use inductive reasoning objectives 1.to form conjectures through inductive reasoning 2.to disprove a...
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Use Inductive Reasoning
Objectives
1. To form conjectures through inductive reasoning
2. To disprove a conjecture with a counterexample
3. To avoid fallacies of inductive reasoning
Example 1
You’re at school eating lunch. You ingest some air while eating, which causes you to belch. Afterward, you notice a number of students staring at you with disgust. You burp again, and looks of distaste greet your natural bodily function. You have similar experiences over the course of the next couple of days. Finally, you conclude that belching in public is socially unacceptable. The process that lead you to this conclusion is called inductive reasoning.
Inductive Reasoning
Inductive reasoning is the process of observing data, recognizing patterns, and making generalizations based on your observations.
Generalization
Generalization: statement that applies to every member of a group
• Science = hypothesis
• Math = conjecture
Conjecture
A conjecture is a general, unproven statement believed to be true based on investigation or observation
Inductive Reasoning
Inductive reasoning can be used to make predictions about the future based on the past or to make conjectures about the past based on the present.
Example 2
A scientist takes a piece of salt, turns it over a Bunsen burner, and observes that it burns with a yellow flame. She does this with many other pieces of salt, finding they all burn with a yellow flame. She therefore makes the conjecture: “All salt burns with a yellow flame.”
Inductive Reasoning
Inductive ReasoningInductive Reasoning
Example 6
(An allegory) Student A neglected to do his/her homework on numerous occasions. When Student A's mean teacher popped a quiz on the class, Student A failed. After the quiz, Student A had several other HW assignments that he/she also neglected to complete. When test time rolled around, Student A failed the exam . Students B-F behaved in a similar, academically deplorable manner. Use inductive reasoning to make a conjecture about the relationship between homework and test/quiz performance.
Example 7
Inductive reasoning does not always lead to the truth. What are some famous examples of conjectures that were later discovered to be false?
To Prove or To Disprove
In science, experiments are used to prove or disprove an hypothesis.
In math, deductive reasoning is used to prove conjectures and counterexamples are used to disprove them.
Counterexample
A counterexample is a single case in which a conjecture is not true.
Example 8
On her first road trip, Little Window Watcher Wilma observes a number of vehicles. Each one she observes has four wheels. She conjectures “All vehicles have four wheels.” What is wrong with her conjecture? What counterexample will disprove it?
Conjecture: All vehicles have 4 wheels
COUNTEREXAMPLE
Example 10
Kenny makes the following conjecture about the sum of two numbers. Find a counterexample to disprove Kenny’s conjecture.
Conjecture: The sum of two numbers is always greater than the larger number.
Example 11
Joe has a friend who just happens to be a Native American named Victor. One day Victor gave Joe a CD. The next day Victor decided that he wanted the CD back, and so he confronted Joe. After reluctantly giving the CD back to his friend, Joe made the conjecture: “Victor, like all Native Americans, is an Indian Giver.” What is wrong with his conjecture? What does this example illustrate?
Inductive Fallacies
The previous example illustrated an inductive fallacy, where a reliable conjecture cannot be justifiably made. Joe was guilty of a Hasty Generalization, basing a conclusion on too little information. Here are some others:
• Unrepresentative Sample• False Analogy• Slothful Induction• Fallacy of Exclusion
Inductive Fallacies
As a group, match each inductive fallacy definition with the corresponding example. Be sure to take some notes, as this priceless information is not in your textbook.
Apply Deductive Reasoning
Objectives:
1. To recognize deductive reasoning and use it to arrive at a true conclusion.
History
When the architects designed this school building, they were approached by an ancient secret society whose members make up numerous Texas dignitaries. They convinced the architects to add several secret passages and hidden conference rooms to their design plans.
History
Sometimes when I stay after school late into the evening grading papers, planning lessons, and contacting parents, I hear strange and inauspicious sounds emanating from behind one of my walls.
Conspiracy?
Thus, it is my conjecture that one of the secret society's hidden passages lies between the walls of Room D202 and D204. This is a bold and perhaps conspiratorial conjecture, but I am confident that it is true. (You should hear the sounds--Oh, my!)
Stop Making Fun of Me!
I have told few people of my theory, and they unanimously dismiss my conviction with ridicule. (Then they ask me if I frequently watch re-runs of the X-Files with the notion that the story lines are largely nonfiction!)
Redemption
To convince the skeptics and to redeem my reputation, I need absolute and conclusive proof that there exists a hidden passage between these classrooms.
Proof
The Principle of Laplace:
The weight of evidence for an extraordinary claim must be proportional to its strangeness.
In Other Words…
“Extraordinary claims require extraordinary evidence.”
-Carl Sagan
Example 1
In your group, come up with a nondestructive method for proving or disproving the extraordinary claim that there’s a secret tunnel between D202 and D204.
Example 2
In the Sudoku puzzle shown, what number must be written in the blue box? Why?
?
Deductive Reasoning
The process of demonstrating that if certain statements are accepted as true, then other statements can be shown to follow from them.
