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We see numbers on automobile license plates, addresses, weather
reports, and, of course, on our smartphones. Yet we look at these
numbers for their role as descriptors, not as an entity in and unto
themselves. Each number has its own history of meaning, usage, and
connotation in the larger world. The Secret Lives of Numbers takes
readers on a journey through integers, considering their
numerological assignments as well as their significance beyond
mathematics and in the realm of popular culture. Of course we all
know that the number 13 carries a certain value of unluckiness with
it. The phobia of the number is called Triskaidekaphobia; Franklin
Delano Roosevelt was known to invite and disinvite guests to
parties to avoid having 13 people in attendance; high-rise
buildings often skip the 13th floor out of superstition. There are
many explanations as to how the number 13 received this negative
honor, but from a mathematical point of view, the number 13 is also
the smallest prime number that when its digits are reversed is also
a prime number. It is honored with a place among the Fibonacci
numbers and integral Pythagorean triples, as well as many other
interesting and lesser-known occurrences. In The Secret Lives of
Numbers, popular mathematician Alfred S. Posamentier provides short
and engaging mini-biographies of more than 100 numbers, starting
with 1 and featuring some especially interesting numbers -like
6,174, a number with most unusual properties -to provide readers
with a more comprehensive picture of the lives of numbers both
mathematically and socially.
The most ubiquitous, and perhaps the most intriguing, number
pattern in mathematics is the Fibonacci sequence. In this simple
pattern beginning with two ones, each succeeding number is the sum
of the two numbers immediately preceding it (1, 1, 2, 3, 5, 8, 13,
21, ad infinitum). Far from being just a curiosity, this sequence
recurs in structures found throughout nature - from the arrangement
of whorls on a pinecone to the branches of certain plant stems. All
of which is astounding evidence for the deep mathematical basis of
the natural world. With admirable clarity, two veteran math
educators take us on a fascinating tour of the many ramifications
of the Fibonacci numbers. They begin with a brief history of a
distinguished Italian discoverer, who, among other accomplishments,
was responsible for popularizing the use of Arabic numerals in the
West. Turning to botany, the authors demonstrate, through
illustrative diagrams, the unbelievable connections between
Fibonacci numbers and natural forms (pineapples, sunflowers, and
daisies are just a few examples). In art, architecture, the stock
market, and other areas of society and culture, they point out
numerous examples of the Fibonacci sequence as well as its
derivative, the "golden ratio." And of course in mathematics, as
the authors amply demonstrate, there are almost boundless
applications in probability, number theory, geometry, algebra, and
Pascal's triangle, to name a few.Accessible and appealing to even
the most math-phobic individual, this fun and enlightening book
allows the reader to appreciate the elegance of mathematics and its
amazing applications in both natural and cultural settings.
We all learned that the ratio of the circumference of a circle to
its diameter is called pi and that the value of this algebraic
symbol is roughly 3.14. What we weren't told, though, is that
behind this seemingly mundane fact is a world of mystery, which has
fascinated mathematicians from ancient times to the present. Simply
put, pi is weird. Mathematicians call it a transcendental number
because its value cannot be calculated by any combination of
addition, subtraction, multiplication, division, and square root
extraction. In this delightful layperson's introduction to one of
math's most interesting phenomena, Drs. Posamentier and Lehmann
review pi's history from prebiblical times to the 21st century, the
many amusing and mind-boggling ways of estimating pi over the
centuries, quirky examples of obsessing about pi (including an
attempt to legislate its exact value), and useful applications of
pi in everyday life, including statistics. This enlightening and
stimulating approach to mathematics will entertain lay readers
while improving their mathematical literacy.
