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Beginning and Intermediate Algebra; 2nd Ed
Beginning and Intermediate Algebra was designed to reduce textbook costs to students while not reducing the quality of materials. This text includes many detailed examples for each section along with several problems for students to practice and master concepts. Complete answers are included for students to check work and receive immediate feedback on their progress.
Topics covered include: pre-algebra review, solving linear equations, graphing linear equations, inequalities, systems of linear equations, polynomials, factoring, rational expressions and equations, radicals, quadratics, and functions including exponential, logarithmic and trigonometric.
Each lesson also includes a World View Note which describes how the lesson fits into math history and into the world, including China, Russia, Central America, Persia, Ancient Babylon (present day Iraq) and more! |
Introductory And Intermediate Algebra - 3rd edition
ISBN13:978-0321279224 ISBN10: 0321279220 This edition has also been released as: ISBN13: 978-0321292735 ISBN10: 0321292731
Summary: Lial/Hornsby/McGinnis's Introductory and Intermediate Algebra, 3e gives students the necessary tools to succeed in developmental math courses and prepares them for future math courses and the rest of their lives. The Lial developmental team creates a pattern for success by emphasizing problem solving skills, vocabulary comprehension, real-world applications, and strong exercise sets. In keeping with its proven track record, this revision includes an effective new des...show moreign, many new exercises and applications, and increased Summary Exercises to enhance comprehension and challenge students' knowledge of the subject matter. ...show less
PAPERBACK Fair 0321279220 Student Edition. Missing up to 10 pages. Heavy wrinkling from liquid damage. Does not affect the text. Heavy wear, wrinkling, creasing, Curling or tears on the cover and spi...show morene May be missing front or back cover. May have used stickers or residue. Poor binding causing loose and9.99 +$3.99 s/h
Good
harambee Kansas city, MO
2005 Paperback Good Minimum jacket wear.
$11.21 |
Mathematics
As
mathematics is both a cultural and a technical field of
study, the curriculum is planned with the following objectives:
(1) to offer students an introduction to mathematics as
an important area of human thought; (2) to prepare students
for graduate study in pure or applied mathematics, and in
such related fields as statistics and operations research;
(3) to serve the needs of students in fields that rely substantially
on mathematics, such as the physical, biological, social
and information sciences, engineering, and business administration;
and (4) to provide liberal arts students with an introduction
to the kinds of mathematical and quantitative thinking important
in the contemporary world.
Individual
guidance in the selection of courses and the design of course
sequences to serve particular needs and interests is offered
by all members of the Department to all students, but the
following information will provide a preliminary basis for
making plans and choices.
Advanced
Placement. Students who have taken one of the two College
Board Advanced Placement Program examinations in calculus,
or the examination in statistics, will receive credit as
follows. Students scoring 4 or 5 on the BC examination in
calculus receive eight hours credit, equivalent to Mathematics
133 and 134. Students scoring 3 on the BC examination in
calculus with an AB sub-score of 4 or 5 receive four hours
credit, equivalent to Mathematics 133. Students scoring
4 or 5 on the AB examination in calculus receive four hours
credit, equivalent to Mathematics 133. Students scoring
4 or 5 on the examination in statistics receive four hours
credit, equivalent to Mathematics 113.
Students
given credit for one or more courses in this way do not
need to take a Mathematics Placement Exam. They are encouraged
to place themselves at the appropriate level in the mathematics
curriculum according to the guidelines given below (see
Initial Placement and Course Sequence Suggestions) in consultation
with a member of the Mathematics Department.
Mathematics
Placement Exams. Students wishing to enroll in an entry-level
calculus course (Mathematics 131, 132, or 133) must take
the Calculus Readiness Exam. Likewise, students wishing
to enroll in an entry-level statistics course (Mathematics
113 or 114) must take the Statistics Readiness Exam. The
placement exams are given twice during orientation. At other
times they may be taken by arrangement with the Mathematics
Department secretary. Please note that all students, regardless
of their examination scores, are encouraged to consult with
a member of the Mathematics Department concerning their
placement in the mathematics curriculum.
Initial
Placement and Course Sequence Suggestions. Students
who wish to continue their study of mathematics can choose
among the following courses:
Courses
Without Prerequisites. Students who wish to satisfy
the quantitative proficiency requirement, or who, simply
out of curiosity, want to take a course in mathematics are
encouraged to consider the courses numbered 100 and below.
Entry-level
Statistics Courses. Students whose primary interest
is in the social orbehavioral
sciences and who have no need for calculus are encouraged
to consider enrolling in Mathematics 113 - Statistical Methods
for the Social and Behavioral Sciences or Mathematics 114
- Statistical Methods for the Biological Sciences. These
courses presuppose good algebra skills and require an appropriate
score on the Statistics Readiness Exam. Students with less
background are encouraged to consider enrolling in Mathematics
100 - Elementary Statistics.
Entry-level
Calculus Courses. Students whose interests are in mathematics,
or in a field requiring calculus, will normally enroll in
Mathematics 131 - Calculus Ia: Limits, Continuity, and Differentiation,
or in Mathematics 133 - Calculus I: Limits, Continuity,
Differentiation, Integration, and Applications. The particular
course, Mathematics 131 or Mathematics 133, depends on the
student's score on the Calculus Readiness Exam. Note that
students who wish to continue with calculus after completing
Mathematics 131 should take its sequel, Mathematics 132
- Calculus Ib: Integration and Applications. The two-semester
sequence Mathematics 131, 132 is equivalent to the more
intensive single semester course, Mathematics 133.
Courses
Following Entry-level Calculus. Students whose secondary-school
preparation includes satisfactory work in calculus obtained
in the College Board Advanced Placement Program or in another
comparable course of study, as well as students who have
completed either Mathematics 132 or 133, can continue their
study of calculus with Mathematics 134 - Calculus II: Special
Functions, Integration Techniques, and Power Series. This
course completes the standard introduction to the calculus
of functions of one variable.
Courses
Following Calculus. Students who have completed Mathematics
134 or have been granted credit for this course through
the College Board Advanced Placement Program or another
comparable course of study can register for Mathematics
220 - Discrete Mathematics, or Mathematics 231 - Multivariable
Calculus, or Mathematics 232 - Linear Algebra. Students
planning to major in mathematics are strongly encouraged
to enroll first in Mathematics 220, and thereafter in Mathematics
231 and Mathematics 232. Students planning a concentration
in Applied Mathematics will also need to take Mathematics
113 - Statistical Methods for the Social and Behavioral
Sciences or Mathematics 114 - Statistical Methods for the
Biological Sciences. First-year students should not register
for a 300-level mathematics course without consulting with
a member of the Mathematics Department.
Major.
A major in mathematics consists of thirty-four hours, including
Mathematics 220, 231 and 232. In addition, students select
one of the following two concentrations:
Concentration
in applied mathematics. Students selecting this concentration
must take either Mathematics 113 or Mathematics 114, and
at least 12 hours of advanced mathematics courses numbered
300 and above, including either Mathematics 301 or 327,
and three courses from among 331, 335, 336, 337, 338, and
340.
Concentration
in pure mathematics. Students selecting this concentration
must take at least 12 hours of advanced mathematics courses
numbered 300 and above, including both Mathematics 301 and
327, and at least one of the following two-course sequences:
Mathematics 301/302, 301/356, 301/358, 327/328 or 327/329.
Important
note: Students planning to pursue graduate work in mathematics,
or a closely related field, need to complete more than the
minimum requirements for the mathematics major. Such students
should plan their major carefully with the advice of a member
of the Mathematics Department.
It
is strongly urged that students specializing in mathematics
also obtain substantial background in some field that uses
mathematics. In particular, students majoring in mathematics
are encouraged to gain some experience with computing. To
that end, credit for one computer science course (that would
also count toward a Computer Science major) may also be
counted toward the thirty-four hour requirement for the
major in mathematics. Private readings are also available,
with the consent of an instructor, in any area of mathematics
appropriate for a student's major. Finally, interdisciplinary
majors involving a coherent program of work in mathematics
and a related field can be arranged through the College
Individual Majors Committee to suit special student interests
and needs.
Minor.
A minor in mathematics consists of at least fifteen hours
of course work, including any three of Mathematics 220,
231, 232, 234, and at least six hours of courses numbered
300 and above.
Honors.
At the end of their junior year, students with outstanding
records are invited to participate in the Mathematics honors
program. Seniors in the program normally elect three hours
of independent study each semester. This special study is
supervised by a faculty advisor who works closely with the
student. Honors students take a comprehensive examination,
written and oral, at the end of the senior year. This honors
examination is conducted by an outside examiner and is designed
to test both the candidate's knowledge of undergraduate
mathematics and mastery of the subjects emphasized in his
or her independent honors study.
Winter
Term. Most members of the Mathematics Department will
be participating in Winter Term 2002, and will be available
to sponsor projects.
Avocational
interests of Department members which could form the basis
for a sponsored Winter Term project include electronic composition
and synthesis of music, games of strategy, and juggling.
For further information regarding these possibilities, inquire
in the Mathematics Department office.
Distinguished
Visiting Scholar. Thanks to the generosity of alumni,
the Mathematics Department is able to sponsor an annual
visit by an eminent mathematical scientist who will conduct
classes and deliver a public lecture.
John
D. Baum Memorial Prize in Mathematics. Established by
the Mathematics Department, this $100 prize is awarded annually
to the Oberlin College student who has achieved the highest
score on the William Lowell Putnam Mathematical Competition.
Rebecca
Cary Orr Memorial Prize in Mathematics. Established
by the family and friends of Rebecca Cary Orr, this $2000
prize is awarded annually by the Mathematics Department
on the basis of scholastic achievement and promise for future
professional accomplishment.
Introductory
Courses
030. Topics
in Contemporary Mathematics 3 hours 3NS, QPf
The interaction of mathematics with the social sciences
is the central theme. Topics are drawn from: graph theory,
game theory, linear programming, coding theory, exploratory
data analysis, and combinatorics. Applications are given
to social choice, decision-making, management and ecological
modeling. Prerequisites: A working knowledge of elementary
algebra and geometry. Notes: This course does not
count toward a major in Mathematics. It is intended for
students who have not satisfied the quantitative proficiency
requirement. Enrollment Limit: 30. Sem 1 MATH-030-01 MWF
3:30-4:20 Mr. Henle Sem 2 MATH-030-01 MWF
9:00-9:50 Mr. Balasuriya
080. Lies,
Damned Lies, and Decisions 3 hours 3NS, QPf
An introduction to the use of data in everyday life, particularly
decision-making. Topics include descriptive statistics and
graphics, data collection methods, probability trees, utility
theory, and decision trees. Notes: This course is
not equivalent to elementary statistics (MATH 100, MATH
113, or MATH 114) and does not count toward a major in mathematics.
It is intended for students who have not satisfied the quantitative
proficiency requirement. Students may not receive credit
for both MATH 080 and any of MATH 100, MATH 113, or MATH
114. Enrollment Limit: 30.
Sem 1 MATH-080-01 MWF 1:30-2:20 Mr. Witmer
090. Environmental
Mathematics 3 hours 3NS, QPh
This course focuses on the application of mathematics to
problems concerning the environment. Topics include simulation
(models of population growth, predator-prey relationships,
and epidemics); optimization (applications to groundwater
hydrology, herbivore foraging, and transportation of hazardous
wastes); and decision analysis (applications to management
of endangered species and resolution of environmental disputes).
Notes: This course does not count toward a major
in mathematics. It is intended for students who have not
satisfied the quantitative proficiency requirement. Not
open to any student who has received credit for a course
in mathematics numbered 133 or higher. Enrollment Limit:
20. Sem 2 MATH 090-01 MWF
2:30-3:20 Mr. Bosch
100. Elementary
Statistics 4 hours 4NS, QPf
An introduction to the statistical analysis of data. Topics
include exploratory data analysis, probability, sampling,
estimation, and hypothesis testing. Statistical software
is introduced, but no prior computer experience is assumed.
This course focuses on statistical ideas and downplays mathematical
formulas. It is intended for students in the social sciences
and humanities with minimal mathematical experience who
have not satisfied the quantitative proficiency requirement.
Notes: MATH 100 does not count toward a mathematics
major and is not open to students who have completed a semester
of calculus. Students may not receive credit for more than
one of MATH 100, MATH 113, and MATH 114. Enrollment Limit:
36. Sem 1 MATH-100-01 MTuThF
1:30-2:20 Mr. Bosch Sem 2 MATH-100-01 MTuThF
1:30-2:20 Mr. Andrews, Mr. Bosch
113. Statistical
Methods for the Social and Behavioral A broad
spectrum of examples is employed. Statistical software is
introduced, but no prior computer experience is assumed.
Prerequisite: An appropriate score on the Statistics
Readiness Exam. Notes: The statistical content of this course
is largely the same as MATH 114; the applications are different.
Students may not receive credit for more than one of MATH
100, MATH 113, and MATH 114. Consent of instructor required. Sem 1 MATH-113-01 MWF
10:00-10:50 Mr. Andrews Limit: 36 Laboratories
114. Statistical
Methods for the Biological Biological
and medical examples are emphasized. Statistical software
is introduced, but no prior computer experience is assumed.
Prerequisites: An appropriate score on the Statistics
Readiness Exam. Notes: The statistical content of
this course is largely the same as MATH 113; the applications
are different. Students may not receive credit for more
than one of MATH 100, MATH 113, and MATH 114. Consent
of instructor required. Sem 1 MATH-114-01 MWF
9:00-9:50 Mr. Andrews Limit: 36 Laboratories
234.
Differential Equations 3 hours
3NS, QPf
An introduction to analytic, qualitative and numerical methods
for solving ordinary differential equations. Topics include
general first order equations, linear first and second order
equations, numerical methods (Euler, Runge-Kutta), systems
of first order equations, phase plane analysis, and Laplace
Transforms. There is emphasis throughout the course on geometric
and qualitative interpretations of differential equations,
as well as applications to the natural sciences. Prerequisite:
MATH 231. Enrollment Limit: 32. Sem 1 MATH-234-01 TuTh
11:00-12:20 Mr. Walsh
240. Applied
Mathematics 3 hours 3NS, QPf
This course will cover several areas of applied mathematical
modeling including linear programming and optimization,
multiple regression, and dynamical systems. Knowledge of
these topics will permit students to construct models that
accurately represent data in a wide range of fields from
image compression to economic theory to population dynamics.
Students will use a variety of software packages to analyze
real world data. Prerequisites: MATH 133 and either
MATH 113 or MATH 114. Enrollment Limit: 20. Sem 2 MATH-240-01 MWF
11:00-11:50 Mr. Andrews, Mr. Bosch
Advanced
Courses
301. Advanced
Calculus 3 hours 3NS, QPf
A rigorous examination of the basic elements of analysis.
The structure of the real number system, continuity, differentiability,
uniform continuity, integrability of functions of a single
variable, sequences, series, and uniform convergence are typical
topics to be explored. Prerequisite: MATH 231. MATH
220 is also highly recommended. Sem 1 MATH-301-01 MWF
1:30-2:20 Mr. Young
302. Topics
in Advanced Calculus: Chaos, Fractals and Dynamics 3
hours 3NS, QPf
This course applies the techniques of Advanced Calculus to
the study of chaotic dynamical systems. One and two dimensional
dynamics, attractors, iterated function systems, and fractal
dimension are typical topics to be explored. Application to
the physical sciences, computer graphics and other branches
of mathematics will be given. Prerequisite: MATH 301
or consent of instructor. Note: Given in alternate
years only. Sem 2 MATH-302-01 MWF
11:00-11:50 Mr. Walsh
317. Number
Theory 3 hours 3NS, QPf
This course is an introduction to number theory. Topics include
primality, divisibility, modular arithmetic, finite fields,
cryptography, and elliptic curves. Emphasis will be placed
both on theoretical questions and on algorithms for computation.
Prerequisite: MATH 301 or consent of the instructor.
Note: Given in alternate years only. Sem 2 MATH-317-01 MWF
2:30-3:20 Mr. Schirokauer
331. Optimization 3
hours 3NS, QPf
An introduction to linear, integer, and nonlinear programming.
Emphasis is placed on the theory of mathematical programming
and the analysis of optimization algorithms. These are applied
to significant problems in the fields of medicine, finance,
public policy, transportation, and telecommunications. Prerequisites:
MATH 231 and MATH 232. Sem 1 MATH-331-01 MWF
3:30-4:20 Mr. Bosch
337. Data
Analysis 3 hours 3NS, QPf In this course students will be given an introduction
to the theory and use of regression graphics. Special regression
and graphics software will be used to study relationships
among several variables. Topics will include regression smoothing,
residual plots, scatterplot matrices, three-dimensional plots,
transformations of predictors and of response variables, and
added-variable plots. Note: Given in alternate years
only. Prerequisites: MATH 113 or 114 and MATH 232 or
consent of the instructor. Sem 2 MATH-337-01 MWF
3:30-4:20 Mr. Witmer
350. Geometry 3
hours 3NS, QPf
This course explores some of the mathematics used to describe
space and distance. Topics will include some or all of the
following: comparison of Euclidean and non-Euclidean geometries,
models of hyperbolic spaces, constructibility problems, and
the differential geometry of curves and surfaces in three-space.
Notes: Given in alternate years only. Prerequisite:
MATH 220.
MATH 231 and MATH 232 are also strongly recommended. Sem 1 MATH-350-01 MWF
10:00-10:50 Mr. Henle |
What does it mean to know mathematics? How does meaning in mathematics education connect to common sense or to the meaning of mathematics itself? How are meanings constructed and communicated and what are the dilemmas related to these processes? There are many answers to these questions, some of which might appear to be contradictory. Thus understanding... more...
Deals with the specific characteristics of mathematical communication in the classroom. This book offers a presentation and an application of the fundamental research method in mathematics education that establishes a reciprocal relationship between everyday classroom communication and epistemological conditions of mathematical knowledge. more...
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Graph Theory, Combinatorics and Algorithms: Interdisciplinary Applications focuses on discrete mathematics and combinatorial algorithms interacting with real world problems in computer science, operations research, applied mathematics and engineering. The book contains eleven chapters written by experts in their respective fields, and covers a wide... more...
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Computing the algebra of secondary cohomology operations, that enrich the structure of the Steenrod algebra, this book solves a long-standing problem on the algebra of secondary cohomology operations. It develops a different algebraic theory of such operations. The results have an impact on the Adams spectral sequence. more...
Presents treatment of the structure theory of association schemes. This book presents generalization of Sylow's group theoretic theorems to scheme theory and also the algebraic proof of Tits' theorem on buildings of spherical type. It also includes characterization of Glauberman's, Z-involutions and is aimed at advanced undergraduate students. more...
Geometric Topology is a foundational component of modern mathematics, involving the study of spacial properties and invariants of familiar objects such as manifolds and complexes. This volume, which is intended both as an introduction to the subject and as a wide ranging resouce for those already grounded in it, consists of 21 expository surveys... more... |
Fundamentals of Precalculus is designed to review the fundamental topics that are necessary for success in calculus. Containing only five chapters, this text contains the rigor essential for building a strong foundation of mathematical skills and concepts, and at the same time supports student...
With its lively, conversational tone and practical focus, this new edition mixes applied and theoretical aspects for a solid introduction to cryptography and security, including the latest significant advancements in the field.
Introduction to Logic is a combined text and workbook for students beginning their study of logic. The workbook style allows students to proceed at their own pace, checking their progress in the end-of-chapter exercises.
The text covers propositional logic and predicate logic with identity, th...
Strong algebra skills are crucial to success in applied calculus. This text is designed to bolster these skills while students study applied calculus. As students make their way through the calculus course, this supplemental text shows them the relevant algebra topics and points out potential proble... |
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Visual Mathematics is a highly interactive visualization software (containing -at least- 67 Visual Mathematics, a member of the Virtual Dynamics Mathematics Virtual Laboratory, is an Intuitively-Easy-To-Use software.
Visual Mathematics modules include the theory necessary to understand every theme, they include very many solved examples. Every student should have this powerful tool at home.
Teachers use Visual Mathematics to prepare homeworks and tests in a short time.
With Visual Mathematics the student solves homework problems while he/she really learns and enjoys mathematics. Visual Mathematics may be used (1) in the classroom, to very easily make clear the topics the teacher covers, (2) in the school library, as reference to review themes covered in classes (3) at home, for the student to study at his own pace and understand while solving and visualizing hundreds of problems. Teachers may use Visual Mathematics to prepare classes. <>
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Calc 3D is a collection of mathematical tools for highschool and university. |
. So, coming nearer the end of the course, this lecture will be a mixture of the linear algebra that comes with a change of basis.
And a change of basis from one basis to another basis is something you really do in applications.
And, I would like to talk about those applications.
I got a little bit involved with compression.
Compressing a signal, compressing an image.
And that's exactly change-of-basis.
And then, the main theme in this chapter is th- the connection between a linear transformation, which doesn't have to have coordinates, and the matrix that tells us that transformation with respect to coordinates. So the matrix is the coordinate-based description of the linear transformation. Let me start out with the nice part, which is just to tell you something about image compression. Those of you -- well, everybody's going to meet compression, because you know that the amount of data that we're getting -- well, these lectures are compressed. So that, actually, probably you see my motion as jerky? Shall I use that word? Have you looked on the web? I should like to find a better word.
Compressed, let's say. So the complete signal is, of course, in those video cameras, and in the videotape, but that goes to the bottom of building nine, and out of that comes a jumpy motion because it uses a standard system for compressing images.
And, you'll notice that the stuff that sits on the board comes very clearly, but it's my motion that needs a whole lot of bits, right? So, and if I were to run up and back up there and back, that would need too many bits, and I'd be compressed even more. So, what does compression mean? Let me just think of a still image.
And of course, satellites, and computations of the climate, computations of combustion, the computers and sensors of all kinds are just giving us overwhelming amounts of data. The Web is, too.
Now, some compression can be done with no loss.
Lossless compression is possible just using, sort of, the fact that there are redundancies.
But I'm talking here about lossy compression.
So I'm talking about -- here's an image.
And what does an image consist of? It consists of a lot of little pixels, right? Maybe five hundred and twelve by five hundred and twelve.
Two to the ninth by two to the ninth pixels, and so this is pixel number one, one, so that's a pixel.
And if we're in black and white, the typical pixel would tell us a gray-scale, from zero to two fifty five.
So a pixel is usually a value of one of the xi, so this would be the i-th pixel, is -- it's usually a real number on a scale from zero to two fifty five.
In other words, two to the eighth possibilities. So usually, that's the standard, so that's eight -- eight bits.
But then we have that for every pixel, so we have five hundred and twelve squared pixels, we're really operating x is a vector in R^n, but what is n? n is five hundred and twelve squared.
That's our problem, right there.
A pixel is a vector that gives us the information about the image. I'm sorry.
The image that comes through is a vector of that length that -- that's the information that we have about the image, if it's a color image, we would have three times that length, because we'd need three coordinates to get color.
So it would be three times five hundred and twelve squared.
It's an enormous amount of information, and we couldn't send out the image for these lectures without compressing it.
It would overload the system. So it has to be compressed.
The standard compression, and still used with lectures is, called JPEG. I think that stands for Joint Photographic Experts Group. They established a system of compression. And I just want to tell you what it's about. It's a change-of-basis.
What basis do we have? The current basis we have is, you could say, the standard basis is, every pixel, give a value.
So that's like we have a vector x which is five hundred and twelve squared long and, in the i-th position, we get a number like one twenty one or something.
The pixel next to it might be one twenty four, maybe where my tie begins to enter, so if it was mostly blue shirt, this would be a slight difference in shading, but pretty close, then the tie would be a different color, so we might have quite a few pixels for the blue shirt, and a whole lot more for the blackboard, that are very close. And that's what are very correlated. And that's what gives us the possibility of compression. For example, before the lecture starts, if we had a blank blackboard, then there's an image, but it would make no sense to take that image and tell you what it is pixel by pixel.
I mean, there's a case in which all pixel values, all gray levels are the same -- or practically the same, depending on the erasing of the board, but extremely close -- and, so that's an image where the standard basis is lousy.
That's the basic fact, that the standard basis which gives the value of every pixel makes no use of the fact that we're getting a whole lot of pixels whose gray levels -- the neighboring pixels tend to have the same gray level as their neighbors. So how do we take advantage of that fact? Well, one basis vector that would be extremely nice to include in the basis would be a vector of all ones. That's not in our standard basis, so let me just write again, the standard basis is our one, and all the rest zeroes, zero, one, and all the rest, zeroes, everybody knows what these standard basis is.
Now, any other basis for R -- so this is -- for this very high-dimensional space -- now I'm going to speak about a better basis. Better basis -- and let me just emphasize, one vector that would be extremely nice to have in that basis is the vector of all ones.
Why is that? Let me just say again, because that vector of all ones, by itself, one vector is able to completely give the information on a solid image. Of course, our image won't be solid, it will have a mix of solid and signal.
So having that one vector in the basis is going to save us a whole lot. Now, the question is, what other vectors should be in the basis? The extreme vector in the basis might be a vector of one minus one, one minus one, one minus one.
That would be a vector that shows -- I mean, that's like a checkerboard vector, right? That's a vector that would, if the image was like a huge checkerboard of plus, minus, plus, minus, plus, minus, that vector would carry the whole signal.
But much more common would be maybe to have half the image, darker and the other half lighter.
So another vector that might be quite useful in here would be half ones and half minus ones. I'm just trying to get across the idea of that a basis could be where, that first of all, we've got the bases at our disposal.
Like, we're free to choose that.
And it's a billion-dollar decision what we choose.
So, and TV people would rather pre- would prefer one basis based on the way the signal is scanned, and movie people would prefer another, I mean, there's giant politics in this question that really reduces to a linear algebra problem, what basis to choose. I'll just mention the best known basis, which JPEG uses, -- let me put that here -- is the Fourier basis. So when you use the Fourier basis, that includes -- this is the constant vector, the D C vector if we're electrical engineers, the l- vector of all ones, so it would include one, one, one, one. Often eight by eight is a good choice. Eight by eight is a good choice. So, what do I mean by this eight by eight? I mean that the big signal, which is five twelve by five twelve, gets broken down, and JPEG does this, into eight by eight blocks. And we -- sort of, this is too much to deal with at once. So what JPEG does is take this eight by eight block, which is sixty four coefficients, sixty four, pixels, and changes the basis on that piece.
And then, now, let's see, I was going to write down Fourier, so you remember Fourier as this vector of all ones, and then, the vector -- oh, well, actually, I gave a lecture earlier about the Fourier matrix, this matrix whose columns are powers of a complex number w. I won't repeat that, because I don't want to go into the details of the Fourier basis, just to tell you how compression works.
So what happens in JPEG? What happens to the video, to each image, of these lectures? It gets broken into eight by eight blocks.
