Transcript Developmental Mathematics

Section 1.1
Thinking Mathematically
HAWKES LEARNING
Students Count. Success Matters.
Copyright © 2015 by Hawkes
Learning/Quant Systems, Inc.
All rights reserved.
Objectives
o Understand mathematical reasoning
o Distinguish between inductive and deductive
reasoning
o Identify arithmetic and geometric sequences
HAWKES LEARNING
Students Count. Success Matters.
Copyright © 2015 by Hawkes
Learning/Quant Systems, Inc.
All rights reserved.
Thinking Mathematically
Mathematical thinking is important for decisions we all
make every day. Possessing the ability to think
mathematically makes one a better problem solver for
all occasions.
Reasoning is defined by the Merriam-Webster
dictionary as the drawing of inferences or conclusions
through the use of statements offered as explanation
or justification.
We begin our discussion of reasoning with inductive
and deductive reasoning.
HAWKES LEARNING
Students Count. Success Matters.
Copyright © 2015 by Hawkes
Learning/Quant Systems, Inc.
All rights reserved.
Inductive Reasoning
Inductive Reasoning
Inductive reasoning is a line of reasoning that arrives at
a general conclusion based on the observation of
specific examples. Inductive reasoning can be
considered a generalization.
HAWKES LEARNING
Students Count. Success Matters.
Copyright © 2015 by Hawkes
Learning/Quant Systems, Inc.
All rights reserved.
Example 1: Using Inductive Reasoning
Consider the following argument.
In New York City, it snowed 30 inches during January
2010 and 35 inches during January 2011. Therefore,
New York City will receive at least 30 inches of snow
every January.
Does this argument use inductive reasoning?
HAWKES LEARNING
Students Count. Success Matters.
Copyright © 2015 by Hawkes
Learning/Quant Systems, Inc.
All rights reserved.
Example 1: Using Inductive Reasoning (cont.)
Solution
Notice that in this argument, the example is specific to
January 2010 and January 2011, and then a very
general conclusion is made. Obviously, the likelihood of
it snowing exactly the same amount each January is not
very practical, but nevertheless, the argument here is
inductive.
HAWKES LEARNING
Students Count. Success Matters.
Copyright © 2015 by Hawkes
Learning/Quant Systems, Inc.
All rights reserved.
Example 2: Using Inductive Reasoning
Consider the following sequence of numbers.
1, 4, 9, 16, 25, ...
If the number pattern continues, can you conclude
what the next number will be? What about the 15th
number?
HAWKES LEARNING
Students Count. Success Matters.
Copyright © 2015 by Hawkes
Learning/Quant Systems, Inc.
All rights reserved.
Example 2: Using Inductive Reasoning (cont.)
Solution
If we look closely, we can see that the numbers are
simply the squares of the natural numbers:
12  1, 22  4, 32  9, etc. Therefore, the next number in
the pattern would be 62  36, then 72  49, and so on.
That means the 15th number in the pattern would be
152 or 225. Notice again that we begin with a specific
example and continue to a generalized answer.
HAWKES LEARNING
Students Count. Success Matters.
Copyright © 2015 by Hawkes
Learning/Quant Systems, Inc.
All rights reserved.
Inductive Reasoning
Counterexample
A counterexample is a contradictory example that does
not satisfy our conclusion, therefore making the
argument invalid. Note that one counterexample is
enough to prove that a line of reasoning is false, but
one positive example is never enough to prove that it is
true.
HAWKES LEARNING
Students Count. Success Matters.
Copyright © 2015 by Hawkes
Learning/Quant Systems, Inc.
All rights reserved.
Example 3: Counterexamples
Consider the statement:
You must have a degree in computer science to become
wealthy in the computer industry.
Is this a valid argument?
HAWKES LEARNING
Students Count. Success Matters.
Copyright © 2015 by Hawkes
Learning/Quant Systems, Inc.
All rights reserved.
Example 3: Counterexamples (cont.)
Solution
For this example, we need to look no further than two
of the most famous people in the computer industry:
Steve Jobs (cofounder of Apple Computers) and Bill
Gates (cofounder of Microsoft). Neither Jobs nor Gates
have a degree in computer science, yet both became
quite wealthy in the computer industry. Hence, we
have found a counterexample to our conclusion, and
the argument is invalid.
HAWKES LEARNING
Students Count. Success Matters.
Copyright © 2015 by Hawkes
Learning/Quant Systems, Inc.
All rights reserved.
Example 4: Reasoning with Sequences
Identify a pattern in each of the following sequences of
numbers, then use the established pattern to find the
next term in the sequence.
a. 4, 9, 14, 19, ____
b. 2, 6, 18, 54, ____
c. 5, 6, 8, 11, ____
HAWKES LEARNING
Students Count. Success Matters.
Copyright © 2015 by Hawkes
Learning/Quant Systems, Inc.
All rights reserved.
Example 4: Reasoning with Sequences (cont.)
Solution
When trying to identify a pattern in a sequence of
numbers, there is not a set method to follow. We will
introduce you to several techniques in this example.
