CONDITIONALS: IF…, THEN…. STANDARDS 1,2,3 What is a CONJECTURE? CONDITIONALS: IF…, THEN…. CONVERSE OF A CONDITIONAL NEGATION OF A CONDITIONAL INVERSE OF A CONDITIONAL CONTRAPOSITIVE OF A CONDITIONAL LAW OF DETACHMENT LAW OF SYLLOGISM DEDUCTIVE VS INDUCTIVE? ELEMENTS TO CONSTRUCT PROOFS GEOMETRIC PROOF 1 GEOMETRIC PROOF 2 GEOMETRIC PROOF 3 GEOMETRIC PROOF 4 GEOMETRIC PROOF 5 END SHOW PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

Students write geometric proofs, including proofs by contradiction. Standard 1: Students demonstrate understanding by identifying and giving examples of undefined terms, axioms, theorems, and inductive and deductive reasoning. Standard 2: Students write geometric proofs, including proofs by contradiction. Standard 3: Students construct and judge the validity of a logical argument and give counterexamples to disprove a statement. PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

Estándar 1: Los estudiantes demuestran entendimiento en identificar ejemplos de términos indefinidos, axiomas, teoremas, y razonamientos inductivos y deductivos. Standard 2: Los estudiantes escriben pruebas geométricas, incluyendo pruebas por contradicción. Standard 3: Los estudiantes construyen y juzgan la validéz de argumentos lógicos y dan contra ejemplos para desaprobar un estatuto. PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

I will hit the target with this angle and pulling this way…yes! STANDARDS 1,2,3 I will hit the target with this angle and pulling this way…yes! PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

STANDARDS 1,2,3 There it goes…! PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

STANDARDS 1,2,3 Go ahead arrow…! PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

I didn’t think about the wind! STANDARDS 1,2,3 WIND I didn’t think about the wind! PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

I didn’t think about the wind! STANDARDS 1,2,3 WIND I didn’t think about the wind! A CONJECTURE is an educated guess, and sometimes may be wrong. PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

They form a square! ? STANDARDS 1,2,3 What conjecture may be made from the given information? Given: K(1,1), L(1,3), M(3,3), N(3,1) x y 1 Conjecture: ? They form a square! L M K N PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

What conjecture may be made from the given information? STANDARDS 1,2,3 What conjecture may be made from the given information? Given: A B D E Conjecture: Point E noncollinear with points A, B, and D. PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

STANDARDS 1,2,3 What conjecture may be made from the given information? Given: A(1,1), B(1,3), C(3,3), D(3,1), E(2,2) ?! x y 1 Conjecture: B C Or… E A D PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

STANDARDS 1,2,3 What conjecture may be made from the given information? Given: A(1,1), B(1,3), C(3,3), D(3,1), E(2,2) x y 1 Conjecture: B C mhh!..or E A D PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

What conjecture may be made from the given information? STANDARDS 1,2,3 What conjecture may be made from the given information? Given: A(1,1), B(1,3), C(3,3), D(3,1), E(2,2) x y 1 Conjecture: Guah! this also works…? B C E A D PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

STANDARDS 1,2,3 ?! x y A B D C E What conjecture may be made from the given information? Given: A(1,1), B(1,3), C(3,3), D(3,1), E(2,2) ?! x y A B D C E Conjecture: x y A B D C E ?! Sometimes we may reach to more than one conjecture! PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

STANDARDS 1,2,3 Determine the validity of the conjecture and give a counterexample should the conjecture be false. Given: Points A, B, C, D B C Conjecture: They only form a square. A D False! Counterexample: They could form an isosceles trapezoid as well! B C A D PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

p q IF…, THEN … p q If p, then q If p, then q STANDARDS 1,2,3 CONDITIONAL STATEMENTS OR CONDITIONALS: IF…, THEN … p q Where: If p, then q p = hypothesis q = conclusion Convert to conditional statements: HYPOTHESIS CONCLUSION Students study to get good grades If p, then q If students study, then they get good grades. HYPOTHESIS CONCLUSION Athletes train hard to win competitions. q p If athletes train hard, then they win competitions. PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