OK. Within each block, we have sixty four coefficients, sixty four basis vectors, sixty four pixels, and we change basis in sixty four dimensional space using these Fourier vectors. Just note, that was a lossless step. Let me emphasize.
In comes the signal x. We change basis.
This is the basis change. Change basis.
Choose a better basis. So it produces, the coefficients c. So sixty four pixels come in, sixty four coefficients come out.
Now comes the compression. Now come -- this was lossless.
It's just -- we know that R -- R sixty four has plenty of bases, and we've chosen one. Now, in that basis, we write the signal in that basis, and that's what my lecture -- that's the math part of my lecture.
Now here's the application part.
The next part is going to be the compression step.
And that's lossy. We're going to lose information. And what will actually happen at that step? Well, one thing we could do is just throw away the small coefficients.
So that's called thresholding, we set some threshold.
Every coefficient, every basis vector that's not in there more than the threshold value, and we set them threshold so that our eye can't see the difference, or can hardly see the difference, whether we throw away that little bit of that basis vector or keep it. So this compression step produces a compressed set of coefficients. I'll just keep going here.
So it keeps going, this compression step produces some coefficient c hat. And with many zeroes.
So that's where the compression came. Probably, there is enough of this vector of all ones -- we very seldom throw that away. Usually, its coefficient will be large. But the coefficient of something like this, that quickly alternative vector, there's probably very little of that in any smooth signal. That's high-frequency -- this is low-frequency, zero frequency.
This stuff is the highest frequency we could have, and if the noise, the jitter is producing that sort of output, but a smooth lecture like this one is, has very little of that highest frequency, very little noise in this lecture.
OK, so we throw away whatever there is, and we're left with just a few coefficients, and then we reconstruct a signal using those coefficients. We take those coefficients, times their basis vectors, but this sum doesn't have sixty four terms any more. Probably, it has about two or three terms. So that would -- say it has three terms. From sixty four down to three, that's compression of twenty one to one.
That's the kind of compression you're looking for.
And everybody is looking for that sort of compression.
Let's see, I guess I met the problem with the FBI and fingerprints. So there's a whole lot of still images. You know, with your thumb, you make these inky marks which go somewhere.
it used to go to Washington and get stored in a big file. So Washington had a file of thirty million murderers, cheaters on quizzes, other stuff, and actually, there was no way to retrieve them in time. So suppose you're at the police station, they say, OK, this person may have done this, check with Washington, have they got -- are his or her fingerprints on file? Well, Washington won't know the answer within a week if it's got filing cabinets full of fingerprints. So of course, the natural step is digitizing. So all fingerprints are now digitized, so now it's at least electronic, but still there's too much information in each one.
I mean, you can't search through that many, fingerprints if the digital image is five twelve squared by five twelve squared, if it's that many pixels.
So you get compressed. So the FBI had to decide what basis to choose for compression of fingerprints.
And then they built a big new facility in West Virginia, and that's where fingerprints now are sent.
So I think, if you get your fingerprints done now at the police station, if it's an up-to-date police station, it happens digitally, and the signal is sent digitally, and then in West Virginia, it's compressed and indexed. And then, if they want to find you, they can do it within minutes instead of within a week. OK.
So this compression comes up for signals, for images, for video -- which is, like these lectures -- there's another aspect. You could treat the video as one still image after another one, and compress each one, and then run them and make a video.
But that misses -- well, you can see why that's not optimal. In a video thing, you have a sequence of images, so video is really a sequence of images but what about one image to the next image? They're extremely correlated. I mean that I'm getting an image every split-second, and also, I'm moving slightly.
That's what's producing the, jumpy motion on the video. But I'm not, like, you know -- each image in the sequence is pretty close to the one before.
So you have to use, like, prediction and correction. I mean, the image of me one instant -- one time-step later, you would assume would be the same, and then plus a small correction. And you would only code and digitize the correction, and compress the correction. So a sequence of images that's highly correlated and the problem in compression is always to use this correlation, this fact that, in time, or in space, things don't change instantly, they're very often smooth changes, and, you can predict one value from the previous value.
OK. So those are applications which are pure linear algebra. I could, well, maybe you'll allow me to tell you, and the book describes, the new basis that's the competition for Fourier.
So the competition for Fourier is called wavelets, and I can describe what that basis is like, say, in the eight by eight case.
So the eight by eight wavelet basis is the vector of all ones, eight ones, then the vector of four ones and four minus ones, then the vector of two ones, and two minus ones, and four zeroes. And also the vector of four zeroes and two ones and two minus ones.
So now I'm up to four, and I need four more, right? For R^8? The next basis vector will be one minus one and six zeroes, and then three more like that, with the one minus one there, and there, and there. So those are eight vectors in eight-dimensional space, those are called wavelets, and it's a very simple wavelet choice, it's a more sophisticated choice.
This is a little jumpy, to jump between one and minus one. And, actually, you can see, now, suppose you compare the wavelet basis with the Fourier basis above.
How could I write this guy, which is in the Fourier basis, it's an eight -- it's a vector in R^8. How would I write that as a combination of the wavelet basis? Have I told you enough about the wavelet basis that you can see, how does this very fast guy -- what combination of the wavelet basis is that very fast guy? It would be this one -- it would be the sum of these four, right? That very fast guy will be that one minus one, and the next one, and the next one, and the next one. So this is the sum of those last four wavelets. This one, we've kept, and so on. So, each -- well, every -- well, that's what a basis does.
Every vector in R^8 is some combination of those, and for the linear algebra -- so the linear algebra is this step, find the coefficient. That's the step we want to take. What if I give you the basis, like this wavelet basis, and I give you the pixel -- so here are the pixel values, P1, P2, down to P8 -- what's the job? What's the linear algebra here? So these are the values, this is in the standard basis, right? Those are just the values at eight successive points. I guess I'm dropping down to one dimension, instead of eight by eight, I'm just going to take eight pixel values along that first top row. So what do I want to do? In standard basis, here are the pixel values.
I want to write that as a combination of c1 times this guy, plus c2 times this guy, plus c3, these are the coefficients, plus c4 times this one -- do you see what I'm doing? I want to write this vector P as a combination of c1 times the first wavelet plus c8 times the eighth wavelet. That's the transform step.
That's the lossless step. That's the step from P -- oh, I'm calling it P here, and I called it x there, so let me -- at the risk of moving, and therefore making this jumpy -- suppose the signal I'm now calling P, that a pixel values, and I'm looking for the coefficients. OK, tell me how to do it.
If I give you eight basis vectors, and I give you the input signal, and I ask for the coefficients, what do I do? What's the step? I'm trying to solve this, I want to know the eight coefficients, so I'm changing from the standard basis, which is just the eight gray-scale values to the wavelet basis, where the same vector is represented by eight numbers. It's got to take eight numbers to tell you a vector in R^8, and those eight numbers are the coefficients of the basis. Look, we've done this thing before. There is the equation in vector notation, we want to see it as a matrix.
This is a combination of columns of the wavelet matrix, right? This is P equals c1, c2, down to c8, and these guys are the columns.
I mean, this is the step that we're constantly taking in this course, the first basis vector goes in the first column, the second basis vector goes in the second column, and so on, the eight columns of this wavelet matrix are the eight basis vectors. This is a wavelet matrix W. So, the step to change basis -- so now I'm finally coming to this change-of-basis, so the change of basis that, let me stay with this board, but -- well, let me just go above it, here.
So the standard basis, we know, the wavelet basis we have here, and the transform is simply, solve the equations, P=W C. So the coefficients are W inverse P. Right.
This shows a critical point. A good basis has a nice, fast, inverse. So good basis means what? Eh? So this is like the billion-dollar competition, and it's not over yet. People are going to come up with better bases than these.
So a good basis will be, first good thing would be fast.
I have to be able to multiply by W fast, and multiply by W -- by its inverse fast. That's -- if a basis doesn't allow you to do that fast, then it's going to take so much time that you can't afford it. So these bases -- the Fourier basis, everybody said, OK, I know how to deal quickly with the Fourier basis, because we have something called the Fast Fourier Transform. So there's a FFT that came in my earlier lecture, and comes in the last chapter of the book, so change-of-basis is done -- if, for the Fourier basis, it's done fast by the FFT and there's a fast wavelet transform. I can change, for this wavelet example, this matrix is easy to invert.
It's just somebody had a smart idea in choosing that wavelet basis and inverting it, it has a nice inverse.
Actually, you can see why it has a nice inverse.
Do you see any property of these eight basis vectors? Well, I've only written five of them, but if you see that property for those five, you'll see it for the three remaining. Well, if I give you those eight vectors and ask, what's a nice property? Well, you would say, first, they're all ones and minus ones and zeroes. So every multiplication is very fast using -- just in binary. But what's the other great property of those vectors? Anybody see it? So, of course, when I think about a basis, one nice property -- I don't have to have it, but I'm happy if it's there -- is that they're orthogonal.
If the basis vectors are orthogonal, then I'm in good shape. And these are...
do you see? Take the dot product of that with that, you get four plus ones and four minus ones, you get zero. Take the dot product of that with that. You get two plus ones and two minus ones. Or the dot product of that with that. Two plus ones and two minus ones. You can easily check that that's an orthogonal basis. It's not orthonormal.
To fix it up, I should divide by the length, to make them unit vectors. Let's suppose I do that.
So somewhere in here, I've got to account for the fact that this has length square root of eight, that has length square root of four, that has length square root of two. But that's just a constant factor that's easy to -- so suppose we've done that.
Then, tell me what's W inverse? That's what chapter four, section four point four was about.
If we have orthonormal columns then the inverse is the same as the transpose. So if we have a fast way to multiply by W, which we do, the inverse is going to look just the same, and we'll have a fast way to do W inverse.
So that's the wavelet basis passes this requirement for fast. We can use it fast.
But there's a second requirement, is it any good? Because the the very fastest thing we could do is not to change basis at all. Right? The fastest thing would be, OK, stay with the standard basis, stay with eight pixel values.
But that was poor from compression point of view, right? Those eight pixel values, if I just took those eight numbers, I can't throw some of those away. If I throw away ninety percent -- if I compress ten to one, and throw away ninety percent of my pixel values, well, my picture's just gone dark. Whereas, the basis that was good, the wavelet basis or the Fourier basis, if I throw away c5, c6, c7, and c8, all I'm throwing away is little blips that are probably there in very small amounts. So the second property that we need is good compression. So first, it has to be fast, and secondly, a few basis vectors should come close to the signal. So a few is enough.
Can I write it that way? A few basis vectors are enough to reproduce the image just exactly as on a video of these 18.06 lectures. Uh, I don't know what the compression rate is, I'll ask, David, who does the compression -- and, by the way, I'll try to get the lectures, that are relevant for the quiz up onto the Web in time. So I'll send them a message today. So, he's using the Fourier basis because the JPEG -- so JPEG two thousand, which will be the next standard for image compression, will include wavelets. So, I mean, you're actually getting a kind of up-to-date, picture of where this big world of signal and image processing is.
That Fourier is what everybody knew, and what people automatically used, and the new one is wavelets, where this is the simplest set of wavelets.
And this isn't the one that the FBI uses, by the way, the FBI uses a smoother wavelet, instead of jumping from one to minus one, it's a smooth, Cutoff. and, that's what we'll be in in JPEG two thousand. OK, so that's that application.
Now, let me come to the math, the linear algebra part of the lecture. Well, we've actually seen a change-of-basis. So let -- let me just review that eh-eh change-of-basis idea, and then the i- and then the transformation to a matrix.
OK. So this, I hope you see that these applications are really big.
Now, I have to talk a little about change-of-basis, and a little about that. The matrix.
OK. OK.
OK. So change-of-basis.
Basically, forgive that put, OK, I have, I have my vector in one basis, and I want to change to a different one.
Actually, you saw it for the wavelet case.
So I need the -- let the matrix W, and the columns of W be the new basis vectors. Then the change-of-basis involves, just as it did there, W inverse.
So we have the vector, say, x, in the old basis, and that converts to a vector, let's say, c, in the new basis, and the relation is exactly what we had there, that x is W c. That's the step we have to take.
OK. So, with respect to a first basis, say v1 up to v8, it has a matrix A.
I'm just setting up letters here.
With respect to a second basis, say, I'll make it u1 up to -- or w1, since I've used (w)s, w1 up to w8, it has a matrix B. And my question is, what's the connection between A and B? How is the matrix -- the transformation T is settled.
We could say, it's a rotation, for example. So that would be one transformation of eight-dimensional space, just spin it a little. Or project it. Or whatever linear transformation we've got.
Now, we have to remember -- my first step is to remind you how you create that matrix A. Then my second step is, we would use the same method to create B, but because it came from the same transformation, there's got to be a relation between A and B. What's the relation between A and B? And let me jump to the answer on that one. That if I have the same transformation, and I'm compute on its matrix in one basis, and then I computer it in another basis, those two matrices are similar.
So these two matrices are similar.
Now, do you remember what similar matrices meant? Similar. A is similar to -- the two matrices are similar. Similar.
And what do I mean by that? I mean that I take the matrix B, and I can compute it from the matrix A using some similarity, some matrix M on one side, and M inverse on the other.
And this M will be the change-of-basis matrix.
This part of the lecture is, admittedly, compressed. What I wanted you to -- it's really the conclusion that I want you to spot. Now, I have to go back and say, what does it mean for A to be the matrix of this transformation T. So I have to remind you what that meant, that was in the last lecture.
Then this is the conclusion that if I change to a different basis, we now know -- see, if I change to a different basis, two things happen. Every vector has new coordinates. There, the rule is this one, between the old coordinates and the new ones. Every matrix changes, every transformation has a new matrix. And the new matrix is related this way, the M could be the same as the W.
The M there would be the W here.
OK. So, can I, in the remaining minutes, recapture my lecture -- the end of my lecture that was just before Thanksgiving, about the matrix? OK. What's the matrix? And I'll just take one basis. So now this part is going to go onto this board here. What is the matrix? What is A? OK.
Using a basis v1 up to v8. Mm.
OK. What's the point? The point is, if I know what the transformation does to those eight basis vectors, I know it completely. I know T, I know everything about T, I know T completely from knowing T of V -- what T does to v1, what T does to v2, what T does to v8.
Why is that? It's because T is a linear transformation. So that if I know what these outputs are -- so these are the inputs v1 up to v8, these are the outputs from the transformation, like everyone rotated, everyone projected, whatever transformation I've done, then why is it that I know everything? How does linearity work? Why? This is because every x is some combination of these basis vectors, right? c1v1, c2v2, c8v8, they were a basis.
That's the whole point of a basis, that every vector is a combination of the basis vectors in exactly one way.
And then, what is T of x? The point is, I claim that we know T of x completely for every x, because every x is a combination of those -- and now we use the linear transformation part to say that the output from x has to be c1 times the output from v1 plus v2 times the output from v2, and so on.
Up through c8 times the output from v8.
So this is like just saying, OK.
We know everything when we know what T does to each basis vector. OK.
So those are the eight things we need. Now -- but we need these answers in this basis.
So this first output is some combination of the eight basis vectors. So write T acting on the first input -- in other words, write the first output as a combination of the basis vectors, say a11 v1 + a21 v2 and so on a81 v8. Write T of v2 as some combination a12 of v1, a22 of v2 and so on. I'm creating the matrix A, column by column. Those numbers go in the first column, these numbers go in the second column, the matrix A that thi- this -- this is our matrix that represents T in this basis is these numbers.
a11 down to a18, a21 down to a28, and so on. OK.
That's the recipe. In other words, if I give you a transformation, and a basis.
So that's what I have to give you.
The inputs are the basis and to tell you what the transformation is. And then, you tell me -- you compute T for each basis, expand that result in the basis, and that gives you the sixty four numbers that go into the matrix A. Let me suppose -- let's close with the best example of all. Suppose v1 to v8, this basis, is the eigenvectors.
Suppose we have an eigenvector basis so that T(vi) is in the same direction of vi. Now, my question is, what is A? Can you carry through the steps? Let's do them together, because we can do it in one minute.
So, we've chosen this perfect basis.
And, actually, with signal image processing, they might look for the eigenvectors.
But that would take more calculation time that just saying, OK, we'll use the wavelet basis.
Or, OK, we'll use the Fourier basis.
But the very best basis is the eigenvector basis.
OK, what's the matrix? So, what's the first column of the matrix? How do I get the first column? I take the first basis vector v1.
I opt -- I look to see, what does the transformation do to it? The output is lambda one v1.
I express that output as a combination so the first input is v1. Its output is lambda one v1.
Now write lambda one v1 as a combination of the basis vectors, well, it's already done.
It's just lambda one times the first basis vector and zero times the others. So this first column will have lambda one and zeroes. OK.
Second input is v2. Output is lambda two v2.
OK, write that output as a combination of the (v)s.
It's already done. It's just lambda two times the second v. So we need, in the second column, we have lambda two times the second v.
Well, you see what's coming, that in that basis, in the eigenvector basis, the matrix is diagonal.
So that's the perfect basis, that's the basis we'd love to have for image processing, but to find the eigenvectors of our pixel matrix would be too expensive.
So we do something cheaper and close, which is to choose a good basis like wavelets. OK, thanks |
Course Outline for Precalculus
Course Description: This course provides a working
knowledge of precalculus and its applications.
It begins with a review of algebraic operations . Emphasis is on solving and
graphing equations that involve
linear, polynomial, exponential, and logarithmic functions . Students learn to
graph trigonometric and inverse
trigonometric functions and learn to use the family of trigonometric identities.
Other topics include conic
sections, arithmetic and geometric sequences, and systems of equations.
Course Objectives: After completing this course, students will be able
to:
• Perform operations on real numbers and polynomials.
• Simplify algebraic, rational, and radical expressions .
• Solve linear and quadratic equations and inequalities.
• Solve word problems involving linear and quadratic equations and inequalities.
• Solve polynomial, rational, and radical equations and applications.
• Solve and graph linear, quadratic, absolute value, and piecewise-defined
functions.
• Perform operations with functions as well as find composition and inverse
functions.
• Graph quadratic, the square root , cubic, and cube root functions.
• Graph and find zeroes of polynomial functions.
• Graph quadratic functions by completing the square, using the vertex formula ,
and using
transformations.
• Solve and graph exponential and logarithmic equations.
• Express angle measure in degrees or radians.
• Evaluate and simplify trigonometric expressions.
• Know the six trigonometric functions and how to evaluate those trigonometric
functions using
positions on the unit circle with respect to the right triangle.
• Graph trigonometric and inverse trigonometric functions.
• Use trigonometric functions to solve a right triangle and apply the Law of
Sines and the Law of
Cosines to solve triangles that are acute or obtuse.
• Solve systems of linear equations and inequalities.
• Model and solve applications using linear systems.
• Evaluate and find partial sums of a series .
• Evaluate and find sums of an arithmetic sequence and a geometric sequence.
• Solve application problems involving arithmetic and geometric sequences and
series.
• Define, identify, and graph conic sections including circle, ellipse,
parabola, and hyperbola.
• Rectangular
Coordinates and the
Graph of a Line
• Relations, Functions,
and Graphs
• Linear Functions
• Use a table of values to graph linear
equations.
• Determine when lines are parallel or
perpendicular.
• Use linear graphs in an applied context.
• Identify functions and state their domain
and range.
• Use function notation.
• Write a linear equation in function form.
• Use function form to identify the slope .
• Use slope-intercept form to graph linear
functions.
• Write a linear equation in point-intercept
form.
• Use these forms to solve applications.
5
Operations on
Functions and
Analyzing Graphs
• The Algebra and
Composition of
Functions
• One-to-One and Inverse
Functions
• Transformations and
Symmetry
• Compose two functions and find the
domain.
• Identify one-to-one functions.
• Find inverse functions using an algebraic method .
• Graph a function and its inverse.
• Use symmetry as an aid to graphing.
• Perform stretches and compressions on
a basic graph.
• Perform vertical and horizontal shifts and
reflections of a basic graph.
• Correctly use vocabulary associated with
a study of angles and triangles.
• Convert between degrees and radians for nonstandard angles.
• Define the six trigonometric functions in
terms of a point on the unit circle or in
terms of a real number.
• Identify and discuss important
characteristics of tangent and cotangent.
• Solve applications of trigonometric
functions.
• Find values of the six trigonometric
functions from their ratio definition.
• Graph the basic trigonometric functions.
10
Trigonometric Identities
• Transformations and
Applications of
Trigonometric Graphs
• Family of Trigonometric
Identities
• The Inverse
Trigonometric Functions
and Their Applications
• Use fundamental identities to express a
given trigonometric function in terms of
the other five.
• Solve applications using these identities.
• Find the inverse trigonometric functions
and evaluate related expressions.
• Apply the definition and notation of inverse trigonometric functions
to simplify
expressions.
• Graph sine and cosine functions with
various amplitudes and periods.
• Write the equation for a given graph.
11
Applications of
Trigonometry
• The Law of Sines
• The Law of Cosines
• More Applications of
Trigonometry
• Solve ASA and AAS triangles.
• Use the Law of Sines to solve
applications.
• Apply the Law of Cosines when two sides
and an included angle are known (SAS).
• Apply the Law of Cosines when three
sides are known (SSS).
• Solve applications using the Law of
Cosines.
• Solve more applications involving trigonometric functions.
• Solve the SSA case, including the
ambiguous case.
• Solve linear systems by graphing, by
substitution, and by elimination.
• Use system of equations to mathematically model and solve
applications.
• Form the augmented matrix of a system
of equations.
• Solve a system of equations using row operations.
• Recognize inconsistent and dependent systems.
• Use system of equations to
mathematically model and solve
applications.
• Define and identify a parabola.
• Graph a parabola.
• Solve applications of parabolas.
• Define and identify an ellipse and a
circle.
• Graph an ellipse and a circle.
• Solve applications of ellipses and circles.
• Define and identify a hyperbola.
• Graph a hyperbola.
• Solve applications of hyperbolas
14
Sequences and Series
• Sequences and Series
• Arithmetic Sequences
• Geometric Sequences
• Write the terms of a sequence given the
general term.
• Determine the general term of a
sequence.
• Find the partial sum of a series.
• Use summation notation to write and
evaluate series.
• Solve applications involving arithmetic
sequences.
• Find the sum of a geometric series.
• Solve application problems involving
geometric sequences and series. |
Focusing on deterministic models, this book is designed for the first half of an operations research course. A subset of Winston's best-selling Operations Research, Introduction to Mathematical Programing offers self-contained chapters that make it flexible enough for one- or two-semester courses ranging from advanced beginning to intermediate in level. Appropriate for undergraduate majors, MBAs, and graduate students, it emphasizes model-formulations and model-building skills as well as interpretation of computer software output. LINDO, GINO, and LINGO software packages are available with the book in Windows, Macintosh, or DOS versions. Linear algebra prerequisite.
Bring your data to life and add meaning to your information with Maps Made Easy Using SAS. Abundant real-world examples and a tutorial approach help new users create maps easily and quickly. You will learn the basic mapping components, including map and response data sets as well as simple SAS/GRAPH statements. With in-depth examples you will move on to more advanced mapping techniques, such as annotating maps and producing customized maps and output. The process used to annotate maps is demystified and described in clear, easy-to-follow steps. You will produce data-driven, updatable maps in GIF format for use in Web-based presentations and other applications. Also presented are details on creating more complicated choropleth maps. These include maps that combine geographic areas with internal boundaries removed, maps that display multiple geographic areas, and clipped maps. Enhance your data presentations with this well-organized guide.
"The book's focus on imaging problems is very unique among the competing books on inverse and ill-posed problems. …It gives a nice introduction into the MATLAB world of images and deblurring problems." — Martin Hanke, Professor, Institut für Mathematik, Johannes-Gutenberg-Universität. When we use a camera, we want the recorded image to be a faithful representation of the scene that we see, but every image is more or less blurry. In image deblurring, the goal is to recover the original, sharp image by using a mathematical model of the blurring process. The key issue is that some information on the lost details is indeed present in the blurred image, but this "hidden" information can be recovered only if we know the details of the blurring process. Deblurring Images: Matrices, Spectra, and Filtering describes the deblurring algorithms and techniques collectively known as spectral filtering methods, in which the singular value decomposition—or a similar decomposition with spectral properties—is used to introduce the necessary regularization or filtering in the reconstructed image. The concise MATLAB® implementations described in the book provide a template of techniques that can be used to restore blurred images from many applications. This book's treatment of image deblurring is unique in two ways: it includes algorithmic and implementation details; and by keeping the formulations in terms of matrices, vectors, and matrix computations, it makes the material accessible to a wide range of readers. Students and researchers in engineering will gain an understanding of the linear algebra behind filtering methods, while readers in applied mathematics, numerical analysis, and computational science will be exposed to modern techniques to solve realistic large-scale problems in image processing. With a focus on practical and efficient algorithms, Deblurring Images: Matrices, Spectra, and Filtering includes many examples, sample image data, and MATLAB codes that allow readers to experiment with the algorithms. It also incorporates introductory material, such as how to manipulate images within the MATLAB environment, making it a stand-alone text. Pointers to the literature are given for techniques not covered in the book. Audience This book is intended for beginners in the field of image restoration and regularization. Readers should be familiar with basic concepts of linear algebra and matrix computations, including the singular value decomposition and orthogonal transformations. A background in signal processing and a familiarity with regularization methods or with ill-posed problems are not needed. For readers who already have this knowledge, this book gives a new and practical perspective on the use of regularization methods to solve real problems. Preface; How to Get the Software; List of Symbols; Chapter 1: The Image Deblurring Problem; Chapter 2: Manipulating Images in MATLAB; Chapter 3: The Blurring Function; Chapter 4: Structured Matrix Computations; Chapter 5: SVD and Spectral Analysis; Chapter 6: Regularization by Spectral Filtering; Chapter 7: Color Images, Smoothing Norms, and Other Topics; Appendix: MATLAB Functions; Bibliography; Index.
