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cornerstone of Elementary Linear Algebra is the authors' clear, careful, and concise presentation of material--written so that students can fully understand how mathematics works. This program balances theory with examples, applications, and geometric intuition for a complete, step-by-step learning system.The Sixth Edition incorporates up-to-date coverage of Computer Algebra Systems (Maple/MATLAB/Mathematica); additional support is provided in a corresponding technology guide. Data and applications also reflect current statistics and examples to engage students and demonstrate the link between theory and practice.
Editorial Reviews
About the Author
Dr 2013 Text and Academic Authors Association Award for CALCULUS, the 2012 William Holmes McGuffey Longevity Award for CALCULUS: AN APPLIED APPROACH, the 2011 William Holmes McGuffey Longevity Award for PRECALCULUS: REAL MATHEMATICS, REAL PEOPLE,
Most Helpful Customer Reviews
I'm taking Linear Algebra right now, and have long since left the lectures behind. The book pulls you along with exciting, unifying concepts for its whole length. And since that's exactly what I've done, I can say that this book is perfect for people who want to study the subject on their own without taking a class; it is comprehensive and clear, covering all of its material on its own without relying on a prof to interpret.
And although linear algebra is probably the most pure math a student just coming off the Calculus 1/2/3 track has seen, the book is loaded with real-world applications. In fact, every chapter ends with an entire section of nothing but applications!
Larson relies heavily on proofs to support and also teach concepts. The important theorems are all proven very clearly. Larson seems to keep in mind that his readers are there because they're still learning the subject matter- something most math books forget when proving theorems for nothing but rigor's sake. There are a few exercises in each section called "guided proofs" which require you to prove some interesting consequence of the section's material and outline basic steps for you to take. These exercises are really a brilliant device for helping students to really think instead of following step-by-step formulas. I would complain, however, that there are many un-guided proof exercises that are difficult and aren't realistic to expect elementary linear algebra students to complete.
Still, reading this textbook is a real pleasure. It's concise and powerful and, importantly, always illustrates all concepts with examples, in addition to the half-dozen-to-dozen general examples for each section.
Very often math books can be hard to read or follow. This book is very readable and the examples are well written and explained. I liked the layout of the book and how the authors either reviewed the material or referred to its location earlier in the book. There are a lot of questions, which are laid out in a progressive manner from the basics to the mastery level. The authors also provided resources to assist with the use of technology (though not for the n-Spire yet). Finally, as one who generally needs to see (i.e. classroom) to learn, I believe that with this text I could have taught my self Linear Algebra fairly easily.
This textbook paired with a good teacher can make you learn a lot during your course of Linear Algebra. I would say it would also be a good book to self-study from because there are numerous exercises and from experience, there is a lot of pencil whipping to it. Most of the proofs in this book are not exceedingly complicated and can be solved with medium difficulty. Just have to put some thought into it because some proofs can turn out long when they really only take a few steps. My course wasn't taught really with applications, 90% theory, but the book has a decent amount of problems for those doing engineering, physics, and other things. I never bought the solutions manual, but it was useful to check the true and false answers for more explanation and I would recommend getting it. I would say don't use it often though because it can be a crutch. With a decent amount of studying, this would be an ideal text for a first course in Linear Algebra at a sophomore level.
I've gone through at least 4 other linear algebra textbooks (Strang, Anton, Beauregard, and Bretscher). The first two were "okay" for my purposes, and the last two were terrible. I was in need of some serious hand-holding for this class, and Falvo's book did just that -- I mean that in a good way.
In each section, there are lots of examples, where all of the steps are laid out in great detail. The author keeps the language as simple as possible to communicate each step to the reader. At the end of each section, there is a nice balance of straightforward computational problems and proofs. I would recommend getting a solutions manual as an extension of the textbook... not simply to copy, but to learn from more problems.
Along with this book, I would also recommend Paul's online linear algebra notes. They were also an immense help for me.
If you're struggling in linear algebra like I was, get this book. You will go from dreading this subject to at least tolerating it, or even enjoying it!
This text was definitely straight to the point. Like many, I did not get to pick a Linear Algebra text, it was required for the class. I am a math major, but I do not think that this text is beyond anyone in college. The sections that require calculus are clearly marked and not essential to the subject matter.
Linear algebra has a reputation of being very dull, but I have found the subject to be immensely rewarding. I am teaching with this text for the first time and it helps me bring an excitement to the subject that the professor who taught me missed.
Bought this book prior to taking LA. The text book for my 300 level class was the Leon text book. I ended up using this book much more. For computations it was much better, for theory and proofs about the same. Very well done.
This book was a great finding! It is one of the only linear algebra textbooks (actually one of the only Math books I've come across) that is easy to follow, offer nice real world examples and has a good theoretical basis to back everything up. Definitely one of the books I would recommend if you had or have problems understanding linear algebra.
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More About
This Book
systems; symbols and commands; first steps in algebra and algebraic notation; common fractions and equations; irrational numbers; algebraic functions; analytical geometry; differentials and integrals; the binomial theorem; maxima and minima; logarithms; and much more. Upon reaching the conclusion, readers will possess the fundamentals of mathematical operations, and will undoubtedly appreciate the compelling magic behind a subject they once dreaded.
From Simple Numbers to the Calculus
Dover Publications, Inc.
A PATIENT is sitting in the doctor's waiting-room. He suspects that he won't get away very quickly so he decides to while away the time by reading. There are all kinds of brochures and advertisements of resorts and cruises on the table. He is particularly attracted by a picture which reveals all the wonder of southern seas and tropical townships. His interest is aroused and he opens the booklet–but he is very disappointed. He can hardly understand a word; it is written in Portuguese. He can understand what the pictures are about and there is something else too that he can follow without a translation, that is the columns of figures, the tables of statistics and the departure and arrival times.
You will think that this is rather a childish example of something which is quite self-evident. Who has ever doubted that nowadays almost all civilised countries use a common system of numbers? What is there so remarkable or worrying about this booklet? The number 3 in a Portuguese book means exactly the same as 3 in an English one, and so does a sum like 521 × 7 = 3,647. You would think that there is nothing more to be said about it. On the contrary; if we look more closely at the rather trivial example we shall come at once on the deepest puzzles of mathematics and we shall grasp a number of most important fundamental principles.
There is one thing that has so far been overlooked. When a Portuguese or a German reads the numbers in the prospectus, each uses different words for them. The way in which the numbers are pronounced in various languages goes to the very root of the mysteries of the number system. The English say "twenty-four," the Germans "four and twenty." Instead of saying "septante" for 70, which would be the logical sequence of "quarante, cinquante, soixante ..." the Frenchman says rather surprisingly "soixantedix" and for 80 he actually undertakes some multiplication in "quatrevingt," or 4 × 20.
The first thing we must note is that there is only a very slight connection between our international system of numbers and the alphabet system. The two are based on quite different principles. Numbers are each in themselves symbols standing for an idea; the letters of the alphabet do not in themselves represent ideas but only sounds; only when they are strung together as words can they become symbols for ideas. The concept "3" needs one sign when written as a number. When written as the word "three," 5 letters are used.
All this is only the beginning. We have been talking up till now of figures and numbers, not of the great number system which is said to be the greatest pride of the human intellect. At this point some readers will say, "We know all about the number system if what is meant by that is simply a system scaled in tens. We use it ourselves every day and if this book is going to theorise instead of sticking to necessary explanations we will just shut it up and fling it away." In defence it must be said that no attempt will be made to put forward a theory of numbers or to explain why three hundred is written "300." Because nothing should be taken for granted however, we must seize hold of such matters which are generally known and accepted, in order to be able to make some of the higher concepts of mathematics intelligible right from the start.
The reader would do wrong to speak slightingly of our system of numbers scaled in tens. One of its greatest merits is that it can be learnt by a Primary School child. But there was a time in history when what a child can now do at school was a difficult problem for the greatest mathematicians. For the well-nigh automatic working of our present number system had not yet been discovered or developed. The technique of calculation only became commonly known in the West during the eleventh century A.D. At that time two "schools of reckoning" were struggling for precedence; the one used the abacus, the other the devices of Muhammad ibn Ahmad al-Khwarizmi. The abacus is an ancient counting frame. Imagine that we have before us a board divided up by vertical lines. Each column represents a step in the system, a single number, a ten, a hundred, a thousand, etc. To use the abacus to reckon with, we place the required number of counters in each column. Suppose we had to add 504,723 and 609,802. We place the first number on the board in the correct columns, using white counters. Then we do the same with the black counters for the second number. We have to count up the total of black and white counters to arrive at the result. There is no zero (0) in this abacus system. When adding up the counters we must not forget that 1,500 equals 1 thousand, 5 hundred; 104,000 equals 1 hundred-thousand and 4 thousand. It will soon be obvious that al-Khwarizmi's system of numbers is superior to the abacus method of reckoning. This Arabian mathematician came from Khorassan and later lived in Baghdad. Between A.D. 800 and 825 he wrote among other things a work in which he describes the foundations of reckoning with the Hindu or so-called Arabian system of figures, using place values. He knew all about zero and wrote it as a small circle. By devious ways, from the crusaders and from the Moorish universities of Toledo, Seville and Granada, the Arabian works in Latin translation came to be known to scholars of Western Europe. Amongst these works was al-Khwarizmi's on Arabic numerals.
No longer were the clumsy abacus counting boards necessary for reckoning. Now an almost magical system of numbers allowed long and complicated calculations to be carried out with complete assurance. All that was needed to accomplish this was the ten digits from 0 to 9, a pen, a piece of paper and a knowledge of the multiplication table. It is difficult for us to imagine the feelings of the people who first realised that they could throw away their counting boards and in future do a complicated operation on a scrap of paper.
But we must leave these somewhat romantic realms of history and return to the commonplace. This Arabic system of numbers which we now use is a system of writing down certain methods of reckoning by means of certain symbols. These methods are used within the limits of the system in such a way that they do a part of the work of thinking for us. Thus they enable our minds to soar in regions which our powers of imagination could not reach, or at least, in which we could very easily wander aimlessly. So we must examine this Arabic system, the so-called system of tens, more closely to understand wherein its strength lies.
Before we go on to the next chapter the budding mathematician should be given a few practical hints. Write your symbols down as neatly and as carefully as possible. Don't tinker with them, don't write things down in a muddle, don't scribble workings in the margin or in odd spaces. Further, never, as a beginner or as an experienced mathematician, leave out steps in the process of your mathematical operations because you have done them in your head. The whole process must be written down on paper. If you like you can work the whole thing out in your head in order to exercise your mental powers, but you had better make notes on it before you go to bed and then next day write it neatly step by step, checking it according to the rules.
CHAPTER 2
THE SYSTEM OF TENS
How extraordinarily simple our system of reckoning is. Really it consists of very little more than the arranging of ten symbols. If we add to these a few connecting signs, like the sign for "plus," "minus," "multiplied by" and "divided by" (+-×÷), and finally the "equals" sign (=), we have almost the whole system at our command. There is one more important thing to be mentioned, the system of place values. By this we mean that a symbol has a value which depends on the place in the number in which it finds itself. Its value increases the further to the left it is placed. Take the number 3333; the extreme right-hand symbol's value is simply 3, the next to the left 30, the next 300, and so on. The value of a number increases tenfold as its position is moved one place from right to left. That is why we speak of the system of tens; 10 is the foundation or base of it.
The Romans used a system of numbers without place values. This made reckoning a difficult business. Suppose we had to add the numbers CCCXLIX and MMCXXIV. The position of the "I" in the numbers does not tell us that its value is in the range between ten and ninety. It tells us something about the value of the last sign. "IV" means "take 1 away from 5." This makes matters very complicated and it is not surprising that reckoning with Roman numerals entailed the use of a counting board. For us who use the Arabic numerals it is comparatively simple to write down 349 under 2124 and add the sum quickly in our heads.
If we are to examine more closely this system of tens which we use nowadays we had better be introduced to a new idea which will simplify the examination for us. This is the idea of a "power." When we speak of raising a number to a power we really only mean that we multiply it by itself as often as is indicated by the little number attached to the figure. For example, if we see 54 printed in a book, this merely means that we are expected to multiply 5 × 5 × 5 × 5; 56 means 5 × 5 × 5 × 5 × 5 × 5; 103 means 10 × 10 × 10. We speak of "5 to the power of 4," "5 to the power of 6," "10 to the power of 3." A number raised to the power of 2, e.g., 102, is usually said to be "squared." We do not normally write 51, or "the power of I." We leave a number unadorned in this case. But we can write a number with the power 0, like this: 100, 250, 30. All these have one value, namely 1. Any number raised to the power of 0 equals 1. The reason for this will be explained in a later chapter, it cannot be given here. In the meantime the fact must be accepted.
Anyone looking at these numbers raised to powers will soon see that if 104 means multiply 10 by itself 4 times, the little number 4 (which is called the index number because it indicates the number of times to multiply), represents the number of noughts in the answer. This is only true when the index number is attached to the number 10, because 10 is the base of our number system.
Let us examine in greater detail any number in this system, say 3206. If we were putting this on to an abacus or counting board we should have to split it up into
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
But there must be some method of writing this down on paper so that we can see from it how each number is constructed, without having recourse to a counting board. It is not so very difficult to discover such a method. We can write the number 3206 thus,
6 × 1 + 0 × (10) + 2 × (10 × 10) + 3 × (10 × 10 × 10),
noting as we do so that any number multiplied by 0 equals 0.
We already know that, when a number is multiplied by itself, a short way of writing this is to put a small index number against it to show that it is being raised to a power. Also, a dot is an accepted alternative sign for ×, the multiplication sign. Therefore our example above could be written
6 . 100 + 0.101 + 2 . 102 + 3 . 103.
(Reminder : 100 always equals 1.)
Now we can see on paper the inner structure of a number in our system, that is, in the scale of ten. We can see the values of the places in which the separate figures are positioned in a number, that is to say their "place value." Our example 3206, when written 6 . 100 + 0. 101 + 2 . 102 + 3 . 103, is called a "series" of powers of 10. The 6, 0, 2, 3 are called the "coefficients" of the powers. Do not be put off by these technical terms; we have used this example in which to introduce them because they will be useful in the next chapter.
Before we finish this one, however, and at the risk of boring the reader, we will have another look at our number system. We can see first of all that the system of place values is maintained in the series. The index numbers of 10 follow each other, 100, 101, 102, 103 (103 being a larger quantity than any of the others); the size of the coefficients makes no difference for 9 . 100 is always smaller than 2 . 101 or even than 0. 101; for 9 . 100 represents 9 only, while 2 . 101 equals 20. Suppose we take the number 109 and write it as a further example of a series :–
9 . 100 + 0 . 101 + 1 . 102
or if we like it in the reverse order, as it makes no difference to the sum of the series,
1 . 102 + 0 . 101 + 9 . 100.
We see that even the nought has a place in the series; it shows that the next greatest place value is going to be filled. Theoretically this kind of series could be continued as long as we like. There is no number which is not capable of being expressed in rising powers of 10, each qualified by a coefficient. A perfect number system really requires that each step in the system of place values, each raising of 10 to a further power, should have a name of its own like ten, hundred, thousand. In our system there is not a complete set of names. There are special words to denote only 101, 102, 103 and 106, that is ten, hundred, thousand and a million. Ten thousand and a hundred thousand are simply multiplication expressions. This irregularity in naming the steps probably has its origin in some practical requirements of man in past ages. In very early times it is likely that money and armies only needed numbers up to thousands. It is said that the wealth of Marco Polo first made the concept of "million" necessary. Large numbers like a billion (1012 in England, 109 in America) are often called astronomical numbers, though they are rarely used in exact science.
Before we leave this chapter introducing our number system in the scale of ten, there are probably some questions which have occurred to the reader and which we must try to answer. Why do we use the words "eleven" and "twelve," only to follow them with "thirteen," "fourteen," "fifteen," and so on? What does the peculiar French word for 80, "quatrevingt," mean? There is no doubt that these upset the regular picture we have drawn of our system of tens. "Quatrevingt" (4 × 20) is very similar in construction to "forty" (4 × 10). "Eleven" and "twelve" look suspiciously like a continuation of the numbers one to ten. They are not apparently composite numbers. Why do not we say "oneten," "twoten," "thirteen" (3-ten), "fourteen" (4-ten), instead of eleven, twelve, etc.? Why indeed is ten the base of our system? Is there anything about the number ten which makes it to be preferred before all others? Is this system based on ten something God-given, descended from heaven? Or is the reason for our preference merely the fact that we have ten fingers and that our ancestors used to count on theirs?
There is absolutely no logical reason for preferring a system based on ten to one based on any other number. In the course of history there have been systems based on 60, 50, 20 and 12. In the year 1690 Leibniz, the great mathematician and philosopher, described the most remarkable of all systems, the binary, which uses nothing but the two numbers 0 and 1. In modern times this number system has been found to be convenient for use in electronic computing machines. Finally, the puzzling "quatrevingt" is nothing but a relic of a Celtic system based on 20 (fingers plus toes!) which has slipped into the French
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Precise Calculator has arbitrary precision and can calculate with complex numbers, fractions, vectors and matrices. Has more than 150 mathematical functions and statistical functions and is programmable (if, goto, print, return, for).
| 677.169 | 1 |
Introduces the theory of commutative Banach algebras. This book offers chapters on Gelfand's theory, regularity and spectral synthesis. It emphasises on applications in abstract harmonic analysis and on treating many special classes of commutative Banach algebras, such as uniform algebras, group algebras and Beurling algebras, and tensor products. more...
Offers an introduction to quantum field theory which addresses both mathematicians and physicists ranging from advanced undergraduate students to professional scientists. This book concerns a detailed study of the mathematical and physical aspects of the quantum theory of light. more...
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Contact
Exercises
This section is similar to a collection of exercises with clues and examples that help you understand the concepts. When solving the exercises from the homework, there won't be given any clues.
The exercises are generated with programming language, which is why the same exercise won't be met twice (at least not very often).
When solving them, you have two buttons – Check the exercise which tells you answered wrong or correctly and New exercise.
Homework
Homework contains exercises from the Exercises section. You can solve as many exercises as you want by choosing a number starting from one. If you do not want that the homework to contain certain types of exercises you can fill in with 0 or leave the box empty. In order to star solving the exercises you have to press the button Start solving the homework.
When going through the homework, you have three buttons: Check the exercise – which tells you if you answered wrong or correctly, Next exercise and I don't know .
The homework also has some information concerning its status – how may exercise it has, how many exercises are left until you finish solving the homework and how many times you chose I don't know button.
IQ tests
IQ tests are divided into three levels of difficulty. Each IQ test is unique; you won't be given the same test twice, except maybe for some questions that may appear again (because each question, respectively each test is generated through programming algorithms).
When going through the IQ test you can see the Next button which will take to next question. The information about the test status refer to the correctitude of the answers, how many questions you have answered, the number of correct answers, the number of wrong answers, and a graph expressing the percentage of exercises solved from the total number of exercises.
The test has 20 questions and its purpose is measure your mathematical intelligence. Your IQ is measured depending on the number of questions solved correctly and the time in which the test is finished. The IQ test result has 7 levels of inteigence , the better and faster you give your answer there are more chances of getting a better result.
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Algebra Concepts is a tool for introducing many of the difficult concepts that are necessary for success in higher level math courses. This program includes a special feature, the Algebra Tool Kit, wh... More: lessons, discussions, ratings, reviews,...
Algebra Concepts is an interactive learning system designed to provide instruction in mathematics at the 7th grade enrichment through adult levels. The instructional goals for Algebra Concepts include... More: lessons, discussions, ratings, reviews,...
This application is a basic tool for learning algebra. Users try to solve a system of linear equations with two equations and two unknown variables. It is based on the same problem used in the free ap... More: lessons, discussions, ratings, reviews,...
iCrosss allows you to build a cross-section of each available polyhedra by your defined plane. The plane can be defined by three points (taps) on polyhedron faces. The application supports regular pol...
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student progression through STEM programs of study and into STEMrelated careers.
PERSPECTIVES ON MATHEMATICS CURRICULUM IN THE TWO-YEAR COLLEGE
A useful framework for understanding two-year mathematics curriculum comes from Cullinane and Treisman (2010), who label the mathematics curriculum in the United States the "normative mathematics course sequence" (pp. 7-8), which they claim is ubiquitous to the P-20 (primary through grade 20) education system. The normative mathematics course sequence extends from basic arithmetic, to pre-algebra, algebra, and intermediate algebra on to trigonometry, pre-calculus, calculus, and other calculus-based courses, with a fuzzy demarcation between precollege and college-level mathematics that starts with college algebra. Geometry may be part of the sequential mathematics continuum, or it may be omitted, to the detriment of students' advancement into calculus and calculus-based sciences such as physics. Because this framework represents the dominant schema for which mathematics is taught and for which student competence is assessed at the secondary and postsecondary levels, I use this framework as the basis for discussing the literature. Later, in my discussion of reforms of the two-year college mathematics curriculum, I again cite Cullinane and Treisman (2010) who are studying alternatives to the normative mathematics course sequence. First, however, I provide a brief historical foundation and then move to contemporary developments in two-year college mathematics.
