.3 Use representation as a tool for exploring and understanding mathematical ideas. G.G.27 Write a proof arguing from a given hypothesis to a given conclusion. G.G.29 Identify corresponding parts of congruent triangles. G.G.49 Investigate, justify, and apply theorems regarding chords of a circle. G.G.50 Investigate, justify, and apply theorems about tangent lines to a circle. G.G.51 Investigate, justify, and apply theorems about the arcs determined by the rays of angles formed by two lines intersecting a circle. G.G.52 Investigate, justify, and apply theorems about arcs of a circle cut by two parallel lines. G.G.53 Investigate, justify, and apply theorems regarding segments intersected by a circle. Materials and Equipment: Resources: Geometer’s Sketchpad Computers Projector Graphing Calculators Compasses Mira’s Rulers Protractors String Cardboard Circles Empty Cans Textbook: “New York Math A/B: An Integrated Approach – Volume 2” Bass, Hall, Johnson, and Wood. “New York Math A/B: An Integrated Approach – Volume 2.” Teacher’s Edition. Prentice Hall, 2001. Chapter 12. Pages 584-631. Bennett, Dan. “Exploring Geometry with The Geometer’s Sketchpad.” 4 th Edition. Key Curriculum Press, 2002. Chapter 6. Pages 117 – 130. Unit Description: This unit is designed to cover chapter 12 sections one through five. Although the chapter has 6 sections, this remaining section could be covered similar to the previous five. The five lessons are intended to be inquiry based, though some of the teacher’s instructions may not seem so. The teacher should give only as much help as is needed to get the students thinking through a situation. Also, GSP is used throughout the unit; sometimes as a worksheet in which students fill in answers while looking at problems, sometimes as a tool for experimenting and conjecturing, and other times for modeling and solving problems. Additionally, most lessons start with a problem that leads into the lesson so that students see how the need for more sophisticated ways of thinking arise from real life problems. (Note: Although the theorems are stated on the last page
Page 1: The Geometry of a Circle Geometry (
Page 5 and 6: Lesson 1: Introduction to Circles D
Page 7 and 8: Notice that some areas of Bob’s f
Page 9 and 10: 15) Fill in the blank: The equation
Page 11 and 12: Lesson 2: Tangents Discovering and
Page 13 and 14: Satellites There are thousands of s
Page 15 and 16: Lesson 3: Properties of Chords and
Page 17 and 18: Locating the Center of the Satellit
Page 19 and 20: 8. Class will discus why they think
Page 21 and 22: 8. For rest of period, students wil
Page 23 and 24: 80 70 60 50 m CE on AB = 60.00 ° m
Page 25: Corollary 3 The opposite angels of

The Geometry of a Circle - By: Dennis Kapatos

Create successful ePaper yourself

Delete template?

Save as template?