This book constitutes the refereed proceedings of the 5th International Workshop on Algorithms in Bioinformatics, WABI 2005, held in Mallorca, Spain, in September 2005 as part of the ALGO 2005 conference meetings. The 34 revised full papers presented were carefully reviewed and selected from 95 submissions. All current issues of algorithms in bioinformatics are addressed with special focus on statistical and probabilistic algorithms in the field of molecular and structural biology. The papers are organized in topical sections on expression (hybrid methods and time patterns), phylogeny (quartets, tree reconciliation, clades and haplotypes), networks, genome rearrangements (transposition model and other models), sequences (strings, multi-alignment and clustering, clustering and representation), and structure (threading and folding). |
In a differential equations course, students learn to use integrating factors to solve first order linear differential equations, and in the process reinforce learning of key concepts from their calculus courses. This capsule offers some differential equations solved by the originators of the technique of using an integrating factor, though they did not use that expression. Solving differential equations via integrating factors is difficult for some students, particularly those who try to memorize a formula. We advocate that students learn to derive the method and solve differential equations using the product rule and the fundamental theorem of calculus, as advocated in a number of modern texts [2, 3]. Memorizing the formula would not be in the spirit of the originators of the method, Johann Bernoulli (1667–1748) and Leonhard Euler (1707–1783), nor does formula memorization lead to deep learning of fundamental mathematical processes. Understanding why integrating factors work, as offered in this historical perspective, can deepen student understanding of calculus topics such as the product rule, the fundamental theorem of calculus, and basic integration techniques. This capsule provides some historical information about the work of Bernoulli and Euler, and we offer student activities that will connect that history to enable more thorough learning of the method of integrating factors.
Historical preliminaries
Johann Bernoulli was a colleague of Gottfried Leibniz (1646–1716) and is acknowledged as one of the foundational figures in the development of the calculus. In the early 1690's he prepared lectures in the nascent calculus for Guillaume de l'Hôpital (1661–1704), who is credited with writing the first text on the calculus |
Thinking Mathematically
.Blitzer continues to raise the bar with his engaging applications developed to motivate readers from diverse majors and backgrounds.Thinking Mathematically, Fifth Edition, draws from the authorrsquo;s unique background in art, psychology, and math to present math in the context of real-world applications. |
Educator - Mathematics: College Calculus Level I with Professor Switkes
English | VP6 650x350 29.970 fps | MP3 128 Kbps 44.1 KHz | 8.37 GB
Genre: eLearning
Dr. Jenny Switkes will help you master the intricacies of Calculus from Limits to Derivatives to Integrals. In Educator's Calculus 1 course, Professor Switkes covers all the important topics with detailed explanations and analysis of common student pitfalls. Calculus can be difficult, but Professor Switkes will show you how to reap the rewards of your hard work, all while showing you the beauty and importance of math. Whether you just need to brush up on your calculus skills or need to cram the night before the final, Professor Switkes has taught mathematics for 10+ years and knows exactly how to help.
Application-oriented introduction relates the subject as closely as possible to science. In-depth explorations of the derivative, the differentiation and integration of the powers of x, theorems on differentiation and antidifferentiation, the chain rule and examinations of trigonometric functions, logarithmic and exponential functions, techniques of integration, polar coordinates, much more. Examples. 1967 edition. Solution guide available upon requestStochastic analysis is not only a thriving area of pure mathematics with intriguing connections to partial differential equations and differential geometry. It also has numerous applications in the natural and social sciences (for instance in financial mathematics or theoretical quantum mechanics) and therefore appears in physics and economics curricula as well.
Calculus is about the very large, the very small, and how things change. The surprise is that something seemingly so abstract ends up explaining the real world. Calculus plays a starring role in the biological, physical, and social sciences. By focusing outside of the classroom, we will see examples of calculus appearing in daily life.
Calculus does not have to be difficult. Dr. William Murray knows what it takes to excel in math and will show you everything you need to know about calculus. Dr. Murray demonstrates his extensive teaching experience by clarifying complicated topics with a wide array of examples, helpful tips, and time-saving tricks. Topics range from Advanced Integration Techniques and Applications of Integrals to Sequences/Series. Dr. Murray received his Ph.D from UC Berkeley, B.S. from Georgetown University, and has been teaching in the university setting for 10+ years.
Professor Raffi Hovasapian helps students develop their Multivariable Calculus intuition with in-depth explanations of concepts before reinforcing an understanding of the material through varied examples. This course is appropriate for those who have completed single-variable calculus. Topics covered include everything from Vectors to Partial Derivatives, Lagrange Multipliers, Line Integrals, Triple Integrals, and Stokes' Theorem. Professor Hovasapian has degrees in Mathematics, Chemistry, and Classics and over 10 years of teaching experience.
For over three decades, this best-selling classic has been used by thousands of students in the United States and abroad as a must-have textbook for a transitional course from calculus to analysis. It has proven to be very useful for mathematics majors who have no previous experience with rigorous proofs. Its friendly style unlocks the mystery of writing proofs, while carefully examining the theoretical basis for calculus.
This twelve-lesson series will cover the ins and outs of vector calculus and the geometry of R^2 and R^3.
The first 8 video lessons look specifically at vectors and the geometry of R^2 and R^3. This set of videos will cover Coordinate Geometry in Three Dimensional Space, Vectors in R2 and R3, the Dot Product, Orthogonal Projections, the Cross Product, Geometry of the Cross Product, and Equations of Lines and Planes in R3.
Functional analysis owes much of its early impetus to problems that arise in the calculus of variations. In turn, the methods developed there have been applied to optimal control, an area that also requires new tools, such as nonsmooth analysis. This self-contained textbook gives a complete course on all these topics. It is written by a leading specialist who is also a noted expositor. |
Methods of solving polynomial equations lie at the heart of classical algebra. There are two interpretations of the problem of solving an equation, leading to two different approaches to its solution. In most courses, the emphasis is on the structure of the equation and finding a way to express the roots as a formula in terms of the coefficients. The simplest example of such a formula is the quadratic formula, which gives the solution of the equation ax2 + bx + c = 0 as
This approach is elegant and leads to some exceedingly profound mathematics. However, for one who actually needs to know a number that satisfies the equation, this approach leaves something to be desired. It works with maximum efficiency in the case of the quadratic equation, but even in that case, if the quantity under the radical is not the square of a rational number, one is forced to resort to approximations in order to get a usable number. For cubic and quartic equations, there are formulas, but they work even less well, since they often involve taking the cube root of a complex number, which is a problem just as complicated as the original equation was, if not more so. Once again, one is forced to resort to numerical approximations. Beyond the fourth degree, the only formulas involve non-algebraic expressions, and are of little practical use. Higher-degree equations are the realm of numerical methods. To understand how numerical methods work, it is useful to begin with the simplest cases and the simplest methods. That is what we are about to do |
Take it step-by-step for pre-calculus success! The quickest route to learning a subject is through a solid grounding in the basics. So what you won't find in Easy Pre-calculus Step-by-Step is a lot of… |
Each year The Math League sponsors contests for grades 4, 5, 6, 7, 8, Algebra Course 1, and High School. Math League's Math Contests are now available for homeschoolers. These are the same contests used by schools, in a non-competitive format for the homeChalk Dust Company offers mathematics instruction on videotape to homeschooled students and a variety of other users. Textbooks used in Chalk Dust programs are published by Houghton Mifflin Company and most are authored by Ron Larson. Offers solutions guides and personal help when needed via telephone or the internet, providing a comprehensive and effective distance-learning environment.
Classmate Math's homeschool programs—The Pre-Algebra Classmate, The Algebra 1 Classmate, and The Geometry Classmate—are complete CD-Rom-based courses that come with an actual instructor built into the curriculum. The "instructor" is a video image that pops up with each lesson. The instructor teaches at a whiteboard such as one would in presenting a new concept to a classroom of students. But in this case, the students can turn on or replay the instructor whenever it is needed.There are approximately 400 lessons in Classmate Math's curriculum, and every lesson features: video example problems, practice problems with animated step-by step audio explanations, multiple choice self-tests, printable extra problems, notes, and pages, and tests and solutions manual.
Key Curriculum Press offers the popular Key to...® Series, with the following subjects: Algebra, Decimals, Fractions, Geometry, Measurement, Metric Measurement, and Percents. They publish high school mathematics textbooks, mathematics software, supplementary materials and workbooks, videos, and manipulative materials. They also provide professional-development services to teachers across the United States. Key Curriculum Press also publishes the textbooks, "Discovering Algebra" and "Discovering Geometry," along with numerous other math textbooks through Calculus.
Learn algebra by watching and completing lessons on videotape. Leonard Firebaugh guides students through all levels of algebra in these academically-oriented tapes. They teach algebra clearly and effectively to students at home or school, and are easily adapted to any educational setting. The tapes are divided into three phases, or levels, of instruction, constituting one full year of coursework. Math Relief offers video programs in Algebra I, Algrebra II, Pre-Algebra, and Geometry. Courses are designed for junior high to high school levels.
Mathematics: A Human Endeavor is technically a textbook, but it is also excellent reading for anyone who is interested in math. In this book, Harold R. Jacob demonstrates that math is all around us, with clear, yet complex, explanations. This is a broad introduction to the mathematical sciences, discussing geometry, probability, combinations, statistics, topology, and more. Learn more about this book here.
Elementary Algebra is full of interesting elements, including cartoons, puzzles, and more, that make learning Algebra fun for students. It is suitable for either classroom use or self-paced study, and combines real-life examples, carefully structured exercises, and humor to help students learn and remember. For information on ordering this textbook, go here.
Geometry: Seeing, Doing, Understanding is also user-friendly, offering students a chance to enjoy geometry as they learn it. This textbook can be found here.
The teacher's guides for these texts can be ordered by calling 888-330-8477.
This series of mathematics texts was written by Joseph Ray in the late 19th century. This is a reprinting of these books for a modern age, with all of them available in CD format, available for you to print at home.
Saxon Math is a skills-based mathematics program for grades K-12. Saxon's unique pedagogical approach is based on instruction, practice and assessment distributed across the grade level. It systematically distributes instruction and practice and assessment throughout the academic year as opposed to concentrating, or massing, the instruction, practice and assessment of related concepts into a short period of time—usually within a unit or chapter.
Singapore Math is a mathematics curriculum modeled after the highly successful mathematics teaching method and texts used in Singapore. They offer math texts from pre-K to 12th grades. This series challenges children to think through and understand mathematical concepts instead of simply memorizing facts and algorithms. One of the benefits of using this program is its affordability. The textbooks are inexpensive and are reusable. The consumable workbooks are priced so that even families with multiple children using this program will find it affordable. |
The word "algebra" comes from the title of a book written by
the ninth-century Arabian mathematician Al-Khowarizmi. In this title, al-jebr w'
almuqabala, the word al-jebr meant transposing a quantity from one side of an
equation to another or "rejoining." Muqabala meant simplification of the
resulting expression. Algebra, then, was the application of a series of techniques,
including reductions, simplifications, transpositions, which manipulated mathematical
expressions. The reason algebra became so powerful was because the resulting expressions
applied to a large number of cases. Arithmetic, on the other hand, dealt with (and applied
to) one case at a time. (Cf., Kline, p. 69)
Suppose I wanted to solve for x in the following second-degree
equations:
x
2 – 4 = 0
x
2 – 5x + 6 = 0
x
2 – 5x + 3 = 0
By adding 4 to each side of equation (1), we quickly obtain a much
clearer expression, one that can be "solved" quite easily:
x2 = 4
So x = 2 and x = -2
The second equation is not quite that straightforward, but a trained
eye would recognize that it is equivalent to the expression
(x – 3)(x – 2) = 0
The solutions, or "roots," will then be clear here: when x
is 2 or 3, the equation would obtain. Lastly, equation (3) is the hardest of the three
because there is no immediate manipulation that can simplify the expression any more.
Eventually, mathematicians devised a formula by which to solve any equation of the
form
ax2 + bx + c = 0
The formula is now called the "quadratic equation" (the
actual formulation is not necessary here).
So this is what algebra was (and is) really all about: solving for
general cases. In order to do this, it is necessary to substitute some numbers with any
number, called a "variable." Before this operation, mathematical problems were
handled more piecemeal: each particular problem or situation was handled separately.
There are a group of problems, however, called the aha problems,
that I believe are the precursor to algebra. Aha in Egyptian means
"something" or "a quantity." Whenever a problem called for an unknown,
the Egyptians would simply call it "something" or aha. Take for instance
example 40 from the Rhind papyrus.(Cf., Wilson, p. 28.) Aahmes writes:
We want to divide 100 loaves between five men in the following way. The
second man receives a certain amount more than the first. The third man receives an equal
amount more than the second, and so on. Also one seventh of the sum of the three largest
shares shall be equal to the sum of the smallest two shares. What is the difference
between the shares?
The Egyptians figured out that one could call the unknowns
"something," in this case, the unknowns being the smallest share and the difference.
We can write the shares received by the five men as:
Man #1: smallest share
Man #2: smallest share + difference
Man #3: smallest share + 2 differences
Man #4: smallest share + 3 differences
Man #5: smallest share + 4 differences
which adds up to:
(a) 5 x (smallest share) + 10 x (difference) = 100
which summarizes the first condition of the problem. The sum of the
three largest shares is "3 x smallest share + 9 differences" and that of the two
smallest shares "2 x smallest share + difference." The second condition
specifies that the seventh part of the first expression be equal to the second expression,
yielding:
1/7 [3(smallest share) + 9(difference)]
= 2(smallest share) + difference
Multiplying through by seven, and simplifying terms (doing al-jebr
w' almuqabala!) we reach:
2 (difference) = 11 (smallest share)
which, together with (a) is a simple system of two equations with two
unknowns. The problem is, Egyptians did not know how to solve these directly.
Whenever a mathematical problem reached this point, the Egyptians would
guess at the answer(s). This seems a bit impractical, indeed this is the start of a
"trial and error" exercise, but the answer they arrived at was not only close
to the right answer: it was used to reach the right answer. In other words, they
would have to "try" just once, and use the "error" to get at the right
answer. For the above problem, then, we could guess that the difference was, say
(conveniently) 11. From (c), we quickly see that the smallest share has to be two.
Plugging these values into (a), we have
5 x 2 + 10 x 11 = 120
which is not equal to 100, but close. To reach the right answer,
Aahmes instructs: "as many times as 120 must be multiplied to give 100, so many times
must 11 be multiplied to give the true difference in shares." The true answer must
therefore be given by difference = 100/120 x 11 = 55/6 = 9 1/6
The Babylonian aha problems are similar. Here is one of the
simpler ones: "Find two numbers whose sum is 14 and whose product is 45." (Cf.,
Wilson, p. 64ff.) If the numbers were equal, then the first condition would be met with
the number 7, but 7x7=49, not 45. The Babylonians got around this problem by postulating
that there must be a number that they could both add to and subtract from 7 such
that the second condition obtained. Calling this number, appropriately, aha,
we get
(7 + aha) x (7 – aha) = 45 multiplying
out
49 + 7aha – 7aha – (aha x aha) =
45 simplifying
49 – aha2 = 45
which obviously leads to aha being 2. So 5+9=14, and 5x9=45.
Notice there was no guessing here.
The Babylonians developed a more useful application of the aha
problems: they were able to extract the square root of any number. Suppose we take
the number Ö 27 (our aha). Since we know that 5 squared
is 25, and that 6 squared is 36, we can infer that Ö 27 is
between 5 and 6. In addition, the Babylonians noticed that since 5 is less than Ö 27, 27/5 must be greater than Ö 27
since
5 x 27/5 = Ö 27 x Ö
27 = 27
Therefore, a better approximation of Ö 27
is halfway between 5 and 27/5. So Ö 27 »
½(5 + 27/5) = 5.2. As it turns out, 5.2 squared is 27.04, a surprisingly close answer.
Further, this method could be used again, i.e., knowing that 5.2 is slightly greater than Ö 27, then 27/5.2 has to be less than Ö
27, so an average of these two numbers would yield an even more accurate answer (better
than six significant figures!).
Interestingly (and finally), the Chinese had a different way of solving
the exact same aha problem. Instead of Ö 27 being the aha,
the offset from 5 became the unknown. This is expressed as follows:
(5 + aha) x (5 + aha) = 27 multiplying out
(5 x 5) + 2(5 x aha) + (aha x aha) = 27 so
that
10aha + aha2 = 2 or aha (10
+ aha) = 2
We know that aha is less than one, so that replacing 10 + aha
by 10 gives an error of less than one in ten. Therefore, we can safely make our first
iteration (aha1) fit the equation like this:
aha1 x 10 = 2, i.e., aha1 = 0.2
This approximation gives us the same answer to which the Babylonians
arrived. Subsequent iterations are produced by making the aha in the bracketed term
the same as the previous aha1 resulting above, obtaining a new aha2:
aha2 x (10 + aha1) = aha2 x (10.2) = 2
Which gives us an aha2 of 0.196078, and 5.1590782 =
26.9992.
The Chinese method of calculating roots is not as fast as the
Babylonian method, but it does have the advantage of being able to be extended to treat
higher roots. Suppose we were looking for the cubed root of two. Since the number will be
obviously between 1 and 2, we can express the problem as: (1 + aha)3 =
2. Extending the expression out to its simplest form, we obtain that
1 + 3aha + 3aha2 + aha3 = 2
Solving for one of the ahas, we get that
aha = 1 / (3 + 3aha + aha2)
And setting the ahas inside the bracket to 0, we get that aha1
is 1/3. 1.33332 = 2.3704. Not too good for a first run, but subsequent runs do
show a marked improvement.
The Chinese continued to expand these problems to higher orders, which
led them to discover that there was a pattern developing. They noticed that there was
symmetry to the resulting expanded equation, no matter how high the root. Further, the
symmetry seemed to expand in a unique way! Consider the first few iterations:
(1 + aha)0 = 1
(1 + aha)1 = 1 + aha
(1 + aha)2 = 1 + 2aha + aha2
(1 + aha)3 = 1 + 3aha + 3aha2
+ aha3
(1 + aha)4 = 1 + 4aha + 6aha2
+ 4aha3 + aha4
(1 + aha)5 = 1 + 5aha + 10aha2
+ 10aha3 + 5aha4 + aha5
If we pay attention to the coefficients, we notice (as the
Chinese evidently did) that they add up in a peculiar way. For example, the 10 in 10aha2
is obtained by adding the coefficients directly above and to the left, i.e., 4 (of 4aha)
plus 6 (of 6aha2). This is the case for all coefficients! What is
most remarkable about this pattern is that it is now know as Pascal's Triangle,
named after French mathematician Blaise Pascal (1623-1662). Clearly, the Chinese knew of
this pattern before Pascal did.
We have seen how the Egyptians, the Babylonians and the Chinese worked
out mathematical problems by substituting unknowns with a "something," an aha.
The word algebra, however, did not appear in the limelight until the Arabs
introduced it. Does this mean that algebra did not exist until then? No, clearly not if by
algebra we mean the process of reducing and rearranging mathematical terms. It is
true that mathematics for the sake of manipulating numbers (or for the sake of
solving for general cases) did not seem to appear in the cultures we have examined, and it
is these theoretical branches of mathematics that we most often find the use of variables
instead of numbers. Pure algebra enabled mathematicians to find a general solution
to specific mathematical problems. In order to find particular solutions to many
mathematical problems, however, the variables (or ahas) used and the algebra-like techniques
here examined sometimes made things a lot easier to calculate, given that the alternative
is a much more inefficient process of trial and error. |
Course Learning Outcomes The student will: • Use the concepts of definite integrals to solve problems involving area, volume, work, and other physical applications. • Use substitution, integration by parts, trigonometric substitution, partial fractions, and tables of anti-derivatives to evaluate definite and indefinite integrals. • Define an improper integral. • Apply the concepts of limits, convergence, and divergence to evaluate some classes of improper integrals. • Determine convergence or divergence of sequences and series. • Use Taylor and MacLaurin series to represent functions. • Use Taylor or MacLaurin series to integrate functions not integrable by conventional methods. • Use the concept of parametric equations and polar coordinates to find areas, lengths of curves, and representations of conic sections. • Apply L'hôpital's Rule to evaluate limits of indeterminate forms.
Graphing Calculator required. TI 83, TI 84 or TI 86 series calculators recommended. Calculators capable of symbolic manipulation will not be allowed on tests. Examples include, but are not limited to, TI 89, TI 92, and Nspire CAS models and HP 48 models. Neither cell phones nor PDA's can be used as calculators. Calculators may be cleared before tests.
Differentiation and Integration Formulas: Students are expected to memorize the differentiation formulas on the last page inside the back cover of the text and integration formulas 1- 20 in the attached chart. |
This book covers the content prescribed for the New Zealand Diploma in Engineering course DE4102 Engineering Mathematics. Some foundation level material is also provided to help those students whose preparation for tertiary mathematics study is patchy, whether that be due to gaps in recent secondary...
Essential Maths and Stats provides a comprehensive overview of tertiary level mathematics and statistics and is the only definitive New Zealand text for mathematics and statistics at entry level. It is also an excellent 'extension' text for secondary school students.
Divided into six ke... |
FUNDAMENTALS OF MATHEMATICS, 9th Edition offers a comprehensive review of all basic mathematics concepts and prepares students for further coursework. The clear exposition and the consistency of presentation make learning arithmetic accessible for all. Key concepts are presented in section objectives and further defined within the context of How and Why; providing a strong foundation for learning. The predominant emphasis of the book focuses on problem-solving, skills, concepts, and applications based on "real world" data, with some introductory algebra integrated throughout. The authors feel strongly about making the connection between mathematics and the modern, day-to-day activities of students. This textbook is suitable for individual study or for a variety of course formats: lab, self-paced, lecture, group or combined formats.
Though the mathematical content of FUNDAMENTALS OF MATHEMATICS is elementary, students using this textbook are often mature adults, bringing with them adult attitudes and experiences and a broad range of abilities. Teaching elementary content to these students, therefore, is effective when it accounts for their distinct and diverse adult needs. Using Fundamentals of Math meets three needs of students which are: students must establish good study habits and overcome math anxiety; students must see connections between mathematics and the modern day-to-day world of adult activities; and students must be paced and challenged according to their individual level of understanding |
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The Mathematical Contest in Modeling is an international college-level competition in applied mathematics in which teams of up to three students must analyze, clarify, and propose solutions to open-ended problems. Students spend an intense weekend in early February analyzing their problem and writing their solution paper. In recent years, more than 2,000 teams from throughout the world have participated. The faculty judging team for the MCM has included OSU's Professor Marvin Keener since its inception.
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Many professors have served the Department of Mathematics at Oklahoma State University until retirement or death. They worked hard to develop a supportive, dynamic environment in which both faculty and students could grow professionally. This award is named for them in recognition of their vitality and excellence.
The intent of the award is to recognize and encourage students who have the insight, interest, and dedication to pursue a career in mathematics. It is presented in the spring semester to outstanding undergraduates who have declared a major in mathematics.
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If you intend to pursue an advanced degree, it is strongly recommended that your undergraduate degree program include Advanced Calculus I (MATH 4143), Advanced Calculus II (MATH 4153), Modern Algebra I (MATH 4613), and Modern Algebra II (MATH 5013). |
already helped many students in this area. MatLab is a computer software program using the general syntaxing of both C and Java. MatLab allows one the ability to either write the code for the problem at hand or to use built-in programs that can assist to get the calculation complete. |
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Pre-Algebra Study and Test-Taking Techniques Help
Study and Test-Taking Techniques
Having an understanding of math anxiety and being able to reduce stress are not enough to be successful in mathematics. You need to learn the basic skills of how to study mathematics. These skills include how to use the classroom, how to use the textbook, how to do your homework, how to review for exams, and how to take an exam.
The Classroom
In order to learn mathematics, it is absolutely necessary to attend class . You should never miss class unless extreme circumstances require it. If you know ahead of time that you will be absent, ask your instructor for the assignment in advance, then read the book, and try to do the homework before the next class. If you have an emergency or are ill and miss class, call your instructor or a friend in the class to get the assignment. Again, read the material and try the problems before the next class, if possible. Be sure to tell your instructor why you were absent.
Finally, if you are going to be absent for an extended period of time, let your instructor know why and get the assignments. If you cannot call him or her, have a friend or parent do it for you.
Come to each class prepared. This means to have all the necessary materials, including homework, notebook, textbook, calculators, pencil, computer disk, and any other supplies you may need.
Always select a seat in front of the classroom and near the center. This assures that you will be able to see the board and hear the instructor.
Pay attention at all times and take good notes. Write down anything your instructor writes on the board. If necessary, bring a tape recorder to class and record the sessions. Be sure to ask your instructor for permission first, though.
You can also ask you instructor to repeat what he or she has said or to slow down if he or she is going too fast. But remember, don't become a pest. You must be reasonable.
Be sure to ask intelligent questions when you don't understand something. Now I know some of you are thinking, "How can I ask intelligent questions when I don't understand?" or, "I don't want to sound stupid."
If you really understand the previous material and you are paying attention, then you can ask intelligent questions.
You must also remember that many times the instructor will leave out steps in solving problems. This is not to make it difficult for you, but you should be able to fill them in. You must be alert, active, and pay complete attention to what's going on in class.
Another important aspect you should realize is that in a class with ten or more students, you cannot expect private tutoring. You cannot expect to understand everything that is taught. But what you must do is copy down everything you can. Later, when you get home, apply the information presented in the chapters of this book.
Be alert, active, and knowledgeable in class.
The Textbook
The textbook is an important tool in learning, and you must know how to use it.
It is important to study your book. Note that I did not say "read" your book, but I purposely said study .
How do you study a mathematics book? It is different from studying a psychology book. First, look at the chapter title. It will tell you what you will be studying in the chapter. For example, if Chapter 5 is entitled "Solving Equations," this means that you will be doing something (solving) to what are called "equations." Next, read the chapter's introduction: it will tell you what topics are contained in the chapter.
Now look at the section headings. Let's say Section 5.3 is entitled "Solving Equations by Using the Multiplication Principle." This tells you that you will be using multiplication to solve a certain type of equation.
Take a pencil and paper and underline in the book all definitions, symbols, and rules. Also, write them down in your notebook.
Now, actually work out each problem that is worked out in the textbook. Do not just copy the problems, but actually try to solve them, following the author's solutions. Fill in any steps the author may have left out. If you do not understand why the author did something, write a note in the book and ask your instructor or a friend to explain it to you. Also, notice how each problem is different from the previous one and what techniques are needed to find the answer. After you have finished this, write the same problems on a separate sheet of paper and try to solve them without looking in your book. Check the results against the author's solutions.
Don't be discouraged if you cannot understand something the first time you read it. Read the selection at least three times. Also, look at your classroom notes. You may find that your instructor has explained the material better than your book. If you still cannot understand the material, do not say, "This book is bad. I can't learn it." What you can do is go to the library and get another book and look up the topic in the table of contents or appendix. Study this author's approach and try to do the problems again.
There is no excuse. If the book is bad, get another one.
Remember that I didn't ever say that learning mathematics was easy. It is not, but it can be done if you put forth the effort!
After you have studied your notes and read the material in the textbook, try to do the homework exercises.
Homework
Probably the single most important factor which determines success in mathematics is doing the homework. There's an old saying that "Mathematics is not a spectator sport." What this means is that in order to learn mathematics, you must do the homework. As stated previously, it is like learning to play an instrument. If you went to music class but never practiced, you could never learn how to play your instrument. Also, you must practice regularly or you will forget or be unable to play your instrument. Likewise, with mathematics, you must do the homework every day it is assigned. Here are my suggestions for doing your homework:
First and most important: DO YOUR HOMEWORK AS SOON AS POSSIBLE AFTER CLASS . The reason is that the material will still be fresh in your mind.
