Inductive Reasoning

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Inductive Reasoning

  1. 1. Inductive Reasoning <ul><li>Observation (Given) </li></ul><ul><li>Conjecture(s) (educated guess) </li></ul><ul><li>True Example(s) </li></ul><ul><li>One Counterexample </li></ul>
  2. 2. Writing Conjectures <ul><li>She refused your request for a date </li></ul><ul><li>WMD were NOT found in Iraq </li></ul><ul><li>No terrorist acts in U.S. for last 3 years </li></ul><ul><li>Mars rover found rounded, smooth stones </li></ul><ul><li>The sun is near the eastern horizon </li></ul><ul><li>The moon is brightly visible </li></ul><ul><li>The grass is wet </li></ul><ul><li>The bread popped up, but was not toasted </li></ul><ul><li>The car won’t start </li></ul>Given:
  3. 3. Writing Conjectures <ul><li>Someone has more than 3 exceptions in this class </li></ul><ul><li>Joe scored a level 5 on his FCAT math exam </li></ul><ul><li>Molly is a cheerleader </li></ul><ul><li>Gary is absent often </li></ul><ul><li>Billy is a goat </li></ul><ul><li>Henry is sleeping in art class </li></ul><ul><li>Max has a positive, “can do” attitude </li></ul><ul><li>I imagine 3 noncollinear points </li></ul><ul><li>The polygon has 7 sides </li></ul>Given:
  4. 4. <ul><li>Given: Points A, B, C are collinear </li></ul><ul><ul><li>Conjecture 1: B is between A and C </li></ul></ul><ul><ul><li>Conjecture 2: Only 1 plane can be constructed </li></ul></ul><ul><ul><li>Conjecture 3: Exactly 1 line with point B can be constructed </li></ul></ul>Are the conjectures true? Find one counterexample
  5. 5. Are the conjectures true? <ul><li>Given: 2 intersecting lines </li></ul><ul><ul><li>Conjecture 1: Exactly 3 angles are formed </li></ul></ul><ul><ul><li>Conjecture 2: Adjacent angles are linear pairs </li></ul></ul><ul><ul><li>Conjecture 3: Exactly 1 pair of vertical angles are formed </li></ul></ul>Find one counterexample
  6. 6. <ul><li>Given: a perimeter of 80 feet </li></ul><ul><ul><li>Conjecture 1: A rectangle of length 25 and width 20 can be constructed </li></ul></ul><ul><ul><li>Conjecture 2: 2 squares with different side lengths can be constructed. </li></ul></ul><ul><ul><li>Conjecture 3: The largest quadrilateral area that can be enclosed is 25 X 15 = 375 sq ft </li></ul></ul>Are the conjectures true? Find one counterexample
  7. 7. Inductive Reasoning <ul><li>Given a fact </li></ul><ul><li>State/write a conjecture (educated guess) </li></ul><ul><li>Find true examples </li></ul><ul><li>Find One Counterexample </li></ul>
  8. 8. <ul><li>Given: 3 noncollinear point </li></ul><ul><ul><li>Conjecture 1: Exactly one line can be drawn </li></ul></ul><ul><ul><li>Conjecture 2: 2 Exactly one plane can be drawn </li></ul></ul>Are the conjectures true? Find one counterexample
  9. 9. Conjunctions and Disjunctions <ul><li>It’s raining It’s Monday </li></ul>p q <ul><li>It’s raining and It’s Monday </li></ul><ul><li>It’s raining or It’s Monday </li></ul>p p q q Λ ν
  10. 10. Negations <ul><li>It’s raining It’s Monday </li></ul>p q <ul><li>It’s NOT raining and It’s Monday </li></ul><ul><li>It’s raining or It’s NOT Monday </li></ul>p p q q Λ ν ~ ~
  11. 11. Paper 50 Cans 30 10 Communities that recycle VENN DIAGRAM
