Lesson 2-1 Inductive Reasoning and Conjecture

5-Minute Check on Chapter 1 Transparency Find the value of x if R is between Q and T, QR = 3x + 5, RT = 4x – 9, and QT = Find the distance between A( – 3, 7) and B(1, 4). 3. Find m  C if  C and  D are supplementary, m  C = 3y – 5, and m  D = 8y __ 4. Find SR if R is the midpoint of SU. 5. Find n if WX bisects  VWY. ___ 6. Find the coordinates of the midpoint of MN if M(3, 6) and N(9, -4). Standardized Test Practice: ACBD (12, 2) (6, 5) (1, 6) (6, 1)

5-Minute Check on Chapter 1 Transparency Find the value of x if R is between Q and T, QR = 3x + 5, RT = 4x – 9, and QT = Find the distance between A( – 3, 7) and B(1, 4) Find m  C if  C and  D are supplementary, m  C = 3y – 5, and m  D = 8y __ 4. Find SR if R is the midpoint of SU Find n if WX bisects  VWY. 10 ___ 6. Find the coordinates of the midpoint of MN if M(3, 6) and N(9, -4). Standardized Test Practice: ACB (12, 2) (6, 5) (1, 6) (6, 1) D

Objectives Make conjectures based on inductive reasoning Find counterexamples

Vocabulary Conjecture – an educated guess based on known information Inductive reasoning – reasoning that uses a number of specific examples to arrive at a plausible generalization or prediction Counterexample – a false example

See the pattern in the following series: a. 1, 4, 7, 10, 13, ______ b. 1, -2, 4, -8, 16, ____ Find the pattern in the following series: a. 2, 4, 8, 16, 32, ____ b. 1, 4, 9, 16, 25, _____ c. 3, 1, 5, 3, 7, 5, ____ d. 1, 2, 3, 5, 7, 11, _____ e. 10, 12, 6, 8, 2, 4, ____f. 5, 3, 10, 6, 15, 9, ____ +3 (-2) Series

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 Example 2

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.” CountyCivilian Labor ForceRate Shawnee90,2543.1% Jefferson 9,9373.0% Jackson 8,9152.8% Douglas55,7303.2% Osage10,1824.0% Wabaunsee 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 cont

DRIVING The table below 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.” StatePopulationLicensed Drivers per 1000 Alabama 4,447, California33,871, Texas20,851, Vermont 608, West Virginia 1,808, Wisconsin 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. Example 4

Summary & Homework Summary: –Conjectures are based on observations and patterns –Counterexamples can be used to show that a conjecture is false Homework: –pgs. 64-5: 4,5,11,13,15,17,21,23,29