Make a habit of studying your mathematics regularly – say, three times a week, five times a week, etc.
Get your book, notes, and all previous homework problems, calculator, pencil, paper (everything you will need) before you start.
Do not dally around. Get started at once and do not let yourself be interrupted after you start.
Concentrate on mathematics only!
Write the assignment at the top of your paper.
Read the directions for the problems carefully.
Copy each problem on your homework paper and number it.
Do not use scratch paper. Show all of your work on your homework paper.
Write neatly and large enough. Don't do sloppy work.
Check your answers with the ones in the back of the book. If no answers are provided, check your work itself.
See if your answers sound reasonable.
Write out any questions you have about the homework problems and ask your instructor or another student when you can.
Draw pictures when possible. This is especially important in courses such as geometry and trigonometry.
If you have made a mistake, try to locate it. Do not depend on the teacher to find all your mistakes. Make sure that you have copied the problem correctly.
Don't give up. Doing a problem wrong is better than not doing it at all. (Note: don't spend an exorbitant amount of time on any one problem though.)
Use any of the special study aids such as summaries, lists of formulas, and symbols that you have made.
Don't skip steps.
Finally, if you cannot get the correct answer to a problem, don't stop. Try the next step.
Review
In order to learn mathematics, it is necessary to review before the tests. It is very important to realize one fact. You cannot cram in mathematics. You cannot let your studying go until the night before the exam. If you do, forget it. I have had students who have told me that they spent 3 hours studying before the exam, and then failed it. If those were the only 3 hours they spent studying, there is no way they could learn the material.
Some teachers provide written reviews. Make sure you do them. If this is the case, you can use the review as a practice test. If not, you can make up your own review. Many books have practice tests at the beginning and the end of chapters. If so, you can use these exams as reviews. Some books have extra problems at the end of each chapter. By doing these problems, you have another way to review.
Finally, if there is no review in the book, you can make up your own review by selecting one or two problems from each section in the chapter. Use these problems to make up your own practice test. Be sure not to select the first or second problem in each unit because most mathematics books are arranged so that the easy problems are first.
When you review, it is important to memorize symbols, rules, procedures, definitions, and formulas. In order to memorize, it is best to make a set of cards as shown here:
On the front of the card, write the name of the property, and on the back, write the property. Then when you are studying, run the cards through the front side and then on the other side. This way you can learn both the property and also the name of the property.
Finally, you must be aware that a review session is not a study session. If you have been doing your work all along, then your review should be short.
Test-Taking Tips
There are three types of exams that mathematics instructors give. They are closed book exams, open book exams, and take home exams.
First, let's talk about closed book exams given in the classroom. Make sure that you show up 5–10 minutes before the class begins. Bring all the necessary materials such as a pencil with an eraser, your calculator, paper, and anything else that you may need. Look over and study materials before class. After you receive your exam paper, look over the entire test before getting started. Read the directions carefully. Do all of the problems that you know how to do first, and then go back and try the others. Don't spend too much time on any one problem. Write down any formulas that you may need. Show all necessary steps if a problem requires it.
If there is any time left after you finish the exam, go back and check it for mistakes.
If you don't understand something, ask your instructor. Remember you cannot expect your instructor to tell you whether or not your answer is correct or how to do the problem.
In open book exams, you should remember that the book is your tool. Do not plan to study the book while you are taking the test. Study it before class. Make sure you know where all the tables, formulas, and rules are in the book. If you are permitted, have the formulas, rules, and summaries written down. Also, have sample problems and their solutions written down.
Don't be elated when your instructor gives you a take home exam. These exams are usually the hardest. You may have to go to the library and get other books on the subject to help you do the problems, or you may have to ask other students for help. The important thing is to get the correct solution. Show all of your work.
When your instructor returns the exam, make a note of the types of problems you missed and go back and review them when you get home.
See your instructor for anything that you are not sure of.
A Final Note
There is still one problem left to discuss: What happens if you are in over your head – i.e., the material you are studying is still too difficult for you?
First, try getting a math tutor. Many schools provide tutors through the learning center or math lab. Usually the tutoring is free. If your school does not provide free tutors, then seek out one on your own and pay him or her to help you learn.
There are other things you can do to help yourself if tutoring doesn't work. You can drop the course you are in and sign up for a lower-level course next semester. You can also audit the course, i.e., you can take the course but do not receive a grade. This way you can learn as much as you can but without the pressure. Of course, you will have to enroll in the course again next semester for a grade.
Many colleges and some high schools offer non-credit brush-up courses such as algebra review or arithmetic review. If you need to learn the basic skills, check your school for one of these courses. Usually no tests or grades are given in these courses.
I hope that you have found the suggestions in this program helpful, and I wish you success in your mathematical endeavor. |
Algebra 1
Open to: All students
Length: Year-long
This is the first year of a four year sequential curriculum of theory and problem based mathematics. This
is an introductory course stressing problem solving using the language of algebra. Algebraic solutions
communicate mathematical ideas with clarity and precision. Topics include: signed numbers, real
numbers, data analysis and probability, formulas, sets, factoring, linear and quadratic equations,
graphing, radicals, and problem solving. In this course, students will be introduced to the graphing
calculator. This algebra course is required for college entrance exams, high school chemistry and
physics, and careers in science, math, law, and most trades and businesses.
Geometry
Open to: All students
Prerequisite: Successful completion of Algebra 1
Length: Year-long
This is the second year of the sequential curriculum in which the relationships between lines, points, and
solids is studied as a mathematical system based on Euclidean geometry. Topics include the deductive
methods of proof, geometric constructions, and the practical applications of plane and solid geometric
principals. Success on college entrance examinations requires a geometry background. Connections
between Algebra and Geometry will be explored and Algebra 1 skills will be reviewed. Students will be
retested in the spring to make sure their algebra skills are adequate before being recommended for a
second year of algebra.
Algebra 2
Open to: All students
Prerequisite: Successful completion of Geometry and Algebra 1
Length: Year-long
This is the third year of the sequential curriculum. Topics covered include various mathematical functions
– linear, quadratic, polynomial, exponential – and their graphs. Also included are developing skills with
radicals, factoring, complex numbers, sequences and series, and matrices and determinants.
Trigonometric ratios, functions, identities, equations, solving triangles and problem solving by
trigonometry are covered in the trigonometry units. Graphing calculators are used as tools for learning
and problem solving. This course is a prerequisite for most colleges and universities.
Junior/Senior Review
Open to: All 11th and 12th grade students
Prerequisite: Successful completion of Algebra 2
Length: Year-long
This course is intended for juniors and seniors who have completed Algebra 2/Trigonometry and want to
take one more year of math. The course reviews the Algebra 1, Geometry, and Algebra 2/Trigonometry
curriculum with emphasis on skill maintenance and practical problem solving. First semester topics from
Algebra/Trig are reviewed and SAT/ACT tests are studied. During semester two, topics from Algebra 2,
Geometry and Trigonometry are studied and reviewed. The purpose of this course is to help students
strengthen a weak math background and increase confidence in their mathematics ability. Students may
enter either or both semesters.
Pre-Calculus
Open to: All students
Prerequisite: Successful completion of Algebra 2
Length: Year-long
This is the fourth year of the sequential curriculum. Analysis is a year-long pre-calculus course, which
uses all prerequisite mathematics in the further study of algebra and trigonometry. The general topics
included in this course are: polynomial functions, inequalities, logarithms, trigonometric functions,
complex numbers, sequence and series, and analytic geometry.
Data Analysis
Open to: 12th grade students with teacher recommendation
Length: Year-long
In this course students will learn to analyze data using various types of graphs, and they will learn to
interpret various types of graphs to get information. Topics included are dot, stem-leaf, box and scatter
plots, exploring surveys, gathering data, and probability. Emphasis will be on consumer topics and use of
data analysis.
Advanced Placement Calculus 1
Open to: All students
Prerequisite: Successful completion of Math Analysis or Pre-approval
Length: Year-long
This is a one-year course in differential and integral calculus. A college level textbook is used. Students
in the course prepare for the College Entrance Examination for Advanced Placement in Calculus AB that
is given in May. Some colleges or universities may grant credit and/or advanced placement based on the
score received on the exam. Applications using graphing calculators will be included.
Advanced Placement Calculus 2
Open to: All students
Prerequisite: Successful completion of Advanced Placement Calculus 1 or Pre-approval
Length: Year-long
This one-year course covers the topics of infinite series, parametric equations, and polar graphs that are
necessary to prepare the student to take the Advanced Placement Calculus BC examination given in
May. Some college and universities may grant credit and/or advanced placement based on the score
received on the exam.
Advanced Placement Statistics
Open to: All students
Prerequisite: Successful completion of Algebra 2 or Pre-approval
Length: Year-long
Advanced Placement Statistics acquaints students with the major concepts and tools for collecting and
analyzing data and drawing conclusions from that data. Students will frequently work on projects
involving the hands-on gathering and analysis of real world data. Ideas and computations presented in
this course have immediate links and connections with actual events. Computers and calculators will
allow students to focus deeply on the concepts involved in statistics. This course prepares students for
the Advanced Placement Examination in Statistics.
Algebra 1 Support
* ELECTIVE CREDIT ONLY
Open to: Students enrolled in Algebra 1
Length: Year-long
This course provides students additional instruction and clarification of concepts taught in the Algebra 1. Note: Elective Credit will be
given for this course.
Geometry Support
* ELECTIVE CREDIT ONLY
Open to: Students enrolled in Geometry
Length: Year-long
This course provides students additional instruction and clarification of concepts taught in the Geometry as needed. Note: Elective Credit
will be given for this course. |
Carnegie Learning's Bridge to Algebra
Article reviews Bridge to Algebra from Carnegie Learning, which is a math tutorial program that features practice activities that can help students master topics such as decimals, percent changes, two-step equations, and scientific notation. The author contends that the program suffers from limited graphics and a complete lack of multimedia, though an artificial intelligence component called Cognitive Tutor provides helpful hints via dialogue box. The program also features effective visual representations of abstract ideas, as well as logical instructional methods. The author contends that the program could provide valuable help for students who need extra support. Published by:Technology & Learning |
This National Strategies booklet describes teaching approaches that can be used to develop mental mathematics abilities beyond level five.
The activities described in this supplement build upon and develop activities suggested in Teaching mental mathematics from level five: algebra and are designed to support pupils in developing their sense of variables and applying their knowledge and understanding of algebraic conventions and solving equations in the context of functions |
039587615Intermediate Algebra: Graphs and Functions
Intermediate Algebra: Graphs and Functions, Third Edition, designed specifically for courses that incorporate early graphing and emphasize problem solving and real-life applications. The use of calculators is integrated throughout the text, but remains optional. The authors' proven approach combines proven pedagogy, innovative features, high-interest applications, and a wide range of technology options that add flexibility for instructors and enhance the learning process.
Selected examples are presented with side-by-side algebraic, graphical, and numerical solutions , a format that shows students how different solution methods can be used to arrive at the same answer.
Each chapter now opens with The Big Picture, an objective based overview of the chapter concepts and Key Terms, a list of the mathematical vocabulary integral to the learning objectives.
Every section opens with "What you should learn" objectives to focus students on the main concepts, and "Why you should learn it," highlighting a relevant, real-life application to motivate student learning.
Collaborate! appearing at the end of selected sections, gives students the opportunity to think, talk, and write about mathematics in a group environment. These activities can be assigned for small group work or for whole class discussions.
Looking Further, at the end of each section exercise set, expands upon mathematical concepts presented in the section. These multi-part explorations and applications enhance the development of critical-thinking and problem-solving |
Fields and Geometry III
Description
This first part of this course generalizes the real numbers to a mathematical structure called a field. Finite fields have many applications, particularly in Information Security where the understanding of finite fields is fundamental to many codes and cryptosystems. Properties and constructions of fields will be investigated in detail. The second part of the course considers projective geometries. Projective geometry is one of the important modern geometries introduced in the 19th century. Projective geometry is more general than our usual Euclidean geometry, and it has useful applications in Information Security, Statistics, Computer Graphics and Computer Vision. The focus of this course will be primarily on projective planes.
Objective
To provide an introduction to the areas of Fields and
Projective Geometry with particular emphasis on the links between the
two areas. At the end of this course students should:
have a knowledge of the structure of finite fields and be able to
perform basic calculations in finite fields.
understand the ideas in projective geometry, and how projective
geometry relates to Euclidean geometry.
have enough tools to study objects and transformations in
projective planes corresponding to fields.
Graduate attributes
Linkage past
Prerequisite is MATHS 1007A/B Mathematics I (Pass
Div I) or both MATHS 1007A/B Mathematics I (Pass Div II) and MATHS
2004 Mathematics IIM (Pass Div I). It will be an advantage to have
done PURE MTH 2002 Algebra II, although the necessary material is
revised at the start of the course.
Linkage present
This course complements the first semester
course PURE MTH 3007 Groups and Rings III. It also contains
concepts that are useful for the course PURE MTH 3006 Coding and
Cryptology III
Linkage future
This course is one of the core Pure Mathematics
courses, and provides a strong foundation for further study in the
areas of Algebra and Projective Geometry. Finite fields have many
applications, and an understanding of their structure is essential
to students who want to further their knowledge of codes and
cryptosystems. |
Discrete Mathematics for Computer Scientists (2nd Edition)
This is a new edition of a successful introduction to discrete mathematics for computer scientists, updated and reorganised to be more appropriate for the modern day undergraduate audience. Discrete mathematics forms the theoretical basis for computer science and this text combines a rigorous approach to mathematical concepts with strong motivation of these techniques via examples. Key Features Thorough coverage of all area of discrete mathematics, including logic, natural numbers, coding theory, combinatorics, sets, algebraic functions, partially ordered structures, graphs, formal & complexity theory Special emphasis on the central role of propositional & predicate logic Full chapters on algorithm analysis & complexity theory Introductory coverage of formal machines & coding theory Over 700 exercises Flexible structure so that the material can be easily adapted for different teaching styles. New to this Edition Improved treatment of induction Coverage of more 'basic' algebra List of symbols including page references for definition/explantion Modern text design and new exercises to aid student comprehension
The two-volume textbook Comprehensive Mathematics for the Working Computer Scientist, of which this is the second volume, is a self-contained comprehensive presentation of mathematics including sets, ...
Metal Detecting for the Beginner: 2nd Edition is an expanded version of the original best-selling book on metal detecting. We've added more pages, more photographs, and a vastly enhanced discussion ...
Worldwide the automotive industry is challenged to make dramatic improvements in vehicle fuel economy, some already legislated and in some cases by new regulations. In Europe there are CO2 emissions ...
Helps you understand the mathematical ideas used in computer animation, virtual reality, CAD, and other areas of computer graphics. This work also helps you to rediscover the mathematical techniques ... |
David Lay, author of the currently used Linear Algebra
textbook, has provided a convenient way for faculty and students to access the
data in homework problems. With the Lay Linear Algebra toolbox installed, at
the command line you type "c2s3" for chapter 2, section 3. MATLAB responds with
a list of homework problems with data and prompts for an exercise number. After
entering the number, MATLAB responds by defining a matrix or matrices
containing the data. In addition, the Lay toolbox has various tools which
manipulate matrices in a way parallel to the presentation in the book. For
example, there are commands for each of the elementary row operations. The
author has included boxed MATLAB subsections in the Study Guide which
demonstrate how to use these tools. To get a hard copy of the Study Guide, you
may contact the publisher. The electronic version is on the CD-ROM that comes
with the book. I can provide you with an electronic copy if you do not have the
CD. I have made use of the slides provided by the publisher. Depicting examples
in 3 dimensions is difficult and the author has done a reasonably good job.
In the MATLAB toolbox folder (e.g.,
C:\MATLAB7\toolbox), create a new folder "Lay"
Extract the files from the zipped file (about 50, all
ending with .m) and put into the "Lay" folder
Open up MATLAB. Go to Set Path under the File menu and
add the "Lay" folder.
Go to the command line window in MATLAB and enter
"c1s1". You should get the following response and prompt: "Exercise number
(1-4,7-18,29-32,34)?"
In addition to the author's tools,
there is an extensive set of material online for creating labs, illustrating
concepts or as places for students to explore. A good place to start is MAA's
Digital Library
[Thomas Hern was a coauthor of a seminal article that I handed out
to attendees a week after the seminar: "Viewing Some Concepts and
Applications in Linear Algebra" from Visualization in Teaching and
Learning Mathematics (1991) of the MAA Notes Series.]
Most tools on the MAA Digital
Library site are stand alone, meaning that they do not need costly software
such as MATLAB, Maple or Mathematica. However, since
we do have access to this software, we may explore what is available for any or
each of them. A good start is the ATLAST project:
The files can be downloaded here
and documentation is available. However, there is a lab manual and guide
published by Prentice Hall. Since Prentice Hall has merged with Addison-Wesley,
we may get them to bundle the guide with the Lay text. I have asked Pearson
(Fred Speers) to send us a review copy or 2. |
Description
This course is a (short) elementary introduction tomodern geometry (in the spirit of Klein's "Erlanger Programm"). Its aim is to expose students to some useful/important geometric ideas and structures of modern mathematics through geometric transformations.
The fundamental concept is that of transformation. A geometry is viewed as the study of those properties of a "space" that are invariant under a certain group of transformations acting on that space. |
The Student Union. The INFORMS Student Union aims to be the preeminent Web site for students and recent graduates in OR/MS and related fields. It seeks to fill their most important needs insofar as that can be done on-line.
What is Operations Research? This site maintained by the US Department of Labor, describes Operations Research as a profession and gives statistics on job opportunities.
WORMS. (World of Operations Research and Management Science). An excellent general site for Operations Research maintained at the University of Melbourne.
Excel and Excel Solver
Excel Tutorial. This is a tutorial developed by Professor Paula Ecklund of the Fuqua School at Duke. It's great, whether you are a beginner, or whether you want to learn more about advanced features in Excel. It includes tutorials on pivot tables, data tables, solver, analysis tools, graphing including scatter plots, array formulae and much much more.
Linear Programming
Frequently Asked Questions on Linear and Nonlinear Programming. These are the FAQs initiated by John Gregory, now maintained by Bob Fourer as part of the NEOS Guide. They are excellent guides, and provides resources beyond linear and non-linear programming. For example, the non-linear guide provides lots of information about heuristic search.
Optimization
Algorithm Animations. These animations are based on algorithms from the textbook Sedgewick, Robert. Algorithms in C++. Addison Wesley Longman, 1992. It also includes pointers to sites with other good animations.
What to do on fourth down. A technical paper by David Romer that uses dynamic programming to show that football teams should go for it on fourth down in a wide range of situations. (PDF - 2.9MB)
Heuristics
The GA Archives. This web site contains a wide range of information on Genetic Algorithms.
GAlib. GAlib contains a set of C++ genetic algorithm objects. The library includes tools for using genetic algorithms to do optimization in any C++ program using any representation and genetic operators.
Ant Colony Optimization. We won't actually cover this in class, and it is an overhyped idea. Nevertheless, the idea of using virtual ants to solve optimization problems is intriguing.
Graph Algorithms and Network Flows
Graph Algorithms. This is a page maintained by Dr. Thomas Emden-Weinert. It has a wide range of interesting links.
Graph Theory Terminology. This is a collection of terms and their definitions as used in graph theory. It is maintained by Stephen Locke.
Graph Theory Glossary: This is another collection of terms used in graph theory, maintained by Chris Caldwell.
Network Flow Algorithms. This is a collection of algorithms developed by Andrew Goldberg and collaborators and maintained by Andrew Goldberg. They are public domain, and known for being very efficient.
Stanford GraphBase. This is a system that was developed by Donald Knuth, of "Art of Programming" and "TeX" fame. |
More About
This Textbook
Overview
Discrete geometry investigates combinatorial properties of configurations of geometric objects. To a working mathematician or computer scientist, it offers sophisticated results and techniques of great diversity and it is a foundation for fields such as computational geometry or combinatorial optimization.
This book is primarily a textbook introduction to various areas of discrete geometry. In each area, it explains several key results and methods, in an accessible and concrete manner. It also contains more advanced material in separate sections and thus it can serve as a collection of surveys in several narrower subfields. The main topics include: basics on convex sets, convex polytopes, and hyperplane arrangements; combinatorial complexity of geometric configurations; intersection patterns and transversals of convex sets; geometric Ramsey-type results; polyhedral combinatorics and high-dimensional convexity; and lastly, embeddings of finite metric spaces into normed spaces.
Jiri Matousek is Professor of Computer Science at Charles University in Prague. His research has contributed to several of the considered areas and to their algorithmic applications. This is his third |
Book Description: It provides a clear summary and outline in algebra. Each topic begins with concepts, formulas, and problem-solving steps, followed by well-designed examples. There are 18,000 examples and exercises in this book-7,000 are real-life word problems modeled after typical questions on standardized tests that are given across the United States. |
Traditionally, linear algebra, vector analysis, and the calculus of functions of several variables are taught as separate subjects. This text explores their close relationship and establishes the underlying links. A rigorous and comprehensive introductory treatment, it features clear, readable proofs... read moreProduct Description:
Traditionally, linear algebra, vector analysis, and the calculus of functions of several variables are taught as separate subjects. This text explores their close relationship and establishes the underlying links. A rigorous and comprehensive introductory treatment, it features clear, readable proofs that illustrate the classical theorems of vector calculus, including the inverse and implicit function theorems. Prerequisites include a knowledge of elementary linear algebra and one-variable calculus. Starting with basic linear algebra and concluding with the integration theorems of Green, Stokes, and Gauss, the text pays particular attention to the relationships between different parametrizations of curves and surfaces, and it surveys their application in line and surface integrals. Concepts are amply illustrated with figures, worked examples, and physical applications. Numerous exercises, with hints and answers, range from routine calculations to theoretical |
Algebra II Literacy Advantage provides a curriculum that builds on the algebraic concepts covered in Algebra I. Through a "Discovery-Confirmation-Practice"-based exploration of intermediate algebra concepts, students are challenged to work toward a mastery of computational skills, deepen their conceptual understanding of key ideas and solution strategies, and Within each lesson, students are supplied with a direct-instruction Study and a scaffolded note-taking guide called a Study Sheet, as well as a Checkup activity that appears after the Study. This sequence provides them with the opportunity to hone their computational skills by working through low-stakes problem sets before moving on to a formal assessment. Success in math is not all about numbers. Algebra II provides extensive scaffolding to help below-proficient readers understand academic math content and make the leap to higher-order thinking. Mathematical vocabulary is supported with rollover definitions and usage examples that feature aural and graphical representations of terms. A feature called Support Cards identifies concrete uses of active reading strategies that facilitate recognition and understanding of key ideas. To further assist students for whom language presents a barrier to learning or who are not reading at grade level, Algebra II Literacy Advantage includes audio resources in both Spanish and English. The content is based on the National Council of Teachers of Mathematics (NCTM) standards and is aligned with state standards. |
ntroduction to English for 8th Grade In this segment of English lesson for 8th grade Math students you will
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8th grade math helpprovides students with all the support required with solving problems. Grade 8 Math help
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Rational and Irrational Numbers :- Numbers appear like dancing letters to
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Short description
Study & Master Mathematics Grade 8 covers and integrates all LOs as stated in the NCS.
Long description
Study & Master Mathematics Grade 8 is an exciting new course that covers and integrates all the Learning Outcomes for Mathematics stated in the National Curriculum Statement. The material is presented in a user-friendly way that will not only boost learners' confidence, but will also show them how to enjoy Mathematics. |
This book covers the content prescribed for the New Zealand Diploma in Engineering course DE4102 Engineering Mathematics. Some foundation level material is also provided to help those students whose preparation for tertiary mathematics study is patchy, whether that be due to gaps in recent secondary...
Essential Maths and Stats provides a comprehensive overview of tertiary level mathematics and statistics and is the only definitive New Zealand text for mathematics and statistics at entry level. It is also an excellent 'extension' text for secondary school students.
Divided into six ke...Mathematics for Engineers and Technologists provides students of engineering and technology with a wide range of analytical and numerical techniques for making quantitative assessments of engineering and technological phenomena. The book provides a foundation for the solution of equations that arise... |
Mathematics - Algebra (356 results)
Elementary Algebra. The author has endeavored to prepare a course sufficiently advanced for the best High Schools and Academies, and at the same time adapted to the requirements of those who are preparing for admission to college. Particular attention has been given to the selection of examples and problems, a sufficient number of which have been given to afford ample practice in the ordinar processes of Algebra, especially in such as are most likely to be met with in the higher branches of mathematics. Problems of a character too difficult for the average student have been purposely excluded, and great care has been taken to obtain accuracy in the answers. The author acknowledges his obligations to the elementary text-books of Todhunter and Hamblin Smith, from which much material and many of the examples and problems have been derived. He also desires to express his thanks for the assistance which he has received from experienced teachers, in the way of suggestions of practical value. Webster Wells. Boston, 1886.
In the selection of materials, those articles liave been taken which have a practical application, and which are preparatory to succeeding parts of the mathematics, philosophy, and astronomy. Tlie object has not been to introduce original matler. In the mathematics, which have been cultivated with success from the days of Pythagoras, and in which the principles already established are sufficient to occupy the most active mind for years, the parts to which the student ought first to attend, are not those recently discovered. Free use has been made of the works of Newton, Maclaurin, Saunderspn, Simpson, Euler, Emerson, Lacroix, and others, but in a way that rendered it inconvenient to refer to them, in particular instances. The proper field for the display of mathematical genius is in the region of invention. But what is requisite for an elementary work, is to collect, arrange and illustrate, materials already provided. However humble this employment, he ought patiently to submit to it, whose object is to instruct, not those who have made considerable progress in the mathematics, but those who are just commencing the study. Original discoveries are not for the benefit of beginners though they may be of great importance to the advancement of science. The arrangement of the parts is such, that the explanation of one is not made to depend on another which is to follow. The addition, multiplication, and division of powers for instance, is placed after involution. In the statement of general rules, if they are reduced to a small number, their applications to particular cases may not, always, be readily understood. On the other hand, if they are very numerous, they become tedious and burdensome to the memory. The rules given in this introduction, are most of them comprehensive; but they are explained and applied, in subordinate articles. A particular demonstration is sometimes substituted for a general one, Avhen the application of the principle to other cases is obvious. The examples are not often taken fi om philosophical subjects, as the learner is supposed to be familiar with none of the sciences except arithmetic. In treating of negative quantities, frequent references are made to mercantile concerns, to debt, and credit, c.These are merely for the purpose of illustration.
