inductive and deductive reasoning

MATH FOUNDATIONS 11

Inductive and Deductive Reasoning

Let’s play a little game

Pick the number of days per week that you like to eat chocolate

Multiply this number by 2Now, add 5Multiply 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 bornWhat do you notice about your answer?The first digit is your original #, and the last two digits

your age

Here’s a Fun Quiz

Think of a number from 1 -10Multiply that number by 9If the number is 2 digits, add the digits togetherNow, subtract 5Convert your number to a letter: A=1, B=2, C=3,

D=4, …Think of a country that starts with that letterRemember the last letter of the name of your country,

and think of an animal that starts with that letterRemember the last letter of your animal, and think of

a fruit that starts with that letterAre you thinking about …..

Orange Kangaroo from Denmark?

Proof that 2 = 1

Let a=bSo, a2 = ab

a2 - b2 = ab - b2 (a-b)(a+b) = b(a-b)

(a+b) = bb + b = b

2b = b2 = 1

4 – 9’s

Apparently, all the numbers from 0 to 100 can be created from four nines

Ex: 9 + 9 – 9 - 9 = 0Ex: 9/9 + 9 – 9 = 1Ex 9/9 + 9/9 = 2

Are there any numbers that can’t be done?

1.1 Inductive ReasoningInductive 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.

Example

1(8) + 1 = 912(8) + 2 = 98

123(8) + 3 = 9871234(8) +4 = 9876

12345(8) + 5 = ____

Does this pattern continue forever?

No!

Verbal example

This swan is whiteI’ve seen 100 white swans

All swans are white

Is this always true?

No!

Assignment – Page 12 # 3 – 17 odd

Section 1.2

Section 1.2

Section 1.2

Section 1.2

1.3 Using Reasoning to Find a counterexample to a Conjecture

A conjecture can be proven false by finding a counter example

Statement: A number that is not negative is positive

Statement: A number that is not negative is positive

Counter example :

This statement is proven false by finding a counter example

o

Statement: All prime numbers are odd

Statement: All prime numbers are odd

Counter example :

This statement is proven false by finding a counter example

2

Statement: All NBA basketball players are tall

Statement: All NBA basketball players are tall

Counter example :

This statement is proven false by finding a counter example

Nate Robinson(5’7”)

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

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

Statement: As you travel north, the climate gets colder

Statement: As you travel north, the climate gets colder

Counter example :

This statement is proven false by finding a counter example

Southern hemisphere

Statement: The sum of 2 numbers is always greater

than the greater of the 2 numbers.

Statement: The sum of 2 numbers is always greater than either of the 2 numbers.

Counterexample: -3 + -5

This statement is proven false by finding a counter example

Assignment

Page 22 #2 – 9, 12, 17

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.

Example

All poodles are dogsAll dogs are mammals

Example

All poodles are dogsAll dogs are mammals

All poodles are mammals

Example

Lynn is a math studentAll math students have a calculator

Example

Lynn is a math studentAll math students have a calculator

Lynn has a calculator

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

“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

Example

Prove that the square of any odd integer is odd

Let = 1st odd integer

Example

Prove that the product of any 2 even integers is even

Let 2k = 1st even integerLet 2j = 2nd even integer

Product = (1st integer)(2nd integer) = (2k)(2j) = 4kj

This number is divisible by 2, therefore it is EVEN

AssignmentPage 31 # 1, 2, 4 – 8, 10, 12,

17, 20

1.5 Proofs that are not valid

Jean Chretien: http://www.youtube.com/watch?v=aX6XMIldkRU

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.

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?

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?

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.

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?

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

1.6 Reasoning to Solve ProblemsMr. 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

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.

1.6 Reasoning to Solve Problems

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).

1.6 Reasoning to Solve Problems

Page 49 #

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 Game20 pennies.notebook

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

1.7 Analyzing Puzzles and Games

Handout: Logic Puzzles, Sudokuhttp://www.sudoku9x9.com/