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Class Greeting Objective: The students will learn the vocabulary inductive reasoning, conjecture and counterexample. They will learn how to use inductive reasoning to identify patterns and make conjectures. Chapter 2 – Lesson 1 Inductive Reasoning and Conjecture Proof is important! This is what Geometry is all about! Vocabulary • Conjecture A guess based on observed or known information For example. You have seen Convent Road flooded when it rains. Now it’s raining, you make the conjecture that Convent Road will flood. Vocabulary • general Related to a group or collection For example. In general, January is cool season in Thailand. Vocabulary • specific Related to an individual For example. In 2010, specifically, January was warm in Thailand. Vocabulary Inductive Reasoning Can you see that inductive a way to reach a general conclusion using reasoning canone or more specific facts sometimes give you the For example. wrong idea? Assuming that it is ALWAYS warm in January in Thailand because you remember that it was warm in 2010 is an example of inductive reasoning. Inductive Reasoning Specific facts + observed conclusion = general rule Example of Inductive Reasoning a way to reach a general conclusion using one or more specific facts Make a conjecture about the next number based on the pattern. 2, 4, 12, 48, 240 Find a pattern: 2 4 ×2 12 ×3 48 ×4 240 ×5 The numbers are multiplied by 2, 3, 4, and 5. Answer: Conjecture: The next number will be multiplied by 6. So, it will be or 1440. Make a conjecture about the next number based on the pattern. Answer: Conjecture: The next number will be For collinear points L, M, and N, and make a conjecture and draw a figure to illustrate your conjecture. Given: points L, M, and N; Since the points are collinear with point N between points L and M. we can make the following conjecture: Answer: Conjecture: L, M, and N ar.. ACE is a right triangle with make a conjecture. Draw a figure and Answer: Conjecture: In ACE, C is a right angle and hypotenuse. is the Prove it FALSE It only takes ONE EXAMPLE to prove a conjecture is false. An example that proves a conjecture false is called a Counterexample. For example. Conjecture: It is ALWAYS warm in January in Thailand. Counterexample: It was cold in Thailand in January 2009. UNEMPLOYMENT Based on the table showing unemployment rates for various counties in Kansas, find a counterexample for the following statement. The unemployment rate is highest in the counties with the most people. County Civilian Labor Force Rate Shawnee 90,254 3.1% Jefferson 9,937 3.0% Jackson 8,915 2.8% Douglas 55,730 3.2% Osage 10,182 4.0% 3,575 3.0% 11,025 2.1% Wabaunsee Pottawatomie Answer: Osage has fewer people than Shawnee and it has a higher rate of unemployment than Shawnee. DRIVING The table shows the 2000 population of selected states, and the number of people per 1000 residents who are licensed drivers in each state. Based on the table, find a counterexample for the following statement. The greater the population of a state, the lower the number of drivers per 1000 residents. State Population Licensed Drivers per 1000 Alabama 4,447,100 792 California 33,871,648 627 Texas 20,851,820 646 608,827 831 West Virginia 1,808,344 745 Wisconsin 5,363,675 703 Vermont Answer: Alabama has a greater population than West Virginia, and it has more drivers per 1000 than West Virginia. Lesson Summary Objective: The students will learn the vocabulary inductive reasoning, conjecture and counterexample. They will learn how to use inductive reasoning to identify patterns and make conjectures. Preview of the Next Lesson: Objective: The students will learn the key vocabulary: conditional statement, hypothesis, and conclusion and how to write conditional statements while identifying the hypothesis and conclusion. Homework Geometry 2-1