What works in math and why has never been the issue; the research
is all out there. Where teachers struggle is the "how"-something
the research rarely manages to tackle. That's the big service What
Successful Math Teachers Do provides. It's a powerful portal to
what the best research looks like in practice, strategy by
strategy-aligned in this new edition to both the Common Core and
the NCTM Standards. How exactly does What Successful Math Teachers
Do work? It couldn't be easier to navigate. The book's eleven
chapters organize clusters of strategies around a single aspect of
a typical instructional program. For each of the 75 strategies, the
authors present: A brief description of that strategy A summary of
supporting research The NCTM and Common Core Standards it
meets--and how Classroom applications, with examples Precautions
and possible pitfalls Primary sources for further reading and
research
An entertaining history of mathematics as chronicled through fifty
short biographies. Mathematics today is the fruit of centuries of
brilliant insights by men and women whose personalities and life
experiences were often as extraordinary as their mathematical
achievements. This entertaining history of mathematics chronicles
those achievements through fifty short biographies that bring these
great thinkers to life while making their contributions
understandable to readers with little math background. Among the
fascinating characters profiled are Isaac Newton (1642-1727), the
founder of classical physics and infinitesimal calculus--he
frequently quarreled with fellow scientists and was obsessed by
alchemy and arcane Bible interpretation; Sophie Germain (1776 -
1831), who studied secretly at the Ecole Polytechnique in Paris,
using the name of a previously enrolled male student--she is
remembered for her work on Fermat's Last Theorem and on elasticity
theory; Emmy Noether (1882 - 1935), whom Albert Einstein described
as the most important woman in the history of mathematics--she made
important contributions to abstract algebra and in physics she
clarified the connection between conservation laws and symmetry;
and Srinivasa Ramanujan (1887-1920), who came from humble origins
in India and had almost no formal training, yet made substantial
contributions to mathematical analysis, number theory, infinite
series, and continued fractions. The unusual behavior and life
circumstances of these and many other intriguing personalities make
for fascinating reading and a highly enjoyable introduction to
mathematics.
. 100 ways to get students hooked on math It happens to the best of
us: that one question that's got you stumped. Or maybe you have the
answer, but it's not all that compelling or convincing. Al
Posamentier and his coauthors to the rescue with this handy
reference containing fun answers to students' 100 most frequently
asked math questions. Even if you already have the answers, Al's
explanations are certain to keep kids hooked-and that's what it's
all about. The questions are all organized around the Common Core's
math content standards and relate directly to Numbers and Quantity,
Functions, Algebra, Geometry, and Statistics and Probability. The
big benefits? You'll discover high-interest ways to: Teach inquiry
and process in mathematical thinking Encourage flexibility in
problem solving Emphasize efficient test-taking strategies Provide
practical applications from mathematics, education, and human
development research Build students' procedural skills and
conceptual understanding Use this complete resource to save time,
anticipate questions, promote process and thinking, and present
yourself as the math expert we know you are.
Whenever the topic of mathematics is mentioned, people tend to
indicate their weakness in the subject as a result of not having
enjoyed its instruction during their school experience. Many
students unfortunately do not have very positive experiences when
learning mathematics, which can result from teachers who have a
tendency 'to teach to the test'. This is truly unfortunate for
several reasons. First, basic algebra and geometry, which are taken
by almost all students, are not difficult subjects, and all
students should be able to master them with the proper motivational
instruction. Second, we live in a technical age, and being
comfortable with basic mathematics can certainly help you deal with
life's daily challenges. Other, less tangible reasons, are the
pleasure one can experience from understanding the many intricacies
of mathematics and its relation to the real world, experiencing the
satisfaction of solving a mathematical problem, and discovering the
intrinsic beauty and historical development of many mathematical
expressions and relationships. These are some of the experiences
that this book is designed to deliver to the reader.The book offers
101 mathematical gems, some of which may require a modicum of high
school mathematics and others, just a desire to carefully apply
oneself to the ideas. Many folks have spent years encountering
mathematical terms, symbols, relationships and other esoteric
expressions. Their origins and their meanings may never have been
revealed, such as the symbols +, -, =, . oo, , , and many others.
This book provides a delightful insight into the origin of
mathematical symbols and popular theorems such as the Pythagorean
Theorem and the Fibonacci Sequence, common mathematical mistakes
and curiosities, intriguing number relationships, and some of the
different mathematical procedures in various countries. The book
uses a historical and cultural approach to the topics, which
enhances the subject matter and greatly adds to its appeal. The
mathematical material can, therefore, be more fully appreciated and
understood by anyone who has a curiosity and interest in
mathematics, especially if in their past experience they were
expected to simply accept ideas and concepts without a clear
understanding of their origins and meaning. It is hoped that this
will cast a new and positive picture of mathematics and provide a
more favorable impression of this most important subject and be a
different experience than what many may have previously
encountered. It is also our wish that some of the fascination and
beauty of mathematics shines through in these presentations.
