Thinking Mathematically

Описание

Problem Solving and Critical Thinking
• develop an understanding of the fundamentals of statistics and how these
numbers are used to make decisions
• understand the mathematical paradoxes of voting in a democracy, increasing
your ability to function as a more fully aware citizen
• use graph theory to examine how mathematics is used to solve problems in the
business world
Mathematics and Your Career
Generally speaking, the income of an occupation is related to the amount of
education required. This, in turn, is usually related to the skill level required in
language and mathematics. With our increasing reliance on technology, the more
mathematics you know, the more career choices you will have.
Mathematics and Your World
Mathematics is a science that helps us recognize, classify, and explore the hidden
patterns of our universe. Focusing on areas as different as planetary motion, animal
markings, shapes of viruses, aerodynamics of figure skaters, and the very origin of
the universe, mathematics is the most powerful tool available for revealing the
underlying structure of our world. Within the last 30 years, mathematicians have
even found order in chaotic events such as the uncontrolled storm of noise in the
nerve cells of the brain during an epileptic seizure.
Inductive Reasoning
Mathematics involves the study of patterns. In everyday life, we frequently rely on
patterns and routines to draw conclusions. Here is an example:
The last six times I went to the beach, the traffic was light on Wednesdays
and heavy on Sundays. My conclusion is that weekdays have lighter traffic
than weekends.
This type of reasoning process is referred to as inductive reasoning, or induction.
“It is better to take what may seem to
be too much math rather than too
little. Career plans change, and one of
the biggest roadblocks in undertaking
new educational or training goals is
poor preparation in mathematics.
Furthermore, not only do people
qualify for more jobs with more
math, they are also better able to
perform their jobs.”
Occupational
Outlook Quarterly
1 Understand and use inductive
reasoning.
INDUCTIVE REASONING
Inductive reasoning is the process of arriving at a general conclusion based on
observations of specific examples.
Although inductive reasoning is a powerful method of drawing conclusions,
we can never be absolutely certain that these conclusions are true. For this reason,
the conclusions are called conjectures, hypotheses, or educated guesses. A strong
inductive argument does not guarantee the truth of the conclusion, but rather
provides strong support for the conclusion. If there is just one case for which the
conjecture does not hold, then the conjecture is false. Such a case is called a
counterexample.
Finding a Counterexample
The ten symbols that we use to write numbers, namely 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9,
are called digits. In each example shown below, the sum of two two-digit numbers
is a three-digit number.
Two-digit
numbers
Three-digit
sums
EXAMPLE 1

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