p q q p IF…, THEN … IF…, THEN … If p, then q If q, then p CONVERSE: STANDARDS 1,2,3 CONDITIONAL STATEMENTS OR CONDITIONALS: IF…, THEN … p q Where: If p, then q p = hypothesis q = conclusion CONVERSE: IF…, THEN … q p Where: If q, then p p = conclusion q = hypothesis PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

q p p q IF…, THEN … CONVERSE: STANDARDS 1,2,3 Write the CONVERSE of the following conditional: HYPOTHESIS CONCLUSION Athletes train hard to win competitions. First convert to If…, then… statement q p If athletes train hard, then they win competitions. Now get the converse: IF…, THEN … p q CONVERSE: If they win competitions, then athletes train hard. PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

q p IF…, THEN … If p, then q CONVERSE: STANDARDS 1,2,3 Write the CONVERSE of the following conditional: HYPOTHESIS CONCLUSION Students study to get good grades First convert to If…, then… statement If p, then q If students study, then they get good grades. Now get the converse: IF…, THEN … p q CONVERSE: If they get good grades, then students study. PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

A linear pair has adjacent angles. STANDARDS 1,2,3 Write the converse of the following true statement, and determine if true or false. If it is false, give a counterexample: A linear pair has adjacent angles. Explore: a) Obtain converse b) Is it true or false? c) If false find a counterexample Plan: Write the given statement as a conditional: If a linear pair, then angles are adjacent. Solve: a) Converse: If angles are adjacent, then they are a linear pair. b) It is false 35° 55° c) Counterexample: Both angles in the figure at the right are adjacent but not a linear pair. The converse of a true conditional, not necessarily is true. PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

~p is “not p” or the negation of p. STANDARDS 1,2,3 NEGATION: The negation of a statement is its denial. ~p is “not p” or the negation of p. An angle is right An angle is not right p ~p An angle is not right An angle is right p ~p PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

~p ~q INVERSE: STANDARDS 1,2,3 The inverse of a conditional statement is when both the hypothesis and the conclusion are denied. ~p ~q PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

a)Writing the conditional in If-Then form: STANDARDS 1,2,3 For the true conditional: a linear pair has supplementary angles; write the inverse and determine if true or false. If false give a counterexample: a)Writing the conditional in If-Then form: If a linear pair, then it has supplementary angles. HYPOTHESIS p CONCLUSION q b) Negating both the hypothesis and the conclusion: If not a linear pair then it doesn’t have supplementary angles. INVERSE Negated HYPOTHESIS ~p Negated CONCLUSION ~q c) Is it true? The inverse of this conditional is false, as shown in the following counterexample: A C B D E 40° 140° In the figure at the left both angles ABC and EBD aren’t a linear pair but they are supplementary. 140° + 40° = 180° PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

IF…, THEN … p q If p, then q IF…, THEN … IF…, THEN … ~q ~p q p STANDARDS 1,2,3 CONTRAPOSITIVE of a conditional statement: The contrapositive of a conditional statement is the negation of the hypothesis and conclusion of its converse. IF…, THEN … q p If p, then q IF…, THEN … p q If q, then p CONVERSE: IF…, THEN … ~p ~q If not q, then not p CONTRAPOSITIVE: PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

STANDARDS 1,2,3 Find the contrapositive of the true conditional if two points lie in a plane, then the entire line containing those points lies in that plane. Is the contrapositive true or false? a) converse: If the entire line containing those points lies in that plane, then the two points lie in a plane. b) contrapositive: If the entire line containing those points does not lie in that plane, then the two points do not lie in a plane. Counterexample: FALSE. Line AB containing points A and B doesn’t lie in plane Q, but A and B do lie in plane R. A B R Q PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

If p q is a true statement and p is true, then q is true. STANDARDS 1,2,3 LAW OF DETACHMENT If p q is a true statement and p is true, then q is true. If two numbers are even, then their sum is a real number is a true conditional, and 4 and 6 are even numbers. Try to reach a logical conclusion using the Law of Detachment. If two numbers are even, then their sum is a real number p q p q is true 4 and 6 are even is true p Conclusion? 4 + 6 = 10, 10 is a real number. is true By Law of Detachment q PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

