Download 1.1 Solving Problems by Inductive Reasoning

1.1 Solving Problems by Inductive Reasoning
The development of mathematics can be traced to the Egyptian and Babylonian cultures (3000 B.C. – A.D.
260) as a necessity for problem solving. To solve a problem or perform an operation, a cookbook-like recipe
was given, and it was performed repeatedly to solve similar problems. By observing that a specific method
worked for a certain type of problem, the Egyptians and the Babylonians concluded that the same method
would work for any similar type of problem. Such a conclusion is called a
conjecture.
A ____________________ is an educated guess based on repeated observations of a particular process or
pattern.
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.
In testing a conjecture obtained by inductive reasoning, it takes only one example, called a
________________________, that does not work in order to prove the conjecture false.
 Inductive reasoning offers a powerful method of drawing conclusions, but there is no guarantee that
the observed conjecture will always be true. The conjecture is not accepted as absolute proof until
the conjecture is proved using ________________
____________________
reasoning
reasoning .
is characterized by applying general principles to specific examples.
Ex 1) We refer to 1,2, 3, … as the natural or counting numbers. The general rule for continuing the pattern
Ex 2) Use inductive reasoning to determine the probable next number.
a) 7, 11, 15, 19, 23
b) 1, 1, 2, 3, 5, 8, 13, 21
c) 3, 9, 27, 81, 243
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A ____________________ can be an assumption, law, rule, widely held idea, or observation. Then reason
inductively or deductively from the premises to obtain a ____________________. The premises and
conclusion make up a ____________________ ____________________.
Ex 3) Identify each premise and the conclusion in each of the following arguments. Then tell whether each
argument is an example of inductive or deductive reasoning.
a) Our house is made of rocks. Both of my next door neighbors have rock houses. Therefore, all
houses in our neighborhood are made of rocks.
b) All calculators can type the symbol +. I have a calculator. I can type the symbol +.
c) Today is Tuesday. Tomorrow will be Wednesday.
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Ex 4) A list of equations is given. Use the list and inductive reasoning to predict the next equation, and then
verify your conjecture.
15873 7  111,111
a) (#34)
15873 14  222,222
15873 21  333,333
15873 28  444,444
1
1

1 2 2
1
1
2


1 2 2  3 3
b) (#42)
1
1
1
3



1 2 2  3 3  4 4
1
1
1
1
4




1 2 2  3 3  4 4  5 5
Ex 5) (#51 page 9) Find a pattern in the figure and use inductive reasoning to predict the next figure.
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1.2 An Application of Inductive Reasoning: Number Patterns
1.2 Book HW: 31, 33, 51, 52, 54
A ____________________ is an ordered list of numbers.
A _________________ _________________ is a list of numbers having a first number, a second
number, a third number, and so on, called the ____________________ of the sequence.
Arithmetic Sequence: A sequence a , a
1
2
, a 3 , … an , … is called an arithmetic sequence if there is a
_________________ _________________ d between successive terms.
Geometric Sequence: A sequence a , a
1
2
, a 3 , … an , … is called an geometric sequence if there is a
real number r  0 , called the _________________ _________________ between successive terms.
Ex 1) Use the method of successive differences to determine the next number in each sequence.
d) (#2) 3, 14, 31, 54, 83, 118, …
e) (#8) 3, 19, 165, 771, 2503, 6483, 14409
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Observe the following pattern:
So
1 3  5  7  9 =
and
1  3  5  7  9  11 =
1  12
1  3  22
1  3  5  32
1  3  5  7  42
We cannot prove this without a method of proof called mathematical induction. But mathematicians have.
So we can generalize this for any counting number n.
Ex 2) Several equations are given illustrating a suspected number pattern. Determine what the next
equation would be, and verify that it is indeed a true statement. Then find a general formula, if possible.
a)
12  13
1  22  13  2 3
1  2  32  13  2 3  33
1  2  3  42  13  2 3  33  4 3
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1 2
2
23
1 2 
2
b)
3 4
1 2  3 
2
45
1 2  3  4 
2
1
Triangular Numbers:
Ex 3) Identify the eighth triangular number.
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Ex 4) The first five pentagonal numbers are shown below. Identify the formula to find the nth pentagonal
number. Then find the 23rd pentagonal number.
Ex 5) (#26) Use the formula S = n2 to find the sum: 1 + 3 + 5 + … + 49
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1.3 Strategies for Problem Solving
1.3 Book HW: 2, 15, 27, 40, 58
In the first two sections of this chapter we stressed pattern recognition and the use of inductive reasoning in
problem solving. Now we will use probably the most famous study of problem solving techniques developed
by George Polya. Polya proposed a four-step method for problem solving.
Pólya’s First Principle: Understand the Problem
This seems so obvious that it is often not even mentioned, yet students are often hindered in their efforts to
solve problems simply because they don’t understand it fully, or even in part.


State the problem you are trying to solve in your own words.
Clearly write the question(s).
Pólya’s Second Principle: Devise a plan
There are many reasonable ways to solve problems. he skill at choosing appropriate strategies is best
learned by solving many problems. You will find choosing strategies increasingly easy. A partial list of
strategies is included:
 Work backward
 Make a table or a chart
 If a formula applies, use it.
 Look for a pattern
 Guess and check.
 Solve a similar simpler problem
 Use trial and error.
 Draw a picture
 Use common sense.
 Use inductive reasoning
 Look for a “catch” if an answer seems too obvious or
 Write an equation and solve it.
impossible.

Make a list of the strategies (two or more, please).
Pólya’s third Principle: Carry out the plan
Now is the time to actually solve the problem! The solution should be included. You may run into “dead
ends” and unforeseen roadblocks, but be persistent.
Pólya’s Fourth Principle: Look back
Much can be gained by taking the time to reflect and look back at what you have done; what worked and
what didn’t. Doing this will enable you to predict what strategies to use to solve future problems.
Rabbit Problem
In the year 1202, Fibonacci became interested in the reproduction of rabbits. He created an imaginary set of
ideal conditions under which rabbits could breed, and posed the question, "How many pairs of rabbits will
there be a year from now?" The ideal set of conditions was a follows:
1)
2)
3)
4)
5)
6)
You begin with one male rabbit and one female rabbit. These rabbits have just been born.
A rabbit will reach sexual maturity after one month.
The gestation period of a rabbit is one month.
Once it has reached sexual maturity, a female rabbit will give birth every month.
A female rabbit will always give birth to one male rabbit and one female rabbit.
Rabbits never die.
Ex 1) So how many male/female rabbit pairs are there after one year (12 months)?
Step 1: Understand the Problem:
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Step 2: Devise a Plan
Use a chart/table because there is a pattern to how the rabbits will reproduce.
Month
Number of Pairs at Start
Number of New Pairs Produced
Number of Pairs at End of Month
1
2
3
4
5
6
7
8
9
10
11
12
Step 3: Carry out the Plan
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Ex 2) The number of dogs and chickens on a farm add up to 12. The number of legs between them is 28.
How many dogs and how many chickens are on the farm if there are at least twice as many chickens as
dogs? Use Polya’s four steps.
Ex 3) (#10) Cindy was asked by her teacher to subtract 3 from a certain number and then divide the result
by 9. Instead, she subtracted 9 and then divided the result by 3, giving an answer of 43. What would her
answer have been if she had worked the problem correctly?
Ex 4) (#42) How many squares are in the following figure?
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