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2.2 2.3 notes a
- 1. Opener: Study the following pattern. Row 1: 1 Row 2: 1 1 1. If the pattern is extended, find Row 3: 1 2 1 the terms in row 7. Row 4: 1 3 3 1 Row 5: 1 4 6 4 1 Row 6: 1 5 10 10 5 1 2. Make a conjecture for the pattern. 3. Make a conjecture for the sum of the terms in each row. 1
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- 3. 2‐2: Conditional Statements What are they? Hypothesis Conclusion Conditional Statements Notation Venn Diagram Examples: 1) If you are 16, then you are a teenager. 2) You live in Illinois, if you live in Northbrook. Venn Diagrams to represent conditionals: 1) If you are 16, then you are a teenager. 2) You live in Illinois, if you live in Northbrook. 3
- 4. Writing Conditional Statements: 1) Tarantulas are spiders. 2) A 25 degree angle is acute. Analyzing truth values: 1) If this is September, then the next month is October. 2) If 2 angles are obtuse, then they are congruent. 3) If an even number greater than 2 is prime, then 5 + 4 = 8. Important conclusion: If p is true, then ___________________________________ If p is false, then __________________________________ 4
- 5. Negation Converse Related Conditionals Inverse Contrapositive Examples: 1) FACT: Glenbrook North students are Spartans. Conditional: Converse: Inverse: Contrapositive: 2) Conditional:If m A = 95 degrees, then A is obtuse. TRUE or FALSE Converse: TRUE or FALSE Inverse: TRUE or FALSE Contrapositive: TRUE or FALSE Look for the pattern (what kind of reasoning?): Which statements always have the same truth value? 5
- 6. 2‐3: Deductive Reasoning A process of using logic to draw conclusions from given facts, definitions, and properties How is this different from inductive reasoning? Inductive or Deductive? 1) 2) There is a myth that birds will abandon their young if you touch them. 6
- 7. Types of Deductive Reasoning Law of Detachment Law of Syllogism If p q is a true If p q and q r, are statement,and p is true, true statements, then then q is also true. p r is a true statement. Example: If I oversleep, Example: If I get over then I will miss the bus. a 90%, then I get an A. If I miss the bus, then I I got a 96%. will have to walk to school. Conjecture: I got an A. Conjecture: If I oversleep then I will have to walk to school. 7
- 8. With your partner, decide whether each of the following examples is valid using Deductive Reasoning. Then decide which law it is demonstrating. Hold up the appropriate response cards. A) Given: If side lengths of a triangle are 5, 12, and 13, then the area of the triangle is 30. The area of PQR is 30. Conjecture: The side lengths of the triangle are 5, 12, and 13. B) Given: In the World Series, if a team wins 4 games, the team wins the Series. In 2004, the Red Sox won 4 games in the World Series. Conjecture: The Red Sox won the World Series. C) Given: If a figure is a kite, then it is a quadrilateral. If a figure is a quadrilateral, then it is a polygon. Conjecture: If a figure is a kite, then it is a polygon. D) Given:If a number is divisible by 2, then it is even. If a number is even, then it is an integer. Conjecture: If a number is an integer, then it is divisible by 2. 8
- 9. Draw a conclusion using Deductive Reasoning: A) Given: If 2y = 4, then z = ‐1. If x + 3 = 12, then 2y = 4. x + 3 = 12. Conclusion: B) Given: If the sum of two angles is 180 degrees, then the angles are supplementary. If 2 angles are supplementary, then they are not the angles in a triangle. m<A = 135 degrees and m<B = 45 degrees Conclusion: 9
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- 11. Conditional: If an animal is a brown bear, then the animal sleeps for 3 months each year. Related Statement: If an animal does not sleep for 3 months each year, then the animal is not a brown bear. 11
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- 13. Yard Enclosed by a fence Has a swimming pool Conditional: If a yard has a swimming pool, then it is enclosed by a fence. Related Statement: If a yard does not have a swimming pool, then it is not enclosed by a fence. 13
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