Do NOW: 1.) Solve 2(x – 15) = 30. Write a justification for each step. 2.) Find the next number in this sequence: 3, 5, 8, … How is the reasoning different.

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Do NOW: 1.) Solve 2(x – 15) = 30. Write a justification for each step. 2.) Find the next number in this sequence: 3, 5, 8, … How is the reasoning different in each problem?

2.4 Deductive Reasoning HW: p. 121 #1, 2, 5, 6, 8

OBJECTIVES Familiarize you with the deductive reasoning process Learn the relationship between inductive and deductive reasoning

Deductive vs. Inductive Reasoning The difference: inductive reasoning uses patterns to arrive at a conclusion (conjecture) deductive reasoning uses facts, rules, definitions or properties to arrive at a conclusion.

Deductive reasoning is the process of reasoning logically from given statements to a conclusion.

Examples of Inductive Reasoning 1)Every quiz has been easy. Therefore, the test will be easy. 2)The teacher used PowerPoint in the last few classes. Therefore, the teacher will use PowerPoint tomorrow. 3)Every fall there have been hurricanes in the tropics. Therefore, there will be hurricanes in the tropics this coming fall.

Example of Deductive Reasoning The catalog states that all entering freshmen must take a mathematics placement test. Conclusion: You will have to take a mathematics placement test. You are an entering freshman. An Example:

Inductive or Deductive Reasoning? Geometry example… 60 ◦ x Triangle sum property – the sum of the angles of any triangle is always 180 degrees. Therefore, angle x = 30° Deductive Reasoning – conclusion is based on a property

Inductive or Deductive Reasoning? Geometry example… What comes next? Colored triangle rotating 90° CW in the corners of the square Is there a rule? Inductive Reasoning

What professions do you think commonly use Deductive Reasoning?

This example shows inductive reasoning: Acute But now on the board let’s prove this using DEDUCTIVE REASONING:

If it is snowing, then the temperature is less than or equal to 32˚F. The temperature is 20˚F. The converse of the original assumption is not true, so It is not possible to conclude that it is snowing.

If M is the midpoint of a segment, then it divides the segment into two congruent segments. M is the midpoint of. You can conclude that M divides into two congruent segments, or

If an angle is obtuse, then it is not acute. is not obtuse. The converse of the assumption is not true, so It is not possible to conclude that is acute.

If a road is icy, then driving conditions are hazardous. Driving conditions are hazardous. The converse of the assumption is not true, so It is not possible to conclude that the road is icy.

There are three switches downstairs. Each corresponds to one of the three light bulbs in the attic. You can turn the switches on and off and leave them in any position. How would you identify which switch corresponds to which light bulb, if you are only allowed one trip upstairs?

Answer Turn one light switch one. Wait about 20 minutes, then turn it off. Turn another light on, and proceed upstairs. One light should be on, Two lights are off- however, one light is hot and one is cold.

Conditional Statements 2 parts are called the hypothesis & conclusion Can be written in “if-then” form; such as, “If…, then…” If you are a brunette, then you have brown hair. Is the converse of this statement true?