The more practice you have with these techniques, the
easier it will become to identify patterns.
a. When considering the sequence 4, 9, 14, 19, ____,
we should try to determine whether the difference
between the consecutive terms is constant (that is,
if the difference is the same for each consecutive
pair of terms)
HAWKES LEARNING
Students Count. Success Matters.
Copyright © 2015 by Hawkes
Learning/Quant Systems, Inc.
All rights reserved.
Example 4: Reasoning with Sequences (cont.)
or if the difference between the consecutive pairs of
terms varies.
4
9
945
14
14  9  5
19
19  14  5
We can see here that there is a common difference
between each number in the sequence. The common
difference is 5. Thus, the next term in the sequence will
be 19 + 5 = 24.
HAWKES LEARNING
Students Count. Success Matters.
Copyright © 2015 by Hawkes
Learning/Quant Systems, Inc.
All rights reserved.
Example 4: Reasoning with Sequences (cont.)
b. The sequence 2, 6, 18, 54, __ does not have a
common difference between terms as in part a.
Since there is no common difference, we need
another approach. Instead of a common difference,
perhaps there is a common ratio between the
consecutive pairs of terms.
2
6
6
3
2
HAWKES LEARNING
Students Count. Success Matters.
18
18
3
6
54
54
3
18
Copyright © 2015 by Hawkes
Learning/Quant Systems, Inc.
All rights reserved.
Example 4: Reasoning with Sequences (cont.)
We can see that each successive number is the
product of the previous number and 3. So, 2 ⋅ 3 = 6,
6 ⋅ 3 = 18, 18 ⋅ 3 = 54, etc. This means that the next
term would be 54 ⋅ 3 = 162.
c. For the sequence, 5, 6, 8, 11, ____, there is no
common difference or common ratio between the
numbers. We need to ask ourselves, What do we
have to do to the first term in order to obtain the
second? In this case, we have to add 1 to 5 in order
to get 6.
HAWKES LEARNING
Students Count. Success Matters.
Copyright © 2015 by Hawkes
Learning/Quant Systems, Inc.
All rights reserved.
Example 4: Reasoning with Sequences (cont.)
For the next term, we see that the sum is 2 + 6 = 8. We
can now see a pattern developing that we need to
investigate further. We added 1 to the first term and 2
to the second term. If we continue in this manner by
adding 3 to the third term, 4 to the fourth term, etc.,
does the pattern continue? For the next term, 8 + 3 =
11, so indeed it does. Thus, to find the next term in the
sequence, we add 4 to 11 to get 15.
HAWKES LEARNING
Students Count. Success Matters.
Copyright © 2015 by Hawkes
Learning/Quant Systems, Inc.
All rights reserved.
Arithmetic Sequences
Arithmetic Sequences
When the common difference between any two
consecutive terms in a number sequence is the same,
we call this an arithmetic sequence.
HAWKES LEARNING
Students Count. Success Matters.
Copyright © 2015 by Hawkes
Learning/Quant Systems, Inc.
All rights reserved.
Geometric Sequences
Geometric Sequences
When the common ratio between any two consecutive
terms in a number sequence is the same, we call this a
geometric sequence.
HAWKES LEARNING
Students Count. Success Matters.
Copyright © 2015 by Hawkes
Learning/Quant Systems, Inc.
All rights reserved.
Example 5: Reasoning with Patterns
A certain type of human cell reproduces in the
following manner: 1 cell, 4 cells, 9 cells, 16 cells.
Determine the number of cells present on the next
production of cells.
HAWKES LEARNING
Students Count. Success Matters.
Copyright © 2015 by Hawkes
Learning/Quant Systems, Inc.
All rights reserved.
Example 5: Reasoning with Patterns (cont.)
Solution
The second example we did concerning inductive
reasoning asked us to consider the sequence of
numbers 1, 4, 9, 16, . . . . Notice that the first iteration
contains 1 cell and the second iteration contains 22  4
cells.
The third and fourth iterations contain 32  9 cells and
42  16 cells, respectively. So, the next iteration in the
sequence will contain 52  25 cells.
HAWKES LEARNING
Students Count. Success Matters.
Copyright © 2015 by Hawkes
Learning/Quant Systems, Inc.
All rights reserved.
Skill Check #1
Skill Check #1
Consider the given figures. To build the figures, the first
figure requires 16 line segments, the second figure
requires 28 line segments, and the third figure requires
40 line segments. How many line segments would be
required
a. in the 8th figure?
HAWKES LEARNING
Students Count. Success Matters.
b. in the nth figure?
Copyright © 2015 by Hawkes
Learning/Quant Systems, Inc.
All rights reserved.
Skill Check #1
Answer: 1. a. 100 b. 4 + 12n
HAWKES LEARNING
Students Count. Success Matters.
Copyright © 2015 by Hawkes
Learning/Quant Systems, Inc.
All rights reserved.
Deductive Reasoning
Deductive Reasoning
Deductive reasoning is a process that begins with
commonly accepted facts and logically arrives at a
specific conclusion.
