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2.1 inductive reasoning and conjecture ink.notebook Page 8 Fill in your pretest and post test results September 09, 2016 Post test Pretest number correct: total number of problems: number correct: total number of problems: Percentage correct: Percentage correct: difference Change in percentage: original Page 57 = Page 58 Ch 2 Reasoning and Proofs 2.1 Inductive Reasoning Cut 1 2.1 inductive reasoning and conjecture ink.notebook September 09, 2016 Page 59 Lesson Objectives Standards Page 60 Lesson Notes Lesson Objectives 2.1 Inductive Reasoning and Conjecture 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. Press the tabs to view details. Press the tabs to view details. 2 2.1 inductive reasoning and conjecture ink.notebook Lesson Objectives Standards Lesson Notes G.MG.3 Apply geometric methods to solve design problems. September 09, 2016 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. G.CO.9 Prove theorems about lines and angles. Press the tabs to view details. 3 2.1 inductive reasoning and conjecture ink.notebook September 09, 2016 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, ____ Rule: 2. 1, 10, 100, 1000, _____ Rule: Rule: Write a conjecture about each value or geometric relationship. (What must be true?) 4. Û1 and Û2 form a right angle. 5. ÛABC and ÛDBE are vertical angles. 6. ÛE and ÛF are right angles. 4 2.1 inductive reasoning and conjecture ink.notebook September 09, 2016 • 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. 5 2.1 inductive reasoning and conjecture ink.notebook September 09, 2016 Example:4 sided figures are rectangles Counterexample : Determine whether each conjecture is true or false. Give a counterexample for any false conjecture. 7. If points A, B, and C are collinear, then AB + BC = AC. 8. If ÛR and ÛS are supplementary, and ÛR and ÛT are supplementary, then ÛT and ÛS are congruent. 6 2.1 inductive reasoning and conjecture ink.notebook Determine whether each conjecture is true or false. Give a counterexample for any false conjecture. 9. If ÛABC and ÛDEF are supplementary, then ÛABC and ÛDEF form a linear pair. September 09, 2016 Determine whether each conjecture is true or false. Give a counterexample for any false conjecture. 11. If ÛABC and ÛCBD form a linear pair, then ÛABC ¤ ÛCBD. Determine whether each conjecture is true or false. Give a counterexample for any false conjecture. 13. If AB + BC = AC, then AB = BC. 14. If Û1 is complementary to Û2, and Û1 is complementary to Û3, then Û2 ¤ Û3. 7 2.1 inductive reasoning and conjecture ink.notebook September 09, 2016 Practice 2.1 Practice WS on Inductive Reasoning and Conjecture Make a conjecture about the next item in each sequence. Make a conjecture about the next item in each sequence. 1. Next: Rule: Rule: Rule: Rule: 8 2.1 inductive reasoning and conjecture ink.notebook Make a conjecture about the next item in each sequence. September 09, 2016 Make a conjecture about the next item in each sequence. 5. Next: Next: Rule: Rule: 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. Rule: Rule: Make a conjecture about each value or geometric relationship. (State something that is true.) 11. Û1 and Û4 form a linear pair. 12. Û3 ¤ Û4 9 2.1 inductive reasoning and conjecture ink.notebook Determine whether each conjecture is true or false. Give a counterexample for any false conjecture. September 09, 2016 Determine whether each conjecture is true or false. Give a counterexample for any false conjecture. 14. If Û1 and Û2 are adjacent angles, then Û1 and Û2 form a linear pair. 16. PATTERNS The figure shows a sequence of squares each made out of identical square tiles. 17. WRITING Explain what a counterexample is… __________________________________________________ __________________________________________________ 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. 10 2.1 inductive reasoning and conjecture ink.notebook September 09, 2016 Answers: Add one dot to each side of the previous figure 1. 1 3. 16 5. 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 11. Û1 and Û4 are supplementary 13. true 15. True 17. A specific case where a conjecture is false. 11