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.

a)

__________________

4444222266676666

444222667666

44226766

4276

b)

415___9

___439

21329

11219

1109

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

n-3

n 1 2 3 4 5 6 15

2n-1

n 1 2 3 4 5 6 15

3[(n-1)+n]

1. Find the formula for the nth term, then find the 15th term.

2. Square in a cross pattern.

3.

First factor

Second factor

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.

3. This gives us two new sequences.

The first factors The second factors

4. Put them together this is your nth term. Now find the 20th term.

1.

2.

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 2 3 4 5 6 30

0 1 3 6 10

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:

mm

Two angles are SUPPLEMENTARY ANGLES if___________________________________________

Draw an example:

mm

VERTICAL ANGLES are angles that are opposite each other when two lines cross.

Draw an example:

mm and mm

Two angles form a LINEAR PAIR if ____________________________________________________

Draw an example:

mm

The sum of a Linear pair of angles is____________________________________________

50

100

30

1. Find the missing angles.

1 2

3 4

5 6

7 8

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_______________________

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 = ______

d

j

b

f g

h

e

r

s

t

w

6 2 °

2 8 ° c a