(a) If you read novels, then you like mystery books. p q is true p q STANDARDS 1,2,3 Determine if statement (c) goes after statements (a) and (b) by the Law of Detachment. If it does not follow, then write invalid [suppose (a) and (b) true]: (a) If you read novels, then you like mystery books. p q is true p q (b) Juan read a novel. is true p (c) He likes mystery books. Yes, it follows by Law of Detachment. q PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

(a) If two angles add up to 90° then they are complementary p q STANDARDS 1,2,3 Determine if statement (c) goes after statements (a) and (b) by the Law of Detachment. If it does not follow, then write invalid [suppose (a) and (b) true]: (a) If two angles add up to 90° then they are complementary p q is true p q (b) m A + m B = 90° is true p (c) A and B are complementary Yes, it follows by Law of Detachment. q PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

(a) If two angles are vertical, then they are congruent p q is true p STANDARDS 1,2,3 Determine if statement (c) goes after statements (a) and (b) by the Law of Detachment. If it does not follow, then write invalid [suppose (a) and (b) true]: (a) If two angles are vertical, then they are congruent p q is true p q (b) 1 and 2 are vertical. is true p (c) 1 and 2 oppose by the vertex. Invalid q What should follow to be true? (c) 1 and 2 are congruent. PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

p q p r q r p q p r q r STANDARDS 1,2,3 LAW OF SYLLOGISM: If p q and q r are true conditionals, then p r is true as well. p q q r p r If is true, then is true. If a vehicle has four wheels, then it is a car If it is a car, then you can drive it. Using the Law of Syllogism, what conclusion may be reached? If the vehicle has four wheels, then you can drive it. p q p r q r PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

STANDARDS 1,2,3 Determine if statement (c) follows from statements (a) and (b) by the Law of Syllogism. In case this is not true, write INVALID. (a) If a mammal, then it has warm blood. p q (b) If it has warm blood then it drinks milk. q r (c) If a mammal, then it drinks milk. p r Yes, by the Law of Syllogism. PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

p q p q q r q r p r p r STANDARDS 1,2,3 Determine if statement (c) follows from statements (a) and (b) by the Law of Syllogism. In case this is not true, write INVALID. Each statement could be read as: (a) A ABC p q If A, then congruent to ABC p q (b) ABC is a right angle. If ABC, then it is a right angle. q r q r (c) A is a right angle. p r If A, then it is a right angle. p r Yes, by the Law of Syllogism. PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

STANDARDS 1,2,3 Can a conclusion be reached using the Law of Detachment or the Law of Syllogism from (a) and (b) (a) ABC is an obtuse angle. p q (b) An obtuse angle is greater than an acute angle. q r CONCLUSION: ABC is greater than an acute angle by Law of Syllogism p r PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

Logical Reasoning Deductive Reasoning Inductive Reasoning ? STANDARDS 1,2,3 Deductive Reasoning Inductive Reasoning - Uses a set of rules to prove a statement. - Finding a general rule based on observation of data, patterns, and past performance. 4x + 2 = 22 Given: Prove: x = 5 4 5 3 2 1 Squares Step Proof: 4x + 2 = 22 Subtraction Property of Equality -2 -2 4x = 20 Division Property of Equality ? 7 4 4 Substitution Property of Equality Rule: We add 2 squares per step. x = 5 PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

PROPERTIES OF REAL NUMBERS For any real numbers a, b, and c: STANDARDS 1,2,3 ALGEBRAIC REVIEW PROPERTIES OF REAL NUMBERS For any real numbers a, b, and c: COMMUTATIVE PROPERTY: Addition: a + b = b + a 5 + 7 = 7 + 5 Multiplication: a b = b a 9 6 = 6 9 PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

PROPERTIES OF REAL NUMBERS For any real numbers a, b, and c: STANDARDS 1,2,3 PROPERTIES OF REAL NUMBERS For any real numbers a, b, and c: ASSOCIATIVE PROPERTY: Addition: (a + b) + c = a + (b + c) (3 + 4) +1 = 3 + (4 + 1) Multiplication: a b c= a b c 34 45 6 = 34 45 6 PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

