Beginning Algebra Tutorial 35

Beginning Algebra
Tutorial 35: Reasoning Skills

Learning Objectives

After completing this tutorial, you should be able to:
1. Use inductive reasoning to solve problems.
2. Find the next terms in a sequence.
3. Use deductive reasoning to solve problems.

Introduction

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

Inductive Reasoning

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.

Looking for a Pattern
(Sequences)

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.

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.

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.

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.

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.

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.

Looking for a Pattern
Involving Figures

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.

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.

Example 4:   Write the next three figures in the pattern

...

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:

Example 5:   Write the next two figures in the pattern

...

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

Deductive Reasoning

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.

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.

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.

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?

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.

Practice Problems

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.

Practice Problems 1a - 1c: Write the next three numbers in the sequence.

1a.   1, 1, 3, 15, 105, ...
1b.   1000, 200, 40, 8, 1.6, ...

1c.     5, 5, 10, 15, 25, ...

Practice Problem 2a: Write the next five figures in the pattern.

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.

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?

Need Extra Help on these Topics?

Go to Get Help Outside the Classroom found in Tutorial 1: How to Succeed in a Math Class for some more suggestions.

Last revised on August 7, 2011 by Kim Seward.