Chapter 1: Inductive and Deductive Reasoning Section 1.1 Chapter 1: Inductive and Deductive Reasoning Section 1.1: Making Conjectures: Inductive Reasoning Terminology: Conjecture: A testable expression that is based on available evidence but is not yet proven. A conjecture can be a true conclusion or it can be false! Inductive Reasoning: A form of reasoning in which a conclusion is reached based on a pattern present in numerous observations. The premises makes the conclusion likely, but does not guarantee it to be true. TRUE PREMISE + TRUE PREMIS = PROBABLY TRUE CONCLUSION Deductive Reasoning: The process of coming up with a conclusion based on facts that have already been shown to be true. The facts that can be used to prove your conclusion deductively may come from accepted definitions, properties, laws or rules. The truth of the premises guarantees the truth of the conclusion. TRUE PREMISE + TRUE PREMISE = TRUE CONCLUSION MAKING CONJECTURES For each situation below, make a conjecture based on your observations 1. Determine the pattern and make a conjecture about which shape comes next. CONJECTURE: The next shape in the pattern is:
Chapter 1: Inductive and Deductive Reasoning Section 1.1 2. Complete the conjecture below that is true for all the equations: 3 + 7 = 10 11 + 5 = 16 9 + 13 = 22 7 + 11 = 18 CONJECTURE: The sum of two odd natural numbers is always. 3. A pattern is created between the figure number, n, and the number of triangles, t, required to make it. Make a conjecture between the figure number and the number of triangles creating it. Figure number (n) 1 2 3 Numbers of Triangles Making up the Figure (t) 1 4 9 CONJECTURE:
Chapter 1: Inductive and Deductive Reasoning Section 1.1 4. Make a conjecture about the number of sides in the polygon and the number of interior regions formed by connecting opposite vertices to one point. Number of Sides (n) Number of Regions Formed (r) 3 4 5 6 1 2 3 4 CONJECTURE:
Chapter 1: Inductive and Deductive Reasoning Section 1.1 INDUCTIVE REASONING When we make a conjecture, we often use inductive reasoning. In these scenarios, we use a series of true statements and make a conjecture based on these statements. The conjecture that we make is based solely on the statements and is not necessarily true. Ex. Make a conjecture based on inductive reasoning: 1. Emily is between 15 and 17 years old Makenzie is between 15 and 17 years old Isaiah is between 15 and 17 years old CONJECTURE: 2. A dog is a mammal with 4 legs A cat is a mammal with 4 legs A horse is a mammal with 4 legs A mouse is a mammal with 4 legs CONJECTURE: 3. The sum of the interior angles of an equilateral triangle is 180 The sum of the interior angles of an isosceles triangle is 180 The sum of the interior angles of a scalene triangle is 180 CONJECTURE:
Chapter 1: Inductive and Deductive Reasoning Section 1.1 4. Use inductive reasoning to make a conjecture about the sum of any two consecutive numbers.
Chapter 1: Inductive and Deductive Reasoning Section 1.2 Section 1.2: Validity of Conjectures and Counterexamples Terminology: Counter Example: An example that shows that a conjecture is not always true, thus proving a conjecture to be invalid. Ex. Conjecture: All prime numbers are odd. Counter Example: 2 is a prime number, 2 is an even number, therefore not all prime numbers are odd. Using Counter Examples to Invalidate Conjectures Ex. In each situation, use a counter example to prove the conjecture to be invalid. Conjecture 1: The product of two integers is a positive number. Conjecture 2: Every student at Dorset Collegiate takes the bus to school. Conjecture 3: Every quadrilateral with four right angles is a square. Conjecture 4: Adam scored twice as well as John.
Chapter 1: Inductive and Deductive Reasoning Section 1.3 Section 1.3: Proving Conjectures using Deductive Reasoning Terminology: Deductive Reasoning: The process of coming up with a conclusion based on facts that have already been shown to be true. The facts that can be used to prove your conclusion deductively may come from accepted definitions, properties, laws or rules. The truth of the premises guarantees the truth of the conclusion. TRUE PREMISE + TRUE PREMISE = TRUE CONCLUSION Proof: A mathematical proof is an argument that shows that a statement is always valid and that no counterexample exists. Transitive Property: If two quantities are equal to the same quantity, then they are equal to each other. In other words, if A implies B and B implies C, than A also implies C. If A B and B C, then A C OR If A = B and B = C, then A = C Properties of Equality: If two quantities are equal, then whenever you perform any operation to one of the quantities, you must also perform the same operation to the other quantity. Ex. If A = B, then: A + 2 = B + 2 A 5 = B 5 A 6 = B 6 A 7 = B 7
Chapter 1: Inductive and Deductive Reasoning Section 1.3 Using Deductive Reasoning to Prove Conjectures When using deductive reasoning we can employ a variety of strategies to prove a conjecture. Some that we will use in this chapter include: Visual Representations 1. Visual Representations 2. Algebraic Expressions and Equations 3. Two-Column Proof Visual Representations involve using Venn Diagrams and/or Tables to help support your conjecture. Ex. All dogs are mammals. All mammals are vertebrates. Shaggy is a dog. What can be deduced about Saggy? Ex. Science is considered a Core subject. Chemistry is a science course. All Core courses in grade 12 have public exams. What can be deduced about the grade 12 chemistry class?
