Patterns and Inductive Reasoning

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Presentation transcript:

Patterns and Inductive Reasoning SpringBoard Lesson 1-2 Patterns and Inductive Reasoning Objectives: 1) To use inductive reasonings to make conjectures.

Ex. 1: Find the next term in the sequence: Inductive Reasoning – Is reasoning that is based on patterns you observe. If you observe a pattern in a sequence you can use inductive reasoning to find the next term. Ex. 1: Find the next term in the sequence: A) 3, 6, 12, 24, ___, ___ B) 1, 2, 4, 7, 11, 16, 22, ___, ___ C) 48 96 Rule: x2 29 37 Rule: +1, +2, +3, +4, …

Conjecture – A conclusion you reach using inductive reasoning. Inductive Reasoning assumes that an observed pattern will continue. This may or may not be true. Ex: x = x • x This is true only for x = 0 and x = 1 Conjecture – A conclusion you reach using inductive reasoning.

Ex. 2: Make a conjecture about the sum of the first 30 odd numbers. = 12 = 22 = 32 = 42 = 52 = 302 1 = 1 1 + 3 = 4 1 + 3 + 5 = 9 1 + 3 + 5 + 7 = 16 1 + 3 + 5 + 7 + 9 = 25 1 + 3 + 5 +...+ 30 = 900

To show that a conjecture is always true, you must prove it. To show that a conjecture is false, you have to find only one example in which the conjecture is not true. This case is called a counterexample. A counterexample can be a drawing, a statement, or a number.

Counter Example – To a conjecture is an example for which the conjecture is incorrect. Ex.1-3: The first 3 odd prime numbers are 3, 5, 7. Make a conjecture about the 4th. 3, 5, 7, ___ One would think that the rule is add 2, but that gives us 9 for the fourth prime number. Is that true? What is the next odd prime number? 11

Each year the price increased by $1.50. GEOMETRY LESSON 1-1 The price of overnight shipping was $8.00 in 2000, $9.50 in 2001, and $11.00 in 2002. Make a conjecture about the price in 2003. Write the data in a table. Find a pattern. 2000 $8.00 2001 2002 $9.50 $11.00 Each year the price increased by $1.50. A possible conjecture is that the price in 2003 will increase by $1.50. If so, the price in 2003 would be $11.00 + $1.50 = $12.50. 1-1

Inductive Reasoning 1. Look for a pattern. 2. Make a conjecture. 3. Prove the conjecture or find a counterexample.