This text is prepared to meet the needs of the student who will continue his mathematics as far as the calculus, and is written in the spirit of applied mathematics. This does not imply that algebra for the engineer is a different subject from algebra for the college man or for the secondary student who is prepared to take such a course. In fact, the topics Avhich the engineer must emphasize, such as numerical com)utations, checks, graphical methods, use of tables, and the solution of specific problems, are among the most vital features of the subject for any student. But important as these topics are, they do not comprise the substance of algebra, which enables it to serve as part of the foundation for future work. Rather they furnish an atmosphere in which that foundation may be well and intelligently laid. The concise review contained in the first chapter covers the topics which have direct bearing on the work which follows. No attempt is made to repeat all of the definitions of elementary algebra. It is assumed that the student retains a certain residue from his earlier study of the subject. The quadratic equation is treated with unusual care and thoroughness. This is done not only for the purpose of review, but because a mastery of the theory of this equation is absolutely necessary for effective work in analytical geometry and calculus. Furthermore, a student who is well grounded in this particular is in a position to appreciate the methods and results of the theory of the general equation with a minimum of eii ort. The theory of equations forms the keystone of most courses in higher algebra. The chapter on this subject is developed gradually, and yet with pointed directness, in the hope that the processes which students often perform in a perfunctory manner will take on additional life and interest.
They feel themselves continually handicapped by this ignorance. Their critical faculty is eager to submit, alike old established beliefs and revolutionary doctrines, to the test of science. But they lack the necessary knowledge. Equally serious is the fact that another generation is at this moment growing up to a similar ignorance. The child, between the ages of six and twelve, lives in a wonderland of discovery; he is for ever asking questions, seeking explanations of natural phenomena. It is because many parents have resorted to sentimental evasion in their replies to these questionings, and because children are often allowed either to blunder on natural truths for themselves or to remain unenlightened, that there exists the body of men and women already described. On all sides intelligent people are demanding something more concrete than theory; on all sides they are turning to science for proof and guidance. To meet this double need the need of the man who would teach himself the elements of science, and the need of the child who shows himself every day eager to have them taught him is the aim of the Thresholds of Science series. This series consists of short, simply written monographs by competent authorities, dealing with every branch of science mathematics, zoology, chemistry and the like. They are well illustrated, and issued at the cheapest possible price.
Advantage has been taken of the issue of a new edition of the Intermediate Algebra to revise the text and to make a number of changes., To meet the wishes of teachers who have used, or propose to use, the book in the advanced classes of the secondary schools, additions have been made in order that the prescribed courses may be formally covered. To the chapter on equations there have been added exercises bearing on or developing further the theory. A chapter on Scales of Notation has been introduced. The note on Annuities, incidental to the Geometrical Progression, has been expanded to constitute a chapter in which are considered the simpler problems of finance related to annuities, and the use of the fundamental tables of Interest and Annuities provided for and explained. The Miscellaneous Examples that were given in the earlier edition have been retained in the hope that the more adventurous students, in particular candidates for Honours at Matriculation, may find in them a help and a stimulus. The chapter on Exponential and Logarithmic Series has been enlarged in order to give prominence to the concrete problem of the construction of tables of logarithmics, it being felt that in this way the significance of the theory is best brought out. Alfred T.DeLURY. Toronto, July 15,
In preparing this second edition the earlier portions of the book have been partly re-written, while the chapters on recent mathematics are greatly enlarged and almost wholly new. The desirability of having a reliable one-volume history for the use of readers who cannot devote themselves to an intensive study of the history of mathematics is generally recognized. On the other hand, it is a difficult task to give an adequate bird s-eye-view of the development of mathematics from its earliest beginnings to the present time. In compiling this history the endeavor has been to use only the most reliable sources. Nevertheless, in covering such a wide territory, mistakes are sure to have crept in. References to the sources used in the revision are given as fully as the limitations of space would permit. These references will assist the reader in following into greater detail the history of any special subject. Frequent use without acknowledgment has been made of the following publications: Annuario Biografico del Circolo MaknuUico di Palermo 1914; Jakrhuch uber die Fortschritte der Mathematiky Berlin;. C.Poggendorffs Biographisch-Literarisckes Handworterbuch, Leipzig; Gedenkkigebuch fur McUhenuUikeTf von Felix Miiller, 3. Aufl., Leipzig und Berlin, i()i 2 Revue SemestrieUe des Publications MathinuUigues, Amsterdam. The author is indebted to Miss Falka M.Gibson of Oakland, Cal. for assistance in the reading of the proofs. Floman Cajori. University of California March, 1919.
The purpose of this book, as implied in the introduction, is as follows: to obtain a vital, modern scholarly course in introductory mathematics that may serve to give such careful training in quantitative thinking and expression as wellinformed citizens of a democracy should possess. It is, of course, not asserted that this ideal has been attained. Our achievements are not the measure of our desires to improve the situation. There is still a very large safety factor of deud wood in this text. The material purposes to present such simple and significant principles of algebra, geometry, trigonometry, practical drawing, and statistics, along with a few elementary notions of other mathematical subjects, the whole involving numerous and rigorous applications of arithmetic, as the average man (more accurately the modal man) is likely to remember and to use. There is here an attempt to teach pupils things worth knowing and to discipline them rigorously in things worth doing. The argument for a thorough reorganization need not be stated here in great detail. But it will be helpful to enumerate some of the major errors of secondary-mathematics instruction in current practice and to indicate briefly how this work attempts to improve the situation. The following serve to illustrate its purpose and program:1. The conventional first-year algebra course is characterized by excessive formalism; and there is much drill work largely on nonessentials.
El Preface This book is the result of twenty years of patient experiment in actual teaching. It is intended to be completed in the first year of the high school. It presents algebraic equations primarily as a device for the solution of problems stated in words, and gives a complete treatment of numerical equations such as are usually included in high-school algebra one-letter and two-letter equations, integral and fractional, including one-letter quadratics and the linear-quadratic pair. So much of algebraic manipulation is included as is necessary for the treatment of these equations. The arithmetic in the book is presented from a new point of view that of approximate computation and is utilized in the evaluation of formulas and in the solution of equations throughout the succeeding pages. Geometrical facts are introduced as the basis of many algebraic and arithmetic problems, and wherever they are not intuitively accepted by the pupils they are accompanied by adequate logical demonstration. Proofs, and parts of proofs, are avoided when they seem to the pupils of an unnecessary and hair-splitting kind. Ah problems are carefully graded, for it is by means of problems that each successive algebraic difficulty is introduced. A great deal of pains has been taken to present new topics clearly and concretely, often dividing them into sub-topics each of which is separately illustrated and apphed to practice. Definitions are generally prepared for by such advance work as will cause the student to feel the need of them; and where no need exists, they are omittedIx this text the authors have endeavored to present a course in algebra for the first year of high school which shall be simple, comprehensible to the students, and of high educational and mathematical value. They have made the solution of equations and problems the core of the course; they have emphasized the essentials, avoiding little-used complexities of algebra; they have taught new ideas inductively; they have emphasized thoughtful rather than mechanical solutions of exercises; they have tried to make the course maintain and increase the students efficiency in arithmetic; they have tried to make the course interesting by including varied problem material and historical notes, and valuable by including practical applications. The essential features of the course have been tried out in the classroom by many teachers. The text contains sufficient material to meet the needs of schools whose pupils have studied algebra before entering the high school; the topics have been arranged, however, so that a class may easily cover the essentials of the course in one school year. Attention is directed to the following devices that have been employed to attain the desired ends: Each topic that is taken up is used in the solution of equations. (See 9, 10, 12, 41, 51, 60, 107, etc.) This makes the study of the various topics purposeful, allows for good gradation in the book as a whole, and emphasizes the equation. Problems are introduced at short intervals. Informational, geometric, and physics problems in reasonable number are used. New types of problems are introduced gradually, appearing first in classified lists, are taught with extreme care, and are used thereafter in miscellaneous lists. Experimental verification is suggested for some of the facts from geometry and physics that are used. (See Exercises 7, 25, 28, 29, 38, 39, 49, 106; 13, 142, 143, 190, etc.
The main object in preparing this new Algebra has been to simplify principles and give them interest, by showmg niunbers. Each successive process is taken up for the sake of the economy or new power which it gives as compared with previous processes. This treatment should not only make each principle clearer to the pupil, but should give increased unity to the subject as a whole. We believe also that this treatment of algebra is better adapted to the practical American spirit, and gives the study of the subject a larger educational value. Among the special features of this Introductory Algebra, the following may be mentioned: A large nmnber of written problems are given in the early part of the book, and these are grouped in types which correspond in a measure to the groups used in treating original exercises in the authors Geometry. Many informational facts are used in the written problems. The central and permanent numerical facts in various departments of knowledge have been collected and tabulated on pages280-286 for use in niaking problems.
Thi 8 work was commenced sixteen years ago at the earnest solicitation of numerous teachers, who were dissatisfied with the textbooks then in use. That they were not alone in their opinion is evidenced by the number of new treatises, or revisions of old ones, printed since that time, and now used in the schools of this country. The crudeness of even the best Algebras of a quarter-century ago was mainly owing to the fact that, as a rule, mathematicians neglected the elementary branches for the more attractive fields of Higher and Applied Mathematics; hence blunders and inconsistencies were allowed which otherwise would not have been tolerated. The wonderful progress made in the Natural Sciences, and the extended use of Algebra in the treatment of Geometrical Magnitudes, have finally called the attention of educators to the necessity of improving the elementary treatises, and more rigidly limiting the meaning of the signs. That this agitation comes none too soon is evident to every thoughtful teacher, and can be readily seen by auy one who compares the various text-books used in our schools. Note the following inconsistencies: In some text-books now before me, 6 : 7 equals f;in others, 6 : 7 equals. In some, 6 -f 4 X 2 = 20;in others, 6 -- 4 X 2 = 14.Of course, the meaning and use of a sign depend upon agi eement, but it is of extreme importance that we do agree in such matters. In the same work, too, statements incompatible with each other are made; thus, a -i-bc and a -i-b Xc are said to have different values, and yet be and bXc are, in all woi ks, said to have one and the same meaning. Since a-h be and a -ib Xe differ only in She use of bXc for be, it is plainly necessary that one or the other of these two statements be changed. One of the objects in writing this book is to urge the adoption of the following law for Numerical Values; viz.,(l) Find the value of each term separately; thus, 6-f-4X 2 = 6 -f8= 14. (2)In finding the m, lue of a term, begin at the Right and use the signs in their oi der; thus, 6-f-4x 2 = 6-r-8= f.In other words, the jm tion of the term to the left of the division sign is the Dividend, and the part to the right is the divisor.
The main object in preparing this new Algebra has been to simplify principles This treatment should not only make each principle dearer to the pupil, but should give increased unity to the subject as a whole. We beheve also that this treatment of algebra is better adapted to the practical American spirit, and gives the study of the subject a larger educational value. Among the special features of this Introductory Algebra, the following may be mentioned: A large nimiber of written problems are given in the early part of the book, and these are grouped in types which correspond in a measure to the groups used in treating original exercises in the authors Geometry. Many informational facts are used in the written problems. The central and permanent niunerical facts in various departments of knowledge have been collected and tabulated on pages280-286 for use in making problems.
There are two classes of men who might be benefited by a work of this kind, viz., teachers of the elements, who have hitherto confined their pupils to the working of rules, without demonstration, and students, who, having acquired some knowledge under this system, find their further progress checked by the insufficiency of their previous methods and attainments. To such it must be an irksome task to recommence their studies entirely; I have therefore placed before them, by itself, the part which has been omitted in their mathematical education, presuming throughout in my reader such a knowledge of the rules of algebra, and the theorems of Euclid, as is usually obtained in schools. It is needless to say that those who have the advantage of University education will not find more in this treatise than a little thought would enable them to collect from the best works now in use 1831, both at Cambridge and Oxford. Nordo I pretend to settle the many disputed points on which I have necessarily been obliged to treat. The perusal of the opinions of an individual, offered simply as such, may excite many to become inquirers, who would otherwise have been workers of rules and followers of dogmas. They may not ultimately coincide in the views promulgated by the work which first drew their attention, but the benefit which they will derive from it is not the less on that account. I am not.
The present volume contains a Second Course in Algebra adapted to the latter part of the high school curriculum. It covers the topics usually included in Interme Idiate and Advanced Algebra in secondary schools. Hence pupils who have completed it will be prepared in algebra vfor scientific and engineering schools as well as for the rx ordinary academic college. The methods which are characteristic of the authors Algebra, Book One are here continued and developed. The chief aim is to simplify principles and give them interest, by showing more plainly, if possible, than has been done heretofore, the practical or common-sense reason for each step or process. Each new process, for instance, is introduced by what may be termed the efficiency-inductive method. In the Exercises also there are special examples which cause the pupil to realize the efficiency meaning of processes from various points of view. As in the authors other mathematical texts, pivotal and permanently valuable number facts and laws from other branches of study are introduced in various ways. This gives a correlation of algebra with geography, history, and other subjects. A further correlation with physics and engineering is obtained by the use of some of the most important formulas in these branches, and also by familiarizing the pupil with their fundamental concepts and number facts.
The present book is an enlargement of the authors Elements of Algebra. To the end of Chapter XXVIII. it is identical with the latter and the School Algebra. Some revision of the later chapters in the Elements has been incorporated in this book, and a number of new chapters have been added. The scope of the books, amply justified by their successful use in high and normal schools and colleges, is stated in the preface to the Elements: The aim has been to make the transition from ordinary Arithmetic to Algebra natural and easy. Ko efforts.have been spared to present the subject in a simple and clear manner. Yet nothing has been slighted or evaded, and all difficulties have been honestly faced and explained. New terms and ideas have been introduced only when the development of the subject made them necessary. Special attention has been paid to making clear the reason for every step taken. Each principle is first illustrated by particular examples, thus preparing the mind of the student to grasp the meaning of a formal statement of the principle and its proof. Directions for performing the different operations are, as a rule, given after these operations have been illustrated by particular examples. The importance of mental discipline to every student of mathematics has also been fully recognized.
Colleges and Scientific Schools. The first part is simply a review of the principles of Algebra preceding Quadratic Equations, with just enough examples to illustrate and enforce these principles. By this brief treatment of the first chapters, sufficient space is allowed, without making the book cumbersome, for a full discussion of Quadratic Equations, The Binomial Theorem, Choice, Chance, Series, Determinants, and The General Properties of Equations. Every effort has been made to present in the clearest light each subject discussed, and to give in matter and methods the best training in algebraic analysis at present attainable. The work is designed for a full-year course. Sections and problems marked with a star can be omitted, if necessary; and for a half-year course many chapters must be omitted. The author gratefully acknowledges his obligation to Mr. G.W. Sawin of Harvard College, who has contributed the excellent chapter on Determinants, and been of invaluable assistance in revising every chapter of the book. Answers to the problems are bound separately in paper covers, and will be furnished free to pupils when teachers apply to the publishers for them. Any corrections or suggestions relating to the work will be thankfully received. G.A. Wentworth. Phillips Exeter Academy, September, 1888.
The present volume contains a second course in algebra adapted to the latter part of the High School curriculum. The book is divided into two parts, Part One being meant for use in such classes as give only a half year to the second course in algebra, while the entire volume is to be used by classes giving a whole year to the second course. In half-year classes, Part Two will constitute a reservoir of extra work for bright pupils. The features which characterize the authors First Book in Algebra are continued and dievelopediuible numbers, facts and laws from other branches of study are introduced in various ways. This gives a correlation of algebra with geography, history, economics, and other school studies.
The present volume contains a First Course in Algebra adapted to the early part of the high school curriculum. The main purpose in writing the book has been to simplify prinr Among the special features of this Algebra, the following may be mentioned: A large number of written problems are given in the ben collected and tabulated on pages387-390 for use in making problems. Similariy the most important formulas in arithmetic, geometry, physics, and engineering have been tabulated for use by teacher and pupil (pp. 385, 386).
TlHE following manual was prepared for the use- of the students of Columbia College, and in its original form it has been employed as a text-book, not only in that institution, but in various Colleges, Academies, High Schools, and other institutions of learning. The flattering manner in which it has been received by our most successful teachers of Mathematics, has induced the Author to publish it in its present revised form. In preparing it anew for the press, such alterations and improvements have been made as have been suggested by the authors practical experience in its use as a college text-book. The opening chapters have been somewhat simplified, the chapter on logarithms has been extended, a section on inequalities has been added, and the whole has been carefully corrected and revised.
As regards the method of teaching algebra, I would make it, in the earlier stages, as much a generalized arithmetic as possible. Results obtained by algebra would be verified by arithmetical instances; and the use of a formula would be indicated as including any number of instances. Elaborate (and to my mind wearisome) processes, useful for solving artificial combinations of difiiculties, would be at least deferred. With a comparative beginner, progress towards new ideas or new stages of old ideas can, I think, best be made by the simplest instances, and it is on this account that I would build algebra entirely on arithmetical foundations so far as concerns the teaching of beginners. Professor Forsyth, M.A,, D, Sc,, F.Rs., Cambridge. It is assumed that pupils will be required throughout the course to solve numerous problems which involve putting questions into equations. Some oJE these problems should be chosen from mensuration, from physics, and from commercial life. The use of graphical methods and illustrations, particularly in connection with the solution of equations, is also expected. Extract from the Report of the American Mathematical Society,
The present volume contains a First Course in Algebra adapted to the early part of the high school curriculum. The main purpose in writing the book has been to simplify pririr jwwer which it gives as compared with previous processes. Among the special features of this Algebra, the following may be mentioned: A large number of written problems are given in tjije been collected and tabulated on pages387-390 for use in making problems. Similarly the most important formulas in arithmetic, geometry, physics, and engineering have been tabulated for use by teacher and pupil (pp. 385, 386).
The aim of this little book is to provide an introductory course as a foundation to elementary algebra. A minimum number of definitions, an early introduction of the literal symbol in its simplest form, a clear conception of the opposition of positive and negative quantity, and a gradual introduction to the early processes are believed to be the first essentials to successful later work. New elements are introduced as the result of some natural process, the exponent, for example, not being mentioned or used until, in multiplication, the pupil meets the operation that produces it. Certain important topics are given a more extended treatment than is customary in most books prepared for beginners. The application of the equation to the problem is made in a form that experience has shown to give excellent results, and the reasoning powers developed by the limited classifications have been equal to the demands of the general problem. Substitution has a much more important position than is usual in elementary teaching, and its constant applications are designed to meet an actual need felt by teachers in higher grades of work.
The present volume contains a second course in algebra adapted to the latter part of the High School curriculum. Many schools give only one half year to the study of the second course in algebra,, and it is the object of this book to supply material adapted to the needs of such schools. In different parts of the country, tentative syllabi have recently. been worked out for such a briefer study, and these syllabi have been carefully considered by the authors in writing the present volume. The features which characterize the authors First Book in Algebra are continued and developediLable number facts, and laws from other branches of study are introduced in various ways. |
Synopsis
A one-volume, one-day algebra course.
Alpha Teach Yourself Algebra I in 24 Hours provides readers with a structured, self-paced, straightforward tutorial on algebra. It's the perfect textbook companion for students struggling with algebra, a solid primer for those looking to get a head start on an upcoming class, and a welcome refresher for parents tasked with helping out with homework. The book provides 24 one-hour lessons, with each chapter designed to build on the previous one.
? Covers classifying number sets, expressions, polynomials, factoring, radicals, exponents and logarithms, and much more
? Each chapter ends with a quiz so readers can identify where they may need more |
Math 3041 – Introduction to Mathematics I
Course Objectives
have achieved competency in the fundamental mathematical concepts necessary for the teaching of mathematics up to the sixth grade.
have become familiar with the main definitions, concepts, results and methods of proof relating to each part of the syllabus.
be able to quote the definitions and results, and to reproduce the proofs of some key results.
be able to solve problems relating to the material covered and present the solutions in a good mathematical style.
Catalog Description
Concepts of Arithmetic and Algebra Three credits. Three hours of lecture per week. Prerequisite: For Elementary School Teachers.
Algebraic properties of the integers, the natural numbers and rational numbers. The decimal number system. Ordering of numbers. The number line. Solution of simple equations and inequalities. Measurement and approximation. Divisibility rules. Maximum common divisor and minimum common multiple. Percentages, ratio and proportion. Word problems. Graphs. Correspondences. The real numbers. |
Amazing resource that give you everything you'd need to make a wicked math library. Also presents example problems, covers the math basics of how to solve them, and then presents how it would be solved using pseudo code. Amazing learning tool. From v...moreAmazing resource that give you everything you'd need to make a wicked math library. Also presents example problems, covers the math basics of how to solve them, and then presents how it would be solved using pseudo code. Amazing learning tool. From vectors to advanced matrix math, this book is awesome.(less) |
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Wouldn't it be nice to have your very own algebra tutor to teach this often-confusing topic? David Chandler has developed a creative approach to teaching algebra in a computer program called Home
Study Companion: Algebra I ($59). His course is based on the textbook Algebra
I: Expressions, Equations, and Applications by Paul Foerster, published by Prentice Hall (which is easily available online). Mr. Chandler provides the "classroom presentation" portion of the topics as he works through 126 sections that cover standard first-year algebra topics, with each section set up as approximately one day's worth of work. He utilizes whiteboard technology so that he can both talk through the lessons and at the same time illustrate anything on the computer just as he would on a chalkboard. This interactive method enhances the learning by utilizing visual and auditory input at the same time. A handy calculator pops up to help with calculations as Mr. Chandler works through the problems with the student. Each lesson is approximately 15 minutes long each. The program is designed as a self-study for both homeschooled students and adults who want to brush up on this important topic.
There were many things we really liked about this program. Perhaps most importantly,
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way the use of auditory and visual instruction, which gives auditory learners
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and understandable, it's great to be able to pause or replay any section as
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Please Note: Some people initially have trouble with the audio if using Quicktime
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This product needs to be purchased along with Paul
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the lessons in the textbook. Since I was not provided
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I was never able to get the sound part of this video
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in the textbook.
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I. In fact, if I hadn't already purchased the entire
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Please Note: Some people initially have trouble with the audio if using Quicktime on a Mac. This is an easily fixed problem and Mr. Chandler willingly walks customers through the simple fix to the problem: downloading and using the VLC media player.
Product Review by Christine Hindle, The
Old Schoolhouse® Magazine,
LLC, February 2012 |
The Number System (Dover Books on Mathematics)
Book Description: This book explores arithmetic's underlying concepts and their logical development. It offers an informal and intuitive understanding of the rigorous logical approach, in addition to a detailed, systematic construction of the number systems of rational, real, and complex numbers. Numerous exercises help students test their progress and practice concepts. 1956 edition.
Buyback (Sell directly to one of these merchants and get cash immediately)
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Mount Wilson CalculusIt is essential to understand the students? thought processes because each student is uniquely suited to a particular method. In order to lead students in a right direction, I ask them what problems they are really having difficulty with and take the necessary time to explain the concepts to them. However, understanding the theories is fundamental but is not enough |
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Sorry I have all the materials listed in Cambridge Econ intranet but I don't have the materials for maths and stats as it seems like there is nowhere to get them Actually, my friends told me that he was given some handouts for the maths and stats and no digital materials.
The materials are very big. I don't know how to post them. They cannot be sent by emails. |
With Algebra, you get to work puzzles by playing with letters, numbers, and symbols. Algebra problems are pretty abstract, and learning the proper algebraic procedures gives you tools to actually solve these abstract problems instead of guessing. |
ChiliMATH is here! This site contains free online math tutorials created to supplement class lectures and to guide students in solving math problems in a straightforward way. My goal is for students to build confidence as they develop their own mathematical skills and knowledge in the process. One secret to succeed in Math is doing a lot of practice. ChiliMATH offers many worked examples which can be printed easily for offline use. I hope that you find these resources helpful in your studies.
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The vast majority of students will not become scientists, mathematicians or engineers. Why does a strong foundation in basic mathematics skills matter? The answer has much to do with our shrinking job base in manufacturing and low skill jobs. The jobs of our new economy increasingly demand the use and understanding of technological tools and concepts. These tools and concepts are better put to use by those who have a sound grounding in elementary and high school mathematics.
Many jobs now require employees who can understand project data, including interpreting graphs and charts. Some positions require understanding of how the concepts of expenditures and income affect a budget. A price increase of, let's say, five percent may make a project infeasible after all the numbers are analyzed. Job seekers who possess these basic skills will have an advantage in the marketplace. Those who don't possess these skills will face increasingly difficult odds.
Aside from the purely defensive point of view of getting left behind by increasing job market competition, these basic skills will make the individual a better citizen. Participation in civic life will be more rich, meaningful and rewarding for those who have passed a threshold of basic mathematics skills. The government's tax policies, national energy policy and budget deficits affect everybody. A full understanding of these issues hinge on a grasp of basic mathematics.
Mathematics teaches some basic functional skills such as reasoning and problem-solving. Innovation and discovery are suppressed and in many cases choked to death without the proper exposure to instruction in basic skills.
The world we live in today with electronic tools at almost everyone's disposal may give some a false sense of security. Even with tools such as calculators and computers to handle much of the computation, being able to identify the necessary operations is still important.
Finally, those a with a decent background in basic arithmetic and mathematical concepts are less likely to be conned by sophisticated marketing that attempts to mislead by the clever use of numbers. |
hrc969
07-24-2007, 04:04 PM
If you are still ooking for help just post and I will help you.
Nick_Roge
08-03-2007, 03:52 AM
Yes, I'm still looking for help.
hrc969
08-03-2007, 04:00 AM
Yes, I'm still looking for help.
Ok. Please answer the following:
How much of Algebra 1 do you remember?
In your own words give the definition of a variable.
Lastly what kind of format were you looking for? As in you as questions and I answer or I give you material to think about or what?
Nick_Roge
08-03-2007, 04:08hrc969
08-03-2007, 04:26If I can find a book for you to download would you like me to tell you what sections to read?
Or do you want me to just give you stuff from my head?
Well either way, I'll start to post material tomorrow.
Nick_Roge
08-03-2007, 02:35 PM
Well, I would prefer something I could use as a reference but either way is fine.
hrc969
08-03-2007, 03:53 PM
Well, I would prefer something I could use as a reference but either way is fine.
Are you in the US. If so what state?
Nick_Roge
08-03-2007, 04:06 PM
Yes, I'm in Texas.
hrc969
08-03-2007, 04:47 PM
Yes, I'm in Texas.
Here are the Algebra 2 standards for texas:
§111.33. Algebra II (One-Half to One Credit).
(a) Basic understandings.
(1) Foundation concepts for high school mathematics. As presented in Grades K-8, the basic understandings of number, operation, and quantitative reasoning; patterns, relationships, and algebraic thinking; geometry; measurement; and probability and statistics are essential foundations for all work in high school mathematics. Students continue to build on this foundation as they expand their understanding through other mathematical experiences.