Liberal arts and sciences courses, including mathematics courses, have been part of the two-year college curriculum since creation of junior colleges in the early 1900s. Cohen and Brawer (1982) observed that, by the time two-year colleges arrived on the U.S. higher education scene, the academic disciplines were already "codified" (p. 284) by the rest of the educational system. Junior colleges that emerged to fill the void between high schools and universities adopted the prevailing curriculum structure advocated by the mathematics discipline and were therefore caught in between the K-12 sector and the four-year college sector from the start. To this end, Cohen and Brawer observed that, "the liberal arts [courses of two-year colleges] were captives of the disciplines; the disciplines dictated the structure of the courses; [and] the courses encompassed the collegiate function" (1982, p. 285). To facilitate the acceptance of college credits at the university level, two-year colleges reproduced the curriculum as well as the pedagogical methods used by universities to which their students sought entry
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This Encyclopedia is a single-volume reference work for readers and researchers investigating national and international aspects of mathematics education at the elementary, secondary and post-secondary levels. It consists of more than 400 entries, arranged alphabetically by headings of greatest pertinence to mathematics education. The volume also gives users a starting point for in depth research and fills a gap as there is no other volume that unites major aspects (concepts, pedagogy and history) of mathematics education. (source: Nielsen Book Data)
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is based on the use of graphing calculators by students enrolled in calculus. There is enough material in the book to cover precalculus review, as well as first year single variable calculus topics. Intended for use in workshop-centered calculus courses. Developed as part of the well-known NSF-sponsored project, Workshop Mathematics, the text is intended for use with students in a math laboratory, instead of a traditional lecture course. There are student-oriented activities, experiments and graphing calculator exercises found throughout the text. The authors are well-known teachers and innovative thinkers about ways to improve undergraduate mathematics teaching.
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Overview
The author adopts the martingale theory as his main theme in this introduction to the modern theory of probability, which is, perhaps, at a practical level, one of the most useful mathematical theories ever devised.
More About
This Book
Overview
The author adopts the martingale theory as his main theme in this introduction to the modern theory of probability, which is, perhaps, at a practical level, one of the most useful mathematical theories ever devised.
Editorial Reviews
From the Publisher
"Williams, who writes as though he were reading the reader's mind, does a brilliant job of leaving it all in. And well that he does, since the bridge from basic probability theory to measure theoretic probability can be difficult crossing. Indeed, so lively is the development from scratch of the needed measure theory, that students of real analysis, even those with no special interest in probability, should take note." D.V. Feldman,
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Learning is an individual endeavor. Some ideas come easily; others take time--sometimes lots of time--to grasp. In addition, individual students learn the same idea in different ways and at different rates. The authors of this textbook designed the classroom lessons and homework to give students time--often weeks and months--to practice an idea and to use it in various settings. This section of the textbook offers students a brief review of 33
geometry and algebra topics followed by additional practice with answers. Not all students will need extra practice. Some will need to do a few topics, while others will need to do many of the sections to help develop their understanding of the ideas. This resource of the text may also be useful to prepare for tests, especially final examinations. How these problems are used will be up to your teacher, your parents, and yourself. In classes where a topic needs additional work by most students, your teacher may assign work from one of the extra practice sections that follow. In most cases, though, the authors expect that these resources will be used by individual students who need to do more than the textbook offers to learn an idea. This will mean that you are going to need to do some extra work outside of class. In the case where additional practice is necessary for you individually or for a few students in your class, you should not expect your teacher to spend time in class going over the solutions to the extra practice problems. After reading the examples and trying the problems, if you still are not successful, talk to your teacher about getting a tutor or extra help outside of class time. Warning! Looking is not the same as doing. You will never become good at any sport just by watching it. In the same way, reading through the worked out examples and understanding the steps are not the same as being able to do the problems yourself. An athlete only gets good with practice. The same is true of developing your algebra and geometry skills. How many of the extra practice problems do you need to try? That is really up to you. Remember that your goal is to be able to do problems of the type you are practicing on your own, confidently and accurately. Two other sources for help with the geometry and algebra topics in this course are the
Geometry Connections Parent Guide
and the
Algebra Connections Parent Guide.
These resources are available free from the internet at
. There is also an order form there in the event you wish to purchase them.
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There is a newer edition of this item:
Foundations of Algorithms, Fourth Edition offers a well-balanced presentation of algorithm design, complexity analysis of algorithms, and computational complexity. The volume is accessible to mainstream computer science students who have a background in college algebra and discrete structures. To support their approach, the authors present mathematical concepts using standard English and a simpler notation than is found in most texts. A review of essential mathematical concepts is presented in three appendices. The authors also reinforce the explanations with numerous concrete examples to help students grasp theoretical concepts.
Customer Reviews
Most Helpful Customer Reviews on Amazon.com (beta)
Amazon.com:
9 reviews
5 of 5 people found the following review helpful
AmazingFeb. 7 2012
By
Oliver
- Published on Amazon.com
Format: Hardcover
This book is truly something. Everything is explained in great amount of detail and several examples of the different techniques used for algorithms are really well explained, and the best part is that you can actually understand it. While everything they say is demonstrated, they dont use as much mathematical notation as other books, instead, they take their time to explain it with words and examples (yet they do use just the right amount of math notation to prove everything they say).
The pseudocode is c++like and that helped me understand everything so much better. They explain complexity, divide and conquer, dynamic programming, greedy algotirithms, and so much more in a way thats easy to understand and well documented. They also include several books after every chapter where you can expand what you have learned. They also include a good amount of pictures.
All in all this is the only book which actually helped me understand prim, kruskal, and dijkstra in a way I could actually code it myself. So I am extremely thankful for that. Also, beware that I found one typo in prim which changed everything. :)
2 of 2 people found the following review helpful
Great Book to OwnJuly 3 2012
By
coco
- Published on Amazon.com
Format: Hardcover
This book is to anyone from beginner to advanced CS students because the book is delivered in a sequential order from basics to advanced concepts, ease of comprehension, and very well illustrated examples. A Must have book.
Nice bookApril 15 2014
By
DT Pham
- Published on Amazon.com
Format: Hardcover
Verified Purchase
This is my required textbook. I think it's nice with the examples and explanation. I bought the used book (like new), and it's really new.
SolidMarch 6 2013
By
OD
- Published on Amazon.com
Format: Hardcover
Verified Purchase
Very well written and explained throughout. This book is pretty math-heavy, but they go to great effort to keep it readable, and this is new to me.
GreatJuly 4 2012
By
masato naka
- Published on Amazon.com
Format: Hardcover
Verified Purchase
In short, I like this book. The contents are well-organized. Graphs, tables and examples are effectively used as well. Each section has its specific example, which helps beginners easily grasp its concept.
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Working Knowledge
STEM Essentials for the 21st Century
Hess, Karl
2013, XVII, 313 p. 93ognizes curricular guidelines set forth by the National Research Council
Offers examples for Mathematica throughout the book
Includes guide to online resources with numerous examples
Features over 100 illustrations
Working Knowledge: STEM Essentials for the 21st Century is designed to inspire a wide range of readers from high school and undergraduate students with an interest in Science, Technology, Engineering, and Mathematics (STEM) to STEM teachers and those who wish to become teachers. Written by renowned scientist and teacher Dr. Karl Hess of the University of Illinois at Urbana, a member of both the National Academy of Sciences and the National Academy of Engineering, the book presents a critical collection of timeless STEM concepts and connects them with contemporary research advances in addition to the needs of our daily lives. With an engaging and accessible style not requiring a formal background in STEM, Dr. Hess takes the reader on a journey from Euclidean Geometry and Cartesian Coordinates up through 21st Century scientific topics like the global positioning system, nanotechnology, and super-efficient alternative energy systems.
Working Knowledge: STEM Essentials for the 21st Century at once serves as an almanac on the fascinating physical, chemical, quantitative features of the natural world and built environment, as well as a need-to-know list of topics for students, teachers, and parents interested in STEM education.
Table of contents
Mathematics: the Study of Quantity, Structure, Space and Change.- Science: the Process of Understanding the Natural World and its Possibilities.- Engineering and Technology: Math and Science Meet Creativity and Design.- STEM in Our Daily Lives.- Some More Advanced STEM Problems.
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Find a BowieLet me help you!In the beginning, Algebra can seem like a foreign language. Do you find yourself asking, "What are real numbers, rational numbers, integers?" Or maybe, "What does it mean when a + 7 = 13?" Perhaps you're saying, "My graph looks nothing like the graph in the book!"
I can expl...
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Symmetry in Mechanics
A Gentle, Modern Introduction
Singer, Stephanie Frank
2004, XII, 193p. 30 illus..Symmetry in Mechanics is directed to students at the undergraduate level and beyond, and offers a lovely presentation of the subject . . . The first chapter presents a standard derivation of the equations for two-body planetary motion. Kepler's laws are then obtained and the rule of conservation laws is emphasized. . . . Singer uses this example from classical physics throughout the book as a vehicle for explaining the concepts of differential geometry and for illustrating their use. These ideas and techniques will allow the reader to understand advanced texts and research literature in which considerably more difficult problems are treated and solved by identical or related methods. The book contains 122 student exercises, many of which are solved in an appendix. The solutions, especially, are valuable for showing how a mathematician approaches and solves specific problems. Using this presentation, the book removes some of the language barriers that divide the worlds of mathematics and physics."
—Physics Today
"This is a very interesting book. Those educated in traditional mechanics will acquire [from reading it] knowledge of modern mathematics hidden beyond traditional concepts in the realm of celestial mechanics, [and] . . . pure mathematicians will understand how their discipline enters into practical problems. The author shows how fundamental concepts of symplectic geometry implicitly occur in mechanics . . . the mathematical presentation is ingenious and subtle. There are a lot of exercises for the reader and the solutions of most of them are given in a separate chapter. I can highly recommend this book to undergraduate and PhD students . . . it is ideally suited for teaching a course on the subject."
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Summary
KEY BENEFIT:George Woodburyrs"s Algebra Seriesempowers students for success in college-level math courses through its early-and-often functions and graphing approach, integrated study strategies, and quality exercise sets that encourage true conceptual understanding. One of the ways George prepares his students for future mathematics courses is by introducing the fundamental algebraic concepts of functions and graphing early and then consistently incorporating these concepts throughout the text. Strong exercise sets that provide both volume and variety are critical to student success, so George has written extensive sets that incorporate not only a substantial amount of skill and drill, but also plenty of writing exercises that encourage critical thinking and creativity. The key to success for developmental students is often not just about the math: itrs"s about making sure that students understand how to study and prepare for a math class. To give students the confidence they require to succeed, George fully integrates study strategies throughout the text and also provides further guidance via an additionalStudy Skills Workbook, written specifically for his texts by Alan Bass. KEY TOPICS: Review of Real Numbers, Linear Equations, Graphing Linear Equations, Systems of Equations, Exponents and Polynomials, Factoring and Quadratic Equations, Rational Expressions and Equations, Radical Expressions and Equations, Quadratic Equations. MARKET: For all readers interested in algebra.
Author Biography
George Woodbury earned his bachelor's degree in mathematics from the University of California—Santa Barbara in 1990 and his master's degree in mathematics from California State University—Northridge, in 1994. He currently teaches at College of the Sequoias in Visalia, CA–just outside of Fresno. He has been honored as an instructor by both his students and his colleagues. Aside from teaching and writing, George served as the department chair of the math/engineering division from 1999—2004. George's Elementary and Intermediate Algebra, First Edition, was published in 2006.
Table of Contents
Review of Real Numbers
Integers, Opposites, and Absolute Value
Operations with Integers
Fractions
Operations with Fractions
Decimals and Percents
Representing Data Graphically
Exponents and Order of Operations
Introduction to Algebra
Linear Equations
Introduction to Linear Equations
Solving Linear Equations: A General Strategy
Problem Solving
Applications of Linear Equations
Applications Involving Percentages
Ratio and Proportion
Linear Inequalities
Graphing Linear Equations
The Rectangular Coordinate System
Equations in Two Variables
Graphing Linear Equations and Their Intercepts
Slope of a Line
Linear Functions
Parallel and Perpendicular Lines
Equations of Lines
Linear Inequalities
Systems of Equations
Systems of Linear Equations
Solving Systems by Graphing
Solving Systems of Equations by Using the Substitution Method for
Solving Systems of Equations by Using the Addition Method
Applications of Systems of Equations
Systems of Linear Inequalities
Exponents and Polynomials
Exponents
Negative Exponents
Scientific Notation
Polynomials
Addition and Subtraction of Polynomials
Multiplying Polynomials
Dividing Polynomials
Factoring and Quadratic Equations
An Introduction to Factoring
The Greatest Common Factor
Factoring by Grouping
Factoring Trinomials of the Formx 2 +bx+c
Factoring Trinomials of the Formax 2 +bx+c, wherea≠1
Factoring Special Binomials
Factoring Polynomials: A General Strategy
Solving Quadratic Equations by Factoring
Quadratic Functions
Applications of Quadratic Equations and Quadratic Functions
Rational Expressions and Equations
Rational Expressions and Functions
Multiplication and Division of Rational Expressions
Addition and Subtraction of Rational Expressions That Have the Same Denominator
Addition and Subtraction of Rational Expressions That Have Different Denominators
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A common problem when students learn about the slope-intercept equation y = mx + b is that they mechanically substitute for m...
see more
A common problem when students learn about the slope-intercept equation y = mx + b is that they mechanically substitute for m and b without understanding their meaning. This lesson is intended to provide students with a method for understanding that m is a rate of change and b is the value when x = 0. This kinesthetic activity allows students to form a physical interpretation of slope and y-intercept by running across a football field. Students will be able to verbalize the meaning of the equation to reinforce understanding and discover that slope (or rate of movement) is the same for all sets of points given a set of data with a linear relationship.
In this lesson, students use remote-controlled cars to create a system of equations. The solution of the system corresponds...
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In this lesson, students use remote-controlled cars to create a system of equations. The solution of the system corresponds to the cars crashing. Multiple representations are woven together throughout the lesson, using graphs, scatter plots, equations, tables, and technological tools. Students calculate the time and place of the crash mathematically, and then test the results by crashing the cars into each other.
Solving linear equations is a cornerstone of Algebra and other higher level math classes. The skills involved are critically...
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Solving linear equations is a cornerstone of Algebra and other higher level math classes. The skills involved are critically important to the students' confidence and success within high school mathematics. In this project, students develop a board game that help their peers review solving linear equations. This project requires internet access as the students will use various websites to collect sample problems of varying degrees of difficulty. The entire project should use about 230 minutes of instruction time. Adjust as needed.
This resource has multiple concepts for geometry and trigonometry. The concepts are divided among chapters with links on...
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This resource has multiple concepts for geometry and trigonometry. The concepts are divided among chapters with links on common unknown concepts to help students understand the text. This resource also provides exercises that can be done by the students (answers are provided).
This lesson offers a pair of puzzles to enforce the skills of identifying equivalent trigonometric expressons. Addtional...
see more
This lesson offers a pair of puzzles to enforce the skills of identifying equivalent trigonometric expressons. Addtional worksheets enhance students' abilities to appreciate and use trigonometry as a tool in problem solving. This lesson is adapted from an article by Mally Moody, which appeared in the March 1992 edition of Mathematics Teacher.
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What sets this volume apart from other mathematics texts is its
emphasis on mathematical tools commonly used by scientists and
engineers to solve real-world problems. Using a unique approach, it
covers intermediate and advanced material in a manner appropriate
for undergraduate students. Based on author Bruce Kusse's course at
the Department of Applied and Engineering Physics at Cornell
University, Mathematical Physics begins with essentials such as
vector and tensor algebra, curvilinear coordinate systems, complex
variables, Fourier series, Fourier and Laplace transforms,
differential and integral equations, and solutions to Laplace's
equations. The book moves on to explain complex topics that often
fall through the cracks in undergraduate programs, including the
Dirac delta-function, multivalued complex functions using branch
cuts, branch points and Riemann sheets, contravariant and covariant
tensors, and an introduction to group theory. This expanded second
edition contains a new appendix on the calculus of variation -- a
valuable addition to the already superb collection of topics on
offer.
This is an ideal text for upper-level undergraduates in physics,
applied physics, physical chemistry, biophysics, and all areas of
engineering. It allows physics professors to prepare students for a
wide range of employment in science and engineering and makes an
excellent reference for scientists and engineers in industry.
Worked out examples appear throughout the book and exercises follow
every chapter. Solutions to the odd-numbered exercises are
available for lecturers at
Bruce Kusse is Professor of Applied and Engineering Physics at
Cornell University, where he has been teaching since 1970. He holds
a PhD from the MIT in electrical engineering with a specialty in
plasma physics.
Erik Westwig is a software engineer with Palisade Corporation, New
Jersey. He holds an MS in applied physics from Cornell University.
"Any lecturer on mathematical methods is also looking for worked
examples and numerous exercises. This book passes these tests
admirably. [...] In summary, a welcome addition to the good books
in this area."
Australian PHYSICS
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Discovering Geometry FAQ
The Discovering Geometry textbook is currently published by Kendall Hunt Publishing. In August 2011, Key Curriculum Press sold their high school mathematics textbooks to Kendall Hunt Publishing. One year later, Key Curriculum was purchased by McGraw Hill.
Who publishes the Discovering Geometry® Video Series, Patty Paper® Geometry, and other books by Michael Serra?
In August 2012, all rights for Michael Serra's books reverted back to him except Discovering Geometry. They are published by Playing It Smart, which is owned by Angie and Michael Serra and available on this website.
How well does DG align to the Common Core StateStandards for Geometry?
Discovering Geometry is an inquiry based learning program and developed before the CCSS were mandated yet relevant today at every level. When DG was first introduced, traditional textbooks taught about proof without investigations. DG is unique because geometric proofs are not introduced until students are ready. Students experience geometry through a series of engaging hands on activities and develop their own conclusions to support proofs. Students work in small cooperative groups on projects using take out cartons, patty paper, and string eventually using compass and straightedge and software. They must defend their positions and are presented with investigations which may or may not be true. When reviewing the eight standards, compare them to our FAQ below. More information can be found here
In Discovering Geometry we begin with students discovering their vocabulary. Students perform investigations that lead not only to the definitions of the basic terms but do so with depth of understanding. For example, students observe examples of figures that are labeled trapezoids and selectively chosen figures that are not trapezoids. Then students discuss with their group members what makes for a trapezoid being a trapezoid. The group creates their definition and then tests and fine-tunes their definition against the examples and non-examples until confident of their definition. The groups then present their definitions to the rest of the class and the class reaches consensus. Each definition along with its illustration is then added to the student's vocabulary list in their notebook.
Almost every lesson involves students working in groups performing geometric investigations with manipulatives. The investigations use a wide range of readily available manipulatives including compasses, straightedges, rulers, protractors, scissors, heavy stock paper, graph paper, patty paper, square dot paper, isometric dot paper, glue stick, tape, string, wooden or plastic cubes, mirrors, and pattern blocks, to name the most common. View this photo
We recommend commercially available geometric solids for discovering volume formulas and surface area formulas for prisms, pyramids, cylinders, cones, and spheres. Students typically use graphing calculators and geometry software such as The Geometer's Sketchpad for investigations and extra explorations. Projects and Explorations in the text and supplementary materials also encourage the use of hands-on materials. Students build arches out of Chinese take-out cartons, build kaleidoscopes out of paper towel tubes, and build clinometers and hypsometers out of cardboard and straws.
How does DG move students from concrete models to abstract ideas?
Moving students from the concrete to the abstract is at the heart of Discovering Geometry. We begin with students discovering all the basic vocabulary and then all the basic properties of geometry through hands-on investigations. Students work with the tools of geometry, build models, perform geometric investigations, then make and test conjectures about their experiences. As they become more and more experienced with the terms and concepts of geometry, gaining confidence, we gradually begin to ask them to explain what they did and why they think what they discovered is true. See comments on van Hiele levels and development of proof here. View this photo
How does DG teach construction with a straight edge and compass?
Right from the start (Chapter 0 Geometric Art) students use the ruler and compass to create geometric art. Then on an almost daily basis in Chapters One and Two students use the ruler, protractor, and compass to perform investigations to discover geometric properties (vertical angles and the properties of parallel lines). In Chapter Three we introduce the classic Euclidean compass and straightedge constructions. This is done inductively with students discovering the construction methods rather than the constructions being a set of rules or steps to memorize. The Euclidean constructions are blended with paper folding constructions (patty paper geometry) to reinforce the properties being discovered. We also recommend that teachers use The Geometer's Sketchpad at this time as a third way to reinforce the concepts they are learning. The use of the compass and straightedge as tools for construction exercises as well as tools for discovery continue throughout the remainder of the text. View this photo
What is DG's philosophy on teaching students to write proofs?
Discovering Geometry's primary focus is getting students to work together to discover and discuss geometry concepts in a student-centered environment. In Discovering Geometry students are guided to discover on their own not only the basic vocabulary of geometry but even the construction techniques they are going to use to discover their own geometric properties.
At the same time students are inductively discovering geometric properties we are carefully developing a full breadth of reasoning skills, both inductive and deductive. There is a heavy emphasis in Discovering Geometry on using higher order thinking as opposed to memorization and drill. In this investigative approach to geometry, reasoning, concepts, and skills are developed with two basic principles in the background – the research of the van Hieles (van Hiele Model) and the work of Michael de Villiers and others on purposes of proof.
At the beginning of the course we assume a low level of geometric reasoning in students. (van Hiele Research suggests that over 70% of all high school students enter geometry at level 0 or 1 on the van Hiele scale.) Through this concrete hands-on discovery approach we attempt to move them up the stages to higher levels of reasoning over the course of the term.
At the same time we recognize that one of the traditional purposes of a proof in geometry, to convince someone that a particular property is true, is not the best use of proofing at the early stages. With most beginning geometry students, conviction comes upon seeing and doing not through proof. The proof does not convince. In fact it often creates more doubt. (See Rethinking Proof by Michael de Villiers.) What we see as a better purpose for proof, in the early stages, is proof as a means of explaining why something they discovered through hands-on investigations is indeed true. Proof as a means of explanation is our initial and primary focus. First students are asked to explain their reasoning to their group members orally then in short paragraphs. In addition to the more typical exercises where students apply the concepts discovered, we also utilize three unique types of exercises that ask students to explain their reasoning.
