“When are we ever going to use this?”
This plaintive question from frustrated mathematics students is heard in schools
around our country as they wrestle with pages of abstract mathematics and learn
algorithms that appear to go nowhere. They study real numbers, but don’t find
any reason to believe that they are real. Thousands of American students still
work from textbooks that limit applications to
age problems and mixtures of
nuts. Despite the call from the National Council of Teachers of Mathematics in
the
Principles and Standards for School Mathematics (2000) for meaningful
learning through study of realistic applications, many students will find that the
only modernization of content over their grandparents’ math books is that jet
planes have replaced the trains that used to travel at different rates between cities.
The twentieth century saw an explosion of applications of mathematics. It is
now hard to find a field of study that does
not use mathematical tools. Biologists
use differential equations. Chemists use solid geometry to describe molecules.
Set designers in theaters use trigonometry to determine
the best lighting for a
play. Historians determine authorship of obscure documents through statistical
analysis of words. Governments, international corporations, and individual in-
vestors use mathematical rules to determine production, employment, and prices.
Everybody uses computers. Unfortunately, even good students don’t know how
mathematics affects their lives. Few understand the power of compound interest.
Few realize that the compound interest embedded in credit
cards can bring adults
to bankruptcy. Few know the mathematical implications of public policies that
will affect their lives. Even fewer know how to make best decisions based on the
probabilities of risk rather than blind gambles.
The secondary-school mathematics curriculum is faced with multiple chal-
lenges. What should students know and be able to do? Proficiency in some algo-
rithms is important. Abstraction in mathematics—stripping concepts of all but
Introduction
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their bare structures—is a feature that makes mathematics a powerful intellectual
tool. But these are not sufficient. Much of the mathematics taught in grades 7 to
12 is there because it is important outside the math classroom. Foundation appli-
cations, like paths of projectiles,
should not be stripped away, but rather should
be used to motivate the arithmetic, algebraic, or geometric concepts. Further, stu-
dents should have an opportunity to see a broad expanse of math applications so
they can find links between their interests and aspirations and their mathematics
coursework.
This book is an effort to promote real-world connections as they are applied
in people’s daily lives and careers. It is an account of the mathematical applica-
tions that we have learned and shared with people in our teaching careers. We
hope this reference guide helps you enjoy and appreciate the use and application
of mathematics in our culture and environment. We
hope you will find some
answers to the question, “When are we ever going to use this?”
audience
This book is intended to be a reference guide for anyone interested in under-
standing how some high school mathematics concepts are applied in nature and
society. We hope that high school students, teachers, and librarians use these
ideas to enhance their learning, teaching, and appreciation for mathematics. The
mathematics described here cover concepts that are
found in courses from pre-
algebra through introductory calculus. Each of the concepts is presented so that
the reader can gain different levels of understanding due to the varying levels of
mathematical complexity. A student or parent referencing the term
angle will
learn through descriptive text and diagrams that it is used for a variety of pur-
poses in navigation and road construction. A student who has learned trigonom-
etry may gain a deeper understanding as to
how an engineer might use the math-
ematics to make predictions by viewing different formulas and calculations. Our
intent is to make the content readable by all levels and ages of students, thereby
hoping that they will recognize value in the
applications of mathematics, regard-
less of their backgrounds.
purpose
This reference guide is an effort to provide exposure to mathematical appli-
cations, and should not be regarded as a primary tool for learning and instruction.
Since we do not intend to teach mathematical concepts here, there are occasions
in which mathematics is discussed without reference as to how an equation is
formed or how it was solved. Instead, each concept is informally described so
that primary emphasis can be placed on its applications. We do not intend for
teachers to teach mathematics in the way it is presented here. Instead, the text
should be used as a tool to enhance current
instructional practices, or to spark
student interest in math, or to create a classroom activity grounded in a particu-
lar application. Therefore, we feel that a more cohesive learning environment
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