Obj. 9 Inductive Reasoning
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Obj. 9 Inductive Reasoning

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The student is able to (I can): ...

The student is able to (I can):

Use inductive reasoning to identify patterns and make conjectures
Find counterexamples to disprove conjectures
Identify, write, and analyze the truth value of conditional statements.
Write the inverse, converse, and contrapositive of a conditional statement.

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  • 1. Obj. 9 Inductive Reasoning Objectives: The student is able to (I can): • Use inductive reasoning to identify patterns and make conjectures • Find counterexamples to disprove conjectures • Identify, write, and analyze the truth value of conditional statements. • Write the inverse, converse, and contrapositive of a conditional statement.
  • 2. Find the next item in the sequence: 1. December, November, October, ... SeptemberSeptemberSeptemberSeptember 2. 3, 6, 9, 12, ... 15151515 3. , , , ... 4. 1, 1, 2, 3, 5, 8, ... 13131313 ———— This is called the FibonacciThis is called the FibonacciThis is called the FibonacciThis is called the Fibonacci sequence.sequence.sequence.sequence.
  • 3. inductive reasoning conjecture Reasoning that a rule or statement is true because specific cases are true. A statement believed true based on inductive reasoning. Complete the conjecture: The product of an odd and an even number is ______ . To do this, we consider some examples: (2)(3) = 6 (4)(7) = 28 (2)(5) = 10 eveneveneveneven
  • 4. counterexample If a conjecture is true, it must be true for every case. Just one exampleJust one exampleJust one exampleJust one example for which the conjecture is false will disprove it. A case that proves a conjecture false. Example: Find a counterexample to the conjecture that all students who take Geometry are 10th graders.
  • 5. Examples To Use Inductive Reasoning 1. Look for a pattern. 2. Make a conjecture. 3. Prove the conjecture or find a counterexample to disprove it. Show that each conjecture is false by giving a counterexample. 1. The product of any two numbers is greater than the numbers themselves. ((((----1)(5) =1)(5) =1)(5) =1)(5) = ----5555 2. Two complementary angles are not congruent. 45º and 45º45º and 45º45º and 45º45º and 45º
  • 6. Sometimes we can use inductive reasoning to solve a problem that does not appear to have a pattern. Example: Find the sum of the first 20 odd numbers. Sum of first 20 odd numbers? 1 1 + 3 1 + 3 + 5 1 + 3 + 5 + 7 1 4 9 16 12 22 32 42 202 = 400
  • 7. These patterns can be expanded to find the “nth” term using algebra. When you complete these sequences by applying a rule, it is called a functionfunctionfunctionfunction. Examples: Find the missing terms and the rule. To find the pattern, the coefficient of n is the difference between each term, and the value at 0 is what is added or subtracted. 1 2 3 4 5 … 8 … 20 … n -3 -2 -1 0 1 4 16 n — 4 1 2 3 4 5 … 8 … 20 … n 32 39 46 53 60 81 165 7n+25
  • 8. conditional statement hypothesis conclusion A statement that can be written as an “if-then” statement. Example: IfIfIfIf today is Saturday, thenthenthenthen we don’t have to go to school. The part of the conditional following the word “if”. “today is Saturday” is the hypothesis. The part of the conditional following the word “then”. “we don’t have to go to school” is the conclusion.
  • 9. Notation Examples Conditional statement: p → q, where p is the hypothesis and q is the conclusion. Identify the hypothesis and conclusion: 1. If I want to buy a book, then I need some money. 2. If today is Thursday, then tomorrow is Friday. 3. Call your parents if you are running late.
  • 10. Examples To write a statement as a conditional, identify the sentence’s hypothesis and conclusion by figuring out which part of the statement depends on the other. Write a conditional statement: • Two angles that are complementary are acute. If two angles are complementary, then theyIf two angles are complementary, then theyIf two angles are complementary, then theyIf two angles are complementary, then they are acute.are acute.are acute.are acute. • Even numbers are divisible by 2. If a number is even, then it is divisible by 2.If a number is even, then it is divisible by 2.If a number is even, then it is divisible by 2.If a number is even, then it is divisible by 2.
  • 11. truth value T if a conditional is true, F if a conditional is false. The statement is false only when theThe statement is false only when theThe statement is false only when theThe statement is false only when the hypothesis is true and the conclusion ishypothesis is true and the conclusion ishypothesis is true and the conclusion ishypothesis is true and the conclusion is false.false.false.false. To show that a conditional is false, you need only find one counterexample where the hypothesis is true and the conclusion is false. Hypothesis Conclusion Statement T T T TTTT FFFF FFFF F T T F F T
  • 12. Examples Determine if each conditional is true. If false, give a counterexample. 1. If your zip code is 76012, then you live in Texas. TrueTrueTrueTrue 2. If a month has 28 days, then it is February. September also has 28 days, whichSeptember also has 28 days, whichSeptember also has 28 days, whichSeptember also has 28 days, which proves the conditional false.proves the conditional false.proves the conditional false.proves the conditional false. 3. If 14 is evenly divisible by 3, then tomorrow is Tuesday. The hypothesis is false, so theThe hypothesis is false, so theThe hypothesis is false, so theThe hypothesis is false, so the conditional isconditional isconditional isconditional is truetruetruetrue.... Texas 76012
  • 13. negation of p “Not p” Notation: ~p Example: The negation of the statement “Blue is my favorite color,” is “Blue is notnotnotnot my favorite color.” Related Conditionals Symbols Conditional p → q Converse q → p Inverse ~p → ~q Contrapositive ~q →~p
  • 14. Example Write the conditional, converse, inverse, and contrapositive of the statement: “A cat is an animal with four paws.” Type Statement Truth Value Conditional (p → q) If an animal is a cat, then it has four paws. T Converse (q → p) If an animal has four paws, then it is a cat. F Inverse (~p → ~q) If an animal is not a cat, then it does not have four paws. F Contrapos- itive (~q → ~p) If an animal does not have four paws, then it is not a cat. T
  • 15. Example Write the conditional, converse, inverse, and contrapositive of the statement: “When n2 = 144, n = 12.” Type Statement Truth Value Conditional (p → q) If n2 = 144, then n = 12. F (n = —12) Converse (q → p) If n = 12, then n2 = 144. T Inverse (~p → ~q) If n2 ≠ 144, then n ≠ 12 T Contrapos- itive (~q → ~p) If n ≠ 12, then n2 ≠ 144 F (n = —12)