Inductive Reasoning Courage is resistance to fear, mastery of fear, not absence of fear. Mark Twain
Inductive Reasoning O Inductive Reasoning is the process of observing a pattern and making a conjecture about the pattern. O A conjecture is an unproven statement based on observations. O So, inductive reasoning is the process of observing a pattern and making an unproven statement about the pattern.
O Ex. 1: Describe the pattern and either draw the next shape or write the next three numbers. 1) 5, 15, 45, 135,.. 1) 405, 1215, 3645 2) -2, 1, 4, 7, 3) 10, 13, 16 4)
O Ex. 2: Make and test conjectures for each problem below. 1) The sum of two odd numbers. 3+5=8 7+9=16-5+7=2 The sum of two odd numbers is even. 1) The product of two even numbers. 4x6=24 8x8=64-4x4=-16 The product of two even numbers is even. 1) The sum of three consecutive odd numbers. 3+5+7=3(5) 1+3+5=3(3) 15+17+19=3(17) The sum of three consecutive odd numbers is three times the middle number.
Counter-example O A counter-example is the case that shows a conjecture to be false. O Conjecture: The product of any two numbers is always positive. O Counter-example: -5 x 4 = -20 O Conjecture: We have school each weekday. O Counter-example: October 10th
O Ex. 3: Find the counter example to each conjecture. 1) All prime numbers are odd. 1) 2 2) If the sum of two numbers is even, then both of those numbers are also even. 1) 3+5=8 1) The sum of three consecutive numbers is always odd. 1) 3+4+5=12
Summary O You should now be able to: O Identify patterns and make conjectures about those patterns. O Provide counter-examples to false conjectures.
Conditional Statements The great use of life is to spend it for something that will outlast it. William James
Conditional statements O Each conditional statement has a condition and a consequence. O Hypothesis: is the condition of the statement (also the if portion of the if-then format). O Conclusion: is the consequence of the statement (also the then portion of the if-then format) O Example: All mammals have hair. O If-then form: If an animal is a mammal, then it has hair.
Ex. 1: For each conditional statement, write it in if-then form. 1) Every student at Westfield has to take Physical Science their freshman year. 1) If a student is a freshman at Westfield, then they have to take Physical Science. 2) Two angles are complementary if their measures add up to 90. 1) If two angle measures sum to be 90, then they are complementary angles 3) Vertical angles have two pairs of opposite rays. 1) If a pair of angles are vertical angles, then they have two pairs of opposite rays. 4) 2x+5=2, because x=-6 1) If x=-6, then 2x+5=2
Negation O A negation is the opposite of the original statement. O The apple is red. O Negation: the apple is NOT red.
Converse, Inverse, and Contrapositive. O Conditional statement: If I forget to put my name on a paper, then I get a zero for that paper. O Converse: switch the hypothesis and conclusion of the original conditional statement. O If I get a zero for a paper, then I forget to put my name on the paper. O Inverse: Negate BOTH the hypothesis and the conclusion of the original conditional statement. O If I remember to put my name on a paper, then I will not get a zero for that paper. O Contrapositive: Negate BOTH the hypothesis and the conclusion of the converse. O If I get greater than a zero on a paper, then I remember to put my name on the paper.
Ex. 2: For each conditional statement, write its converse, inverse, and contrapositive and decide each statements truth value. 1) If you watch this video, then you take notes for geometry. 1) Converse: If you take notes for geometry, then you watch this video. 2) Inverse: If you don t watch this video, then you don t take notes for geometry. 3) Contrapositive: If you don t take notes for geometry, then you don t watch this video.
Equivalent and Biconditional O Equivalent statements are statements that are both true or are both false. The conditional statement and contrapositive are always equivalent. The inverse and converse are always equivalent. O Biconditional statements are statements that the original conditional statement and its converse are both true. O For example: If two angle measures sum to be 90, then they are complementary. O The definition could be written as: Two angles are complementary if and only if (iff) their measures sum to 90.
Ex. 3: For each conditional statement, write its converse, inverse, and contrapositive and decide each statements truth value. If both the conditional and its converse are true, write a biconditional statement. 1) If four points are coplanar, then they lie in the same plane. 1) Converse: If four points lie in the same plane, then they are coplanar. 2) Inverse: If four points aren t coplanar, then they don t lie in the same plane,. 3) Contrapositive: If four points don t lie in the same plane, then they aren t coplanar. 4) Biconditional: Four points are coplanar iff they lie in the same plane.
Summary O At this point, you should be able to: O Write a converse, inverse, and contrapositive to a conditional statement. O Know the requirements for a statement to be biconditional. O Know how to negate a statement.
