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Inductive Reasoning Drawing a conclusion based upon repeated observations Deductive Reasoning Drawing a conclusion based upon facts or rules and NOT repeated observations Four times, John was late to class and four times his teacher gave him a detention John was late to class again today Conclusion: John will get a detention This conclusion was based on INDUCTIVE reasoning One of the class rules is that anyone who is late will receive a detention John was late to class Conclusion: John will get a detention This conclusion was based on DEDUCTIVE reasoning In the first inning of a baseball game, Dave Mover of the Adirondack Avalanche struck out. The next inning the coach benched him. In the second inning, Scott Sponsor struck out. The next inning the coach benched him. In the third inning, Mike Disoda struck out. The next inning the coach benched him. In the fourth inning, Cohen Suzuki struck out. Conclusion: The next inning, the coach benched him. inductive This conclusion was made using ___________ reasoning This sign hangs in the Appalachia Apple Pickers baseball team’s locker room. If you strike out, then in the next inning you will get benched In the first inning, Adam struck out Conclusion: The next inning, the coach benched him. deductive This conclusion was made using ___________ reasoning The Law of Detachment You are told that a given conditional statement is true You are told that the hypothesis of the conditional statement is true. Therefore you can conclude that the conclusion is also true The Law of Detachment If a car is out of gas, then it will not start Sarah’s car is out of gas Therefore Sarah’s car will not start The Law of Detachment If two angles form a linear pair, then their noncommon sides are opposite rays AED and AEB form a linear pair Can we conclude that ED and EB are opposite rays? YES A D E B The Law of Detachment If Mika goes to the beach, then she will wear sunscreen Mika is wearing sunscreen Can we conclude that Mika is at the beach? NO The Law of Syllagism If a, then b If b, then c Therefore, if a, then c The Law of Syllagism ab bc therefore a c The Law of Syllagism If you get a job, then you will earn money If you earn money, then you will buy a car If you get a job, then you will buy a car The Law of Syllagism If you eat food in the hallway, then my kids will go to jail Here’s why Law of Syllogism If you eat food in the hallway, then you will spill something on the floor If you spill something on the floor, someone will slip and fall. Law of Syllogism If you spill something on the floor, someone will slip and fall If someone slips and falls, then they will sue the school Law of Syllogism If someone slips and fall, then they will sue the school If someone sues the school, then the school will have to pay them a lot of money Law of Syllogism If someone sues the school, then the school will have to pay them a lot of money If the school has to pay someone a lot of money, then the school will blame me for not stopping someone from eating in the hall Law of Syllogism If the school has to pay someone a lot of money, then they will blame me for not stopping someone for eating in the hall If the school blames me for not stopping someone from eating in the hall, then I will get upset. Law of Syllogism If the school blames me for not stopping someone from eating in the hall, then I will get upset If I get upset, then someone is going to record me being upset Law of Syllogism If I get upset, then someone will record me being upset If someone records me being upset, then they will post the recording on youtube. Law of Syllogism If someone records me being upset, then they will post the recording on youtube. If someone posts a video of me being upset, then everyone in the world will be able to watch it Law of Syllogism If someone posts a video of me being upset, then everyone in the world will be able to watch it If everyone in the world is able to watch the video of me being upset, then my kids will be too embarrassed to go to school. Law of Syllogism If everyone in the world is able to watch the video of me being upset, then my kids will be too embarrassed to go to school. If my kids are too embarrassed to go to school, then they will never get a high school diploma. Law of Syllogism If my kids are too embarrassed to go to school, then they will never get a high school diploma If my kids never get a high school diploma, then they will never go to college. Law of Syllogism If my kids never get a high school diploma, then they will never go to college If my kids don’t go to college, then they will not get the kind of job that they want. Law of Syllogism If my kids don’t go to college, then they will not get the kind of job that they want. If my kids do not get the kind of job that they want, then they will not be able to buy the things that they want. Law of Syllogism If my kids do not get the kind of job that they want, then they will not be able to buy the things that they want. If my kids are not able to buy the things that they want, then they will just try to steal them from a store. Law of Syllogism If my kids are not able to buy the things that they want, then they will just try to steal them from a store. If my kids try to steal the things they want from a store, then they will get caught. Law of Syllogism If my kids try to steal the things they want from a store, then they will get caught. If my kids get caught stealing