Ms. Andrejko 2-1 Inductive Reasoning and Conjecture.

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Presentation transcript:

Ms. Andrejko 2-1 Inductive Reasoning and Conjecture

Real - Life

Vocabulary Inductive Reasoning- reasoning that uses a number of specific examples to arrive at your conclusion Conjecture- a concluding statement reached using inductive reasoning Counterexample- a false example that can be a number, drawing, or a statement.

Steps to making a conjecture 1. Find a pattern in the sequence that you are given, or write out examples to find a pattern based on what you are given. 2. Write the pattern that you have found in the form of a conjecture. IF GIVEN A CONJECTURE: Find a counter example, to prove that it is false

Examples 1. Conjecture: Each figure grows by increasing by 2 shaded diamonds and 1 shaded diamond. The next figure will have a total of 8 shaded diamonds and 5 white diamonds , -1, 2, 5, 8 Conjecture: Each number is increasing by 3 every time. The next number will be , -4, -8, -16, -32 Conjecture: Each number is two times as many as the previous number. The next number will be -64.

PRACTICE 1. Conjecture: Each figure grows by 2 dots each time. The next figure will contain 12 dots (5 on each side, and one on the top and bottom) 2. -5, -10, -15, -20 Conjecture: Each number is decreasing by 5 more each time. The next number in the series will be Conjecture: Each number is being multiplied by - ½. The next number will be 1/16.

Examples – Make a conjecture about the geometric relationships 4. N is the midpoint of QP Conjecture: QN ≅ NP 5. <3 ≅ <4 Conjecture: < 3 and < 4 are vertical angles 6. <1, <2, <3, <4 form 4 linear pairs Conjecture: <1 and < 3 are vertical angles, <2 and angle 4 are vertical angles

PRACTICE– Make a conjecture about the geometric relationships 4. <ABC is a right angle. Conjecture: < ABC = 90° 5. ABCD is a parallelogram Conjecture: ABCD has 4 sides 6. Conjecture: PQRS is a square

Examples – T or F - Counterexamples 7. If <ABC and <CBD form a linear pair, then <ABC ≅ <CBD 8. If AB, BC, and AC are congruent, then A, B, and C are collinear 9. If AB + BC = AC, then AB = BC Counterexample: A B C = 30, but 10 ≠ 20.A B C

PRACTICE– T or F - Counterexamples 7. If <1 and <2 are adjacent angles, then <1 and <2 form a linear pair 8. If S,T, and U are collinear, and ST = TU, then T is the midpoint of SU 9. If n is a real number, then n 2 > n Counterexample: 0 2 = 0 ; 1 2 = 1 ; ( ½) 2 = (¼) 1 2 Counterexample: TRUE