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WeBWorK Links
General Information about WeBWorK
I. What is WeBWorK ?
WeBWorK is a system that allows students to do their homework
in an interactive web-based environment.
Using WeBWorK, students may try to answer homework problems
more than once. After each try, a message appears telling the student
whether the answer is correct or not. This provides immediate
feedback to the students, allows them to discover what they did wrong,
and hopefully alows them to understand the topic of the question
better.
Each WeBWorK problem set is individualized. Each student has
a different version of a problem generated from a fixed template; for
example the numerical values in the formulas may be slightly
different).
II. How to use WeBWorK to do your homework
Using WeBWorK is quite simple.
Below are the basic steps on how to get started.
NOTE: Most pages of WeBWorK also contain
directions. Therefore, if you are ever unsure of what you should
do, try reading the directions and descriptions on the page at which
you are looking.
You can use any computer with a browser like Firefox or Opera. In
general, it is not possible to use a text-based browser like lynx
since most of the mathematics is rendered using a graphic format.
This will get you to the main page of your course. This page
includes necessary information about logging in.
To log in, click on the 'Login' button.
This will take you to a login page. Enter your
login name and password, and click on the 'Continue' button.
Your login name is the usually just your last name,
followed possibly by a digit for people in the class with the same
last name (e.g. smith, smith2, ...)). Possible exceptions include
compound last name like "den Hartog" or "De La Huerta" which would
be rendered Hartog and Huerta respectively.
Your initial password is (unless you gave me
something else) your six-character student ID ``number''.
If your login is incorrect, you will be told so, and you can
return to the login page and try again.
If you are registered for the course, you should receive and email
from your instructor confirming your login and password.
If your login is
correct you will see a page where you can do following:
Look at and do the problems in a set via Firefox.
To do the first days assignment, for example, click on the line in
the box that starts with 'Section 1.1'. Then click on the 'Do problem
set' button. On each line in the box, where all the sets are
listed, after the set number, you can see whether the set is open or
closed. If the set is open, that means that when you solve a
problem, it will be counted towards your grade. If the set is
closed, you can still solve problems, but your results will not be
recorded. After the indication of whether the set is open or
closed, there is additional information about the due date (if the
set is open), or whether the answers are available (if the set is
closed).
Get a printout of the problem set.
To print out Section 1.1, for example, first choose the download type.
THe default of pdf is probably what you want, although you may have
to configure your browser to use Adobe's Acrobat Reader to view (and
print) the files.
After choosing one of the download types, click on the line in
the box that starts with 'Section 1.1...'. Then click on the 'Get hard
copy' button. A couple of things may happen at this point. If your
browser is set up to handle pdf documents, your browser window will
fill with the homework set in it as a document. In this case, go to
the icon 'file' and choose the 'print' option. If your browser is
not properly configured, you may get an alert message asking what to
with the file. In this case, you can save the file to disk (so that
you can print it at a later time), choose an application to view or
print the file (e.g. GSview or dropPS for postscript or Acrobat
Reader for pdf). Any of the computer consultants show be able to
help you with this kind of problem.
Look at a summary of your WeBWorK homework
scores. This is the second section of the page. If you
click on the button 'Get Summary', you will see your current scores
for all available problem sets.
If you are printing out a problem set or looking at a summary of
your homework scores, you are done. If you are viewing a problem
set via Firefox, you will see a page with the problems in the set
you chose. To view and/or answer a problem, click on the number of
the problem and click on the 'Get Problem' button. Notice that
there are four modes of viewing the problem: 'text',
'formatted-text' and 'typeset', adn typeset2. It is best to view
the problem using 'typeset2' mode, which should be the default. Once
you choose a problem and click on the 'Get Problem' button, you will
see the text of the problem with boxes for your answers. Enter your
answers and click on the 'Submit Answer' button. If you are working
on a problem set that is already closed, you will have the option to
see the correct answer or a solution if one is available (currently
only a few problems have solutions available). To see the correct
answer and/or solution, just check the box(es) and click on the
'Submit Answer' button (you do not have to enter an answer to see
the correct answer or solution). Once you have submitted an answer,
you will be told whether your answer is correct or not. If not, you
can try again. After you've tried a problem, you can either go to
the next problem, the previous problem, or see the list of the
problems again.
If you want to check the status of your
problems (e.g. to double check that your answers have been
recorded), use the "Prob. List" button at the top of the
page to see the problem list page.
When you are finished, log
out using the "Logout" button at the bottom of the page.
That's all, folks!
IV. Important facts to know
What to do if you have problems with WeBWorK:
If you have a problem logging in, contact your instructor.
If you have a problem printing out a set, ask a consultant
at a computer lab. If you don't get sufficient help,
contact your instructor or TA.
If you have questions on specific homework problems or if you have
comments about WeBWorK that you think can help us
make WeBWorK better, click on the 'Feedback'
button on any of the pages of WeBWorK.
If you are logged on to WeBWorK for longer that 30 minutes
without any activity,
you will be asked to log in again. This is a security measure. You
can resume your work after you logged back in. All your results from
the last log in will be saved.
|
The derivative and the integral are the fundamental notions of calculus. Though there is essentially only one derivative, there are a variety of integrals, developed over the years for a variety of purposes, and this book describes them. No other single source treats all of the integrals of Cauchy, Riemann, Riemann-Stieltjes, Lebesgue, Lebesgue-Steiltjes, Henstock-Kurzweil, Weiner, and Feynman. The basic properties of each are proved, their similarities and differences are pointed out, and the reason for their existence and their uses are given. Historical information is plentiful. Advanced undergraduate mathematics majors, graduate students, and faculty members are the audience for the book. Even experienced faculty members are unlikely to be aware of all of the integrals in the Garden of Integrals and the book provides an opportunity to see them and appreciate the richness of the idea of integral. Professor Burke's clear and well-motivated exposition makes this book a joy to read.
A Guide to Complex Variables gives the reader a quick and accessible introduction to the key topics. While the coverage is not comprehensive, it certainly gives the reader a solid grounding in this fundamental area. There are many figures and examples to illustrate the principal ideas, and the exposition is lively and inviting. An undergraduate wanting to have a first look at this subject or a graduate student preparing for the qualifying exams, will find this book to be a useful resource. In addition to important ideas from the Cauchy theory, the book also includes the Riemann mapping theorem, harmonic functions, the argument principle, general conformal mapping and dozens of other central topics. Readers will find this book to be a useful companion to more exhaustive texts in the field. It is a valuable resource for mathematicians and non-mathematicians alikeSink or Float: Thought Problems in Math and Physics is a collection of problems drawn from mathematics and the real world. Its multiple-choice format forces the reader to become actively involved in deciding upon the answer. The book's aim is to show just how much can be learned by using everyday common sense. The problems are all concrete and understandable by nearly anyone, meaning that not only will students become caught up in some of the questions, but professional mathematicians, too, will easily get hooked. The more than 250 questions cover a wide swath of classical math and physics. Each problem's solution, with explanation, appears in the answer section at the end of the book. A notable feature is the generous sprinkling of boxes appearing throughout the text. These contain historical asides or little-known facts. The problems themselves can easily turn into serious debate-starters, and the book will find a natural home in the classroom.
Inequalities permeate mathematics, from the Elements of Euclid to operations research and financial mathematics. Yet too often, especially in secondary and collegiate mathematics, the emphasis is on things equal to one another rather than unequal. While equalities and identities are without doubt important, they don't possess the richness and variety that one finds with inequalities. The objective of this book is to illustrate how the use of visualization can be a powerful tool for better understanding some basic mathematical inequalities. Drawing pictures is a well-known method for problem solving, and the authors will convince you that the same is true when working with inequalities. They show how to produce figures in a systematic way for the illustration of inequalities and open new avenues to creative ways of thinking and teaching. In addition, a geometric argument cannot only show two things unequal, but also help the observer see just how unequal they are. The concentration on geometric inequalities is partially motivated by the hope that secondary and collegiate teachers might use these pictures with their students. Teachers may wish to use one of the drawings when an inequality arises in the course. Alternatively, When Less Is More might serve as a guide for devoting some time to inequalities and problem solving techniques, or even as part of a course on inequalities.
A Guide to Advanced Real Analysis is an outline of the core material in the standard graduate-level real analysis course. It is intended as a resource for students in such a course as well as others who wish to learn or review the subject. On the abstract level, it covers the theory of measure and integration and the basics of point set topology, functional analysis, and the most important types of function spaces. On the more concrete level, it also deals with the applications of these general theories to analysis on Euclidean space: the Lebesgue integral, Hausdorff measure, convolutions, Fourier series and transforms, and distributions. The relevant definitions and major theorems are stated in detail. Proofs, however, are generally presented only as sketches, in such a way that the key ideas are explained but the technical details are omitted. In this way a large amount of material is presented in a concise and readable form.
A Guide to Real Variables provides aid and conceptual support for the student studying for the qualifying exam in real variables. Beginning with the foundations of the subject, the text moves rapidly but thoroughly through basic topics like completeness, convergence, sequences, series, compactness, topology and the like. All the basic examples like the Cantor set, the Weierstrass nowhere differentiable function, the Weierstrass approximation theory, the Baire category theorem, and the Ascoli-Arzela theorem are treated. The book contains over 100 examples, and most of the basic proofs. It illustrates both the theory and the practice of this sophisticated subject resourceA Guide to Topology is an introduction to basic topology. It covers point-set topology as well as Moore-Smith convergence and function spaces. It treats continuity, compactness, the separation axioms, connectedness, completeness, the relative topology, the quotient topology, the product topology, and all the other fundamental ideas of the subject. The book is filled with examples and illustrations researchCHOICE Award winner! A Guide to Elementary Number Theory is a 140-page exposition of the topics considered in a first course in number theory. It is intended for those who may have seen the material before but have half-forgotten it, and also for those who may have misspent their youth by not having a course in number theory and who want to see what it is about without having to wade through a traditional text, some of which approach 500 pages in length. It will be especially useful to graduate student preparing for the qualifying exams. Underwood Dudley received the Ph.D. degree (number theory) from the University of Michigan in 1965. He taught at the Ohio State University and at DePauw University, from which he retired in 2004. He is the author of three books on mathematical oddities, The Trisectors, Mathematical Cranks, and Numerology all published by the Mathematical Association of America. He has also served as editor of the College Mathematics Journal, the Pi Mu Epsilon Journal, and two of the Mathematical Association of America's book series.
Theorems and their proofs lie at the heart of mathematics. In speaking of the purely aesthetic qualities of theorems and proofs, G. H. Hardy wrote that in beautiful proofs 'there is a very high degree of unexpectedness, combined with inevitability and economy.' Charming Proofs present a collection of remarkable proofs in elementary mathematics that are exceptionally elegant, full of ingenuity, and succinct. By means of a surprising argument or a powerful visual representation, the proofs in this collection will invite readers to enjoy the beauty of mathematics, to share their discoveries with others, and to become involved in the process of creating new proofs. Charming Proofs is organized as follows. Following a short introduction about proofs and the process of creating proofs, the authors present, in twelve chapters, a wide and varied selection of proofs they consider charming, Topics include the integers, selected real numbers, points in the plane, triangles, squares, and other polygons, curves, inequalities, plane tilings, origami, colorful proofs, three-dimensional geometry, etc. At the end of each chapter are some challenges that will draw the reader into the process of creating charming proofs. There are over 130 such challenges. Charming Proofs concludes with solutions to all of the challenges, references, and a complete index. As in the authors' previous books with the MAA (Math Made Visual and When Less Is More), secondary school and college and university teachers may wish to use some of the charming proofs in their classrooms to introduce their students to mathematical elegance. Some may wish to use the book as a supplement in an introductory course on proofs, mathematical reasoning, or problem solving. sports. The section on football includes an article that evaluates a method for reducing the advantage of the winner of a coin flip in an NFL overtime game; the section on track and field examines the ultimate limit on how fast a human can run 100 meters; the section on baseball includes an article on the likelihood of streaks; the section on golf has an article that describes the double-pendulum model of a golf swing, and an article on modeling Tiger Wood's career. The articles provide source material for classroom use and student projects. Many students will find mathematical ideas motivated by examples taken from sports more interesting than the examples selected from traditional sources.
Linear algebra occupies a central place in modern mathematics. This book provides a rigorous and thorough development of linear algebra at an advanced level, and is directed at graduate students and professional mathematicians. It approaches linear algebra from an algebraic point of view, but its selection of topics is governed not only for their importance in linear algebra itself, but also for their applications throughout mathematics. Students in algebra, analysis, and topology will find much of interest and use to them, and the careful treatment and breadth of subject matter will make this book a valuable reference for mathematicians throughout their professional lives. Topics treated in this book include: vector spaces and linear transformations; dimension counting and applications; representation of linear transformations by matrices; duality; determinants and their uses; rational and especially Jordan canonical form; bilinear forms; inner product spaces; normal linear transformations and the spectral theorem; and an introduction to matrix groups as Lie groups. The book treats vector spaces in full generality, though it concentrates on the finite dimensional case. Also, it treats vector spaces over arbitrary fields, specializing to algebraically closed fields or to the fields of real and complex numbers as necessary.
Icons of mathematics are certain geometric diagrams that play a crucial role in visualizing mathematical proofs, and in the book the authors present 20 of them and explore the mathematics that lies within and that can be created. The authors devote a chapter to each icon, illustrating its presence in real life, its primary mathematical characteristics and how it plays a central role in visual proofs of a wide range of mathematical facts. Among these are classical results from plane geometry, properties of the integers, means and inequalities, trigonometric identities, theorems from calculus, and puzzles from recreational mathematics.
This book can be used in a one semester undergraduate course or senior capstone course, or as a useful companion in studying algebraic geometry at the graduate level. This Guide is a friendly introduction to plane algebraic curves. It emphasizes geometry and intuition, and the presentation is kept concrete. You'll find an abundance of pictures and examples to help develop your intuition about the subject, which is so basic to understanding and asking fruitful questions. Highlights of the elementary theory are covered, which for some could be an end in itself, and for others an invitation to investigate further. Proofs, when given, are mostly sketched, some in more detail, but typically with less. References to texts that provide further discussion are often included. Computer algebra software has made getting around in algebraic geometry much easier. Algebraic curves and geometry are now being applied to areas such as cryptography, complexity and coding theory, robotics, biological networks, and coupled dynamical systems. Algebraic curves were used in Andrew Wiles' proof of Fermat's Last Theorem, and to understand string theory, you need to know some algebraic geometry. There are other areas on the horizon for which the concepts and tools of algebraic curves and geometry hold tantalizing promise. This introduction to algebraic curves will be appropriate for a wide segment of scientists and engineers wanting an entrance to this burgeoning subject.
This Guide offers a concise overview of the theory of groups, rings, and fields at the graduate level, emphasizing those aspects that are useful in other parts of mathematics. It focuses on the main ideas and how they hang together. It will be useful to both students and professionals. In addition to the standard material on groups, rings, modules, fields, and Galois theory, the book includes discussions of other important topics that are Those looking for a way to review and refresh their basic algebra will benefit from reading this Guide, and it will also serve as a ready reference for mathematicians who make use of algebra in their work.
The purpose of A Guide to Functional Analysis is to introduce the reader with minimal background to the basic scripture of functional analysis. Readers should know some real analysis and some linear algebra. Measure theory rears its ugly head in some of the examples and also in the treatment of spectral theory. The latter is unavoidable and the former allows us to present a rich variety of examples. The nervous reader may safely skip any of the measure theory and still derive a lot from the rest of the book. Apart from this caveat, the book is almost completely self-contained; in a few instances we mention easily accessible references. A feature that sets this book apart from most other functional analysis texts is that it has a lot of examples and a lot of applications. This helps to make the material more concrete, and relates it to ideas that the reader has already seen. It also makes the book more accessible to a broader audience.
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Suitable for the GCSE Modular Mathematics, this book covers different concepts through artwork and diagrams.
Synopsis:
This book is revised in-line with the 2007 GCSE Modular Mathematics specification. This Student Book is delivered in colour giving clarity to different concepts through artwork and diagrams. Worked examples, practice exercises and examiners tips ensure students are fully prepared for their exams. It is written by an experienced author team, including Senior Examiners, which means you can trust that the 2007 specification is covered to ensure exam success
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Appendix C
Open-ended Questions for Automotive Airbag Example
The GE fund grant will allow integration of the existing airbag module
with interactive math and physics exercises that explore the mathematical and
physical relationships behind the rate of inflation of the airbag, as well as
the force with which the airbag and the body of the driver collide. Sample
open-ended questions the students might be asked in mathematics and physics are
provided below:
(1) Curve shape analysis to determine optimum rate of gas production.
The rate at which the airbag opens is critical if it is to function properly.
The bag must be full within 100 milliseconds and must not inflate too rapidly
so as not to exert too much force on the driver's body. A graphical depiction
of the volume of gas produced as a function of time is an excellent way to have
students think about the rate of change in volume required to meet both
criteria. Students' conceptual understanding of the meaning of the plots will
be developed by having them provide a description of the appearance of the
airbag inflations from the driver's point of view in the scenarios described by
the different curves.
(2) Assuming that the airbag approximates a sphere, for each curve students
will determine the maximum velocity with which a driver would collide with
the airbag. They will need to explore the diameter of the airbag D =
D(t) and its derivative d D/ dt (t) as
functions of time in order to obtain the velocity at which the bag approaches
the driver. The volume "V" of a spherical ball is related to the
diameter by the standard formula V = ([[pi]]/ 6) D3.
Students will explore how to relate the derivatives dV/dt and
dD/dt and how to calculate one from the other. Symbolically,
this is an illustration of the chain rule and the technique of implicit
differentiation, both of which are very important topics in beginning
calculus. Furthermore, students will use their computers to explore the four
functions V, D, dV/dt, and dD/dt and
their relationships graphically and numerically by displaying their graphs,
observing how the graphs change as the parameters of the chemical reaction
change, and by computing actual values of the functions. Finally they will
relate this graphical and numerical information to the symbolic calculations.
This work is a simple but effective illustration of the basic pedagogical
principle in reformed calculus of using a concrete example to stress the
multiple ways of representing functional relationships (symbolic, graphical,
numerical, and verbal). The example emphasizes the interplay between these
different representations and how information about the function, its
derivatives and integrals is transported from one representation to another.
The use of computers in the class-laboratory/studio is absolutely essential to
achieve these pedagogical goals.
(3) Students can then examine how the airbag alters the collision process by
determining the amount the force of impact is lessened by having the driver
collide with the airbag rather than the steering wheel of the car. They can
also compute the force of impact of the exploding airbag with a stationary
driver, to determine why it is so important to avoid false triggers.
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id: 05776637
dt: j
an: 2010e.00699
au: Biaglow, Andrew; Erickson, Keith; McMurran, Shawnee
ti: Enzyme kinetics and the Michaelis-Menten equation.
so: PRIMUS, Probl. Resour. Issues Math. Undergrad. Stud. 20, No. 2, Special
Issue: Application activities to enhance learning in the
mathematics-biology interface, 148-168 (2010).
py: 2010
pu: Taylor \& Francis, Philadelphia, PA
la: EN
cc: M65 I75
ut: difference equations; elementary differential equations; Euler's method;
linearization; parameter estimation; enzyme catalysis; Michaelis-Menten
kinetics; mass balance; law of mass action; conservation of mass;
biology; mathematical applications
ci:
li: doi:10.1080/10511970903486491
ab: Summary: The concepts presented in this article represent the cornerstone
of classical mathematical biology. The central problem of the article
relates to enzyme kinetics, which is a biochemical system. However, the
theoretical underpinnings that lead to the formation of systems of
time-dependent ordinary differential equations have been applied widely
to any biological system that involves modeling of populations. In this
project, students first learn about the general balance equation, which
is a statement of conservation within a system. They then learn how to
simplify the balance equation for several specific cases involving
chemically reacting systems. Derivations are reinforced with a concrete
experiment in which enzyme kinetics are illustrated with pennies. While
a working knowledge of differential equations and numerical techniques
is helpful as a prerequisite for this set of activities, all of the
requisite mathematical skills are introduced in the project, so the
methods would also serve as an introduction to these techniques. It is
also helpful if students have some basic understanding of chemical
concepts such as concentration and reaction rate, as typically covered
in high school or college freshman chemistry courses.
rv:
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Please note, this offer price only applies to individual customers when ordering direct from Oxford University Press, while stock lasts. No further discounts will apply. If you are a bookseller, please contact your OUP sales representative.
The Chemistry Maths Book
Broad coverage captures all the key mathematical concepts and theories with which a chemist should be familiar, making it the ideal reference throughout your studies.
The author's unfussy approach lets the subject speak for itself, using a combination of straightforward explanations and reinforcing examples.
The much-praised lucid writing style leads the student through even the most challenging topics in a steady, engaging way.
Extensive range of worked examples demonstrate every important concept and method in the text, to help the student grasp the material being presented.
End of chapter exercises encourage the student to learn through hands-on practice.
Online Resource Centre features additional resources for registered adopters of the text, to facilitate its use in teaching.
New to this edition
Contents reorganized to link the text and examples more closely with the exercises at the end of each chapter: it is now easier for the reader to try for themselves the exercises that are directly relevant to the topic they have been reading.
Material within some chapters has been reorganized to make the development of the subject more logical.
Extensive changes to chapter 1, retitled 'Numbers, Variables and Units' include a new section, Factorization, factors, and factorials, which fills a gap in the coverage of elementary topics and a rewritten and much enlarged section on units.
Chapter 2 has been revised to accommodate more discussion of the factorization and manipulation of algebraic expressions.
In chapter 9, the section on line integrals has been rewritten to clarify the relevance of line integrals to change of state in thermodynamics.
In chapter 13, a revision of the section on the Frobenius method, with new and more demanding examples and exercises.
In chapter 19, revised treatment of eigenvalues and eigenvectors, with new examples and exercises, to improve the flow and clarity of the discussion.
A full set of worked solutions to the end of chapter exercises is now available in the book's Online Resource Centre.
The Chemistry Maths Book provides a complete course companion suitable for students at all levels. All the most useful and important topics are covered, with numerous examples of applications in chemistry and the physical sciences.
Taking a clear, straightforward approach, the book develops ideas in a logical, coherent way, allowing students progressively to build a thorough working understanding of the subject.
Topics are organized into three parts: algebra, calculus, differential equations, and expansions in series; vectors, determinants and
matrices; and numerical analysis and statistics. The extensive use of examples illustrates every important concept and method in the text, and are used to demonstrate applications of the mathematics in chemistry and several basic concepts in physics. The exercises at the end of each chapter, are an essential element of the development of the subject, and have been designed to give students a working understanding of the material in the text.
Online Resource Centre: The Online Resource Centre features the following resources: - Figures from the book in electronic format, ready to download - Full worked solutions to all end of chapter exercises
Readership:
Students of chemistry at all levels.
Erich Steiner, Honorary University Fellow and former senior lecturer at the University of Exeter, UK
Review(s) from previous edition
"It seems well suited both for its stated purpose and as a "brush-up" book for undergraduates, graduate students, and others. The mathematics are carried out briskly and with very little dressing ... there is much material to cover here and it works well through Steiner's particularly lucid presentation. The notation is standard and clear ... I am impressed with this book, I am sure that it will remain open on my desk and will become well worn in short order.
- C. Michael McCallum, University of the Pacific, Journal of Chemical Education, Vol. 74 No. 12 December 1997
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This book contains 25 full color transparencies, an activity worksheet for each transparency, and a teaching suggestions page for each transparency.
This book offers "a unique way to involve students in real life problem solving. Comprehensive Student Activity Masters and Teaching Suggestions provide a convenient way to connect mathematics with real-world situations."
Included for each chapter of the student text:
1) Two or more full color transparencies
2) Student Activity Master
3) Teaching Suggestions
This book is brand new and is in mint condition. The transparencies and worksheets are perforated and three-hole punched so that the pages can easily be removed and placed in a three ring binder. Comes from a smoke free home.
|
Intermediate Algebra, An Individualized Approach
Online Intermediate Algebra Overview
This course assumes a degree of proficiency with Beginning and Elementary Algebra. Each new topic is introduced with a brief review of the needed knowledge from earlier courses, but the review is intended only as a refresher.
The remainder of the course extends the topics of Elementary Algebra and begins a solid development of relations, functions, and their graphing.
Every objective is thoroughly explained and developed. Numerous examples illustrate concepts and procedures. Students are encouraged to work through partial examples. Each unit ends with an exercise specifically designed to evaluate the extent to which the objectives have been learned. The student is always informed of any skills that were not mastered.
Topics include:
simplifying radical expressions and fractions
rational number exponents
polynomials
equation solving
inequalities and absolute values
linear functions
quadratic functions and relations (the conics)
systems of equations
The instruction depends only upon reasonable reading skills and conscientious study habits. With those skills and attitudes, the student is assured a successful math learning experience.
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Find an Alpha, NJ Calculus combat this by helping my students see what the algebraic notation is describing (on a graph for example) so that it makes sense to them. In this way, graphing an inequality makes sense. Algebra 2 students are tasked with putting their previous knowledge to the test's available, they can experiment with math and science concepts and get instant feedback.
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The Math Forum Professional Development
The Math Forum is a community of mathematicians, teachers, and researchers working
together to improve math education. We recognize the opportunities and challenges
inherent in mathematics instruction and so, have developed courses and workshops to
support the teacher-student experience, many in conjunction with the Math Forum's popular
Problem of the Week.
The programs below are offered throughout the year and also may be customized to meet the
specific needs of educator groups. A complete schedule is available on the Math Forum's website.
Course Descriptions
The Math Forum's Problem-Solving Process
The course aligns well with the Math Forum's Problems of the Week but also could be used
to develop techniques to use with problem-solving prompts as well. Registration Information
Moving Students from Arithmetic to Algebra
This course examines a continuum of student work from the Math Forum's Problems of
the Week and how to move students' thinking along the continuum productively. Registration Information
Problem of the Week (PoW) Class Membership: Resources and Strategies for Effective Implementation
Designed for current PoW subscribers, this six-week course provides a basis of understanding allof the
features and resources associated with PoW membership. Registration Information
Problem Solving Strategies
Participants solve challenging middle school and high school algebra, geometry, and probability problems
and develop a supplemental curriculum online supporting the development of mathematical approaches to problems.
