1. 2.3 Deductive Reasoning 2.4a Reasoning in Algebra

2. Today we will… Learning Target: Making inferences using the Law of Detachment and the Law of Syllogism.

3. Purpose (Please read/don’t write): You hear screeching tires, then a loud crash and breaking glass. You see nothing, but you infer that there has been a car accident. We all know the sounds of screeching tires and a crash. We know that these sounds almost always mean a car accident. But there could be some other reason, and therefore another explanation, for the sounds. Perhaps it was not an accident involving two moving vehicles. Maybe an angry driver rammed a parked car. Or maybe someone played the sound of a car crash from an iPod or a recording. Making inferences means choosing the most likely explanation from the facts at hand.

4. Types of Reasoning With your group, find the next two terms of the patterns: 1, 3, 7, 13, 21, … 2.1, -2.11, 2.111, -2.1111, … Inductive reasoning – observing what has happened and making a conjecture about what will happen. Deductive Reasoning (logical reasoning): the process of reasoning logically from given statements (ex: definition and theorems) to a conclusion. If the given statements are true in the situation, then the conclusion is also true. Ex: a physician sees a patient with certain symptoms and uses that information to diagnose the illness

5. Law of Detachment Ex 1: If a car’s battery is dead, then it will not start. A mechanic works on a car and finds that its battery is dead. What can we conclude? Ex 2: If a baseball player is a pitcher, then he should not play complete games on consecutive days. A baseball player pitches a complete game on Monday. What can we conclude about Tuesday’s game? Law of Detachment: If a conditional is true and its hypothesis is true, then its conclusion is true. Symbolic: If p q is a true statement and p is true, then q is true. Be careful: If it is snowing, then is the temperature is 32 o or lower. It is 20 o. Can I conclude that it is snowing?

6. Law of Syllogism Law of Syllogism: If p q and q r are true statements, then p r is a true statement. Ex 1: If Megan eats candy, then she will choose purple candy. If candy is purple, then it is grape flavored candy. Conclude: If Megan eats candy, then. Ex 2: If a figure is a square, then it is a rectangle. If a figure is a rectangle, then it is a parallelogram. Conclude: If a figure is a square, then. Be careful: If a number ends in 6, then it is divisible by 2. If a number ends in 4, then it is divisible by 2. Conclude: impossible – the conclusion of one statement is not the hypothesis of the other statement

7. All Together Ex: Use the Law of Detachment and the Law of Syllogism to draw conclusions… a. If you live in Fort Collins, then you live in Larimer County. If you live in Larimer County, then you live in Northern Colorado. b. If you live in Fort Collins, then you live in Larimer County. Annie lives in Fort Collins. From LOS: From LOD:

8. Properties (on theorem sheet)

9. Properties (on theorem sheet)

10. 2.3 Deductive Reasoning 2.4a Reasoning in Algebra HW: p. 96 #2-20 even, 32, 40