Thinking Mathematically

Problem Solving and Critical Thinking

by

Nittaya Noinan

Inductive ReasoningInductive reasoning is the process of arriving at a general conclusion based on observations of specific examples.

ExampleYou purchased textbooks for 4 classes. Each book cost more than $50.00.Conclusion: All college textbooks cost more than $50.00.

Specific General

Ex. 1: Describing a Visual Pattern

• Sketch the next figure in the pattern.

1 2 3 4 5

Ex. 1: Describing a Visual Pattern - Solution

• The sixth figure in the pattern has 6 squares in the bottom row.

5 6

Ex. 2: Describing a Number Pattern

• Describe a pattern in the sequence of numbers. Predict the next number.

a. 141664

• How do you get to the next number?

• That’s right. Each number is 4 times the previous number. So, the next number is

• 256, right!!!

Ex. 2: Describing a Number Pattern

• Describe a pattern in the sequence of numbers. Predict the next number.

b. -5-2413

• How do you get to the next number?

• That’s right. You add 3 to get to the next number, then 6, then 9. To find the fifth number, you add another multiple of 3 which is +12 or

• 25, That’s right!!!

Goal 2: Using Inductive Reasoning

• Much of the reasoning you need in geometry consists of 3 stages:

1. Look for a Pattern: Look at several examples. Use diagrams and tables to help discover a pattern.

2. Make a Conjecture. Use the example to make a general conjecture. Okay, what is that?

Goal 2: Using Inductive Reasoning

• A conjecture is an unproven statement that is based on observations. Discuss the conjecture with others. Modify the conjecture, if necessary.

3. Verify the conjecture. Use logical reasoning to verify the conjecture is true IN ALL CASES.

Ex. 3: Making a Conjecture

First odd positive integer:1 = 12 1 + 3 = 4 = 22 1 + 3 + 5 = 9 = 32 1 + 3 + 5 + 7 = 16 = 42 The sum of the first n odd positive integers is n2.

Note:

• To prove that a conjecture is true, you need to prove it is true in all cases. To prove that a conjecture is false, you need to provide a single counter example. A counterexample is an example that shows a conjecture is false.

Ex. 4: Finding a counterexample

• Show the conjecture is false by finding a counterexample.

Conjecture: For all real numbers x, the expressions x2 is greater than or equal to x.

Ex. 4: Finding a counterexample- Solution

Conjecture: For all real numbers x, the expressions x2 is greater than or equal to x.

• The conjecture is false. Here is a counterexample: (0.5)2 = 0.25, and 0.25 is NOT greater than or equal to 0.5. In fact, any number between 0 and 1 is a counterexample.

Note:

• Not every conjecture is known to be true or false. Conjectures that are not known to be true or false are called unproven or undecided.

Ex. 5: Examining an Unproven Conjecture

• In the early 1700’s, a Prussian mathematician names Goldbach noticed that many even numbers greater than 2 can be written as the sum of two primes.

• Specific cases:4 = 2 + 2 10 = 3 + 7 16 = 3 + 136 = 3 + 3 12 = 5 + 7 18 = 5 + 138 = 3 + 5 14 = 3 + 11 20 = 3 + 17

Ex. 5: Examining an Unproven Conjecture

• Conjecture: Every even number greater than 2 can be written as the sum of two primes.

• This is called Goldbach’s Conjecture. No one has ever proven this conjecture is true or found a counterexample to show that it is false. As of the writing of this text, it is unknown if this conjecture is true or false. It is known; however, that all even numbers up to 4 x 1014 confirm Goldbach’s Conjecture.

Ex. 6: Using Inductive Reasoning in Real-Life

• Moon cycles. A full moon occurs when the moon is on the opposite side of Earth from the sun. During a full moon, the moon appears as a complete circle.

Ex. 6: Using Inductive Reasoning in Real-Life

• Use inductive reasoning and the information below to make a conjecture about how often a full moon occurs.

• Specific cases: In 2005, the first six full moons occur on January 25, February 24, March 25, April 24, May 23 and June 22.

Ex. 6: Using Inductive Reasoning in Real-Life - Solution

• A full moon occurs every 29 or 30 days.• This conjecture is true. The moon revolves around

the Earth approximately every 29.5 days.• Inductive reasoning is very important to the study of

mathematics. You look for a pattern in specific cases and then you write a conjecture that you think describes the general case. Remember, though, that just because something is true for several specific cases does not prove that it is true in general.

Ex. 6: Using Inductive Reasoning in Real-Life - NOTE

• Inductive reasoning is very important to the study of mathematics. You look for a pattern in specific cases and then you write a conjecture that you think describes the general case. Remember, though, that just because something is true for several specific cases does not prove that it is true in general.

Deductive ReasoningDeductive reasoning is the process of proving a specific conclusion from one or more general statements.

A conclusion that is proved true by deductive reasoning is called a theorem.

ConclusionPremise

Logical ArgumentCalculation

Proof

Deductive ReasoningExample:

The catalog states that all entering freshmen must take a mathematics placement test.

Conclusion: You will have to take a mathematics placement test.

You are an entering freshman.

Deductive ReasoningExample:

If it rains today, I'll carry an umbrella.

Conclusion: I'm not carrying an umbrella.

It's not raining.

