2.1 inductive reasoning and conjecture ink.notebook. September 07, Page 55. Ch 2. Reasoning. Page 56. and Proofs. 2.1 Inductive.

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1 2.1 inductive reasoning and conjecture ink.notebook Page 55 Ch 2 Reasoning and Proofs Page Inductive Reasoning Lesson Objectives Page 57 Standards Lesson Notes Page Inductive Reasoning and Conjecture Press the tabs to view details. 1

2 Lesson Objectives Standards Lesson Notes Lesson Objectives Standards Lesson Notes After this lesson, you should be able to make conjectures based on inductive reasoning.you will find counterexamples and represent patterns with variables and math symbols. G.MG.3 G.CO.9 Apply geometric methods to solve design problems. Prove theorems about lines and angles. Press the tabs to view details. Press the tabs to view details. conjecture = unproven statement based on observation inductive reasoning = when you find a pattern in specific cases and then write a conjecture for the general case. 2

3 2.1 inductive reasoning and conjecture ink.notebook Write a conjecture that describes the pattern in each sequence. Then use your conjecture to find the next item in the sequence. 1. 5, 10, 20, 40, 2. 1, 10, 100, 1000, 3

4 Write a conjecture about each value or geometric relationship. (What must be true?) 4. 1 and 2 form a right angle. 5. ABC and DBE are vertical angles. 6. E and F are right angles. To show that a conjecture is true, you must show that it is true for all cases. To show that a conjecture is false, you must find one counterexample. A counterexample is a specific case for which the conjecture is false. 4

5 Example:4 sided figures are rectangles Counterexample : 7. If points A, B, and C are collinear, then AB + BC = AC. 8. If R and S are supplementary, and R and T are supplementary, then T and S are congruent. 5

6 2.1 inductive reasoning and conjecture ink.notebook 9. If ÛABC and ÛDEF are supplementary, then ÛABC and ÛDEF form a linear pair. 11. If ÛABC and ÛCBD form a linear pair, then ÛABC ÛCBD. 6

7 13. If AB + BC = AC, then AB = BC. 14. If 1 is complementary to 2, and 1 is complementary to 3, then Practice WS on Inductive Reasoning and Conjecture Make a conjecture about the next item in each sequence. 1. Next: Practice 7

8 Make a conjecture about the next item in each sequence. 5. Make a conjecture about the next item in each sequence. Next: Make a conjecture about the next item in each sequence. Next: Make a conjecture about each value or geometric relationship. (State something that is true.) 9. Points A, B, and C are collinear, and D is between B and C. 8

9 Make a conjecture about each value or geometric relationship. (State something that is true.) and 4 form a linear pair If 1 and 2 are adjacent angles, then 1 and 2 form a linear pair. 16. PATTERNS The figure shows a sequence of squares each made out of identical square tiles. a) Starting from zero tiles, how many tiles do you need to make the first square? How many tiles do you have to add to the first square to get the second square? How many tiles do you have to add to the second square to get the third square? b) Make a conjecture about the list of numbers you started writing in your answer to Exercise a. c) Make a conjecture about the sum of the first n odd numbers. 9

10 17. WRITING Explain what a counterexample is Answers: Add one dot to each side of the previous figure multiply the previous number by -1/2 Add one shape to each row of the previous figure, which means adding 2 shaded and 1 unshaded diamond to each new figure. 9. A, B, C, and D are all collinear and 4 are supplementary 13. true 15. True 17. A specific case where a conjecture is false. 10