Multiply math mastery and interest with these inspired teaching
tactics! Invigorate instruction and engage students with this
treasure trove of "Great Ideas" compiled by two of the greatest
minds in mathematics. From commonly taught topics in algebra,
geometry, trigonometry and statistics, to more advanced
explorations into indirect proofs, binomial theorem, irrationality,
relativity and more, this guide outlines actual equations and
techniques that will inspire veteran and new educators alike. This
updated second edition offers more proven practices for bringing
math concepts to life in the classroom, including 114 innovative
strategies organized by subject area User-friendly content
identifying "objective," "materials," and "procedure" for each
technique A range of teaching models, including hands-on and
computer-based methods Specific and straightforward examples with
step-by-step lessons Written by two distinguished leaders in the
field-mathematician, author, professor, university dean and popular
commentator Alfred S. Posamentier, along with mathematical pioneer
and Nobel Prize recipient Herbert A. Hauptman-this guide brings a
refreshing perspective to secondary math instruction to spark
renewed interest and success among students and teachers.
A veteran math educator reveals the hidden fascinations of geometry
and why this staple of math education is important. If you remember
anything about high school geometry class, it's probably doing
proofs. But geometry is more than axioms, postulates, theorems, and
proofs. It's the science of beautiful and extraordinary geometric
relationships--most of which is lost in high school classrooms
where the focus is on the rigor of logically proving those
relationships. This book will awaken readers to the appeal of
geometry by placing the focus squarely on geometry's visually
compelling features and intrinsic elegance. Who knew that straight
lines, circles, and area could be so interesting? Not to mention
optical illusions. So get out the rulers, compasses, or even a
software program, and discover geometry for the first time.
Our physical world is embedded in a geometric environment. Plane
geometry has many amazing wonders beyond those that are briefly
touched on in school curriculums. The triangle, one of the basic
instruments in geometry, has a plethora of unexpected curiosities.
Geometric Gems presents one of the largest collections of triangle
curiosities currently available, which the authors discuss in an
easily understood fashion, requiring nothing more of readers other
than the very basics of school geometry to appreciate these
curiosities and their justifications or proofs.The book is intended
to be widely appreciated by a general audience, and their love for
geometry should be greatly enhanced through exploring these many
unexpected relationships in geometry. Geometric Gems is also
suitable for mathematics teachers, to enhance the education of
their students with these highly motivating triangle properties.
Our physical world is embedded in a geometric environment. Plane
geometry has many amazing wonders beyond those that are briefly
touched on in school curriculums. The triangle, one of the basic
instruments in geometry, has a plethora of unexpected curiosities.
Geometric Gems presents one of the largest collections of triangle
curiosities currently available, which the authors discuss in an
easily understood fashion, requiring nothing more of readers other
than the very basics of school geometry to appreciate these
curiosities and their justifications or proofs.The book is intended
to be widely appreciated by a general audience, and their love for
geometry should be greatly enhanced through exploring these many
unexpected relationships in geometry. Geometric Gems is also
suitable for mathematics teachers, to enhance the education of
their students with these highly motivating triangle properties.
Mathematics is a subject taught from kindergarten through to high
school, and yet it is the one subject that most adults are almost
proud to admit to not having been very good at, and, therefore,
tend to avoid it where they can. However, one of the key factors in
mathematics is its ability to enable us to solve everyday problems.
When we consider 'the worst-case scenario' of the situation, it is
analogous to solving a mathematical problem by considering
extremes. Or, we might consider the best path to take from point A
to point B, where geometric relationships can be helpful. This book
is intended to demonstrate a variety of neglected aspects of
mathematics, in order to demonstrate the power and beauty of the
field of mathematics beyond where most people, students, and
teachers believe is possible.The chapters of the book explore a
multitude of topics: unusual arithmetic calculations and shortcuts,
entertaining and instructional problem-solving strategies, unusual
applications of algebra, and how geometry allows us to better
appreciate physical relationships. Only a basic mathematical
knowledge is needed to understand these topics and problems;
however, the book also demonstrates that, armed with even this
level of understanding, our mathematical skills far exceed what we
learned at school! The final chapter is the most challenging, and
explores a curious problem-solving technique.
Mathematics is a subject taught from kindergarten through to high
school, and yet it is the one subject that most adults are almost
proud to admit to not having been very good at, and, therefore,
tend to avoid it where they can. However, one of the key factors in
mathematics is its ability to enable us to solve everyday problems.
