2.1 Inductive Reasoning and Conjecture

Objectives Make conjectures based on inductive reasoning Make conjectures based on inductive reasoning Find counterexamples Find counterexamples

Making Conjectures Conjecture – an educated guess based on known information Conjecture – an educated guess based on known information Inductive Reasoning – using a number of specific examples to arrive at a plausible generalization or prediction Inductive Reasoning – using a number of specific examples to arrive at a plausible generalization or prediction We use inductive reasoning to create a conjecture. We use inductive reasoning to create a conjecture.

Make a conjecture about the next number based on the pattern. 2, 4, 12, 48, 240 Answer: 1440 Find a pattern: ×2×2 The numbers are multiplied by 2, 3, 4, and 5. Conjecture: The next number will be multiplied by 6. So, it will be or ×3×3×4×4×5×5 Example 1:

Make a conjecture about the next number based on the pattern. Answer: The next number will be Your Turn:

Given: points L, M, and N; Examine the measures of the segments. Since the points can be collinear with point N between points L and M. Answer: Conjecture: L, M, and N are collinear. For points L, M, and N, and, make a conjecture and draw a figure to illustrate your conjecture. Example 2:

ACE is a right triangle with Make a conjecture and draw a figure to illustrate your conjecture. Answer: Conjecture: In  ACE,  C is a right angle and is the hypotenuse. Your Turn:

Counterexamples Recall, conjectures are based on multiple observations. Whenever we are able to find an instance in which the conjecture is false, the entire conjecture is untrue. This false example is referred to as a counterexample.

UNEMPLOYMENT Based on the table showing unemployment rates for various cities in Kansas, find a counterexample for the following statement. The unemployment rate is highest in the cities with the most people. County Civilian Labor Force Rate Shawnee90,2543.1% Jefferson 9,937 9,9373.0% Jackson 8,915 8,9152.8% Douglas55,7303.2% Osage10,1824.0% Wabaunsee 3,575 3,5753.0% Pottawatomie11,0252.1% Source: Labor Market Information Services– Kansas Department of Human Resources Example 3:

Examine the data in the table. Find two cities such that the population of the first is greater than the population of the second while the unemployment rate of the first is less than the unemployment rate of the second. Shawnee has a greater population than Osage while Shawnee has a lower unemployment rate than Osage. Answer: Osage has only 10,182 people on its civilian labor force, and it has a higher rate of unemployment than Shawnee, which has 90,254 people on its civilian labor force. Example 3:

DRIVING The table on the next screen shows selected states, the 2000 population of each state, 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. Your Turn:

StatePopulation Licensed Drivers per 1000 Alabama 4,447,100 4,447, California33,871, Texas20,851, Vermont 608, , West Virginia 1,808,344 1,808, Wisconsin 5,363,675 5,363, Source: The World Almanac and Book of Facts 2003 Answer: Alabama has a greater population than West Virginia, and it has more drivers per 1000 than West Virginia. Your Turn:

Assignment Geometry: Geometry: Pg. 64 – 65 #11 – 36 #11 – 36 Pre-AP Geometry: Pre-AP Geometry: Pg. 64 – 65 #11 – 36,