Transcript Inductive Reasoning

Lesson 2.1
Inductive Reasoning in Geometry
HOMEWORK: Lesson 2.1/1-15 odds,
22-24, 31-40, 42
EC: Due Wednesday
Page 104 – “Improve Reasoning Skills” #1-8
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
Objectives:
1)
Use inductive reasoning to find the next term in a number or
picture pattern
2)
To use inductive reasoning to make conjectures.
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?
• 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?
Inductive Reasoning –
reasoning that is based on
patterns you observe.
Ex. 1: Find the next term in the sequence:
A) 3, 6, 12, 24, ___, ___
B) 1, 2, 4, 7, 11, 16, 22, ___, ___
C)
,___, ___
Solutions
Ex. 1: Find the next term in the sequence:
A) 3, 6, 12, 24, ___,
48 ___
96
Rule: x2
B) 1, 2, 4, 7, 11, 16, 22, ___,
29 ___
37 Rule: +1, +2, +3, +4, …
C)
Rule: divide each
section by half
Steps of Inductive Reasoning
1. Process of observing data
2. Recognizing patterns
3. Making generalizations based on
those patterns
Identify the pattern and find the next 3
numbers:
1) 1, 4, 9, 16, ____, ____, ____
2) 1, 3, 6, 10, ____, ____, ____
3) 1, 1, 2, 3, 5, 8, ____, ____, ____
Solutions
Identify the pattern and find the next 3
numbers:
25
36
49
1) 1, 4, 9, 16, ____,
____,
____
sequence of perfect squares
15
21
28
2) 1, 3, 6, 10, ____, ____, ____
+2, +3, +4, +5, …
13
21
34
3) 1, 1, 2, 3, 5, 8, ____,
____,
____
Fibonacci – add the 2 previous
numbers to get the next.
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 …
Solution
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
Identify the pattern and find the next
item in the pattern.
January, March, May, ...
Observe the data..
Identify the pattern..
Make a generalization
Solution
January, March, May, ...
Alternating months of the year make up the pattern.
(skip every other month)
The next month is July.
Identifying a Pattern
Identify the pattern and find the next item in
the pattern.
7, 14, 21, 28, …
Observe the data..
Identify the pattern..
Make a generalization
Solution
7, 14, 21, 28, …
Multiples of 7 make up the pattern.
(add 7 to each term to get the next)
The next multiple is 35.
Identifying a Pattern
Identify the pattern and find the next item in
the pattern.
Solution
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.
10, 7, 9, 6, 8, 5, 7, . .
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, …
Be very careful with the wording/terms you use
to describe the pattern
Solution
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.
The next item would be 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 describe the pattern and
find the next two figures in the pattern.
Solution
Following the pattern: blue L, red +, green T…
the next figures would be
the red + and the green T
Geometric Patterns
Use inductive reasoning to describe the pattern and
find the next two figures in the pattern.
Solution
Following the pattern: green triangle is moving
CCW 120° (or rotating CCW every other side of
the hexagon)… the next figures would be
Green triangle on the bottom and then two
sides CCW
Geometric Patterns
Describe the figure that goes in the missing boxes.
Describe the next three figures in the pattern below.
Solutions
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.
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.
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.