UNIT 2 Geometric Reasoning 2.1 What is the next item in each sequence? 2, 4, 6, 8, 10, ____ 2, 3, 5, 7, 11, 13, _____ J, F, M, A, M, J, J, _____ O, T, T, F, F, S, S, E, N, ______ 1, 1, 2, 3, 5, 8, 13, 21, ______ , , , , _______ How did you determine the next item in the sequence?

UNIT 2 Geometric Reasoning 2.1 Inductive reasoning is the process of reasoning that a rule or statement is true because specific cases are true. You may use inductive reasoning to draw a conclusion from a pattern. A statement you believe to be true based on inductive reasoning is called a conjecture.

UNIT 2 Geometric Reasoning 2.1 Complete the conjecture: The sum of any two positive integers is _____. List some examples and look for a pattern. 1 + 1 = 2 3.14 + 0.01 = 3.15 3,900 + 1,000,017 = 1,003,917 The sum of two positive numbers is positive.

UNIT 2 Geometric Reasoning 2.1 Make a conjecture for each of the following statements: The product of two odd numbers is ? . The number of lines formed by 4 points, no three of which are collinear, is ? . The number of non overlapping segments formed by n collinear points is _____.

UNIT 2 Geometric Reasoning 2.1 The cloud of water leaving a whale’s blowhole when it exhales is called its blow. A biologist observed blue-whale blows of 25 ft, 29 ft, 27 ft, and 24 ft. Another biologist recorded humpback-whale blows of 8 ft, 7 ft, 8 ft, and 9 ft. Make a conjecture based on the data. Heights of Whale Blows Height of Blue-whale Blows 25 29 27 24 Height of Humpback-whale Blows 8 7 9

UNIT 2 Geometric Reasoning 2.1 All conjectures have a truth value associated with them. If the truth value is TRUE, then the statement is true all of the time. If the truth value is FALSE, then there exists at least one case where the conjecture statement is not true. 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. Find counterexamples for each statement: If the driveway is wet, then it is raining. If it is February, then the month only has 28 days. If it is a weekend, then it is Saturday. All odd numbers less than 10 are prime. If two angles are complimentary, then they are not ≌.

If _____________, then ______________ UNIT 2 Conditional Statements 2.2 A conditional statement is a statement written with the form If _____________, then ______________ hypothesis conclusion P→Q P Q The inner oval represents the hypothesis, and the outer oval represents the conclusion.

UNIT 2 Conditional Statements 2.2 Identify the hypothesis and conclusion of each conditional. A. If today is Thanksgiving Day, then today is Thursday. Hypothesis: Today is Thanksgiving Day. Conclusion: Today is Thursday. B. A number is a rational number if it is an integer. Hypothesis: A number is an integer. Conclusion: The number is a rational number.

UNIT 2 Conditional Statements 2.2 Many sentences without the words if and then can be written as conditionals. To do so, identify the sentence’s hypothesis and conclusion by figuring out which part of the statement depends on the other. Write a conditional statement from the following. An obtuse triangle has exactly one obtuse angle. If a triangle is obtuse, then it has exactly one obtuse angle. Identify the hypothesis and the conclusion. A triangle is obtuse It has exactly one obtuse angle.

UNIT 2 Conditional Statements 2.2 Write a conditional statement for each: “Two angles that are complementary are acute.” If two angles are complementary, then they are acute. Identify the hypothesis and the conclusion. Two angles that are complementary They are acute. If an animal is a blue jay, then it is a bird. An animal is a blue jay It is a bird

UNIT 2 Conditional Statements 2.2 The Conditional Statement (If p, then q) has related logical statements A conditional is a statement that can be written in the form "If p, then q." p  q The converse is the statement formed by exchanging the hypothesis and conclusion. q  p The inverse is the statement formed by negating the hypothesis and conclusion. ~p  ~q The contrapositive is the statement formed by negating and exchanging the hypothesis and conclusion. ~q  ~p

UNIT 2 Conditional Statements 2.2 Write the conditional, converse, inverse and contrapositive for “If two angles are adjacent, then they share a vertex”. If two angles are adjacent, then they share a vertex p  q If they share a vertex, then two angles are adjacent. q  p If two angles are not adjacent, then they do not share a vertex. ~p  ~q If they do not share a vertex, then two angles are not adjacent. ~q  ~p Evaluate the truth value for each statement.

UNIT 2 Deductive Reasoning 2.3 Consider the following TRUE conditional statement… If you miss the bus, then you will be late for school. Tony missed the bus. What can you conclude? Gina was late for school. What can you conclude?

UNIT 2 Deductive Reasoning 2.3 You only need one counter example to show a statement is false. To prove the statement is TRUE using inductive logic, you must show it is true for every possible case! Mathematicians turn to deductive reasoning to prove conjectures are true. Deductive logic uses facts, definitions and properties to draw conclusions.

UNIT 2 Deductive Reasoning 2.3 The Law of Detachment is one valid form of deductive reasoning. Law of Detachment If p  q is a true statement and p is true, then q is true. Consider the conditional statement: If you miss the bus, then you will be late for school. Tony missed the bus. Gina was late for school.

UNIT 2 Deductive Reasoning 2.3 Conditional: If a team wins four games in the World Series, then the team will be world champions. The St. Louis Cardinals won four games in the 2006 World Series. What can you conclude? Conditional: If it is raining, then the garden is watered. Mr. Mack looks out the window and sees it is raining.

UNIT 2 Deductive Reasoning 2.3 Determine if the conjecture is valid by the Law of Detachment. Given: If a student passes his classes, then the student is eligible to play sports. Ramon passed his classes. Conjecture: Ramon is eligible to play sports. Given: If it is raining, then the garden is watered. The garden is watered. Conjecture: Mrs. Mack watered the garden.

UNIT 2 Deductive Reasoning 2.3 Law of Syllogism: Allows you to state a conclusion from two true conditional statements when the conclusion of one statement is the hypothesis of the other statement. Law of Syllogism If p  q and q  r are true statements, then p  r is a true statement. p  q q  r p  r If a quadrilateral is a square, then it contains four right angles. If a quadrilateral contains four right angles, then it is a rectangle.

UNIT 2 Deductive Reasoning 2.3 The law of syllogism allows you to effectively “combine” two conditional statements into one conditional statement. Combine each pair of conditional statements into one conditional statement. If you liked the movie then you saw a good movie If you saw a good movie then you enjoyed yourself. If two lines are not parallel then they intersect. If two lines intersect then they intersect at a single point. If you like the ocean then you will like Florida. If you vacation at the beach then you like the ocean.

UNIT 2 Deductive Reasoning 2.3 What conclusion(s) can be made from each set of statements If a person visits the zoo they will see animals. Karla goes to the zoo. If it snows then we will miss school. If we miss school then practice is cancelled. Practice is canceled. If you drive over the speed limit, then you will receive a ticket Tom is driving over the limit. Sarah received a ticket.

UNIT 2 Deductive Reasoning 2.3 If you are late for practice then you won’t start the game. If you don’t start the game, then your name won’t be announced. Marcus didn’t start the game If a person goes to the zoo, they will see animals. Richard sees animals.

HOMEWORK: 2.1(77): 24-28,42 2.2(84): 14-17,23,31-35 (odd) 2.3(91): 9-19,22