The Geometry of a Circle - By: Dennis Kapatos

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

Lesson 4: Inscribed Angles Exploring Properties of Inscribed Angles and Quadrilaterals Using GSP Materials and Handouts: Teacher’s Computer with GSP Projector Student’s Computers with GSP GSP Worksheet Files Textbooks Graphing Calculators Compasses Rulers Mira’s Protractors Name: Dennis Kapatos Grade: 11 Subject: Geometry Lesson Objectives: ● Students will be able to write/describe and prove Theorem 12.10 and its corollaries. ● Students will be able to prove Theorem 12.11, and explain how it can be thought of as a corollary to Theorem 12.12. ● Given a geometric figure with missing measurements, students will be able to apply the theorems learned so far to find them. ● Students will be able to work cooperatively to make and test conjectures of geometric figures. Reviewing Homework: 1. Answers to the homework questions 1-13, and 18 will be shown with the projector. Students will correct their own homework. 2. Teacher will haves students go over their solutions to question 20 and any questions that the majority of students had trouble with. 3. Teacher will ask one student to present their proof of Theorem 12.9 to the class for discussion. Anticipatory Set: 4. Teacher will ask student to construct any quadrilateral using compasses, rulers, and/or Mira’s. 5. Teacher will ask students to measure all the angles of their quadrilateral. 6. Students will share what their angle measurements are and conclude that any angle can have any measure. Developmental Activity: 6. Teacher will ask students to construct a large circle with any quadrilateral in it, with its vertexes on the circle (an inscribed quadrilateral). 7. Again, students will measure their angles and look for a relationship (opposite angles are supplementary).
8. Class will discus why they think this is the case. 9. Teacher will ask students to go to their computers and start up GSP. 10. Teacher will ask class to start drawing quadrilateral but stop after creating just 2 adjacent sides. 12. Teacher will have students construct central angle to this arc, measure its arc angle, measure the arc angle of the inscribed angle, and conjecture the relationship between the inscribed and central angles (this is Theorem 12.10, the inscribe should be half the central angle. Teacher will have students come up with definitions for these terms as he introduces them. 13. Students will move the points around to test this conjecture and enter Theorem 12.10 into their notes (see first picture). 14. The teacher will ask the class what if a second angle AEC where drawn or A”any-point”C were drawn (it would have the same arc angle)(see second picture). 15. Students will experiment with their sketches, discuss what they think, discover Corollary 1 to Theorem 12.10, and enter this into their notes. 16. Teacher will ask students to hide angle AEC 17. Teacher will ask students what if A and C were opposite each other on circle, that is, what if they were on opposites sides of a diameter (angle ABC would always be equal to 90)(see third picture). 18. Students will move points around on their sketches, discuss what they think, discover Corollary 2 to Theorem 12.10, and enter this into their notes (see third picture). 19. Teacher will ask students to make angle ADC less than 180 again. 20. Teacher will ask student to see if Theorem12.10 holds Hide as Raypoint B gets really close to point C (at this point BC will become a tangent to the circle at point C). 21. Students will try this, discuss what they see and think (see picture). 22. Students will discuss how line BC becomes a tangent, discover Theorem 12.11, see how it is really just another corollary to Theorem 12.10, and enter it into their notes as well. 23. Teacher will asks students to return to their original question about inscribed quadrilaterals. 24. Students should be able to do this part completely on their own and with m! A B C = 4 6 . 2 0 ° B E m! A E C = 4 6 . 2 0 ° m ! A B C = 4 1 . 2 2 ° B m ! A B C = 9 0 . 0 0 ° B D m ! A B C = 4 8 .1 5 ° D D C D A m! A D C = 9 2 . 3 9 ° C A m ! A D C = 8 2 . 4 4 ° C A m ! A D C = 1 8 0 . 0 0 ° m! A D C = 9 6 .3 0 ° A B C little difficulty (see final picture). 25. Teacher will have them enter this as into their notes as officially, Corollary 3 to Theorem 12.10. 26. For the rest of the class, students will prove Theorems 12.10 and 12.11 working in pairs and with the teachers assistance. Students will present their proofs as time allows. m ! A B C = 4 8 . 1 5 ° m! A H C = 1 3 1 . 8 5 ° m AHC = 96.30 ° m ABC = 263.70 ° A Homework: B D H Questions 1-12, 15, 16, 24-27, and 28 on page 610 of the textbook. Students will also be asked to come up with and draw and label 1 new real life application (not one from the book or class) of anything the have learned from this unit so far. They should involve as many measurements as possible or necessary. C
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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: Locating the Center of the Satellit
Page 21 and 22: 8. For rest of period, students wil
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Page 25: Corollary 3 The opposite angels of

The Geometry of a Circle - By: Dennis Kapatos

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