Jun 15
  1.  Precalculus

Trigonometry is a sine of the times. -Unknown 



College Algebra Eleftherios Gkioulekas University of Texas Pan American (G) Very nice and carefully handwritten presentation of the elements of mathematics-sans Euclidean geometry-which were once expected to be known by any person who successfully graduated high school in America. These days, this would be what’s usually called a “precalculus” course with basic algebra as the prerequisite. In this age of a dying American public education-where it’s clear in another generation or so, a quality education will be available only to the children of the wealthy in this country-having such sources available is paramount.Sets, equations and inequalities, systems of equations, functions graphing functions, polynomial functions and he ends with exponentials and logarithms Gkiouslekas’s notes are one of the best sources I know for this purpose-he uses mature mathematical notation from jump, precise definitions and lots of good examples.These notes should be downloaded immediately by any student needing to learn this material right the first time or any instructor for young students.


College  Mathematics (804-107) Kevin Mirus Madison College, Fall 2010      (G) The fact a course like this is all too common at today’s American universities is depressing beyond words. The fact that it’s too often necessary is heartbreaking to the point of contemplating suicide if you’re a young college  mathematics instructor. Before I go off on a rant here, let me return to the business at hand. The course description sums up nicely the course’s content:    This course is designed to review and develop fundamental concepts of mathematics pertinent to the areas of: 1) arithmetic and algebra; 2) geometry and trigonometry; and 3) probability and statistics.Special emphasis is placed on problem solving, critical thinking and logical reasoning, making connections, and using calculators. Topics include performing arithmetic operations and simplifying algebraic expressions, solving linear equations and inequalities in one variable, solving proportions and incorporating percent applications, manipulating formulas, solving and graphing systems of linear equations and inequalities in two variables, finding areas and volumes of geometric figures, applying similar and congruent triangles, converting measurements within and between U.S. and metric systems, applying Pythagorean Theorem, solving right and oblique triangles, calculating probabilities, organizing data and interpreting charts, calculating central and spread measures, and summarizing and analyzing data.   This is a course for nonmathematics majors who have completely forgotten all their basic mathematical skills except for how to count to 10. Ok, I’m being snarky, I apologize. But seriously-I can understand the need to review basic plane geometry and algebraic equations for weak freshman, especially for those who are beginning or returning to college after a 20-30 year absence from the classroom. That I understand. I even understand the usefulness of learning basic inequalities, which really aren’t taught correctly from the beginning at most current American high schools and colleges. But do students entering college really need to learn fractions, scientific notation and decimals? Seriously?!?Still,it does contain some nice useful stuff for advanced grade and high school students as well as a lot of nice practice problems and labs and for the aforementioned rusty returning students, it does give a good review with lots of practice.  I’m just upset our system has degenerated so far that such a course is necessary. .


CK-12 Foundations | Browse Mathematics / Algebra (G) This is quickly becoming the preeminent site for teachers at the pre-college level to find well written,high tech and structured classroom resources of all kinds for both courses and self-study. It is part of the awesome STEM (The Science, Technology, Engineering, and Mathematics (STEM) Education Coalition ) project attempting to revive America’s virtually nonexistent scientific education of our children. STEM also aims to foster early encouragement of scientific careers. The materials at this site are as beautifully literate and instructive as they are diverse in nature.They range from a full set of“Flex Books” to quizzes and exams for students and instructors  alike to a very active blog receiving input from all users. The subjects of the texts cover an equally  vast range of topics ranging from the life sciences to geometry and basic algebra to physics and engineering, all at the most basic of levels.

This particular entry, by Andrew and Anne Gloag, is a  beautifully written full scope textbook on basic algebra suitable for all pre-university level courses on the subject. There are lots of insights and examples. Even more, a customizable edition for experimental classes is available at the site as well. I think any basic algebra teacher at any pre-university level would find it not only a joy to teach from, but a joy for their students as well. I strongly recommend adding this site to your bookmarks and browsing the entire site intently if you’re a serious student or instructor at this level-it really is one of the best free sources currently available.