Registration Information
Teaching Math with the Problems of the Week
This course is designed for current subscribers of the Problem of the Week who want to make the most of their
membership. Course activities include submitting your own answers to and analyzing math in the Math
Fundamentals Problem of the Week, guiding your students through the solution and submission process,
and sharing ideas and reflections. Registration Information
Problem Solving in Geometry and Measurement, Course 1
This course provides teachers an opportunity to deepen their understanding of topics and student
learning in geometry and measurement and the problem solving process. Registration Information
Resources & Strategies for Effective Math in Context (MiC) Implementation
A sequence of four courses offered during the school year for each grade covered by the MiC
curriculum. (NOTE: A certain number of courses are available at no additional cost for schools using MiC.)
Registration Information
The Math Forum Online Workshops
Tools for Building Math Concepts
Explore how technology can help students develop fundamental concepts of multiplication,
fractions, division and area through the process of generating data and examining patterns. Registration Information
Technology Tools for Thinking and Reasoning about Probability
Investigate mathematics topics common to middle school curricula within the theme of probability.
Registration Information
Using Technology and Problem Solving to Build Algebraic Reasoning
Investigate mathematics topics common to middle school curricula within the theme of algebraic reasoning. Registration Information
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The book is intended to be an introductory text for mathematics and computer science students at the second and third year level in universities. It gives an introduction to the subject with sufficient theory for that level of student, with emphasis on algorithms and applications.
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It would really be great if you could tell us about a software that can offer both. If you could get us a resource that would give a step-by-step solution to our problem, it would really be good. Please let us know the genuine websites from where we can get the tool.
I remember having problems with dividing fractions, difference of squares and algebra formulas. Algebra Buster is a truly great piece of algebra software. I have used it through several math classes - Pre Algebra, Intermediate algebra and College Algebra. I would simply type in the problem from a workbook and by clicking on Solve, step by step solution would appear. The program is highly recommended.
Thanks for the details. I have purchased the Algebra Buster from and I happened to read through linear equations this evening. It is pretty cool and easily readable. I was attracted by the descriptive explanations offered on distance of points. Rather than being test preparation oriented, the Algebra Buster aims at educating you with the basic principles of Pre Algebra. The payment guarantee and the unimaginable discounts that they are currently giving makes the purchase particularly appealing
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Detailed Description of this Course Algebra seems hard to many students simply because every topic builds upon one another. What usually happens is that the student will understand the first few topics in the course, then fall behind once variables are introduced. After this point many students don't really understand the purpose of variables in algebra but continue moving along in the course until a point is reached where it is impossible to understand further topics with out a solid grasp of the basics.
The Algebra 1 Tutor series is designed assuming that you know absolutely nothing about Algebra. We begin at the very beginning of the sequence of topics with a review of fractions and exponents and gradually move through variables, expressions, and equations. What sets our tutorials apart is that every single topic is taught by showing fully worked example problems in a step-by-step fashion.
This technique has proven to be extremely effective and has helped thousands of students from around the world learn Algebra with our video lessons. What you'll find is that once you begin to understand the topics, your confidence will improve in Algebra. Then, instead of fearing the subject you'll truly begin to see that it is very logical, approachable, and that you can check your work on virtually every single problem.
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Precalculus
posted on: 10 Dec, 2011 | updated on: 29 Sep, 2012
Precalculus is a vast concept as it covers large number of topics and other concepts. Precalculus does not include topics from Calculus but it covers the topics and concepts which will be useful in calculus. Precalculus is studied from primary schools to research schools to help students in understanding important topics that will be used to solve problems in calculus. So we can say it teaches the basics of calculus.
Some important concepts of precalculus are:
• Curve – Plotting equations on a graph and the shape formed so is called the curve.
• Polar coordinates – This is the system of coordinates in which coordinates ate defined with the help of angles.
• Plane – Plane can be defined as a two dimensional surface which defines Linear Equations.
• Tangents – Tangent is a line which touches the outer surface of the Circle or circular object at exactly one Point.
• Complex Numbers – Complex numbers are the numbers which consist of two parts one is real part and other is imaginary part.
• Conic sections – Conic sections are the curves which are generated when a plane intersects or cuts the parts of a cone. There are different types of conic sections like ellipse, hyperbola, and Parabola.
• Logarithms – Logarithm can be defined as the power to which another number is raised to get another number. It contains base as 10.
• Natural Logs – These are the logarithms which have base 'e'.
• Functions – A function consists of variables with operations defined on them and these Functions depend on particular variables.
• Vectors – Vectors can be defined as quantity which has both magnitude and direction.
There are many other sub concepts of precalculus like Sets, real numbers, composite Functions etc.
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Westley Calculus...As the student to grasp the concept
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Engineering mathematics 4ed. - Solution manual
Engineering mathematics 4ed. - Solution manual
Engineering Mathematics is a comprehensive textbook for vocational courses and foundation modules at degree level. John Bird's approach, based on numerous worked examples supported by problems, is ideal for students of a wide range of abilities, and can be worked through at the student's own pace. Theory is kept to a minimum, placing a firm emphasis on problem-solving skills, and making this a thoroughly practical introduction to the core mathematics needed for engineering studies and practice.
The book presents a logical topic progression, rather than following the structure of a particular syllabus. However, coverage has been carefully matched to the two mathematics units within the new BTEC National specifications, and AVCE specifications. New sections on Boolean algebra, logic circuits matrices and determinants have been added to ensure full syllabus match.
Includes: 900 worked examples, 1700 further problems, 234 multiple choice questions (answers provided), and 16 assessment papers - ideal for use as tests or homework. These are the only problems where answers are not provided in the book. Full worked solutions are available to lecturers only as a free download from
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Product Details:
Advantage provides an interactive learning experience and the tools students need to gain learning confidence and improve their grades. Help your child build a solid academic foundation with High School Advantage a complete student resource center that combines lessons exercises and quizzes with additional learning resources to support and motivate your child in reaching his or her full academic potential.The ADVANTAGE Approach to Effective Learning:. ENGAGE: Interactive learning experience engages the student.. REINFORCE: Standards-driven lesson plan reinforces classroom learning.. SUPPORT: Study aids & reference tools support the learning experience.. MOTIVATE: Brain-building games add fun to studying.Student Benefits. Learn At Your Own Pace.. Overcome Challenging Topics.. Review Lessons As Needed.. Prepare For College.MathSolidify critical math skills and prepare for college with detailed lessons that simplify complex concepts theories and equations.. Algebra II. Geometry & Trigonometry.. Plus: Calculus iPod study materials.ScienceEstablish a thorough understanding of all areas in Science with the help of multimedia presentations interactive lessons and practice exercises.. Biology.. Chemistry.. Physics.Social StudiesStudy landmark historical events to understand their impact on the development of the contemporary United States. Learn basic supply and demand principles and how they affect the marketplace.. US History.. US Government.. Economics.EnglishDevelop an aptitude for organizing and presenting information in a clear and concise manner whether on a research paper college essay or cover letter. . Research.. Writing.. Presenting.Foreign LanguageBuild and perfect basic conversational skills in Spanish French Ger
Description:
Interactive Learning...Individual Training Product Information Learning Microsoft Windows XP is
the fastest and easiest way to install use and enjoy the power of Windows XP. Learn the basics of set up and personalizing the desktop. Master the new look ...
Description:
Help your child build Math skills with fun and engaging
activities. Packed with lessons and exercises from Elementary School to High School, each designed to prepare students for success in state standards testing. Build fundamental skills with Basic Math, ...
Description:
High School Advantage® 2008 was specially developed to supplement classroom
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8th Grade
8th Grade
I am a student going into 8th grade. I would like to know what I should review so I can have a headstart before the school year.
Thanks.
July 19, 2009, 2:27 pm by Riemer111
Well im a freshman. so i know what you will need to know. Well your going to need to know Ordered pairs and tables as functions. And your going to learn how to find solutions. than you will need to know how to find intercepts from graphs stateing the x-intercept and the y-intercept of each line.Then there will be a lesson that talks about rise and run, well your going to use rise and run to find slope.and just remember there are 4 types of slope there is positive slope,negative slope,zero slope and undefined slope. Then you will have to know how to find a rate of change and compare rate of change. Than its going to get harder you will have to learn how to find the slope in the y-intercept. Then it will get a little easy you will learn how to make predictions from best fit lines. which aint all that hard. Then there is this thing in 8th grade called solve by graphing which is easy.
July 23, 2009, 9:53 pm by Guest
well my opinion is that u should review division of exponents, because when u get there it will get harder.
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Description
For combined differential equations and linear algebra courses teaching students who have successfully completed three semesters of calculus.
This complete introduction to both differential equations and linear algebra presents a carefully balanced and sound integration of the two topics. It promotes in-depth understanding rather than rote memorization, enabling students to fully comprehend abstract concepts and leave the course with a solid foundation in linear algebra. Flexible in format, it explains concepts clearly and logically with an abundance of examples and illustrations, without sacrificing level or rigor. A vast array of problems supports the material, with varying levels from which students/instructors can choose.
CourseSmart textbooks do not include any media or print supplements that come packaged with the bound book.
Table of Contents
1. First-Order Differential Equations
2. Matrices and Systems of Linear Equations
3. Determinants
4. Vector Spaces
5. Linear Transformation
6. Linear Differential Equations of Order n
7. Systems of Differential Equations
8. The Laplace Transform and Some Elementary Applications
9. Series Solutions to Linear Differential Equations
Appendices
A. Review of Complex Numbers
B. Review of Partial Fractions
C. Review of Integration Techniques
D. Linearly Independent Solutions to x2yn + xp(x)y1 + q(x)y = 0
E. Answers to Odd-Numbered
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Synopsis
A chief requirement in the study of relativity is absolute differential calculus, which Einstein used to mathematically develop his ideas. This classic was written by a founder in the field, offering a clear, detailed exposition. It examines introductory theories, the fundamental quadratic form and the absolute differential calculus, and physical applications. 1926
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What is SINGAPORE MATH WIKIPEDIA?
In the United States, ' Singapore Math is a teaching method based on the primary textbooks and syllabus from the national curriculum of Singapore. These textbooks have a consistent and strong emphasis on problem solving and model drawing, with a focus on in-depth understanding of the essential ...
I know Singapore Math is frequently championed by traditional educators so I thought I better explain this problem on the talk page before making changes. (Believe it or not, some reform educators think highly of Singapore math too.) --seberle 12:58, 18 November 2009 (UTC) I have ...
The Singapore Mathematical Olympiad (SMO) is a mathematics competition organised by the Singapore Mathematical Society. It comprises three sections, Junior, Senior and Open, each of which is open to all pre-university students studying in Singapore who meet the age requirements for the ...
There are four standard subjects taught to all students: English, the mother tongue, mathematics, and science. Secondary school lasts from four to five years and is divided between Special, Express, Normal (Academic), and Normal ...
Singapore Math is a general term used in U.S. classrooms to describe math teaching strategies and materials modeled after the math curriculum used in Singaporean schools. If you're wondering about the philosophy and history of the Singapore Math method, here's some important information.
Singapore Mathematics is a an approach to mathematics education. This curriculum has generated considerable interest among Western APEC members because of Singapore's high performance on international assessments and the availability of Singapore's mathematics textbooks in English.
Kerrums: Singapore Math is the country of Singapore's national math curriculum. It covers primary and secondary school. Singapore is first in the world for math and second in the world for science according to the Third International Mathematics and Science study in 1999.
What Is Singapore Math? Singapore Math emphasizes the development of conceptual understanding prior to the teaching of procedures. A powerful, hands-on, visual approach—a progression from concrete to pictorial to abstract—is used to introduce concepts, which at the core include strong number ...
Answer (1 of 2): Singapore math, as a program, has a consistent and strong emphasis on problem solving. Other elements that contribute to the program's success include the program's focus on and support for building skills, concepts, and processes and its attention to developing students ...
The Puget Sound region has been involved in a series of math wars as various groups of stakeholders debate how math will be taught in school districts. In The Math Kerfuffle, Another Reason Washington Needs Charter Schools I said: There will continue to be battles between those who ...
I feel for the teachers and administrators who must tolerate hyper-competitive parents in the hyper-competitive Bay Area. Having lived here for more than 10 years, I have had my fair share of debates over curriculum, private vs. public school and the finer points of having my preschooler learn ...
Geometry is a strand that appears in Singapore's Primary Mathematics Syllabus. Singapore's approach to mathematics education has become important to many Western APEC members because their students perform well on international assessments.
A number bond is a mental picture of the relationship between a number and the parts that combine to make it. The concept of number bonds is very basic, an important foundation for understanding how numbers work.
What do two eights equal in math? 8+8=16 What are the eight steps to escape from vorkuta? 1: Secure the Keys 2: Ascend from the Darkness 3: Rain Fire 4: Unleash the Horde 5: Skewer the Winged Beast 6: Weild a Fist of Iron 7: Raise Hell 8: Freedom
This blog has been created to provide a means to communicate and collaborate with parents and community members as we strive to provide a mathematics education in Warsaw Community Schools that will allow each student to rise to their own excellence.
Elementary students at Reynolds School District share, in their own words, how new strategies learned during their first year using Math in Focus™, have given them confidence and a new appreciation for mathematics.
Recently, a day-long seminar featuring Newark Mayor Cory Booker was exclusively organized for administrators of Newark Public Schools. The seminar was an opportunity for local administrators to interact with leading educators from Singapore and gain a deeper understanding of the concepts and ...
Did you know that the tiny country of Singapore scores highest in math education? It's true. Every four years a prestigious study is done that's based in Massachusetts. It's called the TIMSS. Every four years this highly regarded study takes a look at math...
With a click of a stopwatch, 24 sixth graders were off – racing down a worksheet, figuring out as many basic subtraction problems as they could in 60 seconds.When the "sprint" ended, teacher Bill Davidson called out the answers in rapid-fire bursts: "6.1, 4.3, 2.2. . . .
Closing the Achievement Gap: Teaching Content ... The following guest post is from Barry Garelick, co-founder of the U.S. Coalition for World Class Math, an education advocacy organization that addresses mathematics education in U.S. schools.
Answer: > The Bar Model method requires students to draw diagrams in the form of rectangular bars to represent known and unknown quantities, as well as the relationships between these quantities. > > The example above would be used to solve a word problem such as: Mr. Lim read 10 pages from
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Pre-Calculus Homework Help
Pre-calculus is an interesting area of math for students because of its multi-purpose nature. It reviews previously learned topics like trigonometry, introduces new topics like matrices and determinants, and prepares students for a formal course in calculus for the following year.
Typical content for a regular pre-calculus course includes:
functions and their graphs
polynomial and rational functions
exponential and logarithmic functions
trigonometry
systems of equations and inequalities
matrices and determinants
sequences, series, and probablility
While all these topics are important, it is fair to say that the topic of exponential and logarithmic functions is the most widespread in various fields of study, from business math to theoretical physics. As an interesting example, consider the following equation where we would like to solve for x:
ex + 2e-x = 3
A quick review of three basic logarithm rules:
product rule: ln (ab) = ln a + ln b
quotient rule: ln (a/b) = ln a - ln b
power rule: ln an = n ln a
shows that none will be helpful here. One technique for solving this equation is to multiply both sides by ex. The result is:
e2x + 2 = 3ex
After subtracting 3ex from both sides, we have:
e2x - 3ex + 2 = 0
After careful consideration of this equation, you may recognize it as a quadratic, with ex taking the place of x. We can therefore factor this just as we would any quadratic equation:
(ex-2)(ex-1) = 0
Setting each factor equal to zero, we have:
ex-2 = 0 and ex-1 = 0
Adding 2 and adding 1 to the left and right equations, respectively gives:
ex = 2 and ex = 1
Now, finally, in each case we can take the natural log of both sides:
ln ex = ln 2 and ln ex = ln 1
Remembering that ln ex = x, we have our two solutions:
x = ln 2 and x = 0
This example has served to review some of the basic properties of logarithms and to illustrate an early creative twist that was necessary for solving the given equation.
To fulfill our mission of educating students, our online tutoring centers are standing by 24/7, ready to assist students who need extra practice in pre-calculus.
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Graphing
Author:
ISBN-13:
9780768202335
ISBN:
0768202337
Publisher: Schaffer Publications, Frank
Summary: Help students succeed in math! Math Minders provide students with the self-confidence they need to succeed in math. Students learn one step at a time, reviewing skills learned in earlier grades, then moving to skills appropriate for their grade level. They progress gradually, giving them the constant feeling of success! Vocabulary is kept at a level appropriate for each grade level to help ensure success. Fun and sim...ple formats help maintain a high level of student interest. Perfect for home or school, or to reinforce any existing math program.[read more]
Rating:(0)
Ships From:Denver, COShipping:StandardComments: 0768202337 Top of book wavy, as if was in humid environment, but no stains and covers are shiny ... [more] 07. [less]
0768202337 Top of book wavy, as if was in humid environment, but no stains and covers are shiny and bright. Binding tight, pages crisp and clean. Some dents, light scuffs and [more]
07.[less]
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Courses in Mathematics (Division 428)
Only open to designated summer half-term Bridge students. IIIb. (2 in the half-term). (Excl).
Review of elementary algebra; rational and quadratic equations; properties of relations, functions, and their graphs; linear and quadratic functions; inequalities, logarithmic and exponential functions and equations. Equivalent to the first year of Math. 105/106.
Students with credit for Math. 103 can elect Math. 105 for only 2 credits. No credit granted to those who have completed any Mathematics course numbered 110 or higher. (4). (MSA). (QR/1).
ThisSee Elementary Courses above. Enrollment in Math. 110 is by recommendation of Math. 115 instructor and override only. No credit granted to those who already have 4 credits for pre-calculus mathematics courses. (2). (Excl).
A condensed half-term version of Math 105. Offered as a self-study course through the Math Lab and directed toward students who are unable to complete a first calculus course successfully. Students study on their own and consult with tutors in the Math Lab whenever needed.
Four years of high school mathematics. See Elementary Courses above. Credit usually is granted for only one course from among Math. 112, 115, 185, and 295. No credit granted to those who have completed Math. 175. (4). (MSA). (BS). (QR/1).
This course presents calculus from three points of view: geometric (graphs); numerical (tables); and algebraic (formulas). Students develop their reading, writing, and questioning skill. Topics include functions and graphs, derivatives and their application to real-life problems in various fields, and definite integrals.
Three years of high school mathematics including a geometry courseThis course introduces students to the ideas and some of the basic results in Euclidean and non-Euclidean geometry. The course begins with an historical perspective of how the ancient Greeks influenced the study of geometry prior to the 19th century. Then notions of non-Euclidean geometry and geometry in higher dimensions are introduced and studied.
High school mathematics through at least Analytic GeometryDesigned for non-science concentrators and students with no intended concentration who want to learn how to think mathematically without having to take calculus first. Students are introduced to the ideas of Number Theory through lectures and experimentation by using software to investigate numerical phenomena, and to make conjectures that they try to prove.
Math. 112 or 115. No credit granted to those who have completed a 200- (or higher) level mathematics course. (3). (MSA). (BS).
An introduction to the mathematical concepts and techniques used by financial institutions. Topics include rates of simple and compound interest and present and accumulated values; annuity functions and applications to amortization, sinking funds and bond values; depreciation methods; introduction to life tables, life annuities and life insurance value.
Permission of the Honors advisor. Credit is granted for only one course from among Math. 112, 113, 115, 185, and 295. No credit granted to those who have completed Math. 175. (4). (MSA). (BS). (QR/1).
Topics covered include functions and graphs, derivatives, differentiation of algebraic and trigonometric functions and applications, definite and indefinite integrals and applications. Other topics are included at the discretion of the instructor.
Permission of the Honors advisor. Credit is granted for only one course from among Math. 114, 116, 119, 156, 176, 186, and 296. (4). (MSA). (BS). (QR/1).
Topics covered include transcendental functions, techniques of integration, introduction to differential equations, conic sections, and infinite sequences and series. Other topics included at the discretion of the instructor.
Math. 115 and 116. Credit can be earned for only one of Math. 214, 217, 417, or 419. No credit granted to those who have completed or are enrolled in Math. 513. (4). (MSA). (BS).
This course is an alternative to Math 216 (Differential Equations) in that it emphasizes linear algebra including the geometry of two, three and n-dimensional space, and has a lighter treatment of differential equations. It is particularly designed for students who are planning to take a course in linear programming.
Math. 116, 119, 156, 176, 186, or 296. Credit can be earned for only one of Math. 215, 255, or 285. (4). (MSA). (BS). (QR/1).
Topics software.
Math. 116, 119, 156, 176, 186, or 296. Credit can be earned for only one of Math. 216, 256, 286, or 316. (4). (MSA). (BS).
MathFocuses on the development of mathematical problem-solving skills. Problems are taken from classical analysis, elementary number theory, and geometry. For students with outstanding problem-solving ability.
Prior knowledge of first year calculus and permission of the Honors advisor. Credit is granted for only one course from among Math. 112, 113, 115, 185, and 295. (4). (MSA). (BS). (QR/1).
Introduction to mathematical analysis with emphasis on proofs and theory. Covers such topics as set theory, construction of the real number field, limits of sequences and functions, continuity, elementary functions, derivatives and integrals. Additional topics may include countability, topology of real numbers, infinite series, uniform continuity.
This course follows the historical evolution of three fundamental mathematical ideas, in geometry, analysis and algebra – Euclid's parallels postulate and the development of non-Euclidean geometries, the notion of limit and infinitesimals, and the development of the theory of equations culminating with Galois theory.
Math. 385 and enrollment in the Elementary Program in the School of Education. (1-3). (Excl). (EXPERIENTIAL). May be repeated for a total of three credits.
An experiential mathematics course for elementary teachers. Students would tutor elementary (Math. 102) or intermediate (Math. 104) algebra in the Math. Lab. They would also participate in a weekly seminar to discuss mathematical and methodological questions.
Formulation and solution of some of the elementary initial- and boundary-value problems relevant to aerospace engineering. Application of Fourier series, separation of variables, and vector analysis to problems of forced oscillations, wave motion, diffusion, elasticity, and perfect-fluid theory.
A survey course of the basic numerical methods which are used to solve scientific problems. In addition, concepts such as accuracy, stability and efficiency are discussed. The course provides an introduction to MATLAB, an interactive program for numerical linear algebra as well as practice in computer programming.
Math. 215, 255, or 285; and 217. No credit granted to those who have completed or are enrolled in 512. Students with credit for 312 should take 512 rather than 412. One credit granted to those who have completed 312. (3). (Excl). (BS).
Sets, functions (mapping, relations, and the common number systems (integers to complex numbers). These are then applied to the study of groups and rings. These structures are presented as abstractions from many examples. Notions such as generator, subgroup, direct product, isomorphism and homomorphism.
Not open to freshmen, sophomores or mathematics concentrators. (3). (Excl). (BS).
Review of algebra, functions, and graphs followed by an intuitive introduction to the rudiments of calculus. Applications of differentiation and integration to problems in the social sciences. Designed especially for seniors and graduate students with minimal mathematics background.
Four terms of college mathematics beyond Math. 110. Credit can be earned for only one of Math. 214, 217, 417, or 419. No credit granted to those who have completed or are enrolled in Math. 513. I and II. (3). (Excl). (BS).
Finite dimensional linear spaces and matrix representation of linear transformations; bases, subspaces, determinants, eigenvectors, and canonical forms; and structure of solutions of systems of linear equations. Applications to differential and difference equations. The course provides more depth and content than Mathematics 417. Mathematics 513 is the proper election for students contemplating research in mathematics.
We explore how much insurance affects the lives of students (automobile insurance, social security, health insurance, theft insurance) as well as the lives of other family members (retirements, life insurance, group insurance). While the mathematical models are important, an ability to articulate why the insurance options exist and how they satisfy the consumer's needs are equally important. In addition, there are different options available (e.g., in social insurance programs) that offer the opportunity of discussing alternative approaches.
Rates used in compound interest theory; annuities-certain and their application to amortization, sinking funds, and bond values; and introduction to life annuities and life insurance. Both the discrete and the continuous approach are used.
The development of employee benefit plans, both public and private. Particular emphasis is laid on modern pension plans and their relationships to current tax laws and regulations, benefits under the federal social security system, and group insurance.
Investigation of Euclidean geometry based on the Birkhoff or SMSG metric axiom system and reference to contemporary high school texts. Comparison with synthetic Euclid-Hilbert foundations. Historical development of absolute and hyperbolic geometry. Other non-Euclidean geometries. New directions in high school geometry, transformation groups especially isometries of the Euclidean plane as generated by reflections, similarities, and affine transformations.
In the first third of the course the notion of a formal language is introduced and propositional connectives, tautologies and tautological consequences are studied. The heart of the course is the study of 1st order predictive languages and their models. New elements here are quantifiers. Notions of truth, logical consequences and probability lead to completeness and compactness. Applications.
One year of high school algebra. No credit granted to those who have completed or are enrolled in 385. I and IIIb. (3; 2 in the half-term). (Excl). (BS). May not be included in a concentration plan in mathematicsMath. 385 or 485. May not be used in any graduate program in mathematics. (3). (Excl).
The second course in a two-course sequence required of elementary teaching certificate candidates. Topics covered include: the real-number system, probability, and statistics, geometry and measurement.
Math. 489. (3). (Excl). (BS). May be repeated for a total of six credits.
Selected topics in geometry, algebra, computer programming, logic, and combinatorics for prospective and in-service teachers of elementary, middle, or junior high school mathematics. Content may vary from term to term.
At least two 300 or above level math courses, and graduate standing; Qualified undergraduates with permission of instructor only. (1). (Excl). (BS). Offered mandatory credit/no credit. May be repeated for a total of 6 credits.
Math 501 is an introductory and overview seminar course in the methods and applications of modern mathematics. The seminar has two key components: (1) participation in the Applied & Interdisciplinary Math Research Seminar; and (2) preparatory and post-seminar discussions based on these presentations. Topics vary by term.