When we consider 'the worst-case scenario' of the situation, it is
analogous to solving a mathematical problem by considering
extremes. Or, we might consider the best path to take from point A
to point B, where geometric relationships can be helpful. This book
is intended to demonstrate a variety of neglected aspects of
mathematics, in order to demonstrate the power and beauty of the
field of mathematics beyond where most people, students, and
teachers believe is possible.The chapters of the book explore a
multitude of topics: unusual arithmetic calculations and shortcuts,
entertaining and instructional problem-solving strategies, unusual
applications of algebra, and how geometry allows us to better
appreciate physical relationships. Only a basic mathematical
knowledge is needed to understand these topics and problems;
however, the book also demonstrates that, armed with even this
level of understanding, our mathematical skills far exceed what we
learned at school! The final chapter is the most challenging, and
explores a curious problem-solving technique.
This easy-to-navigate guide offers research-based teaching
strategies for introducing nursery and primary school students to
the content and skills they will need to teach mathematics
effectively. Using the popular format of the What Successful
Teachers Do books, the authors present 47 dynamic learning
activities, each including: * A concise statement of the teaching
strategy * Research-based validations for the strategy * Classroom
applications and vignettes * Precautions and possible pitfalls *
Primary sources for further reading
'This is a very interesting book and one that can only help our
battle to make algebra come alive and help our students see the
useful and interesting problems that algebra can help us to deal
with' - Peter Hall, Imberhorne School, East Grinstead Making
Algebra Come Alive! provides versatile enrichment exercises for
teaching a broad range of mathematical topics and applications.
Each activity is presented as a reproducible student investigation.
It is followed by guidelines and notes for the teacher. Each
activity is keyed to the National Council of Teachers of
Mathematics (NCTM) Standards, Revised. This link to the NCTM
standards allows teachers to facilitate linking classroom
activities to specific state and school district content standards.
First and foremost, the activities are meant to be motivational. As
much as possible, we want this book to achieve the goal of being
attractive to people who thought they didn't like mathematics. To
accomplish this, it is necessary for the activities to be quite
different from what students encounter in their basal
texts-different in both substance and form. This seems especially
critical; no matter how excellent a basal text is being used,
nearly every class experiences the "blahs." Unfortunately, this
sort of boredom is often well entrenched long before the teacher
and perhaps even the students are aware of it. Presenting
activities on a regular basis gives the variety and change of pace
needed to sustain interest in any subject.
The book presents a comprehensive overview of various aspects of
three-dimensional geometry that can be experienced on a daily
basis. By covering the wide range of topics - from the psychology
of spatial perception to the principles of 3D modelling and
printing, from the invention of perspective by Renaissance artists
to the art of Origami, from polyhedral shapes to the theory of
knots, from patterns in space to the problem of optimal packing,
and from the problems of cartography to the geometry of solar and
lunar eclipses - this book provides deep insight into phenomena
related to the geometry of space and exposes incredible nuances
that can enrich our lives.The book is aimed at the general
readership and provides more than 420 color illustrations that
support the explanations and replace formal mathematical arguments
with clear graphical representations.
The book presents a comprehensive overview of various aspects of
three-dimensional geometry that can be experienced on a daily
basis. By covering the wide range of topics - from the psychology
of spatial perception to the principles of 3D modelling and
printing, from the invention of perspective by Renaissance artists
to the art of Origami, from polyhedral shapes to the theory of
knots, from patterns in space to the problem of optimal packing,
and from the problems of cartography to the geometry of solar and
lunar eclipses - this book provides deep insight into phenomena
related to the geometry of space and exposes incredible nuances
that can enrich our lives.The book is aimed at the general
readership and provides more than 420 color illustrations that
support the explanations and replace formal mathematical arguments
with clear graphical representations.
Many schools throughout the United States are now employing coaches
to support teachers in the teaching of mathematics. Very often,
these are professionals selected from the current teaching ranks
who did not receive any special training to serve as a coach. This
book provides the skills, knowledge, and lessons from experience
that lead such a mentor to function effectively. Aside from
describing the basic duties of an effective math coach, we also
provide a plethora of resources to enrich instruction, improve
problem-solving direction, and provide teachers with a wide variety
of techniques to enhance their teaching effectiveness.
Two veteran math educators demonstrate how some "magnificent
mistakes" had profound consequences for our understanding of
mathematics' key concepts.