  12. 12. TRUTH TABLE <ul><li>It’s raining It’s Monday </li></ul>p q F F F F T F T F T F F T T T T T p ν q p Λ q q p
  13. 13. TRUTH TABLE <ul><li>I broke curfew I’m grounded </li></ul>p q F F F F T F T F T F F T T T T T p ν q p Λ q q p
  14. 14. TRUTH TABLE <ul><li>It’s raining It’s Monday It’s 3rd period </li></ul>p q r F F F F F T F T F F F F F T F T F T T F F F F F T T F T F T T T F T T T T T T T ( p Λ q ) ν r p Λ q r q p
  15. 15. Conditionals <ul><li>If it rained , then the grass is wet. </li></ul><ul><li>If there was life on Mars , then there was water on Mars. </li></ul><ul><li>If it’s a duck , then it quacks. </li></ul><ul><li>If 2 angles are supplementary , then their sum is 180 degrees. </li></ul><ul><li>If a polygon is a triangle , then it has 3 </li></ul><ul><li>sides. </li></ul>Hypothesis Conclusion IF THEN
  16. 16. Write Conditionals <ul><li>Ducks Quack </li></ul><ul><li>She only dates handsome men </li></ul><ul><li>It’s dark during the night </li></ul><ul><li>2 Perpendicular lines form 4 right angles </li></ul><ul><li>Linear pairs are supplementary </li></ul><ul><li>3 noncollinear points determine a plane </li></ul><ul><li>Vertical angles are congruent </li></ul>Hypothesis Conclusion IF THEN
  17. 17. Conditional <ul><li>If it rained , then the grass is wet . </li></ul><ul><li>If </li></ul><ul><ul><li>It rained </li></ul></ul><ul><li>Then </li></ul><ul><ul><li>The grass is wet </li></ul></ul>Hypothesis Conclusion
  18. 18. Converse <ul><li>If it’s a duck , then it flies . </li></ul><ul><li>If it flies , then it’s a duck </li></ul>Conditional Converse
  19. 19. Inverse <ul><li>If it’s a duck , then it flies . </li></ul><ul><li>If it’s NOT a duck , then it does NOT fly. </li></ul>Conditional Inverse
  20. 20. Contrapositive <ul><li>If it’s a duck , then it flies . </li></ul><ul><li>If it does NOT fly , then it’s NOT a duck </li></ul>Conditional Contrapositive
  21. 21. Converse <ul><li>If it rained , then the grass is wet . </li></ul><ul><li>If the grass is wet , then it rained </li></ul>Conditional Inverse
  22. 22. Inverse <ul><li>If it rained , then the grass is wet . </li></ul><ul><li>If it did NOT rain , then the grass is NOT wet </li></ul>Conditional Inverse
  23. 23. Contrapositive <ul><li>If it rained , then the grass is wet . </li></ul><ul><li>If the grass is NOT wet , then it did NOT rain . </li></ul>Conditional Contrapositive
  24. 24. p Λ q p ν q p q p q
  25. 25. Write Conditionals <ul><li>Given: Ducks are birds </li></ul><ul><li>Write the conditional: </li></ul><ul><li>Write the converse: </li></ul><ul><li>Write the inverse: </li></ul><ul><li>Write the contrapositive: </li></ul>
  26. 26. Law of Detachment <ul><li>1. If it’s a duck then it flies </li></ul><ul><li>2. It’s a duck </li></ul><ul><li>3. CONCLUSION: it flies </li></ul>Valid p q p q IF THEREFORE THEN
  27. 27. Law of Detachment <ul><li>1. If it’s a duck then it flies </li></ul><ul><li>2. It flies </li></ul><ul><li>3. CONCLUSION: it’s a duck </li></ul>Invalid p q q p IF THEREFORE THEN
  28. 28. Law of Detachment <ul><li>1. If then </li></ul><ul><li>2. </li></ul><ul><li>3. CONCLUSION: </li></ul>p q q p IF THEREFORE THEN It's a duck It flies It flies It's a duck
