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Geometric Structure
Inductive & Deductive Reasoning
Introduction
Instruction
In this lesson you will
• Perform geometry investigations
and make some discoveries by
observing common features or
patterns
• Use your discoveries to solve
problems through a process called
inductive reasoning.
• Use inductive reasoning to
discover patterns.
• Learn to use deductive
reasoning.
• Make conjectures.
• See how some conjectures are
logically related to each other.
Examples
Practice
EQ:
What is reasoning?
Introduction
Instruction
Examples
Please go back or choose a
topic from above.
Practice
List of Instructional Pages
1. Inductive Reasoning
2. Patterns in all Places
3. More inductive
reasoning
4. One Possibility
5. Conjecture
6. Goldbach's
Conjecture
7. More Goldbach
8. Examples
9. Deductive Reasoning
10. Algebra Example
11. Number Trick
12. Number Trick
Deduction
Introduction
Instruction
Examples
Practice
Definition: Inductive reasoning
is the process of observing data,
recognizing patterns, and making
generalizations about those
patterns. These generalizations
are often called conjectures.
Next view the presentation called
Patterns in all Places. Pay close
attention (maybe take notes?) to
the use of inductive reasoning to
arrive at a conjecture. You will be
expected to follow a similar
process on your own in
subsequent examples.
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presentation
Patterns in all Places
Connecting Geometric and
Numeric Patterns
©2005 Valerie Muller
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To find an algebraic rule for a pattern…
•Examine the figure closely
•Count the number of suns
•Record the number of suns
in the table
Figure
Number
1
Intro
Figure 1
Number of
Suns
7
Next
End
Vocabulary
To find an algebraic rule for a pattern…
•Examine the new figure
closely
•Focus on the “new”
objects added to the
previous figure
•Count the suns
•Record the number
Figure
Number
1
2
Intro
Number of
Suns
Figure 2
7
7+4=11
Next
End
Vocabulary
To find an algebraic rule for a pattern…
•Examine the new figure
•Focus on the “new”
•Count
•Record
Figure
Number
1
Number of Suns
2
7
11
3
11+4=15
Intro
Next
Figure 3
End
Vocabulary
What is the next figure in the pattern?
•What is the next
term in the pattern?
Figure
Number
1
Number of Suns
2
7
11
3
15
Figure 4 – What must be
added to the third figure to
create the fourth figure?
4
Intro
Next
End
Vocabulary
What is the next figure in the pattern?
•What is the next
term in the pattern?
Figure
Number
Number of Suns
1
2
3
7
11
15
4
19
Intro
add 4
add 4
add 4
Next
Figure 4 – What must be
added to the third figure to
create the fourth figure?
End
Vocabulary
What is the 10th term in the pattern?
•The table may be
extended
Figure
Number
Suns
1
7
2
11
3
15
4
19
•The function rule
may be found
…
10
?
“Shortcut” for repeatedly
adding 4 is multiplying by 4
Intro
Next
End
Vocabulary
What is the 10th term in the pattern?
•The table may be
extended:
Figure
Number
Suns
1
7
2
11
3
15
4
19
7, 11, 15, 19, 23,
27, 31, 35, 39, 43
•The function rule
may be found:
…
10
?
4*Figure # + 3 =
suns
“Shortcut” for repeatedly
adding 4 is multiplying by 4
Intro
End
Next
Vocabulary
What is the nth term in the pattern?
Figure
Number
Suns
1
7
2
11
7, 11, 15, 19, 23, 27,
31, 35, … , 4x + 3
3
15
4
19
•The function rule is
10
43
x
4x + 3
•The table may be
extended:
f(x) = 4x + 3
Multiply figure by 4 and add 3
Intro
Next
End
Vocabulary
Associated terms…
•Relation: pairing between
two sets of numbers to
create a set of ordered pairs
•Function: special relation
where pairs are formed such
that each element of the first
set is paired with exactly one
element of the second set
Intro
Next
End
nth term
Introduction
Instruction
Examples
Practice
Definition: Inductive reasoning
is the process of forming
conjectures that are based on
observations.
Example:
The numbers 72, 963, 10854,
and 7236261 are all divisible by
9.
Add the digits in each number.
Do you see a pattern?
Can you make a conjecture?
One possibility
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Introduction
Instruction
Examples
Practice
One Possibility
•Remember:
72: 7 + 2 = 9 This is
•INDUCTIVE
963: 9 + 6 + logic.
3 = 18 =
1+8=9
You
• 10,854: 1 + 0 + 8 + 5 + 4 = 18 =
observed
a pattern and made
1+8=9
conjecture that you believe
•a 7,236,261:
to7be
+ 2true
+ 3 +for
6 +all
2 +examples.
6 + 1 = 27 =
2+7=9
CONJECTURE: In order
for a number to be divisible
by 9, the sum of the digits
must be divisible by 9.
