Lesson 2 1 Inductive Reasoning and Conjecture Transparency

Lesson 2 -1 Inductive Reasoning and Conjecture

Transparency 2 -1 5 -Minute Check on Chapter 1 1. Find the value of x if R is between Q and T, QR = 3 x + 5, RT = 4 x – 9, and QT = 17. 2. Find the distance between A(– 3, 7) and B(1, 4). 3. Find m C if C and D are supplementary, m C = 3 y – 5, and m D = 8 y + 20. __ 4. Find SR if R is the midpoint of SU. 5. Find n if WX bisects VWY. 6. Standardized Test Practice: ___ Find the coordinates of the midpoint of MN if M(3, 6) and N(9, -4). A (12, 2) B (6, 5) C (1, 6) D (6, 1)

Transparency 2 -1 5 -Minute Check on Chapter 1 1. Find the value of x if R is between Q and T, QR = 3 x + 5, RT = 4 x – 9, and QT = 17. 3 2. Find the distance between A(– 3, 7) and B(1, 4). 5 3. Find m C if C and D are supplementary, m C = 3 y – 5, and m D = 8 y + 20. 40 __ 4. Find SR if R is the midpoint of SU. 22 5. Find n if WX bisects VWY. 10 6. Standardized Test Practice: ___ Find the coordinates of the midpoint of MN if M(3, 6) and N(9, -4). A (12, 2) B (6, 5) C (1, 6) D (6, 1)

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

Series See the pattern in the following series: 16 a. 1, 4, 7, 10, 13, ______ -32 b. 1, -2, 4, -8, 16, ____ +3 +3+3+3 • (-2) Find the pattern in the following series: a. 64 2, 4, 8, 16, 32, ____ 36 b. 1, 4, 9, 16, 25, _____ c. 9 3, 1, 5, 3, 7, 5, ____ 13 d. 1, 2, 3, 5, 7, 11, _____ e. -2 10, 12, 6, 8, 2, 4, ____ f. 20 5, 3, 10, 6, 15, 9, ____

Example 1 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. Conjecture: The next number will be multiplied by 6. So, it will be or 1440. Answer: 1440

Example 2 Make a conjecture about the next number based on the pattern. Answer: The next number will be

Example 3 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 Source: Labor Market Information Services–Kansas Department of Human Resources 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. ”

Example 3 cont 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 4 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: 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 Source: The World Almanac and Book of Facts 2003 “The greater the population of a state, the lower the number of drivers per 1000 residents. ” Answer: Alabama has a greater population than West Virginia, and it has more drivers per 1000 than West Virginia.

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