Beginning Algebra
Tutorial 35:
Reasoning Skills
 Learning Objectives
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After completing this tutorial, you should be able to:
- Use inductive reasoning to solve problems.
- Find the next terms in a sequence.
- Use deductive reasoning to solve problems.
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Introduction
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| In this tutorial we will be looking at basic concepts
of reasoning
skills. We will be looking at both deductive and inductive
reasoning. One thing that this can be helpful with is
looking
for patterns. Looking for patterns to find a solution can be
found
in a variety of fields. Teachers can use patterns to determine a
course of direction for a student. For example, if a student is
exhibiting
the same kind of learning pattern that a teacher has seen in a student
with dyslexia before, they can act upon that accordingly.
Psychologists
and law enforcement study behavioral patterns to solve some of their
problems.
Scientific researchers study patterns to determine end results in their
experiments. Doctors and Veterinarians use patterns to help
diagnose
a patient's illness. Weather forecasters use patterns in weather
to predict temperature, tornadoes, hurricanes, etc. In fact some
aspects of weather forecasting uses Chaos
Theory
- the science of seeing order and pattern where formerly only the
random,
the erratic, and the unpredictable had been observed. Patterns of
all kinds are lurking everywhere around us. I think you are ready
to forge ahead into the wonderful world of reasoning skills. |
Tutorial
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| Inductive reasoning is used when you need to draw a
general conclusion
from specific instances.
For example, when a detective puts together specific
clues to solve
a mystery.
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Looking for a Pattern
(Sequences)
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| In math, an example of inductive reasoning would be
when you are given
a pattern and you need to come up with the rule for the pattern.
A lot of what we will be working with in this lesson are
sequences.
In general, a sequence is an ordered arrangement of numbers,
figures,
or objects.
Specifically, sequences of math are a string of
numbers that
are tied together with some sort of consistent rule, or set of rules,
that
determines the next number in the sequence.
The following are some specific types of sequences of
math:
Arithmetic sequence:
a sequence
such that each successive term is obtained from the previous term by
addition
or subtraction of a fixed number called a difference. The
sequence
4, 7, 10, 13, 16, ... is an example of an arithmetic
sequence.
The pattern is that we are always adding a fixed number of three to the
previous term to get to the next term. Be careful that you don't think
that every sequence that has a pattern in addition is arithmetic. It
is arithmetic if you are always adding the SAME number each time.
Geometric sequence:
a sequence
such that each successive term is obtained from the previous term by
multiplying
by a fixed number called a ratio. The sequence 5, 10, 20, 40, 80,
.... is an example of a geometric sequence. The pattern is
that we are always multiplying by a fixed number of 2 to the previous
term
to get to the next term. Be careful that you don't think that
every
sequence that has a pattern in multiplication is geometric. It
is geometric if you are always multiplying by the SAME number each
time.
Fibonacci sequence:
a basic
Fibonacci sequence is when two numbers are added together to get the
next
number in the sequence. 1, 1, 2, 3, 5, 8, 13, .... is
an example of a Fibonacci sequence where the starting numbers (or
seeds)
are 1 and 1, and we add the two previous numbers to get the next
number
in the sequence.
Note that not all sequences fit
into the specific
patterns that are described above. Those are just the
more common ones. So as you look at patterns, look for those
as
a possibility, but if it doesn't fit one listed above, don't assume it
doesn't have a pattern.
In general, when looking for a pattern start simple and
then go from
there. For example, see if there is some pattern in adding,
subtracting,
multiplying, or dividing. Maybe you are always adding the same
number
to the previous term to get the new term. Or maybe you are
subtracting
the next multiple of three from the previous number. Or you are
multiplying
by a sequence of even numbers. Perhaps, you are always adding or
subtracting the two previous terms to get to the next one.
Exponential
growth is another good pattern to look for. Maybe you are always
squaring or cubing the term number to get your result. Also,
don't
forget that sometimes the pattern of a sequence is a combination of
operations.
Maybe you have to multiply by 2 and then add 5 to get to the next
number
in a sequence or the output of a function. If a problem seems
like
it is taking forever to work, try a different approach - a
different
kind of sequence.
Once you find your pattern, you can use it to find
the next terms
in the sequence.
|
 Example
1: Write the next three numbers in the sequence
5, 7, 11, 17, 25, ... |
| My first inclination is to see if there is some pattern
in addition.
Well, we are not adding the same number each time to get to the next
number.
But, it looks like we have 5 +2, 7
+4,
11 +6, 17 +8,
25, .... I see a pattern in addition - do you see it? We are
always
adding the next even number.
Final Answer:
The pattern is to add the next even number.
The next three
terms would have to be 35, 47, and 61, since 25+10
= 35, 35 +12 =
47,
47 +14 = 61.
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| Example
2: Write the next three numbers in the sequence
7, -7, 14, -42, 168, ... |
| Since we are bouncing back and forth between positive
and negative
numbers, a pattern in addition doesn't look promising. Let's
check
out multiplication. At first glance, I would say that a negative
number is probably what we are looking for here, since it does
alternate
signs. It doesn't appear to be the same number each time, because
7 times -1 is -7, but -7 times -2 equals 14. It looks like we
have
7 (-1), -7 (-2),
14 (-3), -42 (-4), 168, ... Aha, we have a
pattern in multiplication - we are
multiplying
by the next negative integer.
