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Beginning Algebra
Tutorial 35: Reasoning Skills

Learning Objectives

 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.

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)

 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.

 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.

 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.

 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, ...  (answer/discussion to 1a) 1b.   1000, 200, 40, 8, 1.6, ...  (answer/discussion to 1b)

 1c.     5, 5, 10, 15, 25, ...  (answer/discussion to 1c)

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? (answer/discussion to 3a)

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.