One type of exercise we call Angle Chase (In DG3 see page 188 exercise #14, page 224 exercise #20, and page 292 Exercise #25, in DG4 see page 190 exercise #14, page 226 exercise #20, and page 297 Exercise #25). In these exercises, which we recommend that students work on in pairs in class, students take turns finding the measure of an angle and then explaining why that is the measure.
A second novel type of exercise that provides an opportunity for students to explain their reasoning (develop proof skills) is the "What's Wrong With This Picture?" exercises (In DG3 see page 251 exercise #26, page 323 exercise #16, and page 436 exercise #21, In DG4 see page 253 exercise #26, page 328 exercise #16, and page 452 exercise #21). In these exercises students are asked to explain why the figure is incorrect. View this photo
In the third type of exercise that helps students develop their deductive reasoning we simply ask students to explain why something is true (In DG3 see exercises page 436 exercise #20, page 466 exercise #18, page 472 exercise #22, (In DG4 see exercises page 452 exercise #20, page 482 exercise #18, page 488 exercise #22). In the early instances of these exercises we ask them to explain why something is true for a particular case (numbers are provided as in exercise #20 page 436 in DG3) later we begin to ask students to explain their reasoning in more general cases (letters rather than numbers are provided as in DG3 exercise #22 page 472, DG4 exercise #22 page 488).
Proof will also be developed as a means of communication and discovery as student's progress through Discovering Geometry. Eventually as students move to more sophisticated reasoning abilities we introduce geometry as a mathematical system in the final chapter. Systematization then becomes the purpose of proof in Chapter 13.
In Discovering Geometry we develop reasoning skills in four general ways:
1.First, when students perform investigations and make conjectures in the form of formulas (e.g., a formula for the number of diagonals of an n-gon or a formula for the sum of the measures of the interior angles of an n-gon) students must use inductive reasoning while doing geometry.This occurs throughout Discovering Geometry.Chapter Two: ReasoningIn Geometry introduces this type of reasoning skill but this type of reasoning continues throughout the text.It is the type of reasoning mathematicians and scientists primarily do. It is the type of reasoning essential to an investigative discovery approach to learning. It is a critical life skill.
2.The second general way in which reasoning is developed in Discovering Geometry is in the development of logical arguments and proof, or deductive reasoning. This development is based on classroom experiences and research, workshops on reasoning and proof, and particularly the research of the van Hieles, and de Villiers.
•Deductive reasoning is also introduced in Chapter Two: ReasoningIn Geometry and itbegins informally with exercises that ask students to "explain why." This occurs in numerous examples, exercises, and special activities mentioned above and in sections called "Take Another Look."
•Examples of algebraic, paragraph, and flow chart proofs are modeled early. The concepts of conditional statements, converses and counterexamples are emphasized and reinforced. (It is recommended however that students in a regular geometry course not be graded on their proofing abilities at this early stage of geometric reasoning.)
•By Chapter Four, Discovering and Proving Triangle Properties, students are asked to follow the reasoning in paragraph proofs and flow chart proofs. They are eventually asked to create their own flow chart proofs, testing the level of their reasoning.
As students move through the traditional synthetic geometry they move to higher levels of geometric reasoning and critical thinking.Their "explaining why" becomes more fully developed. They are expected to present more detailed paragraph and flow chart proofs. At the same time students are developing a firmer grasp of the content of geometry.
•To mark a transition from discovering geometric properties through inductive reasoning and informally explaining why they think their conjectures are true, more formal deductive reasoning is introduced through a series of three Logic Explorations. These explorations begin with the logical reasoning found in Sherlock Holmes stories.This is followed with four properties of symbolic logic: Modus Ponens, Modus Tollens, Law of Syllogism, and Law of the Contrapositive.Students then create direct, conditional and indirect proofs using these logic properties.
•Finally students develop geometry as a mathematical system (de Villiers Systematization).After establishing definitions, properties of algebra and properties of congruence, and the postulates of geometry, students begin to use these premises to establish theorems and to see the connections between groups of theorems.
3.The third general way in which higher order thinking is developed in Discovering Geometry is in solving word problems.After students have completed their investigations and made conjectures, they are asked to apply their new properties to not only real applications but also in novel situations.Many problems in the exercise sets come from recreational mathematics or are modeled after SAT type questions and mathematics competitions.
4.Finally, a fourth way in which reasoning is fine-tuned in Discovering Geometry is in the puzzles called Improving Algebra Skills, Improving Reasoning Skills, and Improving Visual Thinking Skills.There are 14 Improving Algebra Skills, 22 Improving Reasoning Skills, and 44 Improving Visual Thinking Skills puzzles that are spread throughout the text.They occur after each lesson in every chapter.These puzzles employ a wide variety of higher order thinking skill and problem solving skills.Students are often found working these puzzles without them even being assigned! View this photo
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MATLAB is a powerful programme, which naturally lends itself to the rapid implementation of most numerical algorithms. This text, which uses MATLAB, gives a detailed overview of structured programming and numerical methods for the undergraduate student.
The book covers numerical methods for solving a wide range of problems, from integration to the numerical solution of differential equations or the stimulation of random processes. Examples of programmes that solve problems directly, as well as those that use MATLAB's high-level commands are given.
Each chapter includes extensive examples and tasks, at varying levels of complexity. For practice, the early chapters include programmes that require debugging by the reader, while full solutions are given for all the tasks. The book also includes:
a glossary of MATLAB commands
appendices of mathematical techniques used in numerical methods.
Designed as a text for a first course in programming and algorithm design, as well as in numerical methods courses, the book will be of benefit to a wide range of students from mathematics and engineering, to commerce.
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Course
3 Unit 3 - Symbol Sense and Algebraic Reasoning 1st Edition
In Courses 3 and
4, the mathematical strands in the Contemporary Mathematics in Context
program become increasingly blended within units. For example, in Course
3, the content in Unit 2, Modeling Public Opinion, is primarily
from the discrete mathematics and statistics and probability strands.
Another example is Course 3, Unit 7, Discrete Models of Change.
This unit contains content from both the discrete mathematics and the
algebra and functions strands. (See the descriptions of Course
3 Units.)
In Courses 1 and
2 of the Contemporary Mathematics in Context program, students
have developed a robust understanding of linear, exponential, power, inverse
power, and quadratic models. They have also studied Course 3, Unit 1,
Multiple-Variable Models. This unit develops student ability to
construct and reason with linked quantitative variables and relations
involving several variables and constraints.
Unit Overview
Symbol Sense and
Algebraic Reasoning develops student ability to represent and draw
inferences about algebraic relations and functions using symbolic expressions
and manipulations.
Unit Objectives
To develop a more formal understanding of functions and function
notation
To reason about algebraic expressions by applying the basic
algebraic properties of commutativity, associativity, identity,
inverse, and distributivity
To develop greater facility with algebraic operations with
polynomials, including adding, subtracting, multiplying, factoring,
and solving
To solve linear and quadratic equations and inequalities by
reasoning with their symbolic forms
To prove important mathematical patterns by writing algebraic
expressions, equations, and inequalities in equivalent forms
and applying algebraic reasoning
Sample Overview
Lesson 4 of this
unit develops student ability to solve linear equations by applying
the
field properties and to solve quadratic equations and inequalities by
factoring, by using the quadratic formula, and by using technology.
Students
also learn how to identify whether a quadratic function has 0, 1, or
2 zeroes and how to determine the axis of symmetry and the distance
between the zeroes for a quadratic function.
Instructional
Design
Throughout the curriculum,
interesting problem contexts serve as the foundation for instruction.
As lessons unfold around these problem situations, classroom instruction
tends to follow a common pattern as elaborated under Instructional
Design.
View Sample
Material
Contact
Adobe with any technical questions
about their software or its installation.
How the Algebra
and Functions Strand Continues
In Course 3, students
will study one additional algebra and functions unit, Families of Functions.
This unit reviews and extends student understanding of the basic function
families and develops student ability to adjust function rules to match
patterns in tables, graphs, and problem conditions
Four units in Course
4 extend student understanding of algebra and function concepts in preparation
for post-secondary education. Students develop understanding of the fundamental
concepts underlying calculus, of inverse functions, and of logarithmic
functions and their use in modeling and analyzing problem situations.
Students also extend their ability to use polynomial and rational functions
to solve problems and to manipulate symbolic representations of exponential,
logarithmic, and trigonometric functions.
A unit that develops
understanding and skill in the use of standard spreadsheet operations
while reviewing and extending many of the basic algebra topics from Courses
1-3 is included for students intending to pursue programs in social, management,
and some of the health sciences or humanities.
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Math Department, Mission College, Santa Clara, California
Mr. Hobbs' Math Corner
Welcome to the Home Page for Mission College students who are taking a mathematics course from Rick Hobbs.
Whether you think you can or think you can't -- you are right.
~ Henry Ford
On this page you will find links to your course assignments, course materials, course information and a wide range of web sites related to the wonderful world of mathematics.
Just click on the course that you are currently taking or the other topics below.
Courses:
Other useful links:
Math 902: Prealgebra
Math 903: Elementary Algebra
Math Online Resources
__________________________________
Extra Credit Options
Why Study Math?
AMATYC Math Contest
Mathematical Habits of Mind
Avoid Errors on Tests
Mission College Orientation material
Where To Buy Textbooks???
Print planners, calendars, graph paper and more
Math Study Skills!
College safety video: Watch it!
When Will I Use Math?
Do you want to diagnose your current math abilities? Do you want to review some math concepts from previous courses? Then check out the ALEKS website. Here you will find a great online diagnostic and tutorial tool. ALEKS provides a great way to find any gaps in your knowledge, and provides a personalized plan to help you improve your skills.
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The Faculty of Mathematics
The Mathematics Faculty provides further information about the mathematics courses, including a description of the first-year options. It also provides information specifically about applications to read mathematics.
The Guide to Admissions outlines the procedures and requirements of individual Colleges, and contains information about the examination called STEP (Sixth Term Examination Papers). Most admissions offers for mathematics are conditional on both A-level and STEP results.
There is a page listing STEP resources including the new NRICH STEP prep website.
There is also information especially for Scottish candidates and a reading list containing a selection of books which should appeal to any students interested in mathematics, not just to those planning to come to Cambridge.
In addition there is also a workbook, which is a booklet of fairly straightforward questions intended for students to work through shortly before they arrive in Cambridge to check that there are no gaps in their mathematical knowledge.
The Colleges
Specific information concerning mathematics at an individual College is best obtained by contacting the College directly (there is a central list of college contact details). Most Colleges provide specific information about mathematics:
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General Relativity Without quite original approach to Relativity, in that it tries to convey nontrivial, quantitative ideas about geometry and general relativity using elementary mathematics only
Offers a short, but mathematically correct introduction to the general theory of relativity for first-year students of astronomy, mathematics or physics Paves the way for more advanced texts Over 75 exercises with full solutions help the reader understand the matter
"General Relativity Without Calculus" offers a compact but mathematically correct introduction to the general theory of relativity, assuming only a basic knowledge of high school mathematics and physics. Targeted at first year undergraduates (and advanced high school students) who wish to learn Einstein's theory beyond popular science accounts, it covers the basics of special relativity, Minkowski space-time, non-Euclidean geometry, Newtonian gravity, the Schwarzschild solution, black holes and cosmology. The quick-paced style is balanced by over 75 exercises (including full solutions), allowing readers to test and consolidate their understanding.
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Should College Classes Ditch the Calculator?
Should College Classes Ditch the Calculator?
According to Samuel King, postdoctoral student in the University of Pittsburgh's Learning Research and Development Center, using calculators in college math classes may be doing more harm than good. In a limited study conducted with undergraduate engineering students and published in the British Journal of Educational Technology, King has determined that our use of calculators may be serving as an alternative to an actual, deep understanding of mathematical material.
"We really can't assume that calculators are helping students," says King. "The goal is to understand the core concepts during the lecture. What we found is that use of calculators isn't necessarily helping in that regard."
King, along with co-author and director of the Mathematics Education Centre at Loughborough University, Carol Robinson, conducted the study by interviewing 10 second-year undergraduate students who were enrolled in a competitive engineering program. The students were given a number of mathematical questions dealing with sine waves, which are mathematical curves that describe a smooth repetitive oscillation. To help solve the problems, the students were given the option of using a calculator instead of completing the work entirely by hand. Over half of the students questioned opted to utilize their calculators in order to solve the problems and plot the sine waves.
"Instead of being able to accurately represent or visualize a sine wave, these students adopted a trial-and-error method by entering values into a calculator to determine which of the four answers provided was correct," says King. "It was apparent that the students who adopted this approach had limited understanding of the concept, as none of them attempted to sketch the sine wave after they worked out one or two values."
After completing the work, King and Robinson interviewed the students about how they approached the material. One student who used the calculator stated that she had trouble remembering the rules for how sine waves operate, and found it generally easier to use a calculator instead. In contrast, however, a student who opted to complete the work without a calculator stated that they couldn't see why anyone would have trouble completing the question, but did admit that it would likely be easier with a calculator.
"The limited evidence we collected about the largely procedural use of calculators as a substitute for the mathematical thinking presented indicates that there might be a need to rethink how and when calculators may be used in classes—especially at the undergraduate level," says King. "Are these tools really helping to prepare students or are the students using the tools as a way to bypass information that is difficult to understand? Our evidence suggests the latter, and we encourage more research be done in this area."
Given the small sample size used in the study, it is entirely possible that King's findings are largely anecdotal in how our usage of calculators and understanding of mathematical concepts may positively or negatively correlate. However, King does stress that while all the evidence may not be in, his study does raise important questions regarding how, when and why students choose to use calculators, and in doing so, we may develop a more holistic approach to math instruction
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$ 12.99
Word problems are the most difficult part of any math course -- and the most important to both the SATs and other standardized tests. This book teaches proven methods for analyzing and solving any type of math...
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Concise and authoritative, this monograph is geared toward advanced undergraduate and graduate students. The main theorems whose proofs are given here were first formulated by Lefschetz and have since turned...
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This book presents methods for the computational solution of some important problems of linear algebra: linear systems, linear least squares problems, eigenvalue problems, and linear programming problems. The...
$ 47.99This monograph by a distinguished mathematician constitutes the first systematic summary of research concerning partially ordered groups, semigroups, rings, and fields. The high-level, self-contained treatment...
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A thorough first course in linear algebra, this two-part treatment begins with the basic theory of vector spaces and linear maps, including dimension, determinants, eigenvalues, and eigenvectors. The second...
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Designed to acquaint students of particle physics already familiar with SU(2) and SU(3) with techniques applicable to all simple Lie algebras, this text is especially suited to the study of grand unification...
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The design of experiments holds a central place in statistics. The aim of this book is to present in a readily accessible form certain theoretical results of this vast field. This is intended as a textbook for...
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Algebra adds value to mathematical biology education
Jul 30, 2009
As mathematics continues to become an increasingly important component in undergraduate biology programs, a more comprehensive understanding of the use of algebraic models is needed by the next generation of biologists to facilitate new advances in the life sciences, according to researchers at Sweet Briar College and the Virginia Bioinformatics Institute (VBI) at Virginia Tech.
In the paper, "Mathematical Biology Education: Beyond Calculus," which is featured in the July 31, 2009 issue of Science, VBI Professor Reinhard Laubenbacher and Sweet Briar College Mathematical Sciences Professor Raina Robeva highlight algebraic models as one of the diverse mathematical tools needed in the professional development of up-and-coming life scientists. Despite this critical need, the authors explain, algebraic models have played a less substantial role in undergraduate curricula than other methods.
Future generations of biologists will routinely use mathematical and computational approaches to develop and frame hypotheses, design experiments, and analyze results. Sound mathematical models are essential for this purpose and are currently used in the field of systems biology to understand complex biological networks. Two types of mathematical models, in particular, have been successfully used in biology to reproduce network structure and dynamics: Continuous-time models derived from differential equations (DE models) focus on the kinetics of biochemical reactions, while discrete-time algebraic models built from functions of finite-state variables focus on the logic of the connections of network variables. According to Laubenbacher and Robeva, while DE models have been included more often in undergraduate curricula integrating mathematics and biology, algebraic models should also be viewed as an important training component for students at all education levels.
"Discrete-time algebraic models created from finite-state variables, such as Boolean networks, are increasingly being used to model a variety of biochemical networks, including metabolic, gene regulatory, and signal transduction networks," says Laubenbacher. "Often, researchers do not have enough of the information required to build detailed quantitative models. Algebraic models need less information about the system to be modeled, making them useful for instances where quantitative information may be missing. All the work that goes into building them can then be used to construct detailed kinetic models, when additional information becomes available. In addition, algebraic models are much more intuitive than differential equations models, which makes them more easily accessible to life scientists."
Using algebraic models is a relatively quick, easy and reliable way for students to integrate mathematical modeling into their life sciences coursework. Creating algebraic models of biochemical networks requires only a modest mathematical background, which is usually provided in a college algebra course. Without the complexities involved in teaching students how to construct more complicated models, algebraic models make the introduction of mathematical modeling into life sciences courses more accessible for faculty members as well.
According to Robeva, "The exciting thing about algebraic models from an educational perspective is that they highlight aspects of modern-day biology and can easily fit in both the biology and mathematics curricula. At the introductory level, they provide a quick path for introducing biology students to constructing and using mathematical models in the context of contemporary problems such as gene regulation. At the more advanced level, the general study and analysis of such models often require sophisticated mathematical theories. This makes them perfect for inclusion into mathematics courses, where the biology can provide a meaningful framework for many of the abstract structures. As educators, we should actively be looking for the best ways to seize this opportunity for advancing mathematical biology."
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MATLAB Demystified [NOOK Book] ...
More About
This Book functions, solve algebraic equations, and compute integrals. You'll also learn how to solve differential equations, generate numerical solutions of ODEs, and work with special functions. Packed with hundreds of sample equations and explained solutions, and featuring end-of-chapter quizzes and a final exam, this book will teach you MATLAB essentials in no time at all.
This self-teaching guide offers:
The quickest way to get up and running on MATLAB
Hundreds of worked examples with solutions
Coverage of MATLAB 7
A quiz at the end of each chapter to reinforce learning and pinpoint weaknesses
A final exam at the end of the book
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Simple enough for a beginner, but challenging enough for an advanced user, MATLAB Demystified is your shortcut to computational precision
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At my school you couldn't get a BSEE without passing quite a few courses that required programming. To save money (computer time was charged at $5 per CPU second) the senior-level classes programmed in a low-level but high-speed language "BCPL" by Martin Richards -- the programming language that gave birth to "C" by Dennis Ritchie and Ken Thompson. We ran these programs on an Amdahl mainframe; remember them? They were IBM-370 compatible but faster and cheaper. Sophomore level classes programmed in APL; junior level classes programmed in PL/I. Remember them? We punched cards on IBM model 029s; seniors got to use timesharing terminals called DECwriters (tractor feed paper + dot matrix printheads). Sysops used portable Texas Instruments "Silent 700 Terminals" that printed on thermal paper.
Our Logic Design labs used 7400-series TTL chips. Our microcontroller design labs used 6802 microprocessors. Our analog design labs used 2N4123+4125 BJTs because TI gave us a bunch of those for free.
The crusty old bastards running that MIT course, wrote one of the world's leading computer algebra systems in the 1970s. However, they were so early that they never considered that useful software could get sold for money, and so they let theirs die out when the old Digital Equipment Corp PDP-10s went away (and when DARPA money for computer algebra dried up).
So it is COMPLETELY THE MIT WAY, to solve this problem using computer algebra. You just, um, mention it obliquely. I suggest a line of patter sort of like this:
I solved this problem using pencil and paper, then found this really cool program "MACSYMA free edition online" that let me check my solution. Sure enough I got it right; I even choose the correct root of the quadratic equation and discarded the incorrect root. Whoever this guy "Joel Moses" is, his software worked very well, at least for me!
Joel Moses, of course, was the MIT guy who got the lion's share of the credit for MACSYMA, even though hundreds of people did the actual work.
You can obtain the answer to your Why question, a couple of different ways. First, you can infer it from the title of that diyaudio discussion thread. Or you can read the accompanying .pdf file (attached to post 1), the answer is explicitly stated on page 4, paragraph 2.
So you're going to need to swing the input of that emitter follower output stage by ±32 volts. Your classmate /u/MrZeratul needs to swing the input of his emitter follower output stage by ±46 volts.
Fortunately there are a couple of real-world opamps that can do this. Whether these opamps are available as OrCad models, I have no idea. Analog Devices' ADA4700 (datasheet#1) will do it, and so will Texas Instruments' OPA454 (datasheet#2).
Another way to approach the problem, suggested by /u/fatangaboo, is to abandon the emitter follower output stage and instead implement a different output stage whose voltage gain is in the neighborhood of 5X to 10X.
A third way is to build a discrete component opamp that operates on ±50 volt supplies. Configure it as a noninverting amplifier with a gain of +5.0X. The input to this high voltage discrete opamp comes from ordinary, garden variety, IC opamps that operate on the more usual ±15 volt supplies. The discrete opamp doesn't have to be fancy; just a pair of PNPs as a differential pair with resistive collector loads, an NPN common emitter second stage, and a class B complementary emitter follower output stage. Negative feedback reduces distortion and sets the gain.
Buy a Canada-legal stun gun for CAD 6.00. Disassemble it, remove the oscillator PCB, the high voltage step up transformer, and the electrodes. Hot glue these to the interior of your tuber bazooka combustion chamber. Voila, 15 megavolt spark from a 9V battery, engineered by professionals, for 6 looney. And you don't have to design or solder together anything.