Advanced High-School Mathematics David B. Surowski Shanghai American School Singapore American School January 29,2011 (PG) The Shanghai American School in China is one of the preeminent private primary schools in the world with one of the highest average graduate academic achievement scores in the world. These notes have the sad distinction of being one of the last academic acheivements of Professor Surowski, who died shortly thereafter in China after a 3 year battle with pancreatic cancer. Surowski left behind a distinguished career as an educator in mathematics, both at Kansas State University and the large number of advanced high school cirricula he both taught and designed during his career. KSU continues to maintain-at least at this writing-his webpage, it can be found here. These are Surowski’s notes for an honors cirricula for mathematics students in their final year at SAS, most of whom had learned the basic elements of mathematics up to and including AP calculus. Surowski claims in the preface that, ..the brain power represented in this class of 11 blue-chip surely rivaled that of any assemblage of high school students anywhere and at any time!  Reading these wonderful 400 plus pages of notes and the incredible range of material that,  one needs to keep in mind, was taught to seniors in high school- one can completely believe it. The notes consist of incredibly clear and detailed presentations of the elements of advanced Euclidean geometry (classical constructions of triangles, circles and their intersections ), basic combinatorics and graph theory, inequalities and extreme points, basic abstract algebra,infinite series and differential equations and the elements of probability theory and inferential statistics. Any high school student capable of handling this stellar a regimen is certainly ready for a top undergraduate program and/or one of the many international competitions for gifted students. For those of us lucky enough to have such gifted kids Destined For Great Things as our charges, this should be mandatory lecture material after AP calculus to prepare them for entering Harvard or Oxford. For mere mortals, this would certainly be excellent recommended reading for freshman or sophomores over the summer to strengthen their grasp of the basics before beginning to take serious mathematics courses. And of course, this is necessary material for high school teachers in training.  Very highly recommended.

Trigonometry - Wikibooks, open books for an open world (G) Of course, the ubiquitous Wikibooks, which no bibliography of free online sources would be complete without.  These particular notes are simple, to the point and very visual-they give simple and clear proofs of all the basic results. They also have some surprises, like applications of the trigonometric functions to formulate simple harmonic motion, which is usually covered in a differential equations course. These notes take what I call the Joe Friday Philosophy of Mathematical Education: “Just the facts, Ma’am.” And that’s what you get here, done well and to the point.

Basic  Mathematics lecture notes by Thomas Ward (PG) This is a concise and brief set of notes on basic mathematics for entering university students at The University of Durham in the “Oxford” style-stating many results without proof. On the pluse side, despite its terseness, it does have good examples illustrating algebra and trigonometry calculations. If you have strong students for which this kind of course is basically a review-like they generally do at universities in the U.K., that’s not a problem.  But as a general purpose source, it’s probably too concise and omits too many topics. Good notes, but for strong students or instructors only.

Trigonometry Michael Corral Schoolcraft College (G) A full blown online trigonometry textbook for a course given at the small Schoolcraft College, presuming basic algebra and geometry. The author claims in the preface he wrote the book for 2 reasons: a) He wanted to present a more geometric approach to the subject then most high school texts and  college precalculus courses take and b) to create a course that taught the concepts and formulas that students will actually need in other courses, like physics or engineering. I think the author has succeeded very well on both counts. His prose is very clear and inviting, there are many pictures on every page giving geometric derivations and applications of all the main results and many well conceived and challenging problems from both geometry and physics appropriate for this level. The book is also very well organized from a pedagogical standpoint, concepts are built in order of increasing difficulty from more elementary concepts while maintaining the book’s geometric framework. For example, the book develops right triangle properties in depth and then develops the  properties of general triangles-like the laws of sine and cosine-by partitioning  general triangles into right triangles and using the basic geometry of bisectors and angles. This not only very natural, it also mirrors the actual history of how such problems were originally solved.   This is a book I’d be very happy to use to teach my students trigonometry with at any level.