Final Answer:
The pattern is multiplying by the next negative integer.
The next three terms are -840, 5040, and -35280, since 168(-5)
= -840, -840(-6) =
5040, 5040(-7) = -35280.
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| Example
3: Write the next three numbers in the sequence
100, 97, 88, 61, ... |
| Since the numbers are decreasing that should tell you
that you are
not adding a positive number or multiplying. So we want to check
out subtraction or division. At first glance it looks like it is
some pattern in subtraction. We are not subtracting by the same
number
each time. We have 100 -3,
97 -9, 88 -27,
61, .... Note how we are always subtracting the next power
of 3. We have our pattern.
Final Answer:
The pattern is we are subtracting by the next power of
three.
The next three terms would be -20, -263, and -992, since 61 -
81 = -20, -20 - 243 =
-263,
-263 - 729 = -992.
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Looking for a Pattern
Involving Figures
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Here are some things to look for when trying to
figure out a pattern
involving figures:
| Look for counter clockwise and clockwise changes.
Count sides of figures.
Count lines in figures.
Note changes in direction and figures.
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As with the numeric patterns, this is not all the
possible types of
patterns involving figures. However, it does give you a way to
approach
the problem.
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| Example
4: Write the next three figures in the pattern
 ...
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| It looks like several things change throughout the
pattern. One
thing is that it alternates between a square with a line in it and a
circle.
Also the line in the square alternates from horizontal to
vertical.
With all of that in mind, I believe the next three
figures would
be a square with a vertical line, then a circle, then a square with a
horizontal
line:
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| Example
5: Write the next two figures in the pattern
 ...
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| It looks like one row of asterisks is added at the
bottom of each figure.
The row that is added contains the next counting number of
asterisks.
There are 2 in the row added in the second term, there are 3 in the row
added in the 3rd term and 4 in the row added to the fourth term.
With all of that in mind, I believe the next two
figures would be
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| Deductive reasoning is used when you have a general
rule and you
want to draw on that to get a specific solution.
For example, if you were needing to find the area of a
specific rectangle.
You would use the general formula for the area of the rectangle and
apply
it to the specific rectangle.
Here are some ideas that might help you approach a
problem requiring
deductive reasoning:
| Watch for key words like no or all.
Use process of elimination.
Draw a picture or a diagram if it helps.
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| Example
6: Use the statements below to answer the
question
that follows:
1. All people wearing hats have blonde hair.
2. Some of the people have red hair.
3. All people who have blonde hair like hamburgers.
4. People who have red hair like pizza.
5. Keith has blonde hair.
Which of the following statements MUST be true?
a. Keith likes hamburgers.
b. Keith has red hair.
c. Keith likes pizza.
d. Keith is wearing a hat.
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| Well what do you think?
On deductive reasoning, you need to be a 100%
sure. There can’t
be any doubt.
Since statement 3 says that ALL people who have blonde
hair like hamburgers
and Keith has blonde hair, then statement a,
Keith
likes hamburgers, is a 100% guarantee.
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| Example
7: Jerry, Kevin, Todd and Mark all live on the
first
floor of an apartment complex. One is a manager, one is a
computer
programmer, one is a singer, and the other is a teacher. Use the
statements below to answer the question that follows.
A. Jerry and Todd eat lunch with the singer.
B. Kevin and Mark carpool with the manager.
C. Todd watches CSI with the manger and the singer.
Question: Which is the manager?
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| You can use a process of elimination on this
problem.
Statement A, Jerry and Todd eat lunch with the
singer, doesn’t
let us definitively eliminate anyone from being the manager.
However, statement B, Kevin and Mark carpool with
the manager,
eliminates Kevin and Mark from being the manager. And statement
C,
Todd watches CSI with the manger and the singer, eliminates Todd.
The only one that could be (100%, without a doubt)
the manager is
Jerry.
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Practice Problems
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| These are practice problems to help bring you to the
next level.
It will allow you to check and see if you have an understanding of
these
types of problems. Math works just like
anything
else, if you want to get good at it, then you need to practice
it.
Even the best athletes and musicians had help along the way and lots of
practice, practice, practice, to get good at their sport or instrument.
In fact there is no such thing as too much practice.
To get the most out of these, you should work the
problem out on
your own and then check your answer by clicking on the link for the
answer/discussion
for that problem. At the link you will find the answer
as well as any steps that went into finding that answer.
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 Practice
Problems 1a - 1c:
Write the next three numbers in the
sequence.
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1a. 1, 1, 3, 15, 105, ...
(answer/discussion
to 1a) |
1b. 1000, 200, 40, 8, 1.6, ...
(answer/discussion
to 1b) |
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Practice
Problem 2a:
Write the next five figures in the
pattern.
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Practice
Problem 3a:
Four friends - Suzy, John, Sally, and
Tom - each has
his or her own hobby. One collect coins, one sews, one cooks, and
one plays in a band, not necessarily in that order.
Use the statements below to answer
the question that
follows.
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3a.
A. Suzy and John always eat lunch with the friend that plays
in the band.
B. Sally and Tom carpool with the one who likes to sew.
C. John and the friend that likes to cook visited the one who
likes to sew.
Question: Who is the friend that likes to sew?
(answer/discussion
to 3a)
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All contents copyright (C) 2001 - 2008, WTAMU and Kim Seward. All rights reserved.
Last revised on Jan. 10, 2002 by Kim Seward. |
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