Protip: use the Mexican vegetable "jicama" instead of potato. It's like an enormous oversized radish, 15 cm in diameter, with twice the density ( kg/m3 ) of potato, AND it's got a lot more internal moisture. Thus it's self-lubricating as it slides down the barrel, and it's impact is greater because the projectile's mass is doubled. Wictionary entry for jicama
Consider the current mirror M2-M1. Draw a small signal equivalent circuit of this 2-MOSFET circuit. Each transistor is represented by a dependent current source in parallel with a resistor. The resistors model the finite output conductance (nonflat IV curves in the saturation region). The SPICE parameter "LAMBDA" encapsulates this phenomenon for the LEVEL=1 model equations.
/u/fatangaboo is suggesting that you pretend each of these resistors' value happens to be an easy-to-compute-with value: R is such that (1/sqrt(2)) of the drain-to-source current flows in the dependent source and the rest flows in the resistor.
Voila, (1/sqrt(2)) for the input transistor M2, and (1/sqrt(2)) for the output transistor M1. We rejoice that (1/sqrt(2))2 is 0.5. Half of the small signal input current, appears as small signal output current.
On the other hand, you may be designing with ideal (make-believe) MOSFETs whose IV curves are perfectly flat in the saturation region. Plot IDS versus VDS at 6 different values of VGS and check the flatness. If they're flat then you can ignore output conductance for this assignment. Not in the real world, of course.
It will become crystal clear when you plot the transfer function: Vout on the Y axis, Vin on the X axis. You will see that Wikipedia's article entitled Schmitt trigger is correct; the transfer function includes hysteresis. That's why YOUR time domain plot shows that Vin= 0 volts produces two different Vout voltages -- highlighted in yellow (HERE).
You've asked SPICE "what is Vout when Vin=0?". The real circuit in real life gives two different output voltages; the problem has two solutions. Why get upset when SPICE makes one arbitrary choice and some textbook author makes the opposite arbitrary choice?
Then you will need to keep the 5.00 ampere current source alive and operational at all times, so its power dissipation remains constant & unchanging. Then you will current-switch it: either into the load, or into a dumper. Just exactly like good old bipolar ECL logic used in good old supercomputers like the Crays. Their power dissipation was constant, in fact it was so constant that they didn't even need or use voltage regulators (!!!).
The opamp at left, guarantees that the current flowing in M1 is equal to (Vref1 / R1). By judicious choice and adjustment of Vref1 and/or R1, you can set this to exactly 5.0 amperes. This current is "steered" either into the LED load, or else the dummy load, by M2 and M3.
Choose Vref_gg such that the common source node where M2 and M3 join, is about the same voltage when the digital input is "1" as when it is "0". This ensures the power dissipation in M1 is constant.
R2+C2 are a filter network that prevents M3's gate voltage from bouncing around when the current is steered left and right. Set C2 > {100 x (Qg_tot / Vsupply)} for excellent filtering.
What you call "convention", what /u/naval_person calls "standard practice", is the evolutionary result of tens of thousands of designers working across a >40 year span of time, attempting to make tens of thousands of amplifiers, and then observing which design topology has won the competition. You know, Survival Of The Fittest.
It appears you have tried one topology alternative (NPN diffamp stage -> NPN CE stage) and have been disappointed in its performance. If you try another alternative (NPN diffamp -> PNP CE) and are pleased by its performance, your reason for choosing candidate B is clear: It Works Better. You have not mindlessly copycatted "convention/standard practice"; rather, you have experimentally confirmed its superiority for your application, with your array of semiconductor device choices.
You need a voltmeter with high input impedance (> 1megohm); when you have one, you can measure that solderless breadboard and find the exact source of your difficulty. I think I see it now, but (a) it's your project not mine, and (b) I'm not 100% sure, just 85% confident.
You can use a test jig like this to measure the input impedance of your meter. Measure the voltages at test point 1, then 2, then 3, ... until the indicated voltage on your meter falls. Now use KVL to solve for Rin. Is it > 1Meg?
Once you've got the hiZ voltmeter, measure the voltage at every node and make a table. (The reason why you're confused, is that you haven't measured the right node yet).
Presumably you connected your Joule Thief to a variable DC power supply and wrote some notes in your lab notebook, such as:
Or just use four plain ordinary rectifier diodes. If the solenoid current is <= 1 amp, use 1 amp diodes (1N4002). If the solenoid current is <= 5 amps, use 5 amp diodes (6A4). If the solenoid current is <= 10 amps, use 10 amp diodes (MBR1045). You get the idea.
Have you checked to see whether you've got the correct number of (-1)'s in there? What with it being an inverting topology and all, plus the happy fact that s2 = (-1)*w2 , it's pretty easy to get the sign wrong.
Design with high fT NPN transistors, these will let you maximize the gain bandwidth product of your amplifying stages.
Keep the per-stage gain low, something like 3X per stage. Since gain and bandwidth are inversely proportional, low gain means high bandwidth. Then build a cascade of three of these stages, presto the output is 0.5v x 3 x 3 x 3 = 13.5v pk-pk. Naturally you're going to need power supplies significantly higher than 13.5v, to operate these gain stages with adequate headroom above the peaks and below the troughs. I'll bet you'll wind up with more than 21 volts between VCC and VEE.
Your final stage will probably operate at significant bias current and significant DC power dissipation. Dinky little TO-92 plastic packaged jellybean transistors may not be able to handle this amount of dissipation; TO-92s are usually rated for 600 mW max. You may want to use a "medium power" transistor like the 2SD882 (TO-126, fT= 100MHz) or even a high speed "power transistor" like the D44H11 (TO-220, fT= 60MHz) in the final stage.
It has four states, represented by circles. The name of the first state is "Normal" and the name of the fourth state is "Locked".
Sally Student might perform State Assignment, and she might decide that Normal=00, Pause=01, Probation=10, Locked=11. Sally has two state bits, so she requires two flipflops, one FF to hold each of the two state bits. Now she is ready to draw truth tables and Karnaugh maps.
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ALGEBRA 2N1
FREQUENTLY ASKED QUESTIONS
Why is Algebra 2N1 being offered?
Algebra 2N1 is inspired by similar algebra "immersion" programs throughout California, which have demonstrated that when students focus primarily on mathematics in a supportive environment for a whole semester they are substantially more successful than students who spread out the same courses over two or more semesters. Algebra 2N1 is an intensive 15-unit learning community, consisting of MATH 50, 60, 47 and COUN 110. This learning community not only allows you to complete both MATH 50 and MATH 60 in a single semester, but also provides ample time and support for you to learn algebra in a deep, thorough and meaningful way.
Who should take Algebra 2N1?
Students who have completed MATH 15 with a "B" or better, or placed into MATH 50 (Beginning Algebra), who need to complete MATH 60 for their AA degree or as a prerequisite for college-level mathematics courses for their majors. For example, anyone who is interested in a STEM major or business degree is a good candidate for this course.
It is highly recommended that every student who registers in Algebra 2N1 not enroll in any other academic courses during that semester.
Why should I take Algebra 2N1?
If you need to take Math 50 and Math 60, Algebra 2N1 allows you to complete both and thus get to a college-level math course in a single semester. Also, if you have struggled with Math in the past, or if you just want a solid introduction or refresher, Algebra 2N1 is your golden opportunity to boost your mathematical competence (and confidence!) by leaps and bounds. As mentioned above, it has been demonstrated that students who immerse themselves in mathematics for a semester have much greater success rates than those who spread out the courses. Even if you are not a Math major, you will reap the benefits of the skills and understanding that Algebra 2N1 will provide because being strong in mathematics is often the key to success in other fields.
What will I do in Algebra 2N1?
The Math classes meet 12 hours per week and provide a combination of: sessions with your instructor, learning activities with your peers in small groups, tutoring support, and online Math lab work. The Counseling class meets one day a week for 3 hours and provides support for time management, study skills and college success skills.
How many units is Algebra 2N1? This 15-unit learning community consists of three 4-unit classes-MATH 50, MATH 60, MATH 47 and one 3-unit class – COUN 110.
Does Algebra 2N1 meet the general education MATH requirement for an AA degree?
Yes.
What is The MATH 47 component of Algebra 2N1?
MATH 47, Explorations in Algebra, provides activities designed to support successful completion of both MATH 50 and MATH 60 in one semester.
Can I take Algebra 2N1if I already took MATH 50?
If you have taken MATH 50 and received a "C" or better you are not eligible for Algebra 2N1.
Can I take Algebra 2N1 if I placed into MATH 60 or MATH 56?
Yes.
Will Algebra 2N1 prepare me for college level MATH for a STEM field (science, technology, engineering or MATH) or Business? Yes, successful completion of Algebra 2N1 will prepare you for any introductory college level math course.
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Authorized users only:
Added 09/20/2006
Purplemath contains lessons, links, and homework tips, all designed to help the high school or college algebra student find success. The "how to" lessons include tips and hints, point out common errors, and contain cross-links to related materials. The tone of the lessons is informal, and is directed toward students rather than instructors.
Added 09/20/2006
A video instructional series on algebra for college and high school classrooms and adult learners; 26 half-hour video programs and coordinated books
In this series, host Sol Garfunkel explains how algebra is used for solving real-world problems and clearly explains concepts that may baffle many students. Graphic illustrations and on-location examples help students connect mathematics to daily life. The series also has applications in geometry and calculus instruction.
Added 09/20/2006Added 09/20/2006
This virtual manipulative allows you to solve simple linear equations through the use of a balance beam. Unit blocks (representing 1s) and X-boxes (for the unknown, X), are placed on the pans of a balance beam. Once the beam balances to represent the given linear equation, you can choose to perform any arithmetic operation, as long as you DO THE SAME THING TO BOTH SIDES, thus keeping the beam balanced. The goal, of course, is to get a single X-box on one side, with however many unit blocks needed for balance, thus giving the value of X."
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Problem Solving Approach to Mathematics - With CD - 10th edition
Summary: The new edition of this best-selling text includes a new focus on active and collaborative learning, while maintaining its emphasis on developing skills and concepts. With a wealth of pedagogical tools, as well as relevant discussions of standard curricula and assessments, this book will be a valuable textbook and reference for future teachers. With this revision, two new chapters are included to address the needs of future middle school teachers, in accordance to the NCTM Focal Poin...show morets document23 +$3.99 s/h
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This guide may be a bit rough for you. Math may be the bane of your existence, the thorn in your heel, and the frustration that keeps you awake at night. Or, it may simply be that pestering little course you must take in order to graduate from high school and complete your college degree. The good news is that Math will not plague you forever. You will eventually take the minimum classes in college in order to graduate and then you are finished with it forever. However, until you get to that point, you must take math classes in high school and at least one or two in college.
We've tried to put some ideas together that might make you lose your hatred for the old mathematical skills. Try some. You may surprise yourself and learn that you like punching numbers after all. Math has some elements that are fun and exciting. You may understand why mathematicians spend so much time with their calculators and theorems.
Right vs. Wrong: In math, there is a definitive answer. You will never be caught in the subjective world wondering why your grade is low if you did all the work. There is always a right answer. No matter how hard you work, you will be able to find the answer eventually. Because of this establishment, many people learn to love math, for they will never be caught in a world of subjectivity and possibility.
Games: Math also brings along with it several games. Math is one of the only subjects that will allow students to solve puzzles and play with numbers. It can even be fun during group class settings when you have timed questions for games, not for grades.
Calculators: Calculators are fun little gadgets that accompany math classes. They ease the frustration for some people and bring entertainment for others. If bored in class, you can always punch the buttons on your calculator to keep your busy. And now with such modern technology, calculators even come with games to play.
Terminal Classwork: If you simply abhor all subjects related to math and everything about it, then fear not. There is a light at the end of the tunnel. You will have to take the bare minimum math classes in high school. However, there are so many levels of math in high school, that you may not have to ever take advanced math classes. And in college, you may only have one math class to take to satisfy the requirement for graduation. So, even if you hate math and know you will never like it, you will not have to spend all your time with it. You know that you will just have to get through those few hours until you can eventually concentrate on the coursework you love.
Grand Selection: Once you get to college, most schools will offer a large amount of freedom in how you must satisfy that one math requirement. Some people can graduate from college without even taking a basic math class. Math classes will be designed in college for people like you. Therefore, they will be tailored in a different fashion and designed to be taught in a way for people who already hate math. Consequently, you won't hate math entirely.
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First Course in Mathematics Modeling - With CD - 4th edition
Summary: Offering a solid introduction to the entire modeling process, A FIRST COURSE IN MATHEMATICAL MODELING, 4th Edition delivers an excellent balance of theory and practice, and gives you relevant, hands-on experience developing and sharpening your modeling skills. Throughout, the book emphasizes key facets of modeling, including creative and empirical model construction, model analysis, and model research, and provides myriad opportunities for practice. The authors apply a proven six-ste...show morep problem-solving process to enhance your problem-solving capabilities -- whatever your level. In addition, rather than simply emphasizing the calculation step, the authors first help you learn how to identify problems, construct or select models, and figure out what data needs to be collected. By involving you in the mathematical process as early as possible -- beginning with short projects -- this text facilitates your progressive development and confidence in mathematics and modeling. ...show less
''The examples, projects, and exercises are excellent. Very varied and interesting, with a good mix of topics. It is great to be able to pick and choose from among the topics and examples. At the same time, there is nothing superfluous about the coverage.'' Libby Krussel, The University of Montana
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The latest effort to help those new to Mathematica comes from the O'Reilly folks, known for their programming texts with the zoological covers. This one has the shell of a marine snail, for those of you into zoology. For those not, let's take a look at the pedagogical value of this book.
As readers of this column probably know, Mathematica is software that does mathematics. Its symbolic code offers not only math, modeling and simulation, but a complete documentation and deployment tool. This cookbook assumes a basic knowledge of the program and is not a beginner's guide. Rather it 'jumps right in' with code and examples. Lots of them!
To give you an overview, the chapter titles from the Table of Contents appear below:
Two thoughts come to mind in perusing the above list: it is heavy on programming and also on applications other than strictly math and science. As a bumbling programmer, your editor is not too bothered, as Mathematica is actually coded for most problems, actually crossing over into the sophistication of programming for more advanced applications. The emphasis of the book on a lot of non-math areas is a bit more bothersome, but the inclusion of much code for many problems is the saving grace. For those interested in applications in special areas, go to to see a complete listing.
The first chapter on numerics was fascinating, and I learned quite a few interesting facts and techniques (e.g., approximations, precision and accuracy, numeric types, working with intervals). For reasons of personal interest, I then very quickly glossed over chapters 2 through 9 to get right into the all-to-brief chapters on math.
The chapter on algebra quickly separates the two main classes, i.e., elementary algebra and abstract algebra. The former is our good friend from high school where we manipulate and solve equations, while the latter is concerned with the description and behavior of groups, rings and fields. The author serves up many recipes for solving algebraic equations that are both lucid and powerful. Their big advantages are, of course, to relieve the drudgery of tedious hand calculations. With only 11 pages, this left me begging for more. However, as previously implied, this is not a beginner's guide, but rather a cookbook for the intermediate types.
The chapter on calculus manages, in 29 pages, to cover the more important topics met in an undergraduate course in this area. Here, we not only get recipes for the more common operations (calculating limits, differentiation, integration, max/min problems, vectors and, very briefly, differential equations) but also clear explanations of the use of nomenclature, symbols and algorithms by Mathematica.
The chapter on statistics and data analysis was most interesting to me, due to long years in the field. Here, the book charmed your editor by mentioning that, while most of us use statistical software such as SAS, SPSS and R, Mathematica has greatly beefed up this feature in version 7. It also directs us to another O'Reilly book that I had previously reviewed and highly regard (Boslaugh and Watters' Statistics in a Nutshell).
While beautifully walking us through the various techniques, and demonstrating the most useful and frequently used descriptors, graphics and texts, it leaves the applied statistician wondering why we are doing all that typing! For the mathematical statistician, the available tools in the areas of probability and distributions are superior to those available in most menu-driven programs, and the author delves into this with several useful recipes.
This book is not recommended for the new user. However, for anyone with cursory knowledge of the software, it supplies a number of very nice examples with which to extend user expertise. Also, it serves as a very nice overview of Mathematica's capabilities and concentrates heavily on the useful, versus the abstract.
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Algebra and Trigonometry - 4th edition
Beecher, Penna, and Bittinger'sAlgebra and Trigonometryis known for enabling students to see the math through its focus on visualization and early introduction to functions. With theFourth Edition, the authors continue to innovate by incorporating more ongoing review to help students develop their understanding and study effectively. Mid-chapter Review exercise sets have been added to give students practice in synthesizing the conc...show moreepts, and new Study Summaries provide built-in tools to help them prepare for tests. The MyMathLab course (access kit required) has been expanded so that the online content is even more integrated with the text's approach, with the addition of Vocabulary, Synthesis, and Mid-chapter Review exercises from the text as well as example-based videos created by the authors BOOK ONLY-no supplemental materials included. Covers show moderate wear. Pages show moderate marking/ highlighting. THIS ITEM IS OVERSIZED. PLEASE, NO INTERNATIONAL ORDERS. Very...show more slight pink staining to edge of last few pages. ...show less
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Hardcover 4th Edition text. Hardcover. NO HIGHLIGHTING/WRITING Book is in good condition, book only, has little to no writing/highlighting. Used books may have stickers and varying degrees of shelfwea...show morer. If a book is ordered after noon on Saturday it will not ship until Monday.. Ships fast. Expedited shipping 2-4 business days; Standard shipping 7-14 business days. Ships from USA! ...show less
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Arvind90
Arvind Ravichandran
Popular Science - A Nerd's Guide to Reading. A magnificent, profoundly humane "biography" of cancer—from its first documented appearances thousands of years ago through the epic battles in the twentieth century to cure, control, and conquer it to a radical new understanding of its essence.
Seen in all of the TOOOL educational materials and slide decks, these files are all released under the Creative Commons license. You are free to use any of them for non-commercial purposes, as long as they are properly attributed and the same freedom for others is maintained in all derivative works. Please note that these diagrams have evolved and developed over time.
V.5.5] - All Databases Home. 110.201 Linear Algebra. General Information Spring 2012 Lectures MWF 10:00AM-10:50AM, Shaffer, Room 3 (sections 1,2,3,4,5,6) MWF 11:00AM-11:50AM, Shaffer, Room 303 (sections 7,8,9) First day of class is Monday, January 30th.
Instructor: Nitu Kitchloo Office: Krieger 214 Email: nitu(at)math(dot)jhu(dot)edu Office Hours: Mon,Wed: 12-1PM Section arrangements: Text: Linear Algebra with Applications, Otto Bretscher, Fourth Edition Course Description Vector spaces, matrices, and linear transformations.
Circadian rhythm. Some features of the human circadian (24-hour) biological clock History[edit] The earliest recorded account of a circadian process dates from the 4th century B.C.E., when Androsthenes, a ship captain serving under Alexander the Great, described diurnal leaf movements of the tamarind tree.[1] The observation of a circadian or diurnal process in humans is mentioned in Chinese medical texts dated to around the 13th century, including the Noon and Midnight Manual and the Mnemonic Rhyme to Aid in the Selection of Acu-points According to the Diurnal Cycle, the Day of the Month and the Season of the Year.[2] The first recorded observation of an endogenous circadian oscillation was by the French scientist Jean-Jacques d'Ortous de Mairan in 1729.
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More About
This Textbook
Overview
This book, modern in its writing style as well as in its applications, contains numerous exercises—both skill oriented and applications—, real data problems, and a problem solving method. The book features exercises based on data form the World Wide Web, technology options for those who wish to use a graphing calculator, review boxes, strategic checkpoints, interactive activities, section summaries and projects, and chapter openers and reviews. For anyone who wants to see and understand how mathematics are used in everyday life.
Problem Solving: Probability and Statistics. Sets and Set Operations. Principles of Counting. Introduction to Probability. Computing Probability using the Addition Rule. Computing Probability using the Multiplication Rule. Bayes' Theorem and Its
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This site contains several hundred articles concerned with mathematics and physics. General topics include Number Theory,...
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This site contains several hundred articles concerned with mathematics and physics. General topics include Number Theory, Combinatorics, Geometry, Algebra, Calculus & Differential Equations, Probability & Statistics, Set Theory & Foundations, Reflections on Relativity, History, and Physics. The articles under each general heading are highly varied, many are quite advanced, and there is no apparent organizational scheme. For example, under Calculus & Differential Equations there is a proof that pi is irrational, a examination of the Limit Paradox, a discussion of Ptolemy's Orbit, and an historical review of the cycloid among many other articles. Visitors can browse by topics or search by keyword. (Anyone with information on the identity of the site author please contact the MERLOT submitter.)The goal of MA114 is to demonstrate some of the applicationsof non-calculs mathematical techniques to real-worldproblems. The...
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The goal of MA114 is to demonstrate some of the applicationsof non-calculs mathematical techniques to real-worldproblems. The main components of the course includeQuickTime movies available via the Web, Web-basedassignments (WebAssignments), and a Textbook(Topics in Finite Mathematics) by Page and Paur.
This site is part of the NCTM's Student i-Math Investigations website. It uses algebra and discrete mathematics to analyze...
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This site is part of the NCTM's Student i-Math Investigations website. It uses algebra and discrete mathematics to analyze population changes in a trout pond. Included are applets for numerical and graphical analysis.
We will apply insights from game theory to explain human social behavior, focusing on novel applications which have...
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We will apply insights from game theory to explain human social behavior, focusing on novel applications which have heretofore been the realm of psychologists and philosophers—for example, why people speak indirectly, in what sense beauty is socially constructed, and where our moral intuitions come from—and eschewing traditional economic applications such as industrial organization or auctions. We will employ standard games such as the prisoners dilemma, coordination, hawk-dove, and costly signaling, and use standard game theory tools such as Nash Equilibria, Subgame Perfection, and Perfect Bayesian Equilibria. These tools will be taught from scratch and no existing knowledge of game theory, economics, or mathematics is required. At the same time, students familiar with these games and tools will not find the course redundant because of the focus on non-orthodox applications.
