2-1 Patterns and Inductive Reasoning

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Presentation transcript:

2-1 Patterns and Inductive Reasoning

Inductive Reasoning Inductive Reasoning – Process of observing data, recognizing patterns and making generalizations about those patterns.

Look for a pattern. What are the next two terms in each sequence? b)

1. Look for a pattern. What are the next two terms in each sequence? b)

Conjecture – A conclusion reached using inductive reasoning Conjecture – A conclusion reached using inductive reasoning. An educated guess, hypothesis, based on observation, experimentation, data collection, etc.

1. Look at the circles, what conclusions can you make about the number of regions 20 diameters form?

1. Look at the circles, what conclusions can you make about the number of regions 20 diameters form?

1. Look at the circles, what conclusions can you make about the number of regions 20 diameters form? Conjecture: 20 diameters 20(2) = 40 40 regions

2. Points A, B, and C are collinear and B is between A and C.

2. Points A, B, and C are collinear and B is between A and C. Conjecture: AB + BC = AC

Not all conjectures turn out to be true Not all conjectures turn out to be true. You should test your conjecture multiple times. You can prove that a conjecture is false by finding a counterexample. Counterexample – An example that shows a conjecture is incorrect.

1. Conjecture: All parallelograms have 2 acute angles and 2 obtuse angles. Is this conjecture true or false? If false, provide a counterexample.

1. Conjecture: All parallelograms have 2 acute angles and 2 obtuse angles. Is this conjecture true or false? If false, provide a counterexample. False, a rectangle.

2. Conjecture: If P, Q, and R are collinear, then Q is between P and R. Is this conjecture true or false? If false, provide a counterexample.

Is this conjecture true or false? If false, provide a counterexample. 2. Conjecture: If P, Q, and R are collinear, then Q is between P and R. Is this conjecture true or false? If false, provide a counterexample. False Q P R

3. Conjecture: If A, B, and C are collinear and B is between A and C, then B is the midpoint of . Is this conjecture true or false? If false, provide a counterexample.

Is this conjecture true or false? If false, provide a counterexample. 3. Conjecture: If A, B, and C are collinear and B is between A and C, then B is the midpoint of . Is this conjecture true or false? If false, provide a counterexample. False A B C

4. Conjecture: If A, B, and C are collinear and B is the midpoint of , then AB = BC. Is this conjecture true or false? If false, provide a counterexample.

4. Conjecture: If A, B, and C are collinear and B is the midpoint of , then AB = BC. Is this conjecture true or false? If false, provide a counterexample. True

Problems with inductive reasoning: Arrive at a generalization before every possible case is examined. Reasons are not provided for why things are true.

We use inductive reasoning to make new discoveries and generalizations, then a proof or explanation is sought. Inductive reasoning is the first step in discovering new mathematical facts.

Deductive Reasoning Deductive Reasoning – Used to prove conjectures. Process of showing certain statements follow logically from agreed assumptions and proven facts.