  29. 29. Law of Detachment <ul><li>1. If it rained , then the grass is wet . </li></ul><ul><li>2. It rained </li></ul><ul><li>3. CONCLUSION: the grass is wet </li></ul>Valid p q p q IF THEREFORE THEN
  30. 30. Law of Detachment <ul><li>1. If then </li></ul><ul><li>2. </li></ul><ul><li>3. CONCLUSION: </li></ul>It's a bird It has wings It has wings It's a bird It's a bird It has wings
  31. 31. Law of Detachment <ul><li>1. If it rains, then the grass will get wet. </li></ul><ul><li>2. The grass is wet </li></ul><ul><li>3. CONCLUSION: it rained </li></ul>Invalid A B B A IF THEREFORE THEN
  32. 32. Law of Syllogism <ul><li>1. If it rains , then the grass is wet </li></ul><ul><li>2. If the grass is wet , then I won’t mow . </li></ul><ul><li>3. It rained </li></ul><ul><li>4. CONCLUSION: I won’t mow </li></ul>Valid p q p r IF THEREFORE THEN q r IF THEN
  33. 33. Law of Syllogism <ul><li>1. If it rains, then the grass is wet </li></ul><ul><li>2. If the grass is wet, then I won’t mow. </li></ul><ul><li>3. I didn’t mow </li></ul><ul><li>4. CONCLUSION: The grass is wet </li></ul>Invalid A B C B IF THEREFORE THEN B C IF THEN
  34. 34. Law of Syllogism <ul><li>1. If it’s a duck , then it flies . </li></ul><ul><li>2. If it flies , then it has wings . </li></ul><ul><li>3. It’s a duck </li></ul><ul><li>4. CONCLUSION: it has wings </li></ul>Valid p q p r IF THEREFORE THEN q r IF THEN
  35. 35. Law of Syllogism <ul><li>1. If it’s a duck , then it flies . </li></ul><ul><li>2. If it flies , then it has wings . </li></ul><ul><li>3. It has wings </li></ul><ul><li>4. CONCLUSION: It’ a duck </li></ul>Invalid p q r p IF THEREFORE THEN q r IF THEN
  36. 36. Law of Syllogism <ul><li>1. If it rains, then the grass will get wet. </li></ul><ul><li>2. If the grass is wet, I won’t mow </li></ul><ul><li>3. It rained </li></ul><ul><li>4. CONCLUSION: I won’t mow </li></ul>Valid A B A B IF THEREFORE THEN B C IF THEN
  37. 37. IF IF THEN THEN It's a bird It has wings It has wings It's a bird It's a bird It has wings It lays eggs It lays eggs It lays eggs THEREFORE
  38. 38. Deductive Reasoning <ul><li>1. If Alex takes the car to the store, he will stop at the post office. </li></ul><ul><li>2. If Alex stops at the post office, he will buy stamps. </li></ul><ul><li>What can you conclude using Law of Syllogism? </li></ul>
  39. 39. Deductive Reasoning <ul><li>If the circus is in town, then there are tents at the fairground. If there are tents at the fairground, then Paul is working as a night watchman. </li></ul><ul><li>The circus is in town </li></ul><ul><li>There are tents at the fairgrounds </li></ul><ul><li>Write your conclusions: </li></ul><ul><li>If a. above is true </li></ul><ul><li>If b. above is true </li></ul>
  40. 40. Postulates <ul><li>Thru any 2 points, there is exactly one line </li></ul><ul><li>Thru any 3 noncollinear points, there is exactly one plane </li></ul><ul><li>A line contains at least 2 points </li></ul><ul><li>A plane contains at least 3 noncollinear points </li></ul>
  41. 41. Postulates <ul><li>If 2 points lie in a plane, then the line containing those points lies in the plane </li></ul><ul><li>If 2 lines intersect, then their intersection is exactly one point </li></ul><ul><li>If 2 planes intersect, then their intersection is a line </li></ul>
