Foundations of Math 11
Substantive Assignment Resource Package
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Deductive and Inductive Reasoning
Inductive and deductive reasoning are two methods of logic used to arrive at a
conclusion based on information assumed to be true. Both are used in research to
establish hypotheses.
Deductive reasoning arrives at a specific conclusion based on generalizations.
Inductive reasoning takes events and makes generalizations
Deductive reasoning is reasoning that involves a hierarchy of statements or truths.
Starting with a limited number of simple statements or assumptions, more complex
statements can be built up from the more basic ones. For example, you have
probably studied deductive geometry in mathematics; in it you start with a few
principles and prove various propositions using those principles. To prove more
complicated propositions, you may use propositions that you have already proved
plus the original principles. In more formal logic terms deductive reasoning is
reasoning from stated premises to conclusions formally or necessarily implied by
such premises.
Deductive reasoning can be described as reasoning of the form if A then B.
Deduction is in some sense the direct application of knowledge in the production
of new knowledge.
If-then deductive reasoning is how scientists (and other people!) can test alternate
hypotheses. Making deductions is important when we cannot directly observe a
cause, and can only observe its consequences. This kind of reasoning can be
modeled by the following:
If ...
Then...
But...
Therefore...
For example, we might hypothesize that "The color of a mineral is determined by
its crystal structure."
And so we could test this hypothesis using deductive reasoning:
If the color of a mineral is determined by its crystal structure; then all purple
minerals should have the same crystal structure. But purple amethyst has a
hexagonal structure and purple fluorite has an isometric structure (determined by
observations). Therefore, the hypothesis is not supported or strengthened.
Inductive reasoning is essentially the opposite of deductive reasoning. It involves
trying to create general principles by starting with many specific instances. For
example, in inductive geometry you might measure the interior angles of a group
of randomly drawn triangles. When you discover that the sum of the three angles is
180° regardless of the triangle, you would be tempted to make a generalization
about the sum of the interior angles of a triangle. Bringing forward all these
separate facts provides evidence in order to help support your general statement
about the interior angles.
This is the kind of reasoning used if you have gradually built up an understanding
of how something works. Rather than starting with laws and principles and making
deductions, most people collect relevant experience and try to construct principles
from it.
Again the distinction between the two types of reasoning is not always sharp. In
mathematics it is important to know which kind of formal system you are using
and to stick to it. Inductive proofs are not allowed in a deductive system.
Inductive reasoning progresses from observations of individual cases to the
development of a generality. (Inductive reasoning, or induction, is often confused
with deductive thinking; in the latter, general principles or conditions are applied to
specific instances or situations.) If a child puts his or her hand into a bag of candy
and withdraws three pieces, all of which are red, he or she may conclude that all
the candy is red. Inductive reasoning, or induction, is the process by which a
general conclusion is reached from evaluating specific observations or situations.
Many people distinguish between two basic kinds of argument: inductive and
deductive. Induction is usually described as moving from the specific to the
general, while deduction begins with the general and ends with the specific;
arguments based on experience or observation are best expressed inductively,
while arguments based on laws, rules, or other widely accepted principles are best
expressed deductively.
Inductive reasoning is characterized by drawing a general conclusion (making a
conjecture) from repeated observations of specific examples. The conjecture may or
may not be true. If we encounter even one example (the counterexample) for which the
conjecture does not work, then the conjecture is false.
PLO: C1 Analyze and prove conjectures using inductive and deductive reasoning to
solve problems.
Example 3
For each of the following questions 1 to 4, choose which of examples
PLO: C1 Analyze and prove conjectures using inductive and deductive reasoning
to solve problems.
A, B, C, D, or E would be the next figure in the sequence.
The answers to the above patterns are as follows:
Question 1 E
Question 2 B
Question 3 D
Question 4 A
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
PLO: C1 Analyze and prove conjectures using inductive and deductive
reasoning to solve problems.
...
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.
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
PLO: C1 Analyze and prove conjectures using inductive and deductive
reasoning to solve problems.
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.
The next two figures would be
Using Inductive Reasoning to determine sequences
The previous examples of inductive reasoning involved being given a
pattern and then needing to come up with the rule for the pattern to
continue the pattern.
Another example is number sequences. In general, a sequence is an
ordered arrangement of numbers, figures, or objects.
Specifically, sequences of numbers 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. Work through the following examples of number sequences.
Example 1:
PLO: C1 Analyze and prove conjectures using inductive and deductive
reasoning to solve problems.
Write the next three numbers in the sequence 5, 7, 11, 17, 25, ...
The 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, .... 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:
PLO: C1 Analyze and prove conjectures using inductive and deductive
reasoning to solve problems.
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, 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 is equal to 14. It
looks like we have 7 (-1), -7 (-2), 14 (-3), -42 (-4), 168, ... So 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:
PLO: C1 Analyze and prove conjectures using inductive and deductive
reasoning to solve problems.
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.
Using 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 needed 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 4: Use the statements below to answer the question that
follows:
PLO: C1 Analyze and prove conjectures using inductive and
deductive reasoning to solve problems.
1.
2.
3.
4.
5.
All people wearing hats have blonde hair.
Some of the people have red hair.
All people who have blonde hair like hamburgers.
People who have red hair like pizza.
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.
With 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 5: 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.
PLO: C1 Analyze and prove conjectures using inductive and
deductive reasoning to solve problems.
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: Who 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. Jerry is the manager.
Example 6: 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.
PLO: C1 Analyze and prove conjectures using inductive and deductive
reasoning to solve problems.
Use the statements below to answer the question that follows
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
You can use a process of elimination on this problem. Statement A: Suzy
and John always eat lunch with the friend that plays in the band, doesn't let
us definitively eliminate anyone from being the one who likes to sew.
However, statement B: Sally and Tom carpool with the one who likes to
sew, eliminates Sally and Tom from being the one who likes to sew.
Statement C: John and the friend that likes to cook visited the one who
likes to sew, eliminates John.
The only one that could be (100%, without a doubt) the one who likes
to sew is Suzy.
Example 7: Fill in the empty cells with numbers between one and nine in the
following Sudoku puzzle using the following rules.
PLO: C2 Analyze puzzles and games that involve spatial reasoning using problem
solving strategies.
a. Each number can only appear once in each row
b. Each number can only appear once in each column
c. Each number can only appear once in each 3 by 3 section
Answer
Example 8: How many squares are found in the following diagram.
PLO: C2 Analyze puzzles and games that involve spatial reasoning using problem
solving strategies.
Answer
There are 14 squares in the diagram
Example 9: Use the following clues to pick the correct day on the calendar
PLO: C1 Analyze and prove conjectures using inductive and deductive reasoning to
solve problems.
Clues
The date is not a prime number
The date is not a multiple of 4 or 5
The date is a multiple of 2
The date is divisible by 6
If you have eliminated the dates correctly, you should have two dates left.
It is the greater of those two dates
Answer: The 12th day correct