Chapter 1: Inductive and Deductive Reasoning Section 1.3 Algebraic Expressions and Equations Algebraic expressions and equations are often used to prove questions involving patterns and numbers in which we use variables in the place of numbers, therefore generalizing the context of a conjecture and proving it true for all settings. Some useful information for using Algebra and Deductive Reasoning include: o If n is any integer, than 2n must be an even integer o If n is any integer, than 2n + 1 must be an odd integer o If n is any integer, than n, n + 1, and n + 2 are three consecutive integers. o A number with digits abc can be written using place value such that abc = 100a + 10b + c Ex. In each situation prove the conjecture is true using only deductive reasoning: Conjecture 1: The sum of two consecutive numbers is an odd number Conjecture 2: The product of any two perfect squares is another perfect square
Chapter 1: Inductive and Deductive Reasoning Section 1.3 Conjecture 3: The sum of any two digit number with another two digit number with its digits being the reverse of the first, will be divisible by 11. Conjecture 4: The product of any fraction and its reciprocal will equal 1.
Chapter 1: Inductive and Deductive Reasoning Section 1.3 Two Column Proof A two column proof is a form of deductive reasoning in which the statements of the argument are written in one column and the justifications for the statements are written in the second column. Ex. Prove that the measure of the exterior angle of a triangle is equal to the sum of the two non-adjacent interior angles. STATEMENTS JUSTIFICATIONS
Chapter 1: Inductive and Deductive Reasoning Section 1.3 Proof by Inductive and Deductive Reasoning Sometimes a conjecture will need to be proven by both inductive and deductive reasoning. In such situations we use inductive reasoning to provide several examples where the conjecture is true, then we use deductive reasoning to prove the conjecture true in all settings. Ex. Chuck made the conjecture that the sim of any five consecutive integers is equal to 5 times the median. Use inductive and deductive reasoning to prove the conjecture. Ex. Minnie made a conjecture that the sum of any three consecutive integers is divisible by three. Use inductive and deductive reasoning to prove the conjecture. Ex. Kenny made a conjecture that the difference between the square of any two consecutive numbers is equal to an odd number. Use inductive and deductive reasoning to prove the conjecture.
Chapter 1: Inductive and Deductive Reasoning Section 1.3 Proving Mathematical Tricks Ex. Isaac created a step-by-step number trick: Choose any number Multiply it by 4 Add 10 Divide by 2 Subtract 11 Divide by 2 Add 3 The answer is the number you started with. Use inductive and deductive reasoning to prove that the result will always be the number you started with
Chapter 1: Inductive and Deductive Reasoning Section 1.3 Ex. Jamie created the following step-by-step math trick: Choose any positive number Double it Subtract 2 Double it again Add 8 Divide by 4 Subtract the number you started with Your answer is 1 Use inductive and deductive reasoning to prove this trick is true for any number
Chapter 1: Inductive and Deductive Reasoning Section 1.4 Section 1.4: Proofs that are Not Valid Terminology: Invalid Proof: A proof that contains an error in reasoning or contains an invalid assumption. Premise: A statement that is assumed to be true. Circular Reasoning: An argument that is incorrect because it makes use of the conclusion that is intended to be proved. Invalid Proofs A single error in reasoning will break down the logical argument of a deductive proof. This will result in an invalid conclusion or a conclusion that is not supported by the proof. Division by zero always creates an error in a proof, leading to an invalid conclusion. Circular reasoning must be avoided. Be careful not to assume a result that follows from what you are trying to prove. The reason you are writing a proof is so that others can read and understand it. After you write a proof, have someone else who has not seen your proof read it. If this person gets confused, your proof may need to be clarified.
Chapter 1: Inductive and Deductive Reasoning Section 1.4 Ex. Brie says that she can prove that 2 = 0. Here is her proof. Let a and b both be equal to 1. a = b a 2 = b 2 a 2 b 2 = 0 (a + b)(a b) = 0 (a + b)(a b) 0 = (a b) (a b) a + b = 0 1 + 1 = 0 2 = 0 Transitive Property Square Both Sides Subtract b 2 from both sides Factor using difference of squares Divide both sides by a b Simplification Substitution (since a = b = 1) Explain whether each statement in Brie s proof is valid.
Chapter 1: Inductive and Deductive Reasoning Section 1.4 Ex. Lee claims to have a proof that shows 5 = 5. Lee s Proof I assumed that 5 = 5 Then I squared both sides: ( 5) 2 = 5 2 I got a true statement:25 = 25 This means that my assumption that 5 = 5 must be correct Where is the error in Lee s Proof? Ex. Brad says he can prove that $1 = 1 Brad s Proof $1 can be converted to 100 100 can be expressed as (10) 2 10 is one tenth of a dollar so this becomes (0.1) 2 (0.1) 2 = 0.01 One one hundreth of a dollar is one cent, so $1 = 1 Where is the problem is Brad s reasoning?