Topics include utility theory, application to buying general insurance to reduce risk, compound distribution models for risk portfolios, application of stochastic processes to the ruin problem and to reinsurance.
Isometries and congruences in the Euclidean plane as generated by reflections, translations, and half-turns. Tilings, affine and hyperbolic geometries, and Poincaré model of the hyperbolic plane. Selected applications to ornamental design, crystallography, and regular polytopes.
The course covers topics in discrete and applied geometry which change from year to year. Possible topics include: crystals and quasi-crystals; best packing of spheres and applications; convex geometry and optimization problems; geometric combinatorics and applications in computer science.
Centers on the construction and use of agent-based adaptive models. Course begins with classical differential equation and game theory approaches, and then focuses on the theory and application of particular models of adaptive systems such as models of neural systems, genetic algorithms, classifier systems, and cellular automata.
Intended primarily for students of engineering and of other cognate subjects. Doctoral students in mathematics elect Mathematics 596. Complex numbers, continuity, derivative, conformal representation, integration, Cauchy theorems, power series, singularities, and applications to engineering and mathematical physics.
A study of some of the differential equations of mathematical physics and methods for their solution. Separation of variables for heat, wave, Laplace's and Schrödinger's equations; special functions and their integral representations and asymptotic properties; and eigenvalues as solutions of variational problems.
Elementary distributions, Green's functions and integral solutions for nonhomogeneous problems, Fourier and Hankel transforms, and Fredholm alternative and elementary methods of solution of integral equations, with additional topics as time permits.
Formulation of problems from the private and public sectors using the mathematical model of linear programming. Development of the simplex algorithm; duality theory and economic interpretations. Postoptimality (sensitivity) analysis and applications and interpretations. Introduction to transportation and assignment problems and special purpose algorithms and advanced computational techniques. Students have an opportunity to formulate and solve models developed from more complex case studies and to use various computer programs.
The main topics are set algebra (union, intersection), relations and functions, orderings (partial-, linear-, well-), the natural numbers, finite and denumberable sets, the Axiom of Choice, and ordinal and cardinal numbers.
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Unfortunately,
for much of this subject, there is no substitute for brute memorization.
This course is often used as pre-calculus. "Algebra masters"
will have a much smaller memorization load than other students.
[I mean College Algebra, not the higher-math version.] Much
of this subject is highly dependent on either calculators+ or
tables [slide rules have been passé since the electronic calculator,
but could be used.] Traditionally, this subject relies on some
familiarity with "College Algebra", but not an intensive mastery.
Note: in
many places, I will mention algebraic identities. For serious
students who are either algebra-deficient or have to use the
material under time pressure, I highly recommend working through
"all of the plausible forms" and memorizing them. This usually
means algebraic isolation of various variables in the equations,
and figuring out how to use multiple equations in one problem.
Currently, I am not providing the homework that is required
to learn the material. I would like to partially remedy this
deficiency. [No timetables!]
Note: There
are several kludge notations lurking in this crash review:
SQRT
for "square root" (correctly
notated for 2, 3, 5, and 6)
pi
for p (correctly notated outside
of popup text for pictures)
I will remove
instances of these kludge notations, without advance warning,
when it is convenient in my schedule to do so.
These functions
have a single argument, and take angles. By the heuristic of
"wrapping the real number line around the unit circle," we can
think of these functions as taking real numbers as arguments.
Note that in "stratosphere" math, the above approach is very
painful; other techniques are used to define these functions.
These techniques are the ones which calculators are programmed
with -- but they're not appropriate for this section.
Incidentally,
there are three angle units listed on my HP-48SX: degrees, radians,
and gradients. [Gradients are not commonly used in the U.S.A.]
To rebuild the conversion formulae, remember:
1 circle
= 360 degrees = 2p radians = 400 gradients.
We inherited
degrees from Babylonia. There are 60 minutes in a degree,
and 60 seconds in a minute. [When dealing with the Earth's
rotation in astronomy, the Earth rotates "about" 1 angular
degree in 4 solar-day minutes. The Earth rotates 1 degree
in 4 sidereal-day minutes.] Also, a nautical mile is "about"
1 arc-minute on the Earth's surface. This would be exact
if the Earth was a perfect sphere. A common notation for,
say, 3 degrees 4 minutes 5 seconds is 3°4'5".
Radians
are the "natural" angle unit when dealing with "stratosphere"
math. The defining formulae for sin and cos are much simpler
then. Radians also behave better if you have to deal with
calculus [deriviatives and integrals].
I don't
know much about the history of gradients.
A right
angle is 1/4th of a circle; i.e. 90°, p/2 radians, or 100 gradients.
I will use the common symbol
for a right angle, in the rest of this crash review. An angle
larger than 0, but smaller than a
, is called an acute
angle. An angle larger than a
, but smaller than half a circle, is called an obtuse angle.
is a standard symbol in the literature (although found more
often in geometry texts than trigonometry texts). When it shows
up in a diagram, it designates the labeled angle as a right
angle.
For the
record: Triangles are constructed by picking three points, then
joining each pair by a straight line segment. We assume we can
neglect curvature of the surface. That is, we assume the geometry
is Euclidean. [You will not get the same kind of triangles on
a sphere as on a flat desk or piece of paper. The sphere is
visibly curved, the desk and the flat piece of paper are not
supposed to be curved.] I will occasionally use the fact the
geometry is Euclidean.
The sum
of the three (interior) angles of a triangle is half a circle,
i.e. 180° or p radians or 200 gradients. I will sometimes denote
this by 2
in the rest of this crash review; this is *not* standard notation.
An implicit exercise will be translating 2
to angular measure [degrees, radians, and/or gradients.].
If we pick
a point on a straight line, the two parts of the straight line
formed by taking out the point form an angle of half a circle,
i.e. 180° i.e. p radians i.e. 200 gradients. In general, I will
omit "interior" when talking about angles of a triangle.
We say a
triangle, that contains a right angle, is a right triangle.
If a triangle has two sides of equal length and one side of
different length, we call it an isosceles triangle. If a triangle
has three equal sides, we say it is an equilateral triangle.
Note:
For
an isosceles triangle, the two angles opposite the sides
with equal length (i.e., the two angles whose bounding sides
have one of the two sides of equal length, and the side
with different length) are equal.
For
an equilateral triangle, all three angles are equal, and
equal 60° i.e. p/3 radians i.e. 200/3 gradients.
The angle
measure in radians is simply arc length on a circle of radius
1. In general, for a circle with radius R and an arc with central
angle X in radians,
(arc
length)=RX.
There are
two pictures to keep in mind here, the generic right triangle:
First of
all, all physical angles have some size. We cannot visualize
an angle with negative physical size. They do not exist in anything
sufficiently similar to the physical space we live in.
However,
(especially when dealing with the unit circle), it is often
convenient to measure angles in a specific direction: counterclockwise.
[I'm not going to digress into "stratosphere" math now, but
that is dictating the convention here.]
Then, a
negative sign means we are measuring the angle "unconventionally",
i.e. clockwise.
This will
simplify the use of some of the trigonometric identities we
are going to look at.
Why
should I know the generic right triangle?
Given
a right triangle, the trig function values for the two acute
angles [angles smaller than a right angle] can be computed without
knowing the angles. I prefer to remember the formulae this way
[X is an angle]:
A mnemonic
for the formulae for sin(X), cos(X), and tan(X) is the (fictitious)
Indian Chief Soh-cah-toa, who had no problems with this part
of trigonometry. [I got this from Mr. Coole, a long time ago
-- I was in grade school then.]
The way
I wrote the formulae, above, emphasizes the following identities:
sin(X)csc(X)=1
cos(X)sec(X)=1
tan(X)cot(X)=1
These identities
do work hold when both functions involved are defined for the
angle X, regardless of size.
Another
fact, of some use, is the Pythagorean theorem: H²=A²+O². If
we relabel A as a, O as b, and H as c, we get the familiar form
of the Pythagorean theorem: c²=a²+b². This *only* works for
right triangles. The generalization of the Pythagorean theorem to non-right
triangles is called the law of cosines.
A common
strategy is to memorize how to compute tan(X), cot(X), sec(X),
and csc(X) in terms of sin(X) and cos(X), and then reduce everything
to this. This is not necessarily the least painful way to do
a trig problem, but it is often more important to start the
problem, than figure out how to do it elegantly. [EXERCISE:
derive this from the A, H, O formulation for acute angles. The
formulae work for arbitrary angles.]
The generic
right triangle also motivates some terminology [which we inherited
from twelfth century Arabian mathematics].
First of
all, we say two angles are complementary if they
add up to a right angle. That is, for an angle X, we say -X
is complementary to X. For instance,
the angle
in the upper right-hand corner is complementary to, i.e. the
complement of, the marked angle in the lower left-hand corner.
A related
piece of terminology is supplementary angles:
two angles are said to be supplementary if they add up to half
a circle, i.e. 180° i.e. p radians i.e. 200 gradients. That
is, 2
- X is supplementary to the angle X. To get an intuition for
this, draw a unit circle with the horizontal axis.. Draw an
arbitrary radius from the center to somewhere on the unit circle.
The two angles formed between the radius, and the horizontal
axis, are supplementary.
We say that
sin and cos [sine and cosine], tan and cot [tangent and cotangent],
and sec and csc [secant and cosecant] are cofunctions, and that
the trig function of the complement of an angle X is equal to
the trig cofunction of the angle. This is explicitly in the
function names: cosine is the cofunction of sine, cotangent
is the cofunction of tangent, and cosecant is the cofunction
of secant.
These identities
do work for arbitrary angles. If one side is undefined, both
sides are undefined.
What
are the reference triangles?
The reference
triangles are right triangles that are "easy to construct".
They provide easily-memorized values for the angles with measure
30°, 45°, and 60°, i.e. p/6, p/4, and p/3 radians, i.e. 100/3,
50, and 200/3 gradients.
There are
two reference triangles. They [in the U.S.A.] are known by their
degree names:
The 45-45-90
(degree) right triangle can be constructed from a square with
sides of length one. The hypotenuse is the line connecting two
opposite vertices of the square; it has length by the
Pythagorean Theorem.
[Exercise: compute this!] The legs of the triangle are sides
of the square. Note: 45° is its own complementary angle.
The 30-60-90
triangle is constructed from an equilateral triangle with sides
of length 2. We put one side on the horizontal axis, and bisect
the angle opposite this side. [Note that an equilateral triangle
has three vertex angles, all of which are 60°.]
We now
have a hypotenuse of length 2, one leg of length 1 [from bisecting
the horizontal side], and one side with length [from
the bisecting line].
We get:
sin(30°)=cos(60°)=½
cos(30°)=sin(60°)=/2
tan(30°)=cot(60°)=1/=/3
cot(30°)=tan(60°)=
sec(30°)=csc(60°)=2/=(2)/3
csc(30°)=sec(60°)=2
[Again,
do it.]
Why
should I know the unit circle?
The unit
circle provides a picture on which to memorize reference values
of the trig functions.
Think of
it as the set of possible endpoints for a length 1 hypotenuse
H, with one endpoint of H fixed at the origin. We can construct
generic right triangles with hypotenuse 1 in it. Pick a point
on the circumference, draw a line segment from it to the origin,
and then draw a perpendicular line segment down to the x-axis.
Notice that
the coordinates of the vertex, in Cartesian coordinates, is
(cos(X), sin(X)), where X is the central angle. The horizontal
side (on the x-axis) is A, and the vertical side (parallel to
the y-axis) is O. The radius (length 1) is H. The slope of the
hypotenuse H is tan(X).
However,
observe the four quadrants. Our example triangle has its hypotenuse
in the upper-right quadrant [both coordinates positive]. Horizontal
and vertical hypotenuses create line segments rather than triangles.
Also, at least one of the coordinates go negative in the other
three quadrants [upper-left, lower-left, lower-right].
We proceed
by assuming that the coordinates of the vertex, in Cartesian
coordinates, is (cos(X), sin(X)), regardless of where the vertex
is on the unit circle. This immediately leads us to the Pythagorean
identities:
sin²(X)+cos²(X)=1
[standard; if you memorize only one, learn this one]
tan²(X)+1=sec²(X) [divide by cos²(X)]
1+cot²(X)=csc²(X) [divide by sin²(X)]
If we interpret
undefined as equal to undefined, these identities hold for arbitrary
angles.
Incidentally,
the computation of the slope of the hypotenuse H via tan(X)
also works for arbitrary angles. [Recall that an undefined slope
corresponds to a vertical line].
The next
point is that the signs of the various trigonometric functions
are controlled by the quadrant the function is evaluated in.
We number the quadrants I through IV [Roman numerals] counterclockwise,
as follows:
Now, the
trigonometric function signs are controlled by the quadrants
as follows:
As should
be clear, the boundaries between the quadrants do not behave
this way. In fact, they are reference values for the trigonometric
functions. [For ease of memorization, I will put all of the
reference values for the first quadrant and its boundaries in
one place, elsewhere in this document.]
Now, let's
look at another way to reconstruct the cofunction identities
i.e. the identities relating complementary angles.
What happens
if we reflect the unit circle about the line y=x? We are swapping
the x and y coordinates. That is, (letting X be the central
angle), we are swapping sin(X) and cos(X). We are also physically
reflecting the central angle X to the angle -X.
So, we find that (here are the cofunction identities again):
It is no
coincidence that the slope of the line y=x is 1. This gives
a central angle between the line y=x, and the x-axis, of 45°
i.e. p/4 i.e. 50 gradients [think of the 45-45-90 triangle];
note that this is exactly half of a right angle.
Now, let
us consider reflecting the unit circle about the x-axis:
Note that
the x-coordinate [cos(X)] is unaffected, while the y-coordinate
[sin(X)] is negated. The resulting point is the point we get
from rotating through the angle -X from the angle 0. This gives
the following formulae:
sin(-X)=-sin(X)
"sin(X) is an odd function"
cos(-X)=cos(X) "cos(X) is an even function"
tan(-X)=-tan(X) "tan(X) is an odd function"
cot(-X)=-cot(X) "cot(X) is an odd function"
sec(-X)=sec(X) "sec(X) is an even function"
csc(-X)=-csc(X) "csc(X) is an odd function"
[Exercise:
derive the last four [tan, cot, sec, and csc versions] from
the first two [sin and cos versions.]
These formulae
classify the trigonometric functions into even functions and
odd functions. As you may recall from College Algebra, even
functions are those functions whose value is unchanged by negating
the argument, and odd functions are those functions whose value
is negated by negating the argument.
It is no
coincidence that the slope of the x-axis is 0. The central angle
of the x-axis with the x-axis is clearly 0 [degrees, radians,
gradients, it matters not which unit]. 0 is also exactly half
of 0.
Now, let
us consider reflecting the unit circle about the y-axis:
Note that
the y-coordinate [sin(X)] is unaffected, while the x-coordinate
[cos(X)] is negated. The resulting point is the point we get
from rotating through the angle -X from the angle 0. This gives
the following formulae:
[Exercise:
derive the last four (tan, cot, sec, and csc versions) from
the first two (sin and cos versions.)]
It is no
coincidence that the slope of the y-axis is undefined. The central
angle of the y-axis with the x-axis is clearly .
This is also exactly half of 2 .
Now, we
can directly compute the reference values of cos and sin for
45° i.e. p/4 radians i.e. 50 gradients, i.e. ½
, without using the 45-45-90 reference triangle. 45° is the
angle that is its own complementary angle. Thus,
i.e. [solve
for cos(45°); we know we need the positive root because 45°
is in the first quadrant, so I can omit ±]
cos(45°)
= 1/ =
/2
We can also
directly compute the reference values of cos and sin for 60°,
i.e. p/3 radians, i.e. 200/3 gradients, i.e. [2/3]
. To do this, we first consider inscribing an equilateral hexagon
in the unit circle:
[Just
like an equilateral triangle, an equilateral hexagon has equal
lengths for all of its sides. In general, an equilateral polygon
(regardless of how many sides it has) has equal lengths for
all of its sides.]
Note that
the inscribed hexagon can be considered as the 'splicing together'
of six equilateral triangles.
[If we draw
the line segments from the vertices to the center of the inscribed
hexagon, since the sides have the same length, the angles as
viewed from the center will have the same size. 360/6° = 60°.
(The prior two sentences use Euclidean geometry). We know that
the triangle we just created is isosceles, (two of its sides
are radii), so the two angles opposite the radii have the same
size, and add up to 120°: they are both 60°. All three angles
are equal. Thus all three sides of the triangle have equal length.
(The very last sentence also uses Euclidean geometry)]
Now, observe
that the horizontal side "above the x-axis" is bisected by the
y-axis. Thus, the length of this side in the first quadrant
of the unit circle is ½ [the triangle is equilateral, so the
side being bisected has the length of the radius, i.e. is length
1]. That is, cos(60°)=½.
[The y-axis
has an angle of 90° i.e. p/2 radians i.e. 100 gradients. The
angle created by drawing line segments from the vertices of
this side, to the center, starts at 60° and ends at 120°. So,
the y-axis is 90° - 60° = 30° into the above angle, which means
the y-axis bisects this angle.]
i.e. [solve
for sin(60°); we know we need the positive root because 60°
is in the first quadrant, so I can omit ±]
sin(60°)
= /2
Since 30°
is the complementary angle to 60°, we also have computed sin(30°)
and cos(30°).
How
does "wrapping the real number line around the unit circle"
work?
The unit
circle has a circumference of 2p. "Thus", all trig functions
will have the same value when evaluated 2p radians apart. We
say that all trig functions have a period. [In "stratosphere"
math, this period is arrived at in a very different way.]
The trig
function period identities are [X is an angle, n is a positive
integer]:
These identities
hold even in the undefined case [if one side is undefined, they
both are.]
To formally
demonstrate the n part, I would use natural induction, which
should be buried somewhere in College Algebra. If you don't
recall this term clearly, don't worry about it. However, I'm
only going to explain it for n=1.
We read
these from the unit circle immediately:
sin(X+[circle])
= sin(X)
cos(X+[circle]) = cos(X)
We use the
rewrite of sec(X) and csc(X) in cos(X) and sin(X) to derive
these:
sec(X+[circle])
= sec(X)
csc(X+[circle]) = csc(X)
Now, to
deal with tan(X) and cot(X), we have to be a little more clever.
For tan(X), write:
i.e. tan(X+[½][circle])=tan(X).
The identity cot(X+[½][circle]) = cot(X) is similar, but works
with the multiplicative reciprocals throughout.
What
are the triangle area formulae?
There are
two basic formulae, and one "impractical" one.
"½
base times height"
(Area)=(½)(length
of base)(length of height)
To use this
formula, pick one side of the triangle as the "base". Note its
length. Then, draw a perpendicular line segment from the vertex
of the triangle not in the "base", to the "base", and note this
line segment's length (this is the height). [This formula does
work for obtuse triangles. Extend the base to where it would
hit the perpendicular line.].
This formula
is easily visualized for a right triangle [a rectangle with
a line segment between two opposite vertices gives two right
triangles, both clearly with half the original area since they
are congruent]. It behaves reasonably for line segments [it
gives zero area, which is correct for a line segment; either
base or height is zero for a line segment.]
EXERCISE:
learn to use this formula by applying it to the reference triangles.
The 45-45-90 triangle with hypotenuse should
have area ½, and the 30-60-90 triangle with hypotenuse length
2 should have area .
This formula
has useful analogies in higher dimensions. The volume of either
a pyramid or a cone, for instance, is given by:
(Volume)=(1/3)(area
of base)(length of height)
The "obvious
generalization" for n-dimensional Euclidean space (n a positive
integer) is
(n-d
hypervolume)=(1/n)(n-1 hypervolume of base)(length of
height)
and for
4-d Euclidean space, English permits a simplification:
(hypervolume)=(¼)(volume
of base)(length of height)
The n-dimensional
formula is clearly dimensionally consistent: using a length
unit, both sides have dimension (length)n. The n-dimensional
formula can be directly computed if one is familiar with iterated
integrals (say, from Calculus III or a decent physics course).
"½
of the product of the lengths of two sides, and the sine of
the included angle"
(Area)=[½](length
of side 1)(length of side 2)sin(angle between sides 1, 2)
(Area)=[½]absin(C) (Area)=[½]acsin(B) (Area)=[½]bcsin(A)
We can see
this, from the immediately prior formula, by taking side 1 to
be the base, side 2 to be the hypotenuse of the right triangle
formed between the [extended, if necessary] base and the height,
and then solving for the height in terms of side 2 and the angle
between sides 1 and 2 [an acute angle of the right triangle
we just constructed].
We get:
(length
of height)=(length of side 2)sin(angle between sides 1,2)
EXERCISE:
Learn to use this formula by applying it to the reference triangles.
[The correct areas are the same as before.]
EXERCISE:
Now, learn to use this formula by applying it to the equilateral
triangle with all sides length 2. This triangle is essentially
two copies of the 30-60-90 triangle referred to earlier, so
its area is twice as large -- 2.
I do not
know how to generalize the above formula to n-dimensional Euclidean
geometry.
Heron's
area formula for a triangle:
(Area)²
= s(s-a)(s-b)(s-c) where s=a+b+c and a, b, c are the lengths
of the sides. [Take the positive square root to get area.]
This formula
gives zero when the length of one side is the same as the length
of the other two sides, but will malfunction when one side has
length zero. The Heron referred to here is a Greek mathematician
(B.C.), so the formula can (or should be able to, at least)
be derived in straight Euclidean geometry without coordinates.
However, I have not read this, so I cannot explain it.
EXERCISE:
Learn to use this formula by applying it to the reference triangles,
and also to the equilateral triangle with all sides length 2.
The mathematician
Cartan generalized this formula to n-dimensional Euclidean geometry,
using matrix determinants [this is the determinant of a certain
3x3 matrix]. Cartan's generalization is definitely beyond the
scope of this crash review.
What
is the law of sines?
sin(A)/a
= sin(B)/b = sin(C)/c
Note that
the angles A, B, and C are strictly between 0 and 2
in angular measure. This means that solving for the sine of
an angle by the law of sines does *not* strictly determine the
angle, normally. [It does if the angle is a right angle; then
sin(angle)=1].
If the solved-for
sin(angle) is strictly between 0 and 1, then some work is required
to determine the actual angles. The inverse sine function on
a calculator, or spreadsheet, is programmed to give an acute
angle [strictly between 0 and
in angular measure]. However, since the sine of an angle X is
equal to the sine of its supplementary angle 2-X
[sin(X)=sin(2-X)],
the supplementary angle is *also* a viable choice.
[EXERCISE:
Learn to use the sine law by explicitly writing out the equalities
for the 45-45-90 triangle with hypotenuse , the
30-60-90 triangle with hypotenuse 2, and the equilateral triangle
with all sides length 2.]
What
is the law of cosines?
This is
the generalization of the Pythagorean theorem to
non-right triangles. I'm going to present it "deus ex machina".
c²=a²+b²-2abcos(C)
b²=a²+c²-2accos(B)
a²=b²+c²-2bccos(A)
However,
this formula does behave correctly in the extreme cases [worked
for C, others are similar]:
The
last equation is physically correct: if C=2,
then the triangle is really a line segment, and the
side c physically has length a+b. This is what the algebra
states:
c²=(a+b)²
i.e.
|c|=|a+b| i.e. [a, b, c are all guaranteed to be non-negative,
since they represent physical lengths]
c=a+b
[EXERCISE:
Learn to use the cosine law by explicitly writing out the equalities
for the 45-45-90 triangle with hypotenuse , the 30-60-90 triangle
with hypotenuse 2, and the equilateral triangle with all sides
length 2.]
[EXERCISES:
Learn to use these formulae with the reference values we already
have: 0°, 30°, 45°, 60°, and 90°. Namely: try A=0° [should reduce
to trigonometric function of B or -B, respectively] and B=0°
[trigonometric function of A]. Also, try 2*0°=0°, 2*45°=90°,
2*30°=60°, and 30°+60° = 90°. Ranging into the second quadrant:
try 90°+(another reference angle), 2*60° = 120°, and 2*90° =
180°. The tangent identities should not be usable when one of
the angles is a right angle.]
Note: several
entries in Trigonometry Survival 201 are/will be based on this.
What
are the angle-halving formulae?
The angle-halving
formulae are easily derived from the Pythagorean Identity and
the formula for cos(2A). In general, we need to know which quadrant
the angle A/2 is in to decide on the correct sign.
All are
trig(onometric) sum or difference formulae. The first four are
also product formulae.
EXERCISE:
Derive these from the angle sum and difference formulae, as
follows:
Set
X=A+B and Y=A-B
Substitute
in the angle sum and difference formulae for the affected
functions, and simplify. Solve
for A and B in terms of X and Y, and then replace A, B with
their expressions in X, Y.
EXERCISE:
Numerically use the sum and difference identities for X=A+B,
Y=A-B where A, B are reference angles. The tangent ones will
break down when 90° is A, B, X, or Y.
EXERCISE:
Numerically use the product identities for A=(X+Y)/2, B=(X-Y)/2
where X, Y are reference angles.
Wait!
I'm not completely sure how to solve for A and B in terms of
X and Y!
Until I
get a College Algebra page going, here's a quick summary on
how to solve systems of linear equations. (This domain can use
several.) [While not all systems of linear equations
are solvable, the one we want to is solvable.]
Substitution
We
hope that by cleverly adding multiples of pairs of equations,
we can get equations with reasonably isolated variables.
This works fairly well with two variables.
How
this works in practice:
X=A+B,
Y=A-B.
To
solve for A, eliminate B from the resulting sum
and then solve. Since 1+(-1)=0, we simply add both
equations: X+Y=2A. Dividing by 2 yields (X+Y)/2=A.
To
solve for B, eliminate A from the resulting sum
and then solve. Since 1+(-1)*1=0, we subtract Y=A-B
from X=A+B: X-Y=2B. Dividing by 2 yields (X-Y)/2=B.
Gaussian
Elimination (named after Gauss, the mathematician)
Taking
variables from left to right (in our case, A, then B):
Pick
a linear equation using the "leading variable".
If this variable's coefficient is not 1, divide
the equation by this coefficient. This equation
is now the "topmost equation".
