WARMUP Find the next three items in the pattern 0.01, 0.01001, 0.010010001….. 1,3,5,7…… 1,4,9,16,25…. Monday, Wednesday, Friday….

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WARMUP Find the next three items in the pattern 0.01, , ….. 1,3,5,7…… 1,4,9,16,25…. Monday, Wednesday, Friday….

2-1 INDUCTIVE REASONING

What is INDUCTIVE REASONING? INDUCTIVE REASONING: Process of reasoning that a rule or statement is true because specific patterns are true. CONJECTURE: Statement that is believed to be true based on inductive reasoning. Examples of Conjectures….. The sum of two odd numbers is always ______________________. The product of an even number and an odd number is always ___________________.

CONJECTURES CONTINUED… To show that a conjecture is always true, you must prove it. Conjecture: The product of an even number and an odd number is always even. Proof: Let n be an integer. (Any integer, it does not matter!!) If we multiply n by 2, then it must be an even number 2n. Let 2n+1 be an odd number. Let’s multiply our two numbers together 2n(2n+1)

PROOF CONTINUED… 2n(2n+1)= ???????????????????? How do we know this number is even? Therefore, an even number multiplied by an odd number is always even.

Your turn!! Make a conjecture about the following, then find the next two items of the sequence. 0,1,4,9,16,25,36,49… 0,1,8,27,64… Arrival Times at an airport: 3:00pm, 12:30pm, 10:00am…

With a partner… A farmer is traveling with a fox, a chicken, and a bag of corn seeds. He comes to a river that he must cross by boat. The boat only holds the farmer and one other item/animal. If you leave the chicken with the fox, then the fox will eat the chicken. If you leave the chicken with the corn seeds, the chicken will eat them. How do you get all three across safely?

COUNTEREXAMPLES In order to show that a conjecture is false, we only need to find one counterexample. This can be in the form of a drawing, statement, or a number. INDUCTIVE REASONING IDENTIFY A PATTERN MAKE A CONJECTURE PROVE THE CONJECTURE OR FIND A COUNTER EXAMPLE.

Find a counterexample CONJECTURE: ALL RECTANGLES ARE SQUARES COUNTEREXAMPLE: A rectangle with a length of 12 and a width of 10. CONJECTURE: Let x be a number, then COUNTEREXAMPLE: THINK ABOUT FRACTIONS!!!!

Your turn!! Determine if each conjecture is true or false. If false, provide a counter example

WARMUP Can you find the solution… There are 3 boxes. One is labeled “APPLES,” another is labeled “ORANGES.” The last one is labeled “APPLES AND ORANGES.” You know that each box is labeled incorrectly. You have one chance to pick a fruit from any one box. Which one should you pick from to allow you to correctly label the boxes?

2-2 CONDITIONAL STATEMENTS

If-then statements…. If today is Thursday, then_____________________________. If an animal is a whale then___________________________. If a number is an integer then it is a natural number.

Conditional statements A conditional statement is written in the form of “If p, then q.” The letter p represents our ______________________. The letter q represents our ______________________.

Let’s practice! Rewrite each of the following as a conditional statement. 1. A student must have a grade of 90 to make an A. 2. Water will boil at 212 degrees Farenheit. 3. I will be cheering for the Gamecocks if it is Saturday.

How did you do? If a student has a grade of 90, then that student has an A. If it is 212 degrees Farenheit, then water will boil. If it is Saturday, then I will be cheering for the Gamecocks.

NEGATION Negation of a statement, p, is “not p.” The negation of a true statement is false. Similarly, the negation of a false statement is true…..

Negation examples Conditional Negate the following statements: 1. Mr. Thomas is not the coolest teacher in the world. 2. Fox Creek will win on Friday. 3. The Gamecocks are better than Clemson 4. M is the midpoint of segment AB. Negation 1. Mr. Thomas is the coolest teacher in the world. 2. Fox Creek will not win on Friday. 3. The Gamecocks are not better than Clemson 4. M is not the midpoint of segment AB.

DEFINITIONSYMBOLS Conditional “if p then q.”p---->q Converse: formed by switching hypothesis and conclusion q---->p Inverse: Negate the hypothesis and conclusion. ~p---->~q Contrapositive: Switch hypothesis and conclusion. Then negate both. ~q---->~p

LOGICALLY EQUIVALENT STATEMENTS All conditionals have two possible truth values: True or False If related conditional statements all have the same truth value then they are logically equivalent statements. Conditionals are only false when the hypothesis (p) is true and the conclusion (q) is false.