(2) Algebraic thinking and symbolic reasoning. Symbolic reasoning plays a critical role in algebra; symbols provide powerful ways to represent mathematical situations and to express generalizations. Students study algebraic concepts and the relationships among them to better understand the structure of algebra.
(3) Functions, equations, and their relationship. The study of functions, equations, and their relationship is central to all of mathematics. Students perceive functions and equations as means for analyzing and understanding a broad variety of relationships and as a useful tool for expressing generalizations.
(4) Relationship between algebra and geometry. Equations and functions are algebraic tools that can be used to represent geometric curves and figures; similarly, geometric figures can illustrate algebraic relationships. Students perceive the connections between algebra and geometry and use the tools of one to help solve problems in the other.
(5) Tools for algebraic thinking. Techniques for working with functions and equations are essential in understanding underlying relationships. Students use a variety of representations (concrete, numerical, algorithmic, graphical), tools, and technology, including, but not limited to, powerful and accessible hand-held calculators and computers with graphing capabilities and model mathematical situations to solve meaningful problems.
(6) Underlying mathematical processes. Many processes underlie all content areas in mathematics. As they do mathematics, students continually use problem-solving, computation in problem-solving contexts, language and communication, connections within and outside mathematics, and reasoning, as well as multiple representations, applications and modeling, and justification and proof.
(b) Foundations for functions: knowledge and skills and performance descriptions.
(1) The student uses properties and attributes of functions and applies functions to problem situations. Following are performance descriptions.
(A) For a variety of situations, the student identifies the mathematical domains and ranges and determines reasonable domain and range values for given situations.
(B) In solving problems, the student collects data and records results, organizes the data, makes scatterplots, fits the curves to the appropriate parent function, interprets the results, and proceeds to model, predict, and make decisions and critical judgments.
(2) The student understands the importance of the skills required to manipulate symbols in order to solve problems and uses the necessary algebraic skills required to simplify algebraic expressions and solve equations and inequalities in problem situations. Following are performance descriptions.
(A) The student uses tools including matrices, factoring, and properties of exponents to simplify expressions and transform and solve equations.
(B) The student uses complex numbers to describe the solutions of quadratic equations.
(C) The student connects the function notation of y = and f(x) =.
(3) The student formulates systems of equations and inequalities from problem situations, uses a variety of methods to solve them, and analyzes the solutions in terms of the situations. Following are performance descriptions.
(A) The student analyzes situations and formulates systems of equations or inequalities in two or more unknowns to solve problems.
(1) The student understands that quadratic functions can be represented in different ways and translates among their various representations. Following are performance descriptions.
(A) For given contexts, the student determines the reasonable domain and range values of quadratic functions, as well as interprets and determines the reasonableness of solutions to quadratic equations and inequalities.
(B) The student relates representations of quadratic functions, such as algebraic, tabular, graphical, and verbal descriptions.
(C) The student determines a quadratic function from its roots or a graph.
(2) The student interprets and describes the effects of changes in the parameters of quadratic functions in applied and mathematical situations. Following are performance descriptions.
(B) The student uses the parent function to investigate, describe, and predict the effects of changes in a, h, and k on the graphs of y = a(x - h)2 + k form of a function in applied and purely mathematical situations.
(3) The student formulates equations and inequalities based on quadratic functions, uses a variety of methods to solve them, and analyzes the solutions in terms of the situation. Following are performance descriptions.
(B) The student analyzes and interprets the solutions of quadratic equations using discriminants and solves quadratic equations using the quadratic formula.
(C) The student compares and translates between algebraic and graphical solutions of quadratic equations.
(D) The student solves quadratic equations and inequalities.
(4) The student formulates equations and inequalities based on square root functions, uses a variety of methods to solve them, and analyzes the solutions in terms of the situation. Following are performance descriptions.
(A) The student uses the parent function to investigate, describe, and predict the effects of parameter changes on the graphs of square root functions and describes limitations on the domains and ranges.
(C) For given contexts, the student determines the reasonable domain and range values of square root functions, as well as interprets and determines the reasonableness of solutions to square root equations and inequalities.
(e) Rational functions: knowledge and skills and performance descriptions. The student formulates equations and inequalities based on rational functions, uses a variety of methods to solve them, and analyzes the solutions in terms of the situation. Following are performance descriptions.
(1) The student uses quotients to describe the graphs of rational functions, describes limitations on the domains and ranges, and examines asymptotic behavior.
(2) The student analyzes various representations of rational functions with respect to problem situations.
(3) For given contexts, the student determines the reasonable domain and range values of rational functions, as well as interprets and determines the reasonableness of solutions to rational equations and inequalities.
(5) The student analyzes a situation modeled by a rational function, formulates an equation or inequality composed of a linear or quadratic function, and solves the problem.
(6) The student uses direct and inverse variation functions as models to make predictions in problem situations.
(f) Exponential and logarithmic functions: knowledge and skills and performance descriptions. The student formulates equations and inequalities based on exponential and logarithmic functions, uses a variety of methods to solve them, and analyzes the solutions in terms of the situation. Following are performance descriptions.
(1) The student develops the definition of logarithms by exploring and describing the relationship between exponential functions and their inverses.
(2) The student uses the parent functions to investigate, describe, and predict the effects of parameter changes on the graphs of exponential and logarithmic functions, describes limitations on the domains and ranges, and examines asymptotic behavior.
(3) For given contexts, the student determines the reasonable domain and range values of exponential and logarithmic functions, as well as interprets and determines the reasonableness of solutions to exponential and logarithmic equations and inequalities.
(4) The student solves exponential and logarithmic equations and inequalities using graphs, tables, and algebraic methods.
(5) The student analyzes a situation modeled by an exponential function, formulates an equation or inequality, and solves the problem.
Since you say there is nothing you don't remember from Algebra 1 I'll let you look at the standards and choose something you want to study. Try to find it in one of the above or if you are having trouble tell me to do it. Then you can read it and ask questions.
Algebra 2 is a course which is a review of algebra 1 and also a preview of precalculus. Alot of the stuff in algebra 2 you have already seen but the problems will be harder now. |
Specification
Aims
Brief Description of the unit
Algebraic geometry studies objects called varieties defined by polynomial equations.
A very simple example is the hyperbola defined by the equation xy = 1 in the plane.
There is a way of associating rings to varieties, and then the geometric properties can be studied
using algebra, for example points correspond to maximal ideals, or the geometry of the variety can
give information about certain algebraic properties of the ring.
Algebraic geometry originated in nineteenth century Italy, but it is still a very active area of
research. It has close connections with algebra, number theory, topology, differential geometry and
complex analysis. |
This is a lecture course in elementary algebra with a review of topics that will be used in science and engineering classes. This class will meet for a total of six hours per week with a focus on student-centered learning techniques. Review topics include fractions and mixed numerals, operations with polynomials, scientific notation, ratio and proportion, basic statistical measures, geometric formulas and unit conversions.
Topics include factoring polynomials, solving quadratic equations, applications and problem solving, and simplifying complex rational expressions. Additional topics are radical expressions, radical equations and applications, the quadratic formula, graphs of quadratic equations, and functions.
Students must achieve a C- or better to pass the course.
This is a developmental course. Credit for this course will not be counted toward fulfilling graduation requirements at STCC. |
Lesson 1 Limits What is calculus? The body of mathematics that we call calculus resulted from the investigation of two basic questions by mathematicians in the 18th century. 1. How can we find the line tangent to a curve at a given point on the curve?y 1Bus 247 Homework Set 1 Summer 2009 Queens College Professor Bradbury This assignment is due at the beginning of class on Monday June 15. The assignment must be typed and stapled in order to receive credit. (though graphical solutions may be handwritten in
Bus 24s7 Homework II Fall 2008This assignment will be due on October, 28 in class. The assignment must be typed and stapled in order to receive credit. No late assignment will be accepted unless the professor has approved an exception. Please show all wo
Home Work II Exam II Preparation Assignment This assignment is due in class Tuesday April 29. The assignment will not be accepted if it is turned in late. The assignment will not be accepted unless it is typed. 1) Consider the following Game: Compete Comp |
This book is intended to serve as a one-semester introductory course in number theory and it includes a wealth of exercises. A historical perspective has been adopted and emphasis is given to some of the subject's applied aspects. For students new to number theory, whatever their background, this is a stimulating and entertaining introduction to the... more...
This textbook is intended to serve as a one-semester introductory course in number theory and in this second edition it has been revised throughout and many new exercises have been added. For students new to number theory, whatever their background, this is a stimulating and entertaining introduction to the subject. more... |
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Math Workshops
Math Workshops for Students at NVC
The Math Advocacy Center sponsors free workshops for students on a variety of topics to aid in their math classes. Come by the Advocacy Center in JH308 or email us at [email protected] for more information or to register for one of our workshops. Students must register at least 24 hours in advance to participate to give the instructor time to prepare for the workshop. Walk-ins are generally not guaranteed. Space is limited! Most workshops have room for no more than 10 students so please register as soon as possible to guarantee a seat. Workshops are offered at various times on different topics. Check with the Advocacy Center for dates and times. A list of workshops is provided below.
What Students Are Saying
List of Math Workshops Offered
Description:Will teach students how to effectively take notes, manage their
study time, and use effective study habits to aid in learning the
material of their math class
Math Test Anxiety
Duration: 1 hour
Target Students:0300-1314
Description:Meant for students who struggle with test taking.Will give students test taking strategies useful to ease test anxiety.Special focus on each student's unique needs.
Fractions
Duration: 1 hour
Target Students:0300-0301
Description:Provides a more comprehensive review of fractions and their properties including, adding, subtracting, multiplying, and dividing them.
Integers
Duration: 1 hour
Target Students:0300
Description:Provides a more thorough explanation of integers and how to add and subtract them.Positive and negative numbers will be given more context through number line activities.If time permits, an explanation of absolute value will surface.
Order of Operations
Duration: 1 hour
Target Students: 0300
Description:Will give students a more in-depth review on the order of operations using appropriate PEMDAS strategies.
Multiplying Binomials
Duration: 1 hour
Target Students: 0302
Description:Will provide students with the skills necessary to successfully multiply binomials. "FOIL" is explained in detail as well as other common multiplication strategies for binomials.
Factoring Strategies
Duration: 2 hours
Target Students: 0302-0303
Description:Will provide students with an understanding of what factoring really is with a rundown of the different factoring techniques taught by various professors.
Graphing Strategies for Linear Equations
Duration: 1.5 hours
Target Students: 0301-0302
Description:Explanation of slope, slope-intercept form equation of a line, and how to effectively plot points and graph linear equations.
Graphing Strategies for Quadratic Equations (Parabolas)
Duration: 1.5 hours
Target Students: 0303-1314
Description:Explanation of graphing parabolas in standard or quadratic form.Explanation of how to find axis of symmetry, vertex, intercepts, and orientation will be discussed.May review completing the square as necessary to do so.
Complex Numbers
Duration: 1 hour
Target Students: 0303-1314
Description:Provides students with a better understanding of what a complex number is.An explanation of how to add, subtract, multiply, and divide complex numbers as well as conjugates will be discussed.
Number Sense and Mental Math
Duration: 2 hours
Target Students: 0300-0301
Description:Provides students with an intuitive understanding of numbers, their magnitude, relationships, and how they are affected by operations.Will also provide students with the skills to perform mental arithmetic operations without a calculator.
Accuplacer Review
Duration: 1 hour
Target Students: 0300-0303
Description:Provides students with a refresher on topics pertaining to the Elementary Algebra portion of the Accuplacer exam adapted to the needs of the students. Depending on student's previous math background, this review workshop aims to place students at least one math class above the one in which they are currently registered. This workshop will place realistic expectations of the students. Students will need to make arrangements to show up several times to this workshop in order to be ready for the exam.
MATH 0303 Review
Duration: 1 hour
Target Students: 0303
Description: Provides students with a refresher on topics typically covered in MATH 0303
MATH 1350/1351 Review
Duration: 1 hour
Target Students: 1350-1351
Description: Provides students with a refresher on topics typically covered in MATH 1350 or 1351 |
Review Algebra with Two Sessions RefresherTake a quick review of Algebra in this 2 sessions long course, taught by an expert Mathematics teacher!
This course is a non-graded online review for those of you who want to learn Algebra. This two classes refresher course is specially designed for those of you who want to take a review of Algebra and its various applications. Short, detailed, and precise explanations of Algebra will be provided in this course. Course instructor, Dr. Rose is a PhD holder who has taught Math for 20 years. She teaches Math in an interactive, engaging, detailed, and very thorough manner which keeps students interested in the subject. Thorough review of concepts and applications of Algebra, you will also be prepared for the advanced level course in algebraic sequence.
In this course you will:
Review Algebraic terminology
Review basic algebraic operations
Review algebraic fractions
Review algebraic expressions
Review literal, linear, and systems of equations
Review graphing inequalities
Review basic word problem structure
NOTE: Course will be conducted December 12th (Monday) & December 13th (Tuesday) at 7:00 PM (PST)
This course will be helpful for:
Any student who wants to take a quick review of basics of Algebra
What's included in this course:
2 LIVE interactive online classes + Access to online classes
Course timings: Classes will be held on Dec 12th (Monday) and Dec 13th (Tuesday) at 7:00 PM (PST)
Online access to study material - Word Docs and PPTs
2 online tests to review your performance
Course outline:
Problem Solving Skills
Word Problems
Rules and Basic Operations
Algebraic Terminology
Basic Literal, Linear, Non-linear Equations and Inequalities
Graphing Linear Equations and Inequalities
About course instructor:
Jacquinita A. Rose holds a Ph.D. from University of Oklahoma. She has 20 years in teaching mathematics to students. Dr. Rose teachers in an interactive, engaging, detailed, and very thorough manner which keeps students interested in the subject. Dr Rose enjoys exploring the countryside, writing, sports, cooking, and doing research on alternative medicines and health. A great fan of country music she hopes to meet George Strait in person some day. She lives in Port Hueneme, USA.
About the Instructor
Dr. Jacquinita A. Rose PORT HUENEME, United States
I love mathematics and science. I enjoy the rigor, the challenge, and the discovery. Most importantly, I enjoy teaching, learning, and sharing mathematics with students. I also realize that not everyone shares my enthusiasm for mathematics. So, when I teach, I try to make it fun and enjoyable, while still emphasizing the theoretical concepts and practical applications. One of the basic premises is that we together as a "group" will make it through this math class. |
More About
This Textbook
Overview
Linear algebra and matrix theory have long been fundamental tools in mathematical disciplines as well as fertile fields for research. In this book the authors present classical and recent results of matrix analysis that have proved to be important to applied mathematics. Facts about matrices, beyond those found in an elementary linear algebra course, are needed to understand virtually any area of mathematical science, but the necessary material has appeared only sporadically in the literature and in university curricula. As interest in applied mathematics has grown, the need for a text and reference offering a broad selection of topics in matrix theory has become apparent, and this book meets that need. This volume reflects two concurrent views of matrix analysis. First, it encompasses topics in linear algebra that have arisen out of the needs of mathematical analysis. Second, it is an approach to real and complex linear algebraic problems that does not hesitate to use notions from analysis. Both views are reflected in its choice and treatment of |
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Abstract
Computer algebra systems (CAS) were originally designed for mathematicians, scientists, and engineers, and their implementation into education, and especially into schools, is still very much in its infancy. This situation is reflected in the research on CAS and hence our ability to describe the influence of CAS on student learning in algebra. This chapter synthesises key results from research related to using CAS in algebra from two broad perspectives: firstly, the issues of curriculum, assessment, and teaching; and secondly, that of student learning. Our analysis supports the view that CAS has much to offer the teaching and learning of algebra, but that real benefits may accrue only from thoughtful and structured approaches which take into account the perspectives of the student and teacher, and the intricacies of the relationships between student, teacher, and CAS. In writing the chapter we are mindful that many more questions than answers have emerged and we have included a significant number of these in the hope that they may give impetus and direction to research in the area.
Computer-Based learning environments in mathematics |
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Addressing the Common Core with Fathom
Common Core Statistics and Probability standards are easily addressed with Fathom.
With Fathom, modeling comes alive.
For 40 years, Key Curriculum has created tools and curriculum that incorporate problem solving, real-world applications, conceptual understanding, and mathematics as sensemaking. These pedagogical approaches are at the heart of our dynamic mathematics software, and central to the Mathematical Practice Standards of the Common Core State Standards.
Additionally, the Common Core State Standards include a strong and coherent data analysis, statistics, and probability strand, which starts in kindergarten with basic graphical representations of categorical data. This evolves into deeper concepts in grades 9–12, such as calculating (with technology) and interpreting correlation coefficients, and understanding, calculating, and modeling conditional probability. And, the modeling strand emphasizes the value of technology in building mathematical models.
The current version of Fathom is 2.13—read release notes here. To update for free from an earlier version of Fathom 2, choose Check for Updates from Fathom's Help menu.
Note that Fathom 2.13 is the latest version of Fathom 2L, which is a downloadable version of Fathom that must be registered with a License Name and Authorization Code. Users who have installed Fathom 2 from a CD need to call McGraw-Hill Digital Technical Support at 800-437-3715 and register their software in order to upgrade to Fathom 2.13. |
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exact values in the form of variables such as x and y, radicals and pi when doing step-by-step arithmetic, algebraic and calculus calculations.
Related Products
TI's graphing calculator that operates like a computer. Simpler keypad with separated alpha keys makes navigation through screens, menus and the math objects easier. It includes computer algebra system (CAS) that performs symbolic and numeric calculations seamlessly |
Anyone out there have any input on the new Math Editor? Usability, functionality, any info from 9.1 would be greatly helpful from a faculty perspective. We had a lot of critics for the version in 8, and few faculty were using it because of its limitations. How does 9.1 compare?
I'm interested in knowing what the advantages/challenges are of the new Bb 9.1 Math Editor but am an instructional designer, not a math faculty member. Any thoughts on advantages or sticky points (other than the one mentioned in your post.) |
The main goal of Algebra 1 is to develop fluency in working with linear equations. Students will extend their experiences with tables, graphs, and equations and solve linear equations and inequalities and systems of linear equations and inequalities. Students will extend their knowledge of the number system to include irrational numbers. Students will generate equivalent expressions and use formulas. Students will simplify polynomials and begin to study quadratic relationships. Students will use technology and models to investigate and explore mathematical ideas and relationships and develop multiple strategies for analyzing complex situations. Students will analyze situations verbally, numerically, graphically, and symbolically. Students will apply mathematical skills and make meaningful connections to life's experiences.
-Graphing Calculator for homework (TI-84 will be provided for use in class and for testing)
Grading Policy:Grade Scale:90% – 100%…………A
80% – 89%…………B
75% – 79%…………C
70% – 74%…………D
Below 70%…………F
Tests/Projects/Presentations ............……………………..…………….……...50%
Quizzes/Engagement/Homework/Assignments/Notebook……………………….50%
Semester Grade: Semester 1:75% - 1st 9-Weeks + 2nd 9-Weeks
25% - 1st Semester Exam
Semester 2:75% - 3rd 9-Weeks + 4th 9-Weeks
25% - Final Exam
Test/Projects/Presentations:
You will have at least one unit test after each unit. Unit tests will cover any concepts that have been taught in this math class.Retesting is available for test scores that are below a 70%.Retesting will only be allowed during the timeframe before the next unit test.An additional option to retesting is making corrections to a test grade that is below a 70. The corrections will add points to the test grade, however,every incorrect problem must be corrected on the "correction sheet(s)" provided by the teacher and corrections must be done step-by-step. The highest possible grade you may obtain is a 70.Several projects will be assigned throughout the school year.Projects that are assigned will be given ample time to complete.Projects will lose 10 points for each school day that the project is late.
Quizzes/Engagement
Students need to be involved and occupied with mastering the concept of this course. 5 question quizzes will help student focus on key concepts of the unit.Engagement grades come from participating in activities, warm-ups, decisions, class work, demonstrations, games, etc…
Homework/Assignments:
Homework/Assignments is an essential part of mathematics.You will be required to demonstrate your knowledge of middle school math and algebra concepts during this course.In turn, you must show all necessary work step-by-step to receive credit for Assignments.In addition, all problems must be attempted. If not, I have the discretion to refuse to accept the assignment. Any Assignments turned in after the due day will result in a loss of 10 points per day, up to three day.
Attendance:
If you are absent it is your responsibility to collect missing assignment(s) and notes. Class notes should be obtained from a fellow classmate.Note and assignments will be updated on teacher webpage once a week.For each day of an approved (excused) absence, one additional day will be given to complete any missing assignment(s).If you are absence on the day of a quiz or test, it is yourresponsibility to arrange with the teacher a time and date to make-up the missed quiz or test.
Scholastic Integrity assignment, homework assignment, quiz, or test.In addition, a phone call home will be made to your parents/guardians. |
Investigating Pick's Theorem
Unit Overview
Lesson 1
Lesson 2
Lesson 3
Many students may have seen Pick's Theorem in middle school. In this set of lessons, students rediscover the theorem, use algebra to determine the coefficients of the equation, and explore the concept of change as a mechanism for finding the coefficients of Pick's Theorem.
Math Content
Systems of Equations
Rates of Change
Although no single lesson in this unit addresses connections and representation by itself, the entire unit focuses on the Connections and Representation Standards by allowing students to make connections among mathematical ideas and asking students to use various representations to organize their work |
Mathematics is an essential aspect of scientific
communication and education. Therefore, to realize the potential of the Web
for science, it must be possible to use mathematics on the Web. Mathematical
expressions must move seamlessly between the Web and a wide range of related
environments including authoring tools and content management systems,
XML-based publishing work flows, e-learning environments, and scientific
computing software.
To address this need, W3C brought together key players and major stake
holders to devise a solution. The Math Working Group created the Mathematical
Markup Language (MathML), a highly-structured, information-rich, XML encoding
for mathematical expressions, and is chartered to maintain it.
MathML facilitates the authoring and presentation of mathematical
expressions in print and on the screen, and forms the basis for machine to
machine communication of mathematics on the Web. Designed as an XML
application, MathML provides two sets of tags, one for the presentation of
mathematics and the other associated with the meaning behind equations.
MathML is not designed for people to enter by hand; specialized tools provide
the means for typing in and editing mathematical expressions.
The MathML 1.0 Recommendation first appeared on 7 April 1998. Four
subsequent revisions have followed, culminating with the MathML 2.0,
Second Edition on 21 October 2003. MathML 2, Second Edition is fully
synchronized with Unicode 4.0. It is also fully integrated with XHTML and SVG (Scalable Vector Graphics), and
interoperates well with other W3C technologies such as XSL (the Extensible Stylesheet Language),
CSS (Cascading Style Sheets), and XML Schema.
Development of the next update of MathML, called MathML 3.0,
started in 2006. It will support, among other things additional
notations, features for online assessment (online learning) and
right-to-left formulas (in particular for Arabic).
Highlights Since the Previous Advisory Committee Meeting
The Math Working Group has been working on several drafts besides
MathML 3.0, including a subset of MathML that can be rendered on
any browser that supports CSS, a MathML Primer and an alternative
syntax that might be used for entering MathML in wikis.
The Math Working Group works together with the CSS Working Group, the CDF Working Group and the
HTML Working Group on a common framework for compound documents that supports
typographically correct mathematics.
The Math Working Group publishes a roadmap, a live
document with the status of the various deliverables.
Upcoming Activity Highlights
The group is writing an informative
MathML Primer and is cooperating with the OpenMath
community to integrate the OpenMath Content Dictionaries into MathML
(replacing the old dictionaries in MathML 2). |
Using Logarithms to Determine Relationships
This unit from the Continuing Mathematics Project goes into detail on how logarithms can be used to determine the laws which connect two variables on which experimental data has been collected. The unit follows naturally from the unit entitled The Theory of Logarithms.
The objectives of the unit are that students:
(i) understand the meanings of the terms exponential growth and decay, as they are used in relation to natural phenomena;
(ii) perceive that money invested at 'compound interest' also grows exponentially in denominational terms (even if it decays exponentially in purchasing power);
(iii) be able, when the relationship between two variables x and y is of the form y = C.ax, to plot values of log y against values of x on ordinary graph paper, or to plot values of y against values of x on log-linear graph paper, and so determine the constants C and a;
(iv) be able, when the relationship between two variables x and y is of the form y = C.xa , to plot values of log y against values of log x on ordinary graph paper, or to plot the values of y against the values of x on log-log graph paper, and so determine C and a |
Beginning and Intermediate Algebra, 5th Edition
Description all the tools they need to achieve success.
With this revision, the Lial team has further refined the presentation and exercises throughout the text. They offer several exciting new resources for students that will provide extra help when needed, regardless of the learning environment (classroom, lab, hybrid, online, etc)–new study skills activities in the text, an expanded video program available in MyMathLab and on the Video Resources on DVD, and more!
Table of Contents
Chapter 1 The Real Number System
1.1 Fractions
Study Skills: Reading Your Math Textbook
1.2 Exponents, Order of Operations, and Inequality
Study Skills: Taking Lecture Notes
1.3 Variables, Expressions, and Equations
1.4 Real Numbers and the Number Line
Study Skills: Tackling Your Homework
1.5 Adding and Subtracting Real Numbers
Study Skills: Using Study Cards
1.6 Multiplying and Dividing Real Numbers
Summary Exercises on Operations with Real Numbers
1.7 Properties of Real Numbers
1.8 Simplifying Expressions
Study Skills: Reviewing a Chapter
Chapter 2 Linear Equations and Inequalities in One Variable
2.1 The Addition Property of Equality
2.2 The Multiplication Property of Equality
2.3 More on Solving Linear Equations
Summary Exercises on Solving Linear Equations
Study Skills: Using Study Cards Revisited
2.4 An Introduction to Applications of Linear Equations
2.5 Formulas and Applications from Geometry
2.6 Ratio, Proportion, and Percent
2.7 Further Applications of Linear Equations
2.8 Solving Linear Inequalities
Study Skills: Taking Math Tests
Chapter 3 Linear Equations in Two Variables
3.1 Linear Equations in Two Variables: The Rectangular Coordinate System |
Initially, the CALM Project built a computerised tutorial system to enhance the teaching of calculus to students of the Heriot-Watt University. CALM started in 1985 following funding from the Computers in Teaching Initiative (CTI). The award from CTI provided both the computing expertise to produce the CALM courseware and a laboratory of 32 networked Research Machines microcomputers for student use. The first CALM Project was completed on time by October 1988.
CALM courseware is the result of strong teamwork and dedicated teaching. It has already exceeded our expectation in its impact. There is an enthusiastic response to it by each new group of students. Lessons learnt over the last decade have been built into the courseware wherever possible. We are still listening to what the students tell us from their experiences and we continue to develop courseware which aims to help students learn more effectively and more efficiently.