PROPERTIES OF REAL NUMBERS For any real numbers a, b, and c: STANDARDS 1,2,3 PROPERTIES OF REAL NUMBERS For any real numbers a, b, and c: IDENTITY PROPERTY: Addition: a + 0 = 0 + a=a 5 + 0 = 0 + 5 = 5 Multiplication: a 1 = 1 a = a 9 1 = 1 9 = 9 PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

PROPERTIES OF REAL NUMBERS For any real numbers a, b, and c: STANDARDS 1,2,3 PROPERTIES OF REAL NUMBERS For any real numbers a, b, and c: INVERSE PROPERTY: Addition: a + (-a) = (-a) + a=0 5 + (-5) = (-5) + 5 = 0 1 5 1 5 = = 1 If a=0 then Multiplication: a = a = 1 1 a 3 5 = 5 3 = 1 PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

PROPERTIES OF REAL NUMBERS For any real numbers a, b, and c: STANDARDS 1,2,3 PROPERTIES OF REAL NUMBERS For any real numbers a, b, and c: DISTRIBUTIVE PROPERTY: Distributive: a(b+c) = ab + ac and (b+c)a = ba + ca 3(5+1) = 3(5) + 3(1) and (5+1)3 = 5(3) + 1(3) PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

Name the property shown at each equation: STANDARDS 1,2,3 Name the property shown at each equation: 1 45 = 45 a) Identity property (X) 56 + 34 = 34 + 56 b) Commutative property (+) (-3) + 3 = 0 c) Inverse property (+) 5(9 +2) = 45 + 10 d) Distributive property (2 + 1) +b= 2 + (1 + b) e) Associative property (+) -34(23) = 23(-34) f) Commutative property (X) PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

PROPERTIES OF EQUALITY: ALGEBRAIC REVIEW STANDARDS 1,2,3 PROPERTIES OF EQUALITY: ALGEBRAIC REVIEW SUBSTITUTION PROPERTY OF EQUALITY: If b=2 and 3b +1=7 If a=b, then a may be replaced by b. then 3( )+1=7 2 ADDITION AND SUBTRACTION PROPERTIES OF EQUALITY: For any numbers a, b, and c, if a=b then a+c=b+c and a-c=b-c 10 = 10 22 = 22 + 6 +6 -5 -5 16 = 16 17 = 17 PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

PROPERTIES OF EQUALITY STANDARDS 1,2,3 PROPERTIES OF EQUALITY MULTIPLICATION AND DIVISION PROPERTIES OF EQUALITY: For any real numbers a, b, and c, if a=b, then a c=b c and if c=0, = a c b 28 = 28 2 15 = 15 7 7 30 = 30 4 = 4 36 = 36 3 24 = 24 12 12 72 = 72 3 = 3 PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

Deductive Reasoning: Algebra STANDARDS 1,2,3 FORMAL INFORMAL Two column proofs: Proof: Statements Reasons Given: 4(x + 2) = 2x + 18 Prove: x = 5 4(x + 2) = 2x + 18 4x + 8= 2x + 18 -8 -8 4x = 2x + 10 -2x -2x 2x = 10 (1) given 2 2 (1) 4(x + 2) = 2x + 18 x = 5 (2) 4x + 8= 2x + 18 (2) Distributive prop. (3) 4x = 2x + 10 (3) Subtraction prop. of equality (4) 2x = 10 (4) Subtraction prop. of equality (5) x = 5 (5) Division Prop. of equality. PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

Deductive Reasoning (GEOMETRY) STANDARDS 1,2,3 Deductive Reasoning (GEOMETRY) Conjecture - a statement or conditional trying to prove. Elements to construct proofs: a) Undefined terms - Terms that are so obvious that don’t require to be proven. point, line, etc. b) Definitions - Statements defined using other terms. Triangle is a 3 sided polygon. c) Axioms (Postulates) - Statements or properties that don’t need to be proven to be used in proofs. If two planes intersect their intersection is a line. d) Theorems - Statements or properties that require to be proven to be used in proofs. If two angles form a linear pair, then they are supplementary angles. PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