Deductive Reasoning• The previous example was wrong because I never said what I'd do if it wasn't raining.

– That's what makes deductive reasoning so difficult; it's easy to get fooled.

– We'll learn more about this when we study logic.

Deductive vs. Inductive Reasoning

• The difference between these two types of reasoning is that –inductive reasoning tries to generalize

something from some given information, –deductive reasoning is following a train

of thought that leads to a conclusion.

This is called a Venn Diagram for a statement. In the diagram the outside box or rectangle is used to represent everything. Circles are used to represent general collections of things. Dots are used to represent specific items in a collection. There are 3 basic ways that categories of things can fit together.

women

democrats

women democrats

women democrats

All democrats are women. Some democrats are women. No democrats are women.

One circle inside another. Circles overlap. Circles do not touch.

democrats

Laura Bush

womenGeorge Bush

Laura Bush is a democrat. George Bush is not a woman.

What statement do each of the following Venn Diagrams depict?

Putting more than one statement in a diagram. (A previous example)

Hypothesis:

Dr. Daquila voted in the last election.

Only people over 18 years old vote.

Conclusion:

Dr. Daquila is over 18 years old.

Dr. Daquila

voters

people over 18

IF__THEN__ Statement Construction

Many categorical statements can be made using an if then sentence construction. For example if we have the statement: All tigers are cats.

catstigers

This can be written as an If_then_ statement in the following way:

If it is a tiger then it is a cat.

This is consistent with how we have been thinking of logical statements. In fact the parts of this statement even have the same names:

If it is a tiger then it is a cat.

The phrase “it is a tiger” is called the hypothesis. The phrase “it is a cat” is called the conclusion.

Can the following statement be deduced?

Hypothesis:

If you are cool then you sit in the back.

If you sit in the back then you can’t see.

Conclusion

If you are cool then you can’t see.

There is only one way this can be drawn! So it can be deduced and the argument is valid!

people who can’t see

people who sit in back

cool people

One method that can be used to determine if a statement can be deduced from a collection of statements that form a hypothesis we draw the Venn Diagram in all possible ways that the hypothesis would allow. If any of the ways we have drawn is inconsistent with the conclusion the statement can not be deduced.

Example

Hypothesis:

All football players are talented people.

Pittsburg Steelers are talented people.

Conclusion:

Pittsburg Steelers are football players. (CAN NOT BE DEDUCED and Invalid!)

talented people

football players

Pittsburg Steelers

talented people

football players

Pittsburg Steelers

talented people

football players

Pittsburg Steelers

Even though one of the ways is consistent with the conclusion there is at least one that is not so this statement can not be deduced and is not valid.

Example

• Construct a Venn Diagram to determine the validity of the given argument.

# All smiling cats talk. The Cheshire Cat smiles.

Therefore, the Cheshire Cat talks.

VALID OR INVALID???

ExampleValid argument; x is Cheshire Cat

Things that talk

Smiling cats

X

Examples

• # No one who can afford health insurance is unemployed.

All politicians can afford health insurance.

Therefore, no politician is unemployed.

VALID OR INVALID?????

Examples

X=politician. The argument is valid.

People who can affordHealth Care.

PoliticiansX

Unemployed

Example• # Some professors wear glasses. Mr. Einstein wears glasses.

Therefore, Mr. Einstein is a professor. Let the purple oval be professors, and the blue oval be glass wearers.

Then x (Mr. Einstein) is in the blue oval, but not in the overlapping region. The argument is invalid.

X

Section Quiz

Problem 1: Is the following an example of deductive reasoning or inductive reasoning?“People like taking vacations. When I asked a bunch

of students in the parking lot, they all said they like vacations. Therefore EVERYBODY likes taking vacations.”

Section Quiz

• Problem 2: “If people don’t like to eat, they don’t like cookies. John doesn’t like cookies. This means that John doesn’t like to eat.”– Is it valid to draw the conclusion above?

Section Quiz

Problem 3: What comes next in this series?1, 1, 2, 3, 5, 8, 13, 21, 34, ….?a) 43 b) 55 c) 68 d) 8

Section Quiz

• Problem 4: What comes next in this sequence?

Section Quiz

Problem 1: Is the following an example of deductive reasoning or inductive reasoning?“People like taking vacations. When I asked a bunch of

students in the parking lot, they all said they like vacations. Therefore EVERYBODY likes taking vacations.”

We are making a generalization based on a sample of answers. This is inductive reasoning.

Section Quiz

Problem 2: “If people don’t like to eat, they don’t like cookies. John doesn’t like cookies. This means that John doesn’t like to eat.”Is it valid to draw the conclusion above?

No. Obviously people who don’t like eating won’t like cookies. But there certainly can be people (perhaps John) who like to eat but don’t happen to like cookies!

Section Quiz

Problem 3: What comes next in this series?1, 1, 2, 3, 5, 8, 13, 21, 34, ….?a) 43 b) 55 c) 68 d) 8

After the first two numbers, notice that each number is the sum of the previous two. For example 13 = 5 + 8 or 34 = 13 + 21.So the next number is 21 + 34, or 55. (Fibonacci series)

Section Quiz

• Problem 4: What comes next in this sequence?

We are alternating between blue and pink. The sequence of shapes is square, circle, triangle.So we would expect a pink triangle.

Good Luck