Download Inductive Reasoning

Teach Me to Think:
Developing Thinking Skills,
It’s what sets us apart.
What is Critical Thinking?
• Focused thinking
• Thinking with a definite purpose (goal)
– Can be a complex & involved process
• An active process that involves constant
questioning
Why we need
Critical Thinking students
• “The significant problems we face
cannot be solved at the same level of
thinking we were at when we created
them.”
• An Albert Einstein Quote on Creativity
What are the Goals of Critical
Thinking?
•
•
•
•
•
•
Finding Meaning
Seeking Logic
Searching for reason
Looking answers
Developing facts and opinions
Appreciating different points of view
How Can I develop my Critical
Thinking Skills?
• Think about your thinking
• Think about why you make your choices
and decisions
• Think about why the world is the way it is
• Practice every day!
– Word problems
– Math problems
– Puzzles
– Games of strategy
Critical Thinking and Reasoning
Chapter 2 introduces
INDUCTIVE and DEDUCTIVE Reasoning
These types of Reasoning are essential to
CRITICAL THINKING
Lesson 2.1
Inductive Reasoning in Geometry
HOMEWORK: 2.1/1-15 odds
Objectives:
1) Use inductive reasoning to find the next
term in a number or picture pattern
2) To use inductive reasoning to make
conjectures.
Inductive reasoning:
• make conclusions based on patterns you
observe
Conjecture:
• conclusion reached by inductive reasoning
based on evidence
Geometric Pattern:
• arrangement of geometric figures that repeat
Inductive Reasoning – Is reasoning that is
based on patterns you observe.
If you observe data, then recognize a pattern (the
rule) in a sequence you can use inductive reasoning
to find the next term.
Ex. 1: Find the next term in the sequence:
48 ___
96
A) 3, 6, 12, 24, ___,
Rule: x2
29 ___
37 Rule: +1, +2, +3, +4, …
B) 1, 2, 4, 7, 11, 16, 22, ___,
C)
Rule: divide each
section by half
Inductive Reasoning
1. Process of observing data
2. Recognizing patterns
3. Making generalizations based on
those patterns
An example of inductive reasoning
Suppose your history teacher likes to give
“surprise” quizzes.
You notice that, for the first four chapters of
the book, she gave a quiz the day after she
covered the third lesson.
Based on the pattern in your observations,
you might generalize that you will have a
quiz after the third lesson of every
chapter.
Identifying a Pattern
Find the next item in the pattern.
January, March, May, ...
Observe the data..
identify the pattern..
state the pattern.
Alternating months of the year make up the pattern.
(skip every other month)
The next month is July.
Identifying a Pattern
Find the next item in the pattern.
7, 14, 21, 28, …
Observe the data..
identify the pattern..
state the pattern.
Multiples of 7 make up the pattern.
(add 7 to each term to get the next)
The next multiple is 35.
Identifying a Pattern
Find the next item in the pattern.
In this pattern, the figure rotates 90°
counter-clockwise each time.
The next figure is
.
Inductive reasoning can be used to
make a conjecture about a number
sequence
Consider the sequence 10, 7, 9, 6, 8, 5, 7, . . .
Make a conjecture about the rule for
generating the sequence.
Then find the next three terms.
Solution
10, 7, 9, 6, 8, 5, 7, . .
Look at how the numbers change from term
to term
The 1st term in the sequence is 10.
You subtract 3 to get the 2nd term.
Then you add 2 to get the 3rd term.
You continue alternating between
subtracting 3 and adding 2 to generate the
remaining terms.
The next three terms are 4, 6, and 3.
Identifying a Pattern
Find the next item in the pattern
0.4, 0.04, 0.004, …
Rules & descriptions can be stated in many different
ways:
Multiply each term by 0.1 to get the next.
Divide each term by 10 to get the next.
When reading the pattern from left to right, the next
item in the pattern has one more zero after the
decimal point.
The next item would have 3 zeros after the decimal
point, or 0.0004.
Geometric Patterns
Arrangement of geometric figures that repeat
Use inductive reasoning and make conjecture as to the next
figure in a pattern
Use inductive reasoning to find the next two
figures in the pattern.
Geometric Patterns
Use inductive reasoning to find the next two figures in the pattern.
Geometric Patterns
Describe the figure that goes in the missing boxes.
Describe the next three figures in the pattern below.