Game Theory, also known as Multiperson Decision Theory, is the analysis of situations in which the payoff of a decision maker...
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Game Theory, also known as Multiperson Decision Theory, is the analysis of situations in which the payoff of a decision maker depends not only on his own actions but also on those of others. Game Theory has applications in several fi…elds, such as economics, politics, law, biology, and computer science. In this course, I will introduce the basic tools of game theoretic analysis. In the process, I will outline some of the many applications of Game Theory, primarily in economics.
This course is an introduction to the fundamentals of game theory and mechanism design. Motivations are drawn from...
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This course is an introduction to the fundamentals of game theory and mechanism design. Motivations are drawn from engineered/networked systems (including distributed control of wireline and wireless communication networks, incentive-compatible/dynamic resource allocation, multi-agent systems, pricing and investment decisions in the Internet), and social models (including social and economic networks). The course emphasizes theoretical foundations, mathematical tools, modeling, and equilibrium notions in different environments.
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STCC Course Descriptions
Topics include the real number system, operations of real numbers, simplification of algebraic expressions, solving equations and inequalities. Topics also include graphing of linear equations, slopes, equations of lines, and graphing inequalities in two variables, systems of linear equations, applications and problem solving. Additional topics are exponents, scientific notation, and operations with polynomials. Credit for this course will not be counted toward fulfilling graduations requirements at STCC.
Prerequisites: ARTH-078 or ARTH-073 (minimum grade C-) or placement at Algebra I on the math placement test.
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Kinematics in Two Dimensions
Discussion
synthesis vs. analysis
The geometry that puts algebra on the coordinate plane is called analytic geometry — the topic of the nexttwo sections of this book. Analytic geometry makes a quiet appearance in high school algebra classes. This is when students are introduced to the coordinate plane and when they learn how to graph functions. The topic is rarely identified as analytic geometry, probably because most math teachers don't want to hear the wise guy comment, "This is algebra class. Why are you teaching us geometry?" Analytic geometry is a relatively modern branch of mathematics, going back only as far as the Seventeenth Century. It made its way into the secondary math curriculum sometime in the Nineteenth Century. Analytic geometry is also known as coordinate geometry, because of the central role played by the coordinate plane, or cartesian geometry, in honor of the French philosopher René Descartes (1596–1650) who proposed the coordinate plane in Discourse on the Method — the book famous for the line "I think, therefore I am".
The geometry that uses logic to test proofs is called synthetic geometry — the topic of this section. Synthetic geometry is a large part of secondary school mathematics. It is usually taught as a full year course in the US. Synthetic geometry was invented in the river civilizations of Egypt, Mesopotamia, and the Indus Valley five thousand years ago. It has been a part of formal education for a very long time. The only thing older is arithmetic, which probably predates writing. Synthetic geometry is more commonly known as plane geometry, since the foundations of the subject were laid on two dimensional applications, or euclidean geometry, in honor of the Greek mathematician Euclid of Alexandria (323–283 BCE) who wrote The Elements — the most popular textbook of all time.
Synthesis (joining separated parts into a unified entity) is the opposite of analysis (breaking something up into its constituent parts). Before the Seventeenth Century, mathematicians used geometry to solve sets of related problems from which generalized conclusions could be made. After the Seventeenth Century, mathematicians applied algebra to generalized problems from which sets of related conclusions could be drawn. Synthesis requires a lot of work to arrive at a few conclusions, while analysis produces a lot of conclusions from a little bit of work. This is a huge oversimplification of the situation, but you get the idea. Mathematics improved dramatically when algebra switched from a tool of geometry to its driving engine.
Synthetic geometry still has its usefulness. I think students react more positively to the "easy" geometry of Euclid's Elements than to the "hard" geometry of Descartes' Discourse. Analytic geometry with its strict adherence to the grid just doesn't give them the warm fuzzy feeling they get when they draw a triangle. Descartes wins out in the end, however. Computers do most of the math in the Twenty-first century and computers love coordinates.
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Providing students who need a solid understanding of algebra with an excellent start, this textbook encourages student understanding of algebra through the use of modelling techniques and real-data applications.Providing students who need a solid understanding of algebra with an excellent start, this textbook encourages student understanding of algebra through the use of modelling techniques and real-data applications.Read Less
Very good. Hardcover. Has minor wear and/or markings. SKU: 9781133963028-3-0-3 Orders ship the same or next business day. Expedited shipping within U.S. will arrive in 3-5 days. Hassle free 14 day return policy. Contact Customer Service for questions. ISBN: 9781133963028.
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Mathematics
Mathematics, "The Queen of Sciences" as called by Carl Friedrich Gauss, is the science of number, quantity, and space, either as abstract concepts or as applied to other disciplines (such as physics and engineering).
The distinguished authors of the top-quality books and textbooks listed under Research and Markets' Mathematics category are the world's leading researchers. These publications cover all the key areas in today's research. They are invaluable references, comprehensive and
readily accessible. When available, pre-publication titles are also included, so you can be sure not to miss the latest developments in your research field.
The readership of this category includes both graduate and undergraduate students, as well as researchers and mature mathematics.
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Mathematical Geoscience a comprehensive reference to the mathematical modelling of environmental and geophysical problems, where the physical model is treated as seriously as the mathematical analysis
Provides an abundance of mathematical models, both simple and complex, and elaborates them with approximation techniques. These are of particular use to engineers and environmental scientists
Includes plenty of examples, exercises and references
Mathematical Geoscience is an expository textbook which aims to provide a comprehensive overview of a number of different subjects within the Earth and environmental sciences. Uniquely, it treats its subjects from the perspective of mathematical modelling with a level of sophistication that is appropriate to their proper investigation. The material ranges from the introductory level, where it can be used in undergraduate or graduate courses, to research questions of current interest. The chapters end with notes and references, which provide an entry point into the literature, as well as allowing discursive pointers to further research avenues.
The introductory chapter provides a condensed synopsis of applied mathematical techniques of analysis, as used in modern applied mathematical modelling. There follows a succession of chapters on climate, ocean and atmosphere dynamics, rivers, dunes, landscape formation, groundwater flow, mantle convection, magma transport, glaciers and ice sheets, and sub-glacial floods.
This book introduces a whole range of important geoscientific topics in one single volume and serves as an entry point for a rapidly expanding area of genuine interdisciplinary research. By addressing the interplay between mathematics and the real world, this book will appeal to graduate students, lecturers and researchers in the fields of applied mathematics, the environmental sciences and engineering.
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Algebra and Trigonometry - 7th edition
Summary: For undergraduate courses in Algebra and Trigonometry with optional Graphing Calculator usage.
The Seventh Edition of this dependable text retains its best features -- accuracy, precision, depth, strong student support, and abundant exercises -- while substantially updating content and pedagogy. After completing the book, students will be prepared to handle the algebra found in subsequent courses such as finite mathematics, business mathematics, and engine...show moreering calculus. ...show less
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Upper Saddle River, NJ 2004 Hard cover 7th ed. Very good. No dust jacket. Sewn binding. Cloth over boards. 1184 p. Contains: Illustrations. Audience: General/trade. (10713)FOLLOWING SPECIAL DESCRIPT...show moreION ...show less
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Chapter 3.Differentiation [4 weeks] Tangents and the derivative at a point. The derivative as a function. Differentiation rules: for polynomials and exponentials; for products and quotients. The derivative as a rate of change. Derivatives of trigonometric functions. The chain rule. Implicit Differentiation. Derivatives of inverse functions and logarithms. Inverse trigonometric functions. Related Rates. Linearization and differentials. Additional material (§§11.1, 11.2): parametric equations and their derivatives. Derivatives of Hyperbolic Functions.
Chapter 5.Integration [4 weeks] Area estimates with finite sums. Sigma notation and limits of finite sums. The definite integral. The fundamental theorem of calculus. Indefinite integrals and the substitution method. Substitution and area between curves.
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A guide to concept mapping in mathematics. It provides the reader with an understanding of how the meta-cognitive tool, namely, hierarchical concept maps, and the process of concept mapping can be used innovatively and strategically to improve planning, teaching, learning, and assessment at different educational levels. more...
Word problems are the most difficult part of any math course ?- and the most important to both the SATs and other standardized tests. This book teaches proven methods for analyzing and solving any type of math word problem. more...
As a result of the editors' collaborative teaching at Harvard in the late 1960s, they produced a ground-breaking work -- The Art Of Problem Posing -- which related problem posing strategies to the already popular activity of problem solving. It took the concept of problem posing and created strategies for engaging in that activity as a central theme... more...
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Intermediate Algebra for College Students (5th Edition)
9780136007623
ISBN:
0136007627
Edition: 5 Pub Date: 2008 Publisher: Prentice Hall
Summary: The goal of this book is to provide readers with a strong foundation in algebra by developing the problem-solving and critical thinking abilities. Topics are presented in an interesting format, incorporating real world sourced data and encouraging modeling and problem-solving.
Robert F. Blitzer is the author of Intermediate Algebra for College Students (5th Edition), published 2008 under ISBN 9780136007623 a...nd 0136007627. Twenty Intermediate Algebra for College Students (5th Edition) textbooks are available for sale on ValoreBooks.com, thirteen used from the cheapest price of $11.62, or buy new starting at $105.30wear to hard covers. no cd. some pages bent. hinges compromised but pages intact
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CK-12 Algebra I - Second Edition
Table of Contents
This chapter covers evaluating algebraic expressions, order of operations, using verbal models to write equations, solving problems using equations, inequalities, identifying the domain and range of a function, and graphs of functions.
This chapter covers solving one-step equations, solving two-step and multi-step equations, using ratios and proportions, solving problems using scale drawings, using similar figures to measure, and finding the percent of a number.
This chapter covers linear equations in slope-intercept form and point-slope form, standard form for linear equations, equations of parallel and perpendicular lines, and problem solving using linear models.
This chapter introduces students to linear systems of equations and inequalities as well as probability and combinations. Operations on linear systems are covered, including addition, subtraction, multiplication, and division.
This chapter covers graphing quadratic functions, indentifying the number of solutions of quadratic equations, solving quadratic equations using the quadratic formula, and finding the discriminant of a quadratic equation.
This chapter covers graphing and comparing square root functions, solving radical equations, using the Pythagorean theorem and its converse, using the distance formula, and making & interpreting stem-and-leaf plots & histograms.
An introduction to theoretical probability and data organization including events, conditions, random variables, and graphs/tables.
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CK-12's Algebra I Second Edition is a clear presentation of algebra for the high school student. Topics include: Equations and Functions, Real Numbers, Equations of Lines, Solving Systems of Equations and Quadratic Equations.
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College Geometry : Using the Geometer's Sketchpad - 12 edition
Summary: From two authors who embrace technology in the classroom and value the role of collaborative learning comes College Geometry Using The Geometer's Sketchpad. The book's truly discovery-based approach guides readers to learn geometry through explorations of topics ranging from triangles and circles to transformational, taxicab, and hyperbolic geometries. In the process, readers hone their understanding of geometry and their ability to write rigorous mathematical proofs49.00 +$3.99 s/h
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THE PROGRAM STUDENTS NEED; THE FOCUS TEACHERS WANT! Glencoe Pre-Algebra is a key program in our vertically aligned high school mathematics series developed to help all students achieve a better understanding of mathematics and improve their mathematics scTHE PROGRAM STUDENTS NEED; THE FOCUS TEACHERS WANT! Glencoe Pre-Algebra is a key program in our vertically aligned high school mathematics series developed to help all students achieve a better understanding of mathematics and improve their mathematics sc
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Math.NET aims to provide a self contained clean framework for symbolic mathematical (Computer Algebra System) and numerical/scientific computations, including a parser and support for linear algebra, complex differential analysis, system solving and more
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Without a basic understanding of maths, students of any science discipline are ill-equipped to tackle new problems or to apply themselves to novel situations. In this book, Keith Gregson covers a few essential topics that will help encourage an understanding of mathematics so that the student can build on their understanding and apply it to their... more...
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Intermediate Trades Math is intended to give students the necessary arithmetic knowledge and competence to successfully enter vocational training. It is not acceptable as a prerequisite for ABE Math 040 or Advanced Chemistry or Physics.
Learning Outcomes: Upon successful completion of this course, learners will be able to:
- Add, subtract, multiply, and divide whole numbers
- Add, subtract, multiply, and divide decimals
- Use metric measurements
- Add, subtract, multiply, and divide common fractions and mixed numbers
- Demonstrate ratio and proportion
- Use ratio and proportion to solve percent problems
- Calculate perimeter and area
- Calculate surface area and volume
- Analyze and extract information from bar, line and circle graphs and tables
- Recognize and use positive and negative numbers
- Use basic algebraic expressions and equations
- Use Pythagorean rule and trigonometry to solve problems
- Use a calculator to calculate, solve and check problems
- Solve vocational word problems
Grading System: Letters
Passing Grade: D (50%)
Percentage of Individual Work: 100
Additional Comments:
- a grade of B (75%) is required to progress to Math 041, 042, or 044.
- MATH 031 is not considered a sufficient prerequisite for Math 040 or 043
Text Books:
Textbooks are subject to change. Please contact the bookstore at your local campus for current book lists.
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Functions of One Complex Variable as a textbook for a first course in the theory of functions of one complex variable for students who are mathematically mature enough to understand and execute E - 8 arguments. The actual pre requisites for reading this book are quite minimal; not much more than a stiff course in basic calculus and a few facts about partial derivatives. The topics from advanced calculus that are used (e.g., Leibniz's rule for differ entiating under the integral sign) are proved in detail. Complex Variables is a subject which has something for all mathematicians. In addition to having applications to other parts of analysis, it can rightly claim to be an ancestor of many areas of mathematics (e.g., homotopy theory, manifolds). This view of Complex Analysis as "An Introduction to Mathe matics" has influenced the writing and selection of subject matter for this book. The other guiding principle followed is that all definitions, theorems, etc.
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This is a guest post by Nathan, who recently finished graduate school in math, and will begin a post-doc in the fall. He loves teaching young kids, but is still figuring out how to motivate undergraduates.
The question
Like most mathematicians in academia, I'm teaching calculus in the fall. I taught in grad school, but the syllabus and assignments were already set. This time I'll be in charge, so I need to make some design decisions, like the following:
Are calculators/computers/notes allowed on the exams?
Which purely technical skills must students master (by a technical skill I mean something like expanding rational functions into partial fractions: a task which is deterministic but possibly intricate)?
Will students need to write explanations and/or proofs?
I have some angst about decisions like these, because it seems like each one can go in very different directions depending on what I hope the students are supposed to get from the course. If I'm listing the pros and cons of permitting calculators, I need some yardstick to measure these pros and cons.
My question is: what is the goal of a college calculus course?
I'd love to have an answer that is specific enough that I can use it to make concrete decisions like the ones above. Part of my angst is that I've asked many people this question, including people I respect enormously for their teaching, but often end up with a muddled answer. And there are a couple stock answers that come to mind, but each one doesn't satisfy me for one reason or another. Here's what I have so far.
The contenders.
To teach specific tasks that are necessary for other subjects.
These tasks would include computing integrals and derivatives, converting functions to power series or Fourier series, and so forth.
Intuitive understanding of functions and their behavior.
This is vague, so here's an example: a couple years ago, a friend in medical school showed me a page from his textbook. The page concerned whether a certain drug would affect heart function in one way or in the opposite way (it caused two opposite effects), and it showed a curve relating two involved parameters. It turned out that the essential feature was that this curve was concave down. The book did not use the phrase "concave down," though, and had a rather wordy explanation of the behavior. In this situation, a student who has a good grasp of what concavity is and what its implications are is better equipped to understand the effect described in the book. So if a student has really learned how to think about concavity of functions and its implications, then she can more quickly grasp the essential parts of this medical situation.
To practice communicating with precision.
I'm taking "communication" in a very wide sense here: carefully showing the steps in an integral calculation would count.
Not Satisfied
I have issues with each of these as written. I don't buy number 1, because the bread and butter of calculus class, like computing integrals, isn't something most doctors or scientists will ever do again. Number 2 is a noble goal, but it's overly idealistic; if this is the goal, then our success rate is less than 10%. Number 3 also seems like a great goal, relevant for most of the students, but I think we'd have to write very different sorts of assignments than we currently do if we really want to aim for it.
I would love to have a clear and realistic answer to this question. What do you think?
There's been a movement to make primary and secondary education run more like a business. Just this week in California, a lawsuit funded by Silicon Valley entrepreneur David Welch led to a judge finding that student's constitutional rights were being compromised by the tenure system for teachers in California.
The thinking is that tenure removes the possibility of getting rid of bad teachers, and that bad teachers are what is causing the achievement gap between poor kids and well-off kids. So if we get rid of bad teachers, which is easier after removing tenure, then no child will be "left behind."
The problem is, there's little evidence for this very real achievement gap problem as being caused by tenure, or even by teachers. So this is a huge waste of time.
As a thought experiment, let's say we did away with tenure. This basically means that teachers could be fired at will, say through a bad teacher evaluation score.
An immediate consequence of this would be that many of the best teachers would get other jobs. You see, one of the appeals of teaching is getting a comfortable pension at retirement, but if you have no idea when you're being dismissed, then it makes no sense to put in the 25 or 30 years to get that pension. Plus, what with all the crazy and random value-added teacher models out there, there's no telling when your score will look accidentally bad one year and you'll be summarily dismissed.
People with options and skills will seek other opportunities. After all, we wanted to make it more like a business, and that's what happens when you remove incentives in business!
The problem is you'd still need teachers. So one possibility is to have teachers with middling salaries and no job security. That means lots of turnover among the better teachers as they get better offers. Another option is to pay teachers way more to offset the lack of security. Remember, the only reason teacher salaries have been low historically is that uber competent women like Laura Ingalls Wilder had no other options than being a teacher. I'm pretty sure I'd have been a teacher if I'd been born 150 years ago.
So we either have worse teachers or education doubles in price, both bad options. And, sadly, either way we aren't actually addressing the underlying issue, which is that pesky achievement gap.
People who want to make schools more like businesses also enjoy measuring things, and one way they like measuring things is through standardized tests like achievement scores. They blame teachers for bad scores and they claim they're being data-driven.
I'm tempted to conclude that we should just go ahead and get rid of teacher tenure so we can wait a few years and still see no movement in the achievement gap. The problem with that approach is that we'll see great teachers leave the profession and no progress on the actual root cause, which is very likely to be poverty and inequality, hopelessness and despair. Not sure we want to sacrifice a generation of students just to prove a point about causation.
On the other hand, given that David Welch has a lot of money and seems to be really excited by this fight, it looks like we might have no choice but to blame the teachers, get rid of their tenure, see a bunch of them leave, have a surprise teacher shortage, respond either by paying way more or reinstating tenure, and then only then finally gather the data that none of this has helped and very possibly made things worse.
I had a little secret about my survival in grad school, and that secret has a name, and that name is Jordan Ellenberg. We used to meet every Tuesday and Thursday to study schemes at the CallaLily Cafe a few blocks from the Science Center on Kirkland Street, and even though that sounds kind of dull, it was a blast. It was what kept me sane at Harvard.
You see, Jordan has an infectious positivity about him, which balances my rather intense suspicions, and moreover he's hilariously funny. He's really somewhere between a mathematician and a stand-up comedian, and to be honest I don't know which one he's better at, although he is a deeply talented mathematician.
The reason I'm telling you this is that he's written a book, called How Not To Be Wrong, and available for purchase starting today, which is a delight to read and which will make you understand why I survived graduate school. In fact nobody will ever let me complain again once they've read this book, because it reads just like Jordan talks. In reading it, I felt like I was right back at CallaLily, singing Prince's "Sexy MF" and watching Jordan flirt with the cashier lady again. Aaaah memories.
So what's in the book? Well, he talks a lot about math, and about mathematicians, and the lottery, and in fact he has this long riff which starts out with lottery math, then goes to error-correcting codes and then to made-up languages and then to sphere packing and then arrives again at lotteries. And it's brilliant and true and beautiful and also funny.
I have a theory about this book that you could essentially open it up to any page and begin to enjoy it, since it is thoroughly enjoyable and the math is cumulative but everywhere so well explained that it wouldn't take long to follow along, and pretty soon you'd be giggling along with Jordan at every ridiculous footnote he's inserted into his narrative.
In other words, every page is a standalone positive and ontological examination of the beauty and surprise of mathematical discovery. And so, if you are someone who shares with Jordan a love for mathematics, you will have a consistently great time with this book. In fact I'm imagining that you have an uncle or a mom who loves math or science, in which case this would be a seriously perfect gift to them, but of course you could also give that gift to yourself. I mean, this is a guy who can make nazi jokes funny, and he does.
Having said that, the magic of the book is that it's not just a collection of wonderful mathy tidbits. Jordan also has a point about the act of scrutinizing something in a logical and mathematical fashion. That act itself is courageous and should be appreciated, and he explains why, and he tells us how much we've already benefited from people in the past who have had the bravery to do so. He appreciates them and we should too.
And yet, he also sends the important message that it's not an elitist crew of the usual genius suspects, that in fact we can all do this in our own capacity. It's a great message and, if it ends up allowing people to re-examine their need for certainty in an uncertain world, then Jordan will really end up doing good. Fingers crossed.
That's not to say it's a perfect book, and I wanted to argue with points on basically every other page, but mostly in a good, friendly, over-drinks kind of way, which is provocative but not annoying. One exception I might make came on page 256: no, Jordan, municipal bonds do not always get paid back, and no, stocks do not always go up, not even in expectation. In fact to the extent that both of those statements seem true to many people is the result of many cynical political acts and is damaging, mostly to people like retired civil servants. Don't go there!