Deductive Reasoning Only the person who has faith in himself is able to be faithful to others. Erich Fromm
Deductive Reasoning O Deductive reasoning uses facts, definitions, properties, and laws of logic to form a logical argument.
Laws of Logic O Law of detachment: If the hypothesis of a true conditional statement is true, then the conclusion is also true. O If the flipped method proves to improve learning, then Mr. H will continue it for the whole year. O If Mr. H gets a haircut, then pigs can fly.
Law of Syllogism (dominoes) For want of a nail the shoe was lost. For want of a shoe the horse was lost. For want of a horse the rider was lost. For want of a rider the message was lost. For want of a message the battle was lost. For want of a battle the kingdom was lost. O These statements could be combined to be: For want of a nail, the kingdom was lost.
Law of Syllogism (dominoes) O If q, then r. O If r, then s. O If q, then s.
Ex. 1: Use the Law of Detachment to make a valid conclusion. 1) If two angles have the same measure, then they are congruent. The measure of angle A is 90 and the measure of angle B is 90. 1) Angle A is congruent to angle B. 2) Pythagoras takes a nap at 4pm. It is 4pm on Saturday. 1) Pythagoras is taking a nap.
Ex. 2: Use the Law of Syllogism to make a valid conclusion. 1) If two angles are both right angles, then they have the same measure. If two angles have the same measure, then they are congruent. 1) If two angles are both right angles, then they are congruent. 2) If Jesse get a job, then he can afford a car. If Jesse can afford a car, then he buys a car. 1) If Jesse gets a job, then he buys a car.
Ex. 3: Determine whether each statement is the result of inductive or deductive reasoning. Explain why. 1) For the last two weeks Mr. H has gone around helping students during the class period. You conclude that Mr. H will help students during the class period on Monday. 1) Inductive because you are making a conjecture based on previous observations. 2) The rule at work is that you have to work the full week to get paid on Friday. You were paid on Friday. Therefore, you went to all of your classes. 1) Deductive because you use rules and facts to make a conclusion.
Summary O You should be able to use the laws of logic to make valid conclusions. O You should be able to determine the difference between deductive and inductive reasoning.
Using Postulates and Diagrams A hero is no braver than an ordinary man (or woman), but he (/she) is brave five minutes longer. Ralph Waldo Emerson
Postulates 5-11 O 5: Through any two points there exists exactly one line. O 6: A line contains at least two points. O 7: If two lines intersect, then their intersection is exactly one point. O 8: Through any three noncollinear points there exists exactly one plane.
Postulates 5-11 O 9: A plane contains at least three noncollinear points. O 10: If two points lie in a plane, then the line containing them lies in the plane. O 11: If two planes intersect, then their intersection is a line.
O Ex. 1: State the postulate illustrated by the diagram. 1) A B 2) A B A C C B
O Ex. 2: Use the diagram to write examples of postulate 5 and 7. Through points C and B there is one line called line l. Line DE and Line BF intersect at point D.
O Ex. 3: Use the diagram to determine if each statement is true or false. 1) Line AB lies in plane R 2) Line FH lies in plane R 3) Line AC and Line FG will intersect. 4) Line GH is perpendicular to plane R. 5) Angle LGH is a right angle. 6) Angle LGH and angle LGF are supplementary angles. L
Summary O You should be able to identify the postulate used in drawing a diagram.
Reasoning using Algebra. Certain signs precede certain events. Cicero
Algebraic Properties of Equality. Let a, b, and c be real numbers. 1) Addition Property 2) Subtraction Property 3) Multiplication Property 4) Division Property 5) Substitution Property 6) Distributive Property 1) If a = b, then a + c = b + c. 2) If a = b, then a c = b c. 3) If a = b, then ac = bc. 4) If a = b and c 0, then a c = b c. 5) If a=b, then a can be sustitud ed in for any equation or expression. 6) a b + c = ab + ac
Example 1: Solve 2x + 30 = 75 3x. Write a reason for each step. Equation 2x + 30 = 75 3x 5x + 30 = 75 5x = 45 x = 9 Reason Given Addition Property Subtraction Property Division Property
Example 2: Solve 2(x + 30) = 2(70 3x). Write a reason for each step. Equation 2(x + 30) = 2(70 3x) 2x 60 = 140 6x 4x 60 = 140 4x = 200 x = 50 Reason Given Distributive Property Addition Property Addition Property Division Property
Reflexive Properties of Equality. 1) Real Numbers 2) Segment Length 3) Angle Measure 1) For any real number a, a = a. 2) For any segment AB, AB = AB 3) For any angle A, m A = m A.
Symmetric Properties of Equality. 1) Real Numbers 2) Segment Length 3) Angle Measure 1) For any real number a and b, a = b, then b = a. 2) For any segment AB and CD, if AB = CD, then CD = AB. 3) For any angle A and B, if m A = m B, then m B = m A.