By
adding a suitable multiple of the "topmost equation"
to the other equations, remove A from the resulting
equations.
We
are now done with the "topmost equation". Set it
aside.
When
the only equations left have a multiple of a single
variable equal to a constant, solve those variables.
Then replace those variables in the equations that
have been set aside. Repeat until all variables
have been explicitly solved.
How
this works in practice:
X=A+B,
Y=A-B: A is "leading variable". Both X=A+B and Y=A-B
have coefficient 1 for A.
However,
the multiple of the "topmost equation" I am subtracting
is 1[=1/1; numerator is from the equation I am subtracting
from, denominator is from the "topmost equation".
I arbitrarily choose my topmost equation to be Y=A-B.
We
subtract Y=A-B (i.e. 1*[Y=A-B]) from X=A+B to get
X-Y=2B.
We
solve for B: B=(X-Y)/2.
We
then subsitute this into the "topmost equation"
X=A+B, getting X=A+(X-Y)/2
Isolating
A, we end up at (X+Y)/2=A.
Cramer's
Rule: This is theoretically interesting, since it directly
informs you when the system of linear equations does not
have a unique solution. However, it requires the introduction
of even more terminology. I won't cover it in this refresher.
Angle
classifications
Acute
-- strictly between 0 and
strictly
between 0 and 90° strictly
between 0 and p/2 radians strictly
between 0 and 100 gradients
Right
--
exactly
90° exactly
p/2 radians exactly
100 gradients
Obtuse
-- strictly between
and 2
strictly
between 90 and 180° strictly
between p/2 and p radians strictly
between 100 and 200 gradients
Suggestions:
Note that 15° is computable either with the half-angle or the
difference-angle formula from the standard reference table.
Use both of these methods. Also, 75° can be computed by the
sum formula from the standard reference table; use this method.
Once you are confident that these tables are correct, and if
you want more practice with the angle sum, difference, halving,
and doubling formulae, use these additional reference values
in combination with the earlier ones [0°, 30°, 45°, 60°, and
90°, i.e. the standard table].
[EXERCISE:
translate everything into radians. If you plan to use gradients,
also translate everything into gradients.]
Suggestions:
First, let X be an angle such that 5X works out to be 0, 1,
2, 3, or 4 full circles.
[That is,
we are directly interested in 72°, 144°, 216°, or 288°. I also
included 0°, but that is a standard reference value. We will
recover:
18°
from 72° and the cofunction identities
36° from 144° and the supplementary angle identities
54° from 36° and the cofunction identities
It may provide
some intuition about the following equations to subtract 1 circle
off 216° [-144°] and 288° [-72°].]
Next, use
the angle addition formulae
to rewrite in terms of sin(X) and cos(X)
sin(5X)=0
cos(5X)-1=0 [a convenient rewrite of cos(5X)=1, for this
problem]
These are
the equations that describe an inscribed equilateral pentagon.
They *are* solvable using nothing more than the quadratic formula(!),
and what we already are supposed to know.
This would
occur to me from sin(X) being an (algebraically) odd function, combined
with the angle comments above:
sin(0°)=0,
so we can factor sin(X) out of our expanded version of sin(5X)=0.
Then, use the Pythagorean identity to replace cos²(X) with 1-sin²(X).
Expand the results. This should give a quartic in sin(X) [oops],
which is also a quadratic in sin²(X) [great]. Use the quadratic
formula to solve for sin²(X). Then take square roots on the
solutions we get for sin²(X); we need both the positive and
negative roots. One of these root sets is for 72° and -72°,
and the other one is for 144° and -144°.
Geometrically,
the larger positive root goes with 72°, and the smaller positive
root goes with 144°. Solve for cosine of 72° and 144° with the
Pythagorean identity [be
sure to use the quadrants to force the
correct sign]. Then fill in the table as summarized initially.
Also: when
completing the table, tan(18°) and tan(54°) [and their corresponding
cofunction values, cot(72°) and cot(36°)] are technically difficult
to algebraically compute directly from the sine and cosine values.
[The denominator needs two stages to cancel out correctly.]
I tried that three times in a row, and got three different answers,
all of them wrong. The method I used for the table was to compute
cot(18°) and cot(54°) respectively, and then take the multiplicative
inverse algebraically to get tan(18°) and tan(54°).
Instant
numerical trigonometric tables for sin, cos, and tan near 0
This isn't
really "fair", since it relies on the "stratosphere" math approach.
The first mention of the basis for this, traditionally, is in
the middle of a Calculus series [business or conventional].
However,
it is very practical. (Especially if you plan to use trigonometry
in an engineering course.) I'm going to present how the result
is arrived at in just enough detail, that those readers who
actually know the relevant material will be able to see that
I'm doing it right.
[If you
know what a power series is, and the alternating series test
for convergence, you will immediately recognize what I'm doing.
If not -- well, I said this wasn't "fair". Put these equations
and inequalities into a spreadsheet to see why the above is
plausible. If you don't have access to a spreadsheet, then use
a calculator. If you don't have even that, at least do it for
the numbers 0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, and 0.9.]
For three
decimal place accuracy (the least we are interested in, for
practical purposes), we need to control the error to less than
5*10-4 i.e. [½]10-3. This error term is:
for
sin(X), X³/6
for cos(X), (X4)/24
Now, let's
solve for X:
sin(X)
X³/6<5*10-4
i.e.
X³<30*10-4 i.e.
X³<3*10-3 i.e.
X<[31/3]/10 which is between 0.144 and
0.145
cos(X)
(X4)/24<5*10-4
i.e.
X4<120*10-4 i.e. [yes, there's
a reason I'm not reducing here!]
X<[1201/4]/10 which is between 0.330 and
0.331
That is,
for X in radians:
when
0<X<0.144, sin(X)=X to three decimal places
when 0<X<0.330, cos(X)=1-X² to three decimal places
What if
you want the "instant trig table" to more decimal places? Just
work the above bound calculations with 5*10-(number of
decimal places+1) instead of 5*10-4. Note that
at the final stage, I took the decimal my TI-36SX Solar calculator
gave out, and found the three-decimal place numbers that bracketed
it. For a different number of decimal places, bracket the result
with decimals to as many places as you need precision.
Computing
tan(X) and sec(X) from the results of this "instant trig table"
will give reasonably accurate numerical results. Computing cot(X)
and csc(X) will not work reasonably "near 0"; in general, you
cannot get more significant digits out than you put in, and
sin(X) will lose significant digits as X ends up near 0. This
means you are dividing a large number by a small, imprecise
number, which will *not* give reasonable numerical results.
[The zero decimal places between the decimal point and the first
non-zero digit are "lost significant digits". For example, sin(0.021)=0.021
to three decimal places -- but there are only two significant
digits.]
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Algebra
The branch of mathematics dealing with the solution of equations. These equations involve unknowns, or variables, along with fixed numbers from a...
Algebra (in 'Group theory' article) Here, the basic idea is to associate a group (the so-called Galois group, named after the mathematician E. Galois) with algebraic equation (6) in 6 su
Applications to algebra and geometry (in 'Lattice (mathematics)' article) Lattices, like groups and rings, can be defined as abstract algebras, that is, as systems of elements combined by universally defined operations. The
Lie algebra (in 'Lie group' article) Let G be a Lie group. The identity component of G is an open subset of G and is a smooth manifold. Thus it makes sense to speak of tangent spaces.
Multilinear algebra and tensors (in 'Linear algebra' article) Linear algebra is generally regarded as including also multilinear algebra, the basic notion of which is that of a bilinear mapping of the pair of vec
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Software Categories
Math software downloads
by HydeSoft Computing Graphing software for scientists, engineers, and students. It features multiple scaling types, including linear, logarithmic, and probability scales, as well as several special purpose XY graphs and contour plots of 3D data.
by Clay Pot Software Learn basic math with deep understanding easier and faster by simply practicing using tables (ex. New Multiplication Table with Pictures) that continually expose students to the big picture of how and why math works
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undergraduate-level introduction describes those mathematical properties of prime numbers that can be deduced with the tools of calculus. The capstone of the book is a brief presentation of the Riemann zeta function and of the significance of the Riemann Hypothesis.
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EDUC 6520: Advanced Pedagogical Content Knowledge:Math 4-5
The purpose of this course is to introduce integral components of the intermediate (4-5 grade) mathematics curriculum. While the focus is on mathematical content, teaching methods including the use of multiple representations and technology will be underscored throughout the semester. The major thrust of the course will be development of the real number system and arithmetic operations, measurement, probability, data analysis, and geometry. Prerequisites: Early Childhood License or Permission of Instructor... more »
Credits:2
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Synopsis
Do you find yourself in math clazz hearing terms like polynomials and rules of operation, but not being able to make sense of what they all mean? We've all been there! And this book is for you. It breaks math down in a way that's easy for beginners.
This book starts by reviewing the essence of arithmetic (fractions, divisions, square roots, etc.), then moves on to expressions, operations, equations and function. It goes slow and along the way gives you dozens of exercises to practice.
This book takes some of algebra's most complex equations, and puts them in a language anyone can understand.
The "Plain and Simple English" series is part of BookCaps™ growing library of book and history recaps.
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ISBN: 1230000001111
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Next: Linear Systems by Graphing
Previous: Linear Inequalities in Two Variables
Chapter 7: Solving Systems of Equations and Inequalities
Chapter Outline
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Chapter Summary
Description
This chapter introduces students to linear systems of equations and inequalities as well as probability and combinations. Operations on linear systems are covered, including addition, subtraction, multiplication, and division.
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Why are textbooks in math and science so bad?
Quote by kant
I am studying mathematics at ucla. I am talk this upper dividion analysis course, and the professor only teachs one hour every day for 4 days a week. Can you tell me what kind of workload does he have?
Yeah -- he probably doesn't do anything but prepare for those 4 hours of work.
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Develop student understanding with the Discovering Math series. This 2-pack addresses the correspondence of algebra and geometry, the Pythagorean theorem, and basic geometric concepts and constructions. It also discusses trigonometric ratios, polar coordinates, inductive and deductive reasoning, and geometric proofs.
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Autograph Activities: Students Investigations for 16-19
Autograph is an excellent tool for investigation, and mathematics is at its strongest and most appealing when students can embark upon such journeys of self-discovery.
The ten activities are designed to allow students to fully utilise Autograph's power to explore, investigate and ultimately understand concepts at a depth which the normal classroom setting would not allow.
Areas covered include vectors, differentiation, integration and trigonometry.
Students are equipped with the tools to learn and then encouraged to set off alone on their epic quest for answers.
Both books come with a CD with support material for each chapter and a 30-day trial of Autograph.
£25, by C N Barton, 2009
There's also a discount bundle of 5 of each book for £160
Dynamic Apple Mac software for teaching and learning secondary and college mathematics, for one single user. It's okay to install on your desktop and laptop provided you're the only user. Supplied by download (CD-ROM option available.)
Dynamic software for teaching and learning secondary and college mathematics. For academic Apple Mac users only. We will contact you for proof of original licence, if not previously purchased from Chartwell-Yorke. Supplied by download (CD-ROM option available.)
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Weavil, Karen
We will learn to visualize two and three-dimensional shapesand apply Algebra concepts to them. We will also learn to think logically, sowe can explain a problem step by step. Projects will be given throughout theyear and will count as a test grade.
The final exam counts 25% of you overall final grade forthe year.
Course Syllabus:
1stQuarter: Chapters 1 – 3
(Basic Formulas,Theorems, Postulates, Proofs, and Parallel Lines)
2ndQuarter:Chapters 4 - 6
(Proving Triangles Congruent, Properties ofPolygons)
3rdQuarter:Chapters 7 – 9
Transformations,Similarity, Radicals, Special Right Triangles, Trigonometry)
4thQuarter:Chapters 10 - 12
(Circles, Area &Perimeter of Plane figures, Surface Area & Volume of Solids, Review)
Materials:
Materials need to be brought every day to class
-2 inch Binder with loose leaf paper
-Calculator (TI-84 Plus recommended)
-Pencils, Highlighter (Work done in Pen will notbe graded)
-Notecards / Flashcards
Scale:
Grades:A93 - 100
Tests & Projects..…………………50%B85 – 92
Quizzes…………..........................30%C77 - 84
Homework / Classwork&D70 - 76
Attendance……………….20%FBelow 70
üTests will be given after every chapter. Alltests will be announced. If you are in the room when a test is given, you willtake the test, even if you were out for the review.
üQuizzes will be given throughout each chapter.(Beware "Pop" Quizzes will happen)
üA test or quiz or both will be given each week.So expect it!
üTest corrections will be available on allchapter tests. You must stay after during tutoring times to make corrections
üTextbook / Homework: A textbook will be issuedto each student. Students will have homework nearly every night, either fromthe workbook or from a worksheet. It will be graded daily, sometimes forattempt and sometimes for correctness. Notes must be taken each day and studiedthoroughly each night. If a student falls behind, it will be hard for him/herto catch back up. A large part of being successful in Geometry is memorizingthe postulates, definitions, and theorems given each day in class.
Make-up work:
You may make up work due to an absence ONLY!! I willprovide a box with the make-up work available. Please get the worksheet out ofthe box and ask a friend what was turned in. Also students may look at theirassignment sheets to see what they missed if they were out. It's imperative towork on the assignment even when absent to not get behind. Make sure you find afriend in class to exchange numbers with to get the information from class.(Good skill for college too!!)
Late Work:
Late work will be accepted with penalties at my discretion.
Expectations:
I expect you to behave like ladies and gentlemen. Youshould abide by the following rules at all times while you are in my classroom.
vCome to class on time and prepared
vNo cell phones and other electronic devices.
vRespect others and their property
vDo all work in pencil
Consequences: 1. Warning
2.Teacher/Student Conference
3. Parent contact / Conference
4.Discipline referral to office
Tutoring:
Tutoring will be available Tuesdays and Thursdays afterschool until 4:30.Please come promptly as if no one shows up I may decide toleave early. If you would like to attend but are unable to come during thistime then please speak with me and we will work something out.
Progress Reports:
Progress reports will go out approximately every 3 weeks.You should receive a paper copy on that Wednesday. Dates will be available onthe school website.
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AIEEE, IIT-JEE Mathematics - Permutation & Combination Class I
In this class we will cover the fundamental concepts of Permutation & Combination for IIT-JEE, AIEEE and other Engineering Entrance Exams. This class has been designed for Engineering Aspirants and will cover curriculum of Mathematics for CBSE, ISC and various state board examinations.
This session will gradually take you to various pattern of question with different level of difficulty. The session will cover various concepts with their application in couple of numerical problems.
About Learners Planet . (Teacher)
Learner's Planet is a rich source of online content, video lectures, mock tests, educational games and much more in kindergarten to Grade-12 segment. A subscription for Learner's Planet would bring tons of learning material at your mouse click. Print thousands of worksheets, try online quizzes and see instant results. Brush up your concepts with the help of recorded lectures by experienced teachers. Learn anytime anywhere at your own pace !
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Summary: This tried-and-true text from the pioneer of the basic technical mathematics course now has Addison-Wesley's amazing math technologies MyMathLab and MathXL helping students to develop and maintain the math skills they will need in their technical careers.
Technical mathematics is a course pioneered by Allyn Washington, and the eighth edition of this text preserves the author's highly regarded approach to technical math, while enhancing the integration of te...show morechnology in the text. The primary strength of the text is the heavy integration of technical applications, which aids the student in pursuit of a technical career by showing the importance of a strong foundation in algebraic and trigonometric math.
Allyn Washington defined the technical math market when he wrote the first edition of Basic Technical Mathematics over forty years ago. His continued vision is to provide highly accurate mathematical concepts based on technical applications. The course is designed to allow the student to be simultaneously enrolled in allied technical areas, such as physics or electronics. The material in the text can be easily rearranged to fit the needs of both instructor and students. Above all, the author's vision of this book is to continue to enlighten today's students that an understanding of elementary math is critical in many aspects of life.
Special Caution and Note indicators identify and aid students on difficult topics throughout the text.
Flexibility of Material Coverage. An important and critical feature to the Washington approach to technical math is the flexibility of the table of contents. The chapters of the text are easily adapted to the specific needs of the students as well as the instructor. Certain sections or chapters may be omitted without loss of continuity, and chapters may be reorganized for a customized syllabus. Notes and suggestions on how to reorganize the material are contained in the Answer Book.
Graphing Calculator.The graphing calculator is integrated and emphasized throughout the text, though it is still not required for the course. This integration includes over 160 graphing calculator screens pictured in the text.
Design.The open design is an important aspect that continues in the eighth edition. The spacious layout allows for additional graphing calculator screen graphics in the margin, which helps students visualize the graphing technology.
Word Problems and Solutions.Throughout the text, approximately 120 examples present complete solutions to word problems. These examples are clearly indicated in the margin by the phrase ''solving a word problem.'' In addition, the text includes over 900 word problems within the exercise sets.
Exercises and Figures.The eighth edition includes over 1500 new exercises. Over 200 new figures have been added to help students visualize applications and concepts.
Writing Exercises. The number of writing exercises has been increased. An icon highlights the writing exercises in the text. These exercises reinforce student understanding, as they require students to verbalize their answersCD Missing. Appearance of only slight previous use. Cover and binding show a little wear. All pages are undamaged with potentially only a few, small markings. Help save a tree. Buy all your used book...show mores from Green Earth Books. Read. Recycle and Reuse! International shipping not available on this itemHardcover Good 0321306899
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You might want to check to see if your school's algorithms course uses CLRS. If they use the book (which wouldn't be completely out of the ordinary), well then you're going to have to buy it eventually anyway, right?
CLRS doesn't really require much beyond the algebra you get in high school. There's some stuff on series and sets that you may or may not have gotten in high school, but I found it very readable even if you weren't a math geek (Disclosure: I am a math geek, so perhaps that isn't accurate).
If you have a proofs class in college that might be helpful before reading it, but very little other college math is required.
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Summary: The fundamental goal in Tussy and Gustafson's BASIC MATHEMATICS FOR COLLEGE STUDENTS, Third Edition is to teach students to read, write, and think about mathematics through building a conceptual foundation in the language of mathematics. The book blends instructional approaches that include vocabulary, practice, and well-defined pedagogy, along with an emphasis on reasoning, modeling, communication, and technology skills. Also students planning to take an introductor...show morey algebra course in the future can use this text to build the mathematical foundation they will need.
Tussy and Gustafson understand the challenges of teaching developmental students and this book reflects a holistic approach to teaching mathematics that includes developing study skills, problem solving, and critical thinking alongside mathematical concepts. New features in this edition include a pretest for students to gauge their understanding of prerequisite concepts, problems that make correlations between student life and the mathematical concepts, and study skills information designed to give students the best chance to succeed in the course. Additionally, the text's widely acclaimed Study Sets at the end of every section are tailored to improve students' ability to read, write, and communicate mathematical ideas.
New to the Edition
Check Your Knowledge: Pretests, situated at the beginning of every chapter, have been added to this edition as a way to gauge a student's knowledge base for the upcoming chapter. An instructor may assign the pretest to see how well prepared their students are to understanding the chapter; thereby, allowing the instructor to teach accordingly to their students' abilities from the results of the pretest. Students may also take the pretest by themselves and check their answers at the back of the book, which gives them the opportunity to identify what they already know and on what concepts they need to concentrate.
Study Skills Workshop: At the beginning of each chapter is a one-page study skills guide. This complete mini-course in math study skills provides extra help for developmental students who may have weak study skills, as well as additional assistance and direction for any student. These workshops provide a guide for students to successfully pass the course. For example, students learn how to use a calendar to schedule study times, how to take organized notes, best practices for study groups, and how to effectively study for tests. This helpful reference can be used in the classroom or assigned as homework and is sequenced to match the needs of students as they move through the semester.
Think It Through: Each chapter contains either one or two problems that make the connection between mathematics and student life. These problems are student-relevant and require mathematics skills from the chapter to be applied to a real-life situation. Topics include tuition costs, statistics about college life and many more topics directly connected to the student experience.
New Chapter Openers with TLE Labs: TLE (The Learning Equation) is interactive courseware that uses a guided inquiry approach to teaching developmental math concepts. Each chapter opens with a lab that has students construct their own understanding of the concept to build their problem-solving skills. Each lab addresses a real-world application, with the instruction progressing the student through the concepts and skills necessary for solving the problem. TLE enhances the learning process and is perfect for any instructor wanting to teach via a hybrid course.
ThomsonNOW with HOMEWORK FUNCTIONALITY. Assigned from the instructor, the enhanced iLrn functionality provides direct tutorial assistance to students solving specified questions pulled from the textbook's Problem Sets. This effective and beneficial assistance gives students opportunity to try similar, algorithmically-generated problems, detailed tutorial help, the ability to solve the problem in steps and helpful hints in solving the problem.
iLrn/MathNOW a personalized online learning companion that helps students gauge their unique study needs and makes the most of their study time by building focused personalized learning plans that reinforce key concepts. Completely tailored to the Tussy/Gustafson text, this new resource will help your students diagnose their concept weaknesses and focus their studies to make their efforts efficient and effective. Pre-Tests give students an initial assessment of their knowledge. Personalized Learning Plans, based upon the students' performance on the pre-test quiz, outline key learning needs and organize materials specific to those needs. Post-Tests assess student mastery of core chapter concepts; results can be emailed to the instructor!
Features
STUDY SETS are found at the end of every section and feature a unique organization, tailored to improve students' ability to read, write, and communicate mathematical ideas; thereby, approaching topics from a variety of perspectives. Each comprehensive STUDY SET is divided into six parts: VOCABULARY, CONCEPTS, NOTATION, PRACTICE, APPLICATIONS, and REVIEW.
VOCABULARY, NOTATION, and WRITING problems help students improve their ability to read, write, and communicate mathematical ideas.
The CONCEPT problems section in the STUDY SETS reinforces major ideas through exploration and foster independent thinking and the ability to interpret graphs and data.
PRACTICE problems in the STUDY SETS provide the necessary drill for mastery while the APPLICATIONS provide opportunities for students to deal with real-life situations. Each STUDY SET concludes with a REVIEW section that consists of problems randomly selected from previous sections.
SELF CHECK problems, adjacent to most worked examples, reinforce concepts and build confidence. The answer to each Self Check is printed adjacent to the problem to give students instant feedback.
The KEY CONCEPT section is a one-page review found at the end of each chapter that reinforces important concepts.
REAL-LIFE APPLICATIONS are presented from a number of disciplines, including science, business, economics, manufacturing, entertainment, history, art, music, and mathematics.
ACCENT ON TECHNOLOGY sections introduce keystrokes and show how scientific calculators can be used to solve application problems, for instructors who wish to integrate calculators into their course.
CUMULATIVE REVIEW EXERCISES at the end of Chapters 2, 4, 6, 8 and 10 help students retain what they have learned in prior chapters 0495188956New
One Planet Books Columbia, MO
Ships out same day or next
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Elementary Algebra - 9th edition
Summary: Ideal for lecture-format courses taught at the post-secondary level, ELEMENTARY ALGEBRA, Ninth Edition, makes algebra accessible and engaging. Author Charles ''Pat'' McKeague's passion for teaching mathematics is apparent on every page. With many years of experience teaching mathematics, he knows how to write in a way that you will understand and appreciate. His attention to detail and exceptionally clear writing style help you to move through each new concept with ease, and real-wor...show moreld applications in every chapter highlight the relevance of what you are learning
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GCSE Maths:Linking sequences, functions and graphs
Algebra Study Unit 9: Linking sequences, functions and graphs is for individual teachers or groups of teachers in secondary schools who are considering their teaching of algebra. It discusses some stimulating activities to help pupils to link sequences, functions and graphs.
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Course Communities
Sequences and Series of Constants Plotter
The applet shows graphically and numerically consecutive terms of a sequence or consecutive partial sums of a series. The user enters a formula for a sequence or a series and the terms are plotted. Many examples and practice problems are given. The applet illustrates many concepts associated with convergence and divergence of sequences and series, including the speed of convergence, subsequences, and Weyl's Theorem on Uniform Distribution. It provides a quick and easy illustration that finding limits of sequences by evaluating a few terms can be misleading.
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User ratings
Review: Euclid's Elements
This is going to be a long term project to get through this. We'll see if I come out on the other side.Read full review
Review: Euclid's Elements
User Review - Rlotz - Goodreads
Euclid's Elements is one of the oldest surviving works of mathematics, and the very oldest that uses axiomatic deductive treatment. As such, it is a landmark in the history of Western thought, and has ...Read full review
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02121
Requirements
Prerequisites
Pre-Calculus, volume, and surface area. Finally, they will apply integration to determine work, center of mass, and fluid force. The use of a graphing calculator is considered an integral part of the course and students will use a graphing calculator throughout this course.
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except... read more
Basic Algebra II: Second Edition by Nathan Jacobson This classic text and standard reference comprises all subjects of a first-year graduate-level course, including in-depth coverage of groups and polynomials and extensive use of categories and functors. 1989
The Red Book of Mathematical Problems by Kenneth S. Williams, Kenneth Hardy Handy compilation of 100 practice problems, hints, and solutions indispensable for students preparing for the William Lowell Putnam and other mathematical competitions. Preface to the First Edition. Sources. 1988 edition.
Product Description:
exceptions, their solution requires little more than some knowledge of elementary algebra, though a dash of ingenuity may help. Readers will find here thought-provoking posers involving equations and inequalities. Diophantine equations, number theory, quadratic equations, logarithms, combinations and probability, and much more. The problems range from fairly easy to difficult, and many have extensions or variations the author calls "challenges." By studying these nonroutine problems, students will not only stimulate and develop problem-solving skills, they will acquire valuable underpinnings for more advanced work in mathematics.
Bonus Editorial Feature:
Dr. Alfred S. Posamentier, Professor Emeritus of Mathematics Education at New York's City College and, from 1999 to 2009, the Dean of City College's School of Education, has long been a tireless advocate for the importance of mathematics in education. He is the author or co-author of more than 40 mathematics books for teachers, students, and general readers including The Fascinating Fibonacci Numbers (Prometheus, 2007) and Mathematical Amazements and Surprises: Fascinating Figures and Noteworthy Numbers (Prometheus, 2009).