Another quibble: Jordan talks about how public policy makers make proclamations in the face of uncertainty, and he has a lot of sympathy and seems to think the should keep doing this. I'm on the other side on this one. Telling people to avoid certain foods and then changing stances seems more damaging than helpful and it happens constantly. And it's often tied to industry and money, which also doesn't impress.
Even so, even when I strongly disagree with Jordan, I always want to have the conversation. He forces that on the reader because he's so darn positive and open-minded.
A few more goodies that I wanted to adore without giving too much away. Jordan does a great job with something he calls "The Great Square of Men" and Berkson's Fallacy: it will explain to many many women why they are not finding the man they're looking for. He also throws out a bone to nerds like me when he almost proves that every pig is yellow, and he absolutely kills it, stand-up comedian style, when comparing Ross Perot to a small dark pile of oats. Holy crap he was on a roll there.
So here's one thing I've started doing since reading the book. When I give my 5-year-old son his dessert, it's in the form of Hershey Drops, which are kind of like fat M&M's. I give him 15 and I ask him to count them to make sure I got it right. Sometimes I give him 14 to make sure he's paying attention. But that's not the new part. The new part is something I stole from Jordan's book.
The new part is that some days I ask him, "do you want me to give you 3 rows of 5 drops?" And I wait for him to figure out that's enough and say "yes!" And the other days I ask him "do you want me to give you 5 rows of 3 drops?" and I again wait. And in either case I put the drops out in a rectangle.
And last night, for the first time, he explained to me in a slightly patronizing voice that it doesn't matter which way I do it because it ends up being the same, because of the rectangle formation and how you look at it. And just to check I asked him which would be more, 10 rows of 7 drops or 7 rows of 10 drops, and he told me, "duh, it would be the same because it couldn't be any different."
Today's post is an email interview with Fawn Nguyen, who teaches math at Mesa Union Junior High in southern California. Fawn is on the leadership team for UCSB Mathematics Project that provides professional development for teachers in the Tri-County area. She is a co-founder of the Thousand Oaks Math Teachers' Circle. In an effort to share and learn from other math teachers, Fawn blogs at Finding Ways to Nguyen Students Over. She also started VisualPatterns.org to help students develop algebraic thinking, and more recently, she shares her students' daily math talks to promote number sense. When Fawn is not teaching or writing, she is reading posts on mathblogging.org as one of the editors. She sleeps occasionally and dreams of becoming an architect when all this is done.
Importantly for the below interview, Fawn is not being measured via a value-added model. My questions are italicized.
——
I've been studying the rhetoric around the mathematics Common Core State Standard (CCSS). So far I've listened to Diane Ravitch stuff, I've interviewed Bill McCallum, the lead writer of the math CCSS, and I've also interviewed Kiri Soares, a New York City high school principal. They have very different views. Interestingly, McCallum distinguished three things: standards, curriculum, and testing.
What do you think? Do teachers see those as three different things? Or is it a package deal, where all three things rolled into one in terms of how they're presented?
I can't speak for other teachers. I understand that the standards are not meant to be the curriculum, but the two are not mutually exclusive either. They can't be. Standards inform the curriculum. This might be a terrible analogy, but I love food and cooking, so maybe the standards are the major ingredients, and the curriculum is the entrée that contains those ingredients. In the show Chopped on Food Network, the competing chefs must use all 4 ingredients to make a dish – and the prepared foods that end up on the plates differ widely in taste and presentation. We can't blame the ingredients when the dish is blandly prepared any more than we can blame the standards when the curriculum is poorly written.
Similary, the standards inform testing. Test items for a certain grade level cover the standards of that grade level. I'm not against testing. I'm against bad tests and a lot of it. By bad, I mean multiple-choice items that require more memorization than actual problem solving. But I'm confident we can create good multiple-choice tests because realistically a portion of the test needs to be of this type due to costs.
The three – standards, curriculum, and testing – are not a "package deal" in the sense that the same people are not delivering them to us. But they go together, otherwise what is school mathematics? Funny thing is we have always had the three operating in schools, but somehow the Common Core State Standands (CCSS) seem to get the all the blame for the anxieties and costs connected to testing and curriculum development.
As a teacher, what's good and bad about the CCSS?
I see a lot of good in the CCSS. This set of standards is not perfect, but it's much better than our state standards. We can examine the standards and see for ourselves that the integrity of the standards holds up to their claims of being embedded with mathematical focus, rigor, and coherence.
Implementation of CCSS means that students and teachers can expect consistency in what is being in taught at each grade level across state boundaries. This is a nontrivial effort in addressing equity. This consistency also helps teachers collaborate nationwide, and professional development for teachers will improve and be more relevant and effective.
I can only hope that textbooks will be much better because of the inherent focus and coherence in CCSS. A kid can move from Maine to California and not have to see different state outlines on their textbooks as if he'd taken on a new kind of mathematics in his new school. I went to a textbook publishers fair recently at our district, and I remain optimistic that better products are already on their way.
We had every state create its own assessment, now we have two consortia, PARCC and Smarter Balanced. I've gone through the sample assessments from the latter, and they are far better than the old multiple-choice items of the CST. Kids will have to process the question at a deeper level to show understanding. This is a good thing.
What is potentially bad about the CCSS is the improper or lack of implementation. So, this boils down to the most important element of the Common Core equation – the teacher. There is no doubt that many teachers, myself included, need sustained professional development to do the job right. And I don't mean just PD in making math more relevant and engaging, and in how many ways we can use technology, I mean more importantly, we need PD in content knowledge.
It is a perverse notion to think that anyone with a college education can teach elementary mathematics. Teaching mathematics requires knowing mathematics. To know a concept is to understand it backward and forward, inside and outside, to recognize it in different forms and structures, to put it into context, to ask questions about it that leads to more questions, to know the mathematics beyond this concept. That reminds me just recently a 6th grader said to me as we were working on our unit of dividing by a fraction. She said, "My elementary teacher lied to me! She said we always get a smaller number when we divide two numbers."
Just because one can make tuna casserole does not make one a chef. (Sorry, I'm hungry.)
What are the good and bad things for kids about testing?
Testing is only good for kids when it helps them learn and become more successful – that the feedback from testing should inform the teacher of next moves. Testing has become such a dirty word because we over test our kids. I'm still in the classroom after 23 years, yet I don't have the answers. I struggle with telling my kids that I value them and their learning, yet at the end of each quarter, the narrative sum of their learning is a letter grade.
Then, in the absence of helping kids learn, testing is bad.
What are the good/bad things for the teachers with all these tests?
Ideally, a good test that measures what it's supposed to measure should help the teacher and his students. Testing must be done in moderation. Do we really need to test kids at the start of the school year? Don't we have the results from a few months ago, right before they left for summer vacation? Every test takes time away from learning.
I'm not sure I understand why testing is bad for teachers aside from lost instructional minutes. Again, I can't speak for other teachers. But I do sense heightened anxiety among some teachers because CCSS is new – and newness causes us to squirm in our seats and doubt our abilities. I don't necessarily see this as a bad thing. I see it as an opportunity to learn content at a deeper conceptual level and to implement better teaching strategies.
If we look at anything long and hard enough, we are bound to find the good and the bad. I choose to focus on the positives because I can't make the day any longer and I can't have fewer than 4 hours of sleep a night. I want to spend my energies working with my administrators, my colleagues, my parents to bring the best I can bring into my classroom.
Is there anything else you'd like to add?
The best things about CCSS for me are not even the standards – they are the 8 Mathematical Practices. These are life-long habits that will serve students well, in all disciplines. They're equivalent to the essential cooking techniques, like making roux and roasting garlic and braising kale and shucking oysters. Okay, maybe not that last one, but I just got back from New Orleans, and raw oysters are awesome.
I'm excited to continue to share and collaborate with my colleagues locally and online because we now have a common language! We teachers do this very hard work – day in and day out, late into the nights and into the weekends – because we love our kids and we love teaching. But we need to be mathematically competent first and foremost to teach mathematics. I want the focus to always be about the kids and their learning. We start with them; we end with them.
How are high school math teachers in New York City currently evaluated?
Teachers are now evaluated on 2 things:
First, measures of teacher practice, which are based on observations, in turn based on some rubric. Right now it's the Danielson Rubric. This is a qualitative measure. In fact it is essentially an old method with a new name.
Second, measures of student learning, that is supposed to be "objective". Overall it is worth 40% of the teacher's score but it is separated into two 20% parts, where teachers choose the methodology of one part and principals choose the other. Some stuff is chosen for principals by the city. Any time there is a state test we have to choose it. In terms of the teachers' choices, there are two ways to get evaluated: goals or growth. Goals are based on a given kid, and the teachers can guess they will get a certain slightly lower score or higher score for whatever reason. Otherwise, it's a growth-based score. Teachers can also choose from an array of assessments (state tests, performance tests, and third party exams). They can also choose the cohort (their own kids/ the grade/the school). The city also chose performance tasks in some instances.
Can you give me a concrete example of what a teacher would choose as a goal?
At the beginning of year you give diagnostic tests to students in your subject. Based on what a given kid scored in September, you extrapolate a guess for their performance in the June test. So if a kid has a disrupted homelife you might guess lower. Teacher's goal setting is based on these teachers' guesses.
So in other words, this is really just a measurement of how well teachers guess?
Well they are given a baseline and teachers set goals relative to that, but yes. And they are expected to make those guesses in November, possibly well before homelife is disrupted. It definitely makes things more complicated. And things are pretty complicated. Let me say a bit more.
The first three weeks of school are all testing. We test math, social studies, science, and English in every grade, and overall it depending on teacher/principal selections it can take up to 6 weeks, although not in a given subject. Foreign language and gym teachers also getting measured, by the way, based on those other tests. These early tests are diagnostic tests.
Moreover, they are new types of tests, which are called performance-based assessments, and they are based on writing samples with prompts. They are theoretically better quality because they go deeper, the aren't just bubble standardized tests, but of course they had no pre-existing baseline (like the state tests) and thus had to be administered as diagnostic. Even so, we are still trying to predict growth based on them, which is confusing since we don't know how to predict performance on new tests. Also don't even know how we can consistently grade such essay-based tests- despite "norming protocols", which is yet another source of uncertainty.
How many weeks per year is there testing of students?
The last half of June is gone, a week in January, and 2-3 weeks in the high school in the beginning per subject. That's a minimum of 5 weeks per subject per year, out of a total of 40 weeks. So one eighth of teacher time is spent administering tests. But if you think about it, for the teachers, it's even more. They have to grade these tests too.
I've been studying the rhetoric around the CC. So far I've listened to Diane Ravitch stuff, and to Bill McCallum, the lead writer of the math CC. They have very different views. McCallum distinguished three things, which when they are separated like that, Ravitch doesn't make sense.
Namely, he separates standards, curriculum, and testing. People complain about testing and say that CC standards make testing easier, and we already have too much testing, so CC is a bad thing. But McCallum makes this point: good standards also make good testing easier.
What do you think? Do teachers see those as three different things? Or is it a package deal, where all three things rolled into one in terms of how they're presented?
It's much easier to think of those three things as vertices of a triangle. We cannot make them completely isolated, because they are interrelated.
So, we cannot make the CC good without curriculum and assessment, since there's a feedback loop. Similarly, we cannot have aligned curriculum without good standards and assessment, and we cannot have good tests without good standards and curriculum. The standards have existed forever. The common core is an attempt to create a set of nationwide standards. For example, without a coherent national curriculum it might seem OK to teach creationism in place of evolution in some states. Should that be OK?
CC is attempting to address this, in our global economy, but it hasn't even approached science for clear political reasons. Math and English are the least political subjects so they started with those. This is a long time coming, and people often think CC refers to everything but so far it's really only 40% of a kid's day. Social studies CC standards are actually out right now, but they are very new.
Next, the massive machine of curriculum starts getting into play, as does the testing. I have CC standards and the CC-aligned test, but not curriculum.
Next, you're throwing into the picture teacher evaluation aligned to CC tests. Teachers are freaking out now – they're thinking, my curriculum hasn't been CC-aligned for many years, what do I do now? By the way, importantly, none of the high school curriculum in NY State is actually CC-aligned now. DOE recommendations for the middle school happened last year, and DOE people will probably recommend this year for high school, since they went into talks with publication houses last year to negotiate CC curriculum materials.
The real problem is this: we've created these new standards to make things more difficult and more challenging without recognizing where kids are in the present moment. If I'm a former 5th grader, and the old standards were expecting something from me that I got used to, and it wasn't very much, and now I'm in 6th grade, and there are all these raised expectations, and there's no gap attention.
Bottomline, everybody is freaking out – teachers, students, and parents.
Last year was the first CC-aligned ELA and math tests. Everybody failed. They rolled out the test before any CC curriculum.
From the point of view of NYC teachers, this seems like a terrorizing regime, doesn't it?
Yes, because the CC roll-out is rigidly tied to the tests, which are in turn rigidly tied to evaluations of teachers. So the teachers are worried they are automatically going to get a "failure" on that vector.
Another way of saying this is that, if teacher evaluations were taken out of the mix, we'd have a very different roll-out environment. But as it is, teachers are hugely anxious about the possibility that their kids might fail both the city and state tests, and that would give the teacher an automatic "failure" no matter how good their teacher observations are.
So if I'm a special ed teacher of a bunch of kids reading at 4th and 5th grade level even through they're in 7th grade, I'm particularly worried with the introduction of the new and unknown CC-aligned tests.
So is that really what will happen? Will all these teachers get failing evaluation scores?
That's the big question mark. I doubt it there will be massive failure though. I think given that the scores were so clustered in the middle/low muddle last year, they are going to add a curve and not allow so many students to fail.
So what you're pointing out is that they can just redefine failure?
Exactly. It doesn't actually make sense to fail everyone. Probably 75% of the kids got 2's or 1's out of a 4 point scale. What does failure mean when everyone fails? It just means the test was too hard, or that what the kids were being taught was not relevant to the test.
Let's dig down to the the three topics. As far as you've heard from the teachers, what's good and bad about CC?
My teachers are used to the CC. We've rolled out standards-based grading three years ago, so our math and ELA teachers were well adjusted, and our other subject teachers were familiar. The biggest change is what used to be 9th grade math is now expected of the 8th grade. And the biggest complaint I've heard is that it's too much stuff – nobody can teach all that. But that's always been true about every set of standards.
Did they get rid of anything?
Not sure, because I don't know what the elementary level CC standards did. There was lots of shuffling in the middle school, and lots of emphasis on algebra and algebraic thinking. Maybe they moved data and stats to earlier grades.
So I believe that my teachers in particular were more prepared. In other schools, where teachers weren't explicitly being asked to align themselves to standards, it was a huge shock. For them, it used to be solely about Regents, and also Regents exams are very predictable and consistent, so it was pretty smooth sailing.
Let's move on to curriculum. You mentioned there is no CC-aligned curriculum in NY. I also heard NY state has recently come out against the CC, did you hear that?
Well what I heard is that they previously said they this year's 9th graders (class of 2017) would be held accountable but now the class of 2022 will be. So they've shifted accountability to the future.
What does accountability mean in this context?
It means graduation requirements. You need to pass 5 Regents exams to graduate, and right now there are two versions of some of those exams: one CC-aligned, one old-school. The question is who has to pass the CC-aligned versions to graduate. Now the current 9th grade could take either the CC-aligned or "regular" Regents in math.
I'm going to ask my 9th grade students to take both so we can gather information, even though it means giving them 3 extra hours of tests. Most of my kids pass 2 Regents in 9th grade, 2 in 10th, and 3 in 11th, and then they're supposed to be done. They only take those Regents tests in senior year that they didn't pass earlier.
What are the good and bad things about testing?
What's bad is how much time is lost, as we've already said. And also, it's incredibly stressful. You and I went to school and we had one big college test that was stressful, namely the SAT. In terms of us finishing high school, that was it. For these kids it's test, test, test, test. I don't think it's actually improved the quality of college students across the country. 20 years ago NY was the only one that had extra tests except California achievement tests, which I guess we sometimes took as well.
Another way to say it is that we did take some tests but it didn't take 5 weeks.
And it wasn't high stakes for the teacher!
Let's go straight there: what are the good/bad things for the teachers with all these tests?
Well it definitely makes the teachers more accountable. Even teachers think this: there is a cadre of protected teachers in the city, and the principals didn't want to take the time to get rid of them, so they'd excess them out of the schools, and they would stay in the system.
Now with testing it has become much more the principal's responsibility to get rid of bad teachers. The number of floating teachers is going down.
How did they get rid of the floaters?
A lot of different ways. They made them go into the schools, take interviews, they made their quality of life not great, and a lot if them left or retired or found jobs. Principals took up the mantle as well, and they started to do due diligence.
Sounds like the incentive system for over-worked principals was wrong.
Yes, although the reason it became easier for the principals is because now we have data. So if you're coming in as ineffective and I also have attendance data and observation data, I can add my observational data (subjective albeit rubric based) and do something.
If I may be more skeptical, it sounds like this data gathering was used as a weapon against teachers. There were probably lots of good teachers that have bad numbers attached to them that could get fired if someone wanted them to be fired.
Correct, except those good teachers generally have principals who protect them.
You could give everyone a bad number and then fire the people you want, right?
Correct.
Is that the goal?
Under Bloomberg it was.
Is there anything else you want to mention?
I think testing needs to be dialed down but not disappear. Education is a bi-polar pendulum and it never stops in the middle. We're on an extreme but let's not get rid of everything. There is a place for testing.
Let's get our CC standards, curriculum, and testing reasonable and college-aligned and let's keep it reasonable. Let's do it with standards across states and let's make sure it makes sense.
Also, there are some new tests coming out, called PARCC assessments, that are adaptive tests aligned to the CC. They are supposed to replace Regents down the line and be national.
Here's what bothers me about that. It's even harder to investigate the experience of the student with adaptive tests.
I'm not sure there's enough technology to actually do this anyway very soon. For example, we were given $10,000 for 500 student. That's not going to go far unless it takes 2 weeks to administer the test. But we are investing in our technology this year. For example, I'm looking forward to buying textbooks and get my updates pushed instead of having to buy new books every year.
Last question. They are redoing the SAT because rich kids are doing so much better. Are they just trying to get in on the test prep game? Because, here's the thing, there's no test that can't be gamed that's also easy to grade. It's gotta depend on the letters and grades. We keep trying to shortcut that.
Listen, this is what I tell the kids. What's going to matter to you is the letter of recommendation, so don't be an jerk to your fellow students or to the teachers. Next, are you going to be able to meet the minimum requirements? That's what the SAT is good for. It defines a lower bound.
Is it a good lower bound though?
Well, I define the lower bound as 1000 in total. My kids can target that. It's a reasonable low bar.
To what extent do your students – mostly inner-city, black girls interested in math and science – suffer under the wholly gamed SAT system?
It serves to give them a point of self-reference with the rest of the country. You have to understand, they, like most kids in the nation, don't have a conception of themselves outside of their own experience. The SAT serves that purpose. My kids, like many others, have the dream of Ivy League minus the understanding of where they actually stand.
So you're saying their estimates of their chances are too high?
Yes, oftentimes. They are the big fish in a well-defined pond. At the very least, The SAT helps give them perspective.
My favorite part of Broad's article is the caption of the video at the top, which sums it up nicely:
Funding the Future: As government financing of basic science research has plunged, private donors have filled the void, raising questions about the future of research for the public good.
In his article Broad makes a bunch of great points.
First, the fact that rich people generally ask for research into topics they care about ("personal setting of priorities") to the detriment of basic research. They want flashy stuff, bang for their buck.
Second, academics interested in getting funding from these rich people have to learn to market themselves. From the article:
The availability of so much well-financed ambition has created a new kind of dating game. In what is becoming a common narrative, researchers like to describe how they begged the federal science establishment for funds, were brushed aside and turned instead to the welcoming arms of philanthropists. To help scientists bond quickly with potential benefactors, a cottage industry has emerged, offering workshops, personal coaching, role-playing exercises and the production of video appeals.
If you think about it, the two issues above are kind of wrapped up together. Flashy academic content goes hand in hand with flashy marketing. Let's say goodbye to the true nerd who doesn't button up their cardigan correctly. And I don't know about you but I like those nerds. My mom is one of them.
This morning I thought of another way to express this issue, from the point of view of the individual scientist or mathematician, that might have profound resonance where the above just sounds annoying.
Namely, I believe that academic freedom itself is at stake. Let me explain.
I'm the last person who would defend our current tenure system. It's awful for women, especially those who want kids, and it often breeds a kind of arrogant laziness post-tenure. Even so, there are good things about it, and one of them is academic freedom.
And although theoretically you can have academic freedom without tenure, it is certainly easier with it (example from this piece: "In Oklahoma, a number of state legislators attempted to have Anita Hill fired from her university position because of her testimony before the U.S. Senate. If not for tenure, professors could be attacked every time there's a change in the wind.").
The dwindling tenured positions means there are increasing number of people trying to do research dependent upon outside grants and funding, and without the safety net of tenure. These people often lose their jobs when their funding flags, as we've recently seen at Columbia.
Now let's put these two trends together. We've got fewer and fewer tenure jobs, which are precariously dependent on outside funding, and we've got rich people funding their own tastes and proclivities.
Where does academic freedom shake out in that picture? I'm going to say nowhere.
I am back from Berkeley where I attended a couple of hours of conversations about MOOCs last Friday up at MSRI.
It was a panel discussion given mostly by math and stats people who themselves run MOOCs, and I was wondering if the people who are involved have a better sense of the side effects and feedback loops involved in the process. After all, I'm claiming that the MOOC Revolution will lead to the end of math research, and I wanted to be proven wrong.
