Download INDUCTIVE REASONING

INDUCTIVE REASONING
Inductive Reasoning--- sometimes called inductive logic, is the process to make a generalization on a few
recurring patterns or observations.
1. Brenda has just gotten a job as the plumber’s assistant. Her first task is to open all the water valves to
release the pressure on the lines. The first four valves she discovered opened when turning
counterclockwise.
a) What is her conjecture?
b) Is this a good example of inductive reasoning?
c) What counterexample would disprove her conjecture?
2. Wile E. Coyote is setting up a trap for the roadrunner. Every previous trap has destroyed some part of his
body. However, after he blows him self up he seems to be healed by the next seen.
a) What are two possible conjectures?
b) Is this a good example of inductive reasoning?
c) What counterexample would disprove your conjectures?
3. Ralphy noticed that of the last four days he ate bacon for breakfast Lisa ran from him. Ralphy concluded that
if he had cereal for breakfast, then Lisa would not run from him.
a) Is this a good example of inductive reasoning? Why?
b) What would be a good conjecture?
2.1 Number patterns
Use inductive reasoning to find the next two terms.
1) 2, 20, 200, 2000,
2) 2, 5, 10, 17, 26, 37,
3) a, 6, c, 12, e, 18,
4) 1, 4, 9, 16, 25,
5) 0, 10, 21, 33, 46,
6) 3, 5, 11, 29, 83, 245
7) Fill in the blank to complete the pattern.
6  7  42
90 1  1
66  67  4422
9  1  2  11
a) 666  667  444222
b) 9  2  3  21
6666  6667  44442222
9  3  4  ___
____ _____  _________
9  ___  5  41
Picture patterns
1) Find the next picture in the pattern
a)
c)
c)
2.2 Finding the nth term
n
1
2
3
4
5
6
15
1
2
3
4
5
6
15
1
2
3
4
5
6
15
n-3
n
2n-1
n
3[(n-1)+n]
1. Find the formula for the nth term, then find the 15th term.
2. Square in a cross pattern.
3.
Figurate numbers-Numbers that correspond to geometric figures.
1. Find the differences between the numbers in the series.
2. Sequences that have a constant difference of this type have there patterns in the factors of each term. See if
you can find a pattern in the factors.
First factor
Second factor
3. This gives us two new sequences.
The first factors
4. Put them together this is your nth term. Now find the 20th term.
1.
2.
The second factors
Doubling
Since we doubled it, we know have to do the opposite and dive the nth term.
2.3 Mathematical Modeling
Creating the sequences from events.
1. How many ways can you connect 30 dots on a piece of paper if no three are in a line? Don't
try to draw it, try drawing smaller versions and find the pattern.
n
1
0
2
1
3
3
4
6
5
10
6
30
2. If there are 20 people sitting around a table, how many different pairs of people can have
conversations during dinner? What geometric figures can you use to represent this?
3.If a polygon has 20 diagonals leaving a vertex, then how many sides does it have. Hint: a
polygon has at least 3 sides.
4. Area of a square
2.4 Deductive Reasoning
Deductive Reasoning-- is the process of showing that certain statements follow logically from
agreed-upon assumptions and proven facts. Using true statements/properties to prove an
argument.
EX/ It is a fact that Jimbo the mechanic is always the cheapest mechanic in town.
Using deductive reasoning: "If I go to see a mechanic other than Jimbo, then I will pay more
than I would with Jimbo."
EX/ It is agreed upon that every Monday it rains.
Wrong use: "If it is raining, than it is Monday." Give a counter example.
Using deductive reasoning you could say, "It is Monday, so it will rain."
When you use __________________reasoning, you are generalizing from careful
observations/patterns that something is probably going to happen or is true.
When you use __________________reasoning, you are using established/accepted properties to
show something is true.
2.5 Angle relationships
Two angles are COMPLEMENTARY ANGLES if_________________________________________
Draw an example:
m
 m

Two angles are SUPPLEMENTARY ANGLES if___________________________________________
Draw an example:
m
 m

VERTICAL ANGLES are angles that are opposite each other when two lines cross.
Draw an example:
m
 m
and
m
 m
Two angles form a LINEAR PAIR if ____________________________________________________
Draw an example:
m
 m

The sum of a Linear pair of angles is____________________________________________
1. Find the missing angles.

50
100

30

2.6 Properties of Parallel Lines
Two or more lines are ___________________________if and only if they are coplanar and do not intersect
A line intersecting two or more coplanar lines is called a ____________________________
Using the picture below give one pair of:
Corresponding angels____________________
Alternate interior angles_______________________
Alternate exterior angles_______________________
1
3
5
7
2
4
6
8
If two parallel lines are cut by a transversal,
then corresponding angles are ______________________,
alternate interior angles are __________________________,
and alternate exterior angles are ________________________
Complete each statement.
1. If two angles are vertical angles, then they are ______________.
2. If two angles are a linear pair of angles, then they are______________.
3. If two angles are both equal in measure and ______________________then each angle measures 90°.
4. Find the measure of each angle indicated in the figure below.
a =
______ b =
______
c =
______
d =
______ e =
______
f =
______
g =
______ h =
______
j =
______
r
62° f
d
g
h
c
j
b
a
s
28°
e
t
w