HAWKES LEARNING
Students Count. Success Matters.
Copyright © 2015 by Hawkes
Learning/Quant Systems, Inc.
All rights reserved.
Example 6: Using Deductive Reasoning
Consider the following statement.
If you are a mammal, then you have lungs.
How can this statement be evaluated as a deductive
argument?
Solution
This statement has two parts: determining if you are a
mammal and whether or not you have lungs. Breaking
down the individual components of the statement, we
can observe the following.
HAWKES LEARNING
Students Count. Success Matters.
Copyright © 2015 by Hawkes
Learning/Quant Systems, Inc.
All rights reserved.
Example 6: Using Deductive Reasoning (cont.)
a. I am a mammal (premise is true).
I have lungs (conclusion is true).
b. I am a mammal (premise is true).
I do not have lungs (conclusion is false).
Notice that, in instance b., when the conclusion is false,
the argument does not seem to follow “logical”
thought. There is a much more detailed discussion of
deductive reasoning in Chapter 3.
HAWKES LEARNING
Students Count. Success Matters.
Copyright © 2015 by Hawkes
Learning/Quant Systems, Inc.
All rights reserved.
Example 7: Inductive versus Deductive
Reasoning
To illustrate the difference between inductive and
deductive reasoning, consider the following process.
Choosing a positive integer, multiplying it by 2, and
adding 1 to the product will result in an odd number.
a. Evaluate this conclusion as an inductive argument.
b. Evaluate this conclusion as a deductive argument.
HAWKES LEARNING
Students Count. Success Matters.
Copyright © 2015 by Hawkes
Learning/Quant Systems, Inc.
All rights reserved.
Example 7: Inductive versus Deductive
Reasoning (cont.)
Solution
a. We will begin by looking at some specific examples.
Table 1 – Process Examples
Positive Integer
Arithmetic
3
32  1
7
6
62  1
13
8
82  1
17
HAWKES LEARNING
Students Count. Success Matters.
Result
Copyright © 2015 by Hawkes
Learning/Quant Systems, Inc.
All rights reserved.
Example 7: Inductive versus Deductive
Reasoning (cont.)
If we make the conclusion that choosing a positive
integer, multiplying it by 2, and adding 1 to the product
will result in an odd number simply by looking at these
three examples, we are using inductive reasoning. We
are making a generalization based on three specific
examples. Although we know this to be valid, whenever
we make a generalization, we are taking a risk that the
generalized form will not always hold and therefore will
be an invalid conclusion.
HAWKES LEARNING
Students Count. Success Matters.
Copyright © 2015 by Hawkes
Learning/Quant Systems, Inc.
All rights reserved.
Example 7: Inductive versus Deductive
Reasoning (cont.)
b. Now we will let the variable x represent the chosen
integer and use deductive reasoning to show that
we have indeed made a valid conclusion.
Number: x
Multiply by 2: 2x
Add 1: 2x + 1
This means that, in algebraic form, the problem may
be expressed as 2x + 1 (multiplying the chosen
integer by 2 and adding 1).
HAWKES LEARNING
Students Count. Success Matters.
Copyright © 2015 by Hawkes
Learning/Quant Systems, Inc.
All rights reserved.
Example 7: Inductive versus Deductive
Reasoning (cont.)
Using deductive reasoning, we know that when we
multiply any number by 2, we will always get an even
number. Thus, whenever we have an even number, as
in 2x, and add 1, we will always get an odd number.
Therefore, we can assert that, given any positive
integer x, when x is multiplied by 2 and we add 1, the
result will be an odd number. We can then use this
general statement (which we have just proven to be
true) and apply it to specific numbers.
HAWKES LEARNING
Students Count. Success Matters.
Copyright © 2015 by Hawkes
Learning/Quant Systems, Inc.
All rights reserved.
Skill Check #2
Skill Check #2
Consider the following process.
Select a number and multiply the number by 10. Now
subtract 25 from that product. Then divide by 5. Finally,
subtract the original number from the result.
Can you determine the general solution using
deductive reasoning? Test the general solution with
specific numbers.
HAWKES LEARNING
Students Count. Success Matters.
Copyright © 2015 by Hawkes
Learning/Quant Systems, Inc.
All rights reserved.
Skill Check #2
Answer: 2. To determine the general solution: Select a
number: x; Multiply it by 10: 10x; Subtract 25
from the product: 10x  25; Divide by 5: 2x  5;
Subtract the original value: 2x  5  x; This gives
that the general result is x  5.
Now we test the general solution with specific
numbers. Select three numbers: 2, 10, 15;
Multiply them by 10: 20, 100, 150; Subtract 25
from the product: 5, 75, 125; Divide by 5: 1,
15, 25; Subtract the original value: 3, 5, 10;
Result is always 5 less than the original number.
HAWKES LEARNING
Students Count. Success Matters.
Copyright © 2015 by Hawkes
Learning/Quant Systems, Inc.
All rights reserved.