Deductive Reasoning
The “accepted” statements are sometimes premises or assumptions, and all deductive arguments must have them.
Deductive reasoning uses logical inference to build on these assumptions.
Unlike inductive reasoning, deductivereasoning will always lead to the truthas long as the assumptions are true.
Example 2
All humans have skeletons is a reasonable assumption. So, since Mr. Asake is a human, what must be true about him?
Deductive Reasoning
Deductive Reasoning Deductive Reasoning
Inductive vs. Deductive
1. We use inductive reasoning to investigate and discover things about our world.
2. Since the conjectures we make using our inductive reasoning is based on our fallible observation skills, we can be wrong.
3. We can search for a counterexample to disprove our conjectures.
4. In mathematics, we use our deductive reasoning to prove our conjectures beyond all uncertainty.
Flavors of Deductive ReasoningDeductive reasoning comes in a variety of
flavors, and just to make things confusing, each flavor is know by a number of different names.
1. Law of Detachment = Modus Ponens = Affirming the Antecedent
2. Denying the Consequent = Modus Tollens
3. Law of Syllogism = Chain Rule
Law of Detachment
Symbols Example
If Watson had chalk on his fingers, then he had been playing billiards.
Watson had chalk between his fingers upon returning from the club.
Therefore Watson had been playing billiards.
p q
p
q
Denying the Consequent
Symbols Example
If Watson wished to invest his money in S. African securities with Thurston, then he would have had his check book when playing billiards with Thurston.
Watson did not have his checkbook when he played billiards with Thurston.
Therefore Watson did not wish to invest his money in S. African securities with Thurston.
p q
~ q
~ p
Law of Syllogism
Symbols Example
If I eat pizza after midnight, then I will have nightmares.
If I have nightmares, then I will get very little sleep.
Therefore, if I eat pizza after midnight, then I will get very little sleep.
p q
q r
p r
Example 5
Use one of the laws of deductive reasoning to make a valid conclusion.
If two segments have the same length, then they are congruent. You know that BC = XY.
Example 6
Use one of the laws of deductive reasoning to make a valid conclusion.
If x2 > 25, then x2 > 20.
If x > 5, then x2 > 25.
Example 7
Use one of the laws of deductive reasoning to make a valid conclusion.
If a polygon is regular, then it is both equilateral and equiangular.
Pentagon ABCDE is not equilateral or equiangular.
Use Postulates & Diagrams
Objectives:
1. To illustrate and understand postulates about lines and planes
2. To accurately interpret geometric diagrams
3. To use properties of special pairs of angles to find angle measurements
6 cm
20 cm
S
M
A
Example 1
What is the length of ?SM
26 cm
S
M
A
You basically used the Segment Addition Postulate to get the length of the segment, where SA + AM = SM.
Example 1
Postulates
As you build a deductive system like geometry, you demonstrate that certain statements are logical consequences of other previously accepted or proven statements.
Postulates
This chain of logical reasoning must begin somewhere, so every deductive system must contain some statements that are never proved. In geometry, these are called postulates.
Postulates and Theorems
• Postulates are statements in geometry that are so basic, they are assumed to be true without proof.– Sometimes called
axioms.
• Theorems are statements that were once conjectures but have since been proven to be true based on postulates, definitions, properties, or previously proven conjectures.
Both postulates and theorems are ordinarily written in conditional form.
Repeat Times Seven!
Example 1
State the postulate illustrated by the diagram.
Example 2
How does the diagram shown illustrate one or more postulates?
Interpreting Diagrams
When you interpret a diagram, you can assume only information about size or measure if it is marked.
Interpreting Diagrams
Interpreting Diagrams
Example 3
Sketch and carefully label a diagram with plane A containing noncollinear points R, O, and W, and plane B containing noncollinear points N, W, and R.
Perpendicular Figures
A line is perpendicular to a plane if and only if the line intersects the plane in a point and is perpendicular to every line in the plane that intersects it at that point.
Example 4
Which of the following cannot be assumed from the diagram?
1. A, B, and F are collinear.
2. E, B, and D are collinear.
3. AB plane S
Example 4
Which of the following cannot be assumed from the diagram?
4. CD plane T
5. AF intersects BC at point B.
Example 5a
1. Identify all linear pairs of angles.
2. Identify all pairs of vertical angles.
4
3
2
1
Example 5b
3. If m<1 = 40°, find the measures of the other angles in the diagram.
4
3
2
1
Click me!
Linear Pair Postulate
If two angles form a linear pair, then they are supplementary.
Do we have to prove this?
Vertical Angle Congruence Theorem
Vertical angles are congruent.
Example 6
Find the missing measure of each angle.
65
60
Example 7
Find the value of x and y.
3y - 1 2x +5
35
Example 8
Find the value(s) of x.
Example 9: SAT
For the two intersecting lines, which of the following must be true?
I. a > c
II. a = 2b
III. a + 60 = b + cb
c
a60
Example 10: SAT
In the figure, what is the value of y?
2x
3x
y
x