A Semester Course in Trigonometry Marcel B. Finan Arkansas Tech University (G) This is another of Finan’s fine lecture notes, this one geared for freshman at ATU which have relatively weak mathematics backgrounds-which unfortunately, is the average incoming freshman at most American universities these days. They’re pitched at about the same level as Corral's notes and have a similar geometric slant, but they’re quite a bit broader in scope and assume less background in geometry. “A Semester Course in Analytic Geometry and Trigonometry” would probably be a more accurate title for them as coordinate geometry is developed in detail first. Also, some topics are covered which are usually reserved for later courses, such as DeMoive’s theorem. To cover a semester with Finan will require a bit more selectivity then with Corral or  Surowski ‘s notes, but given the unusual flexibility this gives this text, it’s well worth considering and it’s fantastic for self study or review

.S.O.S. Mathematics-Basic High School and Undergraduate Mathematics Study Resources (G)  This is another free site the serious student or teacher at any level should become familiar with. This is a site designed for quick assistance and/or review in basic topics in mathematics ranging from grade school to early undergraduate mathematics. The lessons at the site aren’t  really detailed enough for use as classroom or independent primary source materials, but that’s not really what the site’s for. It’s for intensive drill and review for students who have a shaky grasp of the basics.  For that purpose,the site has few peers and it should be put on the bookmarks list of all teachers and professors as a source of exercises for your students.

Geometry CK-12 Victor Cifarelli, Andrew Gloag, Dan Greenberg, Jim Sconyers, and Bill Zahner (G)  This is the elementary  geometry textbook for the STEM site discussed above and this is probably the best “book” in the collection-at 800 plus pages, it’s certainly the most comprehensive. It covers all the basics of a good high school elementary geometry course, which is to say it’s basically a rewrite of Euclid’s The Elements with some additional modern concepts.  Most significant of those modern additions are the concepts of metric geometry such as distance as a real number map.  Lines,angles, congruence and similarity relations, triangles, quadrilaterals, complementary and supplementary lines and angles, the trigonometry of right triangles, circles, areas and perimeters are all covered in detail and with great clarity, many examples and lots of exercises. No, it’s not a completely rigorous or logical presentation-although most of the postulates and basic theorems are presented with full proofs and logical thinking is encouraged. But then, neither was Euclid and people got along with it just fine in their formative stages for centuries. This is a course by educators, not mathematicians, and they realize for the average student, it’s much more important they assimilate the visual and intuitive aspects of geometry at this stage then to master completely logical axiomatic thinking. The result is a comprehensive and accessible text for all high school level students in geometry and will serve as a good foundation for more rigorous follow-up courses at the university level. As a mathematics professional, I would prefer more rigor and creativity, of course-and there are some sources listed here I favor personally that supply just that. But it’s debatable whether or not the average high school student or college freshman could appreciate such texts. Indeed, students about to take a college level course in axiomatic geometry could do a lot worse then carefully study this text in the summer months before the course as preparation/review.

Boundless Algebra Open Textbook B (G) This is an open source textbook from another online education source, Boundless. From the site: Boundless’ free textbooks and study tools makes education more affordable, accessible,and effective for students and professors.  That certainly puts them right at home at this website. The organization is pretty interesting-it’s an interactive GUI format where chapters and sub-chapters are delineated by tabs opened by mouse clicks,definitions,examples and basic glossary terms are clearly separated and formatted in boldface title. There are also 2 ways links to related topics either earlier or later in the book, giving the book a holistic content that allows back and forth reference and learning. The topics are pretty standard for the most part: the real line, fractions, decimals, scientific notation, equations and polynomials, basic functions, inequalities and conic sections. The presentation isn’t standard, though-it’s unusually rigorous and careful, with some topics not usually covered in usual courses, like an entire section introducing real entry matrices and their operations. I’m not entirely comfortable recommending it as a textbook for self study-I think the unusual nonlinear organization of the book could throw some students. But I’d certainly recommend it as a supplement to such a course for instructors and for students who want to be challenged.

UNDERSTANDING ALGEBRA JAMES BRENNAN Boise State University  2002 (G) Brennan’s notes for a high school level algebra course for incoming freshman who are weak. It’s well written and relatively careful, but it’s also pretty  cookie cutter.  There’s no inventiveness or originality here at all. It’s not bad, but no reason to make a special effort to read them, either.