His incisive views on aspects of mathematics education may often be encountered in the Letters columns and on the op-ed pages of The New York Times and other newspapers and periodicals. For Dover he provided, with co-author Charles T. Salkind, something very educational and also fun, two long-lived books of problems: Challenging Problems in Geometry and Challenging Problems in Algebra, both on the Dover list since 1996. Why solve problems? Here's an excerpt from a letter Dr. Posamentier sent to The New York Times following an article about Martin Gardner's career in 2009
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Algebra for Dummies - 01 edition
Summary: Algebra touches everyone's lives, from calculating mortgage interest to going Dutch at a restaurant -- not to mention the millions of high school students taking algebra classes. This friendly guide covers everything from fractions to quadratic equations. It includes real-world examples and story problems that will help even the most entrenched algebra-phobes approach the subject with ease.
Mary Jane Sterling has been an educator since graduating from college. Teaching at the junior high, high school, and college levels, she has had the full span of experiences and opportunities while working in education. She has been teaching at Bradley University in Peoria, Illinois, for the past twenty years.
View Table of Contents
Introduction. About This Book. What Not to Read. Foolish Assumptions. How This Book Is Organized. Where to Go from Here.
Part I: Starting Off with the Basics.
Chapter 1: Assembling Your Tools.
Beginning with the Basics: Numbers. Varying Variables. Speaking in Algebra. Taking Aim at Algebra Operations. Playing by the Rules.
Chapter 2: Assigning Signs: Positive and Negative Numbers.
Showing Some Signs. Going In for Operations. Operating with Signed Numbers. Working with Nothing: Zero and Signed Numbers.
Associating and Commuting with Expressions.
Chapter 3: Figuring Out Fractions and Dealing with Decimals. Pulling Numbers Apart and Piecing Them Back Together. Following the Sterling Low-Fraction Diet. Fitting Fractions Together. Putting Fractions to Task. Dealing with Decimals.
Chapter 4: Exploring Exponents and Raising Radicals.
Multiplying the Same Thing Over and Over and . . . . Exploring Exponential Expressions. Multiplying Exponents. Dividing and Conquering. Testing the Power of Zero. Working with Negative Exponents. Powers of Powers. Squaring Up to Square Roots.
Squaring Up to Quadratics. Rooting Out Another Result from Quadratic Equations. Factoring for a Solution. Solving Quadratics with Three Terms. Applying Quadratic Solutions. Figuring Out the Quadratic Formula.
Chapter 15: Distinguishing Equations with Distinctive Powers.
Queuing Up to Cubic Equations. Working Quadratic-Like Equations. Rooting Out Radicals. Dividing Synthetically.
Chapter 16: Fixing Inequalities.
Operating on Inequalities. Solving Linear Inequalities. Working with More Than Two Expressions. Solving Quadratic Inequalities. Working with Absolute Value Inequalities.
Part IV: Applying Algebra.
Chapter 17: Making Formulas Behave.
Measuring Up. Spreading Out: Area Formulas. Pumping Up with Volume Formulas. Going the Distance with Distance Formulas. Calculating Interest and Percent. Working Out the Combinations and Permutations. Formulating Your Own Formulas.
Chapter 18: Sorting Out Story Problems.
Getting Up to Solve Story Problems. Working Around Perimeter, Area, and Volume. Making Up Mixtures. Going the Distance. Righting Right Triangles. Going 'Round in Circles.
Missing Middle Term. Distributing. Breaking Up Fractions. Breaking Up Radicals. Order of Operations. Fractional Exponents. Multiplying Bases Together. A Power to a Power. Reducing. Negative Exponents.
Chapter 21: Ten Ways to Factor a Polynomial.
Two Terms with a GCF. The Difference of Two Squares. The Difference of Two Cubes. The Sum of Two Cubes. Three Terms with a GCF. Three Terms with unFOIL. Changing to Quadratic-Like. Four or More Terms with a GCF. Four or More Terms with Equal Grouping. Four or More Terms with Unequal Grouping.
Chapter 22: Ten Divisibility Rules.
Divisibility by 2. Divisibility by 3. Divisibility by 4. Divisibility by 5. Divisibility by 6. Divisibility by 8. Divisibility by 9. Divisibility by 10. Divisibility by 11. Divisibility by 12.
Chapter 23: Ten Tips for Dealing with Story Problems.
Draw a Picture. Make a List. Assign Variables to Represent Numbers. Translate Conjunctions and Verbs. Look at the Last Sentence. Find a Formula. Simplify by Substituting. Solve an Equation. Check for Sense. Check for Accuracy. Glossary. Index. Book Registration Information
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Information
Required Materials:
A pencil
A scientific calculator.Calculators may be borrowed.A note signed by a parent/guardian must
be filled out prior to receiving the calculator.Students who do not bring a calculator to class will not be
provided with a loaner.
An organized notebook.Each student will need a separate
three-ring binder used solely for Algebra class.It can be on the small side, 1" to 1 ½".Notebook will be assessed throughout
the year.The notebook should be
divided into five sections:Syllabus/Reference,Warm-ups, Notes, Homework,
Tests/Quizzes.
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4. Give examples of how the same absolute error can be problematic in one situation but not in another; e.g., compare "accurate to the nearest foot" when measuring the height of a person versus when measuring the height of a mountain.
9. Show and describe the results of combinations of translations, reflections and rotations (compositions); e.g., perform compositions and specify the result of a composition as the outcome of a single motion, when applicable.
2. Describe and compare characteristics of the following families of functions: square root, cubic, absolute value and basic trigonometric functions; e.g., general shape, possible number of roots, domain and range.
2. Represent and analyze bivariate data using appropriate graphical displays (scatterplots, parallel box-and-whisker plots, histograms with more than one set of data, tables, charts, spreadsheets) with and without technology.
8. Differentiate and explain the relationship between the probability of an event and the odds of an event, and compute one given the other
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5TH/7TH GRADE MATH: GEOMETRY
--- DISCONTINUED. (Verified 8/1999) RETAINED IN DATABASE FOR REFERENCE. --- Geometry is a 7-disk math skills tutorial program geared toward fifth through seventh grade levels. This program may be suitable for mainstreamed students and students with learning, emotional, sensory, or physical disabilities, or with traumatic brain injury. The program covers finding perimeters of polygons; recognizing angles, circles, triangles, rectangles, squares, spheres, cubes, rectangular solids, cylinders, and cones; finding rectangular areas; identifying the radius and diameter of a circle; and identifying parallel and perpendicular lines. COMPATIBILITY: For use on Apple computers.
Notes: Duplication rights are included in the purchase price. ** Apple is a trademark of Apple Computer, Inc. ** This is one in a series of programs on 5th/7th Grade Math Competencies. (See separate entries.)
Price:
250
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Help students do their best on standardized tests in mathematics by familiarizing them with the formats and skills they will need for success
The problems developed for this series are based on standards from the National Council of Teachers of Mathematics and state standards from across the nationThis book includes practice pages on numbers/operations, linear relations and functions, mathematical models, symbol manipulations and change analysisEach section features a test for assessment; 1 word problem is included on each page that requires a written response on a separate piece of paperThis book follows the same style and format as the other Mastering the Standards: Mathematics grades K-6, but topics are slightly different to reflect the more advanced level in algebra The activities may be used at any time of the year to assess understanding, for additional practice or for test preparation
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Real Analysis. A Constructive Approach. Pure and Applied Mathematics: A Wiley Series of Texts, Monographs and Tracts
John Wiley and Sons Ltd, April 2012, Pages: 324
A unique approach to analysis that lets you apply mathematics across a range of subjects
This innovative text sets forth a thoroughly rigorous modern account of the theoretical underpinnings of calculus: continuity, differentiability, and convergence. Using a constructive approach, every proof of every result is direct and ultimately computationally verifiable. In particular, existence is never established by showing that the assumption of non-existence leads to a contradiction. The ultimate consequence of this method is that it makes sense—not just to math majors but also to students from all branches of the sciences.
The text begins with a construction of the real numbers beginning with the rationals, using interval arithmetic. This introduces readers to the reasoning and proof-writing skills necessary for doing and communicating mathematics, and it sets the foundation for the rest of the text, which includes: - Early use of the Completeness Theorem to prove a helpful Inverse Function Theorem - Sequences, limits and series, and the careful derivation of formulas and estimates for important functions - Emphasis on uniform continuity and its consequences, such as boundedness and the extension of uniformly continuous functions from dense subsets - Construction of the Riemann integral for functions uniformly continuous on an interval, and its extension to improper integrals - Differentiation, emphasizing the derivative as a function rather than a pointwise limit - Properties of sequences and series of continuous and differentiable functions - Fourier series and an introduction to more advanced ideas in functional analysis
Examples throughout the text demonstrate the application of new concepts. Readers can test their own skills with problems and projects ranging in difficulty from basic to challenging.
This book is designed mainly for an undergraduate course, and the author understands that many readers will not go on to more advanced pure mathematics. He therefore emphasizes an approach to mathematical analysis that can be applied across a range of subjects in engineering and the sciences.
Preface
Acknowledgements
Introduction
0 Preliminaries
0.1 The Natural Numbers
0.2 The Rationals
1 The Real Numbers and Completeness
1.0 Introduction
1.2 Interval Arithmetic
1.3Fine Families
1.4Definition of the Reals
1.5 Real Number Arithmetic
1.6 Rational Approximations
1.7 Real Intervals and Completeness
1.8 Limits and Limiting Families
Appendix: The Goldbach Number and Trichotomy
2 An Inverse Function Theorem and its Application
2.0 Introduction
2.1 Functions and Inverses
2.2 An Inverse Function Theorem
2.3 The Exponential Function
2.4 Natural Logs and the Euler Number 3
3 Limits, Sequences and Series
3.1 Sequences and Convergence
3.2 Limits of Functions
3.3 Series of Numbers
Appendix I: Some Properties of Exp and Log
Appendix II: Rearrangements of Series
4 Uniform Continuity
4.1 Definitions and elementary Properties
4.2 Limits and Extensions
Appendix I: Are there Non-Continuous Functions?
Appendix II: Continuity of Double-Sided Inverses
Appendix III: The Goldbach Function
5 The Riemann Integral
5.1 Definition and Existence
5.2 Elementary Properties
5.3 Extensions and Improper Integrals
6 Differentiation
6.1 Definitions and Basic Properties
6.2 The Arithmetic of Differentiability
6.3 Two Important Theorems
6.4 Derivative Tools
6.5 Integral Tools
7 Sequences and Series of Functions
7.1 Sequences and Functions
7.2 Integrals and Derivatives o Sequences
7.3 Power Series
7.4 Taylor Series
7.5 The Periodic Functions
Appendix : Binomial Issues
8 The Complex Numbers and Fourier Series
8.0 Introduction
8.1 The Complex Numbers C
8.2 Complex Functions and Vectors
8.3 Fourier Series Theory
References
Index
MARK BRIDGER, PHD, is Associate Professor of Mathematics at Northeastern University in Boston, Massachusetts. The author of numerous journal articles, Dr. Bridger's research focuses on constructive analysis, the philosophy of science, and the use of technology in mathematics education.
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Mathematics All Around, CourseSmart eTextbook, 4th Edition
Description
Mathematics All Around, Fourth Edition, is the textbook for today's liberal arts mathematics students. Tom Pirnot presents math in a way that is accessible, interesting, and relevant. Like having a teacher on call, its clear, conversational writing style is enjoyable to read and focuses on helping students understand the math, not just get the correct answers on the test. Useful features throughout the book enable students to become comfortable with thinking about numbers and interpreting the numerical world around them. CourseSmart textbooks do not include any media or print supplements that come packaged with the bound book
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mathematical knowledge needed for computer and information sciences including, particularly, the binary number system, logic circuits, graph theory, lineary systems, probability and scatistics get clear and concise coverage in this invaluable study guide. Basic high school math is all that's needed to follow the explanations and learn from hundreds of practical problems solved step-by-step. Hundreds of review questions with answers help reinforce learning and increase
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Preface:
The pleasure and profit which the translator has received from the great work here presented, have induced him to lay it before his fellow-teachers and students of Mathematics in a more accessible form than that in which it has hitherto appeared. The want of a comprehensive map of the wide region of mathematical science—a bird's-eye view of its leading features, and of the true bearings and relations of all its parts—is felt by every thoughtful student. He is like the visitor to a great city, who gets no just idea of its extent and situation till he has seen it from some commanding eminence. To have a panoramic view of the whole district—presenting at one glance all the parts in due co-ordination, and the darkest nooks clearly shown—is invaluable to either traveller or student. It is this which has been most perfectly accomplished for mathematical science by the author whose work is here presented.
Author Auguste Comte
Available in these Formats Click on the Button to Download
Downloaded: 2
Date Added: August 20, 2012, 7:20
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This book illustrates connections between various courses taken by undergraduate mathematics majors. As such it can be used as a text for a capstone course. The chapters are essentially independent, and the instructor can choose the topics that will form the course and thus tailor the syllabus to suit the backgrounds and abilities of the students. At the end of such a course the graduating seniors should glimpse mathematics not as a series of independent courses but as something more like an integrated body of knowledge. The book has numerous exercises and examples so that the student has many opportunities to see the material illustrated and fleshed out.
Readership
Undergraduate and graduate students interested in all areas of mathematics.
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ASCI course
a20: Computational Geometry
The ASCI course on computational geometry is a `capita selecta' course.
This means that a number of different, independent topics within the
area of computational geometry will be presented. You may find some
topics more interesting than others. It should be useful to know something
about all topics, though, which is why this course is given this way.
The topics and lecturers are given below.
Lecturers:
Place and time:
April 16 - April 20 (Monday - Friday), 2007.
The course will be held at the Utrecht University
campus (De Uithof), just East of Utrecht city. For information on the campus
and how to reach it, click here.
All rooms are in BBL, the Buys Ballot Laboratorium.
Classes start each day at 10.00 and will continue
until roughly 17.00. Classes will consist of both lecturing time and problem
solving
time. There will be a lunch break. Attendance at all five days is obligatory.
Examination:
All participating students are asked to read a
paper and make an assignment. This will be done a few weeks
after the end of the course.
Course material
Please select one of the topics below for your presentation or
survey. Then send an e-mail to Remco, Marc or Leo (depending on the choice)
to confirm your choice (and avoid duplicate choice), and specify if you
want to do a survey or a presentation. In any case, send a cc of the e-mail
to Marc (so that he knows who want to do a survey and who a presentation).
In any case, the survey or presentation needs to have high quality,
in the sense of well-thought out content, a good structure, and a high
level of care with respect to layout, grammar and spelling. This holds
for presentation and survey. The length of a presentation should be
30 minutes. The length of a survey should be 8-12 pages. Quality is more
important than length!
More information and material for the geometric algebra
part of this course here.
If you choose to take the Geometric Algebra part of the course as your
subject, any option on doing a presentation or report can be discussed. It
is of course most interesting if you can select something related to your
own work. General suggestions are:
Comparison of methods for rotation (matrices, quaternions, versors),
in terms of speed, convenience, etc. A hybrid method may result.
New operations in Euclidean geometry: expand the techniques sketched
in the RIMS paper quoted above.
Non-Euclidean geometry: spherical geometry is also included in the
conformal model. It would be good to embed some of the standard techniques
as a straighforward but useful exercise.
Non-Euclidean geometry: hyperbolic geometry is also included in the
conformal model. It would be good to embed some of the standard techniques
as a straighforward but useful exercise.
Interpolating Euclidean motions: even in GA, Euclidean motions cannot
be covariantly interpolated due to a fundamental mathematical
impossibility on the metrics for the Euclidean group. Help me understand
this issue.
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Book Description: MATHEMATICS FOR THE TRADES: A GUIDED APPROACH, 9/e focuses on the fundamental concepts of arithmetic, algebra, geometry and trigonometry needed by learners in technical trade programs. A wealth of exercises and applications, coded by trade area, include such trades as machine tool, plumbing, carpentry, electrician, auto mechanic, construction, electronics, metal-working, landscaping, drafting, manufacturing, HVAC, police science, food service, and many other occupational and vocational programs. The authors interviewed trades workers, apprentices, teachers, and training program directors to ensure realistic problems and applications and added over 100 new exercises to this edition. geometry, triangle trigonometry, and advanced algebra. For individuals who will need technical math skills to succeed in a wide variety of trades.
Featured Bookstore
New
$51.00
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assignment-1
Course: PH 20, Winter 2008 School: Caltech Rating:
Word Count: 1475
Document Preview this assignment you are introduced to Mathematica, a computing environment developed to do pretty much any kind of mathematics on a computer. Among other things, Mathematica allows you to manipulate symbols, numbers, data, and graphics. Thus, it is a very general program; however, speed and ease of use (and yes, elegance) sometimes leave something to be desired. Such computing environments are already used quite extensively by researchers for a wide range of serious scientific calculations. More immediately, you may find Mathematica useful for a wide range of homework sets. Beware: using Mathematica does not mean that you can turn your brain to idle; you will still need to understand your problem in depth, and you will need to adapt to a different style of work from what you would do with pencil and paper. This assignment gives you a feel for the range of capabilities found in Mathematica: the best way to learn about it is to try out a few examples. Your TA will show you how to run the application on the physlab workstations. Several Mathematica manuals are available in the lab (they are not all for the the last version, 5.1, but the program has not changed much across the last few releases); you can also bring up the manual on your screen from Mathematica's Help menu.
The Assignment, Part 1 (try to work on this during your first lab session)
1. Take a quick tour of Mathematica: visit tour, or (even better) run mathematica at the shell prompt, then launch the Help Browser from the Help bar menu, and select "Tour." Continue with "A Practical Introduction to Mathematica" (get it from APracticalIntroductionToMathematica, or select "The Mathematica Book/A Practical Introduction. . . " in the Help Browser), which presents the same material with more detail and order (you can skim the parts that seem uninteresting or trivial). 2. Explore the functionality of Mathematica on your own, inventing your own simple problems to solve. Be a little creative with your home-made problems, and try to experiment with a large range (or even expand the range) of the functions you have seen in part 1.
Where did that grapefruit come from?
Rumor (history) has it that Caltech undergrads used a kerosene cannon to lob grapefruits onto Pasadena City College, at a distance of 1000 m. Of course, Caltech students would never do something like this, at least not the students we get these days, but the physics of this venerable legend is a good problem to feed to Mathematica. Mathematically (and Mathematically), the problem is that of a projectile subject to gravity, and to the resistance of air (atmospheric drag). Leaving apart drag for the moment, the equations of motion should be quite familiar to you: dvx = 0, dt 1 dvy = -g; dt (1)
As you can imagine, the physics of atmospheric drag is very complicated. Enter approximations! We know for a fact that the drag force is null for an object at rest, and that it grows with velocity. Under certain mathematical assumptions (of smoothness, for instance), we can then write Fdrag = -B1 v - B2 v 2 - B3 v 3 - The linear term (and other odd terms) vanish because drag does not depend on the sign of the velocity. For reasonable velocities, it turns out that the quadratic term is dominant. The resulting differential equation for the velocity is P B2 v 2 dv = - . dt mv m We estimate the coefficient B2 by the following argument: to overcome atmospheric drag, the projectile must push out of the way the volume of air directly in front of it. In a time t, the mass of air moved is mair Avt, where is the density of air, and A is the projectile's frontal area. This air is given a velocity of order v, and therefore a momentum mair v over a time t. It follows that the instantaneous force exerted by the projectile on the air (and therefore, because of Newton's third law, by the air on the projectile as a drag force) is approximately Fdrag = -Av 2 . For a spherical projectile, one finds empirically that the coefficient /2 r2 works better than the nominal area 4r2 .
Box 1: A justification of the quadratic dependence of atmospheric drag. dx dy , vy = ; dt dt which (as you know well) can be solved to yield vx = x = x0 + vx0 t, 1 y = y0 + vy0 t - gt2 . 2
(2)
(3)
Of course, real projectiles don't move quite so simply, because of drag. It turns out that the appropriate expression for the drag force in the case of spherical projectiles, with reasonable velocities, is (see Box 1) 1 (4) Fdrag - r2 v 2 , 2 where is the density of air at sea level, or approximately 1.3 kg/m3 , and where r is the radius of the projectile. This drag force can then be incorporated into the equations of motion, |Fdrag | vx dvx =- , dt m v |Fdrag | vy dvy = -g - , dt m v where v =
2 2 vx + vy .
(5)
These equations are much harder to solve by pencil and paper; we will use Mathematica.
The Assignment, Part 2 (due Oct 7 2005)
1. Using Mathematica, prove the well known result that, in the absence of drag, the optimal firing angle (the angle that yields the longer range for a given initial velocity) is 45 . Then compute the velocity components necessary to reach PCC from Caltech using the optimal firing angle, still not including drag.
2
2. In Mathematica, write a routine to integrate numerically the equations for motion without drag, and verify the above result. Superimpose several plots of the trajectory, with the same initial velocity but different firing angles, to show visually that 45 is optimal. This is your first Ph20 Beautiful PlotTM : try to arrange it so that it makes your point as clearly and boldly as possible. 3. In Mathematica, assume the grapefruit has mass 0.5 kg and radius 0.05 m. Include the acceleration due to drag in the equations of motion, and now predict where the grapefruit will land if you use the initial velocity you have just found. 4. In Mathematica, try increasing the initial velocity components and changing the firing angle until you can reach the range calculated in the drag-free case. Find the optimal firing angle (an approximate solution obtained by trial and error is acceptable). Show a visual comparison of trajectories with the same range (Caltech to PCC), but different firing angles and correspondingly different initial velocities. Again, make this graph beautiful and information-dense. 5. To go beyond trial and error, implement the following hierarchy of Mathematica functions (if needed, enlist the substantial help of your TA): (a) write a function that returns the time when the grapefruit lands (y = 0) as a function of initial angle and speed; (b) using the result of item a, write another function that returns the range (x at y = 0 and t = 0), given the initiph 20.4 Numerical Solution of Ordinary Differential EquationsIntroductionSo far, your assignments have tried to familiarize you with the hardware and software in the Physics Computing Lab, and to introduce you to the basic process of computational
Errors and Form ValidationCSci 1121Errors in JS codeCalling a function that does not exist Browser-dependent code that may not work for other browsers Trying to open a file that does not exist Generating our own errors and handling themDefault
UNIVERSITY OF MINNESOTA, MORRISMultiple Course RevisionsRoute this form to: UMM Deans Office 315 Behmler HallUMMMultiple Course RevisionsRev: 07/2004USE FOR CATALOG YEAR CHANGES ONLYThis form is for presenting changes to Curriculum Commit
UNIVERSITY OF MINNESOTA, MORRISMultiple Course RevisionsRoute this form to: UMM Dean's Office 315 Behmler HallUMMMultiple Course RevisionsRev: 02/2008USE FOR CATALOG YEAR CHANGES ONLYThis form is for presenting changes to Curriculum Commi
UNIVERSITY OF MINNESOTA, MORRISMultiple Course RevisionsRoute this form to: UMM Dean's Office 315 Behmler HallUMMMultiple Course RevisionsRev: 07/2004USE FOR CATALOG YEAR CHANGES ONLYThis form is for presenting changes to Curriculum Commi
MATH 4242: Applied Linear Algebra Fall'08 Prototype for the final examSection 0101. Compute the LU decomposition of a 33 matrix. Use it to compute the determinant of the matrix and to solve a linear system. [Midterm #1, Problem 2. Please have a
Math 4202Applied Linear Algebra Homework 5 SolutionsSpring 20072.4.32 Can you nd a 3 2 matrix A and a 2 3 matrix B such that the product AB is I3 ? The answer is no. Since B has more columns than rows, notice that rref(B) must have at least o
Armet - 1 Author: David Paul Armet Mentor: Jerrold Marsden Editor: Gilliain PierceThe Physics of SailingSailboats have been one of the most important inventions ever made by human beings. They enabled ancient peoples to travel long distances with
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China Hanson Editor: Karen Hurst Mentor: Thanos Siapas Word Count: approx. 3,350Big Brains: Who needs em, anyway?abc Figure 1: Show me your monkey face! Two eyes, a nose, and a mouth are common features of all these primates but there is some
Erik Granstedt Mentor: Prof. James Eisenstein Editor: Pierce, 3033 wordsElectric Propulsion One Step Closer to Warp DriveGoing the Distance Things in space are just so far away Space the final frontier. The scale of space is enormous, with dist
The Astrophysical Journal, 678:1099Y1108, 2008 May 10# 2008. The American Astronomical Society. All rights reserved. Printed in U.S.A.ENRICHMENT OF THE DUST-TO-GAS MASS RATIO IN BONDI/JEANS ACCRETION/CLOUD SYSTEMS DUE TO UNEQUAL CHANGES IN DUST AN
JPL-Electronic Nose for Air Quality Monitoring Mario Blanco, Abhijit Shevade, Margaret A. Ryan, M. L. Homer, W. A. Goddard IIIAmong the goals of the Life Support and Habitation Program (LSH) is to provide new technologies that contribute to the next
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Elementary Number Theory: An Algebraic Approach by Ethan D. Bolker This text uses the concepts usually taught in the first semester of a modern abstract algebra course to illuminate classical number theory: theorems on primitive roots, quadratic Diophantine equations, and more.
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Number Theory and Its History by Oystein Ore A prominent mathematician presents the principal ideas and methods of number theory within a historical and cultural framework. Fascinating, accessible coverage of prime numbers, Aliquot parts, linear indeterminate problems, congruences, Euler's theorem, and more.
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A Course in Algebraic Number Theory by Robert B. Ash GraduateElementary Number Theory: Second Edition by Underwood Dudley Written in a lively, engaging style by the author of popular mathematics books, this volume features nearly 1,000 imaginative exercises and problems. Some solutions included. 1978 edition.
