Math Foundations 20 Unit 1 Lesson 1 Inductive Reasoning PDF

Math Foundations 20 | Unit 1 | Lesson 1 Unit 1 Lesson 1 Inductive Reasoning Purpose ▪ Identify examples of inductive reasoning. ▪ Make conjectures by observing patterns. ▪ Discuss the limitations of inductive reasoning. ▪ Use a counterexample to prove that a conjectur...

Math Foundations 20 | Unit 1 | Lesson 1 Unit 1 Lesson 1 Inductive Reasoning Purpose ▪ Identify examples of inductive reasoning. ▪ Make conjectures by observing patterns. ▪ Discuss the limitations of inductive reasoning. ▪ Use a counterexample to prove that a conjecture is false. Inductive Reasoning Inductive reasoning is a type of logical reasoning that involves making specific observations, recognizing patterns, and drawing a general conclusion. Since inductive reasoning is based on a limited number of specific observations, it can be quite unreliable. Typically, a greater number of observations leads to a more reliable conclusion. Inductive reasoning can have drawbacks. For example, in daily life, stereotypes are often formed as a result of inductive reasoning with few observation. Making a Conjecture Inductive reasoning involves the making of conjectures. A conjecture is a statement that appears to be true based on available data and observations but has not been conclusively proven. A conjecture cannot be proven through inductive reasoning. If it has been proven through another, more rigorous, form of reasoning, it is no longer called a conjecture. 1  Math Foundations 20 | Unit 1 | Lesson 1 Make a conjecture about the sum of two odd integers. To begin, we can list several examples: 1+3=4 5 + 9 = 14 11 + 23 = 34 As we generate examples, it is important to be aware of the possibility that a pattern arises as a result of the choice of examples. The first three examples suggest that the sum of two odd integers ends in 4. We then try to find examples that do not end in 4. 23 + 3 = 26 −15 + 7 = −8 −117 + (−19) = −136 A possible conjecture is: “The sum of two odd integers is an even integer.” Example 1: Make a Conjecture a) Use inductive reasoning to make a conjecture about the sum of the squares of any two consecutive even integers. 2  Math Foundations 20 | Unit 1 | Lesson 1 b) Use inductive reasoning to make a conjecture about the product of an odd integer and an even integer. Using a Counterexample to Disprove a Conjecture It is not possible to prove a conjecture using inductive reasoning. However, it is possible to prove a conjecture false by finding an example that contradicts the conjecture. An example that meets the conditions of the conjecture but does not lead to the conjecture’s conclusion is called a counterexample. The existence of a counterexample does not necessarily mean that the conjecture must be discarded completely. The conjecture can often be revised based on the new evidence. The following examples were used to develop a conjecture about the square of a number: 𝟐𝟐 = 𝟒 Conjecture: 𝟏𝟏𝟐 = 𝟏𝟐𝟏 The square of a number is greater than the number itself. (−𝟑)𝟐 = 𝟗 Find a counterexample to disprove the conjecture, then rewrite the conjecture so that it cannot be disproven by the counterexample. One counterexample for this conjecture is the square of the number 0.5. (0.5)2 = 0.25 In this case, the square of the number is less than the number itself. This is true for numbers between 0 and 1. 3  Math Foundations 20 | Unit 1 | Lesson 1 We can restrict the conjecture by changing the word “numbers” to “integers”, but we must consider whether there are still obvious exceptions. Both 0 and 1 are integers, and both are examples where the square of the number is equal to the number itself. 02 = 0 12 = 1 The revised conjecture is as follows: The square of an integer is greater than or equal to the integer itself. Example 2: Find a Counterexample to a Conjecture a) Find a counterexample to the following conjecture. “The sum of any two numbers is greater than the larger of the two numbers.” b) Steve claims that the difference between any two positive odd integers is always a positive even integer. Do you agree or disagree? Justify your decision. 4