PROPERTIES OF EQUALITY: ALGEBRAIC REVIEW STANDARDS 1,2,3 PROPERTIES OF EQUALITY: ALGEBRAIC REVIEW REFLEXIVE PROPERTY OF EQUALITY: For any real number a, a=a 5=5 -10=-10 SYMMETRIC PROPERTY OF EQUALITY: For all real numbers a and b, if a=b, then b=a X=5 5=X 6X-12=8 8=6X-12 9Y -2Y +1= 3X 2 3X= 9Y -2Y+1 2 TRANSITIVE PROPERTY OF EQUALITY: For all real numbers a, b, and c, if a=b, and b=c then a=c If X=6 and Y= 6 then X=Y If Y=2X+2 and Y=6-3X then 2X+2=6-3X PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

STANDARDS 1,2,3 Congruence in segments and angles is Reflexive, Symmetric and Transitive: For all segments and angles, their measures comply with these same properties. of segments is reflexive. of s is reflexive LM ECA of segments is symmetric. of s is symmetric KL LM LM KL BCE FGH BCE FGH of segments is transitive. of s is transitive KL LM BCE FGH LM AB FGH ECA KL AB BCE ECA PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

DEDUCTIVE REASONING: GEOMETRY (formal) STANDARDS 1,2,3 Given: Prove: M L K B A L is midpoint of KM LM AB KL Two Column Proof: Statements Reasons (1) L is midpoint of KM Given (2) KL LM Definition of Midpoint (3) LM AB Given (4) KL AB of segments is transitive. PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

DEDUCTIVE REASONING: GEOMETRY (formal) STANDARDS 1,2,3 B C F D A E B C F D A E Given: EFD is right Prove: AFB CFB and are complementary. Two Column Proof: Statements Reasons (1) EFD is right Given (2) EC AD Definition of lines (3) AFC is right lines form 4 right s (4) AFC= m 90° Definition of right s (5) AFB + m AFC CFB = addition postulate (6) AFB + m CFB = 90° Substitution prop. of (=) (7) AFB CFB and are complementary. Definition of complementary s PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

DEDUCTIVE REASONING: GEOMETRY (formal) STANDARDS 1,2,3 Given: B H CE bisects BCA FGH ECA Prove: FGH + m BCD = 180° 2( ) Two Column Proof: Statements Reasons (1) CE bisects BCA Given (2) BCE ECA Definition of bisector (3) BCE= m ECA Definition of s (4) FGH ECA Given (5) FGH= m ECA Definition of s (6) BCE= m FGH of s is transitive (7) BCE + m BCD = 180° ECA + addition postulate (8) FGH + m BCD = 180° Substitution prop. of (=) (9) FGH + m BCD = 180° 2( ) Adding like terms PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

DEDUCTIVE REASONING: GEOMETRY (formal) STANDARDS 1,2,3 B C A D F E B C A D Given: FBD is right Prove: ABF CBD and are complementary. Two Column Proof: Statements Reasons (1) FBD is right Given (2) FBD= m 90° Definition of right s (3) FBD + m CBD = 180° ABF + addition postulate (4) CBD = 180° m ABF + 90° + Substitution prop. of (=) (5) ABF + m CBD = 90° Subtraction prop. of (=) (6) ABF CBD and are complementary. Definition of complementary s PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

DEDUCTIVE REASONING: GEOMETRY (formal) STANDARDS 1,2,3 Given: C A B H G D E F C A B H G D E F AC and DF are GE is a transversal Prove: GBC FEH and are supplementary. Two Column Proof: Statements Reasons (1) GE is a transversal AC and DF are Given (2) GBC CBE and are a linear pair Definition of linear pair (3) GBC + m CBE = 180° s in a linear pair are supplementary (4) CBE FEH In lines cut by a transversal CORRESPONDING s are (5) CBE= m FEH Definition of s (6) GBC + m FEH = 180° Substitution prop. of (=) (7) GBC FEH and are supplementary. Definition of supplementary s PRESENTATION CREATED BY SIMON PEREZ. All rights reserved