Geometry CK-12 Victor Cifarelli, Andrew Gloag, Dan Greenberg, Jim Sconyers, and Bill Zahner (G)  This is the elementary  geometry textbook for the STEM site discussed above and this is probably the best “book” in the collection-at 800 plus pages, it’s certainly the most comprehensive. It covers all the basics of a good high school elementary geometry course, which is to say it’s basically a rewrite of Euclid’s The Elements with some additional modern concepts.  Most significant of those modern additions are the concepts of metric geometry such as distance as a real number map.  Lines,angles, congruence and similarity relations, triangles, quadrilaterals, complementary and supplementary lines and angles, the trigonometry of right triangles, circles, areas and perimeters are all covered in detail and with great clarity, many examples and lots of exercises. No, it’s not a completely rigorous or logical presentation-although most of the postulates and basic theorems are presented with full proofs and logical thinking is encouraged. But then, neither was Euclid and people got along with it just fine in their formative stages for centuries. This is a course by educators, not mathematicians, and they realize for the average student, it’s much more important they assimilate the visual and intuitive aspects of geometry at this stage then to master completely logical axiomatic thinking. The result is a comprehensive and accessible text for all high school level students in geometry and will serve as a good foundation for more rigorous follow-up courses at the university level. As a mathematics professional, I would prefer more rigor and creativity, of course-and there are some sources listed here I favor personally that supply just that. But it’s debatable whether or not the average high school student or college freshman could appreciate such texts. Indeed, students about to take a college level course in axiomatic geometry could do a lot worse then carefully study this text in the summer months before the course as preparation/review.

Intermediate Algebra by David Arnold and Bruce Wagner (G)This is a nice, solid algebra text available for free through a Creative Commons Share Alike License. I’m hoping this is the direction of the future, where more and more books at a more and more sophisticated level will be available for free access online. In thisparticular case, this is a book in “intermediate algebra” at the high school and freshman college level-that is to say, it’s a book for students with an understanding of basic arithmetic and equation solving. While the topics are fairly standard ( number systems, inequalities, functions and graphs,linear functions, polynomial functions, radical functions, etc. ) , the emphasis is on careful definition and motivation as well as examples and exercises. There are also complete solutions for the exercises. Useful for self study.

Algebra I By Wing-Suet Li and John E. McCarthy Draft Homepage More advanced students shouldn't read this and mistake it for an abstract algebra source they've never heard of. No,this is merely a high school algebra book covering the usual basics. But high school teachers and thier students should take note as this is a very nice if incomplete source for them on the subject: It's very readable with clear explanations and good exercises. Unlike most books written at this level today, it presents its subject as mathematics and doesn't dumb it down to the point where it looks more like a Sesame Street sketch. It's a very nice resource for such courses and a very good first mathematics book for teenagers.

Intermediate Algebra Text by Jon Blakely College of the Sequoias(G) These are the basic algebra online textbooks written for The College of The Sequoias Yeah, don’t look at me, I never heard of it either.Apparently it’s some kind of community college in California somewhere and apparently it has a pre-algebra course for incoming students-which doesn’t speak highly of its selectivity. Which is surprising in California’s much vaunted public education system, which has mostly managed to keep its standards high despite all the fiscal chaos in the state of late.  Be that as it may, the union of the 2 books cover essentially a review of grade school arithmetic in the context of algebra and all the basic high school algebra: whole numbers, integers, real numbers, linear and quadratic equations, inequalities, rational expressions and radicals, functions, conic sections and linear systems of equations. It’s all very standard stuff presented in a very standard manner, albeit clearly and with nice examples. Worth a look for students at this level, but nothing to mark in your blog.

A Reform Approach to College Algebra Marcel B. Finan Arkansas Tech University 2011  (G) Yet another of Finan’s fine free online textbooks for his students, this one pitched for a beginning algebra course for high school students or freshman college students. Finan’s text covers all the basic topics-but what’s interesting here is that he a) covers them in somewhat greater depth then is normal in a course at this level and b) the presentation downplays the mechanical algebraic aspects, like rearrangements and roots of equations and emphasizes the aspects of this material that is of importance in geometry and calculus. For example, exponential and logarithmic functions are covered with both an arbitrary real number base and the parameters a and b in the form f(x) = b ax  It’s clear Finan is expecting his students to go on to study serious mathematics after this. The presentation is, as with the earlier books, very visual and insightful at all times. It serves as an outstanding prequel to his trigonometry textbook.Highly recommended.