Calculus is an ideal medium to bring applications of mathematics to life. The 22 units of CALM courseware produced cover the syllabus of a typical course on calculus with differentiation, integration, an introduction to numerical methods and elements of ordinary differential equations. Each unit includes the topics first encountered in approximately two lectures of the course. Mathematical modelling and the development of mathematics in Engineering and Applied Physics is an important feature of CALM. For example:
State the Rate invites the students to work through a problem involving the filling of a cup from a coffee dispenser encouraging design considerations;
Fireman is a model of the trajectory of water from a hose which properly directed puts out a fire; and
Escape from Colditz asks the students to work with calculus and numerical methods to solve an optimisation problem.
From the outset our teaching
strategy for each unit has been constructed around:
Test sections, to enable students to assess their own strengths and weaknesses and to allow the teacher to monitor individual progress.
The units are designed to allow the students complete control over their route through the tutorial. The course for which these units are part, is mainly assessed in a conventional way. The test section of each unit can be taken at any one of three levels to cater for a spectrum of ability. The design of the multi-level test was guided by specific requests from the students themselves. The test offers different types of help at each level. Students can start a topic at the easiest level and progress to the hardest level as they gain in confidence. They are free to use the CALM tutorials in whatever way they choose.
The students' marks and test answers are recorded so that the teacher can monitor their progress. Students who are working well are sent encouraging messages. Those who are in difficulty are detected early in the year and given extra attention. By viewing the recorded answers, the teacher is able to identify the source of a student's problem and send an appropriate message.
We have worked as a team throughout
with regular meetings a central feature. We have presented our
results and demonstrated our courseware at national and
international meetings (a list is provided elsewhere in these
screens).
In the course of the project, software tools like evaluation routines, test-making libraries and mathematical display procedures were produced. These have been packaged together and used by other developers of CAL courseware in the University and beyond. The details of these tools are described in our book "Software Tools for Computer Aided Learning in Mathematics" published in April 1991.
In 1986/7 and 1987/8 we compared the examination results of a pilot group with those of another group taking the same final examination, in categories of similar school qualification. On average, students in the computerised tutorial system performed 15% better in the common examination. It is estimated that the introduction of CALM has reduced student failure rates by 5% per annum.
Printed instructions and advice
gathered from previous CALM users are given out to all new
students. Their comments include:
"Great idea
--- you can come and learn when you want."
"Learning via the computer is less embarrassing."
"It is as if there is a tutor in front of you all the time."
"The tests make you work harder."
"The software is always under your control."
"I look forward to the CALM tutorials."
Information from students has been
collected through questionnaires, interviews, structured recall
and informal contacts. We assessed the advantages of the CALM
courseware in two ways:
through feedback from the students
and their observations on the quality of learning that CALM
provides; and
"The program is oriented towards multi-pupil use in a school environment and for this reason may prove very useful as year-round practice, with the teacher being able to monitor each student's progress."
Educate Online Review
"This is a very useful tool for preparing for examinations. It is well worth considering for installation on single machines or more widely across a network."
"The interface is well suited to examination preparation activity."
BECTA CD-ROM Review
"The InputTool represents a considerable advance in providing a flexible and accessible user interface for supporting correct mathematical notation."
"Students who are prepared to spend time and effort with this professionally produced piece of software would enhance their revision programme and the content.
CTI Mathematics Note: This document is in Adobe PDF Format and may have to be downloaded before viewing.
"The package is well designed, easy to use and can be configured to the needs of individual students, with all the topics in core advanced mathematics covered."
"The Computer Aided Learning in Mathematics team at Heriot-Watt University, who developed this program, have clearly used their expertise to good effect."
The Times Educational Supplement, December 19th 1997
"The software was extremely engaging to use and very satisfying. Much better than ploughing through past papers with no idea as you do them whether you are on the right track or not."
"This was considered to be good exam practice, easy to use and with lots of questions. Its other advantage is that it teaches layout and how much you need to write down in exams. It was possibly better than a set of past papers because of the instant feedback"
Mathematics Multimedia Courseware Review |
Other Courses
Home Schooling Pure Maths A level – The Course
The Pure MathsThe AS Level has ten tutor-marked assignments (known as TMAs). The A2 has a further ten TMAs.
Key Topics Covered
AS Level
MPC1: algebra, trigonometry, integration, etc
MPC2: exponentials, logarithms, etc
MFP1: complex numbers, linear equations,
A2 Level
MPC3: algebraic functions, coordinate geometry etc.
MPC4: vectors, further coordinate geometry, etc
MFP2: hyperbolic functions, matrices, etc
The Syllabus
This course prepares candidates for the AQA Mathematics AS level syllabus 5366, for examination in 2013 and later years. Most candidates will then study the A2 syllabus 6366. The full Advanced Level qualification comprises AS and A2. We have chosen this syllabus as it is the most suited for home schooling.
Assessment is by three written papers for the AS Level and three written papers for the A2 level. |
Mathematics
The
mathematics department prepares students to think logically and creatively
about patterns, figures, numbers, functions, and applications and teaches
skills necessary for success in an increasingly technical world. The department
believes each student learns best and feels most successful and confident when
she covers the material at an appropriately challenging pace.
Mathematics is a required subject for all
students in Classes I through VII. All students in Class I and Class II
have math class in heterogeneously mixed groups. Starting in Class III, students
are grouped homogeneously. Algebra 1 is also offered to qualifying Class III
Students. The offered math courses for
Class VIII satisfy the quantitative course requirement. The chart below
summarizes the courses taught starting in Class III through Class VIII.
Standard-Pace
Fast-Pace
Accelerated
III
Introduction to Algebra
Introduction to Algebra
Algebra 1
IV
Algebra
Algebra 1
Algebra 2
V
Algebra 2
Algebra 2
Geometry
VI
Geometry
Geometry
Honors PreCalculus
VII
PreCalculus
Honors PreCalculus
AP Calculus
VIII
Honors Calculus and/or Statistics
AP Calculus and/or Statistics
Statistics or Online Course
Mathematics Courses
Class I
Students study arithmetic using whole numbers,
fractions, and decimals to understand concepts and strengthen skills.
Manipulatives are used to help illustrate some concepts. Other topics include
measurement, geometry, estimation, and problem solving. Mental and written
computation is emphasized, but calculators or computers are used for
appropriate activities. Students work individually as well as in groups when
appropriate. Students are enriched with Weekly Challenges where the language of
mathematics is emphasized as well as communicating problem solving strategies.
Class II
The
mathematics curriculum includes fractions, decimals, percents, number theory,
order of operations, measurement, two-dimensional geometry, data analysis, and
an introduction to negative numbers. Students develop basic financial literacy
through a computer simulation project. All topics involve individual and group
activities and solving a variety of problems. Scientific calculators are
introduced. Computer spreadsheets are also introduced and used during the study
of financial literacy and data analysis. Students participate in the Elementary
School Math Olympiads.
Class III
The course includes writing algebraic expressions and
solving equations, three-dimensional geometry, ratio and proportion, and
probability. Application of these topics is built-in throughout the course. Students
also learn creative problem solving by participating in the Middle School Math
Olympiad.
Class III & Class IV
The
course includes solving and graphing linear equations and inequalities,
exponents, polynomials, solving and graphing quadratic equations and solving
systems of equations. During the year students develop and refine their
problem-solving and critical thinking skills, and a variety of word problems
and applications are introduced. Graphing calculators are introduced. Students participate
in the Math Olympiad.
Class IV & V
All students will study the core topics of Algebra 2:
functions (linear, quadratic, polynomial, radical, exponential, and
logarithmic) and systems of equations. Some analytic geometry and data analysis
will be included if time permits. The applications of the TI-84 Plus Silver
Edition graphing calculator will be introduced. The algebraic and graphical
aspects of each topic will be emphasized.
Class V & VI
In this
course students study Euclidean geometry, including the properties of
triangles, polygons, parallel lines, and circles. They solve problems involving
similar figures, areas, volumes, symmetry, and triangle trigonometry, and learn
the principles of logic by writing formal deductive geometric proofs. This
course may also include the study of probability, combinatorics, and statistics.
Class VI & VII
In
this course students continue to study functions, including polynomial,
rational, exponential, logarithmic, and circular functions, and their
applications. Graphing calculators are used throughout the course. More
traditional analytic and algebraic methods are also emphasized so that students
will understand different approaches and techniques. Other topics covered
include transformations of graphs and inverses of functions, solving equations
and inequalities, and trigonometry. In addition, polar coordinates, conic
sections, sequences and series, and parametric equations are introduced if time
permits. The department usually offers Precalculus sections that move at three
different paces; the fastest pace is designated Honors Precalculus.
Class
VII & VIII
AB and BC
Calculus are both AP courses, and students are required to take the AP test in
May. BC Calculus is a faster paced more demanding course. Beginning with the
concepts of limits and continuity, students go on to learn the definition of
the derivative and its applications, and progress to theory, techniques, and
applications of integration. Additional topics are included as prescribed by
the AP syllabus. Graphing calculators are used as a tool to enhance
understanding of the concepts and facilitate problem solving.
Class
VII & VIII
Students study the crucial concepts of the derivative
and the integral, including their meaning in relation to both graphs and
formulas. They also study applications of both concepts to a variety of
real-world situations.
Class VIII
The purpose of this course is to
introduce students to the major concepts and tools for collecting, analyzing,
and drawing conclusions from data. Students are exposed to four broad
conceptual themes: describing data patterns and departures from patterns,
sampling and experimentation by planning and conducting a study, anticipating
patterns by exploring random phenomena using probability and simulation, and
statistical inference by estimating population parameters and testing
hypotheses. In the AP course the AP Exam will be administered.
Introduction
to Web Design
Class
VII & VIII
This
course introduces a student to web design through the language of the web:
HTML. Students will begin the
semester creating very basic web pages using Notepad and eventually graduate to
designing sites of multiple pages using more robust designing
environments. Other topics such as CSS, PHP, and JavaScript will be
introduced as time permits. The course will be taught in a PC lab and it
is recommended that students enrolled in this course have access to a PC
computer at home. No previous experience is required or expected of
students entering this course.
Introduction to Computer Programming
Class
VII & VIII
This
course provides the student a strong foundation in the basics of computer
programming and introduces her to algorithmic processes. Through a series of
programming projects, the student will learn to code various mathematical
statements, variable assignments, selection and iteration statements, sub
procedures and functions, as well as other programming constructs. Students will have ample opportunity to explore and
experiment with the programming environment, adding their own creative flair to
their projects. The programming language used for this course is Microsoft
Visual Basic .NET and is available for free. This language allows students to
create full Windows programs complete with standard controls (menus, buttons,
textboxes, etc.). The course will be taught in a PC lab and it is recommended
that students enrolled in this course have access to a PC at home. This
class is open to classes VII and VIII. No previous programming experience is
required or expected of students entering this course. |
MAA Review
[Reviewed by Donald L. Vestal, on 09/10/2006]
A professor once told me, in class, that algebraic geometry is a beautiful subject. He also said, outside of class, that algebraic geometry is a hard subject. So it's no surprise that this is a topic generally reserved for graduate students. But could it, or at least some of it, be presented, at the undergraduate level? This book attempts to do that. It introduces the subject at a very concrete level. The polynomial equations studied are at most degree three, so the curves are all lines, conics, or cubics.
In the first chapter, projective geometry is introduced to study how curves intersect (not just in the Euclidean plane, but also at infinity). The intersection of lines and curves is studied as a precursor to Bezout's Theorem. The second chapter focuses on conics and the idea of transforming these curves into a given conic. More work is done with the intersection of curves, including the notion of "peeling off a conic" from the intersection of two curves. At the end, projective geometry is used to develop the duality of a conic and its envelope. In the third chapter, the author proves that all irreducible cubics can be reduced an elliptic curve y2 = x3 + fx2 + gx + h. The group structure of rational points on this elliptic curve is also (briefly) explored. Complex numbers are introduced and Bezout's Theorem and the Fundamental Theorem of Algebra are proven. The final chapter uses parameterization to determine intersection multiplicities of curves, and the duality of conics and their envelopes is extended to higher dimensions.
At the beginning of each of the four chapters the author provides a synopsis of the historical development of the subject. And within each section, many exercises are provided for further discussion and illumination. They tend to be one of three kinds: a very concrete problem involving specific polynomials, a concrete problem involving geometry, or a problem asking for a proof. Below are three examples.
Prove that the cubic C:x2y = x3 + 1 is irreducible. Prove that C is singular at the point Q at infinity on vertical lines. Determine how many lines through Q intersect C three times there.
Consider the following theorem: In the projective plane, let A, C, D, E, F be five points on a conic. Then the points Q = tan A ∩ DE, R = AC ∩ EF, and S = CD ∩ FA are collinear. State the version of this theorem that holds in the Euclidean plane when A is the only point at infinity named.
Let L = 0 be the tangent line to a cubic C = 0 at a flex P. If C is reducible, prove that L is a factor of C.
How accessible is the material to an undergraduate? The material in this book requires a decent background in algebra and geometry. (A little intuition is helpful too.) Very little knowledge of calculus is needed: just the concept of a tangent line. And the author manages to keep things concrete (especially in the exercises). So the end result is a book which is accessible to a motivated undergrad. |
MATHEMATICS DEPARTMENT
A minimum of two years of mathematics is required for graduation from Santa Monica High School (30 units). A minimum of 1 year (10 units) must be taken in grades 10-12. However, students are encouraged to take as many math courses as they wish, while attending Samohi.
Samohi's college prep sequence includes the following; Algebra, Geometry (P & HP), Algebra II (P & HP), and Pre-Calculus (P & HP). Successful completion of these courses prepares students for admission to the UC/CSU system as well as other universities.
For those who want to continue their studies in mathematics, Samohi offers the following; Calculus A/B, B/C, and D/E, Statistics and Statistics AP.
Prior to enrollment transfer students whose previous mathematics course does not clearly communicate an equivalent math level will be administered a math placement test. |
Publisher review: GeoGebra is a very useful mathematics tool for education in secondary schools, which brings together geometry, algebra and calculus.
GeoGebra is also a dynamic geometry system, meaning you can do constructions with points, segments, vectors, lines, conic sections as well as functions and change them dynamically afterwards.
On the other hand, equations and coordinates can be entered directly. Thus, GeoGebra has the ability to deal with variables for numbers, vectors and points, finds derivatives and integrals of functions and offers commands like Root or Extremum.
These two views are characteristic of GeoGebra: an expression in the algebra window corresponds to an object in the geometry window and vice versa. GeoGebra key features:
Captions enabled for all objects
New option "Force Reflex Angle" forces angles to be between 180 and 360 degrees
Pressing toggles the focus between the Input Bar and the Graphics View
Comparing objects of different types doesn\'t return an error, can now compare Text and Image objects
If the Points Export_1 and Export_2 exist, they will be used to define the export rectangle (Export_1 and Export_2 must be within the visible area)
Checkbox now consistent across all platforms
Options -> Checkbox Size -> Regular/Large
Perpendicular check added to "Relation between two objects" Tool
Messages from "Relation between two objects" Tool rewritten
Angular Bisector Command and Tool renamed to Angle Bisector
Line Bisector Command and Tool renamed to Perpendicular Bisector
BMP import
Unicode fonts used in LaTeX equations
LaTeX equations exported at full resolution
in SVG and PDF export, option to export text as editable text or shapes. Stores the text either as text (lets you edit the text in eg InkScape) or as bezier curves (ie guaranteed to look the same even if the correct font is not installed). |
... george mason academy online middle high school course catalog ... number sense data analysis and probability patterns and algebra discrete math and ... they must be consummate problem solvers with the well-honed ability to teach ...
... for solving equations various equation solvers are available which helps to solve such ... problems we can solve our math problems online by using various homework solvers and ... problems related to statistics and probability the algebra homework help tool allows ...
... with problems related to conditional probability but most appropriate one is baye s ... method as it includes the conditional probability of an event a with both possible ... to this theorem the conditional probability of event a is given as pa pa|bpb pa| ... lots of topics students can prefer the online statistics help provided by tutorvista ... and its problems read more on onlineprobabilitysolvers ... |
Mathematics scares and depresses most of us, but politicians, journalists and everyone in power use numbers all the time to bamboozle us. Most maths is really simple - as easy as 2+2 in fact. Better still it can be understood without any jargon, any formulas - and in fact not even many numbers. Most of it is commonsense, and by using a few really simple... more... The result is a must-have for all those needing to apply the methods in... more...
Assuming only basic algebra and Galois theory, this book develops the method of 'algebraic patching' to realize finite groups and, more generally, to solve finite split embedding problems over fields. The method succeeds over rational function fields of one variable over 'ample fields'. Among others, it leads to the solution of twoThis book concentrates on the mathematics of photonic crystals, which form an important class of physical structures investigated in nanotechnology. Photonic crystals are materials which are composed of two or more different dielectrics or metals, and which exhibit a spatially periodic structure, typically at the length scale of hundred nanometers.... more...
Delves into the world of ideas, explores the spell mathematics casts on our lives, and helps you discover mathematics where you least expect it. Be spellbound by the mathematical designs found in nature. Learn how knots may untie the mysteries of life. Be mesmerized by the computer revolution. Discover how the hidden forces of mathematics -hold architectural... more...
Part of the joy of mathematics is that it is everywhere-in soap bubbles, electricity, da Vinci's masterpieces, even in an ocean wave. Written by the well-known mathematics teacher consultant, this volume's collection of over 200 clearly illustrated mathematical ideas, concepts, puzzles, and games shows where they turn up in the "real" world. You'll... more... fields of physics, engineering and chemistry with an interest in fluid dynamics... more...
This volume is an introduction to nonlinear waves and soliton theory in the special environment of compact spaces such a closed curves and surfaces and other domain contours. It assumes familiarity with basic soliton theory and nonlinear dynamical systems. The first part of the book introduces the mathematical concept required for treating the manifolds |
Dunstable Al provides a review and extension of the concepts taught in Algebra1. Topics include, but are not limited to equations and inequalities, coordinates and graphs, general functions, polynomials and rational functions, exponential and logarithmic functions, trigonometric functions of angles |
Sketchpad 3
introduces many tools for integrating dynamic geometry with dynamic
algebra. The sketches in this gallery illustrate some possibilities, from the
equation for a line, to integration of a cubic. They all make use of dynamic
plotting of points from measured or calculated quanitities, and the ones that
plot curves make use of constructed loci.
The sketches on this page are available individually (below) or can be downloaded
as a package.
Slope and Intercept
Use this sketch as an electronic blackboard or exploratory environment for
looking at the relationship between the equation for a line, the angle the line
makes with the x-axis, and the tangent of that angle.
Integration A cubic equation of the form y = a (x - p) (x - q) (x - r) is plotted. The
curve responds dynamically to changes in the coefficients a, p, q, and r. Two
limits of integration, x1 and x2, are given, and between these limits, eight
rectangles are constructed to approximate the area under the curve. The total
integral is computed both by adding the areas of the rectangles and by an exact
computation. It is interesting to compare the results of the two computations.
Sine Graph
This sketch provides an interactive demonstration of the effect of varying
the amplitude, period, and phase of a sinusoidal curve. Dragging points within
the sketch changes the coefficients.
Complex Numbers
Two complex numbers and their product are shown in the complex plane. As the
given numbers are changed by dragging, their product changes accordingly.
Some questions to investigate include: What number when squared equals -1? Is
this the only number? What is the geometric result of multiplying a number by i?
By -i? Can you find a number which, when squared, equals i? |
Resources
Here are a number of resources Algebra Worksheets thinks you might be interested in.
Algebra related websites
An avid algebra fan would really appreciate the algebra research being done at UC Berkeley. They have a list of people actively studying a variety of areas the the algebra sphere. You can also see the different algebra courses they have.
Every state in the United States should have a place on their website reviewing the expected curriculum of students. The California example of this is their curriculum section on what students should know and be able to do in mathematics, emphasizing computational and procedural skills, conceptual understanding, and problem solving.
A leading school in math is MIT and more specifically the MIT mathematics department. Find out the latest news and stay updated on the schedule of seminars and events.
Wolfram MathWorld is the web's most extensive mathematics resource. It was developed by Wolfram Research built with Mathematica technology. |
18 2011 | Series: Math WorkbooksThis has been a big help for my daughter who was having problems remembering the sides, and names of the shapes in beginning 4th grade geometry.
5.0 out of 5 starsvery nice bookDec 30 2011
By R. C. Rathore - Published on Amazon.com
Amazon Verified Purchase
Once a student is introduced to basic concepts such as shapes and angles, he or she can complete all exercises of this book without much help. Highly recommended to build a good foundation in geometry. |
ElementaryElementary Algebra"is typically a 1-semester course that provides a solid foundation in algebraic skills and reasoning for students who have little or no previous experience with the topic. The goal is to effectively prepare students to transition into Intermediate Algebra. |
Hey dudes, I am about two weeks through the semester, and getting a bit worried about my course work. I just don't seem to comprehend the stuff I am learning, especially things to do with system of equations graph. Could somebody out there please enlighten me with trinomials, binomials and multiplying matrices. I can't afford to look for a tutor, but if anyone knows about other ways of improving topics like radical expressions or system of equations painlessly, please get back to me Much appreciated
Algebra Buster is one of the most powerful resources that can render a helping hand to people like you. When I was a newbie, I took help from Algebra Buster. Algebra Buster covers all the principles of Intermediate algebra. Rather than using the Algebra Buster as a line-by-line tutor to work out all your math assignments, you can use it as a coach that can give the fundamental principles of parallel lines, conversion of units and equivalent fractions. Once you assimilate the principles, you can go ahead and work out any tough problem on Algebra 1 within minutes.
Some professors really don't know how to explain that well. Luckily, there are programs like Algebra Buster that makes a great substitute teacher for algebra subjects. It might even be better than a real teacher because it's more accurate and quicker!
Wow, that's amazing news! I was so frustrated but now I am quite thrilled that I will be able to improve upon my grades! Thank you for the reply guys! So then I just have to get the software and do my homework for tomorrow. Where can I find out more about it and purchase it?
factoring, equivalent fractions and complex fractions were a nightmare for me until I found Algebra Buster, which is really the best algebra program that I have ever come across. I have used it through many algebra classes – Remedial Algebra, Basic Math and Remedial Algebra. Simply typing in the math problem and clicking on Solve, Algebra Buster generates step-by-step solution to the problem, and my algebra homework would be ready. I truly recommend the program. |
MATH
1001
- Survey of Math
(M/SR)
(4.0 cr; Prereq-2 yrs high school math; spring, every year) Introductory topics in mathematics, such as number system, geometry, algebra, discrete mathematics, statistics, logic, and the history of mathematics, including applications in today's world.
MATH
1012
- PreCalculus I: Functions
(4.0 cr; Prereq-placement; no credit for students who have received credit for Math 1014; fall, spring, every year) Linear and quadratic functions, power functions with modeling; polynomial functions of higher degree with modeling; real zeros of polynomial functions; rational functions; solving equations in one variable; solving systems of equations; exponential and logarithmic functions, and the graphs of these functions.
MATH
1021
- Survey of Calculus
(M/SR)
(4.0 cr; Prereq-1012 or placement; spring, every year) Short course for students in social sciences, biological sciences, and other areas requiring a minimal amount of calculus. Topics include basic concepts of functions, derivatives and integrals, exponential and logarithmic functions, maxima and minima, partial derivatives; applications.
MATH
1101
- Calculus I
(M/SR)
(5.0 cr; Prereq-1012, 1013 or placement; fall, spring, every year) Limits and continuity; the concepts, properties, and some techniques of differentiation, antidifferentiation, and definite integration and their connection by the Fundamental Theorem. Partial differentiation. Some applications. Students learn the basics of a computer algebra system.
MATH
1102
- Calculus II
(M/SR)
(5.0 cr; Prereq-1101; fall, spring, every year) Techniques of integration. Further applications involving mathematical modeling and solution of simple differential equations. Taylor's Theorem. Limits of sequences. Use and theory of convergence of power series. Students use a computer algebra system.
MATH
1993
- Directed Study
(1.0 - 5.0 cr [max 102211
- History of Mathematics
(4.0 cr; Prereq-Math course above 1100 or #; fall, even years) Historical development of various areas in mathematics and important figures in mathematics from ancient to modern times.
MATH
29933221
- Analysis
(M/SR)
(4.0 cr; Prereq-1102, 2202 or #; fall, every year) Introduction to real and complex analysis. The main topics of calculus-convergence, continuity, differentiation, integration, and series-applied and extended in advanced settings with emphasis on precise statements and rigorous proofs. Concept of metric space. Other topics and applications.
MATH
3231
- Abstract Algebra I
(M/SR)
(4.0 cr; Prereq-2111, 2202 or #; spring, every year) Systematic study of groups and rings, making use of linear algebra. Groups as codifying symmetry throughout mathematics and its applications. The Euclidean algorithm and its consequences, both for integers and polynomials. Other selected topics and applications.
MATH
3994901
- Senior Seminar
(2.0 cr; Prereq-sr; full year course begins fall sem; fall, every year) This is a full-year course, required for all mathematics majors in their senior year. Students must attend year round and present one of the seminars.
MATH
499 |
Description
The aim of this book is to describe the underlying principles of algebraic geometry, some of its important developments in the twentieth century, and some of the problems that occupy its practitioners today. It is intended for the working or the aspiring mathematician who is unfamiliar with... Expand algebraic geometry but wishes to gain an appreciation of its foundations and its goals with a minimum of prerequisites.
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Sewn binding. Cloth over boards. 184 p. Contains: Illustrations. Universitext. Audience: General/trade. 2004 EXCELLENT INTRODUCTION TO ALGEBRAIC GEOMETRY FOCUSING ON THE MORE ANALYTIC ASPECTS AS OPPOSED TO RIED'S MORE ALGEBRAIC APPROACH; WONDERFULLY WRITTEN TEXT FOR BEGINNERS IN THIS VERY DIFFICULT SUBJECT; SOME LOSS OF SURFACE LUSTER BUT OTHERWISE BRAND NEW WITH NO VISIBLE DEFECTS OF ANY KIND, GREAT FOR CLASS! SHIPS IMMEDIATELY FROM NEW YORK CITY, USA UPON PAYMENT |
PreCalc - Fifth Period Class
This course is a college course (4 credits through SCCC) designed to prepare students for learning calculus.
Major emphasis is on the concept of functions. The students will study polynomial, rational, exponential, logarithmic and trigonometric functions, their properties, graphs and related equations and applications. Additional topics include conics, matrices, systems of equations, sequences and series, and probabilities. |
Math
Professional Development
You can master new strategies to address students' needs in various grade levels. From exploring environments that promote effective learning to examining the skills students need to read content material successfully, this course will introduce you to essential techniques that support more independent reading and learning.