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Product Description:
and Mann's theorem, and a solution to Waring's problem. Proofs and explanations of the answers included
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preliminary study of calculus completed
i'll finish the 14th chapter soon and i have 16 chapters so i think i will just finish the other two chapters later. i started reviewing my notes last night and it's not quite as exciting as moving onto a new chapter. but i do eventually need to review everything i've learned, as hard as they may be. i want to get a solid database of notes so that at any time i can go back and relearn a problem that i've forgotten. it's possible that i can keep moving forward and never look back, after all i did that for the first 14 chapters and whenever i went back to review something i was able to access the needed information rather quickly. still, it's kind of bewildering confronting so many new facts and i want to try to make sense of it all. midway through the book, maybe around chapter 8 i realized that my note-taking strategy was inadequate. what i need for the more complicated algorithms, that is, those that involve more than 5 steps is for me to write out how to do each of the steps, since the book is so completely incompetent at explaining things.
throughout the book i understood very few theorems. for instance, i've seen kepler's laws explained in algebra and i understood them but when this book tried to do the same thing in calculus i didn't know what was going on. this comes as a severe disappointment since, as a philosopher, theorems were the one thing that i really wanted to understand. i've decided that doing problems mechanically is so much easier than understanding why one is doing the steps in the first place. for example, anyone can take the derivative of 2x^2 but it is much more difficult to understand why one is taking the derivative. it got to the point where i just gave up trying to understand the theorems. then again, i could rarely understand that books' explanations about anything. for some reason there is something about mathematicians and the printed word that do not go together. when mathematicians (or anyone explaining math) are forced to explain things off the cuff, orally, things for some reason become so much more clear. perhaps it's because it's easier to not know what you're talking about and write, then it is to speak and not know what you're talking about. when you're speaking, you're looking at someone and you're seeing if whether or not they are holding a straight face, moreover, you have to speak fluenty. you don't have to be thinking fluently when you're writing. you can just come up with whatever you think is clear, regardless if it takes 5 minutes to complete a sentence or not. for that reason i had to rely heavily on youtube to explain these calculus concepts. i'm afraid if i ever get up to higher math there simply won't be any youtube videos to explain things and i don't believe in taking courses in order to learn things. i learn things much better by myself. however, i can easily foresee that certain math and physics concepts become so difficult that i will really need a human who knows what they're talking about to explain it to me. it's just a fact of life that it's easier to transmit clarity in speech than in writing.
i'm rather troubled by my inability to understand theorems. i have a feeling that this is not a way to learn math. doing steps mechanically does not imply real understanding as anyone who has read searle's chinese room knows. one of these days my inability to understand theorems is going to haunt me. i'm just hoping that after reading enough math texts, getting more comfortable with the jargon, knowing more and more facts that sooner or later a light bulb will go off and it will come clear to me.
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Develop Algebraic Thinking 3-5 - MAT-925Engage your students with the thinking skills and concepts foundational to Algebra. Teachers will explore ways to introduce intermediate students (grades 3-5) to growth patterns, variables, and coordinate graphs. The included text and research-based journal articles will support teachers as they integrate the suggested strategies and activities into their own classroom practice.
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200676,"ASIN":"1841465445","isPreorder":0},{"priceBreaksMAP":null,"buyingPrice":4.85,"ASIN":"1841465585","isPreorder":0},{"priceBreaksMAP":null,"buyingPrice":2,"ASIN":"184146581X","isPreorder":0}],"shippingId":"1841465445::cmPNQvYQ%2FoPHi8KnI2OBiPXqjweMjI5yISUYyF8w7AjeNJ%2FmGAT3FPW3diuoVVwWS7cHXXJZs%2BcUU%2BQQDgmFuvYQGGqV5vlf,1841465585::Pbp5eHZ6oeTeHda0rKF7oqQN8fCUy2uDefsZKd3sGFPC46cznCILS2WerZ3q1BwG6fdoqtxMU6EfrP9KAomBJldpwIaBmaD7,184146581X::laYrmzeDte55xtB%2B%2FHuYyPN%2FFIytmIj712yCmj3%2BstQ%2B19bheKPeLRL0k7xHGM%2B57WjXdPFsTxV838dDPM2zwLPx8VKskV series of GCSE revision books are good revision aid to benefit all levels of GCSE candidate. The book incorporates pages and pages of review exercises including every aspect of the GCSE Mathematics curriculum which are all easily accessible with handy hints and tips along the way. They include colourful and non-complicated examples and include the major points of every topic in a suitable framework. I would have liked to see a more thorough review of algebra and maybe more exercises to complete along side the notes. I feel this would be essential reading to brush up on the more less familiar concepts.
I bought this guide in the months before my GCSE exam - in my opinion, this book is the only reason I passed! It covers so much information in such a short book, and in an easy-to-remember format. The pages are colourful and make revision easier to handle, and although the jokes aren't quite as funny as they are intended to be, they keep you involved in what you're reading. The summaries qnd questions at the end of each chapter are also very useful.
This book really helped my concentrate on revising for my exam, i recomend it to all those who are unsure of the exam. It uses easy to understand words and easy to follow diagrams, if you have this book your guaranteed to pass with flying colours!
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Algebra II Mathematics
Course Grade Level Description
ALGEBRA II - 1 CREDIT
Algebra II Goal Statement
The goal of Integrated Algebra II is to build upon the concepts taught in Integrated Algebra I and Integrated Geometry while adding new concepts to the students' repertoire of mathematics. In Integrated Algebra I, students studied the concept of functions in various forms such as linear, quadratic, polynomial, and exponential. Integrated Algebra II continues the study of exponential and logarithmic functions and further enlarges the catalog of function families to include rational and trigonometric functions. In addition to extending the algebra strand, Integrated Algebra 2 extends the numeric and logarithmic ideas of accuracy, error, sequences, and iteration. The topic of conic sections fuses algebra with geometry. Students will also extend their knowledge of univariate statistical applications.
It is also the goal of this model to help students see the connections in the mathematics that they have already learned. For example, students will not only gain an in-depth understanding of circular trigonometry, but will also understand its connections to triangular trigonometry. Throughout Integrated Algebra I & II, students will experience mathematics generally, and algebra in particular, not only as the theoretical study of mathematical patterns and relationships, but also as a language that allows us to make sense of mathematical symbols. Finally, students will develop an understanding that algebraic thinking is an accessible and powerful tool that can be used to model and solve real-world problems.
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This course introduces some basic notions of Algebra, and provides examples of how these methods apply to real-life problems. We examine linear equations and their graphs, systems of linear equations and linear inequalities in two variables, with application to linear programming. Next, we introduce matrices and their inverses, with applications to cryptography. Finally, we study some basic set theory, techniques of counting, permutations, and combinations, with applications to elementary probability.
Attendance
It is essential that you attend class regularly. The easiest way for you to learn the material, and to know what material has been covered, is to come to class each day. Students are responsible for finding out what material has been covered or what announcements have been made on days that they miss class.
Additional Contacts
If you have concerns/problems in the course, and are not comfortable discussing them with your instructor, please contact either of the following:
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WileyPLUS for Mathematics and Statistics offers a wealth of assessment tools, outcomes reporting, algorithmically – generated problems, and remedial tutorials that allow you to extend the learning experience beyond the classroom and keep students on-task seven days a week. Instant feedback and automatic grading
by Maple give you more time to teach and help you be more effective in the classroom.
With an emphasis always on clarity and practical applications, this edition continues to provide real-world, technical applications that promote intuitive student learning. It includes all the mathematical topics needed by students in vo-tech programs. Computer projects are given when appropriate, and the graphing calculator fully integrated and calculator screens are given to introduce computations.
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UNIT 1 : SETS, RELATIONS AND FUNCTIONS : Sets and their representation; Union, intersection and complement of sets and their algebraic properties; Power set; Relation, Types of relations, equivalence relations, functions;. one – one, into and onto functions, composition of functions.
UNIT 2 : COMPLEX NUMBERS AND QUADRATIC EQUATIONS : Complex numbers as ordered pairs of reals, Representation of complex numbers in the form a+ib and their representation in a plane, Argand diagram, Algebra of complex numbers, modulus and argument (or amplitude) of a complex number, square root of a complex number, triangle inequality, Quadratic equations in real and complex number system and their solutions.
Relation between roots and co – efficients, nature of roots, formation of quadratic equations with given roots.
UNIT 3 : MATRICES AND DETERMINANTS :UNIT 10: Differential Equations : Ordinary differential equations, their order and degree. Formation of differential equations. Solution of differential equations by the method of separation of variables, solution of homogeneous and linear differential equations
UNIT 11 : CO – ORDINATE GEOMETRY : Cartesian system of rectangular co-ordinates in a plane, distance formula, section formula, locus and its equation, translation of axes, slope of a line, parallel and perpendicular lines, intercepts of a line on the coordinate axes.
Straight lines : : : THREE DIMENSIONAL GEOMETRY : : VECTOR ALGEBRA : Vectors and scalars, addition of vectors, components of a vector in two dimensions and three dimensional space, scalar and vector products, scalar and vector triple product.
UNIT 12 : GENERAL PRINCIPLES AND PROCESSES OF ISOLATION OF METALS : Modes of occurrence of elements in nature, minerals, ores; steps involved in the extraction of metals – concentration, reduction (chemical and electrolytic methods) and refining with special reference to the extraction of Al, Cu, Zn and Fe; Thermodynamic and electrochemical principles involved in the extraction of metals.
UNIT 13 : HYDROGEN :UNIT 14 : S – BLOCK ELEMENTS (ALKALI AND ALKALINE EARTH METALS) : Group – 1 and 2 Elements General introduction, electronic configuration and general trends in physical and chemical properties of elements, anomalous properties of the first element of each group, diagonal relationships.
UNIT 3 : LAWS OF MOTION :UNIT 4 : WORK, ENERGY AND POWER : Work done by a constant force and a variable force; kinetic and potential energies, work – energy theorem, power. Potential energy of a spring, conservation of mechanical energy, conservative and non – conservative forces; Elastic and inelastic collisions in one and two dimensions.
UNIT 5 : ROTATIONAL MOTION : Centre of mass of a two – particle 8 : THERMODYNAMICS : Thermal equilibrium, zeroth law of thermodynamics, concept of temperature. Heat, work and internal energy. First law of thermodynamics. Second law of thermodynamics : reversible and irreversible processes. Carnot engine and its efficiency.
UNIT 9 : KINETIC THEORY OF GASES : Equation of state of a perfect gas, work done on compressing a gas. Kinetic theory of gases – assumptions, concept of pressure. Kinetic energy and temperature: rms speed of gas molecules; Degrees of freedom, Law of equipartition of energy, applicationsUNIT 11 : ELECTROSTATICS : Electric Charges : Conservation of charge, Coulomb's law – forces di electric medium between the plates, Energy stored in a capacitor.
Three dimensional – perception : Understanding and appreciation of scale and proportion of objects, building forms and elements, colour texture, harmony and contrast. Design and drawing of geometrical or abstract shapes and patterns in pencil.
Transformation of forms both 2 D and 3 D union, substraction, rotation, development of surfaces and volumes, Generation of Plan, elevations and 3 D views of objects. Creating two dimensional and three dimensional compositions using given shapes and forms part in any form or medium without express written permission of EDUWEB SOLUTION AND SYSTEMS prohibited,
WARNING:Copying information from This website without permission of EDUWEB SOLUTION AND SYSTEMS is Illegal.
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Prerequisite: acceptable placement score (or ACT math score of at least 28), or at least 3 years of high school algebra and trigonometry with at least a B average, or a grade of C or better in MATH 180. Recommended as a general education liberal studies elective course.
6.... use numerical methods to evaluate derivatives and use calculators and computers efficiently as tools.
COURSE POLICIES AND PROCEDURES:
Probably the best single piece of wisdom I can pass on to you as you begin this course is: "Mathematics is not a spectator sport!" You need to view yourself as the LEARNER – and "learn" is an active verb, not a passive verb. I will do what I can to help structure things so that you have an appropriate sequence of topics and a useful collection of problems, but it is up to YOU to DO the problems and to READ the book and THINK ABOUT the topics.
You must develop a system that works for you, but let me suggest that it might include finding a study group or coming to me with your questions or going to tutoring sessions in the learning center. In any case you should expect to spend at least the traditional expectation of 2 hours outside of class for each hour in class – this is important! Class time is for exploring the topics and answering questions you might have, but you simply can't master the material without putting in the time alone to really engage in the mathematics.
We are in the process of phasing in a new textbook and more than ever it is important that you actually READ the BOOK! The authors attempt to force the reader to think about the material and to develop an intuitive sense of what is going on; there is much emphasis on solving problems and much reliance on graphing technology as well as on symbolic manipulation.
READING THE TEXT: Because it is so important that you read the book, I am going to ask you to OUTLINE the text. You can't read a mathematics text like you might read a novel – it is important that you actually understand what you are reading! Each week I will take a brief look at your growing outline and will check off your name for acceptable work. This will be worth 5 points per week.
HOMEWORK: In a nutshell, working problems is one of the key ways you will learn Calculus. Attending class is important, of course, but without doing problems you will not develop a solid foundation in the material. I will give you daily assignments and will expect that you will do as many as time allows (which I take to be roughly 2 hours per class period). I will not generally collect these assignments but I do see them as testing your understanding and as raising questions for you to ask in class.
LABS: Throughout the course there will be an occasional "lab", a problem set you will work on during class time, and in a group setting. There are many ways we learn, and one way many people find helpful is working in a group, which allows discussion of the issues involved. A very good way to test your understanding of some concept is to try to explain it to a colleague. I hope you find these labs helpful.
EXAMS: There will be exams after each of the first 3 chapters – these will be in two parts, a group practice problem set worth 20 points and then an individual exam worth 80 points, 100 points in total. The final exam will be cumulative and worth 150 points (25 on the group part, 125 on the individual part). Because chapter 4 will take us up to the end of the course, there will not be a separate exam on that chapter, but the final will more heavily emphasize that final chapter than the previous three – this is very reasonable in a course like this, in which the material builds in such a sequential manner.
By the way, I do not expect you to memorize the various formulas – you are allowed a page of notes for each exam, and you can bring all four pages in for the final exam.
GRADING POLICY: In general I use the rather traditional 90% of possible points for an "A", 80% for a "B", 70% for a "C", and 60% for a "D". I will try to make enough points available in non-test situations that "test-anxiety" should not entirely kill your chances for success, but I am a very firm believer in putting students through the exam experience so that I can see whether you, not your study group, understand the material.
AMERICANS WITH DISABILITIES ACT:
I want to include taking exams in the learning center under this category; you will need a written request from Wayne Wojciechowski before I will allow you to take exams there.
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Calculators are used extensively in the Algebra curriculum, and to a lesser extent, in the Geometry curriculum. Students are provided a calculator for classroom use. Calculators are also available from the DHS library for a two-day check out period. Students are not required to purchase calculators. If you wish to purchase a calculator, we recommend the TI-84 series. This calculator is the one that students will use on the TAKS test and is suitable for all High School and most college math courses. However, please do not bring personal calculators to school.
Spiral notebook of your choice to keep notes. You will keep notes for the entire year. The spiral will encourage you to keep them all in one place. You will need a folder or binder to keep all of your daily warm-ups and daily assignments, pop quizzes, etc.
Compass
Protractor
Ruler, marked in inches and centimeters
Lots of paper for homework
Map colors or crayons or markers to aid in marking diagrams to solve problems.
Dry erase markers for use on small white boards. (2 in contrasting colors should be adequate)
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Precalculus
9780471756842
ISBN:
0471756849
Pub Date: 2010 Publisher: Wiley
Summary: This title offers a clear writing style that helps reduce any maths anxiety readers may have while developing their problem-solving skills. It incorporates parallel word and math boxes that provide detailed annotations which follow a multi-modal approach.
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From two experienced teachers, here are four books of problems that follow the school year. Activities include order of operations, signed number factoring, quadratic formula, linear and quadratic function problems. Book C/Grades
An informative introduction to the "world records" held by fourteen members of the animal kingdom. Each spread portrays an animal that is the largest, slowest, longest lived. Readers can see the animal's
This book fits the Business Mathematics course in high schools. It is structured around a three-pronged approach: Basic math review, personal finance and business mathematics. Build and strengthens students' basic skills in
The Human Body contains a collection of true-to-life drawings which show the systems of the human body. A number of related topics such as artificial respiration, nutrition, heredity, and more are also
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You are permitted to bring: one
8.5"x11" sheet of notes (both sides are OK), a ruler, and a
calculator. No other electronics are allowed (for instance, music players).
Please bring a photo ID and be aware
that your TA may spot-check some IDs.
There are no makeup exams. If a serious
emergency occurs (such as illness) and you miss the midterm, you need to
contact your professor as soon as possible and provide documentation. Note that Hall Health provides one acute illness visit per quarter free
for UW students, so even if you do not have medical insurance you can visit a
doctor's office.
To study for the midterm:
1)REVIEW:
First, review all the basic concepts, formulas and methods we covered so far
(use the posted Review File as a starting point, and look through your class
notes and text). For each concept or method, make sure you can define it or
describe it, and give examples. Recall in which homework problems or activities
you had to apply it.
Also, review and practice any algebra skills you had trouble with during
the quarter.
2)UPDATE
YOUR SHEET OF NOTES: As you work through (1) above, make sure your
sheet of notes for the exam contains all the important points. Your sheet
should be clean, neat, and organized, so you can find things easily. Do not
write too small, or make the sheet too busy/messy, or it will confuse you more
than it will help. Remember that you will have limited time during the test! In
particular, writing down entire homework problems is counter-productive, since
you will not get exactly the same problem anyway.
3)PRACTICE:
Once you are comfortable with the material and the homework
problems, print out a few previous exams from the Exam Archive (Exam 1), and
attempt them in test-like conditions: 50 minutes, quiet location, with no help
but your sheet of notes and calculator. When finished (or when the time is up),
compare your work to the posted solutions, and see which parts you missed.
Review again the parts you missed, or bring questions to our review sessions.
During
the exam:
·Start by looking over the test quickly to see
how long it is and about how long you can spend on each problem.
·Start with the problems that look easiest to
you. Read each question very carefully before writing down anything, to make
sure you are answering the correct question and you are not wasting time on
something that was not asked.
·Do not spend too much time on any one problem,
and do not panic. If you get stuck, move on and come back later. If you studied
regularly and well, you should be able to do the entire exam, but sometimes the
pressure can make you unable to think straight. If you return to the same
question later you may notice something you missed the first time.
·Do not cheat. You will get a better grade on your
own work anyway, plus it is almost certain to get caught. There will be
different versions of the exam, and we take cheating very seriously. The
consequences for cheating are outlined in the university policy on academic
misconduct.
·Before handing in your exam, take a quick look
and make sure you answered as much as possible every question. If you have time
left, review your answers.
·MOST IMPORTANTLY: unless otherwise stated,
SHOW ALL YOUR WORK and use the methods learned in this class!
This includes drawing lines and points
on graphs, labeling each of
them, describing the method you use,
showing all computations you
performed, etc. Guessing and checking will not get full
credit if there was a step-by-step procedure to determine the answer. In
particular:
oIf we cannot tell how you got your answer or if
you use an incorrect method, you will not get much credit, even if your answer
happens to be correct.
oOn the other hand, if you show work and your
answer is wrong, you will get some partial credit for the steps that are
correct in your solution. The amount of partial credit will depend on the
grading scheme.
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Solving Quadratic Equations: Cutting Corners
This lesson unit is intended to help teachers assess how well students are able to solve quadratics in one variable. In particular, the lesson will help teachers identify and help students who have the following difficulties: making sense of a real life situation and deciding on the math to apply to the problem; solving quadratic equations by taking square roots, completing the square, using the quadratic formula, and factoring; and interpreting results in the context of a real life situation.Mathematics and Statistics2013-04-26T15:06:37Course Related MaterialsSteps to Solving Equations
This lesson unit is intended to help teachers assess how well students are able to: form and solve linear equations involving factorizing and using the distributive law. In particular, this unit aims to help teachers identify and assist students who have difficulties in: using variables to represent quantities in a real-world or mathematical problem and solving word problems leading to equations of the form px + q = r and p(x + q) = r.Mathematics and Statistics2013-04-26T15:06:34Course Related MaterialsStudent-Generated Questions for Exam Prep
Math teacher Jennifer Giudice takes a unique approach to preparing her students for their upcoming assessment. She assigns her students a homework assignment that requires them to develop 2 possible assessment questions. She uses these questions to both determine where different students are at in their understanding of quadratics and also to select questions that help her best assess their level of understanding.Ms. Giudice shares a few of the questions the following day and asks students to consider what they need to do to prepare for each of the questions.Mathematics and Statistics2013-02-26T16:56:31Course Related MaterialsAlgebra Team: Teacher Collaboration
Algebra teachers, Juliana Jones and Marlo Warburton, have common philosophies and expectations in their algebra classrooms but use their own unique teaching styles and structures to create consistent experiences for students. Collaboration is an important part of that process and allows teachers to provide common learning experiences in their classrooms despite different teaching styles and structures. This video takes a look at the warm-up, lesson, strategies for group work, classroom expectations and routines to discuss commonalities but also identify ways in which each teacher personalizes their classrooms based on their own teaching style and personality.Marlo Warburton, Juliana Jones,Mathematics and Statistics2012-11-02T13:06:40Course Related MaterialsAlgebra Team: Strategies for Group Work Jones' Warburton's Overview of Teaching StylesMy Favorite No: Learning From MistakesMathematics and Statistics2012-11-01T12:47:10Course Related MaterialsThe Factor Game (i-Math Investigations)
An online, interactive, multimedia math investigation. The Factor Game engages students in a friendly contest in which winning strategies involve distinguishing between numbers with many factors and numbers with few factors. Students are then guided through an analysis of game strategies and introduced to the definitions of prime and composite numbers.Mathematics and Statistics2012-09-07T13:53:35Course Related MaterialsDifference of Squares
This lesson uses a series of related arithmetic experiences to prompt students to generalize into more abstract ideas. In particular, students explore arithmetic statements leading to a result that is the factoring pattern for the difference of two squares. An excellent teaching idea on how to help students walk the bridge from arithmetic to algebra.Mathematics and Statistics2012-09-07T13:53:35Course Related MaterialsQuadratic forms
This applet is an exploratory exercise for students to determine the best form for a quadratic. What do the coefficients do? Why do we spend so much time learning how to factor; is that form any better?Mathematics and Statistics2012-07-23T09:10:40Course Related MaterialsFactoring the Sum of Cubes or Difference of Cubes
Factoring the difference or sum of cubes.Mathematics and Statistics2012-07-23T09:10:17Course Related MaterialsFactorize Quadratic Polynomials by Table (Draft version)
Factorize the given quadratic polynomials. 3 levels of difficulties are provided.Mathematics and Statistics2012-07-13T14:53:49Course Related MaterialsFactoring simpleFactoringAngry Birds in Standard Form
Angry Birds in Standard FormMathematics and Statistics2012-07-13T14:53:21Course Related MaterialsAngry Birds in Factored Form
Angry Birds in Factored FormMathematics and Statistics2012-07-13T14:53:21Course Related MaterialsFactoring Quadratics with a Coefficient of One
The applet randomly generates quadratics with a coefficient of one with values of factors between -10 and 10. It also includes a check box to check the factors and a button to generate a new quadratic.Mathematics and Statistics2012-07-09T08:14:10Course Related MaterialsSolving a Quadratic by Factoring
This classroom tested applet demonstrates a step-by-step procedure for solving a quadratic equation (that will need to put into standard form) by factoring. The zero-product property may not be how you like it, but you and your students will get the idea.Mathematics and Statistics2012-07-06T21:33:02Course Related Materials
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Beginning Algebra With ApplicationsBeginning Algebra With Applications Book Description
One of a series of developmental maths textbooks, this volume focuses on basic algebra, with applications. It provides a learning system organized by objectives, around which all lessons, exercises, end-of-chapter review tests and ancillaries are arranged. The last objective in every section is, where applicable, devoted to applications, and a specific strategy is suggested for each major application problem, encouraging students to plan problem-solving strategies before addressing the problems.
Popular Searches
The book Beginning Algebra With Applications by Richard N Aufmann, Vernon C Barker, Joanne S Lockwood
(author) is published or distributed by Houghton Mifflin Harcourt (HMH) [0395969794, 9780395969793].
This particular edition was published on or around 1999-10-1 date.
Beginning Algebra With Applications has Hardcover binding and this format has 558
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Geometric Vectors
In this lesson our instructor talks about geometric vectors. He discusses magnitude and direction. He talks about describing quantities, William Rowan Hamilton, and James Maxwell. He talks about representing vector. He talks about algebraically and geometrically representing vectors. He also discusses adding and subtracting vectors and multiplying vectors. Lastly, he talks about unit vectors and the standard vectors along x and y axis. Four extra example videos round up this lesson.
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Geometric Vectors
Lecture Slides are screen-captured images of important points in the lecture. Students can download and print out these lecture slide images to do practice problems as well as take notes while watching the lecture.
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If you have any doubts about legality of content or you have another suspicions - click here and read DMCA
Mathematics: Applications and Concepts is a three-course middle school series intended to bridge the gap from elementary mathematics to Algebra 1. The program is designed to motivate your students, enable them to see the usefulness of mathematics in the world around them, enhance their fluency in the language of mathematics, and prepare them for success in algebra and geometry.
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Our goal is to develop your capacity for reading, writing, and speaking mathematics.
The first two-thirds of this course concentrates on a core which investigates basic
mathematical paradigms and objects: sets, proof techniques, functions, and relations
(chapters 1--4 of our text).
We follow this with an foray into cardinality
and the topology of the reals
(chapter 5 and supplementary notes).
A rough approximation for our progress toward address these topics is shown in the
Schedule Guesstimate below.
We hope that your experience in this course will serve as a gentle initiation
into the larger mathematical community.
One technical aspect of this initiation will be an introduction to LaTeX,
a markup system for producing mathematical documents on a computer.
Other ways this will be fostered will be through outside readings and participation
in activities of the Bernard Mathematics Society, including Math Coffees.
Your involvement in some of these activities may have tangible rewards
towards your course grade, but others will be expected of you
as fledgling members of the fellowship of mathematicians.
We will have two take-home writs, one in-class review,
and a self-scheduled final examination.
Homework will be assigned regularly and collected weekly.