A Problem-Solving Approach to College Algebra Marcel B. Finan Arkansas Tech University 2002 (PG) This is a fascinating alternate version of the College Algebra text Finan authored here.. This is the same course given entirely as a directed set of exercises. There are of course, many “problem course” textbooks in mathematics on many subjects from Moise’s Problem Courses in Elementary Topology and Analysis to Ian Adamson’s A General Topology Workbook to Lovalz’s  Combinatorial Problems And Exercises . But to my knowledge, there’s never-repeat, never-been one pitched at this basic a level. I’m not sure how I feel about that-the main reason problem course books to date have been designed for upper level undergraduates and up is because problem courses consist mostly of theorems and studens are required to supply their proofs.  Finan’s book has no proofs per se, all calculations and graphs for students to produce, as well as testing them on definitions-which, of course, is the most one can expect in a course at this level. I can’t imagine students not having a very strong background in elementary algebra and their critical thinking skills being much improved after tackling this book. It’ll also be a great intensive refresher for students that have been away from the classroom for awhile. If you think you’re up for the challenge or you’re an instructor and you think your students are, give it a shot, by all means. But don’t expect a soft course by any means.

Reasonable Algebraic Functions (Terse Edition) Alain Schremmer FreeMathTexts.org Version 5.1 — Thursday 21 st July, 2011(PG) This is a continuation of Schremmer’s highly original and ambitiously perspicacious “basic” online textbooks attempting to give rigorous math textbooks for elementary students at the high school level. This is his version of a precalculus textbook, looking to give students the same kind of foundation for calculus the earlier books did. Schremmer complains in the preface that most precalculus books do a very sorry job of preparing students for calculus because they don’t really develop the machinery upon which a modern calculus course should be based and instead rehash the earlier material on algebra and geometry. I completely agree, but it’s hard to see how to alleviate this problem without defeating the  purpose of the course, which is to prepare students for using the algebra and geometry learned in earlier courses in the context of calculus. Schremmer’s solution is a stunning one-he actually develops  the basic ideas of rigorous analysis in an entirely pictorial manner where only certain key concepts are carefully defined- everything else is built from these definitions. For example, he defines relations as “boxes” that take an input and return an output and functions are relations where every input is assigned to one and only one output. This is not quite a rigorous definition in the set theoretic manner, but the basic concept is clearly present and the transition from such a definition to the rigorous one should be quite easy. Another original approach he uses is to describe “local analysis” via a graphical description of “neighborhoods” of a point or number on the line. Using constant distances from the central point of the nieghborhood, he’s able to give a fully intuitive presentation of the topology of the real line (although he doesn’t refer to it that way) and uses it to describe each of the standard library of functions in calculus. Again, it’s another terrific book by Schremmer that manages to be mathematically careful without losing sight of the very elementary level of the students it’s directed at. I think the people that will bemost impressed by Schremmer’s books will be young professors and graduate  students who are frustrated with teaching trick and handwaving laden “service courses” from standard books-I think these books will show them such courses don’t have to be taught that way and can be presented in mathematically satisfying ways that don’t overwhelm the students.

Basic Algebra -Wikibooks (G) Once again, the ubiquitious Wikibook option , this one for algebra. Nothing special here-at least, not in the edition currently online as of this writing-but readily available and readable, so won’t do any harm to check it out.

Math Textbook for the Community College Denny Brown (G) A basic arithmetic and algebra book for high school students. Looks ok, but nothing special.

Elementary Algebra CLASS NOTES James L. Christensen Cutamacka College JAN 8, 2009 3 rd EDITION  (G)A very standard set of notes for a high school algebra course and not particularly well written or inventive. I wouldn’t waste my time, but it’s there and it’s free, so why not take a look?