Lesson 2.1 – Inductive
Reasoning
Objectives:
1) Use inductive reasoning to find the next
term in a number or picture pattern
2) To use inductive reasoning to make
conjectures.
Homework: WS 2.1
Mathematicians use Inductive Reasoning
to find patterns which will then allow
them to conjecture.
We will be doing this ALOT this year!!
Conjectures
A generalization made with inductive
reasoning (Drawing conclusions)
EXAMPLES:
• Bell rings M, T, W, TH at 7:40 am
Conjecture about Friday?
The bell will ring at 7:40 am on Friday
• Chemist puts NaCl on flame stick and puts into
flame and sees an orange-yellow flame. Repeats
for 5 other substances that also contain NaCl also
producing the same color flame.
Conjecture?
All substances containing NaCl will
produce an orange-yellow flame
Finding Patterns
• 2, 4, 7, 11, ...
– Rule?
Add the next consecutive integer
– Next 3 terms?
• 16, 22, 29
• 1, 1, 2, 3, 5, 8, 13, ...
– Rule?
Add previous two terms
(Fibonacci Sequence)
– Next 3 terms?
• 21, 34, 55
• 1, 4, 9, 16, 25, 36, ...
– Rule?
Add consecutive odd numbers OR
the perfect squares
– Next 3 terms?
• 49, 64, 81
Make a conjecture about the sum of the
first 30 odd numbers.
1
1+3
1+3+5
1+3+5+7
1+3+5+7+9
=1
= 12
=4
= 22
=9
= 32
= 16
= 42
= 25
= 52
..
..
1 + 3 + 5 +...+ 61
= 900
= 302
cont.: Make a conjecture about the sum of
the first 30 odd numbers.
Conjecture:
Sum of the first 30 odd numbers = 302
𝒏 = the amount of numbers added
Sum of the first 𝒏 odd numbers = 𝒏𝟐
Truth in Conjectures
To show that a conjecture is always true, you
must prove it.
To show that a conjecture is false, you have to find
only one example in which the conjecture is not
true.
This case is called a counterexample.
A counterexample can be a drawing, a statement,
or a number.
Inductive Reasoning assumes that an
observed pattern will continue.
This may or may not be true.
Ex: x = x • x
This is true only for x = 0 and x = 1
Conjecture – A conclusion you reach using
inductive reasoning.
Counter Example – To a conjecture is an
example for which the conjecture is incorrect.
The first 3 odd prime numbers are
3, 5, 7. Make a conjecture about the 4th.
11
3, 5, 7, ___
One would think that the rule is add 2, but that
gives us 9 for the fourth prime number.
Is that true? No
What is the next odd prime number?
Finding a Counterexample
Show that the conjecture is false by finding a
counterexample.
For every integer n, n3 is positive.
Pick integers and substitute them into the expression
to see if the conjecture holds.
Let n = 1. Since n3 = 1 and 1 > 0, the conjecture holds.
Let n = –3. Since n3 = –27 and –27  0, the
conjecture is false.
n = –3 is a counterexample.
Finding a Counterexample
Show that the conjecture is false by finding a
counterexample.
Two complementary angles are not congruent.
45° + 45° = 90°
If the two congruent angles both measure 45°,
the conjecture is false.
Finding a Counterexample
Show that the conjecture is false by finding a
counterexample.
The monthly high temperature in Abilene is
never below 90°F for two months in a row.
Monthly High Temperatures (ºF) in Abilene, Texas
Jan
Feb
Mar
Apr
May
Jun
Jul
Aug Sep
Oct Nov Dec
88
89
97
99
107
109
110
107 106 103
92
89
The monthly high temperatures in January and February
were 88°F and 89°F, so the conjecture is false.
Finding a Counterexample
Show that the conjecture is false by finding a
counterexample.
Supplementary angles are adjacent.
23°
157°
The supplementary angles are not adjacent,
so the conjecture is false.
Finding a Counterexample
Show that the conjecture is false by finding a
counterexample.
The radius of every planet in the solar system is
less than 50,000 km.
Planets’ Diameters (km)
Mercury Venus Earth
4880
12,100
12,800
Mars
Jupiter
Saturn
Uranus
Neptune
6790
143,000
121,000
51,100
49,500
Since the radius is half the diameter, the radius of
Jupiter is 71,500 km and the radius of Saturn is
60,500 km. The conjecture is false.