Inductive and Deductive Reasoning

1. Inductive and Deductive Reasoning Math Foundations 11

2. Let’s play a little game Pick the number of days per week that you like to eat chocolate Multiply this number by 2 Now, add 5 Multiply this new number by 50 If you’ve already had your birthday this year, add 1764, if not, add 1763 Now, subtract the four digit year that you were born What do you notice about your answer? The first digit is your original #, and the last two digits your age

3. Here’s a Fun Quiz Think of a number from 1 -10 Multiply that number by 9 If the number is 2 digits, add the digits together Now, subtract 5 Convert your number to a letter: A=1, B=2, C=3, D=4, … Think of a country that starts with that letter Remember the last letter of the name of your country, and think of an animal that starts with that letter Remember the last letter of your animal, and think of a fruit that starts with that letter Are you thinking about …..

4. Orange Kangaroo from Denmark?

5. Proof that 2 = 1 Let a=b So, a2 = ab a2 - b2 = ab - b2 (a-b)(a+b) = b(a-b) (a+b) = b b + b = b 2b = b 2 = 1

6. 4 – 9’s Apparently, all the numbers from 0 to 100 can be created from four nines Ex: 9 + 9 – 9 - 9 = 0 Ex: 9/9 + 9 – 9 = 1 Ex 9/9 + 9/9 = 2 Are there any numbers that can’t be done?

7. 1.1 Inductive Reasoning Inductive reasoning – drawing a general conclusion by observing patterns and identifying properties in specific examples. Conjecture – A testable expression that is based on available evidence but is not yet proved.

8. Example 1(8) + 1 = 9 12(8) + 2 = 98 123(8) + 3 = 987 1234(8) +4 = 9876 12345(8) + 5 = ____ Does this pattern continue forever? No!

9. Verbal example This swan is white I’ve seen 100 white swans All swans are white Is this always true? No!

10. Assignment – Page 12 # 3 – 17 odd

11. Section 1.2

12. Section 1.2

13. Section 1.2

14. Section 1.2

15. 1.3 Using Reasoning to Find a counterexample to a Conjecture A conjecture can be proven false by finding a counter example

16. Statement: A number that is not negative is positive

17. Statement: A number that is not negative is positive Counter example : This statement is proven false by finding a counter example o

18. Statement: All prime numbers are odd

19. Statement: All prime numbers are odd Counter example : This statement is proven false by finding a counter example 2

20. Statement: All NBA basketball players are tall

21. Statement: All NBA basketball players are tall Counter example : This statement is proven false by finding a counter example Nate Robinson (5’7”)

22. Statement: The square root of a positive number is always less than the number itself

23. Statement: The square root of a number is always less than the number itself Counter example : This statement is proven false by finding a counter example 1

24. Statement: As you travel north, the climate gets colder

25. Statement: As you travel north, the climate gets colder Counter example : This statement is proven false by finding a counter example Southern hemisphere

26. Assignment Page 22 #2 – 9, 12

27. 1.4 Deductive Reasoning • Deductive reasoning is a process where we draw conclusions using logic that is based on facts we accept as true • A conjecture is proved true only when it is true for every case. This is done by creating a proof for general cases.

28. Example All poodles are dogs All dogs are mammals

29. Example All poodles are dogs All dogs are mammals All poodles are mammals

30. Example Lynn is a math student All math students have a calculator

31. Example Lynn is a math student All math students have a calculator Lynn has a calculator

32. Integer Property Proof • Every integer is either ODD or EVEN • Let j, k be random integers • Need an even integer? Let = an even integer • Need an odd integer? Let = an odd integer

33. “Let” statements • Need consecutive integers? • Let k = 1st integer • = 2nd integer • = 3rd integer • Need consecutive even integers? • Let = 1st even integer • Let = 2nd even integer • Need consecutive odd integers? • Let =1st odd integer • Let = 2nd odd integer