  42. 42. Properties of Equality <ul><li>Reflexive: a = a </li></ul><ul><li>Symmetric: If a = b, then b = a </li></ul><ul><li>Transitive: If a = b, and b = c, then a = c </li></ul><ul><li>Addition & Subtraction: If a = b, then a + c = b + c. If a = b , then a – c = b – c </li></ul><ul><li>Multiplication & Division: If a = b, then ac = bc. If a = b, then a/c = b/c (c  0) </li></ul><ul><li>Substitution: If a = b, then a may be replaced with b </li></ul><ul><li>Distributive: a(b + c) = ab + ac </li></ul>
  43. 43. Name the property <ul><li>1 If 3x = 120, then x = 40 </li></ul><ul><li>If 13 = AB, then AB = 13 </li></ul><ul><li>If y = 75, and y = m  A, then m  A = 75 </li></ul><ul><li>If AB = BC, and BC = CD, then AB = CD </li></ul>
  44. 44. 2 Column Proof Given: 3x + 5 2 Prove: x = 3 = 7 Statements Reasons
  45. 45. Properties of Segments <ul><li>Reflexive: AB = AB </li></ul><ul><li>Symmetric: If AB = CD, then CD = AB </li></ul><ul><li>Transitive: If AB = CD, and CD = EF, then AB = EF </li></ul>
  46. 46. Properties of Angles <ul><li>Reflexive: m  1 = m  1 </li></ul><ul><li>Symmetric: If m  1 = m  2, then m  2 = m  1 </li></ul><ul><li>Transitive: If m  1 = m  2, and m  2 = m  3, then m  1 = m  3 </li></ul>
  47. 47. Segment Postulates <ul><li>Ruler Postulate: Any segment can be measured </li></ul><ul><li>Segment Addition Postulate: If B is between A and C, then AB + BC = AC </li></ul>
  48. 48. Properties of Segment Congruence <ul><li>Reflexive: AB  AB </li></ul><ul><li>Symmetric: If AB  CD, then CD  AB </li></ul><ul><li>Transitive: If AB  CD, and CD  EF, then AB  EF </li></ul>
  49. 49. Angle Postulates <ul><li>Protractor Postulate: Any angle can be measured </li></ul><ul><li>Angle Addition Postulate: If R is inside  PQS, then m  PQR + m  RQS = m  PQS </li></ul>
  50. 50. Properties of Angles <ul><li>Reflexive: m  1  m  1 </li></ul><ul><li>Symmetric: If m  1  m  2, then m  2  m  1 </li></ul><ul><li>Transitive: If m  1  m  2, and m  2  m  3, then m  1  m  3 </li></ul>
  51. 51. Angle Theorems <ul><li>Linear pairs are supplementary </li></ul><ul><li>Adjacent angles that form a right angle are complementary </li></ul><ul><li>Angles supplementary to the same angle or to congruent angles are congruent. </li></ul><ul><li>Angles complementary to the same angle or to congruent angles are congruent </li></ul><ul><li>Vertical angles are congruent </li></ul>
  52. 52. Angle Theorems <ul><li>Perpendicular lines intersect to form 4 right angles </li></ul><ul><li>All right angles are congruent </li></ul><ul><li>Perpendicular lines form congruent adjacent angles </li></ul><ul><li>If 2 angles are congruent and supplementary, then each angle is a right angle </li></ul><ul><li>If 2 congruent angles form a linear pair, then they are right angles </li></ul>
  53. 53. Name the property <ul><li>1 m  1 = m  1 </li></ul><ul><li>If AB + BC = DE + BC, then AB = DE </li></ul><ul><li>If XY = PQ and X Y = R S, then PQ = RS </li></ul><ul><li>If ⅓ x = 5, then x = 15 </li></ul><ul><li>If 2x = 9, then x = 9/2 </li></ul>
  54. 54. 2 Column Proof Given: PR = QS Prove: PQ = RS Statements Reasons P Q R S

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