Auxiliary activities may include the D. H. Hill Problem Contest, Math Coffees,
the Bernard Lecture and other mathematics events on campus.
A culminating activity in the course (along with the final examination)
will be the compilation of a proof portfolio.
The portfolio will include at least a dozen examples of your proof-writing from homework,
each showing your progression from initial effort to polished revision.
Your choices for inclusion in the portfolio should
provide a broad representation of topics and proof techniques from our course.
A rough recipe for the proportions in which these events combine
to produce your evaluation is
2 parts writs,
2 parts review,
3 parts examination,
2 parts portfolio,
2 part homework and other considerations.
(An explanation of my grading system is available in a web-based
memo.)
The portfolio, writs, review and final examination are pledged events;
you are expected to be vigilant in upholding the Honor Code.
Collaboration on homework is encouraged.
However, anything you present or turn in should represent your own
understanding of the material.
If you have any questions regarding ground rules for individual events,
do not hesitate to ask for clarification.
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Algorithmic Puzzles
Interprets puzzle solutions as illustrations of general methods of algorithmic problem solving
Contains a tutorial explaining the main ideas of algorithm design and analysis for a general reader
Algorithmic puzzles are puzzles involving well-defined procedures for solving problems. This book will provide an enjoyable and accessible introduction to algorithmic puzzles that will develop the reader's algorithmic thinking.
The first part of this book is a tutorial on algorithm design strategies and analysis techniques. Algorithm design strategies — exhaustive search, backtracking, divide-and-conquer and a few others — are general approaches to designing step-by-step instructions for solving problems. Analysis techniques are methods for investigating such procedures to answer questions
about the ultimate result of the procedure or how many steps are executed before the procedure stops. The discussion is an elementary level, with puzzle examples, and requires neither programming nor mathematics beyond a secondary school level. Thus, the tutorial provides a gentle and entertaining introduction to main ideas in high-level algorithmic problem solving.
The second and main part of the book contains 150 puzzles, from centuries-old classics to newcomers often asked during job interviews at computing, engineering, and financial companies. The puzzles are divided into three groups by their difficulty levels. The first fifty puzzles in the Easier Puzzles section require only middle school mathematics. The sixty puzzle of average difficulty and forty harder
puzzles require just high school mathematics plus a few topics such as binary numbers and simple recurrences, which are reviewed in the tutorial.
All the puzzles are provided with hints, detailed solutions, and brief comments. The comments deal with the puzzle origins and design or analysis techniques used in the solution. The book should be of interest to puzzle lovers, students and teachers of algorithm courses, and persons expecting to be given puzzles during job interviews.
Readership: Students and teachers of algorithm courses, puzzle enthusiasts, and anyone wishing to learn more about how to solve puzzles and/or develop algorithmic thinking
Anany Levitin is a professor of Computing Sciences at Villanova University. He is the author of a popular textbook on design and analysis of algorithms, which has been translated into Chinese, Greek, Korean, and Russian. He has also published papers on mathematical optimization theory, software engineering, data management, algorithm design techniques, and computer science education.
Maria Levitin is an independent consultant specializing in web applications and data compression. She has previously worked for several leading software companies
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Algebra 1 Advanced is an in-depth study of algebraic symbolism, systems of equations, graphing, problem-solving, and probability and statistics. The students will build upon their previous knowledge to further understand the characteristics and representations of various functions and relations, including first degree equations and inequalities, polynomials, exponential and radical expressions, quadratic equations complex numbers, and rational algebraic expressions. This course is designed for highly motivated and mathematically talented students.
Homework/Assignments
If you are not receiving the 8th grade weekly parent email which lists the planned homework assignments for the week and any special announcements, please email Ellen.DeBacker@bvsd.org to get your email address added to the mailing list.
Please check upcoming events (see above right or above left) to ensure I do not have a conflict/appointment at the time you are thinking of coming in. I am usually at school by 7:45 am and here until at least 4:30 pm.
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Department of Mathematics
MATH 1RA. Developmental Mathematics I
(3 units)
The first semester in a two semester sequence preparing students for college
level mathematics. See the online Schedule
of Courses for restrictions on enrollment based on the Entry Level Math
test. Properties of ordinary arithmetic, integers, rational numbers and
linear equations. CR/NC grading only; not applicable towards baccalaureate
degree requirements. F
MATH 10A. Structure and Concepts in Mathematics
I (3 units)
Prerequisite: students must meet the ELM requirement. Designed for prospective
elementary school teachers. Development of real numbers including integers,
rational and irrational numbers, computation, prime numbers and factorizations,
and problem-solving strategies. Meets B4 G.E. requirement only for liberal
studies majors. FS
MATH 45. What Is Mathematics? (3 units)
Prerequisite: students must meet the ELM requirement. Covers topics from
the following areas: (I) The Mathematics of Social Choice; (II) Management
Science and Optimization; (III) The Mathematics of Growth and Symmetry;
and (IV) Statistics and Probability. G.E. Foundation B4. FS
MATH 70. Calculus for Life Sciences (4
units)
No credit if taken after MATH 75 or 75A and B. Prerequisite: students must
meet the ELM requirement. Functions and graphs, limits, derivatives, antiderivatives,
differential equations, and partial derivatives with applications in the
Life Sciences. FS
MATH 90. Directed Study (1-3; max total
3 units)
Independently arranged course of study in some limited area of mathematics
either to remove a deficiency or to investigate a topic in more depth. (1-3
hours, to be arranged)
MATH 111. Transition to Advanced Mathematics
(3 units)
Prerequisite: MATH 76. Introduction to the language and problems of mathematics.
Topics include set theory, symbolic logic, types of proofs, and mathematical
induction. Special emphasis is given to improving the student's ability
to construct, explain, and justify mathematical arguments. FS
MATH 133. Number Theory for Liberal Studies
(3 units)
Prerequisite: MATH 10B or permission of instructor. The historical development
of the concept of number and arithmetic algorithms. The magnitude of numbers.
Basic number theory. Special numbers and sequences. Number patterns. Modular
arithmetic. F
MATH 134. Geometry for Liberal Studies
(3 units) Prerequisite: MATH 10B or permission of instructor. The use of computer
technology to study and explore concepts in Euclidean geometry. Topics include,
but are not restricted to, properties of polygons, tilings, and polyhedra.
S
MATH 137. Exploring Statistics (3 units)
Prerequisite: MATH 10B or permission of instructor. Descriptive and inferential
statistics with a focus on applications to mathematics education. Use of
technology and activities for student discovery and understanding of data
organization, collection, analysis, and inference. F
MATH 138. Exploring Algebra (3 units)
Prerequisite: MATH 10B or permission of instructor. Designed for prospective
school teachers who wish to develop a deeper conceptual understanding of
algebraic themes and ideas needed to become competent and effective mathematics
teachers. S
MATH 143. History of Mathematics (4 units)
Prerequisite: MATH 75 or 75A and B. History of the development of mathematical
concepts in algebra, geometry, number theory, analytical geometry, and calculus
from ancient times through modern times. Theorems with historical significance
will be studied as they relate to the development of modern mathematics.
S
MATH 149. Capstone Mathematics for Teachers
(4 units)
Prerequisites: MATH 151, 161, and 171. (MATH 161 and MATH 171 may be taken
concurrently.) Secondary school mathematics from an advanced viewpoint.
Builds on students' work in upper-division mathematics to deepen their understanding
of the mathematics taught in secondary school. Students will actively explore
topics in number theory, algebra, analysis, geometry.
MATH 161. Principles of Geometry (3 units)
Prerequisite: MATH 111. The classical elliptic, parabolic, and hyperbolic
geometries developed on a framework of incidence, order and separation,
congruence; coordinatization. Theory of parallels for parabolic and hyperbolic
geometries. Selected topics of modern Euclidean geometry. S
MATH 165. Differential Geometry (3 units)
Prerequisite: MATH 77 and 111 or permission of instructor. Study of geometry
in Euclidean space by means of calculus, including theory of curves and
surfaces, curvature, theory of surfaces, and intrinsic geometry on a surface.
F
MATH 232. Mathematical Models with Technology
(3 units)
Prerequisite: graduate standing in mathematics or permission of instructor.
A technology-assisted study of the mathematics used to model phenomena in
statistics, natural science, and engineering.
MATH 250. Perspectives in Algebra (3 units)
Prerequisite: graduate standing in mathematics or permission of instructor.
Study of advanced topics in algebra, providing a higher perspective to concepts
in the high school curriculum. Topics selected from, but not limited to,
groups, rings, fields, and vector spaces.
MATH 260. Perspectives in Geometry (3
units)
Prerequisite: graduate standing in mathematics or permission of instructor.
Geometry from a transformations point of view. Euclidean and noneuclidean
geometries in two and three dimensions. Problem solving and proofs using
transformations. Topics chosen to be relevant to geometrical concepts in
the high school curriculum.
MATH 270. Perspectives in Analysis (3 units)
Prerequisite: graduate standing in mathematics or permission of instructor.
An overview of the development of mathematical analysis, both real and complex.
Emphasizes interrelation of the various areas of study , the use of technology,
and relevance to the high school mathematics curriculum.
MATH 298. Research Project in Mathematics
(3 units)*
Prerequisite: graduate standing. Independent investigation of advanced character
as the culminating requirement for the master's degree. Approved for RP
grading.
MATH 299. Thesis in Mathematics (3 units)
Prerequisite: See Criteria for Thesis
and Project. Preparation, completion, and submission of an acceptable
thesis for the master's degree. Approved for RP grading.
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Mathematics for Elementary
Teachers
Welcome to Mth125, in which we explore the
mathematics behind the concepts and procedures commonly
encountered in elementary school math curricula.
Catalog Description: A study of the mathematical concepts and techniques that are
fundamental to, and form the basis for, elementary school
mathematics. Topics include: problem solving, inductive and
deductive reasoning, sets, number systems through the real
numbers, number theory, measurement, and 2- and 3-dimensional
geometry. Prerequisite: MTH
051 or satisfactory placement test score.
News and Updates (Spring 2008)
Looking for solutions? The podcasts and
all other solutions are available on the
Mth125 podcast page.
Looking for homework & handouts?
The homework and handouts are available on the
Mth125 homework page.
Need Help? If you are working hard but
don't feel like it's clicking for you, then you might benefit from a
visit to UW-L's Office
of Counseling and Testing Services. They can help you develop more
effective study habits and test-taking strategies. Give them a call, or
stop by for a visit. They're here to help!
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RESOURCES
In addition to the help available through your instructor, you have access to many resources to help you with your work in this class. Please also see the Materials page for additional resources specific to the College Algebra textbook.
Here is where you would go to get free professional tutoring in math and other subjects. Tutoring centers are located at all the major campuses, with hours available 7 days a week at some locations. Students are helped individually or in a group on a first-come-first-served basis. Registration is not required, but you may be asked to show your college ID. Tutoring is also provied by phone, email, or fax by the textbook publisher and details are given in the College Algebra materials page.
The instructional videos that accompany your textbook are available for viewing or checkout at the main campus libraries. The videos are available in VHS tape format or on CD. If you are a distance learning student, the library provides special services to allow you the opportunity to use the libraries through telephone, computer, and mail. Call them at 223-2030 or 1-877-613-6005 or visit the LRS Open/Off-Campus web site.
Each ACC campus offers support services for students with documented physical or psychological disabilities. Students with disabilities must request reasonable accommodations through the Office of Students with Disabilities on the campus where they expect to take the majority of their classes. Students are encouraged to do this three weeks before the start of the semester. Students who are requesting accommodation must provide the instructor with a letter of accommodation from the Office of Students with Disabilities (OSD) at the beginning of the semester. Accommodations can only be made after the instructor receives the letter of accommodation from OSD.
Go here for information on the computer classrooms maintained by the math department, including where they are and what software is available for student use. You will also find links to information and help with math software used at ACC, as well as information on checking out a graphing calculator and which Learning Labs and Testing Centers have graphing calculators students can use. Finally, there are helpful links to some internet resources on graphing calculators.
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WINTER 2011
Math 11
FINITE MATH
MATH 11 UPDATE
Textbook: Applied Finite Mathematics 2nd Edition, by R Sekhon.
Graphing Calculator Required: TI-83, 83+ or 84+ graphing calculator
are recommended for this class You can rent a TI-83 at a low monthly cost at
TI-86 is also acceptable but instructor will not demonstrate it in class.
Math 11 Syllabus Math 11 Calendar
Math 11 Finite Mathematics is an introduction to a variety of mathematical
techniques that are used to model and solve problems in business.
Topics covered include linear models, matrices, linear programming, finance,
combinatorics, probability, Markov chains and game theory.
Students will use technology as appropriate, including the graphing calculator and Excel.
Which calculator is best to use? The instructor will demonstrate
the TI-83, 83+, 84 in class.
These calculators will be the easiest to learn. TI-86 is also acceptable, but the instructor will not demonstrate it.
The TI-89 is acceptable for use in Math 11 but the TI-89 is NOT recommended.
The instructor will not teach in class how to use a TI-89. It is harder to use and
you will need to be independent and responsible to learn how to use it on your own from your instruction manual.
The instructor has a program that we will use during the quarter that she
will download to student's TI 83, 83+, 84, 86 calculators and this program may not
be available for other calculator models.
Older models such as the TI-82 and 85 have some of the same functionality
as the TI-83,84,86, but may lack some of the required functions or operations.
The instructor can not help with calculators from other manufacturers,
which may not be able to do the work, and can not run any programs that will
be given to students using the TI-83, 83+, 84, 86.
For this chapter you need to get the APIVOT program for your calculator from your instructor. Program IS available for TI-83, 84, 86 For other models of calculator, program may not be available - you were informed about this at start of quarter when TI-83, 84, 86 were listed as the acceptable calculators. Please see instructor if you are using a different TI calculator to see if there might be a program that may work on your calculator. For other brands of calculator, the instructor definitely does NOT have a version of the program for your calculator - borrow an acceptable calculator from a friend or the library or do the individual row operations to perform the pivots.
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Course: F10PD2, Pure Mathematics D (2012-13)
Aims
The aim of this module is to examine basic techniques for the
spectral analysis of linear operators. We focus on applications
of the theory to the study of the Laplace operator.
The first part of the course
will be an overview of the basics of spectral theory and the theory of
operators. As the course progress we will focus on methods and tricks
that have been successfully used to solve some long standing problems, several
of which are among the most important ones in mathematical analysis in the
last century.
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Mathematics
Level 1 and 2 courses
We offer a range of appropriate courses suitable for all those students who have not yet achieved a GCSE in Mathematics at grade C or above.
GCSE – Course Outline
The course is linear. Students who have achieved grade D are given the opportunity to sit the exam in November, with a chance to resit in June. Students who have achieved a grade E will do the linear exam in June.
Foundation Course
Students with grades F or below will follow a course on Functional Skills, taking the level 1 exam in January. They will then study a Proficiency in Number and Measure course, taking a level 1 exam in June. If this course is completed successfully then there is an opportunity to progress to the GCSE course the following year. A variety of learning and teaching approaches are used in this course, with regular collaborative tasks and group work.
Level 3 courses
Mathematics at this level is very dependent upon good algebraic skills and students enrolling on these courses will be expected to do some preparation work during the summer. This preparation work can be downloaded from the bottom of this page. There will be a short skills check on enrolment day which will be used to help decide the best course for you.
Mathematics AS/A2 • EDEXCEL
This is for those who enjoy the challenges of mathematics and want an A level which is highly thought of by both employers and universities.
Course Outline
6 units — 3 for AS and 3 for A2 level.
AS Level
You will study two pure maths modules and one applied maths module. The pure maths modules will include topics such as algebra, trigonometry, basic calculus and coordinate geometry. The applied maths module will be chosen from either statistics, mechanics or decision. This choice will be based on your other A level subjects, your career aspirations and the results of your skills check.
The statistics module will include topics such as descriptive statistics, probability, normal distribution and correlation & regression. The mechanics module will include topics such as vectors, straight line motion, forces and moments. The decision module will included topics such as linear programming, sorting and route inspection.
In order to qualify for the A2 course we expect you to achieve a pass at AS level including passes in both of the core modules.
A2 Level
You will study a further two pure maths modules and one more applied maths module. The pure maths modules will include topics such as partial fractions, vectors, calculus and functions The applied modules will generally follow on from the modules that you studied at AS level. However, there may be an opportunity to study a module in decision maths.
Each of the units is examined by a 1.5 hour exam. You will sit one exam in January and two exams in June in Year 13.
To be successful on this course, you will need to be hardworking and diligent, as each unit builds on earlier work. There is plenty of additional support provided to help students with the transition from GCSE to AS level.
Further Mathematics Mathematics (Double Mathematics) AS/A2 • EDEXCEL
This is for those whose favourite subject is maths and who are hoping to pursue a course in mathematics or engineering at university. The class time each week is doubled and you can qualify in two A level subjects, Mathematics and Further Mathematics.
Course Outline
12 units — 3 for AS Mathematics + 3 for A2 level Mathematics + 3 for AS Further Maths + 3 for A2 Further Maths.
Year 12
You will study three pure maths modules and three applied maths modules. The pure modules will include topics such as algebra, trigonometry, basic calculus, geometry, series, logarithms, functions, complex numbers and matrices. The applied modules will include statistics (with topics such as probability, normal distribution and correlation), mechanics (with topics such as vectors, forces and moments) and decision (with topics such as critical path analysis and sorting).
At the end of the lower sixth you will have two AS qualifications, one in Mathematics and one in Further Mathematics.
Year 13
You will study a further four pure maths modules and a selection of applied maths modules taken from statistics, mechanics and decision maths. The pure modules will include topics such as advanced calculus. complex numbers, differential equations, polar coordinates, coordinate systems, hyperbolic functions, matrices and proof. The statistics modules will include the binomial distribution and hypothesis testing. The mechanics modules will include projectiles, centres of mass and direct collisions. The decision module will include game theory, linear programming and dynamic programming.
Each of the units is examined by a 1.5 hour exam. You will sit two exams in January and four exams in June in both Year 12 and Year 13. It is possible to drop the further element of the course in year 13 and simply complete the A level in Mathematics.
Use of Mathematics AS/A2 AQA
If you enjoy maths and using mathematical methods to solve real world problems, you will enjoy this course. It will provide the numerical background for other subjects such as Psychology, Geography, Biology, Business Studies or Physics. You will employ appropriate IT such as graphical calculators and software packages (e.g. Excel and Autograph) to support your mathematical analysis.
This course will enable you to:
model real life situations — eg. spread of an epidemic
investigate real data — eg. economic growth
make sense of situations in your other studies — eg. statistics in Psychology experiments
Course Outline
6 units – 3 at AS level and 3 at A2 level.
AS Level
Each unit is assessed by a 1 hour examination with questions based on a pre-release data sheet. The three units we cover are:
Algebra
Data Analysis
Dynamics
A2 Level
The three units we cover are:
Modelling with Calculus (1 hour exam with pre-release data sheet)
Mathematical Comprehension (1 hour exam with pre-release data sheet)
Mathematical Applications – a portfolio assessment comprising of two pieces of work based on the AS level units or the A2 Calculus unit. Topics may include: the design of a skateboard ramp, the effect of temperature on marathon times, and car depreciation.
All the AS units (except for Algebra) and the Modelling with Calculus unit are Free Standing Maths Qualifications (FSMQs) – they are qualifications in their own right and worth up to 20 UCAS points.
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If you have read through these web pages you will
have a better of idea of how mathematics is studied at
Oxford (Part 1).
Further, you now know that not only will you learn
about many new areas of mathematics, but you will
develop entirely new ways of thinking (Part 2).
As mentioned before, this study guide does not set
out to answer the What? question (What is
studied?). At the most basic level this is outlined in
the
Undergraduate Course Handbook.
Another very important question left unanswered here
is the Why? question (Why study mathematics?).
No one answer will satisfy everyone. But some sources
of information which may help you answer this question
for yourself are listed below. This list barely
scratches the surface of what is available... Enjoy
your search! And enjoy your time studying Mathematics
at Oxford!
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Explorations in Calculus has been written as an enrichment
supplement to an AP calculus course. Graphing calculators have revolutionized the teaching and learning of mathematics. Graphs are a very important component of calculus and are crucial in helping students visualize and gain insight into traditional calculus topics. Calculations can reduce time spent on "paper and pencil" methods and increase time spent on concepts, problem solving and application.
The exploratory exercises incorporate the TI-84 or the TINspire
CAS calculator and were created to guide and encourage
students to:
• discover and explore patterns and then describe them;
• do a problem analytically (paper and pencil) and support results numerically and/or
graphically (with graphing calculator);
• do a problem numerically/graphically (with graphing calculator) and then confirm the results
analytically (paper and pencil);
• do a problem numerically and/or graphically beause paper and pencil methods are impractical
(too time consuming) or impossible.
Each of the 17 explorations starts with a set of Solved Problems followed by a set of Exercises.
The Solved problems are examples that provide a context for instructions that one might
encounter in the exercises, but they also contain useful mathematical hints and lessons as well.
The explorations in this book are not designed to teach you how to use your calculator.
However, when an exploration requires you to perform a complicated calculator operation, you
will be referred to the APPENDIX, where the specific keystrokes are described in detail for the or the
TI-84 in APPENDIX A and for the TI-Nspire in APPENDIX B.
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Abstract
The National Council of Teachers of Mathematics has stated that computer technology is an important factor in the improvement of mathematics instruction and an important tool in mathematics instruction. Computer spreadsheets are readily adaptable for problem solving, can enhance the user's insight into the development and use of algorithms and models, free students from being hampered by laborious manipulation of numbers, and allow students to see the progression of calculations on the screen as they are generated. This set of materials is an attempt to illustrate by example how the spreadsheet can be used in the secondary school mathematics classroom. Included in this set of materials are data disks for use with Appleworks spreadsheet and Better Working Spreadsheet software, and a teachers' guide. The teachers' guide contains an overview of spreadsheets, and directions and suggestions for use with each of the 30 spreadsheet files on the disks. (CW)
Computer-Based learning environments in mathematics
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Matriculation Sc. Actv
Essential Mathematics is a series of two books for Classes 9 and 10. This series is based on the latest syllabus prescribed by the Council for the Indian School Certificate Examinations, New Delhi.
SALIENT FEATURES of the books
Each chapter has a large number of solved problems to illustrate the concepts and methods.
Stress has been laid on concept building. The text is lucid and to the point.
In the exercises, problems are graded from simple to complex.
A list of important definitions, formulae and results are provided at the end of each chapter in the form of POINTS TO REMEMBER.
TEST YOUR KNOWLEDGE at the end of each chapter tests the child's learning.
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Free Coursera Calculus course with hand-drawn animated materials
Robert Ghrist from University of Pennsylvania wrote in to tell us about his new, free Coursera course in single-variable Calculus, which starts on Jan 7. Calculus is one of those amazing, chewy, challenging branches of math, and Ghrist's hand-drawn teaching materials look really engaging.
Calculus is one of the grandest achievements of human thought, explaining everything from planetary orbits to the optimal size of a city to the periodicity of a heartbeat. This brisk course covers the core ideas of single-variable Calculus with emphases on conceptual understanding and applications. The course is ideal for students beginning in the engineering, physical, and social sciences. Distinguishing features of the course include:
the introduction and use of Taylor series and approximations from the beginning;
Watch these next
Great post, and I am likely being pedantic, but I think it's important to distinguish between *describing* and *explaining*.
DevinC
I signed up for this course, rather than another introductory calculus course offered through Coursera, because Ghrist's approach seems radically different than the industry standard. His Funny Little Calculus Textbook starts off with functions, but immediately jumps to Taylor series, assuming the reader knows how to take the derivatives of simple polynomials. In other words, it seems more like a course about understanding calculus than doing calculus.
s2redux
Trying to figure out why George Takei agreed to read this script while wearing pants that are 2 sizes too small.
SamSam
Yay, I already signed up for this course a few months ago!
Even as a programmer I've always felt my basic calc was a bit rusty, and while I could probably just take a two-session refresher course and jump straight to Calc 2, this course looked fun. Now… hopefully I can stick with the schedule better than I could with "The History of the World Since 1300." Who would have guessed that 700 years of history would require lots of reading and lectures? (The course was very good, and I made it through four weeks on-schedule, but in the end I didn't have nearly enough time.)
I have all the prereqs; I just wish I remembered half of them. This class looks fun. Oh well.
sburns54
Holy moley! It's still as densely unapproachable to me as I remember it being when I flunked it in high school! Even with cartoons, which always grab my attention! I was lost by 1:03 seconds in! Thank goodness there are other people that can do this stuff and put it to practical use. I'll just stay in the kitchen, if anyone needs a sandwich.
SamSam
There's also a more basic Calculus One course:
The only prereqs are highschool algebra and trig.
Alissa Mower Clough
I'm getting somewhat aroused. In which sense, I really don't know.
penguinchris
I did a computer science course on Coursera in the spring that I thought was very good. I then tried another and didn't like the way it was done and gave up on it. I definitely need a calculus refresher and this looks good so I'm sold on this one, I hope it turns out well.
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Maths is everywhere, often where we least expect it. Award-winning professor Steven Strogatz acts as our guide as he takes us on a tour of numbers that - unbeknownst to the most of us - form a fascinating and integral part of our everyday lives. In The Joy of X, Strogatz explains the great ideas of maths - from negative numbers to calculus, fat tails... more...
The easy way to brush up on the math skills you need in real life Not everyone retains the math they learned in school. Like any skill, your ability to speak "math" can deteriorate if left unused. From adding and subtracting money in a bank account to figuring out the number of shingles to put on a roof, math in all of its forms factors into daily... more...
Forget the jargon. Forget the anxiety. Just remember the math. In this age of cheap calculators and powerful spreadsheets, who needs to know math? The answer is: everyone. Math is all around us. We confront it shopping in the supermarket, paying our bills, checking the sports stats, and working at our jobs. It is also one of the most fascinating-and... more...
This work presents principles of thin plate and shell theories - emphasizing novel analytical and numerical methods of solving linear and nonlinear plate and shell dilemmas, and new theories for the design and analysis of thin plate-shell structures. more...
