Download 1 Section 1.1 – Inductive and Deductive Reasoning Objective #1

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Section 1.1 – Inductive and Deductive Reasoning
Objective #1:
Understanding and Applying Inductive Reasoning
Inductive Reasoning involves making generalizations from some very
specific events or facts. One is making a prediction or a conclusion about
an object or an event that is to occur in the future based upon a pattern or
upon observations of previous occurrences. Such conclusions or
predictions are called conjectures or hypotheses. There is no guarantee
that the prediction or conclusion will be true.
For example, if after noting summer after summer that there is a drought in
Texas, one may conclude that this summer, there will be a drought in
Texas. Another example would be seeing that the New England Patriots
have had winning records season after season, one would predict that they
will have a winning season this year.
In Mathematics, inductive reasoning is used to predict what comes next in
a sequence. In the next few examples, the next two objects in the
sequence will be determined based upon the pattern of the sequence.
Identify a pattern and then use the pattern to predict the next items:
Ex. 1
1, 3, 5, 7, 9, __ , __
Solution:
Since each number is 2 more than the previous number, we would
predict that the next number would be 9 + 2 = 11 and the following
one would be 11 + 2 = 13.
Ex. 2
1, 3, 9, 27, 81, 243, __ , __
Solution:
Since each number is three times the previous number, we would
induce that the next two numbers would be 243•3 = 729 and
729•3 = 2187.
Ex. 3:
1, 4, 9, 16, 25, 36, 49, __ , __
Solution:
This one is a little more involved. But, notice:
1
4
9
16
25
36
49
\ / \ / \ / \
/ \
/ \
/
+ 3 + 5 + 7 + 9 +11 +13
2
To get the next number, we will add 15 to 49, which will gives us 64
and then add 17 to 64 to give us 81. Or, you may have realized that
2
this simple the list of the perfect squares. In other words, 1 = 1,
2
2
2
2
2
2
2 = 4, 3 = 9, 4 = 16, 5 = 25, 6 = 36, 7 = 49, so the next two
2
2
numbers in the sequence are 8 = 64 and 9 = 81. This indicates that
there is more than one correct way of looking at a sequence and
predicting what comes next.
Ex. 4
1, 6, 4, 3, 7, 0, 10, – 3, 13, – 6, __ , __
The hint here is to look at every other number:
1, 6, 4, 3, 7, 0, 10, – 3, 13, – 6
\
/\ /\ / \
/ \
/
+3 +3 +3 +3
+3
So, the next number will be 16. If we look at the other set of numbers
(not underlined), they are decreasing by 3, so the number that comes
after the 16 will be – 6 – 3 = − 9.
1, 6, 4, 3, 7, 0, 10, – 3, 13, – 6, 16
\ /\ / \
/ \
/ \
/
–3 –3 –3
–3
–3
Ex. 5
Solution:
The number of little squares in the corners keeps
increasing by one, so we would predict that there
would be a little square in each of the four corners
in the next item. Also, the little squares alternate
between being shaded and not shaded, so we would
expect the little squares to be unshaded in the next item.
Ex. 6
Solution:
Moving from left to right, the inside figure becomes the outside figure
in the next item. So, the "Plus" sign inside the fourth figure will
become the outside figure for the missing item. Moving from right to
3
left the outside figure becomes the inside
figure so the quadrilateral on the outside
the sixth figure will become the inside
figure for the missing item.
Objective #2: Understanding and Applying Deductive Reasoning.
Deductive Reasoning involves reasoning from some general statements
of fact (the premises) to a specific logical conclusion. Unlike inductive
reasoning, the conclusion has to be valid given the premises. A conclusion
that is proved to be true by deductive reasoning is a called a theorem.
For example, if Ryan Howard struck out every time he went to bat in
today’s game, you could conclude that he did not hit a homerun in today’s
game. Another example would be if everyone in the St. Philip’s College
Choir sings well and Renita Mitchell is a member of the St. Philip’s College
Choir. Then, you could conclude that Renita Mitchell sings well. There is
only one possible conclusion that can be made with deductive reasoning.
Apply the given procedure for several different numbers and make a
conjecture of how to calculate the numbers. Then use the variable n
and deductive reasoning to prove the conjecture.
Ex. 6 Select a number. Add 5. Double the result. Subtract 4. Subtract the
original number.
Solution:
Select a #
4
9
15
25
Add 5
4+5=9
9 + 5 = 14
15 + 5 = 20
25 + 5 = 30
Double the
result
2•9 = 18
2•14 = 28
2•20 = 40
2•30 = 60
Subtract 4
18 – 4 = 14
28 – 4 = 24
40 – 4 = 36
60 – 4 = 56
Subtract the
original #
14 – 4 = 10
24 – 9 = 15
36 – 15 = 21
56 – 25 = 31
Notice that 10 is 6 more than 4, 15 is 6 more than 9, and so forth. We
would guess the rule would be n + 6.
Now, we will prove it using deductive reasoning:
Select a #:
n
4
Add 5:
n+5
Double the result:
2(n + 5) = 2n + 10
Subtract 4:
2n + 10 – 4 = 2n + 6
Subtract the original #: 2n + 6 – n = n + 6
Ex. 7 Select a number. Multiply by 4. Add 8 to the result. Divide by 2.
Subtract 4.
Solution:
Select a #
3
6
11
17
Multiply by 4 4•3 = 12
4•6 = 24
4•11 = 44
4•17 = 68
Add 8 to the
result
12 + 8 = 20
24 + 8 = 32
44 + 8 = 52
68 + 8 = 76
Divide by 2
20 ÷ 2 = 10
32 ÷ 2 = 16
52 ÷ 2 = 26
76 ÷ 2 = 38
Subtract 4
10 – 4 = 6
16 – 4 = 12
26 – 4 = 22
38 – 4 = 34
Notice that 6 is double 3, 12 is double 6, and so forth. We would
guess the rule would be 2n.
Now, we will prove it using deductive reasoning:
Select a #:
n
Multiply by 4:
4n
Add 8 to the result:
4n + 8
Divide by 2:
(4n + 8)/2 = 2n + 4
Subtract 4:
2n + 4 – 4 = 2n
Determine the reasoning process that is shown in the following:
Ex. 8 All of the books written by J.K. Rowling have been made into
movies. Harry Potter and the Sorcerer's Stone was a book written by
J.K. Rowling. Therefore, Harry Potter and the Sorcerer's Stone was
made into a movie.
Solution:
Deductive Reasoning.
Ex. 9 All of the books written by J.K. Rowling have been made into
movies. Therefore, the novel that J.K. Rowling is currently writing will
be made into a movie.
Solution:
Inductive Reasoning.