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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
ab
bc
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=
33-1 
2
3+9 =
3 9-1 
2
3 + 9 +27 =
327-1 
2
3 + 9 +27 + 81 =
3 81-1 
2
3+ 9 +27 +81 +243 =
3243-1
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
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