things from a store, then they will get arrested. Law of Syllogism If my kids get caught stealing things from a store, then they will get arrested. If my kids get arrested, then they will go to trial Law of Syllogism If my kids get arrested, then they will go to trial If my kids go to trial, then they will be found guilty Law of Syllogism If my kids go to trial, then they will be found guilty If my kids are found guilty, then they will go to jail. Law of Syllogism So….. If you eat food in the hallway, my kids will go to jail Use inductive reasoning to draw a conclusion about the probable next number in each list 27 a ) 3, 7,11,15,19, 23, ____ 34 b) 1,1, 2,3,5,8,13, 21, ____ c) 1, 2, 4,8,16, ____ 32 Use deductive reasoning to draw a conclusion about the following: Premise: Every odd integer is followed by an even integer and every even integer is followed by an odd integer even a ) If a is odd, then a 1 is ______ odd b) If b is even, then a 1 is ______ even c) If c is odd, then c 1 is ______, odd and c 2 is ______ Premise: The password for your itunes account must be at least six characters long, must contain only numbers and letters and must contain at least one number and one letter 1. You have a valid password. Draw three conclusions about your password a. Your password is at least 6 characters long Premise: The password for your itunes account must be at least six characters long, must contain only numbers and letters and must contain at least one number and one letter 1. You have a valid password. Draw three conclusions about your password b. Your password is made up of only numbers and letters Premise: The password for your itunes account must be at least six characters long, must contain only numbers and letters and must contain at least one number and one letter 1. You have a valid password. Draw three conclusions about your password c. Your password contains at least one number and at least one letter Premise: The password for your itunes account must be at least six characters long, must contain only numbers and letters and must contain at least one number and one letter 2. Which of the following are valid passwords 5nhth54 2354654 3#4$BH6 K6dL9 LGTFCC ASd5Fg Premise: May 22, 1985 was a Wednesday. Draw a conclusion about May 23, 1985 May 23, 1985 was a Thursday Premise: May 22, 1985 was a Wednesday. Draw a conclusion about May 28, 1985 May 28, 1985 was a Tuesday Premise: The Indy 500 is always run on Memorial Day. Memorial Day is always the last Monday in May On what date was the Indy 500 run in 1985? May 27, 1985 Be Careful When Drawing Your Conclusions A four engine jet is flying from New York to Los Angeles. The estimated arrival time is 1:00 PM. When one of the engines fails, the plane slows down and the new estimated arrival time becomes 2:00 PM. A little while later, a second engine fails, causing the plane to slow down making the new arrival time 3:00 PM. A short time after that, a third engine fails, making the updated arrival time to be about 4:00 PM. Conclusion: If the last engine fails, the plane will arrive at about 5:00 PM FAT-A-WAY Lose 15 lbs a week Guaranteed Conclusion: If you currently weigh 150 pounds, then in 15 weeks you will be invisible Homework Pg 7 #1 – 12 Hand In Tomorrow Warm Up Predict the next multiplication fact a ) 37 3 111 b) 37 6 222 c) 37 9 333 d ) 37 12 444 e) 37 __ __ 15 ____ 555 Verify your conclusion Determine the most probable next term in each list of numbers 38 a ) 13,18, 23, 28,33, ___ 1 2 b) 32,16,8, 4, 2,1, ___ c) 1 3 3 5 5 7 7 9 9 11 11 13 , , , , , ___ d ) 1, 4,9,16, 25, 36 ___ 56 e) 2, 6,12, 20,30, 42, ___ f ) 1, 2, 3, 4, 5, 6, ___ -7 Determine the most probable next term in each list of numbers g ) 8, 2,8, 2, 2,8, 2, 2, 2,8, 2, 2, 2, 2,8, 2, 2, 2, 2, ___ 2 Predict the next equation, then verify your conjecture a) 1 × 9 + 2 = 11 12 × 9 + 3 = 111 123 × 9 + 4 = 1,111 1,234 × 9 + 5 = 11,111 12,345×9+6 =111,111 Predict the next equation, then verify your conjecture b) 15,873 × 7 =111,111 15,873 ×14 = 222,222 15,873 ×21 = 333,333 15,873 ×28 = 444, 444 15,873 ×35 = 555,555 Predict the next equation, then verify your conjecture c) 11 ×11 =121 111 ×111 =12,321 1,111 ×1,111 =1,234,321 11,111×11,111 =123, 454,321 Predict the next equation, then verify your conjecture d) 2 = 4 -2 2 + 4 = 8-2 2 + 4 + 8 =16-2 2 + 4 + 8 +16 = 32-2 2+ 4 +8+16+32 = 64 -2 Predict the next equation, then verify your conjecture e) 3= 33-1 2 3+9 = 3 9-1 2 3 + 9 +27 = 327-1 2 3 + 9 +27 + 81 = 3 81-1 2 3+ 9 +27 +81 +243 = 3243-1 2 Predict the next equation, then verify your conjecture 1 2 f) 1 2 1 2 1 2 1 2 =1 - 12 + 41 =1 - 14 + 41 + 81 =1 - 81 + 41 + 81 + 161 =1 - 161 + 41 + 81 + 161 + 321 =1- 321 Homework Pg 7&8 #15 – 41 odd Hand In Tomorrow Warm Up 1+2+3+4+5+6+7+8+9+10 = 55 __ 1+2+3+…98+99+100 = 5,050 __ Gauss Method for Finding Sums 1+2+3+4+5+6+7+8+9+10 = 55 __ Number Sum of × of Pairs Each Pair 5 × 11 = 55 Find each sum by using the Gauss Method of Sums a) b) c) d) 1+2+3+…+198+199+200= 10,100 ___ 500,500 1+2+3+…+999+1000=_____ 2,001,000 1+2+3+…+2000=_____ 125,250 1+2+3+…+499+500=_____ How could you modify the Gauss method to find the sum below? a) 1+2+3+…+99+100+101= ___ 50 × 101 + 101 5,050 + 101 5,151 Modify the Gauss method to find the sum below. 2+4+6+8+…+96+98+100 25 x 102 2,550 Modify the Gauss method to find the sum below. 5+10+15+20+…+190+195+200 Numbers in sum: 200 ÷ 5=40 Number of pairs: 20 Sum of each pair: 205 20 x 205 =4100 Homework Gauss Sum Worksheet Hand In Tomorrow