Turbulence modelling is critically important for industries dealing with fluid flow and for applied mathematicians. This collection of lecture courses presented at a Newton Institute instructional conference on the title topic by leading researchers, has been edited or rewritten to provide a coherent account suitable for self-study. more...
The Generalized Riemann Problem (GRP) algorithm comprises common schemes of numerical simulation of compressible, inviscid time-dependent flow. This monograph, including examples illustrating the algorithm's applications, presents the GRP methodology beginning with its underlying mathematical principles. The book is accessible to researchers and graduate... more...
This book is a comprehensive introduction to the mathematical theory of vorticity and incompressible flow ranging from elementary introductory material to current research topics. The first half forms an introductory graduate course on vorticity and incompressible flow. The second half comprises a modern applied mathematics graduate course on the weak... more...
This book presents the current state of the art in computational models for turbulent reacting flows, and analyzes carefully the strengths and weaknesses of the various techniques described. The focus is on formulation of practical models as opposed to numerical issues arising from their solution. more...
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25. Use the formula for sin(x+y) to obtain the identity sin(t + /2) = cos t
26. Use the formula for cos(x+y) to obtain the identity cos(t /2) = sin t
27. Show that sin( x) = sin x.
28. Show that cos( x) = cos x.
29. Use the addition formulas to express tan(x+) in terms of tan(x).
30. Use the addition formulas to express cotan(x+) in terms of cotan(x).
Applications
31.†Sales Fluctuations Sales of General Motors cars and light trucks in 1996 fluctuated from a high of $95 billion in October (t = 0) to a low of $80 billion in April (t = 6) Construct a cosine model for the monthly sales s(t) of General Motors.
† See Exercise 23 in Section 1.
32.*Seasonal Fluctuations Sales of Ocean King Boogie Boards fluctuate sinusoidally from a low of 50 units per week each February 1 (t = 1) to a high of 350 units per week each August 1 (t = 7). Use a cosine function to model the weekly sales s(t) of Ocean King Boogie Boards, where t is time in months.
* See Exercise 24 in Section 1.
Music Musical sounds exhibit the same kind of periodic behavior as the trigonometric functions. High pitched notes have short periods (typically several thousands per second) while the lowest audible notes have periods of about 1/100 second. Electronic synthesizers work by superimposing (adding) sinusoidal functions of different frequencies to create different textures. The following exercises, show some examples of superposition can be used to create interesting periodic functions.
33. Saw-Tooth Wave (a) Graph the following functions in a window with 7 x 7 and 1.5 y 1.5.
y1
=
2
cos x
y3
=
2
cos x
+
2 3
cos 3x
y5
=
2
cos x
+
2 3
cos 3x
+
2 5
cos 5x
(b) Now give a formula for y11 and graph it in the same window.
(c) How would you modify y11 to approximate a saw-tooth wave with an amplitude of 3 and a period of 4?
34. Square Wave Repeat Exercise 33 using sine functions in place of cosine functions in order to approximate a square wave.
35. Harmony If we add two sinusoidal functions whose frequencies are exact ratios of each other, the result is a pleasing sound. The following function models two notes an octave apart together with the intermediate fifth.
y = cos(x) + cos(1.5x) + cos (2x).
Graph this function in the window 0 x 20 and 3 y 3, and estimate the period of the resulting wave.
36. Discord If we add two sinusoidal functions with similar, but unequal, frequency, the result is a function that "pulsates," or exhibits "beats." (Piano tuners often use this phenomenon to help them tune an instrument.) Graph the function
y = cos(x) + cos(.9x)
in the window 50 x 50 and 2 y 2, and estimate the period of the resulting wave.
Communication and Reasoning Exercises
37. Your friend is telling everybody that all six trigonometric exercises can be obtained from the single function sin x. Is he correct? Explain your answer.
38. Another friend claims that all six trigonometric exercises can be obtained from the single function cos x. Is she correct? Explain your answer.
39. If weekly sales of a commodity are given by s(t) = A + Bcos(t), what is the largest B can be? Explain your answer.
40. Complete the following sentence. If the cost of an item is given by c(t) = A + Bcos((t)), then the cost fluctuates by ____ with a period of ____ about a base of ____, peaking at time t = ____.
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Okay, so I'm currently in grade 11 and taking grade 12 Advanced Functions. I've always been really strong in Math and most of my friends are constantly wondering how I solve incredibly difficult problems.
The problem is, even though I do tend to be able to solve difficult questions, I seem to fail at the most basic things (for example, factoring, addition, subtraction, etc.).
Does anyone know of any good online sites that generate good problems for my grade level to practice on? Also, does anyone know of any good strategies to make fewer mistakes (aside from 'practice more')?
Konradz
02-12-2010, 06:01 PM
^ I'm a bit confused. Why would you need exercises for your grade when you say your problems are much simpler?
Could you please explain, more precisely, what your difficulties are and in which context?
+nR.Hikari
02-12-2010, 10:18 PMNored
02-12-2010, 11:34 PM
^ I'm a bit confused. Why would you need exercises for your grade when you say your problems are much simpler?
Could you please explain, more precisely, what your difficulties are and in which context?
Well, the issue is that when I'm faced with a simple problem, I can't seem to make sense of it. For example, a few weeks ago I was supposed to factor 2+2x and I didn't realize that I could factor out '2'. Just two days ago I tried factoring (x-2)³-(x-2)² and couldn't come to a conclusion; I had to ask one of my friends to help and she saw it right away. Other examples are just plain carelessness; 2+2=0, 3³=9, 135+35=150, etc.
On the other hand, series, sequences, functions, complex numbers, difficult equations, etc. seem to be obvious to me. I'm not sure that solving a hundred two-digit addition questions will help, and solving even more complex problems probably isn't feasible to help me with basics. The most common advice I've been given is 'practice more', so I'm trying to follow that as well as ask for other pieces of advice people might have. I'm actually not sure why I have these difficulties; I just do.Currently we're doing polynomial functions. I had a quiz today where all the function stuff I understood (ie. given a function, graph it, or given a graph, find x-intercepts, leading coefficient, degree, etc.) but the last question provided a table of values for a function and I was supposed to use finite differences to find out the degree of the function. I checked all my calculations twice and couldn't find a mistake, but I exhausted all the differences and couldn't come up with an answer.
After checking with my friends, it was apparent that I was given the values for a quartic function, yet I went all the way to 6th differences (which was as far as I could go with the given values) and didn't come up with an answer. It's really simple subtraction, but I seem to have a problem with it.
I'm quite sure that my teachers have gone "WTF?" more than once upon seeing my answers; it's not that I don't understand the material, but I just make a LOT of mistakes doing simple math. I've recently taken to checking over my work, but I never seem to catch the most obvious mistakes; there was a "give a final answer only" question and I wrote 5-2 as the answer.
Some of my previous teachers have given me some advice on how to catch my mistakes (ie. put the test down for 10 minutes, think about something else, then look at it again) but even after using such tactics I still miss the most obvious mistakes.
tl;dr version: I suppose I'm looking for tactics that will help me see what I've done wrong/where I've made mistakes.
Konradz
02-13-2010, 11:39 AM
I think that in addition to trying to "catch your mistakes" (which you must also systematically do of course), you should simply practice on some mental calculus. Start off with that. I can't guarantee you it's gonna be very helpful, but just have someone give you, or if you can manage do so some yourself, some easy mental calculus stuff. Perhaps it'll help get things in?
Also, try to do things more slowly. Since you don't seem to have many problems with the complex aspect of things, you may have some time to think your calculus through carefully. I am very far away from your grade level in Math, most likely, but I know that I always spend a lot of time reading my calculus over and over sometimes to check if i haven't made a retarded mistake, especially in long ass boring calculus, that seems so easy you just do it carelessly.
For example, calculating a length using coordinates. AB = root [ (Xa - Xb)² + (Ya - Yb)² ]
I ALWAYS make mistakes using that formula because it's so god damn annoying, having to put in the coordinates each time and calculating with very simple numbers.
For factoring, go back to the basic stuff you learned many years ago. Take your time and just do a lot of factorization exercises until it becomes a habit.
So that's my suggestion :
- A lot of mental calculus
- A lot of basic exercises especially for factoring
- Take your time to do things
ambius
02-13-2010, 02:16 PM
You should check out:
Art of Problem Solving (
It has got some good info. Under resources, it has some links to some websites with some pretty good questions. If you practice problem solving enough, you'll get better at it (as opposed to "knowing a solution" which does not improve your math ability). XD
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I got to review three of the instructional math video CDs produced by MathTV.com: Basic Mathematics, Algebra 2, and Trigonometry. Note: This was at a time when the videos were offered on CDs. As of 2009, all MathTV.com videos are offered completely FREE of charge online, and they have lots more videos available.
MathTV.com videos are played on your computer. The idea is fairly simple: you watch a teacher solve math problems on the whiteboard. It's like sitting in a classroom and following the teacher; however this is better because you can rewind and replay the 'teacher' as much as you wish.
You see problems on the left, and the video clip on the right.
Each of the CDs covers a wide range of math topics and has extensive amount of video instruction. For example, Basic Mathematics is maybe 7th-8th grade level math, covering whole numbers, fractions, decimals, ratio, proportion, percent, integers, and introduction to very elementary equations. Algebra 2 CD has 9 chapters plus a Review chapter of basic properties and definitions. It covers topics from equations, inequalities, and various functions till sequences, series, and conic sections. Trigonometry CD has eight chapters and an appendix chapter, covering for example right triangle trigonometry, the unit circle, trigonometric equations and identities.
Besides these, MathTv.com offers Prealgebra, Algebra 1, and Word Problems videos. The Basic Mathematics and Prealgebra CD's are almost identical (no need to purchase both) - the
difference is in the order of topics. These two CD's are also the best value, as they contain video solutions to the printable tests, as well as a "Study Guide" in PDF format.
Contents of chapter on equations and inequalities in two variables on MathTV.com CD Algebra 2. The clickable video is the motivational/introductory video clip for this chapter.
Each chapter has 5-8 topics and starts with a motivational video clip showing how some key concept from that chapter is used in society or science. There is also a printable chapter test for each chapter. I liked the motivational or introductory videos; they were cute and to the point.
For each topic within a chapter, there are typically 3-5 problems that are solved on the video clip. After viewing the example solutions, you can go to some practice problems.
The Word Problems CD include Motion problems, Variance problems, age problems, geometry problems etc. - all typical word problems one finds in an algebra course. Oftentimes McKeague uses tables for organizing information, which is a crucial aid for many algebra problems. This CD can be very helpful addition to any algebra textbook. You will get the most value out of it if you try solve the problems YOURSELF beforehand, and then watch the 'expert solution'.
These videos can be used along with any mathematics textbook or workbook you might have - though they seem to follow textbooks written by the instructor, Charles P. McKeague. They are 'companions' - not stand-alone instruction. In other words, you cannot learn everything about algebra or trig just by seeing the videos (though you might learn enough to pass a test). You still need your textbook to explain concepts, symbols, and the whys and wherefores of math. The video clips serve to give you examples of how a skilled teacher solves problems - step by step.
MathTV.com videos can be an excellent way for someone to review and practice 'forgotten' math - for example when needing to pass a test, or for whatever reason you might need to refresh your math.
BUT, they can also be an excellent help for homeschoolers, especially on high school level. Many times homeschooling moms struggle with teaching high school math because they might have forgotten it (when was the last time you solved some absolute value equations or played with trigonometric identities), or because they perhaps didn't learn it real well in the first place.
In those occasions, seeing a teacher solve problems step-by-step can be of great value. In textbooks, these steps are often explained in words, which is much more difficult to follow. Seeing and hearing it (repeatedly if you so wish) can help the learning process compared to just reading a textbook and trying to figure out what happened in each step.
The solutions to problems are quite mechanical, using the common rules. You won't probably find any 'creative' solutions. This is good when you're learning those tecniques and rules; but please note that the CDs do not contain explanations why these rules work or where they come from.
The instructor Charles P. McKeague is also an author of several math textbooks and an inspirational speaker. His style of teaching is very clear. The way he writes on the board and uses symbols is also very clear - concise, yet illustrative and helpful. I was delighted by the way he used notation when solving absolute value inequalities, for example.
Website: Note: This review was written at a time when the videos were offered on CDs. As of 2009, all MathTV.com videos are offered completely FREE of charge online, and they have lots more videos available. The e-pass to access their ebooks, all videos, and XYZ homework is $30 for 12 months; print textbooks run $48 - $68 depending on the subject.
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Mathematical modeling and problem solving is one way to allow students to bring together their previous and new mathematical knowledge in a manner that is visible to the instructor. Many of the state algebra standards refer to using mathematics to solve realworld, mathematically significant problems.
IN THIS SESSION TEACHERS:
Learn strategies to implement Model-Eliciting Activities (MEA) in order to facilitate the development of algebraic thinking within a STEM context.
Grade 6: Recognize and represent relationships between varying quantities; translate from one representation to another;
use patterns, tables, graphs and rules to solve real-world and mathematical problems.
Grade 6: Understand and interpret equations and inequalities involving variables and positive rational numbers. Use
equations and inequalities to represent real-world and mathematical problems; use the idea of maintaining equality to
solve equations. Interpret solutions in the original context.
Grade 7: Understand the concept of proportionality in real-world and mathematical situations, and distinguish between
proportional and other relationships.
Grade 7: Recognize proportional relationships in real-world and mathematical situations; represent these and other
relationships with tables, verbal descriptions, symbols and graphs; solve problems involving proportional relationships
and explain results in the original context
Grade 7: Represent real-world and mathematical situations using equations with variables. Solve equations symbolically,
using the properties of equality. Also solve equations graphically and numerically. Interpret solutions in the original context.
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I had the opportunity to teach an undergraduate course in algebraic geometry, and let me tell you that finding a book at that level was not easy. First, be aware that most books will not be self-contained and will farm out the necessary commutative algebra to outside references.
For a first taste, I would go with An Invitation to Algebraic Geometry by Karen Smith et al. (Springer) It's not really designed for undergrads, and I'm not sure how it would work as a textbook for a course; rather, it was written for people with a certain mathematical maturity who don't know anything about the subject and want to learn the basics. I recommend it because it's short, recent, good for self-study, contains a lot of the classical stuff, and it can give you an idea of the flavor of the subject.
There are tons of algebraic geometry books out there, so I'm sure other good recommendations are forthcoming.
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Streeter-Hutchison Series in Mathematics: Basic Mathematical Skills with Geometry
The "Streeter-Hutchison Series in Mathematics: Basic Mathematical Skills with Geometry, 7/e" by Baratto/Bergman is designed for a one-semester basic ...Show synopsisThe "Streeter-Hutchison Series in Mathematics: Basic Mathematical Skills with Geometry, 7/e" by Baratto/Bergman is designed for a one-semester basic math course. This successful worktext series is appropriate for lecture, learning center, laboratory, or self-paced courses. "Basic Mathematical Skills with Geometry" continues with it's hallmark approach of encouraging the learning mathematics by focusing its coverage on mastering math through practice. The "Streeter-Hutchison" series worktexts seek to provide carefully detailed explanations and accessible pedagogy to introduce basic mathematical skills and put the content in context. With repeated exposure and consistent structure of Streeter's hallmark three-pronged approach to the introduction of basic mathematical skills, students are able to advance quickly in grasping the concepts of the mathematical skill at hand
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This text has been placed on the WWW as a public service
by the Department of Mathematics and Computer Science
of Thiel College. It provides an elementary introduction
to programming Hewlett-Packard
graphing calculators. It deals primarily with the programming techniques required in the Numerical
Analysis course at Thiel,
but it also serves as a good starting point for anyone interested in learning to program these
calculators.
The text is available in two formats. This, the interactive
version, is intended to be read on the computer. There is also a PDF version
that is more suitable for printing. The Acrobat Reader is necessary to
read and print the PDF version. This software is available free from
Adobe.
All references in the interactive version to other parts
of the text are links to the other parts. If you follow one of these links,
use the BACK command on your browser to return to your starting point.
There are four data files needed for Section 9
and one for Section 10 that are
available on the Web. To download these files click here.
When you finish this text, there are a few other programming commands that were not included here
that can be found at More Programming Commands. If you
expect to need some statistics commands in your program, Lesson 35 of the Tutorial mentioned below should be
particularly helpful.
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An investigation of topics including the history of mathematics, number systems, geometry, logic, probability, and statistics. There is an emphasis throughout on problem solving. Recommended for General Education.
For some of you this course might serve to satisfy the math competency requirement, for others this will be just one of the mathematics courses required by your major/minor program. In any case, the main goal of this course is to help you develop and strengthen the foundations of your analytical thinking.
Every day, we are faced with numerous questions, such us: Should I run through this yellow light? What should I eat today? Which courses to enroll next semester? . . . We often have to resolve those questions, make appropriate decisions, and then act according to those decisions. The thinking process required for resolving all kinds of questions, puzzles, problems, is known by the name of analytical thinking. We can use mathematics as a convenient tool for working on the analytical thinking skills. In order to achieve this goal of developing and strengthening your analytical thinking skills, via mathematics, our focus will be on the following questions:
• What does it mean to do mathematics?
• What does it mean to think mathematically?
• What does it mean to understand a piece of mathematics?
In other words, I expect you to:
• Do some mathematics;
• Use mathematical reasoning, i.e., ask questions such as:
– What does (something) mean?
1
2 MATH 155 WAY OF THINKING SYLLABUS - FALL 2003
– How did we get from A to B?
– Is this (a statement, claim, formula, . . . ) correct?
– How do I know that it is correct?
• Strive to understand every idea, concept, problem, solution that we encounter in this course.
The mathematical content which we will use to achieve these objectives will expose you to a variety of areas of mathematics, and thus give you an idea of the importance of mathematics in today's world and a multitude of ways it is being used in practice. We will learn some elements of mathematical logic, set theory, geometry, statistics, probability, consumers mathematics, and some basic algebra.
The General Education aspects of this course. The content and the methods of this course are designed in accordance with general education objectives and the work in this course should help you in developing a number of skills included in the NCTM (National Council of Teachers of Mathematics) 'standards" for mathematics education, and also being among the general education objectives at Viterbo. The main emphasis throughout the course will be on problem solving and developing thinking skills. This includes: (a) writing numbers and performing calculations in various numeration system (INTASC 1), (b) solving simple linear equations (INTASC
1), (c) exploring the mathematical model of simple and compounded interest rates, and learning how to use those ideas in solving the problems of loan payments (INTASC 1), (d) exploring a few major concepts of Euclidean
Geometry, focusing especially on the axiomatic-deductive nature of this mathematical system, including a variety of different proofs of the Pythagorean Theorem (INTASC 1), (e) develop an ability to use deductive reasoning, in the context of the rules of logic and syllogisms, i.e., learn how to make/recognize a valid argument (INTASC 1), (f) some basics of probability and statistics . . . (INTASC 1) Potential benefits of the course. Mastering this material requires to learn how to reason mathematically, and also how to communicate mathematics.
In learning how to do so (on exams, essays, portfolio, and in oral presentations), you will also develop a confidence in your ability to do mathematics. This way you will strengthen your ability to solve problems, analyze arguments, understand abstract concepts.
Other benefits of this course include: cultural skills (appreciation of the history of mathematics and its role in today's world, learning how to handle simple loans, etc.), appreciate the beauty and intellectual honesty of deductive reasoning, thereby adding to life value and aesthetic skills.
I encourage you to read the text at: - the Viterbo critical
thinking web page Text: Robert Blitzer, Thinking Mathematically, Prentice-Hall, 2003. Format: Class sessions will consist of lectures, work in small groups, exams, and individual presentations. I expect students to work out the recommended practice problems and ask for help whenever needed.
MATH 155 WAY OF THINKING SYLLABUS - FALL 2003 3
Resources: Please do not hesitate to contact me for any question you might have; do not let a feeling such as "I am lost . . . " to last. Other resources include:
• Internet and the Blackboard software. There is a lot of material on my web page. There will be some quizzes given using the Blackboard.
• The Learning Center.
• The library. Note that both a video set and a CD set that covers your textbook exist.
You can use either of these to hear a lecture again, or just to see/hear another explanation of a particular topic.
Grading: The final grade is based on homework, exams, presentations, portfolio, and a (cumulative) final exam. There will be opportunities for a small amount of extra credit. The following grading scale applies to individual exams, and to the overall grade as well:
A=90%, AB=87%, B=80%, BC=77%, C=70%, CD=67%, D=60%.
The following exceptions to that scale are possible:
• An A on the final exam (more than 180/200 points) will raise your grade up, one letter, i.e., a B will turn into an A, a BC will become AB, . . . .
• An outstanding presentation, or an outstanding portfolio can raise your grade up a half letter, i.e., a C will turn into a BC, . . . .
• If one is failing the course by the end of the semester, but has over 40% average on exams, and earns at least 55% points on the final, he/she can get a D for the final grade.
• If one is passing the course by the time of the final exam, but earns less than 30% points (a score less than 60/200), that will result in an F for the final grade.
Assignments:
• Recommended practice: First 10, middle 5 and the last 5 problems from each Practice Exercises set in each section that we cover; at least one or two of the Application Exercises, at least one of the Writing in Mathematics Exercise, and at least two of the Critical Thinking Exercises.
These practice problems will not be graded. However, fell free to ask me for help with any difficulty you might have with those problems.
• Four essays, 20 points each:
(1) Essay I - Autobiography: Introduce yourself to me in a 1- 2 pages essay. State your name, and where (city/state) you are coming from. The reason you are taking this course, and what mathematics courses you have had before. What was your experience from those courses and what are your expectations, if any, from this course? This assignment is due Friday, August 29.
(2) Essay II - A mathematical story. In order to make the connection of the first assignment (the Autobiography) with the main goals of the course more explicit, I would like you to recall some of your specific experiences of doing mathematics, and tell me a short story (1-2 pages) about it. In particular, I would like you to address the following questions in this story:
– Try to recall an experience of you actually doing mathematics. Give an example. Describe, make a story about it.
– How about an experience of reasoning mathematically? It would be great if you could give a simple example, and even better if you have had an opportunity to communicate your mathematical reasoning to somebody else.
– Did you ever truly understand a piece of mathematics? Give an example. Describe. Explain.
If your answer to any of the questions above is negative, i.e., you have not had such an experience, then please try to explain how is that possible.
Due: Due Tuesday, September 2, 2003.
(3) Essay III - World without mathematics: another 1-2 pages
20 points essay.
Try to imagine, and describe, a world without mathematics.
Due: Friday, September 5, 2003.
(4) Essay IV - Me, a Mathematician. In this essay, you should answer same questions as in Essay II, but in relation to the material covered in this course. More precisely, the questions are:
– Did you ever, during the work in this course, have an experience of actually doing mathematics. Give an example. Describe. Explain.
– Did you ever, during the work in this course, have an experience of reasoning mathematically? Give an example. Describe. Explain.
– Did you ever truly understand at least one piece of mathematics encountered in this course? Give an example. Describe. Explain.
This last essay is due Monday, December 1, 2003 (the last week of class).
Homework: At the end of each chapter, there is a Chapter Test. Each one of those tests will be due second class period after the corresponding chapter is covered, and each problem on the "test"is worth 1 point.
This rule is a tentative one. Sometimes, I give a different problem set instead of those Chapter reviews.
Exams: There will be three in-class exams, worth 100 points each. An exam will typically cover three chapters worth of material. The exams will be closed notes, closed book. However, a calculator and a formula sheet (but not any worked out problem) is allowed.
MATH 155 WAY OF THINKING SYLLABUS - FALL 2003 5
Before each exam, I will give you a take-home practice exam, which will be very much like the actual exam coming. I will grade (25 points) the first one of those, i.e., the "Exam 1 - Practice", but not the others. I will also allow a makeup (up to 50%) of the lost credit for one of the exams. It will be Exam 1 this time. This makeup will be oral, and will apply to those under 90/100 points on the test, and is to be done within two weeks after the exam. Final Exam: Final exam is a 2-hour, cumulative exam, and is worth 200 points. Portfolio: It should consist of 5 problems, but no two problems should be of the same type (from the same section). Format: You state a problem, write a complete/correct solution to it, and then write a paragraph (or more) explaining why did you choose that particular problem, what did you learn from it, etc.. The portfolio will be worth 50 points.
The problems you choose for the portfolio should illustrate the progress in learning mathematics, the change of the perception (if any) of what mathematics is about, the change (if any) in your perception about your abilities to do mathematics. In other words, the portfolio should provide you with an opportunity to demonstrate what you have learned in the course and the progress you made in that process.
In-class Presentation: The presentation of a proof of the Pythagorean Theorem found on the Internet. Typically, the explanations you will find on the Internet are a bit sketchy. So, part of your job will be to make sure you really understand the proof you are going to present (including filling in the gaps, i.e., the reasons not entirely spelled out in the Internet write-up), and then to clearly explain that proof to your classmates. Sometimes, some people, may find this part quite difficult. Of course, I am here to help you understand and overcome those difficulties, and so please do not hesitate to ask me for help.
You should also be prepared for the questions from the audience (myself and/or other students), and it is expected that you listen closely to other presentations and ask any question you might have.
The presentation will be worth 35 points. In addition to that, one certain problem for one of the exams, or for the final exam, is going to be: State and prove the Pythagorean Theorem.
Usually, I would reserve the last three weeks of class for the presentations. This time, we will try a different model. Starting the last Monday in September, each Monday will be devoted to the presentations. Then, we will take the last week (or more, if necessary) to do just the presentations. The presenters will be determined by a (computer generated) random drawing.
Project: A project worth 50 points will be assigned some time in the second half of the semester. The details will be given then.
Important University Policies: Those are Viterbo's policies on Attendance, Plagiarism, and Sexual Harassment. You can find the statements at:
Disability: Americans with Disability Act: If you are a person with a disability and require any auxiliary aids, services or other accommodations for
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In Constructing Algebra, Catherine Twomey Fosnot and Bill Jacob help teachers recognize, support, and celebrate their students' capacity to structure their worlds algebraically. They identify for teachers the models, contexts, and landmarks that facilitate algebraic thinking in young students, supporting children as they construct mathematical
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