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Introduction
Instruction
Examples
Practice
Inductive reasoning is
the process of drawing
a general conclusion by
observing a pattern
from specific instances.
This conclusion is
called a hypothesis or
conjecture.
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Introduction
Instruction
Examples
Practice
Goldbach’s Conjecture
In 1742, mathematician
Christian Goldbach made the
conjecture that every integer
greater than 2 could be written
as the sum of two prime
numbers.
For example:
20 = 13 + 7
48 = 11 + 37
To this day, no one has proven
this conjecture but it is
accepted as true because no
one has found a
counterexample.
Christian Goldbach
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Introduction
Instruction
Examples
Practice
Use Goldbach’s
conjecture to express
the following numbers
as the sum of two
primes.
100 = 41 + 59
Page list
68 = 37 + 31
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Introduction
Instruction
Now click on
the Examples
link above to
work more
sample
problems for
inductive
reasoning.
Examples
Practice
Graphic
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Introduction
Instruction
Examples
Practice
Definition:
Deductive reasoning is the
process of drawing logical,
certain conclusions by
using an argument.
In deductive reasoning, we
use accepted facts and
general principles to arrive
at a specific conclusion.
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Introduction
Instruction
Algebra Example:
•You can solve an equation
by adding equal amounts
to each side of the
equation.
•You can solve x – 1 = 9 by
adding 1 to both sides of
the equation to get x = 10.
•This is an example of
deductive reasoning.
•You used the additive
property of equality to
solve the equation.
Examples
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Practice
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Introduction
Instruction
You can use deductive reasoning to
explain a number trick:
1. How many days a week do you
eat out?
2. Multiply this number by 2
3. Add 5 to the number you got in
step 2.
4. Multiply the number you
obtained in step 3 by 50.
5. If you have already had your
birthday this year add 1756; if
you haven’t, add 1755.
6. Subtract the four-digit year that
you were born.
7. You should have a three-digit
number.
Examples
Practice
What
the
The
firstare
digit
is
last
two of
the
number
times
that you
digits?
eat out each
week.
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Introduction
Instruction
You can use deductive reasoning to
explain or prove a number trick:
1. Call the number of days you eat
out n
2. Multiplying this number by 2
gives 2n
3. Adding 5 gives 2n + 5
4. Multiplying by 50 gives 100n +
250
5. Adding 1756 gives 100n + 2006
or 100n + 2005
6. Subtracting the four-digit year
that you were born gives 100n +
2006 – 1987 (for example) or
100n + 19
7. The hundred’s digit is n, because
the 18 (or 19) will have no effect
on the hundred’s digit.
Examples
Practice
Elementary,
my dear
Watson!
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The end
Introduction
Instruction
Examples
You have now completed
the instructional portion of
this lesson.
You may proceed to the
practice assignment.
Practice
Introduction
Instruction
Examples
Practice
Examples
Example 1: points
& segments
Example 2: circles
& regions
Example 1
Example 2
Introduction
Instruction
Examples
Please go back or choose a
topic from above.
Practice
Introduction
Instruction
Examples
Practice
Practice
•Finding Patterns
Gizmo
What is reasoning?
Introduction
Instruction
Examples
Please go back or choose a
topic from above.
Practice
Example 1
Back to main
example page
4
1
1
2
3
2
3
1
6
1 5
2
2
4
3
3
Number of
Points
Number of
Segments
3
3
4
6
5
10
6
15
7
21
?
8
28
?
9
36
?
To complete
the table, try
looking for a
pattern.
Back to main
example page
Example 2
We want to divide a circle
into regions by putting
points on the
circumference and
drawing segments from
each point to every other
point. Make a table with
the number of points and
the number of regions.
One point  One region
Two points  Two regions
Three points  Four regions
Example 2
Back to main
example page
Number of
Points
Number of
Regions
1
2
3
4
5
6
7
1
2
4
Example 2
Back to main
example page
Number of
Points
Number of
Regions
1
2
3
4
5
6
7
1
2
4
8
Back to main
example page
Example 2
•Conjecture:
It appears that the
number of regions doubles
each time a point is added.
•Test the conjecture:
Draw a large circle & put
six points on the
circumference. Connect
each point to every other
point & count the regions.
Example 2
Back to main
example page
Number
of Points
Number
of Regions
1
2
3
4
5
6
7
1
2
4
8
16
Be
careful….after
you make
your
conjecture,
try to actually
count the
regions.
Back to main
example page
Example 2
Test the conjecture?
2
1
10
9
8
14
7
13 23
3
11 12
22 31
15
21
16
30
6 17
20 26 29
18
19
27 28
25
4
24
5
There are only 31
regions, so our
conjecture was false.
Now, how many
regions do you think
7 points would make?
Now return to the instruction.