Patterns and Reasoning. CHAPTER 2 SECTIONS 1-3. Patterns:  What comes next?  Examples: 1. 2. 2, 4, 6, 8, , , __ 3. 5, 7, 14, 16, 32, 34,

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Presentation transcript:

Patterns and Reasoning. CHAPTER 2 SECTIONS 1-3

Patterns:  What comes next?  Examples: , 4, 6, 8, ____, ______, ____ 3. 5, 7, 14, 16, 32, 34, ____, ______, _______

Conjectures  a statement about what you think will happen based on the pattern you observed Pattern Conjecture Every day Sandra-Q wears Sandra-Q’s favorite a blue sweater.color is blue. Javier scores at least 4 pointsJavier is the best player during each basketball game.on the basketball team.

Counterexample:  Proves that a conjecture is false. ConjectureCounterexample All birds fly.Penguin If an animal has a tail then it’s a lobster.Dog

How is it used in Math? Find one counterexample to show that each conjecture is false.  All numbers that are divisible by 3 are also divisible by 6.  All whole numbers are greater than their opposites.  All prime numbers are odd integers

 Find one counterexample to show that each conjecture is false. 1. The result of a number multiplied by a positive integer is always larger than the original number. 2. A four-sided figure with four right angles is a square. 3. February has exactly 28 days every year.

Conditional Statements:  an “if” clause and a “then” clause Example: If Shay does not complete her assignments, then she will fail the class. HypothesisConclusion If DeAngeles eats too much ice cream, then he will have a stomach ache. HypothesisConclusion

Conditionals and Their Possy  Converse  opposite (switch the hypothesis and the conclusion) Example: Conditional: If it is raining, then the grass is wet. Converse: If the grass is wet, then it is raining.

CONVERSE EXAMPLES: Write the converse of the following conjectures: 1. If an animal is grey, then it is an elephant. 2. If its 2:55, then its still not time to leave. 3. If the rain stopped, then you will see a rainbow. Conditionals and Their Possy

 Inverse  adding not to each side of the statement Example: Conditional: If it is raining, then the grass is wet. Inverse: If it is not raining, then the grass is not wet. Conditionals and Their Possy

INVERSE EXAMPLES: Write the inverse of the following conjectures: 1. If an animal is grey, then it is an elephant. 2. If its 2:55, then its still not time to leave. 3. If the rain stopped, then you will see a rainbow. Conditionals and Their Possy

 Contrapositive  Switch the hypothesis and the conclusion AND insert not Example: Conditional: If it is raining, then the grass is wet. Inverse: If the grass is not wet, then it is not raining. Conditionals and Their Possy

Contrapositive EXAMPLES: Write the contrapositive of the following conjectures: 1. If an animal is grey, then it is an elephant. 2. If its 2:55, then its still not time to leave. 3. If the rain stopped, then you will see a rainbow. Conditionals and Their Possy

State whether each conditional is true or false. Write the contrapositive for the conditional and state whether the contrapositive is true or false. 1. If you only have $15, then you can buy a meal that costs $ If you received an “A” in this class, then you received 90%. How is it used in Math?

Biconditionals  When both the conditional and the converse are true.  Uses “If and only if” Steps:  Check to see if the conditional statement is true  Check to see if the converse statement is true  Write it using “if and only if”

Biconditionals EXAMPLES: 1. If a whole number is a multiple of 2, then the whole number is even. 2. Rabbits are animals that eat carrots. 3. Two lines that intersect to form four 90° angles are perpendicular. 4. Mammals are warm-blooded animals.