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Patterns and Inductive Reasoning During this lesson, you will use inductive reasoning to make conjectures. Mrs. McConaughy Geometry 1 Standards/Assessment Anchors: Mrs. McConaughy Geometry 2 Daily Warm Up Inductive reasoning ___________ Conjecture __________________________ Give an example of when you have used inductive reasoning in the real world. Mrs. McConaughy Geometry 3 Vocabulary Inductive reasoning reasoning ________________ based upon ______________________________ patterns you observe. the conclusion you reach Conjecture ______________________ using inductive reasoning _______________________________ an example for which the Counterexample ___________________ conjecture is incorrect _______________________________ Mrs. McConaughy Geometry 4 Finding and Using a Pattern Examples: Number and Letter Patterns Mrs. McConaughy Geometry 5 Finding and Using a Pattern Use inductive reasoning to a. find a pattern for each sequence, then b. use the pattern to find the next term in each sequence below: 20, 18, 16, 14, 12 __ U A, C, F, J, O, __ 1, 3, 6, 10, 15, 21, ___ 28 a, 6, c, 12, e, 18, __ g ½, 9, 2/3, 10, ¾, 11, 4/5 __ 1, 3/2, 9/4, 27/8, 81/16 __ Mrs. McConaughy Geometry 6 Using Inductive Reasoning to Make Conjectures EXAMPLE EXAMPLE 3+5=8 -3 + 5 = 2 -1 + 1 = 0 13 +27 = 40 51 + 85 = 136 Conjecture: The sum of two odd numbers is even integer always an ___________. 3 * 4 = 12 12 * 5 = 60 11 * -4 = -44 -24 * -3 = 72 -7 * 8 = - 56 Conjecture: The product a prime number and of ______________ an_________________ even integer is always an_________________. even integer. Mrs. McConaughy Geometry 7 Finding and Using a Pattern Examples: Picture Patterns Mrs. McConaughy Geometry 8 EXAMPLE: Testing a Conjecture When points on a circle are joined by as many segments as possible, overlapping regions are formed inside the circle as shown above. Use inductive reasoning to make a conjecture about the number of regions formed when five points are connected. Mrs. McConaughy Geometry 9 Did you aguess that thetonumber Now find counterexample show thisof conjecture false. at each stage? regions is doubles Points Mrs. McConaughy Geometry Regions 2 2 3 4 4 8 5 16 6 ? 31 10 Testing a Conjecture Not all conjectures turn out to be true. You can prove that a conjecture is false by finding one counterexample. Mrs. McConaughy Geometry 11 Applying Conjectures to Business Mrs. McConaughy Geometry 12 Final Checks for Understanding 1. Find the next term in the following sequence: 2. Use inductive reasoning to make a conjecture: 3. Counterexample: 4. Draw the next picture in the picture pattern below: Mrs. McConaughy Geometry 13 Homework Assignments: Day 1: Inductive Reasoning WS Day 2: Number Patterns WS Day 3: Picture Patterns WS Mrs. McConaughy Geometry 14