In the nineteenth century, English mathematician William Shanks
spent fifteen years calculating the value of pi, setting a record
for the number of decimal places. Later, his calculation was
reproduced using large wooden numerals to decorate the cupola of a
hall in the Palais de la Decouverte in Paris. However, in 1946,
with the aid of a mechanical desk calculator that ran for seventy
hours, it was discovered that there was a mistake in the 528th
decimal place. Today, supercomputers have determined the value of
pi to trillions of decimal places.
This is just one of the amusing and intriguing stories about
mistakes in mathematics in this layperson's guide to mathematical
principles. In another example, the authors show that when we
"prove" that every triangle is isosceles, we are violating a
concept not even known to Euclid - that of "betweenness." And if we
disregard the time-honored Pythagorean theorem, this is a misuse of
the concept of infinity. Even using correct procedures can
sometimes lead to absurd - but enlightening - results.
Requiring no more than high-school-level math competency, this
playful excursion through the nuances of math will give you a
better grasp of this fundamental, all-important science.
If you've been waiting for a book that will evoke the delight and
intrigue that mathematics has to offer, this is the book for you.
What are the odds of finding two people who share the same birth
date in a room of thirty-five? Most people would guess they're
pretty low. In actuality, the probability is better than 80
percent. This is just one of many entertaining examples of
mathematical curiosities presented. Two veteran math educators have
created the perfect introduction to the wonders of mathematics for
the general reader, requiring only a high school background in the
subject.
Among the entertaining and useful tricks they teach are shortcuts
in arithmetic, such as ways to determine at a glance the exact
divisors of any given number. They also demonstrate how the
properties of certain numbers can lead to infinite loops. What is
particularly exciting is how many correct answers turn out to be
counterintuitive. Exploring all these features will instill
insights into the nature of numbers, improve your ability to
manipulate them, and give you an appreciation for the inherent
elegance of mathematics.
As you marvel at the many unusual relationships and novelties
revealed in this ingenious and delightful presentation, you'll be
learning more math than you ever thought possible - and will be
relishing every moment of it
SNAP Bronze Award Winner! Engaging students in math lessons and
encouraging them to explore math principles is much easier-and lots
more fun-when you use this book's amazing collection of quirks,
illustrations, problems, and surprises. Dazzle your students with
Peculiar number patterns and properties Clever shortcuts and tricks
for avoiding cumbersome mathematic processes Curious problems that
reveal problem-solving strategies Algebra problems that shed light
on patterns in mathematics and the behavior of numbers Activities
and demonstrations that uncover timeless geometric principles Used
as attention-getters, motivators, or enrichment activities, these
math wonders leave students questioning and exploring further,
regardless of their grade level or ability.
Professional mathematicians often speak of the beauty of
mathematics and the elegance of its solutions. Yet the esthetic
appeal of math is rarely conveyed to students at the elementary,
secondary, or even college level. Instead, most of us develop
phobias in school about math's elusive logic and then pass these
negative impressions on to our children. We should all be having
fun with math and helping our kids to do better in life by
encouraging them to appreciate not only its usefulness but
especially its charm. That's just what veteran math educator Alfred
Posamentier sets out to do in this delightful exploration of math's
many intriguing, interesting, and fun qualities.
Beginning with the beauty of the number system, thr author doesn't
just talk mathematics; he entices readers to do math and discover
for themselves just how stimulating the process can be Brief and
entertaining introductions to each chapter invite readers to try
their hands at arithmetic marvels, surprising solutions, algebraic
entertainments, geometric wonders, and fun mathematical paradoxes,
among other topics.
Presented in a reader-friendly, conversational tone, the text is
very accessible and the examples are geared to a beginner's level,
so that even the most math-phobic individual will discover the
hidden joy and inherent appeal of doing math. This is the ideal
book for adults looking for a way to turn their kids on to an
important subject or discover for themselves what they might have
missed in their own math education.
This book will present a collection of mathematical problems -
lighthearted in nature - intended to entertain the general
readership. Problems will be selected largely for the unusual and
unexpected solutions to which they lend themselves. Some
interesting contents included: All in all, the book is meant to
entertain the general readership and to convince them about the
power and beauty of mathematics.
This book introduces ten problem-solving strategies by first
presenting the strategy and then applying it to problems in
elementary mathematics. In doing so, first the common approach is
shown, and then a more elegant strategy is provided. Elementary
mathematics is used so that the reader can focus on the strategy
and not be distracted by some more sophisticated mathematics.
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