Notes from Trigonometry Steven Butler Iowa State University 2003  (G)  I found these notes while compiling the online bibliography component of the website and almost immediately fell in love with them. I was immediately floored. For a second, I thought I’d made a mistake and opened a more advanced set of notes, on college geometry or something else. Nope. This is a set of notes for either a high school course or a freshman course in trigonometry. Most mathematics notes on elementary topics are done in that “high school state cirriculia” style that makes them all look so cookie cutter and shallow. They state the major results, they give a few standard examples, they give the student some dopey mnemonic for remembering the material, etc. They look like they’re structured to get the class over as fast for the instructor as for the students. Not these. These notes are well written, careful, conversational and above all, inquisitively enlightening  Butler approaches trigonometry with the depth and maturity of a trained mathematician, the same way one would expect him to present much more advanced topics like real variables or abstract algebra- but always keeping in mind his intended audience is at the very beginning of their mathematical experience. The very beginning of the notes gives an outstanding example of Butler’s style and approach. The author begins with a section asking why mathematics is important and gives an outline of a methodology for solving mathematical problems inspired by George Polya’s classic How To Solve It.  He then discusses the origins of the term trigonometry and how its methodology arises partially out of the origin of its name:
 The word trigonometry comes from two root words. The first is trigonon which means “triangle” and the second is metria which means “measure.” So literally trigonometry is the study of measuring triangles. Examples of things that we can measure in a triangle are the lengths of the sides, the angles (which we will talk about soon), the area of the triangle and so forth.So this book is devoted to studying triangles. But there aren’t similar books dedicated to studying four-sided objects or five-sided objects or so forth. So what distinguishes the triangle?Let us perform an experiment. Imagine that you made a triangle and a square out of sticks and that the corners were joined by a peg of some sort through a hole so essentially the corners were single points. Now grab one side of each shape and lift it up. What happened? The triangle stayed the same and didn’t change its shape, on the other hand the square quickly lost its “squareness” and turned into a different shape. So triangles are rigid, that is they are not easily moved into a different shape or position. It is this property that makes triangles important. It is this same property that dictates that triangles are used in the construction of houses, skyscrapers,bridges or any structure where stability is desired.  

This rigidity, by the way, is also why triangles factored so prominently in the early days of topology and was the element of choice for subdividing topological space constructions-they remained unchanged under continuous transformations. This idea of the triangle as the natural unit of breaking up geometric objects becomes the central idea of the notes and gives a precision and elegance to his presentation of this very simple and classical subject a very modern feel that will be inspiring to teacher and student alike. Butler has succeeded in making a subject most of us are conditioned to think is very mundane and boring, inventive and exciting again, as it must have been to the ancient Greeks who developed it. But there’s so much more to love about them-inspiring historical references, wonderful pictures and challenging problems. Above all, he succeeds in recasting this tired old standby as real mathematics and he does so in a way that’s accessible to just  about everybody.  A must read for anyone who loves mathematics- and especially to the young researcher/grad student who has to teach this subject soon and is dreading it. Butler’s notes will show you how to teach it in a manner that’s every bit as exciting as current research. And that remarkable achievement alone makes them well worth reading.

Algebra and Trigonometry Jerry Alan Veeh Auburn University 2000 (PG)  These superior notes are intended to not only cover the fundamental mathematics of the title, but to do so with a maximum of ability to formulate precisely in words what practical problems are asking. These notes don’t quite have the breadth of Surowski's notes but they do cover basic algebra and trig with equal clarity and rigor as those notes and even cover quite a bit of stuff not covered in courses at this level, such as complex numbers and the complex exponential and logarithim. A very good supplement to an honors high school course or as Veeh originally taught it, a course for weak college freshman or for review. Good stuff indeed. Recommended.

Precalculus George Schildge State University of New York Clinton Community College (G) These are some brief, concise lectures for the algebra, trigonometry and precalculus courses at Clinton Community College. These notes don’t look very impressive, but there’s more to them then what initially meets the  eye.  They’re almost in “bulletpoint” presentation, with very curt facts presented with relevant diagrams. The look like flashcards that need to be printed out and cut out-indeed, I think they’re intended to be used this way. I was a little confused by their format at first until I saw some interesting horizontal lines on the right side of each page and realized they’re for students to write then own notes adding to what’s in the “bulletpoint” left hand side. Not as complete as Corral's notes or Finan's  algebra or  trigonometry  texts, Butler, or some of the other sources here, but will probably be very useful as a study aid and I’d recommend them for that most of all.  But they really won’t be useful for much else .