34. Example Prove that the square of any odd integer is odd Let = 1st odd integer

35. Example Prove that the product of any 2 even integers is even Let 2k = 1st even integer Let 2j = 2nd even integer Product = (1st integer)(2nd integer) = (2k)(2j) = 4kj This number is divisible by 2, therefore it is EVEN

36. Assignment Page 31 # 1, 2, 4 – 8, 10, 12, 17, 20

37. 1.5 Proofs that are not valid Jean Chretien: http://www.youtube.com/watch?v=aX6XMIldkRU

38. 1.5 Invalid Proofs Invalid Proof – a proof that contains an error in reasoning or that contains invalid assumptions. Circular Reasoning – an argument that is incorrect because it makes use of the conclusion to be proved.

39. 1.6 Reasoning to Solve Problems If 3 cats can catch 3 mice in 3 minutes, how long will it take 100 cats to catch 100 mice?

40. 1.6 Reasoning to Solve Problems A bottle and a cork together cost $1.06. The bottle cost $1 more than the cork. How much does the cork cost?

41. 1.6 Reasoning to Solve Problems Assuming both players to be intelligent, who started this game of X's and O's, player X or player O? Explain.

42. 1.6 Reasoning to Solve Problems Suppose you are lost in the woods for hours. You suddenly come upon a cabin. In the cabin you find a lantern, a candle, a wood stove with wood in it , and a match. What do you light first?

43. 1.6 Reasoning to Solve Problems A man went to town one day with $5 in his pocket, but returned in the evening with $15. He bought a hat at the men's furnishings store and some meat at the meat market. Then he had his eyes tested for glasses. Now, this man got paid every Thursday by check, and the banks in the town are open on Tuesday, Friday, and Saturday only. The eye doctor does not keep his office open on Saturday, and the meat market is not open on Thursday or Friday. What day did the man go to town? Tuesday

44. 1.6 Reasoning to Solve Problems Mr. Jones one day got off the train in Chicago and while passing through the station met a friend he had not seen in years. With his friend was a little girl. "Well, I certainly am glad to see you," said Mr. Jones. "Same here," said his friend. "Since I last saw you I've been married--to someone you never knew. This is my little girl." "I'm glad to meet you," said Mr. Jones. "What's your name? "It's the same as my mother's," answered the little girl. "Oh, then your name is Anne," said Mr. Jones. How did he know? His friend was a lady, named Anne

45. 1.6 Reasoning to Solve Problems There are three light bulbs in a room, and three light switches outside the room. You are outside, and want to match up which switch goes with which light bulb. You can only travel into the room once, and cannot come back in again. You can do anything you want upon entering the room. How can you set the situation so that you will know which switch goes with which light bulb? Turn two of the light switches on. Wait a while, and then turn one of them off, then quickly enter the room. One light will be on, one off, and one off but still hot.

46. Pepsi Problem – quite hard A man has a 12 L jug of pepsi, and wants to split it in half, but only has a 8 L and a 5 L jug. How can he do it? Fill the 8 L jug (then we have (4,8,0) in the 12 L, 8 L, and 5 L) Then fill the 5 L with the 8 L (4,3,5). Pour the five back into the 12 (9,3,0). Then transfer the 3 in the 8 L jug into the 5 L jug (9,0,3). Fill the 8 L with the 12 L (1,8,3). Then Finish filling the 5 L with the 8 L (1,6,5). Finally, pour the 5 L back into the 12 L bottle, for (6,6,0).

47. 1.6 Reasoning to Solve Problems Page 49 #

48. 1.7 Analyzing Puzzles and Games The following is a game for two players. Place a pile of 20 pennies on your desk. Determine the starting player. Players alternate turns removing 1 or 2 pennies per turn from the pile. The player to remove the last penny is the winner. Play The Game 20 pennies.notebook

49. 1.6 Reasoning to Solve Problems Assuming both players to be intelligent, who started this game of X's and O's, player X or player O? Explain. O started

50. 1.7 Analyzing Puzzles and Games • Handout: Logic Puzzles, Sudoku • http://www.sudoku9x9.com/