Integrating writing into mathematics presents both challenges and opportunities. Discover the tools you need to make it work. Use a mathography lesson to gain insight into your students' attitudes and learn how to help your students organize and communicate mathematical concepts, and interact with others. Then, develop a plan of action for implementing writing in a mathematics program.
You can bring writing into your classroom in all content areas with a practical plan developed in this course. Covering the research basis for writing across the curriculum, the course will help you teach students effective writing processes, including finding the time to write. You'll learn sound techniques for evaluating writing skills and discover proven ways to promote writing to students.
Understand more about what confuses your high school students when they encounter the theory of function. You'll examine theories of function as a process, a mathematical object, and a tool for description and prediction, and learn new ways to present the concept to your students |
Analyzing Tables with CASIO Graphing Calculator (Linear FunctionsBasic Linear Functions are a big topic in Pre-Algebra and Algebra 1. If you are looking for a PowerPoint to lead your students through the procedures to enter a table, graph the data, and write the expression (or equation) using the CASIO graphing calculator, HERE IT IS! The examples start leading the students with 'baby' steps, but as the PowerPoint progresses the students are offered the chance to progress more independently. The examples are real-world and engaging. Once the students master the list, graphs, and linear expressions using the CASIO, the world of math using this technology is massive!
Presentation (Powerpoint) File
Be sure that you have an application to open this file type before downloading and/or purchasing.
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Was $1.49've been teaching mathematics, international studies, and media design since 1998 in an urban district located in the southeast region of the USA. Before becoming a teacher, I worked as an electrical engineer for several years. |
This book is designed for teachers of students aged from about 13 to 18 years. It shows how Vedic Mathematics can be used in a school course but does not cover all school topics (see contents). The book can be used for teachers who wish to learn the Vedic system or to teach courses on Vedic mathematics for this level. Non-teachers who have a background knowledge of mathematics may also find it appropriate (see contents).
A- A DESCRIPTIVE PREFATORY NOTE ON THE ASTOUNDING WONDERS OF ANCIENT INDIAN VEDIC MATHEMATICS
This Manual is the third of three self-contained Manuals (elementary, intermediate and advanced) and is designed for adults with a good understanding of basic mathematics to learn or teach the Vedic system. So teachers could use it to learn Vedic Mathematics, or it could be used to teach a course on Vedic Mathematics. It is suitable for teachers of children aged about 13 to 18 years.
The eighteen lessons of this course are based on a series of one week summer courses given at Oxford University by the author to Swedish mathematics teachers between 1990 and 1995. Those courses were quite intensive consisting of eighteen, one and a half hour, lessons. Some of the material here is more advanced than would be given to the average 18 year old student but this is what the teachers wanted on the courses and so the same is given here.
The lessons in this book however probably contain more material than could be given in a one and a half hour lesson. The teacher/reader may wish to omit some sections, go through the material in a different sequence to that shown here or break up some sections.
All techniques are fully explained and proofs and explanations are given, the relevant Sutras are indicated throughout (these are listed at the end of this Manual) and, for convenience, answers are given after each exercise. Cross-references are given showing what alternative topics may be continued with at certain points.
It should also be noted that the Vedic system encourages mental work so we always encourage students to work mentally as long as it is comfortable. In the Cosmic Calculator Course pupils are given a short mental test at the start of most or all lessons, which makes a good start to the lesson, revises previous work and introduces some of the ideas needed in the current lesson. In the Vedic system pupils are encouraged to be creative and use whatever method they like.
Some topics will not be found in this text: for example, there is no section on area and volume. This is because the actual methods are the same as currently taught so that the only difference would be to give the relevant Sutra(s).
LESSON 6 : SOLUTION OF EQUATIONS
6.1 TRANSPOSE AND APPLY
6.1a SIMPLE EQUATIONS
6.1b MORE THAN ONE X TERM
6.2 SIMULTANEOUS EQUATIONS
6.2a GENERAL SOLUTION
6.2b Special Types
6.3 QUADRATIC EQUATIONS
6.4 ONE IN RATIO THE OTHER ONE ZERO
6.5 MERGERS
6.6 WHEN THE SAMUCCAYA IS THE SAME IT IS ZERO
6.6a Samuccaya as a common factor
6.6b Samuccaya as the Product of the Independent Terms
6.6c Samuccaya as the Sum of the Denominators
6.6d Samuccaya as a Combination or Total
Proof/ EXTENSION
6.6 e other types
6.7 THE ULTIMATE AND TWICE THE PENULTIMATE
6.8 ONLY THE LAST TERMS
6.9 SUMMATION OF SERIES
6.10 FACTORISATION
LESSON 7 : SQUARES AND SQUARE ROOTS
7.1 Squaring 2-FIGURE NUMBERS
7.2 Algebraic Squaring
7.3 Squaring Longer Numbers
7.4 Written Calculations
7.4 a Left to Right
7.4 b Right to Left
7.5 Square Roots of Perfect Squares |
MAA Review
[Reviewed by Brian Rogers, on 04/10/2007]
With the current surge of interest in undergraduate proof production, bridge courses, and how best to teach undergraduates to perform proof, the re-issue of Rotman's Journey is certainly timely, since Rotman intended this book to be a text for just such courses and uses. Besides that, as I read Journey, I kept finding things that seemed to reflect current research findings on undergraduate proof schemes, methods of teaching that are effective, and even embodied cogntive research that has only been published in the last year. Impressive for a book originally copyrighted in 1998.
Unlike the other books I have seen which are intended for bridge courses, Journey begins with proofs: proofs for students to read and proofs for them to do. Don't get me wrong; the material on propositional calculus and first-order logic is there; it just is not the first thing introduced. I believe this to be salutary: too often, I think we give students tools before they need them. It's like pneumatic nail guns. Certainly they are essential tools for any professional rough carpenter or even finish carpenter, but the average householder can get by fine with a 16-oz hammer — and be a lot safer, too.
Rotman starts with mathematics the average college sophomore math major already knows, so the only new thing to learn is how to do the proofs that establish and explain such knowledge. Tools are introduced sparingly, and only as students encounter proofs that need them. By the time the book reaches chapter 4, however, the basic concepts of analysis have been explored, as well as those of algebra. Both, by the way, are explored along the lines of the historical development of the concept, so the student is allowed to first learn the less-sophisticated answers upon which today's answers were built.
I've been looking for a textbook or supplement to use in classes and workshops I'm currently designing in a project related to my dissertation, and I believe I've found a top-tier candidate. I recommend this as a textbook or supplemental textbook for such classes, especially if you're less than satisfied with current approaches to teaching undergraduate proof-production. I also suggest it for all of those who are curious about other possible approaches to teaching such classes. Rotman writes in a clear, informal style that will be well within the reach of students, and they will be able to learn some on their own from Journey. Combined with effective teaching, this could be a winning combination.
Brian Rogers is a doctoral student in Educational Mathematics at the University of Northern Colorado, Greeley, Colorado. After nearly a half century of teaching English, programming computers, and teaching others to program computers, he feels he is finally where he belongs. Kind comments or questions may be directed to [email protected]; it has not been decided what to do with other types of communication, so if this applies to you, please forget the e-mail address above. |
Introductory & Intermediate Algebra for College Students Plus NEW MyMathLab with Pearson eText Blitzer Algebra Series combines mathematical accuracy with an engaging, friendly, and often fun presentation for maximum appeal. Blitzer's personality shows in his writing, as he draws readers into the material through relevant and thought-provoking applications. Every Blitzer page is interesting and relevant, ensuring that students will actually use their textbook to achieve success!
The Blitzer Algebra Series combines mathematical accuracy with an engaging, friendly, and often fun presentation for maximum appeal. Blitzerís personality shows in his writing, as he draws readers into the material through relevant and thought-provoking applications. Every Blitzer page is interesting and relevant, ensuring that students will actually use their textbook to achieve success!
7.3 Adding and Subtracting Rational Expressions with the Same Denominator
7.4 Adding and Subtracting Rational Expressions with Different Denominators
Mid-Chapter Check Point Section 7.1–Section 7.4
7.5 Complex Rational Expressions
7.6 Solving Rational Equations
7.7 Applications Using Rational Equations and Proportions
7.8 Modeling Using Variation
Chapter 7 Group Project
Chapter 7 Summary
Chapter 7 Review Exercises
Chapter 7 Test
Cumulative Review Exercises (Chapters 1–7)
8. Basics of Functions
8.1 Introduction to Functions
8.2 Graphs of Functions
8.3 The Algebra of Functions
Mid-Chapter Check Point Section 8.1–Section 8.3
8.4 Composite and Inverse Functions
Chapter 8 Group Project
Chapter 8 Summary
Chapter 8 Review Exercises
Chapter 8 Test
Cumulative Review Exercises (Chapters 1–8)
9. Inequalities and Problem Solving
9.1 Reviewing Linear Inequalities and Using Inequalities in Business Applications
9.2 Compound Inequalities
9.3 Equations and Inequalities Involving Absolute Value
Mid-Chapter Check Point Section 9.1–Section 9.3
9.4 Linear Inequalities in Two Variables
Chapter 9 Group Project
Chapter 9 Summary
Chapter 9 Review Exercises
Chapter 9 Test
Cumulative Review Exercises (Chapters 1–9)
10. Radicals, Radical Functions, and Rational Exponents
10.1 Radical Expressions and Functions
10.2 Rational Exponents
10.3 Multiplying and Simplifying Radical Expressions
10.4 Adding, Subtracting, and Dividing Radical Expressions
Mid-Chapter Check Point Section 10.1–Section 10.4
10.5 Multiplying with More Than One Term and Rationalizing Denominators
10.6 Radical Equations
10.7 Complex Numbers
Chapter 10 Group Project
Chapter 10 Summary
Chapter 10 Review Exercises
Chapter 10 Test
Cumulative Review Exercises (Chapters 1–10)
11. Quadratic Equations and Functions
11.1 The Square Root Property and Completing the Square; Distance and Midpoint Formulas
11.2 The Quadratic Formula
11.3 Quadratic Functions and Their Graphs
Mid-Chapter Check Point Section 11.1–Section 11.3
11.4 Equations Quadratic in Form
11.5 Polynomial and Rational Inequalities
Chapter 11 Group Project
Chapter 11 Summary
Chapter 11 Review Exercises
Chapter 11 Test
Cumulative Review Exercises (Chapters 1–11)
12. Exponential and Logarithmic Functions
12.1 Exponential Functions
12.2 Logarithmic Functions
12.3 Properties of Logarithms
Mid-Chapter Check Point Section 12.1–Section 12.3
12.4 Exponential and Logarithmic Equations
12.5 Exponential Growth and Decay; Modeling Data
Chapter 12 Group Project
Chapter 12 Summary
Chapter 12 Review Exercises
Chapter 12 Test
Cumulative Review Exercises (Chapters 1–12)
13. Conic Sections and Systems of Nonlinear Equations
13.1 The Circle
13.2 The Ellipse
13.3 The Hyperbola
Mid-Chapter Check Point Section 13.1–Section 13.3
13.4 The Parabola; Identifying Conic Sections
13.5 Systems of Nonlinear Equations in Two Variables
Chapter 13 Group Project
Chapter 13 Summary
Chapter 13 Review Exercises
Chapter 13 Test
Cumulative Review Exercises (Chapters 1–13)
14. Sequences, Series, and the Binomial Theorem
14.1 Sequences and Summation Notation
14.2 Arithmetic Sequences
14.3 Geometric Sequences and Series
Mid-Chapter Check Point Section 14.1–Section 14.3
14.4 The Binomial Theorem
Chapter 14 Group Project
Chapter 14 Summary
Chapter 14 Review Exercises
Chapter 14 Test
Cumulative Review Exercises (Chapters 1–14)
Appendices
A. Mean, Median, and Mode
B. Matrix Solutions to Linear Systems
C. Determinants and Cramer's Rule
D. Where Did That Come From? Selected Proofsís his Developmental Algebra Series, Bob has written textbooks covering college algebra, algebra and trigonometry, precalculus, and liberal arts mathematics, all published by Pearson Education. When not secluded in his Northern California writerís cabin, Bob can be found hiking the beaches and trails of Point Reyes National Seashore, and tending to the chores required by his beloved entourage of horses, chickens, and irritable roosters. |
Practical
Linear Algebra -- A Geometry Toolboxforms the basis of a
first year course in Linear Algebra for non-math majors such as
engineers and computer scientists. In addition, this book provides
a solid foundation for work in computer graphics and computer aided
design. The authors emphasize a geometric and intuitive approach
that relies heavily on examples and illustrations rather than the
rigorous theorem-proof format used in standard texts.
Special
features include:
250
figures which are also available in electronic form,
150
numerical examples,
200
problems---many solutions are in the text and additional problems
are on a password-protected instructor's website,
supplementary
materials for instructors,
a
"WYSK" (What You Should Know) section closes each chapter,
providing a concise chapter summary which highlights the most
important points, giving students focus for their approach to
learning.
The figures
are not included as window dressing, in fact they play an
important role in bringing the reader to a robust understanding
of the mathematics. However they are not only instructional, they
are also fun!
For example,
on the right is a crazy Pacman path -- created with linear transformations
and a bit of coloration thanks to Postscript
Below, left
is an instructional tool used to demonstrate a rotation by 45 degree,
and below right is simply a fun example of what we can do with 3D
transformations. |
Little Algebra Book
The idea was to turn 'traditional' maths textbooks on their heads. Out with the dull, heavy-as-a-brick, purely functional books that have barely changed style since your parents were at school; in with something small, beautiful and focussed on helping you understand the single, most important rule of algebra: whatever you do to one side, you have to do to the other.
Colin Beveridge kindly sent me a copy of his preprint book to review and comment on. Being a mathematician, I will just set out my comments as plus points and minus points, but they do not carry equal weight! Then, I'll make some final comments.
Positive points
The design makes clever use of the fact that it is a physical book, and the pages become something that the student can interact with instead of just turning to the next one. The central page-join becomes an analogy for the equals sign, with left-hand and right-hand pages acting as the respective sides of an algebraic equation.
Pages also have fold-out and fold-over elements which help create a sense of mystery – attractive not just to younger readers but also to adults as well! This was a particularly neat approach to showing division.
One book, one concept, with a simple almost minimal approach, certainly helps focus on the idea of balancing equations.
The four basic arithmetic operations are each represented by their own individual images, which is useful for reinforcing the idea of different operations being carried out on each of the example equations – the bird for the subtraction is especially cute
There is no attempt made to explain the order of steps for the combination of operations on the final pages, and this could be revisited with higher-level students to see if changing the order of steps makes a difference to the final result – for example, could you divide by 3 first?
Negative points
I'm not sure that stating "The aim of algebra is to get x on its own" on the opening page is either correct or helpful – mathematically. To me, the aim of algebra is to provide a means of notating an abstract problem, so that you can explore possible solutions algorithmically. Of course, you need something short and snappy, but x is not always the only thing you need to solve for later on in algebra! Maybe more discussion is needed to find a more appropriate statement, but I'm not going to enter into it here. The same thing applies to "In an equation…", where I would substitute "In a simple algebra equation…"
Referring to numbers as "pure numbers" is also potentially confusing, I feel.
Evaluation
If I am being picky, I would change the precision of some of the language used, in future reprints – there is no place for "artistic licence" with terminology in mathematics! Overall, I think this book does achieve what it set out to do with a minimum of effort and an efficiency of approach which is to be commended. I can see students wanting to keep their copy, once they have learned the algebra, just because of its overall attractiveness. Anything which helps students enjoy learning algebra should be welcomed with open arms!
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One Response to Little Algebra Book
Too many folks have already thrown away plenty of good money
on nothing but useless salt tablets being shipped
from South America. It's old news that tracking food intake could lead to losing a few pounds [2].
The institution has persistently offered ideal programs and services for those struggling to achieve certain levels of body weight. |
07923578Complex Analysis through Examples and Exercises (Texts in the Mathematical Sciences (closed))
This volume on complex analysis offers an exposition of the theory of complex analysis via a comprehensive set of examples and exercises. The book is self-contained and the exposition of new notions and methods is introduced step by step. A minimal amount of expository theory is included at the beginning of each section in the Preliminaries, with maximum effort placed on well-selected examples and exercises capturing the essence of the material. The examples contain complete solutions and serve as a model for solving similar problems given in the exercises. The readers are left to find the solution in the exercises; the answers, and occasionally, some hints, are given. Special sections contain so-called Composite Examples which consist of combinations of different types of examples explaining some problems completely and giving the reader an opportunity to check all his previously accepted knowledge. Audience: This volume is intended for undergraduate and graduate students in mathematics, physics, technology and economics interested in |
... --- End quote ---
Neon Genesis:
I think unless you're planning on majoring in a career that requires you to know the math like if you're going to be a scientist or something, then there's not really much point in being required to learn a ridiculously hard math subject that you'll never use in real life.
andrewclunn:
I can see calc and trig being optional maybe, but algebra? Well maybe then all of high school is not needed.
SQ the ΣΛ/IGMд:
Most of my family don't use it at all. I use it daily.
Necessary? Don't know but at my school you didn't take algebra or other forms of advanced math unless you were enrolled in college prep classes. Those not planning on college had a much simpler type. |
XX10198: Mathematics 1
This unit is only available to students in the Department of Electronic & Electrical Engineering.
Description:
Aims: This is the first of two first year units intended to lead to confident and error-free manipulation and use of standard mathematical relationships in the context of engineering mathematics. The unit will consolidate and extend topics met at A-level, so that students may improve their fluency and understanding of the basic techniques required for engineering analysis.
Learning Outcomes: After successfully completing this unit the student should be able to:
Handle circular and hyperbolic functions, and sketch curves. Differentiate and integrate elementary functions, products of functions etc. Use complex numbers. Employ standard vector techniques for geometrical purposes. Determine the Fourier series of a periodic function. |
Wolfram Research Mathematica 5.2.0
From simple calculator operations to large-scale
programming and interactive-document preparation,
Mathematica is the tool of choice at the frontiers
scientific research, in engineering analysis and
modeling, in technical education from high school to
graduate school, and wherever quantitative methods are
used.
You probably know Mathematica by name. Or you may be
one of nearly two million users. But do you really
know the breadth of capabilities Mathematica can o
you? Whatever you're working on--calculating,
programming, learning, documenting or
developing--Mathematica is equipped to help.
Mathematica seamlessly integrates a numeric and
symbolic computational engine, graphics system,
programming language, documentation system, and
advanced connectivity to other applications. It is
this range of capabilities--many world-leading in
their own right--that makes Mathematica uniquely
capable as a "one-stop shop" for you or your
organization's technical work.
Wide Range of Uses
Handling complex symbolic calculations that often
involve hundreds of thousands or millions of terms
Illustrating mathematical or scientific concepts for
students from K-12 to postgraduate levels
Typesetting technical information--for example, for
U.S. patents
Giving technical presentations and seminars
Works at All Levels
Usually Mathematica is used with its notebook
interface directly as it comes out of the box.
However, it is increasingly being used through
alternative interfaces such as a web browser or by
other systems as a back-end computational engine.
Some of these uses require in-depth Mathematica
knowledge, while others do not. Mathematica is unusual
in being operable for less involved tasks as well as
being the tool of choice for leading-edge research,
performing many of the world's most complex
computations. It is Mathematica's complete consistency
in design at every stage that gives it this multilevel
capability and helps advanced usage evolve naturally.
Fully Featured, Fully Integrated
At a superficial level, Mathematica is an amazing, yet
easy-to-use calculator. The world's most comprehen
set of mathematical, scientific, engineering, and
financial functions is ready-to-use--often with just
one mouse click or command. However, Mathematica
functions work for any size or precision of number,
compute with symbols, are easily represented
graphically, automatically switch algorithms to get
the best answer, and even check and adjust the
accuracy of their own results. This sophistication
means trustworthy answers every time, even for those
inexperienced with the mechanics of a particular
calculation.
While working through calculations, a notebook
document keeps a complete report: inputs, outputs, and
graphics in an interactive but typeset form. Adding
text, headings, formulas from a textbook, or even
interface elements is straightforward, making online
slide show, web, XML, or printed presentation
immediately available from the original material. In
fact, with notebook document technology, a fully
customized interface can easily be provided so that
recipients can interact with the content. The notebook
is a fully featured, fully integrated technical
document-creation environment.
Easy Programming, Powerful Results
The move from immediate calculations to programmed
computations can occur evolutionarily. Just one line
makes a meaningful program in Mathematica--the
methodology, syntax, and documents used for input and
output remaining as they are for immediate
calculations.
Mathematica is also a robust software development
environment. Mathematica packages can be debugged,
encapsulated, and wrapped in a custom user interface,
all from within the Mathematica system. Alternatively,
Java, C, or links to a proprietary system can use
Mathematica's power behind the scenes.
One Unifying Idea
Symbolic programming is the underlying technology that
provides Mathematica this unmatched range of
abilities. It enables every type of object and every
operation--be they data, functions, graphics,
programs, or even complete documents--to be
represented in a single, uniform way as a symbolic
expression. This unification has many practical
benefits from ease of learning to broadening the scope
of applicability of each function. The raw algorithmic
power of Mathematica is magnified and its utility
extended. |
About Business Math: This class is designed for Seniors. This is a one year course. Business math can count toward MATH or ELECTIVE Credits (5 per semester). Students will cover the essentials as well as the basics in math. Students learn about real-life scenarios such as Managing Money, Loans, Payments, Interest Rates, Taxes, Insurance, Purchasing an Auto/Home, etc. There is an extensive amount of basic skills review (fractions, decimals, percents, order of operations, etc). A basic four-function calculator is required. Students MAY NOT use cell phones in class and may not enter classroom without valid SFHS ID.
LATE OR INCOMPLETE ASSIGNMENTS AND TESTS MUST BE MADE UP WITHIN 2 DAYS OF DUE DATE (WITH PARENT PHONE CALL); WARM UPS ARE GIVEN DURING THE FIRST 10 MINUTES OF CLASS AND MAY NOT BE MADE UP.
About Guided Study: This class/program is designed for freshman and/or sophomores (selected by SFHS). Students receive a Classroom, a Teacher, Mentors, Resources, and CREDITS to do assignments. Behavior, attendance, and grades are monitored very carefully by the Teacher and the Guided Study Program. EXTREMELY IMPORTANT: ALL STUDENTS MUST BRING ALL MATERIALS FOR ALL CLASSES EVERY DAY (EVEN ON BLOCK DAYS). Required materials: 2+inch 3-ring binder, dividers, lined paper, pencils, pens, School ID. |
ExamConnection® for Remedial Algebra
What is ExamConnection?
ExamConnection® is an
adaptive assessment and learning system. Using adaptive questioning, the
ExamConnection® software quickly and accurately determines a student's aptitude
in the critical competencies established by the AMATYC's "Crossroads in
Mathematics". ExamConnection® then instructs the students on the topics they
need to learn. See below for a step by step explanation of the ExamConnection®
Procedure.
Students will start
by taking the remedial algebra adaptive assessment. This assessment is
taken in the software where the students will be asked a varying number
of questions depending on their proficiency in the different knowledge
standards. The software will determine where the student's strength and
weaknesses lie based on the answers given. The questions all belong to
one of the following three levels of difficulty:
Level one
consists of Pre-algebra level problems
Level two
consists of Introductory Algebra level problems
Level three
consists of Intermediate Algebra level problems
At the end of the
assessment, the student will receive a Personalized Study Plan (PSP).
The Personalized
Study Plan (PSP) is the student's individualized curriculum. It tells
the students which areas require further study and where in the Exercise
Book to find the relevant exercises. This tool is the linchpin on which
the whole ExamConnection relies to produce results for each individual
student. The PSP offers guidance in terms of both exercise types as well
as exercise levels that are required.
The PSP can be saved
on the student's computer and/or printed to be used as the blueprint for
further studies.
The Exercise Book
This Exercise Book has been developed based on the American Mathematical
Association of Two Year Colleges (AMATYC) knowledge standards as found
in "Crossroads in Mathematics". These standards closely aligns with those of the National
Council of Teachers of Mathematics (NCTM). Each of the seven sections contains
a short introduction to the knowledge standard and three levels of
problems. Level one will consist of Pre-algebra level problems, level
two of Introductory Algebra level problems, and level three will consist
of Intermediate Algebra level problems.
After the students
fulfill the recommendations found in the Personalized Study Plan (PSP)
they are referred back to the software for more exercises. These
exercises are contained in seven drills (one for each section of the
book and knowledge standards as defined by the AMATYC). Each Drill is a
mirror image of each section of the book.
Upon completion of a
Drill the students will receive a grade reflecting their knowledge.
Once they have
completed all Drills recommended by the PSP, the students go back to the
beginning to complete Assessment number two. |
Introduction To Algebra
posted on: 16 May, 2012 | updated on: 11 Sep, 2012
Algebra is that branch of mathematics, which includes the study of the expressions containing the variables and the constants. Here we are going to learn about Introduction to Algebra. While talking about Intro to Algebra, we Mean the basic knowledge of the variables and their mathematical Relations with different operators. Algebra Introduction includes the basic study of the use of the operators and how the different values of the variables effect the given equations of algebra.
If we have some equality with the left and the right side of the equation, we say that in order to find the value of the variable, we will use hit and trial method, by which we check the left side of the equation and the right side of the equation are equal or nit. In case we have both the sides of the equations as equal, it indicates that the particular value of the variable satisfies the equation. In case the two sides of the equation are not equal, then we say that the particular value does not satisfy the equation.
Now we look at the following equation, 3x + 5 = 8;
So we say that the variable x is unknown, and then if we need to find the value of the value of the unknown variable in the equation, it means that we go on trying different values of 'x' and check by which value, the left side of the equation is equal to the right side of the equation. So we start with x = 0
3 * 0 + 5 = 0 + 5 = 5,
So this time it is not equal to the right side of the equation. Thus x = 0 is not the solution. Now we try for x = 1, so 3 * 1 + 5 = 3 + 5 = 8.
So LHS = RHS, so x= 1 is the solution.
Topics Covered in Introduction To Algebra
The polynomials can be of the following types: linear polynomial, quadratic polynomial and the cubic Polynomials. If we talk about the Cubic Equations, we say that the standard form of the cubic Equation is ax3 + b.x2 + cx ...Read More
The definition of Algebra, can be given as "a branch of mathematics which is use to put the letters in place of numbers". Algebra definition covers several topics like the Real Numbers, complex numbers, matrices, vectors, etc. We will use letter like (P or Q) in place of the given unknown values. For example: "y + 4 = 9"; Here in the given expression 'y' is unknown...Read More |
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