Precalculus An Investigation of Functions Edition 1. 3 David Lippman Pierce College Melonie Rasmussen Pierce College (G) Here’s another original, substantive choice for a free online class source for a course at this level. It’s interesting how all these “books” have a common theme-the authors are fed up with the cookie cutter system of producing textbooks,especially for basic courses, in America. The resulting work is very similar in topic selection and rigor to Stitz and Zeager , but its’ much less comprehensive and focused in approach then that book. The main theme of Lippman and Rasmussen’s book is to explicitly provide a foundation for a serious calculus course and it emphasizes developing precisely those skills most critical to the study of calculus, such as curve sketching, basic algebraic equation manipulation (particularly rational and exponential functions) and trigonometric functions.Also, it’s an “active learning” text and this manifests in several different ways. Firstly, it’s an extremely example driven-there is at least one example and/or graph on every page. Secondly, there are many exercises and they’re very diverse-they run the gamut from simple computations to graphical analyses to difficult multistep word problems.  Lastly and most interesting, many of these more substantial problems are “calculus preview problems”-that is to say, they’re drawn from many of the areas that calculus applications are drawn, such as physics, chemistry and economics and where “precalculus” methods can be used to solve them. This gets the student to recognizing the basic problems and the tools needed to solve them, such as trajectory graphing and taking  slopes at a point to solve a particle motion problem. It’s not quite as good as Stitz and Zeager, but it might be more practical to use for a semester course where most of the students plan to take calculus next. An excellent work in any event.

PRECALCULUS KEN KUNIYUKI SAN DIEGO MESA COLLEGE 2012-2014   At every department at every university, there's the workhorse member of the faculty whose main interest is teaching and preparing the next generation of stars. We've all known That Guy at some point in our careers, maybe we were fortunate enough at the critical nascent phases of our careers to be mentored by him or her. He's the one who gets buried in paperwork because he didn't publish his quota of papers decades ago as a newly minted PHD or Master's student. He's the one that prepares the course schedules, labors in the dark after everyone else has gone home and get mocked by generations of research stars and their students for their trouble. He's also usually the most gifted teacher in the department,the one who discovers and nurtures the talent. He's the one who inspires the freshman with stories of legendary professors, universities and experiences he's known and advises them while stoking their ambitions. My mentor, Nick Metas, was that guy at Queens College of the City University of New York where I was an undergraduate. Apparently at San Diego Mesa College,  Ken Kuniyuki is That Guy. His website is a treasure trove of lower undergraduate level lecture notes, exercise sets, book drafts and other materials  of wonderful quality and the fact his site has recorded over 37 MB of free downloads to date is the measure of their quality. This gigantic collection of lecture notes is another precalculus textbook in progress from online lecture notes. The author also has a follow up full blown calculus text in progress at his website that we'll get to below. From the looks of it, both of these books are going to be very impressive when they reach their final form. Even better- the notes are available free for download via Creative Commons License. I’m an adamant believer that this is the future of book publishing and we need to defend it with our blood if necessary to break the stranglehold of the publishing companies on educational materials. After all, isn’t that what this website is all about? The notes contain what's usually expected in a precalculus course, namely all the material from high school -except Euclidean geometry-students of a previous generation would be expected to know before beginning calculus. The main difference between Kuniyuki's treatment and the usual such courses is a rigor and sophistication usually reserved for much more advanced courses. Set theory, functions and basic logic, scientific notation,polynomial,rational,exponential and logarithmic functions, an exhaustive and careful treatment of  trigonometry,systems of equations and inequalities,matrices and determinants,discrete math and concludes with the geometry of the conic sections and polar coordinates. Don't take my word for it,just browse the beauty and inventiveness of these notes. Whether you're a beginning student or a seasoned pro, these will become one of your go-to sources bookmarked on your computer, I promise.