Unfortunately, I left feeling like I have even more evidence that my fears will be realized.
I think the critical moment came when Ani Adhikari spoke. Professor Adhikari is in the second semester of giving her basic stats MOOC, and from how she described it, she is incredibly good at it, and there's a social network aspect of the class which seems like it's going really well – she says she spends 30 minutes to an hour a day on it herself, interacting with students. I think she said 28,000 students took it her first semester in addition to her in-class students at Berkeley. I know and respect Professory Adhikari personally, as I taught for her at the Berkeley Mills summer program for women many years ago. I know how devoted she is to good teaching.
Even so, she lost me late in the discussion when she explained that EdX, the platform which hosts her stats MOOC, wanted to offer her class three times a year without her participation. She said something to the effect that MOOC professors had to be "extra vigilant" about this outrageous idea and guard against it at all costs.
After all, she said, at the end of the day the MOOC videos are something like a fancy textbook, and we don't hand out textbooks and claim they are courses, so we by the same token cannot hand out MOOC videos (and presumably the social networks associated with them) and claim they are courses.
When I pressed her in the Q&A session as to how exactly she was going to remain vigilant against this threat, she said she has a legal contract with EdX that prevented them from offering the course without her approval.
And I'm happy for her and her great contract, but here are two questions for her and for the community.
First, how long until someone in math or stats makes a kick-ass MOOC and doesn't remember to have that air-tight legal contract? Or has an actual legal battle with EdX and realized their lawyers are not as expensive? Or believes that "information should be free" and does it with the express intention of letting the MOOC be replayed forever?
Second, how much sense does it make to claim that you and your presence are super critical to the success of a MOOC if 28,000 people took this class and you interacted at most one hour a day? Can you possibly claim that the average student benefitted from your presence? It seems to me that the value proposition for the average MOOC student is very similar whether you are there or not.
Overall the impression I got from the speakers, who were mostly MOOC evangelists and involved with MOOCs themselves, was that they loved MOOCs because MOOCs were working for them. They weren't looking much beyond that point to side effects.
There was one exception, namely Susan Holmes, who listed some side effects of MOOCs including a decreased need for math Ph.D.'s. Unfortunately the conversation didn't dwell on this, though, and it happened at the very end of the day.
Here's what I'd like to see: a conversation at MSRI about the future of math research funding in the context of MOOCs and a reduced NSF, where hopefully we come up with something besides "Jim Simons". It's extra ironic that the conversation, if it happens, would be held in the Simons Theater.
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Holy Grail of mathematics revealed as a truly 17th-century numerical and geometrical proof as a letter by Fermat to a colleague. This will withstand all challenges Advanced Probability and Statistics-Second Edition is a clear presentation of the basic topics in statistics and probability, but finishes with the rigorous topics an advanced placement course requires. Volume 2 includes the last 7 chapters and covers the following topics: Sampling Distributions and Estimations, Hypothesis Testing, Regression and Correlation, Chi-Square, Analysis of Variance and F-Distribution, and Non-Parametric Statistics. It also includes a collection of resources in the final chapter Middle School Math Grade 6 covers the fundamentals of fractions, decimals, and geometry. Also explored are units of measurement, graphing concepts, and strategies for utilizing the book's content in practical situations. Volume 1 includes the first 6 chapters Basic Probability and Statistics – A Short Course is an introduction to theoretical probability and data organization. Students learn about events, conditions, random variables, and graphs and tables that allow them to manage data Geometry FlexBook is a clear presentation of the essentials of geometry for the high school student. Topics include: Proof, Congruent Triangles, Quadrilaterals, Similarity, Perimeter & Area, Volume, and Transformations Algebra FlexBook is an introduction to algebraic concepts for the high school student. Topics include: Equations & Functions, Real Numbers, Equations of Lines, Solving Systems of Equations & Quadratic Equations.'
This book is a guide through a playlist of Calculus instructional videos. The format, level of details and rigor, and...
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This book is a guide through a playlist of Calculus instructional videos. The format, level of details and rigor, and progression of topics are consistent with a semester long college level second Calculus course, or equivalently, together with the first workbook, an AP Calculus BC course. The book further provides simple summary of videos, written definitions and statements, worked out examples--even though fully step-by-step solutions are to be found in the videos-- and an index. The playlist and the book are divided into 16 thematic learning modules. Exercises, some with and some without solutions, and sample tests with solutions are provided in a separate companion manual. The book can be used for self study, or as a textbook for a Calculus course following the "flipped classroom" model.
This book is the exercise companion to A youtube Calculus Workbook (part II). Its structures in modules mirrors that of the...
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This book is the exercise companion to A youtube Calculus Workbook (part II). Its structures in modules mirrors that of the workbook. The book includes, for 31 topics, a worksheet of exercises without solutions, which are typically meant to be either worked out in class with the help of the teacher or assigned, a homework set consisting of exercises similar to those of the worksheet, and the complete solutions of the homework sets. It also contains four mock tests with solutions, and a sample final exam with solutions.Additionally, a brief discussion of the use of the Workbook and the exercise book in a flipped classroom model is included.
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This site is a subsite of a larger site called Hippocampus. It contains collections of presentations on college algebra. The...
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This site is a subsite of a larger site called Hippocampus. It contains collections of presentations on college algebra. The site is partitioned into separate beginning algebra and two parts of intermediate algebra pieces. For each part, there are many topics covered where the student listens to a video and is prompted to intercatively work out problems or steps to a problem.
This tutorial teaches students about statistical power and how it is influenced by a variety of features of the hypothesis...
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This tutorial teaches students about statistical power and how it is influenced by a variety of features of the hypothesis testing situation. Students will be able to manipulate features of the test situation and see the effects on statistical power
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Linear Algebra and Its Applications
9780321385178
ISBN:
0321385179
Edition: 4th Pub Date: 2011 Publisher: Pearson Education (US)
Summary: Linear assimi...late. Since they are fundamental to the study of linear algebra, students'understanding of these concepts is vital to their mastery of the subject. David Lay introduces these concepts early in a familiar, concrete Rn setting, develops them gradually, and returns to them again and again throughout the text so that when discussed in the abstract, these concepts are more accessible.
Lay, David C. is the author of Linear Algebra and Its Applications, published 2011 under ISBN 9780321385178 and 0321385179. Five hundred eighty two Linear Algebra and Its Applications textbooks are available for sale on ValoreBooks.com, ninety used from the cheapest price of $91.20, or buy new starting at $161.20 4e. Book is in overall good condition!! Cover shows some edge wear and corners are lightly worn. Pages have a minimal to moderate amount of markings. FAS [more]
ALTERNATE EDITION: Instructor's Edition, 4 Class was Introduction to Linear Algebra. It was about learning the basics of linear algebra.
The book was very helpful for studying for the tests and trying to get a better understanding of what was being taught in class. The step by step processes outlined in the book definitely helped to make the material easier
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Elementary Linear Algebra
9780030973543
ISBN:
0030973546
Edition: 5 Pub Date: 1994 Publisher: Thomson Learning
Summary: Intended for the first course in linear algebra, this widely used text balances mathematical techniques and mathematical proofs. It presents theory in small steps and provides more examples and exercises involving computations than competing texts.
Grossman is the author of Elementary Linear Algebra, published 1994 under ISBN 9780030973543 and 0030973546. Eleven Elementary Linear Algebra textbooks are availa...ble for sale on ValoreBooks.com, three used from the cheapest price of $4.49, or buy new starting at $236 5th Edition May contain highlighting/underlining/notes/etc. May have used ... [more]
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A plain-English guide to the basics of trig Trigonometry deals with the relationship between the sides and angles of triangles... mostly right triangles. In practical use, trigonometry is a friend to astronomers who use triangulation to measure the distance between stars. Trig also has applications in fields as broad as financial analysis, music... more...
The Table of Integrals, Series, and Products is the essential reference for integrals in the English language. Mathematicians, scientists, and engineers, rely on it when identifying and subsequently solving extremely complex problems. Since publication of the first English-language edition in 1965, it has been thoroughly revised and enlarged on a regular... more...
This is the Proceedings of the ICM 2010 Satellite Conference on "Buildings, Finite Geometries and Groups" organized at the Indian Statistical Institute, Bangalore, during August 29 - 31, 2010. This is a collection of articles by some of the currently very active research workers in several areas related to finite simple groups, Chevalley... more...
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A Field Guide to for a second course in undergraduate abstract algebra. … Along the way, certain very interesting results are covered that are not often seen in books at this level … . includes numerous exercises which significantly extend material in the text." (William M. McGovern, SIAM Review, Vol. 47 (2), 2005)
"The textbook focuses on the structure of fields and is intended for a second course in abstract algebra … The reader will learn about equations, both polynomial and differential … . In explaining these concepts, the author also provides … leads the reader along many interesting paths. … There are exercises at the end of each chapter, varying in degree from easy to difficult. … author has incorporated pictures from the history of mathematics, including scans of mathematical stamps and pictures of mathematicians." (Zentralblatt für Didaktik und Mathematik, February, 2005)
"This nice volume starts with constructions with ruler and compass and uses these old problems as a peg for field theory, especially for Galois Theory. … it also deals with some mathematical pearls which one cannot find so easily in the textbooks and for which I like this text a lot … . The whole manuscript is written very carefully and can be heartily recommended to students and to teachers as well." (J. Schoissengeier, Monatshefte für Mathematik, Vol. 148 (4), 2006)
"The book can … be described as a good second course in abstract algebra focusing on the structure of fields in general and field extensions in particular. … There are many exercises … most of them rather challenging. … it is a well written text which is especially suitable for students about to study abstract algebra seriously." (P. Shiu, The Mathematical Gazette, Vol. 90 (519), 2006)
"The distinctive features of Chambert-Loir's book are as follows: first, it develops many results that should give undergraduate mathematics majors immediate and substantial satisfaction … second, the book concludes with a novel undergraduate-level introduction to the algebraic theory of differential equations. ... Summing Up: Highly Recommended. General Readers; upper-division undergraduates through professionals." (D. V. Feldman, CHOICE, Vol. 42 (10), 2005)
"This book treats mainly Galois theory of finite extensions of fields. All the material necessary for such a study is presented in this book … . There are a lot of exercises … . A nice feature is the inclusion of portraits of mathematicians who made important contributions to the subject of this book." (K. Kiyek, Mathematical Reviews, Issue 2005 h)
"This book is intended as a second course in algebra focusing mainly on Galois theory of the finite extensions of fields. … The book is easy to read and mostly self-contained. The large number of exercises at different levels makes it a valuable source both as the basis for a course or self-study." (G. Teschl, Internationale Mathematische Nachrichten, Issue 203, 2006)
"Toward advanced undergraduate students, the book focuses on those parts of abstract algebra which primarily deal with the structure of fields … and the related algebraic theory of differential equations. … the central theme of the text is field theory, together with its relations to some other areas in abstract algebra and to analysis. … No doubt, this fairly unique introduction to some central aspects of modern abstract algebra and its applications is a highly welcome and valuable complement to … literature in the field." (Werner Kleinert, Zentralblatt MATH, Vol. 1155, 2009)
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More About
This Textbook
Overview
Teaching Secondary Mathematics, Third Edition is practical, student-friendly, and solidly grounded in up-to-date research and theory. This popular text for secondary mathematics methods courses provides useful models of how concepts typically found in a secondary mathematics curriculum can be delivered so that all students develop a positive attitude about learning and using mathematics in their daily lives.
A variety of approaches, activities, and lessons is used to stimulate the reader's thinking—technology, reflective thought questions, mathematical challenges, student-life based applications, and group discussions. Technology is emphasized as a teaching tool throughout the text, and many examples for use in secondary classrooms are included. Icons in the margins throughout the book are connected to strands that readers will find useful as they build their professional knowledge and skills: Problem Solving, Technology, History, the National Council of Teachers of Mathematics Principles for School Mathematics, and "Do" activities asking readers to do a problem or activity before reading further in the text. By solving problems, and discussing and reflecting on the problem settings, readers extend and enhance their teaching professionalism, they become more self-motivated, and they are encouraged to become lifelong learners.
New in the Third Edition:
*All chapters have been thoroughly revised and updated to incorporate current research and thinking.
*The National Council of Teachers of Mathematics Standards 2000 are integrated throughout the text.
*Chapter 5, Technology, has been rewritten to reflect new technological advances.
*A Learning Activity ready for use in a secondary classroom has been added to the end of each chapter.
*Two Problem-Solving Challenges with solutions have been added at the end of each chapter.
*Historical references for all mathematicians mentioned in the book have been added within the text and in the margins for easy reference.
*Updated Internet references and resources have been incorporated to enhance the use of the text.
Editorial Reviews
Booknews
This text/CD-ROM package covers general fundamentals, mathematics education fundamentals, and content and strategies, with emphasis on technology as a teaching tool. This second edition incorporates recent research and technological advancements, and contains a new chapter on probability and statistics. The CD-ROM contains MathXpert Plus Calculus Assistant, which mathematics teachers can use with secondary students. Brumbaugh teaches college and K-12. Rock is professor of mathematics education and the coordinator of secondary education at the University of Mississippi
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...If you got high C's or solid B's in math and now you're mystified by some of the new stuff in calculus and precalculus, or if you're baffled by how much algebra and geometry you forgot during the summer (or during those years when you didn't
| 677.169 | 1 |
9780321287137
ISBN:
0321287134
Edition: 3 Pub Date: 2005 Publisher: Addison Wesley
Summary: Linear algebra is relatively easy for students during the early stages of the course, when the material is presented in a familiar, concrete setting. But when abstract concepts are introduced, students often hit a brick wall. Instructors seem to agree that certain concepts (such as linear independence, spanning, subspace, vector space, and linear transformations), are not easily understood, and require time to assimi...late. Since they are fundamental to the study of linear algebra, students' understanding of these concepts is vital to their mastery of the subject. Lay introduces these concepts early in a familiar, concrete Rn setting, develops them gradually, and returns to them again and again throughout the text so that when discussed in the abstract, these concepts are more accessible.
David C. Lay is the author of Linear Algebra and Its Applications, 3rd Updated Edition (Book & CD-ROM), published 2005 under ISBN 9780321287137 and 0321287134. Two hundred thirty two Linear Algebra and Its Applications, 3rd Updated Edition (Book & CD-ROM) textbooks are available for sale on ValoreBooks.com, six used from the cheapest price of $27.75, or buy new starting at $123.97 the text. includes new still sealed disc.Every heavytail order ships with a sweet! We carefully hand clean and reinspect each and every item we ship. Our qualit [more]
no markings in the text. includes new still sealed disc
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Details about Functions Modeling Change:
This is a new edition of the precalculus text developed by the Consortium based at Harvard University and funded by a National Science Foundation Grant. The text is thought-provoking for well-prepared students while still accessible to students with weaker backgrounds. It provides numerical and graphical approaches as well as algebraic approaches to give students another way of mastering the material. This approach encourages students to persist, thereby lowering failure rates. A large number of real-world examples and problems enable students to create mathematical models that will help them understand the world in which they live.
The focus is on those topics that are essential to the study of calculus and these topics are treated in depth. Linear, exponential, power, and periodic functions are introduced before polynomial and rational functions to take advantage of their use to model physical phenomena. Building on the Consortium's Rule of Four: Each function is represented symbolically, numerically, graphically, and verbally where appropriate.
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Meaning in Mathematics 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 the complexity of meaning in mathematics education is a matter of huge importance. There are twin directions in which discussions have developed—theoretical and practical—and this book seeks to move the debate forward along both dimensions while seeking to relate them where appropriate. A discussion of meaning can start from a theoretical examination of mathematics and how mathematicians over time have made sense of their work. However, from a more practical perspective, anybody involved in teaching mathematics is faced with the need to orchestrate the myriad of meanings derived from multiple sources that students develop of mathematical knowledge.
This book presents a wide variety of theoretical reflections and research results about meaning in mathematics and mathematics education based on long-term and collective reflection by the group of authors as a whole. It is the outcome of the work of the BACOMET (BAsic COmponents of Mathematics Education for Teachers) group who spent several years deliberating on this topic. The ten chapters in this book, both separately and together, provide a substantial contribution to clarifying the complex issue of meaning in mathematics education.
This book is of interest to researchers in mathematics education, graduate students of mathematics education, under graduate students in mathematics, secondary mathematics teachers and primary teachers with an interest in mathematics.
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Meanings of Meaning of Mathematics.- "Meaning" and School Mathematics.- The Meaning of Conics: Historical and Didactical Dimensions.- Reconstruction of Meaning as a Didactical Task: The Concept of Function as an Example.- Meaning in Mathematics Education.- Collective Meaning and Common Sense.- Mathematics Education and Common Sense.- Communication and Construction of Meaning.- Making Mathematics and Sharing Mathematics: Two Paths to Co-Constructing Meaning?.- The Hidden Role of Diagrams in Students' Construction of Meaning in Geometry.- What's a Best Fit? Construction of Meaning in a Linear Algebra Session.- Discoursing Mathematics Away.- Meaning and Mathematics.
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books.google.com - The present volume is a translation, revision and updating of our book (pub lished in French) with the title "Geometrie Algebrique Reelle". Since its pub lication in 1987 the theory has made advances in several directions. There have also been new insights into material already in the French edition.... Algebraic Geometry
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A+ National Pre-apprenticeship Maths and Literacy for Hospitality by Andrew Spencer
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Pre-apprenticeship Maths and Literacy for Hospitality is a write-in workbook that helps to prepare students seeking to gain a Hospitality Apprenticeship. It combines practical, real-world scenarios and terminology specifically relevant to the Hospitality industry, and provides students with the mathematical skills they need to confidently pursue a career in the Hospitality trade. Mirroring the format of current apprenticeship entry assessments, Pre-apprenticeship Maths and Literacy for Hospitality includes hundreds of questions to improve students' potential of gaining a successful assessment outcome of 75-80% and above. This workbook will therefore help to increase students' eligibility to obtain a Hospitality Apprenticeship. Pre-apprenticeship Maths and Literacy for Hospitality also supports and consolidates concepts that students studying VET (Vocational Educational Training) may use, as a number of VCE VET programs are also approved pre-apprenticeships. This workbook is also a valuable resource for older students aiming to revisit basic literacy and maths in their preparation to re-enter the workforce at the apprenticeship level.
Buy A+ National Pre-apprenticeship Maths and Literacy for Hospitality book by Andrew Spencer from Australia's Online Bookstore, Boomerang Books.
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Fundamentals of Discrete Mathematics
Course
description
The main goal of the
course is to present some basic facts and
techniques about "counting". We will deal with finite sets and their
subsets, permutations, factorials, binomial coefficients, Pascal's
triangle, the Binomial Theorem, prime numbers and divisors, linear
recurrences (with particular attention to Fibonacci numbers),
generating functions. We will also study some discrete structures such
as graphs and trees, and discuss some classical problems and algorithms
in this setting. Finally, we will discuss some applications of discrete
mathematics to the analysis of certain basic algorithms (e.g. classical
sorting algorithms) and their computational complexity.
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Delving Deeper Delving Deeper: Sequences and Polynomials Part I: Guidelines for Finding a Next Term and a General Term for Any Given Finite Sequence and Part II
Dorothy Meserve, Bruce Meserve, Kathleen Jackson
February 2007, Volume 100, Issue 6, Page 426
Abstract: This issue: Sequences. The aim of Delving Deeper is for teachers to pose and solve novel math problems, expand on mathematical connections, or offer new insights into familiar math concepts. Delving Deeper focuses on mathematics content appealing to secondary school teachers. It provides a forum that allows classroom teachers to share their mathematics from their work with students, their classroom investigations and products, and their other experiences. Delving Deeper is a regular department of Mathematics Teacher.
The National Council of Teachers of Mathematics is the public voice of mathematics education, supporting teachers to ensure equitable mathematics learning of the highest quality for all students through vision, leadership, professional development, and research.
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Mathematical ModellingOver the past decade there has been an increasing demand for suitable material in the area of mathematical modelling as applied to science, engineering, business and management. Recent developments in computer technology and related software have provided the necessary tools of increasing power and sophistication which have significant implications for the use and role of mathematical modelling in the above disciplines. In the past, traditional methods have relied heavily on expensive experimentation and the building of scaled models, but now a more flexible and cost effective approach is available through greater use of mathematical modelling and computer simulation. In particular, developments in computer algebra, symbolic manipulation packages and user friendly software packages for large scale problems, all have important implications in both the teaching of mathematical modelling and, more importantly, its use in the solution of real world problems. Many textbooks have been published which cover the art and techniques of modelling as well as specific mathematical modelling techniques in specialist areas within science and business. In most of these books the mathematical material tends to be rather tailor made to fit in with a one or two semester course for teaching students at the undergraduate or postgraduate level, usually the former. This textbook is quite different in that it is intended to build on and enhance students' modelling skills using a combination of case studies and projects.
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solve equations, and simplify expressions and inequalities. You will know how to represent functions and linear equations on a graph.
Read More
You will be able to calculate equations of two or three variables. You will learn how to operate with matrices and simplify polynomials and radical expressions. This course will demonstrate how to graph a quadratic function, solve conic sections, polynomial functions, rational expressions, exponential and logarithmic functions. You will understand how to solve geometric sequences and series. You will learn about the Binomial theorom. You will be more aware of permutations, combinations, probabilities and trigonometric functions such as angles, sines, cosines, circular functions and inverse functions.
Read Less
Alyssa ComptonUnited States of America
Good explanations, but is there a way we can do practice problems and submit those? Maybe have quizzes in addition to the assessment at the end? I think it would be cool if we could have about a 5-8 problem quiz and then have video explanations that can only be accessed after one has completed the quiz. I'm also finding the first module rather boring- the topic is "advanced algebraic concepts and applications", and I admit that I found that first bit very rudimentary. However, I can see that it picks up as one progresses, so I know it won't stay that way. You might consider making that first module an optional review that isn't graded for completion, so those who don't need it don't have to waste time.
