Greenwood press
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book-20600
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- Introduction
MATHEMATICAL CONCEPTS
Angle, 1 Asymptote, 7 Cartesian Coordinates, 10 Circles, 12 Circumference, 16 Complex Numbers, 16 Conic Sections, 18 Counting, 21 Derivative, 23 Equations, 23 Expected Value, 26 Exponential Decay, 28 Exponential Growth, 30 Fibonacci Sequence, 35 Imaginary Numbers, 37 Integration, 37 Inverse (Multiplicative), 43 Inverse Function, 45 Inverse Square Function, 47 Contents ▲ ▼ ▲ Linear Functions, 49 Logarithms, 55 Logistic Functions, 58 Matrices, 61 Perimeter, 64 Periodic Functions, 67 Plane, 70 Polar Coordinates, 72 Polynomial Functions, 75 Probability, 77 Proportions, 82 Pythagorean Theorem, 87 Quadratic Functions, 89 Quadrilaterals, 93 Rates, 96 Ratio, 102 Reflections, 107 Rotations, 111 Sequences, 114 Series, 117 Similarity, 121 Slope, 124 Square Roots, 124 Standard Deviation, 127 Step Functions, 130 Surface Area, 133 Symbolic Logic, 136 Symmetry, 138 Tangent, 141 Translations, 144 Triangle Trigonometry, 146 Variation, 150 Vectors, 154 Volume, 159 Bibliography, 163 viii CONTENTS “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 ▲ ▼ ▲ 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 Download 1.81 Mb. Do'stlaringiz bilan baham: |
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