Transitive Properties of Equality. 1) Real Numbers 2) Segment Length 3) Angle Measure 1) If a = b and b = c, then a = c. 2) IfAB = CD and CD = EF, then AB = EF. 3) If m A = m B and m B = m C, then m A = m C.
Example 3: Determine if m ABC= m FBD. Show your reasoning. A 1 F 2 C B 3 Equation m 1= m 3 m ABC= m 1+ m 2 m FBD= m 2 +m 3 m FBD= m 2+ m 1 m ABC= m FBD D Reason Given Angle Addition Postulate Angle Addition Postulate Substitution Property Transitive Property
Summary O You should be able to use properties to justify your reasoning. O You should be able to solve a problem and provide reasons for each step.
Prove Statements about segments and angles Anxiety is fear of one s self. Wilhelm Stekel
Congruence of Segments and Angles. (Theorem 2.1 and 2.2) 1) Reflexive Property of Congruence 2) Symmetric Property of Congruence 3) Transitive Property of Congruence For any segment AB and any angle A. 1) AB AB or A A 2) If AB CD, then CD AB or If A B, then B A 3) If AB CD and CD EF, then AB EF or If A B and B C, then A C.
Example 1: Use a two column proof to show that AC BD. A B C D Statements 1) AB = CD 2) AC = AB + BC 3) BD = CD + BC 4) BD = AB + BC 5) AC = BD 6) AC BD Reasons 1) Given 2) Segment Addition Postulate 3) Segment Addition Postulate 4) Substitution Property 5) Transitive Property 6) Definition of Congruence
Ex. 2: Name the property illustrated by the statement. 1) If F G and G H, then F H. 2) If EF GH, then GH EF. 3) AB AB
Example 1:Prove that AB=2AM. You know that M is the midpoint of AB. A M B Statements 1) M is a midpoint of AB. 2) AM MB 3) AM = MB 4) AB = AM + MB 5) AB = AM + AM 6) AB = 2AM Reasons 1) Given 2) Definition of Midpoint 3) Definition of congruence 4) Segment Addition Postulate 5) Substitution Property 6) Simplify
Summary O You should be able to prove statements about segments and angles. O You should be able to write a two column proof.
Prove Angle Pair Relationships. Remember that happiness is a way of travel not a destination. Roy M. Goodman
Theorem 2.3-5 O Theorem 2.3: Right Angles Congruence Theorem. O All right angles are congruent. O Theorem 2.4: Congruent Supplements Theorem. O If two angles are supplementary to the same angle (or to congruent angles), then they are congruent. O Theorem 2.5: Congruent Complements Theorem. O If two angles are complementary to the same angle (or to congruent angles), then they are congruent.
1) Ex. 1: Prove that 1 3, given that 1 and 2 are supplementary and 3 and 2 are supplementary. 2 3 1 Statements 1) 1 and 2 are supp. 3 and 2 are supp. 2) m 1 + m 2 = 180 3) m 3 + m 2 = 180 4) m 1 + m 2 = m 3 + m 2 5) m 1 = m 3 6) 1 3 1) Given Reasons 2) Definition of Supplementary 3) Definition of Supplementary 4) Transitive Property 5) Subtraction Property 6) Definition of Congruence
Postulate 12 and Theorem 2.6 O Postulate 12: Linear Pair Postulate. O If two angles form a linear pair, then they are supplementary. O Theorem 2.6: Vertical Angles Congruence Theorem. O Vertical Angles are Congruent.
O Ex. 2: Given that angle 3 and angle 4 are a linear pair and measure of angle 4 is 112, find the measure of angle 3. Statements 1) 3 and 4 form a linear pair and m 4 = 112 2) 3 and 4 are supp. 3) m 3 + m 4 = 180 4) m 3 + 112 = 180 5) m 3 = 68 1) Given Reasons 2) Linear Pair Postulate 3) Definition of Supplementary 4) Substitution Property 5) Subtraction Property
Ex. 3: Find the value of x if m 1 = (3x 4) and m 4 = (6x 184). Statements 1) m 1 = 3x 4 2) m 4 = (6x 184) 3) 1 4 4) m 1 = m 4 5) 3x 4 = (6x 184) 6) 4 = 3x 184 7) 180=3x 8) 60=x 1 2 3 5 4 Reasons 1) Given 2) Given 3) Vertical Angle Congruence Theorem 4) Definition of Congruence 5) Transitive Property 6) Subtraction Property 7) Addition Property 8) Division Property
Summary O You should be able to identify complementary and supplementary angles. O You should be able to identify linear pairs and vertical angles. O You should be able to use the above definitions, postulates, and theorems to write a proof.