I like the examples, wish there was more practice. I am enjoying the class, I like the module layout and I can tell it will be successful. 2014-08-08 15:08:50
Abubakar AlhassanGhana
Course Module: Module 1: Equations and inequalitiesCourse Topic: Solve equations and simplify expressionsComment: the course is good but i have finished learning and needs exam to write for my certificate 2013-04-28 09:04:00
Jan DeurwaarderBotswana
Course Module: Module 2: How to graph functions and linear equations Course Topic: Functions and linear equations Comment: Looked at and went through Module 1 and 2 of Adv Algebraic concepts and applications in Maths - however my progress states that each (sub)section is incomplete. What haven't I covered?? 2013-01-16 09:01:49
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Calculators Are for Calculating, Mathematica Is for Calculus
Andy Dorsett
In this Wolfram Mathematica Virtual Conference 2011 course, learn different ways to use Mathematica to enhance your calculus class, such as using interactive models and connecting calculus to the real world with built-in datasets
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How to Ace the Rest of Calculus : The Streetwise Guide : Including Multi-Variable Calculus - 01 edition
Summary: Do you remember being hopelessly confused in calculus class? Afterwards, you asked your brainy friend over a cup of coffee, "What was going on in that class?" Your friend then explained it all to you in five minutes flat, making it crystal clear. "Oh," you said, "is that all there is to it?" Later, you wished that friend was around to explain all the lectures to you.
The original How to Ace Calculus played the role of that fri...show moreend for a first-semester calculus class. Now meet your new buddy, How to Ace the Rest of Calculus: The Streetwise Guide. Written by three gifted teachers, it provides humorous and highly readable explanations of the key topics of second and third semester calculus--such as sequences and series, polar coordinates and multivariable calculus--without the technical details and fine print that would be found in a formal text.
Funny, irreverent, and flexible, How to Ace the Rest of Calculus shows why learning calculus can be not only a mind-expanding experience but also fantastic fun. ...show less
Has very little wear on the cover.Has clean, unmarked pages. Binding is tight and in excellent condition.Buy with confidence! Experienced Seller. Fast Shipping!
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Over 100 math formulas at high school level. The covered areas include algebra, geometry, calculus, trigonometry, probability and statistics. Most of the formulas come with examples for better understa
Math Ref Free is a free version of the award winning education app Math Ref. This app gives you just a sample (over 700) of the over 1,400 helpful formulas, figures, tips, and examples that are include
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The concept of the Euclidean simplex is important in the study of n-dimensional Euclidean geometry. This book introduces for the first time the concept of hyperbolic simplex as an important concept in n-dimensional hyperbolic geometry.
Following the emergence of his gyroalgebra in 1988, the author crafted gyrolanguage, the algebraic language... more...
Convexity is an ancient idea going back to Archimedes. Used sporadically in the mathematical literature over the centuries, today it is a flourishing area of research and a mathematical subject in its own right. Convexity is used in optimization theory, functional analysis, complex analysis, and other parts of mathematics.
Convex AnalysisA perennial bestseller by eminent mathematician G. Polya, How to Solve It will show anyone in any field how to think straight. In lucid and appealing prose, Polya reveals how the mathematical method of demonstrating a proof or finding an unknown can be of help in attacking any problem that can be ?reasoned? out?from building a bridge to winning... more...
Differential Forms and the Geometry of General Relativity provides readers with a coherent path to understanding relativity. Requiring little more than calculus and some linear algebra, it helps readers learn just enough differential geometry to grasp the basics of general relativity.
The book contains two intertwined but distinct halves. ... more...
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applications of maths in real life ppt
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lution of some of the fundamental problems in commutative algebra
(polynomial ideal theory, algebraic geometry )
The method (theory plus algorithms) of Gröbner Bases provides a uniform approach to
solving a wide range of problems expressed in terms of sets of multivariate polynomials.
ons of the equations in a reduced way)
The forward kinematic problem
Problem: We have 3 dimensions:
Looking for a point with coordinates (u, v, w)
But we have 6 variables! (x, y, z, u, v, w)
We need to get rid of (x, y, z)
Solution:
Elimination
The forward kinematic problem
The Groebner basis " one of the polynomials looks like this:
u2 + v2 + w2= 5 " equation of a sphere
Problem:
Number of points presented by (x, y, z, u, v, w) is not equal to the number of points presented by
the above equation
Not all points that lie in that sphere can be actually reached
..................[:=> Show Contents <=:]
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Product Description
Learning--and teaching!--math does not have to be difficult! Give your students and yourself the tools to succeed with this Saxon Teacher and Saxon 7/6 kit combination! Introduce your middle-schoolers to the concepts they'll need for upper-level algebra and geometry, including functions and coordinate graphing; integers; multiplying decimals and fractions; radius, circumference, and pi; compound interest; exponential expressions; prime factorization; statistics and probability; and complementary and supplementary angles. This kit includes Saxon's 4th Edition Math 7/6 textbook, solutions manual, and tests/worksheets book.
Give your Saxon Math 7/6 students support and reinforcement! Comprehensive lesson instructions feature complete solutions to every practice problem, problem set, and test problem with step-by-step explanations and helpful hints. These user-friendly CD-ROMs contain hundreds of hours of instruction, allowing students to see and hear actual textbook problems being worked on a digital whiteboard. A slider button allows students to skip problems they don't need help on, or rewind, pause, or fast-forward to get to the sections they'd like to access. Problem set questions can be watched individually after the being worked by the student; the practice set is one continuous video that allows for easy solution review. For use with the 4th Edition. Four Lesson CDs and 1 Test Solutions CD included.
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Need Help with Algebra?
We have assembled a collection of Algebra resources to help you master your high school algebra or College Algebra courses, and to help you review as you progress to more challenging math and science classes.
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Algebra and Trigonometry : Graphs and Models -Text Only - 4th edition
Summary: The authors help students "see the math" through their focus on functions; visual emphasis; side-by-side algebraic and graphical solutions; real-data applications; and examples and exercises. By remaining focused on today's students and their needs, the authors lead students to mathematical understanding and, ultimately, success in classGood
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A lesson designed to abstract the idea of a line; introduce other methods of solving simultaneous equations besides graphing; and to develop problem solving skills. From the Algebra and Trigonometry section of a collection of almost 200 single concept lessons by the Science and Mathematics Initiative for Learning Enhancement.
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Summary: Provides completely worked-out solutions to all odd-numbered exercises within the text, giving you a way to check your answers and ensure that you took the correct steps to arrive at an answer
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clear as possible.
The early chapters provide students with background by investigating the basic properties of groups, rings, fields, and modules. Later chapters examine the relations between groups and sets, the fundamental theorem of Galois theory, and the results and methods of abstract algebra in terms of algebraic number theory, algebraic geometry, noncommutative algebra, and homological algebra, including categories and functors. An extensive supplement to the text delves much further into homological algebra than most introductory texts, offering applications-oriented results. Solutions to all problems appear
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Cliffs Quick Review for Geometry - 01 edition
Summary: When it comes to pinpointing the stuff you really need to know, nobody does it better than CliffsNotes. This fast, effective tutorial helps you master core geometry concepts -- from perimeter, area, and similarity to parallel lines, geometric solids, and coordinate geometry -- and get the best possible grade.
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Ed Kohn, MS is an outstanding educator and author with over 33 years experience teaching mathematics. Currently, he is the testing coordinator and math department chairman at Sherman Oaks Center for Enriched Studies.
2001A used copy at a fantastic price.GuthrieBooks Spring Branch, TX
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Announcements
Information For
Mathematics
Mission
The mission of the Department of Mathematics of Eugenio Maria de Hostos Community College is to provide for our multicultural student population, a majority of whom are female, minorities and from non-traditional backgrounds, a supportive learning environment, a strong foundation of basic knowledge in mathematics, and to prepare them for a variety of careers in mathematics and related fields.
To advance this mission, the Department of Mathematics emphasizes for all its students a conceptual understanding of mathematics together with problem solving and higher order analytic skills. The department strives to develop students' ability to think precisely, creatively and critically, to speak clearly and persuasively, and to be aware of the intellectual power and significance of mathematical reasoning in today's technological society. The Department of Mathematics is committed to the highest standards of excellence in teaching and service.
Goals
To provide students with the mathematical knowledge and skills they need to pursue careers in Computer Information Systems, the Natural Sciences, Engineering Sciences, Mathematics, Allied Health, Business Administration, Accounting, Public Administration, Health and Human Services, Paralegal, and Office Administration and Technology.
To provide students in the liberal arts programs with a broader understanding of the foundation of mathematics, permeating different topics and transcending mere computation, with emphasis on logic and systematic constructions leading to more sophisticated mathematical models.
The mathematics curriculum provides a variety of offerings that survey the meaning of mathematics as a logical system. The particular models chosen to exemplify these logical principles will vary from time to time depending on the current interests of our students and faculty. As such models are meant to be illustrations only, the choice can be selective without any change of purpose.
Although the language of instruction is English, a few sections of some courses in the Mathematics Department are offered in Spanish, depending upon student needs. Language-enhanced materials are used in all developmental courses to support students' linguistic needs.
Effective Fall 2003, no student may be placed in a college-level Mathematics course who has not passed or been exempted from the CUNY Mathematics Skills Test.
Students planning to continue study in mathematics or mathematics related areas are advised to consult with the Mathematics Department Chairperson.
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Mathematics and Statistics
Math Placement Exam
The math placement exam is comprised of an adaptive elementary algebra (EA) test. All examinees begin testing in the elementary algebra section, and will then branch out into different EA questions based upon their performance. Each test is adaptive (interactive), which means that as you complete an item, the computer evaluates your skill level and gives you another problem of either increasing or decreasing difficulty. Questions are automatically chosen to yield the most information about your skill level based upon your answers.
Each section will offer you a description of the skills to be tested, accompanied by practice problems and answers. Feel free to scroll down to the section you would like to practice, and see how your skills measure up!
Elementary Algebra
The elementary algebra version of the placement test will determine your placement into either MATH 009, MATH 012 or MATH 106/107.
The first set of problems, comprised of twelve questions and equal to about 8–17 percent of your grade, will quiz you on:
Operations with integers and rational numbers
Computations with absolute value
Ordering
The second group of problems, adding up to around 42–67 percent of your grade, will cover:
Operations with algebraic expressions
Evaluation of simple formulas and expressions
Addition, subtraction, multiplication, and division of monomials and polynomials
Evaluation of positive rational roots and exponents
Simplifying algebraic fractions and factoring
The third segment amounts to around 17–50 percent of the test and will cover these topics:
Solutions of equations, inequalities and word problems
Solving of equations by factoring
Solving word problems presented in an algebraic context, including geometric reasoning and graphing
Translation of written phrases into algebraic expressions
Practice Test
We encourage you to review the Practice Math Placement Test to help you better prepare for the elementary algebra section of the math placement exam. Some of these examples may or may not be similar to the problems on the actual exam. We recommend that if you have difficulty completing these problems, you should probably review the skills generally covered in a first-year high school algebra class.
Additional Information
For more information on placement tests at UMUC—including where and when to take them—please visit the Placement Testing page from Exams and Testing Services.
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Bull Valley, IL PhysicsDiscrete Math is hard to define, as it often means "something advanced which is not calculus." It is often very similar to Finite Math, which may cover basic counting and probability, or even a bit of linear algebra. But discrete math and combinatorics is also a great enrichment for advanced stu
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Hardcover Good 818593195 is part two of a two-volume introduction to real analysis and is intended for honours undergraduates who have already been exposed to calculus. The emphasis is on rigour and on foundations. The material starts at the very beginning--the construction of the number systems and set theory--then goes on to the basics of analysis (limits, series, continuity, differentiation, Riemann integration), through to power series, several variable calculus and Fourier analysis, and finally to the Lebesgue integral. These are almost entirely set in the concrete setting of the real line and Euclidean spaces, although there is some material on abstract metric and topological spaces. There are also appendices on mathematical logic and the decimal system. The entire text (omitting some less central topics) can be taught in two quarters of twenty-five to thirty lectures each. The course material is deeply intertwined with the exercises, as it is intended that the student actively learn the material (and practice thinking and writing rigorously) by proving several of the key results in the theory. In the third edition, several typos and other errors have been corrected and a few new exercises have been added
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Calculus
Calculus
Study Guide for Stewart's Multivariable Calculus, 6th
Student Solutions Manual for Stewart's Multivariable Calculus, 6th
CalcLabs for Maple for Stewart's Multivariable Calculus, 6th
Single Variable Calculus- Study Guide
Student Solutions Manual for Stewart's Single Variable Calculus, 6th
Calclabs With Maple for Stewart's Single Variable Calculus
CalcLabs with Mathematica for Stewart's Multivariable Calculus, 6th
CalcLabs with Mathematica for Stewart's Single Variable Calculus, 6th
Summary
Reading a calculus textbook is different from reading a newspaper or a novel, or even a physics book. Don't be discouraged if you have to read a passage more than once in order to understand it. You should have pencil and paper at hand to make a calculation or sketch a diagram.
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Web Site Webmath.com This is a dynamic math website where students enter problems and where the site's math engine solves the problem. Students in most cases are given a step-by-... Curriculum: Mathematics Grades: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12
8.
Web Site Prentice Hall Math Textbook Resources This site has middle school and high school lesson quizzes, vocabulary, chapter tests and projects for most chapters in each textbook. In some sections, ther... Curriculum: Mathematics Grades: 6, 7, 8, 9, 10, 11, 12
9.
Web Site Dave's Short Trig Course Check out the short trigonometry course and learn the new way of learning trig. This short course breaks into sections and allows user to learn at his/her o... Curriculum: Mathematics Grades: 9, 10, 11, 12
By Resource Type:
Web Site Document or Handout Image Template Book Video
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Humble Geometry almost guarantee that you will have ?Aha! So that?s how it works!? moments as algebra becomes more familiar and understandable. Algebra 2 builds on the foundation of algebra 1, especially in the ongoing application of the basic concepts of variables, solving equations, and manipulations such as factoring.
...And if you/your child do not know "how", then you are at a disadvantage. This level of knowledge is a necessity for anyone that wants to do well in the world, no matter what level they are or that they aspire to. I really enjoy opening up the world of "How To..." to kids, seeing that look on their faces of "maybe I can do this, I really think I can do this" to "I CAN do this!".
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Choose a function from a list or enter your own, and see the way its equation changes as the function is translated horizontally and vertically, reflected over various lines, stretched, and compressed... More: lessons, discussions, ratings, reviews,...
The applet plots consecutive terms of a user-defined sequence or a series of functions. Those can be, in particlular, Taylor series and Fourier series. A piecewise defined limit function can also be eEnter a set of data points and a function or multiple functions, then manipulate those functions to fit those points. Manipulate the function on a coordinate plane using slider bars. Learn how each co... More: lessons, discussions, ratings, reviews,...
This game explores functions in a different way: a and b = f(a) are drawn in a unique numerical line. When the user changes a, b = f(a) changes following a rule. The objective of the game is to discov... More: lessons, discussions, ratings, reviews,...
Highlight the language of domain and range, and the ideas of continuity and discontinuity, with this tool that links symbolic and graphic representations of each interval of a piecewise linear functio
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Table of Contents
THE MYTHS
Public acceptance of deficient standards contributes significantly to poor performance in mathematics education.
—Everybody Counts, 1989
The source of many of these difficulties can be found in public (and parental) attitudes about mathematics that are rooted more in myth than in realityMoving Beyond Myths: Revitalizing Undergraduate Mathematics
THE MYTHS
Public acceptance of deficient standards contributes significantly to poor performance in mathematics education.
—Everybody Counts, 1989The source of many of these difficulties can be found in public (and parental) attitudes about mathematics that are rooted more in myth than in reality:
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Moving Beyond Myths: Revitalizing Undergraduate Mathematics
Figure 1
Total undergraduate enrollment in mathematical sciences departments at U.S. colleges and universities.
Building Confidence
At Spelman College in Atlanta, 8 percent of the graduates major in mathematics—a rate far greater than the national average of 1.6 percent.
The success of the mathematics and natural science program at Spelman is due to the special attention given to students that builds their confidence in their own ability to master mathematics. All natural science students participate in an eight-week summer program prior to the beginning of their first year, during which study skills are developed and role models are established. This careful mentoring is continued throughout the undergraduate program and develops into opportunities for research experiences and special honors sections. The faculty devotes a great deal of energy to advising, since student motivation is the most powerful factor in learning.
Not only does Spelman produce a large number of minority mathematicians and scientists, but its NASA Program for Women in Science and Engineering graduates a higher-than-average percentage of women mathematics majors who go on to pursue graduate studies in mathematics.
Myth: Success in mathematics depends more on innate ability than on hard work.
Reality: Sustained effort can carry most students to a satisfactory level of achievement in mathematics. Compare music and mathematics: although in both areas genetic factors clearly play a role at the very highest levels of creative achievement, parents and teachers generally believe that children can learn to play music at a reasonable level if only they exert sufficient effort. As a consequence, many
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Moving Beyond Myths: Revitalizing Undergraduate Mathematics
students achieve success and personal satisfaction from their study of music. Whenever parents or teachers believe that genetic ability is the primary factor contributing to success in mathematics, students are likely to fail before they begin; when expectations of success are high, so is the resulting performance.
Myth: Women and members of certain ethnic groups are less capable in mathematics.
Reality: The popular notion that women, Blacks, and Hispanics "can't do math" is just an expression of ignorance or prejudice. Ample evidence shows such beliefs to be false. Experiences of countries such as Holland and Japan belie this myth, as do results from numerous innovative programs in the United States. Such examples demonstrate unequivocally that most college students can succeed in mathematics when learning takes place in an appropriate structure and context.
Myth: Most jobs require little mathematics.
Reality: The truth is just the opposite: more and more jobs—especially those involving the use of computers—require the capability to employ sophisticated quantitative skills. Although a working knowledge of arithmetic may have sufficed for jobs of the past, it is clearly not enough for today, for the next decade, or for the next century.
Myth: All useful mathematics was discovered long ago.
Reality: Mathematical discoveries are essential for industrial competitiveness. Without advances in mathematics we would have neither telephones nor computers, neither jet airplanes nor international banking. Technology depends on both old and new mathematics for innovation and power. Indeed, more new mathematics is being created and used each year than ever before in history.
Mathematics in Action
One way to link undergraduate mathematics to industrial research and development is through student projects in mathematical modelling. Many such programs are patterned after the Mathematics Clinic, which began at Harvey Mudd College nearly twenty years ago. In these programs, which now operate in dozens of institutions, a team consisting of one or more faculty and several students works on an unsolved mathematically oriented problem that comes from a company or government agency.
The problems are usually open-ended and must first be cut down to a manageable size. Faculty leaders assist with the mathematical model and give "short courses" on the mathematics that seems to be needed. Different students work on different parts of the problem, parts that suit their interests and expertise, but teamwork is the mode of operation. Students must make formal oral presentations in terms understandable to the client; as a result, they develop strong expository skills. Written reports are submitted to the client at the end of the project, and so the writing involved in these reports is also a part of the students' education.
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Moving Beyond Myths: Revitalizing Undergraduate Mathematics
Texas Prefreshman Engineering Program
The Texas Prefreshman Engineering Program (TexPREP) was started in 1986 as a statewide expansion of the successful San Antonio PREP program begun in 1979 by Manuel P. Berriozabal, Professor of Mathematics at the University of Texas at San Antonio. The purposes of TexPREP are to recruit potential future scientists and engineers by identifying high-achieving middle school and high school students and by providing these students with academic reenforcement to pursue science and engineering fields. The program operates in 14 Texas cities on 19 college campuses.
Of the 4500 students who have participated in TexPREP, more than three-quarters have been minority students and half have been women. Of the college-age participants, nearly 90 percent either are attending college, plan to attend, or have graduated from college. Sixty percent of TexPREP graduates major in science or engineering fields. TexPREP features a strong academic component, with courses in logic, algebra, probability and statistics, problem solving, engineering, computer science, physics, and technical writing. Other activities include field trips, guest speakers, and practice SAT examinations.
Myth: To do mathematics is to calculate answers.
Reality: Rarely do workers or researchers confront mathematical problems requiring primarily calculation. Authentic problems are often ambiguous, admitting many forms and several answers. Mathematical power is revealed as much by the act of identifying and properly posing problems as by application of specific techniques and algorithms.
Myth: Only scientists and engineers need to study mathematics.
Reality: Mathematics is a science of patterns that is useful in many areas. Indeed, the most rapid areas of growth in applications of mathematics have been in the social, biological, and behavioral sciences. Financial analysts, legal scholars, political pollsters, and sales managers all rely on sophisticated mathematical models to analyze data and make projections. Even artists and musicians use mathematically based computer programs to aid in their work. No longer just a tool for the physical sciences, mathematics is a language for all disciplines.
If these myths were benign, with effects limited to the ignorance of those who believe them, they might be safely ignored. But ignorance in parents and teachers begets ignorance in students. Harmful myths about mathematics metastasize to the body politic, spreading ignorance and excusing underachievement throughout society. Efforts to eradicate these pernicious myths will require sustained support at all educational levels, but especially in colleges and universities where society's leaders are educated
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Prealgebra
MAT
0018
This college preparatory course is designed to build basic skills in mathematics to prepare students for elementary algebra. Topics include basic operations with integers, rational numbers, and decimals; percent usage; geometric figures and their measures, simplification of polynomials, and equation solving techniques. This course must be completed with a grade of "C" or higher. This course does not apply toward a degree
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