Elementary Algebra Collection type: Textbook Textbook by: Wade Ellis, Denny Burzynsk (G)  This is a basic algebra course written for the Connexions series of basic online textbooks produced by Rice University Press. It’s well written, but pretty standard textbook on baby algebra for high school students convering all the usual topics in the usual way. It does have a large number of problems with full solutions and many clear graphics. In other words, it’s yet another good plain vanilla algebra text for beginning  students. You may like it-it didn’t exactly raise my pulse thinking about teaching from or studying it.

Algebra is Vital by Leslie Elizabeth Buck (G)  Now here’s a book worth looking at-because, in a sense, it’s not really a book. This is really a set of lectures for a basic high school /freshman algebra course. Their content is relatively standard, but what makes them special is their presentation.. They’re organized via a medium that probably wouldn’t have been possible before the Worldwide Web. The lectures are presented as video files that unfold frame by frame in PowerPoint format. They’re dynamic and strikingly original in organization, interweaving historical  notes, definitions, graphs and many concrete examples from physics, geometry and more directly into the presentation. They’re not as interesting to me as Schremmer’s notes or as comprehensive as Finan’s, but they’re well worth checking out.

Precalculus Version 3 Carl Stitz and Jeff Zeager  Lakeland Community College(G) This is an impressive online textbook specifically designed by the authors to act as a free open source textbook that can be produced  cheaply for their students. The passion of the authors as teachers of beginning mathematics students and their goal to begin the students with proper mathematical training from the very beginning shines through on every page. This is shown from the outset of their lovingly written preface:

Thank you for your interest in our book, but more importantly, thank you for taking the time to read the Preface. I always read the Prefaces of the textbooks which I use in my classes because I believe it is in the Preface where I begin to understand the authors - who they are, what their motivation for writing the book was, and what they hope the reader will get out of reading the text. Pedagogical issues such as content organization and how professors and students should best use a book can usually be gleaned out of its Table of Contents, but the reasons behind the choices authors make should be shared in the Preface. Also, I feel that the Preface of a textbook should demonstrate the authors' love of their discipline and passion for teaching, so that I come away believing that they really want to help students and not just make money. Thus, I thank my fellow Preface-readers again for giving me the opportunity to share with you the need and vision which guided the creation of this book and passion which both Carl and I hold for Mathematics and the teaching of it.

Now if a textbook I’ve picked up begins like that, I know I made a good choice picking it up. The authors have earned my trust as either a student or an instructor using it, whether or not they deliver in the end. In this case, the authors more then deliver. Although the subject matter chosen by Stitz and Zeager is mostly standard:  number systems, rectangular coordinates, functions, polynomial functions,  rational functions,linear systems of equations and their solutions and a very lengthy concluding chapter on trigonometry-their presentation and development of it is most definitely not standard. Firstly, they are not afraid of rigor and work very hard to make the beginner understand the need for careful definitions. The presentation is strictly mathematical. Rather then making enormous concessions and distorting simplifications to simplify it for non-mathematics students, as most books and notes at the beginning level try and do, they try and show the non-mathematics student the enormous vistas of understanding and practical application that open when one does mathematics rigorously and correctly. Secondly, they aren’t afraid to introduce techniques and concepts that one normally doesn’t see in a course at this level in order to create a truly strong mathematical background for later courses, such as matrix arithmetic, regression analysis and the binomial theorem. Lastly.not only are the authors gifted teachers who show the same concept from multiple viewpoints to drive the main concepts home, they have a wonderfully humorous style of writing. It’s actually fun to read this book. Just a few of the gems within its pages:

If we set b= 0 in the above definition of C, we see that every real number is a complex number. In this sense, the setsNWZQR, and are `nested' like Matryoshka dolls.

Chapter 7:Hooked on Conics

If I’d had this book as a teenager, my career might have gone very differently. Between these notes, Schremmer, Butler and the solid online texts of Finan, there are a lot of wonderful free text that achieves its goals in as complete, organized and literate a manner as possible. A must-read for both students and teachers at all levels. Highly recommended.

Precalculus Class Notes Walter Schreiner Christian Brothers University (Tennessee)

College Algebra James Jones Richland University 

Concepts of Algebra Christopher Cooper McQuarrie University

Elementary Algebra and Trigonometry Christopher Cooper McQuarrie University